Lecture Notes in Bioinformatics
5750
Edited by S. Istrail, P. Pevzner, and M. Waterman Editorial Board: A. Apostolico ...

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Lecture Notes in Bioinformatics

5750

Edited by S. Istrail, P. Pevzner, and M. Waterman Editorial Board: A. Apostolico S. Brunak M. Gelfand T. Lengauer S. Miyano G. Myers M.-F. Sagot D. Sankoff R. Shamir T. Speed M. Vingron W. Wong

Subseries of Lecture Notes in Computer Science

Corrado Priami Ralph-Johan Back Ion Petre (Eds.)

Transactions on Computational Systems Biology XI

13

Series Editors Sorin Istrail, Brown University, Providence, RI, USA Pavel Pevzner, University of California, San Diego, CA, USA Michael Waterman, University of Southern California, Los Angeles, CA, USA Editor-in-Chief Corrado Priami The Microsoft Research - University of Trento Centre for Computational and Systems Biology Piazza Manci, 17, 38050 Povo (TN), Italy E-mail: [email protected] Guest Editors Ralph-Johan Back Ion Petre Åbo Akademi University Department of Information Technologies Joukahaisenkatu 3-5, 20520 Turku, Finland E-mail: {backrj,ipetre}@abo.ﬁ

Library of Congress Control Number: 2009933672 CR Subject Classiﬁcation (1998): J.3, F.1, F.2, I.6, I.2, C.1.3 ISSN ISSN ISBN-10 ISBN-13

0302-9743 (Lecture Notes in Computer Science) 1861-2075 (Transactions on Computational Systems Biology) 3-642-04185-X Springer Berlin Heidelberg New York 978-3-642-04185-3 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2009 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientiﬁc Publishing Services, Chennai, India Printed on acid-free paper SPIN: 12743292 06/3180 543210

Preface

Biology is witnessing a transformation towards a more quantitative science, based on the major technological breakthroughs of the past decade. In this transformation, biology is incorporating mathematical modeling techniques and computational approaches towards numerical simulations, model analysis, and quantitative predictions. An important goal is to formalize and analyze the everchanging inter-connections between components (often on diﬀerent time and space scales), their inﬂuence on one another, regulatory patterns, alternative pathways, etc. Formal reasoning rather than empirical observations is the main driving force in this new type of biological research. At the same time, computer science and applied mathematics are faced with considerable methodological challenges in handling an unprecedented level of concurrency, stochastic eﬀects, a mix of large and small populations, combinatorial explosions in the state space, model reﬁnement, and model (de)composition, etc. This special issue of Transactions on Computational Systems Biology on Computational Models for Cell Processes is based on a workshop with the same name that took place in Turku, Finland, on May 27, 2008. The workshop was organized as a satellite event of the 15th International Symposium on Formal Methods that took place in Turku in the period May 28-31, 2008. This special issue however had an open call for paper submissions, with a separate peer-review process. The accepted papers span an interesting mix of approaches to systems biology, ranging from quantitative to qualitative techniques, from continuous to discrete mathematics, from deterministic to stochastic methods, from computational models for biology to computing paradigms inspired by biology. Overall, they give a good glimpse into some of the exciting current research avenues in computational systems biology. This volume also contains three regular submissions that deal with the relationships between ODEs and stochastic concurrent constraint programming (by Bertolussi and Policriti), with the equilibrium points of genetic regulatory networks (by Chesi), and with probability models describing how epigenetic context aﬀects gene expression and organismal development (by Wallace and Wallace). July 2009

Ralph-Johan Back Ion Petre Corrado Priami

LNCS Transactions on Computational Systems Biology – Editorial Board

Corrado Priami, Editor-in-chief Charles Auﬀray Matthew Bellgard Soren Brunak Luca Cardelli Zhu Chen Vincent Danos Eytan Domany Walter Fontana Takashi Gojobori Martijn A. Huynen Marta Kwiatkowska Doron Lancet Pedro Mendes Bud Mishra Satoru Miayano Denis Noble Yi Pan Alberto Policriti Magali Roux-Rouquie Vincent Schachter Adelinde Uhrmacher Alfonso Valencia

University of Trento, Italy Genexpress, CNRS and Pierre & Marie Curie University, France Murdoch University, Australia Technical University of Denmark, Denmark Microsoft Research Cambridge, UK Shanghai Institute of Hematology, China CNRS, University of Paris VII, France Center for Systems Biology, Weizmann Institute, Israel Santa Fe Institute, USA National Institute of Genetics, Japan Center for Molecular and Biomolecular Informatics The Netherlands University of Birmingham, UK Crown Human Genome Center, Israel Virginia Bioinformatics Institute, USA Courant Institute and Cold Spring Harbor Lab, USA University of Tokyo, Japan University of Oxford, UK Georgia State University, USA University of Udine, Italy CNRS, Pasteur Institute, France Genoscope, France University of Rostock, Germany Centro Nacional de Biotecnologa, Spain

Table of Contents

Computational Models for Cell Processes Process Algebra Modelling Styles for Biomolecular Processes . . . . . . . . . . Muﬀy Calder and Jane Hillston

1

Simple, Enhanced and Mutual Mobile Membranes . . . . . . . . . . . . . . . . . . . . Bogdan Aman and Gabriel Ciobanu

26

Bio-PEPA with Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Federica Ciocchetta

45

In Silico Modelling and Analysis of Ribosome Kinetics and aa-tRNA Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

69

Qualitative and Quantitative Analysis of a Bio-PEPA Model of the Gp130/JAK/STAT Signalling Pathway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maria Luisa Guerriero

90

Rule-Based Modelling and Model Perturbation . . . . . . . . . . . . . . . . . . . . . . Vincent Danos, J´erˆ ome Feret, Walter Fontana, Russ Harmer, and Jean Krivine Extended Stochastic Petri Nets for Model-Based Design of Wetlab Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monika Heiner, Sebastian Lehrack, David Gilbert, and Wolfgang Marwan A Projective Brane Calculus with Activate, Bud and Mate as Primitive Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maria Pamela C. David, Johnrob Y. Bantang, and Eduardo R. Mendoza Accepting Networks of Non-inserting Evolutionary Processors . . . . . . . . . J¨ urgen Dassow and Victor Mitrana

116

138

164

187

Discrete Modeling of Biochemical Signaling with Memory Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Jack and Andrei P˘ aun

200

Dynamical Systems and Stochastic Programming: To Ordinary Diﬀerential Equations and Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luca Bortolussi and Alberto Policriti

216

VIII

Table of Contents

Computing Equilibrium Points of Genetic Regulatory Networks . . . . . . . . Graziano Chesi

268

Code, Context, and Epigenetic Catalysis in Gene Expression . . . . . . . . . . Rodrick Wallace and Deborah Wallace

283

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335

Process Algebra Modelling Styles for Biomolecular Processes Muﬀy Calder1 and Jane Hillston2 1

Department of Computing Science, University of Glasgow, Glasgow G12 8QQ, Scotland 2 Laboratory for Foundations of Computer Science and Centre for Systems Biology, Edinburgh The University of Edinburgh, Edinburgh EHA 9AB, Scotland

Abstract. We investigate how biomolecular processes are modelled in process algebras, focussing on chemical reactions. We consider various modelling styles and how design decisions made in the deﬁnition of the process algebra have an impact on how a modelling style can be applied. Our goal is to highlight the often implicit choices that modellers make in choosing a formalism, and illustrate, through the use of examples, how this can aﬀect expressability as well as the type and complexity of the analysis that can be performed.

1

Introduction

Much recent research has considered the problem of providing suitable abstract models to allow biologists to construct mechanistic models to enhance understanding of biomolecular processes. Process algebras, formal modelling languages originally conceived for modelling concurrent computations, have been widely applied, most notably in the area of signalling pathways [RSS01, CGH06, TK08]. This is experimental science and we are currently evaluating the hypothesis that such formal models can add value to the mathematical analysis that is already undertaken within systems biology in terms of ordinary diﬀerential equation (ODE) models or stochastic simulations directly. In exploring this goal, even within work on process algebras, several diﬀerent styles of modelling have emerged. Ultimately we hope to be able to give guidance on how to choose among these modelling styles, or on how to map molecular components and their interactions to processes, process communication and process composition. However, in the ﬁrst instance we investigate how design decisions made in the deﬁnition of the language have an impact on how a modelling style can be applied, and highlight the often implicit choices that modellers make in choosing a formalism. Recent research eﬀort on process algebras for biomolecular processes, e.g. [CGH06, CVOG06, CH08, Car08], has focussed on deﬁning alternative semantics, such as discrete-state (stochastic) or continuous-state (ODE) semantics. These provide important links with the work where mathematical representations are used directly and establish a valid foundation for process algebra models. Based on these semantics, analysis may be carried out by model-checking, C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 1–25, 2009. c Springer-Verlag Berlin Heidelberg 2009

2

M. Calder and J. Hillston

stochastic simulation based on Gillespie’s algorithm or ODE simulations. Our emphasis in this paper is diﬀerent. Here we consider the forms of abstraction supported by process algebra and how the abstraction and the process algebra chosen aﬀect the expressiveness of the model with respect to the biological processes, as well as the type and complexity of the analysis that can be performed. We focus on one of the most important types of interaction between molecular components: chemical reactions. In chemical notation, these may be ﬁrst order k1 reactions, for example A degrades to B: A−→B, or second order reactions, for exk2 k3 ample A and B combine to form C or C and D: A + B −→C, or A + B −→C + D. Typically, k1 . . . k3 are rate constants for kinetic laws (e.g. mass action). A fundamental aspect of the abstraction used in modelling is the nature of the process mapping. In the literature on process algebras for systems biology we ﬁnd predominantly the molecule-as-process [RSS01, Car08] abstraction, but the species-as-process and reaction-as-process mappings have also been proposed [CGH06, CH08, BP08]. The distinction between the ﬁrst two can be understood by appealing to ecology: the former is essentially individuals-based, whereas the latter is population-based. We note that this distinction is less common in distributed computing system modelling, the origins of process algebra, where population-based models are rarely considered. Further stylistic diﬀerentiation was identiﬁed in [CGH06] where the concepts of reagent-centric and pathway-centric models are introduced, in the context of population-based modelling. Reagent-centric models map all reagents in a reaction to processes, whose variation reﬂect decrease through consumption and increase through product formation (consumers and producers). Reagents such as modiﬁers that do not vary species amounts can also be modelled in this approach. Reagent-centric models provide a ﬁne-grained, distributed view of a system. Pathway-centric models provide a more abstract view of a system, tracking serialisations of events, which are then composed concurrently. Here, processes vary according to their biological state rather than their quantity. Whereas in a reagent-centric approach the processes may be molecules or molecular species, in the pathway-centric approach the processes are molecules or sub-pathways. Thus the interactions between processes are between ﬂows of events corresponding to producers, i.e. components on the left hand sides of a reactions. Most modelling approaches map chemical reactions to events in a straightforward way, and map (possibly a subset of) the chemical components to processes. Bortolussi and Policriti’s work on sCCP, using the reaction-as-process abstraction, is an exception to this. When chemical components are mapped to processes within the reagent-centric approach there is a further choice: between associating processes with all components or only with the reagents on the left hand side of equations, i.e. those reagents that are the reactants of the reaction. To distinguish these two cases, we call the former reagent-centric and the latter reactant-centric. This modelling choice is often inﬂuenced by the form of synchronisation available within the algebra: binary or multi-way. If we have only the former, then only the reactant-centric approach is possible and we are left

Process Algebra Modelling Styles for Biomolecular Processes

3

with an interesting dilemma when there are fewer components on the right hand k2 side of the equation than on the left hand side, e.g. A + B −→C. In summary, a number of factors will inﬂuence the structure of a process algebra model of a biomolecular process: – population-based or individuals-based, – reagent-centric, reactant-centric, pathway-centric or reaction-centric, – the form of synchronisation available in the algebra. In this paper we investigate the interplay between these three factors. Our motivation is to explore the extent to which we can build clear and faithful models using current algebras and analysis techniques, and how design decisions with respect to the process algebra determine the mappings available to the modeller. We consider diﬀerent combinations, investigating their advantages and disadvantages. We will use ﬁve process algebras for illustration: π-calculus, Beta-binders, PEPA, Bio-PEPA, and sCCP; these are brieﬂy outlined in the next section. These are chosen as they represent a spectrum of diﬀerent modelling style, including languages that have been adapted (π-calculus, PEPA and sCCP) and designed (Beta-binders and Bio-PEPA) for biological modelling. This is by no means a comprehensive list of process algebras used in systems biology. In particular we do not include any of the process algebras designed to consider spatial aspects of biomolecular processes [CPR+ 04, Car04, V07, BMMT06, CG09] as they are beyond the scope of this paper. The remainder of the paper is organised as follows. Section 2 gives an overview of the process algebras and Section 3 describes the example pathway used throughout for illustration and comparison. In Sections 4 to 8 we consider modelling in PEPA, Bio-PEPA, π-calculus, Beta-binders and sCCP. We discuss the results in Section 9 and give our conclusions in Section 10.

2

Process Algebras

Process algebras were originally deﬁned to give semantics to concurrent processes in a computing context and have enjoyed considerable success over the three decades since they emerged. Classical process algebras such as CCS [Mil80] and CSP [Hoa85] focus on the functional capabilities of processes and all actions are atomic with only relative timing of actions captured. Subsequently there have been many extensions of process algebras to capture more information about the system being modelled, for example the relative probability of alternative actions (probabilistic process algebras) and the expected duration of actions (stochastic process algebras). Each of the process algebras that we consider is based on three fundamental binary operators: action preﬁx, choice, which is associative and commutative, and synchronous composition, which is also associative and commutative.

4

M. Calder and J. Hillston

Note that in the following we omit the cooperation sets for composition in PEPA and Bio-PEPA and assume them to be the intersection of the alphabets of the processes involved (denoted ). We disregard quantitative aspects of actions, ∗ since the representation of kinetics is orthogonal to the expressiveness we consider here. Therefore in our examples, we will assume that the reaction rate for each considered reaction is unique and use this as the name of the corresponding r1 reaction event, i.e. the reaction A + B −→C + D in chemical notation maps to the process algebra event r1 . In seminal work, Regev and Shapiro [RS01] suggested an abstraction of cell-ascomputation and proposed that models formerly used in the study of interacting computational entities, such as Petri nets, process algebras and automata, could be usefully employed for the study of biological processes. In particular they focussed on the π-calculus [Mil99], and subsequently the stochastic π-calculus [Pri95] based on the molecule-as-process abstraction. This work has been hugely inﬂuential with many other authors following the same abstraction in their own work, even when the details of the process algebra diﬀer. However, the π-calculus has some particular characteristics that are independent of the molecule-as-process abstraction that also shape the style in which models are expressed. In this section we give a brief introduction to process algebras, focussing on the features which lead to diﬀerent modelling paradigms. 2.1

Forms of Synchronisation

The original process algebras, CCS and CSP, diﬀer in their interpretation of actions and consequently the meaning of synchronisation. In CCS all actions are assumed to be communications, and therefore conjugate, i.e. actions are paired, corresponding to an input and an output. An action cannot be carried out without its partner, and the pairing of an input and an output becomes a private τ action. This has the consequence that the interaction, or synchronisation, between processes is strictly binary as once an input has been paired with an output both become unavailable for further interaction. In contrast, in CSP no distinction is made between inputs and outputs and there is no notion of complementarity between actions. Instead action type denotes ownership of a channel and synchronisation is assumed to take place whenever processes undertake actions of the same type, i.e. communication over the named channel. This is termed multiway synchronisation as there is no restriction on the number of processes that may own a channel and thus join a synchronisation. Note that in both these cases the parallel composition operator is generic: in CCS any complementary actions which are on either side of the parallel composition may synchronise; in CSP, processes composed by the parallel operator must synchronise on common actions. Synchronisation in PEPA is a subtle variation of the CSP scheme. Here the parallel composition operator, termed cooperation, is decorated by a set of action types (the cooperation set ) and processes are only forced to synchronise on

Process Algebra Modelling Styles for Biomolecular Processes

5

action types within this set, being able to act concurrently and individually on other action types. Thus the parallel composition is not generic, but a family of parameterised operators. The characteristics of this multiway synchronisation are important in the biological context as they allow one copy of a process (molecule) within a set of identical processes to undertake a reaction individually, something that would not be possible in CSP1 . 2.2

π-Calculus

The π-calculus [Mil99] (and its stochastic form [Pri95]) was designed to express mobility, represented by the passing of channel names. It evolved from CCS [Mil80] and includes the operations of a constant, action preﬁx, choice, parallel composition, communication and scope restriction. There are variants of the syntax, here we use the following form with events π and processes P : π ::= τ | x | x | x(y) | xy P ::= 0 | π.P | P |P | P + P | νxP Following CCS [Mil80], τ is the unobservable event. All other events are observable and paired, e.g. x(y) with xy, with x(y) denoting input y on channel x, and xy denoting output y on channel x. 0 is the inactive process and νxP restricts the scope of the name x to P . In the stochastic form, rates are bound to channels, but as with the other process algebras, we will omit rates here. A structural congruence, denoted ≡, determines when two syntactic expressions are equivalent, and an operational semantics is given by a set of reaction rules that deﬁne how a system evolves following communication. We do not give the full deﬁnitions of the congruence and reaction rules, but note two distinguishing features. First, the constant, 0, is an identity for parallel composition, i.e. there is a syntactic equality P | 0 ≡ P . Second, interaction only occurs when there is a complementary pair of input and output events. The relevant reduction rule is (. . . + xy.Q) | (. . . + x(z).P ) → Q | P {y/z}. There have been numerous applications of the π-calculus to biomolecular processes, starting with the work of Regev et al. [RSS01]. An interesting aspect of the application of π-calculus is that it was designed to facilitate modelling mobility and name passing, thus in the original π-calculus events are parameterised, e.g. x(y). Yet, most biological applications do not exploit mobility — the parameter is not relevant, except when modelling compartments, or internal communications. So, in many models unparameterised events are also permitted, e.g. x and x, and we have also included them here. We note the recent work of Cardelli [Car08] on translations between process algebra and chemical reactions that introduces a subset of the π-calculus and CCS suitable for modelling chemical reactions. It is similar to the syntax above, but excludes event parameters and the ν operator. Additionally, it includes an expression of initial components. 1

This might explain why, to the best of our knowledge, there has been no work applying CSP to biomolecular modelling.

6

M. Calder and J. Hillston

A further distinctive aspect of the π-calculus/CCS paradigm for biomolecular modelling is the underlying assumption of two-way synchronous communication. This means that a a binary chemical reaction, e.g. of the form A + B →r C, is modelled by processes A and B oﬀering events r and r, whereas a unary chemical reaction, e.g. of the form A →r B, must be modelled by an unobservable τ event. 2.3

Beta-Binders

Beta-binders [DPPQ06] is a process algebra based on the π-calculus, designed for modelling and simulation of biological processes. A biological process is modelled by a bio-process, which is a π-calculus process encapsulated in a box with interaction capabilities expressed as beta-binders. Each communication channel has a set of associated types and there are three kinds of binder: visible, hidden, and complexed. Additionally, there are rates, but these are omitted here. A bio-process is either a constant or pair of encapsulated π-calculus processes composed with a synchronous parallel operator. The language has evolved over a number of years, here we use the following syntax for boxes B and beta-binders B, assuming π-calculus processes P : B ::= N il | B[P ] | B B B ::= β(x, Γ ) | β h (x, Γ ) | β c (x, Γ ) Further, there is a additional syntactic category for events, which include functions on boxes to join, split, create and destroy boxes; these are called join, split, new and delete, respectively. These functions are only applied when a condition, deﬁned over binders and π processes, is fulﬁlled. Interaction is two-way and is either intra-box, in which case it is standard π-calculus interaction, or it is inter-box in which case it is speciﬁed by the beta-binders and it is between (visible) input/output pairs, but now the types have only to be compatible (rather than identical). There are additional actions (within boxes) that include changing the status of binders (e.g. unhide or change type). There are three structural congruences: ≡p , the standard congruence on π processes, ≡b , a congruence on boxes (e.g. is associative, commutative), and ≡e , a congruence on events (e.g. join, split have substitution property). 2.4

PEPA

Performance Evaluation Process Algebra (PEPA) was introduced in the early 1990s as a formalism for building Markovian-based performance models of computer and communication systems [Hil96]. All actions in PEPA consist of an action type and a rate, which speciﬁes the average duration of the action as an exponentially distributed random variable. The language has a small set of combinators (preﬁx, choice, parallel composition/cooperation, hiding and constant).

Process Algebra Modelling Styles for Biomolecular Processes

7

Recursive behaviour is speciﬁed by mutually recursive deﬁnitions. As PEPA was designed for specifying ergodic continuous time Markov chains (CTMC), a restriction is often placed on model construction via a two level syntax, meaning that models consist of parallel compositions of sequential components (constructed using only preﬁx and choice): S := α.S | S + S | C

P := P P | P/L | S L where S denotes a sequential component, P a model component and C is a constant deﬁned by a declaration such as def

C=S α.S carries out activity α (with an exponentially distributed duration, but omitted here), and it subsequently behaves as S. As discussed above, PEPA supports multi-way cooperations between components: the result of synchronising on an activity α is thus another α, available for further synchronisation. We write P Q to denote cooperation between P and Q over L. The set which is used L as the subscript to the cooperation symbol, the cooperation set L, determines those activities on which the cooperands are forced to synchronise. For action types not in L, the components proceed independently and concurrently with their enabled activities. We write P Q as an abbreviation for P Q when L L is empty. P/L denotes the component P in which all actions with types in L are hidden meaning that their type is no longer visible but is replaced by the distinguished type τ . We do not consider hiding in the remainder of this paper. The stochastic nature of the actions means that the choice becomes a probabilistic choice governed by a race condition between the involved actions. Similarly actions of parallel components that are not forced to cooperate are also subject to a race condition. When components cooperate on actions but have diﬀerent deﬁnitions of the rate of the action, the rate of the synchronised action is deﬁned to be that of the slowest of the components. While these dynamic considerations do not concern us in this paper, and PEPA has been used for modelling a number of biological examples, we note that the form of the dynamics of synchronisation (the rate of the slowest component) is not always appropriate in this context. 2.5

Bio-PEPA

Bio-PEPA [CH08] is a newly deﬁned modiﬁcation of the PEPA formalism that has been speciﬁcally designed for modelling biochemical networks. It shares many features with PEPA but also has some characteristics to tailor it to the biological application. Functional rates: In contrast to PEPA, individual processes are not able to deﬁne their own rates for actions. Instead the rate associated with an action is speciﬁed once, independently of the processes in which the action occurs.

8

M. Calder and J. Hillston

The value of this rate can be speciﬁed to be a function that depends on the current state of the system. Stoichiometry: For each action, as well as its type, the stoichiometry or degree of involvement is also speciﬁed. Parameterised processes: Bio-PEPA has been designed to support the population-based reagent-centric style of modelling and so a model consists of a number of sequential components each representing a distinct species which evolve quantitatively (increasing or decreasing amounts). Thus in order to capture the state of a system each component is parameterised recording its current level. Diﬀerentiated preﬁx: For each action (reaction) that a component is involved in it records its role within that reaction, e.g. reactant, product, inhibitor etc. This enables the appropriate values to be used in the functional rate associated with this reaction. As with PEPA, Bio-PEPA has a two level grammar. The syntax of the sequential (species) components is deﬁned as: S ::= (α, κ) op S | S + S | C

op ::= ↓ | ↑ | ⊕ | | .

In the preﬁx term (α, κ) op S, α is an action name and can be viewed as the name or label of a reaction, κ is the stoichiometry coeﬃcient of the species and the preﬁx combinator op represents the role of the element in the reaction. Speciﬁcally, ↓ denotes the role of reactant, ↑ product, ⊕ activator, inhibitor and generic modiﬁer. The operator + expresses the choice between possible def actions and the constant C is deﬁned by an equation C = S. The syntax of model components is deﬁned as:

P ::= P P | S(x) L The process P Q denotes the synchronisation between components P and Q L and the set L speciﬁes those activities on which the components must synchronise. In the model component S(x), the parameter x ∈ R represents the initial concentration by default, although according to the analysis to be carried out the parameter may also be interpreted as number of molecules or molecular level after appropriate conversion. 2.6

sCCP

In the Concurrent Constraint Programming (CCP) process algebra, rather than components and actions, there are components and constraints [BJG96]; there are also variables. The components evolve by adding constraints to a constraint store (tell) or checking the current state of the constraint store (ask). This leads to an asynchronous form of communication between components (via global variables in the constraint store) and there is no direct synchronisation. In addition to tell and ask components may also have choice, parallel composition, procedure

Process Algebra Modelling Styles for Biomolecular Processes

9

call and local variables. In the stochastic form of CCP, sCCP [Bor06], a stochastic duration is associated with the ask and tell operators in a manner analogous to the durations of actions in other stochastic process algebras. sCCP has been proposed as a modelling formalism for biological networks, and stochastic, deterministic and hybrid semantics have been associated with models in this context [BP08]. The style of modelling is similar to that of BioPEPA in that a population-based view is taken, although here explicit variables record the quantitative state of species, rather than parameterised components. At a high level the abstraction is that measurable entities (molecules etc.) are associated with stream variables, logical entities are associated with processes or control variables and reactions are associated with processes. In general a reaction is modelled as a sequence of interactions with the constraint store: ﬁrst checking that there is suﬃcient amount of the substrates and then updating the amounts of the products. For mass action reactions the ask step of this sequence will be given a rate equal to the product of the kinetic constant and the amounts of the substrates; the tell step is assumed to be instantaneous. Thus an arbitrary mass action reaction R1 + . . . + Rn −→k P1 + . . . + Pm will be represented as reaction(k, [R1 , . . . , Rn ], [P1 , . . . , Pm ]) : − n askrMA (k,R1 ,...,Rn ) (Ri > 0) .

i=1

ni=1

tell∞ (Ri $= Ri − 1) m tell (P $= P + 1) ∞ j j j=1

Here Ri and Pj are stream variables and rMA is a predeﬁned function with the obvious deﬁnition.

3

Example Pathway

We refer to a small synthetic pathway when exploring how design decisions with respect to the the process algebra determine the mappings available to the modeller. The pathway consists of ﬁve representative reactions. The reactions are given in chemical notation in Figure 1, and presented graphically in Figure 2. While the pathway is a synthetic example, it is based on behaviour we have observed in various pathways, including the ubiquitous Raf/MEK/ERK signalling pathway. The equations exhibit various combinations of increasing/decreasing/preserved reagents between the left and right hand sides. Speciﬁcally, r1 and r4 have a decreasing number of reagents, r2 and r5 have an increasing number of reagents, and r3 has the same number of reagents on the left and right hand sides. Note that r5 has no reagent on the left hand side; we might use a reaction like this

10

M. Calder and J. Hillston A+B C B D+E

→r1 → r2 →r3 → r4 →r5

C A+B D B E

Fig. 1. Example pathway in chemical notation

A

r2

B

r4

r5 E

r1 C

r3 D

Fig. 2. Example pathway

to indicate that E is plentiful, or that it is produced by another pathway that is irrelevant to this abstraction. We will ﬁnd it useful to refer to the degree of a chemical reaction, meaning the number of reactants that it has i.e. the number of reagents on the left hand side. We have not included a homeo-reaction [Car08], where the components on the left hand side are identical, as it is only relevant to distinguish this case when rates are determined. In the example pathway, we assume initial concentrations of A, B and E, unless stated otherwise.

4 4.1

PEPA Models Reagent-Centric Style

In the reagent-centric view, ﬁrst proposed in [CGH06], species concentrations are discretised into levels; the granularity of the system is determined by the number of levels n and the concentration step size h, where there is a given maximum concentration max, h = max/n. As the number of levels increases/step size decreases, the granularity of the model increases. For each species, there is a family of processes, each deﬁning the behaviour for that (abstraction of) concentration. The system is deﬁned by the parallel composition of a number of initial components. The simplest abstraction is obtained when the number of levels is two, so that for each species there are two processes, denoting behaviour in the presence and absence of that species, respectively. We often refer to this kind of model as the high/low model. For example, for species A, AH denotes presence and AL denotes absence (alternatively A1 and A0 , respectively). Figure 3 gives the PEPA high/low model for the example pathway, consisting of a set of equations and a system deﬁnition. Figure 4 illustrates the state space for this model.

Process Algebra Modelling Styles for Biomolecular Processes

AH AL BH BL CH CL

def

= = def = def = def = def = def

r1.AL r2.AH r1.BL + r3.BL r2.BH + r4.BH r2.CL r1.CH

def

System = AH

DH DL EH EL

def

= = def = def = def

11

r4.DL r3.DH r4.EL r5.EH

BH CL DL EH ∗ ∗ ∗ ∗

Fig. 3. Example pathway: PEPA reagent-centric high/low model

r3

(AH , B H , C L , D L , E H) r1

(AH , B L , C L , D H, E H)

r5

r2

r4

(AH , B H , C L , D L , E L )

(AL , B L , C H , D L , E H) r5

r1

r5 r3

(AH , B L , C L , D H, E L )

r2

(AL , B L , C H , D L , E L )

Fig. 4. State space of the PEPA high/low model. Note that we use (AX , BX , CX , DX , EX ) to denote the state since the number of components is ﬁxed and the synchronisation structure does not change.

A0 A1 A2 B0 B1 B2 C0 C1 C2

def

= = def = def = def = def = def = def = def = def

def

r2.A1 r1.A0 + r2.A2 r1.A1 r2.B1 + r4.B1 r1.B0 + r3.B0 + r2.B2 + r4.B2 r1.B1 + r3.B1 r1.C1 r2.C0 + r1.C2 r2.C1

System = A2

D0 D1 D2 E0 E1 E2

def

= = def = def = def = def = def

r3.D1 r4.D0 + r3.D2 r4.D1 r5.E1 r4.E0 + r5.E2 r4.E1

B2 C0 D0 E2 ∗ ∗ ∗ ∗

Fig. 5. Example pathway: PEPA reagent-centric model with n = 3

(A0 , B 0 , C 2 , D 0 , E 1 )

(A 1 , B 1 , C 1 , D 0 , E 1 )

(A2 , B 2 , C 0 , D 0 , E 1 )

(A0 , B 0 , C 2 , D 0 , E 0 )

(A1 , B 1 , C 1 , D 0 , E 0 )

(A2 , B 2 , C 0 , D 0 , E 0 )

(A1 , B 0 , C 1 , D 1 , E 2 )

(A2 , B 1 , C 0 , D 1 , E 2 )

(A1 , B 0 , C 1 , D 1 , E 1 )

(A2 , B 1 , C 0 , D 1 , E 1 )

(A1 , B 0 , C 1 , D 1 , E 0 )

(A2 , B 1 , C 0 , D 1 , E 0 )

(A2 , B 0 , C 0 , D 2 , E 2 ) (A2 , B 0 , C 0 , D 2 , E 1 ) (A2 , B 0 , C 0 , D 2 , E 0 )

Fig. 6. State space of the PEPA reagent-centric model with n = 3. To avoid clutter in the diagram reaction labels are omitted, but r1 and r2 are shown in solid lines, r3 in dashed lines and r4 and r5 in dotted lines.

(A0 , B 0 , C 2 , D 0 , E 2 )

(A1 , B 1 , C 1 , D 0 , E 2 )

(A2 , B 2 , C 0 , D 0 , E 2 )

12 M. Calder and J. Hillston

Process Algebra Modelling Styles for Biomolecular Processes

13

As an example of a model with a diﬀerent granularity, Figure 5 contains a reagent-centric model with n = 3 (i.e. levels 0, 1, and 2). The state space is in Figure 6. Note that regardless of the number of levels, the number of (system) components is constant during system evolution, i.e. there are always ﬁve components (the number of species). Process as molecule in reagent-centric style. The granularity of the reagent-centric style depends on the step size h. In the limit, the ﬁnest grained model has a step size of one molecule. In general, it is impractical to increase n to its corresponding limit, but one alternative is to take a reagent-centric model with n = 1 and interpret each process as denoting the presence or absence of a molecule. An approach based on this abstraction has been used for studying the FGF pathway using stochastic model checking in [HKNT06]. For our example, for species A, AH denotes presence of a molecule and AL denotes absence. So, the population based high/low model model in Figure 3 can also be interpreted as an individuals model, with at most one molecule for each species. Similarly, a model consisting of (at most) two molecules for each species, is given by replacing the system deﬁnition of Figure 3 by the system deﬁnition: (AH AH ) (BH BH ) (CL CL ) (DL DL ) (EH EH ). ∗ ∗ ∗ ∗ Figure 7 illustrates a small portion of the corresponding state space (one transition step). Notice that this system describes the possible evolution of every molecule: it is very ﬁne grained. For example, from the initial state there are 8 possible transitions for reaction r1 , because there are two possible molecules of A that can be consumed, two possible molecules of B that can be consumed, and two possible molecules of C that can be produced (23 combinations). Similarly, there are 4 possibilities for reaction r3 . In many cases this degree of granularity is inappropriate. By appealing to symmetry (i.e. composition is commutative), we can use a form of counter abstraction to represent the molecules AH . . . AH by An , AH . . . AH AL n

n−1

by An−1 , and so on. This counter abstraction involves identifying an equivalence class of states in a high/low model of m molecules, with a state in a model n levels, where n = m. In other words, we deﬁne the processes as in the high/low

(AH, A H , B H, B H, C L , C L , D L , D L , E H , E H) r1

r2

r3

(AH, A H , B L , B H, C L , C L , D H, D L , E H , E H)

.. .

(AL , A H , B L , B H, C H, C L , D L , D L , E H , E H)

.. .

Fig. 7. One transition step in PEPA reagent-centric process-as-molecule model with two molecules

14

M. Calder and J. Hillston (A2 , B 2 , C 0 , D 0 , E 2 )

(AH, A H , B H, B H, C L , C L , D L , D L , E H , E H) r1

r2 r1

r2

(AL , A H , B L , B H, C H, C L , D L , D L , E H , E H) (AH, A L , B L , B H, C H, C L , D L , D L , E H , E H)

(A1 , B 1 , C 1 , D 0 , E 2 )

.. ..

Fig. 8. One transition step in the state space of the PEPA counter abstraction model

A

r2

B

r4

r5 E

r1 C

r3 D

Fig. 9. Example set of reactions with pathways indicated

model of Figure 3, then compose multiple copies of each process and interpret An as representing n molecules. This is illustrated in Figure 8, for the example pathway with two molecules for each species. States in the ﬁne-grained individuals model are quotiented and dashed lines indicate how the quotient class relates to a state in the counter abstraction model. 4.2

Pathway-Centric Style

An alternative style of modelling that has been proposed in PEPA is the pathway-centric style. In this style, we specify the sub-pathways that consume and replenish the initial species, which are the species with signiﬁcant initial concentrations. In the example pathway, this involves deﬁning the sub-pathways starting from A, B, and E. Call these P ath1 , P ath2 and P ath3 , respectively. The example pathway is given in Figure 10, with corresponding state space in Figure 11. Notice that although the system deﬁnition has only 3 components, this space is isomorphic to the high/low reagent-centric model (Figure 4). Notice also implicitly, the model has two levels. For example, P ath1 denotes high concentration of both A and B. We could make levels explicit in this style, by composing

Process Algebra Modelling Styles for Biomolecular Processes

15

def

P ath1 = r1 .r2 .P ath1 def P ath2 = r1 .r2 .P ath2 + r3 .r4 .P ath2 def P ath3 = r4 .r5 .P ath3 def

System = P ath1

P ath2 P ath3 ∗ ∗

Fig. 10. Example pathway: PEPA pathway-centric model

r3

(Path1 , Path2 , Path3 ) r1

r2

(Path 1 , r4.Path 2 , Path3 ) r4

r5

(r2.Path 1 , r2.Path2 , Path3 )

(Path 1 , Path2 , r5 .Path3 ) r5

r1

r3

r5 (Path1 , r4.Path2 , Path3 )

r2

(r2.Path 1 , r2.Path2 , r5.Path3 )

Fig. 11. Pathway-centric model state space

multiple copies of each pathway (with parallel composition, no synchronisation). For example the three level model would be: (P ath1 P ath1 ) (P ath2 P ath2 ) (P ath3 P ath3 ) ∗ ∗ In this case, similarly to the individuals reagent-centric style, there are more potential interleavings than in the reagent-centric population-based representation, and so the explicit state space here will be larger. However, again, by appealing to symmetry, we can work at the aggregate level. Thus for a given number of levels, the state space size and structure of both the pathway-centric and the reagent-centric models should be the same, as established in [CGH06]. Note that tools like the PEPA workbench [TDG09] can automatically detect such symmetries. We observe that assuming chemical reactions of at most degree two, we only require binary synchronisation, for this style of model.

5

Bio-PEPA

The Bio-PEPA formulation [CH08] of the reagent-centric style for the example pathway is given in Figure 12. This example does not fully exploit the power of Bio-PEPA, since the stochiometric coeﬃcients are all simple (1) and the functional rates are omitted. However, it does illustrate how the language focuses on the role of each species, in each reaction. Initial concentrations are denoted A0 for species A, etc. An integral part of a Bio-PEPA speciﬁcation (omitted here) is a deﬁnition of h and n, for every species, as well as initial concentrations (expressed as levels). The state space of this model depends upon the levels, for example, if the number of levels is uniformly 2, then the state space is the same as Figure 6.

16

M. Calder and J. Hillston A B C D E

def

= = def = def = def = def

(r1, 1)↓A + (r2, 1)↑A (r1, 1)↓B + (r2, 1)↑B + (r3, 1)↓B (r2, 1)↑C (r4, 1)↓D + (r3, 1)↑D (r4, 1)↓E + (r5, 1)↑E

System = A(A0 ) B(B0 ) C(C0 ) D(D0 ) E(E0 ) ∗ ∗ ∗ ∗ def

Fig. 12. Example pathway: Bio-PEPA model

Note that in the corresponding PEPA model (i.e. Figure 5), the number of levels is “hardwired” into the equations, whereas in the Bio-PEPA model, it is given as a parameter oﬀering more ﬂexibility to the modeller. If the number of levels is set suﬃciently high the model has a state space corresponding to an individuals model (i.e. if n is chosen to be the number of molecules).

6

π-Calculus

Models in the π-calculus and its stochastic variants predominantly follow the reactant style (e.g. [TK08]), based on the molecules-as-processes abstraction. Thus these are individuals based models. Figure 13 gives the π-calculus model in this style for the example pathway; since each reagent in the example also occurs on the left hand side of a chemical equation, there are processes for A . . . E. The example pathway highlights an interesting aspect of this style because in the biochemistry there are 1. equations with a decreasing number of components, and 2. an equation with no left hand side. Consider the former case. Since synchronisations are between reagents on the left hand side of an equation only, there is an arbitrary (and inconsequential) choice between which component is output and which is input. Further, the components on the left hand side, when translated into processes, evolve into components on the right hand side. If the number of components decreases, then we have to nominate one or more to evolve to 0, the null process. For example, A + B →r C could map to A = r.C and B = r.0; equally, it could map to A = r.0 and B = r.C, or A = r.0 and B = r.C, etc. Taking the ﬁrst choice, A | B evolves to C | 0. This is an example of a “trailing 0”, which is removed through application of the syntactic equality P | 0 ≡ P , i.e. A | B evolves to C. Now consider the second case. We cannot model an equation without a left hand side explicitly, e.g. r5 , but since E is present initially, we could represent the inﬁnite supply of E by a τ event, after oﬀering the output event r4 . However, this would constrain the creation of E to occur only after a molecule has been

Process Algebra Modelling Styles for Biomolecular Processes A B C D E Env

= = = = = =

17

r1 .C r1 .0 + τ.D τ.A | B r4 . B r4 .0 τ.Env | E

System = A | B | E | Env Fig. 13. Example pathway: π-calculus model

. .

..

.

. .

A | B | E | E | Env

.

.

A | D | E | E | Env

C | E | E | Env A | B | E | Env

C | E | Env

A | D | E | Env

A | B | Env

A | D | Env

C | Env

Fig. 14. π-calculus model state space

consumed in the reaction r4 . An alternative, which we use, is to introduce a representation of the environment Env and deﬁne it as follows: Env = τ.Env | E This presents the possibility that an unbounded number of E molecules may be introduced into the system, which is true when we represent the system only qualitatively. In the biological reality and when quantitative information is included in the model in the form of rates the system will become pragmatically bounded meaning that the probability for E to grow unboundedly is extremely small. Figure 14 illustrates possible evolutions for the system with one molecule of A, B and E initially, i.e. the evolution of A | B | E | Env. We have not labelled the transitions since events are either unobservable or become so after synchronisation. Notice that in this state space the number of system components ﬂuctuates, it both increases and decreases. Moreover the state space is inﬁnite due to the potentially unbounded number of E, although a graph isomorphic to the state space of the pathway-centric model is embedded within it. An alternative interpretation of this model is therefore a ﬁne-grained pathway-centric view based on

18

M. Calder and J. Hillston

molecules. Or rather, it is a mixture of two styles: equations are deﬁned for each reagent, but the system deﬁnition has the form of a pathway-centric model. While this approach provides a faithful overall system model, it is not compositional. Speciﬁcally, one equation incorporates aspects of the initial system and it would be misleading to a reader who inspected the behaviour only of a process that arbitrarily terminates, e.g. B, which can evolve into 0. Moreover, some reactions are represented explicitly by named events, i.e. r1 and r4 , whereas the unary or nullary reactions r2 , r3 and r5 are represented by the τ event. Thus, there are no occurrences of the reaction names r2 , r3 and r5 in the model.

7

Beta-Binders

There are several ways to map a chemical reaction in this formalism. For example, we could deﬁne a mapping very similar to the π-calculus mapping, with boxes for the processes that are initial, i.e. the system is given by [A] [B] [E], with suitable beta-binders deﬁned for each box, and each encapsulated process is deﬁned as in Figure 13. The authors recommend this mapping when the reaction denotes a collision of entities, the collision being mapped to (inter-box) communication. However, if we use this mapping, we are left with boxes containing the π-calculus constant process (i.e. 0) and we cannot remove them by the structural congruences: we need to introduce an explicit delete event to remove them. Alternatively, instead of representing reactions by inter-box communication, we could represent reactions by events, i.e. by the box operations. In this case, a reaction such as A + B →r C maps to (A, B) join C, where A, B, and C are constant bio-processes. Figure 15 gives a Beta-binders model of the example pathway using events. Notice that there are four events and no communication: the encapsulated processes are constants, except for process B, which changes its interaction type (to that of D). The state space is given in Figure 16; the space is isomorphic to the π-calculus model, though we could bound the occurrences of new E with a condition. The model is also a mixture of styles: equations are deﬁned for each reagent, but it is not reagent-centric: there is no communication and the system deﬁnition has the form of a reaction-centric model. There is a third possible mapping when the reaction denotes a binding (e.g. ligand to receptor); this is usually written in chemical notation as: A+B →r [A+ B]. In this case we could we use the complex/decomplex beta-binder operations to create and delete dedicated communication channels between boxes [A] and (A, B) join C where A = β(x, ΓA ) [nil] C split (A, B) B = β(x, ΓB ) [chtype(x, ΓD ). nil] (D, E) join B C = β(x, ΓC ) [nil] new E D = β(x, ΓD ) [nil] E = β(x, ΓE ) [nil] Fig. 15. Example pathway in Beta-binders

Process Algebra Modelling Styles for Biomolecular Processes

. .. . ..

19

. .. A

B

E

A

E

A C

E

B

D

E

E

A

E

D

E

E C

E

A

B

A

D

C

Fig. 16. State space of Beta-binders model. Note that following the graphical notation for Beta-binders, we omit the parallel composition operation on bio-processes.

[B]. That is, the two boxes [A] and [B] would evolve into a complex of two boxes, instead of into two separate boxes.

8

sCCP

Our last example is a model in sCCP. This is shown in Figure 17. There are ﬁve processes: one for each reaction, with stream variables representing the species. Each process has the form ask (check that there is suﬃcient of a species) followed by the parallel composition of all the possible eﬀects of the reaction (i.e. production or consumption) expressed by tell. The state space of the model is shown in Figure 18. Unsurprisingly this includes the state space which has been retrieved from the other models, such as the reagent-centric PEPA models (shown in the shaded area in the diagram). However note that this model also permits the unbounded growth of the population of E (as in the π-calculus model), leading to an inﬁnite state space unless an explicit guard is inserted which disables reaction r5 when the population of E reaches a given size. This model bears some similarity to the state based PRISM model given in [CVOG06], where species are represented by state variables. This is not surprising, since the PRISM modelling language is essentially the language of reactive modules [AH90]. However, in [CVOG06], there is still explicit synchronisation and commands are grouped by species, rather than by reaction. The reactionsas-processes models of sCCP can therefore be considered to be reaction-centric and in that they are similar to other rule-based formalisms such as the κ-calculus [VFF+ 07] and BIOCHAM [CRCD+ 04].

9

Discussion

The three main abstractions for mapping chemical equations to process algebras are molecule-as-process, species-as-process, and reaction-as-process. We have

20

M. Calder and J. Hillston

reaction(r1 , [A, B], [C]) : − ask(A > 0 ∧ B > 0). (tell(A $= A − 1) tell(B $= B − 1) tell(C $= C + 1)) reaction(r2 , [C], [A, B]) : − ask(C > 0). (tell(C $= C − 1) tell(A $= A + 1) tell(B $= B + 1)) reaction(r3 , [B], [D]) : − ask(B > 0). (tell(B $= B − 1) tell(D $= D + 1)) reaction(r4 , [D, E], [B]) : − ask(D > 0 ∧ E > 0). (tell(D $= D − 1) tell(E $= E − 1) tell(B $= B + 1)) reaction(r5 , [], [E]) : − (tell(E $= E + 1))

5 reaction system : − reaction(r1 , [A, B], [C]) reaction(r2 , [C], [A, B]) reaction(r3 , [B], [D]) reaction(r4 , [D, E], [B]) reaction(r5 , [], [E]) Fig. 17. Example pathway: sCCP model

. .. . ..

. .. r3

(A=1, B=1, C=0, D=0 , E=2)

r1

(A=1, B=0, C=0, D=1 , E=2)

r5

r2

r5

r4

r3

(A=1, B=1, C=0, D=0 , E=1)

(A=0, B=0, C=1, D=0 , E=2)

r5

r1

(A=1, B=0, C=0, D=1 , E=1)

r5

r2

r4

(A=1, B=1, C=0, D=0 , E=0)

(A=0, B=0, C=1, D=0 , E=1)

r5

r1

r5 r3

(A=1, B=0, C=0, D=1 , E=0)

r2

(A=0, B=0, C=1, D=0 , E=0)

Fig. 18. State space of the sCCP model of the example

further deﬁned four styles: reagent-centric, pathway-centric, reactant-centric, and reaction-centric. We have presented reactant-centric π-calculus and Beta-binders models, and (individuals-based) reagent-centric PEPA models as examples of the the molecule-as-process abstraction, (population-based) pathway-centric PEPA and (population-based) reagent-centric Bio-PEPA models as examples of the species-as-process abstraction, and a reaction-centric sCCP model as an example of the reaction-as-process abstraction.

Process Algebra Modelling Styles for Biomolecular Processes

21

The styles of modelling supported by a process algebra is strongly inﬂuenced by the form of synchronisation available. Whilst languages with multiway synchronisation are capable of representing models in reagent-centric, reactantcentric or pathway-centric style, the same is not true for languages with conjugate actions and binary synchronisation. These languages cannot generally represent reactions in the reagent-centric style. Only ﬁrst degree, or unary, reactions could be modelled in this style in these languages. In the example considered we have only considered reactions with degree one and two — indeed there are thermodynamic arguments for restricting consideration to such reactions if we wish to be faithful to biochemistry. However, abstractions which lead to higher degree reactions are often applied by biologists for a variety of reasons. For example, consider the enzyme-enabled association of two smaller molecules (A and B) into a complex C. In terms of elementary reactions this might proceed as follows: A+B+E

r2 ←→

r1

A + B:E −→r3 C + E

where E is the enzyme and B:E is a complex formed from B and the enzyme. E This could abstracted as A + B −→r C. The abstraction has the advantage that the number of reagents considered in the transformation is reduced, and that the number of reaction rates which have to be measured, estimated or ﬁtted is cut from three to one. Moreover this is typically more consistent with what can be observed in the lab as r1 , r2 r3 . It may not even be known whether the enzyme binds with A or B, leaving uncertainty about how to model the reaction without the abstraction. However, representing this in even the reactant-centric style requires three-way synchronisation, and four-way synchronisation in the reagentcentric style, assuming that the enzyme is modelled as both a reactant and a product in the abstracted reaction. In other words, it is not possible to support modelling such biological abstractions using strictly binary synchronisations. As with reagent-centric style, reaction-centric style seems to implicitly assume a multi-way synchronisation. However note that in the way that this style is captured in sCCP, the only process algebra that currently supports reactioncentric modelling to the best of our knowledge, the requirement is not so strong. What is needed is atomic multi-way composition of updates to the constraint store, but this is not necessarily a synchronisation. Whilst sCCP is the only process algebra supporting reaction-centric, or reaction-as-process, modelling, conversely it is diﬃcult to see sCCP being used to construct models in any of the other styles or abstractions. In process algebras with conjugate actions, each partner in an action/reaction must be assigned an input/output role. In general this will be rather arbitrary and somewhat artiﬁcial from the perspective of the biochemistry. Consider the reaction r1 in our example. When A and B form the complex C there does not appear to be a natural way to choose which of A and B should receive input and which provide output. Furthermore, reactions of degree one, such as r5 in the example, must be represented as a τ action. This means that the textual representation of the model does not clearly articulate the biologists’ notion of

22

M. Calder and J. Hillston

the system. This problem becomes even worse at the level of the state space where all transitions are labelled τ and information about the reactions that gave rise to them is lost. If we consider the contrast between population-based and individuals-based modelling we can observe that population-based modelling is more compact both from the point of view of the textual model expression and the underlying state space. This means that for such models it can be feasible to use explicit state space representations and the analysis techniques associated with them such as model checking, equivalence checking and numerical analysis of the continuous time Markov chain. Of course, such techniques reply on the state space being ﬁnite. In contrast individuals-based modelling has a clear association with stochastic simulation as proposed by Gillespie [GP06]. These models can be used in association with explicit state space techniques, such as those listed above, but only for very small systems or in combination with abstractions such as the assumption of single molecules, as discussed in Section 4.1. In the PEPA and Bio-PEPA models, as a consequence of the two level grammar used to deﬁne these languages as compositions of sequential components, the number of system components is constant, regardless of whether individualsbased or population-based. This matches the species-as-process abstraction since the possible species of the pathway will be known and ﬁxed and is particularly natural in the population-based modelling where the state of the system is a count for each species. In contrast, in the π-calculus and Beta-binder models, which are without the syntactic restriction, the number of system components ﬂuctuates throughout system evolution. This is in keeping with the moleculesas-processes abstraction since we would expect the visible molecules within a system to change as complexes are formed and dissociated etc. In sCCP, based on the reaction-as-process abstraction, the number of species is ﬁxed as the variables in the variable store remain ﬁxed. Here as in the PEPA/Bio-PEPA population-based modelling the state of the system is captured in terms of the number of each species so each species must always be present, even if to record that its current count is zero. The conservative nature of the PEPA/Bio-PEPA models (in terms of number of components, and ﬁxed number of levels) also means that the state space underlying such models is necessarily ﬁnite. This is not the case in the other process algebras as we have seen. It can be argued that if we consider the example as presented there is the potential for unbounded numbers of E via reaction r5 and π-calculus, beta binder and sCCP correctly capture this. But on the other hand, in a biological system unbounded growth like this will lead to cell death, and when we introduced the example we explained that this reaction would be used as an abstraction of some more complex, but bounded, situation. The beta binders and sCCP formalisms do oﬀer language mechanisms which allow the number of E to remain bounded by introducing guards on the reaction, but there is no such possibility in the π-calculus. In this paper we have focussed on the standard discrete state spaces. However analysis based on these state spaces is rarely feasible. Therefore for all the

Process Algebra Modelling Styles for Biomolecular Processes

23

languages there are alternative semantics given by ordinary diﬀerential equations (population-based) and/or Gillespie simulations (individuals-based). The discrete state space does of course form the basis of the Gillespie simulation but it is never considered explicitly and the oﬀered semantics avoid the construction. Additionally, PEPA and Bio-PEPA support an alternative representation, which is based on an explicit discrete state space but seeks to avoid the state space explosion. Rather than states representing the count of molecules of each species, the states represent the current level of concentration for each species. In other words, the range of possible concentration values is discretised into intervals, and these intervals constitute the states of the CTMC. In such models the stochastic element of Gillespie’s approach is retained but the resulting CTMCs can be considerably smaller. Keeping the state space manageable means that the CTMCs can be solved explicitly and the repeated runs necessitated by stochastic simulation are avoided. Further, in addition to quantitative analysis on the CTMC, analysis by model checking of stochastic properties is possible, as illustrated in [CVOG06] or [HKNT06].

10

Conclusions

As highlighted by Regev and Shapiro computational abstractions have already brought considerable beneﬁt to the study of biological phenomena [RS01]. For example the DNA-as-string abstraction has been hugely successful and allowed signiﬁcant leaps forward. In the context of biomolecular processes the potential beneﬁt seems equally large. However further work is needed to assess the abstractions that are on oﬀer, and their suitability to the systems under study. Research in this direction has been enthusiastically taken up by theoretical computer scientists as witnessed by the plethora of formal languages currently proposed for modelling such systems. In this paper we have aimed to extract the general paradigms of expression which underlie process algebras which aim to model biomolecular processes. We have discovered that there are genuine diﬀerences in the form of expression used, and this can impact on the form of analysis that is readily applied. In the long term all research on formal description techniques for biomolecular systems has the objective of attracting biological users, and contributing to the growing body of knowledge on how cells function. However in the medium term we need to develop closer links with biologists, not only as users of our formal description techniques, but also in the important work of evaluating them.

References [AH90] [BMMT06]

Alur, R., Henzinger, T.A.: Reactive modules. Formal methods in System Design 15(1), 7–48 (1990) Barbuti, R., Maggiolo-Schettini, A., Milazzo, P., Troina, A.: A Calculus of Looping Sequences for Modelling Microbiological Systems. Fundamenta Informaticae 72(1-3), 21–35 (2006)

24

M. Calder and J. Hillston

[BJG96]

[Bor06]

[BP08] [CGH06]

[Car04]

[Car08] [CPR+ 04]

[CG09] [CH08] [CRCD+ 04]

[CVOG06]

[VFF+ 07]

[DPPQ06]

[GP06]

[HKNT06]

[Hil96] [Hoa85]

Brim, L., Jacquet, J.-M., Gilbert, D.: A process algebra for synchronous concurrent programming. In: Hanus, M., Rodr´ıguez-Artalejo, M. (eds.) ALP 1996. LNCS, vol. 1139, pp. 165–178. Springer, Heidelberg (1996) Bortolussi, L.: Stochastic concurrent constraint programming. In: Proceedings of QAPL 2006: 4th International workshop on quantitative aspects of programming languages, vol. 164, pp. 65–80 (2006) Bortolussi, L., Policriti, A.: Modelling biological systems in stochastic constraint programming. Constraints 13, 66–90 (2008) Calder, M., Gilmore, S., Hillston, J.: Modelling the inﬂuence of RKIP on the ERK signalling pathway using the stochastic process algebra PEPA. In: Priami, C., Ing´ olfsd´ ottir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 1–23. Springer, Heidelberg (2006) Cardelli, L.: Brane Calculus. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 257–278. Springer, Heidelberg (2005) Cardelli, L.: On process rate semantics. Theoretical Computer Science 391(1), 190–215 (2008) Cardelli, L., Panina, E.M., Regev, A., Shapiro, E., Silverman, W.: BioAmbients: An Abstraction for Biological Compartments. Theoretical Computer Science 325(1), 141–167 (2004) Ciocchetta, F., Guerriero, M.L.: Modelling Biological Compartments in Bio-PEPA. ENTCS 227, 77–95 (2009) Ciochetta, F., Hillston, J.: Bio-PEPA: a framework for modelling and analysis of biological systems. Theoretical Computer Science (to appear) Chabrier-Rivier, N., Chiaverini, M., Danos, V., Fages, F., Sch¨achter, V.: Modeling and querying biomolecular interaction networks. Theoretical Computer Science 325(1), 25–44 (2004) Calder, M., Vyshemirsky, V., Orton, R., Gilbert, D.: Analysis of signalling pathways using Continuous Time Markov Chains. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 44–67. Springer, Heidelberg (2006) Danos, V., Feret, J., Fontana, W., Harmer, R., Krivine, J.: Rule-based modelling of cellular signalling. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 17–41. Springer, Heidelberg (2007) Degano, P., Prandi, D., Priami, C., Quaglia, P.: Beta-binders for biological quantitative experiments. Electronic Notes in Computer Science 164, 101–117 (2006) Gillespie, D., Petzold, L.: Numerical Simulation for Biochemical Kinetics. In: System Modelling in Cellular Biology. MIT Press, Cambridge (2006) Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic model checking of complex biological pathways. In: The Proceedings of 4th International Workshop on Computational Methods in Systems Biology 2006, Trento, Italy, October 18-19 (2006) Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press, Cambridge (1996) Hoare, C.A.R.: Communicating Sequential Processes. Prentice-Hall, Englewood Cliﬀs (1985)

Process Algebra Modelling Styles for Biomolecular Processes [Mil80] [Mil99] [Pri95] [RS01] [RSS01]

[TK08]

[TDG09] [V07]

25

Milner, R.: A Calculus for Communicating Systems. LNCS, vol. 92. Springer, Heidelberg (1980) Milner, R.: Communicating and Mobile Systems: the π-Calculus. Cambridge University Press, Cambridge (1999) Priami, C.: Stochastic π-calculus. The Computer Journal 38, 578–589 (1995) Regev, A., Shapiro, E.: Cellular abstractions: cells as computation. Nature 419, 343 (2001) Regev, A., Silverman, W., Shapiro, E.: Representation and simulation of biochemical processes using π-calculus process algebra. In: Paciﬁc Symposium on Biocomputing 2001 (PSB 2001), pp. 459–470 (2001) Tymchyshyn, O., Kwiatkowska, M.: Combining intra- and inter-cellular dynamics to investigate intestinal homeostasis. In: Fisher, J. (ed.) FMSB 2008. LNCS (LNBI), vol. 5054, pp. 63–76. Springer, Heidelberg (2008) Tribastone, M., Duguid, A., Gilmore, S.: The PEPA Eclipse Plug-in. Performance Evaluation Review 36(4), 28–33 (2009) Versari, C.: A Core Calculus for a Comparative Analysis of Bioinspired Calculi. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 411–425. Springer, Heidelberg (2007)

Simple, Enhanced and Mutual Mobile Membranes Bogdan Aman and Gabriel Ciobanu Romanian Academy, Institute of Computer Science, Ia¸si, Romania A.I.Cuza University, 700506 Ia¸si, Romania [email protected], [email protected]

Abstract. The operations governing the movement of biological membranes are endocytosis and exocytosis. New models of computation are inspired by these biological operations. In this paper we present the models deﬁned by simple, enhanced and mutual mobile membranes, together with their biological motivations. Some results concerning their computational power are presented, including the ﬁrst universality result for mutual mobile membranes. In the case of simple and enhanced mobile membranes, we improve the existing results by reducing the number of membranes needed to get computational universality.

1

Introduction

Simple, enhanced and mutual mobile membranes represent new variants of membrane systems. Membrane systems (also called P systems) were introduced in [16]; standard P systems and several variations are presented in the monograph [17]. Membrane systems were introduced as distributed, parallel and nondeterministic computing models inspired by the compartments of eukaryotic cells and by their biochemical reactions. The cellular components are formally represented in the deﬁnition of membrane systems. The structure of the cell is represented by a set of hierarchically embedded regions, each one delimited by a surrounding boundary (called membrane), and all of them contained inside an external special region called the skin membrane. The molecular species (ions, proteins, etc.) ﬂoating inside cellular compartments are represented by multisets of objects described by means of symbols or strings over a given alphabet, objects which can be modiﬁed or communicated between adjacent compartments. Chemical reactions are represented by evolution rules given in the form of rewriting rules which operate on the objects, as well as on the compartmentalized structure (by dissolving, dividing, creating, or moving membranes). A membrane system can perform computations in the following way: starting from an initial conﬁguration which is deﬁned by the multiset of objects initially placed inside the compartmentalized structure, the system evolves by applying the evolution rules of each membrane in a nondeterministic and maximally parallel manner. A rule is applicable when all the objects that appear in its left hand side are available in the region where the rule is placed. The maximal C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 26–44, 2009. c Springer-Verlag Berlin Heidelberg 2009

Simple, Enhanced and Mutual Mobile Membranes

27

parallelism of rule application means that every rule that is applicable inside a region has to be applied in that region. A halting conﬁguration is reached when no rule is applicable. The result is represented by the number of objects from a speciﬁed region. Several variants of membrane systems are inspired by diﬀerent aspects of living cells (symport and antiport-based communication through membranes, catalytic objects, membrane charge, etc.). Their computing power and eﬃciency have been investigated using the approaches of formal languages and grammars, register machines and complexity theory. An updated bibliography can be found at the webpage http://ppage.psystems.eu A ﬁrst deﬁnition of mobile P systems is given in [21] with rules coming from mobile ambients [5]. Inspired by the operations of endocytosis and exocytosis, namely moving a membrane inside a neighbouring membrane (endocytosis) and moving a membrane outside the membrane where it is placed (exocytosis), the P systems with mobile membranes are introduced in [14] as a variant of P systems with active membranes [17]. We use simple mobile membranes instead of P systems with mobile membranes. The computational power of simple mobile membranes is treated in [12,14]: Turing completeness is obtained by using nine membranes together with the operations of endocytosis and exocytosis [14], while only four mobile membranes are enough using additional contextual evolution rules [12]. In this paper we look at certain biological phenomena which motivate and inspire new speciﬁc rules in simple mobile membranes. Endocytosis is a general term for a group of processes that bring macromolecules, large particles, small molecules, and even small cells into another cell. There are three types of endocytosis: phagocytosis (“cellular eating”), pinocytosis (“cellular drinking”), and receptor-mediated endocytosis in which the membrane infolds around materials from the environment, forming a small pocket. The pocket deepens, forming a vesicle which separates from the membrane and migrates with its contents to the cell’s interior. While pinocytosis can be modelled using communication rules of usual P systems, there is no rule capable to model the process of engulﬁng a cell by another one in phagocytosis. This is the reason why we deﬁne the enhanced mobile membranes in Subsection 2.2; an example on how the new rules work is also presented. The enhanced mobile membranes represent a variant of simple mobile membranes; they have been proposed in [3] for describing some biological mechanisms of the immune system. The operations governing the mobility of the enhanced mobile membrane systems are endocytosis (endo), exocytosis (exo), enhanced endocytosis (fendo) and enhanced exocytosis (fexo). The computational power of the enhanced mobile membranes using these four operations was studied in [13] where it is proved that twelve membranes can provide the computational universality. It is worth noting that unlike the results for simple mobile membranes, the context-free evolution of objects is not used in proving any of these results.

28

B. Aman and G. Ciobanu

Receptor-mediated endocytosis is used by animal cells to capture speciﬁc macromolecules from the cell’s environment. This process depends on receptor proteins, i.e., integral membrane proteins that can bind to a speciﬁc molecule in the cell’s environment. The uptake process is similar to nonspeciﬁc endocytosis. However, in receptor-mediated endocytosis, the receptor proteins at particular sites on the extracellular surface of the plasma membrane bind to speciﬁc substances. These sites are called coated pits because they form a slight depression in the plasma membrane. The cytoplasmic surface of a coated pit is coated by proteins, such as clathrin. Strengthened and stabilized by clathrin molecules, this vesicle carries the macromolecule into the cell [23]. SNARE-mediated exocytosis is the movement of materials out of a cell via vesicles. SNARES (Soluble NSF Attachment Protein Receptor)) located on the vesicles (v-SNARES) and on the target membranes (t-SNARES) interact to form a stable complex that holds the vesicle very close to the target membrane. There is no rule capable to model the mutual agreement between membranes for the receptor-mediated endocytosis and SNARE-mediated exocytosis. This is the reason why we deﬁne the mutual mobile membranes in Subsection 2.3; an example on how the new rules work is also presented. The mutual mobile membranes represent a variant of simple mobile membranes in which the endocytosis and exocytosis work whenever the involved membranes “agree” on the movement; this agreement is described by using dual objects a and a in the involved membranes. The operations governing the mobility of the mutual mobile membranes are mutual endocytosis (mutual endo), and mutual exocytosis (mutual exo). In this paper we study the computational power of simple, enhanced and mutual mobile membranes. For simple mobile membranes we obtain the computational universality by using three membranes, and in this way improving the result presented in [12] where four membranes are used. For enhanced mobile membranes we obtain the computational universality by using nine membranes, thus improving the result from [13] where twelve membranes are used. For mutual mobile membranes we show that by using dual objects a and a in the involved membranes, only seven membranes are enough to obtain the computational universality. The structure of the paper is as follows. In Section 2 we formally deﬁne the simple, enhanced and mutual mobile membranes, and give biological motivations for the enhanced and mutual mobile membranes. Section 3 contains a ﬁrst universality result for mutual mobile membranes, and improvements of the existing results for simple and enhanced mobile membranes. Section 4 presents related results in P systems with active membrane from which the simple mobile membranes originate. Conclusions and references end the paper.

2

Mobile Membranes

In this section we deﬁne the simple, enhanced and mutual mobile membranes, describing some biological phenomena inspiring their rules.

Simple, Enhanced and Mutual Mobile Membranes

2.1

29

Simple Mobile Membranes

Definition 1 ([14]). A simple mobile membrane is a construct Π = (V, H, μ, w1 , . . . , wn , R) where: n ≥ 1 (the initial degree of the system); V is an alphabet (its elements are called objects); H is a ﬁnite set of labels for membranes; μ ⊂ H × H describes the membrane structure, such that (i, j) ∈ μ denotes that the membrane labelled by j is contained in the membrane labelled by i; we distinguish the external membrane (usually called the “skin” membrane) and several internal membranes; a membrane without any other membrane inside it is said to be elementary; 5. w1 , . . . , wn are strings over V , describing the multisets of objects placed in the n regions of μ; 6. R is a ﬁnite set of developmental rules, of the following forms: 1. 2. 3. 4.

object evolution ∗

(a) [a → v]m , for m ∈ H, a ∈ V , v ∈ V ; An object a placed inside a membrane labelled m evolves into a multiset of objects v. endocytosis (b) [a]h [ ]m → [[b]h ]m , for h, m ∈ H, a, b ∈ V ; An elementary membrane labelled h enters the adjacent membrane labelled m, under the control of object a; the labels h and m remain unchanged during the process; however the object a may be modiﬁed to b during the operation; m is not necessarily an elementary membrane. exocytosis (c) [[a]h ]m → [b]h [ ]m , for h, m ∈ H, a, b ∈ V ; An elementary membrane labelled h is sent out of a membrane labelled m, under the control of object a; the labels of the two membranes remain unchanged, but the object a of membrane h may be modiﬁed during this operation; membrane m is not necessarily elementary. The rules are applied according to the following principles: 1. Rules are applied in parallel, non-deterministically choosing the rules, the membranes, and the objects in such a way that the parallelism is maximal; this means that in each step we apply a certain set of rules such that no further rule can be added to the set. 2. The membrane m from the rules of type (a) − (c) is said to be passive, while the membrane h is said to be active. In any step of a computation, any object and any active membrane can be involved in at most one rule. However, the passive membranes can be used by several rules at the same time. In a rule [a → v]m of type (a), object a is active, while membrane m is passive.

30

B. Aman and G. Ciobanu

3. When a membrane is moved across another membrane, by endocytosis or exocytosis, its whole contents (its objects) are moved; the inner objects evolve ﬁrst (if rules are applicable for them), and then any membrane is moved with the contents as obtained after its internal evolution. 4. If a membrane exits the system (by exocytosis), then its internal evolution stops, even if there are rules of type (a) which could be applied. 5. The objects and membranes which do not evolve at a given step are passed unchanged to the next conﬁguration of the system. 2.2

Enhanced Mobile Membranes

The enhanced mobile membranes have been introduced in [3] for describing some biological mechanisms of the immune system. The presentation of the immune system is taken from [10], a book which is revised every few years to keep the pace with the new discoveries in this ﬁeld. The cells of the immune system work together with diﬀerent proteins to seek out and destroy anything foreign or dangerous which enters our body. It takes some time for the immune cell to be activated, but once this happens there are very few hostile organisms having a chance. There are several types of immune cells, each of them with its own strength and weakness. Some seek out and engulf the invaders, while other destroy the infected or mutated body cells. A type of immune cells are the B cells which have the ability to release special proteins called antibodies which mark intruders in order to be destroyed by macrophages. The immune system has also the ability to produce some cells able to remember enemies which it fought in the past. In this way, once the immune system recognizes an invader it attacks more quickly and strongly against it. Dendritic cells can engulf bacteria, viruses, and other cells. Once a dendritic cells engulfs a bacterium, it dissolves this bacterium and places portions of

Fig. 1. Immune System Mechanisms [10]

Simple, Enhanced and Mutual Mobile Membranes

31

bacterium proteins on its surface (see Figure 1). These surface markers serve as an alarm to other immune cells, namely helper T cells, which then infer the form of the invader. This mechanism makes sensitive the T cells to recognize the antigens or other foreign agents which triggers a reaction of the immune system. Antigens are often found on the surface of bacterium and viruses. New rules are introduced according to this biological example. We deﬁne a new variant of mobile membranes, namely the enhanced mobile membranes, originally introduced in [3]. The multiset u is the one indicating the membrane which initializes the move in the rules of type (b) − (e). Definition 2 ([3]). An enhanced mobile membrane is a construct = (V, H, μ, w1 , . . . , wn , R), where: 1. n, V , H, μ, w1 , . . . , wn are as in Deﬁnition 1; 2. R is a ﬁnite set of developmental rules of the following forms: local evolution (a) [ [u → v]m ]h for h, m ∈ H, u ∈ V + , v ∈ V ∗ ; These rules are called local because the evolution of a multiset of objects u of membrane m is possible only when membrane m is inside membrane h. If the restriction of nested membranes is not imposed, that is, the evolution of the multiset of objects u in membrane m is allowed wherever membrane m is placed, then we say that we have a global evolution rule, and write it simply as [u → v]m . endocytosis (b) [uv]h [v ]m → [[w]h w ]m for h, m ∈ H; u ∈ V + , v, v , w, w ∈ V ∗ ; An elementary membrane labelled h enters the adjacent membrane labelled m, under the control of the multisets of objects uv and v . The labels h and m remain unchanged during this process; however the multisets of objects uv and v are replaced with the multisets of objects w and w , respectively. exocytosis + ∗ (c) [[uv]h v ]m → [w]h [w ]m , for h, m ∈H; u ∈ V , v, v , w, w ∈ V ; An elementary membrane labelled h is sent out of a membrane labelled m, under the control of the multisets of objects uv and v . The labels of the two membranes remain unchanged, but the multisets of objects uv and v are replaced with the multisets of objects w and w , respectively. enhanced endocytosis + (d) [v]h [uv ]m→[[w]h w ]m for h, m∈H, u ∈ V , v, v , w, w ∈ V ∗ ; An elementary membrane labelled h is engulfed into the adjacent membrane labelled m, under the control of the multisets of objects uv and v. The labels h and m remain unchanged during the process; however, the multisets of objects uv and v are transformed into the multisets of objects w and w, respectively. The eﬀect of this rule is similar to the eﬀect of rule (b); the diﬀerence is that the movement is not controlled by a multiset of objects inside the moving membrane h, but by a multiset

32

B. Aman and G. Ciobanu

of objects uv placed inside the membrane m which engulfs membrane h. This means that the membrane which initiates the movement is membrane m, and not the membrane h as in rule (b). enhanced exocytosis (e) [[v]h uv ]m→[w]h [w ]m for h, m ∈ H, u ∈ V + , v, v , w, w ∈ V ∗ ; An elementary membrane labelled h is pushed out of a membrane labelled m under the control of the multisets of objects uv and v. The labels of the two membranes remain unchanged; however, the multisets of objects uv and v evolve into the multisets of objects w and w, respectively. The eﬀect of this rule is similar to the one of rule (c); the diﬀerence is that the movement is not controlled by an object inside the moving membrane h, but by a multiset of objects uv placed inside the membrane m which expels membrane h. This means that the membrane which initiates the movement is membrane m, and not the membrane h as in rule (c). The rules of enhanced mobile membranes are applied according to the principles of simple mobile membranes. Using the rules of the enhanced mobile membranes we can describe the immune system mechanisms of Figure 1. We associate a membrane to each cell, and objects to the signals, states and parts of molecules. For the steps done by the dendritic cells presented in Figure 1, we use the following encodings: – dendritic cell: [eat]DC An immature dendritic cell is willing to eat any bacterium it encounters, so we translate it into a membrane labelled by DC which has inside an object eat used to engulf the bacterium. – bacterium cell: [antigen]bacterium A bacterium cell contains antigen so we simply represent it as a membrane labelled by bacterium containing a single object antigen that encodes the information of the bacterium. – lymph node: [ ]lymph node The lymph node is the place where the mature dendritic cells migrate in order to start the immune response, so we translate it into a membrane labelled by lymph node. Using these membranes, we describe the system as follows (here body stands for the body skin): [[eat]DC [ ]lymph node ]body [antigen]bacterium The evolution is described by following rules: * [antigen]bacterium [ ]body → [[antigen]bacterium ]body A bacterium enters through the body skin by performing an endocytosis rule in order to infect the body. The bacterium contains an object antigen which represent its signature.

Simple, Enhanced and Mutual Mobile Membranes

33

* [eat]DC [ ]bacterium → [eat[ ]bacterium ]DC Once an immature dendritic cell becomes sibling to a bacterium, it “eats” the bacterium by performing an enhanced endocytosis rule. Until now the bacterium has controlled its own movement; in this step its movement becomes controlled by the dendritic cell which engulfs it. * [[antigen]bacterium ]DC → [antigen]DC Once the bacterium is engulfed into the dendritic cell, it is dissolved and its content is released into the dendritic cell. * [antigen]DC [ ]lymph node → [[antigen]DC ]lymph node Once the dendritic cell contains parts of the antigen, it enters the lymph node in order to activate a special class of T cells, namely the helper T cells. * [[eat]DC ]lymph node → [[ ]DC ]lymph node Once the dendritic cell enters the lymph node, it matures and the capacity to engulf bacteria disappears; the eat object is consumed. 2.3

Mutual Mobile Membranes

In a receptor-mediated endocytosis a cell engulfs a particle of low-density lipoprotein (LDL) from the outside [23]. To do this, the cell uses receptors that speciﬁcally recognize and bind to the LDL particle. The receptors are clustered together. An LDL particle contains one thousand or more cholesterol molecules. A monolayer of phospholipid surrounds the cholesterol and its embedded with proteins called apoB. This apoB proteins are speciﬁcally recognized by receptors on the cell membrane. The receptors of the coated pit bind to the apoB proteins of the LDL particle. The pit is reenforced by a lattice-like network of proteins called clathrin. Additional clathrin molecules are added to the lattice which eventually pinches oﬀ apart from the membranes. SNARE-mediated exocytosis is the movement of materials out of a cell via vesicles. SNARES located on the vesicles (v-SNARES) and SNARES located on the target membranes (t-SNARES) interact to form a stable complex that holds the vesicle very close to the target membrane as in Figure 3.

Fig. 2. Receptor-Mediated Endocytosis [23]

34

B. Aman and G. Ciobanu

Fig. 3. SNARE-Mediated Exocytosis

Starting from these biological examples we see the necessity to introduce new rules. The rules of enhanced mobile membranes allow a membrane to enter, exit, to engulf or to push out another membrane. The second membrane just undergoes the movement; no permission is required from the second membrane which may not even be aware that a movement involving it has taken place. Following an approach described initially in [4], we introduce a new variant of mobile membranes called mutual mobile membranes. In mutual mobile membranes, a movement takes place only if the involved membranes agree on the movement. This can be described by means of objects a and co-objects a present in the membranes involved in such a movement. Since we have the equality a = a, we have that mutual endocytosis is the same as mutual enhanced endocytosis and mutual exocytosis is the same as mutual enhanced exocytosis. The mutual mobile membranes are deﬁned as follows: Definition 3 ([4]). A mutual mobile membrane is a construct = (V, H, μ, w1 , . . . , wn , R), where: 1. n, V , H, μ, w1 , . . . , wn are as in Deﬁnition 1; 2. R is a ﬁnite set of developmental rules of the following forms: local evolution ∗

(a) [ [u → v]m ]h for h, m ∈ H, u, v ∈ V ; These rules are called local because the evolution of a multiset of objects u of membrane m is possible only when membrane m is inside membrane h. If the restriction of nested membranes is not imposed, that is, the evolution of the multiset of objects u in membrane m is allowed wherever membrane m is placed, then we say that we have a global evolution rule, and write it simply as [u → v]m . mutual endocytosis

Simple, Enhanced and Mutual Mobile Membranes

35

(b) [uv]h [uv ]m → [ [w]h w ]m for h, m ∈ H, u, u ∈ V + , v, v , w, w∈ V ∗ ; An elementary membrane labelled h enters the adjacent membrane labelled m under the control of the multisets of objects uv and uv . The labels h and m remain unchanged during this process; however the multisets of objects uv and uv are replaced with the multisets of objects w and w , respectively. mutual exocytosis + (c) [uv [uv]h ]m → [w]h [w ]m for h, m ∈ H, u, u ∈ V , v, v , w, w∈ V ∗ ; An elementary membrane labelled h exits a membrane labelled m, under the control of the multisets of objects uv and uv . The labels of the two membranes remain unchanged, but the multisets of objects uv and uv are replaced with the multisets of objects w and w , respectively. The rules of the mutual mobile membranes are applied according to principles of simple mobile membranes. An object u indicates the membrane which initializes the move in the rules of type (b) − (c), while an object u indicates the membrane which accepts the movement. Using the rules of the mutual mobile membranes we can describe the receptormediated endocytosis of Figure 2. We associate a membrane to each cell, and objects to the signals, states and parts of molecules. For the steps done by the cells presented in Figure 2, we use the following encodings: – LDL particle: [cholesterol . . . cholesterol apoB . . . apoB]LDL An LDL particle contains one thousand or more cholesterol molecules and some apoB proteins. – cell membrane: [receptor . . . receptor clarithin . . . clarithin]cell The cell contains receptors which are able to recognize apoB proteins and also some proteins clathrin which enforce the pit containing the receptors. Using the above membranes, and the equality apoB = receptor, we can describe the membrane system as follows: [cholesterol . . . apoB . . .]LDL [apoB . . . clarithin . . .]cell The evolution is described by applying a rule of type (b): [cholesterol . . . apoB . . .]LDL [apoB . . . clarithin . . .]cell → → [ [cholesterol . . . apoB . . .]LDL apoB . . . clarithin . . .]cell

3

Computability Power of Mobile Membranes

In this section we present some existing results and also new results related to the computational power of simple, enhanced and mutual mobile membranes. First we present some notations from the ﬁeld of formal languages which are used throughout this section. More notions from formal languages can be found in [7] and [22]. For an alphabet V = {a1 , . . . , an }, we denote by V ∗ the set of all strings over V ; λ denotes the empty string. V ∗ is a monoid with λ as its unit element. For

36

B. Aman and G. Ciobanu

a string x ∈ V ∗ , |x|a denotes the number of occurrences of symbol a in x. A multiset over V is represented by a string over V (together with all its permutations), and each string precisely identiﬁes a multiset. For an alphabet V , the Parikh vector is ψV : V ∗ → Nn with ψV (x) = (|x|a1 , . . . , |x|an ), for all x ∈ V ∗ . For a language L, the Parikh vector is ψV (L) = {ψV (x) | x ∈ L}, while for a family F L of languages, it is P sF L = {ψV (L) | L ∈ F L}. A matrix grammars with appearance checking is a construct G = (N, T, S, M, F ) where N , T are disjoint alphabets of non-terminals and terminals, S ∈ N is the axiom, M is a ﬁnite set of matrices of the form (A1 → x1 , . . . , An → xn ) of context-free rules, and F is a set of occurrences of rules in M . For w, z ∈ (N ∪T )∗ , we write w ⇒m z if there is a matrix (A1 → x1 , . . . , An → xn ) in M and the strings wi ∈ (N ∪ T )∗ , 1 ≤ i ≤ n + 1, such that w = w1 , z = wn+1 , and for all i, 1 ≤ i ≤ n, either (1) wi = wi Ai wi , wi+1 = wi xi wi , for some wi , wi ∈ (N ∪ T )∗ , or (2) wi = wi+1 , Ai does not appear in wi , and the rule Ai → xi appears in F. The language generated by G is L(G) = {x ∈ T ∗ | S ⇒∗ x}. A matrix grammar in the strong binary normal form is a construct G = (N , T , S, M , F ), where N = N1 ∪ N2 ∪ {S, #}, with these three sets mutually disjoint, two distinguished symbols B (1) , B (2) ∈ N2 , and the matrices in M of one of the following forms: (1) (2) (3) (4)

(S → XA), with X ∈ N1 , A ∈ N2 , (X → Y, A → x), with X, Y ∈ N1 , A ∈ N2 , x ∈ (N2 ∪ T )∗ , (X → Y, B (j) → #), with X, Y ∈ N1 , j = 1, 2, (X → λ, A → x), with X ∈ N1 , A ∈ N2 , x ∈ T ∗ .

If we ignore the empty string when comparing languages, then the rules of type (4) are of the form (X → a, A → x), with X ∈ N1 , a ∈ T , A ∈ N2 , x ∈ T ∗ . 3.1

Simple Mobile Membranes

The computational power of simple mobile membranes is treated in [14]. P sM Mn (levol, endo, exo) denotes the family of all sets P s(Π) generated by systems using local evolution rules, together with endocytosis and exocytosis rules and at most n membranes. If the number of membrane is not bounded, this is denoted by P sM M∗ (levol, endo, exo). When global evolution rules are used, levol is replaced by gevol. If a type of rules is not used, then its name is omitted from the list of parameters. The number of membranes does not increase during the computation, but it can decrease by sending membranes out of the skin. The following result establishes an universality result using nine membranes and the operations of endocytosis and exocytosis: Theorem 1 ([14]). P sM M9 (endo, exo) = P sRE. A strengthening of the previous universality result is: Corollary 1 ([14]). P sM M∗ (endo, exo) = P sM Mn (endo, exo) = P sM Mn (gevol, endo, exo) = P sM Mn (levol, endo, exo) = P sRE, for all n ≥ 9.

Simple, Enhanced and Mutual Mobile Membranes

37

An improvement of the result presented in Theorem 1 is: Theorem 2 ([12]). P sM M4 (gevol, endo, exo) = P sRE. We improve the previous result by decreasing the number of membranes to three. Theorem 3. P sM M3 (levol, endo, exo) = P sRE. Proof. Consider a matrix grammar G = (N, T, S, M, F ) in the improved strong binary normal form (hence with N = N1 ∪ N2 ∪ {S; #}), having n1 matrices of types (2) and (4) (that is, not used in the appearance checking mode), and n2 matrices of type (3) (with appearance checking rules). Let B (1) and B (2) be the two objects in N2 for which we have rules B (j) → # in matrices of M . The matrices of the form (X → Y, B (j) → #) are labelled by mi , with i ∈ labj , for j ∈ {12}, such that lab1 , lab2 , and lab0 = {1, . . . , n1 } are mutually disjoint sets. We construct a mobile membrane system Π = (V, H, μ, w1 , w2 , w3 , R, 2) of degree three, where: V = N ∪ {X, Xi,j | X ∈ N1 , 1 ≤ i ≤ n1 , 0 ≤ j ≤ n1 } ∪ {a, a | a ∈ T } ∪ {x | x ∈ (N2 ∪ T )∗ } ∪ {A, Ai,j | A ∈ N2 , 1 ≤ i ≤ n1 , 0 ≤ j ≤ n1 } H = {1, 2, 3} μ = [[ ]2 [ ]3 ]1 w2 = XA, where (S → XA) is the initial matrix of G wh = λ, for all h ∈ {1, 3} The set R of rules is constructed as follows: (i) For each (nonterminal) matrix mi : (X → Y, A → x), X, Y ∈ N1 , A ∈ N2 , x ∈ (N2 ∪ T )∗ , with 1 ≤ i ≤ n1 , we consider the rules: 1. [X]2 [ ]3 → [[Xi,0 ]2 ]3 (endo) 2. [[A]2 ]3 → [Ai,0 ]2 [ ]3 (exo) 3. [[Xi,j → Xi,j+1 ]2 ]1 , j < i (levol) 4. [[Ai,j → Ai,j+1 ]2 ]1 , j < i (levol) 5. [[Ai,i Xi,i → xY ]2 ]1 (levol) 6. [[Ai,j Xj,j → #]2 ]1 , j < i (levol) 7. [[Aj,j Xi,j → #]2 ]1 , j < i (levol) In the initial conﬁguration, we have the objects X and A corresponding to the initial matrix in membrane 2. To simulate a matrix of the above type we start by applying the endocytosis rule 1, thus replacing X with Xi,0 , followed by the exocytosis rule 2, thus replacing a single A ∈ N2 with Ai,0 . No other A ∈ N2 can be replaced until membrane 2 enters membrane 3. Rule 3 (for X) and rule 4 (for A) are used to increment the second indices of X and A. This is done to check if the indices of X and A are the same, and in this case to rewrite A according to the matrix mi . Once the indices are equal, rule 5 is applied to complete the simulation of matrix mi . If the indices of X and A are not the same, rule 6 (if the indices of X is lower than the indices of A) or rule 7 (if the indices of X is bigger than the indices of A) is applied, the computation is blocked without producing any output.

38

B. Aman and G. Ciobanu

(ii) For a terminal matrix mi : (X → a, A → x), X ∈ N1 , a ∈ T , A ∈ N2 , x ∈ T ∗ , where 1 ≤ i ≤ n1 , we use the rules 1-7, where the rule 5 is replaced by the rules: 8. [ai,i Xi,i → a Y ]1 (levol) 9. [[a ]2 ]1 → [a]2 [ ]1 (exo) Observe that simulation of a type (4) matrix is along similar steps, except that we have an a in place of Y . During the ﬁnishing stages of a type (4) simulation, we use rule 8 to replace ai,i by a , and then to rewrite it to a when sending the membrane 2 out of the skin membrane, namely membrane 1. (iii) For each matrix mi : (X → Y, B (k) → #), X, Y ∈ N1 , A ∈ N2 , where n1 + 1 ≤ j ≤ n1 + n2 , j ∈ labk , k = 1, 2, we consider the rules: 10. [X]2 [ ]3 → [[Xk ]2 ]3 , for i ∈ labk (endo) 11. [[Xk B (k) → #]2 ]3 , k = 1, 2 (levol) 12. [[Xk ]2 ]3 → [Y ]2 [ ]3 , k = 1, 2 (exo) The simulation of matrices of type (3) begins by a rule of type 10. This is followed by a rule 11 in case B (k) exists, blocking membrane 2 inside membrane 3 and the computation stops without producing any output. If no B (k) exists, then rule 12 can be used to send out membrane 2, successfully completing the simulation. 3.2

Enhanced Mobile Membranes

The operations governing the mobility of the enhanced mobile membranes are endocytosis (endo), exocytosis (exo), enhanced endocytosis (fendo) and enhanced exocytosis (fexo). The interplay between these four operations is quite powerful, and the computational power of a Turing machine is obtained using twelve membranes without using the context-free evolution of objects [13]. The family of all sets P s(Π) generated by systems of degree at most n using rules α ⊆ {exo, endo, f endo, f exo, cevol} is denoted by P sEM Mn (α). Here endo and exo represent endocytosis and exocytosis, f endo and f exo represent enhanced endocytosis and enhanced exocytosis, and cevol represents contextual evolution. The main results are the following. Theorem 4 ([13]). P sEM M12 (endo, exo, f endo, f exo) = P sRE. Theorem 5 ([13]). P sEM M3 (cevol) = P sRE. Theorem 6 ([13]). P sEM M3 (endo, exo) = P sEM M3 (f endo, f exo). We improve the result of Theorem 4 as follows: Theorem 7. P sEM M9 (endo, exo, f endo, f exo) = P sRE.

Simple, Enhanced and Mutual Mobile Membranes

39

Proof. Consider a matrix grammar G = (N, T, S, M, F ) in the improved strong binary normal form (hence with N = N1 ∪ N2 ∪ {S; #}), having n1 matrices m1 , . . . , mn1 of types (2) and (4) (that is, not used in the appearance checking mode), and n2 matrices of type (3) (with appearance checking rules). The initial matrix is m0 : (S → XA). Let B (1) and B (2) be the two objects in N2 for which we have rules B (j) → # in matrices of M . The matrices of the form (X → Y, B (j) → #) are labelled by mi , 1 ≤ i ≤ n2 with i ∈ labj , for j ∈ {12}, such that lab1 , lab2 , and lab0 = {1, 2, . . . , n1 } are mutually disjoint sets. We construct a mobile membrane system Π = (V, H, μ, w1 , . . . , w9 , R, 7) of degree nine, where: V = N ∪ T ∪ {X0i , A0i | X ∈ N1 , A ∈ N2 , 1 ≤ i ≤ n1 } ∪{Xji , Aji | 0 ≤ i, j ≤ n1 } ∪ {Xij , Xj | X ∈ N1 , j ∈ {1, 2}, 1 ≤ i ≤ n2 } H = {1, . . . , 9} μ = [[ ]7 [ ]8 [ ]9 [[ ]3 [ ]4 [ ]5 [ ]6 ]2 ]1 w7 = XA, where (S → XA) is the initial matrix of G wh = λ, for all h ∈ {1, . . . , 9}\{7}

The set R of rules is constructed as follows: (i) For each (nonterminal) matrix mi : (X → Y, A → x), X, Y ∈ N1 , A ∈ N2 , x ∈ (N2 ∪ T )∗ , with 1 ≤ i ≤ n1 , we consider the rules: 1. [X]7 [ ]8 → [[Xi,i ]7 ]8 (endo) 2. [[A]7 ]8 → [Ai,i ]7 [ ]8 (exo) 3. [Xj,i ]7 [ ]9 → [[Xj−1,i ]7 ]9 (endo) 4. [[Aj,i ]7 ]9 → [Aj−1,i ]7 [ ]9 (exo) 5. [ ]8 [X0,i ]7 → [X0,i [ ]8 ]7 (fendo) 6. [ ]9 [A0,i ]7 → [A0,i [ ]9 ]7 (fendo) 7. [ ]8 [X0,i ]7 → [#[ ]8 ]7 (fendo) 8. [[A0,i ]7 ]9 → [#]7 [ ]9 (exo) 9. [X0,i [ ]8 ]7 → [ ]8 [Y ]7 (fexo) 10. [A0,i [ ]9 ]7 → [ ]9 [x]7 (fexo) In the initial conﬁguration, we have the objects X, A corresponding to the initial matrix in membrane 7. To simulate a matrix of type (2), we start by applying the endocytosis rule 1, thus replacing X with Xi,i , followed by the exocytosis rule 2, thus replacing a single A ∈ N2 with Ai,i . Rule 3 (for X) and rule 4 (for A) are used to decrement the ﬁrst indices of X and A. This is done to check if the indices of X and A are the same, and in this case to rewrite A according to the matrix mi . By using fendo rules 5 and 6, membranes 8 and 9 enter membrane 7 replacing X0,i and A0,i with X0,i and A0,i , respectively. This is then followed by rules 9 and 10, when membranes 8 and 9 exit membrane 7 by fexo rules replacing X0,i and A0,i with Y and x, respectively. If i > j, then we obtain A0,j before X0,i . In this case, we have a conﬁguration where membrane 7 is inside membrane 9 containing A0,j . Then rule 8 is used, replacing A0,j with #, and an inﬁnite computation is obtained (rule 17). If j > i, then we obtain X0,i before A0,j .

40

B. Aman and G. Ciobanu

In this case, we reach a conﬁguration with X0,i Ak,j , k > 0 in membrane 7, and membrane 7 is in the skin membrane. Rule 3 cannot be used now, and the only possibility is to use rule 7, which leads to an inﬁnite computation. Thus, if i = j, then we can correctly simulate a matrix of type (2). (ii) For each matrix mi : (X → Y, B (k) → #), X, Y ∈ N1 , A ∈ N2 , where n1 + 1 ≤ j ≤ n1 + n2 , j ∈ labk , k = 1, 2, we consider the rules: 11. 12. 13. 14. 15.

(j)

[X]7 [ ]2 → [[Xi ]7 ]2 , j = 1, 2 (endo) (j) (j) [ ]j+2 [Xi ]7 → [Xi [ ]j+2 ]7 , j = 1, 2 (fendo) [ ]j+4 [B (j) ]7 → [#[ ]j+4 ]7 , j = 1, 2 (fendo) (j) [Xi [ ]j+2 ]7 → [ ]j+2 [Yj ]7 , j = 1, 2 (fexo) [[Yj ]7 ]2 → [Y ]7 [ ]2 , j = 1, 2 (exo)

The simulation of matrices of type (3) begins by a rule of type 11. Inside membrane 2, rules 12 and 13 are used, and so membrane (j + 2) enters membrane 7, and membrane (j + 4) enters membrane 7 if the symbol B (j) is present. In this case, B (j) is replaced with #. Otherwise, membrane (j+2) (j) comes out of the membrane 7 replacing Xi with Yj . Then membrane 7 exits membrane 2, by replacing Yj with Y thus successfully simulating a matrix of type (3). (iii) For a terminal matrix mi : (X → a, A → x), X ∈ N1 , a ∈ T , A ∈ N2 , x ∈ T ∗ , where 1 ≤ i ≤ n1 : 16. [[a ]7 ]1 → [a]7 [ ]1 (exo) 17. [ ]8 [#]7 → [#[ ]8 ]7 (fendo) [#[ ]8 ]7 → [ ]8 [#]7 (fexo) Observe that simulation of a matrix of type (4) matrix is similar to that of a matrix of type (2), except that we have an a in place of Y in rule 9. During the ﬁnishing stages of a matrix of type (4) simulation, we use rule 16 to replace a with a when sending the membrane 7 out of the skin membrane. 3.3

Mutual Mobile Membranes

Similar to other classes of mobile membranes, we try to establish the number of membranes in mutual mobile membranes in order to obtain a system which is equivalent to Turing machines. The following result oﬀers an answer. The minimum number of membranes needed remains an open problem. The family of all sets P s(Π) generated by systems of degree at most n using rules α ⊆ {mutual exo, mutual endo} is denoted by P sM M Mn (α). Here mutual endo and mutual exo represent mutual endocytosis and mutual exocytosis rules. By using objects and co-objects, the computational power is obtained using a lower number of membranes than for enhanced mobile membranes, namely: Theorem 8. P sM M M7 (mutual endo, mutual exo) = P sRE.

Simple, Enhanced and Mutual Mobile Membranes

41

Proof (Sketch). Consider a matrix grammar G = (N, T, S, M, F ) in the improved strong binary normal form (hence with N = N1 ∪ N2 ∪ {S; #}), having n1 matrices m1 , . . . , mn1 of types (2) and (4) (that is, not used in the appearance checking mode), and n2 matrices of type (3) (with appearance checking rules). The initial matrix is m0 : (S → XA). Let B (1) and B (2) be the two objects in N2 for which we have rules B (j) → # in matrices of M . The matrices of the form (X → Y, B (j) → #) are labelled by mi , 1 ≤ i ≤ n2 with i ∈ labj , for j ∈ {1, 2}, such that lab1 , lab2 , and lab0 = {1, 2, . . . , n1 } are mutually disjoint sets. We construct a mobile membrane system Π = (V, H, μ, w1 , . . . , w7 , R, 7) of degree seven, where: V = N ∪ T ∪ {X0i , A0i | X ∈ N1 , A ∈ N2 , 1 ≤ i ≤ n1 } ∪{Xji , Aji | 0 ≤ i, j ≤ n1 } ∪ {Xij , Xj | X ∈ N1 , j ∈ {1, 2}, 1 ≤ i ≤ n2 } ∪{β, β, γ, γ} H = {1, . . . , 7} μ = [[ ]7 [ ]5 [ ]6 [[ ]3 [ ]4 ]2 ]1 w1 = β, w2 = β, w3 = β, β, γ, γ, w4 = w5 = w6 = β, γ w7 = XAβγ, where (S → XA) is the initial matrix of G.

In what follows we present only a part of the set of rules R constructed, namely the ones used to simulate mi matrices. (i) For each matrix mi : (X → Y, B (k) → #), X, Y ∈ N1 , A ∈ N2 , where n1 + 1 ≤ j ≤ n1 + n2 , j ∈ labk , k = 1, 2, we consider the rules: (j) 1. [βX]7 [β]2 → [β[βXi ]7 ]2 , j = 1, 2 (mutual endo) (j) (j) 2. [βXi ]7 [β]j+2 → [[βXi ]7 β]j+2 , j = 1, 2 (mutual endo) 3. [γ]3 [γ]4 → [γ[γ]4 ]3 (mutual endo) 4. [γB (1) ]7 [γ]4 → [γ#[γ]4 ]7 (mutual endo) 5. [γ[γ]4 ]3 → [γ]3 [γ]4 (mutual exo) 6. [β]3 [β]4 → [β[β]3 ]4 (mutual endo) 7. [γB (2) ]7 [γ]3 → [γ#[γ]3 ]7 (mutual endo) 8. [β[β]3 ]4 → [β]3 [β]4 (mutual exo) (j) 9. [β[βXi ]7 ]j+2 → [β]j+2 [βYj ]7 , j = 1, 2 (mutual exo) 10. [β[βYj ]7 ]2 → [βY ]7 [β]2 , j = 1, 2 (mutual exo) The simulation of matrices of type (3) begins by a rule of type 1. Inside membrane 2, rules 2 is used, by which membrane 7 enters membrane (j + 2). Rules 3 and 6 are used to introduce the remaining membrane (3 or 4) near membrane 7. In this case, if B (j) exists this is replaced with # and the computation is stopped. Otherwise, membrane 7 comes out of membrane (j+ (j) 2) replacing Xi with Yj . After the other membrane is removed from (j +2), membrane 7 exits membrane 2, successfully simulating a matrix of type (3).

4

Related Work: P Systems with Active Membranes

The mobile membranes derive from the P systems with active membranes introduced in [18]. P systems with active membranes are a variant of P systems

42

B. Aman and G. Ciobanu

in which each membrane is supposed to have an “electrical polarization” (also called charge): positive, negative, or neutral. A P system with active membranes has a ﬁnite set of developmental rules, of the following forms: ∗ (a) [a → v]α h , for h ∈ H, α ∈ {+, −, 0}, a ∈ V , v ∈ V α2 1 (b) a[ ]α h →[b]h , for h ∈ H, α1 , α2∈{+, −, 0}, a, b ∈ V α2 1 (c) [a]α h →[ ]h b, for h ∈ H, α1 , α2∈{+, −, 0}, a, b ∈ V

(d) [a]α h → b, for h ∈ H, α∈{+, −, 0}, a, b ∈ V

object evolution communication communication dissolving

division of elementary membranes α2 α3 1 (e) [a]α h →[b]h [c]h , for h∈H, α1,α2,α3∈{+, −, 0}, a, b, c ∈ V division of non-elementary membranes α4 α4 α6 3 α5 (f) [ [ ...[ ...[ → [[ . . . [ ]α hk ]h0 [ [ ]hk+1 . . . [ ]hn ]h0 for k ≥ 1, n > k , hi ∈ H, 0 ≤ i ≤ n, and α0 , . . . , α6 ∈ {+, −, 0} with {α1 , α2 } = {+, −} 1 ]α h1

α2 1 ]α hk [ ]hk+1

2 α0 ]α hn ]h0

3 ]α h1

More details about these rules and how they are applied may be found in [18]. By denoting with LP A the family of languages L(Π) generated by P systems with active membranes, we have the following result: Theorem 9 ([18]). P sRE = P sLP A. N P Ard denotes the family of vectors of natural numbers N (Π) computed by non-cooperative systems Π which do not use division rules of type (f ). The subscript rd stands for “restricted division”. This restriction does not decrease the power of P systems with active membranes: Theorem 10 ([20]). P sRE = N P Ard . N P Ar denotes the family of natural numbers N (Π) computed by systems Π which use only rules of types (a), (b) and (c). Theorem 11 ([15]). P sRE = N P Ar . The set of numbers generated in the minimally parallel way by a system Π is denoted by Nmin (Π). The family of sets Nmin (Π) generated by systems with rules of the non-restricted form, having initially at most n1 membranes and using conﬁgurations with at most n2 membranes during any computation is denoted by Nmin OPn1 ,n2 . When a type of rule is not used, it is not mentioned in the notation. If any of the parameters n1 , n2 is not bounded, then it is replaced by . If the system do not use polarizations for membranes, then we write (a0 ), (b0 ), (c0 ), (d0 ), (e0 ) instead of (a), (b), (c), (d), (e). When using the maximal parallel way we replace the subscript min by max.

Simple, Enhanced and Mutual Mobile Membranes

43

Theorem 12 ([1,6,17]) 1. Nmax OP3,3 ((a), (b), (c)) = N RE. [17] 2. Nmax OP, ((a0 ), (b0 ), (c0 ), (d0 ), (e0 )) = N RE. [1] 3. Nmin OP3,3 ((a), (b), (c)) = N RE. [6] Theorem 13 ([8]). Nmax OPn1 , ((a1 ),(b1 ), (c1 ), (d1 ), (e1 ))=N RE, for all n1≥5. Theorem 14 ([8]). Nmin OPn1 , ((a1 ), (b1 ), (c1 ), (d1 ), (e1 ))=N RE, for all n1≥7. When the rules of a given type (α0 ) are able to change the labels of the involved membranes, then we denote that type of rules by (α0 ). Using the power of label changing, the following results are obtained: Theorem 15 ([2]). P sOP (a0 , b0 , c0 , e0 ) = P sOP (a0 , b0 , c0 ) = = P sOP (a0 , b0 , c0 ) = P sOP (a0 , c0 , e0 ) = P sRE. By introducing replicative-distribution rules for nested membranes: (l0 ) [a[ ]h1 ]h2 → [[u]h1 ]h2 v, for h1 , h2 ∈ H, a ∈ V , u, v ∈ V ∗ ; the following result is obtained: Theorem 16 ([9]). P sOP4 (l0 ) = P sRE.

5

Conclusions

Simple, enhanced and mutual mobile membranes are new models of computation inspired from the biological operations governing the movement of biological membranes: endocytosis and exocytosis. After deﬁning these classes of mobile membranes according to their biological motivations, some results concerning their computational power are presented. For mutual mobile membranes this is the ﬁrst universality result, while for simple and enhanced mobile membranes the results are improvements for existing ones by reducing the number of membranes needed.

Acknowledgements Many thanks to the referees for their helpful remarks and comments. This work has been partially supported by research grants CNCSIS IDEI 402/2007 and CNCSIS TD 345/2008.

References 1. Alhazov, A.: P Systems without Multiplicities of Symbol-Objects. Information Processing Letters 100, 124–129 (2006) 2. Alhazov, A., Pan, L., P˘ aun, G.: Trading Polarizations for Labels in P Systems with Active Membranes. Acta Informatica 41(2-3), 111–144 (2004)

44

B. Aman and G. Ciobanu

3. Aman, B., Ciobanu, G.: Describing the Immune System Using Enhanced Mobile Membranes. Electronic Notes in Theoretical Computer Science, vol. 194, pp. 5–18 (2008) 4. Aman, B., Ciobanu, G.: Resource Competition and Synchronization in Membranes. In: Proceedings SYNASC 2008. IEEE Computing Society, Los Alamitos (2009) 5. Cardelli, L., Gordon, A.: Mobile Ambients. In: Nivat, M. (ed.) FOSSACS 1998. LNCS, vol. 1378, pp. 140–155. Springer, Heidelberg (1998) 6. Ciobanu, G., Pan, L., P˘ aun, G., P´erez-Jim´enez, M.J.: P Systems with Minimal Parallelism. Theoretical Computer Science 378, 117–130 (2007) 7. Dassow, J., P˘ aun, G.: Regulated Rewriting in Formal Language Theory. Springer, Heidelberg (1990) 8. Freund, R., P˘ aun, G., P´erez-Jim´enez, M.J.: Polarizationless P Systems with Active Membranes Working in the Minimally Parallel Mode. In: Akl, S.G., Calude, C.S., Dinneen, M.J., Rozenberg, G., Wareham, H.T. (eds.) UC 2007. LNCS, vol. 4618, pp. 62–76. Springer, Heidelberg (2007) 9. Ishdorj, T.-O., Ionescu, M.: Replicative-Distribution Rules in P Systems with Active Membranes. In: Liu, Z., Araki, K. (eds.) ICTAC 2004. LNCS, vol. 3407, pp. 68–83. Springer, Heidelberg (2005) 10. Janeway, C.A., Travers, P., Walport, M., Shlomchik, M.J.: Immunobiology - The Immune System in Health and Disease, 5th edn. Garland Publishing, New York (2001) 11. Krishna, S.N.: On the Eﬃciency of a Variant of P Systems with Mobile Membranes. In: Cellular Computing: Complexity Aspects, Fenix Editora, Sevilla, pp. 237–246 (2005) 12. Krishna, S.N.: The Power of Mobility: Four Membranes Suﬃce. In: Cooper, S.B., L¨ owe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 242–251. Springer, Heidelberg (2005) 13. Krishna, S.N., Ciobanu, G.: On the Computational Power of Enhanced Mobile Membranes. In: Beckmann, A., Dimitracopoulos, C., L¨ owe, B. (eds.) CiE 2008. LNCS, vol. 5028, pp. 326–335. Springer, Heidelberg (2008) 14. Krishna, S.N., P˘ aun, G.: P Systems with Mobile Membranes. Natural Computing 4, 255–274 (2005) 15. P˘ aun, A.: On P Systems with Membrane Division. In: Unconventional Models of Computation, pp. 187–201 (2000) 16. P˘ aun, G.: Computing with Membranes. Journal of Computer and System Sciences 61, 108–143 (2000) 17. P˘ aun, G.: Membrane Computing. An Introduction. Springer, Berlin (2002) 18. P˘ aun, G.: P Systems with Active Membranes: Attacking NP-Complete Problems. Journal of Automata, Languages and Combinatorics 6, 75–90 (2001) 19. P˘ aun, G., Rozenberg, G., Salomaa, A.: Membrane Computing with External Output. Fundamenta Informaticae 41, 259–266 (2000) 20. P˘ aun, G., Suzuki, Y., Tanaka, H., Yokomori, T.: On the Power of Membrane Division in P Systems. Theoretical Computer Science 324, 61–85 (2004) 21. Petre, I., Petre, L.: Mobile Ambients and P Systems. Journal of Universal Computer Science 5, 588–598 (1999) 22. Salomaa, A.: Formal Languages. Academic Press, London (1973) 23. http://bcs.whfreeman.com/thelifewire

Bio-PEPA with Events Federica Ciocchetta Laboratory for Foundations of Computer Science, The University of Edinburgh, Edinburgh EH8 9AB, Scotland

Abstract. In this work we present an extension of Bio-PEPA, a language recently defined for the modelling and analysis of biochemical systems, to handle events. Events are constructs that represent changes in the system due to some trigger conditions. The events considered here are simple, but nevertheless able to describe most of the discontinuous changes in models and experiments. Events are added to our language without any modification to the rest of the syntax in order to keep the specification of the model as straightforward as possible. Some maps are defined from Bio-PEPA with events to analysis tools. Specifically, we map our language to Hybrid Automata (HA) and we consider a modification of Gillespie’s algorithm for stochastic simulation. In order to test our approach, we present the translation in Bio-PEPA of a biochemical network describing the functional properties of the Acetylcholine receptor with the addition of an event that causes the inactivation of some reactions at a given time.

1

Introduction

Computational models play an important role in systems biology. Indeed they help to study, analyze and predict the behaviour of biological systems. In recent years there have been some applications of process algebras for the analysis of biological systems (e.g. [27,25,8,9]). In most cases the analysis is performed using Gillespie’s stochastic simulation algorithm [18]. Other possibilities exist, such as the mapping to diﬀerential equations [7]. Many biological models need to capture both discrete and continuous phenomena [1,4,23]. These models are called hybrid systems. A ﬁrst example of a hybrid system describes the activation of a certain activity when the concentration of enabling quantities is above the desired threshold. A second example considers a signal or stimuli that becomes null after some time leading to some changes in the interactions of the system. Other examples describe some experiments, where it may be necessary to render the possible change to the system, due, for instance, to the introduction or the removal of some reagents. In this work we present an extension of Bio-PEPA [9,10], a language recently deﬁned for the modelling and analysis of biological systems, to handle events. Broadly speaking, events are constructs that represent changes in the system due to some trigger conditions. Here we are interested in simple forms of events. Speciﬁcally, we refer to the deﬁnition of events reported in the SBML speciﬁcation [22]. These kinds of C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 45–68, 2009. c Springer-Verlag Berlin Heidelberg 2009

46

F. Ciocchetta

events can be found in biochemical networks, such as the ones in the BioModels database [24] or deﬁned in some experimental settings. The idea underlying our work is the following: Biochemical networks with events =⇒ Bio-PEPA with events =⇒ Analysis Starting from a biochemical network with one or more events, we want to map it into a Bio-PEPA system. From that, we can then consider diﬀerent kinds of analysis. In this, view Bio-PEPA is a formal, intermediate, compositional representation of the biochemical network. This idea is the one proposed for (the standard) Bio-PEPA. A ﬁrst challenge concerns the modelling: we need to add events to the BioPEPA system. Events are added to our language as a set of elements and the rest of the syntax is unchanged. There are two motivations for this choice. First, we keep the speciﬁcation of the model as simple as possible. Second, this approach is appropriate when we study the same biochemical system but with diﬀerent experimental regimes as we can modify the list of events without any changes to the rest of the system. A second aspect is the analysis. Some maps must be deﬁned from Bio-PEPA to analysis tools. Speciﬁcally, we map our language to Hybrid Automata (HA) [19]. HA are a formalism that consider both continuous and discrete changes. The continuous part is expressed by a set of variables evolving in each state according to a set of diﬀerential equations and the discrete dynamics is given by transitions between states, triggered by some conditions on variables. Furthermore, we can consider a modiﬁcation of Gillespie’s algorithm [18] in order to tackle events. A preliminary version of this work has been presented in [11]. Here we add some deﬁnitions concerning the kind of events and further details concerning the mappings from Bio-PEPA with events to the Hybrid Automata and Gillespie’s algorithm. Furthermore, we consider more general kinds of events, such as simultaneous events or events with a delay diﬀerent from zero. The rest of the paper is organised as follows. Section 2 reports a description of Bio-PEPA. In Section 3 we deﬁne the events we are considering in this work and then we extend Bio-PEPA in order to handle them. Section 4 describes the mapping from our language to Hybrid Automata. The mapping to stochastic simulation is reported in Section 5. After that, Section 6 illustrates the modelling in Bio-PEPA of a biochemical network describing the functional properties of the Acetylcholine receptor with an event that is triggered at a given time and causes the inactivation of some reactions. In Section 7 we overview some related work. Finally, in Section 8, some conclusions are reported.

2

Bio-PEPA

Bio-PEPA [9,10] is a language for the modelling and analysis of biochemical networks. The syntax of Bio-PEPA is deﬁned as: S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | | .

P ::= P P | S(x) I

Bio-PEPA with Events

47

The component S (species component ) abstracts a biological species and the component P (model component ) describes the system and the interactions among components. The preﬁx term (α, κ) op S contains information about the role of the species in the reaction associated with the action type α: κ is the stoichiometry coeﬃcient of the species and the preﬁx combinator “op” represents the role of the element in the reaction. Speciﬁcally, ↓ indicates a reactant, ↑ a product, ⊕ an activator, an inhibitor and a generic modiﬁer. The operator “+” expresses choice between possible actions and the constant C is deﬁned by def an equation C = S. The parameter x ∈ R+ in S(x) represents the initial quan tity (for instance the concentration) of the species. Finally, the process P Q I denotes the cooperation between components: the set I determines those activities on which the operands are forced to synchronize. In Bio-PEPA the rates are not expressed in the syntax of components but are deﬁned as functional rates. These allow us to express any kind of kinetic law. Each action is associated with a speciﬁc functional rate. A possible modelling style supported by Bio-PEPA is in terms of concentration levels. This is the style considered in the derivation of the transition system for Bio-PEPA. The species concentrations can be discretized into a number of levels. The granularity of the system is expressed in terms of the step size h, i.e. the length of the concentration interval representing a level. The information about the step sizes and the number of levels for each species is collected in a set N . Speciﬁcally, the elements of the set N have the form: “C : h = value h, N = value N, M = value M, V = value V, unit = value u”, where C is the species component name, h is the step size, N is the maximum level, M is the maximum concentration, V is the name of the enclosing compartment and unit is the unit for concentration. In order to fully describe a biochemical network in Bio-PEPA we need to deﬁne structures that collect information about the compartments, the maximum concentrations, number of levels for all the species, the constant parameters and the functional rates. The Bio-PEPA system is deﬁned in the following way: Definition 1. A Bio-PEPA system P is a 6-tuple V, N , K, FR , Compon − ents, P , where: V is the set of compartments, N is the set of quantities describing species, K is the set of parameter deﬁnitions, FR is the set of functional rates, Components is the set of deﬁnitions of sequential components, P is the model component describing the system. For details see [9,10]. The behaviour of the system is deﬁned in terms of an operational semantics. This refers to the level-based modelling style and in this context the parameter in the species components stands for the concentration level. We deﬁne two θ relations. The former, called capability relation, is indicated by − →c . The label θ is of the form (α, w), where w := [S : op(l, κ)] | w :: w, with S a species component, op a symbol representing the role of the species in the reaction, l the level and κ the stoichiometry coeﬃcient. This relation is deﬁned as the minimum relation satisfying the rules reported in Table 1.

48

F. Ciocchetta Table 1. Axioms and rules for Bio-PEPA

prefixReac

(α,[S:↓(l,κ)]) ((α, κ)↓S)(l) −−−−−−−−→c S(l − κ) κ ≤ l ≤ N

prefixProd

(α,[S:↑(l,κ)]) ((α, κ)↑S)(l) −−−−−−−−→c S(l + κ) 0 ≤ l ≤ (N − κ)

prefixMod

((α, κ) op S)(l) −−−−−−−−−→c S(l)

(α,[S:op(l,κ)])

with op = , ⊕, and

0 < l ≤ N if op = ⊕, 0 ≤ l ≤ N otherwise (α,w)

choice1

S1 (l) −−−→c S1 (l ) (α,w)

(S1 + S2 )(l) −−−→c S1 (l ) (α,w)

choice2

S2 (l) −−−→c S2 (l ) (α,w)

(S1 + S2 )(l) −−−→c S2 (l ) (α,S:[op(l,κ)])

constant

S(l) −−−−−−−−−→c S (l )

def

with C = S

(α,C:[op(l,κ)])

C(l) −−−−−−−−−→c S (l ) (α,w)

coop1

P1 −−−→c P1 (α,w) P1 P2 −−−→c P1 P2 L L

with α ∈ /L

(α,w)

coop2

P2 −−−→c P2

P1 P2 −−−→c P1 P2 L L (α,w)

(α,w1 )

coop3

P1 −−−−→c P1

with α ∈ /L

(α,w2 )

P2 −−−−→c P2

1 2 P1 P2 −−−−−−−→c P1 P2 L L

(α,w ::w )

with α ∈ L

˜ where P˜ is The latter relation, called stochastic relation, is − →s ⊆ P˜ × Γ × P, 1 the set of well-deﬁned Bio-PEPA systems and Γ is the set of labels γ = (α, r), with α the action type and r the associated rate. This relation is deﬁned as the minimal relation satisfying the rule: 1

In a well-defined Bio-PEPA system each element has to satisfy some conditions. For instance, we have that each species component C ∈ Comp must have subterms of the form “(α, κ) op C” and the action types in each single component must be all distinct. Furthermore, the model component P must be defined in terms of the species components defined in Comp and, for each cooperation set Lj in P , Lj ⊆ A(P ). For details see [12].

Bio-PEPA with Events

49

(αj ,w)

Final

P −−−−→c P (αj ,rα [w,N ,K])

V, N , K, F , Comp, P −−−−−−−−−−→s V, N , K, F , Comp, P

The element rα [w, N , K] is the rate associated with the action α and is deﬁned as: fα [w, N , K] rα [w, N , K] = h where h is the step size for the species involved in the reaction and the notation fα [w, N , K] means that the function fα is evaluated over w and the information about parameters and species components contained in the sets N and K. In this deﬁnition rα represents the parameter of a negative exponential distribution. The dynamic behaviour of processes is determined by a race condition: all activities enabled attempt to proceed but only the fastest succeeds. A Stochastic Labelled Transition System (SLTS) is deﬁned for a Bio-PEPA system. From this we can obtain a continuous time Markov Chain (CTMC). Both the SLTS and the CTMC derived from Bio-PEPA are deﬁned in terms of levels of concentration. We call this Markov chain the CTMC with levels. Bio-PEPA can be seen as an intermediate, formal, compositional representation of biological systems, from which diﬀerent kinds of analysis can be performed. We have deﬁned some mappings from Bio-PEPA to ODEs, CTMC with levels, stochastic simulation and PRISM [26]. Some tools for the analysis of BioPEPA system have been implemented [3]. In the following we report a brief description of the mapping from Bio-PEPA to ODE, as it is used later in the paper. For further details and the other mappings see [10]. 2.1

From Bio-PEPA to ODE System (πODE )

Let πODE be the mapping from Bio-PEPA system to the associated ODE system. The mapping πODE entails three steps: 1. deﬁnition of the stoichiometry (n × m) matrix D, where n is the number of species and m is the number of reactions; 2. deﬁnition of the kinetic law vector (m × 1) vKL containing the kinetic laws of each reaction; 3. deﬁnition of the vector (n × 1) x, with xT = (x1 , x2 , ..., xn ). A crucial part is the derivation of the stoichiometry matrix D = {dij }. The entries of the matrix are obtained as follows: for each sequential component Ci consider the preﬁx subterms Cij representing the contribution of the species i to the reaction j. If the term represents a reactant we write the corresponding stoichiometry κij as −κij in the entry dij . In the case of a product we write +κij . All other cases are null. The kinetic law vector is derived from the functional rates and its deﬁnition is straightforward.

50

F. Ciocchetta

The ODE system thus obtained has the form: dx = D × vKL dt where the vector of initial concentrations is x0 , with xi,0 the initial concentration of the species i, as given in the speciﬁcation of the system. 2.2

Example

In order to show how to model biochemical systems in Bio-PEPA we consider the network presented in Fig. 1 and we translate it into Bio-PEPA. This network is then used as a running example in the rest of the paper.

3

X

1

2

Y

Fig. 1. Biochemical network composed of two proteins X and Y . The numbers indicate the reactions. Reaction 1 is the translation of Y enhanced by X, reaction 2 is the degradation of X and reaction 3 the translation of X.

The network is composed of two proteins, X and Y . These are involved in the following interactions: r

1 – Translation of Y enhanced by X (reaction 1): X −→X +Y. The kinetic law is mass-action with constant parameter r1 = 0.01; r2 – Degradation of the protein X (reaction 2): X −→∅. The kinetic law is mass-action with constant parameter r2 = 0.02; r3 – Translation of the protein X (reaction 3): ∅−→X. The kinetic law is mass-action with constant parameter r3 = 0.01.

Each reaction i is represented by an action type αi . The kinetic laws are represented by the following functional rates: fα1 = f M A(0.01); fα2 = f M A(0.02); fα3 = 0.01; where f M A(r) stands for mass-action kinetic law with rate r. The Bio-PEPA species components2 corresponding to the two proteins are: X = (α1 , 1) ⊕ X + (α2 , 1)↓X + (α3 , 1)↑X def

2

def

Y = (α1 , 1)↑Y

Note that we use X and Y (capital letters) to indicate the names of the species and the name of the Bio-PEPA components, whereas x and y indicate the associated species concentrations.

Bio-PEPA with Events

51

Fig. 2. ODE integration results for the network

whereas the model component is:

} Y (0) X(0) {α 1

where the initial values are zero for both the proteins. The set of compartments and the set N are not reported. Applying the mapping πODE we obtain the ODE system: dx dt dy dt

= −0.02 · x + 0.01 = 0.01 · x

where x and y are the two variables describing X and Y . The result of ODE integration is reported in Fig. 2. The protein X reaches a steady-state whereas Y increases inﬁnitely.

3 3.1

Bio-PEPA with Events SBML-Like Events: Some Definitions

In this work we consider events as deﬁned in the SBML speciﬁcation [22]. SBML events describe explicit discontinuous state changes in the model. Speciﬁcally, an SBML event has the following structure: “event id, if trigger then event assignment list with delay

52

F. Ciocchetta

where – event id is the event identiﬁer, – trigger is a mathematical expression that, when it is evaluated to true, makes the event ﬁre. It can be composed of one or more conditions; – event assignment list is a list of assignments that are made when the event is executed; – delay is the length of time between the time when the event ﬁres and the time when the event assignments are executed. The trigger and the list of assignments are both mandatory and can involve parameters, species concentrations and compartment sizes. All the triggers are initially evaluated to false. An SBML-like event is immediate if delay is equal to zero. Otherwise, the event is called delayed. The deﬁnition of sequential and simultaneous events is reported below. Definition 2. Two or more SBML-like events are sequential if they are ﬁred one after the other in a given order. They are said to be simultaneous if they happen at the same point in time. In most biochemical systems which we are interested in we have sequential events. In the general situation of simultaneous events, sometimes some tie-breaking rules are necessary to decide which of any set of events is simulated ﬁrst. The most common way to do this is to assign a priority to each event [13]: when there are two or more simultaneous events, the event with the highest priority is deﬁned to be the next event to ﬁre. However, the order in which a set of simultaneous events is ﬁred is not always important, for instance when the assignments of the events inﬂuence diﬀerent variables. We have the following deﬁnition: Definition 3. Two simultaneous events are independent if their event assignments do not eﬀect each other. Otherwise, they are called dependent. If we have simultaneous independent events we may abstract them as a single event and the system is reset according to the assignments of all the set of simultaneous events. Simultaneous independent events are dealt with similarly to sequential ones. 3.2

Assumptions

We make the following assumptions for the events considered in this work. 1. Triggers can involve time and species components’ names, while assignments can involve species components (concentrations), compartments (size), parameters (values) and functional rates (function deﬁnitions). 2. Triggers are deterministic, i.e. when they become true they are ﬁred. 3. Triggers are only unidirectional, i.e. describing the change from one mode to another, but not vice versa. Bidirectional triggers can be decomposed into two unidirectional triggers. 4. Events are either sequential or simultaneous and independent.

Bio-PEPA with Events

53

These assumptions are not restrictive. Indeed the events satisfying these assumptions allow us to represent a large number of discontinuous changes that we can ﬁnd in biological systems. 3.3

The Definition of the Language

We can add events to a Bio-PEPA system by introducing a set of elements that have the form (id, trigger, event assignment, delay), where id is the name of the event, trigger is a mathematical expression involving the components of the Bio-PEPA model and time, event assignment is a list of assignments, delay is 0 (immediate events) or positive real value (delayed events). Formally, we have the following deﬁnitions: trigger ::= cond | cond or cond | cond and cond | not cond ¯ eq value | ¯ k) cond ::= t eq value | expression(C, ¯ ¯ ¯ ¯ k) expression(C, k) eq expression(C, eq ::= = | =|>| 0.8 and X = 0.2, r3 ← 0.005, 0) These two events are sequential and clearly satisfy the assumptions discussed above.

4 4.1

Mapping to Hybrid Automata Hybrid Automata

Hybrid automata (HA) [19] combine discrete transition graphs with continuous dynamical systems. They are used to formally model hybrid systems, dynamical systems with both discrete and continuous components. An hybrid automaton consists of a ﬁnite set of real-valued variables {X1 , X2 , ..., Xn } and a ﬁnite labelled graph, whose vertices correspond to control modes (states), described by diﬀerential equations, and whose edges are control switches, corresponding to discrete events. In addition, we have some labels for the edges, specifying the jump conditions (activation conditions) and labels for the vertices, containing information about initial and invariant conditions. The variables evolve continuously in time, apart from some changes induced by events. When an event happens there is a change in the mode. The dynamic behaviour of each mode is described by a set of diﬀerential equations, generally diﬀerent from mode to mode. We can use HA both for simulation (see for instance the SHIFT language [15]) and model checking (see HyTech [20]). In this work we limit our attention to simulation. For a formal deﬁnition and details of the formalism see [19].

Bio-PEPA with Events

4.2

55

Definition of the Mapping

Here we present the map from Bio-PEPA to HA. First, we limit our attention to the case of immediate events and then we show a way to represent delayed events. Indeed, the translation of the delay associated with an event is not straightforward in the usual deﬁnition of HA. Let P0 = V0 , N0 , K0 , F0 , Comp0 , P0 , Events, t0 be the initial Bio-PEPA system and let Nevents be the number of events. We have the following correspondences: 1. Each species component Ci in Comp is associated with a variable Xi . The set of variables is then given by {X1 , X2 , · · · , XNComp , t}, where t is the variable expressing the time and NComp is the number of species components. The evolution of the variable t is described by the trivial diﬀerential equation dt/dt = 1. 2. The initial conditions of the variables are derived from the initial model component P0 . The variable t is initially set to 0. 3. For each event i in Events, we can consider the trigger tri . We use these triggers to deﬁne the jump conditions. In the case we have only sequential events, the number of possible jump conditions Njump is just NEvents . If simultaneous independent events are possible, we may combine them together in order to deﬁne a new jump condition representing the union of the triggers of the simultaneous events. In this case, the system is reset according to the union of the assignment lists of the events involved. 4. Each mode is described by a speciﬁc instance of the Bio-PEPA system. Indeed modes are deﬁned according to either the initial system or the system modiﬁed with the event assignments relative to a trigger. The number of modes is Njump + 1. σ is used to indicate a mode and the Σ the set of all modes. In each mode some invariant conditions are added in order to force the change of mode when the trigger becomes true. We have that: – The initial mode σ0 is deﬁned from the initial system P0 . It is described in terms of an ODE system and this is derived from the Bio-PEPA model by considering the map πODE . Therefore, we have σ0 = πODE (P0 ). – Given a mode σi = πODE (Pi ), let trij be one possible jump condition that can be satisﬁed from it. We deﬁne the Bio-PEPA system Pj = Pi [event assignmentij ] as the modiﬁcation of the previous system Pi according to the event assignments associated with the trigger. The mode σj is then deﬁned as σj = πODE (Pj ). Case of delayed events. The delay associated with an event represents the time interval between when the event is ﬁred and when its assignments are executed. This information cannot be directly translated in any of the components of standard HA. In the following we report as we handle the delay in HA. First, we introduce a new variable tmode representing the time when the system enters in a speciﬁc mode. It is initially set to zero. The diﬀerential equation associated with this new variable is dtmode /dt = 0, i.e. this variable is constant in each mode.

56

F. Ciocchetta

Second, given an event (id, trigger, event assignment, delay), we split it into two immediate events, deﬁned as: 1. (id1 , trigger, tmode ← t, 0); 2. (id2 , t = tmode + delay, event assignment, 0). The role of the former event is to introduce the delay whereas the role of the second is to guarantee that the assignments of the initial event are executed after the given delay. 4.3

Example (Continued 2)

Consider the network presented in Sect. 2.2 with the addition of the set of events (see Sect. 3.4): [(event1 , Y = 0.8, r3 ← 0, 0)]. A schema of the HA associated with this network is reported in Fig. 3. The set of variables is {x, y, t}, where x and y are the two variables representing the two proteins. The initial concentrations, derived from the initial condition in the Bio-PEPA model, are x = 0, y = 0 and t = 0. We have just one event so we have two modes and the jump condition (guard) is y = 0.8. The former mode is described by the invariant condition y < 0.8 and the latter by y ≥ 0.8. The ODE system corresponding to the initial mode (S1) is derived by applying the mapping πODE to the initial Bio-PEPA system (P0 ) and is: dx dt dy dt

= −0.02 · x + 0.01 = 0.01 · x

For the second mode, the ODE system (S2) is obtained as πODE (P0 [r3 ← 0]) and is: dx dt = −0.02 · x dy dt = 0.01 · x If we consider both event1 and event2 , we have the HA represented in Fig. 4. There are three modes, representing the network at the initial state, when y < 0.8 and when y ≥ 0.8 and x ≥ 0.2. Two jumps conditions are deﬁned in terms of the trigger conditions. The ODE systems describing the ﬁrst and second modes are as above, whereas the ODE system for the third mode (S3) is obtained from πODE (P1 [r3 ← 0.005]) (where P1 is the Bio-PEPA system corresponding to the second mode) and is:

[ y = 0.8] x =0, y=0

S1

S2

Fig. 3. HA representation for the network composed of the two proteins X and Y and with an event involving concentrations

Bio-PEPA with Events

57

[ y = 0.8] x =0, y=0

S1

S2

[ y > 0.8 and x= 0.2]

S3 Fig. 4. HA representation for the network composed of the two proteins X and Y and with two sequential events

Fig. 5. Simulation results for the network composed of the two proteins X and Y and with the addition of event1 dx dt dy dt

= −0.02 · x + 0.005 = 0.01 · x

Some results for the network with just event1 are reported in Fig. 5. The protein X increases until time 200 s when Y reaches the value 0.8 and then decreases to 0. The protein Y increases, but after the event, its rate of increase is much lower than the case without the event. The results for the network with both event1 and event2 is reported in Fig. 6. In this case, when the second event is ﬁred, the protein X starts to increase again and this has eﬀect on the production of Y as well.

58

F. Ciocchetta

Fig. 6. Simulation results for the network composed of the two proteins X and Y and with the addition of event1 and event2

5

Stochastic Simulation by Gillespie’s Algorithm

One of the possible kinds of analysis supported by Bio-PEPA is stochastic simulation using Gillespie’s algorithm [10]. When events are considered the algorithm has to be modiﬁed in order to handle them. Broadly speaking, events are tackled by adding some conditions and some checks along the simulation. We start at time t = 0, with the Bio-PEPA system in its initial conditions. We assume that initially all the triggers evaluate to false. When one of the conditions is satisﬁed, the simulation stops and the system is modiﬁed according to the event assignments associated with the trigger. After that, the simulation can start again until another condition becomes true or the simulation time is reached. The use of triggers involving time can be challenging since it can happen that the time of the event does not coincide with any of the simulation time points. Our approach to deal with this case is discussed below. Note that if the events involve species concentrations, we have to change concentrations into number of molecules for stochastic simulation. Speciﬁcally, we have to multiply each concentration by N a V , where N a is the Avogadro number3 and V is the volume of the compartment. In the the rest of this section we assume that the events are in terms of number of molecules. 3

This is the number of “entities” (atoms or molecules) in one mole of substance. Its value is 6.022 × e+23 (mol)−1 .

Bio-PEPA with Events

59

We propose the following procedure for each simulation run. 1. 2. 3. 4.

Let P0 be the initial Bio-PEPA system and ts the maximum simulation time. While t < ts and triggeri = f alse for i = 1, 2, ..., NEvents , simulate. If t ≥ ts then stop. If t < ts and there exists a triggeri such that it is true, we have that: (a) if delay = 0 modify the Bio-PEPA system according to the event assignments associated with that trigger: P (t) = P(t)[event assignmenti]. Go to (2). (b) if delay > 0 go on with the simulation until time t + delay and then proceed as in (a).

Some ﬁnal observations concern how to use the algorithm in two particular situations. – In the case of two or more independent simultaneous events we proceed as observed in Section 3.1: we can abstract these events as a single event, whose trigger is deﬁned in terms of the triggers of the two events and the event assignments are the union of the assignments. Therefore, we modify the system according to the assignments associated with all the events involved. – When we have an event with a trigger involving time t = t˜, the time value t˜ may not correspond to any of the simulation time point obtained by using Gillespie’s simulation algorithm. Speciﬁcally, there exist two consecutive simulation time points tj and tj+1 such that tj < t˜ < tj+1 . If this happens, we have to decide when the system has to be modiﬁed. In order to handle this situation we consider the following approach: 1. if tj < t˜ + delay < tj+1 with delay ≥ 0 consider the system at time t˜+ delay and modify it at that time point. The simulation restarts from t˜ + delay. 2. If delay > 0 and t˜+ delay ≥ tj+1 consider the last simulation time point th ≤ t˜+ delay and run the simulation until th . Then, modify the system at time t˜ + delay and restart the simulation from that time point.

6

The Acetylcholine Receptor Model

This example concerns the functional properties of the nicotin Acetylcholine Receptors (nAChR). These are transmembrane proteins that mediate interconversions between open and closed channel states under the control of neurotransmitters. The detailed description of the model is reported in [16]. A schema of the model is shown in Figure 7. B (Basal state), A (Active state), D (Desensitized state) and I (Inactivable state) represent the diﬀerent states of the Acetylcholine receptors. The numbers 0, 1, 2 associated with the state are the number of ligands (denoted X) bound to a receptor. In the model the ligands are not modelled explicitly. Each column corresponds to a series of ligand binding actions at two identical sites per receptor whereas each row corresponds to a series of transactions between conformational states. All the

60

F. Ciocchetta kf_5

B0 + 2X

kf_9

A0 + 2X

kr_5 kf_0

kr_0

kr_9 kf_3

kf_6

B1 + X

kr_3

A1 + X

kr_6

kf_1 kr_1

B2

kf_7

kf_10

kr_4

A2

kr_7

I1 + X

kr 2

D0 + 2X

kf_12

kf_15

kr_12

D1 + X

kr_15

kf_8

kf_11

kf_14 kr_14

kr_10

kf_4

kf_2

I0 + 2X

kr_8

I2

kr 11

kf_13 kr_13

kf_16

D2

kr 16

Fig. 7. Schema of the Acetylcholine receptor model Table 2. The Acetylcholine receptor model. The list of parameters. The unit is s−1 . parameter kf0 kf2 kf4 kf6 kf8 kf10 kf12 kf14 kf16

value 3000 30000 1500 130 1500 19.85 3000 0.05 0.05

parameter kr0 kr2 kr4 kr6 kr8 kr10 kr12 kr14 kr16

value 8000 700 17.28 2740 8 1.74 4 0.0012 0.0012

parameter kf1 kf3 kf5 kf7 kf9 kf11 kf13 kf15

value 1500 3000 0.54 3000 19.7 20 1500 0.05

parameter kr1 kr3 kr5 kr7 kr9 kr11 kr13 kr15

value 16000 8.64 10800 4 3.74 0.81 8 0.0012

reactions are reversible and the dynamics are described by mass-action laws. For each reaction i, with i = 1, 2, ...16, the rate of the forward direction is kf i and the rate of the reverse direction kr i. In addition to these elements, there is an event to describe the recovery upon removal of free agonist at a given time. This is expressed by constraining the reaction rates of each second-order ligand-receptor reaction to zero. These constraints prevent ligand binding reactions from happening after that time, hence the states evolve as if the free ligands were completely removed from the system. The event is immediate, the trigger is “t = t2 ”, where t2 = 20 s, and the event assignments are kf0 ← 0, kf1 ← 0, kf3 ← 0, kf4 ← 0, kf7 ← 0 , kf8 ← 0 , kf12 ← 0 , kf13 ← 0. The Bio-PEPA system associated with the Acetylcholine receptor model. In the following we report brieﬂy the deﬁnition of the Bio-PEPA system

Bio-PEPA with Events

61

representing the Acetylcholine receptor model. The complete system is reported in the Appendix A. – Deﬁnition of the compartment list V. In the model we have a single threedimensional compartment, deﬁned as “comp1 : 1e-16, l;”, where l is litre. – Deﬁnition of the set N . Each species is associated with a species component. For each species component we have to declare the step size, the number of levels, the initial and maximum concentrations and the compartment where the species is. The ligand is not represented explicitly. For instance, in the case of B0, B1 and B2 we have: B0 : H = h, N = NB0 , M = MB0 , V = comp1, unit = μM ; B1 : H = h, N = NB1 , M = MB1 , V = comp1, unit = μM ; B2 : H = h, N = NB2 , M = MB2 , V = comp1, unit = μM ; where the step size is 1.66e-5, the number of levels NB0 = NB1 = NB2 is 1 (i.e. the species can be present, 1, or absent, 0), the maximum concentration MB0 = MB1 = MB2 is 1.66e-5 and coincides with the initial concentration of channels at the basal state. Note that the information about the step size and the number of levels is not used in this work, as we do not consider CTMC with levels, however we deﬁne them for completeness. – Deﬁnition of functional rates (FR ) and parameters (K). Each reversible reaction i, i = 0, 1, 2, · · · , 16, is decomposed in two irreversible reactions, fi and ri , representing the forward and inverse directions respectively. The associated kinetic laws are fα fi = f M A(kf i); and fα ri = f M A(kr i), where f M A denotes mass-action. All the parameters are deﬁned in the set K. The values are the ones reported in the paper [16]. – Deﬁnition of species components (Comp) and of the model component (P ). In the following we report the deﬁnition for B0, B1 and B2; the other species are dealt with similarly. def

B0 = (α def B1 = (α (α def B2 = (α

f0 , 1)↓B0 + (α f0 , 1)↑B1 + (α f1 , 1)↑B1 + (α f2 , 1)↓B2 + (α

r0 , 1)↑B0 + (α f5 , 1)↓B0 + (α r5 , 1)↑B0 r0 , 1)↓B1 + (α f6 , 1)↓B1 + (α r6 , 1)↑B1+ r1 , 1)↓B1 r2 , 1)↑B2 + (α f1 , 1)↑B2 + (α r1 , 1)↓B2

The system is described as:

B1(0) B2(0) A0(0) A1(0) A2(0) B0(1.66e-5) L1 L2 L3 L3 L4 L5

I1(0) I2(0) D0(0) D1(0) D2(0) I0(0) L6 L7 L8 L9 L10

where Li , i = 1, ..., 10 are the cooperation sets and the initial values for the species are 0 with the exception of the species B0. – Deﬁnition of events. We have only one event, describing a change in the system at time 20 s: [(event, t = 20, kf0 ← 0; kf1 ← 0; kf3 ← 0 kf4 ← 0; kf7 ← 0; kf8 ← 0; kf12 ← 0; kf13 ← 0, 0)]

62

F. Ciocchetta

Fig. 8. Stochastic simulation results for the Acetylcholine receptor model (average over 100 runs)

Analysis results. The HA associated with the Acetylcholine receptor model is similar to the one for the network presented in Sect.2.2 with the addition of the set of events. We have two modes, described by two diﬀerent sets of diﬀerential equations. The trigger condition involves time and is “t = 20 s”. The details of the two systems describing each mode are not reported. Simulation results made using Gillespie’s algorithm are reported in Fig. 8. The initial number of molecules for B0 is given M0 × V × N a = (1.66e-5 μM ) × (1.e16 l)×(6.022×e+23 (mol)−1 ) = 1000, where N a is the Avogadro number. All the other species are initially null. The number of runs is 100. The graph reproduces results in agreement with the ones reported in the paper [16]. Following the ligand removal, the state I2 loses agonist molecules and is transformed to the state B0 very rapidly, while D2 loses ligand molecules to form D0. Since the data occur on a wide range of times we represent the time on a logarithmic scale.

7

Related Works

The use of mathematical formalisms in order to represent discrete changes in biological systems is not new [1,4,23,17,5,6]. In [1] the authors proposed a hybrid system approach to modelling an intra-cellular network using continuous diﬀerential equations to model some part of the system and mode-switching to

Bio-PEPA with Events

63

describe the changes in the underlying dynamics. Some models with hybrid behaviour are presented and described using CHARON [2], a language that allows formal description of hybrid systems. The authors of [23] discussed the use of discrete changes in biological systems and presented some examples using the formalism HybridSAL [21]. Hybrid Concurrent Constraint Programming is used to model some biological systems with both discrete and continuous changes in [4]. In [5] the authors presented a map from stochastic Concurrent Constraint Programming (sCCP) to HA. The HA generated in this way are said to be able to capture some aspects of the dynamics which are lost if standard diﬀerential equations are used. A discussion of hybrid systems and biology is reported in [6]. Finally, in [17] the authors presented HYPE, a process algebra for the modelling of hybrid systems and used it to represent the repressilator, an artiﬁcial genetic network composed of three genes and their respective proteins with oscillatory behaviour. In none of these works are SBML-like events considered explicitly, but the focus is on general hybrid systems. Events have been proposed in the Beta Workbench (BetaWB) [14] and in the associated programming language BlenX [28]. In BlenX events can be considered as global rules of the environment, triggered only when the conditions associated with them are satisﬁed. Each event is the composition of a condition (cond ) and an action (verb) and is associated with a rate. Conditions can involve number of entities, the simulation time or the simulation step. The possible actions are the join of two entities, the split of one entity into two, the update of a variable of the system and the deletion or the creation of a new entity. The concept of events proposed in BlenX is quite similar to the one considered for Bio-PEPA. The BlenX condition and action correspond to the trigger and event assignment in Bio-PEPA events. However, rates in BlenX have a diﬀerent meaning from the delay in Bio-PEPA. Indeed, in Bio-PEPA an events occurs when the trigger is satisﬁed and the role of the delay is to postpone when the event is executed. BlenX events with a ﬁnite rate can happen only when the trigger is satisﬁed but it is in competition with other actions that are enabled contemporaneously (race condition). BlenX events with inﬁnite rate correspond to immediate events in Bio-PEPA. In order to compare the deﬁnition of events in the two languages, we show how the events proposed in this paper can be described in BlenX. The event event1 deﬁned in Sec. 3.4 is represented in BlenX as: when (Y → value) update (r3 , change par) where value is 0.8 · N aV molecules and the function change par is deﬁned as change par : f unction = 0. The operator “→” recognizes when the quantity bound to Y becomes greater than the speciﬁed value, whereas the action “ update (r3 , change par)” means that the parameter r3 is updated according to the function change par (in our case it assigns the value 0). The rate associated with the update action is always inﬁnite and not reported. Concerning the event in the Acetylcholine receptor model, it is not possible to represent this event in BlenX as conditions involving time are not allowed with the action update.

64

F. Ciocchetta

Note that BlenX events represent more general kinds of interactions than BioPEPA events. For instance, they are used to model the formation of a complex (by using the action join) or the split of a complex into two parts (by using the split action). These reactions (as all the other kinds) are represented in Bio-PEPA by synchronization of the species components over the action types abstracting the reactions. Bio-PEPA events have been introduced speciﬁcally to represent experimental situations when there is change in the system due to some conditions.

8

Conclusions

In this work we have presented an extension of Bio-PEPA to handle SBML-like events. Events are constructs that represent changes in the system due to some trigger conditions. The events considered here are simple, but nevertheless able to describe most of the discontinuous changes in models and experiments. Events are added to our language without any modiﬁcation to the rest of the syntax. The main motivation of this choice is that we want to keep the speciﬁcation of the model as simple as possible. Furthermore, this approach is appropriate when we study the same biochemical system but with diﬀerent experimental regimes. A topic for the future concerns the study of more general events and the possible extension to other kinds of hybrid systems in biology. Furthermore, we plan to exploit the possible kinds of analysis involving hybrid systems in the context of systems biology. In this paper we focus on the mapping to Hybrid Automata and stochastic simulation by (a modiﬁcation of) Gillespie’s algorithm. Further investigation will concern the application of model checking for the study of the properties of biological systems.

Acknowledgements The author thanks Jane Hillston, Vashti Galpin and Adam Duguid for their helpful comments. The author is supported by the EPSRC under the CODA project “Process Algebra Approaches for Collective Dynamics” (EP/c54370x/01).

References 1. Alur, R., Belta, C., Ivancic, F., Kumar, V., Mintz, M., Pappa, G., Rubin, H., Schug, J.: Hybrid modeling and simulation of biomolecular networks. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 19–32. Springer, Heidelberg (2001) 2. Alur, R., Grosu, R., Hur, Y., Kumar, V., Lee, I.: Modular Specification of Hybrid Systems in CHARON. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, p. 6. Springer, Heidelberg (2000) 3. Bio-PEPA Workbench Home Page, http://www.dcs.ed.ac.uk/home/stg/software/biopepa/

Bio-PEPA with Events

65

4. Bockmayr, A., Courtois, A.: Using hybrid concurrent constraint programming to model dynamic biological systems. In: Stuckey, P.J. (ed.) ICLP 2002. LNCS, vol. 2401, p. 85. Springer, Heidelberg (2002) 5. Bortolussi, L., Policriti, A.: Hybrid Approximation of Stochastic Concurrent Constraint Programming. Constraints 13(1-2), 66–90 (2008) 6. Bortolussi, L., Policriti, A.: Hybrid Systems and Biology. Continuous and Discrete Modeling for Systems Biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 424–448. Springer, Heidelberg (2008) 7. Calder, M., Gilmore, S., Hillston, J.: Automatically deriving ODEs from process algebra models of signalling pathways. In: Proc. of CMSB 2005, pp. 204–215 (2005) 8. Calder, M., Gilmore, S., Hillston, J.: Modelling the influence of RKIP on the ERK signalling pathway using the stochastic process algebra PEPA. In: Priami, C., Ing´ olfsd´ ottir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 1–23. Springer, Heidelberg (2006) 9. Ciocchetta, F., Hillston, J.: Bio-PEPA: an extension of the process algebra PEPA for biochemical networks. In: Proc. of FBTC 2007. ENTCS, vol. 194(3), pp. 103– 117 (2008) 10. Ciocchetta, F., Hillston, J.: Bio-PEPA: a framework for the modelling and analysis of biological systems. Theoretical Computer Science (to appear) 11. Ciocchetta, F.: Bio-PEPA with SBML-like Events. In: Proc. of the Workshop Computational Models for Cell Processes, TUCS general publication, vol. 47 (2008) 12. Ciocchetta, F., Hillston, J.: Bio-PEPA: a framework for the modelling and analysis of biological systems. School of Informatics University of Edinburgh Technical Report, EDI-INF-RR-1231 (2008) 13. Cota, B.A., Sargent, R.B.: Simultaneous events and distributed simulation. In: Proc. of the Winter Simulation Conference (1990) 14. Dematt´e, L., Priami, C., Romanel, A.: The BlenX Language: a Tutorial. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 313–365. Springer, Heidelberg (2008) 15. Deshpande, A., Gollu, A., Semenzato, L.: SHIFT Programming Language and Run-Time System for Dynamic Networks of Hybrid Automata. PATH Report, http://path.berkeley.edu/SHIFT/publications.html 16. Edelstein, S.J., Schaad, O., Henry, E., Bertrand, D., Changgeux, J.P.: A kinetic mechanism for nicotin acetylcholine receptors based on multiple allosteric transitions. Biol. Cybern. 75, 361–379 (1996) 17. Galpin, V., Hillston, J., Bortolussi, L.: HYPE applied to the modelling of hybrid biological systems. ENTCS, vol. 218, pp. 33–51 (2008); Also in Proceedings of MFPS 2008 18. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry 81, 2340–2361 (1977) 19. Henzinger, T.A.: The Theory of Hybrid Automata. In: The proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science, LICS (1996) 20. Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: HyTech: A Model Checker for Hybrid Systems. Software Tools for Technology Transfer 1, 110–122 (1997) 21. HybridSal home page, http://sal.csl.sri.com/hybridsal/ 22. Hucka, M., Finney, A., Hoops, S., Keating, S., Le Nov´ere, N.: Systems Biology Markup Language (SBML) Level 2: Structures and Facilities for Model Definitions, http://sbml.org/documents/

66

F. Ciocchetta

23. Lincoln, P., Tiwari, A.: Symbolic systems biology: Hybrid modeling and analysis of biological networks. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 660–672. Springer, Heidelberg (2004) 24. Le Nov´ere, N., Bornstein, B., Broicher, A., Courtot, M., Donizelli, M., Dharuri, H., Li, L., Sauro, H., Schilstra, M., Shapiro, B., Snoep, J.L., Hucka, M.: BioModels Database: a Free, Centralized Database of Curated, Published, Quantitative Kinetic Models of Biochemical and Cellular Systems. Nucleic Acids Research 34, D689–D691 (2006) 25. Priami, C., Quaglia, P.: Beta-binders for biological interactions. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 20–33. Springer, Heidelberg (2005) 26. Prism web site, http://www.prismmodelchecker.org/ 27. Priami, C., Regev, A., Silverman, W., Shapiro, E.: Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Information Processing Letters 80, 25–31 (2001) 28. Dematt´e, L., Priami, C., Romanel, A.: The Beta Workbench: a computational tool to study the dynamics of biological systems. Briefings in Bioinformatics 9(5), 437–449 (2008)

A

Appendix: Bio-PEPA System for the Acetylcholine Receptor Model

In this appendix we report the speciﬁcation of the whole Acetylcholine receptor model in Bio-PEPA. Note that, in the deﬁnition of the species component, we use the following notation: >> indicates a product (it corresponds to the operator ↑ in the Bio-PEPA syntax) and >B1 + (alpha_r_0,1)B1 + (alpha_r_6,1)B2 + (alpha_r_1,1)A0 + (alpha_r_5,1)A1 + (alpha_r_3,1)A2 + (alpha_r_2,1)A2 + (alpha_r_4,1)A2 + (alpha_r_11,1)>>A2 I0 = (alpha_f_7,1)I0 + (alpha_f_9,1)>>I0 + (alpha_r_9,1)I1 + (alpha_r_7,1)I2 + (alpha_r_8,1)I2 + (alpha_r_11,1)D0 +

67

68

F. Ciocchetta

(alpha_r_14,1)>>D0 D1 = (alpha_f_12,1)>>D1 + (alpha_r_12,1)D2 + (alpha_r_13,1)D2 + (alpha_r_16,1) k3bn : (s’=2) ; The next forward transition, from state s=3 to state s=4 in the Prism model, is a combination of several detailed steps of the translation mechanism involving the processing of GTP. The transition is one-directional, again with a signiﬁcant diﬀerence in the rate k3fc for a cognate aa-tRNA compared to the rates k3fp and k3fn for pseudo-cognate and near-cognate aa-tRNA, that are equal. // GTPase activation, GTP hydrolysis // and EF-Tu conformation change [ ] (s=3) & cogn -> k3fc : (s’=4) ; [ ] (s=3) & pseu -> k3fp : (s’=4) ; [ ] (s=3) & near -> k3fn : (s’=4) ; In state s=4, the aa-tRNA can either be rejected, after which control moves to intermediate state s=5, or accommodates, i.e. the ribosome reconforms such that the aa-tRNA can hand over the amino acid it carries, so-called peptidyl transfer. In the latter case, control changes to state s=6. As before, rates for cognates and those for pseudo-cognates and near-cognates are of diﬀerent magnitudes. From the intermediate rejection state s=5, with all booleans set to false again, control returns to the start state s=1.

In Silico Modelling and Analysis of Ribosome Kinetics

77

// rejection [ ] (s=4) & cogn -> k4rc : (s’=5) & (cogn’=false) ; [ ] (s=4) & pseu -> k4rp : (s’=5) & (pseu’=false) ; [ ] (s=4) & near -> k4rn : (s’=5) & (near’=false) ; // accommodation, peptidyl [ ] (s=4) & cogn -> k4fc : [ ] (s=4) & pseu -> k4fp : [ ] (s=4) & near -> k4fn :

transfer (s’=6) ; (s’=6) ; (s’=6) ;

After some movement back-and-forth between state s=6 and state s=7, the binding of the EF-G complex becomes permanent. In the detailed translation mechanism a number of (mainly sequential) steps follows, that are summarized in the Prism model by a single transition to a ﬁnal state s=8, that represents elongation of the protein in nascent with the amino acid carried by the aa-tRNA. The synthesis is successful if the aa-tRNA was either a cognate or pseudo-cognate for the codon under translation, reﬂected by either cogn or pseu being true. In case the aa-tRNA was a near-cognate (non-cognates never pass beyond state s=2), an amino acid that does not correspond to the codon in the genetic code has been inserted. Thus, in this case, an insertion error has occurred. // EF-G binding [ ] (s=6) -> k6f : (s’=7) ; [ ] (s=7) -> k7b : (s’=6) ; // GTP hydrolysis, unlocking, tRNA movement // Pi release, rearrangements of ribosome and EF-G // dissociation of GDP [ ] (s=7) -> k7f : (s’=8) ; A number of transitions, linking the dissociation state s=0 and the rejection state s=5 back to the start state s=1, where a race of aa-tRNAs of the four types commences anew, and looping at the ﬁnal state s=8, complete the Prism model. The transitions are deterministically taking, as no other transitions leave these states. Having no biological counterpart the transitions are assigned a high-rate making the time they take negligible. // no entrance, re-entrance at state 1 [ ] (s=0) -> FAST : (s’=1) ; // rejection, re-entrance at state 1 [ ] (s=5) -> FAST : (s’=1) ; // elongation [ ] (s=8) -> FAST : (s’=8) ; Table 2 collects the rates as compiled from the biological literature and used in the Prism model above.

78

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

Table 2. Rates of the Prism model, adapted from [9,26]. Rate k2bx is based on the estimate of the average delay of non-cognate arrivals of 0.5ms. Rates k4fc, k4fp, k4fn and k7f are accumulative rates of sequentially composed transitions. k1f k2f k2b k2bx

140 190 85 2000

k3fc k3fp, k3fn k3bc k3bp, k3bn

260 0.40 0.23 80

k4rc k4rp, k4rn k4fc k4fp, k4fn

60 FAST 166.7 46.1

k6f k7f k7b

150 145.8 140

In the next two sections, we will study the Prism model described above for the analysis of the probability for insertion errors, i.e. extension of the peptidyl chain with a diﬀerent amino acid than the codon codes for, and of the average insertion times, i.e. the average time it takes to process a codon up to elongation.

4

Insertion Errors

In this section we discuss how the model checking features of Prism can be exploited to predict the misreading frequencies for individual codons. The translation of mRNA into a polypeptide chain is performed by the ribosome machinery with high precision. Experimental measurements show that on average, only one in 1,000 to 10,000 amino acids is added wrongly (cf. [12]).3 For a codon under translation, a pseudo-cognate anticodon carries precisely the amino acid that the codon codes for. Therefore, although diﬀerent in codonanticodon bound, successful matching of a pseudo-cognate does not lead to an insertion error, as –accidentally– the right amino acid has been used for elongation. In our model, the main diﬀerence of cognates vs. pseudo-cognates and near-cognates is in the kinetics. At various stages of the peptidyl transfer the rates for true cognates diﬀer from those for pseudo-cognates and near-cognates up to three orders of magnitude. Figure 2 depicts the relevant abstract automaton, derived from the Prism model discussed above. See also Table 1. In case a transition is labeled with two rates, e.g. 0.23/80, the leftmost number, viz. 0.23, concerns the processing of a cognate aa-tRNA, while the rightmost number, viz. 80, that of a pseudo-cognate or near-cognate. In three states a probabilistic choice has to be made: in state 2 leading to state 0 or 3, in state 3 leading back to state 3 or forward to state 4, and in state 4 leading to rejection in state 5 or eventually to success via state 6. The probabilistic choice in state 2 is the same for cognates, pseudo-cognates and near-cognates alike, the ones in state 3 and in state 4 depend on the type of aa-tRNA, cognates and pseudo-cognates vs. near-cognates. A cognate aa-tRNA starting in state 1 will move to state 2 with probability 1. From here, it will dissociate with probability 85/(85 + 190) ≈ 0.309, moving to 3

Our ﬁndings, see Table 5, based on the kinetic rates available and the assumptions made, are well within these boundaries.

In Silico Modelling and Analysis of Ribosome Kinetics

FAST

79

5 60/FAST 0.23/80

1

2

FAST

0

190

3

260/0.40

4

167/46

6

7

8

85

Fig. 2. Abstract automaton summarizing the Prism code. See also Table 1.

state 0, or will be recognized with the complementary probability 190/(85 + 190) ≈ 0.691, moving to state 3. The same holds for pseudo-cognate and nearcognate aa-tRNA. However, after recognition in state 3, a cognate aa-tRNA will go through the hydrolysis phase leading to state 4 for a fraction 0.999 of the cases (computed as 260/(0.23 + 260)), a fraction being close to 1. In contrast, for a pseudo-cognate or near-cognate aa-tRNA this is 0.40/(0.40 + 80) ≈ 0.005 only. A similar diﬀerence can be noted in state 4 itself. Cognates will accommodate and continue to state 6 with probability 0.736, while pseudo-cognates and nearcognates will do so with the small probability 0.044, the constant FAST being set to 1000 in our experiments as in [9]. Since the transition from state 4 to state 6 is irreversible, the rates of the remaining transitions are not of importance here. For cognates, pseudo-cognates and near-cognates, the probability of reaching state 8 in one attempt can be easily computed, solving a small system of equations by hand or by using Prism. In the latter case, we have Prism evaluate the CSL-formula P=? [ (s!=0 & s!=5) U (s=8) {(s=2) & cogn} ] against our model. The formula asks to establish the probability for all paths where s is not set to 0 nor 5, until s have been set to 8, starting from the (unique) state satisfying s=2 & cogn. The expression {(s = 2)&cogn} is a so-called ﬁlter construction as supported by Prism. We obtain psc = 0.508, psp = 0.484 · 10−4 and psn = 0.484 · 10−4, with psc the probability for a cognate to end up in state 8 —and elongate the peptidyl chain— without going through state 0 nor state 5; psp and psn the analogues for success of pseudo- and near-cognates, respectively. Note that these values are the same for every codon. Diﬀerent among codons in E. coli are the concentrations of cognates, pseudocognates and near-cognates.4 Ultimately, the frequencies f c , f p and f n of the types of aa-tRNA in the cell, i.e. the actual number of molecules of the kind, determine the concentration of the aa-tRNA. Hence, under the usual assumption of homogeneous distribution, the frequencies determine the total rates for the arrival process of an anticodon. The probability for an anticodon arriving to be a cognate, pseudo-cognate or near-cognate can then be calculated from this. 4

See Table 4 in the appendix.

80

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

Fig. 3. Correlation of the ratio f n /f c of the frequency of near-cognates over the frequency of cognates vs. the probability of an insertion error. See also Table 5 in the appendix.

As concluded in [9] based on simulation results, the probability for an erroneous insertion, is strongly correlated with the quotient of the number of nearcognate anticodons and the number of cognate anticodons. See Figure 3. As an advantage of the present setting, this correlation actually can be formally derived. This is as follows. We have that an insertion error occurs if a near-cognate succeeds to attach its amino acid. Note that we already have established psp , psn psc . Therefore, P (error) = P (near & elongation | elongation) =

psn · (f n /tot) psc · (f c /tot) + psp · (f p /tot) + psn · (f n /tot)

≈

psn · f n psc · f c

∼

fn fc

with tot = f c + f p + f n , and where we have used that P (elongation) = (f c /tot) · psc + (f p /tot) · psp + (f n /tot) · psn . Note, the ability to precalculate the probabilities psc , psp and psn is instrumental in obtaining the above result. As such, it illustrates the approach of piecewise analysis, ﬁrst establishing quantities for part of the system to obtain a quantity for the system as a whole.

5

Competition and Insertion Times

In this section, we continue the analysis of the Prism model for translation and discuss the correlation of the average insertion time for the amino acid speciﬁed

In Silico Modelling and Analysis of Ribosome Kinetics

81

by a codon, on the one hand, and and the aa-tRNA competition, i.e. the relative abundance of pseudo-cognate and near-cognate aa-tRNAs, on the other hand. The insertion time of a codon is the average time it takes to elongate the protein in nascent with an amino acid. The average insertion time can be computed in Prism using the concept of rewards, also known as costs in Markov theory. Each state is assigned a value as its reward. Further, the reward of each state is weighted per unit of time. Hence, it is computed by multiplication with the average time spent in the state. The cumulative reward of a path in the chain is deﬁned as a sum over all states in the path of such weighted rewards per state. Thus, by assigning to each state the value 1 as reward, we obtain the total average time for a given path. For example, in Prism the cumulative reward formula R=? [ F (s=8) ] which asks to compute the expected time to reach state s=8. Recall, in state s=8 the amino acid is added to the polypeptide chain. The formula returns the average reward of all the paths that lead from the initial state 1 to state 8. As explained above, in order to obtain the average time for insertion, we assign each state the value 1 as a reward, which in Prism can be done using the following code rewards true: 1 endrewards The construct expresses the fact that 1 is assigned to any state that satisfy the condition true (which is trivially satisﬁed by all states). So, a script calling Prism for model checking the above formula then yields the expected insertion time per codon. Table 6 in the appendix lists the results. Although the correlation of cognate frequency and insertion times is limited, the qualitative claim of [25] of ‘rare’ codons being translated slow and ‘frequent’ codons being translated fast is roughly conﬁrmed by the model. E.g., the codons AGC and CCA have amongst the lowest frequencies, 420 and 617, the lowest and two but lowest frequency, respectively, and translates indeed the slowest, 1.4924 and 1.5622 seconds, respectively. However, the codon CCA with an availability of 581, of one but lowest frequency, is translated at a moderate rate of 0.5525 seconds on the average. Thus, in line with our considerations, cognate availability per se does not suﬃciently predict translation time. Comparably, the fastest insertion times, 52.7 and 64.5 milliseconds, are realized by the codons CU G and CGU , of the codons corresponding to amino acids the one and two but most abundant. The codon CU G of the highest frequency 5136, excluding stop codons, though has an average insertion time of 102.8 milliseconds. A little bit more ingenuity is needed to establish average exit times, for example for a cognate to pass from state s=2 to state s=8. The point is that conditional probabilities are involved. However, since dealing with exponential distributions, elimination of transitions in favour of adding their rates to that of the remaining ones, does the trick. Various results, some of them used below, are collected in Table 3. (The probabilities of failure and success for the non-cognates are trivial, pfx = 1 and psx = 0, with a time per failed attempt Tfx = 0.5 · 10−3 seconds.) There is a visible correlation between the quotient of the number of nearcognate aa-tRNA over the number of cognate aa-tRNA and the average insertion time. See Figure 4. In fact, the average insertion time for a codon is

82

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

Table 3. Exit probabilities and exit times (in seconds) for three types of aa-tRNA, superscripts c, p and n for cognate, pseudo-cognate and near-cognate aa-tRNA, respectively. Failure for exit to states s=0 or s=5, subscript f ; success for exit to state s=8, subscript s. psc 0.5079 psp 4.847 · 10−4 psn 4.847 · 10−4

pfc 0.4921 pfp 0.9995 pfn 0.9995

Tsc 0.03182 Tsp 3.251 Tsn 3.251

Tfc 9.342 · 10−3 Tfp 0.3914 Tfn 0.3914

Fig. 4. Correlation of the ratio (f p + f n )/f c of total frequency of pseudo-cognates and near-cognates over the frequency of cognates vs. average insertion times. See also Table 6 in the appendix.

approximately proportional to the near-cognate/cognate ratio. This can be seen as follows. The insertion of the amino acid is completed if state s=8 is reached, either for a cognate, pseudo-cognate or near-cognate. As we have seen, the probability for either of the latter two is negligible, psp , psn = 4.847 · 10−4 . Therefore, the number of cognate arrivals is decisive. With pfc and psc being the probability for a cognate to fail, i.e. exit at state s=0 or s=5, or to succeed, i.e. reach of state s=8, the insertion time Tins can be regarded as a geometric series. (Note the exponent i below.) Important are the numbers of arrivals of the other aa-tRNA types per single cognate arrival, expressed in terms of frequencies. An arrival occurring for the (i + 1)st arrival of a cognate has spent (i × Tfc ) + c Ts processing cognate aa-tRNA. The number of pseudo-cognate, near-cognate and non-cognate arrivals per individual cognate arrival are, on the average, the f f f relative fractions p , n , and x , respectively (with f p , f n , and f c as before in fc f c fc

In Silico Modelling and Analysis of Ribosome Kinetics (p)

(n)

(x)

(cf) (p)

(n)

(x)

(cf) (p)

(n)

83 (x)

(cs)

Fig. 5. Accumulated delay after three cognate arrivals: (p) delay (f p /f c )·Tfp for failing pseudo-cognates, (n) delay (f n /f c )·Tfn for failing near-cognates, (x) delay (f x /f c )·Tfx for non-cognates, (cf) exit time Tfc for a failing cognate, (cs) exit time Tsc for a successful cognate.

Section 4, and f x the frequency of non-cognate aa-tRNA). See Figure 5. Summing over i, the number of failing cognate aa-tRNA for a successful cognate insertion, yields ∞

i

(pfc ) psc · (delay for i failing and 1 successful cognate arrivals) f f f i = (pfc ) psc · (i + 1) · ( p Tfp + n Tfn + x Tfx ) + i · Tfc + Tsc fc fc fc ∞ i ≈ f p +f n psc Tfn i=0 (i + 1) · (pfc ) fc ∼ f p +f n . fc

Tins =

i=0 ∞ i=0

fx x T is fc f n relatively small, from which it follows that f p +f n Tf is the dominant summand. fc Note that the estimate is not accurate for small values of f p + f n . Nevertheless, closer inspection shows that for these values the approximation remains orderpreserving. Again, the results obtained for parts of the systems are pivotal in the derivation.

Here, we have used that Tfc and Tsc are negligible, Tfp equals Tfn , and

6

Concluding Remarks

In this paper, we presented a stochastic model of the translation process based on presently available data of ribosome kinetics [12,9]. We used the model checking facilities of the Prism tool for continuous-time Markov chains. Compared to [9] that uses simulation, our approach is computationally more reliable (independent on the number of simulations) and has faster response times (taking seconds rather then minutes or hours). More importantly, model checking allowed us to perform piecewise analysis of the system, yielding better insight in the model compared to just observing the end-to-end results with a monolithic model. Based on this, we improved on earlier observations, regarding error probabilities and insertion times, by actually deriving the correlation suggested by the data. In [7] a correlation was reported between the number of copies (concentrations) of cognate tRNAs and the frequency of usage of particular codons in the most abundant proteins in E. coli. It is suggested that this optimization is favorable for the cell growth: when they are urgently needed the most used proteins are translated with maximum speed and accuracy. On the other hand, we observed that there is a high correlation (0.86) between the cognate tRNA concentrations

84

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

and the ration near-cognates vs. cognates which, according to our model, determines the error probabilities. Consequently, it would be interesting to check if there exists even better correlation between the near-cognates/cognates ratios and the codon usage frequencies than between the latter and the concentrations. In conclusion, we have experienced aa-tRNA competition as a very interesting biological case study of intrinsic stochastic nature, falling in the category of the well known lambda-phage example [1]. Our model opens a new avenue for future work on biological systems that possess intrinsically probabilistic properties. It would be interesting to apply our method to processes which, similarly to translation, involve small numbers of molecules, like DNA replication [16,19], DNA repair [11,20], charging of the tRNAs with amino acids [8,15], etc., thus rendering approaches based on ordinary diﬀerential equations less attractive. Acknowledgments. We are grateful to Timo Breit, Christiaan Henkel, Erik Luit, Jasen Markovski, and Hendrik Viljoen for fruitful discussions and constructive feedback.

References 1. Arkin, A., Ross, J., McAdams, H.H.: Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected. Escherichia coli cells. Genetics 149, 1633–1648 (1998) 2. Baier, C., Katoen, J.-P., Hermanns, H.: Approximate symbolic model checking of continuous-time Markov chains. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 146–161. Springer, Heidelberg (1999) 3. Boˇsnaˇcki, D., ten Eikelder, H.M.M., Steijaert, M.N., de Vink, E.P.: Stochastic analysis of amino acid substitution in protein synthesis. In: Heiner, M., Uhrmacher, A.M. (eds.) CMSB 2008. LNCS (LNBI), vol. 5307, pp. 367–386. Springer, Heidelberg (2008) 4. Calder, M., Vyshemirsky, V., Gilbert, D., Orton, R.: Analysis of signalling pathways using continuous time Markov chains. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 44–67. Springer, Heidelberg (2006) 5. Chabrier, N., Fages, F.: Symbolic model checking of biochemical networks. In: Priami, C. (ed.) CMSB 2003. LNCS, vol. 2602, pp. 149–162. Springer, Heidelberg (2003) 6. Danos, V., Feret, J., Fontana, W., Harmer, R., Krivine, J.: Rule-based modelling of cellular signalling. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 17–41. Springer, Heidelberg (2007) 7. Dong, H., Nilsson, L., Kurland, C.G.: Co-variation of tRNA abundance and codon usage in Escherichia coli at diﬀerent growth rates. Journal of Molecular Biology 260, 649–663 (1996) 8. Nureki, O., et al.: Enzyme structure with two catalytic sites for double-sieve selection of substrate. Science 280, 578–582 (1998) 9. Fluitt, A., Pienaar, E., Viljoen, H.: Ribosome kinetics and aa-tRNA competition determine rate and ﬁdelity of peptide synthesis. Computational Biology and Chemistry 31, 335–346 (2007)

In Silico Modelling and Analysis of Ribosome Kinetics

85

10. Gilchrist, M.A., Wagner, A.: A model of protein translation including codon bias, nonsense errors, and ribosome recycling. Journal of Theoretical Biology 239, 417– 434 (2006) 11. Goodman, M.F.: Coping with replication ‘train wrecks’ in Escherichia coli using Pol V, Pol II and RecA proteins. Trends in Biochemical Sciences 25, 189–195 (2000) 12. Gromadski, K.B., Rodnina, M.V.: Kinetic determinants of high-ﬁdelity tRNA discrimination on the ribosome. Molecular Cell 13(2), 191–200 (2004) 13. Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic model checking of complex biological pathways. In: Priami, C. (ed.) CMSB 2006. LNCS (LNBI), vol. 4210, pp. 32–47. Springer, Heidelberg (2006) 14. Heyd, A.W., Drew, D.A.: A mathematical model for elongation of a peptide chain. Bulletin of Mathematical Biology 65, 1095–1109 (2003) 15. Ibba, M., S¨ oll, D.: Aminoacyl-tRNAs: setting the limits of the genetic code. Genes & Development 18, 731–738 (2004) 16. Johnson, K.A.: Conformational coupling in DNA polymerase ﬁdelity. Annual Reviews in Biochemistry 62, 685–713 (1993) 17. Karp, G.: Cell and Molecular Biology, 5th edn. Wiley, Chichester (2008) 18. Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model cheking with Prism: a hybrid approach. Journal on Software Tools for Technology Transfer 6, 128–142 (2004), http://www.prismmodelchecker.org/ 19. Martomo, S.A., Mathews, C.K.: Eﬀects of biological DNA precursor pool asymmetry upon accuracy of DNA replication in vitro. Mutation Research 499, 197–211 (2002) 20. Ni, M., Wang, S.-Y., Li, J.-K., Ouyang, Q.: Simulating the temporal modulation of inducible DNA damage response in Escherichia coli. Biophysical Journal 93, 62–73 (2007) 21. Pape, T., Wintermeyer, W., Rodnina, M.: Complete kinetic mechanism of elongation factor Tu-dependent binding of aa-tRNA to the A-site of E. coli. EMBO Journal 17, 7490–7497 (1998) 22. Priami, C., Regev, A., Shapiro, E., Silverman, W.: Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Information Processing Letters 80, 25–31 (2001) 23. Rodnina, M.V., Wintermeyer, W.: Ribosome ﬁdelity: tRNA discrimination, proofreading and induced ﬁt. Trends in Biochemical Sciences 26(2), 124–130 (2001) 24. Savelsbergh, A., et al.: An elongation factor G-induced ribosome rearrangement precedes tRNA–mRNA translocation. Molecular Cell 11, 1517–1523 (2003) 25. Sørensen, M.A., Kurland, C.G., Pedersen, S.: Codon usage determines translation rate in Escherichia coli. Journal of Molecular Biology 207, 365–377 (1989) 26. Viljoen, H.: Private communication (2008) overproduction in 27. Wahab, S.Z., Rowley, K.O., Holmes, W.M.: Eﬀects of tRNALeu 1 Escherichia coli. Molecular Microbiology 7, 253–263 (1993)

A

Appendix: Suplementary Figures and Data

// translation model

stochastic

86

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

// constants const double ONE=1; const double FAST=1000; // tRNA rates const double c_cogn const double c_pseu const double c_near const double c_nonc const const const const const const const const const const const const const const const const const const const

double double double double double double double double double double double double double double double double double double double

; ; ; ;

k1f = 140; k2b = 85; k2bx=2000; k2f = 190; k3bc= 0.23; k3bp= 80; k3bn= 80; k3fc= 260; k3fp= 0.40; k3fn= 0.40; k4rc= 60; k4rp=FAST; k4rn=FAST; k4fc= 166.7; k4fp= 46.1; k4fn= 46.1; k6f = 150; k7b = 140; k7f = 145.8;

module ribosome s : [0..8] init 1 ; cogn : bool init false pseu : bool init false near : bool init false nonc : bool init false

; ; ; ;

// initial binding [ ] (s=1) -> k1f * c_cogn : (s’=2) & (cogn’=true) ; [ ] (s=1) -> k1f * c_pseu : (s’=2) & (pseu’=true) ; [ ] (s=1) -> k1f * c_near : (s’=2) & (near’=true) ; [ ] (s=1) -> k1f * c_nonc : (s’=2) & (nonc’=true) ; [ ] (s=2) & ( cogn | pseu | near ) -> k2b : (s’=0) & (cogn’=false) & (pseu’=false) & (near’=false) ; [ ] (s=2) & nonc -> k2bx : (s’=0) & (nonc’=false) ;

In Silico Modelling and Analysis of Ribosome Kinetics

// codon recognition [ ] (s=2) & ( cogn | pseu | near ) -> k2f : (s’=3) ; [ ] (s=3) & cogn -> k3bc : (s’=2) ; [ ] (s=3) & pseu -> k3bp : (s’=2) ; [ ] (s=3) & near -> k3bn : (s’=2) ; // GTPase [ ] (s=3) [ ] (s=3) [ ] (s=3)

activation, GTP hydrolysis, reconformation & cogn -> k3fc : (s’=4) ; & pseu -> k3fp : (s’=4) ; & near -> k3fn : (s’=4) ;

// rejection [ ] (s=4) & cogn -> k4rc : (s’=5) & (cogn’=false) ; [ ] (s=4) & pseu -> k4rp : (s’=5) & (pseu’=false) ; [ ] (s=4) & near -> k4rn : (s’=5) & (near’=false) ; // accommodation, peptidyl [ ] (s=4) & cogn -> k4fc : [ ] (s=4) & pseu -> k4fp : [ ] (s=4) & near -> k4fn :

transfer (s’=6) ; (s’=6) ; (s’=6) ;

// EF-G binding [ ] (s=6) -> k6f : (s’=7) ; [ ] (s=7) -> k7b : (s’=6) ; // GTP hydrolysis, unlocking, // tRNA movement and Pi release, // rearrangements of ribosome and EF-G, // dissociation of GDP [ ] (s=7) -> k7f : (s’=8) ; // no entrance, re-entrance at state 1 [ ] (s=0) -> FAST*FAST : (s’=1) ; // rejection, re-entrance at state 1 [ ] (s=5) -> FAST*FAST : (s’=1) ; // elongation [ ] (s=8) -> FAST*FAST : (s’=8) ; endmodule rewards true : 1; endrewards

87

88

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

Table 4. Frequencies of cognate, pseudo-cognate, near-cognate and non-cognates for E. coli as molecules per cell [7]. Stop codons UGA, UAG and UAA. codon cognate pseudo- nearnoncognate cognate cognate

codon cognate pseudo- nearnoncognate cognate cognate

UUU UUC UUG UUA UCU UCC UCG UCA UGU UGC UGG UGA∗ UAU UAC UAG∗ UAA∗

1037 1037 2944 1031 2060 764 1296 1296 1587 1587 943 6219 2030 2030 1200 7200

0 0 0 1913 344 1640 764 1108 0 0 0 0 0 0 0 0

2944 9904 2324 2552 0 4654 2856 1250 1162 4993 4063 4857 0 3388 5230 4576

67493 60533 66206 65978 69070 64416 66558 67820 68725 64894 66468 60398 69444 66056 65044 59698

GUU GUC GUG GUA GCU GCC GCG GCA GGU GGC GGG GGA GAU GAC GAG GAA

5105 1265 3840 3840 3250 617 3250 3250 4359 4359 2137 1069 2396 2396 4717 4717

0 3840 1265 1265 617 3250 617 617 2137 2137 4359 5427 0 0 0 0

0 7372 1068 9036 0 8020 1068 9626 0 4278 0 11807 4717 10958 3464 10555

66369 58997 65301 57333 67607 59587 66539 57981 64978 60700 64978 53171 64361 58120 63293 56202

CUU CUC CUG CUA CCU CCC CCG CCA CGU CGC CGG CGA CAU CAC CAG CAA

943 943 5136 666 1301 1913 1481 581 4752 4752 639 4752 639 639 881 764

5136 5136 943 5413 900 943 720 1620 639 639 4752 639 0 0 764 881

4752 1359 2420 1345 4752 2120 5990 1430 0 2302 6251 2011 6397 3308 6648 1886

60643 64036 62975 64050 64521 66498 63283 67843 66083 63781 59832 64072 64438 67527 63181 67943

AUU AUC AUG AUA ACU ACC ACG ACA AGU AGC AGG AGA AAU AAC AAG AAA

1737 1737 706 1737 2115 1199 1457 916 1408 1408 420 867 1193 1193 1924 1924

1737 1737 1926 1737 541 1457 1199 1740 0 0 867 420 0 0 0 0

2632 6432 4435 6339 0 4338 4789 2791 1287 5416 6318 4248 1924 6268 6523 2976

65368 61568 64407 61661 68818 64480 64029 66027 68779 64650 63869 65939 68357 64013 63027 66574

In Silico Modelling and Analysis of Ribosome Kinetics

Table 5. Probabilities per codon for erroneous elongation UUU UUC UUG UUA UCU UCC UCG UCA UGU UGC UGG UGA UAU UAC UAG UAA

27.4e-4 91.2e-4 7.59e-4 23.5e-4 2.81e-10 56.1e-4 20.3e-4 9.09e-4 6.97e-4 30.4e-4 39.8e-4 7.50e-4 2.81e-10 15.7e-4 41.3e-4 6.04e-4

CUU CUC CUG CUA CCU CCC CCG CCA CGU CGC CGG CGA CAU CAC CAG CAA

46.7e-4 13.6e-4 4.49e-4 18.9e-4 34.1e-4 10.4e-4 37.6e-4 22.8e-4 1.21e-10 4.59e-4 88.7e-4 3.98e-4 91.1e-4 47.5e-4 69.4e-4 22.7e-4

GUU GUC GUG GUA GCU GCC GCG GCA GGU GGC GGG GGA GAU GAC GAG GAA

1.12e-10 55.0e-4 2.68e-4 22.3e-4 1.77e-10 12.5e-4 3.187e-4 28.2e-4 1.32e-10 9.40e-4 2.72e-10 100.3e-4 18.6e-4 43.2e-4 7.09e-4 21.4e-4

AUU AUC AUG AUA ACU ACC ACG ACA AGU AGC AGG AGA AAU AAC AAG AAA

14.4e-4 35.0e-4 58.3e-4 34.4e-4 2.73e-10 34.2e-4 31.7e-4 29.1e-4 8.70e-4 37.2e-4 140.7e-4 48.1e-4 15.2e-4 49.3e-4 32.1e-4 14.6e-4

Table 6. Estimated average insertion time per codon in seconds UUU UUC UUG UUA UCU UCC UCG UCA UGU UGC UGG UGA UAU UAC UAG UAA

0.3327 0.8404 0.1245 0.4436 0.0893 0.7409 0.3035 0.2313 0.1432 0.3296 0.4360 0.1098 0.0758 0.2008 0.4319 0.0963

CUU CUC CUG CUA CCU CCC CCG CCA CGU CGC CGG CGA CAU CAC CAG CAA

0.8901 0.6286 0.1028 0.9217 0.4202 0.1992 0.4257 0.5535 0.0645 0.1010 1.3993 0.0962 0.8811 0.5341 0.7425 0.4058

GUU GUC GUG GUA GCU GCC GCG GCA GGU GGC GGG GGA GAU GAC GAG GAA

0.0527 0.7670 0.1041 0.2604 0.0756 1.5622 0.1010 0.3002 0.0924 0.1673 0.2308 1.2989 0.2180 0.4144 0.1106 0.2243

AUU AUC AUG AUA ACU ACC ACG ACA AGU AGC AGG AGA AAU AAC AAG AAA

0.2733 0.4373 0.8115 0.4321 0.0943 0.4658 0.4073 0.5025 0.1636 0.3905 1.4924 0.5517 0.2242 0.4959 0.3339 0.1945

89

Qualitative and Quantitative Analysis of a Bio-PEPA Model of the Gp130/JAK/STAT Signalling Pathway Maria Luisa Guerriero Laboratory for Foundations of Computer Science, The University of Edinburgh, UK

Abstract. Computational modelling of complex biochemical systems has grown in importance over recent years as a tool for supporting biological studies. Consequently, several formal languages have been recently proposed as modelling languages for biology. Among these, process algebras have been proved capable of providing researchers with new hypotheses on the behaviour of biochemical systems. Bio-PEPA is a process algebra recently defined for the modelling and analysis of biochemical systems, which provides modellers with a wide range of analysis techniques: models can be analysed by stochastic simulation, model-checking, and mathematical methods based on ordinary diﬀerential equations. In this work, we use Bio-PEPA for modelling the gp130/JAK/STAT signalling pathway, and we use both stochastic simulation and model-checking to analyse several qualitative and quantitative aspects of the system.

1 Introduction Several modelling approaches have been used over recent years to analyse complex biological systems such as signaling pathways, ranging from traditional mathematical methods based on diﬀerential equations to computational methods based on stochastic simulation and model-checking. Each of these techniques can be more suitable than others in some context or to study some particular features of biological systems. Process algebras are formal languages traditionally used to model distributed systems of concurrent computing devices. Starting from the biochemical π-calculus [1], several other process algebras have been recently adapted in order to model biochemical systems [2,3,4,5], following the “molecules as processes” paradigm introduced in the landmark paper [6]: molecules are modelled as concurrent processes, and biochemical reactions are represented by actions performed by synchronising processes. Bio-PEPA [7,8] is a process algebra specifically defined to model and analyse biochemical networks. Compared to other process algebras, Bio-PEPA uses a more abstract view of biochemical systems, the so-called “species as processes” abstraction: processes represent molecular species instead of single molecules, and multi-way synchronisations of processes represent changes in the amounts of molecular species resulting from biochemical reactions. Such an abstract view enables modellers to deal with analysis techniques which are computationally infeasible when considering the “molecules as processes” abstraction. C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 90–115, 2009. c Springer-Verlag Berlin Heidelberg 2009

Qualitative and Quantitative Analysis of a Bio-PEPA Model

91

The main feature of Bio-PEPA is that it integrates several kinds of analysis techniques. Both discrete stochastic and continuous deterministic models can be automatically generated from Bio-PEPA models, thus allowing modellers to perform time-series analysis via stochastic simulation, Markovian analysis and ordinary diﬀerential equations (ODEs); in addition, system properties can be verified through model-checking and mathematical techniques such as bifurcation, stability and continuation analysis. Moreover, as for the other process algebras, Bio-PEPA is equipped with an operational semantics which supports various kinds of formal analysis (e.g. causality, equivalence, and reachability analysis). In this work, we define a Bio-PEPA model of the gp130/JAK/STAT signalling pathway, a well-studied system which plays a major role in several biological processes both in human and other organisms. A lot of experimental data is available about the molecules in the pathway, and some mathematical and computational models have been already developed. For these reasons, the gp130/JAK/STAT pathway represents a good case study for exploiting some of the possible Bio-PEPA analysis methods in order to study diﬀerent aspects (both qualitative and quantitative) of the system, and compare them with existing models. The rest of the paper is structured as follows. First, the Bio-PEPA language is introduced in Sec. 2, while the pathway and the Bio-PEPA model are described in Sec. 3 and Sec. 4, respectively. The following three sections are devoted to the analysis of the model: in Sec. 5 several qualitative properties are analysed via model-checking, in Sec. 6 we present some stochastic simulation results, and in Sec. 7 model-checking is employed for quantitative analysis. Finally, Sec. 8 is an overview of the related work and Sec. 9 contains some concluding remarks.

2 Bio-PEPA Bio-PEPA [7,8] is a process algebra which has been recently defined for the modelling and analysis of biochemical networks. It is a biologically-inspired language based on PEPA [9] and, diﬀerently from PEPA and other process algebras, it is able to explicitly represent details such as stoichiometric coeﬃcients and the roles of species in reactions, and it supports the definition of general kinetic laws. Bio-PEPA models can be analysed by diﬀerent techniques (stochastic simulation, analysis based on ODEs, numerical solution of the continuous-time Markov chain (CTMC), and probabilistic model-checking), since the mappings of Bio-PEPA models into specifications for those approaches have been defined [10]. The Bio-PEPA language is based on discrete levels of parameterised species: each component represents a species and its parameter may be interpreted as the number of molecules or discrete levels of concentration depending on the type of analysis to be applied. Parametric levels are considered for the definition of the transition system and for the derivation of a CTMC whose states represent the concentration levels of the species. The syntax of Bio-PEPA is defined as: S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | | .

P ::= P P | S (x) I

92

M.L. Guerriero

The component S is called a species component and abstracts a molecular species, whereas the component P, called a model component, describes the system and the interactions among components. The prefix term (α, κ) op S contains information about the role of the species in the reaction associated with the action type α: κ is the stoichiometric coeﬃcient of the species and the prefix combinator “op” represents its role in the reaction. Specifically, ↓ indicates a reactant, ↑ a product, ⊕ an activator, an inhibitor and a generic modifier. The operator “+” expresses the choice between possible acdef tions and the constant C is defined by an equation C = S . The parameter x ∈ IR+ in S (x) represents the concentration of S . Finally, the process P Q denotes the cooperI ation between components: the set I determines those activities on which the operands are forced to synchronise. Reaction rates are defined as functional rates associated with actions. Bio-PEPA supports a modelling style in terms of concentration levels: the species amounts are discretised into a number of levels, from level 0 (i.e. species not present) to a maximum level N (which depends on the maximum concentration of the species). Each level represents an interval of concentration and the granularity of the system is expressed in terms of the step size H (i.e. the length of the concentration interval). Definition 1. A Bio-PEPA system P is a 6-tuple V, N, K, FR , Comp, P , where: V is the set of compartments, N is the set of quantities describing the species (i.e. H and N), K is the set of parameter definitions, FR is the set of functional rates, Components is the set of definitions of species components, P is the model component describing the system. For discrete state space analysis the behaviour of the system is defined in terms of an operational semantics. A Stochastic Labelled Transition System (SLTS) is defined for a Bio-PEPA system. From this we can obtain a Continuous Time Markov Chain (CTMC). Both the SLTS and the CTMC derived from Bio-PEPA are defined in terms of levels of concentration, and the generated Markov chain is called CTMC with levels. For a full description of the language semantics see [10]. The Bio-PEPA language is supported by software tools such as the Bio-PEPA Workbench [11], which automatically processes Bio-PEPA models and generates other representations in forms suitable for simulation and model-checking. For instance, the generated simulation model can be executed using the Dizzy stochastic simulator [12]. The representation which is used for discrete state space generation and analysis by numerical solution of the underlying CTMC is expressed in the reactive modules language supported by the PRISM model-checker [13]. In addition, the Bio-PEPA Workbench generates reward structures and common CSL [14] formulae used in model-checking.

3 The Gp130/JAK/STAT Signalling Pathway The gp130/JAK/STAT signalling pathway is a well-studied biological system, of great clinical interest because of its key role in human fertility, neuronal repair and

Qualitative and Quantitative Analysis of a Bio-PEPA Model

93

haematological development [15,16,17]. Much experimental data is available on this pathway, and a few mathematical and computational models [18,19,20,21] have been developed. The signalling cascade in the gp130/JAK/STAT pathway is triggered by members of the family of IL (interleukin)-6-type cytokines binding to plasma membrane receptor complexes containing the common signal transducing receptor chain gp130 (glycoprotein 130). Among the targets of gp130 signal transduction, we consider the transcription factors of the STAT (signal transducers and activators of transcription) family, in particular STAT3. A key feature of the pathway is the nuclear/cytoplasmic shuttling of STATs: upon activation, STATs can translocate into the nucleus and activate the transcription of downstream gene targets. Diﬀerent cytokines signal through the formation of diﬀerent receptor complexes, all of them containing gp130 and another subunit. We focus here on two diﬀerent cytokines: LIF (leukaemia inhibitory factor) and OSM (oncostatin M). LIF signals through an heterodimeric receptor complex gp130:LIFR. OSM exhibits the uncommon ability to signal through two diﬀerent receptor complexes: the type I OSM receptor complex (gp130:LIFR), and the type II OSM receptor complex (gp130:OSMR). Figure 1 is a graphical representation of the biochemical reactions occurring in the gp130/JAK/STAT pathway. In the inset the diﬀerent types of receptor complexes are shown.

Fig. 1. Gp130/JAK/STAT pathway: graphical representation. Full arrows represent biochemical reactions, dotted arrows represent transports, dashed arrows represent syntheses.

94

M.L. Guerriero

The molecular species we consider in the model are: two ligands (LIF and OSM), three membrane-bound receptors (gp130, LIFR and OSMR), one eﬀector (STAT3), and two inhibitors (SOCS3 and PIAS3). JAK kinase and TC-PTP phosphatase are implicitly modelled. Four compartments are involved in the system: the exosol (the extracellular space, where the two ligands are located), the cell membrane (location of the receptors), the cytosol (initial location of STAT3), and the nucleus (in which STAT3 can translocate). Receptors are activated by ligand bindings, and active receptors dimerise to form receptor complexes (gp130:LIFR or gp130:OSMR) (reaction r1 in Fig. 1). Once the receptor dimeric complex is formed, each receptor subunit (gp130, LIFR and OSMR) can undergo JAK-mediated phosphorylation (r2). STAT3 can bind on receptors’ phosphorylated sites (r3), and the binding of STAT3 leads to its activation (phosphorylation) (r4). Once phosphorylated, STAT3 dissociates from the receptor complex, and its phosphorylated site allows STAT3 to homodimerise (r5). When STAT3 is in dimeric form, it can translocate into the nucleus (r6) where it can carry out its specific functions (not modelled here): STAT3 binds to the DNA, thus activating the transcription of downstream gene targets. Nuclear STAT3 dimers are inactivated through TC-PTP -mediated dephosphorylation, which leads to the dimers’ dissociation (r7) and to STAT3 export to the cytoplasm (r8), where STAT3 can undergo additional cycles of activation. The two inhibition mechanisms considered are due to SOCS3 and PIAS3. SOCS3 is synthesised by STAT3 (r9) and it acts by competing with STAT3 in binding to receptors (r10). PIAS3 acts by binding to active nuclear STAT3 (r11).

4 The Bio-PEPA Model A Bio-PEPA model of the gp130/JAK/STAT pathway has been developed. The full model can be downloaded from [22]. The model and the reaction rates are based on [21], though some diﬀerences are present due to the conceptual diﬀerences in the used modelling languages (see Sec. 8 for a discussion of such diﬀerences). All kinetic laws are assumed to be mass-action (i.e. depending on the amount of reactants and on given kinetic constants). Each possible form of the molecular species is modelled as a distinct Bio-PEPA species component. For instance, STAT3 is modelled by four distinct species components representing, respectively, the cytoplasmic dephosphorylated monomeric form (STAT3 c), the cytoplasmic phosphorylated dimeric form (STAT3-PD c), the nuclear phosphorylated dimeric form (STAT3-PD n), and the nuclear dephosphorylated monomeric form (STAT3 n); further species components are defined for each state of each complex containing STAT3. Reactions and biochemical modifications are represented by reactions over which the involved species components synchronise. For instance, the reaction representing r7 in Fig. 1 is modelled as the reaction dephospho dedimer stat59 , which decreases the amount of STAT3-PD n and increases (with stoichiometry coeﬃcient 2) the amount of STAT3 n.

Qualitative and Quantitative Analysis of a Bio-PEPA Model

95

As an example, the definitions of the species STAT3-PD n and STAT3 n are reported (here we use the simplified syntax of the Bio-PEPA Workbench, in which the trailing S in prefix terms (α, κ) op S can be omitted). S T AT 3-PD n ::= (reloc stat cn58 , 1) ↑ + (synth socs61 , 1) ⊕ + (unbind pias80 , 1) ↑ + (dephospho dedimer stat59 , 1) ↓ + (bind pias stat80 , 1) ↓ S T AT 3 n ::= (dephospho dedimer stat59 , 2) ↑ + (reloc stat nc60 , 1) ↓ For each of the involved reactions, a functional rate specifying its kinetic rate law is defined. The ones used in the species definitions for STAT3-PD n and STAT3 n are defined as follows. 0.693 reloc stat cn58 = · STAT3-PD c ; k58 //STAT3-PD c relocation cytoplasm -> nucleus

dephospho dedimer stat59 = [k59 · STAT3-PD n] ; //STAT3-PD n dephosphorylation & dedimerisation

reloc stat nc60

0.693 = · STAT3 n ; k60

//STAT3-PD n relocation nucleus -> cytoplasm

synth socs61 = [k61 · STAT3-PD n] ; //SOCS3 synthesis by STAT3-PD n

bind pias stat80

k80 = · PIAS3 · STAT3-PD n ; nucleus · NA //PIAS3/STAT3-PD n binding

unbind pias stat80 = [k−80 · PIAS3:STAT3-PD n] ; //PIAS3/STAT3-PD n unbinding

As mentioned above, the Bio-PEPA Workbench [11] allows us to automatically generate representations of the Bio-PEPA model for diﬀerent analysis tools. In the following sections we show some of the analyses performed using these generated models. In particular, we consider the PRISM [23,13] and Dizzy [12,24] models. We use the PRISM model-checker to verify that some desired properties of the system are satisfied, and the Dizzy simulation tool to perform time-series analysis via stochastic simulation.

5 Model-Checking Based Qualitative Analysis As a first step in the analysis of the model we use the PRISM model-checker [23,13] to verify a number of qualitative properties of the system. Such properties are intended to be consistency checks on the model and they allow us to check for the presence of possible human errors in the modelling process. This kind of checks is particularly useful when modelling complex systems such as the pathway we consider here since, due to the size of the models, trivial typing errors are likely to occur and may be hard to identify.

96

M.L. Guerriero

5.1 PRISM Modelling and Specification Language PRISM [23,13] is a probabilistic model-checker, which can be used to verify properties of CTMCs. Models are described using the state-based PRISM language, and it is possible to specify quantitative properties of the system using a property specification language which includes CSL (Continuous Stochastic Logic) [25,26]. The PRISM language is composed of modules and variables. A model is composed of a number of interacting modules and each module contains a number of local variables, whose values constitute the state of the module. The global state of the model is determined by the local state of all modules. The behaviour of the modules is given by a set of guarded commands, each describing a transition which is enabled when the guard is true. A command includes an update which gives new values to the variables. PRISM properties are made up of state properties φ and path properties ψ. The syntax of PRISM properties is given by the following grammar. φ ::= true | false | expr | φ ∧ φ | φ ∨ φ | ¬φ | φ ⇒ φ | Pp [ψ] | P=? [ψ] | Sp [φ] | S=? [φ] ψ ::= X φ | φ UI φ | φ U φ | FI φ | F φ | GI φ | G φ Here expr is a boolean expression (containing literal values, identifiers and the standard arithmetic and relational operators), ∈ { } is a relational parameter, p ∈ [0, 1] is a probability, and I is an interval of IR+ . The operators Pp [ψ] and P=? [ψ] are used to express transient properties (i.e. which depend on time) whereas the operators Sp [φ] and S=? [φ] are used to express steady state properties (i.e. which hold in the long run). The result of the verification of formulae Pp [ψ] (resp. Sp [φ]) is one of the boolean values true or false depending on whether ψ (resp. φ) is satisfied. The result of the verification of formulae P=? [ψ] (resp. S=? [φ]) is the expected probability with which ψ (resp. φ) is satisfied. The operators X, U, F, and G are used to express neXt, Until, Finally, and Globally properties, respectively. Time-bounded formulae are indexed by an interval I. The PRISM language supports the specification and analysis of reward-based properties. Reward structures allow us to associate real values with certain states or transitions of the model. Such values, which can be thought of as “costs” of the specified states/transitions, are taken into account during the solution of the CTMC. In this way it is possible to reason about various quantitative measures such as “expected number of instances of processes”, “expected number of occurrences of reactions”, “expected time until a condition is satisfied”, etc. The PRISM reward language supports the expression of both instantaneous and cumulative rewards. 5.2 Model-Checking the Bio-PEPA Model with PRISM In the PRISM models generated by the Bio-PEPA Workbench, one module is defined for each species, and the module local variables are used to record the current quantity of each species. The transitions correspond to the activities of the Bio-PEPA model and the updates take the stoichiometry into account. Transition rates are specified in an auxiliary module which defines the functional rates corresponding to all the reactions.

Qualitative and Quantitative Analysis of a Bio-PEPA Model

97

Moreover, lower and upper bounds must be defined for each variable (i.e. for the amount of each species). The step size H in the Bio-PEPA model allows us to consider diﬀerent PRISM models with diﬀerent granularity, leading to systems with diﬀerent numbers of levels. As an example, we provide the PRISM definitions relative to the species STAT3-PD n and STAT3 n, which are obtained from the corresponding Bio-PEPA species definitions reported in Sec. 4. First, the lower and upper levels for both species are computed from the defined step size H and the given bounds on species amounts. MIN S T AT 3-PD n = MIN S T AT 3 n = 0 MAX S T AT 3-PD n = MAX S T AT 3 n = 1500

NL S T AT 3-PD n = NL S T AT 3 n

=

MIN S T AT 3-PD n H MIN S T AT 3 n H

NU S T AT 3-PD n = NU S T AT 3 n

=

MAX S T AT 3-PD n H MAX S T AT 3 n H

The specifications of the behaviour of STAT3-PD n and STAT3 n are given by the two following modules. The third module contains the definition of the functional rates for all reactions. module S T AT 3-PD n S T AT 3-PD n : [NL S T AT 3-PD n .. NU S T AT 3-PD n] init 0; [reloc stat cn58 ] (S T AT 3-PD n + 1 ≤ NU S T AT 3-PD n) → 1 : (S T AT 3-PD n = S T AT 3-PD [synth socs61 ] (S T AT 3-PD n + 0 ≤ NU S T AT 3-PD n) → 1 : (S T AT 3-PD n = S T AT 3-PD [dephospho dedimer stat59 ] (S T AT 3-PD n ≥ 1 + NL S T AT 3-PD n) → 1 : (S T AT 3-PD n = S T AT 3-PD [bind pias stat80 ] (S T AT 3-PD n ≥ 1 + NL S T AT 3-PD n) → 1 : (S T AT 3-PD n = S T AT 3-PD [unbind pias stat80 ] (S T AT 3-PD n + 1 ≤ NU S T AT 3-PD n) → 1 : (S T AT 3-PD n = S T AT 3-PD

n + 1); n + 0); n − 1); n − 1); n + 1);

endmodule

module S T AT 3 n S T AT 3 n : [NL S T AT 3 n .. NU S T AT 3 n] init 0; [dephospho dedimer stat59 ] (S T AT 3 n + 2 ≤ NU S T AT 3 n) → 1 : (S T AT 3 n = S T AT 3 n + 2); [reloc stat nc60 ] (S T AT 3 n ≥ 1 + NL S T AT 3 n) → 1 : (S T AT 3 n = S T AT 3 n − 1); endmodule

98

M.L. Guerriero

module Rates 0.693

0.693 k58 ·STAT3-PD c·H > 0 → : true; H H k ·STAT3-PD n·H k ·STAT3-PD n· H [dephospho dedimer stat59 ] 59 > 0 → 59 : true; H 0.693 H 0.693 ·STAT3 n·H ·STAT3 n·H k k [reloc stat nc60 ] 60 H > 0 → 60 H : true; k ·STAT3-PD n·H k ·STAT3-PD n·H [synth socs61 ] 61 > 0 → 61 : true; H Hk80 k80 nucleus·NA ·PIAS3·H·STAT3-PD n·H nucleus·NA ·PIAS3·H·STAT3-PD n·H [bind pias stat80 ] > 0 → : true; H H k ·PIAS3:STAT3-PD n·H k ·PIAS3:STAT3-PD n·H [unbind pias stat80 ] −80 > 0 → −80 : true; H H [reloc stat cn58 ]

k58

·STAT3-PD c·H

endmodule

The PRISM model generated from the Bio-PEPA model of the gp130/JAK/STAT pathway has 63 species and 118 reactions. Because of the well-known state space explosion problem of model-checking, even if we consider only a few levels for each species, the state space for this model is so huge that it makes the numerical solution of the CTMC nearly unmanageable. To overcome this problem, we consider a subdivision of the pathway into two distinct sub-models in such a way that the analysis of the individual sub-models becomes more feasible. In order to find an appropriate modularisation, we adopt the approach proposed in [27,28], based on the identification of sub-systems with no retroactivity. For the considered model of the gp130/JAK/STAT pathway, two modules with low coupling can be easily identified. In the first sub-model, which refers to the bindings of ligands to receptors and the activation of the receptor dimers, we consider all the distinct combinations of ligand/receptor complexes, and we describe in detail the formation of all possible types of active receptor dimers, considering the fact that diﬀerent ligand-receptor pairs have diﬀerent binding aﬃnities. In the second sub-model, which refers to the downstream signalling pathway, we instead consider as a starting point a single “generic” type of active receptor dimer (referred to as rcpt-DP), and we focus on the reactions involving the activation of STAT3 and its cytoplasmic/nuclear shuttling. These two sub-models refer to sub-systems of the gp130/JAK/STAT pathway which act in a rather sequential way and, as a consequence, it is reasonable to assume that, for the downstream STAT3 signalling to occur, the receptor-complexes must have been already activated. The initial number of active receptor dimers in the second sub-model is defined as the sum of the steady-state quantities of all the active receptor dimers in the first sub-model. This assumption is justified by the fact that the activation of the receptors is fast compared with the following reactions, and therefore the amount of initially inactive receptors is negligible when considering the downstream pathway. As discussed in [27,28], the absence of retroactivity ensures that the modularisation has no significant eﬀect on the overall behaviour of the system. This, together with the fact that we use the output of the first sub-model as input of the second sub-model,

Qualitative and Quantitative Analysis of a Bio-PEPA Model

99

ensures that the structural qualitative properties verified for the individual sub-models in the rest of this section also hold for the full model. Particular care should be taken when verifying quantitative temporal properties over sub-models. Here we only consider semi-quantitative analysis (Sec. 7) as we are interested in relative rather than absolute values. Therefore, in this particular case, the absence of retroactivity ensures the validity, in the full model, of the analysis results obtained in the sub-models. In general, however, the actual reaction rates in the composite model (and therefore the analysis results) might be diﬀerent from the ones in the sub-models, and more advanced approaches for modularisation should be applied. In the rest of this section we use H = 200 as the step size for the ligands-receptors sub-model, and H = 300 for the downstream sub-model. See Sec. 7 for a discussion of the choice of step size values. Deadlock Detection. Deadlock states are the ones in which no transition is enabled. In some cases the presence of deadlock states is (correctly) due to the presence of irreversible reactions which lead to the transformation of all reactants into non-reactive proteins. In other cases deadlocks could be due to the scarcity of one of the reactants of a multimolecular reaction; in our model, for instance, all receptors are consumed (i.e. transformed into diﬀerent forms, such as dimers) while still ligands are available. In other cases deadlocks could be caused by modelling errors. PRISM automatically detects deadlock states when building the state space of models, and this feature can be considered the first step in the identification of potential modelling errors. For instance, in the ligands-receptors sub-model, any state in which ligands are present while all gp130 receptors have been consumed is a deadlock. This suggests that gp130 is the bottleneck of the system. Species Invariants. One simple and yet interesting property that can be verified is the presence of invariants in the amount of the involved proteins. Species invariants are commonly present in biochemical systems because of the existence of basic constraints such as the law of conservation of mass, which states that the amount (i.e. mass) of reactants consumed by a reaction must be equal to the amount of products of the reaction. For instance, given the conservation of mass and the absence of synthesis and degradation reactions, we expect that the sum of the amounts of LIFR receptor present in its various possible forms (free, as gp130:LIF:LIFR complex and as gp130:OSM:LIFR complex, with one or both of its subunits phosphorylated) is constant (and equal to the LIFR initial amount). The satisfaction of the following properties confirms the existence of the expected invariants on the total amount of ligands and receptors (as an example, we report the ones for LIF and LIFR). P≥1 [G (LIF + gp130:LIF:LIFR + gp130-P:LIF:LIFR + gp130:LIF:LIFR-P + gp130-P:LIF:LIFR-P = NU LIF)] → true

100

M.L. Guerriero

P≥1 [G (LIFR + gp130:LIF:LIFR + gp130:OSM:LIFR + gp130-P:LIF:LIFR + gp130:LIF:LIFR-P + gp130-P:LIF:LIFR-P + gp130-P:OSM:LIFR + gp130:OSM:LIFR-P + gp130-P:OSM:LIFR-P = NU LIFR)] → true

Here, and in the rest of the section, the notation Pp [ψ] → true (resp. false) means that ψ is satisfied (resp. is not satisfied), while the notation P=? [ψ] → p (with p ∈ IR) means that the result of ψ is the probability p. Reachability Analysis. Reachability properties allow us to verify whether a given state is eventually reached. States of interest can be, for instance, the ones in which some species reaches a threshold or is totally consumed, or when the amounts of two species coincide. We consider here the states in which a certain number of receptors are phosphorylated, and the ones in which a certain amount of active nuclear STAT3 (STAT3-PD n) is present. We consider first the ligands-receptors sub-model. The satisfaction of the first of the following properties guarantees that a state in which one fourth of the total amount of available receptors is phosphorylated is always reached at some time point. On the contrary, the second property, which is not satisfied, proves that we do not necessarily reach a state with one third of receptors phosphorylated. P≥1 [F (gp130-P:LIF:LIFR-P + gp130-P:OSM:LIFR-P + gp130-P:OSM:OSMR-P > (NU OS MR + NU LIFR + NU gp130) / 4)] → true P≥1 [F (gp130-P:LIF:LIFR-P + gp130-P:OSM:LIFR-P + gp130-P:OSM:OSMR-P > (NU OS MR + NU LIFR + NU gp130) / 3)] → false

The next property, instead, guarantees that in general we could reach a system where no gp130:OSMR receptor complex is activated. P≥1 [F (gp130-P:OSM:OSMR-P > 0)] → false

Regarding the downstream sub-model, we check for the following properties, which guarantee that, at some time point, at least half the initial amount of STAT3 has been transported into the nucleus and activated, but not all of it. P≥1 [F (S T AT 3-PD n > NU S T AT 3 c / 2)] → true P≥1 [F (S T AT 3-PD n > NU S T AT 3 c)] → false

Reversibility. A system is called reversible if the initial state is reachable from any other state (i.e. the system is able to self-reinitialise). More generally, a state is called reversible if it can be reached again at some later time point. The following property, if satisfied, guarantees the reversibility of the system: it states that it is always possible to return to the initial state (in the PRISM language “init” is a predefined formula which completely specifies the initial state).

Qualitative and Quantitative Analysis of a Bio-PEPA Model

101

P=? [G (“init” ⇒ P≥1 [X (!“init” ⇒ P≥1 [F (“init”)])])]

For the ligands-receptors sub-system the result of this property is 0, since we have considered bindings to be irreversible and, therefore, the system cannot return to the initial state in which all receptors and ligands are free. The downstream sub-system, instead, is reversible (the result of the property is 1), thanks to the cytoplasmic/nuclear STAT3 shuttling, which enables the system to return to the initial state in which cytoplasmic STAT3 molecules are not phosphorylated and not bound to receptor dimers. Liveness. The notion of liveness of a reaction in a given state refers to the possibility of it occurring in such a state. In particular, it is interesting to know which reactions are live in the initial state. Since PRISM properties are state-based, it is not possible to explicitly check for the occurrence of a given reaction. However, knowing how each model component is aﬀected by the occurrence of a given reaction, we can verify this kind of property by checking for the expected variations in the involved components. We are interested, for instance, in verifying that in the initial state the binding reactions between ligands and receptors can occur, leading to the three possible types of ligand/receptor dimers (gp130:LIF:LIFR, gp130:OSM:LIFR, and gp130:OSM:OSMR). The following three properties are satisfied, confirming that the three known types of complexes can be formed. P≥1 [G (“init” ⇒ P>0 [X (gp130 = NU gp130 − 1 & LIF = NU LIF − 1 & LIFR = NU LIFR − 1)])] → true P≥1 [G (“init” ⇒ P>0 [X (gp130 = NU gp130 − 1 & OS M = NU OS M − 1 & LIFR = NU LIFR − 1)])] → true P≥1 [G (“init” ⇒ P>0 [X (gp130 = NU gp130 − 1 & OS M = NU OS M − 1 & OS MR = NU OS MR − 1)])] → true

The following property, instead, is not satisfied: it states, as desired, that LIF cannot bind to receptors to form gp130:OSMR dimers. P≥1 [G (“init” ⇒ P>0 [X (gp130 = NU gp130 − 1 & LIF = NU LIF − 1 & OS MR = NU OS MR − 1)])] → false

Causality Analysis. Causality relations between given reactions can be expressed and verified by properties which relate the order of “appearance” of relevant molecules. This kind of property can be used, for instance, to verify the order in which intermediate products are formed within a cascade of events. A form of causality relation can be expressed by using the sequence and consequence relations defined in [29]: specifically, while sequence formulae describe ordering relations between events (e.g. “in order to reach a given state, we must first reach another one”), consequence formulae describe causal relations (e.g. “if a given state occurs, it is necessarily followed by a second one”).

102

M.L. Guerriero

For example, the ordering and causality relations between STAT3 phosphorylation, homodimerisation and relocation into the nucleus can be verified by the following pairs of properties (assuming at system initialisation all STAT3 is present in cytoplasmic monomeric form (STAT3-P c). When the result of the first property is 0, such a property states that it is not possible for a STAT3-PD c molecule to be present if in all previous states we had no rcpt-DP:STAT3-DP1 (a complex formed by a receptor dimer and a STAT3 molecule). Similarly, the following property (when it evaluates to 0) states that STAT3-PD c must be produced before STAT3-PD n appears. P=? [(rcpt-DP:STAT3-DP1 = 0) U S T AT 3-PD c > 0] → 0 P=? [(S T AT 3-PD c = 0) U S T AT 3-PD n > 0] → 0

The following two properties complement the previous two, stating that if at least one complex rcpt-DP:STAT3-DP1 is formed, then at least one STAT3-PD c molecule will necessarily be formed. P=? [G (rcpt-DP:STAT3-DP1 > 0 ⇒ P≥1 [F (S T AT 3-PD c > 0)])] → 1 P=? [G (S T AT 3-PD c > 0 ⇒ P≥1 [F (S T AT 3-PD n > 0)])] → 1

As another example, the following two properties verify that the transport of phosphorylated STAT3 dimers can only occur from the cytoplasm to the nucleus, but not vice versa. The result of the first property is 0 (i.e. transport of STAT3-PD can occur from cytoplasm to nucleus), while the result of the second property is 1 (i.e. transport of STAT3-PD cannot occur from nucleus to cytoplasm) for all reachable values of i, j. P=? [F (S T AT 3-PD c = i & S T AT 3-PD n = j & P≤0 [X (S T AT 3-PD c = i − 1 & S T AT 3-PD n = j + 1)])] → 0 P=? [F (S T AT 3-PD c = i & S T AT 3-PD n = j & P≤0 [X (S T AT 3-PD c = i + 1 & S T AT 3-PD n = j − 1)])] → 1

Conversely, the transport of dephosphorylated STAT3 monomers can only occur from the nucleus to the cytoplasm. P=? [F (S T AT 3 c = i & S T AT 3 n = j & P≤0 [X (S T AT 3 c = i − 1 & S T AT 3 n = j + 1)])] → 1 P=? [F (S T AT 3 c = i & S T AT 3 n = j & P≤0 [X (S T AT 3 c = i + 1 & S T AT 3 n = j − 1)])] → 0

6 Simulation Based Time-Series Analysis In the previous section we have used model-checking in order to check for a number of simple formulae which guarantee us that some key properties of the gp130/JAK/STAT

Qualitative and Quantitative Analysis of a Bio-PEPA Model

(a) Full model

(b) no SOCS3 3000

STAT3_c STAT3_n

2500

STAT3-PD_c STAT3-PD_n

2000 1500 1000 500

Number of molecules

Number of molecules

3000

0

STAT3_c STAT3_n

2500

STAT3-PD_c STAT3-PD_n

2000 1500 1000 500 0

0

100 200 300 400 500 600 700 800

0

100 200 300 400 500 600 700 800

Time (minutes)

Time (minutes)

(c) no PIAS3

(d) no TC-PTP 3000

STAT3_c STAT3_n

2500

STAT3-PD_c STAT3-PD_n

2000 1500 1000 500 0

Number of molecules

3000 Number of molecules

103

STAT3_c STAT3_n

2500

STAT3-PD_c STAT3-PD_n

2000 1500 1000 500 0

0

100 200 300 400 500 600 700 800 Time (minutes)

0

100 200 300 400 500 600 700 800 Time (minutes)

Fig. 2. Simulation results: full model vs. no inhibitors

model are satisfied. This analysis allows us to be more confident about the absence of modelling errors. Now we progress our analysis of the model by means of stochastic simulation. We report here some results obtained by simulating the full model (comprising both the ligands-receptors and the downstream sub-systems) using the Gibson-Bruck [30] stochastic simulation engine implemented in Dizzy [12,24]. Figure 2 shows the time-series evolution produced by the model (Fig. 2(a)) versus the ones in which each of the three inhibitors has been removed (Fig. 2(b)–(d)). Each plot refers to average values computed over 1000 simulation runs, and the amounts of the four diﬀerent forms of STAT3 are shown (cytoplasmic and nuclear dephosphorylated monomers, and cytoplasmic and nuclear phosphorylated dimers). In all the performed simulations, at system initialisation STAT3 is only present in cytoplasmic monomeric form. As shown in Fig. 2(a), as time passes, STAT3 is phosphorylated, dimerised, and transported into the nucleus, until the systems reaches a state in which the inhibition of nuclear STAT3 by dephosphorylation and the nuclear/ cytoplasmic shuttling lead nuclear and cytoplasmic STAT3 to be in equilibrium. When the amount of nuclear STAT3 increases significantly, the inhibitory role of SOCS3 (which is under transcription control of STAT3) comes into play (Fig. 2(b)). SOCS3 is responsible for signal attenuation and, hence, after reaching a peak, nuclear STAT3 decreases.

104

M.L. Guerriero

PIAS3 slows down the production of active nuclear STAT3 by binding to it (Fig. 2(c)). Therefore, if PIAS3 is present, part of nuclear STAT3 is bound to it, while, if PIAS3 is knocked down, the amount of available STAT3 increases. A third inhibitor, TC-PTP, allows nuclear STAT3 to translocate back into the cytoplasm, by dephosphorylating it (Fig. 2(d)). If TC-PTP is present, STAT3 nuclear/cytoplasmic shuttling occurs; instead, if TC-PTP is knocked out (i.e. if nuclear STAT3 is not dephosphorylated), STAT3 accumulates in the nucleus, whilst cytoplasmic STAT3 molecules quickly disappear.

7 Semi-quantitative Analysis of the CTMC with Levels In Sec. 5 we have shown how model-checking can be used in order to discover modelling errors by checking for some basic properties which guarantee the model to behave as expected. In this section, instead, we use model-checking also for quantitative analysis, with the purpose of completing the simulation-based analysis in order to provide additional insight on the behaviour of the gp130/JAK/STAT pathway. The main advantage of model-checking with respect to stochastic simulation is the fact that model-checking is exhaustive: it explores all the possible behaviours of the model and it does not require us to compute an average behaviour of a number of stochastic simulation runs. As mentioned before, the main disadvantage of model-checking is the state space explosion problem, which implies that we cannot deal with too many levels for the model components without inducing an intractable model. In has been shown (see [10]) that, as the number of levels increases, the behaviour of the CTMC with levels tends to the behaviour of ODEs (when the number of molecules is large enough to average out the randomness of the system); this result guarantees the theoretical correctness of the approach. However, if the number of levels is too small, the error introduced by the discretisation becomes significant and the numerical solution of the generated CTMC fails to reproduce the correct behaviour. The number of levels for model components is related to the step size H and to the upper NU and lower NL bounds for each species. The step size H represents the granularity of the system, and it directly aﬀects the accuracy of the results; the upper and lower bounds are also relevant to the accuracy, since imposing bounds on the numbers of molecules causes a state space truncation which might potentially have impact on the behaviour of the system. Therefore, when performing CTMC analysis of Bio-PEPA models, the choice of the step size and of the upper and lower bounds is essential: they must be carefully selected so that the number of levels to be used for the model components is a suitable trade-oﬀ between accuracy and eﬃciency. In the following sections we report some of the results obtained by using the PRISM model-checker to perform quantitative analysis. First we consider reward-based properties which allow us to observe the time-series for some of the species of the system (for comparison with the stochastic simulation), and we discuss the error introduced by discretising and bounding the model; afterwards, we define further properties in order to compute additional (semi-)quantitative measures.

Qualitative and Quantitative Analysis of a Bio-PEPA Model

105

Time-series Analysis Using State Rewards. A reward structure is automatically defined by the Bio-PEPA Workbench for each PRISM component, and it can be referred to either by the component name or by an integer value (implicitly assigned to reward structures based on the order in which they are defined). These reward structures associate an instantaneous reward equal to the current amount of the corresponding molecular species with each state. The evaluation of these reward-based properties corresponds to computing an average behaviour for the species at given time points. As an example, the following reward is used to observe the time evolution of the receptor dimer gp130:LIF:LIFR. rewards “gp130:LIF:LIFR” true : gp130:LIF:LIFR · H; endrewards

Figure 3 reports the results obtained by verifying on the ligands-receptors sub-system the reward-based property Ri=? [I = T ]

for time points T ≤ 30 minutes, where i is an integer variable used to index the reward structure of interest. Figure 4, instead, reports the results obtained by verifying the same reward-based property for time points T ≤ 800 minutes on the downstream sub-system. In this figure, we also report the standard deviation of the number of molecules, which is computed by exploiting reward structures associating the square of the number of molecules of each species with each state: the standard deviation is calculated as the square root of the variance E(Y)2 − E(Y 2 ), where Y is the random variable representing a species in

500 450

Number of molecules

400

gp130:LIF:LIFR gp130:OSM:LIFR gp130:OSM:OSMR gp130-P:LIF:LIFR gp130:LIF:LIFR-P gp130-P:LIF:LIFR-P gp130-P:OSM:LIFR gp130:OSM:LIFR-P gp130-P:OSM:LIFR-P gp130-P:OSM:OSMR gp130:OSM:OSMR-P gp130-P:OSM:OSMR-P

350 300 250 200 150 100 50 0 0

5

10 15 20 Time (minutes)

25

30

Fig. 3. Time-series by model-checking: ligands-receptors sub-model

106

M.L. Guerriero

3000 STAT3_c STAT3_n STAT3-PD_c STAT3-PD_n

Number of molecules

2500 2000 1500 1000 500 0 0

100

200

300 400 500 Time (minutes)

600

700

800

Fig. 4. Time-series by model-checking: downstream sub-model. Thick lines represent the expected numbers of molecules; thin lines represent their standard deviation.

the network, whereas E(Y) and E(Y 2 ) indicate the expected values for the amount of the species Y and for its square value. Figures 3 and 4 have been obtained by analysing the sub-models with step sizes H = 200 and H = 300 respectively. In the next section we discuss the considerations which lead us to the choice of such values. Three kinds of approximation errors could have been introduced by our analysis of the CTMC with levels due to, respectively, the discretisation of the amounts (H), their bounding (NL and NU ), and the subdivision into modules. In the next section we discuss the eﬀect of varying the step size H on the behaviour of the system. Instead, we do not report results concerning the variation of the bounds NL and NU since, in this particular system, increasing the bounds does not have a significant eﬀect: the reason for this is that no synthesis and degradation reactions are defined (with the single exception of SOCS3) and, as a consequence, the amount of most molecular species is clearly bounded by the amounts of the molecules present at system initialisation. The choice of how to modularise the system has been carried out in order to minimise the interaction between the two modules. However, the modularisation has certainly an impact on the quantitative behaviour. In the whole system, for instance, STAT3 and SOCS3 molecules can bind to receptor dimers as soon as they start being phosphorylated; in the downstream sub-model, instead, we had to fix an initial amount of phosphorylated receptor dimers. Despite these possible sources of approximation, comparing Fig. 4 and Fig. 2, we notice that the results obtained by analysing the downstream sub-model using PRISM instantaneous rewards do not diﬀer significantly from the behaviour observed by averaging the results obtained by 1000 stochastic simulation runs of the whole model. Both the time-scale and the relative amounts of molecules are the same in both figures,

Qualitative and Quantitative Analysis of a Bio-PEPA Model

107

and the only significant diﬀerence regarding the absolute amounts is the amount of cytoplasmic monomeric STAT3, which is higher in Fig. 4. We can also observe that the standard deviation reported in Fig. 4 is quite high, due to the stochastic noise which has been introduced by using a small number of levels. Experimenting with Step Sizes. As previously stated, the choice of the step size has a great impact on both accuracy and performance of the analysis: the smaller the step size is, the larger the CTMC state space and, hence, the smaller the discretisation error introduced, but also the longer the time needed for solving the CTMC. Before choosing the values to be used for the step size H in the analysis of the models, we have performed a number of experiments varying H in order to find values representing a good trade-oﬀ between accuracy and performance of the analysis. In Fig. 5 and Fig. 6 we report some results which show how changing the step size aﬀects the behaviour of the system (in ligands-receptors and downstream sub-systems, respectively). In Fig. 5, we compare the results obtained by using six diﬀerent values for H (1000, 500, 300, 250, 200, 150) in the analysis of the ligands-receptors sub-model, and we can observe that H in this case does not have a big impact on the results.

(a) H=1000

(b) H=500

300 200 100

500 Number of molecules

400

0

400 300 200 100 0

0

5 10 15 20 25 30 Time (minutes) (d) H=250

100 0 5 10 15 20 25 30 Time (minutes)

5 10 15 20 25 30 Time (minutes) (f) H=150

500

400 300 200 100 0

0

100 0

Number of molecules

200

200

(e) H=200 Number of molecules

300

300

5 10 15 20 25 30 Time (minutes)

500

400

400

0 0

500 Number of molecules

(c) H=300

500 Number of molecules

Number of molecules

500

400 300 200 100 0

0

5 10 15 20 25 30 Time (minutes)

0

5 10 15 20 25 30 Time (minutes)

Fig. 5. Time-series by model-checking: ligands-receptors sub-model. The three types of receptor complexes are shown, gp130:LIF:LIFR (red), gp130:OSM:LIFR (green), and gp130:OSM:OSMR (blue), in the stage when one (full line) or both (dashed line) receptors are phosphorylated.

108

M.L. Guerriero

(a) STAT_c

(b) STAT3-PD_c 120

Simulation MC, H=1000 MC, H=500

2500

MC, H=400 MC, H=300 MC, H=270

Number of molecules

Number of molecules

3000

2000 1500 1000 500 0

Simulation MC, H=1000 MC, H=500

100 80 60 40 20 0

0

100

200

300

400

500

600

700

800

0

100

200

Time (minutes)

300

400

500

600

700

800

700

800

Time (minutes)

(c) STAT3_n

(d) STAT3-PD_n

500

700 Simulation MC, H=1000 MC, H=500 MC, H=400 MC, H=300 MC, H=270

450 400 350 300

Simulation MC, H=1000 MC, H=500 MC, H=400 MC, H=300 MC, H=270

600 Number of molecules

Number of molecules

MC, H=400 MC, H=300 MC, H=270

250 200 150 100

500 400 300 200 100

50 0

0 0

100

200

300

400

500

Time (minutes)

600

700

800

0

100

200

300

400

500

600

Time (minutes)

Fig. 6. Time-series by model-checking: STAT3 sub-model

The first notable diﬀerence is that in Fig. 5(a) the amounts of gp130:LIF:LIFR and gp130:OSM:LIFR are equal: as expected, with H = 1000 (i.e. one single level for each ligand and receptor) we are not able to observe the fact that LIFR has an higher binding aﬃnity with LIF than with OS M. The other interesting thing is that, contrary to what we expected, there is no noticeable increase of accuracy when decreasing H. Instead, after observing the similarities between Fig. 5(b), (d) and (e), and between Fig. 5(c) and (f), respectively, we drew the conclusion that the first group is the “correct” one; the reason is the rounding error introduced when computing the number of levels starting from the initial amounts (remember that NL = MIN/H and NU = MAX/H): when a small numbers of levels is used, this rounding error happens to be more significant than H itself. In Fig. 6, we compare the results obtained by using five diﬀerent values for H (1000, 500, 400, 300, 270) in the analysis of the downstream sub-model; the value obtained by stochastic simulation is also shown. As for the ligands-receptors sub-model, also for this sub-model we notice that when using H = 1000 we obtain a totally wrong behaviour, and we observe a general increase in accuracy when increasing the number of levels. For the smallest values of H, the relative values and the trends for the considered species are correctly reflected compared to the stochastic simulation results: for instance, both the peaks’ amplitude and the time at which they occur are reproduced quite accurately.

Qualitative and Quantitative Analysis of a Bio-PEPA Model

109

Though not exact with respect to the stochastic simulation, these results are satisfactory enough for the kinds of semi-quantitative properties we are interested in analysing in the next section. Exact quantitative analysis via CTMC, instead, is infeasible for systems such as the model we consider here. Indeed, the time needed for obtaining the results shown in Fig. 6 ranges from a couple of seconds to hours, and for H = 250 the size of the CTMC already becomes prohibitively large to analyse. Semi-quantitative Properties. Using again the “trade-oﬀ” step sizes H = 200 and H = 300, we consider here a few more semi-quantitative properties of the two sub-models. For instance, we are interested in analysing the impact that the diﬀerent aﬃnities of ligand/receptor pairs have on the consumption of the diﬀerent ligands and receptors and on the relative amount of type I and type II receptors formed. Though this kind of analysis is clearly quantitative (since it involves calculating probabilities and, hence, numbers of molecules), we consider such properties semiquantitative because we are not interested in computing absolute values, but rather in knowing relative values with respect to each other. The following properties measure the probability with which the amount of each molecular species never changes from the initial amount. P=? [G (LIF = NU LIF)] → 7.53 · 10−2 P=? [G (OS M = NU OS M)] → 1.45 · 10−6 P=? [G (gp130 = NU gp130)] → 0 P=? [G (LIFR = NU LIFR)] → 1.24 · 10−4 P=? [G (OS MR = NU OS MR)] → 4.56 · 10−4

From the obtained results we notice, for instance, that gp130 is always used (indeed, it is necessary to form all receptor dimers), and that it is more likely for OSM to be consumed than LIF (indeed, LIF is only used in the formation of one type of receptor dimers). We measure also the probability with which the amount of each molecular species reaches its lower bound. This group of properties shows that gp130 is totally consumed in any possible evolution of the system, that LIF and OSM are never totally consumed, and that the probability of LIFR being totally consumed is equal to the probability of OSMR not being used at all. These results mean that gp130 is the bottleneck of the system, while LIF and OSM are present in abundance. P=? [F (LIF = NL LIF)] → 0 P=? [F (OS M = NL OS M)] → 0 P=? [F (gp130 = NL gp130)] → 1

110

M.L. Guerriero

P=? [F (LIFR = NL LIFR)] → 4.56 · 10−4 P=? [F (OS MR = NL OS MR)] → 1.24 · 10−4

Finally, we consider the reward-based property Ri=? [C ≤ T ]

and we verify it on the downstream sub-model for time points T ≤ 800 minutes, where i is an integer variable referring to a transition reward structure. In addition to state rewards, in fact, PRISM allows for the definition of reward structures which associate with each transition a cumulative reward equal to its expected number of occurrences up to the considered time. In Fig. 7 and Fig. 8 the expected number of occurrences for some of the reactions of the downstream sub-model is shown. In particular, in Fig. 7 we compare the number of occurrences of receptors/STAT3 and receptors/SOCS3 binding reactions, which shows intuitively the diﬀerent binding aﬃnities of STAT3 and SOCS3 to the receptor dimers. In Fig. 8, instead, we consider the number of occurrences of transport reactions of STAT3 molecules from cytoplasm to nucleus and back. In Fig. 8(a) we compare the number of occurrences of transport in the two directions: we count each transport from cytoplasm to nucleus twice since STAT3 molecules are translocated in the nucleus in dimeric form and, hence, a pair of STAT3 molecules is moved at each reaction occurrence. We observe that, since at system initialisation no STAT3 molecule is present in the nucleus, the diﬀerence between the two curves in Fig. 8(a) (multiplied by the step

50

receptor / STAT3_c binding receptor / SOCS3 binding

45

Number of occurrences

40 35 30 25 20 15 10 5 0 0

100

200

300 400 500 Time (minutes)

600

700

800

Fig. 7. Expected number of occurrences of receptor binding reactions in the downstream submodel. The full red line plots the number of occurrences of reactions bind rcpt DP stat27 and bind rcpt DP stat28 , while the dashed green line plots the number of occurrences of reactions bind rcpt DP socs62 and bind rcpt DP socs63 .

Qualitative and Quantitative Analysis of a Bio-PEPA Model 35

cytoplasm -> nucleus nucleus -> cytoplasm

Number of occurrences

(cytoplasm->nucleus - nucleus->cytoplasm) * H STAT3_n + STAT3-PD_n*2 + PIAS:STAT3-PD_n*2

2500

30

111

2000

25 20

1500

15

1000 10

500

5

0

0 0

100

200

300 400 500 Time (minutes)

(a)

600

700

800

0

100

200

300

400

500

600

700

800

Time (minutes)

(b)

Fig. 8. Expected number of occurrences of transport reactions in the downstream sub-model. In (a) the full red line plots twice the number of occurrences of reaction reloc stat cn58 , while the dashed green line plots the number of occurrences of reaction reloc stat nc60 . In (b) the full red line is the diﬀerence between the lines in (a) multiplied by H, while the dashed green line is the total current amount of STAT3 molecules in the nucleus (S T AT 3 n + S T AT 3-PD n · 2 + PIAS:STAT3-PD n · 2).

size H) must be the number of STAT3 molecules present in the nucleus. This consideration is confirmed by the perfect agreement of the two curves in Fig. 8(b).

8 Related Work Given its significant impact on various cellular processes, the gp130/JAK/STAT pathway has been subject of numerous studies, both experimental and computational. Consequently, a few variants of the pathway model have been developed in order to analyse diﬀerent aspects of it. In [18] the focus is on the shuttling of STATs from nucleus to cytoplasm and back. A more complete model is developed in [19], which also reports the results of a global sensitivity analysis of parameter interaction. The role of inhibitory mechanisms is instead studied in [20]. These three works are based on mathematical modelling and the analysis is performed by ODEs solvers. In [21], a process algebra based computational model of the gp130/JAK/STAT pathway is presented and analysed using the BetaWB tool [31], a stochastic simulator for the BlenX language [32]. The Bio-PEPA model we present here is strongly based on the BlenX model described in [21], and the simulation results of the two models match well. This agreement is particularly interesting in view of the conceptual diﬀerences existing in the two process algebras. One of these diﬀerences concerns the treatment of complexes, which in BlenX are considered as molecular species consisting of the individual molecules composing them, while in Bio-PEPA they are considered as diﬀerent species not explicitly related to the sub-components. Secondly, immediate reactions can be defined in BlenX, while they are not admitted in Bio-PEPA because of Bio-PEPA’s underlying CTMC semantics. Finally, stoichiometric information can be specified in

112

M.L. Guerriero

Bio-PEPA, while they cannot be explicitly coded in BlenX (requiring reactions involving stoichiometry greater than one to be decomposed into multiple steps). In addition to these theoretical diﬀerences between the languages, we mention that the focus in the two works is quite diﬀerent. In [21] the eﬀects of a number of experiments involving quantitative parameters are analysed and compared with experimental data. The aim of the present work, instead, is to exploit model-checking, in addition to stochastic simulation, to analyse both qualitative and quantitative properties of the model behaviour. A few works have recently been published regarding the application of modelchecking techniques to the analysis of biochemical systems. In [33] the authors demonstrate how the PRISM model-checker can be adopted to model and analyse biochemical pathways, using the FGF pathway as a case study. The approach proposed in this work diﬀers from ours in the level of abstraction considered. Instead of taking a variable number of levels into account, the authors of [33] consider an abstraction in which one single copy of each involved molecular species is present and such that module variables represent changes in state of the molecules. This approach has the evident advantage of reducing the CTMC state space, though it might not be quantitatively correct in general: it can be seen as a level of abstraction equivalent to ours when one single level is used for each species. In the same work, the authors also consider a number of state space reduction techniques, some of which (based on lumpability and symmetry reduction) are exact, meaning that the behaviour of the reduced CTMC is preserved. The notion of CTMC with levels of concentrations has been introduced in [34], in which the ERK signalling pathway was used as a case study, and in [35] the PRISM model-checker is used to analyse it. Following these works, the notion of discrete levels of concentrations has been adopted also in IDD-CSL [36], an Interval Decision Diagram based model-checker for stochastic Petri nets, which allows for the verification of CSL properties. In [37] the authors propose a framework, based on Petri nets, in which qualitative and quantitative (stochastic and continuous) analysis of biochemical pathways are integrated. Qualitative properties such as boundedness, liveness and reversibility are considered, in addition to the possibility to check for P- and T-invariants, and behavioural properties are verified by probabilistic model-checking. Finally, BIOCHAM [38,39] is a framework for modelling, simulating and analysing biochemical systems, in which diﬀerent semantics (diﬀerential, stochastic, discrete, and boolean) are considered. BIOCHAM allows for the verification of temporal properties expressed in the Computation Tree Logic (CTL) by using the NuSMV modelchecker [40].

9 Conclusions and Future Work In this work we have used the gp130/JAK/STAT signalling pathway as a case study for modelling and analysis using the Bio-PEPA process algebra. Among the possible analysis methods made available by the Bio-PEPA Workbench, we have considered stochastic simulation and model-checking. The results obtained by simulation agree well with existing mathematical and computational models. The application of the model-checking approach to the analysis of

Qualitative and Quantitative Analysis of a Bio-PEPA Model

113

the pathway model, though limited by the state space explosion problem, provided us with some useful insight. First, it can be used for consistency checking, in order to guarantee the satisfaction of essential properties and, therefore, the absence of modelling errors. Second, it allows us to check for the satisfaction of semi-quantitative behavioural properties over the whole model, without the need for computing average values over a number of stochastic simulation runs. In order to deal with the computational complexity of model-checking, we have subdivided the pathway model into two distinct sub-models. The time-series analysis obtained by analysing the sub-models individually via model-checking shows a reasonably good agreement with the behaviour obtained via stochastic simulation. The issue of modularisation of models of biochemical systems is a complex one. In this work we have adopted a simple approach which is adequate for this particular case study. A general approach for modularisation of models deserves additional study, in particular in view of the possible performance improvement which this technique could bring in model-checking. Finally, in order to fully exploit the framework provided by Bio-PEPA further analysis could be performed on the MATLAB model generated by the Bio-PEPA Workbench using ODEs based methods to perform, for instance, bifurcation, stability, and continuation analysis. Acknowledgments. The author wishes to thank Jane Hillston for her helpful comments. This research is supported by the EPSRC grant EP/E031439/1 “Stochastic Process Algebra for Biochemical Signalling Pathway Analysis”.

References 1. Regev, A., Silverman, W., Shapiro, E.: Representation and simulation of biochemical processes using the π-calculus process algebra. In: Proceedings of Pacific Symposium on Biocomputing (PSB 2001), vol. 6, pp. 459–470 (2001) 2. Regev, A., Panina, E.M., Silverman, W., Cardelli, L., Shapiro, E.Y.: BioAmbients: an Abstraction for Biological Compartments. Theoretical Computer Science 325(1), 141–167 (2004) 3. Cardelli, L.: Brane Calculi - Interactions of Biological Membranes. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 257–278. Springer, Heidelberg (2005) 4. Priami, C., Quaglia, P.: Operational patterns in Beta-binders. In: Priami, C. (ed.) Transactions on Computational Systems Biology I. LNCS (LNBI), vol. 3380, pp. 50–65. Springer, Heidelberg (2005) 5. Danos, V., Laneve, C.: Formal molecular biology. TCS 325(1) (2004) 6. Regev, A., Shapiro, E.: Cells as Computation. Nature 419(6905), 343 (2002) 7. Ciocchetta, F., Hillston, J.: Bio-PEPA: An extension of the process algebra PEPA for biochemical networks. In: Proc. of FBTC 2007. ENTCS, vol. 194, pp. 103–117 (2008) 8. Ciocchetta, F., Hillston, J.: Bio-PEPA: A Framework for the Modelling and Analysis of Biological Systems. Theoretical Computer Science 410(33-34), 3065–3084 (2009) 9. Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press, Cambridge (1996) 10. Ciocchetta, F., Hillston, J.: Calculi for Biological Systems. In: Formal Methods for Computational Systems Biology (SFM 2008). LNCS, vol. 5016, pp. 265–312. Springer, Heidelberg (2008)

114

M.L. Guerriero

11. Bio-PEPA Workbench Home Page: http://www.dcs.ed.ac.uk/home/stg/software/biopepa/ 12. Ramsey, S., Orrell, D., Bolouri, H.: Dizzy: stochastic simulation of large-scale genetic regulatory networks. J. Bioinf. Comp. Biol. 3(2), 415–436 (2005) 13. PRISM Home Page: http://www.prismmodelchecker.org 14. Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Model-checking continuous-time Markov chains. ACM Trans. Comput. Logic 1(1), 162–170 (2000) 15. Underhill-Day, N., Heath, J.: Oncostatin M (OSM) Cytostasis of Breast Tumor Cells: Characterization of an OSM Receptor β-Specific Kernel. Cancer Research 66(22), 10891–10901 (2006) 16. Heinrich, P., Behrmann, I., Haan, S., Hermanns, H., M¨uller-Newen, G., Schaper, F.: Principles od interleukin (IL)-6-type cytokine signalling and its regulation. Biochem. J. 374, 1–20 (2003) 17. Kisseleva, T., Bhattacharya, S., Braunstein, J., Schindler, C.: Signaling through the JAK/STAT pathway, recent advances and future challenges. Gene 285, 1–24 (2002) 18. Swameye, I., M¨uller, T., Timmer, J., Sandra, O., Klingm¨uller, U.: Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling. PNAS 100, 1028–1033 (2003) 19. Mahdavi, A., Davey, R.E., Bhola, P., Yin, T., Zandstra, P.W.: Sensitivity Analysis of Intracellular Signaling Pathway Kinetics Predicts Targets for Stem Cell Fate Control. PLoS Computational Biology 3(7), 1257–1267 (2007) 20. Singh, A., Jayaraman, A., Hahn, J.: Modeling Regulatory Mechanisms in IL-6 Transduction in Hepatocytes. Biotechnology and Bioengineering 95(5), 850–862 (2006) 21. Guerriero, M.L., Dudka, A., Underhill-Day, N., Heath, J.K., Priami, C.: Narrative-based computational modelling of the Gp130/JAK/STAT signalling pathway. BMC Systems Biology 3(1), 40 (2009) 22. Bio-PEPA Home Page: http://www.biopepa.org/ 23. Hinton, A., Kwiatkowska, M., Norman, G., Parker, D.: PRISM: A tool for automatic verification of probabilistic systems. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 441–444. Springer, Heidelberg (2006) 24. Dizzy Home Page: http://magnet.systemsbiology.net/software/Dizzy 25. Aziz, A., Kanwal, K., Singhal, V., Brayton, V.: Verifying continuous time Markov chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996) 26. Baier, C., Katoen, J.P., Hermanns, H.: Approximate Symbolic Model Checking of Continuous-Time Markov Chains. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 146–161. Springer, Heidelberg (1999) 27. Saez-Rodriguez, J., Kremling, A., Gilles, E.: Dissecting the puzzle of life: modularization of signal transduction networks. Computers and Chemical Engineering 29, 619–629 (2005) 28. Conzelmann, H., Saez-Rodriguez, J., Sauter, T., Bullinger, E., Allg¨ower, F., Gilles, E.: Reduction of mathematical models of signal transduction networks: simulation-based approach applied to EGF receptor signalling. Systems Biology 1(1), 159–169 (2004) 29. Monteiro, P., Ropers, D., Mateescu, R., Freitas, A., de Jong, H.: Temporal logic patterns for querying dynamic models of cellular interaction networks. ECCB 24, 227–233 (2008) 30. Gibson, M., Bruck, J.: Eﬃcient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels. The Journal of Chemical Physics 104, 1876–1889 (2000) 31. Dematt´e, L., Priami, C., Romanel, A.: The Beta Workbench: a computational tool to study the dynamics of biological systems. Briefings in Bioinformatics 9(5), 437–449 (2008), http://www.cosbi.eu/Rpty_Soft_BetaWB.php

Qualitative and Quantitative Analysis of a Bio-PEPA Model

115

32. Dematt´e, L., Priami, C., Romanel, A.: The BlenX Language: A Tutorial. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 313–365. Springer, Heidelberg (2008) 33. Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic Model Checking of Complex Biological Pathways. Theoretical Computer Science 319, 239–257 (2008) 34. Calder, M., Gilmore, S., Hillston, J.: Modelling the Influence of RKIP on the ERK Signalling Pathway Using the Stochastic Process Algebra PEPA. In: Priami, C., Ing´olfsd´ottir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 1–23. Springer, Heidelberg (2006) 35. Calder, M., Vyshemirsky, V., Gilbert, D., Orton, R.: Analysis of signalling pathways using continuous time Markov chains. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 44–67. Springer, Heidelberg (2006) 36. The Idd-CSL Home Page: http://www-dssz.informatik.tu-cottbus.de/software/software.html 37. Heiner, M., Gilbert, D., Donaldson, R.: Petri Nets for Systems and Synthetic Biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 215–264. Springer, Heidelberg (2008) 38. The BIOCHAM Home Page: http://contraintes.inria.fr/BIOCHAM/ 39. Fages, F., Soliman, S., Chabrier-Rivier, N.: Modelling and querying interaction networks in the biochemical abstract machine BIOCHAM. Journal of Biological Physics and Chemistry 4(2), 64–73 (2004) 40. NuSMV Home Page: http://nusmv.irst.itc.it/

Rule-Based Modelling and Model Perturbation Vincent Danos1 , J´erˆome Feret2 , Walter Fontana3 , Russ Harmer4 , and Jean Krivine3,5 1

University of Edinburgh 2 INRIA–ENS–CNRS 3 Harvard Medical School 4 CNRS–Universit´e Paris Diderot 5 Institut des Hautes Etudes Scientiﬁques

Abstract. Rule-based modelling has already proved to be successful for taming the combinatorial complexity, typical of cellular signalling networks, caused by the combination of physical protein-protein interactions and modiﬁcations that generate astronomical numbers of distinct molecular species. However, traditional rule-based approaches, based on an unstructured space of agents and rules, remain susceptible to other combinatorial explosions caused by mutated and/or splice variant agents, that share most but not all of their rules with their wild-type counterparts; and by drugs, which must be clearly distinguished from physiological ligands. In this paper, we deﬁne a syntactic extension of Kappa, an established rule-based modelling platform, that enables the expression of a structured space of agents and rules that allows us to express mutated agents, splice variants, families of related proteins and ligand/drug interventions uniformly. This also enables a mode of model construction where, starting from the current consensus model, we attempt to reproduce in numero the mutational—and more generally the ligand/drug perturbational—analyses that were used in the process of inferring those pathways in the ﬁrst place.

1

Introduction

In recent years, there has been extensive development in the use of modelling to understand cellular signalling networks (see [1, 2, 3, 4] among many others). To date, this line of work has focussed almost exclusively on describing wild-type behaviours, i.e. it deals with the interactions between proteins that take place in a normal healthy cell. This is already highly non-trivial since these signalling networks employ a strategy of binding, modiﬁcation and unbinding between proteins that generates astronomical numbers of non-isomorphic molecular species. This poses an essentially unsolvable scalability problem for any modelling approach, such as ODE-based chemical kinetics or Petri nets, based on exhaustively enumerating reactions between fully-speciﬁed molecular species. In recent years, a new modelling approach has been used to tame this combinatorial explosion, namely agent- or rule-based modelling [5]. In this setting, C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 116–137, 2009. c Springer-Verlag Berlin Heidelberg 2009

Rule-Based Modelling and Model Perturbation

117

molecular species are left implicit; instead, agents are used to represent not complexes but their constituent proteins. Each type of agent has a name and a set of sites. Instead of reactions, we write rules that mention names of agent types and some, but not necessarily all, of their respective sites. In this way, and unlike reactions, a rule need only make explicit those aspects of the agents upon which it acts that are actually relevant to the interaction being described by the rule. So reaction-based models leave agents implicit, considering them at best as an aggregation of molecular species, whereas rule-based models make agents explicit but the reactions implicit, instead considering their rules to be aggregations of reactions. It should be noted that such wild-type models, be they reaction- or rule-based, can already handle situations where, typically as a result of gene ampliﬁcation or ablation, a protein is either over- or under-expressed. Under such circumstances, the response of a cell to external conditions may be exaggerated or attenuated as a consequence of the induced perturbation of mass-action kinetics and of the nature and numbers of complexes that exist in the cell’s resting state (cf. [6]). This does not bring about new protein-protein interactions, it only aﬀects the relative importance of the wild-type interactions, e.g. if protein X has a binding partner Y that is over-expressed, X will be attracted to the greater than usual mass of Y s to the detriment of its binding with other partners. However, many disease states are the result of genetic mutations that build incorrect proteins, with aberrant behaviour, rather than the straightforward modulation of protein expression levels (although in some cases the two defects co-exist and synergize). Such mutant proteins may only diﬀer by one or two amino acids from their wild-type cousins and yet have radically diﬀerent behaviour, e.g. erbB1 with the single substitution L858R, which exists in many kinds of solid tumour, has a constitutively active kinase domain, as does B-Raf with the single V600E mutation. The ﬂip side of this is that much of the wildtype behaviour of a protein is actually shared with such mutants, for instance a binding domain far from any site of mutation will quite likely retain its usual functionality. This poses a further serious challenge to modelling since mutated proteins therefore duplicate large chunks of an already highly combinatorial wildtype network, while also potentially adding interactions. To tackle these issues, we introduce a syntactic extension of Kappa that allows the deﬁnition of a structured space of agents. Agents can either be declared ab initio or derived from existing agents in a manner reminiscent of object-oriented programming (particularly the prototype-based approach). In the latter case, the new agent can gain, lose, rename, mutate or duplicate sites of the agent from which it is derived. This organizes the space of agents hierarchically and thus enables us to write generic rules that mention agents that have many descendants in the hierarchy. These generic rules act as shorthand for sets of normal Kappa rules; they capture behaviours shared by splice variants (e.g. p46, p52 and p66 Shc), genetically related proteins (e.g. ERK1 and ERK2) or mutated proteins. In particular, the conciseness of generic rules enables us to write and analyze large Kappa models far more easily. We illustrate this with a small generic rule

118

V. Danos et al.

set (15 rules) for the erbB receptor network that, once expanded into Kappa, has over 300 rules and which grows considerably larger still if we add in drug interventions and mutated erbB agents. In summary, our agent hierarchy allows us to write large models in a comfortable way, to navigate the perturbation space of the model (ligands, mutations and drugs) and investigate the consequences of chosen perturbations, i.e. those for which we have experimental data, with the static and causal analyses of Kappa. This is particularly interesting for mutational perturbations as these enable us to reproduce, in numero, biochemical experiments that employ engineered mutations. In this way, our rules—a formalization of the consensus pathway assembled by many biochemical experiments—can be tested by checking, in numero, whether perturbing them with mutated agents—representing the engineered mutations—matches those experimental results. Of course, this procedure can never “prove” that a rule is correct but it can be used to reject rules that lead to behaviour incompatible with experimental results. It can also point to the existence of missing links in a model if it throws up false negatives with respect to the experimental data, e.g. it predicts some but not all experimentally observed phosphorylated sites. In other words, it enables us to put our assumptions under the microscope and verify that the consensus wild-type pathway behaves as expected when subjected to perturbations—and if it doesn’t, we will need to change our consensus model. Contribution and relation to existing work. Rule-based modelling is one branch of a rich literature based on the idea of representing proteins and their interactions as concurrent processes, thereby viewing a signalling network as a kind of massively distributed system. This was initially expressed in the formalism of π-calculus [7, 8] but, since then, a number of variants of π-calculus [9, 10] and of other languages for distributed systems [11, 12, 13, 14, 15, 16, 17] have also been proposed for representing various aspects of biological processes, notably the importance of causality and compartments. Rule-based modelling, rather like the BWB/BlenX system [18], was developed out of these ideas but, instead of being based on some prior formalism for general distributed systems, is a domain-speciﬁc modelling language for biological processes. Our language Kappa is particularly closely related to BioNetGen (BNG) [19]. Although the original aims of BNG were rather diﬀerent—it was conceived as a language for describing systems of ODEs in a higher-level fashion, rather than as a modelling language in its own right—the two approaches have much in common and, in particular, our agent hierarchy proposal would work just as well in BNG as in Kappa. Despite these many advances, to the best of our knowledge none of the abovecited approaches, including Kappa and BNG, can deal with all the potential sources of combinatorial explosion in signalling models. Our extension of Kappa with agent hierarchies directly addresses this problem in the speciﬁc context of rule-based modelling. Given that mutating agents, via small changes in their sites and thus interaction capabilities, is central to our proposal, it would be interesting to investigate the possible connections of this work with the recent

Rule-Based Modelling and Model Perturbation

119

use of mutations on the structure of BlenX programs in order to evolve networks via genetic algorithms [20]. However, it should be stressed that our work was originally intended to facilitate the construction (and documentation) of large models in a way that makes explicit any underlying uniformities, rather than in directly enabling an evolutionary analysis of networks.

2

Kappa and Agent Variants

A Kappa [21] model consists of a collection of concrete agents and rules. Each agent, or more properly agent type, has a name, an associated set of sites, each with an optional internal state, and a copy number. An atomic rule falls into one of ﬁve classes—a binding between two agents, an unbinding, the modiﬁcation of an agent, the creation of an agent or the deletion of an agent—but a rule can also be non-atomic, combining several actions. Given a Kappa model, its contact map, which is computed statically from the rules, speciﬁes which agents can bind and on which sites. (See e.g. Figs. 1, 4.) On the other hand its influence map, also computed statically, speciﬁes the causal relations of activation and inhibition between rules, that is to say a rule activates (inhibits) another if its application may add (subtract) from the set of instances of the other one. We will make use of the static analysis of rule accessibility [22] which identiﬁes whether a rule is dead, i.e. cannot be applied, or is potentially applicable; in the latter case, we will use the story sampler [23] to extract, from stochastic simulations [24] of the model, the chains of rule ﬁrings that can lead to an actual application of the rule. If we ﬁnd such a story, this conﬁrms that the static analysis didn’t produce a false positive. The concrete syntax we use to present agents, agent variants and rules should be self-explanatory (although we stress that it can be formalized). One key thing to remember, as said earlier, is that in the deﬁnition of a rule one has the option of not mentioning some sites of an agent. In situations where agents have up to a dozen diﬀerent sites (e.g. the members of the EGF receptor family), this is key to obtaining concise models. This, combined with the ability to mention generic agents, allows us to express enough uniformities for also obtaining concise descriptions of perturbed models. 2.1

Agent Variants

A variant on an agent always introduces a new name and can arise in several diﬀerent ways: it can lose or mutate an existing site, gain a new site or rename/duplicate an existing site. To represent these possibilities formally, we need only introduce two perturbation operations on agents, one to add a site, the other to replace a site with a set of sites. The latter operation subsumes site deletion (by replacing a site with the empty set), site renaming (replacing with a singleton set) and duplication. For example, %gen: A(s,t) %gen: B = A[+u s\{} t\{t1,t2}]

120

V. Danos et al.

declares the agent A with sites s and t and derives from it an agent B with sites t1, t2 and u. This deﬁnes a tree of agent variants; most nodes of the tree are labelled ‘gen’ for generic but leaves of the tree can be labelled ‘conc’ for concrete which signals that that agent can be used in a Kappa model. Note that we have a second tree structure that traces site linkages: any site can be traced back to either a site addition or to a site declared ab initio; and conversely, following the linkages the other way, a site in agent A maps to a set of sites in any given descendant agent B (empty if the site has been deleted, singleton if it has just been renamed). This is important for compiling generic rules into a bona ﬁde Kappa model. Mutation of a site is represented by the compound operation of deleting the original site and, if desired, adding a new site to “replace” it. If the desired result of the mutation is simply the loss of certain wild-type interactions, the loss of the site is enough and no such new site need be added; but sometimes mutations result in new interactions becoming possible in which case we would need to introduce a new site in order to write the new rules expressing the novel interactions of the mutated agent, e.g. the tyrosine kinase inhibitor erlotinib binds to the L858R mutated erbB1 with much higher aﬃnity than to the wild-type receptor. 2.2

A First Example

Let us make this more concrete with an example extracted from a larger model of the MAPK cascade. We start with two basic agent types, MAP2K and MAPK, from which we would like to derive some more speciﬁc agent types. Our ﬁrst declarations introduce the starting agents: %gen: MAP2K(D,S~u,ST~u) %gen: MAPK(CD,T~u,Y~u) Formally, these declarations play a role analogous to that of the axioms in any formal language and, as in that kind of setting, we use them as the starting point to introduce more subtle objects. In this case, we wish to consider the three common kinds of MAPK protein—ERKs, JNKs and p38s—and their respective MAP2K upstream activators—MEKs, JNKKs and p38 kinases. To do this, we ﬁrst introduce three variants of MAPK and three of MAP2K: %gen: ERK = MAPK[+FXFP] %gen: JNK = MAPK %gen: p38 = MAPK Note that, while ERK gains a new site FXFP, an ERK-speciﬁc binding site for immediate early gene products such as Fos and Jun [25], JNK and p38 simply inherit the sites of MAPK without making any changes. As we will see shortly, the introduction of these three variants allows us to express concisely the speciﬁcity of binding between these three distinct families of MAPKs and their cognate upstream activators. Note also that these three agents are still

Rule-Based Modelling and Model Perturbation

121

generic as they represent families of proteins: ERK covers two proteins (ERK1 and ERK2), JNK covers three (JNK1, JNK2 and JNK3) and p38 covers four (p38alpha/beta/gamma/delta); and several of those proteins have multiple splice variants. We formalize this by a further layer of variants: %conc: ERK1 = ERK[T\{T202} Y\{Y204}] %conc: ERK2 = ERK[T\{T185} Y\{Y187}] We show only the case of ERK1 and ERK2 as those of JNK and p38 are completely analogous. Recall that we use the ‘conc’ tag (rather than ‘gen’) to make explicit the fact that ERK1 and ERK2 are concrete, not generic, agents and, as such, can be used in a Kappa model. Note that we have renamed (via singleton duplications) the sites of ERK to include speciﬁc information about the exact residue numbers of their phosphorylatable sites; this is not essential, of course, but does illustrate the documentary power of agent variants over and above their role of structuring the space of agents. We must also introduce generic and concrete variants of MAP2K. Each variant covers two proteins: MEK1 and MEK2 for MEK; MEK4 and MEK7 for JNKK; and MEK3 and MEK6 for p38K. (Again, for the sake of simplicity, we only show the concrete variants of MEK.) %gen: MEK = MAP2K %gen: JNKK = MAP2K %gen: p38K = MAP2K %conc: MEK1 = MEK[S\{S218} ST\{S222}] %conc: MEK2 = MEK[S\{S222} ST\{S226}] Already, the simple fact of hierarchically structuring the agents under consideration yields a useful object in its own right that documents, in a completely formal way, a signiﬁcant amount of biological knowledge (about exactly how related proteins relate to each other) that can easily be found in several online databases but which, in that medium, remains informal and purely descriptive, whereas, in this formalized setting, has already been subjected to an initial step of processing and structuring. It also includes a convenient documentation of the speciﬁc sites of interest, e.g. the precise identities of phosphorylation sites, that are otherwise rather cumbersome to keep track of. Moreover, the creation of this agent hierarchy also facilitates the process of writing rules by enabling us to write them at the appropriately generic level. It eases the cognitive burden of writing rules by exposing clearly the similarities and diﬀerences between various agent types. More concretely, it allows us to avoid writing essentially the same rule many times for closely related agents and, as such, also eliminates the risk of forgetting cases (a very common mistake when developing large rule sets). We turn to this in the next subsection where we will complete the MAPK example.

122

V. Danos et al.

2.3

Generic Rules

We have seen how we can structure agents hierarchically with concrete agents at the leaves and generic agents above them. In this context, a normal (or concrete) Kappa rule is a rule that only mentions concrete agents. A generic rule is syntactically just like a normal rule but mentions one or more generic agents. The purpose of such a rule is to be expanded into a set of concrete rules by replacing each generic agent G in the rule with all appropriate concrete agents C below it in the hierarchy. However, this expansion is modulated by the changes made to G’s sites in C; notably, if site s of G is deleted in C, then no rule testing the existence of s can instantiate G to C. And we must also use the site linkages between C and G to deal with any renaming and duplication of G’s sites in C. So, were we to write the single generic rule MAP2K(D), MAPK(CD) MAP2K(D!0), MAPK(CD!0) this would “incorrectly”, i.e. not as we wish, expand to a collection of concrete rules where all concrete descendants (in the agent hierarchy) of MAP2Ks can bind with all concrete descendants of MAPKs, e.g. JNK2 could bind ERK1. This is the reason why, in the previous section, we introduced a second layer of generic agents—ERK, JNK, p38; MEK, JNKK, p38K. Given that, we can write the following three generic rules that properly respect the desired speciﬁcity of binding between MAP2Ks and MAPKs. MEK(D), ERK(CD) MEK(D!0), ERK(CD!0) JNKK(D), JNK(CD) JNKK(D!0), JNK(CD!0) p38K(D), p38(CD) p38K(D!0), p38(CD!0) These three generic rules expand to eighteen concrete rules if we take ERK1/2, JNK1/2/3 and p38α/β/γ/δ as concrete agents. If we included the many splice variants of the JNKs and p38s, the same three generic rules would expand to over thirty concrete rules. This illustrates the ﬂexibility of our approach whereby a given generic rule can expand diﬀerentially depending on the background of concrete agent variants we select. In particular, a single rule set can be seen as existing at many levels of detail—and this is easily tunable by the modeller as a function of his/her current needs. There is, however, an associated cost, over and above the obvious need to recompile one’s generic rules, when changing the level of detail of a model: under certain circumstances, this will lead to a degradation in the performance of stochastic simulation. The reason for this is that the cost of an event in the simulator depends, in part, on the maximum outdegree of the “wake-up map”, a graph derived from the rule set which keeps track of which rules are reactivated when a rule ﬁres [24]. In the worst-case scenario, our generic rule expansion causes a “blow up” of the wake-up map with concomitant degradation in the simulator’s performance. More generally, our mechanism of using an agent hierarchy and generic rules to generate a concrete rule set allows the Kappa modeller a ﬁner control of the granularity of his/her rules. Consider for example an agent A that can bind two

Rule-Based Modelling and Model Perturbation

123

agents, B1 or B2, and that binding with either is suﬃcient (and necessary) for A to bind a further agent C. To express this in Kappa, we would have to write two rules for A binding C; one for the case of B1, the other for B2. This isn’t too bad—but if we have not two but a large number of activating ligands of A, it rapidly becomes tedious and error-prone to write the rule sets. By using a generic agent B, representing the class of A-activating ligands, we write just one generic rule that covers all cases (albeit requiring recompilation after the addition of new concrete descendants of B). Or to put it another way, we think of the generic agent B as generating a coarse-graining of the model’s molecular species that no longer distinguishes between the various concrete descendants of B (i.e. B1, B2, etc). With more complex agent hierarchies, one can express further, more subtle coarse-graining eﬀects such as the MAP2K-MAPK binding speciﬁcity example above. However, it should be admitted that the example of MAPK is particularly conducive to a treatment of this kind (which is why we use it as our initial example!) and that not all signalling pathways exhibit the same degree of sharing of structure found here, as expressed by the highly generic nature of the rules. This in itself is a useful aspect of our language extension in that it enables us to recognize, formally, the fact that a pathway is highly generic or, on the contrary, particularly obtuse and dependent on many speciﬁc details. Indeed, the purpose of this extension is not to obtain a maximal “compression” of a concrete rule set into as few generic rules as possible; rather it is to illuminate the structure of a model by expressing it at an appropriate level of abstraction.

3

The Perturbation Space

Now that we have shown, with the MAPK example, how our parsimonious language extension enables rapid development of large rule sets via the mechanism of generic rules, let us turn to the main problem of interest here which is to build realistic models incorporating multiple erbB ligands and receptors, mutated forms of those receptors and monoclonal antibodies (mAbs) and tyrosine kinase inhibitors (TKIs) targeting those receptors. Unlike the previous MAPK model where the use of agent variants was convenient but hardly indispensable, in this case it would be a nightmarish process to write the rules directly in Kappa. As we will see, the use of agent variants not only helps to structure the model in a human-understandable manner, it also radically tames the combinatorial explosions caused by having multiple ligands and receptors and by the introduction of mutations. We ﬁrst deﬁne our agent hierarchy. It has two roots, erbB for the receptors and erbL for the ligands, each with four children. %gen: erbB(L,CR,N,atp,AS,C,Y~u) %gen: erbBL(L) The next layer of agents splits the space of ligands into four, each with a diﬀerent repertoire of receptors to which it binds.

124

V. Danos et al.

%gen: %gen: %gen: %gen:

erbBL1 = erbBL erbBL14 = erbBL erbBL34 = erbBL erbBL4 = erbBL

Note that a hierarchical presentation of a model has a degree of intensionality and, in particular, is of course not unique—indeed, the compiled model is actually a presentation of itself. This begs the remark that a presentation is both a way to achieve compactness of description and to document knowledge about relationships between agents that disappears in the compilation process. We also need variants for the four erbB receptors. Note that we introduce them as generic agents and only later specialize them as wild-types and mutant ones (only erbB1-WT is shown here). %gen: erbB1 = erbB[Y\{Y1016, Y1092, Y1110, Y1172, Y1197}] %gen: erbB2 = erbB[L\{}] %gen: erbB3 = erbB[N\{}] %gen: erbB4 = erbB %conc: erbB1_WT = erbB1 To keep our presentation uncluttered, we have only shown the full repertoire of phosphorylation sites for erbB1; in the full model, the other receptors also have a similar complement of Y sites. Note though that erbB2 loses the site L and erbB3 the site N . Finally, let us note the concrete ligands. In what follows, we will in fact only consider EGF and HRG. %conc: EGF = erbBL1 %conc: TGFalpha = erbBL1 %conc: AR = erbBL1 %conc: BTC = erbBL14 %conc: HB-EGF = erbBL14 %conc: ER = erbBL14 %conc: HRG = erbBL34 %conc: NRG2 = erbBL34 %conc: NRG3 = erbBL4 %conc: NRG4 = erbBL4 3.1

The Consensus Model

We build our consensus model on the basis of a conservative reading of the literature; see e.g. [26, 27]. Speciﬁcally, we consider that ligands bind monomer receptors which can then externally (on the trans-side of the plasma membrane) dimerize; this in turn enables the formation of asymmetric dimers that lead to receptor binding on the cis-side and cross-phosphorylation.

Rule-Based Modelling and Model Perturbation

125

erbBL1(L), erbB1(L,CR) -> erbBL1(L!0), erbB1(L!0,CR) erbBL14(L), erbB1(L,CR) -> erbBL14(L!0), erbB1(L!0,CR) erbBL14(L), erbB4(L,CR) -> erbBL14(L!0), erbB4(L!0,CR) erbBL34(L), erbB3(L,CR) -> erbBL34(L!0), erbB3(L!0,CR) erbBL34(L), erbB4(L,CR) -> erbBL34(L!0), erbB4(L!0,CR) erbBL4(L), erbB4(L,CR) -> erbBL4(L!0), erbB4(L!0,CR) These six generic rules expand into a signiﬁcant number of concrete Kappa rules in a manner that depends on the level of detail requested in the identities of ligands. For example, although there are three ligands of type erbBL1 and three of type erbBL14, the two ligands of type erbBL34 actually exist in multiple splice variants, as do those of type erbB4. In any case, at the very least these six generic rules give rise to ﬁfteen concrete rules; and, of course, we also need the unbinding rule: erbBL(L!0), erbB(L!0) -> erbBL(L), erbB(L) Note that this generic unbinding rule will generate concrete rules that will never apply, e.g. a descendant of erbBL1 unbinding erbB3. However, these dead rules are detected by our static analysis and so can be removed (if desired) from the generated rule set. erbBL(L!1), erbB(L!1,CR), erbBL(L!2), erbB(L!2,CR) -> \ erbBL(L!1), erbB(L!1,CR!0), erbBL(L!2), erbB(L!2,CR!0) erbBL(L!1), erbB(L!1,CR), erbB2(CR) -> \ erbBL(L!1), erbB(L!1,CR!0), erbB2(CR!0) The ﬁrst of these generic rules deals with most of the cases of (external) dimerization. The only diﬃculty comes from the fact that erbB2 has no ligand binding site L and, as such, cannot ever match the generic erbB agent in that rule which mentions L. For this reason, we must explicitly include the second rule that covers the case of erbB2 dimerizing with a diﬀerent erbB receptor type; we do not consider erbB2 homodimerization. These three rules generate a very large number (well over 150) of concrete rules due not only to the fact that the erbB generic agent is capable of multiple matches but also because the concrete erbB agents can bind multiple ligand agents. We next have the rule for internal (or asymmetric) dimer formation. Here, a receptor binds its (external) dimer partner on a second site; this dimer is asymmetric because the bond is made between the N site of one receptor and the C site of the other. In a dimer not containing erbB3, this asymmetric dimer can ﬂip states; this is the second rule. erbB(CR!1,N,C), erbB(CR!1,C) -> erbB(CR!1,N!0,C), erbB(CR!1,C!0) erbB(CR!1,N!2,C), erbB(CR!1,N,C!2) -> erbB(CR!1,N,C!3), erbB(CR!1,N!3,C)

The ﬁnal rule is for trans phosphorylation of one erbB receptor by its dimer partner. In an asymmetric dimer, the receptor bound on site C is the activator

126

V. Danos et al.

whereas the receptor bound on N is the activated. It is thus the activator that gets phosphorylated. erbB(N!1,atp,AS), erbB(C!1,Y~u) -> erbB(N!1,atp,AS), erbB(C!1,Y~p) This one generic rule expands into many concrete rules for two independent reasons. Firstly, each erbB agent can be multiply instantiated: the ﬁrst erbB can be anything but erbB3 (whose N site was deleted) and the second can be any of the four erbBs. Secondly, each receptor duplicates the site Y, so we get one concrete rule per duplicand. In the context of the purely wild-type model, neither the atp nor the AS site plays any role. This is because the rule tests only for the existence of these sites (which always succeeds) and that they are both unbound (which also always succeeds since we have no rules for binding to either of them). However, as we will see shortly, these two sites do play an important role once we take drug interventions and mutations into account. We can neatly summarize the model so far with its contact map (Fig. 1).

Fig. 1. Contact map of the consensus model: each concrete agent is represented once with all its sites; possible bindings are indicated by an edge joining two nodes, modiﬁable sites are indicated in grey. Only a restricted subset of known EGFR receptors ligands is shown, namely EGF and HRG.

Rule-Based Modelling and Model Perturbation

3.2

127

Ligand Perturbations

The erbB receptor network clearly has a lot of ﬂexibility in its response to ligands. In particular, the receptor dimers that form depend on the available ligands and the presented receptors. In addition, erbB3 has compromised capability to form asymmetric dimers: it can activate the catalytic activity of its dimer partner but cannot be activated by it. This phenomenon adds yet another layer of subtlety to erbB receptor activation. For example, in a cell line expressing erbB2, erbB3 and erbB4, one would expect HRG to promote phosphorylation of erbB2, via erbB2:erbB4 dimers, as well as phosphorylation of erbB3 and erbB4. On the other hand, were erbB4 not expressed, one would expect only erbB3 phosphorylation, via erbB2:erbB3 dimers. However, this kind of reasoning rapidly becomes highly complicated, particularly in the presence of multiple ligands, and we would like some way of deducing, from the rule set and a choice of expression levels of ligands and receptors, which receptors get phosphorylated (and, in some cases, on which sites). We can do this using static analysis of the rule set. We ﬁrst write dummy rules that detect typical molecular species of interest, e.g. erbB2(Y~p) -> erbB2(Y~p) We then ask the static analyser whether or not our dummy rules can ﬁre. It responds in one of two ways: either a categorical ‘no’ or a tentative ‘yes’. In the case of a ‘no’, we know (since the static analysis never produces false negatives) that our rule set cannot create the molecular species in question—starting from the declared initial solution. In the case of a ‘yes’, we have no certainly (since the analysis can give false positives) that the species can arise, but also no proof that it cannot. In an attempt to conﬁrm the ‘yes’, we then use the story sampler to search for pathways leading to a dummy rule; if (at least) one exists, we have proof that the species can arise. For example, the static analysis shows that, with our rule set, erbB2 phosphorylation cannot take place (categorical ‘no’) under the following conditions: – HRG only; erbB2 and erbB3 only [erbB3 cannot phosphorylate erbB2] – HRG only; erbB1, erbB2 and erbB3 only [erbB1 cannot bind HRG] whereas it can potentially take place (tentative ‘yes’) under the following conditions: – HRG only; erbB2, erbB3 and erbB4 only – EGF and HRG; erbB1, erbB2 and erbB3 only. To conﬁrm this claim, we ask for stories leading to the appropriate observables. In both cases, we ﬁnd indeed a story leading to phosphorylated erbB2 which conﬁrms that the static analysis did not give us false positives (Figs. 2, 3). This combination of static analysis and story sampling enables a powerful model development process where, starting from a consensus, perhaps overly restrictive, rule set, we investigate which observables of interest can arise under

128

V. Danos et al.

Fig. 2. Story leading to erbB2 phosphorylation by erbB4

Fig. 3. Story leading to erbB2 phosphorylation by erbB1

Rule-Based Modelling and Model Perturbation

129

which conditions. We then compare these predictions to experimental data in order to judge the accuracy and completeness of the model. If experimental data conﬂicts with the results of our analysis, this means one of two things: the consensus model either has fatal ﬂaws or missing links. A ‘fatal ﬂaw’ means that certain experimentally unobservable species can be generated by the rule set; in other words, that the mechanism described by the rules makes unwarranted assumptions. A ‘missing link’ corresponds to the more likely situation where an experimentally observed species remains inaccessible with our consensus model; this implies that the rule set lacks certain necessary rules. For example, in the above discussion, we noted that, in our rule set with erbB1, erbB2 and erbB3 only, HRG stimulation leads only to erbB3 phosphorylation; whereas the combination of EGF and HRG leads to phosphorylation of all three receptors. This constitutes an experimentally refutable prediction. In the event of such a refutation, e.g. we observe erbB1 phosphorylation upon HRG stimulation, we could freely postulate various new rules, check that they do indeed open up the possibility of erbB1 phosphorylation and then compare and contrast their eﬀects on other observables. If a new rule creates a ‘fatal ﬂaw’, we can discount it; but in general this may still leave us with a choice between several proposed new mechanisms. To decide between these would require us to ﬁnd a new, experimentally refutable prediction and do the experiment (or ﬁnd it in the literature). We stress that this remains a human-directed model development process— we do not consider automatically generated rules in any form—but one in which variant mechanisms can be built and evaluated in an organized fashion. 3.3

Drug Perturbations

In recent years, particularly with the realization that deregulated erbB signalling contributes to the development of multiple cancers, much research has focussed on ﬁnding ways of blocking the activity of this family of receptors via drug intervention. To date, two broad classes of drug have been developed: monoclonal antibodies (mAbs) and tyrosine kinase inhibitors (TKIs). Antibodies typically act as classical competitive inhibitors that exert their function by binding cell surface receptors in a way that physically obstructs their usual ligands from binding. On the other hand, TKIs behave as classical non-competitive inhibitors of kinase activity that do not prevent substrate binding but instead block the ATP binding site of the kinase domain, thus preventing substrate phosphorylation. As an illustration of the ease with which we can incorporate these kinds of pharmaceutical intervention in our modelling framework, we include rules for C225 (cetuximab, a mAb) binding to erbB1’s site L, ZD1839 (geﬁtinib, a TKI) binding to erbB1’s atp site and 4D5 (trastuzumab, another mAb) binding to erbB2’s dimerization site CR. C225(L), erbB1(L) -> C225(L!0), erbB1(L!0) ZD1839(L), erbB1(atp) -> C225(L!0), erbB1(atp!0) 4D5(L), erbB2(CR) -> C225(L!0), erbB2(CR!0)

130

V. Danos et al.

As each of these molecules has a reversible inhibitory eﬀect on the respective receptor, we also need the accompanying unbinding rules: C225(L!0), erbB1(L!0) -> C225(L), erbB1(L) ZD1839(L!0), erbB1(atp!0) -> ZD1839(L), erbB1(atp) 4D5(L!0), erbB2(CR!0) -> 4D5(L), erbB2(CR) The contact map of the system (Fig. 4) including these inhibitors makes it clear that the antibodies (C225 and 4D5) act as competitive inhibitors: C225 competes with EGF for the ligand binding site of erbB1 and 4D5 competes for the dimerization binding site of erbB2. The inhibitory eﬀect of ZD1839 shows up only in the inﬂuence map: the rule binding ZD1839 to erbB1 inhibits all modiﬁcation rules (concretely, the phosphorylations) dependent on erbB1. This is because ZD1839 binds to the site atp of erbB1 which must be free in order for erbB1 to modify its dimer partner. The presence of ZD1839 thus frustrates, without completely preventing, erbB1-dependent phosphorylation. We will return to this later.

Fig. 4. Contact map of the consensus model with antibodies (C225 and 4D5) as competitive inhibitors, and ZD1839 as non-competitive inhibitor.

Rule-Based Modelling and Model Perturbation

131

It should be noted that, in our examples of inhibitors, all agents are concrete. But, in general, drugs would also be organized, much like natural ligands, by an appropriate agent hierarchy. 3.4

The Uses of Mutational Perturbations

So far, we have seen how the use of agent variants allows us to organize agents hierarchically and thus write generic rules at a convenient level of granularity. In particular, this facilitates the development of models with families of related proteins, or proteins with multiple splice variants, that have overlapping functionality. However, agent variants also enable the treatment of mutated agents which likewise share a lot of the functionality of their wild-type cousins but which also potentially lose some of that functionality and/or gain new functionality. This has two immediate applications. Firstly, it allows us to build models with a mixture of wild-type and mutated agents in order to investigate (statically or numerically) the consequences of mutations. This is particularly interesting in the context of models, as described previously, that also include drug interventions. More subtly, it also allows us to cast a critical eye over the assumptions we make in building our wild-type model. After all, a lot of the experimental data from which consensus pathways have been deduced comes from mutation experiments. These typically eliminate one or more phosphorylation sites in a protein and investigate which, if any, pathways suﬀer from this perturbation. However, such data can be diﬃcult to interpret and the deduced wild-type interactions may be incorrect. For example, as explained in [28], one experiment showed that expressing a kinase-dead mutant of PI3K inhibited Ras activation upon EGF stimulation; this led the authors to propose a role for PI3K in activating Ras. But then, a second study demonstrated that a constitutively active (and membrane associated) mutant of PI3K did not promote Ras activation, which contradicted the conclusions of the ﬁrst study. In the end, it turned out [28] that PI3K actually inhibits Ras deactivation; so PI3K sensitizes Ras for activation but cannot by itself actually activate it. In more details: PI3K promotes Gab1 recruitment to the membrane which, on EGF stimulation, strongly recruits Shp2 to the membrane; Shp2 is a tyrosine phosphatase that dephoses the phospho-tyrosine binding sites for RasGAP (and for PI3K!) on erbB receptors and Gab1. So Shp2 inhibits RasGAP recruitment to the membrane which indirectly aids Ras activation (Sos which activates Ras has an easier job). This gives a measure of the daunting complexity of inferring a protein network, and as a consequence a measure of how helpful a methodology such as the one we illustrate here can prove. Indeed, using agent variants, we can express the kinds of (artiﬁcially) mutated proteins used in biochemical studies and so replay numerically such experiments. We can therefore detect, in numero, if the hypothesized wild-type network is in fact incorrect, e.g. if we had a model for Ras activation including a rule for ‘PI3K activates Ras’, we would have been able to predict that a constitutively active PI3K mutant would activate Ras; the fact that, experimentally, this is

132

V. Danos et al.

not observed means that that rule must be wrong. This kind of perturbational analysis is not just useful for postdictive veriﬁcation of inferences, it is also a discipline to build a model upon such data, and to build further data to refute predictions; this could be particularly interesting if two plausible molecular mechanisms (candidate consensus pathways) made divergent predictions. 3.5

Testing the Wild-Type Model

In two recent papers, Kuriyan and coworkers have developed a conceptual model of erbB receptor acivation that depends on the formation of an asymmetric dimer [27, 29]. We have used this when writing the above rules for the wildtype erbB network in the previous section. They developed their model using a combination of structural and mutational data and provide convincing evidence of its correctness by cotransfecting various artiﬁcal erbB constructs that lack one or more of the N, C and AS sites. We can use agent variants, in combination with static analysis, to reproduce these kinds of results in numero. For example, kinase-dead erbB1 is obtained by deﬁning a variant of erbB1 with the AS site deleted; this agent can no longer phosphorylate its dimer partner. Similarly, we can also introduce variants that delete either the N or the C site instead of, or in addition to, the AS site. %conc: %conc: %conc: %conc: %conc:

erbB1_KD = erbB1[AS/{}] erbB1_noN = erbB1[N/{}] erbB1_noC = erbB1[C/{}] erbB1_KDnoN = erbB1[AS/{} ; N/{}] erbB1_KDnoC = erbB1[AS/{} ; C/{}]

These agents inherit all rules from wild-type erbB1 that do not mention the sites that they lack. So erbB1 KD can freely form asymmetric dimers but phosphorylates nothing, whereas erbB1 noN and erbB1 noC are partially compromised in their ability to form asymmetric dimers: the former can activate its partner and get phosphorylated, but cannot be activated and phosphorylate its partner; the latter can be activated by its partner and phosphorylate it, but cannot activate its partner and get phosphorylated. We can now use static analysis, as in the previous section, to analyze the consequences of coexpressing pairs of these variant agents. We do this by checking the accessibility of the rules erbB1(N!1), erbB1(C!1) -> erbB1(N!1), erbB1(C!1) erbB1(Y1197~p) -> erbB1(Y1197~p) (that respectively detect the possibility of an asymmetric dimer forming and an erbB1 receptor becoming phosphorylated) in an initial solution that includes EGF and a choice of any (one or) two of the erbB1 variants. In particular, we can recapitulate the results of [27] (see their Fig. 6; we use the same combination numbers) in completely automatic fashion:

Rule-Based Modelling and Model Perturbation

133

1. wild-type erbB1 only: asymmetric homodimer accessible; phosphorylation accessible 2. erbB1 KD only: asymmetric homodimer accessible; phosphorylation inaccessible 3. erbB1 KD & erbB1 noN: one asymmetric heterodimer accessible; phosphorylation inaccessible 4. erbB1 KD & erbB1 noC: one asymmetric heterodimer accessible; phosphorylation accessible 5. erbB1 KDnoC only: asymmetric homodimer inaccessible; phosphorylation inaccessible 6. erbB1 KDnoC & erbB1 noN: one asymmetric heterodimer accessible; phosphorylation inaccessible 7. erbB1 KDnoC & erbB1 noC: asymmetric heterodimer inaccessible; phosphorylation inaccessible 8. erbB1 KDnoN only: asymmetric homodimer inaccessible; phosphorylation inaccessible 9. erbB1 KDnoN & erbB1 noN: asymmetric heterodimer inaccessible; phosphorylation inaccessible 10. erbB1 KDnoN & erbB1 noC: one asymmetric heterodimer accesible; phosphorylation accessible In [27], this had to be done by hand, a task that soon begins to get rather subtle, particularly if you want to consider doubly-mutated agents and/or coexpression of more than two receptor constructs at a time. It is thus very useful to be able to express this situation in Kappa and rely on static analysis to detect the impossibility/possibility of phosphorylation. Moreover, if the static analysis announces that phosphorylation is not impossible, we can, as above, use the story sampler to search for ways in which this can actually take place. Again, in some cases, this is easy to do by hand but, beyond a certain degree of complexity, it is highly desirable to have an automatic method in order to avoid making mistakes. These results demonstrate that our consensus model is indeed compatible with the experimental data of Kuriyan et al. and, as such, it passes the test. This comforts us, for now, in our choice of rules but of course provides no guarantee that future experimental data will not invalidate some of them. 3.6

The Limits of Perturbation Testing

We mentioned earlier that our language extension shields the modeller from the underlying rule set generated by generic rules. However, we should say that this is only true qualitatively—if we wish to manipulate the rate constants of our model in such a way that diﬀerent concrete instantiations of one generic rule get diﬀerent kinetics, this can only be done by examining and modifying directly the generated rule set.

134

V. Danos et al.

More generally, the modelling methodology advocated above based on static analysis cannot be used to gauge the eﬀect of perturbations, such as drugs, that restrict, but don’t outlaw, the application of other rules. Or, to put it another way, a perturbation that operates entirely at the level of kinetics is undetectable by this method. We would however expect to observe the eﬀects of such perturbations during stochastic simulation and/or story sampling. Indeed, it would be straightforward to observe the inhibition of erbB1’s kinase activity by tracking that rule’s activity in the absence and presence of drugs. More ambitiously, we could compare the relative strengths of each erbB’s kinase activity and the way in which that is disturbed by drugs that target only one receptor; this would require running the story sampler many times at many time points to get a statistical picture of the model’s activity proﬁle over time. We leave this for future work.

4

Conclusions

Even if wild-type pathways are obviously central to a systemic view of molecular biology, modelling is not just about these. It is equally important to be able to navigate the space of derivatives of a model for two complementary reasons. Firstly, one needs to understand diseased conditions as natural perturbations of the wild-type; secondly, one also needs to represent synthetic perturbations (by genetic knock-outs, domain truncations, point mutations, etc) because they are key in the inference of the wild-type. This is a formidable challenge because the space of such model perturbations introduces a second kind of combinatorial explosion. The well-studied example of the EGF receptor family (see §3) is a powerful illustration of this fact. Now, we have to do something if we want our modelling vessel to stay aﬂoat in the sea of perturbations. In other words, just as the passage from reactions to rules tames the ﬁrst binding-caused explosion, we have to ﬁnd a mechanism to tame what one might call the perturbation-caused explosion. The fact that Kappa describes molecular interactions at the level of domain binding and modiﬁcation seems a good start, since this is the granularity at which the engineering of perturbations in protein networks actually happens (e.g. Y to A mutations that disable a modiﬁcation). But to tackle our representation problem, we need another ingredient, namely a syntactic extension of Kappa that enables a clean, uniform treatment of protein families, splice variants and mutated proteins. This is what we have proposed here. The idea is to structure agents hierarchically so that rules can be expressed at an appropriate level of abstraction, as generic rules, which are then automatically compiled into pure Kappa. This eases the pain (and pitfalls) of writing large rule sets (indeed the modeller has no need to ever look at the resulting concrete rule set, unless he/she wishes to modify its rate constants), and as we wanted, this give means to navigate their perturbation space. Of course there is no magic: to work around the explosive generativity of wildtype pathways we capture postulated regularities by using rules (if the universe

Rule-Based Modelling and Model Perturbation

135

of reactions were lacking any regularity no method could describe them anyway, a rather grim perspective for systems biology); to work around the second source of complexity, again we capture regularities of another kind, namely that much of the wild-type behaviour of a protein is actually shared with its mutants and isoforms. We have shown that this strategy works well with our EGF example, as we were able to neatly set a wild-type model together with a selection of derivatives. With this model in place, one can bring the usual analysis tools of Kappa to bear on the rule set. As we have shown further, even in the absence of quantitative information about rates and copy numbers, one can obtain qualitative predictions about the induced perturbed behaviours and thus support on a full-scale the traditional informal inferences that are commonplace in the experimental investigation of protein networks. Acknowledgements. Jean Krivine is supported via grants from the Agence Nationale de la Recherche (ANR-07-PHYSIO-013-01) and the G´enopole Evry held by A. Benecke of the IHES.

References 1. Kholodenko, B.N., Demin, O.V., Moehren, G., Hoek, J.B.: Quantiﬁcation of Short Term Signaling by the Epidermal Growth Factor Receptor. J. Biol. Chem. 274(42), 30169–30181 (1999) 2. Kiyatkin, A., Aksamitiene, E., Markevich, N.I., Borisov, N.M., Hoek, J.B., Kholodenko, B.N.: Scaﬀolding protein GAB1 sustains epidermal growth factorinduced mitogenic and survival signaling by multiple positive feedback loops. J. Biol. Chem. 281, 19925–19938 (2006) 3. Orton, R.J., Sturm, O.E., Vyshemirsky, V., Calder, M., Gilbert, D.R., Kolch, W.: Computational modelling of the receptor tyrosine kinase activated MAPK pathway. Biochemical Journal 392(2), 249–261 (2005) 4. Schoeberl, B., Eichler-Jonsson, C., Gilles, E.-D., M¨ uller, G.: Computational modeling of the dynamics of the map kinase cascade activated by surface and internalized EGF receptors. Nature Biotechnology 20, 370–375 (2002) 5. Hlavacek, W.S., Faeder, J.R., Blinov, M.L., Posner, R.G., Hucka, M., Fontana, W.: Rules for Modeling Signal-Transduction Systems. Science’s STKE 2006(344) (2006) 6. Maslov, S., Ispolatov, I.: Propagation of large concentration changes in reversible protein-binding networks. Proceedings of the National Academy of Sciences 104(34), 13655–13660 (2007) 7. Regev, A., Silverman, W., Shapiro, E.: Representation and simulation of biochemical processes using the π-calculus process algebra. In: Altman, R.B., Dunker, A.K., Hunter, L., Klein, T.E. (eds.) Paciﬁc Symposium on Biocomputing, vol. 6, pp. 459–470. World Scientiﬁc Press, Singapore (2001) 8. Regev, A., Shapiro, E.: Cells as computation. Nature 419 (September 2002) 9. Priami, C., Regev, A., Shapiro, E., Silverman, W.: Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Information Processing Letters (2001)

136

V. Danos et al.

10. Baldi, C., Degano, P., Priami, C.: Causal π-calculus for biochemical modeling. In: Proceedings of the AI*IA Workshop on BioInformatics 2002, pp. 69–72 (2002) 11. Priami, C., Quaglia, P.: Beta Binders for Biological Interactions. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 20–33. Springer, Heidelberg (2005) 12. Cardelli, L.: Brane Calculi Interactions of Biological Membranes. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 257–278. Springer, Heidelberg (2005) 13. Regev, A., Panina, E.M., Silverman, W., Cardelli, L., Shapiro, E.: BioAmbients: an abstraction for biological compartments. Theoretical Computer Science 325, 141–167 (2004) 14. John, M., Ewald, R., Uhrmacher, A.M.: A Spatial Extension to the π Calculus. Electronic Notes in Theoretical Computer Science, vol. 194(3), pp. 133–148 (2008) 15. Calder, M., Gilmore, S., Hillston, J.: Modelling the inﬂuence of RKIP on the ERK signalling pathway using the stochastic process algebra PEPA. In: Priami, C., Ing´ olfsd´ ottir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 1–23. Springer, Heidelberg (2006) 16. Ciocchetta, F., Hillston, J.: Bio-PEPA: an extension of the process algebra PEPA for biochemical networks. Electronic Notes in Theoretical Computer Science, vol. 194(3), pp. 103–117 (2008) 17. Calzone, L., Fages, F., Soliman, S.: BIOCHAM: an environment for modeling biological systems and formalizing experimental knowledge. Bioinformatics 22(14), 1805–1807 (2006) 18. Dematte, L., Priami, C., Romanel, A.: The BlenX language: a tutorial. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 313–365. Springer, Heidelberg (2008) 19. Blinov, M.L., Faeder, J.R., Hlavacek, W.S.: BioNetGen: software for rule-based modeling of signal transduction based on the interactions of molecular domains. Bioinformatics 20, 3289–3292 (2004) 20. Dematt´e, L., Priami, C., Romanel, A., Soyer, O.: Evolving BlenX programs to simulate the evolution of biological networks. Theoretical Computer Science 408(1), 83–96 (2008) 21. Danos, V., Laneve, C.: Formal molecular biology. Theoretical Computer Science 325(1), 69–110 (2004) 22. Danos, V., Feret, J., Fontana, W., Krivine, J.: Abstract Interpretation of Cellular Signalling Networks. In: Logozzo, F., Peled, D.A., Zuck, L.D. (eds.) VMCAI 2008. LNCS, vol. 4905, pp. 83–97. Springer, Heidelberg (2008) 23. Danos, V., Feret, J., Fontana, W., Harmer, R., Krivine, J.: Rule-Based Modelling of Cellular Signalling. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 17–41. Springer, Heidelberg (2007) 24. Danos, V., Feret, J., Fontana, W., Krivine, J.: Scalable Simulation of Cellular Signaling Networks. In: Shao, Z. (ed.) APLAS 2007. LNCS, vol. 4807, pp. 139–157. Springer, Heidelberg (2007) 25. Murphy, L.O., Smith, S., Chen, R.H., Fingar, D.C., Blenis, J.: Molecular interpretation of ERK signal duration by immediate early gene products. Nat. Cell Biol. 4(8), 556–564 (2002) 26. Burgess, A.W., Cho, H.S., Eigenbrot, C., Ferguson, K.M., Garrett, T.P.J., Leahy, D.J., Lemmon, M.A., Sliwkowski, M.X., Ward, C.W., Yokoyama, S.: An Open-andShut Case? Recent Insights into the Activation of EGF/ErbB Receptors. Molecular Cell 12(3), 541–552 (2003)

Rule-Based Modelling and Model Perturbation

137

27. Zhang, X., Gureasko, J., Shen, K., Cole, P.A., Kuriyan, J.: An Allosteric Mechanism for Activation of the Kinase Domain of Epidermal Growth Factor Receptor. Cell 125(6), 1137–1149 (2006) 28. Sampaio, C., Dance, M., Montagner, A., Edouard, T., Malet, N., Perret, B., Yart, A., Salles, J., Raynal, P.: Signal strength dictates phosphoinositide 3-kinase contribution to Ras/extracellular signal-regulated kinase 1 and 2 activation via differential Gab1/Shp2 recruitment: consequences for resistance to epidermal growth factor receptor inhibition. Mol. Cell Biol. 28(2), 587–600 (2008) 29. Zhang, X., Pickin, K.A., Bose, R., Jura, N., Cole, P.A., Kuriyan, J.: Inhibition of the EGF receptor by binding of MIG6 to an activating kinase domain interface. Nature 450(7170), 741 (2007)

Extended Stochastic Petri Nets for Model-Based Design of Wetlab Experiments Monika Heiner1 , Sebastian Lehrack1 , David Gilbert2 , and Wolfgang Marwan3 1

Department of Computer Science, Brandenburg University of Technology Postbox 10 13 44, 03013 Cottbus, Germany {monika.heiner,slehrack}@informatik.tu-cottbus.de 2 School of Information Systems, Computing and Mathematics Brunel University, Uxbridge, Middlesex UB8 3PH, UK [email protected] 3 Otto von Guericke University & Magdeburg Centre for Systems Biology c/o Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany [email protected]

Abstract. This paper introduces extended stochastic Petri nets to model wetlab experiments. The extentions include read and inhibitor arcs, stochastic transitions with freestyle rate functions as well as several deterministically timed transition types: immediate ﬁring, deterministic ﬁring delay, and scheduled ﬁring. The extensions result into nonMarkovian behaviour, which precludes analytical analysis approaches. But there are adapted stochastic simulation analysis (SSA) methods, ready to deal with the extended behaviour. Having the simulation traces, we apply simulative model checking of PLTL, a linear-time temporal logic (LTL) in a probabilistic setting. We present some typical model components, demonstrating the suitability of the introduced Petri net class for the envisaged application scenario. We conclude by looking brieﬂy at a classical example of prokaryotic gene regulation, the lac operon case.

1

Motivation

This paper extends the Markovian stochastic Petri nets SPN Bio as introduced in [GHL07] to model and analyse biochemical networks. Related application scenarios are discussed in [BGHO08], [GBHD09]. Case studies demonstrating a unifying framework to integrate the qualitative, stochastic and continuous paradigms can be found in [HGD08], [GHR+ 08], [HDG10]. Thus, SPN Bio have been proven to be useful in systems and synthetic biology. However, there are limitations in expressivity. Generally, biologists face the problem to design wetlab experiments to validate or contradict the current understanding of the biochemical network under investigation. In order to be better able to do so, they ask for the following advanced features: C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 138–163, 2009. c Springer-Verlag Berlin Heidelberg 2009

Extended Stochastic Petri Nets

139

– stochastic and deterministic ﬁring behaviour within one model, – relative and absolute timing of the transitions’ ﬁring, – construction of arbitrary schedules of programmed interventions. Therefore, we are going to extend SPN Bio belonging to the Markovian world by several features supporting the comfortable modelling of wetlab experiments. The extentions lead to the deﬁnition of biochemically interpreted Generalised Stochastic Petri nets GSPN Bio and Deterministic and Stochastic Petri nets DSPN Bio . They include read and inhibitor arcs, stochastic transitions with freestyle rate functions as well as several deterministically timed transition types: immediate ﬁring, deterministic ﬁring delay, and scheduled ﬁring. The extension go beyond the Markov property, which precludes analytical analysis approaches; but there are adapted stochastic simulation analysis (SSA) methods, ready to deal with the extended behaviour. Having the simulation traces we apply simulative model checking of linear-time temporal logic (LTL) in a probabilistic setting (PLTL). Simulative model checking approximates the probability of a given temporal logic formula by considering ﬁnite sets of ﬁnite paths through the state space. Thus, it works even for systems with inﬁnite state spaces. We discuss in detail some typical model components, demonstrating the suitability of the introduced Petri net class DSPN Bio for the envisaged application scenario. These components will be analysed by checking sets of stochastic simulation traces against PLTL properties. In doing so, a special category of properties, the so-called invariant properties, will be used to prove at the same time the plausibility of the applied simulation algorithm. We conclude by looking brieﬂy at a classical example of prokaryotic gene regulation, the lac operon case.

2

Stochastic Modelling

We assume basic knowledge of the standard notions of qualitative place/transition Petri nets, see e.g. [Mur89], [Rei82], [HGD08]. To be self-contained we start with recalling the fundamentals of (biochemically interpreted) stochastic Petri nets, belonging to the Markovian world, before introducing the extended notions resulting ﬁnally into non-Markovian Petri nets. 2.1

The Markovian Case - Stochastic Petri Nets (SPN Bio )

As with a qualitative Petri net, a stochastic Petri net maintains a discrete number of tokens on its places. But contrary to the time-free case, a ﬁring rate (waiting time) is associated with each transition t, which are random variables Xt ∈ [0, ∞), deﬁned by probability distributions. Therefore, all reaction times can theoretically still occur, but the likelihood depends on the probability distribution. Consequently, the system behaviour is described by the same discrete state space, and all the diﬀerent execution runs of the underlying qualitative

140

M. Heiner et al.

Petri net can still take place. This allows the use of the same powerful analysis techniques for stochastic Petri nets as they are applied for qualitative Petri nets. For a better understanding we describe the general procedure of a particular simulation run for a stochastic Petri net. Each transition gets its own local timer. When a particular transition becomes enabled, meaning that suﬃcient tokens arrive on its preplaces, then the local timer is set to an initial value, which is computed at this time point by means of the corresponding probability distribution. In general, this value will be diﬀerent for each simulation run. The local timer is then decremented at a constant speed, and the transition will ﬁre when the timer reaches zero. If there is more than one enabled transition, a race for the next ﬁring will take place. After the ﬁring of the winning transition, the timers of the others still enabled transitions keep their values or are reset, depending on the speciﬁc type of the net. Technically, various probability distributions can be chosen to determine the random values for the local timers. Biochemical systems are the prototype for exponentially distributed reactions. Thus, for our purposes, the ﬁring rates of all transitions follow an exponential distribution, which can be described by a single parameter λ, and each transition needs only its particular, generally marking-dependent parameter λ to specify its local time behaviour. The following deﬁnition summarises this informal introduction. Definition 1 (Stochastic Petri net, Syntax). A biochemically interpreted stochastic Petri net is a quintuple SPN Bio = (P, T, f, v, m0 ), where – P and T are ﬁnite, nonempty, and disjoint sets. P is the set of places, and T is the set of transitions. – f : ((P × T ) ∪ (T × P )) → IN0 deﬁnes the set of directed arcs, weighted by nonnegative integer values. – v : T → H is a function, which assigns a stochastic hazard function ht to each transition t, whereby |• t| H := t∈T ht | ht : IN0 → IR+ is the set of all stochastic hazard functions, and v(t) = ht for all transitions t ∈ T . – m0 : P → IN0 gives the initial marking. The stochastic hazard function ht deﬁnes the marking-dependent transition rate λt (m) for the transition t, i.e. ht = λt (m). The domain of ht is restricted to the set of preplaces of t, denoted by • t with • t := {p ∈ P |f (p, t) = 0}, to enforce a close relation between network structure and hazard functions. Therefore, λt (m) actually depends on a sub-marking only. Stochastic Petri net, Semantics. Transitions become enabled as usual, i.e. if all preplaces are suﬃciently marked. However there is a time, which has to elapse, before an enabled transition t ∈ T ﬁres. The transition’s ﬁring delay (waiting time) is an exponentially distributed random variable Xt with the probability density function: fXt (τ ) = λt (m) · e(−λt (m)·τ ) ,

τ ≥ 0.

Extended Stochastic Petri Nets

141

The ﬁring itself does not consume time and follows the standard ﬁring rule of qualitative Petri nets. The semantics of a stochastic Petri net (with exponentially distributed ﬁring delays for all transitions) is described by a continuous time Markov chain (CTMC). The CTMC of a stochastic Petri net without parallel transitions is isomorphic to the reachability graph of the underlying qualitative Petri net, while the arcs between the states are now labelled by the transition rates. For more details see [MBC+ 95], [BK02], [HGD08]. Based on this general SPN Bio deﬁnition, specialised biochemically interpreted stochastic Petri nets can be deﬁned by specifying the required kind of stochastic hazard function more precisely. In this paper, we are going to use the molecule semantics with mass action transition rates. Therefore we deploy the stochastic mass-action hazard function, which tailors the general SPN Bio deﬁnition to biochemical mass-action networks, where tokens correspond to molecules: m(p) ht := ct · . f (p, t) • p∈ t

The constant ct is the transition-speciﬁc stochastic rate constant, and m(p) is the current number of tokens on a preplace p of the transition t. The binomial coeﬃcient describes the number of non-ordered combinations of the f (p, t) molecules, required for the reaction, out of the m(p) available ones. In the following we abbreviate this formula by BioMassAction(ct ). See [GHL07] for another example, reading the tokens as concentration levels. 2.2

The Non-markovian Case - Extended Stochastic Petri Nets

We start oﬀ with an overview and brief biochemical motivation before introducing two classes of extended stochastic Petri nets. There are quite a number of various extensions based on the fundamental stochastic Petri net class SPN , see e.g. [MBC+ 95], [Ger01]. The most important additional features concern deterministically timed transitions, or deterministic transitions for short, which come along in diﬀerent types. The crucial point is that the ﬁring delay (waiting time) before an enabled transition ﬁres does not depend anymore on a random variable, but is speciﬁed by a ﬁxed time duration. To avoid confusion, we will call the transitions with a probabilistic ﬁring delay, as introduced in the former subsection, stochastic transitions, if necessary. In summary, our extended stochastic Petri nets support the following features: – – – –

read and inhibitor arcs, programmed transitions (freestyle rate functions), deterministic ﬁring delay, scheduled transitions.

Read and inhibitor arcs. are popular add-ons enhancing modelling comfort. Read arcs (often also called test arcs) allow to specify positive side-conditions, e.g., if the occurrence of a subunit depends on the conformation of a protein

142

M. Heiner et al.

complex, or if a cell’s reaction to a given stimulus depends on the speciﬁc physiological conditions of the cell. Contrary, inhibitor arcs allow to specify negative side-conditions in an abstract way, e.g., if the presence of a given protein or condition inhibits a speciﬁc reaction. Speaking in technical terms, read and inhibitor arcs are directed arcs, going always from places to transitions. The standard ﬁring rule needs to be adapted accordingly. The enabling condition is extended in the following way: if there is an arc a with a weight w = f (p, t) connecting a place p with a transition t, then t can be enabled in a marking m if the following conditions are also satisﬁed: – a is a read arc ∧ m(p) ≥ w, – a is an inhibitor arc ∧ m(p) < w. The token situation on p is not changed by the ﬁring of t, i.e. m (p) = m(p) for t m− → m . Programmed transitions are stochastic transitions with freestyle rate functions. The ﬁring rate can be speciﬁed by arbitrary mathematical functions, stored in lookup tables, if necessary. To give an example, a popular phenomenon in biology is cooperativity. A biochemical reaction may be controlled by an highly non-linear, cooperative mechanism. Simple versions of cooperativity may be represented by complicated Petri net structures, but there are limits. The kinetic mechanisms of a cooperative behaviour are often not completely understood. However, the acquired understanding must be included in the model to get a coherent system model. Deterministic firing delay is the outstanding characteristics of deterministic transitions. The delay is always relative to the time point where the transition gets enabled. There is one popular special case, the zero delay, for which the immediate transitions are introduced. Immediate transitions have always highest priority, which creates a subtle diﬀerence between an immediate transition and a deterministic transition with zero ﬁring delay: if there is a conﬂict between the two, the immediate transition gets priority. We will use the function TimedFiring(delay) to assign the delay constant. Scheduled transitions belong to the deterministic transitions. The deterministic ﬁring occurs according to a schedule specifying absolute points of the simulation time. A schedule can specify just a single time point, or equidistant time points within a given interval, triggering the ﬁring once or periodically. However, transitions only ﬁre at their scheduled time points if they are enabled. Scheduled transitions can dramatically restrict the behaviour, as we will see in Section 4.3, example EX5. Scheduled transitions allow to disturb the core model at well-deﬁned time points as it is done experimentally with the actual biological system under investigation in the wetlab; see Section 5 for an example. We will use two functions to assign the required values: FixedTimedFiring Single(time point), FixedTimedFiring Periodic(begin time point, repetition, end time point).

Extended Stochastic Petri Nets

2.3

143

Generalised Stochastic Petri Nets (GSPN Bio )

Generalised stochastic Petri nets (GSPN Bio ) are stochastic Petri nets SPN Bio extended by inhibitor arcs and immediate transitions. Inhibitor arcs are a powerful modelling feature and are known to bring computational completeness. Consequently, Petri nets of the net class GSPN have the same expressivity as an universal Turing machine [PW03]. However, in terms of construction of the reachability graph (continuous-time Markov chain), they do not establish additional challenges for ﬁnite state spaces, i.e. bounded Petri nets. Immediate transitions are a very special kind of deterministic transitions with zero ﬁring delay, i.e. they ﬁre immediately after getting enabled, and always prior to (general) deterministic and stochastic transitions. Consequently, getting enabled and the ﬁring itself coincide for immediate transitions. A cyclic system behaviour involving only the ﬁring of immediate transitions corresponds to an inﬁnite behaviour without time progress; we get a time deadlock. If a stochastic simulation encounters a situation with more than one immediate transition enabled, one is chosen randomly [Ger01]. However, an analysis approach will consider all possible choices. In terms of the reachability graph (continuous-time Markov chain), induced by a GSPN Bio Petri net, we distinguish between transient and non-transient states. A system never spends time in a transient state before changing into another state. Thus, the time spent (sojourn time) in transient states is always zero, and not exponentially distributed anymore. Consequently, the underlaying semantics is not a continuous-time Markov chain anymore. However, the transient states can be removed such that the reduced reachability graph corresponds again to a continuous-time Markov chain. See [MBC+ 95] for a precise description of the reduction technique and related formal deﬁnitions. In summary this means that GSPN Bio can still be analysed analytically, if the state space, i.e. the continuous-time Markov chain can be constructed. 2.4

Deterministic and Stochastic Petri Nets (DSPN Bio )

Deterministic and Stochastic Petri Nets (DSPN Bio ) are generalised stochastic Petri nets (GSPN Bio ) extended by deterministic transitions. Deterministic transitions possess a deterministic ﬁring delay (waiting time), speciﬁed by a nonnegative real value. When a deterministic transition gets enabled, a count-down timer is started, initialized with the transition’s ﬁring delay. If the transition gets disabled before the timer reaches zero, the timer is switched oﬀ, and the transition will not ﬁre. Otherwise, the transition will ﬁre as soon as the timer reaches zero. The ﬁring itself does not consume time. If we consider stochastic Petri nets without deterministic transitions, the probability of two transitions ﬁring at the same time is practically zero. Contrary,

144

M. Heiner et al.

in stochastic Petri nets with deterministic transitions, it is possible that two transitions want to ﬁre simultaneously. We already discussed the special case of two concurrently enabled immediate transitions. To analyse such a system, all possible choices have to be considered, while in the simulation a random choice takes place. Definition 2 (Deterministic and stochastic Petri net). A biochemically interpreted deterministic and stochastic Petri net is a septuple DSP NBio = (P, T, f, g, v, d, m0 ), where – P und T are ﬁnite, nonempty, and disjoint sets. P is the set of places, and T is the set of transitions. – The set T is the union of three disjunctive transition sets, i.e. T := Tstoch ∪ Tim ∪ Ttimed with: 1. Tstoch , the set of stochastic transitions with exponentially distributed waiting time, 2. Tim , the set of immediate transitions with waiting time zero, and 3. Ttimed , the set of transitions with deterministic waiting time. – f : ((P × T ) ∪ (T × P )) → IN0 deﬁnes the set of directed arcs, weighted by nonnegative integers. – g : (P × T ) → IN0 deﬁnes the set of directed inhibitor arcs, weighted by nonnegative integers. – v : Tstoch → H is a function, which assigns a stochastic hazard function ht to each transition t ∈ Tstoch•, whereby | t| H := t∈Tstoch ht | ht : IN0 → IR+ is the set of all stochastic hazard functions, and v(t) = ht for all transitions t ∈ Tstoch . – d : Ttimed → IR+ assigns to each deterministic transition t ∈ Ttimed a nonnegative deterministic waiting time. – m0 : P → IN0 gives the initial marking. The stochastic transitions correspond to the transitions of the net class SPN Bio , so they have an exponentially distributed waiting time following the deﬁnitions given in Section 2.1. The net class DSPN Bio is a subset of the class eDSPN , introduced in [Ger01]. For details of the subset relation see [Leh07]. Therefore, the theory, which has been developed to analyse eDSPN Petri nets, see [Ger01] and [Haa03], can be deployed to analyse DSPN Bio , too. The remaining two features read arcs and scheduled transitions are not explicitly mentioned in the deﬁnition above, because the just allow a simpliﬁed speciﬁcation using the orthogonal basic concepts in DSPN Bio . Read arcs do not extend the modelling power as long as an interleaving semantics is considered. A read arc and two opposite arcs are indistinguishable in terms of the reachability graph (continuous-time Markov chain). Scheduled transitions can be replaced by net components consisting of immediate and deterministic transitions only; see [Leh07] for construction patterns. Thus, they do not extend the modelling power.

Extended Stochastic Petri Nets

3

145

Stochastic Analysis

The non-Markovian behaviour of DSP NBio precludes the standard analytical approaches belonging to the Markovian world. However, there are adapted stochastic simulation methods, ready to deal with the extended behaviour, see e.g. [Ger01], [Haa03], [Leh07], and many more. A detailed discussion of the necessary adaptions compared to the fundamental Gillespie algorithm [Gil77] is beyond the given space limitations of this paper. Having the simulation traces, we apply simulative model checking of linear-time temporal logic (LTL) in a probabilistic setting (PLTL). Simulative model checking follows the idea of Monte Carlo sampling and handles large or even inﬁnite state spaces through approximating results by analysing only a subset of the state space – a ﬁnite set of ﬁnite outputs (traces) from a stochastic simulation algorithm (SSA), e.g. Gillespie’s exact SSA or any other suitable variations of it. A natural choice of logic to describe properties of sets of traces is linear-time logic. A linear-time logic operates over sets of linear paths through the state space, equivalent to operating on simulation outputs. A given property holds if it holds in all possible paths. Consequently, there are no path quantiﬁers. We apply PLTL, a probabilistic linear-time temporal logic [DG08], [MC208]. This logic extends standard Linear-time Temporal Logic (LTL) [Pnu81] to a stochastic setting with a probability operator and a ﬁlter construct, deﬁning the initial state of the property. LTL is the fragment of full Computational Tree Logic (CTL*) [CGP01] without path quantiﬁers, implicitly quantifying universally over all paths. To be self-contained we brieﬂy recall the PLTL basics. Syntax. PLTL is a logic to create path formulae φ and to ask for their probabilities. The grammar given in Table 1 deﬁnes a PLTL formula ψ. Semantics. The semantics is deﬁned over ﬁnite sets of ﬁnite linear traces of temporal behaviour, in our case by stochastic simulation runs. Each trace is evaluated to a Boolean truth value, and the probability of a property holding true is computed by the fraction of true values in the set over the whole set. It goes without saying, the choice of simulator and simulation parameters used to compute the sequence of states can aﬀect the semantics of the PLTL property and the correctness of the result. Px is any inequality comparison of the probability of the property holding true, for example P≥0.5 . The expression P=? returns the value of the probability of the property holding true. Equality testing of the probability, P=x , is not supported for obvious reasons. PLTL allows the use of ﬁlters over top-level LTL expressions, denoted by {AP }, similar to those used in Probabilistic Computational Tree Logic (PCTL) [HJ94] and Continuous Stochastic Logic (CSL) [ASSB96]. This permits speciﬁcations to refer to the state or states that the property is checked from, rather than default to the initial state. This means that for a query of the form φ {AP }, φ is checked from the ﬁrst state that AP is satisﬁed. This can be a diﬀerent one for each stochastic run. The temporal operators follow the standard LTL semantics:

146

M. Heiner et al.

Table 1. PLTL syntax. Please note that the square and curly brackets are part of PLTL. Px [ φ ] Px [ φ {AP } ] .

ψ

::= |

φ

::= Xφ | G φ | Fφ | φ U φ | φ R φ | ¬ φ | φ∨φ | φ∧φ | φ⇒ φ | AP .

AP

::= ¬ AP | AP ∨ AP | AP ∧ AP | AP ⇒ AP | value comp value | true | f alse .

comp ::= = | = | ≥ | > | < | ≤ . value ::= value op value | variable | max(variable) | d(variable) | Int | Real . op

::= + | − | ∗ | / ,

with ∈ {}, x ∈ [0, 1]. Px can be replaced by Px=? .

– – – –

Next (X) - The property must hold true in the next time point. Globally (G) - The property must hold true always 1 . Finally (F) - The property must hold true sometime in the future. Until (U) - The ﬁrst property must hold true until the second property holds true. – Release (R) - The second property can only ever not hold true if the ﬁrst property becomes true.

The meta term variable stands for any variable in the model, Int is any integer number and Real is any real number. In our case of stochastic Petri net analysis, a variable is going to be a place name, and the formulae refer to the number of tokens on a place in a given state. Additionally, there is a predeﬁned variable time, referring to the simulation time points. Thus we can, for example, express properties which occur after some simulation time has elapsed. The function max operates over all the token values of a place to return the maximum in the given simulation runs, thus the peak of a species’ concentration, modelled by a place, can be checked, e.g. P rotein = max(P rotein). The function d operates on each place in each state individually to return the derivative, thus increasing token numbers can be checked, e.g. d(P rotein) > 0. This approach to simulative model checking incorporates two approximations. The truth value of a single trace is approximated by operating over a ﬁnite sequence of states only; and the probability of the property is approximated through sampling a ﬁnite number of traces only. Thus, a subset of the model’s behaviour is considered only. However, there are two special categories 1

To be precise, in the given setting of model checking by ﬁnite traces, globally means ’always –as far as known’.

Extended Stochastic Petri Nets

147

of properties, where deﬁnitive, i.e. non-approximating answers are possible by simulative model checking. – Monotone properties comply with the following condition: if the property is satisﬁed in any path through the state space, then it is satisﬁed in any extension of the path [HLMP04]. Formulae without the Globally operator are monotone properties. The Globally operator and semantically equivalent descriptions by the other operators are incompatible with the monotony property. Considering longer paths can only increase the probability. – Invariant Properties have to hold true in every state in every path. Thus they comply with the following condition: if the property is satisﬁed in any path through the state space, then it is satisﬁed in any other path. Their probability is independent of the number of considered paths. They are often used as consistency checks, and so do we in this paper. PLTL may be considered as a linear-time counterpart to CSL. It can easily be used to formalise the visual evaluation of diagrams as generated by deterministic/stochastic simulation runs or by recording experimental time series. In the following chapter we are going to use PLTL to analyse sets of stochastic simulation traces of extended stochastic Petri nets, which have been constructed to illustrate the expressiveness of DSP NBio .

4

Typical Components

We present some typical model components, controlling a network’s inﬂow and outﬂow, and thus demonstrating the suitability of the introduced Petri net class DSP NBio for the envisaged application scenarios of model-based design of wetlab experiments. We use the following abbreviations introduced in Section 2: – – – –

BioMassAction(ct ), TimedFiring(delay), FixedTimedFiring Single(time point), FixedTimedFiring Periodic(begin time point, repetition, end time point);

and we apply the following drawing conventions: – – – – –

read arcs: identiﬁed by a black dot, inhibitor arcs: identiﬁed by a hollow dot, stochastic transition: hollow square, deterministically timed transition: black square, immediate transition: black rectangle.

We are going to examine the behaviour of each component by simulative PLTL model checking over 100 (1,000) simulation runs. The individual runs are independent, so generally diﬀerent. We conﬁne ourselves deliberately on introductory formulae to illustrate the key ideas, increasing at the same time our conﬁdence in the accuracy of our simulation algorithm for the non-Markovian setting.

148

4.1

M. Heiner et al.

Time-Controlled Inflow/Outflow

EX1. In our ﬁrst example we consider a closed system, consisting of one reversible reaction A ↔ B, modelled by the two transitions t1 (BioMassAction(0.11)) and t2 (BioMassAction(0.1)). The two deterministically timed transitions input (FixedTimedFiring Periodic(11,1,20)) and output (FixedTimedFiring Periodic(31,1,40)) are responsible for the absolutely timed inﬂow and outﬂow of tokens, see Figure 1. The transition input does not have preplaces, thus it ﬁres for sure at the time points 11, 12, . . . , 20, producing each time 1,000 additional tokens on place A. Contrary, the transition output removes 1,000 tokens from place B at the time points 31, 32, . . . , 40, provided there are enough tokens to enable the ﬁring. Figure 2 shows the ﬁrst 100 time units of a single simulation run. We give some introductory samples of temporal-logic formulae (queries), formalising the visual inspection of the simulation output as it might be done by the expert evaluating former or designing the next wetlab experiments. We apply these queries to a set of 100 stochastic (single) simulation traces. The ratio

t1 1000 input

1000

500 A

B

output

t2 Fig. 1. First example of time-controlled inﬂow/outﬂow (EX1)

8000 A B

7000

Marking

6000 5000 4000 3000 2000 1000 0 0

20

40

60

80

100

Time

Fig. 2. Simulation result of the network given in Figure 1 (single run) (EX1)

Extended Stochastic Petri Nets

149

of traces where the formula holds to the total number gives us a rough estimate of a formula’s probability. We check over exact Gillespie traces, i.e. all single events are logged. There are generally no ”even” time points (like 30.000000 for 30). However, the ﬁring of scheduled transitions at absolute time points (e.g. 20 in this example) causes exact time points in the simulation traces. We have to keep this in mind when refering to absolute time points in the following queries. Please remember, all place names are read as integer variables in the following formulae; and the predeﬁned variable time relates to the simulation time. The probabilities as computed by simulative model checking are given in brackets. – Maxima (probabilities: 1.0, 0.95). P=? [ G(A < 7550) ] P=? [ G(B < 5350) ] – Peaks (probabilities: 0.9, 1.0). P=? [ F(time = 20 ∧ A > 0.9·max(A) ∧ (3000 < B ∧ B < 3500)) ] P=? [ F((29 < time∧time < 30)∧(5000 < A∧A < 5400)∧B > 0.9·max(B)) ] – Steady state, relative statements (probabilities: 0.03, 0.59, 0.8, 0.91). P=? [ time ≥ 50 ⇒ G(A < B) ] P=? [ time ≥ 55 ⇒ G(A < B) ] P=? [ time ≥ 60 ⇒ G(A < B) ] P=? [ time ≥ 70 ⇒ G(A < B) ] – Steady state, absolute statements (probabilities: 0.39, 1.0). P=? [ time ≥ 50 ⇒ G((1500 < A ∧ A < 1800) ∧ (1600 < B ∧ B < 2000)) ] P=? [ time ≥ 60 ⇒ G((1500 < A ∧ A < 1800) ∧ (1600 < B ∧ B < 2000)) ] EX2. We vary the pattern of our ﬁrst example to remove repeatedly all currently available tokens on place B at equidistant time points, see Figure 3. The immediate transition output consumes all tokens on place B, while there is a token on place output on. The token on place output on is controlled by the deterministically timed transition switch output on (FixedTimedFiring Periodic(20,20, SimEnd)) and the immediate transition switch output oﬀ. The transition switch output on initiates every 20 time units the cleaning process. The immediate transition switch output oﬀ switches oﬀ the outﬂow as soon as the place B is clean; otherwise each token arriving on B would be instantly removed and no token accumulation would be possible anymore. A single simulation run is given in Figure 4. We analyse a set of 100 of such stochastic traces by the following temporal-logic queries (all yield probability 1.0). – If the output is switched on, B is cleaned immediately. P=? [ G(output on = 1 ⇒ B = 0) ] – Cleaning of B at time point 20. P=? [ F(time = 20 ∧ B = 0) ]

150

M. Heiner et al.

switch_output_off

t1 1000 input

output_on 500 A

B

switch_output_on

t2

output

Fig. 3. Second example of time-controlled inﬂow/outﬂow (EX2)

8000 A B

7000

Marking

6000 5000 4000 3000 2000 1000 0 0

20

40

60

80

100

Time

Fig. 4. Simulation result of the network given in Figure 3 (single run) (EX2)

– Cleaning of B at time point 20, ensuring that B does not get cleaned earlier. P=? [ F(B > 0 ∧ (B > 0 U (time = 20 ∧ B = 0))) ] – Cleaning of B at time point 40, ensuring that B remains marked inbetween as soon as it got a token. P=? [ F(time = 20 ∧ B = 0) ∧ F(B > 0 ∧ (B > 0 U (time = 40 ∧ B = 0))) ] 4.2

Token-Controlled Inflow

We discuss two examples and start again with a reversible reaction A ↔ B, modelled by the two stochastic transitions t1 (BioMassAction(0.1)) and t2 (BioMassAction(0.005)), which we consider as a closed system, challenged by experimental interventions. EX3. In our ﬁrst example of token-controlled inﬂow, the tokens on place A are raised by 50 as soon as the token amount drops below the threshold 30, see Figure 5. This behaviour is implemented by the immediate transition input, the ﬁring of which is prevented by an inhibitor arc testing A. The weight 30

Extended Stochastic Petri Nets

t1

50 input

30

151

100 A

B t2

Fig. 5. First example of token-controlled inﬂow (EX3)

500 A B

Marking

400 300 200 100 0 0

20

40

60

80

100

Time

Fig. 6. Simulation result of the network given in Figure 5 (single run) (EX3)

of the inhibitor arc prevents the ﬁring of input until the token amount drops below 30. 50 tokens are added to place A as soon as the inhibition condition becomes invalid, preventing again further inﬂow until the next drop occurs. Figure 6 shows a single simulation run. We analyse a set of 1,000 runs against the following formulae. – The tokens on A never fall below the threshold 30 (probability: 1.0). P=? [ ¬ F(A < 30) ] – The transition input tries to keep the tokens on A between 30 and 80. But there are always some tokens on place B, which may return to A (probabilities: 0.946, 0.996, 0.999). P=? [ F(A = 30 ∧ G(30 ≤ A ∧ A ≤ 80)) ] P=? [ F(A = 30 ∧ G(30 ≤ A ∧ A ≤ 82)) ] P=? [ F(A = 30 ∧ G(30 ≤ A ∧ A ≤ 84)) ] – There is a constant inﬂow due to the transition input, and the rate of t1 is (signiﬁcantly) higher than of t2. Therefore, B increases permanently and

152

M. Heiner et al.

t1 input

5

switch_on

A 20

10

output

B

10

input_on

t2

30

input_off switch_off

Fig. 7. Second example of token-controlled inﬂow (EX4)

40 A input_on

35

Marking

30 25 20 15 10 5 0 0

20

40

60

80

100

Time

Fig. 8. Simulation results of the network given in Figure 7 (single run). The input is switched on/oﬀ (place input on) in dependence on the token situation on A (EX4).

without limits. This is true in the averaged case only, e.g. 100 runs. d(B) speciﬁes the derivative. P=? [ G(d(B) ≥ 0) ] EX4. Our second example of token-controlled inﬂow is given in Figure 7. The transitions t1 (BioMassAction(0.2)) and t2 (BioMassAction(0.1)) form again the reversible reaction A ↔ B. We add the deterministically timed transition output (FixedTimedFiring Periodic(5,5, SimEnd)) to get a signiﬁcant consumption of tokens. Each time output gets activated, it removes 10 tokens from B. If the token amount on place A drops below 10, the deterministically timed transition input (TimedFiring(0.5)) starts working and adds by each ﬁring 5

Extended Stochastic Petri Nets

153

30 A B

25

Marking

20 15 10 5 0 0

20

40

60

80

100

Time

Fig. 9. Simulation results of the network given in Figure 7 (100 runs). A and B oscillate due to the repeated switching between inﬂow on/oﬀ (EX4).

tokens with 0.5 units waiting time inbetween, until there are at least 30 tokens on A. This behaviour is controlled by the immediate transitions switch on and switch oﬀ, and the two places input on and input oﬀ, forming a 1-P-invariant1. Switch on can only ﬁre if there are less than 10 tokens on A, and switch oﬀ can only ﬁre, if there are at least 30 tokens on A. We give two related diagrams. The single run in Figure 8 shows how the input is switched on/oﬀ (place input on) in dependence on the token situation on A. Figure 9 gives the average of 100 runs. It highlights the oscillation of A and B, caused by the repeated switching between inﬂow on/oﬀ. We analyse the token-controlled inﬂow component by the following formulae (1,000 runs) (the ﬁrst three yield probability 1.0). – The two places input on and input oﬀ form a 1-P-invariant. P=? [ G((input on = 1 ∧ input of f = 0) ∨ (input on = 0 ∧ input of f = 1)) ] – The transition input is switched on/oﬀ if the token amount on A crosses the threshold 10 or 30, respectively. P=? [ G(A < 10 ⇒ input on = 1) ] P=? [ G(A ≥ 30 ⇒ input of f = 1) ] – There is a delay of 0.5 time units between the on/oﬀ switch and the reaction of the actual inﬂow transition. E.g., after having switched oﬀ the input, 5 additional tokens will arrive by the already triggered ﬁring of the transition input. Thus, even a weaker range than speciﬁed by the threshold values does not get probability 1 (probability: 0.995). P=? [ G(5 ≤ A ∧ A ≤ 40) ] 1

Exactly one of both places carries a token at any point of time.

154

4.3

M. Heiner et al.

Switch between Deterministic and Stochastic Transitions

The following two networks demonstrate how to switch between deterministic and stochastic transitions. We start oﬀ with a time-controlled switch, before discussing a token-controlled switch. In both cases we consider a non-reversible reaction A → B, which is nevertheless modelled by two transitions: the stochastic transition t stoch (BioMassAction(0.1)) and the deterministically timed transition t det (TimedFiring(0.25)). The chosen net structure ensures that always one of these two transitions only is able to transfer tokens from place A to place B; with other words: the token ﬂow occurs either stochastically or deterministically. The mutually exclusive ﬁring is implemented by the two places stochastic on and det on, forming a 1-P-invariant and establishing side-conditions for t stoch or t det, respectively. EX5. The actual time-controlled switch is performed by two deterministically timed transitions: switch to det (FixedTimedFiring Single(10)) and switch to stoch (FixedTimedFiring Single(30)), which ﬁre (each once!) at the absolute time points 10 or 30, respectively, causing a switch in the other operation mode, see Figure 10. In summary, the modelled reaction A → B behaves deterministically between the time points 10 and 30, and stochastically else.

t_stoch

stochastic_on switch_to_stoch

switch_to_det 1000 A det_on

B t_det

Fig. 10. Example of time-controlled switch between deterministic and stochastic behaviour. The semantic functions assigned to the transitions switch to det and switch to stoch allow them to ﬁre only once (EX5).

EX6. We keep the basic principle for the token-controlled switch, but replace the transitions switching between the operation modi by immediate transitions, which depend on the token situation in place A. The immediate transitions switch to det and switch to stoch ﬁre each once as soon as the token amount on place A drops below 700 or 500, see Figure 12. In summary, the modelled reaction A → B behaves deterministically for token amount between 500 and 700, and stochastically else. The diagrams in Figure 11 and 13 show the behaviour of the two patterns for a single run each. For both we conﬁrm the mutually exclusive operation mode of the stochastic and deterministic behaviour by the following query.

Extended Stochastic Petri Nets

155

A B

1200

Marking

1000 800 600 400 200 0 0

20

40

60

80

100

Time

Fig. 11. Simulation result of the network given in Figure 10 (single run). There is a deterministic token ﬂow from A to B between time points 10 and 30, and stochastic ﬂow else (EX5).

stochastic_on t_stoch 700

switch_to_stoch

A 1000

B

switch_to_det t_det 500

det_on

Fig. 12. Example of token-controlled switch between deterministic and stochastic behaviour. The additional preplaces of the immediate transitions bring the equivalence to the net component in Figure 10; i.e. the immediate transitions ﬁre only once (EX6).

– The two places stochastic on and det on form a 1-P-invariant. P=? [ G((stochastic on = 1 ∧ det on = 0) ∨ (stochastic on = 0 ∧ det on = 1)) ] We conclude the analyses with checking the range of deterministic versus stochastic behaviour for the two discussed patterns. – Deterministic token ﬂow from A to B between time points 10 and 30. P=? [ (10 ≤ time ∧ time < 30) ⇒ det on = 1 ]

156

M. Heiner et al.

A B

1200

Marking

1000 800 600 400 200 0 0

20

40

60

80

100

Time

Fig. 13. Simulation result of the network given in Figure 12 (single run). There is a deterministic token ﬂow from A to B for a token amount on A between 500 and 700, and stochastic ﬂow else (EX6).

– Stochastic token ﬂow from A to B from 0 up to time point 10, and starting at time 30 again. P=? [ (time < 10 ∨ 30 ≤ time) ⇒ stochastic on = 1 ] – Deterministic token ﬂow from A to B for a token amount on A between 500 and 700. P=? [ (500 ≤ A ∧ A < 700) ⇒ det on = 1 ] – Stochastic token ﬂow from A to B for a token amount on A less than 500 or greater or equal 700. P=? [ (A < 500 ∨ 700 ≤ A) ⇒ stochastic on = 1 ] All these properties are invariant properties, i.e. they yield probability 1.0, independently of the number and the length of considered simulation traces.

5

Lac Operon Model

We conclude by looking brieﬂy at a classical example of prokaryotic gene regulation, the lac operon case. We follow the simpliﬁed version discussed in [Wil06] and speciﬁed there by a set of reaction equations and in an SBML-shorthand notation. We keep all naming conventions and the initial conditions, and translate the textual representation into a (qualitative) Petri net, reﬂecting explicitly the inherent structure of the regulatory network, compare Figure 14. Finally, we assign the rate equations as speciﬁed in the SBML code, and we get a stochastic Petri net.

Extended Stochastic Petri Nets InhibitorRnaDegradation

157

InhibitorDegradation IOp

RnapBinding/Dissociation InhibitorTranscription InhibitorTranslation I 50 100 Op Rnap Irna Idna InhibitorBinding/Dissociation RnapOp

LactoseInhibitorBinding/Dissociation ILactose

Transcription 20

LactoseInhibitorDegradation

10000

Intervention

Rna RnaDegradation

Lactose

Translation Conversion Z

ZDegradation

Fig. 14. Lac operon model according to [Wil06]. Macro transitions (drawn as two centric squares) indicate reversible reactions.

The core model of the network under consideration is extended by a special transition – an event in SBML terminology – modelling a timed intervention in a wetlab experiment. The transition Intervention (FixedTimedFiring Periodic(50000,50000, SimEnd) 2 ) introduces 10,000 molecules of Lactose every 50,000 time units, compare Figure 15. To increase our conﬁdence in the model we start with a preliminary structural analysis and compute the P-invariants and T-invariants3 . There are input transitions, so the net can not be covered by P-invariants. However, there are three P-invariants, inducing mass-conserving subnetworks (modules) and enjoying obvious biological meaning. The preserved species is given ﬁrst in the following short-hand notation: – pi1 = {Idna}, – pi2 = {Rnap, RnapOp}, – pi3 = {Op, IOp, RnapOp}. Contrary, T-invariants do cover the net, which is a common consistency criteria for well-formed net structures, allowing e.g. a steady state behaviour. Each Tinvariant induces a self-contained, state-repeating subnetwork (module). Besides the expected three trivial T-invariants for the three reversible reactions: 2 3

Here we diﬀer from the model given in [Wil06], where the modelled intervention occurs only once at a speciﬁed point of time. For all notions used in this section, but note introduced in this paper, see [HGD08].

158

M. Heiner et al.

Lactose

12000

Marking

10000 8000 6000 4000 2000 0 0

50000

100000

150000

200000

250000

300000

Time

Fig. 15. Simulation result of the lac operon model: Lactose

– ti1 = {LactoseInhibitorBinding, LactoseInhibitorDissociation}, – ti2 = {InhibitorBinding, InhibitorDissociation}, – ti3 = {RnapBinding, RnapDissociation}, we get the following six non-trivial T-invariants, each input/output behaviour is made of: – ti4 = {InhibitorT ranscription, InhibitorRnaDegradation}, – ti5 = {InhibitorT ranslation, InhibitorDegradation}, – ti6 = {InhibitorT ranslation, LactoseInhibitorBinding, LactoseInhibitorDegradation}, – ti7 = {Intervention, Conversion}, – ti8 = {RnapBinding, T ranscription, RnaDegradation}, – ti9 = {T ranslation, ZDegradation}. There are four transitions (underlined), which are not involved in non-trivial T-invariants. However, they are crucial for the regulation mechanism between Z and Lactose. Please note, each T-invariant is given in a short-hand notation, enumerating the T-invariants’ transitions in an order, which they may follow to reproduce a state, or what has to happen to get the system back in the steady state after some disturbences. Remarkably, the net fulﬁlls the Deadlock Trap Property (DTP), however is beyond the structural net class extended simple. In summary this allows the conclusion that there is no reachable dead state, in which any further system activities would be prevented. Actually, we expect the model to be live, which can not be proven with the analysis techniques available for (qualitatively) unbounded Petri nets.

Extended Stochastic Petri Nets

300

159

Z

250

Marking

200 150 100 50 0 0

50000

100000

150000

200000

250000

300000

Time

Fig. 16. Simulation result of the lac operon model: Z

However, there are property-preserving reduction rules downsizing the net structure, which are supported by the Integrated Net Analyser INA [SR99]. Applying these structural reduction rules, we get a smaller network, consisting of 2 places and 4 transitions. Liveness becomes obvious for this reduced network; see the supplementary material for more details. The place Z models the enzyme β-Galactosidase; its reaction to the repeatedly sudden increase of Lactose molecules is shown in Figure 16. We analyse for the ﬁrst intervention how a peak of Lactose triggers a peak of Z. – The intervention causes Lactose to peak at time point 50,000 (probabilities 1.0, 1.0, 0.65). P=? [ (49, 999 ≤ time ∧ time < 50, 000) ⇒ Lactose ≤ 0.01·max(Lactose) ] P=? [ time = 50, 000 ⇒ Lactose ≥ 0.99·max(Lactose) ] P=? [ (52, 000 ≤ time ∧ time < 52, 001) ⇒ Lactose ≤ 0.1·max(Lactose) ] – Z is highly likely to be at low concentration at time point 50,000 (probability 0.9). P=? [ time = 50, 000 ⇒ Z ≤ 0.1·max(Z) ] – Z will rise to at least 80% of its maximal value within 2,000 time units (probability 0.925). P=? [ F ( (50, 000 < time ∧ time < 52, 000) ∧ Z ≥ 0.6·max(Z) ) ] – In summary, a peak of Lactose triggers a peak of Z within 2,000 time units (probability 0.925). P=? [ time = 50, 000 ∧ Lactose ≥ 0.99·max(Lactose) ∧ Z ≤ 0.1·max(Z) ⇒ F (Z ≥ 0.8·max(Z) ∧ time < 52, 000) ]

160

6

M. Heiner et al.

Tools

The Petri net components and the lac operon model have been constructed using Snoopy [Sno08], [HRS08], a tool to design and animate or simulate hierarchical graphs, among them qualitative and continuous Petri nets, and the extended stochastic Petri nets as used in this paper. Snoopy provides export to various analysis tools as well as import and export of the Systems Biology Markup Language (SBML) [HFS+ 03]. The qualitative analyses of the lac operon model have been made with the Petri net analysis tool Charlie [Cha08], complemented by the structural reduction rules supported by the Integrated Net Analyser INA [SR99]; see the corresponding log ﬁles in the supplementary material. The quantitative analyses have been done by the cooperation of two tools: Snoopy’s build-in simulation algorithm for extended stochastic Petri nets to generate the sets of simulation traces, and MC2 [MC208], a model checker by Monte Carlo sampling, for the simulative PLTL model checking. MC2 reads sets of simulation traces as, e.g., generated by Snoopy and expects additionally a ﬁle with the temporal-logical formulae. As a proof of concept, we conﬁned ourselves to rather small sets of 100 (1,000) runs only, allowing at the same time an aﬀordable repetition of all computational experiments by the reader. A general recommendation is to start with smaller sets of simulation runs, just to check whether one got a formula right, before analysing larger sets, which could actually be done in parallel. None of the computational experiments for the typical components required more than 6 minutes per net example on a standard machine. Simulative model checking of the lac operon model is slightly more expensive. The traces have been generated on a workstation (2.83 GHz, 64 bit). The 100 exact traces (simulation time interval: 300,000) require about 5 GB. The model checking itself consumes less than 30 minutes on a standard machine. Snoopy, Charlie as well as the data and analysis ﬁles of the discussed Petri net examples are available at www-dssz.informatik.tu-cottbus.de/examples/xspn-components.

7

Summary

This paper extends the Markovian stochastic Petri nets SPN Bio as introduced in [GHL07] to model and analyse biochemical networks. The extensions lead to the deﬁnition of Generalised Stochastic Petri nets GSPN Bio and deterministic and stochastic Petri nets DSPN Bio . They include read and inhibitor arcs as well as several time-dependent transition types, which in summary preclude standard Markovian analysis approaches. Therefore we applied simulative model checking, approximating the probability of a given temporal logic formula by considering ﬁnite sets of ﬁnite paths through the state space. These paths are generated by

Extended Stochastic Petri Nets

161

stochastic simulation algorithms, adjusted to deal with the extended modelling features. We discussed some typical net components demonstrating the usability of DSPN Bio for the envisaged application scenario of model-based experiment design and evaluation. These components have been analysed by checking sets of stochastic simulation traces against PLTL properties. Invariant properties have been used to prove at the same time the plausibility of the applied simulation algorithm. We concluded with brieﬂy looking at the lac operon case study, one of the classical examples of prokaryotic gene regulation. Currently we consider some further extensions of our modelling formalism; among them are variable deterministic ﬁring delays speciﬁed by an interval or an arbitrary marking-dependent function, reset and equal arcs as well as markingdependent arc weights. Simulative model checking is an extremely powerful tool. By way of introduction we have deliberately deployed some basic features of PLTL only. There is an interesting extension, PLTLc [DG08], supporting free variables, and thus allowing richer and more elegant properties, which however are also more complicated to write and to interpret. Thus, demonstrating these advanced features to more sophisticated users is beyond the scope and space limits of this paper. Acknowledgements. The stochastic features of the Snoopy tool have been developed by Sebastian Lehrack, which cumulated in his Master thesis [Leh07]. This work has been ﬁnancially supported by MPI Martinsried and MPI Madgeburg. Snoopy’s quality improvements by Christian Rohr are crucial for the computational experiments presented in this paper. We would like to thank Robin Donaldson for his responsive assistance in MC2 issues.

References [ASSB96]

Aziz, A., Sanwal, K., Singhal, V., Brayton, R.K.: Verifying Continuoustime Markov Chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996) [BGHO08] Breitling, R., Gilbert, D., Heiner, M., Orton, R.: A structured approach for the engineering of biochemical network models, illustrated for signalling pathways. Brieﬁngs in Bioinformatics 9(5), 404–421 (2008) [BK02] Bause, F., Kritzinger, P.S.: Stochastic Petri Nets. Vieweg (2002) [CGP01] Clarke, E.M., Grumberg, O., Peled, D.A.: Model checking. MIT Press, Cambridge (2001) (third printing) [Cha08] Charlie Website. A Tool for the Analysis of Place/Transition Nets. BTU Cottbus (2008), http://www-dssz.informatik.tu-cottbus.de/software/ charlie/charlie.html [DG08] Donaldson, R., Gilbert, D.: A model checking approach to the parameter estimation of biochemical pathways. In: Heiner, M., Uhrmacher, A.M. (eds.) CMSB 2008. LNCS (LNBI), vol. 5307, pp. 269–287. Springer, Heidelberg (2008)

162

M. Heiner et al.

[GBHD09] Gilbert, D., Breitling, R., Heiner, M., Donaldson, R.: An introduction to biomodel engineering, illustrated for signal transduction pathways. In: Corne, D.W., Frisco, P., Paun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2008. LNCS, vol. 5391, pp. 13–28. Springer, Heidelberg (2009) [Ger01] German, R.: Performance analysis of communication systems with nonMarkovian stochastic Petri nets. John Wiley and Sons Ltd., Chichester (2001) [GHL07] Gilbert, D., Heiner, M., Lehrack, S.: A unifying framework for modelling and analysing biochemical pathways using Petri nets. In: Calder, M., Gilmore, S. (eds.) CMSB 2007. LNCS (LNBI), vol. 4695, pp. 200–216. Springer, Heidelberg (2007) [GHR+ 08] Gilbert, D., Heiner, M., Rosser, S., Fulton, R., Gu, X., Trybilo, M.: A Case Study in Model-driven Synthetic Biology. In: Proc. 2nd IFIP Conference on Biologically Inspired Collaborative Computing (BICC), IFIP WCC 2008, Milano, pp. 163–175 (2008) [Gil77] Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 81(25), 2340–2361 (1977) [Haa03] Haas, P.J.: Stochastic Petri nets: Modelling, Stability, Simulation. Springer, Heidelberg (2003) [HDG10] Heiner, M., Donaldson, R., Gilbert, D.: Petri Nets for Systems Biology. In: Iyengar, M.S. (ed.) Symbolic Systems Biology: Theory and Methods. Jones and Bartlett Publishers, Inc., USA (in Press, 2010) [HFS+ 03] Hucka, M., Finney, A., Sauro, H.M., Bolouri, H., Doyle, J.C., Kitano, H., et al.: The Systems Biology Markup Language (SBML): A Medium for Representation and Exchange of Biochemical Network Models. J. Bioinformatics 19, 524–531 (2003) [HGD08] Heiner, M., Gilbert, D., Donaldson, R.: Petri nets in systems and synthetic biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 215–264. Springer, Heidelberg (2008) [HJ94] Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994) [HLMP04] H´erault, T., Lassaigne, R., Magniette, F., Peyronnet, S.: Approximate probabilistic model checking. In: Steﬀen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 307–329. Springer, Heidelberg (2004) [HRS08] Heiner, M., Richter, R., Schwarick, M.: Snoopy - a tool to design and animate/simulate graph-based formalisms. In: Proc. PNTAP 2008, associated to SIMUTools 2008. ACM digital library (2008) [Leh07] Lehrack, S.: A tool to model and simulate stochastic Petri nets in the setting of biochemical networks (in German). Master thesis, BTU Cottbus, Dep. of CS (2007) [MBC+ 95] Ajmone Marsan, M., Balbo, G., Conte, G., Donatelli, S., Franceschinis, G.: Modelling with Generalized Stochastic Petri Nets, 2nd edn. Wiley Series in Parallel Computing. John Wiley and Sons, Chichester (1995) [MC208] MC2 Website. MC2 - PLTL model checker. University of Glasgow (2008), http://www.brc.dcs.gla.ac.uk/software/mc2/ [Mur89] Murata, T.: Petri Nets: Properties, Analysis and Applications. Proc.of the IEEE 77(4), 541–580 (1989) [Pnu81] Pnueli, A.: The temporal semantics of concurrent programs. Theor. Comput. Sci. 13, 45–60 (1981)

Extended Stochastic Petri Nets [PW03] [Rei82] [Sno08]

[SR99]

[Wil06]

163

Priese, L., Wimmel, H.: Theoretical Informatics - Petri Nets (in German). Springer, Heidelberg (2003) Reisig, W.: Petri nets; An introduction. Springer, Heidelberg (1982) Snoopy Website. A Tool to Design and Animate/Simulate Graphs. BTU Cottbus (2008), http://www-dssz.informatik.tu-cottbus.de/software/snoopy.html Starke, P.H., Roch, S.: INA - The Intergrated Net Analyzer. Humboldt University Berlin (1999), http://www.informatik.hu-berlin.de/~ starke/ina.html Wilkinson, D.J.: Stochastic Modelling for System Biology, 1st edn. CRC Press, New York (2006)

A Projective Brane Calculus with Activate, Bud and Mate as Primitive Actions Maria Pamela C. David1, , Johnrob Y. Bantang1,2,3,∗ , and Eduardo R. Mendoza1,4 1

Faculty of Physics and Center for Nanoscience, Ludwig-Maximilians-Universit¨ at M¨ unchen, Geschwister-Scholl-Platz 1, D-80539 M¨ unchen, Germany 2 Max-Planck-Institut f¨ ur Dynamik komplexer technischer Systeme, Sandtorstraße 1, D-39106 Magdeburg, Germany 3 National Institute of Physics, College of Science, University of the Philippines, Diliman, Quezon City 1101 Philippines 4 Department of Computer Science, University of the Philippines, Diliman, Quezon City 1101 Philippines

Abstract. We modify and extend Cardelli’s Brane Calculus and Danos and Pradalier’s Projective Brane Calculus (PBC) to improve consistency with biological characteristics of membrane reactions. We propose a Projective Activate-Bud-Mate (PABM) calculus as an alternative to the Phago-Exo-Pino (PEP) basic calculus of L. Cardelli. PABM uses a generalized formalism for Action activation with receptor-ligand type channel construction that incorporates multiple association and aﬃnity similar to Priami’s beta binders. Calculus elements are ﬁnite. Volumes are associated with systems for more realistic compartment-based reaction probabilities. PABM also uses Brane domains that partition membranes into controllable, independent groupings of projective actions. Domains eliminate the need for parameters in Phago and Bud and allow lateral and cross-membrane interactions. We show that PABM can emulate bitonal membrane reactions. PABM also realizes the idea of L. Cardelli (Cardeli, 2004) on modeling molecules as systems.

1

Introduction

Cellular organization plays a key role in biological systems through the physical regulation of reactions. Enzymes, for instance, are typically sequestered in membrane-bound systems to which access is only made possible through cascades of equally regulated and timed signals. Most current formalisms for modeling, however, do not possess an explicit functionality for modeling compartmentalization. In deterministic models, compartmentalization is modeled with the use of additional variables that diﬀerentiate a species S that is within some compartment X from S that is within another compartment Y . While this has been used with some success, S in X is actually not diﬀerent from S in Y , unless it has already reacted with other species in either compartment. It has only been in

The ﬁrst two authors contributed equally to this work.

C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 164–186, 2009. c Springer-Verlag Berlin Heidelberg 2009

A Projective Brane Calculus with Activate, Bud and Mate

165

recent years that several calculi were developed so that membrane compartmentalization: (a) becomes an inherent part of computations and (b) is emphasized in simulating reactions[1, 2, 3, 4]. Brane calculus is a formalism that can be used to describe systems as mem-brane-bound compartments that may contain other systems[3]. These compartments can merge, split or be hierarchically reorganized through uptake (phagocytosis) or extrusion (exocytosis) mechanisms, based on the capabilities, known as actions, of the membranes that enclose them[3, 5]. An important aspect of these actions — directly adapted from pi calculus — is that they are triggered via highly speciﬁc channel-based communication. Nevertheless, the mapping between channels is not necessarily one-to-one, with some channels having more than one communication partner. Although the original concept of channels in pi calculus was for mobile telecommunication systems, it is compatible with the representation of biological interactions, from enzyme-substrate systems that interact to form a chemically distinct product to receptor-mediated intermembrane communication that leads to membrane reorganization. Another formalism that includes compartments is Priami’s beta binders[4]. Here, much emphasis is given to the promiscuousity of the channels (“beta binders”) through which the compartments interact, as well as its eﬀects on the dynamic evolution of the compartment contents, interactions, and interfaces. As in Brane calculus, compartments can merge and split as a result of binder-based communication. While inherent in Brane Calculus, beta binders needed an extension to include hierarchical construction of compartments. Recent extensions, however, only permit intuitive representation of static hierarchical structures, but still forbid the explicit nesting of compartments[6]. The main advantage of beta binders over Brane calculus is its natural representation of aﬃnity to channel pairings, a concept that is adapted in the proposed extension in this paper. The uniqueness of Cardelli’s Brane calculus lies in the representation of all computations on membranes. This is important, particularly since it is actually the dynamic property of membranes that determines its capability to interact with other membranes and its general environment in vivo. Consequently, this property also determines how a membrane-bound system would evolve. Structure hierarchy can likewise be easily represented in Brane calculus, where nested systems are eﬀectively organized in tree structures[3, 5]. Brane calculus has been previously extended by Danos and Pradalier to incorporate the idea that the inner and outer surfaces of a membrane are not identical[5]. In vivo, it could be frequently observed that the membrane protein domains exposed to the extracellular matrix are diﬀerent from the domains exposed to the cytosol. It is even possible for membrane proteins to possess either an extracellular domain or a cytosolic domain. As a result, the deﬁnition of the inner and outer membranes are diﬀerent. Additional physical restrictions are introduced on which reactions could take place, in particular only directed actions on membrane surfaces that could “see” each other are allowed to interact. This extension using directed actions is known as projective Brane calculus (PBC).

166

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Nevertheless, there are a number of aspects in both calculi that involve concepts not observed in biological system. The purpose of this paper is to introduce further modiﬁcations and extensions combining the strength of both Brane calculi, with the aim of making it even more consistent with the biological characteristics of membrane reactions. Speciﬁcally, we introduce the following changes that result to the proposed extension, the Projective Activate-Bud-Mate calculus (PABM): 1. Use of the minimal set Smin ≡ {bud, mate, !, 0} instead of the set S ≡ {phago, exo, pino, mate, bud, drip} for the possible actions a ∈ Smin (see §2). All other actions in S are realized using only the actions in Smin together with directed Actions of PBC. 2. Abstraction of speciﬁc send-receive channel pairing into less speciﬁc channel name equality, eliminating the distinction between input and output channels. Together with the previous revision, it allows generalized representations in the form ax for Actions and Coactions, where x is a named channel. 3. Introduction of Brane domains, which are autonomous groups of directed Actions within a Brane. The use of Brane domains would also allow interdomain interactions within and across the same membrane. 4. Removal of the parameters for Bud and Phago, allowing the dynamic nature of membranes to be reﬂected in the calculus. 5. Inclusion of volume information as a system attribute to reﬂect its eﬀects on the probability at which collisions will occur inside a compartment. 6. Association of rates to channels emulating an aﬃnity feature similar to beta binders. 7. Treatment of Brane constituents and contents as ﬁnite quantities. 8. Elaboration of molecules as systems, a concept previously introduced by L. Cardelli[3]. These modiﬁcations are also geared towards the development of a machine for Brane calculus that can handle large-scale biological models.

2

Modified Notations

Table 1 summarizes the proposed notation and conceptual changes to the current Brane calculus, provided as a quick reference to the detailed explanations for these changes in the succeeding sections. 2.1

Actions

Notations and terms. In the documentation for the design of a machine for Brane calculus[7], stochastic pi calculus notations for input and output channels are used to distinguish between actions and coactions. At this point, it is important to make a distinction between an action (small ‘a’), a and an Action (capital ‘A’), σ. An action is an element of the set, a ∈ S currently deﬁned as: S ≡ {phago, pino, exo, mate, bud, drip};

(1)

A Projective Brane Calculus with Activate, Bud and Mate

167

Table 1. Comparison of the currently-established Brane calculus (Cardelli’s Brane calculus and PBC) with the proposed calculus. Note that a, ai ∈ S σ, σi , τ ∈ A, i = 1, 2, with a ¯ as the coaction of a. Conventions for parallel composition from PBC [5] are used. Definition Channel (x ∈ C) Set of actions (a ∈ S) Action (σ, τ ∈ A) Brane domain Brane Directionality System Parameter (bud and phago) Choice Series Parallel

Replication

Brane/Pi Calculus !x ←→ ?x S ≡ {phago, pino, exo, mate, bud, drip, . . .} a!x ←→ a ¯?x undeﬁned σ1 ; σ2 σ1 is outside, σ2 is inside σ1 ; σ2 (| P |) σ(τ )

Smin

PABM x ←→ x ≡ {0, bud, mate, !}

!x ←→ ax (a = !) ρ ≡ σ1 ; σ2 [ρ] σ1 is outside, σ2 is inside [ ρ ](| P |) τ.σ and ρ (see text for details)

σ1 + σ 2 σ1 .σ2 or σ1 σ2 σ1 |σ2

σ1 + σ2 σ1 .σ2 or σ1 σ2 σ1 , σ2 ρ1 |ρ2 P ◦Q P ◦Q . . !σ = σ, σ, . . . (inﬁnite) (n)σ = σ, σ, . . . , σ; n parallel n . (σ) = σ.σ. . . . .σ; n series . (n)ρ = ρ|ρ| . . . |ρ; n parallel . . !P = P ◦ P ◦ . . . (inﬁnite) (n)P = P ◦ P ◦ . . . ◦ P ; n parallel

while an Action is an element of the set, σ ∈ A currently deﬁned as: A ≡ {ax; a ∈ S, x ∈ C},

(2)

where the set C contains all possible channels. These notations are used throughout the text. In PABM, we use the following (minimal) set: Smin ≡ {m, b, 0, !};

(3)

where m is Mate, b is Bud, and two new actions, 0 and !, as the null and activate actions, respectively. The set A remains the same but with S replaced by Smin . We demonstrate that all other elements a ∈ S (Eq. 1) can be derived from a combination of these modiﬁcations with the directed Actions of PBC. Note that with the changes, the action becomes a passive entity (i.e. an action waits for an activation signal) by default. Since the deﬁnitions of mate and bud were not changed and have been discussed elsewhere [3, 5, 8], we will only review these deﬁnitions brieﬂy. Activate action, ! Cardelli’s Brane calculus requires two levels of matching before an Action could be executed/activated: (a) input and output channels; and (b) action and co-action. The use of an activation signal is expected to

168

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

improve the symmetry of form for the Actions with respect to channel and activation pairings. Here, we introduce an activate action, ‘!’, to approximate the input/output channel functionality of stochastic pi calculus, consequently precluding the need for an explicit distinction between actions and coactions within Brane. The Action in the form !x acts as an initiator of membrane interaction through channel x. This Action may be interpreted as a binding event, analogous to the required output signal from the intiating membrane or molecule before any non-activate Action can be executed/activated. The use of an activation signal, instead of a more strictly-bound actioncoaction pair with matching channels, is based on the fact that a single compound, modeled here as a communication channel, can interact with more than one substance, which may range from proteins to oligosaccharides, on the cell membrane. As discussed in §2.2, typical biological interactions involving receptors, logically corresponding to channels, are one-to-many relationships, rather than one-to-one pairs. Nevertheless, such cardinality does not imply that reaction speciﬁcity is lost. Another biological characteristic taken into account is the dependence of the kind of reaction that occurs on the receptor type, rather than on the ligand (i.e. it is the receiver that determines which eﬀect will occur). This characteristic is particularly marked in cells of the immune system, as well as antibodies, which have diﬀerent eﬀector functions associated with each class and subclass. The proposed form emphasizes that interaction speciﬁcity is conferred by the channel, but the receiver determines the type of action to execute. The null action, 0. The null action, 0, blocks actions that precede it; an Action in the form σ.0x0 can thus be used to model a blocked Action, σ. The null action can be deactivated with !x, making σ accessible. Biologically, blocking occurs in the event of temporary receptor internalization [9], binding-induced conformational changes [10, 11], and binding-induced physical blocking of other available binding sites. The use of 0 will be useful for modeling bind-and-release, molecular functionality switching, and other membrane-bound mechanisms. Bud. Bud refers to the arbitrary splitting of a membrane, resulting in two membrane-bound compartments[3]. Cardelli makes a distinction between bud and drip; bud occurs when the split occurs with one internal membrane, while drip refers to the separation of zero internal membranes. In PABM, this distinction is not made. Mate. Mate causes the irreversible mixing of actions of membranes that fuse either horizontally (i.e. membranes at the same level of nesting) or vertically through an exocytosis-type process[3]. 2.2

Choice, Parallel, and Series

All discussions of choice, parallel and series compositions are made with reference to Actions, unless otherwise indicated. Parallel and series compositions are not valid for actions and channels.

A Projective Brane Calculus with Activate, Bud and Mate

169

Choice. The concepts of parallel composition, choice and preﬁx(series) are retained from pi calculus. The notation for choice will be retained (‘+’). Choice could either be between actions a1 and a2 or channels x1 and x2 . These are equivalent to having a choice between two (or more) Actions in the basic form, ax. In particular, the following choices within action-channel pairs (Actions) would be equivalent to their respective choices between Actions: (a1 + a2 )x ≡ a1 x + a2 x a(x1 + x2 ) ≡ ax1 + ax2 (a1 + a2 )(x1 + x2 ) ≡ a1 x1 + a2 x1 + a1 x2 + a2 x2

(4) (5) (6)

These equivalences remove the need of implementing Actions in their non-basic forms. Hence, implementing choice in actions and/or channels will be unnecessary since all cases can always be reduced to a choice between (at least) two Actions. Aside from simplifying the implementation of Actions, Eqns. 4 and 5 reﬂect biological phenomena. For instance, Eqn. 5 is illustrated by membrane-bound receptors that have multiple ligands, with each ligand binding with a diﬀerent aﬃnity. At least three virus families, Orthomyxoviradae, Paramyxoviradae and Reoviradae, for example, use sialic acid in cell surfaces to enter via the endocytic pathway(s)[14]. A biological phenomenon that illustrates Eqn. 4, on the other hand, is the receptor for advanced glycation of end products (RAGE), expressed in a wide variety of cell types. RAGE is characterized by its ability to recognize numerous ligands, each of which result in diﬀerent eﬀects[15]. This is equivalent to having several actions associated with the same channel. Although the reactions of RAGE do not involve membrane structure deformations, a feature that would allow the direct modeling of such events may be of interest. Furthermore, Eqn. 5 implies that several receptors (or channels), can be used to initiate the same actions. Diﬀerent receptors, for instance, are used by diﬀerent viruses to enter the cell. Equation 6 is included for the purpose of completeness, but may not have any biological signiﬁcance. Parallel. Parallel pi processes and Actions, as indicated in Table 1, are represented following the notations in Danos and Pradelier and Cardelli. Preﬁx/Series. The original notation will be maintained for the series. A recurring series of the same Action would be used instead of replication to indicate the ﬁnite reusability of an Action. 2.3

Rates

PABM incorporates rates by associating a real number, rx , to the channel of each Action ax. When rx is associated with !x, it corresponds to the rate with which !x reacts on average with its receiver — the basic rate. When this real number is instead associated with ax, a = !, rx is a factor of the basic rate which reﬂects the eﬃciency of the reaction. A value of 1.0 indicates that the speciﬁc reaction rate with a particular receiver is the same as the basic rate. Association of rates to channels is adapted from beta binders[4].

170

2.4

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Aﬃnity

Aﬃnity describes the strength of non-covalent interactions between a ligand to its speciﬁc binding site on the receptor surface; this value is independent of the number of binding sites[12]. Higher aﬃnities are associated with factors such as the exposure of large, interactive amino acid side-chains, highly electronegative groups, or the deformability of a surface; these characteristics generally enable a ligand to form more non-covalent bonds with the receptor[13]. Empirically, a value known as the aﬃnity constant (Ka ) is used to approximate aﬃnity for ligand-receptor systems1 . It is determined by measuring the concentration of free ligand required to ﬁll half of the binding sites on the receptor. When half the sites are ﬁlled, [Ligand · Receptor] = [Ligand] and Ka = 1/[Receptor], where ‘[X]’ is used to indicate the molar concentration of X ; common Ka values range from as low as 5 × 104 to as high as 1011 liters/mole[12]. The use of aﬃnity in process calculi for biology has been proposed by C. Priami and P. Quaglia as a feature for beta binders[4]. Aﬃnity is incorporated as a probability P (a, b) that an interaction between two diﬀerent interfaces a and b can take place, eﬀectively relaxing the requirement for an exact matching of interface[6] — a distinct digression from pi calculus, where interactions occur on syntactically identical ports (lock-and-key model). In PABM, aﬃnity is inherent with choice. Since channels in PABM represent receptor-ligand functionality, the execution of a single action a can be associated with its interaction through more than one channel, say x1 , x2 , . . . , and xn , n > 1, resulting to the Action: a(x1 +x2 +. . .+xn ) that reduces to ax1 +ax2 +. . .+axn (Eqn. 5). When a = !, this results in multiple rates of execution, which depends on the Action a xl that is activated (a ≤ ! and l ≤ n). A similar situation occurs when a = ! and a = !, albeit with a diﬀerent biological implication. Table 2 summarizes the diﬀerence between this approach and Priami’s implementation in beta binders. Table 2. Comparison of aﬃnity in beta binders and PABM Beta Binders PABM each reduction each channel reaction probability P (a, b) multiple channels Implementation between two non-identical using choice interfaces a and b Association

2.5

Branes and Systems

The deﬁnition for Systems as sets of nested Branes is retained, and notations for these are adapted from PBC[5]. Null Systems are represented as . Notations for parallel composition of Systems are also retained. The same replication rules are 1

Notably for antibodies and antigens.

A Projective Brane Calculus with Activate, Bud and Mate

171

applied to both Actions and Systems (Table 1). Branes however, are redeﬁned as a composition of Brane domains; a Brane consisting of a single domain reduces to the original deﬁnition (Table 1). Directed Brane domains and directed Actions. In this section, the concept of directed actions in PBC is extended to Brane domains. A Brane domain, represented as a vector, ρ, is a grouping of directed Actions that approximates the occurrence of composition and functional non-homogeneity (“patchiness”) observed in biological membranes. Consequently, a Brane is now deﬁned as a parallel composition of Brane domains. A Brane deﬁned using a single domain is homogenous, and reduces to a Brane in PBC. Brane domains were introduced to facilitate greater control in processes like membrane budding. As opposed to Cardelli’s calculus where a parameter is used to deﬁne the characteristics of the Brane that will be budded out, the proposed calculus makes these characteristics entirely dependent on the current, dynamic state of the parent membrane. Budding processes, however, are highly localized, and the derived system should not have all the characteristics of the parent membrane. In the proposed calculus, only speciﬁc Brane domains are transferred in budding processes, unless the parent membrane is homogenous. Alternately, a Brane domain can be visualized as a set of directed Actions occurring proximally in a membrane. As an example, a system with Brane domains is subsequently represented as follows: [ ρ1 |ρ2 ](| [ ρ3 ](| P |) ◦ Q |)

(7)

where ρn is of the form σ1 ; σ2 (Table 1). As in the original Cardelli calculus, both ρ1 and ρ2 are visible to ρ3 . Using the rules of PBC, only the “outside” Actions of ρ3 can interact with the “outside” Actions of both ρ1 and ρ2 . The advantage of this feature is relevant in modeling competition between parallel membrane processes (§3.6). It is important to note that Brane domains represent active or functional sites on membranes or molecules and not the molecules themselves. Nevertheless, since at least one active site is associated with proteins, these can be represented as Brane domains. Brane domains can be used to model membrane proteins that function together such as lipid rafts[17] and SNARE complexes[18]. Lateral and cross-membrane interaction. Since interactions are now between two Brane domains, apart from the interaction of a domain from one membrane with another in a diﬀerent membrane, domain-domain interactions within a membrane is now possible. Actions can now be activated by Activate Actions on a neighboring domain (lateral membrane interaction). Activations by Actions on opposite sides of a single membrane (cross-membrane) can also be facilitated, provided that one of the Actions is translocated to the other side of the membrane by a mechanism similar to diﬀusion or channel-mediated transport. This capability can be used to model ligands that interact with receptors

172

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

on the same membrane surface or on the opposite side of the same membrane. Spontaneous membrane and in-membrane operations such as pinocytosis, drip, inversion, and fusion of proteins to form rafts and complexes[17, 18] can now be easily modeled. The following equation shows the competition between a1 x and a2 x since lateral- and cross-membrane interactions are allowed. [ !x ; a1 x |a2 x ; − |− ; a3 x ](| · · · |) ⇒

a1

or a2

(8)

Note that a3 cannot be activated since cross-membrane interaction are allowed only within the same Brane domain. With PABM, the transport of functional particles (e.g. molecules) through the membrane without introduction of atonal reduction rules can also be modeled (see §3.7). Volume information. A single enzyme-substrate experiment in a controlled nanoenvironment has shown that the frequency of collisions between two molecules is inversely proportional to the size of the vesicle where these molecules are contained[19]. Consequently, volume information will be associated with each System, representative of a compartment, allowing adjustments to be made in the probabilities at which the contained reactions will occur. 2.6

Replication

For the purpose of a calculus geared towards discrete biological system modeling, PABM uses a more controlled form of replication for Actions (also applicable to Brane domains and Systems), where the cardinality of replication is indicated (see Table 1). For instance, even if the initial counts of cellular components that are in the order of 104 to ∼ 1010 [16] are large enough to warrant the use of ∞, these are still ﬁnite quantities that may be critical determinants of biological system viability, especially in simulations that run for relatively prolonged periods of time (≥ 24 hours). Finite replication also reﬂects the ﬁnite lifetimes, masses and/or energies of both the components of biological systems and the systems themselves, appropriately manifested in the calculus in the form of ﬁnite Brane or Action usage. The numbers representing the ﬁnite number of replications can also be made stochastic to mimic the heterogeneity of membrane domains in terms of the absolute numbers of its constituents. Finite replication is conceptually similar to energy in beta binders [4], since the special entity E j (with j ∈ + ) can be mapped to the cardinality of replication in PABM. 2.7

Sample Notation: Mitogen-Induced Proliferation of Schwann Cells

Cell proliferation induced by an external signal is one of the simplest biological examples that is, nevertheless, diﬃcult to express in an intuitive manner without the use of spatial information. Mitogens, which induce cell division, are typically associated with one or more cognate receptors through which it can enter a cell. In Schwann cells, which form the insulation for vertebrate neurons

A Projective Brane Calculus with Activate, Bud and Mate

173

in the peripheral nervous system and which are critical for axon regeneration, proliferation is induced by the following mitogens in the neonatal stage: glial growth factor (GGF), platelet-derived growth-factor B (PDGF-BB) and basic ﬁbroblast growth factor (bFGF) [32]. For purposes of illustration, a coarse-grained model of the system can be deﬁned in the above notational changes as follows: [ !xG ; !xa ](| X |) ◦ [ !xP ; !xa ](| X |) ◦ [ !xbF ; !xa ](| X |) ◦ [ mxG , mxP , mxbF ; bxb ](| SC ◦ [ !xb .0xa ; − ](| R |) |) where each X represents a growth factor associated with some channel !xs , where s represents the part of X that binds to the GGF receptor (G), the PDGF-BB receptor (P ) or the bFGF receptor (bF ). The corresponding receptors, mxG , mxP , mxbF are all associated with the Schwann cell (SC). R represents the inactive replication machinery of the cell. This can only be activated on the fusion of one of the growth factors with SC, removing 0xa , and making !xb available for interaction. The availability of !xb in R allows SC to bud through its interaction with Action bxb .

3

Projective Activate, Bud, and Mate Calculus

In this section, we demonstrate that all Actions in S (Eq. 1) can be expressed as the actions in Smin (Eq. 3) combined with the directed Actions of PBC. The use of Smin as primitives has similarities to the basic Mate-Bud-Drip (MBD) calculus [8], which is one of two possible basic calculi for membrane interactions, together with the Phago-Exo-Pino (PEP) calculus. It has been shown [3, 8], however, that an encoding of MBD can be obtained with PEP, but not the opposite, because the maximum level of membranes (i.e. the membrane nesting) cannot grow during computation in MBD. Furthermore, the same articles prove that PEP calculus is Turing complete and Turing powerful, as opposed to MBD. Given these limitations, the use of a Bud- and Mate-based basic calculus appears counterintuitive. However, events indicated in the derivation of the MBD primitives using PEP (Fig. 1A) are not observed in biological systems (Fig. 1B). Although it is partly superﬂuous to observe that the derivations of MBD were previously qualiﬁed as performed for computational purposes only, it is clear that in vivo membrane fusion is characterized by membrane perturbances rather than a series of phagocytosis and exocytosis events[20]. Speciﬁcally, the prevalent hypothesis regarding membrane fusion involves the reduction of the distance between the fusing membranes, followed by the local perturbation of the lipid structure and merger of proximal monolayers. Stalk formation and stalk expansion, and ﬁnally, pore formation are postulated to follow. Furthermore, there is a requirement that each of these steps has to be driven by an energy gradient towards lower energies. The stalk hypothesis is mainly based on the observation that the merger of proximal monolayers precedes the merger of distal monolayers. These events are followed by the intravesicular solvent exchange[20].

174

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Fig. 1. PEP derivation of Mate [3] (A) and the latest model of how membrane fusion occurs [20] (B)

The succeeding discussions focus on the proposed PABM calculus as an encompassing calculus that conforms better with biologically observed phenomena. 3.1

Mate and Bud as Inverses of the Other

We consider Bud and Mate membrane actions as the primitives of this calculus, together with the Activate action, which controls their execution. Figure 2(top to bottom) shows a local deformation of the membrane separating the spaces labeled as P and Q ◦ R resulting from its interaction with Q. The increase in local curvature is then followed by the movement of Q towards the newly formed protrusion. On the fusion of the initial points of deformation, a new membranebound space containing Q is formed within P (Fig. 2, bottom), completing Bud. The reverse process, Mate, can be obtained using an opposite perspective. Here, the membrane separating Q from P merges with the membrane separating P and R (bottom to top). Colors are used to indicate tonality; in these processes, bitonality is conserved, as in PBC[5]. Since Bud and Mate are opposite operations, it would be possible to think of these as belonging to a single operation. 3.2

Projective Equivalence

Projective equivalence arose from the introduction of directed actions by Danos and Pradalier [5]. Brieﬂy, projective equivalence refers to the idea that the nature of membrane interactions is such that one does not make a distinction between top and bottom, or in this case, outside and inside. Consequently, by using a simple point-of-view change (i.e. what one considers inside before, which is a bounded space, is now viewed as the outside, which is unbounded), one reverses the process. If one uses a pointed bitonal tree representation for the structure, the equivalence is simply a change in the distinguished vertex [5], which is a change in the root of the tree. One can then generalize phago and bud as a single budding action, and exo and mate as a single mate action.

A Projective Brane Calculus with Activate, Bud and Mate

175

Fig. 2. Mate and Bud as inverse actions of the other. Bud is shown as a sequence from top to bottom while Mate as the reverse. Note that a distinction is not made between “inside” and “outside” spaces. A bilayer is used to illustrate directionality.

3.3

Basic Reduction Rules

The basic reduction rules of PABM are entirely based on Bud and Mate. Reduction rules are applied between interacting Brane domains, where the location of the activation signal with respect to the receiver (i.e. the directionality of the Action) determines if a Bud will be inward or outward, or if a Mate will be horizontal (i.e. membranes at the same level will merge) or vertical (i.e. the membrane of a content will merge with the membrane of its parent); this is conceptually similar to what has been done in PBC[5]. In the design of a Brane model, the directions at which Bud and Mate proceed are naturally integrated.The reduction rules of PABM are as follows, with “∼” used for indicating projective equivalence[5]. – Bud: P ◦ [ ρ1 |σ1 ; σ2 , τ2 .bx ](| [ ρ2 |σ4 , τ4 .!x ; σ3 ](| Q |) ◦ R |) −→ P ◦ [ σ1 ; σ2 , τ2 , τ4 ](| [ ρ2 |σ4 ; σ3 ](| Q |) |) ◦ [ ρ1 ](| R |) ∼

ρ1† |σ2 , τ2 .bx ; σ1 (| P |) ◦ [ ρ2 |σ4 , τ4 .!x ; σ3 ](| Q |) ◦ R −→ ρ1† (| P ◦ [ σ1 ; σ2 , τ2 , τ4 ](| [ ρ2 |σ4 ; σ3 ](| Q |) |) |) ◦ R

(9)

(10)

176

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

– Mate: P ◦ [ ρ2 |σ1 , τ1 .!x ; σ2 ](| Q |) ◦ [ ρ1 |σ1 , τ3 .mx ; σ2 ](| R |) −→ P ◦ [ ρ1 |ρ2 |σ1 ; σ2 |σ1 , τ1 , τ3 ; σ2 ](| Q ◦ R |)

(11)

∼ ; (| P ◦ [ ρ2 |σ1 , τ1 .!x ; σ2 ](| Q |) |) ◦ R ρ1† |ρ2 |σ2 ; σ1 |σ1 , τ1 , τ3 ; σ2 (| P |) ◦ Q ◦ R

(12)

−→

ρ1† |σ2

σ1 , τ3 .mx

Note that Q in Mate is equivalent to [ σ4 , τ4 .!x ](| σ3 |) Q in Bud. Odd-even subscripts and primed Actions are used to illustrate bitonality preservation. For directionality to be conserved, note the need for the reversal of ρ1 to ρ1† when the perspective is changed. In the case of Mate, interchanging the locations of mx and !x results in slightly diﬀerent Brane domains. Eq. 11 results in: P ◦ [ ρ1 |ρ2 |σ1 , τ1 , τ3 ; σ2 |σ1 ; σ2 ](| Q ◦ R |) ;

(13)

while Eq. 12 results in: [ ρ1 |ρ2 |σ1 ; σ2 |σ1 , τ1 , τ3 ; σ2 ](| P |) ◦ Q ◦ R.

(14)

It is only in the absence of τ1 and τ3 that the location of mx and !x does not result in diﬀerent succeeding states. 3.4

Non-primitive Actions with Bud and Mate

As shown in Eqs. 9 to 12, congruence exists between an inward and outward Bud, and between a horizontal and vertical Mate. This is more clearly illustrated in Fig. 2, where one sees that a simple perspective shift makes the same Bud or Mate operation inward or outward, or vertical or horizontal. For instance, when one chooses P as the “inside”, Q can be viewed as budding in towards P , or that the membrane containing Q is mating with the membrane separating P and R. As a result, PABM considers a ∈ S, a ∈ / Smin as membrane operations congruent to either Bud or Mate operation or its speciﬁc cases. Phago and Exo. Fig. 3(top) shows how Q is exocytosed from R or endocytosed into P via Mate and Bud, respectively. This is congruent to the Mate-Bud reactions in Fig. 2, with R as the outside and P as the inside. Phago is expressed as Bud in Eq. 10 and is congruent to the usual bud in Eq. 9. Pino and Drip. Pino and Drip may be spontaneous or induced Bud actions (see Eq. 10), where a null System (Q = null) is created inside or outside the bounded space P . The activation may also be induced by an appropriate Activate Action outside or inside P , or within-membrane activations (see §2.5). Pino and drip are obtained when Q = null in Fig. 3.

A Projective Brane Calculus with Activate, Bud and Mate

177

Fig. 3. Speciﬁc cases of Mate and Bud: (top, forward) Exo and (top, reverse) Phago; (bottom, forward) Cardelli’s Mate and (bottom, reverse) Bud operations

3.5

Enhanced Membrane Dynamics

A fundamental diﬀerence of the proposed calculus from the Brane calculus of Cardelli is the dynamic nature of the reacting membranes. In Cardelli’s version, the properties of budded membranes are speciﬁed as parameters to provide control; the same is true for the “endosomes” formed during Phago. In biological systems, however, the characteristics of the budded membrane are necessarily dependent on the state of the parent membrane at the time of budding or phagocytosis. Fig. 4 reﬂects this particular case of budding, when a sequential action is associated with the activation Action on an initiating membrane (σ1 , green). Note the incorporation of σ1 in the budded membrane. Also note that only a portion of the membrane is budded out. This dynamic property of the membrane generally implies that systems involved in a Mate followed by a Bud (b.m) will not evolve equivalently when Bud is performed before Mate (m.b). For instance, consider the following initial system: Q0 ≡ [ σ4 , τ4 .mxM ; σ3 , τ3 .bxB ](| [ σ1 , τ1 .!xB ; σ2 ](| P2 |) ◦ P3 |) ◦ [ σ6 , τ6 .!xM ; σ7 ](| P1 |)

(15)

Depending on which operation occurs ﬁrst, the system will evolve in two diﬀerent ways. First, on performance of b.m, the system will evolve as: m

Q0 −−−−−−→ Q0 ≡ [σ4 , τ4 , τ6 ; σ3 , τ3 .bxB |σ6 ; σ7 ](| P1 ◦ [ σ1 , τ1 .!xB ; σ2 ](| P2 |) ◦ P3 |) Q0

b

−−−−−−→

Q1

(16)

≡ [ σ6 ; σ7 ](| P1 ◦ P3 |) ◦ [ σ4 , τ4 , τ6 ; σ3 , τ1 , τ3 ](| [ σ1 ; σ2 ](| P2 |) |)

(17)

178

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Fig. 4. Budding as a dynamic process. Note that a new action, σ1 , associated with the ‘!’ is incorporated into the budded membrane. Domains are illustrated as line segments; only selected domains proximal to the activated action are budded out.

Second, with m.b, the system will evolve as: b

Q0 −−−−−−→ Q0 ≡ [ − ](| P3 |) ◦ [ σ4 , τ4 .mxM ; σ3 , τ3 , τ1 ](| [ σ1 ; σ2 ](| P2 |) |) ◦ [ σ6 , τ6 .!xM ; σ7 ](| P1 |) Q0

m

−−−−−−→ Q1 ≡ [ − ](| P3 |)

(18) (19)

◦ [ σ4 , τ4 , τ6 ; σ3 , τ3 , τ1 |σ6 ; σ7 ](| P1 ◦ [ σ1 ; σ2 ](| P2 |) |) Clearly, Q1 = Q1 . This asymmetry example (b.m = m.b) is depicted in Fig. 5. The diﬀerence disappears when the Mate and Bud are placed in separate domains, σ4 , τ4 .mxM ; σ3 and σ4 ; σ3 , τ3 .bxB . 3.6

Competition of Parallel Membrane Processes

Using the concept of Brane domains, competition of two or more parallel membrane processes can be easily modeled. Given the following system (longhand), the interactions of !x is restricted to bx associated with σ1 or that associated with σ3 . If it interacts with bx in σ1 ; bx , then the system reduction will be in the form: [ σ1 ; bx |σ3 ; bx ](| [ !x ; σ5 ](| Q |) ◦ R |) −→ [ σ1 ; − ](| [ − ; σ5 ](| Q |) |) ◦ [ σ3 ; bx ](| R |)

(20)

A Projective Brane Calculus with Activate, Bud and Mate

179

Fig. 5. Diﬀerences in ﬁnal system states based on the order at which reactions occur. (top) Initial conﬁguration; (middle) Mate then Bud; (bottom) Bud then Mate.

On the other hand, if it interacts with bx in σ3 ; bx instead, then the system reduction will be as follows: [ σ1 ; bx |σ3 ; bx ](| [ !x ; σ5 ](| Q |) ◦ R |) −→ [ σ3 ; − ](| [ − ; σ5 ](| Q |) |) ◦ [ σ1 ; bx ](| R |)

(21)

Competition can also be realized in lateral and cross-membrane processes (see Eq. 8). 3.7

Molecules as Systems

Molecules can either serve as ligands or receptors. In this proposed modiﬁcation, Molecules can be modeled as null Systems containing Activators associated with Actions or other Activators. It may also be in the form of blocking functions, σ.0x, which could only be activated by !x. A molecule can be modeled as: Molecule : [ σ1 , (n)(!xname +!xgeneric ) ; σ2 , (m)(!xname +!xgeneric) ](| |)

(22)

where τ may be a null Brane and the “name” could be the name of the molecule making the channel unique for the molecule and “generic” refers to the generic channel of the molecule. For example, “RNA” can be a generic channel having the name of the protein that it encodes for its speciﬁc name.

180

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Together with cross-membrane interactions, molecule diﬀusion through a membrane can be modeled without atonal reduction. This is illustrated in the reduction below, where the molecule on the left-hand side enters the system P . Note the change in the replication coeﬃcient, reﬂecting the reduction of the active sites in both the molecule and the membrane surrounding P . (!x)(n1 ) ; (bx.!y)(n1 ) (| |) ◦ [ (n2 )my ; − ](| P |) −→ (n2 − 1)my ; − |(bx.!y)(n1 −1) .bx ; (!x)(n1 −1) .!x (| P |) −→ [ (n2 − 1)my ; − ] P ◦ (!x)(n1 −1) ; (bx.!y)(n1 −1) (| |) (23)

3.8

Mass and Energy Conservation

Since budding involves direct movement of Brane domains, mass (represented by an Action) conservation is also simulated. The “consumption” of an Action after reduction can be seen as the usage of the available energy used for and/or transfer of mass during the process, i.e. transformation of the structure into new ones.

4

PABM as an Extension of Existing Brane Calculi

Equivalent expressions for the multiple association of the activation action using Cardelli’s original notations can be derived. Suppose there are three systems that can interact via a generic action a, with coaction a. Multiple association can be realized with: a!x (| Q0 |) ◦ a?x (| P1 |) ◦ a?x (| P2 |) ,

(24)

where system Q0 can proceed with the action (a ↔ a) on both systems P1 and P2 through the same channel !x →?x. It is possible to eliminate the use of coactions (a) through the following representation: !x (| Q0 |) ◦ a?x (| P1 |) ◦ a?x (| P2 |)

(25)

with !x possibly activating either P1 or P2 via a?x. This minor notation change is immediately compatible with Pi calculus, and would require a minor code translation for recognition by the Stochastic Pi Machine (SPiM) [21]. However, the proposed calculus also involves the removal of the sender-receiver pairing (viz. !x →?x), apart from the action-coaction pairing. Moreover, in Eq. 25, the notation is asymmetric, with the activator !x having a diﬀerent form from a?x. It is possible to use α!x as a universal activator to conserve symmetry, but α will be underutilized. The use of a single “sender” is proposed for all the other actions in the form !x, while simultaneously making the notation symmetric with the use of the same form (ax, see Table 1). Both conceptually and implementationwise, these major diﬀerences could be seen as improvements over the current representation.

A Projective Brane Calculus with Activate, Bud and Mate

181

Hence, the multiple association expressed as (24) would be written in PABM as: [ !x ; − ](| Q0 |) ◦ [ ax ; − ](| P1 |) ◦ [ ax ; − ](| P2 |)

(26)

with x as the channel; “!” now belongs to the same class as a = ! (Table 1). The choice of the symbol “!” for the activate action is directly inspired by the Pi calculus notation. As indicated previously, the other major departure from Cardelli’s Brane calculus is the utilization of Brane domains in dynamic membranes to eliminate the use of parameters in phagocytosis and budding. With these domains, a Brane in PBC becomes a special case when a Brane in PABM is homogenous (i.e. is comprised of a single Brane domain). For purposes of comparison, the reduction rules of the original calculus [taken from [8]] are shown in Fig. 6.

Fig. 6. Cardelli’s Brane calculus reduction rules taken from [8]

The realization of non-primitive actions that were illustrated utilizes the same concepts as in the projective equivalence of Danos and Pradalier [5], with the exception that no arguments are explicitly used for the Bud action. Furthermore, the simplicity of the current basis and reduction makes the calculus closer to actual biological membrane operations. Finally, PBC becomes a subset of the proposed calculus since PBC Branes can be simulated using homogenous PABM Branes.

5

Summary and Outlook

We end this paper with brief discussions on a potential application of the calculus, as well as a strategy for its possible implementation.

182

5.1

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Application Example: Viral Infection

Inﬂuenza A causes highly contagious respiratory infections in humans that range in severity from acute to lethal. New strains arise annually, which lead to 250,000 to 500,000 deaths worldwide[22, 23, 24]. It is particularly interesting for biologists because of its ability to evolve very quickly, a trait that makes the development of an eﬃcient vaccine against it particularly challenging[25, 26]. To date, a number of qualitative studies have been performed to investigate its life cycle, but most involved separate analyses of steps in the infection process [24, 26]. One of the recent most extensive quantitative models of inﬂuenza A in cell culture is that by Sidorenko and Reichl[24]. It consists of 49 ordinary diﬀerential equations (ODEs) that involve the use of additional parameters to approximate the movement of viruses and its components across cell compartments. The main results obtained from the model include the identiﬁcation of factors that limit the growth rate of viral progeny; these results are particularly useful in molecular engineering, where engineered viruses are created for vaccine production[24]. Nevertheless, it is clear that much is still not known about the inﬂuenza A life cycle, primarily owing to the complexity of the virus. Some details, for instance, that have not been included in the Sidorenko-Reichl model include the following: 1. Distinction between each of the eight strands of genetic material (vRNA), complexed with three proteins (collectively known as vRNP), throughout the replication cycle 2. Distribution of 11 protein-coding genes across the eight vRNAs 3. Indirect genetic material replication (vRNA → cRNA → vRNA), with the intermediate cRNA being able to interact with the same proteins that vRNA interacts with 4. Requirement for precise viral assembly Accordingly, several key issues remain unanswered: 1. time it takes to assemble vRNPs 2. ratio of infective to non-infective viruses 3. instances of ‘infectivity recovery’ in the event that two complementary noninfective viruses enter a cell Since compartments can be naturally represented in Brane, its use for modeling the inﬂuenza A life cycle is probably an elegant, quantitative alternative that would allow the inclusion of details such as those enumerated previously. Fig. 7 is a general illustration for the possible usage of PABM to model the inﬂuenza infection cycle. This particular model is an interesting application for Brane calculus on account of its scale. Note that all operations used for modeling the system are restricted to budding and mating, including the simulation of the bind-and-release action in the nucleus. The position of activation signals are not explicitly indicated in the ﬁgure, but could be deduced from the illustrated processes. In addition to these, it would also be possible to include details that are known to aﬀect inﬂuenza infectivity, as well as eﬃciency[27]:

A Projective Brane Calculus with Activate, Bud and Mate

183

Fig. 7. Inﬂuenza A infection cycle model being implemented using PABM. All operations are performed with Bud and Mate, including nuclear import and export through the use of the NP protein, nuclear localization sequences (NLS) or nuclear export sequences (NES). All processes are conformant with the actual events in inﬂuenza infection.

1. cleavage eﬃciency of HA 2. distinction between transcriptionally active and inactive vRNPs 5.2

Implementation and Compatibility with SPiM

Previous eﬀorts have been made to implement Brane calculus[28, 29]. These implement calculi based on the set S (Eq. 1) and were found useful for studying events having the same scale as the Semliki forest virus life cycle, which was used as the illustrative example in [3]. Nevertheless, these are not powerful enough to handle models having the scale of the inﬂuenza A life cycle. It is consequently of interest to develop an implementation that is both scalable and robust. The stochastic pi machine (SPiM) was developed by Andrew Phillips, and uses a simulation algorithm for stochastic pi calculus that is particularly suited for simulating biological systems involving a large number of molecules. This

184

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

simulation algorithm makes the execution cost dependent on the number of species, rather than the actual number of molecules, unlike in direct implementations of the Gillespie algorithm [21]. SPiM has been used on a number of occasions for a variety of biological problems [21, 30, 31]. Lately, the algorithm in SPiM has been extended to include compartment-based computation, using the Bioambients formalism [7]. SPiM also has a graphical interface, which signiﬁcantly improves its ease of use. PABM should be compatible with SPiM using the following equivalences: !x ≡ (m!x + b!x + 0!x) ax ≡ a?x σ.!x ≡ a!x(σ)

(27) (28) (29)

where a = m, b, 0 and a is the corresponding coaction. Encoding more speciﬁc stochastic pi calculus constructs would only require the use of very speciﬁc channel names. For PABM to be implemented on top of SPiM, compartments, Brane domains, and action directionality have to be appropriately represented. A separate implementation approach that focuses on the rewriting rules of PABM is also currently being explored. Acknowledgments. We wish to thank Luca Cardelli and Andrew Phillips for helpful discussions.

References [1] P˘ aun, G.: Introduction to membrane computing. In: Applications of Membrane Computing, pp. 1–42 (2006) [2] Regev, A., Shapiro, E.: The π-calculus as an abstraction for biomolecular systems. In: Modelling in Molecular Biology (2004) [3] Cardelli, L.: Brane calculi: interactions of biological membranes. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 257–278. Springer, Heidelberg (2005) [4] Priami, C., Quaglia, P.: Beta binders for biological interactions. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 20–33. Springer, Heidelberg (2005) [5] Danos, V., Pradalier, S.: Projective brane calculus. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 134–148. Springer, Heidelberg (2005) [6] Guerriero, M.L., Priami, C., Romanel, A.: Modeling Static Biological Compartments with Beta-binders. In: Anai, H., Horimoto, K., Kutsia, T. (eds.) Ab 2007. LNCS, vol. 4545, pp. 247–261. Springer, Heidelberg (2007) [7] Phillips, A., Cardelli, L.: Eﬃcient, correct simulation of biological processes in the stochastic pi-calculus. In: Calder, M., Gilmore, S. (eds.) CMSB 2007. LNCS (LNBI), vol. 4695, pp. 184–199. Springer, Heidelberg (2007) [8] Busi, N., Zandron, C.: Modelling and analysis of biological processes by (mem)brane calculi and systems. In: Proceedings of the 2006 Winter Simulation Conference, pp. 1646–1655 (2006)

A Projective Brane Calculus with Activate, Bud and Mate

185

[9] Guglielmo, G.D., Drake, P., Baass, P., Authier, F., Posner, B., Bergeron, J.: Insulin receptor internalization and signalling. Mol. Cell. Biochem. 182, 59–63 (1998) [10] Hsu, S., Bonvin, A.: Atomic insight into the CD4 binding-induced conformational changes in HIV-1 gp120. Proteins: structure, function and bioinformatics 3, 582–593 (2004) [11] Keskin, O.: Binding induced conformational changes of proteins correlate with their intrinsic ﬂuctuations: a case study of antibodies. BMC Structural Biology 7, 31 (2007) [12] Alberts, B., Bray, D., Lewis, J., Raﬀ, M., Roberts, K., Watson, J.: Molecular Biology of the Cell, New York (2002) [13] David, M., Asprer, J., Ibana, J., Concepcion, G., Padlan, E.: A study of the structural correlates of aﬃnity maturation: antibody aﬃnity as a function of chemical interactions, structural plasticity and stability. Mol. Immunol. 44, 1342–1351 (2006) [14] Dimitrov, D.: Virus entry: molecular mechanisms and biomedical applications. Nature Reviews Microbiology 2, 109–122 (2004) [15] Kim, W., Hudson, B., Moser, B., Guo, J., Rong, L., Yu, L., Qu, W., Lalla, E., Lerner, S., Chen, Y., Yan, S.D., D’Agati, V., Naka, Y., Ramasamy, R., Herold, K., Yan, S., Schmidt, A.: Receptor for advanced glycation end products and its ligands: A journey from the complications of diabetes to its pathogenesis. Annals of the New York Academy of Sciences 1043, 553–561 (2006) [16] Thulke, S., Radonic, A., Nitsche, A., Siegert, W.: Quantitative expression analysis of HHV-6 cell receptor CD46 on cells of human cord blood, peripheral blood and G-CSF mobilised leukapheresis cells. Virology Journal 3, 77–81 (2006) [17] Simons, K., Vaz, W.L.C.: Model Systems, Lipid Rafts, and Cell Membranes. Annual Review of Biophysics and Biomolecular Structure 33(1) (June 2004) [18] Sutton, R.B., Fasshauer, D., Jahn, R., Brunger, A.T.: Crystal structure of a SNARE complex involved in synaptic exocytosis at 2.4 ˚ A resolution. Nature 395(6700), 347–353 (1998) [19] Chiu, D., Wilson, C., Karlsson, A., Danielsson, A., Lunqvist, A., Stroemberg, A., Ryttsen, F., Davidson, M., Nordholm, S., Orwar, O., Zare, R.: Manipulating the biochemical nanoenvironment around single molecules contained within vesicles. Chem. Phys. 247, 133–139 (1999) [20] Jahn, R., Grubm¨ uller, H.: Membrane fusion. Current Opinion in Cell Biology 14, 488–495 (2002) [21] Phillips, A., Cardelli, L.: A Correct Abstract Machine for the Stochastic Picalculus. In: Concurrent Models in Molecular Biology (2004) [22] Poland, G.A., Tosh, P., Jacobson, R.M.: Requiring inﬂuenza vaccination for health care workers: seven truths we must accept. Vaccine 23(17-18), 2251–2255 (2005); Vaccines and Immunisation. Based on the Fourth World Congress on Vaccines and Immunisation [23] Baccam, P., Beauchemin, C., Macken, C.A., Hayden, F.G., Perelson, A.S.: Kinetics of Inﬂuenza A Virus Infection in Humans. J. Virol. 80(15), 7590–7599 (2006) [24] Sidorenko, Y., Reichl, U.: Structured model of inﬂuenza virus replication in mdck cells. Biotechnology and bioengineering 88, 1–14 (2004) [25] Bardiya, N., Bae, J.: Inﬂuenza vaccines: recent advances in production technologies. Applied Microbiology and Biotechnology 67(3), 299–305 (2005) [26] Genzel, Y., Schulze-Horsel, J., M¨ ohler, L., Sidorenko, Y., Reichl, U.: Inﬂuenza vaccines –challenges in mammalian cell culture technology. Cell Technology for Cell Products, 503–508 (2007)

186

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

[27] Nayak, D.P., Hui, E.K.-W., Barman, S.: Assembly and budding of inﬂuenza virus. Virus Research 106, 147–165 (2004) [28] de Ronde, J.J., Ndjehan, C.P.: Modelling Networks and Pathways in Systems Biology. Technical report, CA545 Practicum, School of Computing, Dublin City University (2005/2006) [29] David, M.P.C.: BCD: Design and implementation of a stochastic brane machine. Master’s thesis, Department of Computer Science, University of the Philippines, Diliman, Quezon City (2008) [30] Segata, N., Blanzieri, E., Priami, C.: Stochastic π-calculus modelling of multisite phosphorylation based signaling: in silico analysis of the Pho4 transcription factor and the PHO pathway in Saccharomyces cerevisiae. Technical report, Center for Computational and Systems Biology, The Microsoft Research – University of Trento (2007) [31] Yap, J.M.: A Pi-Calculus Model of the CD95 Receptor Medicated Pathway of Apoptosis. Philippine Information Techonology Journal 1(1) (2008) [32] Zhang, B.T., Hikawa, N., Horie, H., Takenaka, T.: Mitogen induced proliferation of isolated adult mouse Schwann cells. J. Neurosci. Res., 648–654 (1995)

Accepting Networks of Non-inserting Evolutionary Processors J¨ urgen Dassow1 and Victor Mitrana2, 1

Faculty of Computer Science, University of Magdeburg P.O.Box 4120, 39016 Magdeburg, Germany [email protected] 2 Faculty of Mathematics, University of Bucharest Str. Academiei 14, 70109 Bucharest, Romania Department of Information Systems and Computation Technical University of Valencia, Camino de Vera s/n. 46022 Valencia, Spain [email protected]

Abstract. In this paper we consider four variants of accepting networks of evolutionary processors with in-place computations, that is the length of every word in every node at any step in the computation is bounded by the length of the input word. These devices are called here accepting networks of non-inserting evolutionary processors (ANNIEP shortly). The variants diﬀer in two respects: ﬁlters that are used to control the exchange of information, i.e., we use random context conditions and regular languages as ﬁlters, and the way of accepting the input word, i.e., at least one output node or all output nodes are nonempty at some moment in the computation. The computational power of these devices is investigated. In the case of ﬁlters deﬁned by regular languages, both variants lead to the class of context-sensitive languages. If random context conditions are used for deﬁning ﬁlters, all linear context-free languages and some non-semilinear (even over the one-letter alphabet) can be accepted with both variants. Moreover, some closure properties of the classes of languages ANNIEPs with random context ﬁlters are also given.

1

Introduction

The origin of networks of evolutionary processors (NEP for short) is a basic architecture for parallel and distributed symbolic processing, related to the Connection Machine [9] as well as the Logic Flow paradigm [7], which consists of several processors, each of them being placed in a node of a virtual complete graph, which are able to handle data associated with the respective node. All the nodes send simultaneously their data and the receiving nodes handle also simultaneously all the arriving messages, according to some strategies, see, e.g., [8,9]. Similar ideas may be met in other bio-inspired models like membrane systems [16], evolutionary systems [4], or models from Distributed Computing area

Work supported by the Alexander von Humboldt Foundation.

C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 187–199, 2009. c Springer-Verlag Berlin Heidelberg 2009

188

J. Dassow and V. Mitrana

like parallel communicating grammar systems [15], networks of parallel language processors [3]. In a series of papers (see [14] for an early survey) one considers that each node may be viewed as a cell having genetic information encoded in DNA sequences which may evolve by local evolutionary events, that is point mutations. Each node is specialized just for one of these evolutionary operations. Furthermore, the data in each node are organized in the form of multisets of words (each word appears in an arbitrarily large number of copies), and all the copies are processed in parallel such that all the possible events that can take place do actually take place. Obviously, the computational process just described is not exactly an evolutionary process in the Darwinian sense. But the rewriting operations we have considered might be interpreted as mutations and the ﬁltering process might be viewed as a selection process. Recombination is missing but it was asserted that evolutionary and functional relationships between genes can be captured by taking only local mutations into consideration [17]. In [13] one presents a characterization of the complexity class NP based on accepting networks of evolutionary processors (ANEP for short). This characterization is extended in [12] to PSPACE and P. The work [10] discusses how ANEPs can be considered as problem solvers. In [11], one shows that every recursively enumerable language can be accepted by an ANEP with 24 nodes. More precisely, one proposes a method for constructing, for every NP-language, an ANEP of size 24 deciding that language in polynomial time. While the number of nodes of this ANEP does not depend on the language, the other parameters of the network (rules, symbols, ﬁlters) depend on it. From a computational point of view it is of interest to consider ANEPs with in-place computations, that is the length of every word in every node at any step in the computation is bounded by the length of the input word. This is our main reason to consider here some variants of networks of evolutionary processors without insertion nodes, called here accepting networks of non-inserting evolutionary processors, ANNIEP shortly. The diﬀerences between the variants of ANNIEPs consist in the ﬁlters and in the way of accepting the input word. Besides accepting networks of evolutionary processors, generating networks of such processors have been investigated (see [2], [5], [14]). In the paper [6], the generative power of networks where only two types of point mutations are allowed for the nodes have been investigated. In case of non-inserting processors one only gets the set of all ﬁnite languages. This paper presents the counterpart for accepting networks, where the situation is completely diﬀerent. We study the computational power of accepting networks of non-inserting processors. In the case of ﬁlters deﬁned by regular languages, both variants of accepting lead to the same class of languages, namely the class of contextsensitive languages. If random context conditions are used for deﬁning ﬁlters, all linear context-free languages and some non-semilinear (even over the oneletter alphabet) can be accepted with both variants. Therefore the power of accepting networks is much greater than that of generating networks (both with non-inserting processors). Moreover, some closure properties of the classes of

Accepting Networks of Non-inserting Evolutionary Processors

189

languages accepted by ANNIEPs with ﬁlters deﬁned by random context conditions are also discussed.

2

Some Notations and Definitions

Throughout the paper we assume that the reader is familiar with the basic notions of the theory of formal languages. We here only recall some notation and notions as they are used in the paper. An alphabet is a ﬁnite and nonempty set of symbols. The cardinality of a ﬁnite set A is written card(A). Any sequence of symbols from an alphabet V is called word over V . The set of all words over V is denoted by V ∗ and the empty word is denoted by ε. A language over V is a subset of V ∗ . The length of a word x is denoted by |x| while alph(x) denotes the (with respect to inclusion) minimal alphabet W such that x ∈ W ∗ . A morphism h : V ∗ −→ U ∗ is said to be literal if |h(a)| = 1 for all a ∈ V ; it is weak literal if |h(a)| ≤ 1 for all a ∈ V . In other words a (weak) literal morphism is called (weak) coding. Let V be an alphabet. We say that a rule a → b, with a, b ∈ V ∪ {ε} is a substitution rule if both a and b are not ε; it is a deletion rule if a = ε and b = ε. The set of all substitution and deletion rules over an alphabet V are denoted by SubV and DelV , respectively. Given a rule σ as above and a word w ∈ V ∗ , we deﬁne the following actions of σ on w: {ubv : ∃u, v ∈ V ∗ (w = uav)}, ∗ • If σ ≡ a → b ∈ SubV , then σ (w) = {w}, otherwise {ub : w = ua}, {bv : w = av}, r l σ (w) = σ (w) = {w}, otherwise {w}, otherwise {uv : ∃u, v ∈ V ∗ (w = uav)}, • If σ ≡ a → ε ∈ DelV , then σ ∗ (w) = {w}, otherwise {u : w = ua}, {v : w = av}, r l σ (w) = σ (w) = {w}, otherwise {w}, otherwise The action α ∈ {∗, l, r} expresses the way of applying a substitution or deletion rule to a word, namely at any position (α = ∗), in the left (α = l), or in the right (α = r) end of the word, respectively. For every rule σ, any action α ∈ {∗, l, r}, and any L ⊆ V ∗ , we deﬁne the α-action of σ on L by σ α (L) = σ α (w). w∈L

Given a ﬁnite set of rules M , we deﬁne the α-action of M on the word w and the language L by: M α (w) = σ α (w) and M α (L) = M α (w), σ∈M

respectively.

w∈L

190

J. Dassow and V. Mitrana

If θV ∗ −→ {0, 1} is a predicate and L ⊆ V ∗ , we write: θ(L) = L ∩ θ−1 (1). We are interested in some special predicates. For two disjoint subsets P and F of an alphabet V , a regular set R over V , and a word x over V , we deﬁne the predicates θs,P,F (x) = 1 if and only if P ⊆ alph(x) and F ∩ alph(x) = ∅, θw,P,F (x) = 1 if and only if alph(x) ∩ P = ∅ and F ∩ alph(x) = ∅, θR (x) = 1 if and only if x ∈ R. The ﬁrst two predicates are based on random context conditions deﬁned by the two sets P (permitting contexts/symbols) and F (forbidding contexts/symbols). Informally, the ﬁrst condition requires (s stands for strong) that all permitting symbols are and no forbidding symbol is present in x, while the second (w stands for weak) is a weaker variant such that at least one permitting symbol appears in x but still no forbidding symbol is present in x. We call these two predicates random context predicates. The third predicate asks for membership in a regular set, and is called a regular predicate. A non-inserting evolutionary processor over V is a tuple (M, ϕ, ψ), where: – M is a set of either substitution or deletion rules over the alphabet V ; formally, M ⊆ SubV or M ⊆ DelV . The set M represents the set of evolutionary rules of the processor. As one can see, a processor is “specialized” in one evolutionary operation, only. – ϕ is the input predicate, while ψ is the output predicate of the processor. Informally, these two predicates work as ﬁlters. A word w can enter or leave the processor, if it satisﬁes the predicate ϕ or ψ, respectively. We are interested in two types of processors, random context non-inserting evolutionary processor over V (or short rcNIEPV ) and regular non-inserting evolutionary processor over V (or short regNIEPV ). These processors are deﬁned by the requirement that, – for an rcNIEPV , both predicates are of the form θs,P,F or of the form θw,P,F for certain subsets P and F of V , – for an regNIEPV , both predicates are of the form θR for some regular set R ⊆ V ∗. We want to stress from the very beginning that the evolutionary processor we discuss here is a mathematical object only and the biological hints presented in the introduction are intended to explain in an informal way how some biological phenomena are sources of inspiration for our mathematical computing model. We denote the set of non-inserting evolutionary processors over V by N IEPV . An accepting network of non-inserting evolutionary processors (ANNIEP for short) is a 8-tuple Γ = (V, U, G, N, α, xIn , Out), where:

Accepting Networks of Non-inserting Evolutionary Processors

191

– V and U are the input and network alphabet, respectively, satisfying V ⊆ U . – G = (XG , EG ) is an undirected graph without loops with the set of vertices XG and the set of edges EG . G is called the underlying graph of the network. – N : XG −→ N IEPV is a mapping which associates with each node x ∈ XG the evolutionary processor N (x) = (Mx , ϕx , ψx ). – α : XG −→ {∗, l, r} is a mapping which associates with each node a type of action; α(x) gives the action mode of the rules of node x on the words existing in that node. – xIn ∈ XG is the input node of Γ . – Out ⊂ XG is the set of output nodes of Γ . An ANNIEP is a random context ANNIEP or regular ANNIEP if all its noninserting evolutionary processors are random context or regular non-inserting evolutionary processors, respectively. We say that card(XG ) is the size of Γ . A configuration of an ANNIEP Γ ∗ as above is a mapping C : XG −→ 2Vf which associates a ﬁnite set of words with every node of the graph. A conﬁguration may be understood as the sets of words which are present in any node (or in the associated prozessor) at a given moment. Given a word z ∈ V ∗ , the initial conﬁguration of Γ on z is deﬁned by (z) (z) C0 (xIn ) = {z} and C0 (x) = ∅ for all x ∈ XG \ {xIn }. A conﬁguration can change either by an evolutionary step or by a communication step. When changing by an evolutionary step, each component C(x) of the conﬁguration C is changed in accordance with the set of evolutionary rules Mx associated with the node x and the way of applying these rules α(x). Formally, we say that the conﬁguration C is obtained in one evolutionary step from the conﬁguration C, written as C =⇒ C , iﬀ C (x) = Mxα(x) (C(x)) for all x ∈ XG . When changing by a communication step, each node processor x ∈ XG sends one copy of each word it has, which is able to pass the output ﬁlter of x, to all the node processors connected to x and receives all the words sent by any node processor connected with x provided that they can pass its input ﬁlter. Formally, we say that the conﬁguration C is obtained in one communication step from conﬁguration C, written as C C , iﬀ C (x) = (C(x) − ψx (C(x))) ∪ (ψy (C(y)) ∩ ϕx (C(y))) for all x ∈ XG . {x,y}∈EG

Note that words that cannot pass the output ﬁlter of a node remain in that node and can be further modiﬁed in the subsequent evolutionary steps, while words that can pass the output ﬁlter of a node but cannot pass the input ﬁlter of any node are lost. Let Γ be an ANNIEP, the computation of Γ on the input word z ∈ V ∗ is a (z) (z) (z) (z) sequence of conﬁgurations C0 , C1 , C2 , . . ., where C0 is the initial conﬁgu(z) (z) (z) (z) ration of Γ on z, C2i =⇒ C2i+1 and C2i+1 C2i+2 , for all i ≥ 0. Note that the conﬁgurations are changed by alternative steps. By the previous deﬁnitions, each

192

J. Dassow and V. Mitrana (z)

(z)

conﬁguration Ci is uniquely determined by the conﬁguration Ci−1 . A computation halts (and it is said to be weak (strong) halting) if one of the following two conditions holds: (i) There exists a conﬁguration in which the set of words existing in at least one output node (all output nodes) is non-empty. In this case, the computation is said to be a weak (strong) accepting computation. (ii) There exist two identical conﬁgurations obtained either in consecutive evolutionary steps or in consecutive communication steps. The language weakly (strongly) accepted by Γ are deﬁned as: Lwa (Γ ) = {z ∈ V ∗ | the computation of Γ on z is a weak accepting one} Lsa (Γ ) = {z ∈ V ∗ | the computation of Γ on z is a strong accepting one}. In the theory of networks some types of underlying graphs are common like rings, stars, grids, etc. Networks of evolutionary words processors, seen as language generating or accepting devices, with underlying graphs having these special forms have been considered in several papers, see, e.g., [14] for an early survey. We focus here on complete ANNIEPs i.e., ANNIEPs having a complete underlying graph. Therefore, in what follows we replace the graph G in the deﬁnition of an ANNIEP by the set of its nodes usually denoted by χ. Moreover, we present an evolutionary network by its nodes x and the parameters corresponding to x, where instead of ϕβ,P Ix ,F Ix and ψ β,P Ox ,F Ox , in case of random context processors, and instead of ϕRx and ϕRx for regular processors, we only mention P Ix , F Ix , P Ox , F Ox , β and Rx , Rx , β, respectively. For x ∈ {wa, sa} and y ∈ {rc, reg}, by Lx (yAN N IEP ) we denote the set of all languages which can be accepted by yANNIEPS. The following two notions will be very useful in the sequel. If h is a one-toone mapping from U to W and Γ = (V, U, χ, N, α, xIn , Out) is an ANNIEP, then we denote by Γh the ANNIEP Γh = (h(V ), h(U ), χ, h(N ), α, xIn , Out), where by h(N ) we mean h(N )(x) = (h(Mx ), ϕβ,h(P Ix ),h(F Ix ) , ψ β,h(P Ox ),h(F Ox ) ) for every x ∈ χ, provided that N (x) = (Mx , ϕβ,P Ix ,F Ix , ψ β,P Ox ,F Ox ). Further, h(a → b) = h(a) → h(b) for any evolutionary rule a → b. Now, given two ANNIEPs Γi = (Vi , Ui , χi , Ni , αi , xiIn , Outi ), i = 1, 2, χ1 ∩ χ2 = ∅, we denote by Γ1 Γ2 = (V1 , U1 ∪U2 , χ1 ∪χ2 , N, α, x1In , Out2 ), where ◦ |χi = ◦i for all ◦ ∈ {N, α} and i = 1, 2.

3

Computational Power of Regular ANNIEPs

We start with a relation between the strong and weak acceptance modes. Theorem 1. Lwa (regAN N IEP ) ⊆ Lsa (regAN N IEP ). Proof. Let L ∈ Lwa (regAN N IEP ). Then L = Lwa (Γ ) for some regular AN NIEP Γ = (V, U, χ, N, α, xIn , Out). Let N (x) = (Mx , ϕRx , ψ Rx ) for a node x of

Accepting Networks of Non-inserting Evolutionary Processors

193

χ. Without loss of generality we may assume that Mx = ∅ for all x ∈ Out. We now construct the regular ANNIEP Γ = (V, U ∪ {Z}, χ ∪ {xOut }, N , α , xIn , {xOut }), where N (x) = N (x) for x ∈ χ \ Out, and y : {a → Z | a ∈ U }, Ry , Z ∗ , α (y) = ∗ for y ∈ Out, xOut : ∅, Z ∗ , ∅, α (xOut ) = ∗. Obviously, if there is a non-empty node y of Out in some conﬁguration of Γ , then y contains some word in some conﬁguration of Γ , too. If this word is ε, then ε is not changed and sent to xOut . If the word in y is non-empty, then all its letters are replaced by Z (note that it cannot leave the node as long as it still contains letters diﬀerent than Z) and it is send to xOut . Conversely, if a word eventually arrives in xOut , then it contains only Z’s which means that it was in a node from Out at some previous step. Thus Γ accepts the same language as Γ does. Moreover, since the set of output nodes of Γ is a singleton, we have Lwa (Γ ) = Lwa (Γ ) = Lsa (Γ ). 2 Note that we have shown a stronger result than given in Theorem 1 because we have shown that the number of output nodes of an ANNIEP accepting in the weak mode can be decreased to one only. We now compare the families of languages generated by ANNIEPs with the family of context-sensitive languages denoted here by L(CS). Theorem 2. L(CS) ⊆ Lwa (regAN N IEP ). Proof. Let L be a context-sensitive language. Then L = L(G) for some contextsensitive grammar G = (N, T, P, S) in Kuroda normal form, i.e., all its rules are of the form A → a, A → BC and AD → BC with A, B, C, D ∈ N and a ∈ T . Let P be the set of rules of the form A → BC and AD → BC. For every p ∈ P with its right-hand side BC we set Rp = (N ∪ T )∗ {Bp }(N ∪ T )∗ , Rp = (N ∪ T )∗ {Bp Cp }(N ∪ T )∗ ,

Rp = (N ∪ T )∗ {Cp }(N ∪ T )∗ and R =

Rp . We construct the ANNIEP Γ = (T, U, χ, H, α, xIn , {xOut })

p∈P

with U = N ∪ T ∪ {Bp , Cp | p = AD → BC or p = A → BC}, χ = {xIn , xOut } ∪ {p, p , p | p ∈ P }, xIn : MxIn , (N ∪ T )∗ , R, α = ∗ MxIn = {a → A | A → a ∈ P } ∪ {B → Bp | p = AD → BC or p = A → BC}, p : {C → Cp }, Rp , Rp , α = ∗ for p = AD → BC or p = A → BC,

194

J. Dassow and V. Mitrana

p : {Bp → A}, Rp , Rp , α = ∗ for p = AD → BC or p = A → BC, {Cp → D}, Rp , (N ∪ T )∗ , α = ∗ for p = AD → BC, p : {Cp → ε}, Rp , (N ∪ T )∗ , α = ∗ for p = A → BC, xOut : ∅, {S}, {S}, α = ∗. The network simulates a derivation in G backwards. Let w be the input word; we claim that for any word z ∈ (N ∪ T )+ in xIn at any computation step we have that z =⇒∗ w in G. Initially, this assertion is true as w lies in xIn . Assume that a word z ∈ (N ∪ T )+ is in the node xIn at some step. If we apply a rule a → A to z, the new word remains in xIn and the assertion holds for this new word. Now assume that we apply B → Bp to z for a rule p = AD → BC. Then the obtained word z = z1 Bp z2 , where z = z1 Bz2 , is sent to the node p, where some C is replaced by Cp . If Bp Cp is not a subword, then the word cannot go out from this node; moreover any word further obtained from this word can never go out from the node p. If Bp Cp is a subword, the word is sent out to the node p , where Bp is replaced by A. This new word is sent out to p . There Cp is either replaced by D, provided that p = AD → BC, or deleted provided that p = A → BC. Finally, the obtained word, say z , is sent to xIn . Altogether, we started with z = vBCu and obtained z = vADu, which implies that z =⇒ z =⇒∗ w. Moreover, since a word only reaches xOut , if it is S, we infer that a word is weakly accepted by Γ if and only if it is generated by G. Thus Lwa (Γ ) = L(G). 2 Theorem 3. Lsa (regAN N IEP ) ⊆ L(CS). Proof. For an ANNIEP Γ = (V, U, χ, N, α, xIn , Out), we construct a linearly bounded automaton, which accepts Lsa (Γ ). We do not give a complete formal construction; we only give an informal description of the automaton and leave the details of the construction to the reader. Let r = card(Out). The automaton has r tapes, and on each tape it nondeterministically follows the itinerary of a copy of the input word. The states are vectors of size 2r, each ith entry, 1 ≤ i ≤ r, being associated with the node containing the word on the tape i, and each ith entry, r + 1 ≤ i ≤ 2r, being 0 or 1 that indicates whether the node associated with the (i − r)th entry has ﬁnished its task on the word on tape i (in this case the entry is 1) or not. Initially, all tapes contain the input word w, the ﬁrst r entries of the initial states are associated with the input node xIn , and the last r entries are 0. Let us now consider an arbitrary conﬁguration of the automaton: the ﬁrst r elements of the current state state are associated with the nodes x1 , x2 , . . . xr , the last r elements are 0, and on the i-th tape, 1 ≤ i ≤ r, the word wi stands. Now the automaton performs on each tape i the following actions: – Changes the word wi according to an application of a rule in Mxi ; let vi be the result. – Checks whether vi can pass the output ﬁlter of xi . In the non-aﬃrmative case the automaton blocks the computation. In the aﬃrmative case, the

Accepting Networks of Non-inserting Evolutionary Processors

195

automaton changes the ith entry of the state into an entry associated with the node yi , which is a nondeterministically chosen node among the nodes of χ \ {xi }. – Check whether vi can pass the input ﬁlter of yi . In the non-aﬃrmative case the automaton blocks the computation. In the aﬃrmative case, the i+r entry becomes 1. From now on, no move is observed on the ith tape and no change is made for the entries i and i + r, until all the entries r + 1, r + 2, . . . , 2r are 1. – Checks whether the state with the last r entries 1 has its ﬁrst r entries associated with all output nodes of Γ . In the aﬃrmative case the automaton accepts the input; otherwise it changes the last r entries into 0 and resumes the actions explained above. It is rather plain that the automaton accepts Lsa (Γ ). Since in any evolutionary step one deletes or substitutes one letter, the length of the words on any tape is bounded by the length of the input word. Thus the workspace of this automaton is linearly bounded. 2 By the Theorems 1, 2 and 3, we get immediately the following two statements. Corollary 1 1. Lwa (regAN N IEP ) = Lsa (regAN N IEP ) = L(CS). 2. Every language in LX (regAN N IEP ), X ∈ {wa, sa}, can be weakly/strongly accepted by a regANNIEP Γ such that the action mode of every node of Γ is ∗. 2

4

Computational Power of Random Context ANNIEPs

We start with two statements that immediately follows from Theorems 1 and 3. Theorem 4 1. Lwa (rcAN N IEP ) ⊆ Lsa (rcAN N IEP ). 2. Lsa (rcAN N IEP ) ⊆ L(CS).

2

We do not know whether the second inclusion is proper or equality holds. Thus we give some further relations to other known language families inside L(CS) and some closure properties which give some more information about the classes Lwa (rcAN N IEP ) and Lsa (rcAN N IEP ). Theorem 5 1. Lwa (rcAN N IEP ) includes the class of linear context-free languages. 2. Lwa (rcAN N IEP ) contains non-semilinear languages. Proof. 1. Let G = (N, T, S, P ) be a linear context-free grammar; without loss of generality we may assume that the following conditions hold:

196

J. Dassow and V. Mitrana

– Every rule in P is of one of the following three forms: A → aB, A → Ba, A → a, where A, B ∈ N and a ∈ T , – If both rules A → aC and B → Db belong to P , then A = B, – The set of nonterminals N of G is {A1 , A2 , . . . , An } for some n ≥ 1 and S = A1 , – There is no rule A → aA or A → Aa for any A ∈ N and a ∈ T . We construct the following ANNIEP with the input alphabet T , the working alphabet U = T ∪ {ai , ai | 1 ≤ i ≤ n} ∪ {Z}, and only one output node xOut . ⎧ ⎧ M = {a → a1 | a ∈ T }, M = ∅, ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ P I = T, F I = {ai | a ∈ T, 1 ≤ i ≤ n}, P I={Z}, F I=U \ {Z}, xIn : xOut : P O = ∅, F O = T, P O = U, F O = ∅, ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ α = ∗, β = w, α = ∗, β = s, If there exists Ai → aAj ∈ P for some a ∈ T and 1 ≤ j = i ≤ n, then the node xi is deﬁned by ⎧ ⎪ ⎪ M = {ai → aj | Ai → aAj ∈ P }, ⎨ P I = {ai | a ∈ T }, F I = U \ {ai | a ∈ T }, xi : P O = {aj | a ∈ T, 1 ≤ j = i ≤ n}, F O = ∅, ⎪ ⎪ ⎩ α = l, β = w, If there exists Ai → Aj a ∈ P for some a ∈ T and 1 ≤ j = i ≤ n, then the node xi is deﬁned by ⎧ M = {ai → aj | Ai → aAj ∈ P }, ⎪ ⎪ ⎨ P I = {ai | a ∈ T }, F I = U \ {ai | a ∈ T }, xi : P O = {aj | a ∈ T, 1 ≤ j = i ≤ n}, F O = ∅, ⎪ ⎪ ⎩ α = r, β = w, Moreover, we set ⎧ M = {aj → ai | a ∈ T, 1 ≤ j = i ≤ n}, ⎪ ⎪ ⎨ P I = {ai | a ∈ T }, F I = ∅, xi : for 1 ≤ i ≤ n, = i ≤ n}, ⎪ P O = {ai | a ∈ T }, F O = {aj | a ∈ T, 1 ≤ j ⎪ ⎩ α = ∗, β = w, ⎧ M = {ai → ε | a ∈ T }, ⎪ ⎪ ⎨ P I = {ai | a ∈ T }, F I = {aj | a ∈ T, 1 ≤ j = i ≤ n}, x ¯i : for 1 ≤ i ≤ n, P O = {a | a ∈ T }, F O = {a | a ∈ T }, ⎪ i i ⎪ ⎩ α = l, β = w, ⎧ M = {ai → ε | a ∈ T }, ⎪ ⎪ ⎨ P I = {ai | a ∈ T }, F I = {aj | a ∈ T, 1 ≤ j = i ≤ n}, x ˜i : for 1 ≤ i ≤ n, P O = {a | a ∈ T }, F O = {a | a ∈ T }, ⎪ i i ⎪ ⎩ α = r, β = w, ⎧ M = {a 1 ≤ i ≤ n}, ⎪ ⎪ in → Z | Ai → a ∈ P, a ∈ T, ⎨ n P I = i=1 {ai | a ∈ T }, F I = U \ ( i=1 {ai | a ∈ T }), y: P O = {Z}, F O = ∅, ⎪ ⎪ ⎩ α = r, β = w,

Accepting Networks of Non-inserting Evolutionary Processors

197

The general idea of this construction is that for every 1 ≤ i ≤ n, the following statement holds: Fact: If S =⇒t uAi v =⇒+ uwv = z for some t ≥ 0, with |z| = m, then (z) (z) hi (w) ∈ (C2m(t+1)+2t (xi ) ∩ C2m(t+1)+2t (y)), where hi is a literal morphism from T to {ai | a ∈ T } defined by h(a) = ai for any a ∈ T . This fact can be proved by a standard induction argument on t. Now, if t = m − 1, then w is reduced to a letter from T , say a, therefore after the word ai is transformed into Z in the node y, it arrives in xOut and the computation halts successfully. This means that z is accepted by the network. (z) (z) (z) (z) On the other hand, if C0 , C1 , C2 , . . . , Cp is an accepting computation (z) (z) on z and hi (w) ∈ (Ct (xi ) ∩ Ct (y)) for some t < p, then the derivation S =⇒∗ uAi v =⇒+ uwv = z holds in G, which concludes the proof of the ﬁrst statement of the theorem. 2. The network with the nodes deﬁned by: ⎧ ⎧ M = {a → a ¯}, M = {a → a ˜}, ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ P I = {a}, F I = {¯ a, a ˜}, P I = {¯ a}, F I = {˜ a}, xIn : x1 : P O = {¯ a }, F O = ∅, P O = {˜ a }, F O = ∅, ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ α = ∗, β = s, α = ∗, β = s, ⎧ ⎧ M = {¯ a → ε}, M = {˜ a → a }, ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ P I = {¯ a, a ˜}, F I = ∅, P I = {˜ a}, F I = {¯ a}, x2 : x3 : P O = {˜ a}, F O = ∅, P O = {a }, F O = {˜ a}, ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ α = ∗, β = s, α = ∗, β = s, ⎧ M = {a → a}, ⎪ ⎪ ⎨ P I = {a }, F I = {a, a ¯, a ˜}, x4 : xOut P O = {a}, F O = {a }, ⎪ ⎪ ⎩ α = ∗, β = s,

⎧ M = ∅, ⎪ ⎪ ⎨ P I = {¯ a}, F I = {a, ¯a, a ˜}, : P O = {¯ a }, F O = ∅, ⎪ ⎪ ⎩ α = ∗, β = s, n

weakly accepts the non-semilinear language {a2 | n ≥ 0}. Indeed, the computation of this netwok on every input is divided in two phases. In the ﬁrst phase, the input word looses one occurrence of a and changes another one to a by visiting the nodes xIn , x1 , x2 , x3 . This process resumes until no occurrence of a is observed in the current word. There are three possiblities: (1) it contains only a’s, (2) it contains only a’s excepting an occurrence of a ¯, (3) it equals a ¯. Now the second phase of the computation starts. In the ﬁrst case, the word enters x4 where all a’s are transformed into original a’s and the ﬁrst phase resumes from xIn with a word that is exactly twice shorter than the word present in the input node in the beginning of the previous ﬁrst phase. In this case, we have checked whether the length of that word was an even number. In the second case listed above, the computation cannot continue anymore, hence the network will eventually halt without accepting. In the third case, the computation halts accepting the input word. This means that the length of the input word could be divided iteratively by 2 until the result was one, hence the length of the input word was a power of 2. 2

198

J. Dassow and V. Mitrana

Theorem 6 1. The class Lwa (rcAN N IEP ) is closed under boolean union, literal morphism, inverse weak literal morphism, mirror image. 2. The class Lsa (rcAN N IEP ) is closed under boolean intersection, literal morphism, inverse weak literal morphism, concatenation, mirror image. Proof. 1. We give an informal proof for union that can be easily formalized by the reader. Let Γ1 and Γ2 be to ANNIEPs; we construct a new ANNIEP Γ that contains three subnetworks. In the input node of the ﬁrst subnetwork, an arbitrary symbol of the input word is substituted by either its primed copy or its barred copy. All words containing a primed symbol are received by a speciﬁc node while those containing a barred symbol are received by another speciﬁc node. All symbols of the words arrived in these two nodes are replaced by their primed and barred copies, respectively. When this process is ﬁnished, each of the two nodes contains only one word. The word containing primed symbols only is given as an input word to the subnetwork formed from Γ1 modiﬁed accordingly. The other word is processed analogously by the subnetwork formed from Γ2 modiﬁed accordingly. The set of output nodes of Γ is the union of the sets of output nodes of Γ1 and Γ2 modiﬁed accordingly. Clearly, Lwa (Γ ) = Lwa (Γ1 ) ∪ Lwa (Γ2 ). If h : V −→ U is a literal morphism and Γ is an ANNIEP with the input alphabet V , then let Γ be the ANNIEP with the input alphabet U formed by two subnetworks as follows. In the input node of the ﬁrst subnetwork, each symbol b of the input word is substituted by a symbol a such that a is a copy of a ∈ V that does not appear in V ∪ U and h(a) = b. When all symbols of the input word were substituted, all the words obtained are sent to the input node of the subnetwork formed from Γ modiﬁed accordingly. It is plain that h(Lwa (Γ )) = Lwa (Γ ). The construction for the closure under inverse weak literal morphism is pretty similar and left to the reader. The closure under mirror image follows pretty simple; it suﬃces to interchange all the action modes l and r of the nodes. 2. The closure under intersection, literal morphism and inverse literal morphism follows similarly to the previous case. Note the fundamental role played by the strong acceptance in the case of intersection. 2 It is known that every recursively enumerable language can be written as the image of the intersection of two linear languages through a weak literal morphism. Therefore, the following statement is a consequence of the second statement of Theorem 4 and Theorem 6: Corollary 2 1. Every recursively enumerable language is the weak literal morphic image of a language in Lsa (AN N IEP ). 2. Lsa (AN N IEP ) is not closed under weak literal morphism. 2

5

Final Remarks

As we showed in this note, the computational power of ANNIEPs is very different than that of generating networks of non-inserting processors. The role of

Accepting Networks of Non-inserting Evolutionary Processors

199

evolutionary operations in generating networks of evolutionary processors, that is generating networks with nodes specialized in all three evolutionary operations, in two operations out of these three and in only one operation, was considered in [1]. A similar investigation on ANEPs has already started.

References 1. Alhazov, A., Dassow, J., Rogozhin, Y., Truthe, B.: Personal communication 2. Castellanos, J., Mart´ın-Vide, C., Mitrana, V., Sempere, J.: Networks of evolutionary processors. Acta Informatica 38, 517–529 (2003) 3. Csuhaj-Varj, E., Salomaa, A.: Networks of parallel language processors. In: P˘ aun, G., Salomaa, A. (eds.) New Trends in Formal Languages. LNCS, vol. 1218, pp. 299–318. Springer, Heidelberg (1997) 4. Csuhaj-Varj, E., Mitrana, V.: Evolutionary systems: a language generating device inspired by evolving communities of cells. Acta Informatica 36, 913–926 (2000) 5. Csuhaj-Varj´ u, E., Mart´ın-Vide, C., Mitrana, V.: Hybrid NEPs are computationally complete. Acta Informatica 41, 257–272 (2005) 6. Dassow, J., Truthe, B.: On the power of networks of evolutionary processors. In: Durand-Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 158–169. Springer, Heidelberg (2007) 7. Errico, L., Jesshope, C.: Towards a new architecture for symbolic processing. In: Artiﬁcial Intelligence and Information-Control Systems of Robots, vol. 94, pp. 31–40. World Scientiﬁc, Singapore (1994) 8. Fahlman, S., Hinton, G., Seijnowski, T.: Massively parallel architectures for AI: NETL, THISTLE and Boltzmann Machines. In: Proc. AAAI National Conf. on AI, pp. 109–113. William Kaufman, Los Altos (1983) 9. Hillis, W.: The Connection Machine. MIT Press, Cambridge (1985) 10. Manea, F., Mart´ın-Vide, C., Mitrana, V.: On the size complexity of universal accepting hybrid networks of evolutionary processors. Mathematical Structures in Computer Science 17, 753–771 (2007) 11. Manea, F., Mitrana, V.: All NP-problems can be solved in polynomial time by accepting hybrid networks of evolutionary processors of constant size. Information Processing Letters 103, 112–118 (2007) 12. Manea, F., Margenstern, M., Mitrana, V., Perez-Jimenez, M.: A new characterization of NP, P, and PSPACE with accepting hybrid networks of evolutionary processors (submitted) 13. Margenstern, M., Mitrana, V., Perez-Jimenez, M.: Accepting hybrid networks of evolutionary systems. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA 2004. LNCS, vol. 3384, pp. 235–246. Springer, Heidelberg (2005) 14. Mart´ın-Vide, C., Mitrana, V.: Networks of evolutionary processors: results and perspectives. In: Molecular Computational Models: Unconventional Approaches, pp. 78–114. Idea Group Publishing, Hershey (2005) 15. P˘ aun, G., Sntean, L.: Parallel communicating grammar systems: the regular case. Annals of University of Bucharest, Ser. Matematica-Informatica 38, 55–63 (1989) 16. P˘ aun, G.: Computing with membranes. Journal of Computer and System Sciences 61, 108–143 (2000) 17. Sankoﬀ, D., et al.: Gene order comparisons for phylogenetic inference: evolution of the mitochondrial genome. In: Proceedings of the National Academy of Sciences of the United States of America, vol. 89, pp. 6575–6579 (1992)

Discrete Modeling of Biochemical Signaling with Memory Enhancement John Jack1 and Andrei P˘ aun1,2,3 1

Department of Computer Science/IfM Louisiana Tech University, P.O. Box 10348, Ruston, LA 71272, USA {johnjack,apaun}@latech.edu http://www.latech.edu 2 Departamento de Inteligencia Artiﬁcial, Facultad de Inform´ atica Universidad Polit´ecnica de Madrid, Campus de Montegancedo S/N, Boadilla del Monte, 28660 Madrid, Spain http://www.upm.es 3 Bioinformatics Department, National Institute of Research and Development for Biological Sciences, Splaiul Independent¸ei, Nr. 296, Sector 6, Bucharest, Romania

Abstract. We present an enhancement of the Nondeterministic Waiting Time algorithm. This work is a continuation of our group’s previous modeling eﬀorts. We have improved our algorithm with a “memory enhancement”. Previously, we have used our algorithm to explore the Fas-mediated apoptotic pathway in cells with a particular focus on cancerous or HIV-1-infected T cells. In this paper, we will describe the memory enhancement and give a simple three reaction model to illustrate the diﬀerences between our technique and a continuous, concentration-based approach using a system of ordinary diﬀerential equations. Furthermore, we provide our results from the modeling of two well-known models: the Lotka-Volterra predator-prey and a circadian rhythm model. For these models, we provide the results of our simulation technique in comparison to results from ordinary diﬀerential equations and the Gillespie Algorithm. We show that our algorithm, while being faster than Gillespie’s approach, is capable of generating oscillatory behavior where ordinary diﬀerential equations do not. Keywords: Discrete modeling, Lotka-Volterra, predator - prey, circadian rhythm, Gillespie, ordinary diﬀerential equations.

1

Introduction

Systems biology, the systematic study of biological systems through a combined eﬀort between computational and experimental results, has received a great deal of attention in recent years [17,18]. There has been an expansive eﬀort from mathematicians and computer scientists to use models to unravel the mysteries behind biochemical/biological systems – e.g., signal transduction, viral dynamics, gene transcription. With the ever-increasing wealth of information C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 200–215, 2009. c Springer-Verlag Berlin Heidelberg 2009

Discrete Modeling of Biochemical Signaling with Memory Enhancement

201

ﬂowing in from biological labs around the world on protein dynamics, the challenge remains for computer scientists to develop/reﬁne eﬃcient algorithms for modeling molecular signaling cascades. Computational tools are being applied to interpret biological results and make predictions into the underlying molecular mechanisms involved in cancer, autoimmune disorders, and neurological disorders. We see two important eﬀorts being undertaken by computer scientists with respect to modeling molecular signaling cascades. First, the algorithms developed to interpret molecular interactions need visibility to biochemists and noncomputer scientists. Notably, the authors of [10,12] have made great strides in developing software, designed for biochemists with little to no knowledge on the modeling algorithm, to design, develop, and implement biochemical network simulations based on their experimental observations. The second major eﬀort is the development of more eﬃcient algorithms to drive biochemical simulation software. With many labs focusing on the modeling of individual pathways – e.g., Fas-mediated apoptosis [11], p53 network [19,22] and the EGF-receptor system [24] – the concept of a realistic whole-cell simulation remains a very distant goal. There are too many unknowns biochemical aspects to build an accurate and reliable model. However, while the biochemical questions surrounding whole-cell simulation are being answered in experimental labs, there is still work to be done in modifying (and developing new) algorithms for simulating biochemical systems. 1.1

Motivation Behind the Paper

Many signal transduction models in the literature contain as many (or more) reactions as proteins. Although the human genome contains over three billion base-pairs, it only encodes approximately 20,000-25,000 genes. The proteins encoded by these genes are entangled in an intricate and diverse web of interactions. The dynamics of these proteins – e.g., expression levels and reactions – deﬁne the complexity and physiological characteristics of the human cell. Some reactions in signaling cascades can sometimes share common reactants and compete for resources. These competing reactions typically have diﬀerent kinetic rates – i.e., some of the competing reactions utilize a given reactant faster than other reactions use the same reactant. Hence, when the numbers of some molecules are very small, stochastic (or nondeterministic) methods for biochemical modeling can play an important role in interpreting the results of lab experiments, and oﬀer insight into unknown aspects of the system. When modeling biochemical networks via systems of ordinary diﬀerential equations (ODEs), the data are considered in terms of concentrations instead of numbers of molecules, and the reactions are deterministically applied. While the ODEs are satisfactory for predicting the average behavior of a biochemical system, they are not ideal for extrapolating the diﬀerent cellular responses resulting from molecular signaling cascades – especially ones involving low numbers

202

J. Jack and A. P˘ aun

of molecules. The Gillespie Algorithm [7,8], which is a numerical simulation algorithm for the chemical master equation, has been extensively employed to address these low molecular multiplicity situations. However, even though it has been modiﬁed to run more eﬃciently [9], it does not scale well with respect to the number of reactions. Hence, an algorithm capable of realistic and eﬃcient wholecell simulation, a signiﬁcant goal for systems biology, is still being explored. Our algorithm is designed to be faster than Gillespie’s algorithm and its derivatives – such as, Gibson’s Next Reaction Method – yet more sensitive than ODE-based simulations. Our group has previously argued in [3,13,14] that an approach involving the Membrane Systems paradigm of computing oﬀers a unique perspective on biochemical network simulation. Speciﬁcally, in [13,14] we discuss the advantages of our simulation technique: the Nondeterministic Waiting Time (NWT) algorithm. Our algorithm is distinct from the Gillespie Algorithm. Yet, it is a discrete, nondeterministic technique which can oﬀer a diﬀerent perspective than systems of ordinary diﬀerential equations on the biochemistry of a cell. In this paper, we will describe a modiﬁcation to our algorithm. In order to improve the deterministic aspects of reaction competition for our simulation technique, we have added a memory enhancement to the NWT. This enhances the sensitivity of our algorithm with respect to reaction competition over limited resources. Since our algorithm relies on the law of mass action to drive the population dynamics, fast reactions may be allowed to use up all the resources of a slow reaction. With the modiﬁed algorithm, a slow reaction will remember how long it has waited when a fast reaction uses all available reactants. The memory can be factored into the equation for calculating the next time the slow reaction will occur, once reactants become available again. In Section 2 we provide the necessary background on the NWT algorithm, a simulation technique based on Membrane Systems. We will discuss the speciﬁcs on the memory enhancement in Section 3, as well as results for a simple biochemical model involving fast-slow reaction competition. In Section 4 we show the results of the NWT algorithm for simulating two popular models: The LotkaVolterra predator-prey model and a circadian rhythm model [1]. For both models, we compare the results of the NWT algorithm with an ODE-based simulation and a simulation based on the Gillespie Algorithm. Section 5 contains our ﬁnal remarks and a discussion on the future research interests of our modeling group.

2

The Nondeterministic Waiting Time Algorithm

The NWT algorithm is a discrete, nondeterministic biochemical simulation algorithm. We track the evolution of a Membrane System where the rules (or reactions) occur over discrete time intervals in an asynchronous manner. Before we give a step-by-step description of the algorithm, it is important to discuss the concept of reaction Waiting Times. Our NWT algorithm is driven by the law of mass action – the time a reaction takes to occur is directly proportional to the number of its reactant molecules.

Discrete Modeling of Biochemical Signaling with Memory Enhancement

203

When dealing with concentration-based kinetics we need to calculate a discrete kinetic constant (for molecules instead of nMs, μMs, etc). We initialize the discrete kinetic constants with Equation 1. constR =

V

i−1

kR × N i−1

(1)

where V is the volume of the system, N is Avogadro’s constant (6.0221415×1023) and i is the number of reactants involved in the reaction. Once the kinetic constants are initialized, for every reaction in the system, we calculate the initial Waiting Time – the amount of time required for one instance of a reaction – using Equation 2. W TR1 =

1 constR1 ∗ |A|

(2)

where A is the reactant, constR1 is the discrete kinetic constant, and |A| represents the number of molecules present in the system at the moment of WT calculation. Equation 2 represents the calculation for a ﬁrst order reaction (involving only one reactant). For second and third order reactions (two and three reactants, resp.), we need to use Equations 3 and 4. 1 constR2 ∗ |A| ∗ |B|

(3)

1 constR3 ∗ |A| ∗ |B| ∗ |C|

(4)

W T R2 = and W TR3 =

where A, B, and C are the reactants, constR2 and constR3 are the discrete kinetic constants, and |A|, |B| and |C| represent the numbers of molecules present in the system at the moment of WT calculation. With the calculation of reaction Waiting Time, we have the amount of time it will take for each reaction to occur. If there are insuﬃcient reactants for a reaction, then we set the Waiting Time equal to inﬁnity; this is easily done in the C programming language. We can now provide the following description of the NWT algorithm (n.b., Step 7 is the new memory enhancement step, which will be explained in Section 3): 1. Build Membrane System: Import model information (alphabet, rules, etc.). For every reaction, Ri , calculate the initial Waiting Time, W TRi . Choose simulation end-time τf in . Set current simulation time to zero (τ = 0). 2. Build Heap: Using the reaction Waiting Times, we build a min-heap of all reactions in the system. 3. Select Rule: Choose the reaction with the lowest Waiting Time – the top of the min-heap. Upon selecting the top node, recursively check to see if there are any children nodes sharing the minimum Waiting Time. If such a tie for minimum Waiting Time exists, proceed to Step 4. If no tie exists, then proceed to Step 5.

204

J. Jack and A. P˘ aun

4. Handle Tie: Check the multiplicities of the reactant species for all tied reactions. If there are enough reactants to satisfy all of the reactions with the minimum Waiting Time, implement all tied reactions. If there are not enough reactants to accommodate all the reactions, then nondeterministically apply as many reactions as possible. 5. Apply Rule: Update the multiplicities of the reactant(s) and product(s) for the reaction(s) from Step 3. Aggregate the simulation time (τ = τ + W Tapplied ). 6. Update Rules: Recalculate the Waiting Time for all reactions whose reactants include the products or reactants of the applied reaction(s). That is, we need to see how the multiplicity changes from the applied reaction(s) have aﬀected the Waiting Times for all rules dependent on those proteins with changed multiplicity. For each such reaction compare the new Waiting Time with the existing Waiting Time and keep the smallest of the two (unless the new time is inﬁnity). 7. Memory Enhancement: If the recalculation of a reaction’s Waiting Time results in a value of inﬁnity, then we must store the amount of time waited as a percentage (M emperc ). If the recalculation of a reaction’s Waiting Time results in a real value and the previous value was inﬁnite, then the Waiting Time will need to be adjusted according to the stored memory percentage. 8. Heap Maintenance: Adjust the min-heap, bubbling reaction nodes up or down in order to satisfy the min-heap property, once reaction Waiting Times have been recalculated according to the multiplicity changes. N.B., to accommodate the multiple changes in Waiting Times, we employ nonstandard heap maintenance methods. 9. Termination: If τ = τf in , then terminate the simulation. Output the multiplicity information for entire simulation. Otherwise, go back to Step 3. For a deeper explanation of the algorithm, we refer the interested reader to [13,14]. In the next section, we will clarify Step 7 of the algorithm – the memory enhancement.

3

Memory Enhancement

As we discussed in Section 1.1, there are often situations in biochemical networks, where one protein is a reactant for two or more reactions of diﬀerent kinetic rates (fast vs. slow). In order to explain our memory enhancement, we will consider an example system (see Table 1). The biochemical system in Table 1 involves three reactions (R1 , R2 and R3 ) acting on four proteins (A, B, C, D). We can mathematically describe the model as a system of ordinary diﬀerential equations (Equation 5) d[A] = −(k1 + k2 )[A] + k3 [D] dt d[B] = k2 ∗ [A] dt

Discrete Modeling of Biochemical Signaling with Memory Enhancement

205

Table 1. An example biochemical system Reaction Rate Constant Initial Molecules R1 : A → C k1 (slow) A=1 R2 : A → B k2 (fast) B=0 R3 : D → D + A k3 C=0 D=1

d[C] = k1 ∗ [A] dt d[D] =0 dt

(5)

The system of ordinary diﬀerential equations in Equation 5 was speciﬁcally designed to illustrate the eﬀects of the memory enhancement. We will compare the enhanced NWT algorithm with solutions to the systems of ODEs, providing two diﬀerences cases based on variable kinetics. By selecting diﬀerent kinetic rates, we will show how the memory enhancement leads to agreement between the NWT and ODEs for strictly deterministic runs, but with nondeterministic decisions it can lead to distinct results and a divergence in overall behavior of the biochemical network. For the sake of simplicity, we will assume the rate constants (k1 , k2 and k3 ) are already in discrete form. Therefore, when refer to ki in Ri above, we have constRi . A model similar to the one described in Table 1 could be used to investigate the dynamics of HIV-1 Tat protein, since it is initially transcribed at very low numbers [16]. Once Tat is assembled in the cytosol, it can be exocytised or translocated to the nucleus [20]. When Tat is translocated to the nucleus it can begin upregulating HIV-1 proteins (including itself). Since the downstream eﬀects of Tat translocation to the nucleus has profound impacts on the cell (causing upregulation of the HIV-1 proteins), a discrete and nondeterministic approach is beneﬁcial to the study of the dynamics of the low levels of Tat [23]. In the system, molecules of A are formed from molecules of D. This reaction can basically be viewed as a combined transcription and translation rule with D being the gene and A being the protein encoded by the gene. Once a molecule of A is formed, it has the option of turning into a molecule of B at rate k2 or it can turn into a molecule of C at a rate k1 . If we consider the species A to be analogous to HIV-1 Tat protein, then reaction R1 (A → B) could be the translocation of Tat from the cytosol to the nucleus, and reaction R2 (A → C) could be the translocation of Tat from the cytosol to the extracellular environment. We will next consider two cases for the model described in Table 1 and discuss the implementation of the memory enhancement. The cases are determined by the values for the discrete kinetic rates. The ﬁrst case shows that the memory enhancement can produce similar results between the ODEs and the NWT when no nondeterministic decisions are made. For the second case, we will show how the technique can produce diﬀerent results, illustrating the NWT algorithm’s

206

J. Jack and A. P˘ aun

ability to explore the stochastic nature of molecular signaling cascades. Note, the NWT algorithm remains the same in both cases, the only diﬀerence lies in the initialization of the discrete kinetic constants. 3.1

Case 1: Deterministic Memory Enhancement

If we let k1 = 4, k2 = 10, and k3 = 5, then we can see the results of a simulation using the NWT algorithm plotted against the solution of the system of ordinary diﬀerential equations (Figure 1).

25

Molecules

20

15

10

5

0

0

1

2

3

4

5

6

7

8

9

10

Time

Fig. 1. The graph shows the accumulation of C molecules throughout a 10 second run. The bars are the discrete results from the NWT algorithm and the black line is the solution of the system of ordinary diﬀerential equations. With the kinetic values (k1 = 4, k2 = 10, k3 = 5), there are no nondeterministic decisions for the entire length of the NWT simulation. Therefore, we are pleased to see the NWT algorithm results are similar to the solution to the system of ordinary diﬀerential equations.

At initialization (t = 0), there is exactly one molecule of D and one molecule of A. Therefore, from Equation 2, we see that all three reactions have real (ﬁnite) Waiting Times when the simulation begins. Moreover, we have W TR1 = 0.25, W TR2 = 0.1 and W TR3 = 0.2. According to the reaction Waiting Times, the ﬁrst reaction to occur is R2 , which immediately exhausts the system’s supply of A molecules, yields one molecule of B and a simulation time of t = 0.1. The Waiting Times for the rules aﬀected by the applied rule must be recalculated; since there are no molecules of A in the system, we have W TR1 = W TR2 = ∞. R3 does not use any proteins involved in the applied reaction (A or C), so W TR3 is left unchanged after the ﬁrst reaction is executed. After the heap maintenance, R3 will be at the top. The next reaction to be applied is R3 , which gives us a new molecule of A and a simulation time of t = 0.2. Now the memory enhancement plays a role. In the ﬁrst step, reaction R2 exhausts the supply of A molecules. When this occurs,

Discrete Modeling of Biochemical Signaling with Memory Enhancement

207

R1 has waited for 0.1 seconds of its total W T . The memory enhancement allows the simulator to keep track of the percentage of time waited. In other words, R1 waited for 0.1 seconds out of its required 0.25 seconds, which means it has waited 40% of its Waiting Time. We store the percentage of time left to wait (60%). So, when a new molecule of A is formed (for instance, at τ = 0.2), we can recalculate the W TR1 using the percentage to adjust its Waiting Time. Continuing after R3 is applied at time τ = 0.2, we have a new molecule of A. Since R1 and R2 both use A as a reactant, we must recalculate the Waiting Times of both reactions. The Waiting Time of R2 and R1 are calculated using Equation 2. However, the memory of R1 allows us to take 60% of its recalculated W T . Therefore, the Waiting Time of R2 is calculated as 0.1, but the Waiting Time of R1 is recalculated as 0.15. This number stems from the equation W TR1 = M em

1 k1 ∗ |C|

(6)

where M em is the percentage of time left to wait (60% in the example above). The second and third order reactions follow similarly. Using this implementation, our NWT algorithm agrees with the ODEs for a strictly deterministic run. In the next subsection, we will explore the implications of the memory enhancement in a system requiring reaction competition over low numbers of molecules. Although we agree with ODEs in a deterministic run, we want to explore the nondeterministic eﬀects of the memory enhancement. 3.2

Case 2: Nondeterministic Memory Enhancement

We will now assume diﬀerent kinetic constants to highlight the eﬀects of the nondeterministic component of the NWT algorithm in conjunction with the memory enhancement. Although the kinetics of our sample system are chosen in a deliberate manner in order to illustrate the nondeterministic eﬀects of the algorithm, we will later show in Section 4.2 how our nondeterministic logic can have similar implications in a model reported in the literature. For our next simulations, we assume k1 = 0.1, k2 = 1.0, and k3 = 0.5. The initial Waiting Times are calculated as W TR1 = 10, W TR2 = 1, and W TR3 = 2. In Figure 2, we see the accumulation of B and C molecules. The results of the ODE-based simulation are visibly diﬀerent than the results of the simulation involving the NWT algorithm. The reasons for the diﬀerences are the nondeterministic decisions on reaction competition for A molecules. Based on the initialized WT s, the ﬁrst reaction to be applied is R2 . After R2 is applied, the simulation time is aggregated (t = 1) and there are no more molecules of A present in the system. Hence, the WT s for R2 and R1 are both inﬁnite after recalculation. We store the percentage of time the slow reaction, R1 , had left to wait when the W T changed to inﬁnity – M emR1 = 90%. The next rule to be applied is R3 , since it was unaﬀected by the application of R2 . The simulation time is adjusted (t = 2), and we now have a new molecule of A. With our new A molecule available, we must recalculate the WT s for R1 and R2 .

208

J. Jack and A. P˘ aun

(a)

(b) Fig. 2. In both graphs we see the results of the ODE-based simulation (straight black line) and the results from the modiﬁed NWT algorithm. (a) The accumulation of molecules of B and (b) the accumulation of C are shown. Molecules of B and C both come from A molecules. However, the reaction for B is faster than the reaction for C. In the ODE models, a molecule of A can be used to partially satisfy B and C. Since our NWT algorithm is discrete, the molecules are nondeterministically chosen to satisfy one or the other. The reaction changing A into C ’remembers’ how long it has waited, and uses this information the next time a molecule of A is ready.

Using M emR1 , we calculate the W T for reaction R1 , using the fact that it need wait only 90% of its new Waiting Time. Therefore, when a new molecule of A is formed two seconds into the run, we recalculate W TR1 using Equation 6. In our case, we have W TR1 = 9 and W TR2 = 1. Continuing the calculations for the simulation, we skip ahead to a future event (t = 18). Up until this point, we have been creating molecules of A, and every single one of them has been deterministically chosen to change into molecule B via reaction R2 . But, at t = 18, a molecule of A has been created, and the Waiting Times of reaction R1 and R2 are equal W TR1 = W TR2 = 1, since we have M emR1 = 10%. In other words, R1 and R2 are each attempting to use the A molecule to form a C and B, resp. The ODE-based simulation has no

Discrete Modeling of Biochemical Signaling with Memory Enhancement

209

problems at this time-point, since it is continuously sending a fraction of each A to form a fraction of B and C, whereas our algorithm represents A discretely and only satisﬁes one reaction per molecule. Our algorithm faces the question: at t = 18 should the A molecule be allowed to satisfy R1 or R2 ? The algorithm answers the question by making a nondeterministic choice between R1 and R2 . If R1 is chosen, then it is applied, and our results stay with the ordinary diﬀerential equations results (up to t = 19). Remember, the ordinary diﬀerential equations have been slowly and continuously aggregating fractions of C molecules throughout to reach one full molecule of C by t = 19. However, if R2 is chosen, then our solution diverges from the previous solution. When the eﬀects of the nondeterministic decisions are aggregated over 1000 seconds, we see the diﬀerent results obtained from the NWT algorithm (Figure 2).

4 4.1

Other Models Lotka-Volterra Predator-Prey

The Lotka-Volterra predator-prey model depicts the interactions of two species. There is a prey population and a predator population, where P1 (t) and P2 (t) represent the number of each species respectively at time t. The model can be written as the following pair of ﬁrst-order, nonlinear, diﬀerential equations dP1 = P1 ∗ (a − b ∗ P2 ) dt dP2 = −P2 ∗ (c − d ∗ P1 ) dt

(7)

where prey species are born at a rate of a and consumed at a rate of b. Predator species are born at a rate of d and die at a rate of c. In Figure 3, we see a picture of the predator-prey model. The picture (as well as the SBML code for the model) was generated with CellDesigner [5,6], which we also used to generate the SBML code to initialize our simulator. The way the system is designed, an increase in prey leads to an increase in predator, and an increase in predator leads to a decrease in prey. Total annihilation of prey leads to total extinction of predator, since the food supply of the predator will be exhausted. We used three diﬀerent simulation techniques to model the reactions described in the Lotka-Volterra model: solution to the system of ordinary diﬀerential equations, the Gillespie Algorithm, and our NWT algorithm. The ODEs were solved in MATLAB, while the other two algorithms were both coded in C. The results are given in Figure 4 and Figure 5. The solution to the ordinary diﬀerential equations in Equation 7 shows consistent oscillations throughout the entire simulation run. The NWT shows dampened oscillations over time. The Gillespie Algorithm has diﬃculties producing the oscillations, due to the stochasticity of the algorithm. In this case, our NWT

210

J. Jack and A. P˘ aun

Fig. 3. The Lotka-Volterra model involves two interacting species. Prey species are born at a rate a and are consumed at a rate b by the predator species. The predator species are born at a rate of d if there is available food (prey). The way the system is designed, an increase in prey leads to an increase in predator. Total annihilation of prey leads to total extinction of predator, since there is no longer any food. The model deterministically leads to oscillatory behavior.

Lotka−Volterra (ODE)

Lotka−Volterra (NWT)

500

500 Prey Pred

400 350 300 250 200 150 100

400 350 300 250 200 150 100

50 0

(a)

Prey Pred

450

Molecules of protein R

Molecules of protein R

450

50 0

20

40

60

80

0

100

0

20

40

Time

60

Time

80

100

(b)

Lotka−Volterra (Gillespie) 500 Prey Pred

Molecules of protein R

450 400 350 300 250 200 150 100 50 0

(c)

0

20

40

60

80

100

Time

Fig. 4. Results of three simulation techniques for the Lotka-Volterra model (up to 100 seconds). (a) solution to ordinary diﬀerential equations, (b) the NWT algorithm, and (c) the Gillespie Algorithm.

Discrete Modeling of Biochemical Signaling with Memory Enhancement Lotka−Volterra (ODE)

Lotka−Volterra (NWT)

450

450 Prey Pred

350 300 250 200 150 100 50

(a)

Prey Pred

400

Molecules of protein R

Molecules of protein R

400

0

211

350 300 250 200 150 100 50

0

100

200

300

Time

400

500

0

0

100

200

300

Time

400

500

(b)

Fig. 5. Results of the two simulation techniques for the Lotka-Volterra model (up to 500 seconds). (a) solution to ordinary diﬀerential equations and (b) the NWT algorithm.

algorithm runs deterministically. The system is small enough and the dynamics are such that the NWT makes no nondeterministic decisions due to reaction competition. If we expand the results of the solution to system of ordinary diﬀerential equations and the NWT algorithm, we see further decline in the amplitude for the NWT algorithm. In Figure 5, we expand the simulation run for a total of ﬁve hundred seconds. The results for the solution to the system of ordinary diﬀerential equations and the NWT algorithm simulation are provided. We modeled this classic system to illustrate the diﬀerences in the results of our simulation technique compared to the solution of the system of ordinary diﬀerential equations, the NWT algorithm, and the Gillespie Algorithm simulations. Our system was able to exhibit oscillatory behavior, albeit the oscillations are damped. However, as you can see in Figure 5, the oscillations persist with the NWT algorithm (and the ordinary diﬀerential equations). Yet, the Gillespie Algorithm will always reach a steady state, whereby the predator and prey species will eventually completely disappear. Since there are no nondeterministic decisions made during the run, we can only attribute the dampened oscillations to the fact that the system is discrete. We will next discuss a circadian rhythm model, which will illustrate how our algorithm can produce Gillespie-like results, even though we have a reduced complexity. 4.2

Circadian Rhythm

Circadian rhythm models are often explored in nature. These act as internal clocks which allow organisms to anticipate daily changes in the environment [1] – for instance, when to hunt for food, when to rest, etc. Yet, at the level of cellular biochemistry, circadian rhythms have also been reported [4]. Biological systems run by internal clocks – that is, certain proteins are created at certain parts of the day. Therefore, simulating circadian rhythm models is important in understanding the way DNA is interpreted and pre-existing proteins waiting to be activated are used by the body for daily survival [1]. We have chosen to model the circadian rhythm model described in [21]. The system describes an activator and a repressor gene (A and R). These genes are

212

J. Jack and A. P˘ aun

transcribed into mRNA, which leads into the translation of the proteins. The activator A binds to the promoters for A and R and increases the transcription rate. The system of ordinary diﬀerential equations described in [21] showed that intrinsic biochemical noise enhanced the oscillations. In Equation 8, we see the system of ordinary diﬀerential equations for the model. dDA dt dDR dt dDA dt dDR dt dDMA dt dA dt dMR dt dR dt dC dt

= θA ∗ DA − γA ∗ DA ∗ A = θR ∗ DR − γR ∗ DR ∗ A = γA ∗ DR ∗ A − θA ∗ DA = γR ∗ DR ∗ A − θR ∗ DR = αA ∗ DA + αA ∗ DA − δMa ∗ MA = βA ∗ MA + θA ∗ DA + θR ∗ DR − A ∗ (γA ∗ DA + γR ∗ DR + γC ∗ R + δA ) = αR ∗ DR + αR ∗ DR − δMR ∗ MR

= βR ∗ MR − γC ∗ A ∗ R + δA ∗ C − δR ∗ R = γC ∗ A ∗ R − δA ∗ C

(8)

where A and R represent the number of activator and repressor proteins, DA and DA represent the number of activator genes with or without binding to A, DR and DR represent the number of repressor genes with or without binding to R, MA and MR represent mRNA molecules of A and R, and C represent the corresponding inactivated complex formed by A and R. Deterministic modeling techniques, like the solution to the systems of ordinary diﬀerential equations, for biochemical interactions fail to produce the oscillations of a circadian rhythm model. However, the stochastic noise from a Gillespiebased approach leads to repeated oscillations throughout an entire run. Our NWT algorithm can produce results similar to the Gillespie algorithm – genetic oscillations – but at a considerably reduced computational cost. The results for the simulation of the circadian rhythm model are shown in Figure 7. We present the results from Gillespie’s Algorithm, the solution of the system of ordinary diﬀerential equations (Equation 8), and our NWT algorithm. The NWT algorithm is able to reproduce the oscillations for the perturbed model, as is the case with the Gillespie approach [21]. Similar to Gillespie, the NWT shows some variability in both the amplitude – numbers of molecules – and the periodicity of oscillations. The authors in [21] showed that parameter values can have a profound impact on oscillations. By reduction of the kinetic rate governing R degradation, the deterministic results produce a single peak followed by a steady state, while a stochastic simulation remains oscillating. Our NWT algorithm also produces oscillations instead of a steady state, but at a reduced computational cost from

Discrete Modeling of Biochemical Signaling with Memory Enhancement

213

Fig. 6. The picture of this system was generated using CellDesigner. The system was described in [21]. We have modeled this system with the NWT algorithm, Gillespie’s Algorithm, and as a system of ordinary diﬀerential equations.

3000

3000

2500

2000

2000

1500

1000

500

0

(a)

ODE Gillespie

2500

Molecules

Molecules

ODE NWT

1500

1000

500

0

50

100

150

200

Time

250

300

350

400

0

0

50

100

150

200

Time

250

300

350

400

(b)

Fig. 7. The results for the circadian rhythm model: (a) the NWT algorithm and (b) the Gillespie Algorithm. Both algorithms are plotted against the solution to the system of ordinary diﬀerential equations.

the Gillespie Algorithm approach. This is the beneﬁt of modeling with the NWT algorithm instead of the Gillespie Algorithm. For our simulation to produce oscillations comparable to the Gillespie Algorithm, we require only 50 random numbers to be generated. This stems from the fact that the NWT algorithm relies on deterministic kinetics for the majority of reactions, but when reactants are limited and competition for reactants exists, nondeterministic decisions drive a variable response from the competing reactions.

5

Final Remarks

We have improved the sensitivity of our NWT algorithm through the addition of memory to Waiting Time calculation. We argue that this gives us an edge over ordinary diﬀerential equations in modeling reactions of low molecular multiplicity. The improvements were illustrated with multiple examples, one designed to

214

J. Jack and A. P˘ aun

speciﬁcally discuss the memory enhancement and two other models from the literature. In the ﬁeld of systems biology, there is a strong emphasis on using nondeterministic (or stochastic) techniques in modeling biochemical networks where low numbers of molecules can be found. We are interested in exploring these types of situations. For instance, in HIV-infected T cells, there are initially low levels of Tat protein, which after translocation to the nucleus, bind to receptor sites and cause upregulation of the HIV-1 proteins [16]. We have already published a paper on the eﬀects of HIV-1 proteins on Fas-mediated apoptosis, and will be looking to use our reﬁned algorithm for future development in this pathway. Also, in regards to T cells, it seems that low levels of cytochrome C released from the mitochondria bind to IP3 R. This receptor binding leads to release of Ca+ form the mitochondria, and [2] showed that this was necessary for both the extrinsic (Fas-mediated) and the intrinisic apoptotic pathways. We are exploring this direction via wetlab experimentation, and we will be using the NWT algorithm to elucidate new aspects to Fas-mediated apoptotic events. Acknowledgments. We gratefully acknowledge support in part from the LONI Institute: fellowship for J.J. and state-of-the-art parallel computing facilities, National Science Foundation Grant CCF-0523572, INBRE Program of the NCRR (a division of NIH), support from CNCSIS grant RP-13, support from CNMP grant 11-56 /2007, support from Spanish Ministry of Science and Education (MEC) under project TIN2006-15595, and support from the Comunidad de Madrid (grant No. CCG07-UPM/TIC-0386 to the LIA research group).

References 1. Barkai, N., Leibler, S.: Biological rhythms: Circadian clocks limited by noise. Nature 403, 267–268 (2000) 2. Boehning, D., van Rossem, D.B., Patterson, R.L., Snyder, S.H.: A peptide inhibitor of cytochrome c/inositol 1,4,5-triphosphate receptor binding blocks intrinsic and extrinisc cell death pathways. PNAS 102(5), 1466–1471 (2005) 3. Cheruku, S., P˘ aun, A., Romero-Campero, F., P´erez-Jim´enez, M., Ibarra, O.: Simulating FAS-Induced Apoptosis by Using P Systems. In: Proceedings of Bio-inspired computing: theory and applications (BIC-TA), Wuhan, China, September 18-22 (2006); also extended version published as Progress in Natural Science 17(4), 424–431 (2006) 4. Dunlap, J.: Circadian Rhythms: An End in the Beginning. Science 280(5369), 1548–1549 (1998) 5. Funahashi, A., Morohashi, M., Kitano, H.: CellDesigner: a process diagram editor for gene-regulatory and biochemical networks. BIOSILICO 1(5), 159–162 (2003) 6. Funahashi, A., Matsuoka, Y., Jouraku, A., Morohashi, M., Kikuchi, N., Kitano, H.: CellDesigner 3.5: A Versatile Modeling Tool for Biochemical Networks. Proceedings of the IEEE 96(8), 1254–1265 (2008) 7. Gillespie, D.T.: A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics 22, 403–434 (1976)

Discrete Modeling of Biochemical Signaling with Memory Enhancement

215

8. Gillespie, D.T.: Exact Stochastic Simulation of Coupled Chemical Reactions. Journal of Physical Chemistry 81(25), 2340–2361 (1977) 9. Gibson, M.A., Bruck, J.: Eﬃcient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels. Journal of Physical Chemistry A 104, 1876–1889 (2000) 10. Hoops, S., et al.: COPASI – a Complex Pathway Simulator. Bioinformatics 22(24), 3067–3074 (2006) 11. Hua, F., Cornejo, M., Cardone, M., Stokes, C., Lauﬀenburger, D.: Eﬀects of bcl2 levels on fas signaling-induced caspase-3 activation: molecular genetic tests of computational model predictions. The Journal of Immunology 175(2), 985–995 (2005); Correction 175(9), 6235–6237 (2005) 12. Hucka, M., et al.: The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics 19(4), 524–531 (2003) 13. Jack, J., Romero-Campero, F.J., Perez-Jimenez, M.J., Ibarra, O.H., P˘ aun, A.: Simulating Apoptosis Using Discrete Methods: A Membrane System and a Stochastic Approach. Language Theory in Biocomputing (2007) 14. Jack, J., Rodriguez-Paton, A., Ibarra, O.H., P˘ aun, A.: Discrete Nondeterministic Modeling of the FAS Pathway. Int. J. Found. Comput. Sci. 19(5), 1147–1162 (2008) 15. Jack, J., P˘ aun, A., Rodriguez-Paton, A.: Eﬀects of HIV-1 Proteins on the Fasmediated Apoptotic Signaling Cascade: A Computational Study of T cell Latency. In: Proceedings of WMC9: 2008. LNCS, vol. 5391, pp. 246–259 (2009) 16. Karn, J.: Tackling Tat. Journal of Molecular Biology 2(22), 235–254 (1999) 17. Kitano, H.: Computational Systems Biology. Nature 420 (2002) 18. Kitano, H.: Systems Biology: A Brief Overview. Science 295, 55–60 (2002) 19. Ma, L., Rice, J.J., Hu, W., Levine, A.J., Stolovitzky, G.A.: A plausible model for the digital response of p53 to DNA damage. PNAS 102(40), 14266–14271 (2005) 20. Selliah, N., Finkel, T.: Biochemical mechanisms of HIV induced T cell apoptosis. Cell Death and Diﬀerentiation 8, 127–136 (2001) 21. Vilar, J.M.G., et al.: Mechanisms of noise-resistance in general oscillations. PNAS 99(9), 5988–5992 (2002) 22. Wagner, J., Ma, L., Rice, J.J., Hu, W., Levine, A.J., Stolovitzky, G.A.: p53-Mdm2 loop controlled by a balance of its feedback strength and eﬀective dampening using ATM and delayed feedback. IEE Proc.-Syst. Biol. 152(3), 109–118 (2005) 23. Weinberger, L., Burnett, J., Toettcher, J., Arkin, A., Schaﬀer, D.: Stochastic Gene Expression in a Lentivrial Positive-Feedback Loop: HIV-1 Tat Fluctuations Drive Phenotypic Diversity. Cell 122(2), 169–182 (2005) 24. Wiley, H.S., Shvartsman, S.Y., Lauﬀenburger, D.A.: Computational modeling of EGF-receptor system: a paradigm for systems biology. TRENDS in Cell Biology 13(1), 43–50 (2003)

Dynamical Systems and Stochastic Programming: To Ordinary Diﬀerential Equations and Back Luca Bortolussi1 and Alberto Policriti2,3 1 2

Dept. of Mathematics and Computer Science, University of Trieste, Italy [email protected] Dept. of Mathematics and Computer Science, University of Udine, Italy [email protected] 3 Applied Genomics Institute (IGA), Udine, Italy [email protected]

Abstract. In this paper we focus on the relation between models of biological systems consisting of ordinary diﬀerential equations (ODE) and models written in a stochastic and concurrent paradigm (sCCP stochastic Concurrent Constraint Programming). In particular, we deﬁne a method to associate a set of ODE’s to an sCCP program and a method converting ODE’s into sCCP programs. Then we study the properties of these two translations. Speciﬁcally, we show that the mapping from sCCP to ODE’s preserves rate semantics for the class of biochemical models (i.e. chemical kinetics is maintained) and we investigate the invertibility properties of the two mappings. Finally, we concentrate on the question of behavioral preservation, i.e if the models obtained applying the mappings have the same dynamics. We give a convergence theorem in the direction from ODE’s to sCCP and we provide several well-known examples in which this property fails in the inverse direction, discussing them in detail.

1

Introduction

The systemic approach to biology is nowadays a fertile and growing research area, considered by many as a promising track to the understanding of life [41,1]. A key ingredient of systems biology resides in coupling wet lab experiments with mathematical modeling and analysis of bio-systems [33]. Many mathematical instruments have been used for this purpose, some concerned with qualitative analysis, others encapsulating also quantitative data [32]. Quantitative modeling is essentially dominated by two main mathematical tools: (ordinary) diﬀerential equations on one side and stochastic processes on the other [32]. Both these methods are concerned with the study of dynamical evolution of systems; however, they diﬀer in the description of the quantities of interest: diﬀerential equations represent them as continuous variables, stochastic processes operate, instead, on discrete quantities. Modeling formalisms mixing discrete and continuous ingredients, like hybrid automata [29], have also been used in modeling bio-systems [2]. C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 216–267, 2009. c Springer-Verlag Berlin Heidelberg 2009

Dynamical Systems and Stochastic Programming

217

We will focus here mainly on the two former approaches, although commenting on the theme of the use of hybrid systems for restoring behavioral equivalence throughout the paper. The theory of dynamical systems and diﬀerential equations (ODE’s) is very attractive, being it a mature research area equipped with a huge set of analysis tools, ranging from static analysis of phase space topology to fast simulation via numerical integration [51,43]. However, writing ODE’s for a given system is generally a diﬃcult task, requiring a considerable expertise. In addition, the representation of biological entities as continuous variables is an approximation that can sometimes be too rough, especially for low populations [26]. Stochastic processes like Continuous Time Markov Chains (CTMC) [39], on the other hand, do not suﬀer from these approximation limits, as they represent biological entities as discrete quantities, thus being more adherent to reality. On the other hand, analyzing a stochastic model is much more diﬃcult, both from an analytical and from a computational point of view [53]. Regarding the description of stochastic models, recently we have seen the application of stochastic process algebras (SPA) [48,45], a class of formal languages developed in theoretical computer science as formal tools to analyze (quantitatively) the performances of computing networks. These languages allow to build CTMC-based models following a simple, paradigmatic, identiﬁcation of biological entities with (computing) processes. Moreover, they are compositional, allowing to build models by composing together sub-models. Ideally, one would like to have a modeling technique that collects the advantages both of stochastic process algebras and diﬀerential equations, or, at least, to switch automatically between the two formalisms, depending on the particular task to be performed. In this direction, there are two related problems that must be faced: (a) studying the (mathematical) relation between the two modeling techniques and (b) ﬁnding automatic methods for converting one formalism into the other. More speciﬁcally, we suggest the following workﬂow: ﬁrst deﬁning translation methods (for a speciﬁc process algebra), thus tackling (b), and then studying the mathematical relations intervening between the models obtained applying these translations. In this way we should be able to evaluate the appropriateness of conversion procedures between SPA and ODE’s and to restrict the focus of the analysis required by (a). There are two directions in the conversion between SPA and ODE: the ﬁrst one associating a set of diﬀerential equations to a stochastic process algebra model, and the inverse one, mapping diﬀerential equations to stochastic process algebra programs. The ﬁrst direction can be helpful for the analysis of SPA models, as ODE’s can be solved and analyzed more eﬃciently. Associating SPA to ODE, instead, can help to clarify the logical pattern of interactions that are hidden in the mathematical structure of diﬀerential equations. Generally, as process algebra models can be written much more easily than diﬀerential equations, even by non-experts (possibly via a graphical interface), the ﬁrst direction, from

218

L. Bortolussi and A. Policriti

SPA to ODE, looks potentially more fruitful, though having both mappings helps the study of the relationship between the two formalisms. Supposing to have such transformations at our disposal, a crucial problem is to single out criteria to evaluate and validate them. The ﬁrst possibility is to inspect the relationship intervening between a SPA program and the associated ODE’s only from a mathematical point of view, forgetting any information about the system modeled. As both stochastic processes and diﬀerential equations are dynamical systems, this approach essentially corresponds to require that both models exhibit the same behavior, i.e. the same dynamical evolution. Of course, we may require agreement only from a qualitative point of view (so that the qualitative features of the dynamics are the same) or even from a quantitative one (numerical values agree). The diﬃculty with this approach is that stochastic processes have a noisy evolution, in contrast with the determinism characterizing diﬀerential equations. Hence, we need to remove the noise. One possibility is to look only at qualitative features of the dynamics, deﬁning them in a precise way; we will go back to this problem in Section 3.2 below. Otherwise, we may average out noise from the stochastic models, thus considering the expected evolution of the system and requiring it to be described precisely (i.e. quantitatively) by the ODE’s. Unfortunately, noise cannot be eliminated so easily, as sometimes it is the driving force of the dynamics [26,52]. Therefore, this second form of equivalence is not completely justiﬁed; we will comment more on this point while discussing some examples in the following. A diﬀerent approach in comparing stochastic and diﬀerential models can be deﬁned if we consider some additional information, which is external to the mathematics of the two models. The idea is to validate the translation w.r.t. this additional information. We explain this point with an example. Consider a model of a set of biochemical reactions; there are diﬀerent chemical kinetic theories that can be used to describe such system, the most famous one being the principle of mass action. Using such a kinetic theory, we can build (in a canonical way) both a model based on diﬀerential equations and a model based on stochastic process algebras. If we are concerned with the principle of mass action more than with dynamical behavior, we may ask that our translation procedures preserves the former, meaning that the ODE’s associated to a mass action SPA program are exactly the ODE’s built according to mass action principle, and viceversa. Essentially, this corresponds to requiring that the translation procedures deﬁned are coherent with (some) principles of the system modeled. For instance, in the case of mass action, coherency corresponds to preserve the meaning of rates (the so called rate semantics in [17]). Notice that in this case we are not requiring anything about dynamics, so coherent models may exhibit a divergent behavior, and this is indeed a well known issue, see, for instance, [26] or Section 3.2 below. Therefore, this comparison is essentially diﬀerent from the behavioral-based one, and it is essentially syntactic, in the sense that it is concerned only with how models are written, not with their time evolution. The operation of associating ODE’s to SPA can be seen also as the deﬁnition of an ODE-based semantic for the stochastic processes, as opposed to the

Dynamical Systems and Stochastic Programming

219

CTMC-based one. Consequently, the comparison of the stochastic model with the derived ODE’s can also be seen as an attempt to discover the mathematical relationship between these two semantics. The problem of associating ODE’s to stochastic process algebras has been tackled only recently in literature. The forefather is the work of Hillston [31], associating ODE’s to models written in PEPA [30], a stochastic process algebra originally designed for performance modeling. Successively, similar methods have been developed for stochastic π-calculus [16,11,44] and for stochastic Concurrent Constraint Programming [7,12]. All these methods build the ODE’s performing a syntactic inspection and manipulation of the set of agents deﬁning the SPA model. In fact, they all satisfy the coherency condition staten above, at least for mass action principle (a proof for stochastic π-calculus can be found in [17]). The inverse problem of associating SPA models to ODE’s has received much less attention, the only example being [12], where we use stochastic Concurrent Constraint Programming as target SPA. In this paper, we will retake the work previously done for sCCP in [12], presenting it in a more detailed and formal way. Basically, we will deﬁne two translation procedures: from sCCP to diﬀerential equations and viceversa. sCCP plays here a central role, thanks to some ingredients giving a noteworthy ﬂexibility to it, the presence of functional rates above all. Therefore, in Section 2, we will recall the basics of sCCP and its application as a modeling language for biological systems, as presented in [15]. Further details on the language can be found in [8]. The translation procedure from sCCP to diﬀerential equations is presented formally in Section 3, while Section 4 is devoted to the presentation of the inverse mapping from general ODE’s into sCCP. In Section 3, we will also show coherency conditions for a class of chemical kinetics and we will comment in detail the problem of behavioral equivalence in the conversion from sCCP to ODE’s. This will be done mainly via examples, exhibiting biological systems for which the translation preservers also the behavior and other systems whose stochastic models show a diﬀerent behavior than ODE’s. The problem of behavioral equivalence is not new, and in fact some examples that we will give are famous ones [26]. However, the syntactic structure of process algebras in general, and sCCP speciﬁcally, give a new ﬂavor to these classical examples, and brings the attention into new ones. The issue of preservation of dynamic behavior in the mapping from ODE’s to sCCP is tackled in Section 4. In this case we are able to exploit the structure of the mapping and thus to give a convergence theorem. Throughout the paper, we will encounter several situations in which discreteness is a crucial ingredient for the dynamics of the system. This points to a third class of dynamical systems that is in the middle between SPA and ODE’s and that can be used to approximate them, namely hybrid automata [29]. In [13,14] we deal with the problem of mapping sCCP programs into hybrid automata, showing that in this case we are able to deal correctly from a behavioral viewpoint with a broader class of sCCP systems. The idea of such mapping is that of translating to ODE’s locally while retaining some level of discreteness in the

220

L. Bortolussi and A. Policriti

ﬁnite control of the hybrid automata. By the way, the method of [13] can be extended into a general framework encompassing also the mapping presented in this paper as a particular case.

2

Preliminaries

In this section we brieﬂy recall the basics of stochastic Concurrent Constraint Programming (sCCP, Section 2.1) and its application as a modeling language for biological systems (Section 2.2). The interested reader is referred to [8] for further details. In Section 2.3, we introduce some restrictions on the language that greatly simplify the mapping from and to ODE’s. 2.1

Stochastic Concurrent Constraint Programming

Concurrent Constraint Programming (CCP, [49]) is a process algebra having two distinct entities: agents and constraints. Constraints are interpreted ﬁrst-order logical formulae, stating relationships among variables (e.g. X = 10 or X+Y < 7). Agents in CCP, instead, have the capability of adding constraints (tell) into a sort of global memory (the constraint store) and checking if certain relations are entailed by the current conﬁguration of the constraint store (ask). The communication mechanism among agents is therefore asynchronous, as information is exchanged through global variables. In addition to ask and tell, the language has all the basic constructs of process algebras: non-deterministic choice, parallel composition, procedure call, plus the declaration of local variables. The stochastic version of CCP (sCCP [7,15]) is obtained by adding a stochastic duration to all instructions interacting with the constraint store C, i.e. ask and tell. Each instruction has an associated random variable representing time (thus taking values in the positive reals), exponentially distributed with rate given by a function associating a real number to each conﬁguration of the constraint store: λ : C → R+ . This is a unusual feature in traditional stochastic process algebras like PEPA [30] or stochastic π-calculus [44] (although recently introduced in BioPEPA [18]), and it will be crucially used in the translation mechanisms. The syntax of sCCP can be found in Table 1. Two diﬀerent kind of actions are present in such table: stochastic actions, having a rate attached to them, and instantaneous actions, having an inﬁnite rate. Table 1. Syntax of sCCP P rogram = D.A D = ε | D.D | p(x) : −A π = tellλ (c) | askλ (c) M = π.G | M + M G = 0 | tell∞ (c).G | p(y) | M | ∃x G | G G A = 0 | tell∞ (c).A | M | ∃x A | A A

Dynamical Systems and Stochastic Programming

221

This second class of actions can be used to model the happening of complex atomic events, like a sequence of store updates happening instantaneously. However, only tell actions can happen instantaneously, and moreover they are always guarded by a stochastic action. The same restriction applies to recursive calls. Operational Semantics. The deﬁnition of the operational semantics is given specifying two diﬀerent kinds of transitions: one dealing with instantaneous actions and the other with stochastically timed ones. The basic idea of this operational semantics is to apply the two transitions in an interleaved fashion: ﬁrst we apply the transitive closure of the instantaneous transition, then we do one step of the timed stochastic transition. To identify a state of the system, we need to take into account both the agents that are to be executed and the current state of the store. Therefore, a conﬁguration will be a point in the space P × C, where P is the space of agents and C is the space of all possible conﬁgurations of the constraint store. The instantaneous transition −→⊆ (P × C) × (P × C) and the stochastic transition =⇒⊆ (P × C) × [0, 1] × R+ × (P × C) are deﬁned according to the structural rules of Tables 2 and 3, respectively. The fact that instantaneous actions and recursive calls are guarded by stochastic actions guarantees that −→ can be applied only for a ﬁnite number of steps. −−−→ Moreover, it can be proven to be conﬂuent, see [8]. With the notation A, d of Table 3, we denote by the conﬁguration obtained by applying the transitions −→ as long as it is possible (i.e., by applying the transitive closure of −→). The −−−→ conﬂuence property of −→ implies that A, d is well deﬁned. The stochastic transition =⇒, instead, is labeled by two numbers: intuitively, the ﬁrst one is the probability of the transition, while the second one is its global rate. Note that, after performing one step of the transition =⇒, we apply the transitive closure of −→. This guarantees that all actions enabled after one =⇒ step are timed. Using relation =⇒, we can build a labeled transition system, whose nodes are conﬁgurations of the system and whose labeled edges correspond to derivable Table 2. Instantaneous transition for stochastic CCP

(IR1) tell∞ (c).A, d −→ A, d c (IR2)

p(x), d −→ A[x/y], d

if p(y) : −A

(IR3)

∃x A, d −→ A[y/x], d

with y fresh

(IR4)

A1 , d −→ A1 , d A1 A2 , d −→ A1 A2 , d

222

L. Bortolussi and A. Policriti

Table 3. Stochastic transition relation for stochastic CCP. The function rate : P ×C → R assigns to each agent its global rate. Its eﬀect is to recursively traverse the syntactic tree of agents, adding up the rates of active stochastic actions. Its formal deﬁnition can be found in [15].

(SR1)

−−−−−−→ tellλ (c).A, d =⇒(1,λ(d)) A, d c

(SR2)

−−−→ askλ (c).A, d =⇒(1,λ(d)) A, d

(SR3)

−−−−−→ M1 , d =⇒(p,λ) A1 , d −−−−−→ M1 + M2 , d =⇒(p ,λ ) A1 , d pλ with p = λ+rate(M and λ = λ + rate(M2 , d) 2 ,d)

(SR4)

−−−−−→ A1 , d =⇒(p,λ) A1 , d −−−−−−−−−→ A1 A2 , d =⇒(p ,λ ) A1 A2 , d pλ with p = λ+rate(A and λ = λ + rate(A2 , d) 2 ,d)

if d c

steps of =⇒. As a matter of fact, this is a multi-graph, as we can derive more than one transition connecting two nodes. Starting from this labeled graph, we can build a Continuous Time Markov Chain (cf. [39] and brlow) as follows: substitute each label (p, λ) with the real number pλ and add up the numbers labeling edges connecting the same nodes. Continuous Time Markov Chains. A Continuous Time Markov Chain (CTMC for short) [39] is a continuous-time stochastic process (Xt )t≥0 taking values in a discrete set of states S and satisfying the memoryless property: P {Xtn = sn | Xtn−1 = sn−1 , . . . , Xt1 = s1 } = P {Xtn = sn | Xtn−1 = sn−1 }, (1) for each n, t1 , . . . , tn , s1 , . . . , sn . A CTMC can be represented as a directed graph whose nodes correspond to the states of S and whose edges are labeled by real numbers, which are the rates of exponentially distributed random variables. In each state there are usually several exiting edges, competing in a race condition in such a way that the fastest one is executed. The time employed by each transition is drawn from the random variable associated to it. When the system changes state, it forgets its past activity and starts a new race condition (this is the memoryless property). Therefore, the traces of a CTMC are made by a sequence of states interleaved by variable time delays, needed to move from one state to another. The time evolution of a CTMC can be characterized equivalently by computing, in each state, the normalized rates of the exit transitions and their sum (called the exit rate). The next state is then chosen according to the probability distribution deﬁned by the normalized rates, while the time spent for

Dynamical Systems and Stochastic Programming

223

the transition is drawn from an exponentially distributed random variable with parameter equal to the exit rate. This second characterization is at the basis of several stochastic simulation algorithms for CTMC, like the well-known Gillespie’s one [26]. Stream Variables and Implementation. Some variables of the system, like those used in the deﬁnition of rate functions, need to store a single number that may vary over time. Such variables, for technical reasons, are conveniently modeled as variables of the constraint store, which, however, must be rigid (over time). To deal with this problem we store time varying parameters as growing lists with an unbounded tail variable. In order to avoid heavy symbolism, we will use a natural notation where X = X + 1 has the intended meaning of1 : “extract the last ground element n in the list X, consider its successor n + 1 and add it to the list (instantiating the old tail variable as a list containing the new ground element and a new tail variable)”. We refer to such variables as stream variables. An interpreter for the language is available and can be used for running simulations. This interpreter is written in Prolog and uses standard constraint solver on ﬁnite domains as manager for the constraint store. All simulations of sCCP shown in the paper are performed with it. 2.2

Modeling Biological Systems in sCCP

In [8,15] we argued that sCCP can be conveniently used for modeling biological systems. In fact, while maintaining the compositionality of process algebras, the presence of a customizable constraint store and of variable rates gives a great ﬂexibility to the modeler, so that diﬀerent kinds of biological systems can be easily described within this framework. In [15], we showed that biochemical reactions and genetic regulatory networks are easily handled by sCCP. In [8] we added to this list also formation of protein complexes and the process of folding of a protein, whose description requires knowledge about spatial position of amino acids constituting the protein (a kind of information easily added building on expressive potential of the constraint store). Finally, in [10] we showed how sCCP can be used to encode Kohn maps [34], a graphical formalism capable of describing implicitly biochemical networks subject to combinatorial explosion of the number of diﬀerent kinds of protein complexes. In this case, the power of the constraint store is used to maintain a graph-based representation of complexes, allowing a linear description of Kohn Maps (i.e., the encoding requires a linear number of characters w.r.t. the ones needed to describe a Kohn map). We recall now the modeling in sCCP of biochemical reactions. A general biochemical reaction has the form R1 + . . . + Rn →f (R,X;k) P1 + . . . + Pm , 1

(2)

The use of primed variables to denote values taken at the next time step is typical of model checking and is not to be confused with ﬁrst derivatives (for which we will used dotted variables, as time is the only independent variable).

224

L. Bortolussi and A. Policriti

where R1 , . . . , Rn are the reactants and P1 , . . . , Pm are the products. The realvalued kinetic function of the reaction is f (R, X; k), depending on the reactants R, on other molecules X acting as modiﬁers, and on some parameters k. This function can be one of the many used in biochemistry (cf. [20,50]) and it is required to satisfy the following boundary condition: it must be zero whenever one reactant is less than its amount consumed by the reaction. For instance, if a reactant R appears two times in the left hand size of (2), then f must be zero for R = 0, 1.2 Biochemical networks can be easily modeled in sCCP taking a reaction-centric approach, where each reaction (or action capability) is associated to a process, while molecules, whose concentration varies over time, are represented by integer variables of the constraint store (actually, stream variables). Moreover, the presence of non-constant rates allows to describe reactions with arbitrary chemical kinetics. More speciﬁcally, to each reaction like (2), we associate the following sCCP agent: f -reaction(R, X, P, k) : m n tellf (R,X;k) i=i (Ri − 1) ∧ j=i (Pj + 1) . f -reaction(R, X, P, k) This agent is a simple recursive loop, modifying the value of reactants’ and products’ variables at a speed given by the kinetic law. Note that the boundary conditions for the rate function f imply that no stream variable will ever become negative, as all reactions that may produce this eﬀect have rate zero3 . In Table 4 we give a list of some of the most common kinetics: mass action, MichaelisMenten and Hill kinetics. In order to describe genetic regulatory networks, instead, we use a modeling style mixing the reaction-centric point of view with the more classical molecularcentric one. Essentially, genes are described by sCCP agents, while proteins are associated to stream variables, like for biochemical reactions. An example of a genetic network can be found in Section 3.2. More information and examples on modeling biological systems in sCCP can be found in [15]. 2.3

Restricted sCCP

The mapping between sCCP and ODE’s is not deﬁned for the whole sCCP language, but rather for a restricted version of it, which is, however, suﬃcient to describe biochemical reaction and genetic networks. This restricted version of sCCP will be denoted in the following by restricted(sCCP ), and is formally speciﬁed by the following deﬁnition: 2

3

In case of mass action kinetics, this condition means that the rate for R + R → P must be kR(R − 1) and not kR2 . This is, however, consistent with the deﬁnition of the mass action principle in the stochastic setting. Boundary conditions for f may be relaxed by checking explicitly with ask instructions that variables stay within their domain. For instance, for the reaction R + R → P , we can precede tell by ask(R > 1) . This allows us to use the more common kR2 as rate function.

Dynamical Systems and Stochastic Programming

225

Table 4. List of some of the most common types of biochemical reaction, taken from [50]. The ﬁrst three are ﬁrst and second order mass-action-like reactions. The second arrow corresponds to a reaction with Michaelis-Menten kinetics. The last arrow replaces Michaelis-Menten kinetics with Hill’s one (see [20]).

R →k P1 + . . . + Pm

fma (R; k) = kR

R1 + R2 →k P1 + . . . + Pm

fma (R1 , R2 ; k) = kR1 R2

R + R →k P1 + . . . + Pm

fma (R; k) = kR(R − 1)

S →E k,v P

fM M (S, E; k, v) =

S →E K,V0 ,h P

fHill (S, E; h, k, v) =

vES k+S vES h k+S h

Definition 1. A restricted(sCCP ) program is a tuple (P rog, X, init(X)) satisfying: 1. P rog is an sCCP-program respecting the grammar deﬁned in Table 5. 2. The variables used in the deﬁnition of agents are taken from a ﬁnite set X = {X1 , . . . , Xn } of global stream-variables, each with the same domain D, usually D = N or, more generally, D = Z. 3. The only admissible updates for variables {X1 , . . . , Xn } are constraints of the form Xi = Xi + k or Xi = Xi − k, with k ∈ D constant. 4. Constraints that can be checked by ask instructions are ﬁnite conjunctions of linear equalities and inequalities. 5. The initial conﬁguration of the store is speciﬁed by the formula init(X), consisting in the following conjunction of constraints: (X1 = x01 )∧. . .∧(Xn = x0n ), with the constants x0i ∈ D referred to as the initial values of the sCCPprogram. This deﬁnition can be justiﬁed looking at the sCCP-agent associated to a biochemical reaction and also at the sCCP-model of genes considered in [15]. In fact, in these cases all employed variables are numerical variables of the stream-type4 , while all updates in the store add or subtract them a predeﬁned constant quantity. Guards, instead, usually check if some molecules are present in the system (X > 0), though we consider here the more general case of linear equalities and inequalities. The use of global variables only, instead, can be justiﬁed noting that the existential operator ∃x is never used (neither in the 4

We do not need further types of variables, as we just need to count the number of diﬀerent molecules in the system.

226

L. Bortolussi and A. Policriti Table 5. Syntax of the restricted version of sCCP P rog = Def.N Def = ε | Def.Def | p : −A π = tellλ (c) | askλ (c) M = π.G | M + M G = tell∞ (c).G | p | M A=0|M N =A|AN

reaction agent nor in gene models of [15]), as the scope of molecular interactions is system-wide. The suppression of the operator ∃x , as a side consequence, guarantees that we can avoid to pass parameters to procedure calls: in fact, each procedure can be deﬁned as operating on a speciﬁc subset of global variables. However, parameter passing is used in Section 2.2 to deﬁne parametrically the reaction agent. Therefore, we agree that each instance of a reaction agent, say f -reaction(R, X, P, k), is replaced with the corresponding ground form f -reaction(R,X,P,k). The same trick will be used for other agents. We demand further comments on the restrictions in Section 3.4. In order to ﬁx the notation in the rest of the paper, we give the following deﬁnition: Definition 2. A restricted(sCCP ) agent A not containing any occurrence of the parallel operator is called a sequential component or a sequential agent. A restricted(sCCP ) agent N is called an sCCP-network if it is the parallel composition of sequential agents. Inspecting the grammar of Table 5, we can observe that the initial conﬁguration of a restricted(sCCP ) program is indeed an sCCP-network. The following property is straightforward: Lemma 1. The number of sequential components forming an sCCP-program (P rog, X, init(X)) remains constant at run-time and equals the number of sequential agents in the sCCP-network of the initial conﬁguration. Proof. As sequential components do not contain any parallel operator, no new agents can be forked at run-time.5 In the rest of the paper, for notational convenience, we usually identify an sCCPprogram with the corresponding sCCP-network. Moreover, forbidding the deﬁnition of local variables implies the following property: Lemma 2. The number of variables involved in the evolution of an sCCPnetwork6 is a subset of {X1 , . . . , Xn }, hence ﬁnite. 5 6

We are counting also deadlocked agents. A variable is involved in the evolution of the network if one of the following things happen: it is updated in a tell instruction, it is part of a guard checked in an ask instruction, or it is used in the deﬁnition of a rate function.

Dynamical Systems and Stochastic Programming

227

The restrictions of restricted(sCCP ) are in the spirit of those introduced in [31]: we are forbidding an inﬁnite unfolding of agents and we are considering global interactions only, forcing the speed of each action to depend on the whole state of the system. Indeed, also in [16] we ﬁnd similar restrictions, though the comparison with sCCP is subtler. First of all, the version of π-calculus presented in [16] does not allow the use of the restriction operator, meaning that interactions have a global scope. However, agents in the π-calculus of [16] are not sequential, as each process is associated to a single molecule and the production of new molecules is essentially achieved by forking processes at run-time. This is not necessary in sCCP, as sCCP-agents model reactions, while molecules are identiﬁed by variables of the system. What is ﬁnite in [16], however, is the number of syntactically diﬀerent agents that can be present in a system. On the Restrictions of the Language. restricted(sCCP ) limits the full language in three main aspects: the allowance of sequential agents only, the suppression of local variables and the simpliﬁcations on the constraint store, cf. Deﬁnition 1. We will, however, comment here only on the ﬁrst one, as the other two are forced by the translation to ODEs, hence their discussion will be postponed to Section 3.4. As observed, in restricted(sCCP ) we constrain all the agents to be sequential, i.e. no occurrence of the parallel operator is allowed. Essentially, sequential agents are automata cooperating together, a property that will be exploited in the next section to represent them graphically in a simple way. Indeed, this restriction is only apparent: we can always convert a non-sequential agent into a network of sequential ones using additional (stream) variables of the constraint store. The idea is simply that of identifying all the syntactically diﬀerent terms that are stochastic choices, associating a new variable to each of them. These variables are used to count the number of copies of each term that are in parallel. Each agents is modiﬁed consequently: all agents will only call recursively themselves, while the variations induced in the number of terms by transitions are dealt with by updating the new state variables. Finally, rates are corrected by multiplying them by the multiplicity variable associated to the agent executing the corresponding transition. This is justiﬁed by the fact that in Markovian models, the global rate of a set of actions is computed by adding all basic rates together—ultimately, a consequence of the properties of the exponential distribution [39]. For instance, consider the agents x and y, deﬁned by x :- tell1 (true).(y

y), y :- tell1 (true).x + tell1 (true).0. They can be made sequential by introducing two variables, X and Y , counting the number of copies of x and y respectively and by replacing x by x :- askX (X > 0).tell∞ (X = X − 1 ∧ Y = Y + 2).x and y by y :- askY (Y > 0).tell∞ (X = X + 1 ∧ Y = Y − 1).y + askY (Y > 0).tell∞ (Y = Y − 1).y .

3

From sCCP to ODE

In this section we deﬁne a translation method associating a set of ordinary differential equations to an sCCP program. This translation applies precisely to

228

L. Bortolussi and A. Policriti

restricted(sCCP ), as deﬁned above. The procedure is organized in several simple steps, illustrated in the following paragraphs. Essentially, we ﬁrst associate a ﬁnite graph to each sequential component of an sCCP network and then, analyzing the graph, we deﬁne an interaction matrix similar to the one deﬁned in [31] or to action matrices of (stochastic) Petri nets (see, for instance, [27]). Writing ODE’s from this matrix is then almost straightforward. After deﬁning this translation, in Section 3.1 we investigate how it relates to biochemical kinetics and we show that the ODE’s associated to an sCCP-model of a set of biochemical reactions are the ones generally considered in standard biochemical praxis [17,20]. Some considerations on dynamical properties are then put forward in Section 3.2, while in Section 3.3 the focus is moved on the concept of behavioral equivalence. Finally, in Section 3.4 we reconsider the restrictions applied to sCCP in the light of the described transformation procedure. Step 1: Reduced Transition Systems The ﬁrst step consists in associating a labeled graph, called reduced transition system [8], to each sequential agent composing the network. As a working example, we consider the following simple sCCP agent: RWX :ask1 (X > 0).tell∞ (X = X − 1).RWX + tell1 (X = X + 2).RWX + askf (X) (true).( ask1 (X > 1).tell∞ (X = X − 2).RWX + tell1 (X = X + 1).RWX f (X) =

1 X 2 +1

This agent performs a sort of random walk in one variable, increasing or decreasing its value by 1 or 2 units, depending on its inner state. Inspecting Table 5, where the syntax of restricted(sCCP ) is summarized, we observe that each branch of a stochastic choice starts with a stochastic timed instruction, i.e. an askλ (c) or a tellλ (c), followed by zero or more tell∞(c), followed by a procedure call or by another stochastic choice. The ﬁrst operation that we need to perform, in order to simplify the structure of agents, is that of collapsing each timed instruction with all the instantaneous tell instructions following it and replacing everything with one “action” of the form action(c, d, λ), where c is a guard that must be entailed by the store for the branch to be entered, d is the constraint that will be posted to the store, and λ is the stochastic rate of the branch, i.e. a function λ : C → R+ ∪ {∞}. The presence of ∞ among possible values of λ is needed to simplify the treatment of instantaneous tells. To achieve this goal, we formally proceed by deﬁning a conversion function, named compact, by structural induction on terms. The result of this function

Dynamical Systems and Stochastic Programming

229

Table 6. Syntax of compact(sCCP ) ˆ .Aˆ P rog = Def ˆ ˆ ˆ ˆ Def = ε | Def .Def | p : −A ˆ ˆ ˆ ˆ π ˆ = action(g, c, λ) M =π ˆ ;G | M ⊕ M ˆ ˆ ˆ ˆ G = p | M A = 0 | M ˆ =A ˆ|A ˆN ˆ N

is that of transforming an agent written in restricted(sCCP ) into an agent of a simpler language, called compact(sCCP ), where ask and tell are replaced by the instruction action. In order to distinguish between the two languages, we denote stochastic summation in compact(sCCP ) by “⊕”, sequential composition by “;”, and we surround procedure calls and nil agent occurrences by double square brackets “·”. The syntax of compact(sCCP ) is formally deﬁned in Table 6; its constraint store, instead, follows the same prescriptions of Deﬁnition 1. In deﬁning the function compact, we use a concatenation operator to merge instantaneous tells with the preceding stochastic action. Formally, compact is deﬁned as follows: Definition 3. The function compact: restricted(sCCP) → compact (sCCP) is deﬁned by structural induction through the following rules: 1. 2. 3. 4. 5. 6.

compact(0) = 0; compact(p) = p. compact(askλ (c).G) = action(c, true, λ) compact(G). compact(tellλ (c).G) = action(true, c, λ) compact(G). compact(tell∞ (c).G) = action(true, c, ∞) compact(G). compact(M + M ) =compact(M )⊕compact(M ).

where ∞ : C → R+ ∪ {∞} is deﬁned by ∞(c) = ∞, for all c ∈ C. We now deﬁne the concatenation operator : Definition 4. The operator is deﬁned by: 1. action(g, c, λ) p = action(g, c, λ);p. ˆ = action(g, c, λ);M ˆ. 2. action(g, c, λ) M 3. action(g1 , c1 , λ1 ) action(g2 , c2 , λ2 ) = action(g1 ∧ g2 , c1 ∧ c2 , min(λ1 , λ2 )). where min(λ1 , λ2 ) : C → R+ ∪{∞} is deﬁned by min(λ1 , λ2 )(c)=min{λ1 (c), λ2 (c)}. Going back to the agent RWX previously deﬁned; if we apply the function compact to it, we obtain the following agent: compact(RWX ) :action(X > 0, true, 1) action(true, X = X − 1, ∞) RWX ⊕ action(true, X = X + 2, 1) RWX

230

L. Bortolussi and A. Policriti

⊕ action(true, true, f (X)) ( action(X > 1, true, 1) action(true, X = X − 2, ∞) RWX ⊕ action(true, X = X + 1, 1) RWX ) After the removal of operator according to the rules in Deﬁnition (4), the agent compact(RWX ) becomes a compact(sCCP) agent: compact(RWX ) = RWX :action(X > 0, X = X − 1, 1) ; RWX ⊕ action(true, X = X + 2, 1) ; RWX ⊕ action(true, true, f (X)) ; ( action(X > 1, X = X − 2, 1) ; RWX ⊕ action(true, X = X + 1, 1) ; RWX ) The above example shows clearly that the function compact simply collapses all the actions performed on the store after one execution of the stochastic transition, as deﬁned in [7,8]. It is a simple exercise to deﬁne a stochastic transition relation for compact(sCCP ) similar to the one for restricted(sCCP ) and to successively prove the strong equivalence [30] between agents A and compact(A).7 Hence, from a semantic point of view, the application of function compact is safe, as stated in the following Lemma 3. For each sequential agent A of restricted(sCCP ), A and Aˆ = compact(A) are strongly equivalent. Let Aˆ =compact(A) be an agent of compact(sCCP ). We want to associate a graph to such agent, containing all possible actions that Aˆ may execute. Nodes ˆ i.e. to diﬀerent in such graph will correspond to diﬀerent internal states of A, stochastic branching points. Edges, on the other hand, will be associated to actions: each edge will correspond to one action(g, c, λ) instruction and will be labeled consequently by the triple (g, c, λ). To deﬁne such graph, we proceed in two simple steps: 1. First we deﬁne an equivalence relation ≡c over the set of compact(sCCP ) agents, granting associativity and commutativity to ⊕ and reducing procedure calls to automatic “macro-like” substitutions (a reasonable move as we do not pass any parameter). We will then work on the set A of compact(sCCP ) agents modulo ≡c , called the set of states; notably, all agents in A are stochastic summations. 2. Then, we deﬁne a structural operational semantics [42] on compact(sCCP ), whose labeled transition system (LTS) will be exactly the target graph. Definition 5. The equivalence relation ≡c between compact(sCCP ) agents is deﬁned as the minimal relation closed with respect to the following three rules: 7

Strong equivalence [30] is a form of bisimulation preserving probabilities: two agents are strongly equivalent if their exit rates are the same and transitions of one agent can be matched by transitions of the other having the same probability.

Dynamical Systems and Stochastic Programming

1. 2. 3.

231

ˆ1 ⊕ M ˆ 2 ≡c M ˆ2 ⊕ M ˆ1 ; M ˆ ˆ ˆ ˆ2 ) ⊕ M ˆ3 ; M1 ⊕ (M2 ⊕ M3 ) ≡c (Mˆ1 ⊕ M ˆ ˆ ˆ p ≡c A if p : −A belongs to the declarations D.

The space of compact(sCCP ) agents modulo ≡c is denoted by A, and is referred to as the space of states. Definition 6. The transition relation ⊆ A × (C × C × RC ) × A is deﬁned in the SOS style as the minimal relation closed with respect to the following rule: ˆ⊕M ˆ action(g, c, λ);G

(g,c,λ)

ˆ G.

The transition relation encodes the possible actions that a compact(sCCP ) agent can undertake. Notice that procedure calls are automatically solved as we are working modulo ≡c . The relation induces a labeled graph, its labeled transition system (LTS), whose nodes are agents in A and whose edges are labeled by triples (g, c, λ), where g ∈ C is a guard, c ∈ C is the update of the store, and λ the functional rate of the edge. Definition 7. Let Aˆ be an agent of compact(sCCP ); the portion of the labeled ˆ transition system reachable from the state Aˆ is denoted by LT S(A). ˆ is Theorem 1. For any agent Aˆ of compact(sCCP ) (modulo ≡c ), LT S(A) ﬁnite. Proof. The agents reachable from Aˆ are subagents of Aˆ or subagents of Aˆ , where ˆ The number of p : −Aˆ is a procedure called by an agent reachable from A. subagents of Aˆ (modulo ≡c ) corresponds to the number of summations present ˆ and it is ﬁnite for any deﬁnable agent. The proposition follows because in A, ˆ there is only a ﬁnite number of agents deﬁned in the declarations D. We are ﬁnally ready to deﬁne the reduced transition system for an agent A of restricted(sCCP). Definition 8. Let A be an agent of restricted(sCCP). Its reduced transition system RT S(A) is a ﬁnite labeled multigraph (S(A), T (A), A ) deﬁned by RT S(A) = LT S(compact(A)). Given RT S(A) = (S(A), T (A), A ), S(A) ⊆ A is the set of RTS-states reachable from agent compact(A), ﬁnite for Theorem 1, T (A) is the set of RTS-edges or RTS-transitions and A : T (A) → C × C × RC is the label function assigning to each RTS-edge the triple (g, c, λ), g, c ∈ C, λ : C → R+ . In order to eﬀectively compute the RTS of an agent Aˆ of compact(sCCP ), we can proceed as follows: 1. Given an compact(sCCP ) program, write the syntactic tree of the agent ˆ . Aˆ and of all the agents Aˆ such that p:-Aˆ is in Def

232

L. Bortolussi and A. Policriti

2. Nodes corresponding to action(g, c, λ) instructions will have one single incoming edge and one single outgoing edge. Remove them, connecting the entering and exiting edges and labeling them by the triple c, g, λ). 3. Leaves of the obtained labeled tree correspond either to nil agents or to procedure calls. The latter are replaced according to the following rule: if the syntactic tree of the procedure p has not been added in the current tree, replace the leaf labeled with p with the corresponding syntactic tree; otherwise, remove the leaf and redirect the incoming edge to the root of the copy of the syntactic tree of p. Iterate the application of the rule until no more leaves corresponding to procedure calls are available. The previous procedure always terminates, as the number of diﬀerent procedures ˆ is ﬁnite, hence the algorithm needs to process only a ﬁnite number p in Def of leaves. Going back to our running example, RT S(RWX ) = LT S(compact(RWX )) is shown in the ﬁgure below. Note that it has one RTS-edge for every action that can be performed by compact(RWX ), and just two RTS-states, corresponding to the two summations present in compact(RWX ). Three intermediate steps in the construction of the RTS can be visualized in Figure 1.

Step 2: The Interaction Matrix Our next step consists in encoding all the information about the dynamics in a single interaction matrix and in a rate vector. Consider the initial sCCP-network N = A1 . . . Ah of a restricted(sCCP ) program (P rog, X, init(X)), with sequential components A1 , . . . , Ah . First of all, we construct the reduced transition system for all the components, i.e. RT S(A1 ) = (S(A1 ), T (A1 ), A1 ), . . . , RT S(Ah ) = (S(Ah ), T (Ah ), Ah ). Then we construct the set of RTS-states and RTS-transitions of the network (we agree that states and transitions belonging to diﬀerent components A1 , . . . , Ah are distinct8 ), putting: S(N ) = S(A1 ) ∪ . . . ∪ S(Ah )

(3)

T (N ) = T (A1 ) ∪ . . . ∪ T (Ah ).9

(4)

and 8 9

If the same component is present in multiple copies, we distinguish among them by suitable labels. The labeling function acting on T (N ) will be denoted consistently by N .

Dynamical Systems and Stochastic Programming

233

Fig. 1. Three intermediate steps of the construction of RTS for the agent RW(X), according to the procedure sketched in the main body of the paper. The top one is the outcome of point 1, the middle one is the labeled tree obtained after step 2, while the bottom one is the result of applying twice the rule of step 3.

Suppose now that there are m RTS-states in S(N ) and k RTS-transitions in T (N ). We conveniently ﬁx an ordering of these two sets, say S(N )={σ1 , . . . , σm } and T (N ) = {t1 , . . . , tk }. The variables Y of the diﬀerential equations are of two diﬀerent kinds, Y = X ∪ P. The ﬁrst type corresponds to the global stream variables of the store, i.e X = {X1 , . . . , Xn } (see Deﬁnition 1). In addition, we associate a variable of the second type Pσi = Pi to each RTS-state of S(N ) = {σ1 , . . . , σm }, so P = {P1 , . . . , Pm }. For the manipulations to follow, we assume the existence

234

L. Bortolussi and A. Policriti

of a lexicographic ordering among all variables, so that vectors and matrices depending on this ordering are deﬁned uniquely and manipulated consistently. Moreover, variables will be also used to index of these objects. Consider now an RTS-transition tj ∈ T (N ), connecting RTS-states σj1 and σj2 , and suppose N (tj ) = (gj , cj , λj ). We introduce the following notation: – rateN j (X) = λj (X) is the rate function of tj ; – guardN j (X) is the indicator function of gj (by Deﬁnition 1, gj is a conjunction of linear equalities and inequalities), i.e. 1 if gj is true for X, N guardj (X) = (5) 0 otherwise. We are now able to deﬁne the rate vector : Definition 9. The rate vector rN for transitions T (N ) = {t1 , . . . , tk } is a kdimensional vector of functions, whose components rjN are deﬁned by N rjN (Y) = rateN j (X) · guardj (X) · Pj1 ,

(6)

where Pj1 is the variable associated to the source state σj1 of transition tj . Consider again a transition tj ∈ T (N ), going from σj1 to σj2 and with label N (tj ) = (gj , cj , λj ), and consider the updates cj , a conjunction of constraints of the form Xi = Xi ± k, according to Deﬁnition 1. We can assume that each variable Xi appears in at most one conjunct of cj .10 We are now ready to deﬁne the interaction matrix. N Definition 10. The interaction matrix IY for an sCPP-network N with respect to variables Y is an integer-valued matrix with n+ m rows (one for each variable of Y) and k columns (one for each RTS-transition T (N )), deﬁned by: N N 1. If σj1 = σj2 , then IY [Pj1 , tj ] = −1 and IY [Pj2 , tj ] = 1. 2. If Xh = Xh ± k is a conjunct of cj , then IYN [Xh , tj ] = ±k. 3. All entries not set by points 1,2 above are equal to zero. RWX For the agent RWX , the interaction matrix IY Y = {X, P0 , P1 } is: ⎛ ⎞ −1 +2 0 −2 +1 (X) RWX IY = ⎝ 0 0 −1 +1 +1 ⎠ (P0 ) 0 0 +1 −1 −1 (P1 )

for the variables

Similarly, the rate vector rRWX is T rRWX = P0 X > 0, P0 , f (X)P0 , P1 X > 1, P1 , ,

(7)

(8)

where · denotes the logical value of a formula (i.e., 1 if the formula is true, 0 otherwise). 10

If, for instance, both Xi = Xi + k1 and Xi = Xi + k2 are in cj . Then we can replace these two constraints with Xi = Xi + (k1 + k2 ).

Dynamical Systems and Stochastic Programming

235

Step 3: Writing ODE’s Once we have the interaction matrix, writing the set of ODE’s is very simple: N we just have to multiply matrix IY by the (column) rate vector rN , in order to N obtain the vector odeY : N N odeN (9) Y = IY · r . Each row of the odeN Y vector gives the diﬀerential equation for the corresponding variable. Speciﬁcally, the equation for variable Yi is Y˙i = odeN Y [Yi ] =

k

N IY [Yi , j] · rjN (Y)

j=1

=

k

N IY [Yi , j] · guardj (X) · ratej (X) · Pj1

j=1

For instance, the set of ODE’s associated to the agent RWX is ⎧ ˙ ⎨ X = P0 (2 − X > 0) + P1 (1 − 2X > 1) P˙0 = − X 21+1 P0 + P1 (1 + X > 1) ⎩ ˙ P1 = X 21+1 P0 + P1 (1 + X > 1) In order to solve a set of ODE’s, we need to ﬁx the initial conditions. The variables Y = X ∪ P of odeN Y are of two distinct types: P, denoting states of the reduced transition systems of the components, and X, representing stream variables of the store. The initial conditions for P are easily determined: we set to one all the variables corresponding to the initial states of RTS of each component, and to 0 all the others. Regarding X, instead, initial conditions are given in the formula init(X) of Deﬁnition 1, specifying the values assigned to stream variables before starting the execution of the sCCP program. Elimination of Redundant State Variables Consider an sCCP component A whose reduced transition system RT S(A) has just one RTS-state. Then, the odeN Y vector of an sCCP-network N having A as one of its components will contain a variable corresponding to this RTS-state, say Pi , with equation P˙i = 0 and initial value Pi (0) = 1. Clearly, such variable is redundant, and we can safely remove it by setting Pi ≡ 1 in all equations containing it and by eliminating its equation from the odeN Y vector. From now on, we assume that this simpliﬁcation has always been carried out. As an example, consider the following agent A :- tellf1 (X) (X = X + 1).A + tellf2 (X) (X = X − 1).A

236

L. Bortolussi and A. Policriti

Its RTS contains just one state, corresponding to the only summation present in it, with associated variable P . As the other variable of the agent is X, the vector odeA {X,P } contains two equations, namely

X˙ P˙

=

f1 (X)P − f2 (X)P 0

.

The simpliﬁcation introduced above just prescribes to remove the equation for P , setting its value to 1 in the other equations; therefore we obtain odeA {X} = (f1 (X) − f2 (X)) . Notice that the set of variables Y is updated consistently, i.e. removing the canceled variables from it. We summarize the whole method just presented deﬁning the following operator. Definition 11. Let N be the sCCP-network of an restricted(sCCP ) program. With ODE(N ) we denote the vector odeN Y associated to N by the translation procedure previously deﬁned, after applying the removal of state variables coming from network components with just one RTS-state. Compositionality of the Transformation Operator In order to clearly state formal properties of the transformation, we need a version of the ODE(N ) indicating explicitly the variables X for which the differential equations are given. In the following, the variables for the equations ODE(N ) are indicated by V AR(ODE(N )). Definition 12. Let N be the sCCP-network of an restricted(sCCP ) program and let Y = V AR(ODE(N )) and ODE(N ) = odeN Y . The ordinary diﬀerential equations of N with respect to the set of variables X, denoted by ODE(N, X), is deﬁned as odeN Y [Yj ] ifXi = Yj ∈ Y, ODE(N, X)[Xi ] = 0 otherwise. The operations performed on ODE(N ) by ODE(N, X) simply consist in the elimination of the equations of ODE(N ) for the variables not in X, and in the addition of equations X˙ = 0 for all variables X in X but not in Y. We can also N associate a new interaction matrix IX to ODE(N, X), whose rows are derived according to Deﬁnition 12. As the set of RTS-transitions T (N ) is unaltered by N ODE(N, X), the equation ODE(N, X) = IX · rN continues to hold. We can now prove the following theorem, stating compositionality of ODE operator.

Dynamical Systems and Stochastic Programming

237

Theorem 2. Let N1 , N2 be two sCCP-networks, and let N = N1 N2 be their parallel composition. If Y1 = V AR(ODE(N1 )), Y2 = V AR(ODE(N2 )), and Y = Y1 ∪ Y2 , then11 ODE(N1 N2 , Y) = ODE(N1 , Y) + ODE(N2 , Y). Proof. The components in N1 N2 are the components of N1 plus the components of N2 . Therefore, the set of RTS-transitions (i.e. edges in the RTS of the components) T (N1 N2 ) of N1 N2 is equal to T (N1 ) ∪ T (N2 ). As each N N column of IY 1 2 is either a transition of T (N1 ) or a transition of T (N2 ), it N N Nh clearly holds IY 1 2 [Yi , tj ] = IY [Yi , tj ] if tj ∈ T (Nh ), h = 1, 2. The following chain of equalities then follows easily from the deﬁnitions: N N N N ODE(N1 N2 , Y)[Yi ] = IY 1 2 [Yi , tj ]rj 1 2 tj ∈T (N1 N2 )

=

N1 IY [Yi , tj ]rjN1 +

tj ∈T (N1 )

N2 IY [Yi , tj ]rjN2

tj ∈T (N2 )

= ODE(N1 , Y) + ODE(N2 , Y). 3.1

Preservation of Rate Semantics

In Section 2.2 we discussed how biochemical reaction with general kinetic laws can be modeled in sCCP. Given a list of reactions, the standard praxis in computational chemistry is that of building a corresponding diﬀerential (set of ODE’s) or stochastic (CTMC) model. The deﬁnition of such models is canonical, and it is fully speciﬁed by the reaction list, see [53] for further details. sCCP models of biochemical reactions are generated by associating an agent, call it biochemical agent, to each reaction in the list. These agents are rather simple: they can execute in one single way, namely an inﬁnite loop consisting of activation steps, where the agents compete stochastically for execution, with rate given by the kinetic law speciﬁed in the reaction arrow, and update steps, in which the store is modiﬁed according to the prescriptions of the reaction. This bijective mapping between reactions and sCCP biochemical agents soon implies that the stochastic model for sCCP is identical to the continuous-time Markov chain generated in classical stochastic simulations with Gillespie algorithm [26], for instance like those obtainable with a program like Dizzy [46,19].12 A diﬀerent question is wether the ODE’s that are associated to an sCCP model of biochemical reactions coincide with the canonical ones. In the rest of the section, we show that this is indeed the case. Following the approach by Cardelli in [17], we can then say that the translation from sCCP to ODE’s preserves the rate semantics. The sense of this sentence is better visualized in Figure 2, graphically depicting the correspondence between 11 12

Here “+” denotes the usual sum of vectors. Stochastic simulations with Michaelis-Menten and Hill rate functions have been considered, for instance, in [47].

238

L. Bortolussi and A. Policriti

Fig. 2. Diagram of relations intervening between stochastic and ODE-based semantics of chemical reactions and sCCP agents

stochastic and diﬀerential models of biochemical reactions and of the derived sCCP agents. Preservation of rate semantics essentially means that the arrows in the diagram commute. As a matter of fact, in [17] the author deals only with mass action kinetics, due to the intrinsic properties of π-calculus (all deﬁnable rates are mass-action like). In our setting, instead, functional rates and the constraint store allow us to deal with arbitrary chemical kinetics, including also Michaelis-Menten and Hill ones (cf. [20]). In the following, we formally prove the equivalence of ODE’s obtained from sCCP agents with the corresponding classical ones. In general, the stochastic and the deterministic rate of a reaction are not the same, because ODE’s variables measure concentration, while sCCP variables count the number of molecules. Therefore, in passing from one model to the other, we need to convert numbers to concentrations, dividing by the volume V times the Avogadro number NA (γ = V NA will be referred as system size). Rates need also to be scaled consistently, see [53] for further details. In the rest of this section, however, we get rid of scaling problems simply by assuming γ = 1. In any case, system size can be reintroduced without diﬃculties, by change the scale of rates and variables after the derivation of ODE’s. We now put forward some notation, in order to specify how to formally derive a set of ODE’s given a set of reactions R = {ρ1 , . . . , ρk }, where each ρi denotes a single reaction. Each reaction ρ has some attributes: a multiset of reactants (species can have a speciﬁc multiplicity), denoted by REACT (ρ), a multiset of products, P ROD(ρ), and a real-valued rate function, RAT E(ρ), depending on the variables associated to the molecules involved in the reaction, V AR(ρ). This last function, V AR, can be easily extended to sets of reactions by letting V AR({ρ1 , . . . , ρk }) = V AR(ρ1 ) ∪ . . . ∪ V AR(ρk ). Now let R be a set of reactions and X = V AR(R). The (canonical) diﬀerential equations associated to R w.r.t. variables X, denoted by ODE(R, X)13 are deﬁned for each variable Xi as X˙ i = ODE(R, X)[Xi ], where:14 13 14

We overload here the symbol introduced in Deﬁnitions 11 and 12; however, the two cases can be easily distinguished looking at their ﬁrst argument. We conveniently identify each variable with the molecule it represents.

Dynamical Systems and Stochastic Programming ODE(R, X)[Xi ] =

RAT E(ρ) −

ρ∈R: Xi ∈ P ROD(ρ)

239

RAT E(ρ). (10)

ρ∈R: Xi ∈ REACT (ρ)

If Xi is not involved in any reaction of R, then ODE(R, X)[Xi ] = 0. We observe that the correct stoichiometry is automatically dealt with by the fact that we are using multisets to list reactants and products. Restricting this construction to a ﬁxed variable ordering allows to state the following straightforward compositionality lemma. Lemma 4. Let R = R1 ∪ R2 be a partition of R and let X1 = V AR(R1 ), X2 = V AR(R2 ), and X = X1 ∪ X2 . Then ODE(R1 ∪ R2 , X) = ODE(R1 , X) + ODE(R2 , X). We turn now to formally deﬁne the encoding of reactions into sCCP agents. For each reaction ρ, its sCCP agent SCCP (ρ) is constructed according to Section 2.2. Operator SCCP is extended compositionally to sets of reactions R = {ρ1 , . . . , ρk } by letting SCCP (R) = SCCP (ρ1 ) . . . SCCP (ρk ). We are ﬁnally ready to state the theorem of preservation of rate semantics: Theorem 3 (Preservation of rate semantics). Let R be a set of biochemical reactions, with X = V AR(R). Then ODE(R, X) = ODE(SCCP (R), X)

(11)

Proof. We prove the theorem by induction on the size k of the set of reactions R. For the base case k = 1, consider a reaction ρ R1 + . . . + Rn →f (R,X;k) P1 + . . . + Pm and its associated sCCP agent SCCP (ρ) ⎛ ⎞ n m SCCP (ρ) :- tellf (R,X;k) ⎝ (Ri − 1) ∧ (Pj + 1)⎠ .SCCP (ρ). i=i

j=i

Clearly, the reduced transition system of such agent is

SCCP (ρ)

Let Y = V AR(ρ), then the interaction matrix IY has |Y| rows and 1 column, with entries corresponding to the stoichiometry of the reaction: SCCP (ρ) IY [Yi ] = 1− 1. Yi ∈P ROD(ρ)

Yi ∈REACT (ρ)

240

L. Bortolussi and A. Policriti

The rate vector rSCCP (ρ) , instead, is the scalar f (R, X; k). Hence, the equation for variable Yi is Y˙ i =

f (R, X; k) −

Yi ∈P ROD(ρ)

f (R, X; k),

Yi ∈REACT (ρ)

which is equal to equation (10). The inductive case follows easily from compositionality properties of ODE operators. Suppose the theorem holds for lists up to k − 1 reactions, and let R = R0 ∪ {ρ} be a set of k chemical reactions (hence |R0 | = k − 1). Then ODE(SCCP (R), X) = ODE(SCCP (R0 ∪ {ρ}), X) = ODE(SCCP (R0 ) SCCP (ρ), X) = ODE(SCCP (R0 ), X) + ODE(SCCP (ρ), X) = ODE(R0 , X) + ODE(ρ, X) = ODE(R, X), where the second equality follows from the deﬁnition of SCCP , the third follows from Theorem 2, the fourth is implied by the induction hypothesis on SCCP (R0 ) and by the base case proof on SCCP (ρ), while the last is a consequence of Lemma 4.

3.2

Preservation of Dynamic Behavior

In Theorem 3 we proved that the ODE map, when applied to sCCP-models of biochemical networks, satisﬁes a condition of coherence: it preserves the kinetic principles used in the construction of the model (i.e., the rate semantics). A diﬀerent question is whether an sCCP-network N (evolving stochastically according to the prescriptions of its semantic) shows a dynamic behavior equivalent to the one exhibited by the equations ODE(N ). This problem is the sCCPcounterpart of the famous mathematical issue concerning the relation between stochastic and diﬀerential models [35,36], studied deeply also in the context of biochemical reactions [26,23]. It is well-known that stochastic and diﬀerential models of biochemical reactions are behaviorally equivalent only in some cases. These results are signiﬁcant also for sCCP. Theorem 3, in fact, states that, when biochemical reactions are concerned, the stochastic process underlying the sCCP-models and the associated ODE’s are exactly the classical ones. Therefore, in the mapping from sCCP to ODE’s we have the same phenomenology as in the classical case. However, the logical structure of sCCP-agents makes the problem of behavioral preservation subtler. In the following, we discuss this problem with diﬀerent examples, especially of situations in which an sCCP-network and the corresponding ODE’s show a diﬀerent behavior. In particular, we are interested in sketching a brief, and plausibly incomplete, classiﬁcation of the causes of behavioral divergence.

Dynamical Systems and Stochastic Programming

241

An important issue is the concept of behavioral equivalence itself, which is diﬃcult to formalize, as already discussed in the introduction. We will return on this problem in the next section. Oregonator. The Oregonator is a chemical systems showing an oscillatory behavior, devised by Field and Noyes [40] as a simpliﬁed version of the BelousovZhabotinsky oscillator15. Essentially, Oregonator is composed of three chemical substances, call them A, B, C, subject to the following reactions: B →k1 A + B →k2 A →k3 2A →k4 C →k5

A ∅ 2A + C ∅ B

(12)

Actually, other chemical substances are involved, but they are kept constant in the experiment. The diﬀerential equations associated to (12) are known to possess a stable limit cycle for a wide range of parameter’s values [28], containing an unstable equilibrium. The limit cycling behavior is clearly visible in Figure 3(a), where the numerical solution of Oregonator’s ODE’s is shown. In Figure 3(b), instead, we plot a stochastic simulation of the sCCP model associated to (12) according to prescriptions of Section 2.2. In this case, the stochastic model shows the same pattern as the diﬀerential one. Theorem 3 guarantees that the graph in Figure 3 depicts the numerical solution of ODE’s associated to the sCCP program by the transformation previously deﬁned. In this case, the behavior is preserved. We remark two things regarding Oregonator. First, the size of each molecular species is of the order of thousands, hence the relative variation induced by one reaction in the stochastic model is small. Under this condition, stochastic and deterministic models of biochemical reactions usually coincide [26]. Another property of the Oregonator that can be important for behavioral preservation is that the limit cycle is an attractor in the phase space: nearby trajectories asymptotically converge to it (see [51]). This means that a relatively small perturbation is not willing to change the overall dynamics: stochastic ﬂuctuations have a negligible eﬀect. Things are diﬀerent if we start from the unstable equilibrium of the system. The numerical solution of ODE’s shows a constant evolution (Figure 4(a)), while the stochastic simulation (Figure 4(b)) essentially evolves as the limit cycle of Figure 3. In fact, stochastic ﬂuctuations, in this case, make the sCCP system move away from the instable equilibrium into the basin of attraction of the limit cycle. This shows another well known fact: stochastic and diﬀerential models usually diﬀer near instabilities [26]. Lotka-Volterra system. The Lotka-Volterra system is a famous simple model of population dynamics, see for example [26] and references therein. There are 15

This chemical system is called “Oregonator” because its inventors where working at the University of Oregon.

242

L. Bortolussi and A. Policriti

(a) ODE model of Oregonator

(b) sCCP model of Oregonator Fig. 3. 3(a): Numerical simulation of the diﬀerential equation model of the Oregonator, with parameters determined according to the method presented in [26]. Speciﬁcally, let As = 500, Bs = 1000 and Cs = 2000 be an equilibrium of the system of equations, and let R1 = 2000, R2 = 50000. Then parameters are equal to k1 = R1 /Bs = 2, k2 = R2 /(As Bs ) = 0.1, k3 = (R1 + R2 )/As = 104, k4 = ((2R1 )/(A2s ))/2 = 4e−7 , and k5 = (R1 + R2 )/Cs = 26. The starting point is A0 = As /2, B0 = Bs /2, C0 = Cs /2. The system soon approaches an attractive limit cycle. 3(b): Stochastic simulation with Gillespie’s method of the sCCP network associated to reactions (12). Parameters and initial conditions are those speciﬁed above. The eﬀect of stochastic ﬂuctuations is negligible, and the plot essentially coincide with its deterministic counterpart.

two species: preys and predators. Preys eat some natural resource, supposed unbounded, and reproduce at a rate depending only on their number. Predators, instead, can reproduce only if they eat preys, otherwise they die. To keep the model simple, we admit predation as the only source of prey’s death. The previous hypotheses can be summarized in the following set of reactions, where E refer to preys and C to predators: E →kb 2E C →kd ∅ E + C →kp 2C

(13)

If we consider the standard mass action ODE’s (they coincide with the equations derived from the sCCP model due to Theorem 3), a typical solution shows

Dynamical Systems and Stochastic Programming

243

(a) ODE model of Oregonator from unsta- (b) sCCP model of Oregonator from unble equilibrium stable equilibrium Fig. 4. 4(a): Numerical simulations of ODE’s derived from reactions (12), with parameters given in caption of Figure 3, starting from an unstable equilibrium of the system. 4(b): Stochastic simulation of sCCP model associated to reactions (12), with the same parameters and initial conditions than the diﬀerential counterpart. As we can see, stochastic ﬂuctuations drive the system away from the unstable equilibrium, so that its surrounding limit cycle is approached.

oscillations in which high values of preys and predators alternate. An example of such a solution is given in Figure 5(a). Inspecting equations, it can be shown that the point Es = kd /kp , Cs = kb /kp is an equilibrium of a rather special kind: it is stable (trajectories starting nearby it stay close) but not asymptotically stable (trajectories starting nearby do not converge to it as time approaches inﬁnity). This behavior is easily understood looking at the phase space (Figure 5(b)), in which we can see that trajectories form closed orbits around the equilibrium, whose amplitude increases with distance from equilibrium. More details can be found, for instance, in [51]. What kind of behavior can we expect from the stochastic evolution of the sCCP model for (13)? Stochastic ﬂuctuations will make the system jump from one trajectory to nearby ones, without any force pulling it towards the equilibrium. Therefore, ﬂuctuations can, in the long run, make the system wander in the phase plane, eventually reaching a borderline trajectory (corresponding to E or C axis in the phase plane). Whenever this happens, then both preys and predators go extinct (C-axis trajectory), or just predators do, while preys go to inﬁnity (E-axis trajectory). This intuition is conﬁrmed in Figure 6, where we compare the ODE solution starting from equilibrium (dotted lines), and a trace of the sCCP model, starting from the same initial conﬁguration. As we can see, the stochastic system starts oscillating until both species go extinct. This is another well known case in which stochastic and diﬀerential dynamics diﬀer, again induced by properties of the phase space [26]. A negatively auto-regulated system. The eﬀect of stochastic ﬂuctuations is mostly remarkable in biological phenomena where gene expression is involved. This is because the transcription of a gene is usually a slower process than protein-protein interaction, and often the number of mRNA strands for a given

244

L. Bortolussi and A. Policriti

(a) ODE solution of Lotka-Volterra model

(b) Phase space of Lotka-Volterra model

Fig. 5. 5(a): Numerical solution of ODE’s associated to reactions (13), with parameters kb = 1, kp = 0.1, kd = 0.1 and initial conditions E0 = 4 and C0 = 10. 5(b): phase portrait of the Lotka Volterra system, for the same value of parameters as above. As we can see, all the solutions show an oscillating behaviour. The system has an (instable) equilibrium for E = 1, C = 10, at the center of the circles.

Fig. 6. Eﬀect of stochastic ﬂuctuations for the Lotka-Volterra system. The dotted lines are an equilibrium solution for the ODE model (parameters are as in caption of Figure 5). A stochastic trace of the sCCP model is drawn with solid lines: both species ﬂuctuate around the equilibrium values until they both get extinct.

gene present in the cell is very small, of the order of some units. As the production of one single mRNA is a rare event (compared to other cellular events), stochastic variability in its happening can induce behaviors diﬃcult to capture if mRNA is approximated with its concentration. Stochasticity in gene expression is indeed a phenomenon that has received a lot of attention, see for instance [38,6]. We present here a simple, artiﬁcial example taken from [53] and depicted in Figure 7. The biological network shown represents a simple autoregulatory mechanism in gene expression of a procaryotic cell. Gene g produces, via mRNA r, a

Dynamical Systems and Stochastic Programming

245

Fig. 7. Diagram of a simple self-regulated gene network. Gene g produces mRNA r and, from it, protein P . P can dimerise and its dimer can bind to a promoter region of gene g, downstream of RNA polymerase binding site. The P2 -binding blocks polymerase activity, thus inhibiting gene expression.

protein P that, as a dimer, can bind to a promoter region of gene g, preventing RNA-polymerase activity and thus inhibiting its own production. Following the approach of [5], genes can be modeled as logical gates having a ﬁxed output (the produced mRNA or protein), and several inputs, corresponding to diﬀerent proteins of the system, exerting a positive or negative regulatory function. A gene gate with one inhibitory input is called in [5] neg gate, and can be modeled in restricted(sCCP) simply as: neg gateP,I :tellkp (P = P + 1).neg gateP,I + askkb ·I (I ≥ 1).askku (true).neg gateP,I , where kp is the basic production rate, kb is the binding rate of the repressor to the promoter region of the gene and ku is its unbinding rate. In order to model the system of Figure 7 we can combine one neg gate with some reactions. This is an example of the modeling style mixing the reactioncentric and the molecular-centric point of view, see Section 2.2. The model is the following: neg gater,P2 r →kt r + P P →kdim1 P2 (14) P2 →kdim2 P r →kd1 ∅ P →kd2 ∅ In Figure 8 we compare a stochastic simulation of the sCCP model of reactions (14) with the numerical solution of the associated ODE’s. As we can readily see, the two plots are completely diﬀerent. In particular, in the stochastic simulation, P2 is produced in short bursts; normally it is slowly degraded. The bursts correspond to mRNA production events, shown in Figure 8(a) as blue peaks. The

246

L. Bortolussi and A. Policriti

(a) sCCP model of system (14)

(b) ODE model of system (14)

Fig. 8. 8(a): Simulation of the sCCP model of (14). The red line corresponds to P2 , while the blue line shows the evolution of r, multiplied for a factor 100 (for visualization purposes). Note that the increases in P2 expression immediately follow mRNA production events. Parameters of the models are the following: kp = 0.01, kb = 1, ku = 10, kt = 10, kdim1 = 1, kdim2 = 1, kd1 = 0.1, and kd2 = 0.01. All molecules are set initially equal to 0. 8(b): Numerical simulation of ODE’s associated to the sCCP model of (14), for the same parameters just given. The evolution of P2 is tamer than in the stochastic counterpart, as it converges quickly to an asymptotic value.

ODE’s system, however, presents a much simpler pattern of evolution, in which the quantity of P2 converges to an asymptotic value. This divergence is caused by the fact that, approximating continuously the number of RNA molecules, we lose the discrete information that seems to characterize its dynamics, i.e. the fact that mRNA can be present in one unit of completely absent from the system. Staten otherwise, continuously approximating molecular species present in low quantities may lead to errors inducing a completely divergent observable behavior. Repressilator. The Repressilator [21] is an artiﬁcial biochemical clock composed of three genes expressing three diﬀerent proteins, tetR, λcI, LacI, exerting a regulatory function on each other’s gene expression. In particular, protein tetR represses the expression of protein λcI, protein λcI represses the gene producing protein LacI, and, ﬁnally, protein LacI is a repressor for protein tetR. The expected behavior is an oscillation of the concentrations of the three proteins. A simple stochastic model of Repressilator can be found in [5], where the authors describe it with three neg gates (see the previous paragraph) cyclically connected, in such a way that the product of one gate inhibits the successive gene gate in the cycle. In addition, they introduce degradation mechanisms for the three repressors. More formally, the model is the following neg gateA,C neg gateB,A neg gateC,B A →kd ∅ B →kd ∅ C →kd ∅

(15)

Dynamical Systems and Stochastic Programming

247

In Figure 9(a) we show a trace of the stochastic model generated by a simulator of sCCP based on Gillespie algorithm. The oscillatory behavior is manifest. If we apply the translation procedure discussed in Section 3 to this particular model, we obtain the ODE’s shown in Table 7, while their numerical integration is shown in Figure 9(b). As we can readily see there is no oscillation at all, but rather the three proteins converge to an asymptotic value, after an initial adjustment. Inspecting the ODE’s, we note the presence of six variables (YA , YB , YC and ZA , ZB , ZC ) in addition to those representing the quantity of repressors in the system (A, B, C). Such variables correspond to states of genes gates, and they are used to model the change of conﬁguration of the gates, from active to repressed and vice versa. This scenario seems rather unjustiﬁed here: there is no argument to support the introduction of these variables, especially because we are continuously approximating boolean quantities. An interesting point regarding Repressilator is the relation between the solution of the ODE’s and the average trace of the stochastic system (i.e. E[X(t)], returning the average value of system variables as a function of time). In fact, we may expect that the behavior preserved by the diﬀerential equations is the average dynamics of the stochastic system, rather than that shown by one of its traces. Interestingly, also the average value of the Repressilator model does not oscillate, as can be seen from Figure 10. This can be explained by noticing that the oscillations’ period in the stochastic model is not constant, but it varies considerably. Hence, for every instant (when the Markov chain is at the stationary regime), we will observe one of the proteins at its peak value approximatively only in one third of the traces. Hence its average value will tend to stabilize at one third of the peak value, as conﬁrmed by Figure 10. In fact, when we average Repressilator, we measure the fraction of traces in which a certain gene is

(a) sCCP model of repressilator

(b) sCCP model of repressilator

Fig. 9. 9(a): Stochastic time trace for the Repressilator system of described by reactions 15. Parameters are kp = 1, kd = 0.01, kb = 1, ku = 0.01. 9(b): Solution of the diﬀerential equations of Table 7, automatically derived from sCCP program associated to reactions 15. Parameters are the same as in stochastic simulation. The stochastic simulation lasts longer than the ODE one in order to better underline its oscillatory behavior.

248

L. Bortolussi and A. Policriti

Table 7. ODE’s derived for the Repressilator, generated by the method of Section 3.1 A˙ = kp YA − kd A B˙ = kp YB − kd B C˙ = kp YC − kd C

Y˙ 1 = ku ZA − kb YA C Y˙ 2 = ku ZB − kb YB A Y˙ 3 = ku ZC − kb YC B

Z˙ 1 = kb YA C − ku ZA Z˙ 2 = kb YB A − ku ZB Z˙ 3 = kb YC B − ku ZC

Fig. 10. Average value of the sCCP model for Repressilator, computed using model checker PRISM [37]. See [8] for further details.

active and the fraction of traces in which it is inactive, for every time instant. In this way, however, we lose any information regarding the sequence of gene gate’s state changing. The diﬀerent behavior existing between a trace of a stochastic system and its average trace suggests that the switching dynamics of genes can be the driving force behind oscillations. This implies that another source of nonequivalence between sCCP models and the associated ODE’s can appear due to the representation of RTS-states with continuous RTS-state variables. Indeed, this example suggested us to preserve part of the discrete dynamics, mapping the sCCP Repressilator into an hybrid automaton. The work put forward in [?] shows that this move is enough to maintain oscillations. The translation to hybrid automata opens an entire range of possibilities to combine discreteness and continuity. These will be investigated in detail in the planned second part of this paper. Sources of non-equivalence. In the previous examples we outlined diﬀerent cases in which an sCCP model and its associated ODE’s fail to be equivalent from a dynamical viewpoint. We remark that most of these examples are well known, as they have been studied in detail in theoretical and applicative contexts, like biochemical reactions [26,53] and our main interest here is in their connection with the sCCP translation machinery. For sake of clarity, we summarize the diﬀerent sources of non-equivalence. 1. In some cases, non-equivalence is a direct consequence of properties of the phase space. For instance, instable trajectories are destroyed by small ﬂuctuations, like the equilibrium trajectory of the Oregonator. Also stable but

Dynamical Systems and Stochastic Programming

249

not asymptotically stable trajectories can be troublesome, as stochastic ﬂuctuations are not counterbalanced by any attracting force, and so they can bring the stochastic system far away from the initial trajectory. This is the case of the Lotka-Volterra system. 2. Another well-known problem is related to the approximation by continuous quantities of integer variables having small (absolute) values. In this case, in fact, the eﬀect of a single stochastic ﬂuctuation has a relative magnitude that is relevant, so the dynamics can change quite dramatically. A typical example appearing in Biology is related to the transcription of genes, as shown in the simple example of a self-regulated gene. 3. A ﬁnal source of non-equivalence is, instead, characteristic of the translation procedure deﬁned for sCCP. In fact, in this case we represent each RTSstate of a component of the system with a continuous variable, which can take values in the real interval [0, 1]. RTS-states represent, in some sense, logical structures that control the activity of the system, while a change of state is an event triggered by some condition of the system. Moreover, in each sCCP trace, each component can be in only one state, hence RTS-state variables are boolean quantities. Continuous approximation, in this case, can have dramatic consequences, as the example of Repressilator seems to suggest.

3.3

Behavioral Equivalence

Comparing the dynamical evolution of a deterministic and a stochastic system is a delicate issue, because stochastic processes have a noisy evolution, hence we need to remove noise from their traces, before attempting any comparison with time traces evolving deterministically. In the previous discussion, in fact, we appealed to the concept of “behavioral equivalence” always in a vague sense, essentially leaving to the reader the task of visually comparing plots and recognizing similarities and diﬀerences. Clearly, a mathematical deﬁnition is needed in order to prove theorems and automatize comparisons. We ﬁrst consider the comparison of traces generated by ODE’s with the average trace of the stochastic system, taken as the representative of its whole ensemble of traces. In practice, for each time instant t we need to compute the average value E(X(t)) of each stream variable X w.r.t. the probability distribution on states of the system at time t. This probability can be obtained as the solution of the Chapman-Kolmogorov forward equation [39], a system of diﬀerential equations of the size of the state space. This equation, known in biochemical literature as the chemical master equation [25], can rarely be solved analytically, and it is also very diﬃcult to integrate numerically [26]. A more eﬃcient approach to compute an estimate of the average consists in generating several (thousands of) stochastic traces and in computing pointwise their sample mean. Alternatively, the average value of one or more variables can be computed for a small sample of time points {t1 , . . . , tk } using numerical techniques, as those implemented in the model checker PRISM [37].

250

L. Bortolussi and A. Policriti

Whatever the method chosen, the computation (even approximate) of the average trace of a stochastic system is a diﬃcult matter. Whenever such trace is known, we can compare it with the trace of the ODE’s, generated using standard numerical techniques [43], using quantitative measures (essentially computing a distance between the two curves). Indeed, in [9] it is shown that the ODE associated to a sCCP program is a ﬁrst order approximation of the true equation for the average. However, the average trace of a stochastic system is not necessarily a good representative of its evolution. A paradigmatic example is the Repressilator, whose average trace (sampled with PRISM, see caption of Figure 10) converges to an asymptotic value, while all its stochastic traces show persistent oscillations. Hence, even when averaging a stochastic system, we may lose the characterizing qualitative features of its dynamics. The example of Repressilator suggests that the notion of behavioral equivalence is probably better captured in a qualitative setting. Qualitative comparison requires a formal deﬁnition of the features of dynamical evolution, like oscillations, convergence to a stable value, and so on. A possibility we suggest in this direction is to describe these features as logical formulae of a suitable logical language L, for instance temporal logic, as done in Simpathica [3]. The concepts below are just sketched; this subject is currently under investigation and we will deal with it in detail in future works. Let Φ denote the set of formulae describing all dynamical features of interest. Associating a Kripke structure K1 to the trace of an ODE and another structure K2 to a stochastic trace, then we may declare these traces equivalent whenever their Kripke structures satisfy the same subset of formulae of Φ (possibly restricting the attention to formulae of degree ≤ n). Below we give three examples of temporal logic formulas expressing inﬁnite oscillations: G(Z = zm → F (Z = zM )) ∧ G(Z = zM → F (Z = zm )) ∧ G(zm Z zM ) ∧ F (Z = zm ); G(Z = zm → X(Z > zm U Z = zM )) ∧ G(Z = zM → X(Z < zM U Z = zm )) ∧ G(zm Z zM ) ∧ F (Z = zm ); dZ dZ ¬G >0 ∧ ¬G dZ = 0 ∧ ¬G < 0 . dt dt dt In the above formulas X stands for next, G stands for always (globally), F stands for sometimes (in the future), U stands for until, zm and zM are minimum and maximum values, and the thirds formula uses propositional formulas taking values according to the sign of the ﬁrst derivative.16 This idea seems promising, as it gives a considerable freedom in the deﬁnition of formulae Φ, hence allowing to privilege some aspects of dynamical evolution 16

In order to use meaningfully the notion of “next” for ODE’s we need to consider a discretization of the time and of the state space, such as that performed in [3].

Dynamical Systems and Stochastic Programming

251

more than others. However, the real problem is in the deﬁnition of a reasonable Kripke structure for a stochastic trace (and for sets of traces). In fact, Kripke structures for ODE’s can be constructed starting from one or more traces, as done in Simpathica [3], in the following way: the bounded (product) domain of all variables is divided in small, compact regions; a state of the Kripke automaton consists of one of such regions; edges connect two states if a trajectory crosses the corresponding regions consecutively. This construction, however, is not reasonable for stochastic traces, as noise would force the addition of many edges that may introduce spurious behaviors. Of course, it is possible to model check directly on CTMC formulae written in CSL [4,37]. However, the complexity of this latter approach makes the deﬁnition of non-deterministic Kripke structures interesting also for stochastic traces. We are currently investigating this direction, considering the introduction of a bounded form of memory to tame noise. 3.4

More on the Restrictions of the Language

restricted(sCCP ) restricts the full language in several aspects, see Section 2.3. Actually, these restrictions have been introduced in order to deﬁne in a reasonably simple way the mapping to ODE’s. We discuss them in detail in the following. First, all agents must be sequential, i.e. not containing any occurrence of the parallel operator. As already remarked at the end of Section 2.3, this does not constitute a real limitation, as each non-sequential agent can be transformed into a network of sequential ones. Here we note that the same trick of Section 2.3 can be used to transform each sCCP-network into an equivalent network where each sequential agent has an RTS with one single state; indeed, this is done implicitly by the transformation to ODE’s itself. However, writing programs in this form is less natural. Another syntactic restriction regards the deﬁnition of local variables. Actually, variables in ODE’s have a global scope. Of course, any local variable can be made global by suitably renaming it. There is a problem, however, concerning the fact that at run-time we may generate an unbounded number of local variables. This implies that their use may lead to a set of ODE’s with an inﬁnite number of variables (although each equation will depend only on a ﬁnite number of them). The uprising of an inﬁnite number of variables requires more complex mathematical techniques, and it prevents the use of standard numerical solvers. Finally, the third class of restrictions regards the constraint store. The restriction to numeric variables is obviously necessary, as we are mapping to ODE’s. The restriction on the admissible constraints for the updating of variables, on the other hand, is related to the fact that each update in a sCCP program needs to be considered as a ﬂux acting on some variables. Indeed, even a simple update like X = 0 is diﬃcult to render within ODE’s framework, as it is inherently discrete. A possible way out is to mix the continuous ingredient of ODE’s with discreteness, mapping sCCP programs to hybrid automata [29]. Within this

252

L. Bortolussi and A. Policriti

formalism, updates like X = 0 are perfectly admissible: they are resets associated to discrete transitions.

4

From ODE’s to sCCP

In this section we deﬁne a transformation SCCP , associating an sCCP network to a generic set of ordinary diﬀerential equations, analyzing both its mathematical properties and the relation with the the map ODE deﬁned in the previous section. Before entering into the mathematical details, we need to make a preliminary remark. Essentially, the main obstacle we have to face in deﬁning the map SCCP is the fact that ODE’s are an aggregate description of a system. To be more precise, if a system can be described by a set of ﬂuxes acting on the diﬀerent entities into play (i.e. on the system variables), then the ODE’s hide part of the logical structure of such ﬂuxes by combining them into the equations. To clarify the concept, consider the following two sCCP agents: A :- (tell1 (X = X + 1) + tell1 (Y = Y + 1)).A B :- (tell1 (X = X + 1 ∧ Y = Y + 1)).B When we apply the ODE operator to the networks N1 = A and N2 = B, we obtain, in both cases, the following equations: X˙ 1 = 1 Y˙ Therefore, two diﬀerent sCCP agents can be mapped into the same set of ODE’s. Note that A and B are “semantically” diﬀerent, as they induce two diﬀerent CTMC. The chain associated to A has edges connecting a state (i, j) to (i + 1, j) and (i, j + 1) (hence the exit rate from (i, j) is 2), while the chain of B has transitions only from (i, j) to (i + 1, j + 1) (with exit rate 1). This information pertains the logical structure of the system, which is manifest in the sCCP program, but irremediably lost in the associated ODE’s. An even worse situation happens for the following agent, implementing a onedimensional random walk [39]: C :- (tell1 (X = X + 1) + tell1 (X = X − 1)).C The equation associated to C by ODE is X˙ = 0, as the production and degradation rate cancel out when summed together. This equation predicts a constant evolution for X, thus failing to capture its erratic behavior. Note, however, that the average value of X is constant also in the stochastic model for C. Therefore, the structural information lost in passing from sCCP agents to ODE’s makes impossible to recover the original sCCP network; stated otherwise, the map ODE(·) is not injective. Indeed, the lack of injectivity of ODE(·) means

Dynamical Systems and Stochastic Programming

253

that an sCCP program is more informative than a set of ODE’s: it deﬁnes not only the ﬂuxes, but also their logical relation. The previous discussion suggests that the map SCCP must be deﬁned with care. Given an ODE set x˙ = f (x), there are many sCCP networks that can be associated to it, i.e. at least all those belonging to the set ODE −1 (f (x)). In order to choose one among them, additional discriminating information is required, essentially related to the structure of the ﬂuxes, hence to the logic of the system modeled. As suggested by the previous discussion, we will therefore deﬁne not a single SCCP map, but rather a class of maps, parametric w.r.t. the additional information required to sort out the logical structure of ﬂuxes. We will then show that, independently of this additional information, the transformation scheme satisﬁes properties guaranteeing a form of coherence w.r.t. the ODE mapping and also a form of behavioral equivalence. Finally, we will provide two instantiations of such scheme, assuming speciﬁc conditions on the system modeled. 4.1

The Translation to sCCP

In the conversion from ODE’s to sCCP, we approximate continuous quantities by discrete variables. Therefore, this mapping will depend on an additional parameter, the step δ, specifying the granularity of the approximation of continuous variables. The magnitude of δ has a strong impact on the preservation of dynamical behavior; this point will be the content of Section 4.3. Consider a system of ﬁrst order ODE’s with n variables x = (x1 , . . . , xn ): x˙ = f (x). We will now deﬁne the notion of set of covering functions, which captures the idea of external knowledge required to solve the ambiguity about the logical structure inherent in the ODE’s. Essentially, a set of covering functions corresponds to a plausible choice of a set of ﬂuxes, generating the given ODE’s. Definition 13. A set of covering functions G for the ODE x˙ = f (x) is a set of pairs {(gi , hi ) | i = 1, . . . , k}, such that each gi is a function gi : Rn → R, each hi is a vector of Zn , and, for each x ∈ Rn , k

hi gi (x) = f (x).

i=1

Example 1. Consider the following simple system of ODE’s with two variables: x˙ = a − by (16) y˙ = c + dx One possible covering set is the following: G = {(a, (1, 0)), (by, (−1, 0)), (c, (0, 1)), (dx, (0, 1))},

254

L. Bortolussi and A. Policriti

which corresponds to the choice of disentangling all addends of the equations. Another possibility, instead, is the following: G = {(−a + by, (−1, 1)), (a − by, (0, 1, )), (c + dx, (0, 1, ))}, as easily veriﬁed. In the sCCP program associated to the ODE’s x˙ = f (x), we approximate the continuous variables x with discrete stream variables X. Deﬁnition 1, however, requires variables X to have integer values. In order to set the size of the basic increment to an arbitrary step δ, we can change variables, setting x = δX and expressing f with respect to X (in this way, a unit increment of Xi corresponds to an increment of δ of xi ). The equation for X thus becomes ˙ = 1 f (δX) = F(X; δ). X δ If we are given a set of covering functions G for x˙ = f (x), we can apply the same variable’s substitution to each gi , obtaining new covering functions Gi (X; δ) = 1 δ gi (δX) such that ˙ = F(X; δ) = 1 f (δX) = 1 X gi (δX) = Gi (X; δ). δ δ i=1 i=1 k

k

The translation to sCCP simply proceeds associating an agent to each element of the set of covering functions G: Definition 14. Let x˙ = f (x) be a set of ODE’s, and G be a set of covering functions for it. Let gi ∈ G and δ ∈ R+ . The agent manGi ,δ is deﬁned as17 manGi ,δ :- ask|Gi (X;δ)| (Gi (X; δ) > 0). tell∞ (X = X + hi ).manGi ,δ + ask|Gi (X;δ)| (Gi (X; δ) < 0). tell∞ (X = X − hi ).manGi ,δ

The agent manGi ,δ is a summation with two branches: both have rate equal to the modulus of function Gi , but one is active when Gi > 0, and it increments the value of X according to the vector hi , while the other is active when Gi < 0, decrementing X by hi . In order to construct the sCCP network associated to a set of ODE x˙ = f (x), we simply need to deﬁne an agent manGi ,δ for each function Gi of the covering set G, putting these agents in parallel. We can render this procedure in the following SCCP operator: Definition 15. Let x˙ = f (x) be a set of ODE’s, G be a set of covering functions for f , and δ ∈ R, δ > 0. The sCCP-network associated to f (x), with respect to the set of covering functions G and the increment’s step δ, indicated by SCCP (f (x), G, δ), is SCCP (f (x), G, δ) = manG1 ,δ . . . manGk ,δ , with x = δX. 17

The name “man” stands for manager.

(17)

Dynamical Systems and Stochastic Programming

255

The initial conditions of the sCCP program, given by init(X), are X(0) = 1δ x0 , where x0 are the initial conditions of the ODE’s. Functional rates of sCCP are central in the deﬁnition of this translation: each function of the covering set becomes a rate in a branch of an sCCP summation. This is made possible only due to the freedom in the deﬁnition of rates, because diﬀerential equations and covering functions considered here are general. The possibility of having general rates in sCCP is intimately connected with the presence of the constraint store, which contains information external to the agents. This means that part of the description of interactions can be moved from the logical structure of agents to the functional form of rates. Common stochastic process algebras like stochastic π-calculus [44] or PEPA [30], on the other hand, have simple numerical rates and they rely just on the structure of agents (and on additivity of the exponential distribution [39]) to compute the global rate. This restricts severely the class of functional rates that they can model. Indeed, in a recent work [18] Hillston introduces general rates in PEPA essentially through the addition of information external to the model, an approach similar in spirit to sCCP. 4.2

Invertibility

We turn now to study the relation between the two translations deﬁned, i.e. ODE and SCCP . Speciﬁcally, we will show that (ODE ◦ SCCP ) returns the original diﬀerential equations, independently from the covering set G. The other direction, instead, cannot hold, as pointed out at the beginning of this section. In fact, we have seen that several sCCP agents can be mapped by ODE to the same equations, hence the map ODE cannot be inverted. Theorem 4. Let x˙ = f (x) be a set of diﬀerential equations, with x=(x1 , . . . , xn ) and X = 1δ x, and let G be a set of covering functions for f . Then ODE(SCCP (f (x), G, δ) , X) = f (x). Proof. From Deﬁnition 15 we know that SCCP (f (x), δ) = manG1 ,δ . . .

manGn ,δ , and by Theorem 2, ODE (manG1 ,δ . . . manGn ,δ , X)=ODE(manG1 ,δ , X)+. . .+ODE(manGn ,δ , X). Now, the agent manGi ,δ can modify several variables Xi , according to the vector hi coupled with the function Gi . The RTS of manGi ,δ is easily seen to have the following form

256

L. Bortolussi and A. Policriti

Therefore, the interaction matrix associated to manGi ,δ has just two columns, corresponding to the vectors hi and −hi , and ODE(manGi ,δ , X) is equal to hi |Gi (X; δ)| Gi (X; δ) > 0 − hi |Gi (X; δ)| Gi (X; δ) < 0 . In the previous equation, · denotes, as in Section 3, the logical value of a formula. The previous equation can be simpliﬁed by noting that |Gi (X; δ)| Gi (X; δ) > 0 − |Gi (X; δ)| Gi (X; δ) < 0 = Gi (X; δ), hence ODE(manGi ,δ , X) = hi Gi (X; δ). By applying Theorem 2 we then obtain ODE(SCCP (f (x), G, δ) , X) =

k

hi Gi (X; δ) = F (X),

i=1

which is equal to x˙ = f (x) when changing the variables back to x. 4.3

Behavioral Equivalence

We start this section by presenting an example showing how the translation from ODE’s to sCCP works. In particular we will be concerned with the behavior exhibited by both systems and with the dependence on the step size δ, governing the size of the basic increment or decrement of variables. Intuitively, δ controls the “precision” of the sCCP agents w.r.t. the original ODE’s. Hence, varying the size of δ, we can calibrate the eﬀect of the stochastic ﬂuctuations, reducing or increasing it. This is evident in the following example, where we compare solutions of ODE’s and the simulation of the corresponding sCCP processes. Let’s consider the following system of equations, representing another model of the Repressilator (see Section 3.2), a synthetic genetic network having an oscillatory behavior (see [21,3]): 0.5 x˙ 1 = α1 x−1 3 − β1 x1 , −1 x˙ 2 = α2 x1 − β2 x0.5 2 , 0.5 x˙ 3 = α3 x−1 2 − β3 x3 ,

α1 = 0.2, α2 = 0.2, α3 = 0.2,

β1 = 0.01 β2 = 0.01 β3 = 0.01.

(18)

We ﬁx the following set G of covering functions: g1 = α1 x−1 3 , h1 = (1, 0, 0), g2 = −1 0.5 α2 x−1 , h = (0, 1, 0), g = α x , h = (0, 0, 1), g = β 2 3 3 2 3 4 1 x1 , h4 = (−1, 0, 0), 1 0.5 0.5 g5 = β2 x2 , h5 = (0, −1, 0), g6 = β3 x3 , h6 = (0, 0, −1). The corresponding sCCP process, after changing variables according to Xi = xδi , is: manG1 ,δ manG2 ,δ manG3 ,δ manG4 ,δ manG5 ,δ manG6 ,δ , where, for instance, the agent manG1 ,δ is

(19)

Dynamical Systems and Stochastic Programming

257

manG1 ,δ :- ask| α1 (δX3 )−1 | ( αδ1 (δX3 )−1 > 0).tell∞ (X1 = X1 + 1).manG1 ,δ δ + ask| α1 (δX3 )−1 | ( αδ1 (δX3 )−1 < 0).tell∞ (X1 = X1 − 1).manG1 ,δ δ

In Figure 11, we study the dependence on δ of the sCCP network obtained from equations (19). From the plots, we note that the smaller the δ, the closer the stochastic trace is to the solution of ODE’s. However, increasing δ, the eﬀect of stochastic perturbations gets stronger and stronger, making the system change dynamics radically. Reducing the value of δ seems to be essentially the same as working with a sufﬁciently high number of molecules in standard biochemical networks, see [26,24] and the discussion in Section 3.2. It is thus reasonable to expect that, by taking δ smaller and smaller, the deterministic and the stochastic dynamics will eventually coincide. In fact, reducing δ we are diminishing the magnitude of stochastic ﬂuctuations, hence their perturbation eﬀects.

(a) Solution of ODE’s (19)

(c) SCCP simulation, δ = 0.01

(b) SCCP simulation, δ = 0.001

(d) SCCP simulation, δ = 1

Fig. 11. Diﬀerent simulations of sCCP agent obtained from S-Systems equations of repressilator (19), as basic step δ varies. Speciﬁcally, in Figure 11(a) we show the solution of ODE’s (19), while in Figures 11(b), 11(c), 11(d) we present three simulations of the sCCP agent corresponding to ODE’s (19), for δ = 0.001, 0.01, 1 respectively. In the last diagram, the behavior of S-System’s equations is destroyed. Note that in Figure 11(c) the time axis is stretched by a factor of 100, while in Figure 11(b) the time axis is stretched by a factor of 1000, consistently with the rescaling of variables by 1δ performed in the translation from ODE to sCCP.

258

L. Bortolussi and A. Policriti

This conjecture is indeed true: in the rest of the section we prove that, under mild conditions on the ODE’s and of the functions of G, the trajectories of the stochastic simulation converge to the solution of the ODE’s, independently of the choice of the covering set G. In fact, the set of stochastic traces whose distance from the solution of the ODE’s is greater than a ﬁxed arbitrary constant has zero probability in the limit δ → 0. Kurtz theorem. In 1970 Thomas Kurtz proved a theorem giving conditions for a family of density dependent Continuous Time Markov Chains to converge to a solution of a system of ODE’s [35,36]. In fact, under mild assumptions on the smoothness of functions into play, the trajectories of the CTMC remain, in the limit, close to the solution of a particular set of ODE’s with probability one. Our mapping SCCP easily ﬁts into Kurtz’s framework, with the step δ playing the role of the density. We start by recalling the Kurtz’s theorem. Let V be a positive parameter, playing the role of the “size” of the system, and XV (t) be a family of CTMC with state space Zn , depending on the parameter V . Suppose that there exist a continuous positive real function ϕ : Rn × Zn → R, such that the inﬁnitesimal generator matrix [39] Q = (qX,Y ) for XV (t) is given by 1 qX,X+h = V ϕ( X, h), h = 0. V In addition, let Φ(x) = h∈Zn hϕ(x, h). Theorem 5 (Kurtz [35]). Fix a bounded time interval [0, T ]. Suppose there exists an open set E ⊆ Rn and a constant ME ∈ R+ such that 1. |Φ(x) − Φ(y)| < ME |x − y|, ∀x, y ∈ E (i.e. Φ satisﬁes the Lipschitz condition); 2. supx∈E h∈Zn |h|ϕ(x, h) < ∞; 3. limd→∞ supx∈E |h|>d |h|ϕ(x, h) = 0. Then, for every trajectory x(t) that is a solution of x˙ = Φ(x) satisfying x(0) = x0 and x(t) ∈ E, t ∈ [0, T ], if lim

V →∞

1 XV (0) = x0 , V

then for every ε > 0, 1 lim P sup XV (t) − x(t) > ε = 0. V →∞ t≤T V This theorem states that the trajectories of XV (t) converge, in a bounded time interval, to the solution of x˙ = Φ(x), when V → ∞. The function Φ is essentially the sum of all ﬂuxes of the system.

Dynamical Systems and Stochastic Programming

259

Convergence for SCCP Our framework can be easily adapted to ﬁt this theorem. Consider a system of ODE’s x˙ = f (x) and a set of covering functions G. Denote by Xδ (t) the CTMC associated to the sCCP-network SCCP (f (x), G, δ). Theorem 6. Let x˙ = f (x) be a system of ODE’s, with x ∈ Rn , and [0, T ] a bounded time interval. Let G = {(gi , hi ) | i = 1 . . . , μ} be a set of covering functions for f . If there exists an open set E ⊆ Rn such that f satisﬁes the Lipschitz condition in E and supx∈E |gi (x)| < ∞, for each i = 1 . . . , μ, then for every ε > 0 lim P sup |δXδ (t) − x(t)| > ε = 0,

δ→0

t≤T

where x(t) is the solution of x˙ = f (x) with initial condition x(0) = x0 and δXδ (0) = x0 . Proof. In order to prove the theorem, we simply need to show that we satisfy all the hypothesis of the Kurtz’s theorem. First of all, in this setting the density V is equal to 1δ , so that 1δ → ∞ when δ → 0. Consider now the function gi (x), and deﬁne as customary gi+ (x) = gi (x)gi (x) ≥ 0 and gi− (x) = gi (x)gi (x) ≤ 0, so that gi (x) = gi+ (x) − gi− (x) and |gi (x)| = gi+ (x) + gi− (x), where · denotes the logical value as before. δ Consider now the inﬁnitesimal generator matrix Qδ = (qX,Y ) of the CTMC n Xδ (t). It is straightforward to prove that, for each h ∈ Z , δ qX,X+h =

1 δ

gi+ (δX) +

i | hi =h

1 δ

gi− (δX),

i | hi =−h

where the sum must be intended equal to zero if the index set is empty. Clearly, these are density dependent rates, with density 1δ . Note that there is a ﬁnite δ number of vectors for which qX,X+h is diﬀerent from zero, as the set {h1 , . . . , hμ } is ﬁnite. Therefore, the function ϕ of the Kurtz theorem is simply deﬁned as ϕ(x, h) =

i | hi =h

gi+ (x) +

gi− (x).

i | hi =−h

Then, the function Φ(x) is Φ(x) =

h

hϕ(x, h) =

μ j=1

hj (gj+ (x) − gj− (x)) =

μ j=1

hj gj (x) = f (x).

260

L. Bortolussi and A. Policriti

It only remains to prove that conditions 1–3 of Theorem 5 are satisﬁed. Condition 1 is obvious because Φ = f is Lipschitz by hypothesis, while condition 3 hold because |h| > M implies ϕ(x, h) = 0, where M = max1≤i≤μ |hi |. Finally, condition 2 follows because h

|h|ϕ(x, h) ≤ M

μ μ (gj+ (x) + gj− (x)) = M |gj (x)|, j=1

j=1

hence sup

x∈E

h

|h|ϕ(x, h) ≤ sup M x∈E

μ j=1

|gj (x)| ≤

μ

sup M |gj (x)| ≤ ∞,

j=1 x∈E

due to the condition on gi functions. Comments and examples on Theorem 6. Theorem 6 states that sCCP networks are able to simulate ODE’s with an arbitrary precision. The cost of an exact stochastic simulation of the sCCP-network of Deﬁnition 15, however, is proportional to 1δ , hence accurate stochastic simulations of ODEs are computationally impractical. On the other hand, there is no apparent reason to generate stochastic trajectories indistinguishable from the solution of the ODE’s, as the latter can be generally obtained with much less computational eﬀort. In a work related to ours [22], Hillston et al. used the same Kurtz theorem to prove an analogous result for the equations that can be obtained from a PEPA program. Theorem 6 can be seen as a generalization of their result. Moreover, in [22] the authors suggest that a stochastic approximation of ODE’s can be used together with analysis techniques typical of CTMC, like steady state analysis. This is a promising direction, but extreme care must be used. Kurtz theorem, in fact, guarantees convergence only in a ﬁxed and bounded time interval [0, T ], hence it does say nothing about asymptotic convergence of stochastic trajectories to ODE’s. Intuitively, the step δ may not be the only responsible for asymptotic convergence; an important role should also be played by initial conditions through topological properties of the phase space. If the ODE-trajectory we are considering is stable, i.e. resistant to small perturbations, then we can expect it to be reproduced in sCCP along the whole time axis, given a step δ small enough. On the other hand, if the trajectory is unstable, then even small perturbations can drive the dynamics far away from it; stochasticity, in this case, will unavoidably produce a trace dramatically diﬀerent from the one of ODE’s. Of course, by taking the interval [0, T ] of the theorem big enough (hence δ small enough), we can postpone arbitrarily far away in time the moment in which a stochastic and an unstable deterministic trajectories will diverge. As an example of instability, let’s consider a simple linear system of diﬀerential equations: X˙ X +Y = (20) 4X + Y Y˙

Dynamical Systems and Stochastic Programming

261

The theory of dynamical systems [51] tells us that the point (0, 0) is a saddle node, i.e. an unstable equilibrium whose phase space resembles the one depicted in Figure 12(a). The two straight lines arethe directions spanned by the eigen11 vectors of the matrix of coeﬃcients , and are called stable and unstable 41 manifolds. Motion in the stable manifold converges to the equilibrium (0, 0), while the unstable manifold and all other trajectories diverge to inﬁnity. However, small perturbations applied to the stable manifold can bring the system on a divergent hyperbolic trajectory, so we expect that ODE’s and the associated sCCP agent, when starting from the stable manifold (say from point (1, −2)), will eventually jump on a divergent trajectory. Moreover, we expect that the smaller δ the later this event will happen. This intuition is conﬁrmed in Figures 12(b), 12(c), 12(d).

(a) Phase portrait of a saddle node

(c) sCCP simulation — δ = 0.001

(b) ODE solution

(d) sCCP simulation — δ = 0.00001

Fig. 12. 12(a): Phase space of the linear system (20). The origin is a saddle node; the stable manifold is displayed with arrows pointing towards the origin, while the unstable manifold has arrows diverging from it. 12(b): Solution of the ODE’s (20), starting from (1, −2), a point belonging to the stable manifold. 12(c),12(d): Simulation of the sCCP agent associated to the linear system (20), w.r.t. the set of convering functions {(X, (1, 0)), (Y, (1, 0)), (4X, (0, 1)), (Y, (0, 1))} with initial conditions (1, −2). The step δ is equal respectively to 0.001 and 0.00001. The time in which these trajectories diverge from the solution of the ODE’s increases as δ becomes smaller.

262

L. Bortolussi and A. Policriti

This example shows that convergence issues need to be investigate further. In particular, conditions taking into account the topology of the phase space of the ODE’s are required in order to guarantee also asymptotic convergence. Another interesting direction to investigate is to exploit other results by Kurtz [36] in order to state error bounds in the approximation. Examples of Sets of Covering Functions. All the results of this Section have been given parametrically w.r.t. a set of covering functions G. When we motivated the introduction of such concept, we stated that a choice of a speciﬁc G corresponds to a speciﬁc logical structure of the ﬂuxes generating the ODEs. We discuss now two possible choices of G, one natural in absence of information, and the other tailored for ODEs coming from sets of biochemical reactions for which it is possible to reconstruct the reactions from the ODEs. Example 2. The simplest choice of a covering set is the one in which all the addends of ODEs are treated as independent ﬂux sources. To be more ki speciﬁc, consider a set of ODE f (x) = (f1 (x), . . . , fk (x)), with fi (x) = j=1 fij (x), where fij are the single addends of the ODE. The idea is to treat independently each such fij , so that GD = {(fij , ei ) | 1 ≤ i ≤ k, 1 ≤ j ≤ ki }. Such covering set GD will be called in the following the disentangled covering set. This was the choice adopted, for instance, when discussing the Repressilator example in Section 4.3. This choice is reasonable in absence of any further information on the system modeled by the ODE’s, and it does not preserve structural properties of the system, like mass conservation. For instance, consider the system deﬁned by the single reaction A →k·A B, which preserves the total mass A+B. With the disentangled covering set, however, we would reconstruct the logical structure of the following system of biochemical reactions: A →k·A and →k·A B, which does not preserve the total mass A + B. Example 3. Assume now we have a system generated by a set of mass action reactions such that the left hand side (i.e., the list of reactants) of each such reaction is unique. This has the consequence that each reaction is uniquely identiﬁed by the algebraic structure as a monomial of its rate function. For instance, the only reaction with A and B as reagents, A+ B →k ?, is uniquely identiﬁed by the signature as a monomial of its rate function kAB, i.e. by the monomial AB with coeﬃcient 1. This property has the immediate consequence that, whenever we ﬁnd two addends of the ODE with the same signature as a monomial, we are guaranteed that they are two instances of the same ﬂux. That is to say, if the condition is satisﬁed, we know how to reconstruct the set of reactions that originated the ODE’s. ki Formally, given f (x) = (f1 (x), . . . , fk (x)), with fi (x) = j=1 fij (x), we can construct the covering set GR as follows: list all the diﬀerent support monomials {p1 , . . . , ps } in the set {fij | 1 ≤ i ≤ k, 1 ≤ j ≤ ki }, and, for each l, deﬁne the terms αl,j ∈ Z and βl ∈ R+ such that: 1. if pl occurs in the equation for variable i, then its occurrence is equal to αl,i βl pl ;

Dynamical Systems and Stochastic Programming

263

2. if pl does not occur in the equation for variable i, then αl,i = 0; 3. all αl,j = 0 are prime among them18 ; Then, letting αl = (αl,1 , . . . , αl,k ), the covering set GR can be deﬁned as GR = {(βl pl , αl ) | l = 1, . . . , s}. For instance, consider the ODEs ⎧ ⎨ X˙ = −2kX 2 Y Y˙ = −kX 2Y ⎩ ˙ Z = 3kX 2 Y The associated covering set GR is simply GR = {(kX 2 Y, (−2, −1, 3))}, corresponding to the single reaction 2X + Y →k 3Z.

5

Final Discussion

In this paper we presented a method to associate ordinary diﬀerential equations to sCCP programs (written with a restricted syntax), and also a method that generates an sCCP-network from a set of ODE’s. The translation from sCCP to ODE’s is based on the construction of a graph, called RTS, whose edges represent all possible actions performable by sCCP-agents. Properties of restricted(sCCP ) guarantee that the graph is always ﬁnite. From an RTS, we can construct an interaction matrix containing the modiﬁcations that each action makes to each variable. Writing the corresponding ODE’s is simply a matter of combining the interaction matrix with the rate of each action. The inverse translation, from ODE’s to sCCP, exploits the functional form that rates have in sCCP. In this way, we can associate sCCP-agents to general ODE’s. An important feature of this method is that it is parametric w.r.t. the basic increment of variables, meaning that we can reduce the eﬀect of stochastic ﬂuctuations in the sCCP-model. Actually, we proved in Theorem 6 that, in the limit of an inﬁnitesimal increment, the trajectories of the ODE’s and of the corresponding sCCP-system coincide. In Section 3.1, we showed that the translation from sCCP to ODE’s, when applied to models of biochemical reactions, preserves the rate semantics in the sense of [17]. This condition, however, is not suﬃcient to guarantee that the translation maintains also the dynamical behavior of the sCCP-model. In fact, in Section 3.2, we provided several examples where an sCCP-network and the associated ODE’s manifest a diﬀerent behavior. This divergence can be caused by many factors, all qualitatively diﬀerent. Preserving dynamical behavior, however, is not just a mathematical game, but is is a central property that a translation from sCCP to ODE should have in order to be used as an analysis technique for stochastic process algebras. In this light, also the mapping from ODE to sCCP can be seen as a tool to investigate behavioral preservation. 18

This condition guarantees that αlj are uniquely deﬁned.

264

L. Bortolussi and A. Policriti

In Section 4, when we introduced the notion of set of covering functions, we noted that in the passage to ODE’s we unavoidably lose something of the logic of the sCCP model. This also suggest that the preservation of behavior may be reasonable only from a qualitative point of view. Indeed, this weaker approach ﬁts better with the management of stochastic noise, see the discussion at the end of Section 3.2. The loss of precision in passing to ODE’s is, however, counterbalances by the computational gain: simulating stochastic processes is undoubtedly much more expensive than numerically solving ODE’s [24]. There are several open problems related to the question of behavioral equivalence. We list hereafter some of the most important ones, according to us. – We need to identify the class of sCCP models (and their regions of parameter space/initial conditions) for which the mapping ODE preserves dynamics. Intuitively, according to discussion of Section 3.2, this may happen if all variables have big absolute values and if the phase space of the ODE’s has asymptotically stable trajectories with ample basins of attraction. – The repressilator and the simple self-inhibited genetic network of Section 3.2 suggest that the discrete ingredient cannot be continuously approximated so easily. In particular, associating continuous variables to RTS-states seems rather arbitrary. A possible solution can be that of transforming an sCCP network into a hybrid system, in which continuous and discrete dynamics coexist. In this way, we may be able to preserve part of the discrete structure of an sCCP-network, possibly just that fundamental for maintaining the behavior. We are investigating this direction, mapping sCCP-programs to hybrid automata [29,2]. The ﬁrst results are encouraging, see [13,14] – The notion of behavioral equivalence needs to be speciﬁed formally. At the end of Section 3.2, we suggested an approach based on a suitable temporal logic, in which equivalence would mean equi-satisﬁability of the same set of formulae. As a ﬁnal remark, we would like to consider this work under the perspective of the study of systemic properties. In fact, when we model a biological system, we are concerned mainly with the understanding of its systemic properties, especially what they are and how they emerge from basic interactions. In this direction, a modeler needs a formal language to specify biological systems, possibly provided with diﬀerent semantics, related to one another and stratiﬁed in several layers of increasing approximation and abstraction. For example, sCCP has a natural CTMC-based semantics, but an ODE-based one can be assigned to it via the ODE operator. A possible layer in the middle consists in a semantic based, for instance, on hybrid automata. Finally, we need also a language to specify system’s properties, automatically verifying them on the diﬀerent semantics, or better, on the simpler semantic where answers are correct (i.e., on the simpler semantic showing the same dynamical behavior of the most general one). All these features must clearly be part of the same operative framework (and of the same software tool), hence all the open questions presented above can be seen as steps in this direction.

Dynamical Systems and Stochastic Programming

265

References 1. Converging sciences. Trento (2004), http://www.unitn.it/events/consci/ 2. Alur, R., Belta, C., Ivancic, F., Kumar, V., Mintz, M., Pappas, G., Rubin, H., Schug, J.: Hybrid modeling and simulation of biomolecular networks. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 19–32. Springer, Heidelberg (2001) 3. Antoniotti, M., Policriti, A., Ugel, N., Mishra, B.: Model building and model checking for biochemical processes. Cell Biochemistry and Biophysics 38(3), 271–286 (2003) 4. Aziz, A., Singhal, V., Balarin, F., Brayton, R., Sangiovanni-Vincentelli, A.: Verifying continuous time markov chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102. Springer, Heidelberg (1996) 5. Blossey, R., Cardelli, L., Phillips, A.: A compositional approach to the stochastic dynamics of gene networks. T. Comp. Sys. Biology, 99–122 (2006) 6. Blossey, R., Cardelli, L., Phillips, A.: Compositionality, stochasticity and cooperativity in dynamic models of gene regulation. HFPS Journal (2007) (in print) 7. Bortolussi, L.: Stochastic concurrent constraint programming. In: Proceedings of 4th International Workshop on Quantitative Aspects of Programming Languages (QAPL 2006). ENTCS, vol. 164, pp. 65–80 (2006) 8. Bortolussi, L.: Constraint-based approaches to stochastic dynamics of biological systems. PhD thesis, PhD in Computer Science, University of Udine (2007), http://www.dmi.units.it/~ bortolu/files/reps/Bortolussi-PhDThesis.pdf 9. Bortolussi, L.: A master equation approach to diﬀerential approximations of stochastic concurrent constraint programming. In: Proceedings of QAPL 2008. ENTCS (2008) (to appear) 10. Bortolussi, L., Fonda, S., Policriti, A.: Constraint-based simulation of biological systems described by molecular interaction maps. In: Proceedings of IEEE conference on Bioinformatics and Biomedicine, BIBM 2007 (2007) 11. Bortolussi, L., Policriti, A.: Relating stochastic process algebras and diﬀerential equations for biological modeling. In: Proceedings of PASTA 2006 (2006) 12. Bortolussi, L., Policriti, A.: Stochastic concurrent constraint programming and diﬀerential equations. In: Proceedings of Fifth Workshop on Quantitative Aspects of Programming Languages, QAPL 2007. ENTCS, vol. 16713 (2007) 13. Bortolussi, L., Policriti, A.: Hybrid approximation of stochastic concurrent constraint programming. In: Proceedings of IFAC 2008 (2008) 14. Bortolussi, L., Policriti, A.: The importance of being (a little bit) discrete. In: Proceedings of FBTC 2008. ENTCS (2008) (to appear) 15. Bortolussi, L., Policriti, A.: Modeling biological systems in concurrent constraint programming. Constraints 13(1) (2008) 16. Cardelli, L.: From processes to odes by chemistry (2006), http://lucacardelli.name/ 17. Cardelli, L.: On process rate semantics. Theoretical Computer Science 391(3), 190–215 (2008) 18. Ciocchetta, F., Hillston, J.: Bio-PEPA: an extension of the process algebra PEPA for biochemical networks. In: Proceeding of FBTC 2007. Workshop of CONCUR 2007 (2007) 19. Seattle CompBio Group, Institute for Systems Biology. Dizzy home page 20. Cornish-Bowden, A.: Fundamentals of Chemical Kinetics, 3rd edn. Portland Press (2004)

266

L. Bortolussi and A. Policriti

21. Elowitz, M.B., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403, 335–338 (2000) 22. Geisweiller, N., Hillston, J., Stenico, M.: Relating continuous and discrete pepa models of signalling pathways. Theoretical Computer Science (2008) (in print) 23. Gillespie, D.: The chemical langevin equation. Journal of Chemical Physics 113(1), 297–306 (2000) 24. Gillespie, D., Petzold, L.: Numerical Simulation for Biochemical Kinetics. In: System Modelling in Cellular Biology. MIT Press, Cambridge (2006) 25. Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. of Computational Physics 22 (1976) 26. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. of Physical Chemistry 81(25) (1977) 27. Haas, P.J.: Stochastic Petri Nets. Springer, Heidelberg (2002) 28. Hastings, S.P., Murray, J.D.: The existence of oscillatory solutions in the ﬁeld-noyes model for the belousov-zhabotinskii reaction. SIAM Journal on Applied Mathematics 28(3), 678–688 (1975) 29. Henzinger, T.A.: The theory of hybrid automata. In: LICS 1996: Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science (1996) 30. Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press, Cambridge (1996) 31. Hillston, J.: Fluid ﬂow approximation of PEPA models. In: Proceedings of the Second International Conference on the Quantitative Evaluation of Systems, QEST 2005 (2005) 32. Kitano, H.: Foundations of Systems Biology. MIT Press, Cambridge (2001) 33. Kitano, H.: Computational systems biology. Nature 420, 206–210 (2002) 34. Kohn, K.W., Aladjem, M.I., Weinstein, J.N., Pommier, Y.: Molecular interaction maps of bioregulatory networks: A general rubric for systems biology. Molecular Biology of the Cell 17(1), 1–13 (2006) 35. Kurtz, T.G.: Solutions of ordinary diﬀerential equations as limits of pure jump markov processes. Journal of Applied Probability 7, 49–58 (1970) 36. Kurtz, T.G.: Limit theorems for sequences of jump markov processes approximating ordinary diﬀerential processes. Journal of Applied Probability 8, 244–356 (1971) 37. Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with prism: A hybrid approach. International Journal on Software Tools for Technology Transfer 6(2), 128–142 (2004) 38. Mcadams, H.H., Arkin, A.: Stochastic mechanisms in gene expression. PNAS 94, 814–819 (1997) 39. Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1997) 40. Noyes, R.M., Field, R.J.: Oscillatory chemical reactions. Annual Review of Physical Chemistry 25, 95–119 (1974) 41. Nurse, P.: Understanding cells. Nature 24 (2003) 42. Plotkin, G.D.: A structural approach to operational semantics. J. Log. Algebr. Program., 60-61, 17–139 (2004) 43. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C++: The Art of Scientiﬁc Computing. Cambridge University Press, Cambridge (2002) 44. Priami, C.: Stochastic π-calculus. The Computer Journal 38(6), 578–589 (1995) 45. Priami, C., Regev, A., Shapiro, E.Y., Silverman, W.: Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Inf. Process. Lett. 80(1), 25–31 (2001)

Dynamical Systems and Stochastic Programming

267

46. Ramsey, S., Orrell, D., Bolouri, H.: Dizzy: stochastic simulation of large-scale genetic regulatory networks. Journal of Bioinformatics and Computational Biology 3(2), 415–436 (2005) 47. Rao, C.V., Arkin, A.P.: Stochastic chemical kinetics and the quasi-steady state assumption: Application to the gillespie algorithm. Journal of Chemical Physics 118(11), 4999–5010 (2003) 48. Regev, A., Shapiro, E.: Cellular abstractions: Cells as computation. Nature 419 (2002) 49. Saraswat, V.A.: Concurrent Constraint Programming. MIT press, Cambridge (1993) 50. Shapiro, B.E., Levchenko, A., Meyerowitz, E.M., Wold, B.J., Mjolsness, E.D.: Cellerator: extending a computer algebra system to include biochemical arrows for signal transduction simulations. Bioinformatics 19(5), 677–678 (2003) 51. Strogatz, S.H.: Non-Linear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry and Engeneering. Perseus books, Cambridge (1994) 52. Vilar, J.M.G., Yuan Kueh, H., Barkai, N., Leibler, S.: Mechanisms of noise resistance in genetic oscillators. PNAS 99(9), 5991 (2002) 53. Wilkinson, D.J.: Stochastic Modelling for Systems Biology. Chapman & Hall, Boca Raton (2006)

Computing Equilibrium Points of Genetic Regulatory Networks Graziano Chesi Department of Electrical and Electronic Engineering University of Hong Kong [email protected] http://www.eee.hku.hk/~chesi

Abstract. Computing equilibrium points of genetic regulatory networks is a problem of primary importance for numerous investigations in these systems. This paper addresses this problem for diﬀerential equation models, with the regulation function expressed in a general form which includes both SUM form and PROD form for saturation functions of any type. Speciﬁcally, a recursive algorithm is proposed, which provides at each recursion a region guaranteed to contain all equilibrium points. This region progressively shrinks, and asymptotically converges to the sought set of equilibrium points. Moreover, the proposed algorithm can also allow one to delimit and ﬁnd limit cycles. Some numerical examples are reported to illustrate and validate the proposed algorithm, including examples where standard mathematical tools fail to compute the sought equilibrium points. Keywords: Genetic regulatory network, Diﬀerential equation, Saturation, Equilibrium point, Limit cycle.

1

Introduction

Genetic regulatory networks explain the interactions between genes and proteins to form complex systems that perform complicated biological functions, see for instance [1,2,3,4,5,6,7,8]. Basically, there are two types of genetic regulatory network models, i.e., the Boolean model (or discrete model) and the diﬀerential equation model (or continuous model). In Boolean models, the activity of each gene is expressed in one of two states, ON or OFF, and the state of a gene is expressed by a Boolean function of the states of other related genes. In the diﬀerential equation models, the variables describe the concentrations of gene products, such as mRNAs and proteins, as continuous values of the gene regulation systems. See for example [9,10,11,12,13] and references therein for a wider categorization of genetic regulatory networks models. This paper focuses on genetic regulatory networks described through diﬀerential equation models. In these models the dynamics of each concentration is expressed by a function of all concentrations of the system. This function typically consists of two parts: a linear part which deﬁnes the natural decay rate of C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 268–282, 2009. c Springer-Verlag Berlin Heidelberg 2009

Computing Equilibrium Points of Genetic Regulatory Networks

269

the concentration itself, and a nonlinear part which deﬁnes the inﬂuence by all the other concentrations. The nonlinear part can be either described via sum of saturation functions (in this case the system is said to be in SUM form) or via product of saturation functions (in this case the system is said to be in PROD form). See for instance [14,15,16,17]. A fundamental problem in these networks consists of determining the equilibrium points, i.e. the amounts of concentrations for which the regulation process results complete. This is a necessary step for several investigations, such as steady-state, stability, disturbance rejection, etc. Unfortunately, to determine equilibrium points of genetic regulatory networks is a diﬃcult problem because these systems contain saturation functions, and hence the calculation of the equilibrium points amounts to solving a system of nonlinear equations. Indeed, there do not exist techniques able to guarantee to ﬁnd all solutions of such a system, except in the case of polynomial equations, which however can be addressed only for small degrees and small number of variables, see for instance [18,19,20,21] and references therein. In this paper we address the problem of computing equilibrium points of genetic regulatory networks described through diﬀerential equation models. We consider a general model which includes both SUM form and PROD form for saturation functions of any type. The contribution consists of a recursive algorithm which holds the following properties. First, at each recursion the algorithm provides a region containing all equilibrium points, i.e. no equilibrium is lost. Second, this region progressively shrinks, i.e. the conservatism does not increase. Third, this region asymptotically converges to the set of equilibrium points, i.e. all equilibrium points are found. The proposed algorithm is illustrated and validated through some numerical examples with synthetic and real genetic regulatory networks. In these examples it is also shown that standard mathematical tools for solving systems of nonlinear equations may fail to compute the sought equilibrium points. Moreover, in these examples it is also explained that the proposed algorithm can be useful to delimit and ﬁnd limit cycles. The paper is organized as follows. Section 2 introduces some preliminaries on genetic regulatory networks. Section 3 describes the proposed results. Section 4 presents some numerical examples. Finally, Section 5 reports some concluding remarks.

2

Preliminaries

First of all, let us introduce the notation used throughout the paper: -

R+ : positive real number set, i.e. {x ∈ R : x ≥ 0}; 0n : null vector of size n × 1; In : identity matrix of size n × n; ei : i-th column of In ; diag(x1 , . . . , xn ): diagonal matrix with xi at the (i, i) entry; X T : transpose of vector/matrix X; TF: transcription factor.

270

G. Chesi

The genetic regulatory networks considered in this paper are described by the diﬀerential equation model ⎧ ˙ i (t) = −ai mi (t) + bi (p1 (t), . . . , pn (t)) ⎨m p˙ i (t) = −ci pi (t) + di mi (t) (1) ⎩ i = 1, . . . , n where mi (t), pi (t) ∈ R+ are the concentrations of mRNA and protein of the i-th gene, ai , ci ∈ R+ are the degradation rates, di ∈ R+ expresses the eﬀect of mi (t) on pi (t), and bi : Rn+ → R+ is the regulatory function of the i-th gene. This function is typically nonlinear, and either always increases or always decreases with respect to any component of p(t) whenever its other components are ﬁxed, i.e. (−1)ki bi (p1 , . . . , pi−1 , x2 , pi+1 , . . . , pn ) ≥ (−1)ki bi (p1 , . . . , pi−1 , x1 , pi+1 , . . . , pn ) ∀x1 , x2 : x1 ≤ x2 ∀p1 (t), . . . , pn (t) ∈ R+ ∀i = 1, . . . , n (2) for some k1 , . . . , kn ∈ {0, 1}. In genetic regulatory networks with SUM form, the function bi (p1 (t), . . . , pn (t)) is expressed as the sum of functions of a single variable, i.e. bi (p1 (t), . . . , pn (t)) =

n

αi,j bi,j (pj (t))

(3)

j=1

where αi,j ∈ R+ is the contribution of TF j to the transcriptional rate for gene i, and bi,j : R+ → R+ is a monotonic function, i.e. bi,j (pj (t)) either always increases or always decreases with respect to pj (t). In genetic regulatory networks with PROD form, the function bi (p1 (t), . . . , pn (t)) is expressed as the product of the functions bi,j (pj (t)), i.e. bi (p1 (t), . . . , pn (t)) = αi

n

bi,j (pj (t))

(4)

j=1

where αi ∈ R+ represents the transcriptional rate for gene i. Each function bi,j (pj (t)) in (3) and (4) is typically expressed as ⎧ if TF j is an activator of gene i ⎨ f (pj (t)) bi,j (pj (t)) = 1 − f (pj (t)) if TF j is a repressor of gene i ⎩ γ otherwise

(5)

where γ ∈ R is a constant depending on the model which expresses the independence of gene i on TF j (γ = 0 for SUM form, γ = 1 for PROD form), and the function f (pj (t)) is a saturation function. For saturation function we mean a function satisfying the following properties: ⎧ f : R+ → [0, 1] ⎪ ⎪ ⎨ f (0) = 0 (6) limx→∞ f (x) = 1 ⎪ ⎪ ⎩ f (x2 ) ≥ f (x1 ) ∀x1 , x2 : x1 ≤ x2

Computing Equilibrium Points of Genetic Regulatory Networks

271

Hence, a saturation function is an increasing function between 0 and 1 deﬁned for positive value of the variable. For instance, in the case of regulatory functions with Hill form, the function f (pj (t)) is given by f (pj (t)) =

βH

pj (t)H + pj (t)H

(7)

where β ∈ R+ and H is an integer known as Hill coeﬃcient. In order to describe the results of this paper in a more compact form, we introduce a matrix version of the model (1) according to m(t) ˙ = Am(t) + b(p(t)) (8) p(t) ˙ = Cp(t) + Dm(t) where

m(t) = (m1 (t), . . . , mn (t)) p(t) = (p1 (t), . . . , pn (t))

(9)

are the vectors containing the concentrations of mRNA and protein, and A = diag(−a1 , . . . , −an ) C = diag(−c1 , . . . , −cn ) D = diag(d1 , . . . , dn )

(10)

are diagonal matrices containing the decay rates (matrices A and C) and the eﬀect of m(t) on p(t) (matrix D). The function b : Rn+ → Rn+ is a nonlinear function representing the regulation of the process, whose i-th component bi (p(t)) satisﬁes the monotonicity condition (2). We observe that the model (8) under the assumption (2), which is an equivalent matrix version of the model (1), includes: 1. genetic regulatory networks with SUM form, by choosing the i-th component of b(p(t)) as in (3); 2. genetic regulatory networks with PROD form, by choosing the i-th component of b(p(t)) as in (4); 3. genetic regulatory networks that are neither in SUM form nor in PROD form, provided that (2) holds. For instance, the choice for n = 3 given by ⎛ ⎞ b1,1 (p1 (t)) + b1,2 (p2 (t))3 ⎠ eb2,1 (p1 (t)) b2,3 b(p(t)) = ⎝ (11) (p3 (t)) 3 b3,2 (p2 (t)) + b3,3 (p3 (t)) deﬁnes a genetic regulatory networks which is neither in SUM form nor in PROD form, but which is included in the model (8) under the assumption (2). The problem addressed in this paper consists of determining the equilibrium points of (8), i.e. the solutions of the system of nonlinear equations ⎧ ⎨ Am + b(p) = 0n Cp + Dm = 0n (12) ⎩ m, p ∈ Rn+

272

G. Chesi

Remark 1. Before proceeding let us observe that existing mathematical tools for solving systems of nonlinear equations generally do not guarantee to ﬁnd all solutions of such systems. Indeed, systems of nonlinear equations can be solved via either analytical techniques or numerical techniques. Analytical techniques can be used in the case of polynomial or rational equations, and provides the sought solutions as roots of a one-variable polynomial. Unfortunately, the degree of this polynomial is prohibitive (except for very small systems) since in the worst case coincides with the maximum number of solutions of the system, which is given by the degree of the equations to the power of the number of variables, see for instance [18,22,19,21]. Numerical techniques, which are either based on the numerical minimization of a suitable function via for example Newton’s iterations starting from an initial point, or on homotopy methods which adopt continuation strategies, do not suﬀer of the previous problems. Unfortunately, these techniques cannot guarantee to ﬁnd all sought solutions, see for instance [23,20] and Section 4. Remark 2. Another remark concerns the fact that genetic regulatory networks can be also modeled as stochastic systems, where the input is represented by a stochastic process such as white noise. For instance, such an input could aﬀect (8) according to m(t) ˙ = Am(t) + b(p(t)) + w(t) (13) p(t) ˙ = Cp(t) + Dm(t) where w(t) ∈ Rn is a stochastic process. In these systems there are no equilibrium points in the classic sense since the input is a non-constant function of the time and hence the equation Am + b(p) + w(t) = 0n (14) would not admit solutions where m and p do not depend on the time (which is the classic deﬁnition of equilibrium point). Instead, one can consider equilibrium points corresponding to particular constant values of the stochastic process, such as its mean value, that the algorithm proposed in this paper allows one to compute. Indeed, these equilibrium points are deﬁned analogously to (12) as ⎧ ¯ = 0n ⎨ Am + b(p) + w Cp + Dm = 0n (15) ⎩ m, p ∈ Rn+ where w ¯ ∈ Rn is the the stochastic expectation of w(t).

3

Equilibria Computation

In this section we describe the proposed algorithm. Speciﬁcally, in Theorems 1 and 2 we introduce two preliminary functions and we describe their properties. Then, in Theorem 3 we provide the main algorithm to be used to compute the sought equilibrium points.

Computing Equilibrium Points of Genetic Regulatory Networks

273

Before proceeding, let us observe that the m-component of any solution of (12) is related to its p-component by the relationship Cp + Dm = 0n where C, D are nonsingular diagonal matrices with C negative deﬁnite. This means that (12) can be equivalently rewritten as ⎧ ⎨ −AD−1 Cp + b(p) = 0n m = −D−1 Cp (16) ⎩ p ∈ Rn+ Therefore, in the sequel we will focus on the computation of the vectors p fulﬁlling (16). We indicate the set of such vectors as E = p ∈ Rn+ : − AD−1 Cp + b(p) = 0n

(17)

Theorem 1. Let H be the rectangle defined by H = p ∈ Rn+ : pi ∈ [pi,− , pi,+ ]

(18)

for some p1,− , p1,+ , . . . , pn,− , pn,+ ∈ R+ , and let us define the map A(H) as A(H) = p ∈ Rn+ : pi ∈ [qi,− , qi,+ ]

(19)

where q1,− , q1,+ , . . . , qn,− , qn,+ ∈ R+ are computed according to qi,− = max pi,− , min eTi C −1 DA−1 z z∈Z T −1 −1 qi,+ = min pi,+ , max ei C DA z z∈Z

(20) (21)

where Z is the set given by Z = {b(p) : pi ∈ {pi,− , pi,+ } , i = 1, . . . , n} .

(22)

Then, the following properties hold: - Property P1: A(H) ⊆ H; - Property P2: p∗ ∈ H ∩ E ⇒ p∗ ∈ A(H); - Property P3: H ∩ A(H) = ∅ ⇒ H ∩ E = ∅. Proof. First, the property P1 holds because from (20)–(21) one has qi,− ≥ pi,− and qi,+ ≤ pi,+ ∀i = 1, . . . , n.

(23)

Second, the property P2 holds due to the monotonicity property (2) of bi (p) with respect to each component of p and to the linearity of the function eTi C −1 DA−1 z with respect to z. In fact, we have

274

G. Chesi

p∗ ∈ H ⇒ bi (p∗ ) ∈ [min zi , max zi ].

(24)

p∗ ∈ E ⇒ eTi C −1 DA−1 b(p∗ ) = p∗i .

(25)

p∗ ∈ H ∩ E ⇒ qi,− ≤ p∗i and qi,+ ≥ p∗i .

(26)

z∈Z

Moreover,

z∈Z

Hence, it follows

Lastly, the property P3 holds because, if one suppose for contradiction that H ∩ A(H) = ∅ and H contains a vector p∗ of E, then it would follow from the property P2 that p∗ belongs to A(H), hence contradicting the assumption that H ∩ A(H) = ∅. Let us observe that map A(·) requires trivial computations, i.e. evaluation of a linear function in some given points. In fact, let us observe that the set Z is ﬁnite. From the map A(·) we deﬁne the map B(·) in the following theorem. Theorem 2. Let H be a rectangle in (18), and let us define the map B(H) as follows: -

(Step (Step (Step (Step (Step

1) 2) 3) 4) 5)

set H(0) = H and k = 0 (k denotes the iteration number); if H(k) ∩ A(H(k) ) = ∅, set B(H) = ∅ and exit; if A(H(k) ) is a point, set B(H) = A(H(k) ) and exit; if H(k) = A(H(k) ), set B(H) = H(k) and exit; set H(k+1) = A(H(k) ), k = k + 1, and go to 2.

Then, B(H) returns either a rectangle, a point, or the empty set. Moreover: - Property P4: B(H) ⊆ H; - Property P5: p∗ ∈ H ∩ E ⇒ p∗ ∈ B(H). Proof. First of all, let us observe that the output of B(H) can be either the empty set (output of Step 2), a point (output of Step 3), or a rectangle (output of Step 4). Then, the property P4 follows from the fact that the output of B(H) is a sequence of applications of the map A(·) for which the property P1 ensures that the output is a subset of the input. Lastly, the property P5 holds since B(H) returns either a sequence of applications of the map A(·) for which the property P2 ensures that no vector of H ∩ E can be lost, or the empty set in the case H(k) ∩ A(H(k) ) = ∅ which however guarantees the absence of vectors of E in H(k) (and hence in H) due to the properties P2 and P3. The map B(·) transforms a given rectangle via a sequence of applications of the map A(·), and returns a set which can be either a rectangle, a point, or the empty set. By exploiting the map B(·) we derive the algorithm for the computation of the sought equilibrium points as follows.

Computing Equilibrium Points of Genetic Regulatory Networks

275

Theorem 3. (Algorithm for equilibrium points computation) Let H be a rectangle in (18) and let us define the map C(H) as follows: - (Step 1) if B(H) is either the empty set or a point, then set C(H) = B(H) and exit; - (Step 2) divide the rectangle B(H) in 2k rectangles H1 , . . . , H2k by taking the middle point on eachside of B(H) with nonzero length; - (Step 3) set C(H) = i=1,...,2k C(Hi ) and exit. Then, the algorithm to be launched in C(Rn+ ), for which the following properties hold: - Property P6: the positive octant Rn+ is progressively shrunk without losing any point of E; - Property P7: the set provided by the algorithm asymptotically converges to the set E. Proof. The property P6 holds because B(H) is guaranteed to include any vector in H ∩ E according to the property P5, moreover from the property P4 one has that the set returned by the algorithm cannot increase. Then, property P7 holds because no portion of Rn+ is lost in the division of each rectangle B(H). Hence, the proposed algorithm for computing the equilibrium points of (8) is launched as C(Rn+ ), which means that the positive octant Rn+ is used as initial rectangle H. This because Rn+ is clearly guaranteed to contain all solutions of (16). Then, the initial rectangle is passed to the map B(·). If the output of this map is either the empty set or a point, then the algorithm stops as it is guaranteed that there are no equilibrium points inside the considered rectangle. Otherwise, the output is another rectangle, which is then divided in smaller ones. The rectangles obtained in this division are passed to the map C(·) itself, hence realizing a recursive algorithm. As explained by the properties P6 and P7, the set provided by the algorithm is guaranteed to contain all points of E at each recursion, and to asymptotically converge to E. Remark 3. It is worth to remark that the proposed algorithm diﬀers from existing techniques for computing the solutions of systems of nonlinear equations. A ﬁrst diﬀerence is that the proposed algorithm does not rely on analytical techniques, which can be used only in special cases and typically for small systems. A second diﬀerence is that the proposed algorithm does not consider one possible initial point only contrary to some numerical techniques. Instead, the proposed algorithm consider the whole space of possible solutions, and progressively shrinks this space to the sought set of equilibrium points without losing any portion of it. Remark 4. Lastly, it is interesting to observe that the proposed algorithm can also allow one to investigate limit cycles of (8), which are periodic solutions m(t), p(t) of (8) satisfying the condition

276

G. Chesi

∃T ∈ R :

m(t) = m(t + T ) p(t) = p(t + T )

∀t ≥ 0

(27)

where T represents the period. Indeed, at the ﬁrst recursion of the proposed algorithm one obtains the rectangle B(Rn+ ) which is expected to contain existing limit cycles of (8) as they are periodic solutions of the system of diﬀerential equations. This suggests a strategy which can be useful to establish the existence of limit cycles in (8). In fact, once that B(Rn+ ) has been found at the ﬁrst recursion of the algorithm, one can investigate the trajectories starting along its boundary (for instance, at the vertices) to reveal limit cycles. See for instance Example 3.

4

Illustrative Examples

In this section we present some examples where the proposed algorithm is used. We report only the p-component of each equilibrium point, being the mcomponent directly given by D−1 Cp according to (16). The computational time for all examples is lesser than 5 seconds with an implementation of the proposed algorithm in Matlab 7 running under Windows XP on a personal computer with Pentium IV 2.2 GHz and 2 GB RAM. 4.1

Genetic Regulatory Network in PROD Form with Non-Hill Function

Let us start by considering the genetic regulatory network described in PROD form given by ⎧ m ˙ 1 (t) = −0.17m1(t) + 0.73f (p2 )(1 − f (p3 )) ⎪ ⎪ ⎨ m ˙ 2 (t) = −0.8m2(t) + 0.95(1 − f (p3 )) (28) ˙ 3 (t) = −0.52m3(t) + 0.58(1 − f (p1 )) ⎪m ⎪ ⎩ p˙ i (t) = −pi (t) + mi (t) ∀i = 1, 2, 3 and the saturation function 2

f (pi (t)) = 1 − e−pi (t) .

(29)

This genetic regulatory network is characterized by the fact that TF 1 is a regressor of gene 3, TF 2 is an activator of gene 1, and TF 3 is a regressor of genes 1 and 2. Let us use the algorithm proposed in Theorem 3. At the ﬁrst recursion of the algorithm we obtain that the positive octant R+ 3 is shrunk to the rectangle shown in Figure 1a. At the second recursion, the rectangle previously found is divided in four equal rectangles, one of which is shown in Figure 1b, another one shrinks to the equilibrium point shown in Figure 1b, and the other two converges to the empty set. At the fourth recursion, another equilibrium point is found as shown in Figure 1c, and only one rectangle is left. Then, at the eight recursion the last equilibrium point is found and no rectangle is left as shown in Figure 1d. We

Computing Equilibrium Points of Genetic Regulatory Networks 1.3

1.3

1.2

1.2

1.1

1.1

0.9

p2

1

0.9

p2

1

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0

0.5

1

p1 (a) 1.5

2

2.5

3

0.3

1.3

1.3

1.2

1.2

1.1

1.1

0.9

0

0.5

1

p1 (b)

2

2.5

3

0

0.5

1

p1 (d)

2

2.5

3

1.5

p2

1

0.9

p2

1

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.3

277

0.4

0

0.5

1

p1 (c) 1.5

2

2.5

3

0.3

1.5

Fig. 1. Steps of the proposed algorithm for the example in Section 4.1 (shown in the plane p1 –p2 for clarity of presentation): (a) ﬁrst recursion, R3+ is shrunk to a rectangle; (b) second recursion, an equilibrium point is found (denoted by the “∗” mark); (c) fourth recursion, another equilibrium point is found. (d): ninth recursion, the last equilibrium point is found.

hence conclude that this system has three equilibrium points, in particular the set E in (17) is given by E = (3.246, 1.189, 0.000)T , (0.461, 0.527, 0.902)T , (0.166, 0.366, 1.085)T . (30) For comparison, we attempt to use standard mathematical tools, in particular via Matlab and Mathematica. We hence use the functions “solve” (Matlab function for both analytical and numerical techniques) and ”ﬁndroot” (Mathematica function for numerical technique) which ﬁnd only one solution. This happens because the equations in (12) are neither polynomial nor rational in this case, which means that no analytical technique exist for ﬁnding the solutions in this case. Existing tools therefore apply numerical techniques which allow to ﬁnd a local solution starting from an initial point, but the other solutions are lost.

278

4.2

G. Chesi

Genetic Regulatory Network in SUM Form with Hill Function

In this example we consider the genetic regulatory network in SUM form with ⎧ m ˙ 1 (t) = −2.0m1 (t) + 0.9(1 − f (p2 )) + 0.5f (p3) ⎪ ⎪ ⎪ ⎪ m ˙ 2 (t) = −2.2m2 (t) + 0.9(1 − f (p3 )) + 0.5f (p4) ⎪ ⎪ ⎪ ⎪ m ˙ 3 (t) = −2.4m3 (t) + 0.9(1 − f (p4 )) + 0.5f (p5) ⎪ ⎪ ⎪ ⎨ .. . (31) ⎪ m ˙ (t) = −3.4m8 (t) + 0.9(1 − f (p9 )) + 0.5f (p10 ) ⎪ 8 ⎪ ⎪ ⎪ m ˙ 9 (t) = −3.6m9 (t) + 0.9(1 − f (p10 )) + 0.5f (p1 ) ⎪ ⎪ ⎪ ⎪ m ˙ (t) = −3.8m10 (t) + 0.9(1 − f (p1 )) + 0.5f (p2 ) ⎪ ⎪ ⎩ 10 p˙i (t) = −pi (t) + mi (t) ∀i = 1, . . . , 10 where the saturation function is chosen as the Hill function f (pi (t)) =

1 . 1 + pi (t)6

(32)

This genetic regulatory network is characterized by the cyclic structure where gene i has TF i + 1 as regressor and TF i + 2 as activator. By using the algorithm proposed in Theorem 3 we have that the positive octant R10 + shrinks to the set E = (0.449, 0.408, 0.375, 0.346, 0.321, 0.300, 0.281, 0.267, 0.251, 0.236)T , (33) hence implying that there is one equilibrium point only in this genetic regulatory network. Also in this case we attempt to use standard mathematical tools as done in the previous example. However, by using analytical techniques (which can be used since the equations in (12) are rational for this example) we do not obtain any solution. This happens because the degree of the one-variable polynomial that the analytical techniques allow one to ﬁnd is prohibitive in this case since the equations in (12) have degree 12 (the degree of b(p)) and 10 variables (the p-components of the state), therefore there can be up to 1210 solutions. Also, we attempt to use numerical techniques, and ﬁnd that they return the sought equilibrium point. Unfortunately, these techniques are not able to establish whether this solution is unique or not. 4.3

Repressilator Model in E. Coli

Here we consider the repressilator investigated in Escherichia coli [24]: ⎧ ˙ i (t) = −mi (t) + αrep (1 − f (pj (t))) ⎨m p˙ i (t) = −β rep (pi (t) − mi (t)) ⎩ i = lacl, tetR, cl; j = cl, lacl, tetR where the saturation function is the Hill function

(34)

Computing Equilibrium Points of Genetic Regulatory Networks

279

8

p3

6 4 2 0 0

2

4

6

p2

8

8

7

6

5

4

3

2

0

1

p1

Fig. 2. Found equilibrium point and limit cycle in the example of Section 4.3

f (pi (t)) =

1 1 + pi (t)2

(35)

and αrep , β rep ∈ R+ are positive constants. Let us select the plausible values αrep = 10 and β rep = 1. By using the algorithm proposed in Theorem 3 we ﬁnd that there is a unique equilibrium point, in particular E = (2, 2, 2)T .

(36)

For this example it is interesting to observe that, in addition to the found equilibrium point, there exists a limit cycle that the proposed algorithm can help to ﬁnd. Indeed, as explained in Remark 4, at the ﬁrst recursion of the proposed algorithm one obtains the rectangle B(R3+ ), which is equal to [0.1010, 9.899]3. Then, the limit cycle is revealed by simply computing the trajectory of the system starting at the vertices of this rectangle. Figure 2 shows the projection on the plane p1 -p2 of the found limit cycle.

280

4.4

G. Chesi

Genetic Regulatory Network in SUM Form with Non-Hill Function

As last example, we consider the genetic regulatory network in SUM form described by ⎧ ˙ 1 (t) = −2m1 (t) + 0.5f (p5 ) ⎪ ⎪m ⎪ ⎪ m ˙ 2 (t) = −m2 (t) + 0.1(1 − f (p2 )) + 0.4(1 − f (p4 )) ⎪ ⎪ ⎨ m ˙ 3 (t) = −0.6m3(t) + 0.2f (p1) + 1.1(1 − f (p4 )) (37) m ˙ 4 (t) = −m4 (t) + 0.5(1 − f (p3 )) + 1.5f (p4 ) ⎪ ⎪ ⎪ ⎪ m ˙ 5 (t) = −2m5 (t) + 0.3f (p2 ) + 0.3(1 − f (p5 )) ⎪ ⎪ ⎩ p˙ i (t) = −pi (t) + mi (t) ∀i = 1, . . . , 5 and the saturation function f (pi (t)) =

2 arctan(pi (t)2 ). π

(38)

This genetic regulatory network is characterized by the fact that TF 1 is an activator of gene 3, TF 2 is an activator of gene 5 and a regressor of gene 2, TF 3 is a regressor of gene 4, TF 4 is a regressor of genes 2 and 3 and an activator of gene 4, and TF 5 is an activator of gene 1 and a regressor of gene 5. By using the algorithm proposed in Theorem 3 as done in the previous examples we conclude that this system has three equilibrium points, in particular the set E in (17) is given by E = (0.0037, 0.1961, 0.4518, 1.566, 0.1515)T , (0.0039, 0.3130,1.003, 0.9278, 0.1570)T , (0.0046, 0.4827, 1.821, 0.1035, 0.1691)T .

(39)

However, by using standard mathematical tools, we obtain only one solution similarly to the example in Section 4.1.

5

Conclusion

We have proposed an algorithm which allows one to ﬁnd the equilibrium points of genetic regulatory networks described by diﬀerential equation models and which include both SUM form and PROD form with saturation functions of any type. The proposed algorithm is guaranteed to ﬁnd all sought equilibrium points, moreover as shown by some numerical examples the computation is reasonably fast also in cases where standard mathematical tools for solving systems of nonlinear equations may fail. It is hence expected that the proposed algorithm represents a useful tool for researchers working in the area of genetic regulatory networks. In particular, the proposed algorithm can allow one to investigate issues such as stability, disturbance rejection, and robustness, for which the knowledge of the equilibrium points is required, see for instance [25,26,27,28].

Computing Equilibrium Points of Genetic Regulatory Networks

281

Acknowledgement The author would like to thank the Editor and the Reviewers for their time and useful comments.

References 1. D’haeseleer, P., Wen, X., Fuhrman, S., Somogyi, R.: Mining the gene expression matrix: Inferring gene relationships from large scale gene expression data. In: Paton, R.C., Holcombe, M. (eds.) Information Processing in Cells and Tissues. Plenum Publishing, New York (1998) 2. Davidson, E.H.: The Regulatory Genome: Gene Regulatory Networks In Development And Evolution. Academic Press, London (2006) 3. D’haeseleer, P.: Reconstructing Gene Networks from Large Scale Gene Expression Data. PhD thesis, University of New Mexico (2000) 4. D’haeseleer, P., Liang, S., Somogyi, R.: Genetic network inference: From coexpression clustering to reverse engineering. Bioinformatics 16(8), 707–726 (2000) 5. Li, C., Chen, L., Aihara, K.: A systems biology perspective on signal processing in genetic network motifs. IEEE Signal Processing Magazine 221(3), 136–142 (2007) 6. Yuh, C.H., Bolouri, H., Davidson, E.H.: Genomic cis-regulatory logic: Experimental and computational analysis of a sea urchin gene. Science 279, 1896–1902 (1998) 7. Tsai, H.K., Yang, J.M., Tsai, Y.F., Kao, C.Y.: An evolutionary approach for gene expression patterns. IEEE Transactions on Information Technology in Biomedicine 8(2), 69–78 (2004) 8. Maraziotis, I.A., Dragomir, A., Bezerianos, A.: Gene networks reconstruction and time-series prediction from microarray data using recurrent neural fuzzy networks. IET Systems and Biology 1(1), 41–50 (2007) 9. Smolen, P., Baxter, D.A., Byrne, J.H.: Mathematical modeling of gene networks. Neuron 26(3), 567–580 (2000) 10. Bower, J.M., Bolouri, H. (eds.): Computational Modeling of Genetic and Biochemical Networks. Computational Molecular Biology. MIT Press, Cambridge (2001) 11. Jong, H.D.: Modeling and simulation of genetic regulatory systems: A literature review. Journal of Computation Biology 9, 67–103 (2002) 12. D’haeseleer, P., Liang, S., Somogyi, R.: Gene expression data analysis and modeling. In: Proc. Paciﬁc Symposium on Biocomputing, Hawaii, USA (1999) 13. Aracena, J., Lamine, S.B., Mermet, M.A., Cohen, O., Demongeot, J.: Mathematical modeling in genetic networks: Relationships between the genetic expression and both chromosomic breakage and positive circuits. IEEE Transactions on Systems, Man, and Cybernetics–Part b: Cybernetics 33(5), 825–834 (2003) 14. Bintu, L., Buchler, N.E., Garcia, H.G., Gerland, U., Hwa, T., Kondev, J., Phillips, R.: Transcriptional regulation by the numbers: models. Current Opinion in Genetics and Development 15(2), 116–124 (2005) 15. Li, C., Chen, L., Aihara, K.: Stability of genetic networks with sum regulatory logic: Lure system and lmi approach. IEEE Trans. on Circuits and Systems I 53(11), 2451–2458 (2006) 16. Li, C., Chen, L., Aihara, K.: Stochastic stability of genetic networks with disturbance attenuation. IEEE Transactions on Circuits and Systems II 54(10), 892–896 (2007)

282

G. Chesi

17. Chesi, G., Hung, Y.S.: Stability analysis of uncertain genetic SUM regulatory networks. Automatica 44(9), 2298–2305 (2008) 18. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach. Int. Journal of Robust and Nonlinear Control 13(13), 1239–1257 (2003) 19. Mora, T.: Solving Polynomial Equation Systems II. Cambridge University Press, Cambridge (2005) 20. Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, Heidelberg (2006) 21. Chesi, G.: Optimal representation matrices for solving polynomial systems via LMI. Int. Journal of Pure and Applied Mathematics 45(3), 397–412 (2008) 22. Stetter, H.J.: Numerical Polynomial Algebra. SIAM, Philadelphia (2004) 23. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia (1987) 24. Elowitz, M.B., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403, 335–338 (2000) 25. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Englewood Cliﬀs (2001) 26. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Homogeneous Lyapunov functions for systems with structured uncertainties. Automatica 39(6), 1027–1035 (2003) 27. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Solving quadratic distance problems: an LMI-based approach. IEEE Trans. on Automatic Control 48(2), 200–212 (2003) 28. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems. Lecture Notes in Control and Information Sciences, vol. 390. Springer, London (2009)

Code, Context, and Epigenetic Catalysis in Gene Expression Rodrick Wallace1 and Deborah Wallace2 1

The New York State Psychiatric Institute, 549 W. 123 St., Suite 16F, New York, NY, 10027. Tel.: (212) 865-4766 [email protected] 2 Consumers Union [email protected]

Abstract. We examine a class of probability models describing how epigenetic context aﬀects gene expression and organismal development, using the asymptotic limit theorems of information theory in a highly formal manner. Taking classic results on spontaneous symmetry breaking abducted from statistical physics in groupoid, rather than group, circumstances, the work suggests that epigenetic information sources act as analogs to a tunable catalyst, directing development into diﬀerent characteristic pathways according to the structure of external signals. The results have signiﬁcant implications for epigenetic epidemiology, in particular for understanding how environmental stressors, in a large sense, can induce a broad spectrum of developmental disorders in humans.

1 1.1

Introduction Toward New Tools

Researchers have begun to explore a de-facto cognitive paradigm for gene expression in which contextual factors determine the behavior of what Cohen calls a ‘reactive system’, not at all a deterministic, or even stochastic, mechanical process (e.g., [18, 19, 74]). The diﬀerent approaches, while highly formal, are nonetheless much in the spirit of the pioneering eﬀorts of Maturana and Varela [53, 54] who foresaw the essential role that cognitive process must play in a vast realm of biological phenomena. O’Nuallain [57] has recently placed gene expression ﬁrmly in the realm of complex linguistic behavior, for which context imposes meaning, claiming that the analogy between gene expression and language production is useful both as a fruitful research paradigm and also, given the relative lack of success of natural language processing (nlp) by computer, as a cautionary tale for molecular biology. First O’Nuallain argues that, at the orthographic or phonological level, depending on whether the language is written or spoken, we can map from phonetic elements to nucleotide sequence. His second claim is that Nature has designed highly ambiguous codes in both cases, and left disambiguation to the context. C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 283–334, 2009. c Springer-Verlag Berlin Heidelberg 2009

284

R. Wallace and D. Wallace

He notes that, given our concern with the Human Genome Project (HGP) and its implications for human health, only 2% of diseases can be traced back to a straightforward genetic cause. As a consequence the HGP will have to be redone for a variety of metabolic contexts in order to establish a sound technology of genetic engineering [58]. Here we investigate a broad class of probability models based on the asymptotic limit theorems of information theory that instantiate this perspective, ﬁnding a ‘natural’ means by which epigenetic context ‘farms’ gene expression in an inherently punctuated manner via a kind of tunable catalysis. The models will be used to study how normal developmental modes can be driven by external context into pathological trajectories often expressed, in humans, as comorbid psychiatric and physical disorders, expanding recent work [71]. It appears possible to convert such models to powerful tools for data analysis, much as those based on the Central Limit Theorem can be converted to parametric statistics. A more formal version of the underlying mathematics can be found in [34]. We will begin with a summary of the biological context, then examine the popular spinglass model of development taken from neural network studies that we will ultimately generalize using a cognitive paradigm. The expanded approach permits import of tools and methods from statistical physics via the homology between information source uncertainty and free energy density, and this leads directly to the idea of epigenetic catalysis. It is worth keeping in mind throughout the formal mathematics that Feynman’s basic measure of information is simply the free energy needed to erase it [31]. 1.2

Epigenetic Epidemiology

What we attempt is itself embedded in a large and lively intellectual context. Jablonka and Lamb [41, 42] have long argued that information can be transmitted from one generation to the next in ways other than through the base sequence of DNA. It can be transmitted through cultural and behavioral means in higher animals, and by epigenetic means in cell lineages. All of these transmission systems allow the inheritance of environmentally induced variation. Such Epigenetic Inheritance Systems are the memory structures that enable somatic cells of diﬀerent phenotypes but identical genotypes to transmit their phenotypes to their descendants, even when the stimuli that originally induced these phenotypes are no longer present. In chromatin-marking systems information is carried from one cell generation to the next because it rides with DNA as binding proteins or additional chemical groups that are attached to DNA and inﬂuence its activity. When DNA is replicated, so are the chromatin marks. One type of mark is the methylation pattern a gene carries. The same DNA sequence can have several diﬀerent methylation patterns, each reﬂecting a diﬀerent functional state. These alternative patterns can be stably inherited through many cell divisions. Epigenetic inheritance systems are very diﬀerent from the genetic system. Many variations are directed and predictable outcomes of environmental changes.

Code, Context, and Epigenetic Catalysis in Gene Expression

285

Epigenetic variants are, in the view of [41, 42], often, although not necessarily, adaptive. The frequency with which variants arise and their rate of reversion varies widely and epigenetic variations induced by environmental changes may be produced coordinatedly at several loci. Parenthetically, some authors, e.g., [39], disagree with the assumption of adaptiveness, inferring that input responsible for methylation eﬀects simply produces a phenotypic variability then subject to selection. The matter remains open. Jablonka and Lamb [42] conclude that epigenetic systems may therefore produce rapid, reversible, co-ordinated, heritable changes. However such systems can also underlie non-induced changes, changes that are induced but non-adaptive, and changes that are very stable. What is needed, they feel, is a concept of epigenetic heritability comparable to the classical concept of heritability, and a model similar to those used for measuring the eﬀects of cultural inheritance on human behavior in populations. Following a furious decade of research and debate, this perspective received much empirical conﬁrmation. Backdahl et al. [6], for example, write that epigenetic regulation of gene expression primarily works through modifying the secondary and tertiary structures of DNA (chromatin), making it more or less accessible to transcription. The sum and interaction of epigenetic modiﬁcations has been proposed to constitute an ‘epigenetic code’ which organizes the chromatin structure on diﬀerent hierarchical levels [67]. Modiﬁcations of histones include acetylation, methylation, phosphorylation, ubiquitination, and sumoylation, but also other modiﬁcations have been observed. Some such modiﬁcations are quite stable and play an important part in epigenetic memory although DNA methylation is the only epigenetic modiﬁcation that has maintenance machinery which preserves the marks through mitosis. This argues for DNA methylation to function as a form of epigenetic memory for the epigenome. Codes and memory, of course, are inherent to any cognitive paradigm. Jaenish and Bird [45] argue that cells of a multicellular organism are genetically homogeneous but structurally and functionally heterogeneous owing to the diﬀerential expression of genes. Many of these diﬀerences in gene expression arise during development and are subsequently retained through mitosis. External inﬂuences on epigenetic processes are seen in the eﬀects of diet on long-term diseases such as cancer. Thus, epigenetic mechanisms seem to allow an organism to respond to the environment through changes in gene expression. Epigenetic modiﬁcations of the genome provide a mechanism that allows the stable propagation of gene activity states from one generation of cells to the next. Because epigenetic states are reversible they can be modiﬁed by environmental factors, which may contribute to the development of abnormal responses. What needs to be explained, from their perspective, is the variety of stimuli that can bring about epigenetic changes, ranging from developmental progression and aging to viral infection and diet. Jaenish and Bird conclude that the future will see intense study of the chains of signaling that are responsible for epigenetic programming. As a result, we will

286

R. Wallace and D. Wallace

be able to understand, and perhaps manipulate, the ways in which the genome learns from experience. Indeed, our central interest precisely regards the manner in which the asymptotic limit theorems of information theory constrain such chains of signaling, in the same sense that the Central Limit Theorem constrains sums of stochastic variates. Crews et al. [21, 22] provide a broad overview of induced epigenetic change in phenotype, as do Guerrero-Bosagna et al. [39], who focus particularly on early development. They propose that changes arising because of alterations in early development processes, in some cases environmentally induced, can appear whether or not such changes could become ﬁxed and prosper in a population. They recognize two ways for this to occur, ﬁrst by dramatically modifying DNA aspects in the germ line with transgenerational consequences – mutations or persistent epigenetic modiﬁcations of the genome – or by inducing ontogenetical variation in every generation, although not inheritance via the germ line. From their perspective inductive environmental forces can act to create, through these means, new conformations of organisms which also implies new possibilities within the surrounding environment. Foley et al. [32] take a very general perspective on the prospects for epigenetic epidemiology. They argue that epimutation is estimated to be 100 times more frequent than genetic mutation and may occur randomly or in response to the environment. Periods of rapid cell division and epigenetic remodeling are likely to be most sensitive to stochastic or environmentally mediated epimutation. Disruption of epigenetic proﬁle is a feature of most cancers and is speculated to play a role in the etiology of other complex diseases including asthma, allergy, obesity, type 2 diabetes, coronary heart disease, autism spectrum disorders, bipolar disorders, and schizophrenia. They ﬁnd evidence that a small change in the level of DNA methylation, especially in the lower range in an animal model, can dramatically alter expression for some genes. The timing of nutritional insuﬃciency or other environmental exposures may also be critical. In particular low-level maternal care was associated with developmental dysfunction and altered stress response in the young. Foley et al. emphasize the potential implications of such ﬁndings, given how widely stress is implicated in disease onset and relapse. They especially note that when epigenetic status or change in status over time is the outcome, then models for either threshold-based dichotomies or proportional data will be required. Threshold models, deﬁned by a given level or pattern of methylation or a degree of change in methylation over time, will, in their view, beneﬁt from relevant functional data to identify meaningful thresholds. A special contribution of the approach taken here is that just such threshold behavior leads ‘naturally’ to a language-like ‘dual information source’ constrained by the necessary conditions imposed by information theory’s asymptotic limit theorems, allowing development of statistical models of complicated cognitive phenomena, including but not limited to cognitive gene expression.

Code, Context, and Epigenetic Catalysis in Gene Expression

287

A recent review by Weaver [77] focuses speciﬁcally on the epigenetic eﬀects of glucocorticoids – stress hormones. In mammals, Weaver argues, the closeness or degree of positive attachment in parent-infant bonding and parental investment during early life has long-term consequences on development of interindividual diﬀerences in cognitive and emotional development in the oﬀspring. The long-term eﬀects of the early social experience, he continues, particularly of the mother-oﬀspring interaction, have been widely investigated. The nature of that interaction inﬂuences gene expression and the development of behavioral responses in the oﬀspring that remain stable from early development to the later stages of life. Although enhancing the oﬀspring’s ability to respond according to environmental clues early in life can have immediate adaptive value, the cost, Weaver says, is that these adaptations serve as predictors of ill health in later life. He concludes that maternal inﬂuences on the development of neuroendocrine systems that underlie hypothalamic-pituitary-adrenal (HPA) axis and behavioral responses to stress mediate the relation between early environment and health in the adult oﬀspring. In particular, he argues, exposure of the mother to environmental adversity alters the nature of mother-oﬀspring interaction, which, in turn, inﬂuences the development of defensive responses to threat and reproductive strategies in the progeny. In an updated review of epigenetic epidemiology, Jablonka [43] ﬁnds it clear that the health and general physiology of animals and people can be aﬀected not only by the interplay of their own genes and conditions of life, but also by the inherited eﬀects of the interplay of genes and environment in their ancestors. These ancestral inﬂuences on health, Jablonka says, depend neither on inheriting particular genes, nor on the persistence of the ancestral environment. Signiﬁcantly, Bossdorf et al. [11] invoke ‘contexts’ much like Baars’ model of consciousness [68], and infer a need to expand the concept of variation and evolution in natural populations, taking into account several likely interacting ecologically relevant inheritance systems. Potentially, this may result in a signiﬁcant expansion, though by all means not a negation, of the Modern Evolutionary Synthesis as well as in more conceptual and empirical integration between ecology and evolution. More formally, Scherrer and Jost [62, 63] use information theory arguments to extend the deﬁnition of the gene to include the local epigenetic machinery, something they characterize as the ‘genon’. Their central point is that coding information is not simply contained in the coded sequence, but is, in their terms, provided by the genon that accompanies it on the expression pathway and controls in which peptide it will end up. In their view the information that counts is not about the identity of a nucleotide or an amino acid derived from it, but about the relative frequency of the transcription and generation of a particular type of coding sequence that then contributes to the determination of the types and numbers of functional products derived from the DNA coding region under consideration. From our perspective the formal tools for understanding such phenomena involve asymptotic limit theorems aﬀecting information sources – active systems

288

R. Wallace and D. Wallace

that generate or ‘provide’ information – and these are respectively the Rate Distortion Theorem and its zero error limit, the Shannon-McMillan Theorem, described in the Mathematical Appendix. We begin with a reconsideration of the current de-facto standard systems biology neural network-analog model of development, and proceed to its generalization.

2 2.1

Models of Development The Spinglass Model

Ciliberti et al.[16, 17], culminating a long series of papers, apply the spinglass model from statistical physics to organisimal development in an evolutionary context. We summarize their formalism and look at some of the less obvious topological implications – in particular the mapping of disjoint directed homotopy classes of phenotype paths into interaction matrix space. We then extend the approach by applying a cognitive paradigm for gene expression ﬁrst developed in [74]. Analogs to phase transition arguments in physical systems generate punctuated equilibrium evolutionary transitions in a ‘highly natural’ manner, even for the spinglass treatment, and a hierarchical extension permits incorporation of epigenetic eﬀects as a kind of tunable catalysis. The spinglass model of development assumes that N transcriptional regulators are represented by their expression patterns S(t) = [S1 (t), ..., SN (t)] at some time t during a developmental or cell-biological process and in one cell or domain of an embryo. The transcriptional regulators inﬂuence each other’s expression through cross-regulatory and autoregulatory interactions described by a matrix w = (wij ). For nonzero elements, if wij > 0 the interaction is activating, if wij < 0 it is repressing. w represents, in this model, the regulatory genotype of the system, while the expression state S(t) is the phenotype. These regulatory interactions change the expression of the network S(t) as time progresses according to a diﬀerence equation N Si (t + Δt) = σ[ wij Sj (t)],

(1)

j=1

where Δt is a constant and σ a sigmodial function whose value lies in the interval (−1, 1). In the spinglass limit σ is the sign function, taking only the values ±1. The networks of interest in the spinglass model are those whose expression state begins from a prespeciﬁed initial state S(0) at time t = 0 and converge to a prespeciﬁed stable equilibrium state S∞ . Such a network is termed viable, for obvious reasons. After an elaborate and very diﬃcult simulation exercise, a particular series of results emerges. Reference [16] ﬁnds that viable networks comprise a tiny

Code, Context, and Epigenetic Catalysis in Gene Expression

289

fraction of possible ones. They could be widely scattered in the space of all possible networks and occupy disconnected islands in this space. However, direct computation indicates precisely the opposite. The metagraph of viable networks has one ‘giant’ connected component that comprises most or all viable networks. Any two networks in this component can be reached from one another through gradual changes of one regulatory interaction at a time, changes that never leave the space of viable networks, for this calculation. In general, within the giant component, randomly chosen pairs of networks with the same phenotype will have vastly diﬀerent organization, in terms of the matrix (wij ). Deﬁne 0 ≤ d ≤ 1 as the the fraction of genes that diﬀer in their expression state between S0 and S∞ . A typical result is that for N = 5 genes, 6 ≤ M ≤ 7 total regulatory interactions, and d = 0.4, full enumeration ﬁnds a total of only 37,338 viable networks out of 6.3 × 107 possible ones [16]. Long random walks through the space of viable networks, however, visit all but a very small fraction of the nodes of the metagraph, and this missing fraction decreases as N increases. Large N require elaborate Monte Carlo sampling for simulation, a diﬃcult and computationally intensive enterprise. In w-space [16, 17] deﬁne a metric characterizing the distance between two network topologies as D(w, w ) =

1 |sign(wij ) − sign(wij )|, 2M+ i,j

where M+ is the maximum number of regulatory interactions, and sign(x)=±1 depends on the sign of x, and is 0 for x = 0. Several observations emerge directly. 1. This approach is formally similar to spinglass neural network models of learning by selection, e.g., as proposed by Toulouse et al. [66] nearly a generation ago. Subsequent work [4, 5], summarized in [23], suggests that such models are simply not suﬃcient to the task of understanding high level cognitive function, and these have been largely supplanted by complicated ‘global workspace’ concepts whose mathematical characterization is highly nontrivial [3]. 2. What [16, 17] observe, in another idiom, is that in phenotype space, in S-space, the set of all paths associated with viable networks forms an equivalence class, closely analogous to the directed homotopy equivalence classes in the sense of [36, 37]. Directed homotopy diﬀers from simple homotopy (e.g., [50]) in that one uses paths from one point to another rather than loops, and seeks continuous deformations between them. See [74] for discussion in a biological context. Thus there is, in this spinglass model, a mapping from S-space into (wij ) space, characterized by the metric D, that associates a unique simply connected component with each dihomotopy-like equivalence class of paths connecting two particular phenotype points. Indeed, the w-space component might well be treated according to standard homotopy arguments, i.e., using loops.

290

R. Wallace and D. Wallace

3. What one does with homotopically simply connected components is patch them together to build larger, and more interesting, topological structures, using the Seifert-Van Kampen Theorem (SVKT) (e.g., [50], Ch. 10). If paths within Sspace are not continuously transformable into one another, (if there are ‘holes’), then several distinct dihomotopy classes will exist, e.g., as in ﬁgures 1 and 2 of [74], explored further below in terms of developmental critical periods and their ‘shadows’. The obvious conjecture is that, under such a circumstance, very complex topological objects may lurk in w-space, not just the simply connected component discovered by by [16, 17]. These may, according to the SVKT, intersect as well as exist as isolated and disconnected sets. In particular, if there are dihomotopy ‘holes’ in S-space, consequently reﬂected in disconnected patches in w-space, then punctuated transition events of various sorts may well become an evolutionary norm, as in [38], even for the spinglass model. 4. A large and increasing body of work surrounding coupled cell networks invokes groupoids, a natural generalization of symmetry groups. As [25] remarks, until recently the abstract theory of coupled cell systems has mainly focused on the eﬀects of symmetry in the network and the consequent formation of spatial and spatiotemporal patterns. The formal setting for this theory centers upon the symmetry group of the network. Reference [25] concludes that analysis of robust patterns of synchrony in general coupled cell systems – that is, dynamics in which sets of cells behave identically as a consequence of the network topology – leads to the fruitful notion of the ‘symmetry groupoid’ of a coupled cell network. A groupoid is a generalization of a group, in which products of elements are not always deﬁned. The symmetry groupoid of a coupled cell network is a natural algebraic formalization of the ‘local symmetries’ that relate subsets of the network to each other. In particular ‘admissible’ vector ﬁelds – those speciﬁed by the network topology – are precisely those that are equivariant under the action of the symmetry groupoid. The Appendix provides a summary of standard material on groupoids that will be of later use. 5. Both of these – analogous – approaches can apparently be coarse-grained into a symbolic dynamics associated with (simple) information sources having particular grammar and syntax. The method is straightforward (e.g., [7, 55]). One could, thus, probably translate the spinglass results of Ciliberti et al. into symbolic dynamics, using groupoid methods to study the underlying topological objects. 6. The spinglass model of development is abstracted from longstanding (if ultimately unsucessful) attempts at similar treatments of neural networks involved in high level cognition (e.g., [44, 56, 61, 64]). Thus and consequently [16, 17] are invoking an implicit cognitive paradigm for gene expression (e.g., [18, 19, 74]). Cognitive process, as the philosopher Fred Dretske eloquently argues (e.g., [26]), is constrained by the necessary conditions imposed by the asymptotic limit theorems of information theory. A little work produces a very general cognitive gene expression metanetwork structure recognizably similar to that found

Code, Context, and Epigenetic Catalysis in Gene Expression

291

in [16, 17]. The massively parallel computations are hidden, somewhat, in the required empirical ﬁtting of regression model analogs based on the asymptotic limit theorems of information theory rather than on the central limit theorem. 7. A salient characteristic of high level cognitive process is precisely its inherent punctuation (e.g., [4, 5, 68]), and this emerges directly using an information theory approach via the famous homology between information and free energy (e.g., [31]). ‘Simple’ neural network analogs will inevitably have more diﬃculty replicating such behavior, but as discussed, the mapping of disjoint dihomotopy equivalence classes from phenotype sequence space to disjoint sets in interaction matrix space provides a straightforward example for spinglass models. The next sections use information theory methods to make the transition from crossectional w-space into that of serially correlated sequences of phenotypes, expanding on the results of [74]. 2.2

Shifting Perspective: Cognition as an Information Source

Atlan and Cohen [2], in the context of a study of the immune system, argue that the essence of cognition is the comparison of a perceived signal with an internal, learned picture of the world, and then choice of a single response from a large repertoire of possible responses. Such choice inherently involves information and information transmission since it always generates a reduction in uncertainty, as explained in [1] (p. 21). More formally, a pattern of incoming input – like the S(t) of equation (1) – is mixed in a systematic algorithmic manner with a pattern of internal ongoing activity – like the (wij ) according to equation (1) – to create a path of combined signals x = (a0 , a1 , ..., an , ...) – analogous to the sequence of S(t+Δt) of equation (1), with, say, n = t/Δt. Each ak thus represents some functional composition of internal and external signals. This path is fed into a highly nonlinear decision oscillator, h, a ‘sudden threshold machine’, in a sense, that generates an output h(x) that is an element of one of two disjoint sets B0 and B1 of possible system responses. Let us deﬁne the sets Bk as B0 = {b0 , ..., bk }, B1 = {bk+1 , ..., bm }. Assume a graded response, supposing that if h(x) ∈ B0 , the pattern is not recognized, and if h(x) ∈ B1 , the pattern has been recognized, and some action bj , k + 1 ≤ j ≤ m takes place.

292

R. Wallace and D. Wallace

The principal objects of formal interest are paths x triggering pattern recognition-and-response. That is, given a ﬁxed initial state a0 , examine all possible subsequent paths x beginning with a0 and leading to the event h(x) ∈ B1 . Thus h(a0 , ..., aj ) ∈ B0 for all 0 < j < m, but h(a0 , ..., am ) ∈ B1 . For each positive integer n, let N (n) be the number of high probability grammatical and syntactical paths of length n which begin with some particular a0 and lead to the condition h(x) ∈ B1 . Call such paths ‘meaningful’, assuming, not unreasonably, that N (n) will be considerably less than the number of all possible paths of length n leading from a0 to the condition h(x) ∈ B1 . While the combining algorithm, the form of the nonlinear oscillator, and the details of grammar and syntax are all unspeciﬁed in this model, the critical assumption which permits inference of the necessary conditions constrained by the asymptotic limit theorems of information theory is that the ﬁnite limit H = lim

n→∞

log[N (n)] n

(2)

both exists and is independent of the path x. Deﬁne such a pattern recognition-and-response cognitive process as ergodic. Not all cognitive processes are likely to be ergodic in this sense, implying that H, if it indeed exists at all, is path dependent, although extension to nearly ergodic processes seems possible [73]. Invoking the spirit of the Shannon-McMillan Theorem, whose content is described in more detail in the Appendix, as choice involves an inherent reduction in uncertainty, it is then possible to deﬁne an adiabatically, piecewise stationary, ergodic (APSE) information source X associated with stochastic variates Xj having joint and conditional probabilities P (a0 , ..., an ) and P (an |a0 , ..., an−1 ) such that appropriate conditional and joint Shannon uncertainties satisfy the classic relations log[N (n)] = n→∞ n

H[X] = lim

lim H(Xn |X0 , ..., Xn−1 ) =

n→∞

lim

n→∞

H(X0 , ..., Xn ) . n+1

(3)

See the Mathematical Appendix for a summary of basic information theory results. This information source is deﬁned as dual to the underlying ergodic cognitive process. Adiabatic means that the source has been parametized according to some scheme, and that, over a certain range, along a particular piece, as the parameters vary, the source remains as close to stationary and ergodic as needed for information theory’s central theorems to apply. Stationary means that the system’s probabilities do not change in time, and ergodic, roughly, that the cross

Code, Context, and Epigenetic Catalysis in Gene Expression

293

sectional means approximate long-time averages. Between pieces it is necessary to invoke various kinds of phase transition formalisms, as described more fully in [68, 74]. Using the developmental vernacular of [16, 17], we now examine paths in phenotype space that begins at some S0 and converges n = t/Δt → ∞ to some other S∞ . Suppose the system is conceived at S0 , and h represents (for example) reproduction when phenotype S∞ is reached. Thus h(x) can have two values, i.e., B0 not able to reproduce, and B1 , mature enough to reproduce. Then x = (S0 , SΔt , ..., SnΔt , ...) until h(x) = B1 . Structure is now subsumed within the sequential grammar and syntax of the dual information source rather than within the cross sectional internals of (wij )space, a simplifying shift in perspective. This transformation carries heavy computational burdens, as well as providing deeper mathematical insight. First, the fact that viable networks comprise a tiny fraction of all those possible emerges easily from the spinglass formulation simply because of the ‘mechanical’ limit that the number of paths from S0 to S∞ will always be far smaller than the total number of possible paths, most of which simply do not end on the target conﬁguration. From the information source perspective, which inherently subsumes a far larger set of dynamical structures than possible in a spinglass model – not simply those of symbolic dynamics – the result is what [47] characterizes as the ‘Eproperty’ of a stationary, ergodic information source. This property is that, in the limit of inﬁnitely long output, the classiﬁcation of output strings into two sets: 1. A very large collection of gibberish which does not conform to underlying (sequential) rules of grammar and syntax, in a large sense, and which has nearzero probability, and 2. A relatively small ‘meaningful’ set, in conformity with underlying structural rules, having very high probability. The essential content of the Shannon-McMillan Theorem is that, if N (n) is the number of meaningful strings of length n, then the uncertainty of an information source X can be deﬁned as H[X] = limn→∞ log[N (n)]/n, that can be expressed in terms of joint and conditional probabilities as in equation (3) above. Proving these results for general stationary, ergodic information sources requires considerable mathematical machinery [20, 24, 47]. Second, information source uncertainty has an important heuristic interpretation that [1] describes as follows: ...[W]e may regard a portion of text in a particular language as being produced by an information source. The probabilities P [Xn = an |X0 = a0 , ...Xn−1 = an−1 ] may be estimated from the available data about the language; in this way we can estimate the uncertainty associated with the language. A large uncertainty means, by the [Shannon-McMillan Theorem], a large number of ‘meaningful’ sequences. Thus given two languages with uncertainties H1 and H2 respectively, if H1 > H2 , then in

294

R. Wallace and D. Wallace

the absence of noise it is easier to communicate in the ﬁrst language; more can be said in the same amount of time. On the other hand, it will be easier to reconstruct a scrambled portion of text in the second language, since fewer of the possible sequences of length n are meaningful. This will prove important below. Third, information source uncertainty is homologous with free energy density in a physical system, a matter having implications across a broad class of dynamical behaviors. The free energy density of a physical system having volume V and partition function Z(K) derived from the system’s Hamiltonian – the energy function – at inverse temperature K is (e.g., [49]) F [K] = lim − V →∞

1 log[Z(K, V )] = K V

ˆ log[Z(K, V )] , V →∞ V lim

(4)

where Zˆ = Z −1/K . The partition function for a physical system is the normalizing sum in an equation having the form exp[−Ei /kT ] P [Ei ] = j exp[−Ej /kT ] where Ei is the energy of state i, k a constant, and T the system temperature, and P [Ei ] is the probability of state i. Feynman [31], following the classic arguments of [9] that present idealized machines using information to do work, concludes the information contained in a message is most simply measured by the free energy needed to erase it. The arguments of [9] are clever indeed, and the Feynman treatment of them in [31] is well worth reading. Thus, according to this argument, source uncertainty is homologous to free energy density as deﬁned above, i.e., from the similarity with the relation H = limn→∞ log[N (n)]/n. Ash’s comment above then has an important corollary: If, for a biological system, H1 > H2 , source 1 will require more metabolic free energy than source 2.

3

Symmetry Arguments

A formal equivalence class algebra, in the sense of the groupoid section of the Appendix, can now be constructed by choosing diﬀerent origin and end points S0 , S∞ and deﬁning equivalence of two states by the existence of a high probability meaningful path connecting them with the same origin and end. Disjoint

Code, Context, and Epigenetic Catalysis in Gene Expression

295

partition by equivalence class, analogous to orbit equivalence classes for dynamical systems, deﬁnes the vertices of the proposed network of cognitive dual languages, much enlarged beyond the spinglass example. We thus envision a network of metanetworks, in the sense of [16]. Each vertex then represents a different equivalence class of information sources dual to a cognitive process. This is an abstract set of metanetwork ‘languages’ dual to the cognitive processes of gene expression and development. This structure generates a groupoid, in the sense of [78]. States aj , ak in a set A are related by the groupoid morphism if and only if there exists a high probability grammatical path connecting them to the same base and end points, and tuning across the various possible ways in which that can happen – the diﬀerent cognitive languages – parametizes the set of equivalence relations and creates the (very large) groupoid. There is an implicit hierarchy. First, there is structure within the system having the same base and end points, as in [16]. Second, there is a complicated groupoid structure deﬁned by sets of dual information sources surrounding the variation of base and end points. We do not need to know what that structure is in any detail, but can show that its existence has profound implications. We begin with the simple case, the set of dual information sources associated with a ﬁxed pair of beginning and end states. 3.1

The First Level

The spinglass model of [16, 17] produced a simply connected, but otherwise undiﬀerentiated, metanetwork of gene expression dynamics that could be traversed continuously by single-gene transitions in the highly parallel w-space. Taking the serial grammar/syntax model above, we ﬁnd that not all high probability meaningful paths from S0 to S∞ are actually the same. They are structured by the uncertainty of the associated dual information source, and that has a homological relation with free energy density. Let us index possible dual information sources connecting base and end points by some set A = ∪α. Argument by abduction from statistical physics is direct: Given metabolic energy density available at a rate M , and an allowed development time τ , let K = 1/κM τ for some appropriate scaling constant κ, so that M τ is total developmental free energy. Then the probability of a particular Hα will be determined by the standard expression (e.g., [49]), exp[−Hβ K] P [Hβ ] = , α exp[−Hα K]

(5)

where the sum may, in fact, be a complicated abstract integral. This is just a version of the fundamental probability relation from statistical mechanics, as above. The sum in the denominator, the partition function in statistical physics, is a crucial normalizing factor that allows the deﬁnition of of P [Hβ ] as a probability. A basic requirement, then, is that the sum/integral always converges. K is the inverse product of a scaling factor, a metabolic energy density rate term, and

296

R. Wallace and D. Wallace

a characteristic development time τ . The developmental energy might be raised to some power, e.g., K = 1/(κ(M τ )b ), suggesting the possibility of allometric scaling. Thus, in this formulation, there must be structure within a (cross sectional) connected component in the w-space of [16, 17], determined in no small measure by available energy. Some dual information sources will be ‘richer’/smarter than others, but, conversely, must use more metabolic energy for their completion. 3.2

The Second Level

The next generalization is crucial: While we might simply impose an equivalence class structure based on equal levels of energy/source uncertainty, producing a groupoid in the sense of the Appendix (and possibly allowing a Morse Theory approach in the sense of [52, 59]), we can do more by now allowing both source and end points to vary, as well as by imposing energy-level equivalence. This produces a far more highly structured groupoid that we now investigate. Equivalence classes deﬁne groupoids, by standard mechanisms [13, 35, 78]. The basic equivalence classes – here involving both information source uncertainty level and the variation of S0 and S∞ , will deﬁne transitive groupoids, and higher order systems can be constructed by the union of transitive groupoids, having larger alphabets that allow more complicated statements in the sense of Ash above. Again, given an appropriately scaled, dimensionless, ﬁxed, inverse available metabolic energy density rate and development time, so that K = 1/κM τ , we propose that the metabolic-energy-constrained probability of an information source representing equivalence class Di , HDi , will again be given by exp[−HDi K] P [HDi ] = , j exp[−HDj K]

(6)

where the sum/integral is over all possible elements of the largest available symmetry groupoid. By the arguments of Ash above, compound sources, formed by the union of underlying transitive groupoids, being more complex, generally having richer alphabets, as it were, will all have higher free-energy-densityequivalents than those of the base (transitive) groupoids. Let ZD = exp[−HDj K]. (7) j

We now deﬁne the Groupoid free energy of the system, FD , at inverse normalized metabolic energy density K, as 1 FD [K] = − log[ZD [K]], (8) K again following the standard arguments from statistical physics [31, 49]. The groupoid free energy construct permits introduction of important ideas from statistical physics.

Code, Context, and Epigenetic Catalysis in Gene Expression

3.3

297

Spontaneous Symmetry Breaking

We have expressed the probability of an information source in terms of its relation to a ﬁxed, scaled, available (inverse) metabolic free energy, seen as a kind of equivalent (inverse) system temperature. This gives a statistical thermodynamic path leading to deﬁnition of a ‘higher’ free energy construct – FD [K] – to which we now apply Landau’s fundamental heuristic phase transition argument [49, 59, 65]. The essence of Landau’s insight was that certain phase transitions were usually in the context of a signiﬁcant symmetry change in the physical states of a system, with one phase being far more symmetric than the other. A symmetry is lost in the transition, a phenomenon called spontaneous symmetry breaking. The greatest possible set of symmetries in a physical system is that of the Hamiltonian describing its energy states. Usually states accessible at lower temperatures will lack the symmetries available at higher temperatures, so that the lower temperature phase is less symmetric: The randomization of higher temperatures – in this case limited by available metabolic free energy – ensures that higher symmetry/energy states – mixed transitive groupoid structures – will then be accessible to the system. Absent high metabolic free energy, however, only the simplest transitive groupoid structures can be manifest. A full treatment from this perspective requires invocation of groupoid representations, no small matter (e.g., [10, 14]). Somewhat more rigorously, the biological renormalization schemes of the Appendix to [74] may now be imposed on FD [K] itself, leading to a spectrum of highly punctuated transitions in the overall system of developmental information sources. Most deeply, however, an extended version of Pettini’s Morse-Theory-based topological hypothesis [59] can now be invoked, i.e., that changes in underlying groupoid structure are a necessary (but not suﬃcient) consequence of phase changes in FD [K]. Necessity, but not suﬃciency, is important, as it, in theory, allows mixed groupoid symmetries. The essential insight is that the single simply connected giant component of [16, 17] is unlikely to be the full story, and that more complete models will likely be plagued – or graced – by highly punctuated dynamics. Several matters are worth noting. First, Landau’s spontaneous symmetry breaking arguments are perhaps the simplest approach possible here. The formal mathematical development requires invoking holonomy groups and groupoids, as in [34]. Second, one need not be restricted to terms of the form exp[−Hj K], as any f (Hj , K) such that the sum over j converges will serve, although the resulting ‘thermodynamic’ relations between variates of central interest may then be less elegant. Third, there may be some allometric scaling tradeoﬀ between metabolic energy rate and development time determined by a relation of the form K ∝ (τ M )α .

298

4

R. Wallace and D. Wallace

Tunable Epigenetic Catalysis

Incorporating the inﬂuence of embedding contexts – epigenetic eﬀects – is most elegantly done by invoking the Joint Asymptotic Equipartition Theorem (JAEPT) [20]. For example, given an embedding contextual information source, say Z, that aﬀects development, then the dual cognitive source uncertainty HDi is replaced by a joint uncertainty H(XDi , Z). The objects of interest then become the jointly typical dual sequences y n = (xn , z n ), where x is associated with cognitive gene expression and z with the embedding context. Restricting consideration of x and z to those sequences that are in fact jointly typical allows use of the information transmitted from Z to X as the splitting criterion. One important inference is that, from the information theory ‘chain rule’ [20], H(X, Y ) = H(X) + H(Y |X) ≤ H(X) + H(Y ), while there are approximately exp[nH(X)] typical X sequences, and exp[nH(Z)] typical Z sequences, and hence exp[n(H(x) + H(Y ))] independent joint sequences, there are only about exp[nH(X, Z)] ≤ exp[n(H(X) + H(Y ))] jointly typical sequences, so that the eﬀect of the embedding context, in this model, is to lower the relative free energy of a particular developmental channel. Thus the eﬀect of epigenetic regulation is to channel development into pathways that might otherwise be inhibited by an energy barrier. Hence the epigenetic information source Z acts as a tunable catalyst, a kind of second order cognitive enzyme, to enable and direct developmental pathways. This result permits hierarchical models similar to those of higher order cognitive neural function that incorporate Baars’ contexts in a natural way [73, 74]. It is worth emphasizing that this is indeed a relative energy argument, since, metabolically, two systems must now be supported, i.e., that of the ‘reaction’ itself and that of its catalytic regulator. ‘Programming’ and stabilizing inevitably intertwined, as it were. This elaboration allows a spectrum of possible ‘ﬁnal’ phenotypes, what [33] calls developmental or phenotype plasticity. Thus gene expression is seen as, in part, responding to environmental or other, internal, developmental signals. West-Eberhard [79] argues that any new input, whether it comes from the genome, like a mutation, or from the external environment, like a temperature change, a pathogen, or a parental opinion, has a developmental eﬀect only if the preexisting phenotype is responsive to it. A new input causes a reorganization of the phenotype, or ‘developmental recombination.’ In developmental recombination, phenotypic traits are expressed in new or distinctive combinations during ontogeny, or undergo correlated quantitative change in dimensions. Developmental recombination can result in evolutionary divergence at all levels of organization. Individual development can be visualized as a series of branching pathways. Each branch point, according to [79], is a developmental decision, or switch point, governed by some regulatory apparatus, and each switch point deﬁnes a modular trait. Developmental recombination implies the origin or deletion of a

Code, Context, and Epigenetic Catalysis in Gene Expression

299

branch and a new or lost modular trait. It is important to realize that the novel regulatory response and the novel trait originate simultaneously. Their origins are, in fact, inseparable events. There cannot, [79] concludes, be a change in the phenotype, a novel phenotypic state, without an altered developmental pathway. These mechanisms are accomplished in our formulation by allowing the set B1 in section 2.2 to span a distribution of possible ‘ﬁnal’ states S∞ . Then the groupoid arguments merely expand to permit traverse of both initial states and possible ﬁnal sets, recognizing that there can now be a possible overlap in the latter, and the epigenetic eﬀects are realized through the joint uncertainties H(XDi , Z), so that the epigenetic information source Z serves to direct as well the possible ﬁnal states of XDi . Again, [62, 63] use information theory arguments to suggest something similar to epigenetic catalysis, ﬁnding the information in a sequence is not contained in the sequence but has been provided by the machinery that accompanies it on the expression pathway. That work does not, however, invoke a cognitive paradigm, its attendant groupoid symmetries, or the homology between information source uncertainty and free energy density that drives dynamics. The mechanics of channeling can be made more precise as follows.

5

Rate Distortion Dynamics

Real time problems, like the crosstalk between epigenetic and genetic structures, are inherently rate distortion problems, and the interaction between biological structures can be restated in communication theory terms. Suppose a sequence of signals is generated by a biological information source Y having output y n = y1 , y2 , .... This is ‘digitized’ in terms of the observed behavior of the system with which it communicates, say a sequence of observed behaviors bn = b1 , b2 , .... The bi happen in real time. Assume each bn is then deterministically retranslated back into a reproduction of the original biological signal, bn → yˆn = yˆ1 , yˆ2 , .... Here the information source Y is the epigenetic Z, and B is XDi , but the terminology used here is more standard [20]. Deﬁne a distortion measure d(y, yˆ) which compares the original to the retranslated path. Many distortion measures are possible, as described in the Mathematical Appendix. The distortion between paths y n and yˆn is deﬁned as 1 d(yj , yˆj ). n j=1 n

d(y n , yˆn ) =

A remarkable fact of the Rate Distortion Theorem is that the basic result is independent of the exact distortion measure chosen [20, 24].

300

R. Wallace and D. Wallace

Suppose that with each path y n and bn -path retranslation into the y-language, denoted yˆn , there are associated individual, joint, and conditional probability distributions p(y n ), p(ˆ y n ), p(y n , yˆn ), p(y n |ˆ y n ). The average distortion is deﬁned as D= p(y n )d(y n , yˆn ).

(9)

yn

It is possible, using the distributions given above, to deﬁne the information transmitted from the Y to the Yˆ process using the Shannon source uncertainty of the strings: I(Y, Yˆ ) = H(Y ) − H(Y |Yˆ ) = H(Y ) + H(Yˆ ) − H(Y, Yˆ ),

(10)

where H(..., ...) is the joint and H(...|...) the conditional uncertainty [1, 20]. If there is no uncertainty in Y given the retranslation Yˆ , then no information is lost, and the systems are in perfect synchrony. In general, of course, this will not be true. The rate distortion function R(D) for a source Y with a distortion measure d(y, yˆ) is deﬁned as R(D) =

p(y,ˆ y );

min

(y,y) ˆ

p(y)p(y|ˆ y )d(y,ˆ y)≤D

I(Y, Yˆ ).

(11)

The minimization is over all conditional distributions p(y|ˆ y) for which the joint distribution p(y, yˆ) = p(y)p(y|ˆ y) satisﬁes the average distortion constraint (i.e., average distortion ≤ D). The Rate Distortion Theorem states that R(D) is the minimum necessary rate of information transmission which ensures communication does not exceed average distortion D. Thus R(D) deﬁnes a minimum necessary channel capacity. References [20, 24] provide details. The rate distortion function has been explicitly calculated for a number of simple systems. Recall, now, the relation between information source uncertainty and channel capacity [1, 20]: H[X] ≤ C,

(12)

where H is the uncertainty of the source X and C the channel capacity, deﬁned according to the relation [1, 20] C = max I(X|Y ).

(13)

P (X)

X is the message, Y the channel, and the probability distribution P (X) is chosen so as to maximize the rate of information transmission along a Y .

Code, Context, and Epigenetic Catalysis in Gene Expression

301

Finally, recall the analogous deﬁnition of the rate distortion function above, again an extremum over a probability distribution. Recall, again, equations (4-8), i.e., that the free energy of a physical system at a normalized inverse temperature-analog K = 1/κT is deﬁned as F (K) = − log[Z(K)]/K where Z(K) the partition function deﬁned by the system Hamiltonian. More precisely, if the possible energy states of the system are a set Ei , i = 1, 2, ... then, at normalized inverse temperature K, the probability of a state Ei is determined by the relation P [Ei ] = exp[−Ei K]/ j exp[−Ej K]. The partition function is simply the normalizing factor. Applying this formalism, it is possible to extend the rate distortion model by describing a probability distribution for D across an ensemble of possible rate distortion functions in terms of available free metabolic energy, K = 1/κM τ . The key is to take the R(D) as representing energy as a function of the average distortion. Assume a ﬁxed K, so that the probability density function of an average distortion D, given a ﬁxed K, is then exp[−R(D)K] P [D, K] = Dmax . Dmin exp[−R(D)K]dD

(14)

Thus lowering K in this model rapidly raises the possibility of low distortion communication between linked systems. We deﬁne the rate distortion partition function as just the normalizing factor in this equation:

Dmax

ZR [K] =

exp[−R(D)K]dD,

(15)

Dmin

again taking K = 1/κM τ . We now deﬁne a new free energy-analog, the rate distortion free-energy, as FR [K] = −

1 log[ZR [K]], K

(16)

and apply Landau’s spontaneous symmetry breaking argument to generate punctuated changes in the linkage between the genetic information source XDi and the embedding epigenetic information source Z. Recall that Landau’s insight was that certain phase transitions were usually in the context of a signiﬁcant symmetry change in the physical states of a system. Again, the biological renormalization schemes of the Appendix to [74] may now be imposed on FR [K] itself, leading to a spectrum of highly punctuated transitions in the overall system of interacting biological substructures. Since 1/K is proportional to the embedding metabolic free energy, we assert that 1. The greatest possible set of symmetries will be realized for high developmental metabolic free energies, and 2. Phase transitions, related to total available developmental metabolic free energy, will be accompanied by fundamental changes in the ﬁnal topology of the

302

R. Wallace and D. Wallace

system of interest – phenotype changes – recognizing that evolutionary selection acts on phenotypes, not genotypes. The relation 1/K = κM τ suggests the possibility of evolutionary tradeoﬀs between development time and the rate of available metabolic free energy.

6

More Topology

It seems possible to extend this treatment using standard topological arguments. Taking T = 1/K in equations (6) and (14) as a product of eigenvalues, we can deﬁne it as the determinant of a Hessian matrix representing a Morse Function, f , on some underlying, background, manifold, M, characterized in terms of (as yet unspeciﬁed) variables X = (x1 , ..., xn ), so that 1/K = det(Hi,j ), Hi,j = ∂ 2 f /∂xi ∂xj .

(17)

Again, see the Appendix for a brief outline of Morse Theory. Thus κ, M , and the development time τ are seen as eigenvalues of H on the manifold M in an abstract space deﬁned by some set of variables X . By construction H has everywhere only nonzero, and indeed, positive, eigenvalues, whose product thereby deﬁnes T as a generalized volume. Thus, and accordingly, all critical points of f have index zero, that is, no eigenvalues of H are ever negative at any point, and hence at any critical point Xc where df (Xc ) = 0. This deﬁnes a particularly simple topological structure for M: If the interval [a, b] contains a critical value of f with a single critical point Xc , then the topology of the set Mb deﬁned above diﬀers from that of Ma in a manner determined by the index i of the critical point. Mb is then homeomorphic to the manifold obtained from attaching to Ma an i-handle, the direct product of an i-disk and an (m − i)-disk. One obtains, in this case, since i = 0, the two halves of a sphere with critical points at the top and bottom [52, 59]. This is, as in [16], a simply connected object. What one does then is to invoke the Seifert-Van Kampen Theorem (SVKT, [50]) and patch together the various simply connected subcomponents to construct the larger, complicated, topological object representing the full range of possibilities. The physical natures of κ, M , and τ thus impose constraints on the possible complexity of this system, in the sense of the SVKT.

7

Inherited Epigenetic Memory

The cognitive paradigm for gene expression invoked here requires an internal picture of the world against which incoming signals are compared – algorithmically

Code, Context, and Epigenetic Catalysis in Gene Expression

303

combined according to the rules of Section 2.2 – and then fed into a sharply stepwise decision oscillator that chooses one (or a few) action(s) from a much large repertoire of possibilities. Memory is inherent, and much recent work, as described in the introduction, suggests that epigenetic memory is indeed heritable. The abduction of spinglass and other models from neural network studies to the analysis of development and its evolution carries with it the possibility of more than one system of memory. What Baars called ‘contexts’ channeling high level animal cognition may often be the inﬂuence of cultural inheritance, in a large sense. Our formalism suggests a class of statistical models that indeed greatly generalize those used for measuring the eﬀects of cultural inheritance on human behavior in populations. Epigenetic machinery, as a dual information source to a cognitive process, serves as a heritable system, intermediate between (relatively) hard-wired classical genetics, and a (usually) highly Larmarckian embedding cultural context. In particular, the three heritable systems interact, in our model, through a crosstalk in which the epigenetic machinery acts as a kind of intelligent catalyst for gene expression.

8

Multiple Processes

The argument to this point has, in large measure, been directly abducted from recent formal studies of high level cognition – consciousness – based on a Dretskestyle information theoretic treatment of Bernard Baars’ global workspace model [3, 68]. A deﬁning and grossly simplifying characteristic of that phenomenon is its rapidity: typically the global broadcasts of consciousness occur in a matter of a few hundred milliseconds, limiting the number of processes that can operate simultaneously. Slower cognitive dynamics can, therefore, be far more complex than individual consciousness. One well known example is institutional distributed cognition that encompasses both individual and group cognition in a hierarchical structure typically operating on timescales ranging from a few seconds or minutes in combat or hunting groups, to years at the level of major governmental structures, commercial enterprises, religious organizations, or other analogous large scale cultural artifacts. Reference [73] provides the ﬁrst formal mathematical analysis of institutional distributed cognition. Clearly cognitive gene expression is not generally limited to a few hundred milliseconds, and something much like the distributed cognition analysis may be applied here as well. Extending the analysis requires recognizing an individual cognitive actor can participate in more than one ‘task’, synchronously, asynchronously, or strictly sequentially. Again, the analogy is with institutional function whereby many individuals often work together on several distinct projects: Envision a multiplicity of possible cognitive gene expression dual ‘languages’ that themselves form a higher order network linked by crosstalk. Next, describe crosstalk measures linking diﬀerent dual languages on that meta-meta (MM) network by some characteristic magnitude ω, and deﬁne a topology on the MM network by renormalizing the network structure to zero if

304

R. Wallace and D. Wallace

the crosstalk is less than ω and set it equal to one if greater or equal to it. A particular ω, of suﬃcient magnitude, deﬁnes a giant component of network elements linked by mutual information greater or equal to it, in the sense of [29], as more fully described in [73] (Section 3.4). The fundamental trick is, in the Morse Theory sense [52], to invert the argument so that a given topology for the giant component will, in turn, deﬁne some critical value, ωC , so that network elements interacting by mutual information less than that value will be unable to participate, will be locked out and not active. ω becomes an epigenetically syntactically-dependent detection limit, and depends critically on the instantaneous topology of the giant component deﬁning the interaction between possible gene interaction MM networks. Suppose, now, that a set of such giant components exists at some generalized k system ‘time’ k and is characterized by a set of parameters Ωk = ω1k , ..., ωm . Fixed parameter values deﬁne a particular giant component set having a particular set of topological structures. Suppose that, over a sequence of times the set of giant components can be characterized by a possibly coarse-grained path γn = Ω0 , Ω1 , ..., Ωn−1 having signiﬁcant serial correlations that, in fact, permit deﬁnition of an adiabatically, piecewise stationary, ergodic (APSE) information source Γ . Suppose that a set of (external or internal) epigenetic signals impinging on the set of such giant components can also be characterized by another APSE information source Z that interacts not only with the system of interest globally, but with the tuning parameters of the set of giant components characterized by Γ . Pair the paths (γn , zn ) and apply the joint information argument above, generating a splitting criterion between high and low probability sets of pairs of paths. We now have a multiple workspace cognitive genetic expression structure driven by epigenetic catalysis.

9

‘Coevolutionary’ Development

The model can be applied to multiple interacting information sources representing simultaneous gene expression processes, for example across a spatially diﬀerentiating organism as it develops. This is, in a broad sense, a ‘coevolutionary’ phenomenon in that the development of one segment may aﬀect that of others. Most generally we assume that diﬀerent cognitive developmental subprocesses of gene expression characterized by information sources Hm interact through chemical or other signals and assume that diﬀerent processes become each other’s principal environments, a broadly coevolutionary phenomenon. We write Hm = Hm (K1 ...Ks , ...Hj ...),

(18)

where the Ks represent other relevant parameters and j = m. The dynamics of such a system is driven by a recursive network of stochastic diﬀerential equations, similar to those used to study many other highly parallel dynamic structures (e.g., [83]).

Code, Context, and Epigenetic Catalysis in Gene Expression

305

Letting the Kj and Hm all be represented as parameters Qj , (with the caveat that Hm not depend on itself), one can deﬁne, according to the generalized Onsager development of the Appendix, S m = Hm −

Qi ∂Hm /∂Qi

i

to obtain a complicated recursive system of phenomenological ‘Onsager relations’ stochastic diﬀerential equations, dQjt =

[Lj,i (t, ...∂S m /∂Qi ...)dt + σj,i (t, ...∂S m /∂Qi ...)dBti ],

(19)

i

where, again, for notational simplicity only, we have expressed both the Hj and the external K’s in terms of the same symbols Qj . m ranges over the Hm and we could allow diﬀerent kinds of ‘noise’ dBti , having particular forms of quadratic variation that may, in fact, represent a projection of environmental factors under something like a rate distortion manifold [73, 74]. As usual for such systems, there will be multiple quasi-stable points within a given system’s Hm , representing a class of generalized resilience modes accessible via punctuation. Second, however, there may well be analogs to fragmentation when the system exceeds the critical values of Kc according to the approach of [74]. That is, the K-parameter structure will represent full-scale fragmentation of the entire structure, and not just punctuation within it. We thus infer two classes of punctuation possible for this kind of structure. There are other possible patterns: 1. Setting equation (19) equal to zero and solving for stationary points again gives attractor states since the noise terms preclude unstable equilibria. 2. This system may converge to limit cycle or ‘strange attractor’ behaviors in which the system seems to chase its tail endlessly, e.g., the cycle of climate-driven phenotype changes in persistent temperate region plants. 3. What is converged to in both cases is not a simple state or limit cycle of states. Rather it is an equivalence class, or set of them, of highly dynamic information sources coupled by mutual interaction through crosstalk. Thus ‘stability’ in this extended model represents particular patterns of ongoing dynamics rather than some identiﬁable ‘state’, although such dynamics may be indexed by a ‘stable’ set of phenotypes. Here we become enmeshed in a system of highly recursive phenomenological stochastic diﬀerential equations, but at a deeper level than the standard stochastic chemical reaction model (e.g., [84]), and in a dynamic rather than static manner: the objects of this system are equivalence classes of information sources and their crosstalk, rather than simple ﬁnal states of a chemical system.

306

10

R. Wallace and D. Wallace

Multiple Models

Recent work [75] argues that consciousness may have undergone the characteristic branching and pruning of evolutionary development, particularly in view of the rapidity of currently surviving conscious mechanisms. According to that study, evolution is littered with polyphyletic parallelisms: many roads lead to functional Romes, and consciousness, as a particular form of high order cognitive process operating in real time, embodies one such example, represented by an equivalence class structure that factors the broad realm of necessary conditions information theoretic realizations of Baars’ global workspace model. Many diﬀerent physiological systems, then, can support rapidly shifting, highly tunable, and even simultaneous assemblages of interacting unconscious cognitive modules. Thus [75] concludes the variety of possibilities suggests minds today may be only a small surviving fraction of ancient evolutionary radiations – bush phylogenies of consciousness pruned by selection and chance extinction. Even in the realms of rapid global broadcast inherent to real time cognition, [75] speculates, following a long tradition, that ancient backbrain structures instantiate rapid emotional responses, while the newer forebrain harbors rapid ‘reasoned’ responses in animal consciousness. The cooperation and competition of these two rapid phenomena produces, of course, a plethora of systematic behaviors. Since consciousness is necessarily restricted to realms of a few hundred milliseconds, evolutionary pruning may well have resulted in only a small surviving fraction of previous evolutionary radiations. Processes operating on longer timescales may well be spared such draconian evolutionary selection. That is, the vast spectrum of mathematical models of cognitive gene expression inherent to our analysis here, in the context of development times much longer than a few hundred milliseconds, implies current organisms may simultaneously harbor several, possibly many, quite diﬀerent cognitive gene expression mechanisms. It seems likely, then, that, with some generality, slow phenomena, ranging from institutional distributed cognition to cognitive gene expression, permit the operation of very many quite diﬀerent cognitive processes simultaneously or in rapid succession. One inference is, then, that gene expression and its epigenetic regulation are, for even very simple organisms, far more complex than individual human consciousness, currently regarded as one of the ‘really big’ unsolved scientiﬁc problems. Neural network models adapted or abducted from inadequate cognitive studies of a generation ago are unlikely to cleave the Gordian Knot of scientiﬁc inference surrounding gene expression.

11

Epigenetic Focus

The Tuning Theorem analysis of the Appendix permits an inattentional blindness/concentrated focus perspective on the famous computational ‘no free lunch’ theorem of [81, 82]. Following closely the arguments of [28], [81, 82] have established that there exists no generally superior function optimizer. There is no ‘free

Code, Context, and Epigenetic Catalysis in Gene Expression

307

lunch’ in the sense that an optimizer ‘pays’ for superior performance on some functions with inferior performance on others. If the distribution of functions is uniform, then gains and losses balance precisely, and all optimizers have identical average performance. The formal demonstration depends primarily upon a theorem that describes how information is conserved in optimization. This Conservation Lemma states that when an optimizer evaluates points, the posterior joint distribution of values for those points is exactly the prior joint distribution. Put simply, observing the values of a randomly selected function does not change the distribution: An optimizer has to ‘pay’ for its superiority on one subset of functions with inferiority on the complementary subset. As [28] describes, anyone slightly familiar with the evolutionary computing literature recognizes the paper template ‘Algorithm X was treated with modiﬁcation Y to obtain the best known results for problems P1 and P2 .’ Anyone who has tried to ﬁnd subsequent reports on ‘promising’ algorithms knows that they are extremely rare. Why should this be? A claim that an algorithm is the very best for two functions is a claim that it is the very worst, on average, for all but two functions. It is due to the diversity of the benchmark set of test problems that the ‘promise’ is rarely realized. Boosting performance for one subset of the problems usually detracts from performance for the complement. Reference [28] argues that hammers contain information about the distribution of nail-driving problems. Screwdrivers contain information about the distribution of screw-driving problems. Swiss army knives contain information about a broad distribution of survival problems. Swiss army knives do many jobs, but none particularly well. When the many jobs must be done under primitive conditions, Swiss army knives are ideal. Thus, according to [28], the tool literally carries information about the task optimizers are literally tools-an algorithm implemented by a computing device is a physical entity. Another way of looking at this is to recognize that a computed solution is simply the product of the information processing of a problem, and, by a very famous argument, information can never be gained simply by processing. Thus a problem X is transmitted as a message by an information processing channel, Y , a computing device, and recoded as an answer. By the Tuning Theorem argument of the Appendix there will be a channel coding of Y which, when properly tuned, is most eﬃciently transmitted by the problem. In general, then, the most eﬃcient coding of the transmission channel, that is, the best algorithm turning a problem into a solution, will necessarily be highly problem-speciﬁc. Thus there can be no best algorithm for all equivalence classes of problems, although there may well be an optimal algorithm for any given class. The tuning theorem form of the No Free Lunch theorem will apply quite generally to cognitive biological and social structures, as well as to massively parallel machines. Rate distortion, however, occurs when the problem is collapsed into a smaller, simpliﬁed, version and then solved. Then there must be a tradeoﬀ between allowed average distortion and the rate of solution: the retina eﬀect. In a very fundamental

308

R. Wallace and D. Wallace

sense – particularly for real time systems – rate distortion manifolds present a generalization of the converse of the no free lunch arguments. The neural corollary is known as inattentional blindness [69]. We are led to suggest that there may well be a considerable set of no free lunch-like conundrums confronting highly parallel real-time structures, including epigenetic control of gene expression, and that they may interact in distinctly complicated ways.

12 12.1

Developmental Disorders Network Information Theory

Let U be an information source representing a systematic embedding environmental ‘program’ interacting with the process of cognitive gene expression, here deﬁned as a complicated information set of sources having source joint uncertainty H(Z1 , ..., Zn ) that guides the system into a particular equivalence class of desired developmental behaviors and trajectories. To model the eﬀect of U on development one can, most simply, invoke results from network information theory, ([20], p. 388). Given three interacting information sources, say Y1 , Y2 , Z, the splitting criterion between high and low probability sets of states, taking Z as the external context, is given by I(Y1 , Y2 |Z) = H(Z) + H(Y1 |Z) + H(Y2 |Z) − H(Y1 , Y2 , Z), where, again, H(...|...) and H(..., ..., ...) represent conditional and joint uncertainties. This generalizes to the relation I(Y1 , ..., Yn |Z) = H(Z) +

n

H(Yj |Z) − H(Y1 , ..., Yn , Z).

j=1

Thus the fundamental splitting criterion between low and high probability sets of joint developmental paths becomes I(Z1 , ..., Zn |U ) = H(U ) +

n

H(Zj |U ) − H(Z1 , ..., Zn , U ).

(20)

j=1

Again, the Zi represent internal information sources and U that of the embedding environmental context. The central point is that a one step extension of that system via the results of network information theory [20] allows incorporating the eﬀect of an external environmental ‘farmer’ in guiding cognitive developmental gene expression. 12.2

Embedding Ecosystems as Information Sources

The principal farmer for a developing organism is the ecosystem in which it is embedded, in a large sense. Summarizing brieﬂy the arguments of [74], ecosystems, under appropriate coarse graining, often have reconizable grammar and

Code, Context, and Epigenetic Catalysis in Gene Expression

309

syntax. For example, the turn-of-the-seasons in a temperate climate, for most natural communities, is remarkably similar from year to year in the sense that the ice melts, migrating birds return, trees bud, ﬂowers and grass grow, plants and animals reproduce, the foliage turns, birds migrate, frost, snow, the rivers freeze, and so on in a predictable manner from year to year. Suppose, then, that we can coarse grain an ecosystem at time t according to some appropriate partition of the phase space in which each division Aj represents a particular range of numbers for each possible species in the ecosystem, along with associated parameters such as temperature, rainfall, humidity, insolation, and so on. We examine longitudinal paths, statements of the form x(n) = A0 , A1 , ..., An deﬁned in terms of some ‘natural’ time unit characteristic of the system. Then n corresponds to a time unit T , so that t = T, 2T, ..., nT . Our interest is in the serial correlation along paths. If N (n) is the number of possible paths of length n that are consistent with the underlying grammar and syntax of the appropriately coarse grained ecosystem, for example, spring leads to summer, autumn, winter, back to spring, etc., but never spring to autumn to summer to winter in a temperate climate. The essential assumption is that, for appropriate coarse graining, N (n), the number of possible grammatical paths, is much smaller than the total conceivable number of paths, and that, in the limit of large n, log[N (n)] n→∞ n

H = lim

both exists and is independent of path. Not all possible ecosystem coarse grainings are likely to lead to this result, as is sometimes the case with Markov models. Reference [40] in particular emphasizes that mesoscale ecosystem processes are most likely to entrain dynamics at larger and smaller scales, a process [74] characterizes as mesoscale resonance, a generalization of the Baldwin eﬀect. See that reference for details, broadly based on the Tuning Theorem. 12.3

Ecosystems Farm Organismal Development

The environmental and ecosystem farming of development may not always be benign. Suppose we can operationalize and quantify degrees of both overfocus or inattentional blindness (IAB) and of overall structure or environment distortion (D) in the actions of a highly parallel cognitive epigenetic regulatory system. The essential assumption is that the (internal) dual information source of a cognitive structure that has low levels of both IAB overfocus and structure/environment distortion will tend to be richer than that of one having greater levels. This is shown in ﬁgure 1a, where H is the source uncertainty dual to internal cognitive process, X = IAB, and Y = D. Regions of low X, Y , near the origin,

310

R. Wallace and D. Wallace

have greater source uncertainty than those nearby, so H(X, Y ) shows a (relatively gentle) peak at the origin, taken here as simply the product of two error functions. We are, then, particularly interested in the internal cognitive capacity of the structure itself, as paramatized by degree of overfocus and by the (large scale) distortion between implementation and impact. That capacity, a purely internal quantity, need not be convex in the parameter D, which is taken to characterize interaction with an external environment, and thus becomes a context for internal measures. Such measures need not themselves be convex in D. The generalized Onsager argument, based on the homology between information source uncertainty and free energy, as explained more fully in the Appendix, is shown in ﬁgure 1b. S = H(X, Y )− XdH/dX − Y dH/dY , the ‘disorder’ analog to entropy in a physical system, is graphed on the Z axis against the X − Y plane, assuming a gentle peak in H at the origin. Peaks in S, according to theory, constitute repulsive system barriers, which must be overcome by external forces. In ﬁgure 1b there are three quasi-stable topological resilience modes, in the sense of [71], marked as A, B, and C. The A region is locked in to low levels of both overfocus and distortion, as it sits in a pocket. Forcing the system in either direction, that is, increasing either IAB or D, will, initially, be met by homeostatic attempts to return to the resilience state A, according to this model. If overall distortion becomes severe in spite of homeostatic developmental mechanisms, the system will then jump to the quasi-stable state B, a second pocket. According to the model, however, once that transition takes place, there will be a tendency for the system to remain in a condition of high distortion. That is, the system will become locked-in to a structure with high distortion in the match between structure implementation and structure impact, but one having lower overall cognitive capacity, i.e., a lower value of H in ﬁgure 1a. The third pocket, marked C, is a broad plain in which both IAB and D remain high, a highly overfocused, poorly linked pattern of behavior which will require signiﬁcant intervention to alter once it reaches such a quasi-stable resilience mode. The structure’s cognitive capacity, measured by H in ﬁgure 1a, is the lowest of all for this condition of pathological resilience, and attempts to correct the problem – to return to condition A, will be met with very high barriers in S, according to ﬁgure 1b. That is, mode C is very highly resilient, although pathologically so, much like the eutrophication of a pure lake by sewage outﬂow. See [70, 71] for discussions of ecological resilience and literature references. We can argue that the three quasi-equilibrium conﬁgurations of ﬁgure 1b represent diﬀerent dynamical states of the system, and that the possibility of transition between them represents the breaking of the associated symmetry groupoid by external forcing mechanisms. That is, three manifolds representing three diﬀerent kinds of system dynamics have been patched together to create a more complicated topological structure. For cognitive phenomena, such behavior is likely to be the rule rather than the exception. ‘Pure’ groupoids are abstractions, and the fundamental questions will involve linkages which break the underlying symmetry.

Code, Context, and Epigenetic Catalysis in Gene Expression

311

Fig. 1. a. Source uncertainty, H, of the dual information source of epigenetic cognition, as parametized by degrees of focus, X = IAB and distortion, Y = D, between implementation and actual impact. Note the relatively gentle peak at low values of X, Y . Here H is generated as the product of two error functions. b. Generalized Onsager treatment of ﬁgure 1a. S = H(X, Y ) − XdH/dX − Y dH/dY . The regions marked A, B, and C represent realms of resilient quasi-stability, divided by barriers deﬁned by the relative peaks in S. Transition among them requires a forcing mechanism. From another perspective, limiting energy or other resources, or imposing stress from the outside, driving down H in ﬁgure 1a, would force the system into the lower plain of C, in which the system would then become trapped in states having high levels of distortion and inattentional blindness/overfocus.

In all of this, as in equation (19), system convergence is not to some ﬁxed state, limit cycle, or pseudorandom strange attractor, but rather to some appropriate set of highly dynamic information sources, i.e., behavior patterns constituting, here, developmental trajectories, rather than to some ﬁxed ‘answer to a computing problem’ [72]. What this model suggests is that suﬃciently strong external perturbation can force a highly parallel real-time cognitive epigenetic structure from a normal, almost homeostatic, developmental path into one involving a widespread, comorbid, developmental disorder. This is a well studied pattern for humans and their institutions, reviewed at some length elsewhere [71, 73]. Indeed, this argument provides the foundation of a fairly comprehensive model of chronic developmental dysfunction across a broad class of cognitive systems, including, but not limited to, cognitive epigenetic control of gene expression. One approach might be as follows: A developmental process can be viewed as involving a sequence of surfaces like ﬁgure 1, having, for example, ‘critical periods’ when the barriers between

312

R. Wallace and D. Wallace

the normal state A and the pathological states B and C are relatively low. This might particularly occur under circumstances of rapid growth or long-term energy demand, since the peaks of ﬁgure 1 are inherently energy maxima by the duality between information source uncertainty and free energy density. During such a time the peaks of ﬁgure 1 might be relatively suppressed, and the system would become highly sensitive to perturbation, and to the onset of a subsequent pathological developmental trajectory. To reiterate, then, during times of rapid growth, embryonic de- and re- methylation, and/or other high system demand, metabolic energy limitation imposes the need to focus via something like a rate distortion manifold. Cognitive process requires energy through the homologies with free energy density, and more focus at one end necessarily implies less at some other. In a distributed zero sum developmental game, as it were, some cognitive or metabolic processes must receive more free energy than others, and these may then be more easily aﬀected by external chemical, biological, or social stressors, or by simple stochastic variation. Something much like this has indeed become a standard perspective (e.g., [76]). A structure trapped in region C might be said to suﬀer something much like what [80] describes as the loss of gradient problem, in which one part of a multiple population coevolutionary system comes to dominate the others, creating an impossible situation in which the other participants do not have enough information from which to learn. That is, the cliﬀ just becomes too steep to climb. Reference [80] also characterizes focusing problems in which a two-population coevolutionary process becomes overspecialized on the opponent’s weaknesses, eﬀectively a kind of inattentional blindness. Thus there seems some consonance between our asymptotic analysis of cognitive structural function and current studies of pathologies aﬀecting coevolutionary algorithms (e.g. [30, 72]). In particular the possibility of historic trajectory, of path dependence, in producing individualized failure modes, suggests there can be no one-size-ﬁts-all amelioration strategy. Equation (20) basically enables a kind of environmental catalysis to cognitive gene expression, in a sense closely similar to the arguments of Section 4. This is analogous to, but more general than, the ‘mesoscale resonance’ invoked by [74]: during critical periods, according to these models, environmental signals can have vast impact on developmental trajectory. 12.4

A Simple Probability Argument

Again, critical periods of rapid growth require energy, and by the homology between free energy density and cognitive information source uncertainty, that energy requirement may be in the context of a zero-sum game so that the barriers of ﬁgure 1 may be lowered by metabolic energy constraints or high energy demand. In particular the groupoid structure of equation (5) changes progressively as the organism develops, with new equivalence classes being added to A = ∪α. If metabolic energy remains capped, then exp[−Hβ K] P [Hβ ] = α exp[−Hα K]

Code, Context, and Epigenetic Catalysis in Gene Expression

313

must decrease with increase in α, i.e., with increase in the cardinality of A. Thus, for restricted K, barriers between diﬀerent developmental paths must fall as the system becomes more complicated. A precis of these results can be more formally captured using methods closely similar to recent algebraic geometry approaches to concurrent, i.e., highly parallel, computing [26, 37, 60].

13

Reconsidering Directed Homotopy: Shadows

Here we reconsider directed homotopy in a developmental context, as shadowed by critical developmental periods. First, we restrict the analysis to a two dimensional phenotype space, and begin development at some S0 as in ﬁgure 2. If one requires temporal path dependence – no reverse development – then ﬁgure 2 shows two possible ﬁnal states, S1 and S2 , separated by a critical point C that casts a path-dependent developmental shadow in time. There are, consequently, two separate ‘ways’ of reaching a ﬁnal state in this model. The Si thus represent (relatively) static phenotypic expressions of the solutions to equation (19) that are, of themselves, highly dynamic information sources. Elements of each ‘way’ can be transformed into each other by continuous deformation without crossing the impenetrable shadow cast by the critical period C. These ways are the equivalence classes deﬁning the system’s topological structure, a groupoid analogous to the fundamental homotopy group in spaces that admit of loops [50] rather than time-driven, one-way paths. That is, the closed loops needed for classical homotopy theory are impossible for this kind of system because of the ‘ﬂow of time’ deﬁning the output of an information source – one goes from S0 to some ﬁnal state. The theory is thus one of directed homotopy, dihomotopy, and the central question revolves around the continuous deformation of paths in development space into one another, without crossing the shadow cast by the critical period C. Reference [36] provides another introduction to the formalism. Thus the external signals U of equation (20), as a catalytic mechanism, can deﬁne quite diﬀerent developmental dihomotopies. Such considerations suggest that a multitasking developmental process that becomes trapped in a particular pattern cannot, in general, expect to emerge from it in the absence of external forcing mechanisms or the stochastic resonance/mutational action of ‘noise’. Emerging from such a trap involves largescale topological changes, and this is the functional equivalent of a ﬁrst order phase transition in a physical systems and requires energy. The fundamental topological insight is that environmental context – the U in equation (20) – can be imposed on the ‘natural’ groupoids underlying massively parallel gene expression. This sort of behavior is, as noted in [71], central to ecosystem resilience theory. Apparently the set of developmental manifolds, and its subsets of directed homotopy equivalence classes, formally classiﬁes quasi-equilibrium states, and

314

R. Wallace and D. Wallace

Fig. 2. Given an initial developmental state S0 and a critical period C casting a pathdependent developmental shadow, there are two directed homotopy equivalence classes of deformable paths leading, respectively, to ﬁnal phenotype states S1 and S2 that are expressions of the highly dynamic information source solutions to equation (19). These equivalence classes deﬁne a topological groupoid on the developmental system.

thus characterizes the diﬀerent possible developmental resilience modes. Some of these may be highly pathological. Shifts between markedly diﬀerent topological modes appear to be necessary effects of phase transitions, involving analogs to phase changes in physical systems. It seems clear that both ‘normal development’ and possible pathological states can be represented as topological resilience/phase modes in this model, suggesting a real equivalence between diﬃculties in carrying out gene expression and its stabilization. This mirrors recent results on the relation between programming diﬃculty and system stability in highly parallel computing devices [70].

14

Epigenetic Programming of Artificial Systems for Biotechnology

Reference [72] examines how highly parallel ‘Self-X’ computing machines – selfprogramming, protecting, repairing, etc. – are inevitably coevolutionary in the sense of Section 9 above, since elements of a dynamic structural hierarchy always interact, an eﬀect that will asymptotically dominate system behavior at great

Code, Context, and Epigenetic Catalysis in Gene Expression

315

scale. The ‘farming’ paradigm provides a model for programming such devices, that, while broadly similar to the liquid state machines of [51], diﬀers in that convergence is to an information source, a systematic dynamic behavior pattern, rather than to a computed ﬁxed ‘answer’. As the farming metaphor suggests, stabilizing complex coevolutionary mechanisms appears as diﬃcult as programming them. Suﬃciently large networks of even the most dimly cognitive modules will become emergently coevolutionary, suggesting the necessity of ‘second order’ evolutionary programming that generalizes the conventional Nix/Vose models. Although we cannot pursue the argument in detail here, very clearly such an approach to programming highly parallel coevolutionary machines – equivalent to deliberate epigenetic farming – should be applicable to a broad class of artiﬁcial biological systems/machines for which some particular ongoing behavior is to be required, rather than some ﬁnal state ‘answer’. Examples might include the manufacture, in a large sense, of a dynamic product, e.g., a chemical substance, anti-cancer or artiﬁcial immune search-and-destroy strategy, biological signal detection/transduction process, and so on. Tunable epigenetic catalysis lowers an ‘eﬀective energy’ associated with the convergence of a highly coevolutionary cognitive system to a ﬁnal dynamic behavioral strategy. Given a particular ‘farming’ information source acting as the program, the behavior of the ﬁnal state of interest will become associated with the lowest value of the free energy-analog, possibly calculable by optimization methods. If the retina-like rate distortion manifold has been properly implemented, a kind of converse to the no free lunch theorem, then this optimization procedure should converge to an appropriate solution, ﬁxed or dynamic. Thus we invoke a synergism between the focusing theorem and a ‘tunable epigenetic catalysis theorem’ to raise the probability of an acceptable solution, particularly for a real-time system whose dynamics will be dominated by rate distortion theorem constraints. The degree of catalysis needed for convergence in a real time system would seem critically dependent on the rate distortion function R(D) or on its product with an acceptable reaction time, τ , that is, on there being suﬃcient bandwidth in the communication between a cognitive biological ‘machine’ and its embedding environment. If that bandwidth is too limited, or the available reaction time too short, then the system will inevitably freeze out into what amounts to a highly dysfunctional ‘ground state’. The essential point would seem to be a convergence between emerging needs in biotechnology and general strategies for programming coevolutionary computing devices.

15

Discussion and Conclusions

We have hidden the kind of massive calculations made explicit in [16, 17], burying them as ‘ﬁtting regression-model analogs to data’, possibly at a second order epigenetic hierarchical level. In the real world such calculations would be quite diﬃcult, particularly given the introduction of punctuated transitions that must

316

R. Wallace and D. Wallace

be ﬁtted using elaborate renormalization calculations, typically requiring such exotic objects as Lambert W-functions (e.g., [68, 73, 74]). Analogies with neural network studies suggest, however, intractable conceptual diﬃculties for spinglass-type models of gene expression and development dynamics, much as claimed by [57]. In spite of nearly a century of sophisticated neural network model studies – including elegant treatments like [66] – Atmanspacher [3] claims that to formulate a serious, clear-cut and transparent formal framework for cognitive neuroscience is a challenge comparable to the early stage of physics four centuries ago. Only a very few contemporary approaches, including that of [68], are worth mentioning, in his view. Furthermore, [48] has identiﬁed what might well be described as the suﬃciency failing of neural network models, that is, neural networks can be constructed as Turing machines that can replicate any known dynamic behavior in the same sense that the Ptolemaic Theory of planetary motion, as a Fourier expansion in epicycles, can, to suﬃcient order, mimic any observed orbit. Keplerian central motion provides an essential reduction. The particular characterization of [48] is that ‘neural possibility is not neural plausibility’. Likewise, [8] concludes that neural-centered explanations of high order mental function commit the mereological fallacy, that is, the fundamental logical error of attributing what is in fact a property of an entirety to a limited part of the whole system. ‘The brain’ does not exist in isolation, but as part of a complete biological individual who is most often deeply embedded in social and cultural contexts. Neural network-like models of gene expression and development applied to complex living things inherently commit both errors, particularly in a social, cultural, or environmental milieu. This suggests a particular necessity for the formal inclusion of the eﬀects of embedding contexts – the epigenetic Z and the environmental U – in the sense of [4, 5]. That is, gene expression and development are conditioned by signals from embedding physiological, social, and for humans, cultural, environments. As described above, our formulation can include such inﬂuences in a highly natural manner, as they inﬂuence epigenetic catalysis. In addition, multiple, and quite diﬀerent, cognitive gene expression mechanisms may operate simultaneously, or in appropriate sequence, given suﬃcient development time. Although epigenetic catalysis, as we have explored it here, might seem worthy of special focus, this would be a kind of intellectual optical illusion akin to inattentional blindness. Epigenetic catalysis is only one aspect of a general cognitive paradigm for gene expression, and this larger, and very complicated ‘perceptual ﬁeld’ should remain the center of intellectual attention, rather than any single element of that ﬁeld. This is to take, perhaps, an ‘East Asian’ rather than ‘Western’ perspective on the matter [69]. Developmental disorders, in a broad sense that must include comorbid mental and physical characteristics, emerge as pathological ‘resilience’ modes, in the sense of [71], a viewpoint from ecosystem theory quite similar to that of epigenetic epidemiology [32, 76]. Environmental farming through an embedding

Code, Context, and Epigenetic Catalysis in Gene Expression

317

information source aﬀecting internal epigenetic regulation of gene expression, can, as a kind of programming of a highly parallel cognitive system, place the organism into a quasi-stable pathological developmental pattern converging on a dysfunctional phenotype. The probability models of cognitive process presented here will lead, most fundamentally, to statistical tools based on the asymptotic limit theorems of information theory, in the same sense that the usual parametric statistics are based on the Central Limit Theorem. We have not, then, given ‘a’ model of development and its disorders in cognitive gene expression, but, rather, outlined a possible general strategy for ﬁtting empirically-determined statistical models to real data, in precisely the sense that one would ﬁt the usual parametric statistical models to normally distributed data. The ﬁtting of statistical models does not, of itself, perform scientiﬁc inference. That is done by comparing ﬁtted models for similar systems under diﬀerent, or diﬀerent systems under similar, conditions, and by examining the structure of residuals. One implication of this work, then, is that understanding complicated processes of gene expression and development – and their pathologies – will require construction of data analysis tools considerably more sophisticated than now available, including the present crop of simple models abducted from neural network studies or stochastic chemical reaction theory. Most centrally, however, currently popular (and fundable) reductionist approaches to understanding gene expression must eventually exhaust themselves in the same desert of sand-grain hyperparticularity that appears to have driven James Crick from molecular biology into consciousness studies, a ﬁeld now mature enough to provide tools for use in the other direction.

Acknowledgments The author thanks Dr. C. Guerrero-Bosagna and two anonymous reviewers for comments useful in revision.

References 1. Ash, R.: Information Theory. Dover Publications, New York (1990) 2. Atlan, H., Cohen, I.: Immune information, self-organization, and meaning. International Immunology 10, 711–717 (1998) 3. Atmanspacher, H.: Toward an information theoretical implementation of contextual conditions for consciousness. Acta Biotheoretica 54, 157–160 (2006) 4. Baars, B.: A Cognitive Theory of Consciousness. Cambridge University Press, New York (1988) 5. Baars, B.: Global workspace theory of consciousness: toward a cognitive neuroscience of human experience. Progress in Brain Research 150, 45–53 (2005) 6. Backdahl, L., Bushell, A., Beck, S.: Inﬂammatory signalling as mediator of epigenetic modulation in tissue-speciﬁc chronic inﬂammation. The International Journal of Biochemistry and Cell Biology (2009), doi:10.1016/j.biocel.2008.08.023

318

R. Wallace and D. Wallace

7. Beck, C., Schlogl, F.: Thermodynamics of Chaotic Systems. Cambridge University Press, Cambridge (1995) 8. Bennett, M., Hacker, P.: Philosophical Foundations of Neuroscience. Blackwell Publishing, Malden (2003) 9. Bennett, C.: Logical depth and physical complexity. In: Herkin, R. (ed.) The Universal Turing Machine: A Half-Century Survey, pp. 227–257. Oxford University Press, Oxford (1988) 10. Bos, R.: Continuous representations of groupoids. arXiv:math/0612639 (2007) 11. Bossdorf, O., Richards, C., Pigliucci, M.: Epigenetics for ecologists. Ecology Letters 11, 106–115 (2008) 12. Britten, R., Davidson, E.: Gene regulation for higher cells: a theory. Science 165, 349–357 (1969) 13. Brown, R.: From groups to groupoids: a brief survey. Bulletin of the London Mathematical Society 19, 113–134 (1987) 14. Buneci, M.: Representare de Groupoizi. Editura Mirton, Timisoara (2003) 15. Cannas Da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras. American Mathematical Society, RI (1999) 16. Ciliberti, S., Martin, O., Wagner, A.: Robustness can evolve gradually in complex regulatory networks with varying topology. PLoS Computational Biology 3(2), e15 (2007) 17. Ciliberti, S., Martin, O., Wagner, A.: Innovation and robustness in complex regulatory gene networks. Proceedings of the National Academy of Sciences 104, 13591–13596 (2007) 18. Cohen, I.: Immune system computation and the immunological homunculus. In: Nierstrasz, O., Whittle, J., Harel, D., Reggio, G. (eds.) MoDELS 2006. LNCS, vol. 4199, pp. 499–512. Springer, Heidelberg (2006) 19. Cohen, I., Harel, D.: Explaining a complex living system: dynamics, multi-scaling, and emergence. Journal of the Royal Society: Interface 4, 175–182 (2007) 20. Cover, T., Thomas, J.: Elements of Information Theory. John Wiley and Sons, New York (1991) 21. Crews, D., McLachlan, J.A.: Epigenetics, evolution, endocrine disruption, health, and disease. Endocrinology 147, S4–S10 (2006) 22. Crews, D., Gore, A., Hsu, T., Dangleben, N., Spinetta, M., Schallert, T., Anway, M., Skinner, M.: Transgenerational epigenetic imprints on mate preference. Proceedings of the National Academy of Sciences 104, 5942–5946 (2007) 23. Dehaene, S., Naccache, L.: Towards a cognitive neuroscience of consciousness: basic evidence and a workspace framework. Cognition 79, 1–37 (2001) 24. Dembo, A., Zeitouni, O.: Large Deviations: Techniques and Applications, 2nd edn. Springer, New York (1998) 25. Dias, A., Stewart, I.: Symmetry groupoids and admissible vector ﬁelds for coupled cell networks. Journal of the London Mathematical Society 69, 707–736 (2004) 26. Dretske, F.: The explanatory role of information. Philosophical Transactions of the Royal Society A 349, 59–70 (1994) 27. Emery, M.: Stochastic Calculus on Manifolds. Springer, New York (1989) 28. English, T.: Evaluation of evolutionary and genetic optimizers: no free lunch. In: Fogel, L., Angeline, P., Back, T. (eds.) Evolutionary Programming V: Proceedings of the Fifth Annual Conference on Evolutionary Programming, pp. 163–169. MIT Press, Cambridge (1996) 29. Erdos, P., Renyi, A.: On the evolution of random graphs (1960); reprinted in The Art of Counting, pp. 574–618 (1973), and in Selected Papers of Alfred Renyi, pp. 482–525 (1976)

Code, Context, and Epigenetic Catalysis in Gene Expression

319

30. Ficici, S., Milnik, O., Pollak, J.: A game-theoretic and dynamical systems analysis of selection methods in coevolution. IEEE Transactions on Evolutionary Computation 9, 580–602 (2005) 31. Feynman, R.: Lectures on Computation. Westview Press, New York (2000) 32. Foley, D., Craid, J., Morley, R., Olsson, C., Dwyer, T., Smith, K., Saﬀery, R.: Prospects for epigenetic epidemiology. American Journal of Epidemiology 169, 389–400 (2009) 33. Gilbert, S.: Mechanisms for the environmental regulation of gene expression: ecological aspects of animal development. Journal of Bioscience 30, 65–74 (2001) 34. Glazebrook, J.F., Wallace, R.: Small worlds and red queens in the global workspace: an information-theoretic approach. Cognitive Systems Reserch (2009), doi:10.1016/j.cogsys.2009.01.002 35. Golubitsky, M., Stewart, I.: Nonlinear dynamics and networks: the groupoid formalism. Bulletin of the American Mathematical Society 43, 305–364 (2006) 36. Goubault, E., Raussen, M.: Dihomotopy as a tool in state space analysis. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 16–37. Springer, Heidelberg (2002) 37. Goubault, E.: Some geometric perspectives on concurrency theory. Homology, Homotopy, and Applications 5, 95–136 (2003) 38. Gould, S.: The Structure of Evolutionary Theory. Harvard University Press, Cambridge (2002) 39. Guerrero-Bosagna, C., Sabat, P., Valladares, L.: Environmental signaling and evolutionary change: can exposure of pregnant mammals to environmental estrogens lead to epigenetically induced evolutionary changes in embryos? Evolution and Development 7, 341–350 (2005) 40. Holling, C.: Cross-scale morphology, geometry and dynamicsl of ecosystems. Ecological Monographs 41, 1–50 (1992) 41. Jablonka, E., Lamb, M.: Epigenetic Inheritance and Evolution: The Lamarckian Dimension. Oxford University Press, Oxford (1995) 42. Jablonka, E., Lamb, M.J.: Epigenetic inheritance in evolution. Journal of Evolutionary Biology 11, 159–183 (1998) 43. Jablonka, E.: Epigenetic epidemiology. International Journal of Epidemiology 33, 929–935 (2004) 44. Jaeger, J., Surkova, S., Blagov, M., Janssens, H., Kosman, D., Kozlov, K., Manu, M., Myasnikova, E., Vanario-Alonso, C., Samsonova, M., Sharp, D., Reintiz, J.: Dynamic control of positional information in the early Drosophila embryo. Nature 430, 368–371 (2004) 45. Jaenisch, R., Bird, A.: Epigenetic regulation of gene expression: how the genome integrates intrinsic and environmental signals. Nature Genetics Supplement 33, 245–254 (2003) 46. Kastner, M.: Phase transitions and conﬁguration space topology. ArXiv condmat/0703401 (2006) 47. Khinchin, A.: Mathematical Foundations of Information Theory. Dover, New York (1957) 48. Krebs, P.: Models of cognition: neurological possibility does not indicate neurological plausibility. In: Bara, B., Barsalou, L., Bucciarelli, M. (eds.) Proceedings of CogSci 2005, Stresa, Italy, pp. 1184–1189 (2005), http://cogprints.org/4498/ 49. Landau, L., Lifshitz, E.: Statistical Physics, Part I, 3rd edn., Part I. Elsevier, New York (2007) 50. Lee, J.: Introduction to topological manifolds. Springer, New York (2000)

320

R. Wallace and D. Wallace

51. Maas, W., Natschlager, T., Markram, H.: Real-time computing without stable states: a new framework for neural computation based on perturbations. Neural Computation 14, 2531–2560 (2002) 52. Matsumoto, Y.: An Introduction to Morse Theory. American Mathematical Society, Providence (2002) 53. Maturana, H.R., Varela, F.J.: Autopoiesis and Cognition. Reidel Publishing Company, Dordrecht (1980) 54. Maturana, H.R., Varela, F.J.: The Tree of Knowledge. Shambhala Publications, Boston (1992) 55. McCauly, J.: Chaos, Dynamics, and Fractals. Cambridge Nonlinear Science Series, Cambridge, UK (1994) 56. Mjolsness, E., Sharp, D., Reinitz, J.: A connectionist model of development. Journal of Theoretical Biology 152, 429–458 (1991) 57. O’Nuallain, S.: Code and context in gene expression, cognition, and consciousness. In: Barbiere, M. (ed.) The Codes of Life: The Rules of Macroevolution, ch. 15, pp. 347–356. Springer, New York (2008) 58. O’Nuallain, S., Strohman, R.: Genome and natural language: how far can the analogy be extended? In: Witzany, G. (ed.) Proceedings of Biosemiotics. Tartu University Press, Umweb (2007) 59. Pettini, M.: Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics. Springer, New York (2007) 60. Pratt, V.: Modeling concurrency with geometry. In: Proceedings of the 18th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 311–322 (1991) 61. Reinitz, J., Sharp, D.: Mechanisms of even stripe formation. Mechanics of Development 49, 133–158 (1995) 62. Scherrer, K., Jost, J.: The gene and the genon concept: a functional and information-theoretic analysis. Molecular Systems Biology 3, 87–95 (2007) 63. Scherrer, K., Jost, J.: Gene and genon concept: coding versus regulation. Theory in Bioscience 126, 65–113 (2007) 64. Sharp, D., Reinitz, J.: Prediction of mutant expression patterns using gene circuits. BioSystems 47, 79–90 (1998) 65. Skierski, M., Grundland, A., Tuszynski, J.: Analysis of the three-dimensional timedependent Landau-Ginzburg equation and its solutions. Journal of Physics A (Math. Gen.) 22, 3789–3808 (1989) 66. Toulouse, G., Dehaene, S., Changeux, J.: Spin glass model of learning by selection. Proceedings of the National Academy of Sciences 83, 1695–1698 (1986) 67. Turner, B.: Histone acetylation and an epigeneticv code. Bioessays 22, 836–845 (2000) 68. Wallace, R.: Consciousness: A Mathematical Treatment of the Global Neuronal Workspace Model. Springer, New York (2005) 69. Wallace, R.: Culture and inattentional blindness. Journal of Theoretical Biology 245, 378–390 (2007) 70. Wallace, R.: Toward formal models of biologically inspired, highly parallel machine cognition. International Journal of Parallel, Emergent, and Distributed Systems 23, 367–408 (2008) 71. Wallace, R.: Developmental disorders as pathological resilience domains. Ecology and Society 13(1), 29 (2008), http://www.ecologyandsociety.org/vol13/iss1/art29/ 72. Wallace, R.: Programming coevolutionary machines: the emerging conundrum. International Journal of Parallel, Emergent, and Distributed Systems (in press, 2009)

Code, Context, and Epigenetic Catalysis in Gene Expression

321

73. Wallace, R., Fullilove, M.: Collective Consciousness and its Discontents: Institutional Distributed Cognition, Racial Policy, and Public Health in the United States. Springer, New York (2008) 74. Wallace, R., Wallace, D.: Punctuated equilibrium in statistical models of generalized coevolutionary resilience: how sudden ecosystem transitions can entrain both phenotype expression and Darwinian selection. In: Priami, C. (ed.) Transactions on Computational Systems Biology IX. LNCS (LNBI), vol. 5121, pp. 23–85. Springer, Heidelberg (2008) 75. Wallace, R.G., Wallace, R.: Evolutionary radiation and the spectrum of consciousness. Consciousness and Cognition (2009), doi:10.1016/j.concog.2008.12.002 76. Waterland, R., Michels, K.: Epigenetic epidemiology of the developmental origins hypothesis. Annual Reviews of Nutrition 27, 363–388 (2007) 77. Weaver, I.: Epigenetic eﬀects of glucocorticoids. Seminars in Fetal and Neonatal Medicine (2009), doi:10.1016/j.siny.2008.12.002 78. Weinstein, A.: Groupoids: unifying internal and external symmetry. Notices of the American Mathematical Association 43, 744–752 (1996) 79. West-Eberhard, M.: Developmental plasticity and the origin of species diﬀerences. Proceedings of the National Academy of Sciences 102, 6543–6549 (2005) 80. Wiegand, R.: An analysis of cooperative coevolutionary algorithms. PhD Thesis, George Mason University (2003) 81. Wolpert, D., Macready, W.: No free lunch theorems for search. Santa Fe Institute, SFI-TR-02-010 (1995) 82. Wolpert, D., Macready, W.: No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation 1, 67–82 (1997) 83. Wymer, C.R.: Structural nonlinear continuous-time models in econometrics. Macroeconomic Dynamics 1, 518–548 (1997) 84. Zhu, R., Rebirio, A., Salahub, D., Kaufmann, S.: Studying genetic regulatory networks at the molecular level: delayed reaction stochastic models. Journal of Theoretical Biology 246, 725–745 (2007)

16 16.1

Mathematical Appendix The Shannon-McMillan Theorem

According to the structure of the underlying language of which a message is a particular expression, some messages are more ‘meaningful’ than others, that is, are in accord with the grammar and syntax of the language. The ShannonMcMillan or Asymptotic Equipartition Theorem, describes how messages themselves are to be classiﬁed. Suppose a long sequence of symbols is chosen, using the output of a random variable X, so that an output sequence of length n, with the form xn = (α0 , α1 , ..., αn−1 ) has joint and conditional probabilities P (X0 = α0 , X1 = α1 , ..., Xn−1 = αn−1 ) P (Xn = αn |X0 = α0 , ..., Xn−1 = αn−1 ).

322

R. Wallace and D. Wallace

Using these probabilities we may calculate the conditional uncertainty H(Xn |X0 , X1 , ..., Xn−1 ). The uncertainty of the information source, H[X], is deﬁned as H[X] = lim H(Xn |X0 , X1 , ..., Xn−1 ). n→∞

In general

(21)

H(Xn |X0 , X1 , ..., Xn−1 ) ≤ H(Xn ).

Only if the random variables Xj are all stochastically independent does equality hold. If there is a maximum n such that, for all m > 0 H(Xn+m |X0 , ..., Xn+m−1 ) = H(Xn |X0 , ..., Xn−1 ), then the source is said to be of order n. It is easy to show that H[X] = lim

n→∞

H(X0 , ...Xn ) . n+1

In general the outputs of the Xj , j = 0, 1, ..., n are dependent. That is, the output of the communication process at step n depends on previous steps. Such serial correlation, in fact, is the very structure which enables most of what is done in this paper. Here, however, the processes are all assumed statble in time, that is, the probabilities and serial correlations do not change in time, and the system is stationary. A very broad class of such self-correlated, stationary, information sources, the so-called ergodic sources for which the long-run relative frequency of a sequence converges stochastically to the probability assigned to it, have a particularly interesting property: It is possible, in the limit of large n, to divide all sequences of outputs of an ergodic information source into two distinct sets, S1 and S2 , having, respectively, very high and very low probabilities of occurrence, with the source uncertainty providing the splitting criterion. In particular the Shannon-McMillan Theorem states that, for a (long) sequence having n (serially correlated) elements, the number of ‘meaningful’ sequences, N (n) – those belonging to set S1 – will satisfy the relation log[N (n)] ≈ H[X]. n More formally, lim

n→∞

log[N (n)] = H[X] n

= lim H(Xn |X0 , ..., Xn−1 ) n→∞

(22)

Code, Context, and Epigenetic Catalysis in Gene Expression

= lim

n→∞

H(X0 , ..., Xn ) . n+1

323

(23)

Using the internal structures of the information source permits limiting attention only to high probability ‘meaningful’ sequences of symbols. 16.2

The Rate Distortion Theorem

The Shannon-McMillan Theorem can be expressed as the ‘zero error limit’ of the Rate Distortion Theorem [20, 24] which deﬁnes a splitting criterion that identiﬁes high probability pairs of sequences. We follow closely the treatment of [20]. The origin of the problem is the question of representing one information source by a simpler one in such a way that the least information is lost. For example we might have a continuous variate between 0 and 100, and wish to represent it in terms of a small set of integers in a way that minimizes the inevitable distortion that process creates. Typically, for example, an analog audio signal will be replaced by a ‘digital’ one. The problem is to do this in a way which least distorts the reconstructed audio waveform. Suppose the original stationary, ergodic information source Y with output from a particular alphabet generates sequences of the form y n = y1 , ..., yn . These are ‘digitized,’ in some sense, producing a chain of ‘digitized values’ bn = b1 , ..., bn , where the b-alphabet is much more restricted than the y-alphabet. bn is, in turn, deterministically retranslated into a reproduction of the original signal y n . That is, each bm is mapped on to a unique n-length y-sequence in the alphabet of the information source Y : bm → yˆn = yˆ1 , ..., yˆn . Note, however, that many y n sequences may be mapped onto the same retranslation sequence yˆn , so that information will, in general, be lost. The central problem is to explicitly minimize that loss. The retranslation process deﬁnes a new stationary, ergodic information source, Yˆ . The next step is to deﬁne a distortion measure, d(y, yˆ), which compares the original to the retranslated path. For example the Hamming distortion is d(y, yˆ) = 1, y = yˆ d(y, yˆ) = 0, y = yˆ.

(24)

324

R. Wallace and D. Wallace

For continuous variates the Squared error distortion is d(y, yˆ) = (y − yˆ)2 .

(25)

There are many possibilities. The distortion between paths y n and yˆn is deﬁned as 1 d(yj , yˆj ). n j=1 n

d(y n , yˆn ) =

(26)

Suppose that with each path y n and bn -path retranslation into the y-language and denoted y n , there are associated individual, joint, and conditional probability distributions p(y n ), p(ˆ y n ), p(y n |ˆ y n ). The average distortion is deﬁned as D= p(y n )d(y n , yˆn ).

(27)

yn

It is possible, using the distributions given above, to deﬁne the information transmitted from the incoming Y to the outgoing Yˆ process in the usual manner, using the Shannon source uncertainty of the strings: I(Y, Yˆ ) = H(Y ) − H(Y |Yˆ ) = H(Y ) + H(Yˆ ) − H(Y, Yˆ ). If there is no uncertainty in Y given the retranslation Yˆ , then no information is lost. In general, this will not be true. The information rate distortion function R(D) for a source Y with a distortion measure d(y, yˆ) is deﬁned as R(D) = p(y,ˆ y);

min

(y,y) ˆ

I(Y, Yˆ ).

(28)

p(y)p(y|ˆ y)d(y,ˆ y)≤D

The minimization is over all conditional distributions p(y|ˆ y) for which the joint distribution p(y, yˆ) = p(y)p(y|ˆ y) satisﬁes the average distortion constraint (i.e., average distortion ≤ D). The Rate Distortion Theorem states that R(D) is the maximum achievable rate of information transmission which does not exceed the distortion D. See [20, 24] details. More to the point, however, is the following: Pairs of sequences (y n , yˆn ) can be deﬁned as distortion typical ; that is, for a given average distortion D, deﬁned in terms of a particular measure, pairs of sequences can be divided into two sets, a high probability one containing a relatively small number of (matched) pairs with d(y n , yˆn ) ≤ D, and a low probability one containing most pairs. As n → ∞, the smaller set approaches unit probability, and, for those pairs, p(y n ) ≥ p(ˆ y n |y n ) exp[−nI(Y, Yˆ )].

(29)

Code, Context, and Epigenetic Catalysis in Gene Expression

325

Thus, roughly speaking, I(Y, Yˆ ) embodies the splitting criterion between high and low probability pairs of paths. For the theory of interacting information sources, then, I(Y, Yˆ ) can play the role of H in the dynamic treatment above. The rate distortion function can actually be calculated in many cases by using a Lagrange multiplier method – see Section 13.7 of [20]. 16.3

Groupoids

Basic ideas. Following [78] closely, a groupoid, G, is deﬁned by a base set A upon which some mapping – a morphism – can be deﬁned. Note that not all possible pairs of states (aj , ak ) in the base set A can be connected by such a morphism. Those that can deﬁne the groupoid element, a morphism g = (aj , ak ) having the natural inverse g −1 = (ak , aj ). Given such a pairing, it is possible to deﬁne ‘natural’ end-point maps α(g) = aj , β(g) = ak from the set of morphisms G into A, and a formally associative product in the groupoid g1 g2 provided α(g1 g2 ) = α(g1 ), β(g1 g2 ) = β(g2 ), and β(g1 ) = α(g2 ). Then the product is deﬁned, and associative, (g1 g2 )g3 = g1 (g2 g3 ). In addition, there are natural left and right identity elements λg , ρg such that λg g = g = gρg [78]. An orbit of the groupoid G over A is an equivalence class for the relation aj ∼ Gak if and only if there is a groupoid element g with α(g) = aj and β(g) = ak . Following [15], we note that a groupoid is called transitive if it has just one orbit. The transitive groupoids are the building blocks of groupoids in that there is a natural decomposition of the base space of a general groupoid into orbits. Over each orbit there is a transitive groupoid, and the disjoint union of these transitive groupoids is the original groupoid. Conversely, the disjoint union of groupoids is itself a groupoid. The isotropy group of a ∈ X consists of those g in G with α(g) = a = β(g). These groups prove fundamental to classifying groupoids. If G is any groupoid over A, the map (α, β) : G → A × A is a morphism from G to the pair groupoid of A. The image of (α, β) is the orbit equivalence relation ∼ G, and the functional kernel is the union of the isotropy groups. If f : X → Y is a function, then the kernel of f , ker(f ) = [(x1 , x2 ) ∈ X × X : f (x1 ) = f (x2 )] deﬁnes an equivalence relation. Groupoids may have additional structure. As [78] explains, a groupoid G is a topological groupoid over a base space X if G and X are topological spaces and α, β and multiplication are continuous maps. A criticism sometimes applied to groupoid theory is that their classiﬁcation up to isomorphism is nothing other than the classiﬁcation of equivalence relations via the orbit equivalence relation and groups via the isotropy groups. The imposition of a compatible topological structure produces a nontrivial interaction between the two structures. It is possible to introduce a metric structure on manifolds of related information sources, producing such interaction.

326

R. Wallace and D. Wallace

In essence, a groupoid is a category in which all morphisms have an inverse, here deﬁned in terms of connection to a base point by a meaningful path of an information source dual to a cognitive process. As [78] points out, the morphism (α, β) suggests another way of looking at groupoids. A groupoid over A identiﬁes not only which elements of A are equivalent to one another (isomorphic), but it also parametizes the diﬀerent ways (isomorphisms) in which two elements can be equivalent, i.e., all possible information sources dual to some cognitive process. Given the information theoretic characterization of cognition presented above, this produces a full modular cognitive network in a highly natural manner. Brown [13] describes the fundamental structure as follows: A groupoid should be thought of as a group with many objects, or with many identities... A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, ﬂexibility and range of applications... EXAMPLE 1. A disjoint union [of groups] G = ∪λ Gλ , λ ∈ Λ, is a groupoid: the product ab is deﬁned if and only if a, b belong to the same Gλ , and ab is then just the product in the group Gλ . There is an identity 1λ for each λ ∈ Λ. The maps α, β coincide and map Gλ to λ, λ ∈ Λ. EXAMPLE 2. An equivalence relation R on [a set] X becomes a groupoid with α, β : R → X the two projections, and product (x, y)(y, z) = (x, z) whenever (x, y), (y, z) ∈ R. There is an identity, namely (x, x), for each x ∈ X... [78] makes the following fundamental point: Almost every interesting equivalence relation on a space B arises in a natural way as the orbit equivalence relation of some groupoid G over B. Instead of dealing directly with the orbit space B/G as an object in the category Smap of sets and mappings, one should consider instead the groupoid G itself as an object in the category Ghtp of groupoids and homotopy classes of morphisms. The groupoid approach has become quite popular in the study of networks of coupled dynamical systems which can be deﬁned by diﬀerential equation models, [35]. Global and local symmetry groupoids. Here we follow [78] fairly closely, using the example of a ﬁnite tiling. Consider a tiling of the euclidean plane R2 by identical 2 by 1 rectangles, speciﬁed by the set X (one dimensional) where the grout between tiles is X = H ∪V , having H = R×Z and V = 2Z ×R, where R is the set of real numbers and Z the integers. Call each connected component of R2 \X, that is, the complement of the two dimensional real plane intersecting X, a tile. Let Γ be the group of those rigid motions of R2 which leave X invariant, i.e., the normal subgroup of translations by elements of the lattice Λ = H ∩ V =

Code, Context, and Epigenetic Catalysis in Gene Expression

327

2Z × Z (corresponding to corner points of the tiles), together with reﬂections through each of the points 1/2Λ = Z × 1/2Z, and across the horizontal and vertical lines through those points. As noted in [78], much is lost in this coarsegraining, in particular the same symmetry group would arise if we replaced X entirely by the lattice Λ of corner points. Γ retains no information about the local structure of the tiled plane. In the case of a real tiling, restricted to the ﬁnite set B = [0, 2m] × [0, n] the symmetry group shrinks drastically: The subgroup leaving X ∩ B invariant contains just four elements even though a repetitive pattern is clearly visible. A two-stage groupoid approach recovers the lost structure. We deﬁne the transformation groupoid of the action of Γ on R2 to be the set G(Γ, R2 ) = {(x, γ, y|x ∈ R2 , y ∈ R2 , γ ∈ Γ, x = γy}, with the partially deﬁned binary operation (x, γ, y)(y, ν, z) = (x, γν, z). Here α(x, γ, y) = x, and β(x, γ, y) = y, and the inverses are natural. We can form the restriction of G to B (or any other subset of R2 ) by deﬁning G(Γ, R2 )|B = {g ∈ G(Γ, R2 )|α(g), β(g) ∈ B} 1. An orbit of the groupoid G over B is an equivalence class for the relation x ∼G y if and only if there is a groupoid element g with α(g) = x and β(g) = y. Two points are in the same orbit if they are similarly placed within their tiles or within the grout pattern. 2. The isotropy group of x ∈ B consists of those g in G with α(g) = x = β(g). It is trivial for every point except those in 1/2Λ ∩ B, for which it is Z2 × Z2 , the direct product of integers modulo two with itself. By contrast, embedding the tiled structure within a larger context permits definition of a much richer structure, i.e., the identiﬁcation of local symmetries. We construct a second groupoid as follows. Consider the plane R2 as being decomposed as the disjoint union of P1 = B ∩ X (the grout), P2 = B\P1 (the complement of P1 in B, which is the tiles), and P3 = R2 \B (the exterior of the tiled room). Let E be the group of all euclidean motions of the plane, and deﬁne the local symmetry groupoid Gloc as the set of triples (x, γ, y) in B × E × B for which x = γy, and for which y has a neighborhood U in R2 such that γ(U ∩ Pi ) ⊆ Pi for i = 1, 2, 3. The composition is given by the same formula as for G(Γ, R2 ). For this groupoid-in-context there are only a ﬁnite number of orbits: O1 O2 O3 O4

= = = =

interior points of the tiles. interior edges of the tiles. interior crossing points of the grout. exterior boundary edge points of the tile grout.

328

R. Wallace and D. Wallace

O5 = boundary ‘T’ points. O6 = boundary corner points. The isotropy group structure is, however, now very rich indeed: The isotropy group of a point in O1 is now isomorphic to the entire rotation group O2 . It is Z2 × Z2 for O2 . For O3 it is the eight-element dihedral group D4 . For O4 , O5 and O6 it is simply Z2 . These are the ‘local symmetries’ of the tile-in-context. 16.4

Morse Theory

Morse theory examines relations between analytic behavior of a function – the location and character of its critical points – and the underlying topology of the manifold on which the function is deﬁned. We are interested in a number of such functions, for example information source uncertainty on a parameter space and ‘second order’ iterations involving parameter manifolds determining critical behavior, for example sudden onset of a giant component in the mean number model [74], and universality class tuning in the mean ﬁeld model of the next section. These can be reformulated from a Morse theory perspective. Here we follow closely the elegant treatments of [46, 59]. The essential idea of Morse theory is to examine an n-dimensional manifold M as decomposed into level sets of some function f : M → R where R is the set of real numbers. The a-level set of f is deﬁned as f −1 (a) = {x ∈ M : f (x) = a}, the set of all points in M with f (x) = a. If M is compact, then the whole manifold can be decomposed into such slices in a canonical fashion between two limits, deﬁned by the minimum and maximum of f on M . Let the part of M below a be deﬁned as Ma = f −1 (−∞, a] = {x ∈ M : f (x) ≤ a}. These sets describe the whole manifold as a varies between the minimum and maximum of f . Morse functions are deﬁned as a particular set of smooth functions f : M → R as follows. Suppose a function f has a critical point xc , so that the derivative df (xc ) = 0, with critical value f (xc ). Then f is a Morse function if its critical points are nondegenerate in the sense that the Hessian matrix of second derivatives at xc , whose elements, in terms of local coordinates are Hi,j = ∂ 2 f /∂xi ∂xj , has rank n, which means that it has only nonzero eigenvalues, so that there are no lines or surfaces of critical points and, ultimately, critical points are isolated.

Code, Context, and Epigenetic Catalysis in Gene Expression

329

The index of the critical point is the number of negative eigenvalues of H at xc . A level set f −1 (a) of f is called a critical level if a is a critical value of f , that is, if there is at least one critical point xc ∈ f −1 (a). Again following [59], the essential results of Morse theory are: 1. If an interval [a, b] contains no critical values of f , then the topology of f −1 [a, v] does not change for any v ∈ (a, b]. Importantly, the result is valid even if f is not a Morse function, but only a smooth function. 2. If the interval [a, b] contains critical values, the topology of f −1 [a, v] changes in a manner determined by the properties of the matrix H at the critical points. 3. If f : M → R is a Morse function, the set of all the critical points of f is a discrete subset of M , i.e., critical points are isolated. This is Sard’s Theorem. 4. If f : M → R is a Morse function, with M compact, then on a ﬁnite interval [a, b] ⊂ R, there is only a ﬁnite number of critical points p of f such that f (p) ∈ [a, b]. The set of critical values of f is a discrete set of R. 5. For any diﬀerentiable manifold M , the set of Morse functions on M is an open dense set in the set of real functions of M of diﬀerentiability class r for 0 ≤ r ≤ ∞. 6. Some topological invariants of M , that is, quantities that are the same for all the manifolds that have the same topology as M , can be estimated and sometimes computed exactly once all the critical points of f are known: Let the Morse numbers μi (i = 0, ..., m) of a function f on M be the number of critical points of f of index i, (the number of negative eigenvalues of H). The Euler characteristic of the complicated manifold M can be expressed as the alternating sum of the Morse numbers of any Morse function on M , χ=

m

(−1)i μi .

i=1

The Euler characteristic reduces, in the case of a simple polyhedron, to χ=V −E+F where V, E, and F are the numbers of vertices, edges, and faces in the polyhedron. 7. Another important theorem states that, if the interval [a, b] contains a critical value of f with a single critical point xc , then the topology of the set Mb deﬁned above diﬀers from that of Ma in a way which is determined by the index, i, of the critical point. Then Mb is homeomorphic to the manifold obtained from attaching to Ma an i-handle, i.e., the direct product of an i-disk and an (m − i)-disk. Again, see [52, 59] for details. 16.5

Generalized Onsager Theory

Understanding the time dynamics of groupoid-driven information systems away from phase transition critical points requires a phenomenology similar to the

330

R. Wallace and D. Wallace

Onsager relations of nonequilibrium thermodynamics. This also leads to a general theory involving large-scale topological changes in the sense of Morse theory. If the Groupoid Free Energy (GFE) of a biological process is parametized by some vector of quantities K = (K1 , ..., Km ), then, in analogy with nonequilibrium thermodynamics, gradients in the Kj of the disorder, deﬁned as SG = FG (K) −

m

Kj ∂FG /∂Kj

(30)

j=1

become of central interest. Equation (30) is similar to the deﬁnition of entropy in terms of the free energy of a physical system. Pursuing the homology further, the generalized Onsager relations deﬁning temporal dynamics of systems having a GFE become dKj /dt = Lj,i ∂SG /∂Ki , (31) i

where the Lj,i are, in ﬁrst order, constants reﬂecting the nature of the underlying cognitive phenomena. The L-matrix is to be viewed empirically, in the same spirit as the slope and intercept of a regression model, and may have structure far diﬀerent than familiar from more simple chemical or physical processes. The ∂SG /∂K are analogous to thermodynamic forces in a chemical system, and may be subject to override by external physiological or other driving mechanisms: biological and cognitive phenomena, unlike simple physical systems, can make choices as to resource allocation. That is, an essential contrast with simple physical systems driven by (say) entropy maximization is that complex biological or cognitive structures can make decisions about resource allocation, to the extent resources are available. Thus resource availability is a context, not a determinant, of behavior. Equations (30) and (31) can be derived in a simple parameter-free covariant manner which relies on the underlying topology of the information source space implicit to the development [74]. We will not pursue that development here. The dynamics, as we have presented them so far, have been noiseless, while biological systems are always very noisy. Equation (31) might be rewritten as dKj /dt = Lj,i ∂SG /∂Ki + σW (t) i

where σ is a constant and W (t) represents white noise. This leads directly to a family of classic stochastic diﬀerential equations having the form dKtj = Lj (t, K)dt + σ j (t, K)dBt ,

(32)

where the Lj and σ j are appropriately regular functions of t and K, and dBt represents the noise structure, and we have readjusted the indices. Further progress in this direction requires introduction of methods from stochastic diﬀerential geometry and related topics in the sense of [27]. The obvious inference is that noise – not necessarily ‘white’ – can serve as a tool to

Code, Context, and Epigenetic Catalysis in Gene Expression

331

shift the system between various topological modes, as a kind of crosstalk and the source of a generalized stochastic resonance. Eﬀectively, topological shifts between and within dynamic manifolds constitute another theory of phase transitions [59], and this phenomenological Onsager treatment would likely be much enriched by explicit adoption of a Morse theory perspective. 16.6

The Tuning Theorem

Messages from an information source, seen as symbols xj from some alphabet, each having probabilities Pj associated with a random variable X, are ‘encoded’ into the language of a ‘transmission channel’, a random variable Y with symbols yk , having probabilities Pk , possibly with error. Someone receiving the symbol yk then retranslates it (without error) into some xk , which may or may not be the same as the xj that was sent. More formally, the message sent along the channel is characterized by a random variable X having the distribution P (X = xj ) = Pj , j = 1, ..., M. The channel through which the message is sent is characterized by a second random variable Y having the distribution P (Y = yk ) = Pk , k = 1, ..., L. Let the joint probability distribution of X and Y be deﬁned as P (X = xj , Y = yk ) = P (xj , yk ) = Pj,k and the conditional probability of Y given X as P (Y = yk |X = xj ) = P (yk |xj ). Then the Shannon uncertainty of X and Y independently and the joint uncertainty of X and Y together are deﬁned respectively as H(X) = −

M

Pj log(Pj )

j=1

H(Y ) = −

L

Pk log(Pk )

k=1

H(X, Y ) = −

M L j=1 k=1

Pj,k log(Pj,k ).

(33)

332

R. Wallace and D. Wallace

The conditional uncertainty of Y given X is deﬁned as H(Y |X) = −

M L

Pj,k log[P (yk |xj )].

(34)

j=1 k=1

For any two stochastic variates X and Y , H(Y ) ≥ H(Y |X), as knowledge of X generally gives some knowledge of Y . Equality occurs only in the case of stochastic independence. Since P (xj , yk ) = P (xj )P (yk |xj ), we have H(X|Y ) = H(X, Y ) − H(Y ). The information transmitted by translating the variable X into the channel transmission variable Y – possibly with error – and then retranslating without error the transmitted Y back into X is deﬁned as I(X|Y ) = H(X)− H(X|Y ) = H(X)+ H(Y )− H(X, Y )

(35)

See, for example, [1, 20, 47] for details. The essential point is that if there is no uncertainty in X given the channel Y , then there is no loss of information through transmission. In general this will not be true, and herein lies the essence of the theory. Given a ﬁxed vocabulary for the transmitted variable X, and a ﬁxed vocabulary and probability distribution for the channel Y , we may vary the probability distribution of X in such a way as to maximize the information sent. The capacity of the channel is deﬁned as C = max I(X|Y ) P (X)

(36)

subject to the subsidiary condition that P (X) = 1. The critical trick of the Shannon Coding Theorem for sending a message with arbitrarily small error along the channel Y at any rate R < C is to encode it in longer and longer ‘typical’ sequences of the variable X; that is, those sequences whose distribution of symbols approximates the probability distribution P (X) above which maximizes C. If S(n) is the number of such ‘typical’ sequences of length n, then log[S(n)] ≈ nH(X), where H(X) is the uncertainty of the stochastic variable deﬁned above. Some consideration shows that S(n) is much less than the total number of possible messages of length n. Thus, as n → ∞, only a vanishingly small fraction of all possible messages is meaningful in this sense. This observation, after some considerable development, is what allows the Coding Theorem to work so well. In sum, the prescription is to encode messages in typical sequences, which are sent at very nearly the capacity of the channel. As the encoded messages become

Code, Context, and Epigenetic Catalysis in Gene Expression

333

longer and longer, their maximum possible rate of transmission without error approaches channel capacity as a limit. Again, [1, 20, 47] provide details. This approach can be, in a sense, inverted to give a tuning theorem which parsimoniously describes the essence of the Rate Distortion Manifold. Telephone lines, optical wave, guides and the tenuous plasma through which a planetary probe transmits data to earth may all be viewed in traditional information-theoretic terms as a noisy channel around which we must structure a message so as to attain an optimal error-free transmission rate. Telephone lines, wave guides, and interplanetary plasmas are, relatively speaking, ﬁxed on the timescale of most messages, as are most other signaling networks. Indeed, the capacity of a channel, is deﬁned by varying the probability distribution of the ‘message’ process X so as to maximize I(X|Y ). Suppose there is some message X so critical that its probability distribution must remain ﬁxed. The trick is to ﬁx the distribution P (x) but modify the channel – i.e., tune it – so as to maximize I(X|Y ). The dual channel capacity C ∗ can be deﬁned as C∗ =

max

I(X|Y ).

max

I(Y |X)

P (Y ),P (Y |X)

(37)

But C∗ =

P (Y ),P (Y |X)

since I(X|Y ) = H(X) + H(Y ) − H(X, Y ) = I(Y |X). Thus, in a purely formal mathematical sense, the message transmits the channel, and there will indeed be, according to the Coding Theorem, a channel distribution P (Y ) which maximizes C ∗ . One may do better than this, however, by modifying the channel matrix P (Y |X). Since M P (yj ) = P (xi )P (yj |xi ), i=1

P (Y ) is entirely deﬁned by the channel matrix P (Y |X) for ﬁxed P (X) and C∗ =

max

P (Y ),P (Y |X)

I(Y |X) = max I(Y |X). P (Y |X)

Calculating C ∗ requires maximizing the complicated expression I(X|Y ) = H(X) + H(Y ) − H(X, Y ), that contains products of terms and their logs, subject to constraints that the sums of probabilities are 1 and each probability is itself between 0 and 1. Maximization is done by varying the channel matrix terms P (yj |xi ) within the constraints. This is a diﬃcult problem in nonlinear optimization. However, for the special case M = L, C ∗ may be found by inspection:

334

R. Wallace and D. Wallace

If M = L, then choose

P (yj |xi ) = δj,i ,

where δi,j is 1 if i = j and 0 otherwise. For this special case C ∗ = H(X), with P (yk ) = P (xk ) for all k. Information is thus transmitted without error when the channel becomes ‘typical’ with respect to the ﬁxed message distribution P (X). If M < L, matters reduce to this case, but for L < M information must be lost, leading to Rate Distortion limitations. Thus modifying the channel may be a far more eﬃcient means of ensuring transmission of an important message than encoding that message in a ‘natural’ language which maximizes the rate of transmission of information on a ﬁxed channel. We have examined the two limits in which either the distributions of P (Y ) or of P (X) are kept ﬁxed. The ﬁrst provides the usual Shannon Coding Theorem, and the second a tuning theorem variant, a tunable retina-like Rate Distortion Manifold. It seems likely, however, than for many important systems P (X) and P (Y ) will interpenetrate, to use Richard Levins’ terminology. That is, P (X) and P (Y ) will aﬀect each other in characteristic ways, so that some form of mutual tuning may be the most eﬀective strategy.

Author Index

Aman, Bogdan

26

Bantang, Johnrob Y. 164 Bortolussi, Luca 216 Boˇsnaˇcki, D. 69

Harmer, Russ 116 Heiner, Monika 138 Hillston, Jane 1 Jack, John

200

Calder, Muﬀy 1 Chesi, Graziano 268 Ciobanu, Gabriel 26 Ciocchetta, Federica 45

Krivine, Jean

Danos, Vincent 116 Dassow, J¨ urgen 187 David, Maria Pamela C. de Vink, E.P. 69

Marwan, Wolfgang 138 Mendoza, Eduardo R. 164 Mitrana, Victor 187

Lehrack, Sebastian

164

138

P˘ aun, Andrei 200 Policriti, Alberto 216 Pronk, T.E. 69

Feret, J´erˆ ome 116 Fontana, Walter 116 Gilbert, David 138 Guerriero, Maria Luisa

116

90

Wallace, Deborah Wallace, Rodrick

283 283

5750

Edited by S. Istrail, P. Pevzner, and M. Waterman Editorial Board: A. Apostolico S. Brunak M. Gelfand T. Lengauer S. Miyano G. Myers M.-F. Sagot D. Sankoff R. Shamir T. Speed M. Vingron W. Wong

Subseries of Lecture Notes in Computer Science

Corrado Priami Ralph-Johan Back Ion Petre (Eds.)

Transactions on Computational Systems Biology XI

13

Series Editors Sorin Istrail, Brown University, Providence, RI, USA Pavel Pevzner, University of California, San Diego, CA, USA Michael Waterman, University of Southern California, Los Angeles, CA, USA Editor-in-Chief Corrado Priami The Microsoft Research - University of Trento Centre for Computational and Systems Biology Piazza Manci, 17, 38050 Povo (TN), Italy E-mail: [email protected] Guest Editors Ralph-Johan Back Ion Petre Åbo Akademi University Department of Information Technologies Joukahaisenkatu 3-5, 20520 Turku, Finland E-mail: {backrj,ipetre}@abo.ﬁ

Library of Congress Control Number: 2009933672 CR Subject Classiﬁcation (1998): J.3, F.1, F.2, I.6, I.2, C.1.3 ISSN ISSN ISBN-10 ISBN-13

0302-9743 (Lecture Notes in Computer Science) 1861-2075 (Transactions on Computational Systems Biology) 3-642-04185-X Springer Berlin Heidelberg New York 978-3-642-04185-3 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2009 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientiﬁc Publishing Services, Chennai, India Printed on acid-free paper SPIN: 12743292 06/3180 543210

Preface

Biology is witnessing a transformation towards a more quantitative science, based on the major technological breakthroughs of the past decade. In this transformation, biology is incorporating mathematical modeling techniques and computational approaches towards numerical simulations, model analysis, and quantitative predictions. An important goal is to formalize and analyze the everchanging inter-connections between components (often on diﬀerent time and space scales), their inﬂuence on one another, regulatory patterns, alternative pathways, etc. Formal reasoning rather than empirical observations is the main driving force in this new type of biological research. At the same time, computer science and applied mathematics are faced with considerable methodological challenges in handling an unprecedented level of concurrency, stochastic eﬀects, a mix of large and small populations, combinatorial explosions in the state space, model reﬁnement, and model (de)composition, etc. This special issue of Transactions on Computational Systems Biology on Computational Models for Cell Processes is based on a workshop with the same name that took place in Turku, Finland, on May 27, 2008. The workshop was organized as a satellite event of the 15th International Symposium on Formal Methods that took place in Turku in the period May 28-31, 2008. This special issue however had an open call for paper submissions, with a separate peer-review process. The accepted papers span an interesting mix of approaches to systems biology, ranging from quantitative to qualitative techniques, from continuous to discrete mathematics, from deterministic to stochastic methods, from computational models for biology to computing paradigms inspired by biology. Overall, they give a good glimpse into some of the exciting current research avenues in computational systems biology. This volume also contains three regular submissions that deal with the relationships between ODEs and stochastic concurrent constraint programming (by Bertolussi and Policriti), with the equilibrium points of genetic regulatory networks (by Chesi), and with probability models describing how epigenetic context aﬀects gene expression and organismal development (by Wallace and Wallace). July 2009

Ralph-Johan Back Ion Petre Corrado Priami

LNCS Transactions on Computational Systems Biology – Editorial Board

Corrado Priami, Editor-in-chief Charles Auﬀray Matthew Bellgard Soren Brunak Luca Cardelli Zhu Chen Vincent Danos Eytan Domany Walter Fontana Takashi Gojobori Martijn A. Huynen Marta Kwiatkowska Doron Lancet Pedro Mendes Bud Mishra Satoru Miayano Denis Noble Yi Pan Alberto Policriti Magali Roux-Rouquie Vincent Schachter Adelinde Uhrmacher Alfonso Valencia

University of Trento, Italy Genexpress, CNRS and Pierre & Marie Curie University, France Murdoch University, Australia Technical University of Denmark, Denmark Microsoft Research Cambridge, UK Shanghai Institute of Hematology, China CNRS, University of Paris VII, France Center for Systems Biology, Weizmann Institute, Israel Santa Fe Institute, USA National Institute of Genetics, Japan Center for Molecular and Biomolecular Informatics The Netherlands University of Birmingham, UK Crown Human Genome Center, Israel Virginia Bioinformatics Institute, USA Courant Institute and Cold Spring Harbor Lab, USA University of Tokyo, Japan University of Oxford, UK Georgia State University, USA University of Udine, Italy CNRS, Pasteur Institute, France Genoscope, France University of Rostock, Germany Centro Nacional de Biotecnologa, Spain

Table of Contents

Computational Models for Cell Processes Process Algebra Modelling Styles for Biomolecular Processes . . . . . . . . . . Muﬀy Calder and Jane Hillston

1

Simple, Enhanced and Mutual Mobile Membranes . . . . . . . . . . . . . . . . . . . . Bogdan Aman and Gabriel Ciobanu

26

Bio-PEPA with Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Federica Ciocchetta

45

In Silico Modelling and Analysis of Ribosome Kinetics and aa-tRNA Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

69

Qualitative and Quantitative Analysis of a Bio-PEPA Model of the Gp130/JAK/STAT Signalling Pathway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maria Luisa Guerriero

90

Rule-Based Modelling and Model Perturbation . . . . . . . . . . . . . . . . . . . . . . Vincent Danos, J´erˆ ome Feret, Walter Fontana, Russ Harmer, and Jean Krivine Extended Stochastic Petri Nets for Model-Based Design of Wetlab Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monika Heiner, Sebastian Lehrack, David Gilbert, and Wolfgang Marwan A Projective Brane Calculus with Activate, Bud and Mate as Primitive Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maria Pamela C. David, Johnrob Y. Bantang, and Eduardo R. Mendoza Accepting Networks of Non-inserting Evolutionary Processors . . . . . . . . . J¨ urgen Dassow and Victor Mitrana

116

138

164

187

Discrete Modeling of Biochemical Signaling with Memory Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Jack and Andrei P˘ aun

200

Dynamical Systems and Stochastic Programming: To Ordinary Diﬀerential Equations and Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luca Bortolussi and Alberto Policriti

216

VIII

Table of Contents

Computing Equilibrium Points of Genetic Regulatory Networks . . . . . . . . Graziano Chesi

268

Code, Context, and Epigenetic Catalysis in Gene Expression . . . . . . . . . . Rodrick Wallace and Deborah Wallace

283

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335

Process Algebra Modelling Styles for Biomolecular Processes Muﬀy Calder1 and Jane Hillston2 1

Department of Computing Science, University of Glasgow, Glasgow G12 8QQ, Scotland 2 Laboratory for Foundations of Computer Science and Centre for Systems Biology, Edinburgh The University of Edinburgh, Edinburgh EHA 9AB, Scotland

Abstract. We investigate how biomolecular processes are modelled in process algebras, focussing on chemical reactions. We consider various modelling styles and how design decisions made in the deﬁnition of the process algebra have an impact on how a modelling style can be applied. Our goal is to highlight the often implicit choices that modellers make in choosing a formalism, and illustrate, through the use of examples, how this can aﬀect expressability as well as the type and complexity of the analysis that can be performed.

1

Introduction

Much recent research has considered the problem of providing suitable abstract models to allow biologists to construct mechanistic models to enhance understanding of biomolecular processes. Process algebras, formal modelling languages originally conceived for modelling concurrent computations, have been widely applied, most notably in the area of signalling pathways [RSS01, CGH06, TK08]. This is experimental science and we are currently evaluating the hypothesis that such formal models can add value to the mathematical analysis that is already undertaken within systems biology in terms of ordinary diﬀerential equation (ODE) models or stochastic simulations directly. In exploring this goal, even within work on process algebras, several diﬀerent styles of modelling have emerged. Ultimately we hope to be able to give guidance on how to choose among these modelling styles, or on how to map molecular components and their interactions to processes, process communication and process composition. However, in the ﬁrst instance we investigate how design decisions made in the deﬁnition of the language have an impact on how a modelling style can be applied, and highlight the often implicit choices that modellers make in choosing a formalism. Recent research eﬀort on process algebras for biomolecular processes, e.g. [CGH06, CVOG06, CH08, Car08], has focussed on deﬁning alternative semantics, such as discrete-state (stochastic) or continuous-state (ODE) semantics. These provide important links with the work where mathematical representations are used directly and establish a valid foundation for process algebra models. Based on these semantics, analysis may be carried out by model-checking, C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 1–25, 2009. c Springer-Verlag Berlin Heidelberg 2009

2

M. Calder and J. Hillston

stochastic simulation based on Gillespie’s algorithm or ODE simulations. Our emphasis in this paper is diﬀerent. Here we consider the forms of abstraction supported by process algebra and how the abstraction and the process algebra chosen aﬀect the expressiveness of the model with respect to the biological processes, as well as the type and complexity of the analysis that can be performed. We focus on one of the most important types of interaction between molecular components: chemical reactions. In chemical notation, these may be ﬁrst order k1 reactions, for example A degrades to B: A−→B, or second order reactions, for exk2 k3 ample A and B combine to form C or C and D: A + B −→C, or A + B −→C + D. Typically, k1 . . . k3 are rate constants for kinetic laws (e.g. mass action). A fundamental aspect of the abstraction used in modelling is the nature of the process mapping. In the literature on process algebras for systems biology we ﬁnd predominantly the molecule-as-process [RSS01, Car08] abstraction, but the species-as-process and reaction-as-process mappings have also been proposed [CGH06, CH08, BP08]. The distinction between the ﬁrst two can be understood by appealing to ecology: the former is essentially individuals-based, whereas the latter is population-based. We note that this distinction is less common in distributed computing system modelling, the origins of process algebra, where population-based models are rarely considered. Further stylistic diﬀerentiation was identiﬁed in [CGH06] where the concepts of reagent-centric and pathway-centric models are introduced, in the context of population-based modelling. Reagent-centric models map all reagents in a reaction to processes, whose variation reﬂect decrease through consumption and increase through product formation (consumers and producers). Reagents such as modiﬁers that do not vary species amounts can also be modelled in this approach. Reagent-centric models provide a ﬁne-grained, distributed view of a system. Pathway-centric models provide a more abstract view of a system, tracking serialisations of events, which are then composed concurrently. Here, processes vary according to their biological state rather than their quantity. Whereas in a reagent-centric approach the processes may be molecules or molecular species, in the pathway-centric approach the processes are molecules or sub-pathways. Thus the interactions between processes are between ﬂows of events corresponding to producers, i.e. components on the left hand sides of a reactions. Most modelling approaches map chemical reactions to events in a straightforward way, and map (possibly a subset of) the chemical components to processes. Bortolussi and Policriti’s work on sCCP, using the reaction-as-process abstraction, is an exception to this. When chemical components are mapped to processes within the reagent-centric approach there is a further choice: between associating processes with all components or only with the reagents on the left hand side of equations, i.e. those reagents that are the reactants of the reaction. To distinguish these two cases, we call the former reagent-centric and the latter reactant-centric. This modelling choice is often inﬂuenced by the form of synchronisation available within the algebra: binary or multi-way. If we have only the former, then only the reactant-centric approach is possible and we are left

Process Algebra Modelling Styles for Biomolecular Processes

3

with an interesting dilemma when there are fewer components on the right hand k2 side of the equation than on the left hand side, e.g. A + B −→C. In summary, a number of factors will inﬂuence the structure of a process algebra model of a biomolecular process: – population-based or individuals-based, – reagent-centric, reactant-centric, pathway-centric or reaction-centric, – the form of synchronisation available in the algebra. In this paper we investigate the interplay between these three factors. Our motivation is to explore the extent to which we can build clear and faithful models using current algebras and analysis techniques, and how design decisions with respect to the process algebra determine the mappings available to the modeller. We consider diﬀerent combinations, investigating their advantages and disadvantages. We will use ﬁve process algebras for illustration: π-calculus, Beta-binders, PEPA, Bio-PEPA, and sCCP; these are brieﬂy outlined in the next section. These are chosen as they represent a spectrum of diﬀerent modelling style, including languages that have been adapted (π-calculus, PEPA and sCCP) and designed (Beta-binders and Bio-PEPA) for biological modelling. This is by no means a comprehensive list of process algebras used in systems biology. In particular we do not include any of the process algebras designed to consider spatial aspects of biomolecular processes [CPR+ 04, Car04, V07, BMMT06, CG09] as they are beyond the scope of this paper. The remainder of the paper is organised as follows. Section 2 gives an overview of the process algebras and Section 3 describes the example pathway used throughout for illustration and comparison. In Sections 4 to 8 we consider modelling in PEPA, Bio-PEPA, π-calculus, Beta-binders and sCCP. We discuss the results in Section 9 and give our conclusions in Section 10.

2

Process Algebras

Process algebras were originally deﬁned to give semantics to concurrent processes in a computing context and have enjoyed considerable success over the three decades since they emerged. Classical process algebras such as CCS [Mil80] and CSP [Hoa85] focus on the functional capabilities of processes and all actions are atomic with only relative timing of actions captured. Subsequently there have been many extensions of process algebras to capture more information about the system being modelled, for example the relative probability of alternative actions (probabilistic process algebras) and the expected duration of actions (stochastic process algebras). Each of the process algebras that we consider is based on three fundamental binary operators: action preﬁx, choice, which is associative and commutative, and synchronous composition, which is also associative and commutative.

4

M. Calder and J. Hillston

Note that in the following we omit the cooperation sets for composition in PEPA and Bio-PEPA and assume them to be the intersection of the alphabets of the processes involved (denoted ). We disregard quantitative aspects of actions, ∗ since the representation of kinetics is orthogonal to the expressiveness we consider here. Therefore in our examples, we will assume that the reaction rate for each considered reaction is unique and use this as the name of the corresponding r1 reaction event, i.e. the reaction A + B −→C + D in chemical notation maps to the process algebra event r1 . In seminal work, Regev and Shapiro [RS01] suggested an abstraction of cell-ascomputation and proposed that models formerly used in the study of interacting computational entities, such as Petri nets, process algebras and automata, could be usefully employed for the study of biological processes. In particular they focussed on the π-calculus [Mil99], and subsequently the stochastic π-calculus [Pri95] based on the molecule-as-process abstraction. This work has been hugely inﬂuential with many other authors following the same abstraction in their own work, even when the details of the process algebra diﬀer. However, the π-calculus has some particular characteristics that are independent of the molecule-as-process abstraction that also shape the style in which models are expressed. In this section we give a brief introduction to process algebras, focussing on the features which lead to diﬀerent modelling paradigms. 2.1

Forms of Synchronisation

The original process algebras, CCS and CSP, diﬀer in their interpretation of actions and consequently the meaning of synchronisation. In CCS all actions are assumed to be communications, and therefore conjugate, i.e. actions are paired, corresponding to an input and an output. An action cannot be carried out without its partner, and the pairing of an input and an output becomes a private τ action. This has the consequence that the interaction, or synchronisation, between processes is strictly binary as once an input has been paired with an output both become unavailable for further interaction. In contrast, in CSP no distinction is made between inputs and outputs and there is no notion of complementarity between actions. Instead action type denotes ownership of a channel and synchronisation is assumed to take place whenever processes undertake actions of the same type, i.e. communication over the named channel. This is termed multiway synchronisation as there is no restriction on the number of processes that may own a channel and thus join a synchronisation. Note that in both these cases the parallel composition operator is generic: in CCS any complementary actions which are on either side of the parallel composition may synchronise; in CSP, processes composed by the parallel operator must synchronise on common actions. Synchronisation in PEPA is a subtle variation of the CSP scheme. Here the parallel composition operator, termed cooperation, is decorated by a set of action types (the cooperation set ) and processes are only forced to synchronise on

Process Algebra Modelling Styles for Biomolecular Processes

5

action types within this set, being able to act concurrently and individually on other action types. Thus the parallel composition is not generic, but a family of parameterised operators. The characteristics of this multiway synchronisation are important in the biological context as they allow one copy of a process (molecule) within a set of identical processes to undertake a reaction individually, something that would not be possible in CSP1 . 2.2

π-Calculus

The π-calculus [Mil99] (and its stochastic form [Pri95]) was designed to express mobility, represented by the passing of channel names. It evolved from CCS [Mil80] and includes the operations of a constant, action preﬁx, choice, parallel composition, communication and scope restriction. There are variants of the syntax, here we use the following form with events π and processes P : π ::= τ | x | x | x(y) | xy P ::= 0 | π.P | P |P | P + P | νxP Following CCS [Mil80], τ is the unobservable event. All other events are observable and paired, e.g. x(y) with xy, with x(y) denoting input y on channel x, and xy denoting output y on channel x. 0 is the inactive process and νxP restricts the scope of the name x to P . In the stochastic form, rates are bound to channels, but as with the other process algebras, we will omit rates here. A structural congruence, denoted ≡, determines when two syntactic expressions are equivalent, and an operational semantics is given by a set of reaction rules that deﬁne how a system evolves following communication. We do not give the full deﬁnitions of the congruence and reaction rules, but note two distinguishing features. First, the constant, 0, is an identity for parallel composition, i.e. there is a syntactic equality P | 0 ≡ P . Second, interaction only occurs when there is a complementary pair of input and output events. The relevant reduction rule is (. . . + xy.Q) | (. . . + x(z).P ) → Q | P {y/z}. There have been numerous applications of the π-calculus to biomolecular processes, starting with the work of Regev et al. [RSS01]. An interesting aspect of the application of π-calculus is that it was designed to facilitate modelling mobility and name passing, thus in the original π-calculus events are parameterised, e.g. x(y). Yet, most biological applications do not exploit mobility — the parameter is not relevant, except when modelling compartments, or internal communications. So, in many models unparameterised events are also permitted, e.g. x and x, and we have also included them here. We note the recent work of Cardelli [Car08] on translations between process algebra and chemical reactions that introduces a subset of the π-calculus and CCS suitable for modelling chemical reactions. It is similar to the syntax above, but excludes event parameters and the ν operator. Additionally, it includes an expression of initial components. 1

This might explain why, to the best of our knowledge, there has been no work applying CSP to biomolecular modelling.

6

M. Calder and J. Hillston

A further distinctive aspect of the π-calculus/CCS paradigm for biomolecular modelling is the underlying assumption of two-way synchronous communication. This means that a a binary chemical reaction, e.g. of the form A + B →r C, is modelled by processes A and B oﬀering events r and r, whereas a unary chemical reaction, e.g. of the form A →r B, must be modelled by an unobservable τ event. 2.3

Beta-Binders

Beta-binders [DPPQ06] is a process algebra based on the π-calculus, designed for modelling and simulation of biological processes. A biological process is modelled by a bio-process, which is a π-calculus process encapsulated in a box with interaction capabilities expressed as beta-binders. Each communication channel has a set of associated types and there are three kinds of binder: visible, hidden, and complexed. Additionally, there are rates, but these are omitted here. A bio-process is either a constant or pair of encapsulated π-calculus processes composed with a synchronous parallel operator. The language has evolved over a number of years, here we use the following syntax for boxes B and beta-binders B, assuming π-calculus processes P : B ::= N il | B[P ] | B B B ::= β(x, Γ ) | β h (x, Γ ) | β c (x, Γ ) Further, there is a additional syntactic category for events, which include functions on boxes to join, split, create and destroy boxes; these are called join, split, new and delete, respectively. These functions are only applied when a condition, deﬁned over binders and π processes, is fulﬁlled. Interaction is two-way and is either intra-box, in which case it is standard π-calculus interaction, or it is inter-box in which case it is speciﬁed by the beta-binders and it is between (visible) input/output pairs, but now the types have only to be compatible (rather than identical). There are additional actions (within boxes) that include changing the status of binders (e.g. unhide or change type). There are three structural congruences: ≡p , the standard congruence on π processes, ≡b , a congruence on boxes (e.g. is associative, commutative), and ≡e , a congruence on events (e.g. join, split have substitution property). 2.4

PEPA

Performance Evaluation Process Algebra (PEPA) was introduced in the early 1990s as a formalism for building Markovian-based performance models of computer and communication systems [Hil96]. All actions in PEPA consist of an action type and a rate, which speciﬁes the average duration of the action as an exponentially distributed random variable. The language has a small set of combinators (preﬁx, choice, parallel composition/cooperation, hiding and constant).

Process Algebra Modelling Styles for Biomolecular Processes

7

Recursive behaviour is speciﬁed by mutually recursive deﬁnitions. As PEPA was designed for specifying ergodic continuous time Markov chains (CTMC), a restriction is often placed on model construction via a two level syntax, meaning that models consist of parallel compositions of sequential components (constructed using only preﬁx and choice): S := α.S | S + S | C

P := P P | P/L | S L where S denotes a sequential component, P a model component and C is a constant deﬁned by a declaration such as def

C=S α.S carries out activity α (with an exponentially distributed duration, but omitted here), and it subsequently behaves as S. As discussed above, PEPA supports multi-way cooperations between components: the result of synchronising on an activity α is thus another α, available for further synchronisation. We write P Q to denote cooperation between P and Q over L. The set which is used L as the subscript to the cooperation symbol, the cooperation set L, determines those activities on which the cooperands are forced to synchronise. For action types not in L, the components proceed independently and concurrently with their enabled activities. We write P Q as an abbreviation for P Q when L L is empty. P/L denotes the component P in which all actions with types in L are hidden meaning that their type is no longer visible but is replaced by the distinguished type τ . We do not consider hiding in the remainder of this paper. The stochastic nature of the actions means that the choice becomes a probabilistic choice governed by a race condition between the involved actions. Similarly actions of parallel components that are not forced to cooperate are also subject to a race condition. When components cooperate on actions but have diﬀerent deﬁnitions of the rate of the action, the rate of the synchronised action is deﬁned to be that of the slowest of the components. While these dynamic considerations do not concern us in this paper, and PEPA has been used for modelling a number of biological examples, we note that the form of the dynamics of synchronisation (the rate of the slowest component) is not always appropriate in this context. 2.5

Bio-PEPA

Bio-PEPA [CH08] is a newly deﬁned modiﬁcation of the PEPA formalism that has been speciﬁcally designed for modelling biochemical networks. It shares many features with PEPA but also has some characteristics to tailor it to the biological application. Functional rates: In contrast to PEPA, individual processes are not able to deﬁne their own rates for actions. Instead the rate associated with an action is speciﬁed once, independently of the processes in which the action occurs.

8

M. Calder and J. Hillston

The value of this rate can be speciﬁed to be a function that depends on the current state of the system. Stoichiometry: For each action, as well as its type, the stoichiometry or degree of involvement is also speciﬁed. Parameterised processes: Bio-PEPA has been designed to support the population-based reagent-centric style of modelling and so a model consists of a number of sequential components each representing a distinct species which evolve quantitatively (increasing or decreasing amounts). Thus in order to capture the state of a system each component is parameterised recording its current level. Diﬀerentiated preﬁx: For each action (reaction) that a component is involved in it records its role within that reaction, e.g. reactant, product, inhibitor etc. This enables the appropriate values to be used in the functional rate associated with this reaction. As with PEPA, Bio-PEPA has a two level grammar. The syntax of the sequential (species) components is deﬁned as: S ::= (α, κ) op S | S + S | C

op ::= ↓ | ↑ | ⊕ | | .

In the preﬁx term (α, κ) op S, α is an action name and can be viewed as the name or label of a reaction, κ is the stoichiometry coeﬃcient of the species and the preﬁx combinator op represents the role of the element in the reaction. Speciﬁcally, ↓ denotes the role of reactant, ↑ product, ⊕ activator, inhibitor and generic modiﬁer. The operator + expresses the choice between possible def actions and the constant C is deﬁned by an equation C = S. The syntax of model components is deﬁned as:

P ::= P P | S(x) L The process P Q denotes the synchronisation between components P and Q L and the set L speciﬁes those activities on which the components must synchronise. In the model component S(x), the parameter x ∈ R represents the initial concentration by default, although according to the analysis to be carried out the parameter may also be interpreted as number of molecules or molecular level after appropriate conversion. 2.6

sCCP

In the Concurrent Constraint Programming (CCP) process algebra, rather than components and actions, there are components and constraints [BJG96]; there are also variables. The components evolve by adding constraints to a constraint store (tell) or checking the current state of the constraint store (ask). This leads to an asynchronous form of communication between components (via global variables in the constraint store) and there is no direct synchronisation. In addition to tell and ask components may also have choice, parallel composition, procedure

Process Algebra Modelling Styles for Biomolecular Processes

9

call and local variables. In the stochastic form of CCP, sCCP [Bor06], a stochastic duration is associated with the ask and tell operators in a manner analogous to the durations of actions in other stochastic process algebras. sCCP has been proposed as a modelling formalism for biological networks, and stochastic, deterministic and hybrid semantics have been associated with models in this context [BP08]. The style of modelling is similar to that of BioPEPA in that a population-based view is taken, although here explicit variables record the quantitative state of species, rather than parameterised components. At a high level the abstraction is that measurable entities (molecules etc.) are associated with stream variables, logical entities are associated with processes or control variables and reactions are associated with processes. In general a reaction is modelled as a sequence of interactions with the constraint store: ﬁrst checking that there is suﬃcient amount of the substrates and then updating the amounts of the products. For mass action reactions the ask step of this sequence will be given a rate equal to the product of the kinetic constant and the amounts of the substrates; the tell step is assumed to be instantaneous. Thus an arbitrary mass action reaction R1 + . . . + Rn −→k P1 + . . . + Pm will be represented as reaction(k, [R1 , . . . , Rn ], [P1 , . . . , Pm ]) : − n askrMA (k,R1 ,...,Rn ) (Ri > 0) .

i=1

ni=1

tell∞ (Ri $= Ri − 1) m tell (P $= P + 1) ∞ j j j=1

Here Ri and Pj are stream variables and rMA is a predeﬁned function with the obvious deﬁnition.

3

Example Pathway

We refer to a small synthetic pathway when exploring how design decisions with respect to the the process algebra determine the mappings available to the modeller. The pathway consists of ﬁve representative reactions. The reactions are given in chemical notation in Figure 1, and presented graphically in Figure 2. While the pathway is a synthetic example, it is based on behaviour we have observed in various pathways, including the ubiquitous Raf/MEK/ERK signalling pathway. The equations exhibit various combinations of increasing/decreasing/preserved reagents between the left and right hand sides. Speciﬁcally, r1 and r4 have a decreasing number of reagents, r2 and r5 have an increasing number of reagents, and r3 has the same number of reagents on the left and right hand sides. Note that r5 has no reagent on the left hand side; we might use a reaction like this

10

M. Calder and J. Hillston A+B C B D+E

→r1 → r2 →r3 → r4 →r5

C A+B D B E

Fig. 1. Example pathway in chemical notation

A

r2

B

r4

r5 E

r1 C

r3 D

Fig. 2. Example pathway

to indicate that E is plentiful, or that it is produced by another pathway that is irrelevant to this abstraction. We will ﬁnd it useful to refer to the degree of a chemical reaction, meaning the number of reactants that it has i.e. the number of reagents on the left hand side. We have not included a homeo-reaction [Car08], where the components on the left hand side are identical, as it is only relevant to distinguish this case when rates are determined. In the example pathway, we assume initial concentrations of A, B and E, unless stated otherwise.

4 4.1

PEPA Models Reagent-Centric Style

In the reagent-centric view, ﬁrst proposed in [CGH06], species concentrations are discretised into levels; the granularity of the system is determined by the number of levels n and the concentration step size h, where there is a given maximum concentration max, h = max/n. As the number of levels increases/step size decreases, the granularity of the model increases. For each species, there is a family of processes, each deﬁning the behaviour for that (abstraction of) concentration. The system is deﬁned by the parallel composition of a number of initial components. The simplest abstraction is obtained when the number of levels is two, so that for each species there are two processes, denoting behaviour in the presence and absence of that species, respectively. We often refer to this kind of model as the high/low model. For example, for species A, AH denotes presence and AL denotes absence (alternatively A1 and A0 , respectively). Figure 3 gives the PEPA high/low model for the example pathway, consisting of a set of equations and a system deﬁnition. Figure 4 illustrates the state space for this model.

Process Algebra Modelling Styles for Biomolecular Processes

AH AL BH BL CH CL

def

= = def = def = def = def = def

r1.AL r2.AH r1.BL + r3.BL r2.BH + r4.BH r2.CL r1.CH

def

System = AH

DH DL EH EL

def

= = def = def = def

11

r4.DL r3.DH r4.EL r5.EH

BH CL DL EH ∗ ∗ ∗ ∗

Fig. 3. Example pathway: PEPA reagent-centric high/low model

r3

(AH , B H , C L , D L , E H) r1

(AH , B L , C L , D H, E H)

r5

r2

r4

(AH , B H , C L , D L , E L )

(AL , B L , C H , D L , E H) r5

r1

r5 r3

(AH , B L , C L , D H, E L )

r2

(AL , B L , C H , D L , E L )

Fig. 4. State space of the PEPA high/low model. Note that we use (AX , BX , CX , DX , EX ) to denote the state since the number of components is ﬁxed and the synchronisation structure does not change.

A0 A1 A2 B0 B1 B2 C0 C1 C2

def

= = def = def = def = def = def = def = def = def

def

r2.A1 r1.A0 + r2.A2 r1.A1 r2.B1 + r4.B1 r1.B0 + r3.B0 + r2.B2 + r4.B2 r1.B1 + r3.B1 r1.C1 r2.C0 + r1.C2 r2.C1

System = A2

D0 D1 D2 E0 E1 E2

def

= = def = def = def = def = def

r3.D1 r4.D0 + r3.D2 r4.D1 r5.E1 r4.E0 + r5.E2 r4.E1

B2 C0 D0 E2 ∗ ∗ ∗ ∗

Fig. 5. Example pathway: PEPA reagent-centric model with n = 3

(A0 , B 0 , C 2 , D 0 , E 1 )

(A 1 , B 1 , C 1 , D 0 , E 1 )

(A2 , B 2 , C 0 , D 0 , E 1 )

(A0 , B 0 , C 2 , D 0 , E 0 )

(A1 , B 1 , C 1 , D 0 , E 0 )

(A2 , B 2 , C 0 , D 0 , E 0 )

(A1 , B 0 , C 1 , D 1 , E 2 )

(A2 , B 1 , C 0 , D 1 , E 2 )

(A1 , B 0 , C 1 , D 1 , E 1 )

(A2 , B 1 , C 0 , D 1 , E 1 )

(A1 , B 0 , C 1 , D 1 , E 0 )

(A2 , B 1 , C 0 , D 1 , E 0 )

(A2 , B 0 , C 0 , D 2 , E 2 ) (A2 , B 0 , C 0 , D 2 , E 1 ) (A2 , B 0 , C 0 , D 2 , E 0 )

Fig. 6. State space of the PEPA reagent-centric model with n = 3. To avoid clutter in the diagram reaction labels are omitted, but r1 and r2 are shown in solid lines, r3 in dashed lines and r4 and r5 in dotted lines.

(A0 , B 0 , C 2 , D 0 , E 2 )

(A1 , B 1 , C 1 , D 0 , E 2 )

(A2 , B 2 , C 0 , D 0 , E 2 )

12 M. Calder and J. Hillston

Process Algebra Modelling Styles for Biomolecular Processes

13

As an example of a model with a diﬀerent granularity, Figure 5 contains a reagent-centric model with n = 3 (i.e. levels 0, 1, and 2). The state space is in Figure 6. Note that regardless of the number of levels, the number of (system) components is constant during system evolution, i.e. there are always ﬁve components (the number of species). Process as molecule in reagent-centric style. The granularity of the reagent-centric style depends on the step size h. In the limit, the ﬁnest grained model has a step size of one molecule. In general, it is impractical to increase n to its corresponding limit, but one alternative is to take a reagent-centric model with n = 1 and interpret each process as denoting the presence or absence of a molecule. An approach based on this abstraction has been used for studying the FGF pathway using stochastic model checking in [HKNT06]. For our example, for species A, AH denotes presence of a molecule and AL denotes absence. So, the population based high/low model model in Figure 3 can also be interpreted as an individuals model, with at most one molecule for each species. Similarly, a model consisting of (at most) two molecules for each species, is given by replacing the system deﬁnition of Figure 3 by the system deﬁnition: (AH AH ) (BH BH ) (CL CL ) (DL DL ) (EH EH ). ∗ ∗ ∗ ∗ Figure 7 illustrates a small portion of the corresponding state space (one transition step). Notice that this system describes the possible evolution of every molecule: it is very ﬁne grained. For example, from the initial state there are 8 possible transitions for reaction r1 , because there are two possible molecules of A that can be consumed, two possible molecules of B that can be consumed, and two possible molecules of C that can be produced (23 combinations). Similarly, there are 4 possibilities for reaction r3 . In many cases this degree of granularity is inappropriate. By appealing to symmetry (i.e. composition is commutative), we can use a form of counter abstraction to represent the molecules AH . . . AH by An , AH . . . AH AL n

n−1

by An−1 , and so on. This counter abstraction involves identifying an equivalence class of states in a high/low model of m molecules, with a state in a model n levels, where n = m. In other words, we deﬁne the processes as in the high/low

(AH, A H , B H, B H, C L , C L , D L , D L , E H , E H) r1

r2

r3

(AH, A H , B L , B H, C L , C L , D H, D L , E H , E H)

.. .

(AL , A H , B L , B H, C H, C L , D L , D L , E H , E H)

.. .

Fig. 7. One transition step in PEPA reagent-centric process-as-molecule model with two molecules

14

M. Calder and J. Hillston (A2 , B 2 , C 0 , D 0 , E 2 )

(AH, A H , B H, B H, C L , C L , D L , D L , E H , E H) r1

r2 r1

r2

(AL , A H , B L , B H, C H, C L , D L , D L , E H , E H) (AH, A L , B L , B H, C H, C L , D L , D L , E H , E H)

(A1 , B 1 , C 1 , D 0 , E 2 )

.. ..

Fig. 8. One transition step in the state space of the PEPA counter abstraction model

A

r2

B

r4

r5 E

r1 C

r3 D

Fig. 9. Example set of reactions with pathways indicated

model of Figure 3, then compose multiple copies of each process and interpret An as representing n molecules. This is illustrated in Figure 8, for the example pathway with two molecules for each species. States in the ﬁne-grained individuals model are quotiented and dashed lines indicate how the quotient class relates to a state in the counter abstraction model. 4.2

Pathway-Centric Style

An alternative style of modelling that has been proposed in PEPA is the pathway-centric style. In this style, we specify the sub-pathways that consume and replenish the initial species, which are the species with signiﬁcant initial concentrations. In the example pathway, this involves deﬁning the sub-pathways starting from A, B, and E. Call these P ath1 , P ath2 and P ath3 , respectively. The example pathway is given in Figure 10, with corresponding state space in Figure 11. Notice that although the system deﬁnition has only 3 components, this space is isomorphic to the high/low reagent-centric model (Figure 4). Notice also implicitly, the model has two levels. For example, P ath1 denotes high concentration of both A and B. We could make levels explicit in this style, by composing

Process Algebra Modelling Styles for Biomolecular Processes

15

def

P ath1 = r1 .r2 .P ath1 def P ath2 = r1 .r2 .P ath2 + r3 .r4 .P ath2 def P ath3 = r4 .r5 .P ath3 def

System = P ath1

P ath2 P ath3 ∗ ∗

Fig. 10. Example pathway: PEPA pathway-centric model

r3

(Path1 , Path2 , Path3 ) r1

r2

(Path 1 , r4.Path 2 , Path3 ) r4

r5

(r2.Path 1 , r2.Path2 , Path3 )

(Path 1 , Path2 , r5 .Path3 ) r5

r1

r3

r5 (Path1 , r4.Path2 , Path3 )

r2

(r2.Path 1 , r2.Path2 , r5.Path3 )

Fig. 11. Pathway-centric model state space

multiple copies of each pathway (with parallel composition, no synchronisation). For example the three level model would be: (P ath1 P ath1 ) (P ath2 P ath2 ) (P ath3 P ath3 ) ∗ ∗ In this case, similarly to the individuals reagent-centric style, there are more potential interleavings than in the reagent-centric population-based representation, and so the explicit state space here will be larger. However, again, by appealing to symmetry, we can work at the aggregate level. Thus for a given number of levels, the state space size and structure of both the pathway-centric and the reagent-centric models should be the same, as established in [CGH06]. Note that tools like the PEPA workbench [TDG09] can automatically detect such symmetries. We observe that assuming chemical reactions of at most degree two, we only require binary synchronisation, for this style of model.

5

Bio-PEPA

The Bio-PEPA formulation [CH08] of the reagent-centric style for the example pathway is given in Figure 12. This example does not fully exploit the power of Bio-PEPA, since the stochiometric coeﬃcients are all simple (1) and the functional rates are omitted. However, it does illustrate how the language focuses on the role of each species, in each reaction. Initial concentrations are denoted A0 for species A, etc. An integral part of a Bio-PEPA speciﬁcation (omitted here) is a deﬁnition of h and n, for every species, as well as initial concentrations (expressed as levels). The state space of this model depends upon the levels, for example, if the number of levels is uniformly 2, then the state space is the same as Figure 6.

16

M. Calder and J. Hillston A B C D E

def

= = def = def = def = def

(r1, 1)↓A + (r2, 1)↑A (r1, 1)↓B + (r2, 1)↑B + (r3, 1)↓B (r2, 1)↑C (r4, 1)↓D + (r3, 1)↑D (r4, 1)↓E + (r5, 1)↑E

System = A(A0 ) B(B0 ) C(C0 ) D(D0 ) E(E0 ) ∗ ∗ ∗ ∗ def

Fig. 12. Example pathway: Bio-PEPA model

Note that in the corresponding PEPA model (i.e. Figure 5), the number of levels is “hardwired” into the equations, whereas in the Bio-PEPA model, it is given as a parameter oﬀering more ﬂexibility to the modeller. If the number of levels is set suﬃciently high the model has a state space corresponding to an individuals model (i.e. if n is chosen to be the number of molecules).

6

π-Calculus

Models in the π-calculus and its stochastic variants predominantly follow the reactant style (e.g. [TK08]), based on the molecules-as-processes abstraction. Thus these are individuals based models. Figure 13 gives the π-calculus model in this style for the example pathway; since each reagent in the example also occurs on the left hand side of a chemical equation, there are processes for A . . . E. The example pathway highlights an interesting aspect of this style because in the biochemistry there are 1. equations with a decreasing number of components, and 2. an equation with no left hand side. Consider the former case. Since synchronisations are between reagents on the left hand side of an equation only, there is an arbitrary (and inconsequential) choice between which component is output and which is input. Further, the components on the left hand side, when translated into processes, evolve into components on the right hand side. If the number of components decreases, then we have to nominate one or more to evolve to 0, the null process. For example, A + B →r C could map to A = r.C and B = r.0; equally, it could map to A = r.0 and B = r.C, or A = r.0 and B = r.C, etc. Taking the ﬁrst choice, A | B evolves to C | 0. This is an example of a “trailing 0”, which is removed through application of the syntactic equality P | 0 ≡ P , i.e. A | B evolves to C. Now consider the second case. We cannot model an equation without a left hand side explicitly, e.g. r5 , but since E is present initially, we could represent the inﬁnite supply of E by a τ event, after oﬀering the output event r4 . However, this would constrain the creation of E to occur only after a molecule has been

Process Algebra Modelling Styles for Biomolecular Processes A B C D E Env

= = = = = =

17

r1 .C r1 .0 + τ.D τ.A | B r4 . B r4 .0 τ.Env | E

System = A | B | E | Env Fig. 13. Example pathway: π-calculus model

. .

..

.

. .

A | B | E | E | Env

.

.

A | D | E | E | Env

C | E | E | Env A | B | E | Env

C | E | Env

A | D | E | Env

A | B | Env

A | D | Env

C | Env

Fig. 14. π-calculus model state space

consumed in the reaction r4 . An alternative, which we use, is to introduce a representation of the environment Env and deﬁne it as follows: Env = τ.Env | E This presents the possibility that an unbounded number of E molecules may be introduced into the system, which is true when we represent the system only qualitatively. In the biological reality and when quantitative information is included in the model in the form of rates the system will become pragmatically bounded meaning that the probability for E to grow unboundedly is extremely small. Figure 14 illustrates possible evolutions for the system with one molecule of A, B and E initially, i.e. the evolution of A | B | E | Env. We have not labelled the transitions since events are either unobservable or become so after synchronisation. Notice that in this state space the number of system components ﬂuctuates, it both increases and decreases. Moreover the state space is inﬁnite due to the potentially unbounded number of E, although a graph isomorphic to the state space of the pathway-centric model is embedded within it. An alternative interpretation of this model is therefore a ﬁne-grained pathway-centric view based on

18

M. Calder and J. Hillston

molecules. Or rather, it is a mixture of two styles: equations are deﬁned for each reagent, but the system deﬁnition has the form of a pathway-centric model. While this approach provides a faithful overall system model, it is not compositional. Speciﬁcally, one equation incorporates aspects of the initial system and it would be misleading to a reader who inspected the behaviour only of a process that arbitrarily terminates, e.g. B, which can evolve into 0. Moreover, some reactions are represented explicitly by named events, i.e. r1 and r4 , whereas the unary or nullary reactions r2 , r3 and r5 are represented by the τ event. Thus, there are no occurrences of the reaction names r2 , r3 and r5 in the model.

7

Beta-Binders

There are several ways to map a chemical reaction in this formalism. For example, we could deﬁne a mapping very similar to the π-calculus mapping, with boxes for the processes that are initial, i.e. the system is given by [A] [B] [E], with suitable beta-binders deﬁned for each box, and each encapsulated process is deﬁned as in Figure 13. The authors recommend this mapping when the reaction denotes a collision of entities, the collision being mapped to (inter-box) communication. However, if we use this mapping, we are left with boxes containing the π-calculus constant process (i.e. 0) and we cannot remove them by the structural congruences: we need to introduce an explicit delete event to remove them. Alternatively, instead of representing reactions by inter-box communication, we could represent reactions by events, i.e. by the box operations. In this case, a reaction such as A + B →r C maps to (A, B) join C, where A, B, and C are constant bio-processes. Figure 15 gives a Beta-binders model of the example pathway using events. Notice that there are four events and no communication: the encapsulated processes are constants, except for process B, which changes its interaction type (to that of D). The state space is given in Figure 16; the space is isomorphic to the π-calculus model, though we could bound the occurrences of new E with a condition. The model is also a mixture of styles: equations are deﬁned for each reagent, but it is not reagent-centric: there is no communication and the system deﬁnition has the form of a reaction-centric model. There is a third possible mapping when the reaction denotes a binding (e.g. ligand to receptor); this is usually written in chemical notation as: A+B →r [A+ B]. In this case we could we use the complex/decomplex beta-binder operations to create and delete dedicated communication channels between boxes [A] and (A, B) join C where A = β(x, ΓA ) [nil] C split (A, B) B = β(x, ΓB ) [chtype(x, ΓD ). nil] (D, E) join B C = β(x, ΓC ) [nil] new E D = β(x, ΓD ) [nil] E = β(x, ΓE ) [nil] Fig. 15. Example pathway in Beta-binders

Process Algebra Modelling Styles for Biomolecular Processes

. .. . ..

19

. .. A

B

E

A

E

A C

E

B

D

E

E

A

E

D

E

E C

E

A

B

A

D

C

Fig. 16. State space of Beta-binders model. Note that following the graphical notation for Beta-binders, we omit the parallel composition operation on bio-processes.

[B]. That is, the two boxes [A] and [B] would evolve into a complex of two boxes, instead of into two separate boxes.

8

sCCP

Our last example is a model in sCCP. This is shown in Figure 17. There are ﬁve processes: one for each reaction, with stream variables representing the species. Each process has the form ask (check that there is suﬃcient of a species) followed by the parallel composition of all the possible eﬀects of the reaction (i.e. production or consumption) expressed by tell. The state space of the model is shown in Figure 18. Unsurprisingly this includes the state space which has been retrieved from the other models, such as the reagent-centric PEPA models (shown in the shaded area in the diagram). However note that this model also permits the unbounded growth of the population of E (as in the π-calculus model), leading to an inﬁnite state space unless an explicit guard is inserted which disables reaction r5 when the population of E reaches a given size. This model bears some similarity to the state based PRISM model given in [CVOG06], where species are represented by state variables. This is not surprising, since the PRISM modelling language is essentially the language of reactive modules [AH90]. However, in [CVOG06], there is still explicit synchronisation and commands are grouped by species, rather than by reaction. The reactionsas-processes models of sCCP can therefore be considered to be reaction-centric and in that they are similar to other rule-based formalisms such as the κ-calculus [VFF+ 07] and BIOCHAM [CRCD+ 04].

9

Discussion

The three main abstractions for mapping chemical equations to process algebras are molecule-as-process, species-as-process, and reaction-as-process. We have

20

M. Calder and J. Hillston

reaction(r1 , [A, B], [C]) : − ask(A > 0 ∧ B > 0). (tell(A $= A − 1) tell(B $= B − 1) tell(C $= C + 1)) reaction(r2 , [C], [A, B]) : − ask(C > 0). (tell(C $= C − 1) tell(A $= A + 1) tell(B $= B + 1)) reaction(r3 , [B], [D]) : − ask(B > 0). (tell(B $= B − 1) tell(D $= D + 1)) reaction(r4 , [D, E], [B]) : − ask(D > 0 ∧ E > 0). (tell(D $= D − 1) tell(E $= E − 1) tell(B $= B + 1)) reaction(r5 , [], [E]) : − (tell(E $= E + 1))

5 reaction system : − reaction(r1 , [A, B], [C]) reaction(r2 , [C], [A, B]) reaction(r3 , [B], [D]) reaction(r4 , [D, E], [B]) reaction(r5 , [], [E]) Fig. 17. Example pathway: sCCP model

. .. . ..

. .. r3

(A=1, B=1, C=0, D=0 , E=2)

r1

(A=1, B=0, C=0, D=1 , E=2)

r5

r2

r5

r4

r3

(A=1, B=1, C=0, D=0 , E=1)

(A=0, B=0, C=1, D=0 , E=2)

r5

r1

(A=1, B=0, C=0, D=1 , E=1)

r5

r2

r4

(A=1, B=1, C=0, D=0 , E=0)

(A=0, B=0, C=1, D=0 , E=1)

r5

r1

r5 r3

(A=1, B=0, C=0, D=1 , E=0)

r2

(A=0, B=0, C=1, D=0 , E=0)

Fig. 18. State space of the sCCP model of the example

further deﬁned four styles: reagent-centric, pathway-centric, reactant-centric, and reaction-centric. We have presented reactant-centric π-calculus and Beta-binders models, and (individuals-based) reagent-centric PEPA models as examples of the the molecule-as-process abstraction, (population-based) pathway-centric PEPA and (population-based) reagent-centric Bio-PEPA models as examples of the species-as-process abstraction, and a reaction-centric sCCP model as an example of the reaction-as-process abstraction.

Process Algebra Modelling Styles for Biomolecular Processes

21

The styles of modelling supported by a process algebra is strongly inﬂuenced by the form of synchronisation available. Whilst languages with multiway synchronisation are capable of representing models in reagent-centric, reactantcentric or pathway-centric style, the same is not true for languages with conjugate actions and binary synchronisation. These languages cannot generally represent reactions in the reagent-centric style. Only ﬁrst degree, or unary, reactions could be modelled in this style in these languages. In the example considered we have only considered reactions with degree one and two — indeed there are thermodynamic arguments for restricting consideration to such reactions if we wish to be faithful to biochemistry. However, abstractions which lead to higher degree reactions are often applied by biologists for a variety of reasons. For example, consider the enzyme-enabled association of two smaller molecules (A and B) into a complex C. In terms of elementary reactions this might proceed as follows: A+B+E

r2 ←→

r1

A + B:E −→r3 C + E

where E is the enzyme and B:E is a complex formed from B and the enzyme. E This could abstracted as A + B −→r C. The abstraction has the advantage that the number of reagents considered in the transformation is reduced, and that the number of reaction rates which have to be measured, estimated or ﬁtted is cut from three to one. Moreover this is typically more consistent with what can be observed in the lab as r1 , r2 r3 . It may not even be known whether the enzyme binds with A or B, leaving uncertainty about how to model the reaction without the abstraction. However, representing this in even the reactant-centric style requires three-way synchronisation, and four-way synchronisation in the reagentcentric style, assuming that the enzyme is modelled as both a reactant and a product in the abstracted reaction. In other words, it is not possible to support modelling such biological abstractions using strictly binary synchronisations. As with reagent-centric style, reaction-centric style seems to implicitly assume a multi-way synchronisation. However note that in the way that this style is captured in sCCP, the only process algebra that currently supports reactioncentric modelling to the best of our knowledge, the requirement is not so strong. What is needed is atomic multi-way composition of updates to the constraint store, but this is not necessarily a synchronisation. Whilst sCCP is the only process algebra supporting reaction-centric, or reaction-as-process, modelling, conversely it is diﬃcult to see sCCP being used to construct models in any of the other styles or abstractions. In process algebras with conjugate actions, each partner in an action/reaction must be assigned an input/output role. In general this will be rather arbitrary and somewhat artiﬁcial from the perspective of the biochemistry. Consider the reaction r1 in our example. When A and B form the complex C there does not appear to be a natural way to choose which of A and B should receive input and which provide output. Furthermore, reactions of degree one, such as r5 in the example, must be represented as a τ action. This means that the textual representation of the model does not clearly articulate the biologists’ notion of

22

M. Calder and J. Hillston

the system. This problem becomes even worse at the level of the state space where all transitions are labelled τ and information about the reactions that gave rise to them is lost. If we consider the contrast between population-based and individuals-based modelling we can observe that population-based modelling is more compact both from the point of view of the textual model expression and the underlying state space. This means that for such models it can be feasible to use explicit state space representations and the analysis techniques associated with them such as model checking, equivalence checking and numerical analysis of the continuous time Markov chain. Of course, such techniques reply on the state space being ﬁnite. In contrast individuals-based modelling has a clear association with stochastic simulation as proposed by Gillespie [GP06]. These models can be used in association with explicit state space techniques, such as those listed above, but only for very small systems or in combination with abstractions such as the assumption of single molecules, as discussed in Section 4.1. In the PEPA and Bio-PEPA models, as a consequence of the two level grammar used to deﬁne these languages as compositions of sequential components, the number of system components is constant, regardless of whether individualsbased or population-based. This matches the species-as-process abstraction since the possible species of the pathway will be known and ﬁxed and is particularly natural in the population-based modelling where the state of the system is a count for each species. In contrast, in the π-calculus and Beta-binder models, which are without the syntactic restriction, the number of system components ﬂuctuates throughout system evolution. This is in keeping with the moleculesas-processes abstraction since we would expect the visible molecules within a system to change as complexes are formed and dissociated etc. In sCCP, based on the reaction-as-process abstraction, the number of species is ﬁxed as the variables in the variable store remain ﬁxed. Here as in the PEPA/Bio-PEPA population-based modelling the state of the system is captured in terms of the number of each species so each species must always be present, even if to record that its current count is zero. The conservative nature of the PEPA/Bio-PEPA models (in terms of number of components, and ﬁxed number of levels) also means that the state space underlying such models is necessarily ﬁnite. This is not the case in the other process algebras as we have seen. It can be argued that if we consider the example as presented there is the potential for unbounded numbers of E via reaction r5 and π-calculus, beta binder and sCCP correctly capture this. But on the other hand, in a biological system unbounded growth like this will lead to cell death, and when we introduced the example we explained that this reaction would be used as an abstraction of some more complex, but bounded, situation. The beta binders and sCCP formalisms do oﬀer language mechanisms which allow the number of E to remain bounded by introducing guards on the reaction, but there is no such possibility in the π-calculus. In this paper we have focussed on the standard discrete state spaces. However analysis based on these state spaces is rarely feasible. Therefore for all the

Process Algebra Modelling Styles for Biomolecular Processes

23

languages there are alternative semantics given by ordinary diﬀerential equations (population-based) and/or Gillespie simulations (individuals-based). The discrete state space does of course form the basis of the Gillespie simulation but it is never considered explicitly and the oﬀered semantics avoid the construction. Additionally, PEPA and Bio-PEPA support an alternative representation, which is based on an explicit discrete state space but seeks to avoid the state space explosion. Rather than states representing the count of molecules of each species, the states represent the current level of concentration for each species. In other words, the range of possible concentration values is discretised into intervals, and these intervals constitute the states of the CTMC. In such models the stochastic element of Gillespie’s approach is retained but the resulting CTMCs can be considerably smaller. Keeping the state space manageable means that the CTMCs can be solved explicitly and the repeated runs necessitated by stochastic simulation are avoided. Further, in addition to quantitative analysis on the CTMC, analysis by model checking of stochastic properties is possible, as illustrated in [CVOG06] or [HKNT06].

10

Conclusions

As highlighted by Regev and Shapiro computational abstractions have already brought considerable beneﬁt to the study of biological phenomena [RS01]. For example the DNA-as-string abstraction has been hugely successful and allowed signiﬁcant leaps forward. In the context of biomolecular processes the potential beneﬁt seems equally large. However further work is needed to assess the abstractions that are on oﬀer, and their suitability to the systems under study. Research in this direction has been enthusiastically taken up by theoretical computer scientists as witnessed by the plethora of formal languages currently proposed for modelling such systems. In this paper we have aimed to extract the general paradigms of expression which underlie process algebras which aim to model biomolecular processes. We have discovered that there are genuine diﬀerences in the form of expression used, and this can impact on the form of analysis that is readily applied. In the long term all research on formal description techniques for biomolecular systems has the objective of attracting biological users, and contributing to the growing body of knowledge on how cells function. However in the medium term we need to develop closer links with biologists, not only as users of our formal description techniques, but also in the important work of evaluating them.

References [AH90] [BMMT06]

Alur, R., Henzinger, T.A.: Reactive modules. Formal methods in System Design 15(1), 7–48 (1990) Barbuti, R., Maggiolo-Schettini, A., Milazzo, P., Troina, A.: A Calculus of Looping Sequences for Modelling Microbiological Systems. Fundamenta Informaticae 72(1-3), 21–35 (2006)

24

M. Calder and J. Hillston

[BJG96]

[Bor06]

[BP08] [CGH06]

[Car04]

[Car08] [CPR+ 04]

[CG09] [CH08] [CRCD+ 04]

[CVOG06]

[VFF+ 07]

[DPPQ06]

[GP06]

[HKNT06]

[Hil96] [Hoa85]

Brim, L., Jacquet, J.-M., Gilbert, D.: A process algebra for synchronous concurrent programming. In: Hanus, M., Rodr´ıguez-Artalejo, M. (eds.) ALP 1996. LNCS, vol. 1139, pp. 165–178. Springer, Heidelberg (1996) Bortolussi, L.: Stochastic concurrent constraint programming. In: Proceedings of QAPL 2006: 4th International workshop on quantitative aspects of programming languages, vol. 164, pp. 65–80 (2006) Bortolussi, L., Policriti, A.: Modelling biological systems in stochastic constraint programming. Constraints 13, 66–90 (2008) Calder, M., Gilmore, S., Hillston, J.: Modelling the inﬂuence of RKIP on the ERK signalling pathway using the stochastic process algebra PEPA. In: Priami, C., Ing´ olfsd´ ottir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 1–23. Springer, Heidelberg (2006) Cardelli, L.: Brane Calculus. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 257–278. Springer, Heidelberg (2005) Cardelli, L.: On process rate semantics. Theoretical Computer Science 391(1), 190–215 (2008) Cardelli, L., Panina, E.M., Regev, A., Shapiro, E., Silverman, W.: BioAmbients: An Abstraction for Biological Compartments. Theoretical Computer Science 325(1), 141–167 (2004) Ciocchetta, F., Guerriero, M.L.: Modelling Biological Compartments in Bio-PEPA. ENTCS 227, 77–95 (2009) Ciochetta, F., Hillston, J.: Bio-PEPA: a framework for modelling and analysis of biological systems. Theoretical Computer Science (to appear) Chabrier-Rivier, N., Chiaverini, M., Danos, V., Fages, F., Sch¨achter, V.: Modeling and querying biomolecular interaction networks. Theoretical Computer Science 325(1), 25–44 (2004) Calder, M., Vyshemirsky, V., Orton, R., Gilbert, D.: Analysis of signalling pathways using Continuous Time Markov Chains. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 44–67. Springer, Heidelberg (2006) Danos, V., Feret, J., Fontana, W., Harmer, R., Krivine, J.: Rule-based modelling of cellular signalling. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 17–41. Springer, Heidelberg (2007) Degano, P., Prandi, D., Priami, C., Quaglia, P.: Beta-binders for biological quantitative experiments. Electronic Notes in Computer Science 164, 101–117 (2006) Gillespie, D., Petzold, L.: Numerical Simulation for Biochemical Kinetics. In: System Modelling in Cellular Biology. MIT Press, Cambridge (2006) Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic model checking of complex biological pathways. In: The Proceedings of 4th International Workshop on Computational Methods in Systems Biology 2006, Trento, Italy, October 18-19 (2006) Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press, Cambridge (1996) Hoare, C.A.R.: Communicating Sequential Processes. Prentice-Hall, Englewood Cliﬀs (1985)

Process Algebra Modelling Styles for Biomolecular Processes [Mil80] [Mil99] [Pri95] [RS01] [RSS01]

[TK08]

[TDG09] [V07]

25

Milner, R.: A Calculus for Communicating Systems. LNCS, vol. 92. Springer, Heidelberg (1980) Milner, R.: Communicating and Mobile Systems: the π-Calculus. Cambridge University Press, Cambridge (1999) Priami, C.: Stochastic π-calculus. The Computer Journal 38, 578–589 (1995) Regev, A., Shapiro, E.: Cellular abstractions: cells as computation. Nature 419, 343 (2001) Regev, A., Silverman, W., Shapiro, E.: Representation and simulation of biochemical processes using π-calculus process algebra. In: Paciﬁc Symposium on Biocomputing 2001 (PSB 2001), pp. 459–470 (2001) Tymchyshyn, O., Kwiatkowska, M.: Combining intra- and inter-cellular dynamics to investigate intestinal homeostasis. In: Fisher, J. (ed.) FMSB 2008. LNCS (LNBI), vol. 5054, pp. 63–76. Springer, Heidelberg (2008) Tribastone, M., Duguid, A., Gilmore, S.: The PEPA Eclipse Plug-in. Performance Evaluation Review 36(4), 28–33 (2009) Versari, C.: A Core Calculus for a Comparative Analysis of Bioinspired Calculi. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 411–425. Springer, Heidelberg (2007)

Simple, Enhanced and Mutual Mobile Membranes Bogdan Aman and Gabriel Ciobanu Romanian Academy, Institute of Computer Science, Ia¸si, Romania A.I.Cuza University, 700506 Ia¸si, Romania [email protected], [email protected]

Abstract. The operations governing the movement of biological membranes are endocytosis and exocytosis. New models of computation are inspired by these biological operations. In this paper we present the models deﬁned by simple, enhanced and mutual mobile membranes, together with their biological motivations. Some results concerning their computational power are presented, including the ﬁrst universality result for mutual mobile membranes. In the case of simple and enhanced mobile membranes, we improve the existing results by reducing the number of membranes needed to get computational universality.

1

Introduction

Simple, enhanced and mutual mobile membranes represent new variants of membrane systems. Membrane systems (also called P systems) were introduced in [16]; standard P systems and several variations are presented in the monograph [17]. Membrane systems were introduced as distributed, parallel and nondeterministic computing models inspired by the compartments of eukaryotic cells and by their biochemical reactions. The cellular components are formally represented in the deﬁnition of membrane systems. The structure of the cell is represented by a set of hierarchically embedded regions, each one delimited by a surrounding boundary (called membrane), and all of them contained inside an external special region called the skin membrane. The molecular species (ions, proteins, etc.) ﬂoating inside cellular compartments are represented by multisets of objects described by means of symbols or strings over a given alphabet, objects which can be modiﬁed or communicated between adjacent compartments. Chemical reactions are represented by evolution rules given in the form of rewriting rules which operate on the objects, as well as on the compartmentalized structure (by dissolving, dividing, creating, or moving membranes). A membrane system can perform computations in the following way: starting from an initial conﬁguration which is deﬁned by the multiset of objects initially placed inside the compartmentalized structure, the system evolves by applying the evolution rules of each membrane in a nondeterministic and maximally parallel manner. A rule is applicable when all the objects that appear in its left hand side are available in the region where the rule is placed. The maximal C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 26–44, 2009. c Springer-Verlag Berlin Heidelberg 2009

Simple, Enhanced and Mutual Mobile Membranes

27

parallelism of rule application means that every rule that is applicable inside a region has to be applied in that region. A halting conﬁguration is reached when no rule is applicable. The result is represented by the number of objects from a speciﬁed region. Several variants of membrane systems are inspired by diﬀerent aspects of living cells (symport and antiport-based communication through membranes, catalytic objects, membrane charge, etc.). Their computing power and eﬃciency have been investigated using the approaches of formal languages and grammars, register machines and complexity theory. An updated bibliography can be found at the webpage http://ppage.psystems.eu A ﬁrst deﬁnition of mobile P systems is given in [21] with rules coming from mobile ambients [5]. Inspired by the operations of endocytosis and exocytosis, namely moving a membrane inside a neighbouring membrane (endocytosis) and moving a membrane outside the membrane where it is placed (exocytosis), the P systems with mobile membranes are introduced in [14] as a variant of P systems with active membranes [17]. We use simple mobile membranes instead of P systems with mobile membranes. The computational power of simple mobile membranes is treated in [12,14]: Turing completeness is obtained by using nine membranes together with the operations of endocytosis and exocytosis [14], while only four mobile membranes are enough using additional contextual evolution rules [12]. In this paper we look at certain biological phenomena which motivate and inspire new speciﬁc rules in simple mobile membranes. Endocytosis is a general term for a group of processes that bring macromolecules, large particles, small molecules, and even small cells into another cell. There are three types of endocytosis: phagocytosis (“cellular eating”), pinocytosis (“cellular drinking”), and receptor-mediated endocytosis in which the membrane infolds around materials from the environment, forming a small pocket. The pocket deepens, forming a vesicle which separates from the membrane and migrates with its contents to the cell’s interior. While pinocytosis can be modelled using communication rules of usual P systems, there is no rule capable to model the process of engulﬁng a cell by another one in phagocytosis. This is the reason why we deﬁne the enhanced mobile membranes in Subsection 2.2; an example on how the new rules work is also presented. The enhanced mobile membranes represent a variant of simple mobile membranes; they have been proposed in [3] for describing some biological mechanisms of the immune system. The operations governing the mobility of the enhanced mobile membrane systems are endocytosis (endo), exocytosis (exo), enhanced endocytosis (fendo) and enhanced exocytosis (fexo). The computational power of the enhanced mobile membranes using these four operations was studied in [13] where it is proved that twelve membranes can provide the computational universality. It is worth noting that unlike the results for simple mobile membranes, the context-free evolution of objects is not used in proving any of these results.

28

B. Aman and G. Ciobanu

Receptor-mediated endocytosis is used by animal cells to capture speciﬁc macromolecules from the cell’s environment. This process depends on receptor proteins, i.e., integral membrane proteins that can bind to a speciﬁc molecule in the cell’s environment. The uptake process is similar to nonspeciﬁc endocytosis. However, in receptor-mediated endocytosis, the receptor proteins at particular sites on the extracellular surface of the plasma membrane bind to speciﬁc substances. These sites are called coated pits because they form a slight depression in the plasma membrane. The cytoplasmic surface of a coated pit is coated by proteins, such as clathrin. Strengthened and stabilized by clathrin molecules, this vesicle carries the macromolecule into the cell [23]. SNARE-mediated exocytosis is the movement of materials out of a cell via vesicles. SNARES (Soluble NSF Attachment Protein Receptor)) located on the vesicles (v-SNARES) and on the target membranes (t-SNARES) interact to form a stable complex that holds the vesicle very close to the target membrane. There is no rule capable to model the mutual agreement between membranes for the receptor-mediated endocytosis and SNARE-mediated exocytosis. This is the reason why we deﬁne the mutual mobile membranes in Subsection 2.3; an example on how the new rules work is also presented. The mutual mobile membranes represent a variant of simple mobile membranes in which the endocytosis and exocytosis work whenever the involved membranes “agree” on the movement; this agreement is described by using dual objects a and a in the involved membranes. The operations governing the mobility of the mutual mobile membranes are mutual endocytosis (mutual endo), and mutual exocytosis (mutual exo). In this paper we study the computational power of simple, enhanced and mutual mobile membranes. For simple mobile membranes we obtain the computational universality by using three membranes, and in this way improving the result presented in [12] where four membranes are used. For enhanced mobile membranes we obtain the computational universality by using nine membranes, thus improving the result from [13] where twelve membranes are used. For mutual mobile membranes we show that by using dual objects a and a in the involved membranes, only seven membranes are enough to obtain the computational universality. The structure of the paper is as follows. In Section 2 we formally deﬁne the simple, enhanced and mutual mobile membranes, and give biological motivations for the enhanced and mutual mobile membranes. Section 3 contains a ﬁrst universality result for mutual mobile membranes, and improvements of the existing results for simple and enhanced mobile membranes. Section 4 presents related results in P systems with active membrane from which the simple mobile membranes originate. Conclusions and references end the paper.

2

Mobile Membranes

In this section we deﬁne the simple, enhanced and mutual mobile membranes, describing some biological phenomena inspiring their rules.

Simple, Enhanced and Mutual Mobile Membranes

2.1

29

Simple Mobile Membranes

Definition 1 ([14]). A simple mobile membrane is a construct Π = (V, H, μ, w1 , . . . , wn , R) where: n ≥ 1 (the initial degree of the system); V is an alphabet (its elements are called objects); H is a ﬁnite set of labels for membranes; μ ⊂ H × H describes the membrane structure, such that (i, j) ∈ μ denotes that the membrane labelled by j is contained in the membrane labelled by i; we distinguish the external membrane (usually called the “skin” membrane) and several internal membranes; a membrane without any other membrane inside it is said to be elementary; 5. w1 , . . . , wn are strings over V , describing the multisets of objects placed in the n regions of μ; 6. R is a ﬁnite set of developmental rules, of the following forms: 1. 2. 3. 4.

object evolution ∗

(a) [a → v]m , for m ∈ H, a ∈ V , v ∈ V ; An object a placed inside a membrane labelled m evolves into a multiset of objects v. endocytosis (b) [a]h [ ]m → [[b]h ]m , for h, m ∈ H, a, b ∈ V ; An elementary membrane labelled h enters the adjacent membrane labelled m, under the control of object a; the labels h and m remain unchanged during the process; however the object a may be modiﬁed to b during the operation; m is not necessarily an elementary membrane. exocytosis (c) [[a]h ]m → [b]h [ ]m , for h, m ∈ H, a, b ∈ V ; An elementary membrane labelled h is sent out of a membrane labelled m, under the control of object a; the labels of the two membranes remain unchanged, but the object a of membrane h may be modiﬁed during this operation; membrane m is not necessarily elementary. The rules are applied according to the following principles: 1. Rules are applied in parallel, non-deterministically choosing the rules, the membranes, and the objects in such a way that the parallelism is maximal; this means that in each step we apply a certain set of rules such that no further rule can be added to the set. 2. The membrane m from the rules of type (a) − (c) is said to be passive, while the membrane h is said to be active. In any step of a computation, any object and any active membrane can be involved in at most one rule. However, the passive membranes can be used by several rules at the same time. In a rule [a → v]m of type (a), object a is active, while membrane m is passive.

30

B. Aman and G. Ciobanu

3. When a membrane is moved across another membrane, by endocytosis or exocytosis, its whole contents (its objects) are moved; the inner objects evolve ﬁrst (if rules are applicable for them), and then any membrane is moved with the contents as obtained after its internal evolution. 4. If a membrane exits the system (by exocytosis), then its internal evolution stops, even if there are rules of type (a) which could be applied. 5. The objects and membranes which do not evolve at a given step are passed unchanged to the next conﬁguration of the system. 2.2

Enhanced Mobile Membranes

The enhanced mobile membranes have been introduced in [3] for describing some biological mechanisms of the immune system. The presentation of the immune system is taken from [10], a book which is revised every few years to keep the pace with the new discoveries in this ﬁeld. The cells of the immune system work together with diﬀerent proteins to seek out and destroy anything foreign or dangerous which enters our body. It takes some time for the immune cell to be activated, but once this happens there are very few hostile organisms having a chance. There are several types of immune cells, each of them with its own strength and weakness. Some seek out and engulf the invaders, while other destroy the infected or mutated body cells. A type of immune cells are the B cells which have the ability to release special proteins called antibodies which mark intruders in order to be destroyed by macrophages. The immune system has also the ability to produce some cells able to remember enemies which it fought in the past. In this way, once the immune system recognizes an invader it attacks more quickly and strongly against it. Dendritic cells can engulf bacteria, viruses, and other cells. Once a dendritic cells engulfs a bacterium, it dissolves this bacterium and places portions of

Fig. 1. Immune System Mechanisms [10]

Simple, Enhanced and Mutual Mobile Membranes

31

bacterium proteins on its surface (see Figure 1). These surface markers serve as an alarm to other immune cells, namely helper T cells, which then infer the form of the invader. This mechanism makes sensitive the T cells to recognize the antigens or other foreign agents which triggers a reaction of the immune system. Antigens are often found on the surface of bacterium and viruses. New rules are introduced according to this biological example. We deﬁne a new variant of mobile membranes, namely the enhanced mobile membranes, originally introduced in [3]. The multiset u is the one indicating the membrane which initializes the move in the rules of type (b) − (e). Definition 2 ([3]). An enhanced mobile membrane is a construct = (V, H, μ, w1 , . . . , wn , R), where: 1. n, V , H, μ, w1 , . . . , wn are as in Deﬁnition 1; 2. R is a ﬁnite set of developmental rules of the following forms: local evolution (a) [ [u → v]m ]h for h, m ∈ H, u ∈ V + , v ∈ V ∗ ; These rules are called local because the evolution of a multiset of objects u of membrane m is possible only when membrane m is inside membrane h. If the restriction of nested membranes is not imposed, that is, the evolution of the multiset of objects u in membrane m is allowed wherever membrane m is placed, then we say that we have a global evolution rule, and write it simply as [u → v]m . endocytosis (b) [uv]h [v ]m → [[w]h w ]m for h, m ∈ H; u ∈ V + , v, v , w, w ∈ V ∗ ; An elementary membrane labelled h enters the adjacent membrane labelled m, under the control of the multisets of objects uv and v . The labels h and m remain unchanged during this process; however the multisets of objects uv and v are replaced with the multisets of objects w and w , respectively. exocytosis + ∗ (c) [[uv]h v ]m → [w]h [w ]m , for h, m ∈H; u ∈ V , v, v , w, w ∈ V ; An elementary membrane labelled h is sent out of a membrane labelled m, under the control of the multisets of objects uv and v . The labels of the two membranes remain unchanged, but the multisets of objects uv and v are replaced with the multisets of objects w and w , respectively. enhanced endocytosis + (d) [v]h [uv ]m→[[w]h w ]m for h, m∈H, u ∈ V , v, v , w, w ∈ V ∗ ; An elementary membrane labelled h is engulfed into the adjacent membrane labelled m, under the control of the multisets of objects uv and v. The labels h and m remain unchanged during the process; however, the multisets of objects uv and v are transformed into the multisets of objects w and w, respectively. The eﬀect of this rule is similar to the eﬀect of rule (b); the diﬀerence is that the movement is not controlled by a multiset of objects inside the moving membrane h, but by a multiset

32

B. Aman and G. Ciobanu

of objects uv placed inside the membrane m which engulfs membrane h. This means that the membrane which initiates the movement is membrane m, and not the membrane h as in rule (b). enhanced exocytosis (e) [[v]h uv ]m→[w]h [w ]m for h, m ∈ H, u ∈ V + , v, v , w, w ∈ V ∗ ; An elementary membrane labelled h is pushed out of a membrane labelled m under the control of the multisets of objects uv and v. The labels of the two membranes remain unchanged; however, the multisets of objects uv and v evolve into the multisets of objects w and w, respectively. The eﬀect of this rule is similar to the one of rule (c); the diﬀerence is that the movement is not controlled by an object inside the moving membrane h, but by a multiset of objects uv placed inside the membrane m which expels membrane h. This means that the membrane which initiates the movement is membrane m, and not the membrane h as in rule (c). The rules of enhanced mobile membranes are applied according to the principles of simple mobile membranes. Using the rules of the enhanced mobile membranes we can describe the immune system mechanisms of Figure 1. We associate a membrane to each cell, and objects to the signals, states and parts of molecules. For the steps done by the dendritic cells presented in Figure 1, we use the following encodings: – dendritic cell: [eat]DC An immature dendritic cell is willing to eat any bacterium it encounters, so we translate it into a membrane labelled by DC which has inside an object eat used to engulf the bacterium. – bacterium cell: [antigen]bacterium A bacterium cell contains antigen so we simply represent it as a membrane labelled by bacterium containing a single object antigen that encodes the information of the bacterium. – lymph node: [ ]lymph node The lymph node is the place where the mature dendritic cells migrate in order to start the immune response, so we translate it into a membrane labelled by lymph node. Using these membranes, we describe the system as follows (here body stands for the body skin): [[eat]DC [ ]lymph node ]body [antigen]bacterium The evolution is described by following rules: * [antigen]bacterium [ ]body → [[antigen]bacterium ]body A bacterium enters through the body skin by performing an endocytosis rule in order to infect the body. The bacterium contains an object antigen which represent its signature.

Simple, Enhanced and Mutual Mobile Membranes

33

* [eat]DC [ ]bacterium → [eat[ ]bacterium ]DC Once an immature dendritic cell becomes sibling to a bacterium, it “eats” the bacterium by performing an enhanced endocytosis rule. Until now the bacterium has controlled its own movement; in this step its movement becomes controlled by the dendritic cell which engulfs it. * [[antigen]bacterium ]DC → [antigen]DC Once the bacterium is engulfed into the dendritic cell, it is dissolved and its content is released into the dendritic cell. * [antigen]DC [ ]lymph node → [[antigen]DC ]lymph node Once the dendritic cell contains parts of the antigen, it enters the lymph node in order to activate a special class of T cells, namely the helper T cells. * [[eat]DC ]lymph node → [[ ]DC ]lymph node Once the dendritic cell enters the lymph node, it matures and the capacity to engulf bacteria disappears; the eat object is consumed. 2.3

Mutual Mobile Membranes

In a receptor-mediated endocytosis a cell engulfs a particle of low-density lipoprotein (LDL) from the outside [23]. To do this, the cell uses receptors that speciﬁcally recognize and bind to the LDL particle. The receptors are clustered together. An LDL particle contains one thousand or more cholesterol molecules. A monolayer of phospholipid surrounds the cholesterol and its embedded with proteins called apoB. This apoB proteins are speciﬁcally recognized by receptors on the cell membrane. The receptors of the coated pit bind to the apoB proteins of the LDL particle. The pit is reenforced by a lattice-like network of proteins called clathrin. Additional clathrin molecules are added to the lattice which eventually pinches oﬀ apart from the membranes. SNARE-mediated exocytosis is the movement of materials out of a cell via vesicles. SNARES located on the vesicles (v-SNARES) and SNARES located on the target membranes (t-SNARES) interact to form a stable complex that holds the vesicle very close to the target membrane as in Figure 3.

Fig. 2. Receptor-Mediated Endocytosis [23]

34

B. Aman and G. Ciobanu

Fig. 3. SNARE-Mediated Exocytosis

Starting from these biological examples we see the necessity to introduce new rules. The rules of enhanced mobile membranes allow a membrane to enter, exit, to engulf or to push out another membrane. The second membrane just undergoes the movement; no permission is required from the second membrane which may not even be aware that a movement involving it has taken place. Following an approach described initially in [4], we introduce a new variant of mobile membranes called mutual mobile membranes. In mutual mobile membranes, a movement takes place only if the involved membranes agree on the movement. This can be described by means of objects a and co-objects a present in the membranes involved in such a movement. Since we have the equality a = a, we have that mutual endocytosis is the same as mutual enhanced endocytosis and mutual exocytosis is the same as mutual enhanced exocytosis. The mutual mobile membranes are deﬁned as follows: Definition 3 ([4]). A mutual mobile membrane is a construct = (V, H, μ, w1 , . . . , wn , R), where: 1. n, V , H, μ, w1 , . . . , wn are as in Deﬁnition 1; 2. R is a ﬁnite set of developmental rules of the following forms: local evolution ∗

(a) [ [u → v]m ]h for h, m ∈ H, u, v ∈ V ; These rules are called local because the evolution of a multiset of objects u of membrane m is possible only when membrane m is inside membrane h. If the restriction of nested membranes is not imposed, that is, the evolution of the multiset of objects u in membrane m is allowed wherever membrane m is placed, then we say that we have a global evolution rule, and write it simply as [u → v]m . mutual endocytosis

Simple, Enhanced and Mutual Mobile Membranes

35

(b) [uv]h [uv ]m → [ [w]h w ]m for h, m ∈ H, u, u ∈ V + , v, v , w, w∈ V ∗ ; An elementary membrane labelled h enters the adjacent membrane labelled m under the control of the multisets of objects uv and uv . The labels h and m remain unchanged during this process; however the multisets of objects uv and uv are replaced with the multisets of objects w and w , respectively. mutual exocytosis + (c) [uv [uv]h ]m → [w]h [w ]m for h, m ∈ H, u, u ∈ V , v, v , w, w∈ V ∗ ; An elementary membrane labelled h exits a membrane labelled m, under the control of the multisets of objects uv and uv . The labels of the two membranes remain unchanged, but the multisets of objects uv and uv are replaced with the multisets of objects w and w , respectively. The rules of the mutual mobile membranes are applied according to principles of simple mobile membranes. An object u indicates the membrane which initializes the move in the rules of type (b) − (c), while an object u indicates the membrane which accepts the movement. Using the rules of the mutual mobile membranes we can describe the receptormediated endocytosis of Figure 2. We associate a membrane to each cell, and objects to the signals, states and parts of molecules. For the steps done by the cells presented in Figure 2, we use the following encodings: – LDL particle: [cholesterol . . . cholesterol apoB . . . apoB]LDL An LDL particle contains one thousand or more cholesterol molecules and some apoB proteins. – cell membrane: [receptor . . . receptor clarithin . . . clarithin]cell The cell contains receptors which are able to recognize apoB proteins and also some proteins clathrin which enforce the pit containing the receptors. Using the above membranes, and the equality apoB = receptor, we can describe the membrane system as follows: [cholesterol . . . apoB . . .]LDL [apoB . . . clarithin . . .]cell The evolution is described by applying a rule of type (b): [cholesterol . . . apoB . . .]LDL [apoB . . . clarithin . . .]cell → → [ [cholesterol . . . apoB . . .]LDL apoB . . . clarithin . . .]cell

3

Computability Power of Mobile Membranes

In this section we present some existing results and also new results related to the computational power of simple, enhanced and mutual mobile membranes. First we present some notations from the ﬁeld of formal languages which are used throughout this section. More notions from formal languages can be found in [7] and [22]. For an alphabet V = {a1 , . . . , an }, we denote by V ∗ the set of all strings over V ; λ denotes the empty string. V ∗ is a monoid with λ as its unit element. For

36

B. Aman and G. Ciobanu

a string x ∈ V ∗ , |x|a denotes the number of occurrences of symbol a in x. A multiset over V is represented by a string over V (together with all its permutations), and each string precisely identiﬁes a multiset. For an alphabet V , the Parikh vector is ψV : V ∗ → Nn with ψV (x) = (|x|a1 , . . . , |x|an ), for all x ∈ V ∗ . For a language L, the Parikh vector is ψV (L) = {ψV (x) | x ∈ L}, while for a family F L of languages, it is P sF L = {ψV (L) | L ∈ F L}. A matrix grammars with appearance checking is a construct G = (N, T, S, M, F ) where N , T are disjoint alphabets of non-terminals and terminals, S ∈ N is the axiom, M is a ﬁnite set of matrices of the form (A1 → x1 , . . . , An → xn ) of context-free rules, and F is a set of occurrences of rules in M . For w, z ∈ (N ∪T )∗ , we write w ⇒m z if there is a matrix (A1 → x1 , . . . , An → xn ) in M and the strings wi ∈ (N ∪ T )∗ , 1 ≤ i ≤ n + 1, such that w = w1 , z = wn+1 , and for all i, 1 ≤ i ≤ n, either (1) wi = wi Ai wi , wi+1 = wi xi wi , for some wi , wi ∈ (N ∪ T )∗ , or (2) wi = wi+1 , Ai does not appear in wi , and the rule Ai → xi appears in F. The language generated by G is L(G) = {x ∈ T ∗ | S ⇒∗ x}. A matrix grammar in the strong binary normal form is a construct G = (N , T , S, M , F ), where N = N1 ∪ N2 ∪ {S, #}, with these three sets mutually disjoint, two distinguished symbols B (1) , B (2) ∈ N2 , and the matrices in M of one of the following forms: (1) (2) (3) (4)

(S → XA), with X ∈ N1 , A ∈ N2 , (X → Y, A → x), with X, Y ∈ N1 , A ∈ N2 , x ∈ (N2 ∪ T )∗ , (X → Y, B (j) → #), with X, Y ∈ N1 , j = 1, 2, (X → λ, A → x), with X ∈ N1 , A ∈ N2 , x ∈ T ∗ .

If we ignore the empty string when comparing languages, then the rules of type (4) are of the form (X → a, A → x), with X ∈ N1 , a ∈ T , A ∈ N2 , x ∈ T ∗ . 3.1

Simple Mobile Membranes

The computational power of simple mobile membranes is treated in [14]. P sM Mn (levol, endo, exo) denotes the family of all sets P s(Π) generated by systems using local evolution rules, together with endocytosis and exocytosis rules and at most n membranes. If the number of membrane is not bounded, this is denoted by P sM M∗ (levol, endo, exo). When global evolution rules are used, levol is replaced by gevol. If a type of rules is not used, then its name is omitted from the list of parameters. The number of membranes does not increase during the computation, but it can decrease by sending membranes out of the skin. The following result establishes an universality result using nine membranes and the operations of endocytosis and exocytosis: Theorem 1 ([14]). P sM M9 (endo, exo) = P sRE. A strengthening of the previous universality result is: Corollary 1 ([14]). P sM M∗ (endo, exo) = P sM Mn (endo, exo) = P sM Mn (gevol, endo, exo) = P sM Mn (levol, endo, exo) = P sRE, for all n ≥ 9.

Simple, Enhanced and Mutual Mobile Membranes

37

An improvement of the result presented in Theorem 1 is: Theorem 2 ([12]). P sM M4 (gevol, endo, exo) = P sRE. We improve the previous result by decreasing the number of membranes to three. Theorem 3. P sM M3 (levol, endo, exo) = P sRE. Proof. Consider a matrix grammar G = (N, T, S, M, F ) in the improved strong binary normal form (hence with N = N1 ∪ N2 ∪ {S; #}), having n1 matrices of types (2) and (4) (that is, not used in the appearance checking mode), and n2 matrices of type (3) (with appearance checking rules). Let B (1) and B (2) be the two objects in N2 for which we have rules B (j) → # in matrices of M . The matrices of the form (X → Y, B (j) → #) are labelled by mi , with i ∈ labj , for j ∈ {12}, such that lab1 , lab2 , and lab0 = {1, . . . , n1 } are mutually disjoint sets. We construct a mobile membrane system Π = (V, H, μ, w1 , w2 , w3 , R, 2) of degree three, where: V = N ∪ {X, Xi,j | X ∈ N1 , 1 ≤ i ≤ n1 , 0 ≤ j ≤ n1 } ∪ {a, a | a ∈ T } ∪ {x | x ∈ (N2 ∪ T )∗ } ∪ {A, Ai,j | A ∈ N2 , 1 ≤ i ≤ n1 , 0 ≤ j ≤ n1 } H = {1, 2, 3} μ = [[ ]2 [ ]3 ]1 w2 = XA, where (S → XA) is the initial matrix of G wh = λ, for all h ∈ {1, 3} The set R of rules is constructed as follows: (i) For each (nonterminal) matrix mi : (X → Y, A → x), X, Y ∈ N1 , A ∈ N2 , x ∈ (N2 ∪ T )∗ , with 1 ≤ i ≤ n1 , we consider the rules: 1. [X]2 [ ]3 → [[Xi,0 ]2 ]3 (endo) 2. [[A]2 ]3 → [Ai,0 ]2 [ ]3 (exo) 3. [[Xi,j → Xi,j+1 ]2 ]1 , j < i (levol) 4. [[Ai,j → Ai,j+1 ]2 ]1 , j < i (levol) 5. [[Ai,i Xi,i → xY ]2 ]1 (levol) 6. [[Ai,j Xj,j → #]2 ]1 , j < i (levol) 7. [[Aj,j Xi,j → #]2 ]1 , j < i (levol) In the initial conﬁguration, we have the objects X and A corresponding to the initial matrix in membrane 2. To simulate a matrix of the above type we start by applying the endocytosis rule 1, thus replacing X with Xi,0 , followed by the exocytosis rule 2, thus replacing a single A ∈ N2 with Ai,0 . No other A ∈ N2 can be replaced until membrane 2 enters membrane 3. Rule 3 (for X) and rule 4 (for A) are used to increment the second indices of X and A. This is done to check if the indices of X and A are the same, and in this case to rewrite A according to the matrix mi . Once the indices are equal, rule 5 is applied to complete the simulation of matrix mi . If the indices of X and A are not the same, rule 6 (if the indices of X is lower than the indices of A) or rule 7 (if the indices of X is bigger than the indices of A) is applied, the computation is blocked without producing any output.

38

B. Aman and G. Ciobanu

(ii) For a terminal matrix mi : (X → a, A → x), X ∈ N1 , a ∈ T , A ∈ N2 , x ∈ T ∗ , where 1 ≤ i ≤ n1 , we use the rules 1-7, where the rule 5 is replaced by the rules: 8. [ai,i Xi,i → a Y ]1 (levol) 9. [[a ]2 ]1 → [a]2 [ ]1 (exo) Observe that simulation of a type (4) matrix is along similar steps, except that we have an a in place of Y . During the ﬁnishing stages of a type (4) simulation, we use rule 8 to replace ai,i by a , and then to rewrite it to a when sending the membrane 2 out of the skin membrane, namely membrane 1. (iii) For each matrix mi : (X → Y, B (k) → #), X, Y ∈ N1 , A ∈ N2 , where n1 + 1 ≤ j ≤ n1 + n2 , j ∈ labk , k = 1, 2, we consider the rules: 10. [X]2 [ ]3 → [[Xk ]2 ]3 , for i ∈ labk (endo) 11. [[Xk B (k) → #]2 ]3 , k = 1, 2 (levol) 12. [[Xk ]2 ]3 → [Y ]2 [ ]3 , k = 1, 2 (exo) The simulation of matrices of type (3) begins by a rule of type 10. This is followed by a rule 11 in case B (k) exists, blocking membrane 2 inside membrane 3 and the computation stops without producing any output. If no B (k) exists, then rule 12 can be used to send out membrane 2, successfully completing the simulation. 3.2

Enhanced Mobile Membranes

The operations governing the mobility of the enhanced mobile membranes are endocytosis (endo), exocytosis (exo), enhanced endocytosis (fendo) and enhanced exocytosis (fexo). The interplay between these four operations is quite powerful, and the computational power of a Turing machine is obtained using twelve membranes without using the context-free evolution of objects [13]. The family of all sets P s(Π) generated by systems of degree at most n using rules α ⊆ {exo, endo, f endo, f exo, cevol} is denoted by P sEM Mn (α). Here endo and exo represent endocytosis and exocytosis, f endo and f exo represent enhanced endocytosis and enhanced exocytosis, and cevol represents contextual evolution. The main results are the following. Theorem 4 ([13]). P sEM M12 (endo, exo, f endo, f exo) = P sRE. Theorem 5 ([13]). P sEM M3 (cevol) = P sRE. Theorem 6 ([13]). P sEM M3 (endo, exo) = P sEM M3 (f endo, f exo). We improve the result of Theorem 4 as follows: Theorem 7. P sEM M9 (endo, exo, f endo, f exo) = P sRE.

Simple, Enhanced and Mutual Mobile Membranes

39

Proof. Consider a matrix grammar G = (N, T, S, M, F ) in the improved strong binary normal form (hence with N = N1 ∪ N2 ∪ {S; #}), having n1 matrices m1 , . . . , mn1 of types (2) and (4) (that is, not used in the appearance checking mode), and n2 matrices of type (3) (with appearance checking rules). The initial matrix is m0 : (S → XA). Let B (1) and B (2) be the two objects in N2 for which we have rules B (j) → # in matrices of M . The matrices of the form (X → Y, B (j) → #) are labelled by mi , 1 ≤ i ≤ n2 with i ∈ labj , for j ∈ {12}, such that lab1 , lab2 , and lab0 = {1, 2, . . . , n1 } are mutually disjoint sets. We construct a mobile membrane system Π = (V, H, μ, w1 , . . . , w9 , R, 7) of degree nine, where: V = N ∪ T ∪ {X0i , A0i | X ∈ N1 , A ∈ N2 , 1 ≤ i ≤ n1 } ∪{Xji , Aji | 0 ≤ i, j ≤ n1 } ∪ {Xij , Xj | X ∈ N1 , j ∈ {1, 2}, 1 ≤ i ≤ n2 } H = {1, . . . , 9} μ = [[ ]7 [ ]8 [ ]9 [[ ]3 [ ]4 [ ]5 [ ]6 ]2 ]1 w7 = XA, where (S → XA) is the initial matrix of G wh = λ, for all h ∈ {1, . . . , 9}\{7}

The set R of rules is constructed as follows: (i) For each (nonterminal) matrix mi : (X → Y, A → x), X, Y ∈ N1 , A ∈ N2 , x ∈ (N2 ∪ T )∗ , with 1 ≤ i ≤ n1 , we consider the rules: 1. [X]7 [ ]8 → [[Xi,i ]7 ]8 (endo) 2. [[A]7 ]8 → [Ai,i ]7 [ ]8 (exo) 3. [Xj,i ]7 [ ]9 → [[Xj−1,i ]7 ]9 (endo) 4. [[Aj,i ]7 ]9 → [Aj−1,i ]7 [ ]9 (exo) 5. [ ]8 [X0,i ]7 → [X0,i [ ]8 ]7 (fendo) 6. [ ]9 [A0,i ]7 → [A0,i [ ]9 ]7 (fendo) 7. [ ]8 [X0,i ]7 → [#[ ]8 ]7 (fendo) 8. [[A0,i ]7 ]9 → [#]7 [ ]9 (exo) 9. [X0,i [ ]8 ]7 → [ ]8 [Y ]7 (fexo) 10. [A0,i [ ]9 ]7 → [ ]9 [x]7 (fexo) In the initial conﬁguration, we have the objects X, A corresponding to the initial matrix in membrane 7. To simulate a matrix of type (2), we start by applying the endocytosis rule 1, thus replacing X with Xi,i , followed by the exocytosis rule 2, thus replacing a single A ∈ N2 with Ai,i . Rule 3 (for X) and rule 4 (for A) are used to decrement the ﬁrst indices of X and A. This is done to check if the indices of X and A are the same, and in this case to rewrite A according to the matrix mi . By using fendo rules 5 and 6, membranes 8 and 9 enter membrane 7 replacing X0,i and A0,i with X0,i and A0,i , respectively. This is then followed by rules 9 and 10, when membranes 8 and 9 exit membrane 7 by fexo rules replacing X0,i and A0,i with Y and x, respectively. If i > j, then we obtain A0,j before X0,i . In this case, we have a conﬁguration where membrane 7 is inside membrane 9 containing A0,j . Then rule 8 is used, replacing A0,j with #, and an inﬁnite computation is obtained (rule 17). If j > i, then we obtain X0,i before A0,j .

40

B. Aman and G. Ciobanu

In this case, we reach a conﬁguration with X0,i Ak,j , k > 0 in membrane 7, and membrane 7 is in the skin membrane. Rule 3 cannot be used now, and the only possibility is to use rule 7, which leads to an inﬁnite computation. Thus, if i = j, then we can correctly simulate a matrix of type (2). (ii) For each matrix mi : (X → Y, B (k) → #), X, Y ∈ N1 , A ∈ N2 , where n1 + 1 ≤ j ≤ n1 + n2 , j ∈ labk , k = 1, 2, we consider the rules: 11. 12. 13. 14. 15.

(j)

[X]7 [ ]2 → [[Xi ]7 ]2 , j = 1, 2 (endo) (j) (j) [ ]j+2 [Xi ]7 → [Xi [ ]j+2 ]7 , j = 1, 2 (fendo) [ ]j+4 [B (j) ]7 → [#[ ]j+4 ]7 , j = 1, 2 (fendo) (j) [Xi [ ]j+2 ]7 → [ ]j+2 [Yj ]7 , j = 1, 2 (fexo) [[Yj ]7 ]2 → [Y ]7 [ ]2 , j = 1, 2 (exo)

The simulation of matrices of type (3) begins by a rule of type 11. Inside membrane 2, rules 12 and 13 are used, and so membrane (j + 2) enters membrane 7, and membrane (j + 4) enters membrane 7 if the symbol B (j) is present. In this case, B (j) is replaced with #. Otherwise, membrane (j+2) (j) comes out of the membrane 7 replacing Xi with Yj . Then membrane 7 exits membrane 2, by replacing Yj with Y thus successfully simulating a matrix of type (3). (iii) For a terminal matrix mi : (X → a, A → x), X ∈ N1 , a ∈ T , A ∈ N2 , x ∈ T ∗ , where 1 ≤ i ≤ n1 : 16. [[a ]7 ]1 → [a]7 [ ]1 (exo) 17. [ ]8 [#]7 → [#[ ]8 ]7 (fendo) [#[ ]8 ]7 → [ ]8 [#]7 (fexo) Observe that simulation of a matrix of type (4) matrix is similar to that of a matrix of type (2), except that we have an a in place of Y in rule 9. During the ﬁnishing stages of a matrix of type (4) simulation, we use rule 16 to replace a with a when sending the membrane 7 out of the skin membrane. 3.3

Mutual Mobile Membranes

Similar to other classes of mobile membranes, we try to establish the number of membranes in mutual mobile membranes in order to obtain a system which is equivalent to Turing machines. The following result oﬀers an answer. The minimum number of membranes needed remains an open problem. The family of all sets P s(Π) generated by systems of degree at most n using rules α ⊆ {mutual exo, mutual endo} is denoted by P sM M Mn (α). Here mutual endo and mutual exo represent mutual endocytosis and mutual exocytosis rules. By using objects and co-objects, the computational power is obtained using a lower number of membranes than for enhanced mobile membranes, namely: Theorem 8. P sM M M7 (mutual endo, mutual exo) = P sRE.

Simple, Enhanced and Mutual Mobile Membranes

41

Proof (Sketch). Consider a matrix grammar G = (N, T, S, M, F ) in the improved strong binary normal form (hence with N = N1 ∪ N2 ∪ {S; #}), having n1 matrices m1 , . . . , mn1 of types (2) and (4) (that is, not used in the appearance checking mode), and n2 matrices of type (3) (with appearance checking rules). The initial matrix is m0 : (S → XA). Let B (1) and B (2) be the two objects in N2 for which we have rules B (j) → # in matrices of M . The matrices of the form (X → Y, B (j) → #) are labelled by mi , 1 ≤ i ≤ n2 with i ∈ labj , for j ∈ {1, 2}, such that lab1 , lab2 , and lab0 = {1, 2, . . . , n1 } are mutually disjoint sets. We construct a mobile membrane system Π = (V, H, μ, w1 , . . . , w7 , R, 7) of degree seven, where: V = N ∪ T ∪ {X0i , A0i | X ∈ N1 , A ∈ N2 , 1 ≤ i ≤ n1 } ∪{Xji , Aji | 0 ≤ i, j ≤ n1 } ∪ {Xij , Xj | X ∈ N1 , j ∈ {1, 2}, 1 ≤ i ≤ n2 } ∪{β, β, γ, γ} H = {1, . . . , 7} μ = [[ ]7 [ ]5 [ ]6 [[ ]3 [ ]4 ]2 ]1 w1 = β, w2 = β, w3 = β, β, γ, γ, w4 = w5 = w6 = β, γ w7 = XAβγ, where (S → XA) is the initial matrix of G.

In what follows we present only a part of the set of rules R constructed, namely the ones used to simulate mi matrices. (i) For each matrix mi : (X → Y, B (k) → #), X, Y ∈ N1 , A ∈ N2 , where n1 + 1 ≤ j ≤ n1 + n2 , j ∈ labk , k = 1, 2, we consider the rules: (j) 1. [βX]7 [β]2 → [β[βXi ]7 ]2 , j = 1, 2 (mutual endo) (j) (j) 2. [βXi ]7 [β]j+2 → [[βXi ]7 β]j+2 , j = 1, 2 (mutual endo) 3. [γ]3 [γ]4 → [γ[γ]4 ]3 (mutual endo) 4. [γB (1) ]7 [γ]4 → [γ#[γ]4 ]7 (mutual endo) 5. [γ[γ]4 ]3 → [γ]3 [γ]4 (mutual exo) 6. [β]3 [β]4 → [β[β]3 ]4 (mutual endo) 7. [γB (2) ]7 [γ]3 → [γ#[γ]3 ]7 (mutual endo) 8. [β[β]3 ]4 → [β]3 [β]4 (mutual exo) (j) 9. [β[βXi ]7 ]j+2 → [β]j+2 [βYj ]7 , j = 1, 2 (mutual exo) 10. [β[βYj ]7 ]2 → [βY ]7 [β]2 , j = 1, 2 (mutual exo) The simulation of matrices of type (3) begins by a rule of type 1. Inside membrane 2, rules 2 is used, by which membrane 7 enters membrane (j + 2). Rules 3 and 6 are used to introduce the remaining membrane (3 or 4) near membrane 7. In this case, if B (j) exists this is replaced with # and the computation is stopped. Otherwise, membrane 7 comes out of membrane (j+ (j) 2) replacing Xi with Yj . After the other membrane is removed from (j +2), membrane 7 exits membrane 2, successfully simulating a matrix of type (3).

4

Related Work: P Systems with Active Membranes

The mobile membranes derive from the P systems with active membranes introduced in [18]. P systems with active membranes are a variant of P systems

42

B. Aman and G. Ciobanu

in which each membrane is supposed to have an “electrical polarization” (also called charge): positive, negative, or neutral. A P system with active membranes has a ﬁnite set of developmental rules, of the following forms: ∗ (a) [a → v]α h , for h ∈ H, α ∈ {+, −, 0}, a ∈ V , v ∈ V α2 1 (b) a[ ]α h →[b]h , for h ∈ H, α1 , α2∈{+, −, 0}, a, b ∈ V α2 1 (c) [a]α h →[ ]h b, for h ∈ H, α1 , α2∈{+, −, 0}, a, b ∈ V

(d) [a]α h → b, for h ∈ H, α∈{+, −, 0}, a, b ∈ V

object evolution communication communication dissolving

division of elementary membranes α2 α3 1 (e) [a]α h →[b]h [c]h , for h∈H, α1,α2,α3∈{+, −, 0}, a, b, c ∈ V division of non-elementary membranes α4 α4 α6 3 α5 (f) [ [ ...[ ...[ → [[ . . . [ ]α hk ]h0 [ [ ]hk+1 . . . [ ]hn ]h0 for k ≥ 1, n > k , hi ∈ H, 0 ≤ i ≤ n, and α0 , . . . , α6 ∈ {+, −, 0} with {α1 , α2 } = {+, −} 1 ]α h1

α2 1 ]α hk [ ]hk+1

2 α0 ]α hn ]h0

3 ]α h1

More details about these rules and how they are applied may be found in [18]. By denoting with LP A the family of languages L(Π) generated by P systems with active membranes, we have the following result: Theorem 9 ([18]). P sRE = P sLP A. N P Ard denotes the family of vectors of natural numbers N (Π) computed by non-cooperative systems Π which do not use division rules of type (f ). The subscript rd stands for “restricted division”. This restriction does not decrease the power of P systems with active membranes: Theorem 10 ([20]). P sRE = N P Ard . N P Ar denotes the family of natural numbers N (Π) computed by systems Π which use only rules of types (a), (b) and (c). Theorem 11 ([15]). P sRE = N P Ar . The set of numbers generated in the minimally parallel way by a system Π is denoted by Nmin (Π). The family of sets Nmin (Π) generated by systems with rules of the non-restricted form, having initially at most n1 membranes and using conﬁgurations with at most n2 membranes during any computation is denoted by Nmin OPn1 ,n2 . When a type of rule is not used, it is not mentioned in the notation. If any of the parameters n1 , n2 is not bounded, then it is replaced by . If the system do not use polarizations for membranes, then we write (a0 ), (b0 ), (c0 ), (d0 ), (e0 ) instead of (a), (b), (c), (d), (e). When using the maximal parallel way we replace the subscript min by max.

Simple, Enhanced and Mutual Mobile Membranes

43

Theorem 12 ([1,6,17]) 1. Nmax OP3,3 ((a), (b), (c)) = N RE. [17] 2. Nmax OP, ((a0 ), (b0 ), (c0 ), (d0 ), (e0 )) = N RE. [1] 3. Nmin OP3,3 ((a), (b), (c)) = N RE. [6] Theorem 13 ([8]). Nmax OPn1 , ((a1 ),(b1 ), (c1 ), (d1 ), (e1 ))=N RE, for all n1≥5. Theorem 14 ([8]). Nmin OPn1 , ((a1 ), (b1 ), (c1 ), (d1 ), (e1 ))=N RE, for all n1≥7. When the rules of a given type (α0 ) are able to change the labels of the involved membranes, then we denote that type of rules by (α0 ). Using the power of label changing, the following results are obtained: Theorem 15 ([2]). P sOP (a0 , b0 , c0 , e0 ) = P sOP (a0 , b0 , c0 ) = = P sOP (a0 , b0 , c0 ) = P sOP (a0 , c0 , e0 ) = P sRE. By introducing replicative-distribution rules for nested membranes: (l0 ) [a[ ]h1 ]h2 → [[u]h1 ]h2 v, for h1 , h2 ∈ H, a ∈ V , u, v ∈ V ∗ ; the following result is obtained: Theorem 16 ([9]). P sOP4 (l0 ) = P sRE.

5

Conclusions

Simple, enhanced and mutual mobile membranes are new models of computation inspired from the biological operations governing the movement of biological membranes: endocytosis and exocytosis. After deﬁning these classes of mobile membranes according to their biological motivations, some results concerning their computational power are presented. For mutual mobile membranes this is the ﬁrst universality result, while for simple and enhanced mobile membranes the results are improvements for existing ones by reducing the number of membranes needed.

Acknowledgements Many thanks to the referees for their helpful remarks and comments. This work has been partially supported by research grants CNCSIS IDEI 402/2007 and CNCSIS TD 345/2008.

References 1. Alhazov, A.: P Systems without Multiplicities of Symbol-Objects. Information Processing Letters 100, 124–129 (2006) 2. Alhazov, A., Pan, L., P˘ aun, G.: Trading Polarizations for Labels in P Systems with Active Membranes. Acta Informatica 41(2-3), 111–144 (2004)

44

B. Aman and G. Ciobanu

3. Aman, B., Ciobanu, G.: Describing the Immune System Using Enhanced Mobile Membranes. Electronic Notes in Theoretical Computer Science, vol. 194, pp. 5–18 (2008) 4. Aman, B., Ciobanu, G.: Resource Competition and Synchronization in Membranes. In: Proceedings SYNASC 2008. IEEE Computing Society, Los Alamitos (2009) 5. Cardelli, L., Gordon, A.: Mobile Ambients. In: Nivat, M. (ed.) FOSSACS 1998. LNCS, vol. 1378, pp. 140–155. Springer, Heidelberg (1998) 6. Ciobanu, G., Pan, L., P˘ aun, G., P´erez-Jim´enez, M.J.: P Systems with Minimal Parallelism. Theoretical Computer Science 378, 117–130 (2007) 7. Dassow, J., P˘ aun, G.: Regulated Rewriting in Formal Language Theory. Springer, Heidelberg (1990) 8. Freund, R., P˘ aun, G., P´erez-Jim´enez, M.J.: Polarizationless P Systems with Active Membranes Working in the Minimally Parallel Mode. In: Akl, S.G., Calude, C.S., Dinneen, M.J., Rozenberg, G., Wareham, H.T. (eds.) UC 2007. LNCS, vol. 4618, pp. 62–76. Springer, Heidelberg (2007) 9. Ishdorj, T.-O., Ionescu, M.: Replicative-Distribution Rules in P Systems with Active Membranes. In: Liu, Z., Araki, K. (eds.) ICTAC 2004. LNCS, vol. 3407, pp. 68–83. Springer, Heidelberg (2005) 10. Janeway, C.A., Travers, P., Walport, M., Shlomchik, M.J.: Immunobiology - The Immune System in Health and Disease, 5th edn. Garland Publishing, New York (2001) 11. Krishna, S.N.: On the Eﬃciency of a Variant of P Systems with Mobile Membranes. In: Cellular Computing: Complexity Aspects, Fenix Editora, Sevilla, pp. 237–246 (2005) 12. Krishna, S.N.: The Power of Mobility: Four Membranes Suﬃce. In: Cooper, S.B., L¨ owe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 242–251. Springer, Heidelberg (2005) 13. Krishna, S.N., Ciobanu, G.: On the Computational Power of Enhanced Mobile Membranes. In: Beckmann, A., Dimitracopoulos, C., L¨ owe, B. (eds.) CiE 2008. LNCS, vol. 5028, pp. 326–335. Springer, Heidelberg (2008) 14. Krishna, S.N., P˘ aun, G.: P Systems with Mobile Membranes. Natural Computing 4, 255–274 (2005) 15. P˘ aun, A.: On P Systems with Membrane Division. In: Unconventional Models of Computation, pp. 187–201 (2000) 16. P˘ aun, G.: Computing with Membranes. Journal of Computer and System Sciences 61, 108–143 (2000) 17. P˘ aun, G.: Membrane Computing. An Introduction. Springer, Berlin (2002) 18. P˘ aun, G.: P Systems with Active Membranes: Attacking NP-Complete Problems. Journal of Automata, Languages and Combinatorics 6, 75–90 (2001) 19. P˘ aun, G., Rozenberg, G., Salomaa, A.: Membrane Computing with External Output. Fundamenta Informaticae 41, 259–266 (2000) 20. P˘ aun, G., Suzuki, Y., Tanaka, H., Yokomori, T.: On the Power of Membrane Division in P Systems. Theoretical Computer Science 324, 61–85 (2004) 21. Petre, I., Petre, L.: Mobile Ambients and P Systems. Journal of Universal Computer Science 5, 588–598 (1999) 22. Salomaa, A.: Formal Languages. Academic Press, London (1973) 23. http://bcs.whfreeman.com/thelifewire

Bio-PEPA with Events Federica Ciocchetta Laboratory for Foundations of Computer Science, The University of Edinburgh, Edinburgh EH8 9AB, Scotland

Abstract. In this work we present an extension of Bio-PEPA, a language recently defined for the modelling and analysis of biochemical systems, to handle events. Events are constructs that represent changes in the system due to some trigger conditions. The events considered here are simple, but nevertheless able to describe most of the discontinuous changes in models and experiments. Events are added to our language without any modification to the rest of the syntax in order to keep the specification of the model as straightforward as possible. Some maps are defined from Bio-PEPA with events to analysis tools. Specifically, we map our language to Hybrid Automata (HA) and we consider a modification of Gillespie’s algorithm for stochastic simulation. In order to test our approach, we present the translation in Bio-PEPA of a biochemical network describing the functional properties of the Acetylcholine receptor with the addition of an event that causes the inactivation of some reactions at a given time.

1

Introduction

Computational models play an important role in systems biology. Indeed they help to study, analyze and predict the behaviour of biological systems. In recent years there have been some applications of process algebras for the analysis of biological systems (e.g. [27,25,8,9]). In most cases the analysis is performed using Gillespie’s stochastic simulation algorithm [18]. Other possibilities exist, such as the mapping to diﬀerential equations [7]. Many biological models need to capture both discrete and continuous phenomena [1,4,23]. These models are called hybrid systems. A ﬁrst example of a hybrid system describes the activation of a certain activity when the concentration of enabling quantities is above the desired threshold. A second example considers a signal or stimuli that becomes null after some time leading to some changes in the interactions of the system. Other examples describe some experiments, where it may be necessary to render the possible change to the system, due, for instance, to the introduction or the removal of some reagents. In this work we present an extension of Bio-PEPA [9,10], a language recently deﬁned for the modelling and analysis of biological systems, to handle events. Broadly speaking, events are constructs that represent changes in the system due to some trigger conditions. Here we are interested in simple forms of events. Speciﬁcally, we refer to the deﬁnition of events reported in the SBML speciﬁcation [22]. These kinds of C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 45–68, 2009. c Springer-Verlag Berlin Heidelberg 2009

46

F. Ciocchetta

events can be found in biochemical networks, such as the ones in the BioModels database [24] or deﬁned in some experimental settings. The idea underlying our work is the following: Biochemical networks with events =⇒ Bio-PEPA with events =⇒ Analysis Starting from a biochemical network with one or more events, we want to map it into a Bio-PEPA system. From that, we can then consider diﬀerent kinds of analysis. In this, view Bio-PEPA is a formal, intermediate, compositional representation of the biochemical network. This idea is the one proposed for (the standard) Bio-PEPA. A ﬁrst challenge concerns the modelling: we need to add events to the BioPEPA system. Events are added to our language as a set of elements and the rest of the syntax is unchanged. There are two motivations for this choice. First, we keep the speciﬁcation of the model as simple as possible. Second, this approach is appropriate when we study the same biochemical system but with diﬀerent experimental regimes as we can modify the list of events without any changes to the rest of the system. A second aspect is the analysis. Some maps must be deﬁned from Bio-PEPA to analysis tools. Speciﬁcally, we map our language to Hybrid Automata (HA) [19]. HA are a formalism that consider both continuous and discrete changes. The continuous part is expressed by a set of variables evolving in each state according to a set of diﬀerential equations and the discrete dynamics is given by transitions between states, triggered by some conditions on variables. Furthermore, we can consider a modiﬁcation of Gillespie’s algorithm [18] in order to tackle events. A preliminary version of this work has been presented in [11]. Here we add some deﬁnitions concerning the kind of events and further details concerning the mappings from Bio-PEPA with events to the Hybrid Automata and Gillespie’s algorithm. Furthermore, we consider more general kinds of events, such as simultaneous events or events with a delay diﬀerent from zero. The rest of the paper is organised as follows. Section 2 reports a description of Bio-PEPA. In Section 3 we deﬁne the events we are considering in this work and then we extend Bio-PEPA in order to handle them. Section 4 describes the mapping from our language to Hybrid Automata. The mapping to stochastic simulation is reported in Section 5. After that, Section 6 illustrates the modelling in Bio-PEPA of a biochemical network describing the functional properties of the Acetylcholine receptor with an event that is triggered at a given time and causes the inactivation of some reactions. In Section 7 we overview some related work. Finally, in Section 8, some conclusions are reported.

2

Bio-PEPA

Bio-PEPA [9,10] is a language for the modelling and analysis of biochemical networks. The syntax of Bio-PEPA is deﬁned as: S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | | .

P ::= P P | S(x) I

Bio-PEPA with Events

47

The component S (species component ) abstracts a biological species and the component P (model component ) describes the system and the interactions among components. The preﬁx term (α, κ) op S contains information about the role of the species in the reaction associated with the action type α: κ is the stoichiometry coeﬃcient of the species and the preﬁx combinator “op” represents the role of the element in the reaction. Speciﬁcally, ↓ indicates a reactant, ↑ a product, ⊕ an activator, an inhibitor and a generic modiﬁer. The operator “+” expresses choice between possible actions and the constant C is deﬁned by def an equation C = S. The parameter x ∈ R+ in S(x) represents the initial quan tity (for instance the concentration) of the species. Finally, the process P Q I denotes the cooperation between components: the set I determines those activities on which the operands are forced to synchronize. In Bio-PEPA the rates are not expressed in the syntax of components but are deﬁned as functional rates. These allow us to express any kind of kinetic law. Each action is associated with a speciﬁc functional rate. A possible modelling style supported by Bio-PEPA is in terms of concentration levels. This is the style considered in the derivation of the transition system for Bio-PEPA. The species concentrations can be discretized into a number of levels. The granularity of the system is expressed in terms of the step size h, i.e. the length of the concentration interval representing a level. The information about the step sizes and the number of levels for each species is collected in a set N . Speciﬁcally, the elements of the set N have the form: “C : h = value h, N = value N, M = value M, V = value V, unit = value u”, where C is the species component name, h is the step size, N is the maximum level, M is the maximum concentration, V is the name of the enclosing compartment and unit is the unit for concentration. In order to fully describe a biochemical network in Bio-PEPA we need to deﬁne structures that collect information about the compartments, the maximum concentrations, number of levels for all the species, the constant parameters and the functional rates. The Bio-PEPA system is deﬁned in the following way: Definition 1. A Bio-PEPA system P is a 6-tuple V, N , K, FR , Compon − ents, P , where: V is the set of compartments, N is the set of quantities describing species, K is the set of parameter deﬁnitions, FR is the set of functional rates, Components is the set of deﬁnitions of sequential components, P is the model component describing the system. For details see [9,10]. The behaviour of the system is deﬁned in terms of an operational semantics. This refers to the level-based modelling style and in this context the parameter in the species components stands for the concentration level. We deﬁne two θ relations. The former, called capability relation, is indicated by − →c . The label θ is of the form (α, w), where w := [S : op(l, κ)] | w :: w, with S a species component, op a symbol representing the role of the species in the reaction, l the level and κ the stoichiometry coeﬃcient. This relation is deﬁned as the minimum relation satisfying the rules reported in Table 1.

48

F. Ciocchetta Table 1. Axioms and rules for Bio-PEPA

prefixReac

(α,[S:↓(l,κ)]) ((α, κ)↓S)(l) −−−−−−−−→c S(l − κ) κ ≤ l ≤ N

prefixProd

(α,[S:↑(l,κ)]) ((α, κ)↑S)(l) −−−−−−−−→c S(l + κ) 0 ≤ l ≤ (N − κ)

prefixMod

((α, κ) op S)(l) −−−−−−−−−→c S(l)

(α,[S:op(l,κ)])

with op = , ⊕, and

0 < l ≤ N if op = ⊕, 0 ≤ l ≤ N otherwise (α,w)

choice1

S1 (l) −−−→c S1 (l ) (α,w)

(S1 + S2 )(l) −−−→c S1 (l ) (α,w)

choice2

S2 (l) −−−→c S2 (l ) (α,w)

(S1 + S2 )(l) −−−→c S2 (l ) (α,S:[op(l,κ)])

constant

S(l) −−−−−−−−−→c S (l )

def

with C = S

(α,C:[op(l,κ)])

C(l) −−−−−−−−−→c S (l ) (α,w)

coop1

P1 −−−→c P1 (α,w) P1 P2 −−−→c P1 P2 L L

with α ∈ /L

(α,w)

coop2

P2 −−−→c P2

P1 P2 −−−→c P1 P2 L L (α,w)

(α,w1 )

coop3

P1 −−−−→c P1

with α ∈ /L

(α,w2 )

P2 −−−−→c P2

1 2 P1 P2 −−−−−−−→c P1 P2 L L

(α,w ::w )

with α ∈ L

˜ where P˜ is The latter relation, called stochastic relation, is − →s ⊆ P˜ × Γ × P, 1 the set of well-deﬁned Bio-PEPA systems and Γ is the set of labels γ = (α, r), with α the action type and r the associated rate. This relation is deﬁned as the minimal relation satisfying the rule: 1

In a well-defined Bio-PEPA system each element has to satisfy some conditions. For instance, we have that each species component C ∈ Comp must have subterms of the form “(α, κ) op C” and the action types in each single component must be all distinct. Furthermore, the model component P must be defined in terms of the species components defined in Comp and, for each cooperation set Lj in P , Lj ⊆ A(P ). For details see [12].

Bio-PEPA with Events

49

(αj ,w)

Final

P −−−−→c P (αj ,rα [w,N ,K])

V, N , K, F , Comp, P −−−−−−−−−−→s V, N , K, F , Comp, P

The element rα [w, N , K] is the rate associated with the action α and is deﬁned as: fα [w, N , K] rα [w, N , K] = h where h is the step size for the species involved in the reaction and the notation fα [w, N , K] means that the function fα is evaluated over w and the information about parameters and species components contained in the sets N and K. In this deﬁnition rα represents the parameter of a negative exponential distribution. The dynamic behaviour of processes is determined by a race condition: all activities enabled attempt to proceed but only the fastest succeeds. A Stochastic Labelled Transition System (SLTS) is deﬁned for a Bio-PEPA system. From this we can obtain a continuous time Markov Chain (CTMC). Both the SLTS and the CTMC derived from Bio-PEPA are deﬁned in terms of levels of concentration. We call this Markov chain the CTMC with levels. Bio-PEPA can be seen as an intermediate, formal, compositional representation of biological systems, from which diﬀerent kinds of analysis can be performed. We have deﬁned some mappings from Bio-PEPA to ODEs, CTMC with levels, stochastic simulation and PRISM [26]. Some tools for the analysis of BioPEPA system have been implemented [3]. In the following we report a brief description of the mapping from Bio-PEPA to ODE, as it is used later in the paper. For further details and the other mappings see [10]. 2.1

From Bio-PEPA to ODE System (πODE )

Let πODE be the mapping from Bio-PEPA system to the associated ODE system. The mapping πODE entails three steps: 1. deﬁnition of the stoichiometry (n × m) matrix D, where n is the number of species and m is the number of reactions; 2. deﬁnition of the kinetic law vector (m × 1) vKL containing the kinetic laws of each reaction; 3. deﬁnition of the vector (n × 1) x, with xT = (x1 , x2 , ..., xn ). A crucial part is the derivation of the stoichiometry matrix D = {dij }. The entries of the matrix are obtained as follows: for each sequential component Ci consider the preﬁx subterms Cij representing the contribution of the species i to the reaction j. If the term represents a reactant we write the corresponding stoichiometry κij as −κij in the entry dij . In the case of a product we write +κij . All other cases are null. The kinetic law vector is derived from the functional rates and its deﬁnition is straightforward.

50

F. Ciocchetta

The ODE system thus obtained has the form: dx = D × vKL dt where the vector of initial concentrations is x0 , with xi,0 the initial concentration of the species i, as given in the speciﬁcation of the system. 2.2

Example

In order to show how to model biochemical systems in Bio-PEPA we consider the network presented in Fig. 1 and we translate it into Bio-PEPA. This network is then used as a running example in the rest of the paper.

3

X

1

2

Y

Fig. 1. Biochemical network composed of two proteins X and Y . The numbers indicate the reactions. Reaction 1 is the translation of Y enhanced by X, reaction 2 is the degradation of X and reaction 3 the translation of X.

The network is composed of two proteins, X and Y . These are involved in the following interactions: r

1 – Translation of Y enhanced by X (reaction 1): X −→X +Y. The kinetic law is mass-action with constant parameter r1 = 0.01; r2 – Degradation of the protein X (reaction 2): X −→∅. The kinetic law is mass-action with constant parameter r2 = 0.02; r3 – Translation of the protein X (reaction 3): ∅−→X. The kinetic law is mass-action with constant parameter r3 = 0.01.

Each reaction i is represented by an action type αi . The kinetic laws are represented by the following functional rates: fα1 = f M A(0.01); fα2 = f M A(0.02); fα3 = 0.01; where f M A(r) stands for mass-action kinetic law with rate r. The Bio-PEPA species components2 corresponding to the two proteins are: X = (α1 , 1) ⊕ X + (α2 , 1)↓X + (α3 , 1)↑X def

2

def

Y = (α1 , 1)↑Y

Note that we use X and Y (capital letters) to indicate the names of the species and the name of the Bio-PEPA components, whereas x and y indicate the associated species concentrations.

Bio-PEPA with Events

51

Fig. 2. ODE integration results for the network

whereas the model component is:

} Y (0) X(0) {α 1

where the initial values are zero for both the proteins. The set of compartments and the set N are not reported. Applying the mapping πODE we obtain the ODE system: dx dt dy dt

= −0.02 · x + 0.01 = 0.01 · x

where x and y are the two variables describing X and Y . The result of ODE integration is reported in Fig. 2. The protein X reaches a steady-state whereas Y increases inﬁnitely.

3 3.1

Bio-PEPA with Events SBML-Like Events: Some Definitions

In this work we consider events as deﬁned in the SBML speciﬁcation [22]. SBML events describe explicit discontinuous state changes in the model. Speciﬁcally, an SBML event has the following structure: “event id, if trigger then event assignment list with delay

52

F. Ciocchetta

where – event id is the event identiﬁer, – trigger is a mathematical expression that, when it is evaluated to true, makes the event ﬁre. It can be composed of one or more conditions; – event assignment list is a list of assignments that are made when the event is executed; – delay is the length of time between the time when the event ﬁres and the time when the event assignments are executed. The trigger and the list of assignments are both mandatory and can involve parameters, species concentrations and compartment sizes. All the triggers are initially evaluated to false. An SBML-like event is immediate if delay is equal to zero. Otherwise, the event is called delayed. The deﬁnition of sequential and simultaneous events is reported below. Definition 2. Two or more SBML-like events are sequential if they are ﬁred one after the other in a given order. They are said to be simultaneous if they happen at the same point in time. In most biochemical systems which we are interested in we have sequential events. In the general situation of simultaneous events, sometimes some tie-breaking rules are necessary to decide which of any set of events is simulated ﬁrst. The most common way to do this is to assign a priority to each event [13]: when there are two or more simultaneous events, the event with the highest priority is deﬁned to be the next event to ﬁre. However, the order in which a set of simultaneous events is ﬁred is not always important, for instance when the assignments of the events inﬂuence diﬀerent variables. We have the following deﬁnition: Definition 3. Two simultaneous events are independent if their event assignments do not eﬀect each other. Otherwise, they are called dependent. If we have simultaneous independent events we may abstract them as a single event and the system is reset according to the assignments of all the set of simultaneous events. Simultaneous independent events are dealt with similarly to sequential ones. 3.2

Assumptions

We make the following assumptions for the events considered in this work. 1. Triggers can involve time and species components’ names, while assignments can involve species components (concentrations), compartments (size), parameters (values) and functional rates (function deﬁnitions). 2. Triggers are deterministic, i.e. when they become true they are ﬁred. 3. Triggers are only unidirectional, i.e. describing the change from one mode to another, but not vice versa. Bidirectional triggers can be decomposed into two unidirectional triggers. 4. Events are either sequential or simultaneous and independent.

Bio-PEPA with Events

53

These assumptions are not restrictive. Indeed the events satisfying these assumptions allow us to represent a large number of discontinuous changes that we can ﬁnd in biological systems. 3.3

The Definition of the Language

We can add events to a Bio-PEPA system by introducing a set of elements that have the form (id, trigger, event assignment, delay), where id is the name of the event, trigger is a mathematical expression involving the components of the Bio-PEPA model and time, event assignment is a list of assignments, delay is 0 (immediate events) or positive real value (delayed events). Formally, we have the following deﬁnitions: trigger ::= cond | cond or cond | cond and cond | not cond ¯ eq value | ¯ k) cond ::= t eq value | expression(C, ¯ ¯ ¯ ¯ k) expression(C, k) eq expression(C, eq ::= = | =|>| 0.8 and X = 0.2, r3 ← 0.005, 0) These two events are sequential and clearly satisfy the assumptions discussed above.

4 4.1

Mapping to Hybrid Automata Hybrid Automata

Hybrid automata (HA) [19] combine discrete transition graphs with continuous dynamical systems. They are used to formally model hybrid systems, dynamical systems with both discrete and continuous components. An hybrid automaton consists of a ﬁnite set of real-valued variables {X1 , X2 , ..., Xn } and a ﬁnite labelled graph, whose vertices correspond to control modes (states), described by diﬀerential equations, and whose edges are control switches, corresponding to discrete events. In addition, we have some labels for the edges, specifying the jump conditions (activation conditions) and labels for the vertices, containing information about initial and invariant conditions. The variables evolve continuously in time, apart from some changes induced by events. When an event happens there is a change in the mode. The dynamic behaviour of each mode is described by a set of diﬀerential equations, generally diﬀerent from mode to mode. We can use HA both for simulation (see for instance the SHIFT language [15]) and model checking (see HyTech [20]). In this work we limit our attention to simulation. For a formal deﬁnition and details of the formalism see [19].

Bio-PEPA with Events

4.2

55

Definition of the Mapping

Here we present the map from Bio-PEPA to HA. First, we limit our attention to the case of immediate events and then we show a way to represent delayed events. Indeed, the translation of the delay associated with an event is not straightforward in the usual deﬁnition of HA. Let P0 = V0 , N0 , K0 , F0 , Comp0 , P0 , Events, t0 be the initial Bio-PEPA system and let Nevents be the number of events. We have the following correspondences: 1. Each species component Ci in Comp is associated with a variable Xi . The set of variables is then given by {X1 , X2 , · · · , XNComp , t}, where t is the variable expressing the time and NComp is the number of species components. The evolution of the variable t is described by the trivial diﬀerential equation dt/dt = 1. 2. The initial conditions of the variables are derived from the initial model component P0 . The variable t is initially set to 0. 3. For each event i in Events, we can consider the trigger tri . We use these triggers to deﬁne the jump conditions. In the case we have only sequential events, the number of possible jump conditions Njump is just NEvents . If simultaneous independent events are possible, we may combine them together in order to deﬁne a new jump condition representing the union of the triggers of the simultaneous events. In this case, the system is reset according to the union of the assignment lists of the events involved. 4. Each mode is described by a speciﬁc instance of the Bio-PEPA system. Indeed modes are deﬁned according to either the initial system or the system modiﬁed with the event assignments relative to a trigger. The number of modes is Njump + 1. σ is used to indicate a mode and the Σ the set of all modes. In each mode some invariant conditions are added in order to force the change of mode when the trigger becomes true. We have that: – The initial mode σ0 is deﬁned from the initial system P0 . It is described in terms of an ODE system and this is derived from the Bio-PEPA model by considering the map πODE . Therefore, we have σ0 = πODE (P0 ). – Given a mode σi = πODE (Pi ), let trij be one possible jump condition that can be satisﬁed from it. We deﬁne the Bio-PEPA system Pj = Pi [event assignmentij ] as the modiﬁcation of the previous system Pi according to the event assignments associated with the trigger. The mode σj is then deﬁned as σj = πODE (Pj ). Case of delayed events. The delay associated with an event represents the time interval between when the event is ﬁred and when its assignments are executed. This information cannot be directly translated in any of the components of standard HA. In the following we report as we handle the delay in HA. First, we introduce a new variable tmode representing the time when the system enters in a speciﬁc mode. It is initially set to zero. The diﬀerential equation associated with this new variable is dtmode /dt = 0, i.e. this variable is constant in each mode.

56

F. Ciocchetta

Second, given an event (id, trigger, event assignment, delay), we split it into two immediate events, deﬁned as: 1. (id1 , trigger, tmode ← t, 0); 2. (id2 , t = tmode + delay, event assignment, 0). The role of the former event is to introduce the delay whereas the role of the second is to guarantee that the assignments of the initial event are executed after the given delay. 4.3

Example (Continued 2)

Consider the network presented in Sect. 2.2 with the addition of the set of events (see Sect. 3.4): [(event1 , Y = 0.8, r3 ← 0, 0)]. A schema of the HA associated with this network is reported in Fig. 3. The set of variables is {x, y, t}, where x and y are the two variables representing the two proteins. The initial concentrations, derived from the initial condition in the Bio-PEPA model, are x = 0, y = 0 and t = 0. We have just one event so we have two modes and the jump condition (guard) is y = 0.8. The former mode is described by the invariant condition y < 0.8 and the latter by y ≥ 0.8. The ODE system corresponding to the initial mode (S1) is derived by applying the mapping πODE to the initial Bio-PEPA system (P0 ) and is: dx dt dy dt

= −0.02 · x + 0.01 = 0.01 · x

For the second mode, the ODE system (S2) is obtained as πODE (P0 [r3 ← 0]) and is: dx dt = −0.02 · x dy dt = 0.01 · x If we consider both event1 and event2 , we have the HA represented in Fig. 4. There are three modes, representing the network at the initial state, when y < 0.8 and when y ≥ 0.8 and x ≥ 0.2. Two jumps conditions are deﬁned in terms of the trigger conditions. The ODE systems describing the ﬁrst and second modes are as above, whereas the ODE system for the third mode (S3) is obtained from πODE (P1 [r3 ← 0.005]) (where P1 is the Bio-PEPA system corresponding to the second mode) and is:

[ y = 0.8] x =0, y=0

S1

S2

Fig. 3. HA representation for the network composed of the two proteins X and Y and with an event involving concentrations

Bio-PEPA with Events

57

[ y = 0.8] x =0, y=0

S1

S2

[ y > 0.8 and x= 0.2]

S3 Fig. 4. HA representation for the network composed of the two proteins X and Y and with two sequential events

Fig. 5. Simulation results for the network composed of the two proteins X and Y and with the addition of event1 dx dt dy dt

= −0.02 · x + 0.005 = 0.01 · x

Some results for the network with just event1 are reported in Fig. 5. The protein X increases until time 200 s when Y reaches the value 0.8 and then decreases to 0. The protein Y increases, but after the event, its rate of increase is much lower than the case without the event. The results for the network with both event1 and event2 is reported in Fig. 6. In this case, when the second event is ﬁred, the protein X starts to increase again and this has eﬀect on the production of Y as well.

58

F. Ciocchetta

Fig. 6. Simulation results for the network composed of the two proteins X and Y and with the addition of event1 and event2

5

Stochastic Simulation by Gillespie’s Algorithm

One of the possible kinds of analysis supported by Bio-PEPA is stochastic simulation using Gillespie’s algorithm [10]. When events are considered the algorithm has to be modiﬁed in order to handle them. Broadly speaking, events are tackled by adding some conditions and some checks along the simulation. We start at time t = 0, with the Bio-PEPA system in its initial conditions. We assume that initially all the triggers evaluate to false. When one of the conditions is satisﬁed, the simulation stops and the system is modiﬁed according to the event assignments associated with the trigger. After that, the simulation can start again until another condition becomes true or the simulation time is reached. The use of triggers involving time can be challenging since it can happen that the time of the event does not coincide with any of the simulation time points. Our approach to deal with this case is discussed below. Note that if the events involve species concentrations, we have to change concentrations into number of molecules for stochastic simulation. Speciﬁcally, we have to multiply each concentration by N a V , where N a is the Avogadro number3 and V is the volume of the compartment. In the the rest of this section we assume that the events are in terms of number of molecules. 3

This is the number of “entities” (atoms or molecules) in one mole of substance. Its value is 6.022 × e+23 (mol)−1 .

Bio-PEPA with Events

59

We propose the following procedure for each simulation run. 1. 2. 3. 4.

Let P0 be the initial Bio-PEPA system and ts the maximum simulation time. While t < ts and triggeri = f alse for i = 1, 2, ..., NEvents , simulate. If t ≥ ts then stop. If t < ts and there exists a triggeri such that it is true, we have that: (a) if delay = 0 modify the Bio-PEPA system according to the event assignments associated with that trigger: P (t) = P(t)[event assignmenti]. Go to (2). (b) if delay > 0 go on with the simulation until time t + delay and then proceed as in (a).

Some ﬁnal observations concern how to use the algorithm in two particular situations. – In the case of two or more independent simultaneous events we proceed as observed in Section 3.1: we can abstract these events as a single event, whose trigger is deﬁned in terms of the triggers of the two events and the event assignments are the union of the assignments. Therefore, we modify the system according to the assignments associated with all the events involved. – When we have an event with a trigger involving time t = t˜, the time value t˜ may not correspond to any of the simulation time point obtained by using Gillespie’s simulation algorithm. Speciﬁcally, there exist two consecutive simulation time points tj and tj+1 such that tj < t˜ < tj+1 . If this happens, we have to decide when the system has to be modiﬁed. In order to handle this situation we consider the following approach: 1. if tj < t˜ + delay < tj+1 with delay ≥ 0 consider the system at time t˜+ delay and modify it at that time point. The simulation restarts from t˜ + delay. 2. If delay > 0 and t˜+ delay ≥ tj+1 consider the last simulation time point th ≤ t˜+ delay and run the simulation until th . Then, modify the system at time t˜ + delay and restart the simulation from that time point.

6

The Acetylcholine Receptor Model

This example concerns the functional properties of the nicotin Acetylcholine Receptors (nAChR). These are transmembrane proteins that mediate interconversions between open and closed channel states under the control of neurotransmitters. The detailed description of the model is reported in [16]. A schema of the model is shown in Figure 7. B (Basal state), A (Active state), D (Desensitized state) and I (Inactivable state) represent the diﬀerent states of the Acetylcholine receptors. The numbers 0, 1, 2 associated with the state are the number of ligands (denoted X) bound to a receptor. In the model the ligands are not modelled explicitly. Each column corresponds to a series of ligand binding actions at two identical sites per receptor whereas each row corresponds to a series of transactions between conformational states. All the

60

F. Ciocchetta kf_5

B0 + 2X

kf_9

A0 + 2X

kr_5 kf_0

kr_0

kr_9 kf_3

kf_6

B1 + X

kr_3

A1 + X

kr_6

kf_1 kr_1

B2

kf_7

kf_10

kr_4

A2

kr_7

I1 + X

kr 2

D0 + 2X

kf_12

kf_15

kr_12

D1 + X

kr_15

kf_8

kf_11

kf_14 kr_14

kr_10

kf_4

kf_2

I0 + 2X

kr_8

I2

kr 11

kf_13 kr_13

kf_16

D2

kr 16

Fig. 7. Schema of the Acetylcholine receptor model Table 2. The Acetylcholine receptor model. The list of parameters. The unit is s−1 . parameter kf0 kf2 kf4 kf6 kf8 kf10 kf12 kf14 kf16

value 3000 30000 1500 130 1500 19.85 3000 0.05 0.05

parameter kr0 kr2 kr4 kr6 kr8 kr10 kr12 kr14 kr16

value 8000 700 17.28 2740 8 1.74 4 0.0012 0.0012

parameter kf1 kf3 kf5 kf7 kf9 kf11 kf13 kf15

value 1500 3000 0.54 3000 19.7 20 1500 0.05

parameter kr1 kr3 kr5 kr7 kr9 kr11 kr13 kr15

value 16000 8.64 10800 4 3.74 0.81 8 0.0012

reactions are reversible and the dynamics are described by mass-action laws. For each reaction i, with i = 1, 2, ...16, the rate of the forward direction is kf i and the rate of the reverse direction kr i. In addition to these elements, there is an event to describe the recovery upon removal of free agonist at a given time. This is expressed by constraining the reaction rates of each second-order ligand-receptor reaction to zero. These constraints prevent ligand binding reactions from happening after that time, hence the states evolve as if the free ligands were completely removed from the system. The event is immediate, the trigger is “t = t2 ”, where t2 = 20 s, and the event assignments are kf0 ← 0, kf1 ← 0, kf3 ← 0, kf4 ← 0, kf7 ← 0 , kf8 ← 0 , kf12 ← 0 , kf13 ← 0. The Bio-PEPA system associated with the Acetylcholine receptor model. In the following we report brieﬂy the deﬁnition of the Bio-PEPA system

Bio-PEPA with Events

61

representing the Acetylcholine receptor model. The complete system is reported in the Appendix A. – Deﬁnition of the compartment list V. In the model we have a single threedimensional compartment, deﬁned as “comp1 : 1e-16, l;”, where l is litre. – Deﬁnition of the set N . Each species is associated with a species component. For each species component we have to declare the step size, the number of levels, the initial and maximum concentrations and the compartment where the species is. The ligand is not represented explicitly. For instance, in the case of B0, B1 and B2 we have: B0 : H = h, N = NB0 , M = MB0 , V = comp1, unit = μM ; B1 : H = h, N = NB1 , M = MB1 , V = comp1, unit = μM ; B2 : H = h, N = NB2 , M = MB2 , V = comp1, unit = μM ; where the step size is 1.66e-5, the number of levels NB0 = NB1 = NB2 is 1 (i.e. the species can be present, 1, or absent, 0), the maximum concentration MB0 = MB1 = MB2 is 1.66e-5 and coincides with the initial concentration of channels at the basal state. Note that the information about the step size and the number of levels is not used in this work, as we do not consider CTMC with levels, however we deﬁne them for completeness. – Deﬁnition of functional rates (FR ) and parameters (K). Each reversible reaction i, i = 0, 1, 2, · · · , 16, is decomposed in two irreversible reactions, fi and ri , representing the forward and inverse directions respectively. The associated kinetic laws are fα fi = f M A(kf i); and fα ri = f M A(kr i), where f M A denotes mass-action. All the parameters are deﬁned in the set K. The values are the ones reported in the paper [16]. – Deﬁnition of species components (Comp) and of the model component (P ). In the following we report the deﬁnition for B0, B1 and B2; the other species are dealt with similarly. def

B0 = (α def B1 = (α (α def B2 = (α

f0 , 1)↓B0 + (α f0 , 1)↑B1 + (α f1 , 1)↑B1 + (α f2 , 1)↓B2 + (α

r0 , 1)↑B0 + (α f5 , 1)↓B0 + (α r5 , 1)↑B0 r0 , 1)↓B1 + (α f6 , 1)↓B1 + (α r6 , 1)↑B1+ r1 , 1)↓B1 r2 , 1)↑B2 + (α f1 , 1)↑B2 + (α r1 , 1)↓B2

The system is described as:

B1(0) B2(0) A0(0) A1(0) A2(0) B0(1.66e-5) L1 L2 L3 L3 L4 L5

I1(0) I2(0) D0(0) D1(0) D2(0) I0(0) L6 L7 L8 L9 L10

where Li , i = 1, ..., 10 are the cooperation sets and the initial values for the species are 0 with the exception of the species B0. – Deﬁnition of events. We have only one event, describing a change in the system at time 20 s: [(event, t = 20, kf0 ← 0; kf1 ← 0; kf3 ← 0 kf4 ← 0; kf7 ← 0; kf8 ← 0; kf12 ← 0; kf13 ← 0, 0)]

62

F. Ciocchetta

Fig. 8. Stochastic simulation results for the Acetylcholine receptor model (average over 100 runs)

Analysis results. The HA associated with the Acetylcholine receptor model is similar to the one for the network presented in Sect.2.2 with the addition of the set of events. We have two modes, described by two diﬀerent sets of diﬀerential equations. The trigger condition involves time and is “t = 20 s”. The details of the two systems describing each mode are not reported. Simulation results made using Gillespie’s algorithm are reported in Fig. 8. The initial number of molecules for B0 is given M0 × V × N a = (1.66e-5 μM ) × (1.e16 l)×(6.022×e+23 (mol)−1 ) = 1000, where N a is the Avogadro number. All the other species are initially null. The number of runs is 100. The graph reproduces results in agreement with the ones reported in the paper [16]. Following the ligand removal, the state I2 loses agonist molecules and is transformed to the state B0 very rapidly, while D2 loses ligand molecules to form D0. Since the data occur on a wide range of times we represent the time on a logarithmic scale.

7

Related Works

The use of mathematical formalisms in order to represent discrete changes in biological systems is not new [1,4,23,17,5,6]. In [1] the authors proposed a hybrid system approach to modelling an intra-cellular network using continuous diﬀerential equations to model some part of the system and mode-switching to

Bio-PEPA with Events

63

describe the changes in the underlying dynamics. Some models with hybrid behaviour are presented and described using CHARON [2], a language that allows formal description of hybrid systems. The authors of [23] discussed the use of discrete changes in biological systems and presented some examples using the formalism HybridSAL [21]. Hybrid Concurrent Constraint Programming is used to model some biological systems with both discrete and continuous changes in [4]. In [5] the authors presented a map from stochastic Concurrent Constraint Programming (sCCP) to HA. The HA generated in this way are said to be able to capture some aspects of the dynamics which are lost if standard diﬀerential equations are used. A discussion of hybrid systems and biology is reported in [6]. Finally, in [17] the authors presented HYPE, a process algebra for the modelling of hybrid systems and used it to represent the repressilator, an artiﬁcial genetic network composed of three genes and their respective proteins with oscillatory behaviour. In none of these works are SBML-like events considered explicitly, but the focus is on general hybrid systems. Events have been proposed in the Beta Workbench (BetaWB) [14] and in the associated programming language BlenX [28]. In BlenX events can be considered as global rules of the environment, triggered only when the conditions associated with them are satisﬁed. Each event is the composition of a condition (cond ) and an action (verb) and is associated with a rate. Conditions can involve number of entities, the simulation time or the simulation step. The possible actions are the join of two entities, the split of one entity into two, the update of a variable of the system and the deletion or the creation of a new entity. The concept of events proposed in BlenX is quite similar to the one considered for Bio-PEPA. The BlenX condition and action correspond to the trigger and event assignment in Bio-PEPA events. However, rates in BlenX have a diﬀerent meaning from the delay in Bio-PEPA. Indeed, in Bio-PEPA an events occurs when the trigger is satisﬁed and the role of the delay is to postpone when the event is executed. BlenX events with a ﬁnite rate can happen only when the trigger is satisﬁed but it is in competition with other actions that are enabled contemporaneously (race condition). BlenX events with inﬁnite rate correspond to immediate events in Bio-PEPA. In order to compare the deﬁnition of events in the two languages, we show how the events proposed in this paper can be described in BlenX. The event event1 deﬁned in Sec. 3.4 is represented in BlenX as: when (Y → value) update (r3 , change par) where value is 0.8 · N aV molecules and the function change par is deﬁned as change par : f unction = 0. The operator “→” recognizes when the quantity bound to Y becomes greater than the speciﬁed value, whereas the action “ update (r3 , change par)” means that the parameter r3 is updated according to the function change par (in our case it assigns the value 0). The rate associated with the update action is always inﬁnite and not reported. Concerning the event in the Acetylcholine receptor model, it is not possible to represent this event in BlenX as conditions involving time are not allowed with the action update.

64

F. Ciocchetta

Note that BlenX events represent more general kinds of interactions than BioPEPA events. For instance, they are used to model the formation of a complex (by using the action join) or the split of a complex into two parts (by using the split action). These reactions (as all the other kinds) are represented in Bio-PEPA by synchronization of the species components over the action types abstracting the reactions. Bio-PEPA events have been introduced speciﬁcally to represent experimental situations when there is change in the system due to some conditions.

8

Conclusions

In this work we have presented an extension of Bio-PEPA to handle SBML-like events. Events are constructs that represent changes in the system due to some trigger conditions. The events considered here are simple, but nevertheless able to describe most of the discontinuous changes in models and experiments. Events are added to our language without any modiﬁcation to the rest of the syntax. The main motivation of this choice is that we want to keep the speciﬁcation of the model as simple as possible. Furthermore, this approach is appropriate when we study the same biochemical system but with diﬀerent experimental regimes. A topic for the future concerns the study of more general events and the possible extension to other kinds of hybrid systems in biology. Furthermore, we plan to exploit the possible kinds of analysis involving hybrid systems in the context of systems biology. In this paper we focus on the mapping to Hybrid Automata and stochastic simulation by (a modiﬁcation of) Gillespie’s algorithm. Further investigation will concern the application of model checking for the study of the properties of biological systems.

Acknowledgements The author thanks Jane Hillston, Vashti Galpin and Adam Duguid for their helpful comments. The author is supported by the EPSRC under the CODA project “Process Algebra Approaches for Collective Dynamics” (EP/c54370x/01).

References 1. Alur, R., Belta, C., Ivancic, F., Kumar, V., Mintz, M., Pappa, G., Rubin, H., Schug, J.: Hybrid modeling and simulation of biomolecular networks. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 19–32. Springer, Heidelberg (2001) 2. Alur, R., Grosu, R., Hur, Y., Kumar, V., Lee, I.: Modular Specification of Hybrid Systems in CHARON. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, p. 6. Springer, Heidelberg (2000) 3. Bio-PEPA Workbench Home Page, http://www.dcs.ed.ac.uk/home/stg/software/biopepa/

Bio-PEPA with Events

65

4. Bockmayr, A., Courtois, A.: Using hybrid concurrent constraint programming to model dynamic biological systems. In: Stuckey, P.J. (ed.) ICLP 2002. LNCS, vol. 2401, p. 85. Springer, Heidelberg (2002) 5. Bortolussi, L., Policriti, A.: Hybrid Approximation of Stochastic Concurrent Constraint Programming. Constraints 13(1-2), 66–90 (2008) 6. Bortolussi, L., Policriti, A.: Hybrid Systems and Biology. Continuous and Discrete Modeling for Systems Biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 424–448. Springer, Heidelberg (2008) 7. Calder, M., Gilmore, S., Hillston, J.: Automatically deriving ODEs from process algebra models of signalling pathways. In: Proc. of CMSB 2005, pp. 204–215 (2005) 8. Calder, M., Gilmore, S., Hillston, J.: Modelling the influence of RKIP on the ERK signalling pathway using the stochastic process algebra PEPA. In: Priami, C., Ing´ olfsd´ ottir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 1–23. Springer, Heidelberg (2006) 9. Ciocchetta, F., Hillston, J.: Bio-PEPA: an extension of the process algebra PEPA for biochemical networks. In: Proc. of FBTC 2007. ENTCS, vol. 194(3), pp. 103– 117 (2008) 10. Ciocchetta, F., Hillston, J.: Bio-PEPA: a framework for the modelling and analysis of biological systems. Theoretical Computer Science (to appear) 11. Ciocchetta, F.: Bio-PEPA with SBML-like Events. In: Proc. of the Workshop Computational Models for Cell Processes, TUCS general publication, vol. 47 (2008) 12. Ciocchetta, F., Hillston, J.: Bio-PEPA: a framework for the modelling and analysis of biological systems. School of Informatics University of Edinburgh Technical Report, EDI-INF-RR-1231 (2008) 13. Cota, B.A., Sargent, R.B.: Simultaneous events and distributed simulation. In: Proc. of the Winter Simulation Conference (1990) 14. Dematt´e, L., Priami, C., Romanel, A.: The BlenX Language: a Tutorial. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 313–365. Springer, Heidelberg (2008) 15. Deshpande, A., Gollu, A., Semenzato, L.: SHIFT Programming Language and Run-Time System for Dynamic Networks of Hybrid Automata. PATH Report, http://path.berkeley.edu/SHIFT/publications.html 16. Edelstein, S.J., Schaad, O., Henry, E., Bertrand, D., Changgeux, J.P.: A kinetic mechanism for nicotin acetylcholine receptors based on multiple allosteric transitions. Biol. Cybern. 75, 361–379 (1996) 17. Galpin, V., Hillston, J., Bortolussi, L.: HYPE applied to the modelling of hybrid biological systems. ENTCS, vol. 218, pp. 33–51 (2008); Also in Proceedings of MFPS 2008 18. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry 81, 2340–2361 (1977) 19. Henzinger, T.A.: The Theory of Hybrid Automata. In: The proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science, LICS (1996) 20. Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: HyTech: A Model Checker for Hybrid Systems. Software Tools for Technology Transfer 1, 110–122 (1997) 21. HybridSal home page, http://sal.csl.sri.com/hybridsal/ 22. Hucka, M., Finney, A., Hoops, S., Keating, S., Le Nov´ere, N.: Systems Biology Markup Language (SBML) Level 2: Structures and Facilities for Model Definitions, http://sbml.org/documents/

66

F. Ciocchetta

23. Lincoln, P., Tiwari, A.: Symbolic systems biology: Hybrid modeling and analysis of biological networks. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 660–672. Springer, Heidelberg (2004) 24. Le Nov´ere, N., Bornstein, B., Broicher, A., Courtot, M., Donizelli, M., Dharuri, H., Li, L., Sauro, H., Schilstra, M., Shapiro, B., Snoep, J.L., Hucka, M.: BioModels Database: a Free, Centralized Database of Curated, Published, Quantitative Kinetic Models of Biochemical and Cellular Systems. Nucleic Acids Research 34, D689–D691 (2006) 25. Priami, C., Quaglia, P.: Beta-binders for biological interactions. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 20–33. Springer, Heidelberg (2005) 26. Prism web site, http://www.prismmodelchecker.org/ 27. Priami, C., Regev, A., Silverman, W., Shapiro, E.: Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Information Processing Letters 80, 25–31 (2001) 28. Dematt´e, L., Priami, C., Romanel, A.: The Beta Workbench: a computational tool to study the dynamics of biological systems. Briefings in Bioinformatics 9(5), 437–449 (2008)

A

Appendix: Bio-PEPA System for the Acetylcholine Receptor Model

In this appendix we report the speciﬁcation of the whole Acetylcholine receptor model in Bio-PEPA. Note that, in the deﬁnition of the species component, we use the following notation: >> indicates a product (it corresponds to the operator ↑ in the Bio-PEPA syntax) and >B1 + (alpha_r_0,1)B1 + (alpha_r_6,1)B2 + (alpha_r_1,1)A0 + (alpha_r_5,1)A1 + (alpha_r_3,1)A2 + (alpha_r_2,1)A2 + (alpha_r_4,1)A2 + (alpha_r_11,1)>>A2 I0 = (alpha_f_7,1)I0 + (alpha_f_9,1)>>I0 + (alpha_r_9,1)I1 + (alpha_r_7,1)I2 + (alpha_r_8,1)I2 + (alpha_r_11,1)D0 +

67

68

F. Ciocchetta

(alpha_r_14,1)>>D0 D1 = (alpha_f_12,1)>>D1 + (alpha_r_12,1)D2 + (alpha_r_13,1)D2 + (alpha_r_16,1) k3bn : (s’=2) ; The next forward transition, from state s=3 to state s=4 in the Prism model, is a combination of several detailed steps of the translation mechanism involving the processing of GTP. The transition is one-directional, again with a signiﬁcant diﬀerence in the rate k3fc for a cognate aa-tRNA compared to the rates k3fp and k3fn for pseudo-cognate and near-cognate aa-tRNA, that are equal. // GTPase activation, GTP hydrolysis // and EF-Tu conformation change [ ] (s=3) & cogn -> k3fc : (s’=4) ; [ ] (s=3) & pseu -> k3fp : (s’=4) ; [ ] (s=3) & near -> k3fn : (s’=4) ; In state s=4, the aa-tRNA can either be rejected, after which control moves to intermediate state s=5, or accommodates, i.e. the ribosome reconforms such that the aa-tRNA can hand over the amino acid it carries, so-called peptidyl transfer. In the latter case, control changes to state s=6. As before, rates for cognates and those for pseudo-cognates and near-cognates are of diﬀerent magnitudes. From the intermediate rejection state s=5, with all booleans set to false again, control returns to the start state s=1.

In Silico Modelling and Analysis of Ribosome Kinetics

77

// rejection [ ] (s=4) & cogn -> k4rc : (s’=5) & (cogn’=false) ; [ ] (s=4) & pseu -> k4rp : (s’=5) & (pseu’=false) ; [ ] (s=4) & near -> k4rn : (s’=5) & (near’=false) ; // accommodation, peptidyl [ ] (s=4) & cogn -> k4fc : [ ] (s=4) & pseu -> k4fp : [ ] (s=4) & near -> k4fn :

transfer (s’=6) ; (s’=6) ; (s’=6) ;

After some movement back-and-forth between state s=6 and state s=7, the binding of the EF-G complex becomes permanent. In the detailed translation mechanism a number of (mainly sequential) steps follows, that are summarized in the Prism model by a single transition to a ﬁnal state s=8, that represents elongation of the protein in nascent with the amino acid carried by the aa-tRNA. The synthesis is successful if the aa-tRNA was either a cognate or pseudo-cognate for the codon under translation, reﬂected by either cogn or pseu being true. In case the aa-tRNA was a near-cognate (non-cognates never pass beyond state s=2), an amino acid that does not correspond to the codon in the genetic code has been inserted. Thus, in this case, an insertion error has occurred. // EF-G binding [ ] (s=6) -> k6f : (s’=7) ; [ ] (s=7) -> k7b : (s’=6) ; // GTP hydrolysis, unlocking, tRNA movement // Pi release, rearrangements of ribosome and EF-G // dissociation of GDP [ ] (s=7) -> k7f : (s’=8) ; A number of transitions, linking the dissociation state s=0 and the rejection state s=5 back to the start state s=1, where a race of aa-tRNAs of the four types commences anew, and looping at the ﬁnal state s=8, complete the Prism model. The transitions are deterministically taking, as no other transitions leave these states. Having no biological counterpart the transitions are assigned a high-rate making the time they take negligible. // no entrance, re-entrance at state 1 [ ] (s=0) -> FAST : (s’=1) ; // rejection, re-entrance at state 1 [ ] (s=5) -> FAST : (s’=1) ; // elongation [ ] (s=8) -> FAST : (s’=8) ; Table 2 collects the rates as compiled from the biological literature and used in the Prism model above.

78

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

Table 2. Rates of the Prism model, adapted from [9,26]. Rate k2bx is based on the estimate of the average delay of non-cognate arrivals of 0.5ms. Rates k4fc, k4fp, k4fn and k7f are accumulative rates of sequentially composed transitions. k1f k2f k2b k2bx

140 190 85 2000

k3fc k3fp, k3fn k3bc k3bp, k3bn

260 0.40 0.23 80

k4rc k4rp, k4rn k4fc k4fp, k4fn

60 FAST 166.7 46.1

k6f k7f k7b

150 145.8 140

In the next two sections, we will study the Prism model described above for the analysis of the probability for insertion errors, i.e. extension of the peptidyl chain with a diﬀerent amino acid than the codon codes for, and of the average insertion times, i.e. the average time it takes to process a codon up to elongation.

4

Insertion Errors

In this section we discuss how the model checking features of Prism can be exploited to predict the misreading frequencies for individual codons. The translation of mRNA into a polypeptide chain is performed by the ribosome machinery with high precision. Experimental measurements show that on average, only one in 1,000 to 10,000 amino acids is added wrongly (cf. [12]).3 For a codon under translation, a pseudo-cognate anticodon carries precisely the amino acid that the codon codes for. Therefore, although diﬀerent in codonanticodon bound, successful matching of a pseudo-cognate does not lead to an insertion error, as –accidentally– the right amino acid has been used for elongation. In our model, the main diﬀerence of cognates vs. pseudo-cognates and near-cognates is in the kinetics. At various stages of the peptidyl transfer the rates for true cognates diﬀer from those for pseudo-cognates and near-cognates up to three orders of magnitude. Figure 2 depicts the relevant abstract automaton, derived from the Prism model discussed above. See also Table 1. In case a transition is labeled with two rates, e.g. 0.23/80, the leftmost number, viz. 0.23, concerns the processing of a cognate aa-tRNA, while the rightmost number, viz. 80, that of a pseudo-cognate or near-cognate. In three states a probabilistic choice has to be made: in state 2 leading to state 0 or 3, in state 3 leading back to state 3 or forward to state 4, and in state 4 leading to rejection in state 5 or eventually to success via state 6. The probabilistic choice in state 2 is the same for cognates, pseudo-cognates and near-cognates alike, the ones in state 3 and in state 4 depend on the type of aa-tRNA, cognates and pseudo-cognates vs. near-cognates. A cognate aa-tRNA starting in state 1 will move to state 2 with probability 1. From here, it will dissociate with probability 85/(85 + 190) ≈ 0.309, moving to 3

Our ﬁndings, see Table 5, based on the kinetic rates available and the assumptions made, are well within these boundaries.

In Silico Modelling and Analysis of Ribosome Kinetics

FAST

79

5 60/FAST 0.23/80

1

2

FAST

0

190

3

260/0.40

4

167/46

6

7

8

85

Fig. 2. Abstract automaton summarizing the Prism code. See also Table 1.

state 0, or will be recognized with the complementary probability 190/(85 + 190) ≈ 0.691, moving to state 3. The same holds for pseudo-cognate and nearcognate aa-tRNA. However, after recognition in state 3, a cognate aa-tRNA will go through the hydrolysis phase leading to state 4 for a fraction 0.999 of the cases (computed as 260/(0.23 + 260)), a fraction being close to 1. In contrast, for a pseudo-cognate or near-cognate aa-tRNA this is 0.40/(0.40 + 80) ≈ 0.005 only. A similar diﬀerence can be noted in state 4 itself. Cognates will accommodate and continue to state 6 with probability 0.736, while pseudo-cognates and nearcognates will do so with the small probability 0.044, the constant FAST being set to 1000 in our experiments as in [9]. Since the transition from state 4 to state 6 is irreversible, the rates of the remaining transitions are not of importance here. For cognates, pseudo-cognates and near-cognates, the probability of reaching state 8 in one attempt can be easily computed, solving a small system of equations by hand or by using Prism. In the latter case, we have Prism evaluate the CSL-formula P=? [ (s!=0 & s!=5) U (s=8) {(s=2) & cogn} ] against our model. The formula asks to establish the probability for all paths where s is not set to 0 nor 5, until s have been set to 8, starting from the (unique) state satisfying s=2 & cogn. The expression {(s = 2)&cogn} is a so-called ﬁlter construction as supported by Prism. We obtain psc = 0.508, psp = 0.484 · 10−4 and psn = 0.484 · 10−4, with psc the probability for a cognate to end up in state 8 —and elongate the peptidyl chain— without going through state 0 nor state 5; psp and psn the analogues for success of pseudo- and near-cognates, respectively. Note that these values are the same for every codon. Diﬀerent among codons in E. coli are the concentrations of cognates, pseudocognates and near-cognates.4 Ultimately, the frequencies f c , f p and f n of the types of aa-tRNA in the cell, i.e. the actual number of molecules of the kind, determine the concentration of the aa-tRNA. Hence, under the usual assumption of homogeneous distribution, the frequencies determine the total rates for the arrival process of an anticodon. The probability for an anticodon arriving to be a cognate, pseudo-cognate or near-cognate can then be calculated from this. 4

See Table 4 in the appendix.

80

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

Fig. 3. Correlation of the ratio f n /f c of the frequency of near-cognates over the frequency of cognates vs. the probability of an insertion error. See also Table 5 in the appendix.

As concluded in [9] based on simulation results, the probability for an erroneous insertion, is strongly correlated with the quotient of the number of nearcognate anticodons and the number of cognate anticodons. See Figure 3. As an advantage of the present setting, this correlation actually can be formally derived. This is as follows. We have that an insertion error occurs if a near-cognate succeeds to attach its amino acid. Note that we already have established psp , psn psc . Therefore, P (error) = P (near & elongation | elongation) =

psn · (f n /tot) psc · (f c /tot) + psp · (f p /tot) + psn · (f n /tot)

≈

psn · f n psc · f c

∼

fn fc

with tot = f c + f p + f n , and where we have used that P (elongation) = (f c /tot) · psc + (f p /tot) · psp + (f n /tot) · psn . Note, the ability to precalculate the probabilities psc , psp and psn is instrumental in obtaining the above result. As such, it illustrates the approach of piecewise analysis, ﬁrst establishing quantities for part of the system to obtain a quantity for the system as a whole.

5

Competition and Insertion Times

In this section, we continue the analysis of the Prism model for translation and discuss the correlation of the average insertion time for the amino acid speciﬁed

In Silico Modelling and Analysis of Ribosome Kinetics

81

by a codon, on the one hand, and and the aa-tRNA competition, i.e. the relative abundance of pseudo-cognate and near-cognate aa-tRNAs, on the other hand. The insertion time of a codon is the average time it takes to elongate the protein in nascent with an amino acid. The average insertion time can be computed in Prism using the concept of rewards, also known as costs in Markov theory. Each state is assigned a value as its reward. Further, the reward of each state is weighted per unit of time. Hence, it is computed by multiplication with the average time spent in the state. The cumulative reward of a path in the chain is deﬁned as a sum over all states in the path of such weighted rewards per state. Thus, by assigning to each state the value 1 as reward, we obtain the total average time for a given path. For example, in Prism the cumulative reward formula R=? [ F (s=8) ] which asks to compute the expected time to reach state s=8. Recall, in state s=8 the amino acid is added to the polypeptide chain. The formula returns the average reward of all the paths that lead from the initial state 1 to state 8. As explained above, in order to obtain the average time for insertion, we assign each state the value 1 as a reward, which in Prism can be done using the following code rewards true: 1 endrewards The construct expresses the fact that 1 is assigned to any state that satisfy the condition true (which is trivially satisﬁed by all states). So, a script calling Prism for model checking the above formula then yields the expected insertion time per codon. Table 6 in the appendix lists the results. Although the correlation of cognate frequency and insertion times is limited, the qualitative claim of [25] of ‘rare’ codons being translated slow and ‘frequent’ codons being translated fast is roughly conﬁrmed by the model. E.g., the codons AGC and CCA have amongst the lowest frequencies, 420 and 617, the lowest and two but lowest frequency, respectively, and translates indeed the slowest, 1.4924 and 1.5622 seconds, respectively. However, the codon CCA with an availability of 581, of one but lowest frequency, is translated at a moderate rate of 0.5525 seconds on the average. Thus, in line with our considerations, cognate availability per se does not suﬃciently predict translation time. Comparably, the fastest insertion times, 52.7 and 64.5 milliseconds, are realized by the codons CU G and CGU , of the codons corresponding to amino acids the one and two but most abundant. The codon CU G of the highest frequency 5136, excluding stop codons, though has an average insertion time of 102.8 milliseconds. A little bit more ingenuity is needed to establish average exit times, for example for a cognate to pass from state s=2 to state s=8. The point is that conditional probabilities are involved. However, since dealing with exponential distributions, elimination of transitions in favour of adding their rates to that of the remaining ones, does the trick. Various results, some of them used below, are collected in Table 3. (The probabilities of failure and success for the non-cognates are trivial, pfx = 1 and psx = 0, with a time per failed attempt Tfx = 0.5 · 10−3 seconds.) There is a visible correlation between the quotient of the number of nearcognate aa-tRNA over the number of cognate aa-tRNA and the average insertion time. See Figure 4. In fact, the average insertion time for a codon is

82

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

Table 3. Exit probabilities and exit times (in seconds) for three types of aa-tRNA, superscripts c, p and n for cognate, pseudo-cognate and near-cognate aa-tRNA, respectively. Failure for exit to states s=0 or s=5, subscript f ; success for exit to state s=8, subscript s. psc 0.5079 psp 4.847 · 10−4 psn 4.847 · 10−4

pfc 0.4921 pfp 0.9995 pfn 0.9995

Tsc 0.03182 Tsp 3.251 Tsn 3.251

Tfc 9.342 · 10−3 Tfp 0.3914 Tfn 0.3914

Fig. 4. Correlation of the ratio (f p + f n )/f c of total frequency of pseudo-cognates and near-cognates over the frequency of cognates vs. average insertion times. See also Table 6 in the appendix.

approximately proportional to the near-cognate/cognate ratio. This can be seen as follows. The insertion of the amino acid is completed if state s=8 is reached, either for a cognate, pseudo-cognate or near-cognate. As we have seen, the probability for either of the latter two is negligible, psp , psn = 4.847 · 10−4 . Therefore, the number of cognate arrivals is decisive. With pfc and psc being the probability for a cognate to fail, i.e. exit at state s=0 or s=5, or to succeed, i.e. reach of state s=8, the insertion time Tins can be regarded as a geometric series. (Note the exponent i below.) Important are the numbers of arrivals of the other aa-tRNA types per single cognate arrival, expressed in terms of frequencies. An arrival occurring for the (i + 1)st arrival of a cognate has spent (i × Tfc ) + c Ts processing cognate aa-tRNA. The number of pseudo-cognate, near-cognate and non-cognate arrivals per individual cognate arrival are, on the average, the f f f relative fractions p , n , and x , respectively (with f p , f n , and f c as before in fc f c fc

In Silico Modelling and Analysis of Ribosome Kinetics (p)

(n)

(x)

(cf) (p)

(n)

(x)

(cf) (p)

(n)

83 (x)

(cs)

Fig. 5. Accumulated delay after three cognate arrivals: (p) delay (f p /f c )·Tfp for failing pseudo-cognates, (n) delay (f n /f c )·Tfn for failing near-cognates, (x) delay (f x /f c )·Tfx for non-cognates, (cf) exit time Tfc for a failing cognate, (cs) exit time Tsc for a successful cognate.

Section 4, and f x the frequency of non-cognate aa-tRNA). See Figure 5. Summing over i, the number of failing cognate aa-tRNA for a successful cognate insertion, yields ∞

i

(pfc ) psc · (delay for i failing and 1 successful cognate arrivals) f f f i = (pfc ) psc · (i + 1) · ( p Tfp + n Tfn + x Tfx ) + i · Tfc + Tsc fc fc fc ∞ i ≈ f p +f n psc Tfn i=0 (i + 1) · (pfc ) fc ∼ f p +f n . fc

Tins =

i=0 ∞ i=0

fx x T is fc f n relatively small, from which it follows that f p +f n Tf is the dominant summand. fc Note that the estimate is not accurate for small values of f p + f n . Nevertheless, closer inspection shows that for these values the approximation remains orderpreserving. Again, the results obtained for parts of the systems are pivotal in the derivation.

Here, we have used that Tfc and Tsc are negligible, Tfp equals Tfn , and

6

Concluding Remarks

In this paper, we presented a stochastic model of the translation process based on presently available data of ribosome kinetics [12,9]. We used the model checking facilities of the Prism tool for continuous-time Markov chains. Compared to [9] that uses simulation, our approach is computationally more reliable (independent on the number of simulations) and has faster response times (taking seconds rather then minutes or hours). More importantly, model checking allowed us to perform piecewise analysis of the system, yielding better insight in the model compared to just observing the end-to-end results with a monolithic model. Based on this, we improved on earlier observations, regarding error probabilities and insertion times, by actually deriving the correlation suggested by the data. In [7] a correlation was reported between the number of copies (concentrations) of cognate tRNAs and the frequency of usage of particular codons in the most abundant proteins in E. coli. It is suggested that this optimization is favorable for the cell growth: when they are urgently needed the most used proteins are translated with maximum speed and accuracy. On the other hand, we observed that there is a high correlation (0.86) between the cognate tRNA concentrations

84

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

and the ration near-cognates vs. cognates which, according to our model, determines the error probabilities. Consequently, it would be interesting to check if there exists even better correlation between the near-cognates/cognates ratios and the codon usage frequencies than between the latter and the concentrations. In conclusion, we have experienced aa-tRNA competition as a very interesting biological case study of intrinsic stochastic nature, falling in the category of the well known lambda-phage example [1]. Our model opens a new avenue for future work on biological systems that possess intrinsically probabilistic properties. It would be interesting to apply our method to processes which, similarly to translation, involve small numbers of molecules, like DNA replication [16,19], DNA repair [11,20], charging of the tRNAs with amino acids [8,15], etc., thus rendering approaches based on ordinary diﬀerential equations less attractive. Acknowledgments. We are grateful to Timo Breit, Christiaan Henkel, Erik Luit, Jasen Markovski, and Hendrik Viljoen for fruitful discussions and constructive feedback.

References 1. Arkin, A., Ross, J., McAdams, H.H.: Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected. Escherichia coli cells. Genetics 149, 1633–1648 (1998) 2. Baier, C., Katoen, J.-P., Hermanns, H.: Approximate symbolic model checking of continuous-time Markov chains. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 146–161. Springer, Heidelberg (1999) 3. Boˇsnaˇcki, D., ten Eikelder, H.M.M., Steijaert, M.N., de Vink, E.P.: Stochastic analysis of amino acid substitution in protein synthesis. In: Heiner, M., Uhrmacher, A.M. (eds.) CMSB 2008. LNCS (LNBI), vol. 5307, pp. 367–386. Springer, Heidelberg (2008) 4. Calder, M., Vyshemirsky, V., Gilbert, D., Orton, R.: Analysis of signalling pathways using continuous time Markov chains. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 44–67. Springer, Heidelberg (2006) 5. Chabrier, N., Fages, F.: Symbolic model checking of biochemical networks. In: Priami, C. (ed.) CMSB 2003. LNCS, vol. 2602, pp. 149–162. Springer, Heidelberg (2003) 6. Danos, V., Feret, J., Fontana, W., Harmer, R., Krivine, J.: Rule-based modelling of cellular signalling. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 17–41. Springer, Heidelberg (2007) 7. Dong, H., Nilsson, L., Kurland, C.G.: Co-variation of tRNA abundance and codon usage in Escherichia coli at diﬀerent growth rates. Journal of Molecular Biology 260, 649–663 (1996) 8. Nureki, O., et al.: Enzyme structure with two catalytic sites for double-sieve selection of substrate. Science 280, 578–582 (1998) 9. Fluitt, A., Pienaar, E., Viljoen, H.: Ribosome kinetics and aa-tRNA competition determine rate and ﬁdelity of peptide synthesis. Computational Biology and Chemistry 31, 335–346 (2007)

In Silico Modelling and Analysis of Ribosome Kinetics

85

10. Gilchrist, M.A., Wagner, A.: A model of protein translation including codon bias, nonsense errors, and ribosome recycling. Journal of Theoretical Biology 239, 417– 434 (2006) 11. Goodman, M.F.: Coping with replication ‘train wrecks’ in Escherichia coli using Pol V, Pol II and RecA proteins. Trends in Biochemical Sciences 25, 189–195 (2000) 12. Gromadski, K.B., Rodnina, M.V.: Kinetic determinants of high-ﬁdelity tRNA discrimination on the ribosome. Molecular Cell 13(2), 191–200 (2004) 13. Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic model checking of complex biological pathways. In: Priami, C. (ed.) CMSB 2006. LNCS (LNBI), vol. 4210, pp. 32–47. Springer, Heidelberg (2006) 14. Heyd, A.W., Drew, D.A.: A mathematical model for elongation of a peptide chain. Bulletin of Mathematical Biology 65, 1095–1109 (2003) 15. Ibba, M., S¨ oll, D.: Aminoacyl-tRNAs: setting the limits of the genetic code. Genes & Development 18, 731–738 (2004) 16. Johnson, K.A.: Conformational coupling in DNA polymerase ﬁdelity. Annual Reviews in Biochemistry 62, 685–713 (1993) 17. Karp, G.: Cell and Molecular Biology, 5th edn. Wiley, Chichester (2008) 18. Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model cheking with Prism: a hybrid approach. Journal on Software Tools for Technology Transfer 6, 128–142 (2004), http://www.prismmodelchecker.org/ 19. Martomo, S.A., Mathews, C.K.: Eﬀects of biological DNA precursor pool asymmetry upon accuracy of DNA replication in vitro. Mutation Research 499, 197–211 (2002) 20. Ni, M., Wang, S.-Y., Li, J.-K., Ouyang, Q.: Simulating the temporal modulation of inducible DNA damage response in Escherichia coli. Biophysical Journal 93, 62–73 (2007) 21. Pape, T., Wintermeyer, W., Rodnina, M.: Complete kinetic mechanism of elongation factor Tu-dependent binding of aa-tRNA to the A-site of E. coli. EMBO Journal 17, 7490–7497 (1998) 22. Priami, C., Regev, A., Shapiro, E., Silverman, W.: Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Information Processing Letters 80, 25–31 (2001) 23. Rodnina, M.V., Wintermeyer, W.: Ribosome ﬁdelity: tRNA discrimination, proofreading and induced ﬁt. Trends in Biochemical Sciences 26(2), 124–130 (2001) 24. Savelsbergh, A., et al.: An elongation factor G-induced ribosome rearrangement precedes tRNA–mRNA translocation. Molecular Cell 11, 1517–1523 (2003) 25. Sørensen, M.A., Kurland, C.G., Pedersen, S.: Codon usage determines translation rate in Escherichia coli. Journal of Molecular Biology 207, 365–377 (1989) 26. Viljoen, H.: Private communication (2008) overproduction in 27. Wahab, S.Z., Rowley, K.O., Holmes, W.M.: Eﬀects of tRNALeu 1 Escherichia coli. Molecular Microbiology 7, 253–263 (1993)

A

Appendix: Suplementary Figures and Data

// translation model

stochastic

86

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

// constants const double ONE=1; const double FAST=1000; // tRNA rates const double c_cogn const double c_pseu const double c_near const double c_nonc const const const const const const const const const const const const const const const const const const const

double double double double double double double double double double double double double double double double double double double

; ; ; ;

k1f = 140; k2b = 85; k2bx=2000; k2f = 190; k3bc= 0.23; k3bp= 80; k3bn= 80; k3fc= 260; k3fp= 0.40; k3fn= 0.40; k4rc= 60; k4rp=FAST; k4rn=FAST; k4fc= 166.7; k4fp= 46.1; k4fn= 46.1; k6f = 150; k7b = 140; k7f = 145.8;

module ribosome s : [0..8] init 1 ; cogn : bool init false pseu : bool init false near : bool init false nonc : bool init false

; ; ; ;

// initial binding [ ] (s=1) -> k1f * c_cogn : (s’=2) & (cogn’=true) ; [ ] (s=1) -> k1f * c_pseu : (s’=2) & (pseu’=true) ; [ ] (s=1) -> k1f * c_near : (s’=2) & (near’=true) ; [ ] (s=1) -> k1f * c_nonc : (s’=2) & (nonc’=true) ; [ ] (s=2) & ( cogn | pseu | near ) -> k2b : (s’=0) & (cogn’=false) & (pseu’=false) & (near’=false) ; [ ] (s=2) & nonc -> k2bx : (s’=0) & (nonc’=false) ;

In Silico Modelling and Analysis of Ribosome Kinetics

// codon recognition [ ] (s=2) & ( cogn | pseu | near ) -> k2f : (s’=3) ; [ ] (s=3) & cogn -> k3bc : (s’=2) ; [ ] (s=3) & pseu -> k3bp : (s’=2) ; [ ] (s=3) & near -> k3bn : (s’=2) ; // GTPase [ ] (s=3) [ ] (s=3) [ ] (s=3)

activation, GTP hydrolysis, reconformation & cogn -> k3fc : (s’=4) ; & pseu -> k3fp : (s’=4) ; & near -> k3fn : (s’=4) ;

// rejection [ ] (s=4) & cogn -> k4rc : (s’=5) & (cogn’=false) ; [ ] (s=4) & pseu -> k4rp : (s’=5) & (pseu’=false) ; [ ] (s=4) & near -> k4rn : (s’=5) & (near’=false) ; // accommodation, peptidyl [ ] (s=4) & cogn -> k4fc : [ ] (s=4) & pseu -> k4fp : [ ] (s=4) & near -> k4fn :

transfer (s’=6) ; (s’=6) ; (s’=6) ;

// EF-G binding [ ] (s=6) -> k6f : (s’=7) ; [ ] (s=7) -> k7b : (s’=6) ; // GTP hydrolysis, unlocking, // tRNA movement and Pi release, // rearrangements of ribosome and EF-G, // dissociation of GDP [ ] (s=7) -> k7f : (s’=8) ; // no entrance, re-entrance at state 1 [ ] (s=0) -> FAST*FAST : (s’=1) ; // rejection, re-entrance at state 1 [ ] (s=5) -> FAST*FAST : (s’=1) ; // elongation [ ] (s=8) -> FAST*FAST : (s’=8) ; endmodule rewards true : 1; endrewards

87

88

D. Boˇsnaˇcki, T.E. Pronk, and E.P. de Vink

Table 4. Frequencies of cognate, pseudo-cognate, near-cognate and non-cognates for E. coli as molecules per cell [7]. Stop codons UGA, UAG and UAA. codon cognate pseudo- nearnoncognate cognate cognate

codon cognate pseudo- nearnoncognate cognate cognate

UUU UUC UUG UUA UCU UCC UCG UCA UGU UGC UGG UGA∗ UAU UAC UAG∗ UAA∗

1037 1037 2944 1031 2060 764 1296 1296 1587 1587 943 6219 2030 2030 1200 7200

0 0 0 1913 344 1640 764 1108 0 0 0 0 0 0 0 0

2944 9904 2324 2552 0 4654 2856 1250 1162 4993 4063 4857 0 3388 5230 4576

67493 60533 66206 65978 69070 64416 66558 67820 68725 64894 66468 60398 69444 66056 65044 59698

GUU GUC GUG GUA GCU GCC GCG GCA GGU GGC GGG GGA GAU GAC GAG GAA

5105 1265 3840 3840 3250 617 3250 3250 4359 4359 2137 1069 2396 2396 4717 4717

0 3840 1265 1265 617 3250 617 617 2137 2137 4359 5427 0 0 0 0

0 7372 1068 9036 0 8020 1068 9626 0 4278 0 11807 4717 10958 3464 10555

66369 58997 65301 57333 67607 59587 66539 57981 64978 60700 64978 53171 64361 58120 63293 56202

CUU CUC CUG CUA CCU CCC CCG CCA CGU CGC CGG CGA CAU CAC CAG CAA

943 943 5136 666 1301 1913 1481 581 4752 4752 639 4752 639 639 881 764

5136 5136 943 5413 900 943 720 1620 639 639 4752 639 0 0 764 881

4752 1359 2420 1345 4752 2120 5990 1430 0 2302 6251 2011 6397 3308 6648 1886

60643 64036 62975 64050 64521 66498 63283 67843 66083 63781 59832 64072 64438 67527 63181 67943

AUU AUC AUG AUA ACU ACC ACG ACA AGU AGC AGG AGA AAU AAC AAG AAA

1737 1737 706 1737 2115 1199 1457 916 1408 1408 420 867 1193 1193 1924 1924

1737 1737 1926 1737 541 1457 1199 1740 0 0 867 420 0 0 0 0

2632 6432 4435 6339 0 4338 4789 2791 1287 5416 6318 4248 1924 6268 6523 2976

65368 61568 64407 61661 68818 64480 64029 66027 68779 64650 63869 65939 68357 64013 63027 66574

In Silico Modelling and Analysis of Ribosome Kinetics

Table 5. Probabilities per codon for erroneous elongation UUU UUC UUG UUA UCU UCC UCG UCA UGU UGC UGG UGA UAU UAC UAG UAA

27.4e-4 91.2e-4 7.59e-4 23.5e-4 2.81e-10 56.1e-4 20.3e-4 9.09e-4 6.97e-4 30.4e-4 39.8e-4 7.50e-4 2.81e-10 15.7e-4 41.3e-4 6.04e-4

CUU CUC CUG CUA CCU CCC CCG CCA CGU CGC CGG CGA CAU CAC CAG CAA

46.7e-4 13.6e-4 4.49e-4 18.9e-4 34.1e-4 10.4e-4 37.6e-4 22.8e-4 1.21e-10 4.59e-4 88.7e-4 3.98e-4 91.1e-4 47.5e-4 69.4e-4 22.7e-4

GUU GUC GUG GUA GCU GCC GCG GCA GGU GGC GGG GGA GAU GAC GAG GAA

1.12e-10 55.0e-4 2.68e-4 22.3e-4 1.77e-10 12.5e-4 3.187e-4 28.2e-4 1.32e-10 9.40e-4 2.72e-10 100.3e-4 18.6e-4 43.2e-4 7.09e-4 21.4e-4

AUU AUC AUG AUA ACU ACC ACG ACA AGU AGC AGG AGA AAU AAC AAG AAA

14.4e-4 35.0e-4 58.3e-4 34.4e-4 2.73e-10 34.2e-4 31.7e-4 29.1e-4 8.70e-4 37.2e-4 140.7e-4 48.1e-4 15.2e-4 49.3e-4 32.1e-4 14.6e-4

Table 6. Estimated average insertion time per codon in seconds UUU UUC UUG UUA UCU UCC UCG UCA UGU UGC UGG UGA UAU UAC UAG UAA

0.3327 0.8404 0.1245 0.4436 0.0893 0.7409 0.3035 0.2313 0.1432 0.3296 0.4360 0.1098 0.0758 0.2008 0.4319 0.0963

CUU CUC CUG CUA CCU CCC CCG CCA CGU CGC CGG CGA CAU CAC CAG CAA

0.8901 0.6286 0.1028 0.9217 0.4202 0.1992 0.4257 0.5535 0.0645 0.1010 1.3993 0.0962 0.8811 0.5341 0.7425 0.4058

GUU GUC GUG GUA GCU GCC GCG GCA GGU GGC GGG GGA GAU GAC GAG GAA

0.0527 0.7670 0.1041 0.2604 0.0756 1.5622 0.1010 0.3002 0.0924 0.1673 0.2308 1.2989 0.2180 0.4144 0.1106 0.2243

AUU AUC AUG AUA ACU ACC ACG ACA AGU AGC AGG AGA AAU AAC AAG AAA

0.2733 0.4373 0.8115 0.4321 0.0943 0.4658 0.4073 0.5025 0.1636 0.3905 1.4924 0.5517 0.2242 0.4959 0.3339 0.1945

89

Qualitative and Quantitative Analysis of a Bio-PEPA Model of the Gp130/JAK/STAT Signalling Pathway Maria Luisa Guerriero Laboratory for Foundations of Computer Science, The University of Edinburgh, UK

Abstract. Computational modelling of complex biochemical systems has grown in importance over recent years as a tool for supporting biological studies. Consequently, several formal languages have been recently proposed as modelling languages for biology. Among these, process algebras have been proved capable of providing researchers with new hypotheses on the behaviour of biochemical systems. Bio-PEPA is a process algebra recently defined for the modelling and analysis of biochemical systems, which provides modellers with a wide range of analysis techniques: models can be analysed by stochastic simulation, model-checking, and mathematical methods based on ordinary diﬀerential equations. In this work, we use Bio-PEPA for modelling the gp130/JAK/STAT signalling pathway, and we use both stochastic simulation and model-checking to analyse several qualitative and quantitative aspects of the system.

1 Introduction Several modelling approaches have been used over recent years to analyse complex biological systems such as signaling pathways, ranging from traditional mathematical methods based on diﬀerential equations to computational methods based on stochastic simulation and model-checking. Each of these techniques can be more suitable than others in some context or to study some particular features of biological systems. Process algebras are formal languages traditionally used to model distributed systems of concurrent computing devices. Starting from the biochemical π-calculus [1], several other process algebras have been recently adapted in order to model biochemical systems [2,3,4,5], following the “molecules as processes” paradigm introduced in the landmark paper [6]: molecules are modelled as concurrent processes, and biochemical reactions are represented by actions performed by synchronising processes. Bio-PEPA [7,8] is a process algebra specifically defined to model and analyse biochemical networks. Compared to other process algebras, Bio-PEPA uses a more abstract view of biochemical systems, the so-called “species as processes” abstraction: processes represent molecular species instead of single molecules, and multi-way synchronisations of processes represent changes in the amounts of molecular species resulting from biochemical reactions. Such an abstract view enables modellers to deal with analysis techniques which are computationally infeasible when considering the “molecules as processes” abstraction. C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 90–115, 2009. c Springer-Verlag Berlin Heidelberg 2009

Qualitative and Quantitative Analysis of a Bio-PEPA Model

91

The main feature of Bio-PEPA is that it integrates several kinds of analysis techniques. Both discrete stochastic and continuous deterministic models can be automatically generated from Bio-PEPA models, thus allowing modellers to perform time-series analysis via stochastic simulation, Markovian analysis and ordinary diﬀerential equations (ODEs); in addition, system properties can be verified through model-checking and mathematical techniques such as bifurcation, stability and continuation analysis. Moreover, as for the other process algebras, Bio-PEPA is equipped with an operational semantics which supports various kinds of formal analysis (e.g. causality, equivalence, and reachability analysis). In this work, we define a Bio-PEPA model of the gp130/JAK/STAT signalling pathway, a well-studied system which plays a major role in several biological processes both in human and other organisms. A lot of experimental data is available about the molecules in the pathway, and some mathematical and computational models have been already developed. For these reasons, the gp130/JAK/STAT pathway represents a good case study for exploiting some of the possible Bio-PEPA analysis methods in order to study diﬀerent aspects (both qualitative and quantitative) of the system, and compare them with existing models. The rest of the paper is structured as follows. First, the Bio-PEPA language is introduced in Sec. 2, while the pathway and the Bio-PEPA model are described in Sec. 3 and Sec. 4, respectively. The following three sections are devoted to the analysis of the model: in Sec. 5 several qualitative properties are analysed via model-checking, in Sec. 6 we present some stochastic simulation results, and in Sec. 7 model-checking is employed for quantitative analysis. Finally, Sec. 8 is an overview of the related work and Sec. 9 contains some concluding remarks.

2 Bio-PEPA Bio-PEPA [7,8] is a process algebra which has been recently defined for the modelling and analysis of biochemical networks. It is a biologically-inspired language based on PEPA [9] and, diﬀerently from PEPA and other process algebras, it is able to explicitly represent details such as stoichiometric coeﬃcients and the roles of species in reactions, and it supports the definition of general kinetic laws. Bio-PEPA models can be analysed by diﬀerent techniques (stochastic simulation, analysis based on ODEs, numerical solution of the continuous-time Markov chain (CTMC), and probabilistic model-checking), since the mappings of Bio-PEPA models into specifications for those approaches have been defined [10]. The Bio-PEPA language is based on discrete levels of parameterised species: each component represents a species and its parameter may be interpreted as the number of molecules or discrete levels of concentration depending on the type of analysis to be applied. Parametric levels are considered for the definition of the transition system and for the derivation of a CTMC whose states represent the concentration levels of the species. The syntax of Bio-PEPA is defined as: S ::= (α, κ) op S | S + S | C where op = ↓ | ↑ | ⊕ | | .

P ::= P P | S (x) I

92

M.L. Guerriero

The component S is called a species component and abstracts a molecular species, whereas the component P, called a model component, describes the system and the interactions among components. The prefix term (α, κ) op S contains information about the role of the species in the reaction associated with the action type α: κ is the stoichiometric coeﬃcient of the species and the prefix combinator “op” represents its role in the reaction. Specifically, ↓ indicates a reactant, ↑ a product, ⊕ an activator, an inhibitor and a generic modifier. The operator “+” expresses the choice between possible acdef tions and the constant C is defined by an equation C = S . The parameter x ∈ IR+ in S (x) represents the concentration of S . Finally, the process P Q denotes the cooperI ation between components: the set I determines those activities on which the operands are forced to synchronise. Reaction rates are defined as functional rates associated with actions. Bio-PEPA supports a modelling style in terms of concentration levels: the species amounts are discretised into a number of levels, from level 0 (i.e. species not present) to a maximum level N (which depends on the maximum concentration of the species). Each level represents an interval of concentration and the granularity of the system is expressed in terms of the step size H (i.e. the length of the concentration interval). Definition 1. A Bio-PEPA system P is a 6-tuple V, N, K, FR , Comp, P , where: V is the set of compartments, N is the set of quantities describing the species (i.e. H and N), K is the set of parameter definitions, FR is the set of functional rates, Components is the set of definitions of species components, P is the model component describing the system. For discrete state space analysis the behaviour of the system is defined in terms of an operational semantics. A Stochastic Labelled Transition System (SLTS) is defined for a Bio-PEPA system. From this we can obtain a Continuous Time Markov Chain (CTMC). Both the SLTS and the CTMC derived from Bio-PEPA are defined in terms of levels of concentration, and the generated Markov chain is called CTMC with levels. For a full description of the language semantics see [10]. The Bio-PEPA language is supported by software tools such as the Bio-PEPA Workbench [11], which automatically processes Bio-PEPA models and generates other representations in forms suitable for simulation and model-checking. For instance, the generated simulation model can be executed using the Dizzy stochastic simulator [12]. The representation which is used for discrete state space generation and analysis by numerical solution of the underlying CTMC is expressed in the reactive modules language supported by the PRISM model-checker [13]. In addition, the Bio-PEPA Workbench generates reward structures and common CSL [14] formulae used in model-checking.

3 The Gp130/JAK/STAT Signalling Pathway The gp130/JAK/STAT signalling pathway is a well-studied biological system, of great clinical interest because of its key role in human fertility, neuronal repair and

Qualitative and Quantitative Analysis of a Bio-PEPA Model

93

haematological development [15,16,17]. Much experimental data is available on this pathway, and a few mathematical and computational models [18,19,20,21] have been developed. The signalling cascade in the gp130/JAK/STAT pathway is triggered by members of the family of IL (interleukin)-6-type cytokines binding to plasma membrane receptor complexes containing the common signal transducing receptor chain gp130 (glycoprotein 130). Among the targets of gp130 signal transduction, we consider the transcription factors of the STAT (signal transducers and activators of transcription) family, in particular STAT3. A key feature of the pathway is the nuclear/cytoplasmic shuttling of STATs: upon activation, STATs can translocate into the nucleus and activate the transcription of downstream gene targets. Diﬀerent cytokines signal through the formation of diﬀerent receptor complexes, all of them containing gp130 and another subunit. We focus here on two diﬀerent cytokines: LIF (leukaemia inhibitory factor) and OSM (oncostatin M). LIF signals through an heterodimeric receptor complex gp130:LIFR. OSM exhibits the uncommon ability to signal through two diﬀerent receptor complexes: the type I OSM receptor complex (gp130:LIFR), and the type II OSM receptor complex (gp130:OSMR). Figure 1 is a graphical representation of the biochemical reactions occurring in the gp130/JAK/STAT pathway. In the inset the diﬀerent types of receptor complexes are shown.

Fig. 1. Gp130/JAK/STAT pathway: graphical representation. Full arrows represent biochemical reactions, dotted arrows represent transports, dashed arrows represent syntheses.

94

M.L. Guerriero

The molecular species we consider in the model are: two ligands (LIF and OSM), three membrane-bound receptors (gp130, LIFR and OSMR), one eﬀector (STAT3), and two inhibitors (SOCS3 and PIAS3). JAK kinase and TC-PTP phosphatase are implicitly modelled. Four compartments are involved in the system: the exosol (the extracellular space, where the two ligands are located), the cell membrane (location of the receptors), the cytosol (initial location of STAT3), and the nucleus (in which STAT3 can translocate). Receptors are activated by ligand bindings, and active receptors dimerise to form receptor complexes (gp130:LIFR or gp130:OSMR) (reaction r1 in Fig. 1). Once the receptor dimeric complex is formed, each receptor subunit (gp130, LIFR and OSMR) can undergo JAK-mediated phosphorylation (r2). STAT3 can bind on receptors’ phosphorylated sites (r3), and the binding of STAT3 leads to its activation (phosphorylation) (r4). Once phosphorylated, STAT3 dissociates from the receptor complex, and its phosphorylated site allows STAT3 to homodimerise (r5). When STAT3 is in dimeric form, it can translocate into the nucleus (r6) where it can carry out its specific functions (not modelled here): STAT3 binds to the DNA, thus activating the transcription of downstream gene targets. Nuclear STAT3 dimers are inactivated through TC-PTP -mediated dephosphorylation, which leads to the dimers’ dissociation (r7) and to STAT3 export to the cytoplasm (r8), where STAT3 can undergo additional cycles of activation. The two inhibition mechanisms considered are due to SOCS3 and PIAS3. SOCS3 is synthesised by STAT3 (r9) and it acts by competing with STAT3 in binding to receptors (r10). PIAS3 acts by binding to active nuclear STAT3 (r11).

4 The Bio-PEPA Model A Bio-PEPA model of the gp130/JAK/STAT pathway has been developed. The full model can be downloaded from [22]. The model and the reaction rates are based on [21], though some diﬀerences are present due to the conceptual diﬀerences in the used modelling languages (see Sec. 8 for a discussion of such diﬀerences). All kinetic laws are assumed to be mass-action (i.e. depending on the amount of reactants and on given kinetic constants). Each possible form of the molecular species is modelled as a distinct Bio-PEPA species component. For instance, STAT3 is modelled by four distinct species components representing, respectively, the cytoplasmic dephosphorylated monomeric form (STAT3 c), the cytoplasmic phosphorylated dimeric form (STAT3-PD c), the nuclear phosphorylated dimeric form (STAT3-PD n), and the nuclear dephosphorylated monomeric form (STAT3 n); further species components are defined for each state of each complex containing STAT3. Reactions and biochemical modifications are represented by reactions over which the involved species components synchronise. For instance, the reaction representing r7 in Fig. 1 is modelled as the reaction dephospho dedimer stat59 , which decreases the amount of STAT3-PD n and increases (with stoichiometry coeﬃcient 2) the amount of STAT3 n.

Qualitative and Quantitative Analysis of a Bio-PEPA Model

95

As an example, the definitions of the species STAT3-PD n and STAT3 n are reported (here we use the simplified syntax of the Bio-PEPA Workbench, in which the trailing S in prefix terms (α, κ) op S can be omitted). S T AT 3-PD n ::= (reloc stat cn58 , 1) ↑ + (synth socs61 , 1) ⊕ + (unbind pias80 , 1) ↑ + (dephospho dedimer stat59 , 1) ↓ + (bind pias stat80 , 1) ↓ S T AT 3 n ::= (dephospho dedimer stat59 , 2) ↑ + (reloc stat nc60 , 1) ↓ For each of the involved reactions, a functional rate specifying its kinetic rate law is defined. The ones used in the species definitions for STAT3-PD n and STAT3 n are defined as follows. 0.693 reloc stat cn58 = · STAT3-PD c ; k58 //STAT3-PD c relocation cytoplasm -> nucleus

dephospho dedimer stat59 = [k59 · STAT3-PD n] ; //STAT3-PD n dephosphorylation & dedimerisation

reloc stat nc60

0.693 = · STAT3 n ; k60

//STAT3-PD n relocation nucleus -> cytoplasm

synth socs61 = [k61 · STAT3-PD n] ; //SOCS3 synthesis by STAT3-PD n

bind pias stat80

k80 = · PIAS3 · STAT3-PD n ; nucleus · NA //PIAS3/STAT3-PD n binding

unbind pias stat80 = [k−80 · PIAS3:STAT3-PD n] ; //PIAS3/STAT3-PD n unbinding

As mentioned above, the Bio-PEPA Workbench [11] allows us to automatically generate representations of the Bio-PEPA model for diﬀerent analysis tools. In the following sections we show some of the analyses performed using these generated models. In particular, we consider the PRISM [23,13] and Dizzy [12,24] models. We use the PRISM model-checker to verify that some desired properties of the system are satisfied, and the Dizzy simulation tool to perform time-series analysis via stochastic simulation.

5 Model-Checking Based Qualitative Analysis As a first step in the analysis of the model we use the PRISM model-checker [23,13] to verify a number of qualitative properties of the system. Such properties are intended to be consistency checks on the model and they allow us to check for the presence of possible human errors in the modelling process. This kind of checks is particularly useful when modelling complex systems such as the pathway we consider here since, due to the size of the models, trivial typing errors are likely to occur and may be hard to identify.

96

M.L. Guerriero

5.1 PRISM Modelling and Specification Language PRISM [23,13] is a probabilistic model-checker, which can be used to verify properties of CTMCs. Models are described using the state-based PRISM language, and it is possible to specify quantitative properties of the system using a property specification language which includes CSL (Continuous Stochastic Logic) [25,26]. The PRISM language is composed of modules and variables. A model is composed of a number of interacting modules and each module contains a number of local variables, whose values constitute the state of the module. The global state of the model is determined by the local state of all modules. The behaviour of the modules is given by a set of guarded commands, each describing a transition which is enabled when the guard is true. A command includes an update which gives new values to the variables. PRISM properties are made up of state properties φ and path properties ψ. The syntax of PRISM properties is given by the following grammar. φ ::= true | false | expr | φ ∧ φ | φ ∨ φ | ¬φ | φ ⇒ φ | Pp [ψ] | P=? [ψ] | Sp [φ] | S=? [φ] ψ ::= X φ | φ UI φ | φ U φ | FI φ | F φ | GI φ | G φ Here expr is a boolean expression (containing literal values, identifiers and the standard arithmetic and relational operators), ∈ { } is a relational parameter, p ∈ [0, 1] is a probability, and I is an interval of IR+ . The operators Pp [ψ] and P=? [ψ] are used to express transient properties (i.e. which depend on time) whereas the operators Sp [φ] and S=? [φ] are used to express steady state properties (i.e. which hold in the long run). The result of the verification of formulae Pp [ψ] (resp. Sp [φ]) is one of the boolean values true or false depending on whether ψ (resp. φ) is satisfied. The result of the verification of formulae P=? [ψ] (resp. S=? [φ]) is the expected probability with which ψ (resp. φ) is satisfied. The operators X, U, F, and G are used to express neXt, Until, Finally, and Globally properties, respectively. Time-bounded formulae are indexed by an interval I. The PRISM language supports the specification and analysis of reward-based properties. Reward structures allow us to associate real values with certain states or transitions of the model. Such values, which can be thought of as “costs” of the specified states/transitions, are taken into account during the solution of the CTMC. In this way it is possible to reason about various quantitative measures such as “expected number of instances of processes”, “expected number of occurrences of reactions”, “expected time until a condition is satisfied”, etc. The PRISM reward language supports the expression of both instantaneous and cumulative rewards. 5.2 Model-Checking the Bio-PEPA Model with PRISM In the PRISM models generated by the Bio-PEPA Workbench, one module is defined for each species, and the module local variables are used to record the current quantity of each species. The transitions correspond to the activities of the Bio-PEPA model and the updates take the stoichiometry into account. Transition rates are specified in an auxiliary module which defines the functional rates corresponding to all the reactions.

Qualitative and Quantitative Analysis of a Bio-PEPA Model

97

Moreover, lower and upper bounds must be defined for each variable (i.e. for the amount of each species). The step size H in the Bio-PEPA model allows us to consider diﬀerent PRISM models with diﬀerent granularity, leading to systems with diﬀerent numbers of levels. As an example, we provide the PRISM definitions relative to the species STAT3-PD n and STAT3 n, which are obtained from the corresponding Bio-PEPA species definitions reported in Sec. 4. First, the lower and upper levels for both species are computed from the defined step size H and the given bounds on species amounts. MIN S T AT 3-PD n = MIN S T AT 3 n = 0 MAX S T AT 3-PD n = MAX S T AT 3 n = 1500

NL S T AT 3-PD n = NL S T AT 3 n

=

MIN S T AT 3-PD n H MIN S T AT 3 n H

NU S T AT 3-PD n = NU S T AT 3 n

=

MAX S T AT 3-PD n H MAX S T AT 3 n H

The specifications of the behaviour of STAT3-PD n and STAT3 n are given by the two following modules. The third module contains the definition of the functional rates for all reactions. module S T AT 3-PD n S T AT 3-PD n : [NL S T AT 3-PD n .. NU S T AT 3-PD n] init 0; [reloc stat cn58 ] (S T AT 3-PD n + 1 ≤ NU S T AT 3-PD n) → 1 : (S T AT 3-PD n = S T AT 3-PD [synth socs61 ] (S T AT 3-PD n + 0 ≤ NU S T AT 3-PD n) → 1 : (S T AT 3-PD n = S T AT 3-PD [dephospho dedimer stat59 ] (S T AT 3-PD n ≥ 1 + NL S T AT 3-PD n) → 1 : (S T AT 3-PD n = S T AT 3-PD [bind pias stat80 ] (S T AT 3-PD n ≥ 1 + NL S T AT 3-PD n) → 1 : (S T AT 3-PD n = S T AT 3-PD [unbind pias stat80 ] (S T AT 3-PD n + 1 ≤ NU S T AT 3-PD n) → 1 : (S T AT 3-PD n = S T AT 3-PD

n + 1); n + 0); n − 1); n − 1); n + 1);

endmodule

module S T AT 3 n S T AT 3 n : [NL S T AT 3 n .. NU S T AT 3 n] init 0; [dephospho dedimer stat59 ] (S T AT 3 n + 2 ≤ NU S T AT 3 n) → 1 : (S T AT 3 n = S T AT 3 n + 2); [reloc stat nc60 ] (S T AT 3 n ≥ 1 + NL S T AT 3 n) → 1 : (S T AT 3 n = S T AT 3 n − 1); endmodule

98

M.L. Guerriero

module Rates 0.693

0.693 k58 ·STAT3-PD c·H > 0 → : true; H H k ·STAT3-PD n·H k ·STAT3-PD n· H [dephospho dedimer stat59 ] 59 > 0 → 59 : true; H 0.693 H 0.693 ·STAT3 n·H ·STAT3 n·H k k [reloc stat nc60 ] 60 H > 0 → 60 H : true; k ·STAT3-PD n·H k ·STAT3-PD n·H [synth socs61 ] 61 > 0 → 61 : true; H Hk80 k80 nucleus·NA ·PIAS3·H·STAT3-PD n·H nucleus·NA ·PIAS3·H·STAT3-PD n·H [bind pias stat80 ] > 0 → : true; H H k ·PIAS3:STAT3-PD n·H k ·PIAS3:STAT3-PD n·H [unbind pias stat80 ] −80 > 0 → −80 : true; H H [reloc stat cn58 ]

k58

·STAT3-PD c·H

endmodule

The PRISM model generated from the Bio-PEPA model of the gp130/JAK/STAT pathway has 63 species and 118 reactions. Because of the well-known state space explosion problem of model-checking, even if we consider only a few levels for each species, the state space for this model is so huge that it makes the numerical solution of the CTMC nearly unmanageable. To overcome this problem, we consider a subdivision of the pathway into two distinct sub-models in such a way that the analysis of the individual sub-models becomes more feasible. In order to find an appropriate modularisation, we adopt the approach proposed in [27,28], based on the identification of sub-systems with no retroactivity. For the considered model of the gp130/JAK/STAT pathway, two modules with low coupling can be easily identified. In the first sub-model, which refers to the bindings of ligands to receptors and the activation of the receptor dimers, we consider all the distinct combinations of ligand/receptor complexes, and we describe in detail the formation of all possible types of active receptor dimers, considering the fact that diﬀerent ligand-receptor pairs have diﬀerent binding aﬃnities. In the second sub-model, which refers to the downstream signalling pathway, we instead consider as a starting point a single “generic” type of active receptor dimer (referred to as rcpt-DP), and we focus on the reactions involving the activation of STAT3 and its cytoplasmic/nuclear shuttling. These two sub-models refer to sub-systems of the gp130/JAK/STAT pathway which act in a rather sequential way and, as a consequence, it is reasonable to assume that, for the downstream STAT3 signalling to occur, the receptor-complexes must have been already activated. The initial number of active receptor dimers in the second sub-model is defined as the sum of the steady-state quantities of all the active receptor dimers in the first sub-model. This assumption is justified by the fact that the activation of the receptors is fast compared with the following reactions, and therefore the amount of initially inactive receptors is negligible when considering the downstream pathway. As discussed in [27,28], the absence of retroactivity ensures that the modularisation has no significant eﬀect on the overall behaviour of the system. This, together with the fact that we use the output of the first sub-model as input of the second sub-model,

Qualitative and Quantitative Analysis of a Bio-PEPA Model

99

ensures that the structural qualitative properties verified for the individual sub-models in the rest of this section also hold for the full model. Particular care should be taken when verifying quantitative temporal properties over sub-models. Here we only consider semi-quantitative analysis (Sec. 7) as we are interested in relative rather than absolute values. Therefore, in this particular case, the absence of retroactivity ensures the validity, in the full model, of the analysis results obtained in the sub-models. In general, however, the actual reaction rates in the composite model (and therefore the analysis results) might be diﬀerent from the ones in the sub-models, and more advanced approaches for modularisation should be applied. In the rest of this section we use H = 200 as the step size for the ligands-receptors sub-model, and H = 300 for the downstream sub-model. See Sec. 7 for a discussion of the choice of step size values. Deadlock Detection. Deadlock states are the ones in which no transition is enabled. In some cases the presence of deadlock states is (correctly) due to the presence of irreversible reactions which lead to the transformation of all reactants into non-reactive proteins. In other cases deadlocks could be due to the scarcity of one of the reactants of a multimolecular reaction; in our model, for instance, all receptors are consumed (i.e. transformed into diﬀerent forms, such as dimers) while still ligands are available. In other cases deadlocks could be caused by modelling errors. PRISM automatically detects deadlock states when building the state space of models, and this feature can be considered the first step in the identification of potential modelling errors. For instance, in the ligands-receptors sub-model, any state in which ligands are present while all gp130 receptors have been consumed is a deadlock. This suggests that gp130 is the bottleneck of the system. Species Invariants. One simple and yet interesting property that can be verified is the presence of invariants in the amount of the involved proteins. Species invariants are commonly present in biochemical systems because of the existence of basic constraints such as the law of conservation of mass, which states that the amount (i.e. mass) of reactants consumed by a reaction must be equal to the amount of products of the reaction. For instance, given the conservation of mass and the absence of synthesis and degradation reactions, we expect that the sum of the amounts of LIFR receptor present in its various possible forms (free, as gp130:LIF:LIFR complex and as gp130:OSM:LIFR complex, with one or both of its subunits phosphorylated) is constant (and equal to the LIFR initial amount). The satisfaction of the following properties confirms the existence of the expected invariants on the total amount of ligands and receptors (as an example, we report the ones for LIF and LIFR). P≥1 [G (LIF + gp130:LIF:LIFR + gp130-P:LIF:LIFR + gp130:LIF:LIFR-P + gp130-P:LIF:LIFR-P = NU LIF)] → true

100

M.L. Guerriero

P≥1 [G (LIFR + gp130:LIF:LIFR + gp130:OSM:LIFR + gp130-P:LIF:LIFR + gp130:LIF:LIFR-P + gp130-P:LIF:LIFR-P + gp130-P:OSM:LIFR + gp130:OSM:LIFR-P + gp130-P:OSM:LIFR-P = NU LIFR)] → true

Here, and in the rest of the section, the notation Pp [ψ] → true (resp. false) means that ψ is satisfied (resp. is not satisfied), while the notation P=? [ψ] → p (with p ∈ IR) means that the result of ψ is the probability p. Reachability Analysis. Reachability properties allow us to verify whether a given state is eventually reached. States of interest can be, for instance, the ones in which some species reaches a threshold or is totally consumed, or when the amounts of two species coincide. We consider here the states in which a certain number of receptors are phosphorylated, and the ones in which a certain amount of active nuclear STAT3 (STAT3-PD n) is present. We consider first the ligands-receptors sub-model. The satisfaction of the first of the following properties guarantees that a state in which one fourth of the total amount of available receptors is phosphorylated is always reached at some time point. On the contrary, the second property, which is not satisfied, proves that we do not necessarily reach a state with one third of receptors phosphorylated. P≥1 [F (gp130-P:LIF:LIFR-P + gp130-P:OSM:LIFR-P + gp130-P:OSM:OSMR-P > (NU OS MR + NU LIFR + NU gp130) / 4)] → true P≥1 [F (gp130-P:LIF:LIFR-P + gp130-P:OSM:LIFR-P + gp130-P:OSM:OSMR-P > (NU OS MR + NU LIFR + NU gp130) / 3)] → false

The next property, instead, guarantees that in general we could reach a system where no gp130:OSMR receptor complex is activated. P≥1 [F (gp130-P:OSM:OSMR-P > 0)] → false

Regarding the downstream sub-model, we check for the following properties, which guarantee that, at some time point, at least half the initial amount of STAT3 has been transported into the nucleus and activated, but not all of it. P≥1 [F (S T AT 3-PD n > NU S T AT 3 c / 2)] → true P≥1 [F (S T AT 3-PD n > NU S T AT 3 c)] → false

Reversibility. A system is called reversible if the initial state is reachable from any other state (i.e. the system is able to self-reinitialise). More generally, a state is called reversible if it can be reached again at some later time point. The following property, if satisfied, guarantees the reversibility of the system: it states that it is always possible to return to the initial state (in the PRISM language “init” is a predefined formula which completely specifies the initial state).

Qualitative and Quantitative Analysis of a Bio-PEPA Model

101

P=? [G (“init” ⇒ P≥1 [X (!“init” ⇒ P≥1 [F (“init”)])])]

For the ligands-receptors sub-system the result of this property is 0, since we have considered bindings to be irreversible and, therefore, the system cannot return to the initial state in which all receptors and ligands are free. The downstream sub-system, instead, is reversible (the result of the property is 1), thanks to the cytoplasmic/nuclear STAT3 shuttling, which enables the system to return to the initial state in which cytoplasmic STAT3 molecules are not phosphorylated and not bound to receptor dimers. Liveness. The notion of liveness of a reaction in a given state refers to the possibility of it occurring in such a state. In particular, it is interesting to know which reactions are live in the initial state. Since PRISM properties are state-based, it is not possible to explicitly check for the occurrence of a given reaction. However, knowing how each model component is aﬀected by the occurrence of a given reaction, we can verify this kind of property by checking for the expected variations in the involved components. We are interested, for instance, in verifying that in the initial state the binding reactions between ligands and receptors can occur, leading to the three possible types of ligand/receptor dimers (gp130:LIF:LIFR, gp130:OSM:LIFR, and gp130:OSM:OSMR). The following three properties are satisfied, confirming that the three known types of complexes can be formed. P≥1 [G (“init” ⇒ P>0 [X (gp130 = NU gp130 − 1 & LIF = NU LIF − 1 & LIFR = NU LIFR − 1)])] → true P≥1 [G (“init” ⇒ P>0 [X (gp130 = NU gp130 − 1 & OS M = NU OS M − 1 & LIFR = NU LIFR − 1)])] → true P≥1 [G (“init” ⇒ P>0 [X (gp130 = NU gp130 − 1 & OS M = NU OS M − 1 & OS MR = NU OS MR − 1)])] → true

The following property, instead, is not satisfied: it states, as desired, that LIF cannot bind to receptors to form gp130:OSMR dimers. P≥1 [G (“init” ⇒ P>0 [X (gp130 = NU gp130 − 1 & LIF = NU LIF − 1 & OS MR = NU OS MR − 1)])] → false

Causality Analysis. Causality relations between given reactions can be expressed and verified by properties which relate the order of “appearance” of relevant molecules. This kind of property can be used, for instance, to verify the order in which intermediate products are formed within a cascade of events. A form of causality relation can be expressed by using the sequence and consequence relations defined in [29]: specifically, while sequence formulae describe ordering relations between events (e.g. “in order to reach a given state, we must first reach another one”), consequence formulae describe causal relations (e.g. “if a given state occurs, it is necessarily followed by a second one”).

102

M.L. Guerriero

For example, the ordering and causality relations between STAT3 phosphorylation, homodimerisation and relocation into the nucleus can be verified by the following pairs of properties (assuming at system initialisation all STAT3 is present in cytoplasmic monomeric form (STAT3-P c). When the result of the first property is 0, such a property states that it is not possible for a STAT3-PD c molecule to be present if in all previous states we had no rcpt-DP:STAT3-DP1 (a complex formed by a receptor dimer and a STAT3 molecule). Similarly, the following property (when it evaluates to 0) states that STAT3-PD c must be produced before STAT3-PD n appears. P=? [(rcpt-DP:STAT3-DP1 = 0) U S T AT 3-PD c > 0] → 0 P=? [(S T AT 3-PD c = 0) U S T AT 3-PD n > 0] → 0

The following two properties complement the previous two, stating that if at least one complex rcpt-DP:STAT3-DP1 is formed, then at least one STAT3-PD c molecule will necessarily be formed. P=? [G (rcpt-DP:STAT3-DP1 > 0 ⇒ P≥1 [F (S T AT 3-PD c > 0)])] → 1 P=? [G (S T AT 3-PD c > 0 ⇒ P≥1 [F (S T AT 3-PD n > 0)])] → 1

As another example, the following two properties verify that the transport of phosphorylated STAT3 dimers can only occur from the cytoplasm to the nucleus, but not vice versa. The result of the first property is 0 (i.e. transport of STAT3-PD can occur from cytoplasm to nucleus), while the result of the second property is 1 (i.e. transport of STAT3-PD cannot occur from nucleus to cytoplasm) for all reachable values of i, j. P=? [F (S T AT 3-PD c = i & S T AT 3-PD n = j & P≤0 [X (S T AT 3-PD c = i − 1 & S T AT 3-PD n = j + 1)])] → 0 P=? [F (S T AT 3-PD c = i & S T AT 3-PD n = j & P≤0 [X (S T AT 3-PD c = i + 1 & S T AT 3-PD n = j − 1)])] → 1

Conversely, the transport of dephosphorylated STAT3 monomers can only occur from the nucleus to the cytoplasm. P=? [F (S T AT 3 c = i & S T AT 3 n = j & P≤0 [X (S T AT 3 c = i − 1 & S T AT 3 n = j + 1)])] → 1 P=? [F (S T AT 3 c = i & S T AT 3 n = j & P≤0 [X (S T AT 3 c = i + 1 & S T AT 3 n = j − 1)])] → 0

6 Simulation Based Time-Series Analysis In the previous section we have used model-checking in order to check for a number of simple formulae which guarantee us that some key properties of the gp130/JAK/STAT

Qualitative and Quantitative Analysis of a Bio-PEPA Model

(a) Full model

(b) no SOCS3 3000

STAT3_c STAT3_n

2500

STAT3-PD_c STAT3-PD_n

2000 1500 1000 500

Number of molecules

Number of molecules

3000

0

STAT3_c STAT3_n

2500

STAT3-PD_c STAT3-PD_n

2000 1500 1000 500 0

0

100 200 300 400 500 600 700 800

0

100 200 300 400 500 600 700 800

Time (minutes)

Time (minutes)

(c) no PIAS3

(d) no TC-PTP 3000

STAT3_c STAT3_n

2500

STAT3-PD_c STAT3-PD_n

2000 1500 1000 500 0

Number of molecules

3000 Number of molecules

103

STAT3_c STAT3_n

2500

STAT3-PD_c STAT3-PD_n

2000 1500 1000 500 0

0

100 200 300 400 500 600 700 800 Time (minutes)

0

100 200 300 400 500 600 700 800 Time (minutes)

Fig. 2. Simulation results: full model vs. no inhibitors

model are satisfied. This analysis allows us to be more confident about the absence of modelling errors. Now we progress our analysis of the model by means of stochastic simulation. We report here some results obtained by simulating the full model (comprising both the ligands-receptors and the downstream sub-systems) using the Gibson-Bruck [30] stochastic simulation engine implemented in Dizzy [12,24]. Figure 2 shows the time-series evolution produced by the model (Fig. 2(a)) versus the ones in which each of the three inhibitors has been removed (Fig. 2(b)–(d)). Each plot refers to average values computed over 1000 simulation runs, and the amounts of the four diﬀerent forms of STAT3 are shown (cytoplasmic and nuclear dephosphorylated monomers, and cytoplasmic and nuclear phosphorylated dimers). In all the performed simulations, at system initialisation STAT3 is only present in cytoplasmic monomeric form. As shown in Fig. 2(a), as time passes, STAT3 is phosphorylated, dimerised, and transported into the nucleus, until the systems reaches a state in which the inhibition of nuclear STAT3 by dephosphorylation and the nuclear/ cytoplasmic shuttling lead nuclear and cytoplasmic STAT3 to be in equilibrium. When the amount of nuclear STAT3 increases significantly, the inhibitory role of SOCS3 (which is under transcription control of STAT3) comes into play (Fig. 2(b)). SOCS3 is responsible for signal attenuation and, hence, after reaching a peak, nuclear STAT3 decreases.

104

M.L. Guerriero

PIAS3 slows down the production of active nuclear STAT3 by binding to it (Fig. 2(c)). Therefore, if PIAS3 is present, part of nuclear STAT3 is bound to it, while, if PIAS3 is knocked down, the amount of available STAT3 increases. A third inhibitor, TC-PTP, allows nuclear STAT3 to translocate back into the cytoplasm, by dephosphorylating it (Fig. 2(d)). If TC-PTP is present, STAT3 nuclear/cytoplasmic shuttling occurs; instead, if TC-PTP is knocked out (i.e. if nuclear STAT3 is not dephosphorylated), STAT3 accumulates in the nucleus, whilst cytoplasmic STAT3 molecules quickly disappear.

7 Semi-quantitative Analysis of the CTMC with Levels In Sec. 5 we have shown how model-checking can be used in order to discover modelling errors by checking for some basic properties which guarantee the model to behave as expected. In this section, instead, we use model-checking also for quantitative analysis, with the purpose of completing the simulation-based analysis in order to provide additional insight on the behaviour of the gp130/JAK/STAT pathway. The main advantage of model-checking with respect to stochastic simulation is the fact that model-checking is exhaustive: it explores all the possible behaviours of the model and it does not require us to compute an average behaviour of a number of stochastic simulation runs. As mentioned before, the main disadvantage of model-checking is the state space explosion problem, which implies that we cannot deal with too many levels for the model components without inducing an intractable model. In has been shown (see [10]) that, as the number of levels increases, the behaviour of the CTMC with levels tends to the behaviour of ODEs (when the number of molecules is large enough to average out the randomness of the system); this result guarantees the theoretical correctness of the approach. However, if the number of levels is too small, the error introduced by the discretisation becomes significant and the numerical solution of the generated CTMC fails to reproduce the correct behaviour. The number of levels for model components is related to the step size H and to the upper NU and lower NL bounds for each species. The step size H represents the granularity of the system, and it directly aﬀects the accuracy of the results; the upper and lower bounds are also relevant to the accuracy, since imposing bounds on the numbers of molecules causes a state space truncation which might potentially have impact on the behaviour of the system. Therefore, when performing CTMC analysis of Bio-PEPA models, the choice of the step size and of the upper and lower bounds is essential: they must be carefully selected so that the number of levels to be used for the model components is a suitable trade-oﬀ between accuracy and eﬃciency. In the following sections we report some of the results obtained by using the PRISM model-checker to perform quantitative analysis. First we consider reward-based properties which allow us to observe the time-series for some of the species of the system (for comparison with the stochastic simulation), and we discuss the error introduced by discretising and bounding the model; afterwards, we define further properties in order to compute additional (semi-)quantitative measures.

Qualitative and Quantitative Analysis of a Bio-PEPA Model

105

Time-series Analysis Using State Rewards. A reward structure is automatically defined by the Bio-PEPA Workbench for each PRISM component, and it can be referred to either by the component name or by an integer value (implicitly assigned to reward structures based on the order in which they are defined). These reward structures associate an instantaneous reward equal to the current amount of the corresponding molecular species with each state. The evaluation of these reward-based properties corresponds to computing an average behaviour for the species at given time points. As an example, the following reward is used to observe the time evolution of the receptor dimer gp130:LIF:LIFR. rewards “gp130:LIF:LIFR” true : gp130:LIF:LIFR · H; endrewards

Figure 3 reports the results obtained by verifying on the ligands-receptors sub-system the reward-based property Ri=? [I = T ]

for time points T ≤ 30 minutes, where i is an integer variable used to index the reward structure of interest. Figure 4, instead, reports the results obtained by verifying the same reward-based property for time points T ≤ 800 minutes on the downstream sub-system. In this figure, we also report the standard deviation of the number of molecules, which is computed by exploiting reward structures associating the square of the number of molecules of each species with each state: the standard deviation is calculated as the square root of the variance E(Y)2 − E(Y 2 ), where Y is the random variable representing a species in

500 450

Number of molecules

400

gp130:LIF:LIFR gp130:OSM:LIFR gp130:OSM:OSMR gp130-P:LIF:LIFR gp130:LIF:LIFR-P gp130-P:LIF:LIFR-P gp130-P:OSM:LIFR gp130:OSM:LIFR-P gp130-P:OSM:LIFR-P gp130-P:OSM:OSMR gp130:OSM:OSMR-P gp130-P:OSM:OSMR-P

350 300 250 200 150 100 50 0 0

5

10 15 20 Time (minutes)

25

30

Fig. 3. Time-series by model-checking: ligands-receptors sub-model

106

M.L. Guerriero

3000 STAT3_c STAT3_n STAT3-PD_c STAT3-PD_n

Number of molecules

2500 2000 1500 1000 500 0 0

100

200

300 400 500 Time (minutes)

600

700

800

Fig. 4. Time-series by model-checking: downstream sub-model. Thick lines represent the expected numbers of molecules; thin lines represent their standard deviation.

the network, whereas E(Y) and E(Y 2 ) indicate the expected values for the amount of the species Y and for its square value. Figures 3 and 4 have been obtained by analysing the sub-models with step sizes H = 200 and H = 300 respectively. In the next section we discuss the considerations which lead us to the choice of such values. Three kinds of approximation errors could have been introduced by our analysis of the CTMC with levels due to, respectively, the discretisation of the amounts (H), their bounding (NL and NU ), and the subdivision into modules. In the next section we discuss the eﬀect of varying the step size H on the behaviour of the system. Instead, we do not report results concerning the variation of the bounds NL and NU since, in this particular system, increasing the bounds does not have a significant eﬀect: the reason for this is that no synthesis and degradation reactions are defined (with the single exception of SOCS3) and, as a consequence, the amount of most molecular species is clearly bounded by the amounts of the molecules present at system initialisation. The choice of how to modularise the system has been carried out in order to minimise the interaction between the two modules. However, the modularisation has certainly an impact on the quantitative behaviour. In the whole system, for instance, STAT3 and SOCS3 molecules can bind to receptor dimers as soon as they start being phosphorylated; in the downstream sub-model, instead, we had to fix an initial amount of phosphorylated receptor dimers. Despite these possible sources of approximation, comparing Fig. 4 and Fig. 2, we notice that the results obtained by analysing the downstream sub-model using PRISM instantaneous rewards do not diﬀer significantly from the behaviour observed by averaging the results obtained by 1000 stochastic simulation runs of the whole model. Both the time-scale and the relative amounts of molecules are the same in both figures,

Qualitative and Quantitative Analysis of a Bio-PEPA Model

107

and the only significant diﬀerence regarding the absolute amounts is the amount of cytoplasmic monomeric STAT3, which is higher in Fig. 4. We can also observe that the standard deviation reported in Fig. 4 is quite high, due to the stochastic noise which has been introduced by using a small number of levels. Experimenting with Step Sizes. As previously stated, the choice of the step size has a great impact on both accuracy and performance of the analysis: the smaller the step size is, the larger the CTMC state space and, hence, the smaller the discretisation error introduced, but also the longer the time needed for solving the CTMC. Before choosing the values to be used for the step size H in the analysis of the models, we have performed a number of experiments varying H in order to find values representing a good trade-oﬀ between accuracy and performance of the analysis. In Fig. 5 and Fig. 6 we report some results which show how changing the step size aﬀects the behaviour of the system (in ligands-receptors and downstream sub-systems, respectively). In Fig. 5, we compare the results obtained by using six diﬀerent values for H (1000, 500, 300, 250, 200, 150) in the analysis of the ligands-receptors sub-model, and we can observe that H in this case does not have a big impact on the results.

(a) H=1000

(b) H=500

300 200 100

500 Number of molecules

400

0

400 300 200 100 0

0

5 10 15 20 25 30 Time (minutes) (d) H=250

100 0 5 10 15 20 25 30 Time (minutes)

5 10 15 20 25 30 Time (minutes) (f) H=150

500

400 300 200 100 0

0

100 0

Number of molecules

200

200

(e) H=200 Number of molecules

300

300

5 10 15 20 25 30 Time (minutes)

500

400

400

0 0

500 Number of molecules

(c) H=300

500 Number of molecules

Number of molecules

500

400 300 200 100 0

0

5 10 15 20 25 30 Time (minutes)

0

5 10 15 20 25 30 Time (minutes)

Fig. 5. Time-series by model-checking: ligands-receptors sub-model. The three types of receptor complexes are shown, gp130:LIF:LIFR (red), gp130:OSM:LIFR (green), and gp130:OSM:OSMR (blue), in the stage when one (full line) or both (dashed line) receptors are phosphorylated.

108

M.L. Guerriero

(a) STAT_c

(b) STAT3-PD_c 120

Simulation MC, H=1000 MC, H=500

2500

MC, H=400 MC, H=300 MC, H=270

Number of molecules

Number of molecules

3000

2000 1500 1000 500 0

Simulation MC, H=1000 MC, H=500

100 80 60 40 20 0

0

100

200

300

400

500

600

700

800

0

100

200

Time (minutes)

300

400

500

600

700

800

700

800

Time (minutes)

(c) STAT3_n

(d) STAT3-PD_n

500

700 Simulation MC, H=1000 MC, H=500 MC, H=400 MC, H=300 MC, H=270

450 400 350 300

Simulation MC, H=1000 MC, H=500 MC, H=400 MC, H=300 MC, H=270

600 Number of molecules

Number of molecules

MC, H=400 MC, H=300 MC, H=270

250 200 150 100

500 400 300 200 100

50 0

0 0

100

200

300

400

500

Time (minutes)

600

700

800

0

100

200

300

400

500

600

Time (minutes)

Fig. 6. Time-series by model-checking: STAT3 sub-model

The first notable diﬀerence is that in Fig. 5(a) the amounts of gp130:LIF:LIFR and gp130:OSM:LIFR are equal: as expected, with H = 1000 (i.e. one single level for each ligand and receptor) we are not able to observe the fact that LIFR has an higher binding aﬃnity with LIF than with OS M. The other interesting thing is that, contrary to what we expected, there is no noticeable increase of accuracy when decreasing H. Instead, after observing the similarities between Fig. 5(b), (d) and (e), and between Fig. 5(c) and (f), respectively, we drew the conclusion that the first group is the “correct” one; the reason is the rounding error introduced when computing the number of levels starting from the initial amounts (remember that NL = MIN/H and NU = MAX/H): when a small numbers of levels is used, this rounding error happens to be more significant than H itself. In Fig. 6, we compare the results obtained by using five diﬀerent values for H (1000, 500, 400, 300, 270) in the analysis of the downstream sub-model; the value obtained by stochastic simulation is also shown. As for the ligands-receptors sub-model, also for this sub-model we notice that when using H = 1000 we obtain a totally wrong behaviour, and we observe a general increase in accuracy when increasing the number of levels. For the smallest values of H, the relative values and the trends for the considered species are correctly reflected compared to the stochastic simulation results: for instance, both the peaks’ amplitude and the time at which they occur are reproduced quite accurately.

Qualitative and Quantitative Analysis of a Bio-PEPA Model

109

Though not exact with respect to the stochastic simulation, these results are satisfactory enough for the kinds of semi-quantitative properties we are interested in analysing in the next section. Exact quantitative analysis via CTMC, instead, is infeasible for systems such as the model we consider here. Indeed, the time needed for obtaining the results shown in Fig. 6 ranges from a couple of seconds to hours, and for H = 250 the size of the CTMC already becomes prohibitively large to analyse. Semi-quantitative Properties. Using again the “trade-oﬀ” step sizes H = 200 and H = 300, we consider here a few more semi-quantitative properties of the two sub-models. For instance, we are interested in analysing the impact that the diﬀerent aﬃnities of ligand/receptor pairs have on the consumption of the diﬀerent ligands and receptors and on the relative amount of type I and type II receptors formed. Though this kind of analysis is clearly quantitative (since it involves calculating probabilities and, hence, numbers of molecules), we consider such properties semiquantitative because we are not interested in computing absolute values, but rather in knowing relative values with respect to each other. The following properties measure the probability with which the amount of each molecular species never changes from the initial amount. P=? [G (LIF = NU LIF)] → 7.53 · 10−2 P=? [G (OS M = NU OS M)] → 1.45 · 10−6 P=? [G (gp130 = NU gp130)] → 0 P=? [G (LIFR = NU LIFR)] → 1.24 · 10−4 P=? [G (OS MR = NU OS MR)] → 4.56 · 10−4

From the obtained results we notice, for instance, that gp130 is always used (indeed, it is necessary to form all receptor dimers), and that it is more likely for OSM to be consumed than LIF (indeed, LIF is only used in the formation of one type of receptor dimers). We measure also the probability with which the amount of each molecular species reaches its lower bound. This group of properties shows that gp130 is totally consumed in any possible evolution of the system, that LIF and OSM are never totally consumed, and that the probability of LIFR being totally consumed is equal to the probability of OSMR not being used at all. These results mean that gp130 is the bottleneck of the system, while LIF and OSM are present in abundance. P=? [F (LIF = NL LIF)] → 0 P=? [F (OS M = NL OS M)] → 0 P=? [F (gp130 = NL gp130)] → 1

110

M.L. Guerriero

P=? [F (LIFR = NL LIFR)] → 4.56 · 10−4 P=? [F (OS MR = NL OS MR)] → 1.24 · 10−4

Finally, we consider the reward-based property Ri=? [C ≤ T ]

and we verify it on the downstream sub-model for time points T ≤ 800 minutes, where i is an integer variable referring to a transition reward structure. In addition to state rewards, in fact, PRISM allows for the definition of reward structures which associate with each transition a cumulative reward equal to its expected number of occurrences up to the considered time. In Fig. 7 and Fig. 8 the expected number of occurrences for some of the reactions of the downstream sub-model is shown. In particular, in Fig. 7 we compare the number of occurrences of receptors/STAT3 and receptors/SOCS3 binding reactions, which shows intuitively the diﬀerent binding aﬃnities of STAT3 and SOCS3 to the receptor dimers. In Fig. 8, instead, we consider the number of occurrences of transport reactions of STAT3 molecules from cytoplasm to nucleus and back. In Fig. 8(a) we compare the number of occurrences of transport in the two directions: we count each transport from cytoplasm to nucleus twice since STAT3 molecules are translocated in the nucleus in dimeric form and, hence, a pair of STAT3 molecules is moved at each reaction occurrence. We observe that, since at system initialisation no STAT3 molecule is present in the nucleus, the diﬀerence between the two curves in Fig. 8(a) (multiplied by the step

50

receptor / STAT3_c binding receptor / SOCS3 binding

45

Number of occurrences

40 35 30 25 20 15 10 5 0 0

100

200

300 400 500 Time (minutes)

600

700

800

Fig. 7. Expected number of occurrences of receptor binding reactions in the downstream submodel. The full red line plots the number of occurrences of reactions bind rcpt DP stat27 and bind rcpt DP stat28 , while the dashed green line plots the number of occurrences of reactions bind rcpt DP socs62 and bind rcpt DP socs63 .

Qualitative and Quantitative Analysis of a Bio-PEPA Model 35

cytoplasm -> nucleus nucleus -> cytoplasm

Number of occurrences

(cytoplasm->nucleus - nucleus->cytoplasm) * H STAT3_n + STAT3-PD_n*2 + PIAS:STAT3-PD_n*2

2500

30

111

2000

25 20

1500

15

1000 10

500

5

0

0 0

100

200

300 400 500 Time (minutes)

(a)

600

700

800

0

100

200

300

400

500

600

700

800

Time (minutes)

(b)

Fig. 8. Expected number of occurrences of transport reactions in the downstream sub-model. In (a) the full red line plots twice the number of occurrences of reaction reloc stat cn58 , while the dashed green line plots the number of occurrences of reaction reloc stat nc60 . In (b) the full red line is the diﬀerence between the lines in (a) multiplied by H, while the dashed green line is the total current amount of STAT3 molecules in the nucleus (S T AT 3 n + S T AT 3-PD n · 2 + PIAS:STAT3-PD n · 2).

size H) must be the number of STAT3 molecules present in the nucleus. This consideration is confirmed by the perfect agreement of the two curves in Fig. 8(b).

8 Related Work Given its significant impact on various cellular processes, the gp130/JAK/STAT pathway has been subject of numerous studies, both experimental and computational. Consequently, a few variants of the pathway model have been developed in order to analyse diﬀerent aspects of it. In [18] the focus is on the shuttling of STATs from nucleus to cytoplasm and back. A more complete model is developed in [19], which also reports the results of a global sensitivity analysis of parameter interaction. The role of inhibitory mechanisms is instead studied in [20]. These three works are based on mathematical modelling and the analysis is performed by ODEs solvers. In [21], a process algebra based computational model of the gp130/JAK/STAT pathway is presented and analysed using the BetaWB tool [31], a stochastic simulator for the BlenX language [32]. The Bio-PEPA model we present here is strongly based on the BlenX model described in [21], and the simulation results of the two models match well. This agreement is particularly interesting in view of the conceptual diﬀerences existing in the two process algebras. One of these diﬀerences concerns the treatment of complexes, which in BlenX are considered as molecular species consisting of the individual molecules composing them, while in Bio-PEPA they are considered as diﬀerent species not explicitly related to the sub-components. Secondly, immediate reactions can be defined in BlenX, while they are not admitted in Bio-PEPA because of Bio-PEPA’s underlying CTMC semantics. Finally, stoichiometric information can be specified in

112

M.L. Guerriero

Bio-PEPA, while they cannot be explicitly coded in BlenX (requiring reactions involving stoichiometry greater than one to be decomposed into multiple steps). In addition to these theoretical diﬀerences between the languages, we mention that the focus in the two works is quite diﬀerent. In [21] the eﬀects of a number of experiments involving quantitative parameters are analysed and compared with experimental data. The aim of the present work, instead, is to exploit model-checking, in addition to stochastic simulation, to analyse both qualitative and quantitative properties of the model behaviour. A few works have recently been published regarding the application of modelchecking techniques to the analysis of biochemical systems. In [33] the authors demonstrate how the PRISM model-checker can be adopted to model and analyse biochemical pathways, using the FGF pathway as a case study. The approach proposed in this work diﬀers from ours in the level of abstraction considered. Instead of taking a variable number of levels into account, the authors of [33] consider an abstraction in which one single copy of each involved molecular species is present and such that module variables represent changes in state of the molecules. This approach has the evident advantage of reducing the CTMC state space, though it might not be quantitatively correct in general: it can be seen as a level of abstraction equivalent to ours when one single level is used for each species. In the same work, the authors also consider a number of state space reduction techniques, some of which (based on lumpability and symmetry reduction) are exact, meaning that the behaviour of the reduced CTMC is preserved. The notion of CTMC with levels of concentrations has been introduced in [34], in which the ERK signalling pathway was used as a case study, and in [35] the PRISM model-checker is used to analyse it. Following these works, the notion of discrete levels of concentrations has been adopted also in IDD-CSL [36], an Interval Decision Diagram based model-checker for stochastic Petri nets, which allows for the verification of CSL properties. In [37] the authors propose a framework, based on Petri nets, in which qualitative and quantitative (stochastic and continuous) analysis of biochemical pathways are integrated. Qualitative properties such as boundedness, liveness and reversibility are considered, in addition to the possibility to check for P- and T-invariants, and behavioural properties are verified by probabilistic model-checking. Finally, BIOCHAM [38,39] is a framework for modelling, simulating and analysing biochemical systems, in which diﬀerent semantics (diﬀerential, stochastic, discrete, and boolean) are considered. BIOCHAM allows for the verification of temporal properties expressed in the Computation Tree Logic (CTL) by using the NuSMV modelchecker [40].

9 Conclusions and Future Work In this work we have used the gp130/JAK/STAT signalling pathway as a case study for modelling and analysis using the Bio-PEPA process algebra. Among the possible analysis methods made available by the Bio-PEPA Workbench, we have considered stochastic simulation and model-checking. The results obtained by simulation agree well with existing mathematical and computational models. The application of the model-checking approach to the analysis of

Qualitative and Quantitative Analysis of a Bio-PEPA Model

113

the pathway model, though limited by the state space explosion problem, provided us with some useful insight. First, it can be used for consistency checking, in order to guarantee the satisfaction of essential properties and, therefore, the absence of modelling errors. Second, it allows us to check for the satisfaction of semi-quantitative behavioural properties over the whole model, without the need for computing average values over a number of stochastic simulation runs. In order to deal with the computational complexity of model-checking, we have subdivided the pathway model into two distinct sub-models. The time-series analysis obtained by analysing the sub-models individually via model-checking shows a reasonably good agreement with the behaviour obtained via stochastic simulation. The issue of modularisation of models of biochemical systems is a complex one. In this work we have adopted a simple approach which is adequate for this particular case study. A general approach for modularisation of models deserves additional study, in particular in view of the possible performance improvement which this technique could bring in model-checking. Finally, in order to fully exploit the framework provided by Bio-PEPA further analysis could be performed on the MATLAB model generated by the Bio-PEPA Workbench using ODEs based methods to perform, for instance, bifurcation, stability, and continuation analysis. Acknowledgments. The author wishes to thank Jane Hillston for her helpful comments. This research is supported by the EPSRC grant EP/E031439/1 “Stochastic Process Algebra for Biochemical Signalling Pathway Analysis”.

References 1. Regev, A., Silverman, W., Shapiro, E.: Representation and simulation of biochemical processes using the π-calculus process algebra. In: Proceedings of Pacific Symposium on Biocomputing (PSB 2001), vol. 6, pp. 459–470 (2001) 2. Regev, A., Panina, E.M., Silverman, W., Cardelli, L., Shapiro, E.Y.: BioAmbients: an Abstraction for Biological Compartments. Theoretical Computer Science 325(1), 141–167 (2004) 3. Cardelli, L.: Brane Calculi - Interactions of Biological Membranes. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 257–278. Springer, Heidelberg (2005) 4. Priami, C., Quaglia, P.: Operational patterns in Beta-binders. In: Priami, C. (ed.) Transactions on Computational Systems Biology I. LNCS (LNBI), vol. 3380, pp. 50–65. Springer, Heidelberg (2005) 5. Danos, V., Laneve, C.: Formal molecular biology. TCS 325(1) (2004) 6. Regev, A., Shapiro, E.: Cells as Computation. Nature 419(6905), 343 (2002) 7. Ciocchetta, F., Hillston, J.: Bio-PEPA: An extension of the process algebra PEPA for biochemical networks. In: Proc. of FBTC 2007. ENTCS, vol. 194, pp. 103–117 (2008) 8. Ciocchetta, F., Hillston, J.: Bio-PEPA: A Framework for the Modelling and Analysis of Biological Systems. Theoretical Computer Science 410(33-34), 3065–3084 (2009) 9. Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press, Cambridge (1996) 10. Ciocchetta, F., Hillston, J.: Calculi for Biological Systems. In: Formal Methods for Computational Systems Biology (SFM 2008). LNCS, vol. 5016, pp. 265–312. Springer, Heidelberg (2008)

114

M.L. Guerriero

11. Bio-PEPA Workbench Home Page: http://www.dcs.ed.ac.uk/home/stg/software/biopepa/ 12. Ramsey, S., Orrell, D., Bolouri, H.: Dizzy: stochastic simulation of large-scale genetic regulatory networks. J. Bioinf. Comp. Biol. 3(2), 415–436 (2005) 13. PRISM Home Page: http://www.prismmodelchecker.org 14. Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Model-checking continuous-time Markov chains. ACM Trans. Comput. Logic 1(1), 162–170 (2000) 15. Underhill-Day, N., Heath, J.: Oncostatin M (OSM) Cytostasis of Breast Tumor Cells: Characterization of an OSM Receptor β-Specific Kernel. Cancer Research 66(22), 10891–10901 (2006) 16. Heinrich, P., Behrmann, I., Haan, S., Hermanns, H., M¨uller-Newen, G., Schaper, F.: Principles od interleukin (IL)-6-type cytokine signalling and its regulation. Biochem. J. 374, 1–20 (2003) 17. Kisseleva, T., Bhattacharya, S., Braunstein, J., Schindler, C.: Signaling through the JAK/STAT pathway, recent advances and future challenges. Gene 285, 1–24 (2002) 18. Swameye, I., M¨uller, T., Timmer, J., Sandra, O., Klingm¨uller, U.: Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling. PNAS 100, 1028–1033 (2003) 19. Mahdavi, A., Davey, R.E., Bhola, P., Yin, T., Zandstra, P.W.: Sensitivity Analysis of Intracellular Signaling Pathway Kinetics Predicts Targets for Stem Cell Fate Control. PLoS Computational Biology 3(7), 1257–1267 (2007) 20. Singh, A., Jayaraman, A., Hahn, J.: Modeling Regulatory Mechanisms in IL-6 Transduction in Hepatocytes. Biotechnology and Bioengineering 95(5), 850–862 (2006) 21. Guerriero, M.L., Dudka, A., Underhill-Day, N., Heath, J.K., Priami, C.: Narrative-based computational modelling of the Gp130/JAK/STAT signalling pathway. BMC Systems Biology 3(1), 40 (2009) 22. Bio-PEPA Home Page: http://www.biopepa.org/ 23. Hinton, A., Kwiatkowska, M., Norman, G., Parker, D.: PRISM: A tool for automatic verification of probabilistic systems. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 441–444. Springer, Heidelberg (2006) 24. Dizzy Home Page: http://magnet.systemsbiology.net/software/Dizzy 25. Aziz, A., Kanwal, K., Singhal, V., Brayton, V.: Verifying continuous time Markov chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996) 26. Baier, C., Katoen, J.P., Hermanns, H.: Approximate Symbolic Model Checking of Continuous-Time Markov Chains. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 146–161. Springer, Heidelberg (1999) 27. Saez-Rodriguez, J., Kremling, A., Gilles, E.: Dissecting the puzzle of life: modularization of signal transduction networks. Computers and Chemical Engineering 29, 619–629 (2005) 28. Conzelmann, H., Saez-Rodriguez, J., Sauter, T., Bullinger, E., Allg¨ower, F., Gilles, E.: Reduction of mathematical models of signal transduction networks: simulation-based approach applied to EGF receptor signalling. Systems Biology 1(1), 159–169 (2004) 29. Monteiro, P., Ropers, D., Mateescu, R., Freitas, A., de Jong, H.: Temporal logic patterns for querying dynamic models of cellular interaction networks. ECCB 24, 227–233 (2008) 30. Gibson, M., Bruck, J.: Eﬃcient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels. The Journal of Chemical Physics 104, 1876–1889 (2000) 31. Dematt´e, L., Priami, C., Romanel, A.: The Beta Workbench: a computational tool to study the dynamics of biological systems. Briefings in Bioinformatics 9(5), 437–449 (2008), http://www.cosbi.eu/Rpty_Soft_BetaWB.php

Qualitative and Quantitative Analysis of a Bio-PEPA Model

115

32. Dematt´e, L., Priami, C., Romanel, A.: The BlenX Language: A Tutorial. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 313–365. Springer, Heidelberg (2008) 33. Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic Model Checking of Complex Biological Pathways. Theoretical Computer Science 319, 239–257 (2008) 34. Calder, M., Gilmore, S., Hillston, J.: Modelling the Influence of RKIP on the ERK Signalling Pathway Using the Stochastic Process Algebra PEPA. In: Priami, C., Ing´olfsd´ottir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 1–23. Springer, Heidelberg (2006) 35. Calder, M., Vyshemirsky, V., Gilbert, D., Orton, R.: Analysis of signalling pathways using continuous time Markov chains. In: Priami, C., Plotkin, G. (eds.) Transactions on Computational Systems Biology VI. LNCS (LNBI), vol. 4220, pp. 44–67. Springer, Heidelberg (2006) 36. The Idd-CSL Home Page: http://www-dssz.informatik.tu-cottbus.de/software/software.html 37. Heiner, M., Gilbert, D., Donaldson, R.: Petri Nets for Systems and Synthetic Biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 215–264. Springer, Heidelberg (2008) 38. The BIOCHAM Home Page: http://contraintes.inria.fr/BIOCHAM/ 39. Fages, F., Soliman, S., Chabrier-Rivier, N.: Modelling and querying interaction networks in the biochemical abstract machine BIOCHAM. Journal of Biological Physics and Chemistry 4(2), 64–73 (2004) 40. NuSMV Home Page: http://nusmv.irst.itc.it/

Rule-Based Modelling and Model Perturbation Vincent Danos1 , J´erˆome Feret2 , Walter Fontana3 , Russ Harmer4 , and Jean Krivine3,5 1

University of Edinburgh 2 INRIA–ENS–CNRS 3 Harvard Medical School 4 CNRS–Universit´e Paris Diderot 5 Institut des Hautes Etudes Scientiﬁques

Abstract. Rule-based modelling has already proved to be successful for taming the combinatorial complexity, typical of cellular signalling networks, caused by the combination of physical protein-protein interactions and modiﬁcations that generate astronomical numbers of distinct molecular species. However, traditional rule-based approaches, based on an unstructured space of agents and rules, remain susceptible to other combinatorial explosions caused by mutated and/or splice variant agents, that share most but not all of their rules with their wild-type counterparts; and by drugs, which must be clearly distinguished from physiological ligands. In this paper, we deﬁne a syntactic extension of Kappa, an established rule-based modelling platform, that enables the expression of a structured space of agents and rules that allows us to express mutated agents, splice variants, families of related proteins and ligand/drug interventions uniformly. This also enables a mode of model construction where, starting from the current consensus model, we attempt to reproduce in numero the mutational—and more generally the ligand/drug perturbational—analyses that were used in the process of inferring those pathways in the ﬁrst place.

1

Introduction

In recent years, there has been extensive development in the use of modelling to understand cellular signalling networks (see [1, 2, 3, 4] among many others). To date, this line of work has focussed almost exclusively on describing wild-type behaviours, i.e. it deals with the interactions between proteins that take place in a normal healthy cell. This is already highly non-trivial since these signalling networks employ a strategy of binding, modiﬁcation and unbinding between proteins that generates astronomical numbers of non-isomorphic molecular species. This poses an essentially unsolvable scalability problem for any modelling approach, such as ODE-based chemical kinetics or Petri nets, based on exhaustively enumerating reactions between fully-speciﬁed molecular species. In recent years, a new modelling approach has been used to tame this combinatorial explosion, namely agent- or rule-based modelling [5]. In this setting, C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 116–137, 2009. c Springer-Verlag Berlin Heidelberg 2009

Rule-Based Modelling and Model Perturbation

117

molecular species are left implicit; instead, agents are used to represent not complexes but their constituent proteins. Each type of agent has a name and a set of sites. Instead of reactions, we write rules that mention names of agent types and some, but not necessarily all, of their respective sites. In this way, and unlike reactions, a rule need only make explicit those aspects of the agents upon which it acts that are actually relevant to the interaction being described by the rule. So reaction-based models leave agents implicit, considering them at best as an aggregation of molecular species, whereas rule-based models make agents explicit but the reactions implicit, instead considering their rules to be aggregations of reactions. It should be noted that such wild-type models, be they reaction- or rule-based, can already handle situations where, typically as a result of gene ampliﬁcation or ablation, a protein is either over- or under-expressed. Under such circumstances, the response of a cell to external conditions may be exaggerated or attenuated as a consequence of the induced perturbation of mass-action kinetics and of the nature and numbers of complexes that exist in the cell’s resting state (cf. [6]). This does not bring about new protein-protein interactions, it only aﬀects the relative importance of the wild-type interactions, e.g. if protein X has a binding partner Y that is over-expressed, X will be attracted to the greater than usual mass of Y s to the detriment of its binding with other partners. However, many disease states are the result of genetic mutations that build incorrect proteins, with aberrant behaviour, rather than the straightforward modulation of protein expression levels (although in some cases the two defects co-exist and synergize). Such mutant proteins may only diﬀer by one or two amino acids from their wild-type cousins and yet have radically diﬀerent behaviour, e.g. erbB1 with the single substitution L858R, which exists in many kinds of solid tumour, has a constitutively active kinase domain, as does B-Raf with the single V600E mutation. The ﬂip side of this is that much of the wildtype behaviour of a protein is actually shared with such mutants, for instance a binding domain far from any site of mutation will quite likely retain its usual functionality. This poses a further serious challenge to modelling since mutated proteins therefore duplicate large chunks of an already highly combinatorial wildtype network, while also potentially adding interactions. To tackle these issues, we introduce a syntactic extension of Kappa that allows the deﬁnition of a structured space of agents. Agents can either be declared ab initio or derived from existing agents in a manner reminiscent of object-oriented programming (particularly the prototype-based approach). In the latter case, the new agent can gain, lose, rename, mutate or duplicate sites of the agent from which it is derived. This organizes the space of agents hierarchically and thus enables us to write generic rules that mention agents that have many descendants in the hierarchy. These generic rules act as shorthand for sets of normal Kappa rules; they capture behaviours shared by splice variants (e.g. p46, p52 and p66 Shc), genetically related proteins (e.g. ERK1 and ERK2) or mutated proteins. In particular, the conciseness of generic rules enables us to write and analyze large Kappa models far more easily. We illustrate this with a small generic rule

118

V. Danos et al.

set (15 rules) for the erbB receptor network that, once expanded into Kappa, has over 300 rules and which grows considerably larger still if we add in drug interventions and mutated erbB agents. In summary, our agent hierarchy allows us to write large models in a comfortable way, to navigate the perturbation space of the model (ligands, mutations and drugs) and investigate the consequences of chosen perturbations, i.e. those for which we have experimental data, with the static and causal analyses of Kappa. This is particularly interesting for mutational perturbations as these enable us to reproduce, in numero, biochemical experiments that employ engineered mutations. In this way, our rules—a formalization of the consensus pathway assembled by many biochemical experiments—can be tested by checking, in numero, whether perturbing them with mutated agents—representing the engineered mutations—matches those experimental results. Of course, this procedure can never “prove” that a rule is correct but it can be used to reject rules that lead to behaviour incompatible with experimental results. It can also point to the existence of missing links in a model if it throws up false negatives with respect to the experimental data, e.g. it predicts some but not all experimentally observed phosphorylated sites. In other words, it enables us to put our assumptions under the microscope and verify that the consensus wild-type pathway behaves as expected when subjected to perturbations—and if it doesn’t, we will need to change our consensus model. Contribution and relation to existing work. Rule-based modelling is one branch of a rich literature based on the idea of representing proteins and their interactions as concurrent processes, thereby viewing a signalling network as a kind of massively distributed system. This was initially expressed in the formalism of π-calculus [7, 8] but, since then, a number of variants of π-calculus [9, 10] and of other languages for distributed systems [11, 12, 13, 14, 15, 16, 17] have also been proposed for representing various aspects of biological processes, notably the importance of causality and compartments. Rule-based modelling, rather like the BWB/BlenX system [18], was developed out of these ideas but, instead of being based on some prior formalism for general distributed systems, is a domain-speciﬁc modelling language for biological processes. Our language Kappa is particularly closely related to BioNetGen (BNG) [19]. Although the original aims of BNG were rather diﬀerent—it was conceived as a language for describing systems of ODEs in a higher-level fashion, rather than as a modelling language in its own right—the two approaches have much in common and, in particular, our agent hierarchy proposal would work just as well in BNG as in Kappa. Despite these many advances, to the best of our knowledge none of the abovecited approaches, including Kappa and BNG, can deal with all the potential sources of combinatorial explosion in signalling models. Our extension of Kappa with agent hierarchies directly addresses this problem in the speciﬁc context of rule-based modelling. Given that mutating agents, via small changes in their sites and thus interaction capabilities, is central to our proposal, it would be interesting to investigate the possible connections of this work with the recent

Rule-Based Modelling and Model Perturbation

119

use of mutations on the structure of BlenX programs in order to evolve networks via genetic algorithms [20]. However, it should be stressed that our work was originally intended to facilitate the construction (and documentation) of large models in a way that makes explicit any underlying uniformities, rather than in directly enabling an evolutionary analysis of networks.

2

Kappa and Agent Variants

A Kappa [21] model consists of a collection of concrete agents and rules. Each agent, or more properly agent type, has a name, an associated set of sites, each with an optional internal state, and a copy number. An atomic rule falls into one of ﬁve classes—a binding between two agents, an unbinding, the modiﬁcation of an agent, the creation of an agent or the deletion of an agent—but a rule can also be non-atomic, combining several actions. Given a Kappa model, its contact map, which is computed statically from the rules, speciﬁes which agents can bind and on which sites. (See e.g. Figs. 1, 4.) On the other hand its influence map, also computed statically, speciﬁes the causal relations of activation and inhibition between rules, that is to say a rule activates (inhibits) another if its application may add (subtract) from the set of instances of the other one. We will make use of the static analysis of rule accessibility [22] which identiﬁes whether a rule is dead, i.e. cannot be applied, or is potentially applicable; in the latter case, we will use the story sampler [23] to extract, from stochastic simulations [24] of the model, the chains of rule ﬁrings that can lead to an actual application of the rule. If we ﬁnd such a story, this conﬁrms that the static analysis didn’t produce a false positive. The concrete syntax we use to present agents, agent variants and rules should be self-explanatory (although we stress that it can be formalized). One key thing to remember, as said earlier, is that in the deﬁnition of a rule one has the option of not mentioning some sites of an agent. In situations where agents have up to a dozen diﬀerent sites (e.g. the members of the EGF receptor family), this is key to obtaining concise models. This, combined with the ability to mention generic agents, allows us to express enough uniformities for also obtaining concise descriptions of perturbed models. 2.1

Agent Variants

A variant on an agent always introduces a new name and can arise in several diﬀerent ways: it can lose or mutate an existing site, gain a new site or rename/duplicate an existing site. To represent these possibilities formally, we need only introduce two perturbation operations on agents, one to add a site, the other to replace a site with a set of sites. The latter operation subsumes site deletion (by replacing a site with the empty set), site renaming (replacing with a singleton set) and duplication. For example, %gen: A(s,t) %gen: B = A[+u s\{} t\{t1,t2}]

120

V. Danos et al.

declares the agent A with sites s and t and derives from it an agent B with sites t1, t2 and u. This deﬁnes a tree of agent variants; most nodes of the tree are labelled ‘gen’ for generic but leaves of the tree can be labelled ‘conc’ for concrete which signals that that agent can be used in a Kappa model. Note that we have a second tree structure that traces site linkages: any site can be traced back to either a site addition or to a site declared ab initio; and conversely, following the linkages the other way, a site in agent A maps to a set of sites in any given descendant agent B (empty if the site has been deleted, singleton if it has just been renamed). This is important for compiling generic rules into a bona ﬁde Kappa model. Mutation of a site is represented by the compound operation of deleting the original site and, if desired, adding a new site to “replace” it. If the desired result of the mutation is simply the loss of certain wild-type interactions, the loss of the site is enough and no such new site need be added; but sometimes mutations result in new interactions becoming possible in which case we would need to introduce a new site in order to write the new rules expressing the novel interactions of the mutated agent, e.g. the tyrosine kinase inhibitor erlotinib binds to the L858R mutated erbB1 with much higher aﬃnity than to the wild-type receptor. 2.2

A First Example

Let us make this more concrete with an example extracted from a larger model of the MAPK cascade. We start with two basic agent types, MAP2K and MAPK, from which we would like to derive some more speciﬁc agent types. Our ﬁrst declarations introduce the starting agents: %gen: MAP2K(D,S~u,ST~u) %gen: MAPK(CD,T~u,Y~u) Formally, these declarations play a role analogous to that of the axioms in any formal language and, as in that kind of setting, we use them as the starting point to introduce more subtle objects. In this case, we wish to consider the three common kinds of MAPK protein—ERKs, JNKs and p38s—and their respective MAP2K upstream activators—MEKs, JNKKs and p38 kinases. To do this, we ﬁrst introduce three variants of MAPK and three of MAP2K: %gen: ERK = MAPK[+FXFP] %gen: JNK = MAPK %gen: p38 = MAPK Note that, while ERK gains a new site FXFP, an ERK-speciﬁc binding site for immediate early gene products such as Fos and Jun [25], JNK and p38 simply inherit the sites of MAPK without making any changes. As we will see shortly, the introduction of these three variants allows us to express concisely the speciﬁcity of binding between these three distinct families of MAPKs and their cognate upstream activators. Note also that these three agents are still

Rule-Based Modelling and Model Perturbation

121

generic as they represent families of proteins: ERK covers two proteins (ERK1 and ERK2), JNK covers three (JNK1, JNK2 and JNK3) and p38 covers four (p38alpha/beta/gamma/delta); and several of those proteins have multiple splice variants. We formalize this by a further layer of variants: %conc: ERK1 = ERK[T\{T202} Y\{Y204}] %conc: ERK2 = ERK[T\{T185} Y\{Y187}] We show only the case of ERK1 and ERK2 as those of JNK and p38 are completely analogous. Recall that we use the ‘conc’ tag (rather than ‘gen’) to make explicit the fact that ERK1 and ERK2 are concrete, not generic, agents and, as such, can be used in a Kappa model. Note that we have renamed (via singleton duplications) the sites of ERK to include speciﬁc information about the exact residue numbers of their phosphorylatable sites; this is not essential, of course, but does illustrate the documentary power of agent variants over and above their role of structuring the space of agents. We must also introduce generic and concrete variants of MAP2K. Each variant covers two proteins: MEK1 and MEK2 for MEK; MEK4 and MEK7 for JNKK; and MEK3 and MEK6 for p38K. (Again, for the sake of simplicity, we only show the concrete variants of MEK.) %gen: MEK = MAP2K %gen: JNKK = MAP2K %gen: p38K = MAP2K %conc: MEK1 = MEK[S\{S218} ST\{S222}] %conc: MEK2 = MEK[S\{S222} ST\{S226}] Already, the simple fact of hierarchically structuring the agents under consideration yields a useful object in its own right that documents, in a completely formal way, a signiﬁcant amount of biological knowledge (about exactly how related proteins relate to each other) that can easily be found in several online databases but which, in that medium, remains informal and purely descriptive, whereas, in this formalized setting, has already been subjected to an initial step of processing and structuring. It also includes a convenient documentation of the speciﬁc sites of interest, e.g. the precise identities of phosphorylation sites, that are otherwise rather cumbersome to keep track of. Moreover, the creation of this agent hierarchy also facilitates the process of writing rules by enabling us to write them at the appropriately generic level. It eases the cognitive burden of writing rules by exposing clearly the similarities and diﬀerences between various agent types. More concretely, it allows us to avoid writing essentially the same rule many times for closely related agents and, as such, also eliminates the risk of forgetting cases (a very common mistake when developing large rule sets). We turn to this in the next subsection where we will complete the MAPK example.

122

V. Danos et al.

2.3

Generic Rules

We have seen how we can structure agents hierarchically with concrete agents at the leaves and generic agents above them. In this context, a normal (or concrete) Kappa rule is a rule that only mentions concrete agents. A generic rule is syntactically just like a normal rule but mentions one or more generic agents. The purpose of such a rule is to be expanded into a set of concrete rules by replacing each generic agent G in the rule with all appropriate concrete agents C below it in the hierarchy. However, this expansion is modulated by the changes made to G’s sites in C; notably, if site s of G is deleted in C, then no rule testing the existence of s can instantiate G to C. And we must also use the site linkages between C and G to deal with any renaming and duplication of G’s sites in C. So, were we to write the single generic rule MAP2K(D), MAPK(CD) MAP2K(D!0), MAPK(CD!0) this would “incorrectly”, i.e. not as we wish, expand to a collection of concrete rules where all concrete descendants (in the agent hierarchy) of MAP2Ks can bind with all concrete descendants of MAPKs, e.g. JNK2 could bind ERK1. This is the reason why, in the previous section, we introduced a second layer of generic agents—ERK, JNK, p38; MEK, JNKK, p38K. Given that, we can write the following three generic rules that properly respect the desired speciﬁcity of binding between MAP2Ks and MAPKs. MEK(D), ERK(CD) MEK(D!0), ERK(CD!0) JNKK(D), JNK(CD) JNKK(D!0), JNK(CD!0) p38K(D), p38(CD) p38K(D!0), p38(CD!0) These three generic rules expand to eighteen concrete rules if we take ERK1/2, JNK1/2/3 and p38α/β/γ/δ as concrete agents. If we included the many splice variants of the JNKs and p38s, the same three generic rules would expand to over thirty concrete rules. This illustrates the ﬂexibility of our approach whereby a given generic rule can expand diﬀerentially depending on the background of concrete agent variants we select. In particular, a single rule set can be seen as existing at many levels of detail—and this is easily tunable by the modeller as a function of his/her current needs. There is, however, an associated cost, over and above the obvious need to recompile one’s generic rules, when changing the level of detail of a model: under certain circumstances, this will lead to a degradation in the performance of stochastic simulation. The reason for this is that the cost of an event in the simulator depends, in part, on the maximum outdegree of the “wake-up map”, a graph derived from the rule set which keeps track of which rules are reactivated when a rule ﬁres [24]. In the worst-case scenario, our generic rule expansion causes a “blow up” of the wake-up map with concomitant degradation in the simulator’s performance. More generally, our mechanism of using an agent hierarchy and generic rules to generate a concrete rule set allows the Kappa modeller a ﬁner control of the granularity of his/her rules. Consider for example an agent A that can bind two

Rule-Based Modelling and Model Perturbation

123

agents, B1 or B2, and that binding with either is suﬃcient (and necessary) for A to bind a further agent C. To express this in Kappa, we would have to write two rules for A binding C; one for the case of B1, the other for B2. This isn’t too bad—but if we have not two but a large number of activating ligands of A, it rapidly becomes tedious and error-prone to write the rule sets. By using a generic agent B, representing the class of A-activating ligands, we write just one generic rule that covers all cases (albeit requiring recompilation after the addition of new concrete descendants of B). Or to put it another way, we think of the generic agent B as generating a coarse-graining of the model’s molecular species that no longer distinguishes between the various concrete descendants of B (i.e. B1, B2, etc). With more complex agent hierarchies, one can express further, more subtle coarse-graining eﬀects such as the MAP2K-MAPK binding speciﬁcity example above. However, it should be admitted that the example of MAPK is particularly conducive to a treatment of this kind (which is why we use it as our initial example!) and that not all signalling pathways exhibit the same degree of sharing of structure found here, as expressed by the highly generic nature of the rules. This in itself is a useful aspect of our language extension in that it enables us to recognize, formally, the fact that a pathway is highly generic or, on the contrary, particularly obtuse and dependent on many speciﬁc details. Indeed, the purpose of this extension is not to obtain a maximal “compression” of a concrete rule set into as few generic rules as possible; rather it is to illuminate the structure of a model by expressing it at an appropriate level of abstraction.

3

The Perturbation Space

Now that we have shown, with the MAPK example, how our parsimonious language extension enables rapid development of large rule sets via the mechanism of generic rules, let us turn to the main problem of interest here which is to build realistic models incorporating multiple erbB ligands and receptors, mutated forms of those receptors and monoclonal antibodies (mAbs) and tyrosine kinase inhibitors (TKIs) targeting those receptors. Unlike the previous MAPK model where the use of agent variants was convenient but hardly indispensable, in this case it would be a nightmarish process to write the rules directly in Kappa. As we will see, the use of agent variants not only helps to structure the model in a human-understandable manner, it also radically tames the combinatorial explosions caused by having multiple ligands and receptors and by the introduction of mutations. We ﬁrst deﬁne our agent hierarchy. It has two roots, erbB for the receptors and erbL for the ligands, each with four children. %gen: erbB(L,CR,N,atp,AS,C,Y~u) %gen: erbBL(L) The next layer of agents splits the space of ligands into four, each with a diﬀerent repertoire of receptors to which it binds.

124

V. Danos et al.

%gen: %gen: %gen: %gen:

erbBL1 = erbBL erbBL14 = erbBL erbBL34 = erbBL erbBL4 = erbBL

Note that a hierarchical presentation of a model has a degree of intensionality and, in particular, is of course not unique—indeed, the compiled model is actually a presentation of itself. This begs the remark that a presentation is both a way to achieve compactness of description and to document knowledge about relationships between agents that disappears in the compilation process. We also need variants for the four erbB receptors. Note that we introduce them as generic agents and only later specialize them as wild-types and mutant ones (only erbB1-WT is shown here). %gen: erbB1 = erbB[Y\{Y1016, Y1092, Y1110, Y1172, Y1197}] %gen: erbB2 = erbB[L\{}] %gen: erbB3 = erbB[N\{}] %gen: erbB4 = erbB %conc: erbB1_WT = erbB1 To keep our presentation uncluttered, we have only shown the full repertoire of phosphorylation sites for erbB1; in the full model, the other receptors also have a similar complement of Y sites. Note though that erbB2 loses the site L and erbB3 the site N . Finally, let us note the concrete ligands. In what follows, we will in fact only consider EGF and HRG. %conc: EGF = erbBL1 %conc: TGFalpha = erbBL1 %conc: AR = erbBL1 %conc: BTC = erbBL14 %conc: HB-EGF = erbBL14 %conc: ER = erbBL14 %conc: HRG = erbBL34 %conc: NRG2 = erbBL34 %conc: NRG3 = erbBL4 %conc: NRG4 = erbBL4 3.1

The Consensus Model

We build our consensus model on the basis of a conservative reading of the literature; see e.g. [26, 27]. Speciﬁcally, we consider that ligands bind monomer receptors which can then externally (on the trans-side of the plasma membrane) dimerize; this in turn enables the formation of asymmetric dimers that lead to receptor binding on the cis-side and cross-phosphorylation.

Rule-Based Modelling and Model Perturbation

125

erbBL1(L), erbB1(L,CR) -> erbBL1(L!0), erbB1(L!0,CR) erbBL14(L), erbB1(L,CR) -> erbBL14(L!0), erbB1(L!0,CR) erbBL14(L), erbB4(L,CR) -> erbBL14(L!0), erbB4(L!0,CR) erbBL34(L), erbB3(L,CR) -> erbBL34(L!0), erbB3(L!0,CR) erbBL34(L), erbB4(L,CR) -> erbBL34(L!0), erbB4(L!0,CR) erbBL4(L), erbB4(L,CR) -> erbBL4(L!0), erbB4(L!0,CR) These six generic rules expand into a signiﬁcant number of concrete Kappa rules in a manner that depends on the level of detail requested in the identities of ligands. For example, although there are three ligands of type erbBL1 and three of type erbBL14, the two ligands of type erbBL34 actually exist in multiple splice variants, as do those of type erbB4. In any case, at the very least these six generic rules give rise to ﬁfteen concrete rules; and, of course, we also need the unbinding rule: erbBL(L!0), erbB(L!0) -> erbBL(L), erbB(L) Note that this generic unbinding rule will generate concrete rules that will never apply, e.g. a descendant of erbBL1 unbinding erbB3. However, these dead rules are detected by our static analysis and so can be removed (if desired) from the generated rule set. erbBL(L!1), erbB(L!1,CR), erbBL(L!2), erbB(L!2,CR) -> \ erbBL(L!1), erbB(L!1,CR!0), erbBL(L!2), erbB(L!2,CR!0) erbBL(L!1), erbB(L!1,CR), erbB2(CR) -> \ erbBL(L!1), erbB(L!1,CR!0), erbB2(CR!0) The ﬁrst of these generic rules deals with most of the cases of (external) dimerization. The only diﬃculty comes from the fact that erbB2 has no ligand binding site L and, as such, cannot ever match the generic erbB agent in that rule which mentions L. For this reason, we must explicitly include the second rule that covers the case of erbB2 dimerizing with a diﬀerent erbB receptor type; we do not consider erbB2 homodimerization. These three rules generate a very large number (well over 150) of concrete rules due not only to the fact that the erbB generic agent is capable of multiple matches but also because the concrete erbB agents can bind multiple ligand agents. We next have the rule for internal (or asymmetric) dimer formation. Here, a receptor binds its (external) dimer partner on a second site; this dimer is asymmetric because the bond is made between the N site of one receptor and the C site of the other. In a dimer not containing erbB3, this asymmetric dimer can ﬂip states; this is the second rule. erbB(CR!1,N,C), erbB(CR!1,C) -> erbB(CR!1,N!0,C), erbB(CR!1,C!0) erbB(CR!1,N!2,C), erbB(CR!1,N,C!2) -> erbB(CR!1,N,C!3), erbB(CR!1,N!3,C)

The ﬁnal rule is for trans phosphorylation of one erbB receptor by its dimer partner. In an asymmetric dimer, the receptor bound on site C is the activator

126

V. Danos et al.

whereas the receptor bound on N is the activated. It is thus the activator that gets phosphorylated. erbB(N!1,atp,AS), erbB(C!1,Y~u) -> erbB(N!1,atp,AS), erbB(C!1,Y~p) This one generic rule expands into many concrete rules for two independent reasons. Firstly, each erbB agent can be multiply instantiated: the ﬁrst erbB can be anything but erbB3 (whose N site was deleted) and the second can be any of the four erbBs. Secondly, each receptor duplicates the site Y, so we get one concrete rule per duplicand. In the context of the purely wild-type model, neither the atp nor the AS site plays any role. This is because the rule tests only for the existence of these sites (which always succeeds) and that they are both unbound (which also always succeeds since we have no rules for binding to either of them). However, as we will see shortly, these two sites do play an important role once we take drug interventions and mutations into account. We can neatly summarize the model so far with its contact map (Fig. 1).

Fig. 1. Contact map of the consensus model: each concrete agent is represented once with all its sites; possible bindings are indicated by an edge joining two nodes, modiﬁable sites are indicated in grey. Only a restricted subset of known EGFR receptors ligands is shown, namely EGF and HRG.

Rule-Based Modelling and Model Perturbation

3.2

127

Ligand Perturbations

The erbB receptor network clearly has a lot of ﬂexibility in its response to ligands. In particular, the receptor dimers that form depend on the available ligands and the presented receptors. In addition, erbB3 has compromised capability to form asymmetric dimers: it can activate the catalytic activity of its dimer partner but cannot be activated by it. This phenomenon adds yet another layer of subtlety to erbB receptor activation. For example, in a cell line expressing erbB2, erbB3 and erbB4, one would expect HRG to promote phosphorylation of erbB2, via erbB2:erbB4 dimers, as well as phosphorylation of erbB3 and erbB4. On the other hand, were erbB4 not expressed, one would expect only erbB3 phosphorylation, via erbB2:erbB3 dimers. However, this kind of reasoning rapidly becomes highly complicated, particularly in the presence of multiple ligands, and we would like some way of deducing, from the rule set and a choice of expression levels of ligands and receptors, which receptors get phosphorylated (and, in some cases, on which sites). We can do this using static analysis of the rule set. We ﬁrst write dummy rules that detect typical molecular species of interest, e.g. erbB2(Y~p) -> erbB2(Y~p) We then ask the static analyser whether or not our dummy rules can ﬁre. It responds in one of two ways: either a categorical ‘no’ or a tentative ‘yes’. In the case of a ‘no’, we know (since the static analysis never produces false negatives) that our rule set cannot create the molecular species in question—starting from the declared initial solution. In the case of a ‘yes’, we have no certainly (since the analysis can give false positives) that the species can arise, but also no proof that it cannot. In an attempt to conﬁrm the ‘yes’, we then use the story sampler to search for pathways leading to a dummy rule; if (at least) one exists, we have proof that the species can arise. For example, the static analysis shows that, with our rule set, erbB2 phosphorylation cannot take place (categorical ‘no’) under the following conditions: – HRG only; erbB2 and erbB3 only [erbB3 cannot phosphorylate erbB2] – HRG only; erbB1, erbB2 and erbB3 only [erbB1 cannot bind HRG] whereas it can potentially take place (tentative ‘yes’) under the following conditions: – HRG only; erbB2, erbB3 and erbB4 only – EGF and HRG; erbB1, erbB2 and erbB3 only. To conﬁrm this claim, we ask for stories leading to the appropriate observables. In both cases, we ﬁnd indeed a story leading to phosphorylated erbB2 which conﬁrms that the static analysis did not give us false positives (Figs. 2, 3). This combination of static analysis and story sampling enables a powerful model development process where, starting from a consensus, perhaps overly restrictive, rule set, we investigate which observables of interest can arise under

128

V. Danos et al.

Fig. 2. Story leading to erbB2 phosphorylation by erbB4

Fig. 3. Story leading to erbB2 phosphorylation by erbB1

Rule-Based Modelling and Model Perturbation

129

which conditions. We then compare these predictions to experimental data in order to judge the accuracy and completeness of the model. If experimental data conﬂicts with the results of our analysis, this means one of two things: the consensus model either has fatal ﬂaws or missing links. A ‘fatal ﬂaw’ means that certain experimentally unobservable species can be generated by the rule set; in other words, that the mechanism described by the rules makes unwarranted assumptions. A ‘missing link’ corresponds to the more likely situation where an experimentally observed species remains inaccessible with our consensus model; this implies that the rule set lacks certain necessary rules. For example, in the above discussion, we noted that, in our rule set with erbB1, erbB2 and erbB3 only, HRG stimulation leads only to erbB3 phosphorylation; whereas the combination of EGF and HRG leads to phosphorylation of all three receptors. This constitutes an experimentally refutable prediction. In the event of such a refutation, e.g. we observe erbB1 phosphorylation upon HRG stimulation, we could freely postulate various new rules, check that they do indeed open up the possibility of erbB1 phosphorylation and then compare and contrast their eﬀects on other observables. If a new rule creates a ‘fatal ﬂaw’, we can discount it; but in general this may still leave us with a choice between several proposed new mechanisms. To decide between these would require us to ﬁnd a new, experimentally refutable prediction and do the experiment (or ﬁnd it in the literature). We stress that this remains a human-directed model development process— we do not consider automatically generated rules in any form—but one in which variant mechanisms can be built and evaluated in an organized fashion. 3.3

Drug Perturbations

In recent years, particularly with the realization that deregulated erbB signalling contributes to the development of multiple cancers, much research has focussed on ﬁnding ways of blocking the activity of this family of receptors via drug intervention. To date, two broad classes of drug have been developed: monoclonal antibodies (mAbs) and tyrosine kinase inhibitors (TKIs). Antibodies typically act as classical competitive inhibitors that exert their function by binding cell surface receptors in a way that physically obstructs their usual ligands from binding. On the other hand, TKIs behave as classical non-competitive inhibitors of kinase activity that do not prevent substrate binding but instead block the ATP binding site of the kinase domain, thus preventing substrate phosphorylation. As an illustration of the ease with which we can incorporate these kinds of pharmaceutical intervention in our modelling framework, we include rules for C225 (cetuximab, a mAb) binding to erbB1’s site L, ZD1839 (geﬁtinib, a TKI) binding to erbB1’s atp site and 4D5 (trastuzumab, another mAb) binding to erbB2’s dimerization site CR. C225(L), erbB1(L) -> C225(L!0), erbB1(L!0) ZD1839(L), erbB1(atp) -> C225(L!0), erbB1(atp!0) 4D5(L), erbB2(CR) -> C225(L!0), erbB2(CR!0)

130

V. Danos et al.

As each of these molecules has a reversible inhibitory eﬀect on the respective receptor, we also need the accompanying unbinding rules: C225(L!0), erbB1(L!0) -> C225(L), erbB1(L) ZD1839(L!0), erbB1(atp!0) -> ZD1839(L), erbB1(atp) 4D5(L!0), erbB2(CR!0) -> 4D5(L), erbB2(CR) The contact map of the system (Fig. 4) including these inhibitors makes it clear that the antibodies (C225 and 4D5) act as competitive inhibitors: C225 competes with EGF for the ligand binding site of erbB1 and 4D5 competes for the dimerization binding site of erbB2. The inhibitory eﬀect of ZD1839 shows up only in the inﬂuence map: the rule binding ZD1839 to erbB1 inhibits all modiﬁcation rules (concretely, the phosphorylations) dependent on erbB1. This is because ZD1839 binds to the site atp of erbB1 which must be free in order for erbB1 to modify its dimer partner. The presence of ZD1839 thus frustrates, without completely preventing, erbB1-dependent phosphorylation. We will return to this later.

Fig. 4. Contact map of the consensus model with antibodies (C225 and 4D5) as competitive inhibitors, and ZD1839 as non-competitive inhibitor.

Rule-Based Modelling and Model Perturbation

131

It should be noted that, in our examples of inhibitors, all agents are concrete. But, in general, drugs would also be organized, much like natural ligands, by an appropriate agent hierarchy. 3.4

The Uses of Mutational Perturbations

So far, we have seen how the use of agent variants allows us to organize agents hierarchically and thus write generic rules at a convenient level of granularity. In particular, this facilitates the development of models with families of related proteins, or proteins with multiple splice variants, that have overlapping functionality. However, agent variants also enable the treatment of mutated agents which likewise share a lot of the functionality of their wild-type cousins but which also potentially lose some of that functionality and/or gain new functionality. This has two immediate applications. Firstly, it allows us to build models with a mixture of wild-type and mutated agents in order to investigate (statically or numerically) the consequences of mutations. This is particularly interesting in the context of models, as described previously, that also include drug interventions. More subtly, it also allows us to cast a critical eye over the assumptions we make in building our wild-type model. After all, a lot of the experimental data from which consensus pathways have been deduced comes from mutation experiments. These typically eliminate one or more phosphorylation sites in a protein and investigate which, if any, pathways suﬀer from this perturbation. However, such data can be diﬃcult to interpret and the deduced wild-type interactions may be incorrect. For example, as explained in [28], one experiment showed that expressing a kinase-dead mutant of PI3K inhibited Ras activation upon EGF stimulation; this led the authors to propose a role for PI3K in activating Ras. But then, a second study demonstrated that a constitutively active (and membrane associated) mutant of PI3K did not promote Ras activation, which contradicted the conclusions of the ﬁrst study. In the end, it turned out [28] that PI3K actually inhibits Ras deactivation; so PI3K sensitizes Ras for activation but cannot by itself actually activate it. In more details: PI3K promotes Gab1 recruitment to the membrane which, on EGF stimulation, strongly recruits Shp2 to the membrane; Shp2 is a tyrosine phosphatase that dephoses the phospho-tyrosine binding sites for RasGAP (and for PI3K!) on erbB receptors and Gab1. So Shp2 inhibits RasGAP recruitment to the membrane which indirectly aids Ras activation (Sos which activates Ras has an easier job). This gives a measure of the daunting complexity of inferring a protein network, and as a consequence a measure of how helpful a methodology such as the one we illustrate here can prove. Indeed, using agent variants, we can express the kinds of (artiﬁcially) mutated proteins used in biochemical studies and so replay numerically such experiments. We can therefore detect, in numero, if the hypothesized wild-type network is in fact incorrect, e.g. if we had a model for Ras activation including a rule for ‘PI3K activates Ras’, we would have been able to predict that a constitutively active PI3K mutant would activate Ras; the fact that, experimentally, this is

132

V. Danos et al.

not observed means that that rule must be wrong. This kind of perturbational analysis is not just useful for postdictive veriﬁcation of inferences, it is also a discipline to build a model upon such data, and to build further data to refute predictions; this could be particularly interesting if two plausible molecular mechanisms (candidate consensus pathways) made divergent predictions. 3.5

Testing the Wild-Type Model

In two recent papers, Kuriyan and coworkers have developed a conceptual model of erbB receptor acivation that depends on the formation of an asymmetric dimer [27, 29]. We have used this when writing the above rules for the wildtype erbB network in the previous section. They developed their model using a combination of structural and mutational data and provide convincing evidence of its correctness by cotransfecting various artiﬁcal erbB constructs that lack one or more of the N, C and AS sites. We can use agent variants, in combination with static analysis, to reproduce these kinds of results in numero. For example, kinase-dead erbB1 is obtained by deﬁning a variant of erbB1 with the AS site deleted; this agent can no longer phosphorylate its dimer partner. Similarly, we can also introduce variants that delete either the N or the C site instead of, or in addition to, the AS site. %conc: %conc: %conc: %conc: %conc:

erbB1_KD = erbB1[AS/{}] erbB1_noN = erbB1[N/{}] erbB1_noC = erbB1[C/{}] erbB1_KDnoN = erbB1[AS/{} ; N/{}] erbB1_KDnoC = erbB1[AS/{} ; C/{}]

These agents inherit all rules from wild-type erbB1 that do not mention the sites that they lack. So erbB1 KD can freely form asymmetric dimers but phosphorylates nothing, whereas erbB1 noN and erbB1 noC are partially compromised in their ability to form asymmetric dimers: the former can activate its partner and get phosphorylated, but cannot be activated and phosphorylate its partner; the latter can be activated by its partner and phosphorylate it, but cannot activate its partner and get phosphorylated. We can now use static analysis, as in the previous section, to analyze the consequences of coexpressing pairs of these variant agents. We do this by checking the accessibility of the rules erbB1(N!1), erbB1(C!1) -> erbB1(N!1), erbB1(C!1) erbB1(Y1197~p) -> erbB1(Y1197~p) (that respectively detect the possibility of an asymmetric dimer forming and an erbB1 receptor becoming phosphorylated) in an initial solution that includes EGF and a choice of any (one or) two of the erbB1 variants. In particular, we can recapitulate the results of [27] (see their Fig. 6; we use the same combination numbers) in completely automatic fashion:

Rule-Based Modelling and Model Perturbation

133

1. wild-type erbB1 only: asymmetric homodimer accessible; phosphorylation accessible 2. erbB1 KD only: asymmetric homodimer accessible; phosphorylation inaccessible 3. erbB1 KD & erbB1 noN: one asymmetric heterodimer accessible; phosphorylation inaccessible 4. erbB1 KD & erbB1 noC: one asymmetric heterodimer accessible; phosphorylation accessible 5. erbB1 KDnoC only: asymmetric homodimer inaccessible; phosphorylation inaccessible 6. erbB1 KDnoC & erbB1 noN: one asymmetric heterodimer accessible; phosphorylation inaccessible 7. erbB1 KDnoC & erbB1 noC: asymmetric heterodimer inaccessible; phosphorylation inaccessible 8. erbB1 KDnoN only: asymmetric homodimer inaccessible; phosphorylation inaccessible 9. erbB1 KDnoN & erbB1 noN: asymmetric heterodimer inaccessible; phosphorylation inaccessible 10. erbB1 KDnoN & erbB1 noC: one asymmetric heterodimer accesible; phosphorylation accessible In [27], this had to be done by hand, a task that soon begins to get rather subtle, particularly if you want to consider doubly-mutated agents and/or coexpression of more than two receptor constructs at a time. It is thus very useful to be able to express this situation in Kappa and rely on static analysis to detect the impossibility/possibility of phosphorylation. Moreover, if the static analysis announces that phosphorylation is not impossible, we can, as above, use the story sampler to search for ways in which this can actually take place. Again, in some cases, this is easy to do by hand but, beyond a certain degree of complexity, it is highly desirable to have an automatic method in order to avoid making mistakes. These results demonstrate that our consensus model is indeed compatible with the experimental data of Kuriyan et al. and, as such, it passes the test. This comforts us, for now, in our choice of rules but of course provides no guarantee that future experimental data will not invalidate some of them. 3.6

The Limits of Perturbation Testing

We mentioned earlier that our language extension shields the modeller from the underlying rule set generated by generic rules. However, we should say that this is only true qualitatively—if we wish to manipulate the rate constants of our model in such a way that diﬀerent concrete instantiations of one generic rule get diﬀerent kinetics, this can only be done by examining and modifying directly the generated rule set.

134

V. Danos et al.

More generally, the modelling methodology advocated above based on static analysis cannot be used to gauge the eﬀect of perturbations, such as drugs, that restrict, but don’t outlaw, the application of other rules. Or, to put it another way, a perturbation that operates entirely at the level of kinetics is undetectable by this method. We would however expect to observe the eﬀects of such perturbations during stochastic simulation and/or story sampling. Indeed, it would be straightforward to observe the inhibition of erbB1’s kinase activity by tracking that rule’s activity in the absence and presence of drugs. More ambitiously, we could compare the relative strengths of each erbB’s kinase activity and the way in which that is disturbed by drugs that target only one receptor; this would require running the story sampler many times at many time points to get a statistical picture of the model’s activity proﬁle over time. We leave this for future work.

4

Conclusions

Even if wild-type pathways are obviously central to a systemic view of molecular biology, modelling is not just about these. It is equally important to be able to navigate the space of derivatives of a model for two complementary reasons. Firstly, one needs to understand diseased conditions as natural perturbations of the wild-type; secondly, one also needs to represent synthetic perturbations (by genetic knock-outs, domain truncations, point mutations, etc) because they are key in the inference of the wild-type. This is a formidable challenge because the space of such model perturbations introduces a second kind of combinatorial explosion. The well-studied example of the EGF receptor family (see §3) is a powerful illustration of this fact. Now, we have to do something if we want our modelling vessel to stay aﬂoat in the sea of perturbations. In other words, just as the passage from reactions to rules tames the ﬁrst binding-caused explosion, we have to ﬁnd a mechanism to tame what one might call the perturbation-caused explosion. The fact that Kappa describes molecular interactions at the level of domain binding and modiﬁcation seems a good start, since this is the granularity at which the engineering of perturbations in protein networks actually happens (e.g. Y to A mutations that disable a modiﬁcation). But to tackle our representation problem, we need another ingredient, namely a syntactic extension of Kappa that enables a clean, uniform treatment of protein families, splice variants and mutated proteins. This is what we have proposed here. The idea is to structure agents hierarchically so that rules can be expressed at an appropriate level of abstraction, as generic rules, which are then automatically compiled into pure Kappa. This eases the pain (and pitfalls) of writing large rule sets (indeed the modeller has no need to ever look at the resulting concrete rule set, unless he/she wishes to modify its rate constants), and as we wanted, this give means to navigate their perturbation space. Of course there is no magic: to work around the explosive generativity of wildtype pathways we capture postulated regularities by using rules (if the universe

Rule-Based Modelling and Model Perturbation

135

of reactions were lacking any regularity no method could describe them anyway, a rather grim perspective for systems biology); to work around the second source of complexity, again we capture regularities of another kind, namely that much of the wild-type behaviour of a protein is actually shared with its mutants and isoforms. We have shown that this strategy works well with our EGF example, as we were able to neatly set a wild-type model together with a selection of derivatives. With this model in place, one can bring the usual analysis tools of Kappa to bear on the rule set. As we have shown further, even in the absence of quantitative information about rates and copy numbers, one can obtain qualitative predictions about the induced perturbed behaviours and thus support on a full-scale the traditional informal inferences that are commonplace in the experimental investigation of protein networks. Acknowledgements. Jean Krivine is supported via grants from the Agence Nationale de la Recherche (ANR-07-PHYSIO-013-01) and the G´enopole Evry held by A. Benecke of the IHES.

References 1. Kholodenko, B.N., Demin, O.V., Moehren, G., Hoek, J.B.: Quantiﬁcation of Short Term Signaling by the Epidermal Growth Factor Receptor. J. Biol. Chem. 274(42), 30169–30181 (1999) 2. Kiyatkin, A., Aksamitiene, E., Markevich, N.I., Borisov, N.M., Hoek, J.B., Kholodenko, B.N.: Scaﬀolding protein GAB1 sustains epidermal growth factorinduced mitogenic and survival signaling by multiple positive feedback loops. J. Biol. Chem. 281, 19925–19938 (2006) 3. Orton, R.J., Sturm, O.E., Vyshemirsky, V., Calder, M., Gilbert, D.R., Kolch, W.: Computational modelling of the receptor tyrosine kinase activated MAPK pathway. Biochemical Journal 392(2), 249–261 (2005) 4. Schoeberl, B., Eichler-Jonsson, C., Gilles, E.-D., M¨ uller, G.: Computational modeling of the dynamics of the map kinase cascade activated by surface and internalized EGF receptors. Nature Biotechnology 20, 370–375 (2002) 5. Hlavacek, W.S., Faeder, J.R., Blinov, M.L., Posner, R.G., Hucka, M., Fontana, W.: Rules for Modeling Signal-Transduction Systems. Science’s STKE 2006(344) (2006) 6. Maslov, S., Ispolatov, I.: Propagation of large concentration changes in reversible protein-binding networks. Proceedings of the National Academy of Sciences 104(34), 13655–13660 (2007) 7. Regev, A., Silverman, W., Shapiro, E.: Representation and simulation of biochemical processes using the π-calculus process algebra. In: Altman, R.B., Dunker, A.K., Hunter, L., Klein, T.E. (eds.) Paciﬁc Symposium on Biocomputing, vol. 6, pp. 459–470. World Scientiﬁc Press, Singapore (2001) 8. Regev, A., Shapiro, E.: Cells as computation. Nature 419 (September 2002) 9. Priami, C., Regev, A., Shapiro, E., Silverman, W.: Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Information Processing Letters (2001)

136

V. Danos et al.

10. Baldi, C., Degano, P., Priami, C.: Causal π-calculus for biochemical modeling. In: Proceedings of the AI*IA Workshop on BioInformatics 2002, pp. 69–72 (2002) 11. Priami, C., Quaglia, P.: Beta Binders for Biological Interactions. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 20–33. Springer, Heidelberg (2005) 12. Cardelli, L.: Brane Calculi Interactions of Biological Membranes. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 257–278. Springer, Heidelberg (2005) 13. Regev, A., Panina, E.M., Silverman, W., Cardelli, L., Shapiro, E.: BioAmbients: an abstraction for biological compartments. Theoretical Computer Science 325, 141–167 (2004) 14. John, M., Ewald, R., Uhrmacher, A.M.: A Spatial Extension to the π Calculus. Electronic Notes in Theoretical Computer Science, vol. 194(3), pp. 133–148 (2008) 15. Calder, M., Gilmore, S., Hillston, J.: Modelling the inﬂuence of RKIP on the ERK signalling pathway using the stochastic process algebra PEPA. In: Priami, C., Ing´ olfsd´ ottir, A., Mishra, B., Riis Nielson, H. (eds.) Transactions on Computational Systems Biology VII. LNCS (LNBI), vol. 4230, pp. 1–23. Springer, Heidelberg (2006) 16. Ciocchetta, F., Hillston, J.: Bio-PEPA: an extension of the process algebra PEPA for biochemical networks. Electronic Notes in Theoretical Computer Science, vol. 194(3), pp. 103–117 (2008) 17. Calzone, L., Fages, F., Soliman, S.: BIOCHAM: an environment for modeling biological systems and formalizing experimental knowledge. Bioinformatics 22(14), 1805–1807 (2006) 18. Dematte, L., Priami, C., Romanel, A.: The BlenX language: a tutorial. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 313–365. Springer, Heidelberg (2008) 19. Blinov, M.L., Faeder, J.R., Hlavacek, W.S.: BioNetGen: software for rule-based modeling of signal transduction based on the interactions of molecular domains. Bioinformatics 20, 3289–3292 (2004) 20. Dematt´e, L., Priami, C., Romanel, A., Soyer, O.: Evolving BlenX programs to simulate the evolution of biological networks. Theoretical Computer Science 408(1), 83–96 (2008) 21. Danos, V., Laneve, C.: Formal molecular biology. Theoretical Computer Science 325(1), 69–110 (2004) 22. Danos, V., Feret, J., Fontana, W., Krivine, J.: Abstract Interpretation of Cellular Signalling Networks. In: Logozzo, F., Peled, D.A., Zuck, L.D. (eds.) VMCAI 2008. LNCS, vol. 4905, pp. 83–97. Springer, Heidelberg (2008) 23. Danos, V., Feret, J., Fontana, W., Harmer, R., Krivine, J.: Rule-Based Modelling of Cellular Signalling. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 17–41. Springer, Heidelberg (2007) 24. Danos, V., Feret, J., Fontana, W., Krivine, J.: Scalable Simulation of Cellular Signaling Networks. In: Shao, Z. (ed.) APLAS 2007. LNCS, vol. 4807, pp. 139–157. Springer, Heidelberg (2007) 25. Murphy, L.O., Smith, S., Chen, R.H., Fingar, D.C., Blenis, J.: Molecular interpretation of ERK signal duration by immediate early gene products. Nat. Cell Biol. 4(8), 556–564 (2002) 26. Burgess, A.W., Cho, H.S., Eigenbrot, C., Ferguson, K.M., Garrett, T.P.J., Leahy, D.J., Lemmon, M.A., Sliwkowski, M.X., Ward, C.W., Yokoyama, S.: An Open-andShut Case? Recent Insights into the Activation of EGF/ErbB Receptors. Molecular Cell 12(3), 541–552 (2003)

Rule-Based Modelling and Model Perturbation

137

27. Zhang, X., Gureasko, J., Shen, K., Cole, P.A., Kuriyan, J.: An Allosteric Mechanism for Activation of the Kinase Domain of Epidermal Growth Factor Receptor. Cell 125(6), 1137–1149 (2006) 28. Sampaio, C., Dance, M., Montagner, A., Edouard, T., Malet, N., Perret, B., Yart, A., Salles, J., Raynal, P.: Signal strength dictates phosphoinositide 3-kinase contribution to Ras/extracellular signal-regulated kinase 1 and 2 activation via differential Gab1/Shp2 recruitment: consequences for resistance to epidermal growth factor receptor inhibition. Mol. Cell Biol. 28(2), 587–600 (2008) 29. Zhang, X., Pickin, K.A., Bose, R., Jura, N., Cole, P.A., Kuriyan, J.: Inhibition of the EGF receptor by binding of MIG6 to an activating kinase domain interface. Nature 450(7170), 741 (2007)

Extended Stochastic Petri Nets for Model-Based Design of Wetlab Experiments Monika Heiner1 , Sebastian Lehrack1 , David Gilbert2 , and Wolfgang Marwan3 1

Department of Computer Science, Brandenburg University of Technology Postbox 10 13 44, 03013 Cottbus, Germany {monika.heiner,slehrack}@informatik.tu-cottbus.de 2 School of Information Systems, Computing and Mathematics Brunel University, Uxbridge, Middlesex UB8 3PH, UK [email protected] 3 Otto von Guericke University & Magdeburg Centre for Systems Biology c/o Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany [email protected]

Abstract. This paper introduces extended stochastic Petri nets to model wetlab experiments. The extentions include read and inhibitor arcs, stochastic transitions with freestyle rate functions as well as several deterministically timed transition types: immediate ﬁring, deterministic ﬁring delay, and scheduled ﬁring. The extensions result into nonMarkovian behaviour, which precludes analytical analysis approaches. But there are adapted stochastic simulation analysis (SSA) methods, ready to deal with the extended behaviour. Having the simulation traces, we apply simulative model checking of PLTL, a linear-time temporal logic (LTL) in a probabilistic setting. We present some typical model components, demonstrating the suitability of the introduced Petri net class for the envisaged application scenario. We conclude by looking brieﬂy at a classical example of prokaryotic gene regulation, the lac operon case.

1

Motivation

This paper extends the Markovian stochastic Petri nets SPN Bio as introduced in [GHL07] to model and analyse biochemical networks. Related application scenarios are discussed in [BGHO08], [GBHD09]. Case studies demonstrating a unifying framework to integrate the qualitative, stochastic and continuous paradigms can be found in [HGD08], [GHR+ 08], [HDG10]. Thus, SPN Bio have been proven to be useful in systems and synthetic biology. However, there are limitations in expressivity. Generally, biologists face the problem to design wetlab experiments to validate or contradict the current understanding of the biochemical network under investigation. In order to be better able to do so, they ask for the following advanced features: C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 138–163, 2009. c Springer-Verlag Berlin Heidelberg 2009

Extended Stochastic Petri Nets

139

– stochastic and deterministic ﬁring behaviour within one model, – relative and absolute timing of the transitions’ ﬁring, – construction of arbitrary schedules of programmed interventions. Therefore, we are going to extend SPN Bio belonging to the Markovian world by several features supporting the comfortable modelling of wetlab experiments. The extentions lead to the deﬁnition of biochemically interpreted Generalised Stochastic Petri nets GSPN Bio and Deterministic and Stochastic Petri nets DSPN Bio . They include read and inhibitor arcs, stochastic transitions with freestyle rate functions as well as several deterministically timed transition types: immediate ﬁring, deterministic ﬁring delay, and scheduled ﬁring. The extension go beyond the Markov property, which precludes analytical analysis approaches; but there are adapted stochastic simulation analysis (SSA) methods, ready to deal with the extended behaviour. Having the simulation traces we apply simulative model checking of linear-time temporal logic (LTL) in a probabilistic setting (PLTL). Simulative model checking approximates the probability of a given temporal logic formula by considering ﬁnite sets of ﬁnite paths through the state space. Thus, it works even for systems with inﬁnite state spaces. We discuss in detail some typical model components, demonstrating the suitability of the introduced Petri net class DSPN Bio for the envisaged application scenario. These components will be analysed by checking sets of stochastic simulation traces against PLTL properties. In doing so, a special category of properties, the so-called invariant properties, will be used to prove at the same time the plausibility of the applied simulation algorithm. We conclude by looking brieﬂy at a classical example of prokaryotic gene regulation, the lac operon case.

2

Stochastic Modelling

We assume basic knowledge of the standard notions of qualitative place/transition Petri nets, see e.g. [Mur89], [Rei82], [HGD08]. To be self-contained we start with recalling the fundamentals of (biochemically interpreted) stochastic Petri nets, belonging to the Markovian world, before introducing the extended notions resulting ﬁnally into non-Markovian Petri nets. 2.1

The Markovian Case - Stochastic Petri Nets (SPN Bio )

As with a qualitative Petri net, a stochastic Petri net maintains a discrete number of tokens on its places. But contrary to the time-free case, a ﬁring rate (waiting time) is associated with each transition t, which are random variables Xt ∈ [0, ∞), deﬁned by probability distributions. Therefore, all reaction times can theoretically still occur, but the likelihood depends on the probability distribution. Consequently, the system behaviour is described by the same discrete state space, and all the diﬀerent execution runs of the underlying qualitative

140

M. Heiner et al.

Petri net can still take place. This allows the use of the same powerful analysis techniques for stochastic Petri nets as they are applied for qualitative Petri nets. For a better understanding we describe the general procedure of a particular simulation run for a stochastic Petri net. Each transition gets its own local timer. When a particular transition becomes enabled, meaning that suﬃcient tokens arrive on its preplaces, then the local timer is set to an initial value, which is computed at this time point by means of the corresponding probability distribution. In general, this value will be diﬀerent for each simulation run. The local timer is then decremented at a constant speed, and the transition will ﬁre when the timer reaches zero. If there is more than one enabled transition, a race for the next ﬁring will take place. After the ﬁring of the winning transition, the timers of the others still enabled transitions keep their values or are reset, depending on the speciﬁc type of the net. Technically, various probability distributions can be chosen to determine the random values for the local timers. Biochemical systems are the prototype for exponentially distributed reactions. Thus, for our purposes, the ﬁring rates of all transitions follow an exponential distribution, which can be described by a single parameter λ, and each transition needs only its particular, generally marking-dependent parameter λ to specify its local time behaviour. The following deﬁnition summarises this informal introduction. Definition 1 (Stochastic Petri net, Syntax). A biochemically interpreted stochastic Petri net is a quintuple SPN Bio = (P, T, f, v, m0 ), where – P and T are ﬁnite, nonempty, and disjoint sets. P is the set of places, and T is the set of transitions. – f : ((P × T ) ∪ (T × P )) → IN0 deﬁnes the set of directed arcs, weighted by nonnegative integer values. – v : T → H is a function, which assigns a stochastic hazard function ht to each transition t, whereby |• t| H := t∈T ht | ht : IN0 → IR+ is the set of all stochastic hazard functions, and v(t) = ht for all transitions t ∈ T . – m0 : P → IN0 gives the initial marking. The stochastic hazard function ht deﬁnes the marking-dependent transition rate λt (m) for the transition t, i.e. ht = λt (m). The domain of ht is restricted to the set of preplaces of t, denoted by • t with • t := {p ∈ P |f (p, t) = 0}, to enforce a close relation between network structure and hazard functions. Therefore, λt (m) actually depends on a sub-marking only. Stochastic Petri net, Semantics. Transitions become enabled as usual, i.e. if all preplaces are suﬃciently marked. However there is a time, which has to elapse, before an enabled transition t ∈ T ﬁres. The transition’s ﬁring delay (waiting time) is an exponentially distributed random variable Xt with the probability density function: fXt (τ ) = λt (m) · e(−λt (m)·τ ) ,

τ ≥ 0.

Extended Stochastic Petri Nets

141

The ﬁring itself does not consume time and follows the standard ﬁring rule of qualitative Petri nets. The semantics of a stochastic Petri net (with exponentially distributed ﬁring delays for all transitions) is described by a continuous time Markov chain (CTMC). The CTMC of a stochastic Petri net without parallel transitions is isomorphic to the reachability graph of the underlying qualitative Petri net, while the arcs between the states are now labelled by the transition rates. For more details see [MBC+ 95], [BK02], [HGD08]. Based on this general SPN Bio deﬁnition, specialised biochemically interpreted stochastic Petri nets can be deﬁned by specifying the required kind of stochastic hazard function more precisely. In this paper, we are going to use the molecule semantics with mass action transition rates. Therefore we deploy the stochastic mass-action hazard function, which tailors the general SPN Bio deﬁnition to biochemical mass-action networks, where tokens correspond to molecules: m(p) ht := ct · . f (p, t) • p∈ t

The constant ct is the transition-speciﬁc stochastic rate constant, and m(p) is the current number of tokens on a preplace p of the transition t. The binomial coeﬃcient describes the number of non-ordered combinations of the f (p, t) molecules, required for the reaction, out of the m(p) available ones. In the following we abbreviate this formula by BioMassAction(ct ). See [GHL07] for another example, reading the tokens as concentration levels. 2.2

The Non-markovian Case - Extended Stochastic Petri Nets

We start oﬀ with an overview and brief biochemical motivation before introducing two classes of extended stochastic Petri nets. There are quite a number of various extensions based on the fundamental stochastic Petri net class SPN , see e.g. [MBC+ 95], [Ger01]. The most important additional features concern deterministically timed transitions, or deterministic transitions for short, which come along in diﬀerent types. The crucial point is that the ﬁring delay (waiting time) before an enabled transition ﬁres does not depend anymore on a random variable, but is speciﬁed by a ﬁxed time duration. To avoid confusion, we will call the transitions with a probabilistic ﬁring delay, as introduced in the former subsection, stochastic transitions, if necessary. In summary, our extended stochastic Petri nets support the following features: – – – –

read and inhibitor arcs, programmed transitions (freestyle rate functions), deterministic ﬁring delay, scheduled transitions.

Read and inhibitor arcs. are popular add-ons enhancing modelling comfort. Read arcs (often also called test arcs) allow to specify positive side-conditions, e.g., if the occurrence of a subunit depends on the conformation of a protein

142

M. Heiner et al.

complex, or if a cell’s reaction to a given stimulus depends on the speciﬁc physiological conditions of the cell. Contrary, inhibitor arcs allow to specify negative side-conditions in an abstract way, e.g., if the presence of a given protein or condition inhibits a speciﬁc reaction. Speaking in technical terms, read and inhibitor arcs are directed arcs, going always from places to transitions. The standard ﬁring rule needs to be adapted accordingly. The enabling condition is extended in the following way: if there is an arc a with a weight w = f (p, t) connecting a place p with a transition t, then t can be enabled in a marking m if the following conditions are also satisﬁed: – a is a read arc ∧ m(p) ≥ w, – a is an inhibitor arc ∧ m(p) < w. The token situation on p is not changed by the ﬁring of t, i.e. m (p) = m(p) for t m− → m . Programmed transitions are stochastic transitions with freestyle rate functions. The ﬁring rate can be speciﬁed by arbitrary mathematical functions, stored in lookup tables, if necessary. To give an example, a popular phenomenon in biology is cooperativity. A biochemical reaction may be controlled by an highly non-linear, cooperative mechanism. Simple versions of cooperativity may be represented by complicated Petri net structures, but there are limits. The kinetic mechanisms of a cooperative behaviour are often not completely understood. However, the acquired understanding must be included in the model to get a coherent system model. Deterministic firing delay is the outstanding characteristics of deterministic transitions. The delay is always relative to the time point where the transition gets enabled. There is one popular special case, the zero delay, for which the immediate transitions are introduced. Immediate transitions have always highest priority, which creates a subtle diﬀerence between an immediate transition and a deterministic transition with zero ﬁring delay: if there is a conﬂict between the two, the immediate transition gets priority. We will use the function TimedFiring(delay) to assign the delay constant. Scheduled transitions belong to the deterministic transitions. The deterministic ﬁring occurs according to a schedule specifying absolute points of the simulation time. A schedule can specify just a single time point, or equidistant time points within a given interval, triggering the ﬁring once or periodically. However, transitions only ﬁre at their scheduled time points if they are enabled. Scheduled transitions can dramatically restrict the behaviour, as we will see in Section 4.3, example EX5. Scheduled transitions allow to disturb the core model at well-deﬁned time points as it is done experimentally with the actual biological system under investigation in the wetlab; see Section 5 for an example. We will use two functions to assign the required values: FixedTimedFiring Single(time point), FixedTimedFiring Periodic(begin time point, repetition, end time point).

Extended Stochastic Petri Nets

2.3

143

Generalised Stochastic Petri Nets (GSPN Bio )

Generalised stochastic Petri nets (GSPN Bio ) are stochastic Petri nets SPN Bio extended by inhibitor arcs and immediate transitions. Inhibitor arcs are a powerful modelling feature and are known to bring computational completeness. Consequently, Petri nets of the net class GSPN have the same expressivity as an universal Turing machine [PW03]. However, in terms of construction of the reachability graph (continuous-time Markov chain), they do not establish additional challenges for ﬁnite state spaces, i.e. bounded Petri nets. Immediate transitions are a very special kind of deterministic transitions with zero ﬁring delay, i.e. they ﬁre immediately after getting enabled, and always prior to (general) deterministic and stochastic transitions. Consequently, getting enabled and the ﬁring itself coincide for immediate transitions. A cyclic system behaviour involving only the ﬁring of immediate transitions corresponds to an inﬁnite behaviour without time progress; we get a time deadlock. If a stochastic simulation encounters a situation with more than one immediate transition enabled, one is chosen randomly [Ger01]. However, an analysis approach will consider all possible choices. In terms of the reachability graph (continuous-time Markov chain), induced by a GSPN Bio Petri net, we distinguish between transient and non-transient states. A system never spends time in a transient state before changing into another state. Thus, the time spent (sojourn time) in transient states is always zero, and not exponentially distributed anymore. Consequently, the underlaying semantics is not a continuous-time Markov chain anymore. However, the transient states can be removed such that the reduced reachability graph corresponds again to a continuous-time Markov chain. See [MBC+ 95] for a precise description of the reduction technique and related formal deﬁnitions. In summary this means that GSPN Bio can still be analysed analytically, if the state space, i.e. the continuous-time Markov chain can be constructed. 2.4

Deterministic and Stochastic Petri Nets (DSPN Bio )

Deterministic and Stochastic Petri Nets (DSPN Bio ) are generalised stochastic Petri nets (GSPN Bio ) extended by deterministic transitions. Deterministic transitions possess a deterministic ﬁring delay (waiting time), speciﬁed by a nonnegative real value. When a deterministic transition gets enabled, a count-down timer is started, initialized with the transition’s ﬁring delay. If the transition gets disabled before the timer reaches zero, the timer is switched oﬀ, and the transition will not ﬁre. Otherwise, the transition will ﬁre as soon as the timer reaches zero. The ﬁring itself does not consume time. If we consider stochastic Petri nets without deterministic transitions, the probability of two transitions ﬁring at the same time is practically zero. Contrary,

144

M. Heiner et al.

in stochastic Petri nets with deterministic transitions, it is possible that two transitions want to ﬁre simultaneously. We already discussed the special case of two concurrently enabled immediate transitions. To analyse such a system, all possible choices have to be considered, while in the simulation a random choice takes place. Definition 2 (Deterministic and stochastic Petri net). A biochemically interpreted deterministic and stochastic Petri net is a septuple DSP NBio = (P, T, f, g, v, d, m0 ), where – P und T are ﬁnite, nonempty, and disjoint sets. P is the set of places, and T is the set of transitions. – The set T is the union of three disjunctive transition sets, i.e. T := Tstoch ∪ Tim ∪ Ttimed with: 1. Tstoch , the set of stochastic transitions with exponentially distributed waiting time, 2. Tim , the set of immediate transitions with waiting time zero, and 3. Ttimed , the set of transitions with deterministic waiting time. – f : ((P × T ) ∪ (T × P )) → IN0 deﬁnes the set of directed arcs, weighted by nonnegative integers. – g : (P × T ) → IN0 deﬁnes the set of directed inhibitor arcs, weighted by nonnegative integers. – v : Tstoch → H is a function, which assigns a stochastic hazard function ht to each transition t ∈ Tstoch•, whereby | t| H := t∈Tstoch ht | ht : IN0 → IR+ is the set of all stochastic hazard functions, and v(t) = ht for all transitions t ∈ Tstoch . – d : Ttimed → IR+ assigns to each deterministic transition t ∈ Ttimed a nonnegative deterministic waiting time. – m0 : P → IN0 gives the initial marking. The stochastic transitions correspond to the transitions of the net class SPN Bio , so they have an exponentially distributed waiting time following the deﬁnitions given in Section 2.1. The net class DSPN Bio is a subset of the class eDSPN , introduced in [Ger01]. For details of the subset relation see [Leh07]. Therefore, the theory, which has been developed to analyse eDSPN Petri nets, see [Ger01] and [Haa03], can be deployed to analyse DSPN Bio , too. The remaining two features read arcs and scheduled transitions are not explicitly mentioned in the deﬁnition above, because the just allow a simpliﬁed speciﬁcation using the orthogonal basic concepts in DSPN Bio . Read arcs do not extend the modelling power as long as an interleaving semantics is considered. A read arc and two opposite arcs are indistinguishable in terms of the reachability graph (continuous-time Markov chain). Scheduled transitions can be replaced by net components consisting of immediate and deterministic transitions only; see [Leh07] for construction patterns. Thus, they do not extend the modelling power.

Extended Stochastic Petri Nets

3

145

Stochastic Analysis

The non-Markovian behaviour of DSP NBio precludes the standard analytical approaches belonging to the Markovian world. However, there are adapted stochastic simulation methods, ready to deal with the extended behaviour, see e.g. [Ger01], [Haa03], [Leh07], and many more. A detailed discussion of the necessary adaptions compared to the fundamental Gillespie algorithm [Gil77] is beyond the given space limitations of this paper. Having the simulation traces, we apply simulative model checking of linear-time temporal logic (LTL) in a probabilistic setting (PLTL). Simulative model checking follows the idea of Monte Carlo sampling and handles large or even inﬁnite state spaces through approximating results by analysing only a subset of the state space – a ﬁnite set of ﬁnite outputs (traces) from a stochastic simulation algorithm (SSA), e.g. Gillespie’s exact SSA or any other suitable variations of it. A natural choice of logic to describe properties of sets of traces is linear-time logic. A linear-time logic operates over sets of linear paths through the state space, equivalent to operating on simulation outputs. A given property holds if it holds in all possible paths. Consequently, there are no path quantiﬁers. We apply PLTL, a probabilistic linear-time temporal logic [DG08], [MC208]. This logic extends standard Linear-time Temporal Logic (LTL) [Pnu81] to a stochastic setting with a probability operator and a ﬁlter construct, deﬁning the initial state of the property. LTL is the fragment of full Computational Tree Logic (CTL*) [CGP01] without path quantiﬁers, implicitly quantifying universally over all paths. To be self-contained we brieﬂy recall the PLTL basics. Syntax. PLTL is a logic to create path formulae φ and to ask for their probabilities. The grammar given in Table 1 deﬁnes a PLTL formula ψ. Semantics. The semantics is deﬁned over ﬁnite sets of ﬁnite linear traces of temporal behaviour, in our case by stochastic simulation runs. Each trace is evaluated to a Boolean truth value, and the probability of a property holding true is computed by the fraction of true values in the set over the whole set. It goes without saying, the choice of simulator and simulation parameters used to compute the sequence of states can aﬀect the semantics of the PLTL property and the correctness of the result. Px is any inequality comparison of the probability of the property holding true, for example P≥0.5 . The expression P=? returns the value of the probability of the property holding true. Equality testing of the probability, P=x , is not supported for obvious reasons. PLTL allows the use of ﬁlters over top-level LTL expressions, denoted by {AP }, similar to those used in Probabilistic Computational Tree Logic (PCTL) [HJ94] and Continuous Stochastic Logic (CSL) [ASSB96]. This permits speciﬁcations to refer to the state or states that the property is checked from, rather than default to the initial state. This means that for a query of the form φ {AP }, φ is checked from the ﬁrst state that AP is satisﬁed. This can be a diﬀerent one for each stochastic run. The temporal operators follow the standard LTL semantics:

146

M. Heiner et al.

Table 1. PLTL syntax. Please note that the square and curly brackets are part of PLTL. Px [ φ ] Px [ φ {AP } ] .

ψ

::= |

φ

::= Xφ | G φ | Fφ | φ U φ | φ R φ | ¬ φ | φ∨φ | φ∧φ | φ⇒ φ | AP .

AP

::= ¬ AP | AP ∨ AP | AP ∧ AP | AP ⇒ AP | value comp value | true | f alse .

comp ::= = | = | ≥ | > | < | ≤ . value ::= value op value | variable | max(variable) | d(variable) | Int | Real . op

::= + | − | ∗ | / ,

with ∈ {}, x ∈ [0, 1]. Px can be replaced by Px=? .

– – – –

Next (X) - The property must hold true in the next time point. Globally (G) - The property must hold true always 1 . Finally (F) - The property must hold true sometime in the future. Until (U) - The ﬁrst property must hold true until the second property holds true. – Release (R) - The second property can only ever not hold true if the ﬁrst property becomes true.

The meta term variable stands for any variable in the model, Int is any integer number and Real is any real number. In our case of stochastic Petri net analysis, a variable is going to be a place name, and the formulae refer to the number of tokens on a place in a given state. Additionally, there is a predeﬁned variable time, referring to the simulation time points. Thus we can, for example, express properties which occur after some simulation time has elapsed. The function max operates over all the token values of a place to return the maximum in the given simulation runs, thus the peak of a species’ concentration, modelled by a place, can be checked, e.g. P rotein = max(P rotein). The function d operates on each place in each state individually to return the derivative, thus increasing token numbers can be checked, e.g. d(P rotein) > 0. This approach to simulative model checking incorporates two approximations. The truth value of a single trace is approximated by operating over a ﬁnite sequence of states only; and the probability of the property is approximated through sampling a ﬁnite number of traces only. Thus, a subset of the model’s behaviour is considered only. However, there are two special categories 1

To be precise, in the given setting of model checking by ﬁnite traces, globally means ’always –as far as known’.

Extended Stochastic Petri Nets

147

of properties, where deﬁnitive, i.e. non-approximating answers are possible by simulative model checking. – Monotone properties comply with the following condition: if the property is satisﬁed in any path through the state space, then it is satisﬁed in any extension of the path [HLMP04]. Formulae without the Globally operator are monotone properties. The Globally operator and semantically equivalent descriptions by the other operators are incompatible with the monotony property. Considering longer paths can only increase the probability. – Invariant Properties have to hold true in every state in every path. Thus they comply with the following condition: if the property is satisﬁed in any path through the state space, then it is satisﬁed in any other path. Their probability is independent of the number of considered paths. They are often used as consistency checks, and so do we in this paper. PLTL may be considered as a linear-time counterpart to CSL. It can easily be used to formalise the visual evaluation of diagrams as generated by deterministic/stochastic simulation runs or by recording experimental time series. In the following chapter we are going to use PLTL to analyse sets of stochastic simulation traces of extended stochastic Petri nets, which have been constructed to illustrate the expressiveness of DSP NBio .

4

Typical Components

We present some typical model components, controlling a network’s inﬂow and outﬂow, and thus demonstrating the suitability of the introduced Petri net class DSP NBio for the envisaged application scenarios of model-based design of wetlab experiments. We use the following abbreviations introduced in Section 2: – – – –

BioMassAction(ct ), TimedFiring(delay), FixedTimedFiring Single(time point), FixedTimedFiring Periodic(begin time point, repetition, end time point);

and we apply the following drawing conventions: – – – – –

read arcs: identiﬁed by a black dot, inhibitor arcs: identiﬁed by a hollow dot, stochastic transition: hollow square, deterministically timed transition: black square, immediate transition: black rectangle.

We are going to examine the behaviour of each component by simulative PLTL model checking over 100 (1,000) simulation runs. The individual runs are independent, so generally diﬀerent. We conﬁne ourselves deliberately on introductory formulae to illustrate the key ideas, increasing at the same time our conﬁdence in the accuracy of our simulation algorithm for the non-Markovian setting.

148

4.1

M. Heiner et al.

Time-Controlled Inflow/Outflow

EX1. In our ﬁrst example we consider a closed system, consisting of one reversible reaction A ↔ B, modelled by the two transitions t1 (BioMassAction(0.11)) and t2 (BioMassAction(0.1)). The two deterministically timed transitions input (FixedTimedFiring Periodic(11,1,20)) and output (FixedTimedFiring Periodic(31,1,40)) are responsible for the absolutely timed inﬂow and outﬂow of tokens, see Figure 1. The transition input does not have preplaces, thus it ﬁres for sure at the time points 11, 12, . . . , 20, producing each time 1,000 additional tokens on place A. Contrary, the transition output removes 1,000 tokens from place B at the time points 31, 32, . . . , 40, provided there are enough tokens to enable the ﬁring. Figure 2 shows the ﬁrst 100 time units of a single simulation run. We give some introductory samples of temporal-logic formulae (queries), formalising the visual inspection of the simulation output as it might be done by the expert evaluating former or designing the next wetlab experiments. We apply these queries to a set of 100 stochastic (single) simulation traces. The ratio

t1 1000 input

1000

500 A

B

output

t2 Fig. 1. First example of time-controlled inﬂow/outﬂow (EX1)

8000 A B

7000

Marking

6000 5000 4000 3000 2000 1000 0 0

20

40

60

80

100

Time

Fig. 2. Simulation result of the network given in Figure 1 (single run) (EX1)

Extended Stochastic Petri Nets

149

of traces where the formula holds to the total number gives us a rough estimate of a formula’s probability. We check over exact Gillespie traces, i.e. all single events are logged. There are generally no ”even” time points (like 30.000000 for 30). However, the ﬁring of scheduled transitions at absolute time points (e.g. 20 in this example) causes exact time points in the simulation traces. We have to keep this in mind when refering to absolute time points in the following queries. Please remember, all place names are read as integer variables in the following formulae; and the predeﬁned variable time relates to the simulation time. The probabilities as computed by simulative model checking are given in brackets. – Maxima (probabilities: 1.0, 0.95). P=? [ G(A < 7550) ] P=? [ G(B < 5350) ] – Peaks (probabilities: 0.9, 1.0). P=? [ F(time = 20 ∧ A > 0.9·max(A) ∧ (3000 < B ∧ B < 3500)) ] P=? [ F((29 < time∧time < 30)∧(5000 < A∧A < 5400)∧B > 0.9·max(B)) ] – Steady state, relative statements (probabilities: 0.03, 0.59, 0.8, 0.91). P=? [ time ≥ 50 ⇒ G(A < B) ] P=? [ time ≥ 55 ⇒ G(A < B) ] P=? [ time ≥ 60 ⇒ G(A < B) ] P=? [ time ≥ 70 ⇒ G(A < B) ] – Steady state, absolute statements (probabilities: 0.39, 1.0). P=? [ time ≥ 50 ⇒ G((1500 < A ∧ A < 1800) ∧ (1600 < B ∧ B < 2000)) ] P=? [ time ≥ 60 ⇒ G((1500 < A ∧ A < 1800) ∧ (1600 < B ∧ B < 2000)) ] EX2. We vary the pattern of our ﬁrst example to remove repeatedly all currently available tokens on place B at equidistant time points, see Figure 3. The immediate transition output consumes all tokens on place B, while there is a token on place output on. The token on place output on is controlled by the deterministically timed transition switch output on (FixedTimedFiring Periodic(20,20, SimEnd)) and the immediate transition switch output oﬀ. The transition switch output on initiates every 20 time units the cleaning process. The immediate transition switch output oﬀ switches oﬀ the outﬂow as soon as the place B is clean; otherwise each token arriving on B would be instantly removed and no token accumulation would be possible anymore. A single simulation run is given in Figure 4. We analyse a set of 100 of such stochastic traces by the following temporal-logic queries (all yield probability 1.0). – If the output is switched on, B is cleaned immediately. P=? [ G(output on = 1 ⇒ B = 0) ] – Cleaning of B at time point 20. P=? [ F(time = 20 ∧ B = 0) ]

150

M. Heiner et al.

switch_output_off

t1 1000 input

output_on 500 A

B

switch_output_on

t2

output

Fig. 3. Second example of time-controlled inﬂow/outﬂow (EX2)

8000 A B

7000

Marking

6000 5000 4000 3000 2000 1000 0 0

20

40

60

80

100

Time

Fig. 4. Simulation result of the network given in Figure 3 (single run) (EX2)

– Cleaning of B at time point 20, ensuring that B does not get cleaned earlier. P=? [ F(B > 0 ∧ (B > 0 U (time = 20 ∧ B = 0))) ] – Cleaning of B at time point 40, ensuring that B remains marked inbetween as soon as it got a token. P=? [ F(time = 20 ∧ B = 0) ∧ F(B > 0 ∧ (B > 0 U (time = 40 ∧ B = 0))) ] 4.2

Token-Controlled Inflow

We discuss two examples and start again with a reversible reaction A ↔ B, modelled by the two stochastic transitions t1 (BioMassAction(0.1)) and t2 (BioMassAction(0.005)), which we consider as a closed system, challenged by experimental interventions. EX3. In our ﬁrst example of token-controlled inﬂow, the tokens on place A are raised by 50 as soon as the token amount drops below the threshold 30, see Figure 5. This behaviour is implemented by the immediate transition input, the ﬁring of which is prevented by an inhibitor arc testing A. The weight 30

Extended Stochastic Petri Nets

t1

50 input

30

151

100 A

B t2

Fig. 5. First example of token-controlled inﬂow (EX3)

500 A B

Marking

400 300 200 100 0 0

20

40

60

80

100

Time

Fig. 6. Simulation result of the network given in Figure 5 (single run) (EX3)

of the inhibitor arc prevents the ﬁring of input until the token amount drops below 30. 50 tokens are added to place A as soon as the inhibition condition becomes invalid, preventing again further inﬂow until the next drop occurs. Figure 6 shows a single simulation run. We analyse a set of 1,000 runs against the following formulae. – The tokens on A never fall below the threshold 30 (probability: 1.0). P=? [ ¬ F(A < 30) ] – The transition input tries to keep the tokens on A between 30 and 80. But there are always some tokens on place B, which may return to A (probabilities: 0.946, 0.996, 0.999). P=? [ F(A = 30 ∧ G(30 ≤ A ∧ A ≤ 80)) ] P=? [ F(A = 30 ∧ G(30 ≤ A ∧ A ≤ 82)) ] P=? [ F(A = 30 ∧ G(30 ≤ A ∧ A ≤ 84)) ] – There is a constant inﬂow due to the transition input, and the rate of t1 is (signiﬁcantly) higher than of t2. Therefore, B increases permanently and

152

M. Heiner et al.

t1 input

5

switch_on

A 20

10

output

B

10

input_on

t2

30

input_off switch_off

Fig. 7. Second example of token-controlled inﬂow (EX4)

40 A input_on

35

Marking

30 25 20 15 10 5 0 0

20

40

60

80

100

Time

Fig. 8. Simulation results of the network given in Figure 7 (single run). The input is switched on/oﬀ (place input on) in dependence on the token situation on A (EX4).

without limits. This is true in the averaged case only, e.g. 100 runs. d(B) speciﬁes the derivative. P=? [ G(d(B) ≥ 0) ] EX4. Our second example of token-controlled inﬂow is given in Figure 7. The transitions t1 (BioMassAction(0.2)) and t2 (BioMassAction(0.1)) form again the reversible reaction A ↔ B. We add the deterministically timed transition output (FixedTimedFiring Periodic(5,5, SimEnd)) to get a signiﬁcant consumption of tokens. Each time output gets activated, it removes 10 tokens from B. If the token amount on place A drops below 10, the deterministically timed transition input (TimedFiring(0.5)) starts working and adds by each ﬁring 5

Extended Stochastic Petri Nets

153

30 A B

25

Marking

20 15 10 5 0 0

20

40

60

80

100

Time

Fig. 9. Simulation results of the network given in Figure 7 (100 runs). A and B oscillate due to the repeated switching between inﬂow on/oﬀ (EX4).

tokens with 0.5 units waiting time inbetween, until there are at least 30 tokens on A. This behaviour is controlled by the immediate transitions switch on and switch oﬀ, and the two places input on and input oﬀ, forming a 1-P-invariant1. Switch on can only ﬁre if there are less than 10 tokens on A, and switch oﬀ can only ﬁre, if there are at least 30 tokens on A. We give two related diagrams. The single run in Figure 8 shows how the input is switched on/oﬀ (place input on) in dependence on the token situation on A. Figure 9 gives the average of 100 runs. It highlights the oscillation of A and B, caused by the repeated switching between inﬂow on/oﬀ. We analyse the token-controlled inﬂow component by the following formulae (1,000 runs) (the ﬁrst three yield probability 1.0). – The two places input on and input oﬀ form a 1-P-invariant. P=? [ G((input on = 1 ∧ input of f = 0) ∨ (input on = 0 ∧ input of f = 1)) ] – The transition input is switched on/oﬀ if the token amount on A crosses the threshold 10 or 30, respectively. P=? [ G(A < 10 ⇒ input on = 1) ] P=? [ G(A ≥ 30 ⇒ input of f = 1) ] – There is a delay of 0.5 time units between the on/oﬀ switch and the reaction of the actual inﬂow transition. E.g., after having switched oﬀ the input, 5 additional tokens will arrive by the already triggered ﬁring of the transition input. Thus, even a weaker range than speciﬁed by the threshold values does not get probability 1 (probability: 0.995). P=? [ G(5 ≤ A ∧ A ≤ 40) ] 1

Exactly one of both places carries a token at any point of time.

154

4.3

M. Heiner et al.

Switch between Deterministic and Stochastic Transitions

The following two networks demonstrate how to switch between deterministic and stochastic transitions. We start oﬀ with a time-controlled switch, before discussing a token-controlled switch. In both cases we consider a non-reversible reaction A → B, which is nevertheless modelled by two transitions: the stochastic transition t stoch (BioMassAction(0.1)) and the deterministically timed transition t det (TimedFiring(0.25)). The chosen net structure ensures that always one of these two transitions only is able to transfer tokens from place A to place B; with other words: the token ﬂow occurs either stochastically or deterministically. The mutually exclusive ﬁring is implemented by the two places stochastic on and det on, forming a 1-P-invariant and establishing side-conditions for t stoch or t det, respectively. EX5. The actual time-controlled switch is performed by two deterministically timed transitions: switch to det (FixedTimedFiring Single(10)) and switch to stoch (FixedTimedFiring Single(30)), which ﬁre (each once!) at the absolute time points 10 or 30, respectively, causing a switch in the other operation mode, see Figure 10. In summary, the modelled reaction A → B behaves deterministically between the time points 10 and 30, and stochastically else.

t_stoch

stochastic_on switch_to_stoch

switch_to_det 1000 A det_on

B t_det

Fig. 10. Example of time-controlled switch between deterministic and stochastic behaviour. The semantic functions assigned to the transitions switch to det and switch to stoch allow them to ﬁre only once (EX5).

EX6. We keep the basic principle for the token-controlled switch, but replace the transitions switching between the operation modi by immediate transitions, which depend on the token situation in place A. The immediate transitions switch to det and switch to stoch ﬁre each once as soon as the token amount on place A drops below 700 or 500, see Figure 12. In summary, the modelled reaction A → B behaves deterministically for token amount between 500 and 700, and stochastically else. The diagrams in Figure 11 and 13 show the behaviour of the two patterns for a single run each. For both we conﬁrm the mutually exclusive operation mode of the stochastic and deterministic behaviour by the following query.

Extended Stochastic Petri Nets

155

A B

1200

Marking

1000 800 600 400 200 0 0

20

40

60

80

100

Time

Fig. 11. Simulation result of the network given in Figure 10 (single run). There is a deterministic token ﬂow from A to B between time points 10 and 30, and stochastic ﬂow else (EX5).

stochastic_on t_stoch 700

switch_to_stoch

A 1000

B

switch_to_det t_det 500

det_on

Fig. 12. Example of token-controlled switch between deterministic and stochastic behaviour. The additional preplaces of the immediate transitions bring the equivalence to the net component in Figure 10; i.e. the immediate transitions ﬁre only once (EX6).

– The two places stochastic on and det on form a 1-P-invariant. P=? [ G((stochastic on = 1 ∧ det on = 0) ∨ (stochastic on = 0 ∧ det on = 1)) ] We conclude the analyses with checking the range of deterministic versus stochastic behaviour for the two discussed patterns. – Deterministic token ﬂow from A to B between time points 10 and 30. P=? [ (10 ≤ time ∧ time < 30) ⇒ det on = 1 ]

156

M. Heiner et al.

A B

1200

Marking

1000 800 600 400 200 0 0

20

40

60

80

100

Time

Fig. 13. Simulation result of the network given in Figure 12 (single run). There is a deterministic token ﬂow from A to B for a token amount on A between 500 and 700, and stochastic ﬂow else (EX6).

– Stochastic token ﬂow from A to B from 0 up to time point 10, and starting at time 30 again. P=? [ (time < 10 ∨ 30 ≤ time) ⇒ stochastic on = 1 ] – Deterministic token ﬂow from A to B for a token amount on A between 500 and 700. P=? [ (500 ≤ A ∧ A < 700) ⇒ det on = 1 ] – Stochastic token ﬂow from A to B for a token amount on A less than 500 or greater or equal 700. P=? [ (A < 500 ∨ 700 ≤ A) ⇒ stochastic on = 1 ] All these properties are invariant properties, i.e. they yield probability 1.0, independently of the number and the length of considered simulation traces.

5

Lac Operon Model

We conclude by looking brieﬂy at a classical example of prokaryotic gene regulation, the lac operon case. We follow the simpliﬁed version discussed in [Wil06] and speciﬁed there by a set of reaction equations and in an SBML-shorthand notation. We keep all naming conventions and the initial conditions, and translate the textual representation into a (qualitative) Petri net, reﬂecting explicitly the inherent structure of the regulatory network, compare Figure 14. Finally, we assign the rate equations as speciﬁed in the SBML code, and we get a stochastic Petri net.

Extended Stochastic Petri Nets InhibitorRnaDegradation

157

InhibitorDegradation IOp

RnapBinding/Dissociation InhibitorTranscription InhibitorTranslation I 50 100 Op Rnap Irna Idna InhibitorBinding/Dissociation RnapOp

LactoseInhibitorBinding/Dissociation ILactose

Transcription 20

LactoseInhibitorDegradation

10000

Intervention

Rna RnaDegradation

Lactose

Translation Conversion Z

ZDegradation

Fig. 14. Lac operon model according to [Wil06]. Macro transitions (drawn as two centric squares) indicate reversible reactions.

The core model of the network under consideration is extended by a special transition – an event in SBML terminology – modelling a timed intervention in a wetlab experiment. The transition Intervention (FixedTimedFiring Periodic(50000,50000, SimEnd) 2 ) introduces 10,000 molecules of Lactose every 50,000 time units, compare Figure 15. To increase our conﬁdence in the model we start with a preliminary structural analysis and compute the P-invariants and T-invariants3 . There are input transitions, so the net can not be covered by P-invariants. However, there are three P-invariants, inducing mass-conserving subnetworks (modules) and enjoying obvious biological meaning. The preserved species is given ﬁrst in the following short-hand notation: – pi1 = {Idna}, – pi2 = {Rnap, RnapOp}, – pi3 = {Op, IOp, RnapOp}. Contrary, T-invariants do cover the net, which is a common consistency criteria for well-formed net structures, allowing e.g. a steady state behaviour. Each Tinvariant induces a self-contained, state-repeating subnetwork (module). Besides the expected three trivial T-invariants for the three reversible reactions: 2 3

Here we diﬀer from the model given in [Wil06], where the modelled intervention occurs only once at a speciﬁed point of time. For all notions used in this section, but note introduced in this paper, see [HGD08].

158

M. Heiner et al.

Lactose

12000

Marking

10000 8000 6000 4000 2000 0 0

50000

100000

150000

200000

250000

300000

Time

Fig. 15. Simulation result of the lac operon model: Lactose

– ti1 = {LactoseInhibitorBinding, LactoseInhibitorDissociation}, – ti2 = {InhibitorBinding, InhibitorDissociation}, – ti3 = {RnapBinding, RnapDissociation}, we get the following six non-trivial T-invariants, each input/output behaviour is made of: – ti4 = {InhibitorT ranscription, InhibitorRnaDegradation}, – ti5 = {InhibitorT ranslation, InhibitorDegradation}, – ti6 = {InhibitorT ranslation, LactoseInhibitorBinding, LactoseInhibitorDegradation}, – ti7 = {Intervention, Conversion}, – ti8 = {RnapBinding, T ranscription, RnaDegradation}, – ti9 = {T ranslation, ZDegradation}. There are four transitions (underlined), which are not involved in non-trivial T-invariants. However, they are crucial for the regulation mechanism between Z and Lactose. Please note, each T-invariant is given in a short-hand notation, enumerating the T-invariants’ transitions in an order, which they may follow to reproduce a state, or what has to happen to get the system back in the steady state after some disturbences. Remarkably, the net fulﬁlls the Deadlock Trap Property (DTP), however is beyond the structural net class extended simple. In summary this allows the conclusion that there is no reachable dead state, in which any further system activities would be prevented. Actually, we expect the model to be live, which can not be proven with the analysis techniques available for (qualitatively) unbounded Petri nets.

Extended Stochastic Petri Nets

300

159

Z

250

Marking

200 150 100 50 0 0

50000

100000

150000

200000

250000

300000

Time

Fig. 16. Simulation result of the lac operon model: Z

However, there are property-preserving reduction rules downsizing the net structure, which are supported by the Integrated Net Analyser INA [SR99]. Applying these structural reduction rules, we get a smaller network, consisting of 2 places and 4 transitions. Liveness becomes obvious for this reduced network; see the supplementary material for more details. The place Z models the enzyme β-Galactosidase; its reaction to the repeatedly sudden increase of Lactose molecules is shown in Figure 16. We analyse for the ﬁrst intervention how a peak of Lactose triggers a peak of Z. – The intervention causes Lactose to peak at time point 50,000 (probabilities 1.0, 1.0, 0.65). P=? [ (49, 999 ≤ time ∧ time < 50, 000) ⇒ Lactose ≤ 0.01·max(Lactose) ] P=? [ time = 50, 000 ⇒ Lactose ≥ 0.99·max(Lactose) ] P=? [ (52, 000 ≤ time ∧ time < 52, 001) ⇒ Lactose ≤ 0.1·max(Lactose) ] – Z is highly likely to be at low concentration at time point 50,000 (probability 0.9). P=? [ time = 50, 000 ⇒ Z ≤ 0.1·max(Z) ] – Z will rise to at least 80% of its maximal value within 2,000 time units (probability 0.925). P=? [ F ( (50, 000 < time ∧ time < 52, 000) ∧ Z ≥ 0.6·max(Z) ) ] – In summary, a peak of Lactose triggers a peak of Z within 2,000 time units (probability 0.925). P=? [ time = 50, 000 ∧ Lactose ≥ 0.99·max(Lactose) ∧ Z ≤ 0.1·max(Z) ⇒ F (Z ≥ 0.8·max(Z) ∧ time < 52, 000) ]

160

6

M. Heiner et al.

Tools

The Petri net components and the lac operon model have been constructed using Snoopy [Sno08], [HRS08], a tool to design and animate or simulate hierarchical graphs, among them qualitative and continuous Petri nets, and the extended stochastic Petri nets as used in this paper. Snoopy provides export to various analysis tools as well as import and export of the Systems Biology Markup Language (SBML) [HFS+ 03]. The qualitative analyses of the lac operon model have been made with the Petri net analysis tool Charlie [Cha08], complemented by the structural reduction rules supported by the Integrated Net Analyser INA [SR99]; see the corresponding log ﬁles in the supplementary material. The quantitative analyses have been done by the cooperation of two tools: Snoopy’s build-in simulation algorithm for extended stochastic Petri nets to generate the sets of simulation traces, and MC2 [MC208], a model checker by Monte Carlo sampling, for the simulative PLTL model checking. MC2 reads sets of simulation traces as, e.g., generated by Snoopy and expects additionally a ﬁle with the temporal-logical formulae. As a proof of concept, we conﬁned ourselves to rather small sets of 100 (1,000) runs only, allowing at the same time an aﬀordable repetition of all computational experiments by the reader. A general recommendation is to start with smaller sets of simulation runs, just to check whether one got a formula right, before analysing larger sets, which could actually be done in parallel. None of the computational experiments for the typical components required more than 6 minutes per net example on a standard machine. Simulative model checking of the lac operon model is slightly more expensive. The traces have been generated on a workstation (2.83 GHz, 64 bit). The 100 exact traces (simulation time interval: 300,000) require about 5 GB. The model checking itself consumes less than 30 minutes on a standard machine. Snoopy, Charlie as well as the data and analysis ﬁles of the discussed Petri net examples are available at www-dssz.informatik.tu-cottbus.de/examples/xspn-components.

7

Summary

This paper extends the Markovian stochastic Petri nets SPN Bio as introduced in [GHL07] to model and analyse biochemical networks. The extensions lead to the deﬁnition of Generalised Stochastic Petri nets GSPN Bio and deterministic and stochastic Petri nets DSPN Bio . They include read and inhibitor arcs as well as several time-dependent transition types, which in summary preclude standard Markovian analysis approaches. Therefore we applied simulative model checking, approximating the probability of a given temporal logic formula by considering ﬁnite sets of ﬁnite paths through the state space. These paths are generated by

Extended Stochastic Petri Nets

161

stochastic simulation algorithms, adjusted to deal with the extended modelling features. We discussed some typical net components demonstrating the usability of DSPN Bio for the envisaged application scenario of model-based experiment design and evaluation. These components have been analysed by checking sets of stochastic simulation traces against PLTL properties. Invariant properties have been used to prove at the same time the plausibility of the applied simulation algorithm. We concluded with brieﬂy looking at the lac operon case study, one of the classical examples of prokaryotic gene regulation. Currently we consider some further extensions of our modelling formalism; among them are variable deterministic ﬁring delays speciﬁed by an interval or an arbitrary marking-dependent function, reset and equal arcs as well as markingdependent arc weights. Simulative model checking is an extremely powerful tool. By way of introduction we have deliberately deployed some basic features of PLTL only. There is an interesting extension, PLTLc [DG08], supporting free variables, and thus allowing richer and more elegant properties, which however are also more complicated to write and to interpret. Thus, demonstrating these advanced features to more sophisticated users is beyond the scope and space limits of this paper. Acknowledgements. The stochastic features of the Snoopy tool have been developed by Sebastian Lehrack, which cumulated in his Master thesis [Leh07]. This work has been ﬁnancially supported by MPI Martinsried and MPI Madgeburg. Snoopy’s quality improvements by Christian Rohr are crucial for the computational experiments presented in this paper. We would like to thank Robin Donaldson for his responsive assistance in MC2 issues.

References [ASSB96]

Aziz, A., Sanwal, K., Singhal, V., Brayton, R.K.: Verifying Continuoustime Markov Chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996) [BGHO08] Breitling, R., Gilbert, D., Heiner, M., Orton, R.: A structured approach for the engineering of biochemical network models, illustrated for signalling pathways. Brieﬁngs in Bioinformatics 9(5), 404–421 (2008) [BK02] Bause, F., Kritzinger, P.S.: Stochastic Petri Nets. Vieweg (2002) [CGP01] Clarke, E.M., Grumberg, O., Peled, D.A.: Model checking. MIT Press, Cambridge (2001) (third printing) [Cha08] Charlie Website. A Tool for the Analysis of Place/Transition Nets. BTU Cottbus (2008), http://www-dssz.informatik.tu-cottbus.de/software/ charlie/charlie.html [DG08] Donaldson, R., Gilbert, D.: A model checking approach to the parameter estimation of biochemical pathways. In: Heiner, M., Uhrmacher, A.M. (eds.) CMSB 2008. LNCS (LNBI), vol. 5307, pp. 269–287. Springer, Heidelberg (2008)

162

M. Heiner et al.

[GBHD09] Gilbert, D., Breitling, R., Heiner, M., Donaldson, R.: An introduction to biomodel engineering, illustrated for signal transduction pathways. In: Corne, D.W., Frisco, P., Paun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2008. LNCS, vol. 5391, pp. 13–28. Springer, Heidelberg (2009) [Ger01] German, R.: Performance analysis of communication systems with nonMarkovian stochastic Petri nets. John Wiley and Sons Ltd., Chichester (2001) [GHL07] Gilbert, D., Heiner, M., Lehrack, S.: A unifying framework for modelling and analysing biochemical pathways using Petri nets. In: Calder, M., Gilmore, S. (eds.) CMSB 2007. LNCS (LNBI), vol. 4695, pp. 200–216. Springer, Heidelberg (2007) [GHR+ 08] Gilbert, D., Heiner, M., Rosser, S., Fulton, R., Gu, X., Trybilo, M.: A Case Study in Model-driven Synthetic Biology. In: Proc. 2nd IFIP Conference on Biologically Inspired Collaborative Computing (BICC), IFIP WCC 2008, Milano, pp. 163–175 (2008) [Gil77] Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 81(25), 2340–2361 (1977) [Haa03] Haas, P.J.: Stochastic Petri nets: Modelling, Stability, Simulation. Springer, Heidelberg (2003) [HDG10] Heiner, M., Donaldson, R., Gilbert, D.: Petri Nets for Systems Biology. In: Iyengar, M.S. (ed.) Symbolic Systems Biology: Theory and Methods. Jones and Bartlett Publishers, Inc., USA (in Press, 2010) [HFS+ 03] Hucka, M., Finney, A., Sauro, H.M., Bolouri, H., Doyle, J.C., Kitano, H., et al.: The Systems Biology Markup Language (SBML): A Medium for Representation and Exchange of Biochemical Network Models. J. Bioinformatics 19, 524–531 (2003) [HGD08] Heiner, M., Gilbert, D., Donaldson, R.: Petri nets in systems and synthetic biology. In: Bernardo, M., Degano, P., Zavattaro, G. (eds.) SFM 2008. LNCS, vol. 5016, pp. 215–264. Springer, Heidelberg (2008) [HJ94] Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994) [HLMP04] H´erault, T., Lassaigne, R., Magniette, F., Peyronnet, S.: Approximate probabilistic model checking. In: Steﬀen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 307–329. Springer, Heidelberg (2004) [HRS08] Heiner, M., Richter, R., Schwarick, M.: Snoopy - a tool to design and animate/simulate graph-based formalisms. In: Proc. PNTAP 2008, associated to SIMUTools 2008. ACM digital library (2008) [Leh07] Lehrack, S.: A tool to model and simulate stochastic Petri nets in the setting of biochemical networks (in German). Master thesis, BTU Cottbus, Dep. of CS (2007) [MBC+ 95] Ajmone Marsan, M., Balbo, G., Conte, G., Donatelli, S., Franceschinis, G.: Modelling with Generalized Stochastic Petri Nets, 2nd edn. Wiley Series in Parallel Computing. John Wiley and Sons, Chichester (1995) [MC208] MC2 Website. MC2 - PLTL model checker. University of Glasgow (2008), http://www.brc.dcs.gla.ac.uk/software/mc2/ [Mur89] Murata, T.: Petri Nets: Properties, Analysis and Applications. Proc.of the IEEE 77(4), 541–580 (1989) [Pnu81] Pnueli, A.: The temporal semantics of concurrent programs. Theor. Comput. Sci. 13, 45–60 (1981)

Extended Stochastic Petri Nets [PW03] [Rei82] [Sno08]

[SR99]

[Wil06]

163

Priese, L., Wimmel, H.: Theoretical Informatics - Petri Nets (in German). Springer, Heidelberg (2003) Reisig, W.: Petri nets; An introduction. Springer, Heidelberg (1982) Snoopy Website. A Tool to Design and Animate/Simulate Graphs. BTU Cottbus (2008), http://www-dssz.informatik.tu-cottbus.de/software/snoopy.html Starke, P.H., Roch, S.: INA - The Intergrated Net Analyzer. Humboldt University Berlin (1999), http://www.informatik.hu-berlin.de/~ starke/ina.html Wilkinson, D.J.: Stochastic Modelling for System Biology, 1st edn. CRC Press, New York (2006)

A Projective Brane Calculus with Activate, Bud and Mate as Primitive Actions Maria Pamela C. David1, , Johnrob Y. Bantang1,2,3,∗ , and Eduardo R. Mendoza1,4 1

Faculty of Physics and Center for Nanoscience, Ludwig-Maximilians-Universit¨ at M¨ unchen, Geschwister-Scholl-Platz 1, D-80539 M¨ unchen, Germany 2 Max-Planck-Institut f¨ ur Dynamik komplexer technischer Systeme, Sandtorstraße 1, D-39106 Magdeburg, Germany 3 National Institute of Physics, College of Science, University of the Philippines, Diliman, Quezon City 1101 Philippines 4 Department of Computer Science, University of the Philippines, Diliman, Quezon City 1101 Philippines

Abstract. We modify and extend Cardelli’s Brane Calculus and Danos and Pradalier’s Projective Brane Calculus (PBC) to improve consistency with biological characteristics of membrane reactions. We propose a Projective Activate-Bud-Mate (PABM) calculus as an alternative to the Phago-Exo-Pino (PEP) basic calculus of L. Cardelli. PABM uses a generalized formalism for Action activation with receptor-ligand type channel construction that incorporates multiple association and aﬃnity similar to Priami’s beta binders. Calculus elements are ﬁnite. Volumes are associated with systems for more realistic compartment-based reaction probabilities. PABM also uses Brane domains that partition membranes into controllable, independent groupings of projective actions. Domains eliminate the need for parameters in Phago and Bud and allow lateral and cross-membrane interactions. We show that PABM can emulate bitonal membrane reactions. PABM also realizes the idea of L. Cardelli (Cardeli, 2004) on modeling molecules as systems.

1

Introduction

Cellular organization plays a key role in biological systems through the physical regulation of reactions. Enzymes, for instance, are typically sequestered in membrane-bound systems to which access is only made possible through cascades of equally regulated and timed signals. Most current formalisms for modeling, however, do not possess an explicit functionality for modeling compartmentalization. In deterministic models, compartmentalization is modeled with the use of additional variables that diﬀerentiate a species S that is within some compartment X from S that is within another compartment Y . While this has been used with some success, S in X is actually not diﬀerent from S in Y , unless it has already reacted with other species in either compartment. It has only been in

The ﬁrst two authors contributed equally to this work.

C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 164–186, 2009. c Springer-Verlag Berlin Heidelberg 2009

A Projective Brane Calculus with Activate, Bud and Mate

165

recent years that several calculi were developed so that membrane compartmentalization: (a) becomes an inherent part of computations and (b) is emphasized in simulating reactions[1, 2, 3, 4]. Brane calculus is a formalism that can be used to describe systems as mem-brane-bound compartments that may contain other systems[3]. These compartments can merge, split or be hierarchically reorganized through uptake (phagocytosis) or extrusion (exocytosis) mechanisms, based on the capabilities, known as actions, of the membranes that enclose them[3, 5]. An important aspect of these actions — directly adapted from pi calculus — is that they are triggered via highly speciﬁc channel-based communication. Nevertheless, the mapping between channels is not necessarily one-to-one, with some channels having more than one communication partner. Although the original concept of channels in pi calculus was for mobile telecommunication systems, it is compatible with the representation of biological interactions, from enzyme-substrate systems that interact to form a chemically distinct product to receptor-mediated intermembrane communication that leads to membrane reorganization. Another formalism that includes compartments is Priami’s beta binders[4]. Here, much emphasis is given to the promiscuousity of the channels (“beta binders”) through which the compartments interact, as well as its eﬀects on the dynamic evolution of the compartment contents, interactions, and interfaces. As in Brane calculus, compartments can merge and split as a result of binder-based communication. While inherent in Brane Calculus, beta binders needed an extension to include hierarchical construction of compartments. Recent extensions, however, only permit intuitive representation of static hierarchical structures, but still forbid the explicit nesting of compartments[6]. The main advantage of beta binders over Brane calculus is its natural representation of aﬃnity to channel pairings, a concept that is adapted in the proposed extension in this paper. The uniqueness of Cardelli’s Brane calculus lies in the representation of all computations on membranes. This is important, particularly since it is actually the dynamic property of membranes that determines its capability to interact with other membranes and its general environment in vivo. Consequently, this property also determines how a membrane-bound system would evolve. Structure hierarchy can likewise be easily represented in Brane calculus, where nested systems are eﬀectively organized in tree structures[3, 5]. Brane calculus has been previously extended by Danos and Pradalier to incorporate the idea that the inner and outer surfaces of a membrane are not identical[5]. In vivo, it could be frequently observed that the membrane protein domains exposed to the extracellular matrix are diﬀerent from the domains exposed to the cytosol. It is even possible for membrane proteins to possess either an extracellular domain or a cytosolic domain. As a result, the deﬁnition of the inner and outer membranes are diﬀerent. Additional physical restrictions are introduced on which reactions could take place, in particular only directed actions on membrane surfaces that could “see” each other are allowed to interact. This extension using directed actions is known as projective Brane calculus (PBC).

166

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Nevertheless, there are a number of aspects in both calculi that involve concepts not observed in biological system. The purpose of this paper is to introduce further modiﬁcations and extensions combining the strength of both Brane calculi, with the aim of making it even more consistent with the biological characteristics of membrane reactions. Speciﬁcally, we introduce the following changes that result to the proposed extension, the Projective Activate-Bud-Mate calculus (PABM): 1. Use of the minimal set Smin ≡ {bud, mate, !, 0} instead of the set S ≡ {phago, exo, pino, mate, bud, drip} for the possible actions a ∈ Smin (see §2). All other actions in S are realized using only the actions in Smin together with directed Actions of PBC. 2. Abstraction of speciﬁc send-receive channel pairing into less speciﬁc channel name equality, eliminating the distinction between input and output channels. Together with the previous revision, it allows generalized representations in the form ax for Actions and Coactions, where x is a named channel. 3. Introduction of Brane domains, which are autonomous groups of directed Actions within a Brane. The use of Brane domains would also allow interdomain interactions within and across the same membrane. 4. Removal of the parameters for Bud and Phago, allowing the dynamic nature of membranes to be reﬂected in the calculus. 5. Inclusion of volume information as a system attribute to reﬂect its eﬀects on the probability at which collisions will occur inside a compartment. 6. Association of rates to channels emulating an aﬃnity feature similar to beta binders. 7. Treatment of Brane constituents and contents as ﬁnite quantities. 8. Elaboration of molecules as systems, a concept previously introduced by L. Cardelli[3]. These modiﬁcations are also geared towards the development of a machine for Brane calculus that can handle large-scale biological models.

2

Modified Notations

Table 1 summarizes the proposed notation and conceptual changes to the current Brane calculus, provided as a quick reference to the detailed explanations for these changes in the succeeding sections. 2.1

Actions

Notations and terms. In the documentation for the design of a machine for Brane calculus[7], stochastic pi calculus notations for input and output channels are used to distinguish between actions and coactions. At this point, it is important to make a distinction between an action (small ‘a’), a and an Action (capital ‘A’), σ. An action is an element of the set, a ∈ S currently deﬁned as: S ≡ {phago, pino, exo, mate, bud, drip};

(1)

A Projective Brane Calculus with Activate, Bud and Mate

167

Table 1. Comparison of the currently-established Brane calculus (Cardelli’s Brane calculus and PBC) with the proposed calculus. Note that a, ai ∈ S σ, σi , τ ∈ A, i = 1, 2, with a ¯ as the coaction of a. Conventions for parallel composition from PBC [5] are used. Definition Channel (x ∈ C) Set of actions (a ∈ S) Action (σ, τ ∈ A) Brane domain Brane Directionality System Parameter (bud and phago) Choice Series Parallel

Replication

Brane/Pi Calculus !x ←→ ?x S ≡ {phago, pino, exo, mate, bud, drip, . . .} a!x ←→ a ¯?x undeﬁned σ1 ; σ2 σ1 is outside, σ2 is inside σ1 ; σ2 (| P |) σ(τ )

Smin

PABM x ←→ x ≡ {0, bud, mate, !}

!x ←→ ax (a = !) ρ ≡ σ1 ; σ2 [ρ] σ1 is outside, σ2 is inside [ ρ ](| P |) τ.σ and ρ (see text for details)

σ1 + σ 2 σ1 .σ2 or σ1 σ2 σ1 |σ2

σ1 + σ2 σ1 .σ2 or σ1 σ2 σ1 , σ2 ρ1 |ρ2 P ◦Q P ◦Q . . !σ = σ, σ, . . . (inﬁnite) (n)σ = σ, σ, . . . , σ; n parallel n . (σ) = σ.σ. . . . .σ; n series . (n)ρ = ρ|ρ| . . . |ρ; n parallel . . !P = P ◦ P ◦ . . . (inﬁnite) (n)P = P ◦ P ◦ . . . ◦ P ; n parallel

while an Action is an element of the set, σ ∈ A currently deﬁned as: A ≡ {ax; a ∈ S, x ∈ C},

(2)

where the set C contains all possible channels. These notations are used throughout the text. In PABM, we use the following (minimal) set: Smin ≡ {m, b, 0, !};

(3)

where m is Mate, b is Bud, and two new actions, 0 and !, as the null and activate actions, respectively. The set A remains the same but with S replaced by Smin . We demonstrate that all other elements a ∈ S (Eq. 1) can be derived from a combination of these modiﬁcations with the directed Actions of PBC. Note that with the changes, the action becomes a passive entity (i.e. an action waits for an activation signal) by default. Since the deﬁnitions of mate and bud were not changed and have been discussed elsewhere [3, 5, 8], we will only review these deﬁnitions brieﬂy. Activate action, ! Cardelli’s Brane calculus requires two levels of matching before an Action could be executed/activated: (a) input and output channels; and (b) action and co-action. The use of an activation signal is expected to

168

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

improve the symmetry of form for the Actions with respect to channel and activation pairings. Here, we introduce an activate action, ‘!’, to approximate the input/output channel functionality of stochastic pi calculus, consequently precluding the need for an explicit distinction between actions and coactions within Brane. The Action in the form !x acts as an initiator of membrane interaction through channel x. This Action may be interpreted as a binding event, analogous to the required output signal from the intiating membrane or molecule before any non-activate Action can be executed/activated. The use of an activation signal, instead of a more strictly-bound actioncoaction pair with matching channels, is based on the fact that a single compound, modeled here as a communication channel, can interact with more than one substance, which may range from proteins to oligosaccharides, on the cell membrane. As discussed in §2.2, typical biological interactions involving receptors, logically corresponding to channels, are one-to-many relationships, rather than one-to-one pairs. Nevertheless, such cardinality does not imply that reaction speciﬁcity is lost. Another biological characteristic taken into account is the dependence of the kind of reaction that occurs on the receptor type, rather than on the ligand (i.e. it is the receiver that determines which eﬀect will occur). This characteristic is particularly marked in cells of the immune system, as well as antibodies, which have diﬀerent eﬀector functions associated with each class and subclass. The proposed form emphasizes that interaction speciﬁcity is conferred by the channel, but the receiver determines the type of action to execute. The null action, 0. The null action, 0, blocks actions that precede it; an Action in the form σ.0x0 can thus be used to model a blocked Action, σ. The null action can be deactivated with !x, making σ accessible. Biologically, blocking occurs in the event of temporary receptor internalization [9], binding-induced conformational changes [10, 11], and binding-induced physical blocking of other available binding sites. The use of 0 will be useful for modeling bind-and-release, molecular functionality switching, and other membrane-bound mechanisms. Bud. Bud refers to the arbitrary splitting of a membrane, resulting in two membrane-bound compartments[3]. Cardelli makes a distinction between bud and drip; bud occurs when the split occurs with one internal membrane, while drip refers to the separation of zero internal membranes. In PABM, this distinction is not made. Mate. Mate causes the irreversible mixing of actions of membranes that fuse either horizontally (i.e. membranes at the same level of nesting) or vertically through an exocytosis-type process[3]. 2.2

Choice, Parallel, and Series

All discussions of choice, parallel and series compositions are made with reference to Actions, unless otherwise indicated. Parallel and series compositions are not valid for actions and channels.

A Projective Brane Calculus with Activate, Bud and Mate

169

Choice. The concepts of parallel composition, choice and preﬁx(series) are retained from pi calculus. The notation for choice will be retained (‘+’). Choice could either be between actions a1 and a2 or channels x1 and x2 . These are equivalent to having a choice between two (or more) Actions in the basic form, ax. In particular, the following choices within action-channel pairs (Actions) would be equivalent to their respective choices between Actions: (a1 + a2 )x ≡ a1 x + a2 x a(x1 + x2 ) ≡ ax1 + ax2 (a1 + a2 )(x1 + x2 ) ≡ a1 x1 + a2 x1 + a1 x2 + a2 x2

(4) (5) (6)

These equivalences remove the need of implementing Actions in their non-basic forms. Hence, implementing choice in actions and/or channels will be unnecessary since all cases can always be reduced to a choice between (at least) two Actions. Aside from simplifying the implementation of Actions, Eqns. 4 and 5 reﬂect biological phenomena. For instance, Eqn. 5 is illustrated by membrane-bound receptors that have multiple ligands, with each ligand binding with a diﬀerent aﬃnity. At least three virus families, Orthomyxoviradae, Paramyxoviradae and Reoviradae, for example, use sialic acid in cell surfaces to enter via the endocytic pathway(s)[14]. A biological phenomenon that illustrates Eqn. 4, on the other hand, is the receptor for advanced glycation of end products (RAGE), expressed in a wide variety of cell types. RAGE is characterized by its ability to recognize numerous ligands, each of which result in diﬀerent eﬀects[15]. This is equivalent to having several actions associated with the same channel. Although the reactions of RAGE do not involve membrane structure deformations, a feature that would allow the direct modeling of such events may be of interest. Furthermore, Eqn. 5 implies that several receptors (or channels), can be used to initiate the same actions. Diﬀerent receptors, for instance, are used by diﬀerent viruses to enter the cell. Equation 6 is included for the purpose of completeness, but may not have any biological signiﬁcance. Parallel. Parallel pi processes and Actions, as indicated in Table 1, are represented following the notations in Danos and Pradelier and Cardelli. Preﬁx/Series. The original notation will be maintained for the series. A recurring series of the same Action would be used instead of replication to indicate the ﬁnite reusability of an Action. 2.3

Rates

PABM incorporates rates by associating a real number, rx , to the channel of each Action ax. When rx is associated with !x, it corresponds to the rate with which !x reacts on average with its receiver — the basic rate. When this real number is instead associated with ax, a = !, rx is a factor of the basic rate which reﬂects the eﬃciency of the reaction. A value of 1.0 indicates that the speciﬁc reaction rate with a particular receiver is the same as the basic rate. Association of rates to channels is adapted from beta binders[4].

170

2.4

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Aﬃnity

Aﬃnity describes the strength of non-covalent interactions between a ligand to its speciﬁc binding site on the receptor surface; this value is independent of the number of binding sites[12]. Higher aﬃnities are associated with factors such as the exposure of large, interactive amino acid side-chains, highly electronegative groups, or the deformability of a surface; these characteristics generally enable a ligand to form more non-covalent bonds with the receptor[13]. Empirically, a value known as the aﬃnity constant (Ka ) is used to approximate aﬃnity for ligand-receptor systems1 . It is determined by measuring the concentration of free ligand required to ﬁll half of the binding sites on the receptor. When half the sites are ﬁlled, [Ligand · Receptor] = [Ligand] and Ka = 1/[Receptor], where ‘[X]’ is used to indicate the molar concentration of X ; common Ka values range from as low as 5 × 104 to as high as 1011 liters/mole[12]. The use of aﬃnity in process calculi for biology has been proposed by C. Priami and P. Quaglia as a feature for beta binders[4]. Aﬃnity is incorporated as a probability P (a, b) that an interaction between two diﬀerent interfaces a and b can take place, eﬀectively relaxing the requirement for an exact matching of interface[6] — a distinct digression from pi calculus, where interactions occur on syntactically identical ports (lock-and-key model). In PABM, aﬃnity is inherent with choice. Since channels in PABM represent receptor-ligand functionality, the execution of a single action a can be associated with its interaction through more than one channel, say x1 , x2 , . . . , and xn , n > 1, resulting to the Action: a(x1 +x2 +. . .+xn ) that reduces to ax1 +ax2 +. . .+axn (Eqn. 5). When a = !, this results in multiple rates of execution, which depends on the Action a xl that is activated (a ≤ ! and l ≤ n). A similar situation occurs when a = ! and a = !, albeit with a diﬀerent biological implication. Table 2 summarizes the diﬀerence between this approach and Priami’s implementation in beta binders. Table 2. Comparison of aﬃnity in beta binders and PABM Beta Binders PABM each reduction each channel reaction probability P (a, b) multiple channels Implementation between two non-identical using choice interfaces a and b Association

2.5

Branes and Systems

The deﬁnition for Systems as sets of nested Branes is retained, and notations for these are adapted from PBC[5]. Null Systems are represented as . Notations for parallel composition of Systems are also retained. The same replication rules are 1

Notably for antibodies and antigens.

A Projective Brane Calculus with Activate, Bud and Mate

171

applied to both Actions and Systems (Table 1). Branes however, are redeﬁned as a composition of Brane domains; a Brane consisting of a single domain reduces to the original deﬁnition (Table 1). Directed Brane domains and directed Actions. In this section, the concept of directed actions in PBC is extended to Brane domains. A Brane domain, represented as a vector, ρ, is a grouping of directed Actions that approximates the occurrence of composition and functional non-homogeneity (“patchiness”) observed in biological membranes. Consequently, a Brane is now deﬁned as a parallel composition of Brane domains. A Brane deﬁned using a single domain is homogenous, and reduces to a Brane in PBC. Brane domains were introduced to facilitate greater control in processes like membrane budding. As opposed to Cardelli’s calculus where a parameter is used to deﬁne the characteristics of the Brane that will be budded out, the proposed calculus makes these characteristics entirely dependent on the current, dynamic state of the parent membrane. Budding processes, however, are highly localized, and the derived system should not have all the characteristics of the parent membrane. In the proposed calculus, only speciﬁc Brane domains are transferred in budding processes, unless the parent membrane is homogenous. Alternately, a Brane domain can be visualized as a set of directed Actions occurring proximally in a membrane. As an example, a system with Brane domains is subsequently represented as follows: [ ρ1 |ρ2 ](| [ ρ3 ](| P |) ◦ Q |)

(7)

where ρn is of the form σ1 ; σ2 (Table 1). As in the original Cardelli calculus, both ρ1 and ρ2 are visible to ρ3 . Using the rules of PBC, only the “outside” Actions of ρ3 can interact with the “outside” Actions of both ρ1 and ρ2 . The advantage of this feature is relevant in modeling competition between parallel membrane processes (§3.6). It is important to note that Brane domains represent active or functional sites on membranes or molecules and not the molecules themselves. Nevertheless, since at least one active site is associated with proteins, these can be represented as Brane domains. Brane domains can be used to model membrane proteins that function together such as lipid rafts[17] and SNARE complexes[18]. Lateral and cross-membrane interaction. Since interactions are now between two Brane domains, apart from the interaction of a domain from one membrane with another in a diﬀerent membrane, domain-domain interactions within a membrane is now possible. Actions can now be activated by Activate Actions on a neighboring domain (lateral membrane interaction). Activations by Actions on opposite sides of a single membrane (cross-membrane) can also be facilitated, provided that one of the Actions is translocated to the other side of the membrane by a mechanism similar to diﬀusion or channel-mediated transport. This capability can be used to model ligands that interact with receptors

172

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

on the same membrane surface or on the opposite side of the same membrane. Spontaneous membrane and in-membrane operations such as pinocytosis, drip, inversion, and fusion of proteins to form rafts and complexes[17, 18] can now be easily modeled. The following equation shows the competition between a1 x and a2 x since lateral- and cross-membrane interactions are allowed. [ !x ; a1 x |a2 x ; − |− ; a3 x ](| · · · |) ⇒

a1

or a2

(8)

Note that a3 cannot be activated since cross-membrane interaction are allowed only within the same Brane domain. With PABM, the transport of functional particles (e.g. molecules) through the membrane without introduction of atonal reduction rules can also be modeled (see §3.7). Volume information. A single enzyme-substrate experiment in a controlled nanoenvironment has shown that the frequency of collisions between two molecules is inversely proportional to the size of the vesicle where these molecules are contained[19]. Consequently, volume information will be associated with each System, representative of a compartment, allowing adjustments to be made in the probabilities at which the contained reactions will occur. 2.6

Replication

For the purpose of a calculus geared towards discrete biological system modeling, PABM uses a more controlled form of replication for Actions (also applicable to Brane domains and Systems), where the cardinality of replication is indicated (see Table 1). For instance, even if the initial counts of cellular components that are in the order of 104 to ∼ 1010 [16] are large enough to warrant the use of ∞, these are still ﬁnite quantities that may be critical determinants of biological system viability, especially in simulations that run for relatively prolonged periods of time (≥ 24 hours). Finite replication also reﬂects the ﬁnite lifetimes, masses and/or energies of both the components of biological systems and the systems themselves, appropriately manifested in the calculus in the form of ﬁnite Brane or Action usage. The numbers representing the ﬁnite number of replications can also be made stochastic to mimic the heterogeneity of membrane domains in terms of the absolute numbers of its constituents. Finite replication is conceptually similar to energy in beta binders [4], since the special entity E j (with j ∈ + ) can be mapped to the cardinality of replication in PABM. 2.7

Sample Notation: Mitogen-Induced Proliferation of Schwann Cells

Cell proliferation induced by an external signal is one of the simplest biological examples that is, nevertheless, diﬃcult to express in an intuitive manner without the use of spatial information. Mitogens, which induce cell division, are typically associated with one or more cognate receptors through which it can enter a cell. In Schwann cells, which form the insulation for vertebrate neurons

A Projective Brane Calculus with Activate, Bud and Mate

173

in the peripheral nervous system and which are critical for axon regeneration, proliferation is induced by the following mitogens in the neonatal stage: glial growth factor (GGF), platelet-derived growth-factor B (PDGF-BB) and basic ﬁbroblast growth factor (bFGF) [32]. For purposes of illustration, a coarse-grained model of the system can be deﬁned in the above notational changes as follows: [ !xG ; !xa ](| X |) ◦ [ !xP ; !xa ](| X |) ◦ [ !xbF ; !xa ](| X |) ◦ [ mxG , mxP , mxbF ; bxb ](| SC ◦ [ !xb .0xa ; − ](| R |) |) where each X represents a growth factor associated with some channel !xs , where s represents the part of X that binds to the GGF receptor (G), the PDGF-BB receptor (P ) or the bFGF receptor (bF ). The corresponding receptors, mxG , mxP , mxbF are all associated with the Schwann cell (SC). R represents the inactive replication machinery of the cell. This can only be activated on the fusion of one of the growth factors with SC, removing 0xa , and making !xb available for interaction. The availability of !xb in R allows SC to bud through its interaction with Action bxb .

3

Projective Activate, Bud, and Mate Calculus

In this section, we demonstrate that all Actions in S (Eq. 1) can be expressed as the actions in Smin (Eq. 3) combined with the directed Actions of PBC. The use of Smin as primitives has similarities to the basic Mate-Bud-Drip (MBD) calculus [8], which is one of two possible basic calculi for membrane interactions, together with the Phago-Exo-Pino (PEP) calculus. It has been shown [3, 8], however, that an encoding of MBD can be obtained with PEP, but not the opposite, because the maximum level of membranes (i.e. the membrane nesting) cannot grow during computation in MBD. Furthermore, the same articles prove that PEP calculus is Turing complete and Turing powerful, as opposed to MBD. Given these limitations, the use of a Bud- and Mate-based basic calculus appears counterintuitive. However, events indicated in the derivation of the MBD primitives using PEP (Fig. 1A) are not observed in biological systems (Fig. 1B). Although it is partly superﬂuous to observe that the derivations of MBD were previously qualiﬁed as performed for computational purposes only, it is clear that in vivo membrane fusion is characterized by membrane perturbances rather than a series of phagocytosis and exocytosis events[20]. Speciﬁcally, the prevalent hypothesis regarding membrane fusion involves the reduction of the distance between the fusing membranes, followed by the local perturbation of the lipid structure and merger of proximal monolayers. Stalk formation and stalk expansion, and ﬁnally, pore formation are postulated to follow. Furthermore, there is a requirement that each of these steps has to be driven by an energy gradient towards lower energies. The stalk hypothesis is mainly based on the observation that the merger of proximal monolayers precedes the merger of distal monolayers. These events are followed by the intravesicular solvent exchange[20].

174

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Fig. 1. PEP derivation of Mate [3] (A) and the latest model of how membrane fusion occurs [20] (B)

The succeeding discussions focus on the proposed PABM calculus as an encompassing calculus that conforms better with biologically observed phenomena. 3.1

Mate and Bud as Inverses of the Other

We consider Bud and Mate membrane actions as the primitives of this calculus, together with the Activate action, which controls their execution. Figure 2(top to bottom) shows a local deformation of the membrane separating the spaces labeled as P and Q ◦ R resulting from its interaction with Q. The increase in local curvature is then followed by the movement of Q towards the newly formed protrusion. On the fusion of the initial points of deformation, a new membranebound space containing Q is formed within P (Fig. 2, bottom), completing Bud. The reverse process, Mate, can be obtained using an opposite perspective. Here, the membrane separating Q from P merges with the membrane separating P and R (bottom to top). Colors are used to indicate tonality; in these processes, bitonality is conserved, as in PBC[5]. Since Bud and Mate are opposite operations, it would be possible to think of these as belonging to a single operation. 3.2

Projective Equivalence

Projective equivalence arose from the introduction of directed actions by Danos and Pradalier [5]. Brieﬂy, projective equivalence refers to the idea that the nature of membrane interactions is such that one does not make a distinction between top and bottom, or in this case, outside and inside. Consequently, by using a simple point-of-view change (i.e. what one considers inside before, which is a bounded space, is now viewed as the outside, which is unbounded), one reverses the process. If one uses a pointed bitonal tree representation for the structure, the equivalence is simply a change in the distinguished vertex [5], which is a change in the root of the tree. One can then generalize phago and bud as a single budding action, and exo and mate as a single mate action.

A Projective Brane Calculus with Activate, Bud and Mate

175

Fig. 2. Mate and Bud as inverse actions of the other. Bud is shown as a sequence from top to bottom while Mate as the reverse. Note that a distinction is not made between “inside” and “outside” spaces. A bilayer is used to illustrate directionality.

3.3

Basic Reduction Rules

The basic reduction rules of PABM are entirely based on Bud and Mate. Reduction rules are applied between interacting Brane domains, where the location of the activation signal with respect to the receiver (i.e. the directionality of the Action) determines if a Bud will be inward or outward, or if a Mate will be horizontal (i.e. membranes at the same level will merge) or vertical (i.e. the membrane of a content will merge with the membrane of its parent); this is conceptually similar to what has been done in PBC[5]. In the design of a Brane model, the directions at which Bud and Mate proceed are naturally integrated.The reduction rules of PABM are as follows, with “∼” used for indicating projective equivalence[5]. – Bud: P ◦ [ ρ1 |σ1 ; σ2 , τ2 .bx ](| [ ρ2 |σ4 , τ4 .!x ; σ3 ](| Q |) ◦ R |) −→ P ◦ [ σ1 ; σ2 , τ2 , τ4 ](| [ ρ2 |σ4 ; σ3 ](| Q |) |) ◦ [ ρ1 ](| R |) ∼

ρ1† |σ2 , τ2 .bx ; σ1 (| P |) ◦ [ ρ2 |σ4 , τ4 .!x ; σ3 ](| Q |) ◦ R −→ ρ1† (| P ◦ [ σ1 ; σ2 , τ2 , τ4 ](| [ ρ2 |σ4 ; σ3 ](| Q |) |) |) ◦ R

(9)

(10)

176

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

– Mate: P ◦ [ ρ2 |σ1 , τ1 .!x ; σ2 ](| Q |) ◦ [ ρ1 |σ1 , τ3 .mx ; σ2 ](| R |) −→ P ◦ [ ρ1 |ρ2 |σ1 ; σ2 |σ1 , τ1 , τ3 ; σ2 ](| Q ◦ R |)

(11)

∼ ; (| P ◦ [ ρ2 |σ1 , τ1 .!x ; σ2 ](| Q |) |) ◦ R ρ1† |ρ2 |σ2 ; σ1 |σ1 , τ1 , τ3 ; σ2 (| P |) ◦ Q ◦ R

(12)

−→

ρ1† |σ2

σ1 , τ3 .mx

Note that Q in Mate is equivalent to [ σ4 , τ4 .!x ](| σ3 |) Q in Bud. Odd-even subscripts and primed Actions are used to illustrate bitonality preservation. For directionality to be conserved, note the need for the reversal of ρ1 to ρ1† when the perspective is changed. In the case of Mate, interchanging the locations of mx and !x results in slightly diﬀerent Brane domains. Eq. 11 results in: P ◦ [ ρ1 |ρ2 |σ1 , τ1 , τ3 ; σ2 |σ1 ; σ2 ](| Q ◦ R |) ;

(13)

while Eq. 12 results in: [ ρ1 |ρ2 |σ1 ; σ2 |σ1 , τ1 , τ3 ; σ2 ](| P |) ◦ Q ◦ R.

(14)

It is only in the absence of τ1 and τ3 that the location of mx and !x does not result in diﬀerent succeeding states. 3.4

Non-primitive Actions with Bud and Mate

As shown in Eqs. 9 to 12, congruence exists between an inward and outward Bud, and between a horizontal and vertical Mate. This is more clearly illustrated in Fig. 2, where one sees that a simple perspective shift makes the same Bud or Mate operation inward or outward, or vertical or horizontal. For instance, when one chooses P as the “inside”, Q can be viewed as budding in towards P , or that the membrane containing Q is mating with the membrane separating P and R. As a result, PABM considers a ∈ S, a ∈ / Smin as membrane operations congruent to either Bud or Mate operation or its speciﬁc cases. Phago and Exo. Fig. 3(top) shows how Q is exocytosed from R or endocytosed into P via Mate and Bud, respectively. This is congruent to the Mate-Bud reactions in Fig. 2, with R as the outside and P as the inside. Phago is expressed as Bud in Eq. 10 and is congruent to the usual bud in Eq. 9. Pino and Drip. Pino and Drip may be spontaneous or induced Bud actions (see Eq. 10), where a null System (Q = null) is created inside or outside the bounded space P . The activation may also be induced by an appropriate Activate Action outside or inside P , or within-membrane activations (see §2.5). Pino and drip are obtained when Q = null in Fig. 3.

A Projective Brane Calculus with Activate, Bud and Mate

177

Fig. 3. Speciﬁc cases of Mate and Bud: (top, forward) Exo and (top, reverse) Phago; (bottom, forward) Cardelli’s Mate and (bottom, reverse) Bud operations

3.5

Enhanced Membrane Dynamics

A fundamental diﬀerence of the proposed calculus from the Brane calculus of Cardelli is the dynamic nature of the reacting membranes. In Cardelli’s version, the properties of budded membranes are speciﬁed as parameters to provide control; the same is true for the “endosomes” formed during Phago. In biological systems, however, the characteristics of the budded membrane are necessarily dependent on the state of the parent membrane at the time of budding or phagocytosis. Fig. 4 reﬂects this particular case of budding, when a sequential action is associated with the activation Action on an initiating membrane (σ1 , green). Note the incorporation of σ1 in the budded membrane. Also note that only a portion of the membrane is budded out. This dynamic property of the membrane generally implies that systems involved in a Mate followed by a Bud (b.m) will not evolve equivalently when Bud is performed before Mate (m.b). For instance, consider the following initial system: Q0 ≡ [ σ4 , τ4 .mxM ; σ3 , τ3 .bxB ](| [ σ1 , τ1 .!xB ; σ2 ](| P2 |) ◦ P3 |) ◦ [ σ6 , τ6 .!xM ; σ7 ](| P1 |)

(15)

Depending on which operation occurs ﬁrst, the system will evolve in two diﬀerent ways. First, on performance of b.m, the system will evolve as: m

Q0 −−−−−−→ Q0 ≡ [σ4 , τ4 , τ6 ; σ3 , τ3 .bxB |σ6 ; σ7 ](| P1 ◦ [ σ1 , τ1 .!xB ; σ2 ](| P2 |) ◦ P3 |) Q0

b

−−−−−−→

Q1

(16)

≡ [ σ6 ; σ7 ](| P1 ◦ P3 |) ◦ [ σ4 , τ4 , τ6 ; σ3 , τ1 , τ3 ](| [ σ1 ; σ2 ](| P2 |) |)

(17)

178

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Fig. 4. Budding as a dynamic process. Note that a new action, σ1 , associated with the ‘!’ is incorporated into the budded membrane. Domains are illustrated as line segments; only selected domains proximal to the activated action are budded out.

Second, with m.b, the system will evolve as: b

Q0 −−−−−−→ Q0 ≡ [ − ](| P3 |) ◦ [ σ4 , τ4 .mxM ; σ3 , τ3 , τ1 ](| [ σ1 ; σ2 ](| P2 |) |) ◦ [ σ6 , τ6 .!xM ; σ7 ](| P1 |) Q0

m

−−−−−−→ Q1 ≡ [ − ](| P3 |)

(18) (19)

◦ [ σ4 , τ4 , τ6 ; σ3 , τ3 , τ1 |σ6 ; σ7 ](| P1 ◦ [ σ1 ; σ2 ](| P2 |) |) Clearly, Q1 = Q1 . This asymmetry example (b.m = m.b) is depicted in Fig. 5. The diﬀerence disappears when the Mate and Bud are placed in separate domains, σ4 , τ4 .mxM ; σ3 and σ4 ; σ3 , τ3 .bxB . 3.6

Competition of Parallel Membrane Processes

Using the concept of Brane domains, competition of two or more parallel membrane processes can be easily modeled. Given the following system (longhand), the interactions of !x is restricted to bx associated with σ1 or that associated with σ3 . If it interacts with bx in σ1 ; bx , then the system reduction will be in the form: [ σ1 ; bx |σ3 ; bx ](| [ !x ; σ5 ](| Q |) ◦ R |) −→ [ σ1 ; − ](| [ − ; σ5 ](| Q |) |) ◦ [ σ3 ; bx ](| R |)

(20)

A Projective Brane Calculus with Activate, Bud and Mate

179

Fig. 5. Diﬀerences in ﬁnal system states based on the order at which reactions occur. (top) Initial conﬁguration; (middle) Mate then Bud; (bottom) Bud then Mate.

On the other hand, if it interacts with bx in σ3 ; bx instead, then the system reduction will be as follows: [ σ1 ; bx |σ3 ; bx ](| [ !x ; σ5 ](| Q |) ◦ R |) −→ [ σ3 ; − ](| [ − ; σ5 ](| Q |) |) ◦ [ σ1 ; bx ](| R |)

(21)

Competition can also be realized in lateral and cross-membrane processes (see Eq. 8). 3.7

Molecules as Systems

Molecules can either serve as ligands or receptors. In this proposed modiﬁcation, Molecules can be modeled as null Systems containing Activators associated with Actions or other Activators. It may also be in the form of blocking functions, σ.0x, which could only be activated by !x. A molecule can be modeled as: Molecule : [ σ1 , (n)(!xname +!xgeneric ) ; σ2 , (m)(!xname +!xgeneric) ](| |)

(22)

where τ may be a null Brane and the “name” could be the name of the molecule making the channel unique for the molecule and “generic” refers to the generic channel of the molecule. For example, “RNA” can be a generic channel having the name of the protein that it encodes for its speciﬁc name.

180

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Together with cross-membrane interactions, molecule diﬀusion through a membrane can be modeled without atonal reduction. This is illustrated in the reduction below, where the molecule on the left-hand side enters the system P . Note the change in the replication coeﬃcient, reﬂecting the reduction of the active sites in both the molecule and the membrane surrounding P . (!x)(n1 ) ; (bx.!y)(n1 ) (| |) ◦ [ (n2 )my ; − ](| P |) −→ (n2 − 1)my ; − |(bx.!y)(n1 −1) .bx ; (!x)(n1 −1) .!x (| P |) −→ [ (n2 − 1)my ; − ] P ◦ (!x)(n1 −1) ; (bx.!y)(n1 −1) (| |) (23)

3.8

Mass and Energy Conservation

Since budding involves direct movement of Brane domains, mass (represented by an Action) conservation is also simulated. The “consumption” of an Action after reduction can be seen as the usage of the available energy used for and/or transfer of mass during the process, i.e. transformation of the structure into new ones.

4

PABM as an Extension of Existing Brane Calculi

Equivalent expressions for the multiple association of the activation action using Cardelli’s original notations can be derived. Suppose there are three systems that can interact via a generic action a, with coaction a. Multiple association can be realized with: a!x (| Q0 |) ◦ a?x (| P1 |) ◦ a?x (| P2 |) ,

(24)

where system Q0 can proceed with the action (a ↔ a) on both systems P1 and P2 through the same channel !x →?x. It is possible to eliminate the use of coactions (a) through the following representation: !x (| Q0 |) ◦ a?x (| P1 |) ◦ a?x (| P2 |)

(25)

with !x possibly activating either P1 or P2 via a?x. This minor notation change is immediately compatible with Pi calculus, and would require a minor code translation for recognition by the Stochastic Pi Machine (SPiM) [21]. However, the proposed calculus also involves the removal of the sender-receiver pairing (viz. !x →?x), apart from the action-coaction pairing. Moreover, in Eq. 25, the notation is asymmetric, with the activator !x having a diﬀerent form from a?x. It is possible to use α!x as a universal activator to conserve symmetry, but α will be underutilized. The use of a single “sender” is proposed for all the other actions in the form !x, while simultaneously making the notation symmetric with the use of the same form (ax, see Table 1). Both conceptually and implementationwise, these major diﬀerences could be seen as improvements over the current representation.

A Projective Brane Calculus with Activate, Bud and Mate

181

Hence, the multiple association expressed as (24) would be written in PABM as: [ !x ; − ](| Q0 |) ◦ [ ax ; − ](| P1 |) ◦ [ ax ; − ](| P2 |)

(26)

with x as the channel; “!” now belongs to the same class as a = ! (Table 1). The choice of the symbol “!” for the activate action is directly inspired by the Pi calculus notation. As indicated previously, the other major departure from Cardelli’s Brane calculus is the utilization of Brane domains in dynamic membranes to eliminate the use of parameters in phagocytosis and budding. With these domains, a Brane in PBC becomes a special case when a Brane in PABM is homogenous (i.e. is comprised of a single Brane domain). For purposes of comparison, the reduction rules of the original calculus [taken from [8]] are shown in Fig. 6.

Fig. 6. Cardelli’s Brane calculus reduction rules taken from [8]

The realization of non-primitive actions that were illustrated utilizes the same concepts as in the projective equivalence of Danos and Pradalier [5], with the exception that no arguments are explicitly used for the Bud action. Furthermore, the simplicity of the current basis and reduction makes the calculus closer to actual biological membrane operations. Finally, PBC becomes a subset of the proposed calculus since PBC Branes can be simulated using homogenous PABM Branes.

5

Summary and Outlook

We end this paper with brief discussions on a potential application of the calculus, as well as a strategy for its possible implementation.

182

5.1

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

Application Example: Viral Infection

Inﬂuenza A causes highly contagious respiratory infections in humans that range in severity from acute to lethal. New strains arise annually, which lead to 250,000 to 500,000 deaths worldwide[22, 23, 24]. It is particularly interesting for biologists because of its ability to evolve very quickly, a trait that makes the development of an eﬃcient vaccine against it particularly challenging[25, 26]. To date, a number of qualitative studies have been performed to investigate its life cycle, but most involved separate analyses of steps in the infection process [24, 26]. One of the recent most extensive quantitative models of inﬂuenza A in cell culture is that by Sidorenko and Reichl[24]. It consists of 49 ordinary diﬀerential equations (ODEs) that involve the use of additional parameters to approximate the movement of viruses and its components across cell compartments. The main results obtained from the model include the identiﬁcation of factors that limit the growth rate of viral progeny; these results are particularly useful in molecular engineering, where engineered viruses are created for vaccine production[24]. Nevertheless, it is clear that much is still not known about the inﬂuenza A life cycle, primarily owing to the complexity of the virus. Some details, for instance, that have not been included in the Sidorenko-Reichl model include the following: 1. Distinction between each of the eight strands of genetic material (vRNA), complexed with three proteins (collectively known as vRNP), throughout the replication cycle 2. Distribution of 11 protein-coding genes across the eight vRNAs 3. Indirect genetic material replication (vRNA → cRNA → vRNA), with the intermediate cRNA being able to interact with the same proteins that vRNA interacts with 4. Requirement for precise viral assembly Accordingly, several key issues remain unanswered: 1. time it takes to assemble vRNPs 2. ratio of infective to non-infective viruses 3. instances of ‘infectivity recovery’ in the event that two complementary noninfective viruses enter a cell Since compartments can be naturally represented in Brane, its use for modeling the inﬂuenza A life cycle is probably an elegant, quantitative alternative that would allow the inclusion of details such as those enumerated previously. Fig. 7 is a general illustration for the possible usage of PABM to model the inﬂuenza infection cycle. This particular model is an interesting application for Brane calculus on account of its scale. Note that all operations used for modeling the system are restricted to budding and mating, including the simulation of the bind-and-release action in the nucleus. The position of activation signals are not explicitly indicated in the ﬁgure, but could be deduced from the illustrated processes. In addition to these, it would also be possible to include details that are known to aﬀect inﬂuenza infectivity, as well as eﬃciency[27]:

A Projective Brane Calculus with Activate, Bud and Mate

183

Fig. 7. Inﬂuenza A infection cycle model being implemented using PABM. All operations are performed with Bud and Mate, including nuclear import and export through the use of the NP protein, nuclear localization sequences (NLS) or nuclear export sequences (NES). All processes are conformant with the actual events in inﬂuenza infection.

1. cleavage eﬃciency of HA 2. distinction between transcriptionally active and inactive vRNPs 5.2

Implementation and Compatibility with SPiM

Previous eﬀorts have been made to implement Brane calculus[28, 29]. These implement calculi based on the set S (Eq. 1) and were found useful for studying events having the same scale as the Semliki forest virus life cycle, which was used as the illustrative example in [3]. Nevertheless, these are not powerful enough to handle models having the scale of the inﬂuenza A life cycle. It is consequently of interest to develop an implementation that is both scalable and robust. The stochastic pi machine (SPiM) was developed by Andrew Phillips, and uses a simulation algorithm for stochastic pi calculus that is particularly suited for simulating biological systems involving a large number of molecules. This

184

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

simulation algorithm makes the execution cost dependent on the number of species, rather than the actual number of molecules, unlike in direct implementations of the Gillespie algorithm [21]. SPiM has been used on a number of occasions for a variety of biological problems [21, 30, 31]. Lately, the algorithm in SPiM has been extended to include compartment-based computation, using the Bioambients formalism [7]. SPiM also has a graphical interface, which signiﬁcantly improves its ease of use. PABM should be compatible with SPiM using the following equivalences: !x ≡ (m!x + b!x + 0!x) ax ≡ a?x σ.!x ≡ a!x(σ)

(27) (28) (29)

where a = m, b, 0 and a is the corresponding coaction. Encoding more speciﬁc stochastic pi calculus constructs would only require the use of very speciﬁc channel names. For PABM to be implemented on top of SPiM, compartments, Brane domains, and action directionality have to be appropriately represented. A separate implementation approach that focuses on the rewriting rules of PABM is also currently being explored. Acknowledgments. We wish to thank Luca Cardelli and Andrew Phillips for helpful discussions.

References [1] P˘ aun, G.: Introduction to membrane computing. In: Applications of Membrane Computing, pp. 1–42 (2006) [2] Regev, A., Shapiro, E.: The π-calculus as an abstraction for biomolecular systems. In: Modelling in Molecular Biology (2004) [3] Cardelli, L.: Brane calculi: interactions of biological membranes. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 257–278. Springer, Heidelberg (2005) [4] Priami, C., Quaglia, P.: Beta binders for biological interactions. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 20–33. Springer, Heidelberg (2005) [5] Danos, V., Pradalier, S.: Projective brane calculus. In: Danos, V., Schachter, V. (eds.) CMSB 2004. LNCS (LNBI), vol. 3082, pp. 134–148. Springer, Heidelberg (2005) [6] Guerriero, M.L., Priami, C., Romanel, A.: Modeling Static Biological Compartments with Beta-binders. In: Anai, H., Horimoto, K., Kutsia, T. (eds.) Ab 2007. LNCS, vol. 4545, pp. 247–261. Springer, Heidelberg (2007) [7] Phillips, A., Cardelli, L.: Eﬃcient, correct simulation of biological processes in the stochastic pi-calculus. In: Calder, M., Gilmore, S. (eds.) CMSB 2007. LNCS (LNBI), vol. 4695, pp. 184–199. Springer, Heidelberg (2007) [8] Busi, N., Zandron, C.: Modelling and analysis of biological processes by (mem)brane calculi and systems. In: Proceedings of the 2006 Winter Simulation Conference, pp. 1646–1655 (2006)

A Projective Brane Calculus with Activate, Bud and Mate

185

[9] Guglielmo, G.D., Drake, P., Baass, P., Authier, F., Posner, B., Bergeron, J.: Insulin receptor internalization and signalling. Mol. Cell. Biochem. 182, 59–63 (1998) [10] Hsu, S., Bonvin, A.: Atomic insight into the CD4 binding-induced conformational changes in HIV-1 gp120. Proteins: structure, function and bioinformatics 3, 582–593 (2004) [11] Keskin, O.: Binding induced conformational changes of proteins correlate with their intrinsic ﬂuctuations: a case study of antibodies. BMC Structural Biology 7, 31 (2007) [12] Alberts, B., Bray, D., Lewis, J., Raﬀ, M., Roberts, K., Watson, J.: Molecular Biology of the Cell, New York (2002) [13] David, M., Asprer, J., Ibana, J., Concepcion, G., Padlan, E.: A study of the structural correlates of aﬃnity maturation: antibody aﬃnity as a function of chemical interactions, structural plasticity and stability. Mol. Immunol. 44, 1342–1351 (2006) [14] Dimitrov, D.: Virus entry: molecular mechanisms and biomedical applications. Nature Reviews Microbiology 2, 109–122 (2004) [15] Kim, W., Hudson, B., Moser, B., Guo, J., Rong, L., Yu, L., Qu, W., Lalla, E., Lerner, S., Chen, Y., Yan, S.D., D’Agati, V., Naka, Y., Ramasamy, R., Herold, K., Yan, S., Schmidt, A.: Receptor for advanced glycation end products and its ligands: A journey from the complications of diabetes to its pathogenesis. Annals of the New York Academy of Sciences 1043, 553–561 (2006) [16] Thulke, S., Radonic, A., Nitsche, A., Siegert, W.: Quantitative expression analysis of HHV-6 cell receptor CD46 on cells of human cord blood, peripheral blood and G-CSF mobilised leukapheresis cells. Virology Journal 3, 77–81 (2006) [17] Simons, K., Vaz, W.L.C.: Model Systems, Lipid Rafts, and Cell Membranes. Annual Review of Biophysics and Biomolecular Structure 33(1) (June 2004) [18] Sutton, R.B., Fasshauer, D., Jahn, R., Brunger, A.T.: Crystal structure of a SNARE complex involved in synaptic exocytosis at 2.4 ˚ A resolution. Nature 395(6700), 347–353 (1998) [19] Chiu, D., Wilson, C., Karlsson, A., Danielsson, A., Lunqvist, A., Stroemberg, A., Ryttsen, F., Davidson, M., Nordholm, S., Orwar, O., Zare, R.: Manipulating the biochemical nanoenvironment around single molecules contained within vesicles. Chem. Phys. 247, 133–139 (1999) [20] Jahn, R., Grubm¨ uller, H.: Membrane fusion. Current Opinion in Cell Biology 14, 488–495 (2002) [21] Phillips, A., Cardelli, L.: A Correct Abstract Machine for the Stochastic Picalculus. In: Concurrent Models in Molecular Biology (2004) [22] Poland, G.A., Tosh, P., Jacobson, R.M.: Requiring inﬂuenza vaccination for health care workers: seven truths we must accept. Vaccine 23(17-18), 2251–2255 (2005); Vaccines and Immunisation. Based on the Fourth World Congress on Vaccines and Immunisation [23] Baccam, P., Beauchemin, C., Macken, C.A., Hayden, F.G., Perelson, A.S.: Kinetics of Inﬂuenza A Virus Infection in Humans. J. Virol. 80(15), 7590–7599 (2006) [24] Sidorenko, Y., Reichl, U.: Structured model of inﬂuenza virus replication in mdck cells. Biotechnology and bioengineering 88, 1–14 (2004) [25] Bardiya, N., Bae, J.: Inﬂuenza vaccines: recent advances in production technologies. Applied Microbiology and Biotechnology 67(3), 299–305 (2005) [26] Genzel, Y., Schulze-Horsel, J., M¨ ohler, L., Sidorenko, Y., Reichl, U.: Inﬂuenza vaccines –challenges in mammalian cell culture technology. Cell Technology for Cell Products, 503–508 (2007)

186

M.P.C. David, J.Y. Bantang, and E.R. Mendoza

[27] Nayak, D.P., Hui, E.K.-W., Barman, S.: Assembly and budding of inﬂuenza virus. Virus Research 106, 147–165 (2004) [28] de Ronde, J.J., Ndjehan, C.P.: Modelling Networks and Pathways in Systems Biology. Technical report, CA545 Practicum, School of Computing, Dublin City University (2005/2006) [29] David, M.P.C.: BCD: Design and implementation of a stochastic brane machine. Master’s thesis, Department of Computer Science, University of the Philippines, Diliman, Quezon City (2008) [30] Segata, N., Blanzieri, E., Priami, C.: Stochastic π-calculus modelling of multisite phosphorylation based signaling: in silico analysis of the Pho4 transcription factor and the PHO pathway in Saccharomyces cerevisiae. Technical report, Center for Computational and Systems Biology, The Microsoft Research – University of Trento (2007) [31] Yap, J.M.: A Pi-Calculus Model of the CD95 Receptor Medicated Pathway of Apoptosis. Philippine Information Techonology Journal 1(1) (2008) [32] Zhang, B.T., Hikawa, N., Horie, H., Takenaka, T.: Mitogen induced proliferation of isolated adult mouse Schwann cells. J. Neurosci. Res., 648–654 (1995)

Accepting Networks of Non-inserting Evolutionary Processors J¨ urgen Dassow1 and Victor Mitrana2, 1

Faculty of Computer Science, University of Magdeburg P.O.Box 4120, 39016 Magdeburg, Germany [email protected] 2 Faculty of Mathematics, University of Bucharest Str. Academiei 14, 70109 Bucharest, Romania Department of Information Systems and Computation Technical University of Valencia, Camino de Vera s/n. 46022 Valencia, Spain [email protected]

Abstract. In this paper we consider four variants of accepting networks of evolutionary processors with in-place computations, that is the length of every word in every node at any step in the computation is bounded by the length of the input word. These devices are called here accepting networks of non-inserting evolutionary processors (ANNIEP shortly). The variants diﬀer in two respects: ﬁlters that are used to control the exchange of information, i.e., we use random context conditions and regular languages as ﬁlters, and the way of accepting the input word, i.e., at least one output node or all output nodes are nonempty at some moment in the computation. The computational power of these devices is investigated. In the case of ﬁlters deﬁned by regular languages, both variants lead to the class of context-sensitive languages. If random context conditions are used for deﬁning ﬁlters, all linear context-free languages and some non-semilinear (even over the one-letter alphabet) can be accepted with both variants. Moreover, some closure properties of the classes of languages ANNIEPs with random context ﬁlters are also given.

1

Introduction

The origin of networks of evolutionary processors (NEP for short) is a basic architecture for parallel and distributed symbolic processing, related to the Connection Machine [9] as well as the Logic Flow paradigm [7], which consists of several processors, each of them being placed in a node of a virtual complete graph, which are able to handle data associated with the respective node. All the nodes send simultaneously their data and the receiving nodes handle also simultaneously all the arriving messages, according to some strategies, see, e.g., [8,9]. Similar ideas may be met in other bio-inspired models like membrane systems [16], evolutionary systems [4], or models from Distributed Computing area

Work supported by the Alexander von Humboldt Foundation.

C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 187–199, 2009. c Springer-Verlag Berlin Heidelberg 2009

188

J. Dassow and V. Mitrana

like parallel communicating grammar systems [15], networks of parallel language processors [3]. In a series of papers (see [14] for an early survey) one considers that each node may be viewed as a cell having genetic information encoded in DNA sequences which may evolve by local evolutionary events, that is point mutations. Each node is specialized just for one of these evolutionary operations. Furthermore, the data in each node are organized in the form of multisets of words (each word appears in an arbitrarily large number of copies), and all the copies are processed in parallel such that all the possible events that can take place do actually take place. Obviously, the computational process just described is not exactly an evolutionary process in the Darwinian sense. But the rewriting operations we have considered might be interpreted as mutations and the ﬁltering process might be viewed as a selection process. Recombination is missing but it was asserted that evolutionary and functional relationships between genes can be captured by taking only local mutations into consideration [17]. In [13] one presents a characterization of the complexity class NP based on accepting networks of evolutionary processors (ANEP for short). This characterization is extended in [12] to PSPACE and P. The work [10] discusses how ANEPs can be considered as problem solvers. In [11], one shows that every recursively enumerable language can be accepted by an ANEP with 24 nodes. More precisely, one proposes a method for constructing, for every NP-language, an ANEP of size 24 deciding that language in polynomial time. While the number of nodes of this ANEP does not depend on the language, the other parameters of the network (rules, symbols, ﬁlters) depend on it. From a computational point of view it is of interest to consider ANEPs with in-place computations, that is the length of every word in every node at any step in the computation is bounded by the length of the input word. This is our main reason to consider here some variants of networks of evolutionary processors without insertion nodes, called here accepting networks of non-inserting evolutionary processors, ANNIEP shortly. The diﬀerences between the variants of ANNIEPs consist in the ﬁlters and in the way of accepting the input word. Besides accepting networks of evolutionary processors, generating networks of such processors have been investigated (see [2], [5], [14]). In the paper [6], the generative power of networks where only two types of point mutations are allowed for the nodes have been investigated. In case of non-inserting processors one only gets the set of all ﬁnite languages. This paper presents the counterpart for accepting networks, where the situation is completely diﬀerent. We study the computational power of accepting networks of non-inserting processors. In the case of ﬁlters deﬁned by regular languages, both variants of accepting lead to the same class of languages, namely the class of contextsensitive languages. If random context conditions are used for deﬁning ﬁlters, all linear context-free languages and some non-semilinear (even over the oneletter alphabet) can be accepted with both variants. Therefore the power of accepting networks is much greater than that of generating networks (both with non-inserting processors). Moreover, some closure properties of the classes of

Accepting Networks of Non-inserting Evolutionary Processors

189

languages accepted by ANNIEPs with ﬁlters deﬁned by random context conditions are also discussed.

2

Some Notations and Definitions

Throughout the paper we assume that the reader is familiar with the basic notions of the theory of formal languages. We here only recall some notation and notions as they are used in the paper. An alphabet is a ﬁnite and nonempty set of symbols. The cardinality of a ﬁnite set A is written card(A). Any sequence of symbols from an alphabet V is called word over V . The set of all words over V is denoted by V ∗ and the empty word is denoted by ε. A language over V is a subset of V ∗ . The length of a word x is denoted by |x| while alph(x) denotes the (with respect to inclusion) minimal alphabet W such that x ∈ W ∗ . A morphism h : V ∗ −→ U ∗ is said to be literal if |h(a)| = 1 for all a ∈ V ; it is weak literal if |h(a)| ≤ 1 for all a ∈ V . In other words a (weak) literal morphism is called (weak) coding. Let V be an alphabet. We say that a rule a → b, with a, b ∈ V ∪ {ε} is a substitution rule if both a and b are not ε; it is a deletion rule if a = ε and b = ε. The set of all substitution and deletion rules over an alphabet V are denoted by SubV and DelV , respectively. Given a rule σ as above and a word w ∈ V ∗ , we deﬁne the following actions of σ on w: {ubv : ∃u, v ∈ V ∗ (w = uav)}, ∗ • If σ ≡ a → b ∈ SubV , then σ (w) = {w}, otherwise {ub : w = ua}, {bv : w = av}, r l σ (w) = σ (w) = {w}, otherwise {w}, otherwise {uv : ∃u, v ∈ V ∗ (w = uav)}, • If σ ≡ a → ε ∈ DelV , then σ ∗ (w) = {w}, otherwise {u : w = ua}, {v : w = av}, r l σ (w) = σ (w) = {w}, otherwise {w}, otherwise The action α ∈ {∗, l, r} expresses the way of applying a substitution or deletion rule to a word, namely at any position (α = ∗), in the left (α = l), or in the right (α = r) end of the word, respectively. For every rule σ, any action α ∈ {∗, l, r}, and any L ⊆ V ∗ , we deﬁne the α-action of σ on L by σ α (L) = σ α (w). w∈L

Given a ﬁnite set of rules M , we deﬁne the α-action of M on the word w and the language L by: M α (w) = σ α (w) and M α (L) = M α (w), σ∈M

respectively.

w∈L

190

J. Dassow and V. Mitrana

If θV ∗ −→ {0, 1} is a predicate and L ⊆ V ∗ , we write: θ(L) = L ∩ θ−1 (1). We are interested in some special predicates. For two disjoint subsets P and F of an alphabet V , a regular set R over V , and a word x over V , we deﬁne the predicates θs,P,F (x) = 1 if and only if P ⊆ alph(x) and F ∩ alph(x) = ∅, θw,P,F (x) = 1 if and only if alph(x) ∩ P = ∅ and F ∩ alph(x) = ∅, θR (x) = 1 if and only if x ∈ R. The ﬁrst two predicates are based on random context conditions deﬁned by the two sets P (permitting contexts/symbols) and F (forbidding contexts/symbols). Informally, the ﬁrst condition requires (s stands for strong) that all permitting symbols are and no forbidding symbol is present in x, while the second (w stands for weak) is a weaker variant such that at least one permitting symbol appears in x but still no forbidding symbol is present in x. We call these two predicates random context predicates. The third predicate asks for membership in a regular set, and is called a regular predicate. A non-inserting evolutionary processor over V is a tuple (M, ϕ, ψ), where: – M is a set of either substitution or deletion rules over the alphabet V ; formally, M ⊆ SubV or M ⊆ DelV . The set M represents the set of evolutionary rules of the processor. As one can see, a processor is “specialized” in one evolutionary operation, only. – ϕ is the input predicate, while ψ is the output predicate of the processor. Informally, these two predicates work as ﬁlters. A word w can enter or leave the processor, if it satisﬁes the predicate ϕ or ψ, respectively. We are interested in two types of processors, random context non-inserting evolutionary processor over V (or short rcNIEPV ) and regular non-inserting evolutionary processor over V (or short regNIEPV ). These processors are deﬁned by the requirement that, – for an rcNIEPV , both predicates are of the form θs,P,F or of the form θw,P,F for certain subsets P and F of V , – for an regNIEPV , both predicates are of the form θR for some regular set R ⊆ V ∗. We want to stress from the very beginning that the evolutionary processor we discuss here is a mathematical object only and the biological hints presented in the introduction are intended to explain in an informal way how some biological phenomena are sources of inspiration for our mathematical computing model. We denote the set of non-inserting evolutionary processors over V by N IEPV . An accepting network of non-inserting evolutionary processors (ANNIEP for short) is a 8-tuple Γ = (V, U, G, N, α, xIn , Out), where:

Accepting Networks of Non-inserting Evolutionary Processors

191

– V and U are the input and network alphabet, respectively, satisfying V ⊆ U . – G = (XG , EG ) is an undirected graph without loops with the set of vertices XG and the set of edges EG . G is called the underlying graph of the network. – N : XG −→ N IEPV is a mapping which associates with each node x ∈ XG the evolutionary processor N (x) = (Mx , ϕx , ψx ). – α : XG −→ {∗, l, r} is a mapping which associates with each node a type of action; α(x) gives the action mode of the rules of node x on the words existing in that node. – xIn ∈ XG is the input node of Γ . – Out ⊂ XG is the set of output nodes of Γ . An ANNIEP is a random context ANNIEP or regular ANNIEP if all its noninserting evolutionary processors are random context or regular non-inserting evolutionary processors, respectively. We say that card(XG ) is the size of Γ . A configuration of an ANNIEP Γ ∗ as above is a mapping C : XG −→ 2Vf which associates a ﬁnite set of words with every node of the graph. A conﬁguration may be understood as the sets of words which are present in any node (or in the associated prozessor) at a given moment. Given a word z ∈ V ∗ , the initial conﬁguration of Γ on z is deﬁned by (z) (z) C0 (xIn ) = {z} and C0 (x) = ∅ for all x ∈ XG \ {xIn }. A conﬁguration can change either by an evolutionary step or by a communication step. When changing by an evolutionary step, each component C(x) of the conﬁguration C is changed in accordance with the set of evolutionary rules Mx associated with the node x and the way of applying these rules α(x). Formally, we say that the conﬁguration C is obtained in one evolutionary step from the conﬁguration C, written as C =⇒ C , iﬀ C (x) = Mxα(x) (C(x)) for all x ∈ XG . When changing by a communication step, each node processor x ∈ XG sends one copy of each word it has, which is able to pass the output ﬁlter of x, to all the node processors connected to x and receives all the words sent by any node processor connected with x provided that they can pass its input ﬁlter. Formally, we say that the conﬁguration C is obtained in one communication step from conﬁguration C, written as C C , iﬀ C (x) = (C(x) − ψx (C(x))) ∪ (ψy (C(y)) ∩ ϕx (C(y))) for all x ∈ XG . {x,y}∈EG

Note that words that cannot pass the output ﬁlter of a node remain in that node and can be further modiﬁed in the subsequent evolutionary steps, while words that can pass the output ﬁlter of a node but cannot pass the input ﬁlter of any node are lost. Let Γ be an ANNIEP, the computation of Γ on the input word z ∈ V ∗ is a (z) (z) (z) (z) sequence of conﬁgurations C0 , C1 , C2 , . . ., where C0 is the initial conﬁgu(z) (z) (z) (z) ration of Γ on z, C2i =⇒ C2i+1 and C2i+1 C2i+2 , for all i ≥ 0. Note that the conﬁgurations are changed by alternative steps. By the previous deﬁnitions, each

192

J. Dassow and V. Mitrana (z)

(z)

conﬁguration Ci is uniquely determined by the conﬁguration Ci−1 . A computation halts (and it is said to be weak (strong) halting) if one of the following two conditions holds: (i) There exists a conﬁguration in which the set of words existing in at least one output node (all output nodes) is non-empty. In this case, the computation is said to be a weak (strong) accepting computation. (ii) There exist two identical conﬁgurations obtained either in consecutive evolutionary steps or in consecutive communication steps. The language weakly (strongly) accepted by Γ are deﬁned as: Lwa (Γ ) = {z ∈ V ∗ | the computation of Γ on z is a weak accepting one} Lsa (Γ ) = {z ∈ V ∗ | the computation of Γ on z is a strong accepting one}. In the theory of networks some types of underlying graphs are common like rings, stars, grids, etc. Networks of evolutionary words processors, seen as language generating or accepting devices, with underlying graphs having these special forms have been considered in several papers, see, e.g., [14] for an early survey. We focus here on complete ANNIEPs i.e., ANNIEPs having a complete underlying graph. Therefore, in what follows we replace the graph G in the deﬁnition of an ANNIEP by the set of its nodes usually denoted by χ. Moreover, we present an evolutionary network by its nodes x and the parameters corresponding to x, where instead of ϕβ,P Ix ,F Ix and ψ β,P Ox ,F Ox , in case of random context processors, and instead of ϕRx and ϕRx for regular processors, we only mention P Ix , F Ix , P Ox , F Ox , β and Rx , Rx , β, respectively. For x ∈ {wa, sa} and y ∈ {rc, reg}, by Lx (yAN N IEP ) we denote the set of all languages which can be accepted by yANNIEPS. The following two notions will be very useful in the sequel. If h is a one-toone mapping from U to W and Γ = (V, U, χ, N, α, xIn , Out) is an ANNIEP, then we denote by Γh the ANNIEP Γh = (h(V ), h(U ), χ, h(N ), α, xIn , Out), where by h(N ) we mean h(N )(x) = (h(Mx ), ϕβ,h(P Ix ),h(F Ix ) , ψ β,h(P Ox ),h(F Ox ) ) for every x ∈ χ, provided that N (x) = (Mx , ϕβ,P Ix ,F Ix , ψ β,P Ox ,F Ox ). Further, h(a → b) = h(a) → h(b) for any evolutionary rule a → b. Now, given two ANNIEPs Γi = (Vi , Ui , χi , Ni , αi , xiIn , Outi ), i = 1, 2, χ1 ∩ χ2 = ∅, we denote by Γ1 Γ2 = (V1 , U1 ∪U2 , χ1 ∪χ2 , N, α, x1In , Out2 ), where ◦ |χi = ◦i for all ◦ ∈ {N, α} and i = 1, 2.

3

Computational Power of Regular ANNIEPs

We start with a relation between the strong and weak acceptance modes. Theorem 1. Lwa (regAN N IEP ) ⊆ Lsa (regAN N IEP ). Proof. Let L ∈ Lwa (regAN N IEP ). Then L = Lwa (Γ ) for some regular AN NIEP Γ = (V, U, χ, N, α, xIn , Out). Let N (x) = (Mx , ϕRx , ψ Rx ) for a node x of

Accepting Networks of Non-inserting Evolutionary Processors

193

χ. Without loss of generality we may assume that Mx = ∅ for all x ∈ Out. We now construct the regular ANNIEP Γ = (V, U ∪ {Z}, χ ∪ {xOut }, N , α , xIn , {xOut }), where N (x) = N (x) for x ∈ χ \ Out, and y : {a → Z | a ∈ U }, Ry , Z ∗ , α (y) = ∗ for y ∈ Out, xOut : ∅, Z ∗ , ∅, α (xOut ) = ∗. Obviously, if there is a non-empty node y of Out in some conﬁguration of Γ , then y contains some word in some conﬁguration of Γ , too. If this word is ε, then ε is not changed and sent to xOut . If the word in y is non-empty, then all its letters are replaced by Z (note that it cannot leave the node as long as it still contains letters diﬀerent than Z) and it is send to xOut . Conversely, if a word eventually arrives in xOut , then it contains only Z’s which means that it was in a node from Out at some previous step. Thus Γ accepts the same language as Γ does. Moreover, since the set of output nodes of Γ is a singleton, we have Lwa (Γ ) = Lwa (Γ ) = Lsa (Γ ). 2 Note that we have shown a stronger result than given in Theorem 1 because we have shown that the number of output nodes of an ANNIEP accepting in the weak mode can be decreased to one only. We now compare the families of languages generated by ANNIEPs with the family of context-sensitive languages denoted here by L(CS). Theorem 2. L(CS) ⊆ Lwa (regAN N IEP ). Proof. Let L be a context-sensitive language. Then L = L(G) for some contextsensitive grammar G = (N, T, P, S) in Kuroda normal form, i.e., all its rules are of the form A → a, A → BC and AD → BC with A, B, C, D ∈ N and a ∈ T . Let P be the set of rules of the form A → BC and AD → BC. For every p ∈ P with its right-hand side BC we set Rp = (N ∪ T )∗ {Bp }(N ∪ T )∗ , Rp = (N ∪ T )∗ {Bp Cp }(N ∪ T )∗ ,

Rp = (N ∪ T )∗ {Cp }(N ∪ T )∗ and R =

Rp . We construct the ANNIEP Γ = (T, U, χ, H, α, xIn , {xOut })

p∈P

with U = N ∪ T ∪ {Bp , Cp | p = AD → BC or p = A → BC}, χ = {xIn , xOut } ∪ {p, p , p | p ∈ P }, xIn : MxIn , (N ∪ T )∗ , R, α = ∗ MxIn = {a → A | A → a ∈ P } ∪ {B → Bp | p = AD → BC or p = A → BC}, p : {C → Cp }, Rp , Rp , α = ∗ for p = AD → BC or p = A → BC,

194

J. Dassow and V. Mitrana

p : {Bp → A}, Rp , Rp , α = ∗ for p = AD → BC or p = A → BC, {Cp → D}, Rp , (N ∪ T )∗ , α = ∗ for p = AD → BC, p : {Cp → ε}, Rp , (N ∪ T )∗ , α = ∗ for p = A → BC, xOut : ∅, {S}, {S}, α = ∗. The network simulates a derivation in G backwards. Let w be the input word; we claim that for any word z ∈ (N ∪ T )+ in xIn at any computation step we have that z =⇒∗ w in G. Initially, this assertion is true as w lies in xIn . Assume that a word z ∈ (N ∪ T )+ is in the node xIn at some step. If we apply a rule a → A to z, the new word remains in xIn and the assertion holds for this new word. Now assume that we apply B → Bp to z for a rule p = AD → BC. Then the obtained word z = z1 Bp z2 , where z = z1 Bz2 , is sent to the node p, where some C is replaced by Cp . If Bp Cp is not a subword, then the word cannot go out from this node; moreover any word further obtained from this word can never go out from the node p. If Bp Cp is a subword, the word is sent out to the node p , where Bp is replaced by A. This new word is sent out to p . There Cp is either replaced by D, provided that p = AD → BC, or deleted provided that p = A → BC. Finally, the obtained word, say z , is sent to xIn . Altogether, we started with z = vBCu and obtained z = vADu, which implies that z =⇒ z =⇒∗ w. Moreover, since a word only reaches xOut , if it is S, we infer that a word is weakly accepted by Γ if and only if it is generated by G. Thus Lwa (Γ ) = L(G). 2 Theorem 3. Lsa (regAN N IEP ) ⊆ L(CS). Proof. For an ANNIEP Γ = (V, U, χ, N, α, xIn , Out), we construct a linearly bounded automaton, which accepts Lsa (Γ ). We do not give a complete formal construction; we only give an informal description of the automaton and leave the details of the construction to the reader. Let r = card(Out). The automaton has r tapes, and on each tape it nondeterministically follows the itinerary of a copy of the input word. The states are vectors of size 2r, each ith entry, 1 ≤ i ≤ r, being associated with the node containing the word on the tape i, and each ith entry, r + 1 ≤ i ≤ 2r, being 0 or 1 that indicates whether the node associated with the (i − r)th entry has ﬁnished its task on the word on tape i (in this case the entry is 1) or not. Initially, all tapes contain the input word w, the ﬁrst r entries of the initial states are associated with the input node xIn , and the last r entries are 0. Let us now consider an arbitrary conﬁguration of the automaton: the ﬁrst r elements of the current state state are associated with the nodes x1 , x2 , . . . xr , the last r elements are 0, and on the i-th tape, 1 ≤ i ≤ r, the word wi stands. Now the automaton performs on each tape i the following actions: – Changes the word wi according to an application of a rule in Mxi ; let vi be the result. – Checks whether vi can pass the output ﬁlter of xi . In the non-aﬃrmative case the automaton blocks the computation. In the aﬃrmative case, the

Accepting Networks of Non-inserting Evolutionary Processors

195

automaton changes the ith entry of the state into an entry associated with the node yi , which is a nondeterministically chosen node among the nodes of χ \ {xi }. – Check whether vi can pass the input ﬁlter of yi . In the non-aﬃrmative case the automaton blocks the computation. In the aﬃrmative case, the i+r entry becomes 1. From now on, no move is observed on the ith tape and no change is made for the entries i and i + r, until all the entries r + 1, r + 2, . . . , 2r are 1. – Checks whether the state with the last r entries 1 has its ﬁrst r entries associated with all output nodes of Γ . In the aﬃrmative case the automaton accepts the input; otherwise it changes the last r entries into 0 and resumes the actions explained above. It is rather plain that the automaton accepts Lsa (Γ ). Since in any evolutionary step one deletes or substitutes one letter, the length of the words on any tape is bounded by the length of the input word. Thus the workspace of this automaton is linearly bounded. 2 By the Theorems 1, 2 and 3, we get immediately the following two statements. Corollary 1 1. Lwa (regAN N IEP ) = Lsa (regAN N IEP ) = L(CS). 2. Every language in LX (regAN N IEP ), X ∈ {wa, sa}, can be weakly/strongly accepted by a regANNIEP Γ such that the action mode of every node of Γ is ∗. 2

4

Computational Power of Random Context ANNIEPs

We start with two statements that immediately follows from Theorems 1 and 3. Theorem 4 1. Lwa (rcAN N IEP ) ⊆ Lsa (rcAN N IEP ). 2. Lsa (rcAN N IEP ) ⊆ L(CS).

2

We do not know whether the second inclusion is proper or equality holds. Thus we give some further relations to other known language families inside L(CS) and some closure properties which give some more information about the classes Lwa (rcAN N IEP ) and Lsa (rcAN N IEP ). Theorem 5 1. Lwa (rcAN N IEP ) includes the class of linear context-free languages. 2. Lwa (rcAN N IEP ) contains non-semilinear languages. Proof. 1. Let G = (N, T, S, P ) be a linear context-free grammar; without loss of generality we may assume that the following conditions hold:

196

J. Dassow and V. Mitrana

– Every rule in P is of one of the following three forms: A → aB, A → Ba, A → a, where A, B ∈ N and a ∈ T , – If both rules A → aC and B → Db belong to P , then A = B, – The set of nonterminals N of G is {A1 , A2 , . . . , An } for some n ≥ 1 and S = A1 , – There is no rule A → aA or A → Aa for any A ∈ N and a ∈ T . We construct the following ANNIEP with the input alphabet T , the working alphabet U = T ∪ {ai , ai | 1 ≤ i ≤ n} ∪ {Z}, and only one output node xOut . ⎧ ⎧ M = {a → a1 | a ∈ T }, M = ∅, ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ P I = T, F I = {ai | a ∈ T, 1 ≤ i ≤ n}, P I={Z}, F I=U \ {Z}, xIn : xOut : P O = ∅, F O = T, P O = U, F O = ∅, ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ α = ∗, β = w, α = ∗, β = s, If there exists Ai → aAj ∈ P for some a ∈ T and 1 ≤ j = i ≤ n, then the node xi is deﬁned by ⎧ ⎪ ⎪ M = {ai → aj | Ai → aAj ∈ P }, ⎨ P I = {ai | a ∈ T }, F I = U \ {ai | a ∈ T }, xi : P O = {aj | a ∈ T, 1 ≤ j = i ≤ n}, F O = ∅, ⎪ ⎪ ⎩ α = l, β = w, If there exists Ai → Aj a ∈ P for some a ∈ T and 1 ≤ j = i ≤ n, then the node xi is deﬁned by ⎧ M = {ai → aj | Ai → aAj ∈ P }, ⎪ ⎪ ⎨ P I = {ai | a ∈ T }, F I = U \ {ai | a ∈ T }, xi : P O = {aj | a ∈ T, 1 ≤ j = i ≤ n}, F O = ∅, ⎪ ⎪ ⎩ α = r, β = w, Moreover, we set ⎧ M = {aj → ai | a ∈ T, 1 ≤ j = i ≤ n}, ⎪ ⎪ ⎨ P I = {ai | a ∈ T }, F I = ∅, xi : for 1 ≤ i ≤ n, = i ≤ n}, ⎪ P O = {ai | a ∈ T }, F O = {aj | a ∈ T, 1 ≤ j ⎪ ⎩ α = ∗, β = w, ⎧ M = {ai → ε | a ∈ T }, ⎪ ⎪ ⎨ P I = {ai | a ∈ T }, F I = {aj | a ∈ T, 1 ≤ j = i ≤ n}, x ¯i : for 1 ≤ i ≤ n, P O = {a | a ∈ T }, F O = {a | a ∈ T }, ⎪ i i ⎪ ⎩ α = l, β = w, ⎧ M = {ai → ε | a ∈ T }, ⎪ ⎪ ⎨ P I = {ai | a ∈ T }, F I = {aj | a ∈ T, 1 ≤ j = i ≤ n}, x ˜i : for 1 ≤ i ≤ n, P O = {a | a ∈ T }, F O = {a | a ∈ T }, ⎪ i i ⎪ ⎩ α = r, β = w, ⎧ M = {a 1 ≤ i ≤ n}, ⎪ ⎪ in → Z | Ai → a ∈ P, a ∈ T, ⎨ n P I = i=1 {ai | a ∈ T }, F I = U \ ( i=1 {ai | a ∈ T }), y: P O = {Z}, F O = ∅, ⎪ ⎪ ⎩ α = r, β = w,

Accepting Networks of Non-inserting Evolutionary Processors

197

The general idea of this construction is that for every 1 ≤ i ≤ n, the following statement holds: Fact: If S =⇒t uAi v =⇒+ uwv = z for some t ≥ 0, with |z| = m, then (z) (z) hi (w) ∈ (C2m(t+1)+2t (xi ) ∩ C2m(t+1)+2t (y)), where hi is a literal morphism from T to {ai | a ∈ T } defined by h(a) = ai for any a ∈ T . This fact can be proved by a standard induction argument on t. Now, if t = m − 1, then w is reduced to a letter from T , say a, therefore after the word ai is transformed into Z in the node y, it arrives in xOut and the computation halts successfully. This means that z is accepted by the network. (z) (z) (z) (z) On the other hand, if C0 , C1 , C2 , . . . , Cp is an accepting computation (z) (z) on z and hi (w) ∈ (Ct (xi ) ∩ Ct (y)) for some t < p, then the derivation S =⇒∗ uAi v =⇒+ uwv = z holds in G, which concludes the proof of the ﬁrst statement of the theorem. 2. The network with the nodes deﬁned by: ⎧ ⎧ M = {a → a ¯}, M = {a → a ˜}, ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ P I = {a}, F I = {¯ a, a ˜}, P I = {¯ a}, F I = {˜ a}, xIn : x1 : P O = {¯ a }, F O = ∅, P O = {˜ a }, F O = ∅, ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ α = ∗, β = s, α = ∗, β = s, ⎧ ⎧ M = {¯ a → ε}, M = {˜ a → a }, ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ P I = {¯ a, a ˜}, F I = ∅, P I = {˜ a}, F I = {¯ a}, x2 : x3 : P O = {˜ a}, F O = ∅, P O = {a }, F O = {˜ a}, ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ α = ∗, β = s, α = ∗, β = s, ⎧ M = {a → a}, ⎪ ⎪ ⎨ P I = {a }, F I = {a, a ¯, a ˜}, x4 : xOut P O = {a}, F O = {a }, ⎪ ⎪ ⎩ α = ∗, β = s,

⎧ M = ∅, ⎪ ⎪ ⎨ P I = {¯ a}, F I = {a, ¯a, a ˜}, : P O = {¯ a }, F O = ∅, ⎪ ⎪ ⎩ α = ∗, β = s, n

weakly accepts the non-semilinear language {a2 | n ≥ 0}. Indeed, the computation of this netwok on every input is divided in two phases. In the ﬁrst phase, the input word looses one occurrence of a and changes another one to a by visiting the nodes xIn , x1 , x2 , x3 . This process resumes until no occurrence of a is observed in the current word. There are three possiblities: (1) it contains only a’s, (2) it contains only a’s excepting an occurrence of a ¯, (3) it equals a ¯. Now the second phase of the computation starts. In the ﬁrst case, the word enters x4 where all a’s are transformed into original a’s and the ﬁrst phase resumes from xIn with a word that is exactly twice shorter than the word present in the input node in the beginning of the previous ﬁrst phase. In this case, we have checked whether the length of that word was an even number. In the second case listed above, the computation cannot continue anymore, hence the network will eventually halt without accepting. In the third case, the computation halts accepting the input word. This means that the length of the input word could be divided iteratively by 2 until the result was one, hence the length of the input word was a power of 2. 2

198

J. Dassow and V. Mitrana

Theorem 6 1. The class Lwa (rcAN N IEP ) is closed under boolean union, literal morphism, inverse weak literal morphism, mirror image. 2. The class Lsa (rcAN N IEP ) is closed under boolean intersection, literal morphism, inverse weak literal morphism, concatenation, mirror image. Proof. 1. We give an informal proof for union that can be easily formalized by the reader. Let Γ1 and Γ2 be to ANNIEPs; we construct a new ANNIEP Γ that contains three subnetworks. In the input node of the ﬁrst subnetwork, an arbitrary symbol of the input word is substituted by either its primed copy or its barred copy. All words containing a primed symbol are received by a speciﬁc node while those containing a barred symbol are received by another speciﬁc node. All symbols of the words arrived in these two nodes are replaced by their primed and barred copies, respectively. When this process is ﬁnished, each of the two nodes contains only one word. The word containing primed symbols only is given as an input word to the subnetwork formed from Γ1 modiﬁed accordingly. The other word is processed analogously by the subnetwork formed from Γ2 modiﬁed accordingly. The set of output nodes of Γ is the union of the sets of output nodes of Γ1 and Γ2 modiﬁed accordingly. Clearly, Lwa (Γ ) = Lwa (Γ1 ) ∪ Lwa (Γ2 ). If h : V −→ U is a literal morphism and Γ is an ANNIEP with the input alphabet V , then let Γ be the ANNIEP with the input alphabet U formed by two subnetworks as follows. In the input node of the ﬁrst subnetwork, each symbol b of the input word is substituted by a symbol a such that a is a copy of a ∈ V that does not appear in V ∪ U and h(a) = b. When all symbols of the input word were substituted, all the words obtained are sent to the input node of the subnetwork formed from Γ modiﬁed accordingly. It is plain that h(Lwa (Γ )) = Lwa (Γ ). The construction for the closure under inverse weak literal morphism is pretty similar and left to the reader. The closure under mirror image follows pretty simple; it suﬃces to interchange all the action modes l and r of the nodes. 2. The closure under intersection, literal morphism and inverse literal morphism follows similarly to the previous case. Note the fundamental role played by the strong acceptance in the case of intersection. 2 It is known that every recursively enumerable language can be written as the image of the intersection of two linear languages through a weak literal morphism. Therefore, the following statement is a consequence of the second statement of Theorem 4 and Theorem 6: Corollary 2 1. Every recursively enumerable language is the weak literal morphic image of a language in Lsa (AN N IEP ). 2. Lsa (AN N IEP ) is not closed under weak literal morphism. 2

5

Final Remarks

As we showed in this note, the computational power of ANNIEPs is very different than that of generating networks of non-inserting processors. The role of

Accepting Networks of Non-inserting Evolutionary Processors

199

evolutionary operations in generating networks of evolutionary processors, that is generating networks with nodes specialized in all three evolutionary operations, in two operations out of these three and in only one operation, was considered in [1]. A similar investigation on ANEPs has already started.

References 1. Alhazov, A., Dassow, J., Rogozhin, Y., Truthe, B.: Personal communication 2. Castellanos, J., Mart´ın-Vide, C., Mitrana, V., Sempere, J.: Networks of evolutionary processors. Acta Informatica 38, 517–529 (2003) 3. Csuhaj-Varj, E., Salomaa, A.: Networks of parallel language processors. In: P˘ aun, G., Salomaa, A. (eds.) New Trends in Formal Languages. LNCS, vol. 1218, pp. 299–318. Springer, Heidelberg (1997) 4. Csuhaj-Varj, E., Mitrana, V.: Evolutionary systems: a language generating device inspired by evolving communities of cells. Acta Informatica 36, 913–926 (2000) 5. Csuhaj-Varj´ u, E., Mart´ın-Vide, C., Mitrana, V.: Hybrid NEPs are computationally complete. Acta Informatica 41, 257–272 (2005) 6. Dassow, J., Truthe, B.: On the power of networks of evolutionary processors. In: Durand-Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 158–169. Springer, Heidelberg (2007) 7. Errico, L., Jesshope, C.: Towards a new architecture for symbolic processing. In: Artiﬁcial Intelligence and Information-Control Systems of Robots, vol. 94, pp. 31–40. World Scientiﬁc, Singapore (1994) 8. Fahlman, S., Hinton, G., Seijnowski, T.: Massively parallel architectures for AI: NETL, THISTLE and Boltzmann Machines. In: Proc. AAAI National Conf. on AI, pp. 109–113. William Kaufman, Los Altos (1983) 9. Hillis, W.: The Connection Machine. MIT Press, Cambridge (1985) 10. Manea, F., Mart´ın-Vide, C., Mitrana, V.: On the size complexity of universal accepting hybrid networks of evolutionary processors. Mathematical Structures in Computer Science 17, 753–771 (2007) 11. Manea, F., Mitrana, V.: All NP-problems can be solved in polynomial time by accepting hybrid networks of evolutionary processors of constant size. Information Processing Letters 103, 112–118 (2007) 12. Manea, F., Margenstern, M., Mitrana, V., Perez-Jimenez, M.: A new characterization of NP, P, and PSPACE with accepting hybrid networks of evolutionary processors (submitted) 13. Margenstern, M., Mitrana, V., Perez-Jimenez, M.: Accepting hybrid networks of evolutionary systems. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA 2004. LNCS, vol. 3384, pp. 235–246. Springer, Heidelberg (2005) 14. Mart´ın-Vide, C., Mitrana, V.: Networks of evolutionary processors: results and perspectives. In: Molecular Computational Models: Unconventional Approaches, pp. 78–114. Idea Group Publishing, Hershey (2005) 15. P˘ aun, G., Sntean, L.: Parallel communicating grammar systems: the regular case. Annals of University of Bucharest, Ser. Matematica-Informatica 38, 55–63 (1989) 16. P˘ aun, G.: Computing with membranes. Journal of Computer and System Sciences 61, 108–143 (2000) 17. Sankoﬀ, D., et al.: Gene order comparisons for phylogenetic inference: evolution of the mitochondrial genome. In: Proceedings of the National Academy of Sciences of the United States of America, vol. 89, pp. 6575–6579 (1992)

Discrete Modeling of Biochemical Signaling with Memory Enhancement John Jack1 and Andrei P˘ aun1,2,3 1

Department of Computer Science/IfM Louisiana Tech University, P.O. Box 10348, Ruston, LA 71272, USA {johnjack,apaun}@latech.edu http://www.latech.edu 2 Departamento de Inteligencia Artiﬁcial, Facultad de Inform´ atica Universidad Polit´ecnica de Madrid, Campus de Montegancedo S/N, Boadilla del Monte, 28660 Madrid, Spain http://www.upm.es 3 Bioinformatics Department, National Institute of Research and Development for Biological Sciences, Splaiul Independent¸ei, Nr. 296, Sector 6, Bucharest, Romania

Abstract. We present an enhancement of the Nondeterministic Waiting Time algorithm. This work is a continuation of our group’s previous modeling eﬀorts. We have improved our algorithm with a “memory enhancement”. Previously, we have used our algorithm to explore the Fas-mediated apoptotic pathway in cells with a particular focus on cancerous or HIV-1-infected T cells. In this paper, we will describe the memory enhancement and give a simple three reaction model to illustrate the diﬀerences between our technique and a continuous, concentration-based approach using a system of ordinary diﬀerential equations. Furthermore, we provide our results from the modeling of two well-known models: the Lotka-Volterra predator-prey and a circadian rhythm model. For these models, we provide the results of our simulation technique in comparison to results from ordinary diﬀerential equations and the Gillespie Algorithm. We show that our algorithm, while being faster than Gillespie’s approach, is capable of generating oscillatory behavior where ordinary diﬀerential equations do not. Keywords: Discrete modeling, Lotka-Volterra, predator - prey, circadian rhythm, Gillespie, ordinary diﬀerential equations.

1

Introduction

Systems biology, the systematic study of biological systems through a combined eﬀort between computational and experimental results, has received a great deal of attention in recent years [17,18]. There has been an expansive eﬀort from mathematicians and computer scientists to use models to unravel the mysteries behind biochemical/biological systems – e.g., signal transduction, viral dynamics, gene transcription. With the ever-increasing wealth of information C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 200–215, 2009. c Springer-Verlag Berlin Heidelberg 2009

Discrete Modeling of Biochemical Signaling with Memory Enhancement

201

ﬂowing in from biological labs around the world on protein dynamics, the challenge remains for computer scientists to develop/reﬁne eﬃcient algorithms for modeling molecular signaling cascades. Computational tools are being applied to interpret biological results and make predictions into the underlying molecular mechanisms involved in cancer, autoimmune disorders, and neurological disorders. We see two important eﬀorts being undertaken by computer scientists with respect to modeling molecular signaling cascades. First, the algorithms developed to interpret molecular interactions need visibility to biochemists and noncomputer scientists. Notably, the authors of [10,12] have made great strides in developing software, designed for biochemists with little to no knowledge on the modeling algorithm, to design, develop, and implement biochemical network simulations based on their experimental observations. The second major eﬀort is the development of more eﬃcient algorithms to drive biochemical simulation software. With many labs focusing on the modeling of individual pathways – e.g., Fas-mediated apoptosis [11], p53 network [19,22] and the EGF-receptor system [24] – the concept of a realistic whole-cell simulation remains a very distant goal. There are too many unknowns biochemical aspects to build an accurate and reliable model. However, while the biochemical questions surrounding whole-cell simulation are being answered in experimental labs, there is still work to be done in modifying (and developing new) algorithms for simulating biochemical systems. 1.1

Motivation Behind the Paper

Many signal transduction models in the literature contain as many (or more) reactions as proteins. Although the human genome contains over three billion base-pairs, it only encodes approximately 20,000-25,000 genes. The proteins encoded by these genes are entangled in an intricate and diverse web of interactions. The dynamics of these proteins – e.g., expression levels and reactions – deﬁne the complexity and physiological characteristics of the human cell. Some reactions in signaling cascades can sometimes share common reactants and compete for resources. These competing reactions typically have diﬀerent kinetic rates – i.e., some of the competing reactions utilize a given reactant faster than other reactions use the same reactant. Hence, when the numbers of some molecules are very small, stochastic (or nondeterministic) methods for biochemical modeling can play an important role in interpreting the results of lab experiments, and oﬀer insight into unknown aspects of the system. When modeling biochemical networks via systems of ordinary diﬀerential equations (ODEs), the data are considered in terms of concentrations instead of numbers of molecules, and the reactions are deterministically applied. While the ODEs are satisfactory for predicting the average behavior of a biochemical system, they are not ideal for extrapolating the diﬀerent cellular responses resulting from molecular signaling cascades – especially ones involving low numbers

202

J. Jack and A. P˘ aun

of molecules. The Gillespie Algorithm [7,8], which is a numerical simulation algorithm for the chemical master equation, has been extensively employed to address these low molecular multiplicity situations. However, even though it has been modiﬁed to run more eﬃciently [9], it does not scale well with respect to the number of reactions. Hence, an algorithm capable of realistic and eﬃcient wholecell simulation, a signiﬁcant goal for systems biology, is still being explored. Our algorithm is designed to be faster than Gillespie’s algorithm and its derivatives – such as, Gibson’s Next Reaction Method – yet more sensitive than ODE-based simulations. Our group has previously argued in [3,13,14] that an approach involving the Membrane Systems paradigm of computing oﬀers a unique perspective on biochemical network simulation. Speciﬁcally, in [13,14] we discuss the advantages of our simulation technique: the Nondeterministic Waiting Time (NWT) algorithm. Our algorithm is distinct from the Gillespie Algorithm. Yet, it is a discrete, nondeterministic technique which can oﬀer a diﬀerent perspective than systems of ordinary diﬀerential equations on the biochemistry of a cell. In this paper, we will describe a modiﬁcation to our algorithm. In order to improve the deterministic aspects of reaction competition for our simulation technique, we have added a memory enhancement to the NWT. This enhances the sensitivity of our algorithm with respect to reaction competition over limited resources. Since our algorithm relies on the law of mass action to drive the population dynamics, fast reactions may be allowed to use up all the resources of a slow reaction. With the modiﬁed algorithm, a slow reaction will remember how long it has waited when a fast reaction uses all available reactants. The memory can be factored into the equation for calculating the next time the slow reaction will occur, once reactants become available again. In Section 2 we provide the necessary background on the NWT algorithm, a simulation technique based on Membrane Systems. We will discuss the speciﬁcs on the memory enhancement in Section 3, as well as results for a simple biochemical model involving fast-slow reaction competition. In Section 4 we show the results of the NWT algorithm for simulating two popular models: The LotkaVolterra predator-prey model and a circadian rhythm model [1]. For both models, we compare the results of the NWT algorithm with an ODE-based simulation and a simulation based on the Gillespie Algorithm. Section 5 contains our ﬁnal remarks and a discussion on the future research interests of our modeling group.

2

The Nondeterministic Waiting Time Algorithm

The NWT algorithm is a discrete, nondeterministic biochemical simulation algorithm. We track the evolution of a Membrane System where the rules (or reactions) occur over discrete time intervals in an asynchronous manner. Before we give a step-by-step description of the algorithm, it is important to discuss the concept of reaction Waiting Times. Our NWT algorithm is driven by the law of mass action – the time a reaction takes to occur is directly proportional to the number of its reactant molecules.

Discrete Modeling of Biochemical Signaling with Memory Enhancement

203

When dealing with concentration-based kinetics we need to calculate a discrete kinetic constant (for molecules instead of nMs, μMs, etc). We initialize the discrete kinetic constants with Equation 1. constR =

V

i−1

kR × N i−1

(1)

where V is the volume of the system, N is Avogadro’s constant (6.0221415×1023) and i is the number of reactants involved in the reaction. Once the kinetic constants are initialized, for every reaction in the system, we calculate the initial Waiting Time – the amount of time required for one instance of a reaction – using Equation 2. W TR1 =

1 constR1 ∗ |A|

(2)

where A is the reactant, constR1 is the discrete kinetic constant, and |A| represents the number of molecules present in the system at the moment of WT calculation. Equation 2 represents the calculation for a ﬁrst order reaction (involving only one reactant). For second and third order reactions (two and three reactants, resp.), we need to use Equations 3 and 4. 1 constR2 ∗ |A| ∗ |B|

(3)

1 constR3 ∗ |A| ∗ |B| ∗ |C|

(4)

W T R2 = and W TR3 =

where A, B, and C are the reactants, constR2 and constR3 are the discrete kinetic constants, and |A|, |B| and |C| represent the numbers of molecules present in the system at the moment of WT calculation. With the calculation of reaction Waiting Time, we have the amount of time it will take for each reaction to occur. If there are insuﬃcient reactants for a reaction, then we set the Waiting Time equal to inﬁnity; this is easily done in the C programming language. We can now provide the following description of the NWT algorithm (n.b., Step 7 is the new memory enhancement step, which will be explained in Section 3): 1. Build Membrane System: Import model information (alphabet, rules, etc.). For every reaction, Ri , calculate the initial Waiting Time, W TRi . Choose simulation end-time τf in . Set current simulation time to zero (τ = 0). 2. Build Heap: Using the reaction Waiting Times, we build a min-heap of all reactions in the system. 3. Select Rule: Choose the reaction with the lowest Waiting Time – the top of the min-heap. Upon selecting the top node, recursively check to see if there are any children nodes sharing the minimum Waiting Time. If such a tie for minimum Waiting Time exists, proceed to Step 4. If no tie exists, then proceed to Step 5.

204

J. Jack and A. P˘ aun

4. Handle Tie: Check the multiplicities of the reactant species for all tied reactions. If there are enough reactants to satisfy all of the reactions with the minimum Waiting Time, implement all tied reactions. If there are not enough reactants to accommodate all the reactions, then nondeterministically apply as many reactions as possible. 5. Apply Rule: Update the multiplicities of the reactant(s) and product(s) for the reaction(s) from Step 3. Aggregate the simulation time (τ = τ + W Tapplied ). 6. Update Rules: Recalculate the Waiting Time for all reactions whose reactants include the products or reactants of the applied reaction(s). That is, we need to see how the multiplicity changes from the applied reaction(s) have aﬀected the Waiting Times for all rules dependent on those proteins with changed multiplicity. For each such reaction compare the new Waiting Time with the existing Waiting Time and keep the smallest of the two (unless the new time is inﬁnity). 7. Memory Enhancement: If the recalculation of a reaction’s Waiting Time results in a value of inﬁnity, then we must store the amount of time waited as a percentage (M emperc ). If the recalculation of a reaction’s Waiting Time results in a real value and the previous value was inﬁnite, then the Waiting Time will need to be adjusted according to the stored memory percentage. 8. Heap Maintenance: Adjust the min-heap, bubbling reaction nodes up or down in order to satisfy the min-heap property, once reaction Waiting Times have been recalculated according to the multiplicity changes. N.B., to accommodate the multiple changes in Waiting Times, we employ nonstandard heap maintenance methods. 9. Termination: If τ = τf in , then terminate the simulation. Output the multiplicity information for entire simulation. Otherwise, go back to Step 3. For a deeper explanation of the algorithm, we refer the interested reader to [13,14]. In the next section, we will clarify Step 7 of the algorithm – the memory enhancement.

3

Memory Enhancement

As we discussed in Section 1.1, there are often situations in biochemical networks, where one protein is a reactant for two or more reactions of diﬀerent kinetic rates (fast vs. slow). In order to explain our memory enhancement, we will consider an example system (see Table 1). The biochemical system in Table 1 involves three reactions (R1 , R2 and R3 ) acting on four proteins (A, B, C, D). We can mathematically describe the model as a system of ordinary diﬀerential equations (Equation 5) d[A] = −(k1 + k2 )[A] + k3 [D] dt d[B] = k2 ∗ [A] dt

Discrete Modeling of Biochemical Signaling with Memory Enhancement

205

Table 1. An example biochemical system Reaction Rate Constant Initial Molecules R1 : A → C k1 (slow) A=1 R2 : A → B k2 (fast) B=0 R3 : D → D + A k3 C=0 D=1

d[C] = k1 ∗ [A] dt d[D] =0 dt

(5)

The system of ordinary diﬀerential equations in Equation 5 was speciﬁcally designed to illustrate the eﬀects of the memory enhancement. We will compare the enhanced NWT algorithm with solutions to the systems of ODEs, providing two diﬀerences cases based on variable kinetics. By selecting diﬀerent kinetic rates, we will show how the memory enhancement leads to agreement between the NWT and ODEs for strictly deterministic runs, but with nondeterministic decisions it can lead to distinct results and a divergence in overall behavior of the biochemical network. For the sake of simplicity, we will assume the rate constants (k1 , k2 and k3 ) are already in discrete form. Therefore, when refer to ki in Ri above, we have constRi . A model similar to the one described in Table 1 could be used to investigate the dynamics of HIV-1 Tat protein, since it is initially transcribed at very low numbers [16]. Once Tat is assembled in the cytosol, it can be exocytised or translocated to the nucleus [20]. When Tat is translocated to the nucleus it can begin upregulating HIV-1 proteins (including itself). Since the downstream eﬀects of Tat translocation to the nucleus has profound impacts on the cell (causing upregulation of the HIV-1 proteins), a discrete and nondeterministic approach is beneﬁcial to the study of the dynamics of the low levels of Tat [23]. In the system, molecules of A are formed from molecules of D. This reaction can basically be viewed as a combined transcription and translation rule with D being the gene and A being the protein encoded by the gene. Once a molecule of A is formed, it has the option of turning into a molecule of B at rate k2 or it can turn into a molecule of C at a rate k1 . If we consider the species A to be analogous to HIV-1 Tat protein, then reaction R1 (A → B) could be the translocation of Tat from the cytosol to the nucleus, and reaction R2 (A → C) could be the translocation of Tat from the cytosol to the extracellular environment. We will next consider two cases for the model described in Table 1 and discuss the implementation of the memory enhancement. The cases are determined by the values for the discrete kinetic rates. The ﬁrst case shows that the memory enhancement can produce similar results between the ODEs and the NWT when no nondeterministic decisions are made. For the second case, we will show how the technique can produce diﬀerent results, illustrating the NWT algorithm’s

206

J. Jack and A. P˘ aun

ability to explore the stochastic nature of molecular signaling cascades. Note, the NWT algorithm remains the same in both cases, the only diﬀerence lies in the initialization of the discrete kinetic constants. 3.1

Case 1: Deterministic Memory Enhancement

If we let k1 = 4, k2 = 10, and k3 = 5, then we can see the results of a simulation using the NWT algorithm plotted against the solution of the system of ordinary diﬀerential equations (Figure 1).

25

Molecules

20

15

10

5

0

0

1

2

3

4

5

6

7

8

9

10

Time

Fig. 1. The graph shows the accumulation of C molecules throughout a 10 second run. The bars are the discrete results from the NWT algorithm and the black line is the solution of the system of ordinary diﬀerential equations. With the kinetic values (k1 = 4, k2 = 10, k3 = 5), there are no nondeterministic decisions for the entire length of the NWT simulation. Therefore, we are pleased to see the NWT algorithm results are similar to the solution to the system of ordinary diﬀerential equations.

At initialization (t = 0), there is exactly one molecule of D and one molecule of A. Therefore, from Equation 2, we see that all three reactions have real (ﬁnite) Waiting Times when the simulation begins. Moreover, we have W TR1 = 0.25, W TR2 = 0.1 and W TR3 = 0.2. According to the reaction Waiting Times, the ﬁrst reaction to occur is R2 , which immediately exhausts the system’s supply of A molecules, yields one molecule of B and a simulation time of t = 0.1. The Waiting Times for the rules aﬀected by the applied rule must be recalculated; since there are no molecules of A in the system, we have W TR1 = W TR2 = ∞. R3 does not use any proteins involved in the applied reaction (A or C), so W TR3 is left unchanged after the ﬁrst reaction is executed. After the heap maintenance, R3 will be at the top. The next reaction to be applied is R3 , which gives us a new molecule of A and a simulation time of t = 0.2. Now the memory enhancement plays a role. In the ﬁrst step, reaction R2 exhausts the supply of A molecules. When this occurs,

Discrete Modeling of Biochemical Signaling with Memory Enhancement

207

R1 has waited for 0.1 seconds of its total W T . The memory enhancement allows the simulator to keep track of the percentage of time waited. In other words, R1 waited for 0.1 seconds out of its required 0.25 seconds, which means it has waited 40% of its Waiting Time. We store the percentage of time left to wait (60%). So, when a new molecule of A is formed (for instance, at τ = 0.2), we can recalculate the W TR1 using the percentage to adjust its Waiting Time. Continuing after R3 is applied at time τ = 0.2, we have a new molecule of A. Since R1 and R2 both use A as a reactant, we must recalculate the Waiting Times of both reactions. The Waiting Time of R2 and R1 are calculated using Equation 2. However, the memory of R1 allows us to take 60% of its recalculated W T . Therefore, the Waiting Time of R2 is calculated as 0.1, but the Waiting Time of R1 is recalculated as 0.15. This number stems from the equation W TR1 = M em

1 k1 ∗ |C|

(6)

where M em is the percentage of time left to wait (60% in the example above). The second and third order reactions follow similarly. Using this implementation, our NWT algorithm agrees with the ODEs for a strictly deterministic run. In the next subsection, we will explore the implications of the memory enhancement in a system requiring reaction competition over low numbers of molecules. Although we agree with ODEs in a deterministic run, we want to explore the nondeterministic eﬀects of the memory enhancement. 3.2

Case 2: Nondeterministic Memory Enhancement

We will now assume diﬀerent kinetic constants to highlight the eﬀects of the nondeterministic component of the NWT algorithm in conjunction with the memory enhancement. Although the kinetics of our sample system are chosen in a deliberate manner in order to illustrate the nondeterministic eﬀects of the algorithm, we will later show in Section 4.2 how our nondeterministic logic can have similar implications in a model reported in the literature. For our next simulations, we assume k1 = 0.1, k2 = 1.0, and k3 = 0.5. The initial Waiting Times are calculated as W TR1 = 10, W TR2 = 1, and W TR3 = 2. In Figure 2, we see the accumulation of B and C molecules. The results of the ODE-based simulation are visibly diﬀerent than the results of the simulation involving the NWT algorithm. The reasons for the diﬀerences are the nondeterministic decisions on reaction competition for A molecules. Based on the initialized WT s, the ﬁrst reaction to be applied is R2 . After R2 is applied, the simulation time is aggregated (t = 1) and there are no more molecules of A present in the system. Hence, the WT s for R2 and R1 are both inﬁnite after recalculation. We store the percentage of time the slow reaction, R1 , had left to wait when the W T changed to inﬁnity – M emR1 = 90%. The next rule to be applied is R3 , since it was unaﬀected by the application of R2 . The simulation time is adjusted (t = 2), and we now have a new molecule of A. With our new A molecule available, we must recalculate the WT s for R1 and R2 .

208

J. Jack and A. P˘ aun

(a)

(b) Fig. 2. In both graphs we see the results of the ODE-based simulation (straight black line) and the results from the modiﬁed NWT algorithm. (a) The accumulation of molecules of B and (b) the accumulation of C are shown. Molecules of B and C both come from A molecules. However, the reaction for B is faster than the reaction for C. In the ODE models, a molecule of A can be used to partially satisfy B and C. Since our NWT algorithm is discrete, the molecules are nondeterministically chosen to satisfy one or the other. The reaction changing A into C ’remembers’ how long it has waited, and uses this information the next time a molecule of A is ready.

Using M emR1 , we calculate the W T for reaction R1 , using the fact that it need wait only 90% of its new Waiting Time. Therefore, when a new molecule of A is formed two seconds into the run, we recalculate W TR1 using Equation 6. In our case, we have W TR1 = 9 and W TR2 = 1. Continuing the calculations for the simulation, we skip ahead to a future event (t = 18). Up until this point, we have been creating molecules of A, and every single one of them has been deterministically chosen to change into molecule B via reaction R2 . But, at t = 18, a molecule of A has been created, and the Waiting Times of reaction R1 and R2 are equal W TR1 = W TR2 = 1, since we have M emR1 = 10%. In other words, R1 and R2 are each attempting to use the A molecule to form a C and B, resp. The ODE-based simulation has no

Discrete Modeling of Biochemical Signaling with Memory Enhancement

209

problems at this time-point, since it is continuously sending a fraction of each A to form a fraction of B and C, whereas our algorithm represents A discretely and only satisﬁes one reaction per molecule. Our algorithm faces the question: at t = 18 should the A molecule be allowed to satisfy R1 or R2 ? The algorithm answers the question by making a nondeterministic choice between R1 and R2 . If R1 is chosen, then it is applied, and our results stay with the ordinary diﬀerential equations results (up to t = 19). Remember, the ordinary diﬀerential equations have been slowly and continuously aggregating fractions of C molecules throughout to reach one full molecule of C by t = 19. However, if R2 is chosen, then our solution diverges from the previous solution. When the eﬀects of the nondeterministic decisions are aggregated over 1000 seconds, we see the diﬀerent results obtained from the NWT algorithm (Figure 2).

4 4.1

Other Models Lotka-Volterra Predator-Prey

The Lotka-Volterra predator-prey model depicts the interactions of two species. There is a prey population and a predator population, where P1 (t) and P2 (t) represent the number of each species respectively at time t. The model can be written as the following pair of ﬁrst-order, nonlinear, diﬀerential equations dP1 = P1 ∗ (a − b ∗ P2 ) dt dP2 = −P2 ∗ (c − d ∗ P1 ) dt

(7)

where prey species are born at a rate of a and consumed at a rate of b. Predator species are born at a rate of d and die at a rate of c. In Figure 3, we see a picture of the predator-prey model. The picture (as well as the SBML code for the model) was generated with CellDesigner [5,6], which we also used to generate the SBML code to initialize our simulator. The way the system is designed, an increase in prey leads to an increase in predator, and an increase in predator leads to a decrease in prey. Total annihilation of prey leads to total extinction of predator, since the food supply of the predator will be exhausted. We used three diﬀerent simulation techniques to model the reactions described in the Lotka-Volterra model: solution to the system of ordinary diﬀerential equations, the Gillespie Algorithm, and our NWT algorithm. The ODEs were solved in MATLAB, while the other two algorithms were both coded in C. The results are given in Figure 4 and Figure 5. The solution to the ordinary diﬀerential equations in Equation 7 shows consistent oscillations throughout the entire simulation run. The NWT shows dampened oscillations over time. The Gillespie Algorithm has diﬃculties producing the oscillations, due to the stochasticity of the algorithm. In this case, our NWT

210

J. Jack and A. P˘ aun

Fig. 3. The Lotka-Volterra model involves two interacting species. Prey species are born at a rate a and are consumed at a rate b by the predator species. The predator species are born at a rate of d if there is available food (prey). The way the system is designed, an increase in prey leads to an increase in predator. Total annihilation of prey leads to total extinction of predator, since there is no longer any food. The model deterministically leads to oscillatory behavior.

Lotka−Volterra (ODE)

Lotka−Volterra (NWT)

500

500 Prey Pred

400 350 300 250 200 150 100

400 350 300 250 200 150 100

50 0

(a)

Prey Pred

450

Molecules of protein R

Molecules of protein R

450

50 0

20

40

60

80

0

100

0

20

40

Time

60

Time

80

100

(b)

Lotka−Volterra (Gillespie) 500 Prey Pred

Molecules of protein R

450 400 350 300 250 200 150 100 50 0

(c)

0

20

40

60

80

100

Time

Fig. 4. Results of three simulation techniques for the Lotka-Volterra model (up to 100 seconds). (a) solution to ordinary diﬀerential equations, (b) the NWT algorithm, and (c) the Gillespie Algorithm.

Discrete Modeling of Biochemical Signaling with Memory Enhancement Lotka−Volterra (ODE)

Lotka−Volterra (NWT)

450

450 Prey Pred

350 300 250 200 150 100 50

(a)

Prey Pred

400

Molecules of protein R

Molecules of protein R

400

0

211

350 300 250 200 150 100 50

0

100

200

300

Time

400

500

0

0

100

200

300

Time

400

500

(b)

Fig. 5. Results of the two simulation techniques for the Lotka-Volterra model (up to 500 seconds). (a) solution to ordinary diﬀerential equations and (b) the NWT algorithm.

algorithm runs deterministically. The system is small enough and the dynamics are such that the NWT makes no nondeterministic decisions due to reaction competition. If we expand the results of the solution to system of ordinary diﬀerential equations and the NWT algorithm, we see further decline in the amplitude for the NWT algorithm. In Figure 5, we expand the simulation run for a total of ﬁve hundred seconds. The results for the solution to the system of ordinary diﬀerential equations and the NWT algorithm simulation are provided. We modeled this classic system to illustrate the diﬀerences in the results of our simulation technique compared to the solution of the system of ordinary diﬀerential equations, the NWT algorithm, and the Gillespie Algorithm simulations. Our system was able to exhibit oscillatory behavior, albeit the oscillations are damped. However, as you can see in Figure 5, the oscillations persist with the NWT algorithm (and the ordinary diﬀerential equations). Yet, the Gillespie Algorithm will always reach a steady state, whereby the predator and prey species will eventually completely disappear. Since there are no nondeterministic decisions made during the run, we can only attribute the dampened oscillations to the fact that the system is discrete. We will next discuss a circadian rhythm model, which will illustrate how our algorithm can produce Gillespie-like results, even though we have a reduced complexity. 4.2

Circadian Rhythm

Circadian rhythm models are often explored in nature. These act as internal clocks which allow organisms to anticipate daily changes in the environment [1] – for instance, when to hunt for food, when to rest, etc. Yet, at the level of cellular biochemistry, circadian rhythms have also been reported [4]. Biological systems run by internal clocks – that is, certain proteins are created at certain parts of the day. Therefore, simulating circadian rhythm models is important in understanding the way DNA is interpreted and pre-existing proteins waiting to be activated are used by the body for daily survival [1]. We have chosen to model the circadian rhythm model described in [21]. The system describes an activator and a repressor gene (A and R). These genes are

212

J. Jack and A. P˘ aun

transcribed into mRNA, which leads into the translation of the proteins. The activator A binds to the promoters for A and R and increases the transcription rate. The system of ordinary diﬀerential equations described in [21] showed that intrinsic biochemical noise enhanced the oscillations. In Equation 8, we see the system of ordinary diﬀerential equations for the model. dDA dt dDR dt dDA dt dDR dt dDMA dt dA dt dMR dt dR dt dC dt

= θA ∗ DA − γA ∗ DA ∗ A = θR ∗ DR − γR ∗ DR ∗ A = γA ∗ DR ∗ A − θA ∗ DA = γR ∗ DR ∗ A − θR ∗ DR = αA ∗ DA + αA ∗ DA − δMa ∗ MA = βA ∗ MA + θA ∗ DA + θR ∗ DR − A ∗ (γA ∗ DA + γR ∗ DR + γC ∗ R + δA ) = αR ∗ DR + αR ∗ DR − δMR ∗ MR

= βR ∗ MR − γC ∗ A ∗ R + δA ∗ C − δR ∗ R = γC ∗ A ∗ R − δA ∗ C

(8)

where A and R represent the number of activator and repressor proteins, DA and DA represent the number of activator genes with or without binding to A, DR and DR represent the number of repressor genes with or without binding to R, MA and MR represent mRNA molecules of A and R, and C represent the corresponding inactivated complex formed by A and R. Deterministic modeling techniques, like the solution to the systems of ordinary diﬀerential equations, for biochemical interactions fail to produce the oscillations of a circadian rhythm model. However, the stochastic noise from a Gillespiebased approach leads to repeated oscillations throughout an entire run. Our NWT algorithm can produce results similar to the Gillespie algorithm – genetic oscillations – but at a considerably reduced computational cost. The results for the simulation of the circadian rhythm model are shown in Figure 7. We present the results from Gillespie’s Algorithm, the solution of the system of ordinary diﬀerential equations (Equation 8), and our NWT algorithm. The NWT algorithm is able to reproduce the oscillations for the perturbed model, as is the case with the Gillespie approach [21]. Similar to Gillespie, the NWT shows some variability in both the amplitude – numbers of molecules – and the periodicity of oscillations. The authors in [21] showed that parameter values can have a profound impact on oscillations. By reduction of the kinetic rate governing R degradation, the deterministic results produce a single peak followed by a steady state, while a stochastic simulation remains oscillating. Our NWT algorithm also produces oscillations instead of a steady state, but at a reduced computational cost from

Discrete Modeling of Biochemical Signaling with Memory Enhancement

213

Fig. 6. The picture of this system was generated using CellDesigner. The system was described in [21]. We have modeled this system with the NWT algorithm, Gillespie’s Algorithm, and as a system of ordinary diﬀerential equations.

3000

3000

2500

2000

2000

1500

1000

500

0

(a)

ODE Gillespie

2500

Molecules

Molecules

ODE NWT

1500

1000

500

0

50

100

150

200

Time

250

300

350

400

0

0

50

100

150

200

Time

250

300

350

400

(b)

Fig. 7. The results for the circadian rhythm model: (a) the NWT algorithm and (b) the Gillespie Algorithm. Both algorithms are plotted against the solution to the system of ordinary diﬀerential equations.

the Gillespie Algorithm approach. This is the beneﬁt of modeling with the NWT algorithm instead of the Gillespie Algorithm. For our simulation to produce oscillations comparable to the Gillespie Algorithm, we require only 50 random numbers to be generated. This stems from the fact that the NWT algorithm relies on deterministic kinetics for the majority of reactions, but when reactants are limited and competition for reactants exists, nondeterministic decisions drive a variable response from the competing reactions.

5

Final Remarks

We have improved the sensitivity of our NWT algorithm through the addition of memory to Waiting Time calculation. We argue that this gives us an edge over ordinary diﬀerential equations in modeling reactions of low molecular multiplicity. The improvements were illustrated with multiple examples, one designed to

214

J. Jack and A. P˘ aun

speciﬁcally discuss the memory enhancement and two other models from the literature. In the ﬁeld of systems biology, there is a strong emphasis on using nondeterministic (or stochastic) techniques in modeling biochemical networks where low numbers of molecules can be found. We are interested in exploring these types of situations. For instance, in HIV-infected T cells, there are initially low levels of Tat protein, which after translocation to the nucleus, bind to receptor sites and cause upregulation of the HIV-1 proteins [16]. We have already published a paper on the eﬀects of HIV-1 proteins on Fas-mediated apoptosis, and will be looking to use our reﬁned algorithm for future development in this pathway. Also, in regards to T cells, it seems that low levels of cytochrome C released from the mitochondria bind to IP3 R. This receptor binding leads to release of Ca+ form the mitochondria, and [2] showed that this was necessary for both the extrinsic (Fas-mediated) and the intrinisic apoptotic pathways. We are exploring this direction via wetlab experimentation, and we will be using the NWT algorithm to elucidate new aspects to Fas-mediated apoptotic events. Acknowledgments. We gratefully acknowledge support in part from the LONI Institute: fellowship for J.J. and state-of-the-art parallel computing facilities, National Science Foundation Grant CCF-0523572, INBRE Program of the NCRR (a division of NIH), support from CNCSIS grant RP-13, support from CNMP grant 11-56 /2007, support from Spanish Ministry of Science and Education (MEC) under project TIN2006-15595, and support from the Comunidad de Madrid (grant No. CCG07-UPM/TIC-0386 to the LIA research group).

References 1. Barkai, N., Leibler, S.: Biological rhythms: Circadian clocks limited by noise. Nature 403, 267–268 (2000) 2. Boehning, D., van Rossem, D.B., Patterson, R.L., Snyder, S.H.: A peptide inhibitor of cytochrome c/inositol 1,4,5-triphosphate receptor binding blocks intrinsic and extrinisc cell death pathways. PNAS 102(5), 1466–1471 (2005) 3. Cheruku, S., P˘ aun, A., Romero-Campero, F., P´erez-Jim´enez, M., Ibarra, O.: Simulating FAS-Induced Apoptosis by Using P Systems. In: Proceedings of Bio-inspired computing: theory and applications (BIC-TA), Wuhan, China, September 18-22 (2006); also extended version published as Progress in Natural Science 17(4), 424–431 (2006) 4. Dunlap, J.: Circadian Rhythms: An End in the Beginning. Science 280(5369), 1548–1549 (1998) 5. Funahashi, A., Morohashi, M., Kitano, H.: CellDesigner: a process diagram editor for gene-regulatory and biochemical networks. BIOSILICO 1(5), 159–162 (2003) 6. Funahashi, A., Matsuoka, Y., Jouraku, A., Morohashi, M., Kikuchi, N., Kitano, H.: CellDesigner 3.5: A Versatile Modeling Tool for Biochemical Networks. Proceedings of the IEEE 96(8), 1254–1265 (2008) 7. Gillespie, D.T.: A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics 22, 403–434 (1976)

Discrete Modeling of Biochemical Signaling with Memory Enhancement

215

8. Gillespie, D.T.: Exact Stochastic Simulation of Coupled Chemical Reactions. Journal of Physical Chemistry 81(25), 2340–2361 (1977) 9. Gibson, M.A., Bruck, J.: Eﬃcient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels. Journal of Physical Chemistry A 104, 1876–1889 (2000) 10. Hoops, S., et al.: COPASI – a Complex Pathway Simulator. Bioinformatics 22(24), 3067–3074 (2006) 11. Hua, F., Cornejo, M., Cardone, M., Stokes, C., Lauﬀenburger, D.: Eﬀects of bcl2 levels on fas signaling-induced caspase-3 activation: molecular genetic tests of computational model predictions. The Journal of Immunology 175(2), 985–995 (2005); Correction 175(9), 6235–6237 (2005) 12. Hucka, M., et al.: The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics 19(4), 524–531 (2003) 13. Jack, J., Romero-Campero, F.J., Perez-Jimenez, M.J., Ibarra, O.H., P˘ aun, A.: Simulating Apoptosis Using Discrete Methods: A Membrane System and a Stochastic Approach. Language Theory in Biocomputing (2007) 14. Jack, J., Rodriguez-Paton, A., Ibarra, O.H., P˘ aun, A.: Discrete Nondeterministic Modeling of the FAS Pathway. Int. J. Found. Comput. Sci. 19(5), 1147–1162 (2008) 15. Jack, J., P˘ aun, A., Rodriguez-Paton, A.: Eﬀects of HIV-1 Proteins on the Fasmediated Apoptotic Signaling Cascade: A Computational Study of T cell Latency. In: Proceedings of WMC9: 2008. LNCS, vol. 5391, pp. 246–259 (2009) 16. Karn, J.: Tackling Tat. Journal of Molecular Biology 2(22), 235–254 (1999) 17. Kitano, H.: Computational Systems Biology. Nature 420 (2002) 18. Kitano, H.: Systems Biology: A Brief Overview. Science 295, 55–60 (2002) 19. Ma, L., Rice, J.J., Hu, W., Levine, A.J., Stolovitzky, G.A.: A plausible model for the digital response of p53 to DNA damage. PNAS 102(40), 14266–14271 (2005) 20. Selliah, N., Finkel, T.: Biochemical mechanisms of HIV induced T cell apoptosis. Cell Death and Diﬀerentiation 8, 127–136 (2001) 21. Vilar, J.M.G., et al.: Mechanisms of noise-resistance in general oscillations. PNAS 99(9), 5988–5992 (2002) 22. Wagner, J., Ma, L., Rice, J.J., Hu, W., Levine, A.J., Stolovitzky, G.A.: p53-Mdm2 loop controlled by a balance of its feedback strength and eﬀective dampening using ATM and delayed feedback. IEE Proc.-Syst. Biol. 152(3), 109–118 (2005) 23. Weinberger, L., Burnett, J., Toettcher, J., Arkin, A., Schaﬀer, D.: Stochastic Gene Expression in a Lentivrial Positive-Feedback Loop: HIV-1 Tat Fluctuations Drive Phenotypic Diversity. Cell 122(2), 169–182 (2005) 24. Wiley, H.S., Shvartsman, S.Y., Lauﬀenburger, D.A.: Computational modeling of EGF-receptor system: a paradigm for systems biology. TRENDS in Cell Biology 13(1), 43–50 (2003)

Dynamical Systems and Stochastic Programming: To Ordinary Diﬀerential Equations and Back Luca Bortolussi1 and Alberto Policriti2,3 1 2

Dept. of Mathematics and Computer Science, University of Trieste, Italy [email protected] Dept. of Mathematics and Computer Science, University of Udine, Italy [email protected] 3 Applied Genomics Institute (IGA), Udine, Italy [email protected]

Abstract. In this paper we focus on the relation between models of biological systems consisting of ordinary diﬀerential equations (ODE) and models written in a stochastic and concurrent paradigm (sCCP stochastic Concurrent Constraint Programming). In particular, we deﬁne a method to associate a set of ODE’s to an sCCP program and a method converting ODE’s into sCCP programs. Then we study the properties of these two translations. Speciﬁcally, we show that the mapping from sCCP to ODE’s preserves rate semantics for the class of biochemical models (i.e. chemical kinetics is maintained) and we investigate the invertibility properties of the two mappings. Finally, we concentrate on the question of behavioral preservation, i.e if the models obtained applying the mappings have the same dynamics. We give a convergence theorem in the direction from ODE’s to sCCP and we provide several well-known examples in which this property fails in the inverse direction, discussing them in detail.

1

Introduction

The systemic approach to biology is nowadays a fertile and growing research area, considered by many as a promising track to the understanding of life [41,1]. A key ingredient of systems biology resides in coupling wet lab experiments with mathematical modeling and analysis of bio-systems [33]. Many mathematical instruments have been used for this purpose, some concerned with qualitative analysis, others encapsulating also quantitative data [32]. Quantitative modeling is essentially dominated by two main mathematical tools: (ordinary) diﬀerential equations on one side and stochastic processes on the other [32]. Both these methods are concerned with the study of dynamical evolution of systems; however, they diﬀer in the description of the quantities of interest: diﬀerential equations represent them as continuous variables, stochastic processes operate, instead, on discrete quantities. Modeling formalisms mixing discrete and continuous ingredients, like hybrid automata [29], have also been used in modeling bio-systems [2]. C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 216–267, 2009. c Springer-Verlag Berlin Heidelberg 2009

Dynamical Systems and Stochastic Programming

217

We will focus here mainly on the two former approaches, although commenting on the theme of the use of hybrid systems for restoring behavioral equivalence throughout the paper. The theory of dynamical systems and diﬀerential equations (ODE’s) is very attractive, being it a mature research area equipped with a huge set of analysis tools, ranging from static analysis of phase space topology to fast simulation via numerical integration [51,43]. However, writing ODE’s for a given system is generally a diﬃcult task, requiring a considerable expertise. In addition, the representation of biological entities as continuous variables is an approximation that can sometimes be too rough, especially for low populations [26]. Stochastic processes like Continuous Time Markov Chains (CTMC) [39], on the other hand, do not suﬀer from these approximation limits, as they represent biological entities as discrete quantities, thus being more adherent to reality. On the other hand, analyzing a stochastic model is much more diﬃcult, both from an analytical and from a computational point of view [53]. Regarding the description of stochastic models, recently we have seen the application of stochastic process algebras (SPA) [48,45], a class of formal languages developed in theoretical computer science as formal tools to analyze (quantitatively) the performances of computing networks. These languages allow to build CTMC-based models following a simple, paradigmatic, identiﬁcation of biological entities with (computing) processes. Moreover, they are compositional, allowing to build models by composing together sub-models. Ideally, one would like to have a modeling technique that collects the advantages both of stochastic process algebras and diﬀerential equations, or, at least, to switch automatically between the two formalisms, depending on the particular task to be performed. In this direction, there are two related problems that must be faced: (a) studying the (mathematical) relation between the two modeling techniques and (b) ﬁnding automatic methods for converting one formalism into the other. More speciﬁcally, we suggest the following workﬂow: ﬁrst deﬁning translation methods (for a speciﬁc process algebra), thus tackling (b), and then studying the mathematical relations intervening between the models obtained applying these translations. In this way we should be able to evaluate the appropriateness of conversion procedures between SPA and ODE’s and to restrict the focus of the analysis required by (a). There are two directions in the conversion between SPA and ODE: the ﬁrst one associating a set of diﬀerential equations to a stochastic process algebra model, and the inverse one, mapping diﬀerential equations to stochastic process algebra programs. The ﬁrst direction can be helpful for the analysis of SPA models, as ODE’s can be solved and analyzed more eﬃciently. Associating SPA to ODE, instead, can help to clarify the logical pattern of interactions that are hidden in the mathematical structure of diﬀerential equations. Generally, as process algebra models can be written much more easily than diﬀerential equations, even by non-experts (possibly via a graphical interface), the ﬁrst direction, from

218

L. Bortolussi and A. Policriti

SPA to ODE, looks potentially more fruitful, though having both mappings helps the study of the relationship between the two formalisms. Supposing to have such transformations at our disposal, a crucial problem is to single out criteria to evaluate and validate them. The ﬁrst possibility is to inspect the relationship intervening between a SPA program and the associated ODE’s only from a mathematical point of view, forgetting any information about the system modeled. As both stochastic processes and diﬀerential equations are dynamical systems, this approach essentially corresponds to require that both models exhibit the same behavior, i.e. the same dynamical evolution. Of course, we may require agreement only from a qualitative point of view (so that the qualitative features of the dynamics are the same) or even from a quantitative one (numerical values agree). The diﬃculty with this approach is that stochastic processes have a noisy evolution, in contrast with the determinism characterizing diﬀerential equations. Hence, we need to remove the noise. One possibility is to look only at qualitative features of the dynamics, deﬁning them in a precise way; we will go back to this problem in Section 3.2 below. Otherwise, we may average out noise from the stochastic models, thus considering the expected evolution of the system and requiring it to be described precisely (i.e. quantitatively) by the ODE’s. Unfortunately, noise cannot be eliminated so easily, as sometimes it is the driving force of the dynamics [26,52]. Therefore, this second form of equivalence is not completely justiﬁed; we will comment more on this point while discussing some examples in the following. A diﬀerent approach in comparing stochastic and diﬀerential models can be deﬁned if we consider some additional information, which is external to the mathematics of the two models. The idea is to validate the translation w.r.t. this additional information. We explain this point with an example. Consider a model of a set of biochemical reactions; there are diﬀerent chemical kinetic theories that can be used to describe such system, the most famous one being the principle of mass action. Using such a kinetic theory, we can build (in a canonical way) both a model based on diﬀerential equations and a model based on stochastic process algebras. If we are concerned with the principle of mass action more than with dynamical behavior, we may ask that our translation procedures preserves the former, meaning that the ODE’s associated to a mass action SPA program are exactly the ODE’s built according to mass action principle, and viceversa. Essentially, this corresponds to requiring that the translation procedures deﬁned are coherent with (some) principles of the system modeled. For instance, in the case of mass action, coherency corresponds to preserve the meaning of rates (the so called rate semantics in [17]). Notice that in this case we are not requiring anything about dynamics, so coherent models may exhibit a divergent behavior, and this is indeed a well known issue, see, for instance, [26] or Section 3.2 below. Therefore, this comparison is essentially diﬀerent from the behavioral-based one, and it is essentially syntactic, in the sense that it is concerned only with how models are written, not with their time evolution. The operation of associating ODE’s to SPA can be seen also as the deﬁnition of an ODE-based semantic for the stochastic processes, as opposed to the

Dynamical Systems and Stochastic Programming

219

CTMC-based one. Consequently, the comparison of the stochastic model with the derived ODE’s can also be seen as an attempt to discover the mathematical relationship between these two semantics. The problem of associating ODE’s to stochastic process algebras has been tackled only recently in literature. The forefather is the work of Hillston [31], associating ODE’s to models written in PEPA [30], a stochastic process algebra originally designed for performance modeling. Successively, similar methods have been developed for stochastic π-calculus [16,11,44] and for stochastic Concurrent Constraint Programming [7,12]. All these methods build the ODE’s performing a syntactic inspection and manipulation of the set of agents deﬁning the SPA model. In fact, they all satisfy the coherency condition staten above, at least for mass action principle (a proof for stochastic π-calculus can be found in [17]). The inverse problem of associating SPA models to ODE’s has received much less attention, the only example being [12], where we use stochastic Concurrent Constraint Programming as target SPA. In this paper, we will retake the work previously done for sCCP in [12], presenting it in a more detailed and formal way. Basically, we will deﬁne two translation procedures: from sCCP to diﬀerential equations and viceversa. sCCP plays here a central role, thanks to some ingredients giving a noteworthy ﬂexibility to it, the presence of functional rates above all. Therefore, in Section 2, we will recall the basics of sCCP and its application as a modeling language for biological systems, as presented in [15]. Further details on the language can be found in [8]. The translation procedure from sCCP to diﬀerential equations is presented formally in Section 3, while Section 4 is devoted to the presentation of the inverse mapping from general ODE’s into sCCP. In Section 3, we will also show coherency conditions for a class of chemical kinetics and we will comment in detail the problem of behavioral equivalence in the conversion from sCCP to ODE’s. This will be done mainly via examples, exhibiting biological systems for which the translation preservers also the behavior and other systems whose stochastic models show a diﬀerent behavior than ODE’s. The problem of behavioral equivalence is not new, and in fact some examples that we will give are famous ones [26]. However, the syntactic structure of process algebras in general, and sCCP speciﬁcally, give a new ﬂavor to these classical examples, and brings the attention into new ones. The issue of preservation of dynamic behavior in the mapping from ODE’s to sCCP is tackled in Section 4. In this case we are able to exploit the structure of the mapping and thus to give a convergence theorem. Throughout the paper, we will encounter several situations in which discreteness is a crucial ingredient for the dynamics of the system. This points to a third class of dynamical systems that is in the middle between SPA and ODE’s and that can be used to approximate them, namely hybrid automata [29]. In [13,14] we deal with the problem of mapping sCCP programs into hybrid automata, showing that in this case we are able to deal correctly from a behavioral viewpoint with a broader class of sCCP systems. The idea of such mapping is that of translating to ODE’s locally while retaining some level of discreteness in the

220

L. Bortolussi and A. Policriti

ﬁnite control of the hybrid automata. By the way, the method of [13] can be extended into a general framework encompassing also the mapping presented in this paper as a particular case.

2

Preliminaries

In this section we brieﬂy recall the basics of stochastic Concurrent Constraint Programming (sCCP, Section 2.1) and its application as a modeling language for biological systems (Section 2.2). The interested reader is referred to [8] for further details. In Section 2.3, we introduce some restrictions on the language that greatly simplify the mapping from and to ODE’s. 2.1

Stochastic Concurrent Constraint Programming

Concurrent Constraint Programming (CCP, [49]) is a process algebra having two distinct entities: agents and constraints. Constraints are interpreted ﬁrst-order logical formulae, stating relationships among variables (e.g. X = 10 or X+Y < 7). Agents in CCP, instead, have the capability of adding constraints (tell) into a sort of global memory (the constraint store) and checking if certain relations are entailed by the current conﬁguration of the constraint store (ask). The communication mechanism among agents is therefore asynchronous, as information is exchanged through global variables. In addition to ask and tell, the language has all the basic constructs of process algebras: non-deterministic choice, parallel composition, procedure call, plus the declaration of local variables. The stochastic version of CCP (sCCP [7,15]) is obtained by adding a stochastic duration to all instructions interacting with the constraint store C, i.e. ask and tell. Each instruction has an associated random variable representing time (thus taking values in the positive reals), exponentially distributed with rate given by a function associating a real number to each conﬁguration of the constraint store: λ : C → R+ . This is a unusual feature in traditional stochastic process algebras like PEPA [30] or stochastic π-calculus [44] (although recently introduced in BioPEPA [18]), and it will be crucially used in the translation mechanisms. The syntax of sCCP can be found in Table 1. Two diﬀerent kind of actions are present in such table: stochastic actions, having a rate attached to them, and instantaneous actions, having an inﬁnite rate. Table 1. Syntax of sCCP P rogram = D.A D = ε | D.D | p(x) : −A π = tellλ (c) | askλ (c) M = π.G | M + M G = 0 | tell∞ (c).G | p(y) | M | ∃x G | G G A = 0 | tell∞ (c).A | M | ∃x A | A A

Dynamical Systems and Stochastic Programming

221

This second class of actions can be used to model the happening of complex atomic events, like a sequence of store updates happening instantaneously. However, only tell actions can happen instantaneously, and moreover they are always guarded by a stochastic action. The same restriction applies to recursive calls. Operational Semantics. The deﬁnition of the operational semantics is given specifying two diﬀerent kinds of transitions: one dealing with instantaneous actions and the other with stochastically timed ones. The basic idea of this operational semantics is to apply the two transitions in an interleaved fashion: ﬁrst we apply the transitive closure of the instantaneous transition, then we do one step of the timed stochastic transition. To identify a state of the system, we need to take into account both the agents that are to be executed and the current state of the store. Therefore, a conﬁguration will be a point in the space P × C, where P is the space of agents and C is the space of all possible conﬁgurations of the constraint store. The instantaneous transition −→⊆ (P × C) × (P × C) and the stochastic transition =⇒⊆ (P × C) × [0, 1] × R+ × (P × C) are deﬁned according to the structural rules of Tables 2 and 3, respectively. The fact that instantaneous actions and recursive calls are guarded by stochastic actions guarantees that −→ can be applied only for a ﬁnite number of steps. −−−→ Moreover, it can be proven to be conﬂuent, see [8]. With the notation A, d of Table 3, we denote by the conﬁguration obtained by applying the transitions −→ as long as it is possible (i.e., by applying the transitive closure of −→). The −−−→ conﬂuence property of −→ implies that A, d is well deﬁned. The stochastic transition =⇒, instead, is labeled by two numbers: intuitively, the ﬁrst one is the probability of the transition, while the second one is its global rate. Note that, after performing one step of the transition =⇒, we apply the transitive closure of −→. This guarantees that all actions enabled after one =⇒ step are timed. Using relation =⇒, we can build a labeled transition system, whose nodes are conﬁgurations of the system and whose labeled edges correspond to derivable Table 2. Instantaneous transition for stochastic CCP

(IR1) tell∞ (c).A, d −→ A, d c (IR2)

p(x), d −→ A[x/y], d

if p(y) : −A

(IR3)

∃x A, d −→ A[y/x], d

with y fresh

(IR4)

A1 , d −→ A1 , d A1 A2 , d −→ A1 A2 , d

222

L. Bortolussi and A. Policriti

Table 3. Stochastic transition relation for stochastic CCP. The function rate : P ×C → R assigns to each agent its global rate. Its eﬀect is to recursively traverse the syntactic tree of agents, adding up the rates of active stochastic actions. Its formal deﬁnition can be found in [15].

(SR1)

−−−−−−→ tellλ (c).A, d =⇒(1,λ(d)) A, d c

(SR2)

−−−→ askλ (c).A, d =⇒(1,λ(d)) A, d

(SR3)

−−−−−→ M1 , d =⇒(p,λ) A1 , d −−−−−→ M1 + M2 , d =⇒(p ,λ ) A1 , d pλ with p = λ+rate(M and λ = λ + rate(M2 , d) 2 ,d)

(SR4)

−−−−−→ A1 , d =⇒(p,λ) A1 , d −−−−−−−−−→ A1 A2 , d =⇒(p ,λ ) A1 A2 , d pλ with p = λ+rate(A and λ = λ + rate(A2 , d) 2 ,d)

if d c

steps of =⇒. As a matter of fact, this is a multi-graph, as we can derive more than one transition connecting two nodes. Starting from this labeled graph, we can build a Continuous Time Markov Chain (cf. [39] and brlow) as follows: substitute each label (p, λ) with the real number pλ and add up the numbers labeling edges connecting the same nodes. Continuous Time Markov Chains. A Continuous Time Markov Chain (CTMC for short) [39] is a continuous-time stochastic process (Xt )t≥0 taking values in a discrete set of states S and satisfying the memoryless property: P {Xtn = sn | Xtn−1 = sn−1 , . . . , Xt1 = s1 } = P {Xtn = sn | Xtn−1 = sn−1 }, (1) for each n, t1 , . . . , tn , s1 , . . . , sn . A CTMC can be represented as a directed graph whose nodes correspond to the states of S and whose edges are labeled by real numbers, which are the rates of exponentially distributed random variables. In each state there are usually several exiting edges, competing in a race condition in such a way that the fastest one is executed. The time employed by each transition is drawn from the random variable associated to it. When the system changes state, it forgets its past activity and starts a new race condition (this is the memoryless property). Therefore, the traces of a CTMC are made by a sequence of states interleaved by variable time delays, needed to move from one state to another. The time evolution of a CTMC can be characterized equivalently by computing, in each state, the normalized rates of the exit transitions and their sum (called the exit rate). The next state is then chosen according to the probability distribution deﬁned by the normalized rates, while the time spent for

Dynamical Systems and Stochastic Programming

223

the transition is drawn from an exponentially distributed random variable with parameter equal to the exit rate. This second characterization is at the basis of several stochastic simulation algorithms for CTMC, like the well-known Gillespie’s one [26]. Stream Variables and Implementation. Some variables of the system, like those used in the deﬁnition of rate functions, need to store a single number that may vary over time. Such variables, for technical reasons, are conveniently modeled as variables of the constraint store, which, however, must be rigid (over time). To deal with this problem we store time varying parameters as growing lists with an unbounded tail variable. In order to avoid heavy symbolism, we will use a natural notation where X = X + 1 has the intended meaning of1 : “extract the last ground element n in the list X, consider its successor n + 1 and add it to the list (instantiating the old tail variable as a list containing the new ground element and a new tail variable)”. We refer to such variables as stream variables. An interpreter for the language is available and can be used for running simulations. This interpreter is written in Prolog and uses standard constraint solver on ﬁnite domains as manager for the constraint store. All simulations of sCCP shown in the paper are performed with it. 2.2

Modeling Biological Systems in sCCP

In [8,15] we argued that sCCP can be conveniently used for modeling biological systems. In fact, while maintaining the compositionality of process algebras, the presence of a customizable constraint store and of variable rates gives a great ﬂexibility to the modeler, so that diﬀerent kinds of biological systems can be easily described within this framework. In [15], we showed that biochemical reactions and genetic regulatory networks are easily handled by sCCP. In [8] we added to this list also formation of protein complexes and the process of folding of a protein, whose description requires knowledge about spatial position of amino acids constituting the protein (a kind of information easily added building on expressive potential of the constraint store). Finally, in [10] we showed how sCCP can be used to encode Kohn maps [34], a graphical formalism capable of describing implicitly biochemical networks subject to combinatorial explosion of the number of diﬀerent kinds of protein complexes. In this case, the power of the constraint store is used to maintain a graph-based representation of complexes, allowing a linear description of Kohn Maps (i.e., the encoding requires a linear number of characters w.r.t. the ones needed to describe a Kohn map). We recall now the modeling in sCCP of biochemical reactions. A general biochemical reaction has the form R1 + . . . + Rn →f (R,X;k) P1 + . . . + Pm , 1

(2)

The use of primed variables to denote values taken at the next time step is typical of model checking and is not to be confused with ﬁrst derivatives (for which we will used dotted variables, as time is the only independent variable).

224

L. Bortolussi and A. Policriti

where R1 , . . . , Rn are the reactants and P1 , . . . , Pm are the products. The realvalued kinetic function of the reaction is f (R, X; k), depending on the reactants R, on other molecules X acting as modiﬁers, and on some parameters k. This function can be one of the many used in biochemistry (cf. [20,50]) and it is required to satisfy the following boundary condition: it must be zero whenever one reactant is less than its amount consumed by the reaction. For instance, if a reactant R appears two times in the left hand size of (2), then f must be zero for R = 0, 1.2 Biochemical networks can be easily modeled in sCCP taking a reaction-centric approach, where each reaction (or action capability) is associated to a process, while molecules, whose concentration varies over time, are represented by integer variables of the constraint store (actually, stream variables). Moreover, the presence of non-constant rates allows to describe reactions with arbitrary chemical kinetics. More speciﬁcally, to each reaction like (2), we associate the following sCCP agent: f -reaction(R, X, P, k) : m n tellf (R,X;k) i=i (Ri − 1) ∧ j=i (Pj + 1) . f -reaction(R, X, P, k) This agent is a simple recursive loop, modifying the value of reactants’ and products’ variables at a speed given by the kinetic law. Note that the boundary conditions for the rate function f imply that no stream variable will ever become negative, as all reactions that may produce this eﬀect have rate zero3 . In Table 4 we give a list of some of the most common kinetics: mass action, MichaelisMenten and Hill kinetics. In order to describe genetic regulatory networks, instead, we use a modeling style mixing the reaction-centric point of view with the more classical molecularcentric one. Essentially, genes are described by sCCP agents, while proteins are associated to stream variables, like for biochemical reactions. An example of a genetic network can be found in Section 3.2. More information and examples on modeling biological systems in sCCP can be found in [15]. 2.3

Restricted sCCP

The mapping between sCCP and ODE’s is not deﬁned for the whole sCCP language, but rather for a restricted version of it, which is, however, suﬃcient to describe biochemical reaction and genetic networks. This restricted version of sCCP will be denoted in the following by restricted(sCCP ), and is formally speciﬁed by the following deﬁnition: 2

3

In case of mass action kinetics, this condition means that the rate for R + R → P must be kR(R − 1) and not kR2 . This is, however, consistent with the deﬁnition of the mass action principle in the stochastic setting. Boundary conditions for f may be relaxed by checking explicitly with ask instructions that variables stay within their domain. For instance, for the reaction R + R → P , we can precede tell by ask(R > 1) . This allows us to use the more common kR2 as rate function.

Dynamical Systems and Stochastic Programming

225

Table 4. List of some of the most common types of biochemical reaction, taken from [50]. The ﬁrst three are ﬁrst and second order mass-action-like reactions. The second arrow corresponds to a reaction with Michaelis-Menten kinetics. The last arrow replaces Michaelis-Menten kinetics with Hill’s one (see [20]).

R →k P1 + . . . + Pm

fma (R; k) = kR

R1 + R2 →k P1 + . . . + Pm

fma (R1 , R2 ; k) = kR1 R2

R + R →k P1 + . . . + Pm

fma (R; k) = kR(R − 1)

S →E k,v P

fM M (S, E; k, v) =

S →E K,V0 ,h P

fHill (S, E; h, k, v) =

vES k+S vES h k+S h

Definition 1. A restricted(sCCP ) program is a tuple (P rog, X, init(X)) satisfying: 1. P rog is an sCCP-program respecting the grammar deﬁned in Table 5. 2. The variables used in the deﬁnition of agents are taken from a ﬁnite set X = {X1 , . . . , Xn } of global stream-variables, each with the same domain D, usually D = N or, more generally, D = Z. 3. The only admissible updates for variables {X1 , . . . , Xn } are constraints of the form Xi = Xi + k or Xi = Xi − k, with k ∈ D constant. 4. Constraints that can be checked by ask instructions are ﬁnite conjunctions of linear equalities and inequalities. 5. The initial conﬁguration of the store is speciﬁed by the formula init(X), consisting in the following conjunction of constraints: (X1 = x01 )∧. . .∧(Xn = x0n ), with the constants x0i ∈ D referred to as the initial values of the sCCPprogram. This deﬁnition can be justiﬁed looking at the sCCP-agent associated to a biochemical reaction and also at the sCCP-model of genes considered in [15]. In fact, in these cases all employed variables are numerical variables of the stream-type4 , while all updates in the store add or subtract them a predeﬁned constant quantity. Guards, instead, usually check if some molecules are present in the system (X > 0), though we consider here the more general case of linear equalities and inequalities. The use of global variables only, instead, can be justiﬁed noting that the existential operator ∃x is never used (neither in the 4

We do not need further types of variables, as we just need to count the number of diﬀerent molecules in the system.

226

L. Bortolussi and A. Policriti Table 5. Syntax of the restricted version of sCCP P rog = Def.N Def = ε | Def.Def | p : −A π = tellλ (c) | askλ (c) M = π.G | M + M G = tell∞ (c).G | p | M A=0|M N =A|AN

reaction agent nor in gene models of [15]), as the scope of molecular interactions is system-wide. The suppression of the operator ∃x , as a side consequence, guarantees that we can avoid to pass parameters to procedure calls: in fact, each procedure can be deﬁned as operating on a speciﬁc subset of global variables. However, parameter passing is used in Section 2.2 to deﬁne parametrically the reaction agent. Therefore, we agree that each instance of a reaction agent, say f -reaction(R, X, P, k), is replaced with the corresponding ground form f -reaction(R,X,P,k). The same trick will be used for other agents. We demand further comments on the restrictions in Section 3.4. In order to ﬁx the notation in the rest of the paper, we give the following deﬁnition: Definition 2. A restricted(sCCP ) agent A not containing any occurrence of the parallel operator is called a sequential component or a sequential agent. A restricted(sCCP ) agent N is called an sCCP-network if it is the parallel composition of sequential agents. Inspecting the grammar of Table 5, we can observe that the initial conﬁguration of a restricted(sCCP ) program is indeed an sCCP-network. The following property is straightforward: Lemma 1. The number of sequential components forming an sCCP-program (P rog, X, init(X)) remains constant at run-time and equals the number of sequential agents in the sCCP-network of the initial conﬁguration. Proof. As sequential components do not contain any parallel operator, no new agents can be forked at run-time.5 In the rest of the paper, for notational convenience, we usually identify an sCCPprogram with the corresponding sCCP-network. Moreover, forbidding the deﬁnition of local variables implies the following property: Lemma 2. The number of variables involved in the evolution of an sCCPnetwork6 is a subset of {X1 , . . . , Xn }, hence ﬁnite. 5 6

We are counting also deadlocked agents. A variable is involved in the evolution of the network if one of the following things happen: it is updated in a tell instruction, it is part of a guard checked in an ask instruction, or it is used in the deﬁnition of a rate function.

Dynamical Systems and Stochastic Programming

227

The restrictions of restricted(sCCP ) are in the spirit of those introduced in [31]: we are forbidding an inﬁnite unfolding of agents and we are considering global interactions only, forcing the speed of each action to depend on the whole state of the system. Indeed, also in [16] we ﬁnd similar restrictions, though the comparison with sCCP is subtler. First of all, the version of π-calculus presented in [16] does not allow the use of the restriction operator, meaning that interactions have a global scope. However, agents in the π-calculus of [16] are not sequential, as each process is associated to a single molecule and the production of new molecules is essentially achieved by forking processes at run-time. This is not necessary in sCCP, as sCCP-agents model reactions, while molecules are identiﬁed by variables of the system. What is ﬁnite in [16], however, is the number of syntactically diﬀerent agents that can be present in a system. On the Restrictions of the Language. restricted(sCCP ) limits the full language in three main aspects: the allowance of sequential agents only, the suppression of local variables and the simpliﬁcations on the constraint store, cf. Deﬁnition 1. We will, however, comment here only on the ﬁrst one, as the other two are forced by the translation to ODEs, hence their discussion will be postponed to Section 3.4. As observed, in restricted(sCCP ) we constrain all the agents to be sequential, i.e. no occurrence of the parallel operator is allowed. Essentially, sequential agents are automata cooperating together, a property that will be exploited in the next section to represent them graphically in a simple way. Indeed, this restriction is only apparent: we can always convert a non-sequential agent into a network of sequential ones using additional (stream) variables of the constraint store. The idea is simply that of identifying all the syntactically diﬀerent terms that are stochastic choices, associating a new variable to each of them. These variables are used to count the number of copies of each term that are in parallel. Each agents is modiﬁed consequently: all agents will only call recursively themselves, while the variations induced in the number of terms by transitions are dealt with by updating the new state variables. Finally, rates are corrected by multiplying them by the multiplicity variable associated to the agent executing the corresponding transition. This is justiﬁed by the fact that in Markovian models, the global rate of a set of actions is computed by adding all basic rates together—ultimately, a consequence of the properties of the exponential distribution [39]. For instance, consider the agents x and y, deﬁned by x :- tell1 (true).(y

y), y :- tell1 (true).x + tell1 (true).0. They can be made sequential by introducing two variables, X and Y , counting the number of copies of x and y respectively and by replacing x by x :- askX (X > 0).tell∞ (X = X − 1 ∧ Y = Y + 2).x and y by y :- askY (Y > 0).tell∞ (X = X + 1 ∧ Y = Y − 1).y + askY (Y > 0).tell∞ (Y = Y − 1).y .

3

From sCCP to ODE

In this section we deﬁne a translation method associating a set of ordinary differential equations to an sCCP program. This translation applies precisely to

228

L. Bortolussi and A. Policriti

restricted(sCCP ), as deﬁned above. The procedure is organized in several simple steps, illustrated in the following paragraphs. Essentially, we ﬁrst associate a ﬁnite graph to each sequential component of an sCCP network and then, analyzing the graph, we deﬁne an interaction matrix similar to the one deﬁned in [31] or to action matrices of (stochastic) Petri nets (see, for instance, [27]). Writing ODE’s from this matrix is then almost straightforward. After deﬁning this translation, in Section 3.1 we investigate how it relates to biochemical kinetics and we show that the ODE’s associated to an sCCP-model of a set of biochemical reactions are the ones generally considered in standard biochemical praxis [17,20]. Some considerations on dynamical properties are then put forward in Section 3.2, while in Section 3.3 the focus is moved on the concept of behavioral equivalence. Finally, in Section 3.4 we reconsider the restrictions applied to sCCP in the light of the described transformation procedure. Step 1: Reduced Transition Systems The ﬁrst step consists in associating a labeled graph, called reduced transition system [8], to each sequential agent composing the network. As a working example, we consider the following simple sCCP agent: RWX :ask1 (X > 0).tell∞ (X = X − 1).RWX + tell1 (X = X + 2).RWX + askf (X) (true).( ask1 (X > 1).tell∞ (X = X − 2).RWX + tell1 (X = X + 1).RWX f (X) =

1 X 2 +1

This agent performs a sort of random walk in one variable, increasing or decreasing its value by 1 or 2 units, depending on its inner state. Inspecting Table 5, where the syntax of restricted(sCCP ) is summarized, we observe that each branch of a stochastic choice starts with a stochastic timed instruction, i.e. an askλ (c) or a tellλ (c), followed by zero or more tell∞(c), followed by a procedure call or by another stochastic choice. The ﬁrst operation that we need to perform, in order to simplify the structure of agents, is that of collapsing each timed instruction with all the instantaneous tell instructions following it and replacing everything with one “action” of the form action(c, d, λ), where c is a guard that must be entailed by the store for the branch to be entered, d is the constraint that will be posted to the store, and λ is the stochastic rate of the branch, i.e. a function λ : C → R+ ∪ {∞}. The presence of ∞ among possible values of λ is needed to simplify the treatment of instantaneous tells. To achieve this goal, we formally proceed by deﬁning a conversion function, named compact, by structural induction on terms. The result of this function

Dynamical Systems and Stochastic Programming

229

Table 6. Syntax of compact(sCCP ) ˆ .Aˆ P rog = Def ˆ ˆ ˆ ˆ Def = ε | Def .Def | p : −A ˆ ˆ ˆ ˆ π ˆ = action(g, c, λ) M =π ˆ ;G | M ⊕ M ˆ ˆ ˆ ˆ G = p | M A = 0 | M ˆ =A ˆ|A ˆN ˆ N

is that of transforming an agent written in restricted(sCCP ) into an agent of a simpler language, called compact(sCCP ), where ask and tell are replaced by the instruction action. In order to distinguish between the two languages, we denote stochastic summation in compact(sCCP ) by “⊕”, sequential composition by “;”, and we surround procedure calls and nil agent occurrences by double square brackets “·”. The syntax of compact(sCCP ) is formally deﬁned in Table 6; its constraint store, instead, follows the same prescriptions of Deﬁnition 1. In deﬁning the function compact, we use a concatenation operator to merge instantaneous tells with the preceding stochastic action. Formally, compact is deﬁned as follows: Definition 3. The function compact: restricted(sCCP) → compact (sCCP) is deﬁned by structural induction through the following rules: 1. 2. 3. 4. 5. 6.

compact(0) = 0; compact(p) = p. compact(askλ (c).G) = action(c, true, λ) compact(G). compact(tellλ (c).G) = action(true, c, λ) compact(G). compact(tell∞ (c).G) = action(true, c, ∞) compact(G). compact(M + M ) =compact(M )⊕compact(M ).

where ∞ : C → R+ ∪ {∞} is deﬁned by ∞(c) = ∞, for all c ∈ C. We now deﬁne the concatenation operator : Definition 4. The operator is deﬁned by: 1. action(g, c, λ) p = action(g, c, λ);p. ˆ = action(g, c, λ);M ˆ. 2. action(g, c, λ) M 3. action(g1 , c1 , λ1 ) action(g2 , c2 , λ2 ) = action(g1 ∧ g2 , c1 ∧ c2 , min(λ1 , λ2 )). where min(λ1 , λ2 ) : C → R+ ∪{∞} is deﬁned by min(λ1 , λ2 )(c)=min{λ1 (c), λ2 (c)}. Going back to the agent RWX previously deﬁned; if we apply the function compact to it, we obtain the following agent: compact(RWX ) :action(X > 0, true, 1) action(true, X = X − 1, ∞) RWX ⊕ action(true, X = X + 2, 1) RWX

230

L. Bortolussi and A. Policriti

⊕ action(true, true, f (X)) ( action(X > 1, true, 1) action(true, X = X − 2, ∞) RWX ⊕ action(true, X = X + 1, 1) RWX ) After the removal of operator according to the rules in Deﬁnition (4), the agent compact(RWX ) becomes a compact(sCCP) agent: compact(RWX ) = RWX :action(X > 0, X = X − 1, 1) ; RWX ⊕ action(true, X = X + 2, 1) ; RWX ⊕ action(true, true, f (X)) ; ( action(X > 1, X = X − 2, 1) ; RWX ⊕ action(true, X = X + 1, 1) ; RWX ) The above example shows clearly that the function compact simply collapses all the actions performed on the store after one execution of the stochastic transition, as deﬁned in [7,8]. It is a simple exercise to deﬁne a stochastic transition relation for compact(sCCP ) similar to the one for restricted(sCCP ) and to successively prove the strong equivalence [30] between agents A and compact(A).7 Hence, from a semantic point of view, the application of function compact is safe, as stated in the following Lemma 3. For each sequential agent A of restricted(sCCP ), A and Aˆ = compact(A) are strongly equivalent. Let Aˆ =compact(A) be an agent of compact(sCCP ). We want to associate a graph to such agent, containing all possible actions that Aˆ may execute. Nodes ˆ i.e. to diﬀerent in such graph will correspond to diﬀerent internal states of A, stochastic branching points. Edges, on the other hand, will be associated to actions: each edge will correspond to one action(g, c, λ) instruction and will be labeled consequently by the triple (g, c, λ). To deﬁne such graph, we proceed in two simple steps: 1. First we deﬁne an equivalence relation ≡c over the set of compact(sCCP ) agents, granting associativity and commutativity to ⊕ and reducing procedure calls to automatic “macro-like” substitutions (a reasonable move as we do not pass any parameter). We will then work on the set A of compact(sCCP ) agents modulo ≡c , called the set of states; notably, all agents in A are stochastic summations. 2. Then, we deﬁne a structural operational semantics [42] on compact(sCCP ), whose labeled transition system (LTS) will be exactly the target graph. Definition 5. The equivalence relation ≡c between compact(sCCP ) agents is deﬁned as the minimal relation closed with respect to the following three rules: 7

Strong equivalence [30] is a form of bisimulation preserving probabilities: two agents are strongly equivalent if their exit rates are the same and transitions of one agent can be matched by transitions of the other having the same probability.

Dynamical Systems and Stochastic Programming

1. 2. 3.

231

ˆ1 ⊕ M ˆ 2 ≡c M ˆ2 ⊕ M ˆ1 ; M ˆ ˆ ˆ ˆ2 ) ⊕ M ˆ3 ; M1 ⊕ (M2 ⊕ M3 ) ≡c (Mˆ1 ⊕ M ˆ ˆ ˆ p ≡c A if p : −A belongs to the declarations D.

The space of compact(sCCP ) agents modulo ≡c is denoted by A, and is referred to as the space of states. Definition 6. The transition relation ⊆ A × (C × C × RC ) × A is deﬁned in the SOS style as the minimal relation closed with respect to the following rule: ˆ⊕M ˆ action(g, c, λ);G

(g,c,λ)

ˆ G.

The transition relation encodes the possible actions that a compact(sCCP ) agent can undertake. Notice that procedure calls are automatically solved as we are working modulo ≡c . The relation induces a labeled graph, its labeled transition system (LTS), whose nodes are agents in A and whose edges are labeled by triples (g, c, λ), where g ∈ C is a guard, c ∈ C is the update of the store, and λ the functional rate of the edge. Definition 7. Let Aˆ be an agent of compact(sCCP ); the portion of the labeled ˆ transition system reachable from the state Aˆ is denoted by LT S(A). ˆ is Theorem 1. For any agent Aˆ of compact(sCCP ) (modulo ≡c ), LT S(A) ﬁnite. Proof. The agents reachable from Aˆ are subagents of Aˆ or subagents of Aˆ , where ˆ The number of p : −Aˆ is a procedure called by an agent reachable from A. subagents of Aˆ (modulo ≡c ) corresponds to the number of summations present ˆ and it is ﬁnite for any deﬁnable agent. The proposition follows because in A, ˆ there is only a ﬁnite number of agents deﬁned in the declarations D. We are ﬁnally ready to deﬁne the reduced transition system for an agent A of restricted(sCCP). Definition 8. Let A be an agent of restricted(sCCP). Its reduced transition system RT S(A) is a ﬁnite labeled multigraph (S(A), T (A), A ) deﬁned by RT S(A) = LT S(compact(A)). Given RT S(A) = (S(A), T (A), A ), S(A) ⊆ A is the set of RTS-states reachable from agent compact(A), ﬁnite for Theorem 1, T (A) is the set of RTS-edges or RTS-transitions and A : T (A) → C × C × RC is the label function assigning to each RTS-edge the triple (g, c, λ), g, c ∈ C, λ : C → R+ . In order to eﬀectively compute the RTS of an agent Aˆ of compact(sCCP ), we can proceed as follows: 1. Given an compact(sCCP ) program, write the syntactic tree of the agent ˆ . Aˆ and of all the agents Aˆ such that p:-Aˆ is in Def

232

L. Bortolussi and A. Policriti

2. Nodes corresponding to action(g, c, λ) instructions will have one single incoming edge and one single outgoing edge. Remove them, connecting the entering and exiting edges and labeling them by the triple c, g, λ). 3. Leaves of the obtained labeled tree correspond either to nil agents or to procedure calls. The latter are replaced according to the following rule: if the syntactic tree of the procedure p has not been added in the current tree, replace the leaf labeled with p with the corresponding syntactic tree; otherwise, remove the leaf and redirect the incoming edge to the root of the copy of the syntactic tree of p. Iterate the application of the rule until no more leaves corresponding to procedure calls are available. The previous procedure always terminates, as the number of diﬀerent procedures ˆ is ﬁnite, hence the algorithm needs to process only a ﬁnite number p in Def of leaves. Going back to our running example, RT S(RWX ) = LT S(compact(RWX )) is shown in the ﬁgure below. Note that it has one RTS-edge for every action that can be performed by compact(RWX ), and just two RTS-states, corresponding to the two summations present in compact(RWX ). Three intermediate steps in the construction of the RTS can be visualized in Figure 1.

Step 2: The Interaction Matrix Our next step consists in encoding all the information about the dynamics in a single interaction matrix and in a rate vector. Consider the initial sCCP-network N = A1 . . . Ah of a restricted(sCCP ) program (P rog, X, init(X)), with sequential components A1 , . . . , Ah . First of all, we construct the reduced transition system for all the components, i.e. RT S(A1 ) = (S(A1 ), T (A1 ), A1 ), . . . , RT S(Ah ) = (S(Ah ), T (Ah ), Ah ). Then we construct the set of RTS-states and RTS-transitions of the network (we agree that states and transitions belonging to diﬀerent components A1 , . . . , Ah are distinct8 ), putting: S(N ) = S(A1 ) ∪ . . . ∪ S(Ah )

(3)

T (N ) = T (A1 ) ∪ . . . ∪ T (Ah ).9

(4)

and 8 9

If the same component is present in multiple copies, we distinguish among them by suitable labels. The labeling function acting on T (N ) will be denoted consistently by N .

Dynamical Systems and Stochastic Programming

233

Fig. 1. Three intermediate steps of the construction of RTS for the agent RW(X), according to the procedure sketched in the main body of the paper. The top one is the outcome of point 1, the middle one is the labeled tree obtained after step 2, while the bottom one is the result of applying twice the rule of step 3.

Suppose now that there are m RTS-states in S(N ) and k RTS-transitions in T (N ). We conveniently ﬁx an ordering of these two sets, say S(N )={σ1 , . . . , σm } and T (N ) = {t1 , . . . , tk }. The variables Y of the diﬀerential equations are of two diﬀerent kinds, Y = X ∪ P. The ﬁrst type corresponds to the global stream variables of the store, i.e X = {X1 , . . . , Xn } (see Deﬁnition 1). In addition, we associate a variable of the second type Pσi = Pi to each RTS-state of S(N ) = {σ1 , . . . , σm }, so P = {P1 , . . . , Pm }. For the manipulations to follow, we assume the existence

234

L. Bortolussi and A. Policriti

of a lexicographic ordering among all variables, so that vectors and matrices depending on this ordering are deﬁned uniquely and manipulated consistently. Moreover, variables will be also used to index of these objects. Consider now an RTS-transition tj ∈ T (N ), connecting RTS-states σj1 and σj2 , and suppose N (tj ) = (gj , cj , λj ). We introduce the following notation: – rateN j (X) = λj (X) is the rate function of tj ; – guardN j (X) is the indicator function of gj (by Deﬁnition 1, gj is a conjunction of linear equalities and inequalities), i.e. 1 if gj is true for X, N guardj (X) = (5) 0 otherwise. We are now able to deﬁne the rate vector : Definition 9. The rate vector rN for transitions T (N ) = {t1 , . . . , tk } is a kdimensional vector of functions, whose components rjN are deﬁned by N rjN (Y) = rateN j (X) · guardj (X) · Pj1 ,

(6)

where Pj1 is the variable associated to the source state σj1 of transition tj . Consider again a transition tj ∈ T (N ), going from σj1 to σj2 and with label N (tj ) = (gj , cj , λj ), and consider the updates cj , a conjunction of constraints of the form Xi = Xi ± k, according to Deﬁnition 1. We can assume that each variable Xi appears in at most one conjunct of cj .10 We are now ready to deﬁne the interaction matrix. N Definition 10. The interaction matrix IY for an sCPP-network N with respect to variables Y is an integer-valued matrix with n+ m rows (one for each variable of Y) and k columns (one for each RTS-transition T (N )), deﬁned by: N N 1. If σj1 = σj2 , then IY [Pj1 , tj ] = −1 and IY [Pj2 , tj ] = 1. 2. If Xh = Xh ± k is a conjunct of cj , then IYN [Xh , tj ] = ±k. 3. All entries not set by points 1,2 above are equal to zero. RWX For the agent RWX , the interaction matrix IY Y = {X, P0 , P1 } is: ⎛ ⎞ −1 +2 0 −2 +1 (X) RWX IY = ⎝ 0 0 −1 +1 +1 ⎠ (P0 ) 0 0 +1 −1 −1 (P1 )

for the variables

Similarly, the rate vector rRWX is T rRWX = P0 X > 0, P0 , f (X)P0 , P1 X > 1, P1 , ,

(7)

(8)

where · denotes the logical value of a formula (i.e., 1 if the formula is true, 0 otherwise). 10

If, for instance, both Xi = Xi + k1 and Xi = Xi + k2 are in cj . Then we can replace these two constraints with Xi = Xi + (k1 + k2 ).

Dynamical Systems and Stochastic Programming

235

Step 3: Writing ODE’s Once we have the interaction matrix, writing the set of ODE’s is very simple: N we just have to multiply matrix IY by the (column) rate vector rN , in order to N obtain the vector odeY : N N odeN (9) Y = IY · r . Each row of the odeN Y vector gives the diﬀerential equation for the corresponding variable. Speciﬁcally, the equation for variable Yi is Y˙i = odeN Y [Yi ] =

k

N IY [Yi , j] · rjN (Y)

j=1

=

k

N IY [Yi , j] · guardj (X) · ratej (X) · Pj1

j=1

For instance, the set of ODE’s associated to the agent RWX is ⎧ ˙ ⎨ X = P0 (2 − X > 0) + P1 (1 − 2X > 1) P˙0 = − X 21+1 P0 + P1 (1 + X > 1) ⎩ ˙ P1 = X 21+1 P0 + P1 (1 + X > 1) In order to solve a set of ODE’s, we need to ﬁx the initial conditions. The variables Y = X ∪ P of odeN Y are of two distinct types: P, denoting states of the reduced transition systems of the components, and X, representing stream variables of the store. The initial conditions for P are easily determined: we set to one all the variables corresponding to the initial states of RTS of each component, and to 0 all the others. Regarding X, instead, initial conditions are given in the formula init(X) of Deﬁnition 1, specifying the values assigned to stream variables before starting the execution of the sCCP program. Elimination of Redundant State Variables Consider an sCCP component A whose reduced transition system RT S(A) has just one RTS-state. Then, the odeN Y vector of an sCCP-network N having A as one of its components will contain a variable corresponding to this RTS-state, say Pi , with equation P˙i = 0 and initial value Pi (0) = 1. Clearly, such variable is redundant, and we can safely remove it by setting Pi ≡ 1 in all equations containing it and by eliminating its equation from the odeN Y vector. From now on, we assume that this simpliﬁcation has always been carried out. As an example, consider the following agent A :- tellf1 (X) (X = X + 1).A + tellf2 (X) (X = X − 1).A

236

L. Bortolussi and A. Policriti

Its RTS contains just one state, corresponding to the only summation present in it, with associated variable P . As the other variable of the agent is X, the vector odeA {X,P } contains two equations, namely

X˙ P˙

=

f1 (X)P − f2 (X)P 0

.

The simpliﬁcation introduced above just prescribes to remove the equation for P , setting its value to 1 in the other equations; therefore we obtain odeA {X} = (f1 (X) − f2 (X)) . Notice that the set of variables Y is updated consistently, i.e. removing the canceled variables from it. We summarize the whole method just presented deﬁning the following operator. Definition 11. Let N be the sCCP-network of an restricted(sCCP ) program. With ODE(N ) we denote the vector odeN Y associated to N by the translation procedure previously deﬁned, after applying the removal of state variables coming from network components with just one RTS-state. Compositionality of the Transformation Operator In order to clearly state formal properties of the transformation, we need a version of the ODE(N ) indicating explicitly the variables X for which the differential equations are given. In the following, the variables for the equations ODE(N ) are indicated by V AR(ODE(N )). Definition 12. Let N be the sCCP-network of an restricted(sCCP ) program and let Y = V AR(ODE(N )) and ODE(N ) = odeN Y . The ordinary diﬀerential equations of N with respect to the set of variables X, denoted by ODE(N, X), is deﬁned as odeN Y [Yj ] ifXi = Yj ∈ Y, ODE(N, X)[Xi ] = 0 otherwise. The operations performed on ODE(N ) by ODE(N, X) simply consist in the elimination of the equations of ODE(N ) for the variables not in X, and in the addition of equations X˙ = 0 for all variables X in X but not in Y. We can also N associate a new interaction matrix IX to ODE(N, X), whose rows are derived according to Deﬁnition 12. As the set of RTS-transitions T (N ) is unaltered by N ODE(N, X), the equation ODE(N, X) = IX · rN continues to hold. We can now prove the following theorem, stating compositionality of ODE operator.

Dynamical Systems and Stochastic Programming

237

Theorem 2. Let N1 , N2 be two sCCP-networks, and let N = N1 N2 be their parallel composition. If Y1 = V AR(ODE(N1 )), Y2 = V AR(ODE(N2 )), and Y = Y1 ∪ Y2 , then11 ODE(N1 N2 , Y) = ODE(N1 , Y) + ODE(N2 , Y). Proof. The components in N1 N2 are the components of N1 plus the components of N2 . Therefore, the set of RTS-transitions (i.e. edges in the RTS of the components) T (N1 N2 ) of N1 N2 is equal to T (N1 ) ∪ T (N2 ). As each N N column of IY 1 2 is either a transition of T (N1 ) or a transition of T (N2 ), it N N Nh clearly holds IY 1 2 [Yi , tj ] = IY [Yi , tj ] if tj ∈ T (Nh ), h = 1, 2. The following chain of equalities then follows easily from the deﬁnitions: N N N N ODE(N1 N2 , Y)[Yi ] = IY 1 2 [Yi , tj ]rj 1 2 tj ∈T (N1 N2 )

=

N1 IY [Yi , tj ]rjN1 +

tj ∈T (N1 )

N2 IY [Yi , tj ]rjN2

tj ∈T (N2 )

= ODE(N1 , Y) + ODE(N2 , Y). 3.1

Preservation of Rate Semantics

In Section 2.2 we discussed how biochemical reaction with general kinetic laws can be modeled in sCCP. Given a list of reactions, the standard praxis in computational chemistry is that of building a corresponding diﬀerential (set of ODE’s) or stochastic (CTMC) model. The deﬁnition of such models is canonical, and it is fully speciﬁed by the reaction list, see [53] for further details. sCCP models of biochemical reactions are generated by associating an agent, call it biochemical agent, to each reaction in the list. These agents are rather simple: they can execute in one single way, namely an inﬁnite loop consisting of activation steps, where the agents compete stochastically for execution, with rate given by the kinetic law speciﬁed in the reaction arrow, and update steps, in which the store is modiﬁed according to the prescriptions of the reaction. This bijective mapping between reactions and sCCP biochemical agents soon implies that the stochastic model for sCCP is identical to the continuous-time Markov chain generated in classical stochastic simulations with Gillespie algorithm [26], for instance like those obtainable with a program like Dizzy [46,19].12 A diﬀerent question is wether the ODE’s that are associated to an sCCP model of biochemical reactions coincide with the canonical ones. In the rest of the section, we show that this is indeed the case. Following the approach by Cardelli in [17], we can then say that the translation from sCCP to ODE’s preserves the rate semantics. The sense of this sentence is better visualized in Figure 2, graphically depicting the correspondence between 11 12

Here “+” denotes the usual sum of vectors. Stochastic simulations with Michaelis-Menten and Hill rate functions have been considered, for instance, in [47].

238

L. Bortolussi and A. Policriti

Fig. 2. Diagram of relations intervening between stochastic and ODE-based semantics of chemical reactions and sCCP agents

stochastic and diﬀerential models of biochemical reactions and of the derived sCCP agents. Preservation of rate semantics essentially means that the arrows in the diagram commute. As a matter of fact, in [17] the author deals only with mass action kinetics, due to the intrinsic properties of π-calculus (all deﬁnable rates are mass-action like). In our setting, instead, functional rates and the constraint store allow us to deal with arbitrary chemical kinetics, including also Michaelis-Menten and Hill ones (cf. [20]). In the following, we formally prove the equivalence of ODE’s obtained from sCCP agents with the corresponding classical ones. In general, the stochastic and the deterministic rate of a reaction are not the same, because ODE’s variables measure concentration, while sCCP variables count the number of molecules. Therefore, in passing from one model to the other, we need to convert numbers to concentrations, dividing by the volume V times the Avogadro number NA (γ = V NA will be referred as system size). Rates need also to be scaled consistently, see [53] for further details. In the rest of this section, however, we get rid of scaling problems simply by assuming γ = 1. In any case, system size can be reintroduced without diﬃculties, by change the scale of rates and variables after the derivation of ODE’s. We now put forward some notation, in order to specify how to formally derive a set of ODE’s given a set of reactions R = {ρ1 , . . . , ρk }, where each ρi denotes a single reaction. Each reaction ρ has some attributes: a multiset of reactants (species can have a speciﬁc multiplicity), denoted by REACT (ρ), a multiset of products, P ROD(ρ), and a real-valued rate function, RAT E(ρ), depending on the variables associated to the molecules involved in the reaction, V AR(ρ). This last function, V AR, can be easily extended to sets of reactions by letting V AR({ρ1 , . . . , ρk }) = V AR(ρ1 ) ∪ . . . ∪ V AR(ρk ). Now let R be a set of reactions and X = V AR(R). The (canonical) diﬀerential equations associated to R w.r.t. variables X, denoted by ODE(R, X)13 are deﬁned for each variable Xi as X˙ i = ODE(R, X)[Xi ], where:14 13 14

We overload here the symbol introduced in Deﬁnitions 11 and 12; however, the two cases can be easily distinguished looking at their ﬁrst argument. We conveniently identify each variable with the molecule it represents.

Dynamical Systems and Stochastic Programming ODE(R, X)[Xi ] =

RAT E(ρ) −

ρ∈R: Xi ∈ P ROD(ρ)

239

RAT E(ρ). (10)

ρ∈R: Xi ∈ REACT (ρ)

If Xi is not involved in any reaction of R, then ODE(R, X)[Xi ] = 0. We observe that the correct stoichiometry is automatically dealt with by the fact that we are using multisets to list reactants and products. Restricting this construction to a ﬁxed variable ordering allows to state the following straightforward compositionality lemma. Lemma 4. Let R = R1 ∪ R2 be a partition of R and let X1 = V AR(R1 ), X2 = V AR(R2 ), and X = X1 ∪ X2 . Then ODE(R1 ∪ R2 , X) = ODE(R1 , X) + ODE(R2 , X). We turn now to formally deﬁne the encoding of reactions into sCCP agents. For each reaction ρ, its sCCP agent SCCP (ρ) is constructed according to Section 2.2. Operator SCCP is extended compositionally to sets of reactions R = {ρ1 , . . . , ρk } by letting SCCP (R) = SCCP (ρ1 ) . . . SCCP (ρk ). We are ﬁnally ready to state the theorem of preservation of rate semantics: Theorem 3 (Preservation of rate semantics). Let R be a set of biochemical reactions, with X = V AR(R). Then ODE(R, X) = ODE(SCCP (R), X)

(11)

Proof. We prove the theorem by induction on the size k of the set of reactions R. For the base case k = 1, consider a reaction ρ R1 + . . . + Rn →f (R,X;k) P1 + . . . + Pm and its associated sCCP agent SCCP (ρ) ⎛ ⎞ n m SCCP (ρ) :- tellf (R,X;k) ⎝ (Ri − 1) ∧ (Pj + 1)⎠ .SCCP (ρ). i=i

j=i

Clearly, the reduced transition system of such agent is

SCCP (ρ)

Let Y = V AR(ρ), then the interaction matrix IY has |Y| rows and 1 column, with entries corresponding to the stoichiometry of the reaction: SCCP (ρ) IY [Yi ] = 1− 1. Yi ∈P ROD(ρ)

Yi ∈REACT (ρ)

240

L. Bortolussi and A. Policriti

The rate vector rSCCP (ρ) , instead, is the scalar f (R, X; k). Hence, the equation for variable Yi is Y˙ i =

f (R, X; k) −

Yi ∈P ROD(ρ)

f (R, X; k),

Yi ∈REACT (ρ)

which is equal to equation (10). The inductive case follows easily from compositionality properties of ODE operators. Suppose the theorem holds for lists up to k − 1 reactions, and let R = R0 ∪ {ρ} be a set of k chemical reactions (hence |R0 | = k − 1). Then ODE(SCCP (R), X) = ODE(SCCP (R0 ∪ {ρ}), X) = ODE(SCCP (R0 ) SCCP (ρ), X) = ODE(SCCP (R0 ), X) + ODE(SCCP (ρ), X) = ODE(R0 , X) + ODE(ρ, X) = ODE(R, X), where the second equality follows from the deﬁnition of SCCP , the third follows from Theorem 2, the fourth is implied by the induction hypothesis on SCCP (R0 ) and by the base case proof on SCCP (ρ), while the last is a consequence of Lemma 4.

3.2

Preservation of Dynamic Behavior

In Theorem 3 we proved that the ODE map, when applied to sCCP-models of biochemical networks, satisﬁes a condition of coherence: it preserves the kinetic principles used in the construction of the model (i.e., the rate semantics). A diﬀerent question is whether an sCCP-network N (evolving stochastically according to the prescriptions of its semantic) shows a dynamic behavior equivalent to the one exhibited by the equations ODE(N ). This problem is the sCCPcounterpart of the famous mathematical issue concerning the relation between stochastic and diﬀerential models [35,36], studied deeply also in the context of biochemical reactions [26,23]. It is well-known that stochastic and diﬀerential models of biochemical reactions are behaviorally equivalent only in some cases. These results are signiﬁcant also for sCCP. Theorem 3, in fact, states that, when biochemical reactions are concerned, the stochastic process underlying the sCCP-models and the associated ODE’s are exactly the classical ones. Therefore, in the mapping from sCCP to ODE’s we have the same phenomenology as in the classical case. However, the logical structure of sCCP-agents makes the problem of behavioral preservation subtler. In the following, we discuss this problem with diﬀerent examples, especially of situations in which an sCCP-network and the corresponding ODE’s show a diﬀerent behavior. In particular, we are interested in sketching a brief, and plausibly incomplete, classiﬁcation of the causes of behavioral divergence.

Dynamical Systems and Stochastic Programming

241

An important issue is the concept of behavioral equivalence itself, which is diﬃcult to formalize, as already discussed in the introduction. We will return on this problem in the next section. Oregonator. The Oregonator is a chemical systems showing an oscillatory behavior, devised by Field and Noyes [40] as a simpliﬁed version of the BelousovZhabotinsky oscillator15. Essentially, Oregonator is composed of three chemical substances, call them A, B, C, subject to the following reactions: B →k1 A + B →k2 A →k3 2A →k4 C →k5

A ∅ 2A + C ∅ B

(12)

Actually, other chemical substances are involved, but they are kept constant in the experiment. The diﬀerential equations associated to (12) are known to possess a stable limit cycle for a wide range of parameter’s values [28], containing an unstable equilibrium. The limit cycling behavior is clearly visible in Figure 3(a), where the numerical solution of Oregonator’s ODE’s is shown. In Figure 3(b), instead, we plot a stochastic simulation of the sCCP model associated to (12) according to prescriptions of Section 2.2. In this case, the stochastic model shows the same pattern as the diﬀerential one. Theorem 3 guarantees that the graph in Figure 3 depicts the numerical solution of ODE’s associated to the sCCP program by the transformation previously deﬁned. In this case, the behavior is preserved. We remark two things regarding Oregonator. First, the size of each molecular species is of the order of thousands, hence the relative variation induced by one reaction in the stochastic model is small. Under this condition, stochastic and deterministic models of biochemical reactions usually coincide [26]. Another property of the Oregonator that can be important for behavioral preservation is that the limit cycle is an attractor in the phase space: nearby trajectories asymptotically converge to it (see [51]). This means that a relatively small perturbation is not willing to change the overall dynamics: stochastic ﬂuctuations have a negligible eﬀect. Things are diﬀerent if we start from the unstable equilibrium of the system. The numerical solution of ODE’s shows a constant evolution (Figure 4(a)), while the stochastic simulation (Figure 4(b)) essentially evolves as the limit cycle of Figure 3. In fact, stochastic ﬂuctuations, in this case, make the sCCP system move away from the instable equilibrium into the basin of attraction of the limit cycle. This shows another well known fact: stochastic and diﬀerential models usually diﬀer near instabilities [26]. Lotka-Volterra system. The Lotka-Volterra system is a famous simple model of population dynamics, see for example [26] and references therein. There are 15

This chemical system is called “Oregonator” because its inventors where working at the University of Oregon.

242

L. Bortolussi and A. Policriti

(a) ODE model of Oregonator

(b) sCCP model of Oregonator Fig. 3. 3(a): Numerical simulation of the diﬀerential equation model of the Oregonator, with parameters determined according to the method presented in [26]. Speciﬁcally, let As = 500, Bs = 1000 and Cs = 2000 be an equilibrium of the system of equations, and let R1 = 2000, R2 = 50000. Then parameters are equal to k1 = R1 /Bs = 2, k2 = R2 /(As Bs ) = 0.1, k3 = (R1 + R2 )/As = 104, k4 = ((2R1 )/(A2s ))/2 = 4e−7 , and k5 = (R1 + R2 )/Cs = 26. The starting point is A0 = As /2, B0 = Bs /2, C0 = Cs /2. The system soon approaches an attractive limit cycle. 3(b): Stochastic simulation with Gillespie’s method of the sCCP network associated to reactions (12). Parameters and initial conditions are those speciﬁed above. The eﬀect of stochastic ﬂuctuations is negligible, and the plot essentially coincide with its deterministic counterpart.

two species: preys and predators. Preys eat some natural resource, supposed unbounded, and reproduce at a rate depending only on their number. Predators, instead, can reproduce only if they eat preys, otherwise they die. To keep the model simple, we admit predation as the only source of prey’s death. The previous hypotheses can be summarized in the following set of reactions, where E refer to preys and C to predators: E →kb 2E C →kd ∅ E + C →kp 2C

(13)

If we consider the standard mass action ODE’s (they coincide with the equations derived from the sCCP model due to Theorem 3), a typical solution shows

Dynamical Systems and Stochastic Programming

243

(a) ODE model of Oregonator from unsta- (b) sCCP model of Oregonator from unble equilibrium stable equilibrium Fig. 4. 4(a): Numerical simulations of ODE’s derived from reactions (12), with parameters given in caption of Figure 3, starting from an unstable equilibrium of the system. 4(b): Stochastic simulation of sCCP model associated to reactions (12), with the same parameters and initial conditions than the diﬀerential counterpart. As we can see, stochastic ﬂuctuations drive the system away from the unstable equilibrium, so that its surrounding limit cycle is approached.

oscillations in which high values of preys and predators alternate. An example of such a solution is given in Figure 5(a). Inspecting equations, it can be shown that the point Es = kd /kp , Cs = kb /kp is an equilibrium of a rather special kind: it is stable (trajectories starting nearby it stay close) but not asymptotically stable (trajectories starting nearby do not converge to it as time approaches inﬁnity). This behavior is easily understood looking at the phase space (Figure 5(b)), in which we can see that trajectories form closed orbits around the equilibrium, whose amplitude increases with distance from equilibrium. More details can be found, for instance, in [51]. What kind of behavior can we expect from the stochastic evolution of the sCCP model for (13)? Stochastic ﬂuctuations will make the system jump from one trajectory to nearby ones, without any force pulling it towards the equilibrium. Therefore, ﬂuctuations can, in the long run, make the system wander in the phase plane, eventually reaching a borderline trajectory (corresponding to E or C axis in the phase plane). Whenever this happens, then both preys and predators go extinct (C-axis trajectory), or just predators do, while preys go to inﬁnity (E-axis trajectory). This intuition is conﬁrmed in Figure 6, where we compare the ODE solution starting from equilibrium (dotted lines), and a trace of the sCCP model, starting from the same initial conﬁguration. As we can see, the stochastic system starts oscillating until both species go extinct. This is another well known case in which stochastic and diﬀerential dynamics diﬀer, again induced by properties of the phase space [26]. A negatively auto-regulated system. The eﬀect of stochastic ﬂuctuations is mostly remarkable in biological phenomena where gene expression is involved. This is because the transcription of a gene is usually a slower process than protein-protein interaction, and often the number of mRNA strands for a given

244

L. Bortolussi and A. Policriti

(a) ODE solution of Lotka-Volterra model

(b) Phase space of Lotka-Volterra model

Fig. 5. 5(a): Numerical solution of ODE’s associated to reactions (13), with parameters kb = 1, kp = 0.1, kd = 0.1 and initial conditions E0 = 4 and C0 = 10. 5(b): phase portrait of the Lotka Volterra system, for the same value of parameters as above. As we can see, all the solutions show an oscillating behaviour. The system has an (instable) equilibrium for E = 1, C = 10, at the center of the circles.

Fig. 6. Eﬀect of stochastic ﬂuctuations for the Lotka-Volterra system. The dotted lines are an equilibrium solution for the ODE model (parameters are as in caption of Figure 5). A stochastic trace of the sCCP model is drawn with solid lines: both species ﬂuctuate around the equilibrium values until they both get extinct.

gene present in the cell is very small, of the order of some units. As the production of one single mRNA is a rare event (compared to other cellular events), stochastic variability in its happening can induce behaviors diﬃcult to capture if mRNA is approximated with its concentration. Stochasticity in gene expression is indeed a phenomenon that has received a lot of attention, see for instance [38,6]. We present here a simple, artiﬁcial example taken from [53] and depicted in Figure 7. The biological network shown represents a simple autoregulatory mechanism in gene expression of a procaryotic cell. Gene g produces, via mRNA r, a

Dynamical Systems and Stochastic Programming

245

Fig. 7. Diagram of a simple self-regulated gene network. Gene g produces mRNA r and, from it, protein P . P can dimerise and its dimer can bind to a promoter region of gene g, downstream of RNA polymerase binding site. The P2 -binding blocks polymerase activity, thus inhibiting gene expression.

protein P that, as a dimer, can bind to a promoter region of gene g, preventing RNA-polymerase activity and thus inhibiting its own production. Following the approach of [5], genes can be modeled as logical gates having a ﬁxed output (the produced mRNA or protein), and several inputs, corresponding to diﬀerent proteins of the system, exerting a positive or negative regulatory function. A gene gate with one inhibitory input is called in [5] neg gate, and can be modeled in restricted(sCCP) simply as: neg gateP,I :tellkp (P = P + 1).neg gateP,I + askkb ·I (I ≥ 1).askku (true).neg gateP,I , where kp is the basic production rate, kb is the binding rate of the repressor to the promoter region of the gene and ku is its unbinding rate. In order to model the system of Figure 7 we can combine one neg gate with some reactions. This is an example of the modeling style mixing the reactioncentric and the molecular-centric point of view, see Section 2.2. The model is the following: neg gater,P2 r →kt r + P P →kdim1 P2 (14) P2 →kdim2 P r →kd1 ∅ P →kd2 ∅ In Figure 8 we compare a stochastic simulation of the sCCP model of reactions (14) with the numerical solution of the associated ODE’s. As we can readily see, the two plots are completely diﬀerent. In particular, in the stochastic simulation, P2 is produced in short bursts; normally it is slowly degraded. The bursts correspond to mRNA production events, shown in Figure 8(a) as blue peaks. The

246

L. Bortolussi and A. Policriti

(a) sCCP model of system (14)

(b) ODE model of system (14)

Fig. 8. 8(a): Simulation of the sCCP model of (14). The red line corresponds to P2 , while the blue line shows the evolution of r, multiplied for a factor 100 (for visualization purposes). Note that the increases in P2 expression immediately follow mRNA production events. Parameters of the models are the following: kp = 0.01, kb = 1, ku = 10, kt = 10, kdim1 = 1, kdim2 = 1, kd1 = 0.1, and kd2 = 0.01. All molecules are set initially equal to 0. 8(b): Numerical simulation of ODE’s associated to the sCCP model of (14), for the same parameters just given. The evolution of P2 is tamer than in the stochastic counterpart, as it converges quickly to an asymptotic value.

ODE’s system, however, presents a much simpler pattern of evolution, in which the quantity of P2 converges to an asymptotic value. This divergence is caused by the fact that, approximating continuously the number of RNA molecules, we lose the discrete information that seems to characterize its dynamics, i.e. the fact that mRNA can be present in one unit of completely absent from the system. Staten otherwise, continuously approximating molecular species present in low quantities may lead to errors inducing a completely divergent observable behavior. Repressilator. The Repressilator [21] is an artiﬁcial biochemical clock composed of three genes expressing three diﬀerent proteins, tetR, λcI, LacI, exerting a regulatory function on each other’s gene expression. In particular, protein tetR represses the expression of protein λcI, protein λcI represses the gene producing protein LacI, and, ﬁnally, protein LacI is a repressor for protein tetR. The expected behavior is an oscillation of the concentrations of the three proteins. A simple stochastic model of Repressilator can be found in [5], where the authors describe it with three neg gates (see the previous paragraph) cyclically connected, in such a way that the product of one gate inhibits the successive gene gate in the cycle. In addition, they introduce degradation mechanisms for the three repressors. More formally, the model is the following neg gateA,C neg gateB,A neg gateC,B A →kd ∅ B →kd ∅ C →kd ∅

(15)

Dynamical Systems and Stochastic Programming

247

In Figure 9(a) we show a trace of the stochastic model generated by a simulator of sCCP based on Gillespie algorithm. The oscillatory behavior is manifest. If we apply the translation procedure discussed in Section 3 to this particular model, we obtain the ODE’s shown in Table 7, while their numerical integration is shown in Figure 9(b). As we can readily see there is no oscillation at all, but rather the three proteins converge to an asymptotic value, after an initial adjustment. Inspecting the ODE’s, we note the presence of six variables (YA , YB , YC and ZA , ZB , ZC ) in addition to those representing the quantity of repressors in the system (A, B, C). Such variables correspond to states of genes gates, and they are used to model the change of conﬁguration of the gates, from active to repressed and vice versa. This scenario seems rather unjustiﬁed here: there is no argument to support the introduction of these variables, especially because we are continuously approximating boolean quantities. An interesting point regarding Repressilator is the relation between the solution of the ODE’s and the average trace of the stochastic system (i.e. E[X(t)], returning the average value of system variables as a function of time). In fact, we may expect that the behavior preserved by the diﬀerential equations is the average dynamics of the stochastic system, rather than that shown by one of its traces. Interestingly, also the average value of the Repressilator model does not oscillate, as can be seen from Figure 10. This can be explained by noticing that the oscillations’ period in the stochastic model is not constant, but it varies considerably. Hence, for every instant (when the Markov chain is at the stationary regime), we will observe one of the proteins at its peak value approximatively only in one third of the traces. Hence its average value will tend to stabilize at one third of the peak value, as conﬁrmed by Figure 10. In fact, when we average Repressilator, we measure the fraction of traces in which a certain gene is

(a) sCCP model of repressilator

(b) sCCP model of repressilator

Fig. 9. 9(a): Stochastic time trace for the Repressilator system of described by reactions 15. Parameters are kp = 1, kd = 0.01, kb = 1, ku = 0.01. 9(b): Solution of the diﬀerential equations of Table 7, automatically derived from sCCP program associated to reactions 15. Parameters are the same as in stochastic simulation. The stochastic simulation lasts longer than the ODE one in order to better underline its oscillatory behavior.

248

L. Bortolussi and A. Policriti

Table 7. ODE’s derived for the Repressilator, generated by the method of Section 3.1 A˙ = kp YA − kd A B˙ = kp YB − kd B C˙ = kp YC − kd C

Y˙ 1 = ku ZA − kb YA C Y˙ 2 = ku ZB − kb YB A Y˙ 3 = ku ZC − kb YC B

Z˙ 1 = kb YA C − ku ZA Z˙ 2 = kb YB A − ku ZB Z˙ 3 = kb YC B − ku ZC

Fig. 10. Average value of the sCCP model for Repressilator, computed using model checker PRISM [37]. See [8] for further details.

active and the fraction of traces in which it is inactive, for every time instant. In this way, however, we lose any information regarding the sequence of gene gate’s state changing. The diﬀerent behavior existing between a trace of a stochastic system and its average trace suggests that the switching dynamics of genes can be the driving force behind oscillations. This implies that another source of nonequivalence between sCCP models and the associated ODE’s can appear due to the representation of RTS-states with continuous RTS-state variables. Indeed, this example suggested us to preserve part of the discrete dynamics, mapping the sCCP Repressilator into an hybrid automaton. The work put forward in [?] shows that this move is enough to maintain oscillations. The translation to hybrid automata opens an entire range of possibilities to combine discreteness and continuity. These will be investigated in detail in the planned second part of this paper. Sources of non-equivalence. In the previous examples we outlined diﬀerent cases in which an sCCP model and its associated ODE’s fail to be equivalent from a dynamical viewpoint. We remark that most of these examples are well known, as they have been studied in detail in theoretical and applicative contexts, like biochemical reactions [26,53] and our main interest here is in their connection with the sCCP translation machinery. For sake of clarity, we summarize the diﬀerent sources of non-equivalence. 1. In some cases, non-equivalence is a direct consequence of properties of the phase space. For instance, instable trajectories are destroyed by small ﬂuctuations, like the equilibrium trajectory of the Oregonator. Also stable but

Dynamical Systems and Stochastic Programming

249

not asymptotically stable trajectories can be troublesome, as stochastic ﬂuctuations are not counterbalanced by any attracting force, and so they can bring the stochastic system far away from the initial trajectory. This is the case of the Lotka-Volterra system. 2. Another well-known problem is related to the approximation by continuous quantities of integer variables having small (absolute) values. In this case, in fact, the eﬀect of a single stochastic ﬂuctuation has a relative magnitude that is relevant, so the dynamics can change quite dramatically. A typical example appearing in Biology is related to the transcription of genes, as shown in the simple example of a self-regulated gene. 3. A ﬁnal source of non-equivalence is, instead, characteristic of the translation procedure deﬁned for sCCP. In fact, in this case we represent each RTSstate of a component of the system with a continuous variable, which can take values in the real interval [0, 1]. RTS-states represent, in some sense, logical structures that control the activity of the system, while a change of state is an event triggered by some condition of the system. Moreover, in each sCCP trace, each component can be in only one state, hence RTS-state variables are boolean quantities. Continuous approximation, in this case, can have dramatic consequences, as the example of Repressilator seems to suggest.

3.3

Behavioral Equivalence

Comparing the dynamical evolution of a deterministic and a stochastic system is a delicate issue, because stochastic processes have a noisy evolution, hence we need to remove noise from their traces, before attempting any comparison with time traces evolving deterministically. In the previous discussion, in fact, we appealed to the concept of “behavioral equivalence” always in a vague sense, essentially leaving to the reader the task of visually comparing plots and recognizing similarities and diﬀerences. Clearly, a mathematical deﬁnition is needed in order to prove theorems and automatize comparisons. We ﬁrst consider the comparison of traces generated by ODE’s with the average trace of the stochastic system, taken as the representative of its whole ensemble of traces. In practice, for each time instant t we need to compute the average value E(X(t)) of each stream variable X w.r.t. the probability distribution on states of the system at time t. This probability can be obtained as the solution of the Chapman-Kolmogorov forward equation [39], a system of diﬀerential equations of the size of the state space. This equation, known in biochemical literature as the chemical master equation [25], can rarely be solved analytically, and it is also very diﬃcult to integrate numerically [26]. A more eﬃcient approach to compute an estimate of the average consists in generating several (thousands of) stochastic traces and in computing pointwise their sample mean. Alternatively, the average value of one or more variables can be computed for a small sample of time points {t1 , . . . , tk } using numerical techniques, as those implemented in the model checker PRISM [37].

250

L. Bortolussi and A. Policriti

Whatever the method chosen, the computation (even approximate) of the average trace of a stochastic system is a diﬃcult matter. Whenever such trace is known, we can compare it with the trace of the ODE’s, generated using standard numerical techniques [43], using quantitative measures (essentially computing a distance between the two curves). Indeed, in [9] it is shown that the ODE associated to a sCCP program is a ﬁrst order approximation of the true equation for the average. However, the average trace of a stochastic system is not necessarily a good representative of its evolution. A paradigmatic example is the Repressilator, whose average trace (sampled with PRISM, see caption of Figure 10) converges to an asymptotic value, while all its stochastic traces show persistent oscillations. Hence, even when averaging a stochastic system, we may lose the characterizing qualitative features of its dynamics. The example of Repressilator suggests that the notion of behavioral equivalence is probably better captured in a qualitative setting. Qualitative comparison requires a formal deﬁnition of the features of dynamical evolution, like oscillations, convergence to a stable value, and so on. A possibility we suggest in this direction is to describe these features as logical formulae of a suitable logical language L, for instance temporal logic, as done in Simpathica [3]. The concepts below are just sketched; this subject is currently under investigation and we will deal with it in detail in future works. Let Φ denote the set of formulae describing all dynamical features of interest. Associating a Kripke structure K1 to the trace of an ODE and another structure K2 to a stochastic trace, then we may declare these traces equivalent whenever their Kripke structures satisfy the same subset of formulae of Φ (possibly restricting the attention to formulae of degree ≤ n). Below we give three examples of temporal logic formulas expressing inﬁnite oscillations: G(Z = zm → F (Z = zM )) ∧ G(Z = zM → F (Z = zm )) ∧ G(zm Z zM ) ∧ F (Z = zm ); G(Z = zm → X(Z > zm U Z = zM )) ∧ G(Z = zM → X(Z < zM U Z = zm )) ∧ G(zm Z zM ) ∧ F (Z = zm ); dZ dZ ¬G >0 ∧ ¬G dZ = 0 ∧ ¬G < 0 . dt dt dt In the above formulas X stands for next, G stands for always (globally), F stands for sometimes (in the future), U stands for until, zm and zM are minimum and maximum values, and the thirds formula uses propositional formulas taking values according to the sign of the ﬁrst derivative.16 This idea seems promising, as it gives a considerable freedom in the deﬁnition of formulae Φ, hence allowing to privilege some aspects of dynamical evolution 16

In order to use meaningfully the notion of “next” for ODE’s we need to consider a discretization of the time and of the state space, such as that performed in [3].

Dynamical Systems and Stochastic Programming

251

more than others. However, the real problem is in the deﬁnition of a reasonable Kripke structure for a stochastic trace (and for sets of traces). In fact, Kripke structures for ODE’s can be constructed starting from one or more traces, as done in Simpathica [3], in the following way: the bounded (product) domain of all variables is divided in small, compact regions; a state of the Kripke automaton consists of one of such regions; edges connect two states if a trajectory crosses the corresponding regions consecutively. This construction, however, is not reasonable for stochastic traces, as noise would force the addition of many edges that may introduce spurious behaviors. Of course, it is possible to model check directly on CTMC formulae written in CSL [4,37]. However, the complexity of this latter approach makes the deﬁnition of non-deterministic Kripke structures interesting also for stochastic traces. We are currently investigating this direction, considering the introduction of a bounded form of memory to tame noise. 3.4

More on the Restrictions of the Language

restricted(sCCP ) restricts the full language in several aspects, see Section 2.3. Actually, these restrictions have been introduced in order to deﬁne in a reasonably simple way the mapping to ODE’s. We discuss them in detail in the following. First, all agents must be sequential, i.e. not containing any occurrence of the parallel operator. As already remarked at the end of Section 2.3, this does not constitute a real limitation, as each non-sequential agent can be transformed into a network of sequential ones. Here we note that the same trick of Section 2.3 can be used to transform each sCCP-network into an equivalent network where each sequential agent has an RTS with one single state; indeed, this is done implicitly by the transformation to ODE’s itself. However, writing programs in this form is less natural. Another syntactic restriction regards the deﬁnition of local variables. Actually, variables in ODE’s have a global scope. Of course, any local variable can be made global by suitably renaming it. There is a problem, however, concerning the fact that at run-time we may generate an unbounded number of local variables. This implies that their use may lead to a set of ODE’s with an inﬁnite number of variables (although each equation will depend only on a ﬁnite number of them). The uprising of an inﬁnite number of variables requires more complex mathematical techniques, and it prevents the use of standard numerical solvers. Finally, the third class of restrictions regards the constraint store. The restriction to numeric variables is obviously necessary, as we are mapping to ODE’s. The restriction on the admissible constraints for the updating of variables, on the other hand, is related to the fact that each update in a sCCP program needs to be considered as a ﬂux acting on some variables. Indeed, even a simple update like X = 0 is diﬃcult to render within ODE’s framework, as it is inherently discrete. A possible way out is to mix the continuous ingredient of ODE’s with discreteness, mapping sCCP programs to hybrid automata [29]. Within this

252

L. Bortolussi and A. Policriti

formalism, updates like X = 0 are perfectly admissible: they are resets associated to discrete transitions.

4

From ODE’s to sCCP

In this section we deﬁne a transformation SCCP , associating an sCCP network to a generic set of ordinary diﬀerential equations, analyzing both its mathematical properties and the relation with the the map ODE deﬁned in the previous section. Before entering into the mathematical details, we need to make a preliminary remark. Essentially, the main obstacle we have to face in deﬁning the map SCCP is the fact that ODE’s are an aggregate description of a system. To be more precise, if a system can be described by a set of ﬂuxes acting on the diﬀerent entities into play (i.e. on the system variables), then the ODE’s hide part of the logical structure of such ﬂuxes by combining them into the equations. To clarify the concept, consider the following two sCCP agents: A :- (tell1 (X = X + 1) + tell1 (Y = Y + 1)).A B :- (tell1 (X = X + 1 ∧ Y = Y + 1)).B When we apply the ODE operator to the networks N1 = A and N2 = B, we obtain, in both cases, the following equations: X˙ 1 = 1 Y˙ Therefore, two diﬀerent sCCP agents can be mapped into the same set of ODE’s. Note that A and B are “semantically” diﬀerent, as they induce two diﬀerent CTMC. The chain associated to A has edges connecting a state (i, j) to (i + 1, j) and (i, j + 1) (hence the exit rate from (i, j) is 2), while the chain of B has transitions only from (i, j) to (i + 1, j + 1) (with exit rate 1). This information pertains the logical structure of the system, which is manifest in the sCCP program, but irremediably lost in the associated ODE’s. An even worse situation happens for the following agent, implementing a onedimensional random walk [39]: C :- (tell1 (X = X + 1) + tell1 (X = X − 1)).C The equation associated to C by ODE is X˙ = 0, as the production and degradation rate cancel out when summed together. This equation predicts a constant evolution for X, thus failing to capture its erratic behavior. Note, however, that the average value of X is constant also in the stochastic model for C. Therefore, the structural information lost in passing from sCCP agents to ODE’s makes impossible to recover the original sCCP network; stated otherwise, the map ODE(·) is not injective. Indeed, the lack of injectivity of ODE(·) means

Dynamical Systems and Stochastic Programming

253

that an sCCP program is more informative than a set of ODE’s: it deﬁnes not only the ﬂuxes, but also their logical relation. The previous discussion suggests that the map SCCP must be deﬁned with care. Given an ODE set x˙ = f (x), there are many sCCP networks that can be associated to it, i.e. at least all those belonging to the set ODE −1 (f (x)). In order to choose one among them, additional discriminating information is required, essentially related to the structure of the ﬂuxes, hence to the logic of the system modeled. As suggested by the previous discussion, we will therefore deﬁne not a single SCCP map, but rather a class of maps, parametric w.r.t. the additional information required to sort out the logical structure of ﬂuxes. We will then show that, independently of this additional information, the transformation scheme satisﬁes properties guaranteeing a form of coherence w.r.t. the ODE mapping and also a form of behavioral equivalence. Finally, we will provide two instantiations of such scheme, assuming speciﬁc conditions on the system modeled. 4.1

The Translation to sCCP

In the conversion from ODE’s to sCCP, we approximate continuous quantities by discrete variables. Therefore, this mapping will depend on an additional parameter, the step δ, specifying the granularity of the approximation of continuous variables. The magnitude of δ has a strong impact on the preservation of dynamical behavior; this point will be the content of Section 4.3. Consider a system of ﬁrst order ODE’s with n variables x = (x1 , . . . , xn ): x˙ = f (x). We will now deﬁne the notion of set of covering functions, which captures the idea of external knowledge required to solve the ambiguity about the logical structure inherent in the ODE’s. Essentially, a set of covering functions corresponds to a plausible choice of a set of ﬂuxes, generating the given ODE’s. Definition 13. A set of covering functions G for the ODE x˙ = f (x) is a set of pairs {(gi , hi ) | i = 1, . . . , k}, such that each gi is a function gi : Rn → R, each hi is a vector of Zn , and, for each x ∈ Rn , k

hi gi (x) = f (x).

i=1

Example 1. Consider the following simple system of ODE’s with two variables: x˙ = a − by (16) y˙ = c + dx One possible covering set is the following: G = {(a, (1, 0)), (by, (−1, 0)), (c, (0, 1)), (dx, (0, 1))},

254

L. Bortolussi and A. Policriti

which corresponds to the choice of disentangling all addends of the equations. Another possibility, instead, is the following: G = {(−a + by, (−1, 1)), (a − by, (0, 1, )), (c + dx, (0, 1, ))}, as easily veriﬁed. In the sCCP program associated to the ODE’s x˙ = f (x), we approximate the continuous variables x with discrete stream variables X. Deﬁnition 1, however, requires variables X to have integer values. In order to set the size of the basic increment to an arbitrary step δ, we can change variables, setting x = δX and expressing f with respect to X (in this way, a unit increment of Xi corresponds to an increment of δ of xi ). The equation for X thus becomes ˙ = 1 f (δX) = F(X; δ). X δ If we are given a set of covering functions G for x˙ = f (x), we can apply the same variable’s substitution to each gi , obtaining new covering functions Gi (X; δ) = 1 δ gi (δX) such that ˙ = F(X; δ) = 1 f (δX) = 1 X gi (δX) = Gi (X; δ). δ δ i=1 i=1 k

k

The translation to sCCP simply proceeds associating an agent to each element of the set of covering functions G: Definition 14. Let x˙ = f (x) be a set of ODE’s, and G be a set of covering functions for it. Let gi ∈ G and δ ∈ R+ . The agent manGi ,δ is deﬁned as17 manGi ,δ :- ask|Gi (X;δ)| (Gi (X; δ) > 0). tell∞ (X = X + hi ).manGi ,δ + ask|Gi (X;δ)| (Gi (X; δ) < 0). tell∞ (X = X − hi ).manGi ,δ

The agent manGi ,δ is a summation with two branches: both have rate equal to the modulus of function Gi , but one is active when Gi > 0, and it increments the value of X according to the vector hi , while the other is active when Gi < 0, decrementing X by hi . In order to construct the sCCP network associated to a set of ODE x˙ = f (x), we simply need to deﬁne an agent manGi ,δ for each function Gi of the covering set G, putting these agents in parallel. We can render this procedure in the following SCCP operator: Definition 15. Let x˙ = f (x) be a set of ODE’s, G be a set of covering functions for f , and δ ∈ R, δ > 0. The sCCP-network associated to f (x), with respect to the set of covering functions G and the increment’s step δ, indicated by SCCP (f (x), G, δ), is SCCP (f (x), G, δ) = manG1 ,δ . . . manGk ,δ , with x = δX. 17

The name “man” stands for manager.

(17)

Dynamical Systems and Stochastic Programming

255

The initial conditions of the sCCP program, given by init(X), are X(0) = 1δ x0 , where x0 are the initial conditions of the ODE’s. Functional rates of sCCP are central in the deﬁnition of this translation: each function of the covering set becomes a rate in a branch of an sCCP summation. This is made possible only due to the freedom in the deﬁnition of rates, because diﬀerential equations and covering functions considered here are general. The possibility of having general rates in sCCP is intimately connected with the presence of the constraint store, which contains information external to the agents. This means that part of the description of interactions can be moved from the logical structure of agents to the functional form of rates. Common stochastic process algebras like stochastic π-calculus [44] or PEPA [30], on the other hand, have simple numerical rates and they rely just on the structure of agents (and on additivity of the exponential distribution [39]) to compute the global rate. This restricts severely the class of functional rates that they can model. Indeed, in a recent work [18] Hillston introduces general rates in PEPA essentially through the addition of information external to the model, an approach similar in spirit to sCCP. 4.2

Invertibility

We turn now to study the relation between the two translations deﬁned, i.e. ODE and SCCP . Speciﬁcally, we will show that (ODE ◦ SCCP ) returns the original diﬀerential equations, independently from the covering set G. The other direction, instead, cannot hold, as pointed out at the beginning of this section. In fact, we have seen that several sCCP agents can be mapped by ODE to the same equations, hence the map ODE cannot be inverted. Theorem 4. Let x˙ = f (x) be a set of diﬀerential equations, with x=(x1 , . . . , xn ) and X = 1δ x, and let G be a set of covering functions for f . Then ODE(SCCP (f (x), G, δ) , X) = f (x). Proof. From Deﬁnition 15 we know that SCCP (f (x), δ) = manG1 ,δ . . .

manGn ,δ , and by Theorem 2, ODE (manG1 ,δ . . . manGn ,δ , X)=ODE(manG1 ,δ , X)+. . .+ODE(manGn ,δ , X). Now, the agent manGi ,δ can modify several variables Xi , according to the vector hi coupled with the function Gi . The RTS of manGi ,δ is easily seen to have the following form

256

L. Bortolussi and A. Policriti

Therefore, the interaction matrix associated to manGi ,δ has just two columns, corresponding to the vectors hi and −hi , and ODE(manGi ,δ , X) is equal to hi |Gi (X; δ)| Gi (X; δ) > 0 − hi |Gi (X; δ)| Gi (X; δ) < 0 . In the previous equation, · denotes, as in Section 3, the logical value of a formula. The previous equation can be simpliﬁed by noting that |Gi (X; δ)| Gi (X; δ) > 0 − |Gi (X; δ)| Gi (X; δ) < 0 = Gi (X; δ), hence ODE(manGi ,δ , X) = hi Gi (X; δ). By applying Theorem 2 we then obtain ODE(SCCP (f (x), G, δ) , X) =

k

hi Gi (X; δ) = F (X),

i=1

which is equal to x˙ = f (x) when changing the variables back to x. 4.3

Behavioral Equivalence

We start this section by presenting an example showing how the translation from ODE’s to sCCP works. In particular we will be concerned with the behavior exhibited by both systems and with the dependence on the step size δ, governing the size of the basic increment or decrement of variables. Intuitively, δ controls the “precision” of the sCCP agents w.r.t. the original ODE’s. Hence, varying the size of δ, we can calibrate the eﬀect of the stochastic ﬂuctuations, reducing or increasing it. This is evident in the following example, where we compare solutions of ODE’s and the simulation of the corresponding sCCP processes. Let’s consider the following system of equations, representing another model of the Repressilator (see Section 3.2), a synthetic genetic network having an oscillatory behavior (see [21,3]): 0.5 x˙ 1 = α1 x−1 3 − β1 x1 , −1 x˙ 2 = α2 x1 − β2 x0.5 2 , 0.5 x˙ 3 = α3 x−1 2 − β3 x3 ,

α1 = 0.2, α2 = 0.2, α3 = 0.2,

β1 = 0.01 β2 = 0.01 β3 = 0.01.

(18)

We ﬁx the following set G of covering functions: g1 = α1 x−1 3 , h1 = (1, 0, 0), g2 = −1 0.5 α2 x−1 , h = (0, 1, 0), g = α x , h = (0, 0, 1), g = β 2 3 3 2 3 4 1 x1 , h4 = (−1, 0, 0), 1 0.5 0.5 g5 = β2 x2 , h5 = (0, −1, 0), g6 = β3 x3 , h6 = (0, 0, −1). The corresponding sCCP process, after changing variables according to Xi = xδi , is: manG1 ,δ manG2 ,δ manG3 ,δ manG4 ,δ manG5 ,δ manG6 ,δ , where, for instance, the agent manG1 ,δ is

(19)

Dynamical Systems and Stochastic Programming

257

manG1 ,δ :- ask| α1 (δX3 )−1 | ( αδ1 (δX3 )−1 > 0).tell∞ (X1 = X1 + 1).manG1 ,δ δ + ask| α1 (δX3 )−1 | ( αδ1 (δX3 )−1 < 0).tell∞ (X1 = X1 − 1).manG1 ,δ δ

In Figure 11, we study the dependence on δ of the sCCP network obtained from equations (19). From the plots, we note that the smaller the δ, the closer the stochastic trace is to the solution of ODE’s. However, increasing δ, the eﬀect of stochastic perturbations gets stronger and stronger, making the system change dynamics radically. Reducing the value of δ seems to be essentially the same as working with a sufﬁciently high number of molecules in standard biochemical networks, see [26,24] and the discussion in Section 3.2. It is thus reasonable to expect that, by taking δ smaller and smaller, the deterministic and the stochastic dynamics will eventually coincide. In fact, reducing δ we are diminishing the magnitude of stochastic ﬂuctuations, hence their perturbation eﬀects.

(a) Solution of ODE’s (19)

(c) SCCP simulation, δ = 0.01

(b) SCCP simulation, δ = 0.001

(d) SCCP simulation, δ = 1

Fig. 11. Diﬀerent simulations of sCCP agent obtained from S-Systems equations of repressilator (19), as basic step δ varies. Speciﬁcally, in Figure 11(a) we show the solution of ODE’s (19), while in Figures 11(b), 11(c), 11(d) we present three simulations of the sCCP agent corresponding to ODE’s (19), for δ = 0.001, 0.01, 1 respectively. In the last diagram, the behavior of S-System’s equations is destroyed. Note that in Figure 11(c) the time axis is stretched by a factor of 100, while in Figure 11(b) the time axis is stretched by a factor of 1000, consistently with the rescaling of variables by 1δ performed in the translation from ODE to sCCP.

258

L. Bortolussi and A. Policriti

This conjecture is indeed true: in the rest of the section we prove that, under mild conditions on the ODE’s and of the functions of G, the trajectories of the stochastic simulation converge to the solution of the ODE’s, independently of the choice of the covering set G. In fact, the set of stochastic traces whose distance from the solution of the ODE’s is greater than a ﬁxed arbitrary constant has zero probability in the limit δ → 0. Kurtz theorem. In 1970 Thomas Kurtz proved a theorem giving conditions for a family of density dependent Continuous Time Markov Chains to converge to a solution of a system of ODE’s [35,36]. In fact, under mild assumptions on the smoothness of functions into play, the trajectories of the CTMC remain, in the limit, close to the solution of a particular set of ODE’s with probability one. Our mapping SCCP easily ﬁts into Kurtz’s framework, with the step δ playing the role of the density. We start by recalling the Kurtz’s theorem. Let V be a positive parameter, playing the role of the “size” of the system, and XV (t) be a family of CTMC with state space Zn , depending on the parameter V . Suppose that there exist a continuous positive real function ϕ : Rn × Zn → R, such that the inﬁnitesimal generator matrix [39] Q = (qX,Y ) for XV (t) is given by 1 qX,X+h = V ϕ( X, h), h = 0. V In addition, let Φ(x) = h∈Zn hϕ(x, h). Theorem 5 (Kurtz [35]). Fix a bounded time interval [0, T ]. Suppose there exists an open set E ⊆ Rn and a constant ME ∈ R+ such that 1. |Φ(x) − Φ(y)| < ME |x − y|, ∀x, y ∈ E (i.e. Φ satisﬁes the Lipschitz condition); 2. supx∈E h∈Zn |h|ϕ(x, h) < ∞; 3. limd→∞ supx∈E |h|>d |h|ϕ(x, h) = 0. Then, for every trajectory x(t) that is a solution of x˙ = Φ(x) satisfying x(0) = x0 and x(t) ∈ E, t ∈ [0, T ], if lim

V →∞

1 XV (0) = x0 , V

then for every ε > 0, 1 lim P sup XV (t) − x(t) > ε = 0. V →∞ t≤T V This theorem states that the trajectories of XV (t) converge, in a bounded time interval, to the solution of x˙ = Φ(x), when V → ∞. The function Φ is essentially the sum of all ﬂuxes of the system.

Dynamical Systems and Stochastic Programming

259

Convergence for SCCP Our framework can be easily adapted to ﬁt this theorem. Consider a system of ODE’s x˙ = f (x) and a set of covering functions G. Denote by Xδ (t) the CTMC associated to the sCCP-network SCCP (f (x), G, δ). Theorem 6. Let x˙ = f (x) be a system of ODE’s, with x ∈ Rn , and [0, T ] a bounded time interval. Let G = {(gi , hi ) | i = 1 . . . , μ} be a set of covering functions for f . If there exists an open set E ⊆ Rn such that f satisﬁes the Lipschitz condition in E and supx∈E |gi (x)| < ∞, for each i = 1 . . . , μ, then for every ε > 0 lim P sup |δXδ (t) − x(t)| > ε = 0,

δ→0

t≤T

where x(t) is the solution of x˙ = f (x) with initial condition x(0) = x0 and δXδ (0) = x0 . Proof. In order to prove the theorem, we simply need to show that we satisfy all the hypothesis of the Kurtz’s theorem. First of all, in this setting the density V is equal to 1δ , so that 1δ → ∞ when δ → 0. Consider now the function gi (x), and deﬁne as customary gi+ (x) = gi (x)gi (x) ≥ 0 and gi− (x) = gi (x)gi (x) ≤ 0, so that gi (x) = gi+ (x) − gi− (x) and |gi (x)| = gi+ (x) + gi− (x), where · denotes the logical value as before. δ Consider now the inﬁnitesimal generator matrix Qδ = (qX,Y ) of the CTMC n Xδ (t). It is straightforward to prove that, for each h ∈ Z , δ qX,X+h =

1 δ

gi+ (δX) +

i | hi =h

1 δ

gi− (δX),

i | hi =−h

where the sum must be intended equal to zero if the index set is empty. Clearly, these are density dependent rates, with density 1δ . Note that there is a ﬁnite δ number of vectors for which qX,X+h is diﬀerent from zero, as the set {h1 , . . . , hμ } is ﬁnite. Therefore, the function ϕ of the Kurtz theorem is simply deﬁned as ϕ(x, h) =

i | hi =h

gi+ (x) +

gi− (x).

i | hi =−h

Then, the function Φ(x) is Φ(x) =

h

hϕ(x, h) =

μ j=1

hj (gj+ (x) − gj− (x)) =

μ j=1

hj gj (x) = f (x).

260

L. Bortolussi and A. Policriti

It only remains to prove that conditions 1–3 of Theorem 5 are satisﬁed. Condition 1 is obvious because Φ = f is Lipschitz by hypothesis, while condition 3 hold because |h| > M implies ϕ(x, h) = 0, where M = max1≤i≤μ |hi |. Finally, condition 2 follows because h

|h|ϕ(x, h) ≤ M

μ μ (gj+ (x) + gj− (x)) = M |gj (x)|, j=1

j=1

hence sup

x∈E

h

|h|ϕ(x, h) ≤ sup M x∈E

μ j=1

|gj (x)| ≤

μ

sup M |gj (x)| ≤ ∞,

j=1 x∈E

due to the condition on gi functions. Comments and examples on Theorem 6. Theorem 6 states that sCCP networks are able to simulate ODE’s with an arbitrary precision. The cost of an exact stochastic simulation of the sCCP-network of Deﬁnition 15, however, is proportional to 1δ , hence accurate stochastic simulations of ODEs are computationally impractical. On the other hand, there is no apparent reason to generate stochastic trajectories indistinguishable from the solution of the ODE’s, as the latter can be generally obtained with much less computational eﬀort. In a work related to ours [22], Hillston et al. used the same Kurtz theorem to prove an analogous result for the equations that can be obtained from a PEPA program. Theorem 6 can be seen as a generalization of their result. Moreover, in [22] the authors suggest that a stochastic approximation of ODE’s can be used together with analysis techniques typical of CTMC, like steady state analysis. This is a promising direction, but extreme care must be used. Kurtz theorem, in fact, guarantees convergence only in a ﬁxed and bounded time interval [0, T ], hence it does say nothing about asymptotic convergence of stochastic trajectories to ODE’s. Intuitively, the step δ may not be the only responsible for asymptotic convergence; an important role should also be played by initial conditions through topological properties of the phase space. If the ODE-trajectory we are considering is stable, i.e. resistant to small perturbations, then we can expect it to be reproduced in sCCP along the whole time axis, given a step δ small enough. On the other hand, if the trajectory is unstable, then even small perturbations can drive the dynamics far away from it; stochasticity, in this case, will unavoidably produce a trace dramatically diﬀerent from the one of ODE’s. Of course, by taking the interval [0, T ] of the theorem big enough (hence δ small enough), we can postpone arbitrarily far away in time the moment in which a stochastic and an unstable deterministic trajectories will diverge. As an example of instability, let’s consider a simple linear system of diﬀerential equations: X˙ X +Y = (20) 4X + Y Y˙

Dynamical Systems and Stochastic Programming

261

The theory of dynamical systems [51] tells us that the point (0, 0) is a saddle node, i.e. an unstable equilibrium whose phase space resembles the one depicted in Figure 12(a). The two straight lines arethe directions spanned by the eigen11 vectors of the matrix of coeﬃcients , and are called stable and unstable 41 manifolds. Motion in the stable manifold converges to the equilibrium (0, 0), while the unstable manifold and all other trajectories diverge to inﬁnity. However, small perturbations applied to the stable manifold can bring the system on a divergent hyperbolic trajectory, so we expect that ODE’s and the associated sCCP agent, when starting from the stable manifold (say from point (1, −2)), will eventually jump on a divergent trajectory. Moreover, we expect that the smaller δ the later this event will happen. This intuition is conﬁrmed in Figures 12(b), 12(c), 12(d).

(a) Phase portrait of a saddle node

(c) sCCP simulation — δ = 0.001

(b) ODE solution

(d) sCCP simulation — δ = 0.00001

Fig. 12. 12(a): Phase space of the linear system (20). The origin is a saddle node; the stable manifold is displayed with arrows pointing towards the origin, while the unstable manifold has arrows diverging from it. 12(b): Solution of the ODE’s (20), starting from (1, −2), a point belonging to the stable manifold. 12(c),12(d): Simulation of the sCCP agent associated to the linear system (20), w.r.t. the set of convering functions {(X, (1, 0)), (Y, (1, 0)), (4X, (0, 1)), (Y, (0, 1))} with initial conditions (1, −2). The step δ is equal respectively to 0.001 and 0.00001. The time in which these trajectories diverge from the solution of the ODE’s increases as δ becomes smaller.

262

L. Bortolussi and A. Policriti

This example shows that convergence issues need to be investigate further. In particular, conditions taking into account the topology of the phase space of the ODE’s are required in order to guarantee also asymptotic convergence. Another interesting direction to investigate is to exploit other results by Kurtz [36] in order to state error bounds in the approximation. Examples of Sets of Covering Functions. All the results of this Section have been given parametrically w.r.t. a set of covering functions G. When we motivated the introduction of such concept, we stated that a choice of a speciﬁc G corresponds to a speciﬁc logical structure of the ﬂuxes generating the ODEs. We discuss now two possible choices of G, one natural in absence of information, and the other tailored for ODEs coming from sets of biochemical reactions for which it is possible to reconstruct the reactions from the ODEs. Example 2. The simplest choice of a covering set is the one in which all the addends of ODEs are treated as independent ﬂux sources. To be more ki speciﬁc, consider a set of ODE f (x) = (f1 (x), . . . , fk (x)), with fi (x) = j=1 fij (x), where fij are the single addends of the ODE. The idea is to treat independently each such fij , so that GD = {(fij , ei ) | 1 ≤ i ≤ k, 1 ≤ j ≤ ki }. Such covering set GD will be called in the following the disentangled covering set. This was the choice adopted, for instance, when discussing the Repressilator example in Section 4.3. This choice is reasonable in absence of any further information on the system modeled by the ODE’s, and it does not preserve structural properties of the system, like mass conservation. For instance, consider the system deﬁned by the single reaction A →k·A B, which preserves the total mass A+B. With the disentangled covering set, however, we would reconstruct the logical structure of the following system of biochemical reactions: A →k·A and →k·A B, which does not preserve the total mass A + B. Example 3. Assume now we have a system generated by a set of mass action reactions such that the left hand side (i.e., the list of reactants) of each such reaction is unique. This has the consequence that each reaction is uniquely identiﬁed by the algebraic structure as a monomial of its rate function. For instance, the only reaction with A and B as reagents, A+ B →k ?, is uniquely identiﬁed by the signature as a monomial of its rate function kAB, i.e. by the monomial AB with coeﬃcient 1. This property has the immediate consequence that, whenever we ﬁnd two addends of the ODE with the same signature as a monomial, we are guaranteed that they are two instances of the same ﬂux. That is to say, if the condition is satisﬁed, we know how to reconstruct the set of reactions that originated the ODE’s. ki Formally, given f (x) = (f1 (x), . . . , fk (x)), with fi (x) = j=1 fij (x), we can construct the covering set GR as follows: list all the diﬀerent support monomials {p1 , . . . , ps } in the set {fij | 1 ≤ i ≤ k, 1 ≤ j ≤ ki }, and, for each l, deﬁne the terms αl,j ∈ Z and βl ∈ R+ such that: 1. if pl occurs in the equation for variable i, then its occurrence is equal to αl,i βl pl ;

Dynamical Systems and Stochastic Programming

263

2. if pl does not occur in the equation for variable i, then αl,i = 0; 3. all αl,j = 0 are prime among them18 ; Then, letting αl = (αl,1 , . . . , αl,k ), the covering set GR can be deﬁned as GR = {(βl pl , αl ) | l = 1, . . . , s}. For instance, consider the ODEs ⎧ ⎨ X˙ = −2kX 2 Y Y˙ = −kX 2Y ⎩ ˙ Z = 3kX 2 Y The associated covering set GR is simply GR = {(kX 2 Y, (−2, −1, 3))}, corresponding to the single reaction 2X + Y →k 3Z.

5

Final Discussion

In this paper we presented a method to associate ordinary diﬀerential equations to sCCP programs (written with a restricted syntax), and also a method that generates an sCCP-network from a set of ODE’s. The translation from sCCP to ODE’s is based on the construction of a graph, called RTS, whose edges represent all possible actions performable by sCCP-agents. Properties of restricted(sCCP ) guarantee that the graph is always ﬁnite. From an RTS, we can construct an interaction matrix containing the modiﬁcations that each action makes to each variable. Writing the corresponding ODE’s is simply a matter of combining the interaction matrix with the rate of each action. The inverse translation, from ODE’s to sCCP, exploits the functional form that rates have in sCCP. In this way, we can associate sCCP-agents to general ODE’s. An important feature of this method is that it is parametric w.r.t. the basic increment of variables, meaning that we can reduce the eﬀect of stochastic ﬂuctuations in the sCCP-model. Actually, we proved in Theorem 6 that, in the limit of an inﬁnitesimal increment, the trajectories of the ODE’s and of the corresponding sCCP-system coincide. In Section 3.1, we showed that the translation from sCCP to ODE’s, when applied to models of biochemical reactions, preserves the rate semantics in the sense of [17]. This condition, however, is not suﬃcient to guarantee that the translation maintains also the dynamical behavior of the sCCP-model. In fact, in Section 3.2, we provided several examples where an sCCP-network and the associated ODE’s manifest a diﬀerent behavior. This divergence can be caused by many factors, all qualitatively diﬀerent. Preserving dynamical behavior, however, is not just a mathematical game, but is is a central property that a translation from sCCP to ODE should have in order to be used as an analysis technique for stochastic process algebras. In this light, also the mapping from ODE to sCCP can be seen as a tool to investigate behavioral preservation. 18

This condition guarantees that αlj are uniquely deﬁned.

264

L. Bortolussi and A. Policriti

In Section 4, when we introduced the notion of set of covering functions, we noted that in the passage to ODE’s we unavoidably lose something of the logic of the sCCP model. This also suggest that the preservation of behavior may be reasonable only from a qualitative point of view. Indeed, this weaker approach ﬁts better with the management of stochastic noise, see the discussion at the end of Section 3.2. The loss of precision in passing to ODE’s is, however, counterbalances by the computational gain: simulating stochastic processes is undoubtedly much more expensive than numerically solving ODE’s [24]. There are several open problems related to the question of behavioral equivalence. We list hereafter some of the most important ones, according to us. – We need to identify the class of sCCP models (and their regions of parameter space/initial conditions) for which the mapping ODE preserves dynamics. Intuitively, according to discussion of Section 3.2, this may happen if all variables have big absolute values and if the phase space of the ODE’s has asymptotically stable trajectories with ample basins of attraction. – The repressilator and the simple self-inhibited genetic network of Section 3.2 suggest that the discrete ingredient cannot be continuously approximated so easily. In particular, associating continuous variables to RTS-states seems rather arbitrary. A possible solution can be that of transforming an sCCP network into a hybrid system, in which continuous and discrete dynamics coexist. In this way, we may be able to preserve part of the discrete structure of an sCCP-network, possibly just that fundamental for maintaining the behavior. We are investigating this direction, mapping sCCP-programs to hybrid automata [29,2]. The ﬁrst results are encouraging, see [13,14] – The notion of behavioral equivalence needs to be speciﬁed formally. At the end of Section 3.2, we suggested an approach based on a suitable temporal logic, in which equivalence would mean equi-satisﬁability of the same set of formulae. As a ﬁnal remark, we would like to consider this work under the perspective of the study of systemic properties. In fact, when we model a biological system, we are concerned mainly with the understanding of its systemic properties, especially what they are and how they emerge from basic interactions. In this direction, a modeler needs a formal language to specify biological systems, possibly provided with diﬀerent semantics, related to one another and stratiﬁed in several layers of increasing approximation and abstraction. For example, sCCP has a natural CTMC-based semantics, but an ODE-based one can be assigned to it via the ODE operator. A possible layer in the middle consists in a semantic based, for instance, on hybrid automata. Finally, we need also a language to specify system’s properties, automatically verifying them on the diﬀerent semantics, or better, on the simpler semantic where answers are correct (i.e., on the simpler semantic showing the same dynamical behavior of the most general one). All these features must clearly be part of the same operative framework (and of the same software tool), hence all the open questions presented above can be seen as steps in this direction.

Dynamical Systems and Stochastic Programming

265

References 1. Converging sciences. Trento (2004), http://www.unitn.it/events/consci/ 2. Alur, R., Belta, C., Ivancic, F., Kumar, V., Mintz, M., Pappas, G., Rubin, H., Schug, J.: Hybrid modeling and simulation of biomolecular networks. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 19–32. Springer, Heidelberg (2001) 3. Antoniotti, M., Policriti, A., Ugel, N., Mishra, B.: Model building and model checking for biochemical processes. Cell Biochemistry and Biophysics 38(3), 271–286 (2003) 4. Aziz, A., Singhal, V., Balarin, F., Brayton, R., Sangiovanni-Vincentelli, A.: Verifying continuous time markov chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102. Springer, Heidelberg (1996) 5. Blossey, R., Cardelli, L., Phillips, A.: A compositional approach to the stochastic dynamics of gene networks. T. Comp. Sys. Biology, 99–122 (2006) 6. Blossey, R., Cardelli, L., Phillips, A.: Compositionality, stochasticity and cooperativity in dynamic models of gene regulation. HFPS Journal (2007) (in print) 7. Bortolussi, L.: Stochastic concurrent constraint programming. In: Proceedings of 4th International Workshop on Quantitative Aspects of Programming Languages (QAPL 2006). ENTCS, vol. 164, pp. 65–80 (2006) 8. Bortolussi, L.: Constraint-based approaches to stochastic dynamics of biological systems. PhD thesis, PhD in Computer Science, University of Udine (2007), http://www.dmi.units.it/~ bortolu/files/reps/Bortolussi-PhDThesis.pdf 9. Bortolussi, L.: A master equation approach to diﬀerential approximations of stochastic concurrent constraint programming. In: Proceedings of QAPL 2008. ENTCS (2008) (to appear) 10. Bortolussi, L., Fonda, S., Policriti, A.: Constraint-based simulation of biological systems described by molecular interaction maps. In: Proceedings of IEEE conference on Bioinformatics and Biomedicine, BIBM 2007 (2007) 11. Bortolussi, L., Policriti, A.: Relating stochastic process algebras and diﬀerential equations for biological modeling. In: Proceedings of PASTA 2006 (2006) 12. Bortolussi, L., Policriti, A.: Stochastic concurrent constraint programming and diﬀerential equations. In: Proceedings of Fifth Workshop on Quantitative Aspects of Programming Languages, QAPL 2007. ENTCS, vol. 16713 (2007) 13. Bortolussi, L., Policriti, A.: Hybrid approximation of stochastic concurrent constraint programming. In: Proceedings of IFAC 2008 (2008) 14. Bortolussi, L., Policriti, A.: The importance of being (a little bit) discrete. In: Proceedings of FBTC 2008. ENTCS (2008) (to appear) 15. Bortolussi, L., Policriti, A.: Modeling biological systems in concurrent constraint programming. Constraints 13(1) (2008) 16. Cardelli, L.: From processes to odes by chemistry (2006), http://lucacardelli.name/ 17. Cardelli, L.: On process rate semantics. Theoretical Computer Science 391(3), 190–215 (2008) 18. Ciocchetta, F., Hillston, J.: Bio-PEPA: an extension of the process algebra PEPA for biochemical networks. In: Proceeding of FBTC 2007. Workshop of CONCUR 2007 (2007) 19. Seattle CompBio Group, Institute for Systems Biology. Dizzy home page 20. Cornish-Bowden, A.: Fundamentals of Chemical Kinetics, 3rd edn. Portland Press (2004)

266

L. Bortolussi and A. Policriti

21. Elowitz, M.B., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403, 335–338 (2000) 22. Geisweiller, N., Hillston, J., Stenico, M.: Relating continuous and discrete pepa models of signalling pathways. Theoretical Computer Science (2008) (in print) 23. Gillespie, D.: The chemical langevin equation. Journal of Chemical Physics 113(1), 297–306 (2000) 24. Gillespie, D., Petzold, L.: Numerical Simulation for Biochemical Kinetics. In: System Modelling in Cellular Biology. MIT Press, Cambridge (2006) 25. Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. of Computational Physics 22 (1976) 26. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. of Physical Chemistry 81(25) (1977) 27. Haas, P.J.: Stochastic Petri Nets. Springer, Heidelberg (2002) 28. Hastings, S.P., Murray, J.D.: The existence of oscillatory solutions in the ﬁeld-noyes model for the belousov-zhabotinskii reaction. SIAM Journal on Applied Mathematics 28(3), 678–688 (1975) 29. Henzinger, T.A.: The theory of hybrid automata. In: LICS 1996: Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science (1996) 30. Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press, Cambridge (1996) 31. Hillston, J.: Fluid ﬂow approximation of PEPA models. In: Proceedings of the Second International Conference on the Quantitative Evaluation of Systems, QEST 2005 (2005) 32. Kitano, H.: Foundations of Systems Biology. MIT Press, Cambridge (2001) 33. Kitano, H.: Computational systems biology. Nature 420, 206–210 (2002) 34. Kohn, K.W., Aladjem, M.I., Weinstein, J.N., Pommier, Y.: Molecular interaction maps of bioregulatory networks: A general rubric for systems biology. Molecular Biology of the Cell 17(1), 1–13 (2006) 35. Kurtz, T.G.: Solutions of ordinary diﬀerential equations as limits of pure jump markov processes. Journal of Applied Probability 7, 49–58 (1970) 36. Kurtz, T.G.: Limit theorems for sequences of jump markov processes approximating ordinary diﬀerential processes. Journal of Applied Probability 8, 244–356 (1971) 37. Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with prism: A hybrid approach. International Journal on Software Tools for Technology Transfer 6(2), 128–142 (2004) 38. Mcadams, H.H., Arkin, A.: Stochastic mechanisms in gene expression. PNAS 94, 814–819 (1997) 39. Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1997) 40. Noyes, R.M., Field, R.J.: Oscillatory chemical reactions. Annual Review of Physical Chemistry 25, 95–119 (1974) 41. Nurse, P.: Understanding cells. Nature 24 (2003) 42. Plotkin, G.D.: A structural approach to operational semantics. J. Log. Algebr. Program., 60-61, 17–139 (2004) 43. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C++: The Art of Scientiﬁc Computing. Cambridge University Press, Cambridge (2002) 44. Priami, C.: Stochastic π-calculus. The Computer Journal 38(6), 578–589 (1995) 45. Priami, C., Regev, A., Shapiro, E.Y., Silverman, W.: Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Inf. Process. Lett. 80(1), 25–31 (2001)

Dynamical Systems and Stochastic Programming

267

46. Ramsey, S., Orrell, D., Bolouri, H.: Dizzy: stochastic simulation of large-scale genetic regulatory networks. Journal of Bioinformatics and Computational Biology 3(2), 415–436 (2005) 47. Rao, C.V., Arkin, A.P.: Stochastic chemical kinetics and the quasi-steady state assumption: Application to the gillespie algorithm. Journal of Chemical Physics 118(11), 4999–5010 (2003) 48. Regev, A., Shapiro, E.: Cellular abstractions: Cells as computation. Nature 419 (2002) 49. Saraswat, V.A.: Concurrent Constraint Programming. MIT press, Cambridge (1993) 50. Shapiro, B.E., Levchenko, A., Meyerowitz, E.M., Wold, B.J., Mjolsness, E.D.: Cellerator: extending a computer algebra system to include biochemical arrows for signal transduction simulations. Bioinformatics 19(5), 677–678 (2003) 51. Strogatz, S.H.: Non-Linear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry and Engeneering. Perseus books, Cambridge (1994) 52. Vilar, J.M.G., Yuan Kueh, H., Barkai, N., Leibler, S.: Mechanisms of noise resistance in genetic oscillators. PNAS 99(9), 5991 (2002) 53. Wilkinson, D.J.: Stochastic Modelling for Systems Biology. Chapman & Hall, Boca Raton (2006)

Computing Equilibrium Points of Genetic Regulatory Networks Graziano Chesi Department of Electrical and Electronic Engineering University of Hong Kong [email protected] http://www.eee.hku.hk/~chesi

Abstract. Computing equilibrium points of genetic regulatory networks is a problem of primary importance for numerous investigations in these systems. This paper addresses this problem for diﬀerential equation models, with the regulation function expressed in a general form which includes both SUM form and PROD form for saturation functions of any type. Speciﬁcally, a recursive algorithm is proposed, which provides at each recursion a region guaranteed to contain all equilibrium points. This region progressively shrinks, and asymptotically converges to the sought set of equilibrium points. Moreover, the proposed algorithm can also allow one to delimit and ﬁnd limit cycles. Some numerical examples are reported to illustrate and validate the proposed algorithm, including examples where standard mathematical tools fail to compute the sought equilibrium points. Keywords: Genetic regulatory network, Diﬀerential equation, Saturation, Equilibrium point, Limit cycle.

1

Introduction

Genetic regulatory networks explain the interactions between genes and proteins to form complex systems that perform complicated biological functions, see for instance [1,2,3,4,5,6,7,8]. Basically, there are two types of genetic regulatory network models, i.e., the Boolean model (or discrete model) and the diﬀerential equation model (or continuous model). In Boolean models, the activity of each gene is expressed in one of two states, ON or OFF, and the state of a gene is expressed by a Boolean function of the states of other related genes. In the diﬀerential equation models, the variables describe the concentrations of gene products, such as mRNAs and proteins, as continuous values of the gene regulation systems. See for example [9,10,11,12,13] and references therein for a wider categorization of genetic regulatory networks models. This paper focuses on genetic regulatory networks described through diﬀerential equation models. In these models the dynamics of each concentration is expressed by a function of all concentrations of the system. This function typically consists of two parts: a linear part which deﬁnes the natural decay rate of C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 268–282, 2009. c Springer-Verlag Berlin Heidelberg 2009

Computing Equilibrium Points of Genetic Regulatory Networks

269

the concentration itself, and a nonlinear part which deﬁnes the inﬂuence by all the other concentrations. The nonlinear part can be either described via sum of saturation functions (in this case the system is said to be in SUM form) or via product of saturation functions (in this case the system is said to be in PROD form). See for instance [14,15,16,17]. A fundamental problem in these networks consists of determining the equilibrium points, i.e. the amounts of concentrations for which the regulation process results complete. This is a necessary step for several investigations, such as steady-state, stability, disturbance rejection, etc. Unfortunately, to determine equilibrium points of genetic regulatory networks is a diﬃcult problem because these systems contain saturation functions, and hence the calculation of the equilibrium points amounts to solving a system of nonlinear equations. Indeed, there do not exist techniques able to guarantee to ﬁnd all solutions of such a system, except in the case of polynomial equations, which however can be addressed only for small degrees and small number of variables, see for instance [18,19,20,21] and references therein. In this paper we address the problem of computing equilibrium points of genetic regulatory networks described through diﬀerential equation models. We consider a general model which includes both SUM form and PROD form for saturation functions of any type. The contribution consists of a recursive algorithm which holds the following properties. First, at each recursion the algorithm provides a region containing all equilibrium points, i.e. no equilibrium is lost. Second, this region progressively shrinks, i.e. the conservatism does not increase. Third, this region asymptotically converges to the set of equilibrium points, i.e. all equilibrium points are found. The proposed algorithm is illustrated and validated through some numerical examples with synthetic and real genetic regulatory networks. In these examples it is also shown that standard mathematical tools for solving systems of nonlinear equations may fail to compute the sought equilibrium points. Moreover, in these examples it is also explained that the proposed algorithm can be useful to delimit and ﬁnd limit cycles. The paper is organized as follows. Section 2 introduces some preliminaries on genetic regulatory networks. Section 3 describes the proposed results. Section 4 presents some numerical examples. Finally, Section 5 reports some concluding remarks.

2

Preliminaries

First of all, let us introduce the notation used throughout the paper: -

R+ : positive real number set, i.e. {x ∈ R : x ≥ 0}; 0n : null vector of size n × 1; In : identity matrix of size n × n; ei : i-th column of In ; diag(x1 , . . . , xn ): diagonal matrix with xi at the (i, i) entry; X T : transpose of vector/matrix X; TF: transcription factor.

270

G. Chesi

The genetic regulatory networks considered in this paper are described by the diﬀerential equation model ⎧ ˙ i (t) = −ai mi (t) + bi (p1 (t), . . . , pn (t)) ⎨m p˙ i (t) = −ci pi (t) + di mi (t) (1) ⎩ i = 1, . . . , n where mi (t), pi (t) ∈ R+ are the concentrations of mRNA and protein of the i-th gene, ai , ci ∈ R+ are the degradation rates, di ∈ R+ expresses the eﬀect of mi (t) on pi (t), and bi : Rn+ → R+ is the regulatory function of the i-th gene. This function is typically nonlinear, and either always increases or always decreases with respect to any component of p(t) whenever its other components are ﬁxed, i.e. (−1)ki bi (p1 , . . . , pi−1 , x2 , pi+1 , . . . , pn ) ≥ (−1)ki bi (p1 , . . . , pi−1 , x1 , pi+1 , . . . , pn ) ∀x1 , x2 : x1 ≤ x2 ∀p1 (t), . . . , pn (t) ∈ R+ ∀i = 1, . . . , n (2) for some k1 , . . . , kn ∈ {0, 1}. In genetic regulatory networks with SUM form, the function bi (p1 (t), . . . , pn (t)) is expressed as the sum of functions of a single variable, i.e. bi (p1 (t), . . . , pn (t)) =

n

αi,j bi,j (pj (t))

(3)

j=1

where αi,j ∈ R+ is the contribution of TF j to the transcriptional rate for gene i, and bi,j : R+ → R+ is a monotonic function, i.e. bi,j (pj (t)) either always increases or always decreases with respect to pj (t). In genetic regulatory networks with PROD form, the function bi (p1 (t), . . . , pn (t)) is expressed as the product of the functions bi,j (pj (t)), i.e. bi (p1 (t), . . . , pn (t)) = αi

n

bi,j (pj (t))

(4)

j=1

where αi ∈ R+ represents the transcriptional rate for gene i. Each function bi,j (pj (t)) in (3) and (4) is typically expressed as ⎧ if TF j is an activator of gene i ⎨ f (pj (t)) bi,j (pj (t)) = 1 − f (pj (t)) if TF j is a repressor of gene i ⎩ γ otherwise

(5)

where γ ∈ R is a constant depending on the model which expresses the independence of gene i on TF j (γ = 0 for SUM form, γ = 1 for PROD form), and the function f (pj (t)) is a saturation function. For saturation function we mean a function satisfying the following properties: ⎧ f : R+ → [0, 1] ⎪ ⎪ ⎨ f (0) = 0 (6) limx→∞ f (x) = 1 ⎪ ⎪ ⎩ f (x2 ) ≥ f (x1 ) ∀x1 , x2 : x1 ≤ x2

Computing Equilibrium Points of Genetic Regulatory Networks

271

Hence, a saturation function is an increasing function between 0 and 1 deﬁned for positive value of the variable. For instance, in the case of regulatory functions with Hill form, the function f (pj (t)) is given by f (pj (t)) =

βH

pj (t)H + pj (t)H

(7)

where β ∈ R+ and H is an integer known as Hill coeﬃcient. In order to describe the results of this paper in a more compact form, we introduce a matrix version of the model (1) according to m(t) ˙ = Am(t) + b(p(t)) (8) p(t) ˙ = Cp(t) + Dm(t) where

m(t) = (m1 (t), . . . , mn (t)) p(t) = (p1 (t), . . . , pn (t))

(9)

are the vectors containing the concentrations of mRNA and protein, and A = diag(−a1 , . . . , −an ) C = diag(−c1 , . . . , −cn ) D = diag(d1 , . . . , dn )

(10)

are diagonal matrices containing the decay rates (matrices A and C) and the eﬀect of m(t) on p(t) (matrix D). The function b : Rn+ → Rn+ is a nonlinear function representing the regulation of the process, whose i-th component bi (p(t)) satisﬁes the monotonicity condition (2). We observe that the model (8) under the assumption (2), which is an equivalent matrix version of the model (1), includes: 1. genetic regulatory networks with SUM form, by choosing the i-th component of b(p(t)) as in (3); 2. genetic regulatory networks with PROD form, by choosing the i-th component of b(p(t)) as in (4); 3. genetic regulatory networks that are neither in SUM form nor in PROD form, provided that (2) holds. For instance, the choice for n = 3 given by ⎛ ⎞ b1,1 (p1 (t)) + b1,2 (p2 (t))3 ⎠ eb2,1 (p1 (t)) b2,3 b(p(t)) = ⎝ (11) (p3 (t)) 3 b3,2 (p2 (t)) + b3,3 (p3 (t)) deﬁnes a genetic regulatory networks which is neither in SUM form nor in PROD form, but which is included in the model (8) under the assumption (2). The problem addressed in this paper consists of determining the equilibrium points of (8), i.e. the solutions of the system of nonlinear equations ⎧ ⎨ Am + b(p) = 0n Cp + Dm = 0n (12) ⎩ m, p ∈ Rn+

272

G. Chesi

Remark 1. Before proceeding let us observe that existing mathematical tools for solving systems of nonlinear equations generally do not guarantee to ﬁnd all solutions of such systems. Indeed, systems of nonlinear equations can be solved via either analytical techniques or numerical techniques. Analytical techniques can be used in the case of polynomial or rational equations, and provides the sought solutions as roots of a one-variable polynomial. Unfortunately, the degree of this polynomial is prohibitive (except for very small systems) since in the worst case coincides with the maximum number of solutions of the system, which is given by the degree of the equations to the power of the number of variables, see for instance [18,22,19,21]. Numerical techniques, which are either based on the numerical minimization of a suitable function via for example Newton’s iterations starting from an initial point, or on homotopy methods which adopt continuation strategies, do not suﬀer of the previous problems. Unfortunately, these techniques cannot guarantee to ﬁnd all sought solutions, see for instance [23,20] and Section 4. Remark 2. Another remark concerns the fact that genetic regulatory networks can be also modeled as stochastic systems, where the input is represented by a stochastic process such as white noise. For instance, such an input could aﬀect (8) according to m(t) ˙ = Am(t) + b(p(t)) + w(t) (13) p(t) ˙ = Cp(t) + Dm(t) where w(t) ∈ Rn is a stochastic process. In these systems there are no equilibrium points in the classic sense since the input is a non-constant function of the time and hence the equation Am + b(p) + w(t) = 0n (14) would not admit solutions where m and p do not depend on the time (which is the classic deﬁnition of equilibrium point). Instead, one can consider equilibrium points corresponding to particular constant values of the stochastic process, such as its mean value, that the algorithm proposed in this paper allows one to compute. Indeed, these equilibrium points are deﬁned analogously to (12) as ⎧ ¯ = 0n ⎨ Am + b(p) + w Cp + Dm = 0n (15) ⎩ m, p ∈ Rn+ where w ¯ ∈ Rn is the the stochastic expectation of w(t).

3

Equilibria Computation

In this section we describe the proposed algorithm. Speciﬁcally, in Theorems 1 and 2 we introduce two preliminary functions and we describe their properties. Then, in Theorem 3 we provide the main algorithm to be used to compute the sought equilibrium points.

Computing Equilibrium Points of Genetic Regulatory Networks

273

Before proceeding, let us observe that the m-component of any solution of (12) is related to its p-component by the relationship Cp + Dm = 0n where C, D are nonsingular diagonal matrices with C negative deﬁnite. This means that (12) can be equivalently rewritten as ⎧ ⎨ −AD−1 Cp + b(p) = 0n m = −D−1 Cp (16) ⎩ p ∈ Rn+ Therefore, in the sequel we will focus on the computation of the vectors p fulﬁlling (16). We indicate the set of such vectors as E = p ∈ Rn+ : − AD−1 Cp + b(p) = 0n

(17)

Theorem 1. Let H be the rectangle defined by H = p ∈ Rn+ : pi ∈ [pi,− , pi,+ ]

(18)

for some p1,− , p1,+ , . . . , pn,− , pn,+ ∈ R+ , and let us define the map A(H) as A(H) = p ∈ Rn+ : pi ∈ [qi,− , qi,+ ]

(19)

where q1,− , q1,+ , . . . , qn,− , qn,+ ∈ R+ are computed according to qi,− = max pi,− , min eTi C −1 DA−1 z z∈Z T −1 −1 qi,+ = min pi,+ , max ei C DA z z∈Z

(20) (21)

where Z is the set given by Z = {b(p) : pi ∈ {pi,− , pi,+ } , i = 1, . . . , n} .

(22)

Then, the following properties hold: - Property P1: A(H) ⊆ H; - Property P2: p∗ ∈ H ∩ E ⇒ p∗ ∈ A(H); - Property P3: H ∩ A(H) = ∅ ⇒ H ∩ E = ∅. Proof. First, the property P1 holds because from (20)–(21) one has qi,− ≥ pi,− and qi,+ ≤ pi,+ ∀i = 1, . . . , n.

(23)

Second, the property P2 holds due to the monotonicity property (2) of bi (p) with respect to each component of p and to the linearity of the function eTi C −1 DA−1 z with respect to z. In fact, we have

274

G. Chesi

p∗ ∈ H ⇒ bi (p∗ ) ∈ [min zi , max zi ].

(24)

p∗ ∈ E ⇒ eTi C −1 DA−1 b(p∗ ) = p∗i .

(25)

p∗ ∈ H ∩ E ⇒ qi,− ≤ p∗i and qi,+ ≥ p∗i .

(26)

z∈Z

Moreover,

z∈Z

Hence, it follows

Lastly, the property P3 holds because, if one suppose for contradiction that H ∩ A(H) = ∅ and H contains a vector p∗ of E, then it would follow from the property P2 that p∗ belongs to A(H), hence contradicting the assumption that H ∩ A(H) = ∅. Let us observe that map A(·) requires trivial computations, i.e. evaluation of a linear function in some given points. In fact, let us observe that the set Z is ﬁnite. From the map A(·) we deﬁne the map B(·) in the following theorem. Theorem 2. Let H be a rectangle in (18), and let us define the map B(H) as follows: -

(Step (Step (Step (Step (Step

1) 2) 3) 4) 5)

set H(0) = H and k = 0 (k denotes the iteration number); if H(k) ∩ A(H(k) ) = ∅, set B(H) = ∅ and exit; if A(H(k) ) is a point, set B(H) = A(H(k) ) and exit; if H(k) = A(H(k) ), set B(H) = H(k) and exit; set H(k+1) = A(H(k) ), k = k + 1, and go to 2.

Then, B(H) returns either a rectangle, a point, or the empty set. Moreover: - Property P4: B(H) ⊆ H; - Property P5: p∗ ∈ H ∩ E ⇒ p∗ ∈ B(H). Proof. First of all, let us observe that the output of B(H) can be either the empty set (output of Step 2), a point (output of Step 3), or a rectangle (output of Step 4). Then, the property P4 follows from the fact that the output of B(H) is a sequence of applications of the map A(·) for which the property P1 ensures that the output is a subset of the input. Lastly, the property P5 holds since B(H) returns either a sequence of applications of the map A(·) for which the property P2 ensures that no vector of H ∩ E can be lost, or the empty set in the case H(k) ∩ A(H(k) ) = ∅ which however guarantees the absence of vectors of E in H(k) (and hence in H) due to the properties P2 and P3. The map B(·) transforms a given rectangle via a sequence of applications of the map A(·), and returns a set which can be either a rectangle, a point, or the empty set. By exploiting the map B(·) we derive the algorithm for the computation of the sought equilibrium points as follows.

Computing Equilibrium Points of Genetic Regulatory Networks

275

Theorem 3. (Algorithm for equilibrium points computation) Let H be a rectangle in (18) and let us define the map C(H) as follows: - (Step 1) if B(H) is either the empty set or a point, then set C(H) = B(H) and exit; - (Step 2) divide the rectangle B(H) in 2k rectangles H1 , . . . , H2k by taking the middle point on eachside of B(H) with nonzero length; - (Step 3) set C(H) = i=1,...,2k C(Hi ) and exit. Then, the algorithm to be launched in C(Rn+ ), for which the following properties hold: - Property P6: the positive octant Rn+ is progressively shrunk without losing any point of E; - Property P7: the set provided by the algorithm asymptotically converges to the set E. Proof. The property P6 holds because B(H) is guaranteed to include any vector in H ∩ E according to the property P5, moreover from the property P4 one has that the set returned by the algorithm cannot increase. Then, property P7 holds because no portion of Rn+ is lost in the division of each rectangle B(H). Hence, the proposed algorithm for computing the equilibrium points of (8) is launched as C(Rn+ ), which means that the positive octant Rn+ is used as initial rectangle H. This because Rn+ is clearly guaranteed to contain all solutions of (16). Then, the initial rectangle is passed to the map B(·). If the output of this map is either the empty set or a point, then the algorithm stops as it is guaranteed that there are no equilibrium points inside the considered rectangle. Otherwise, the output is another rectangle, which is then divided in smaller ones. The rectangles obtained in this division are passed to the map C(·) itself, hence realizing a recursive algorithm. As explained by the properties P6 and P7, the set provided by the algorithm is guaranteed to contain all points of E at each recursion, and to asymptotically converge to E. Remark 3. It is worth to remark that the proposed algorithm diﬀers from existing techniques for computing the solutions of systems of nonlinear equations. A ﬁrst diﬀerence is that the proposed algorithm does not rely on analytical techniques, which can be used only in special cases and typically for small systems. A second diﬀerence is that the proposed algorithm does not consider one possible initial point only contrary to some numerical techniques. Instead, the proposed algorithm consider the whole space of possible solutions, and progressively shrinks this space to the sought set of equilibrium points without losing any portion of it. Remark 4. Lastly, it is interesting to observe that the proposed algorithm can also allow one to investigate limit cycles of (8), which are periodic solutions m(t), p(t) of (8) satisfying the condition

276

G. Chesi

∃T ∈ R :

m(t) = m(t + T ) p(t) = p(t + T )

∀t ≥ 0

(27)

where T represents the period. Indeed, at the ﬁrst recursion of the proposed algorithm one obtains the rectangle B(Rn+ ) which is expected to contain existing limit cycles of (8) as they are periodic solutions of the system of diﬀerential equations. This suggests a strategy which can be useful to establish the existence of limit cycles in (8). In fact, once that B(Rn+ ) has been found at the ﬁrst recursion of the algorithm, one can investigate the trajectories starting along its boundary (for instance, at the vertices) to reveal limit cycles. See for instance Example 3.

4

Illustrative Examples

In this section we present some examples where the proposed algorithm is used. We report only the p-component of each equilibrium point, being the mcomponent directly given by D−1 Cp according to (16). The computational time for all examples is lesser than 5 seconds with an implementation of the proposed algorithm in Matlab 7 running under Windows XP on a personal computer with Pentium IV 2.2 GHz and 2 GB RAM. 4.1

Genetic Regulatory Network in PROD Form with Non-Hill Function

Let us start by considering the genetic regulatory network described in PROD form given by ⎧ m ˙ 1 (t) = −0.17m1(t) + 0.73f (p2 )(1 − f (p3 )) ⎪ ⎪ ⎨ m ˙ 2 (t) = −0.8m2(t) + 0.95(1 − f (p3 )) (28) ˙ 3 (t) = −0.52m3(t) + 0.58(1 − f (p1 )) ⎪m ⎪ ⎩ p˙ i (t) = −pi (t) + mi (t) ∀i = 1, 2, 3 and the saturation function 2

f (pi (t)) = 1 − e−pi (t) .

(29)

This genetic regulatory network is characterized by the fact that TF 1 is a regressor of gene 3, TF 2 is an activator of gene 1, and TF 3 is a regressor of genes 1 and 2. Let us use the algorithm proposed in Theorem 3. At the ﬁrst recursion of the algorithm we obtain that the positive octant R+ 3 is shrunk to the rectangle shown in Figure 1a. At the second recursion, the rectangle previously found is divided in four equal rectangles, one of which is shown in Figure 1b, another one shrinks to the equilibrium point shown in Figure 1b, and the other two converges to the empty set. At the fourth recursion, another equilibrium point is found as shown in Figure 1c, and only one rectangle is left. Then, at the eight recursion the last equilibrium point is found and no rectangle is left as shown in Figure 1d. We

Computing Equilibrium Points of Genetic Regulatory Networks 1.3

1.3

1.2

1.2

1.1

1.1

0.9

p2

1

0.9

p2

1

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0

0.5

1

p1 (a) 1.5

2

2.5

3

0.3

1.3

1.3

1.2

1.2

1.1

1.1

0.9

0

0.5

1

p1 (b)

2

2.5

3

0

0.5

1

p1 (d)

2

2.5

3

1.5

p2

1

0.9

p2

1

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.3

277

0.4

0

0.5

1

p1 (c) 1.5

2

2.5

3

0.3

1.5

Fig. 1. Steps of the proposed algorithm for the example in Section 4.1 (shown in the plane p1 –p2 for clarity of presentation): (a) ﬁrst recursion, R3+ is shrunk to a rectangle; (b) second recursion, an equilibrium point is found (denoted by the “∗” mark); (c) fourth recursion, another equilibrium point is found. (d): ninth recursion, the last equilibrium point is found.

hence conclude that this system has three equilibrium points, in particular the set E in (17) is given by E = (3.246, 1.189, 0.000)T , (0.461, 0.527, 0.902)T , (0.166, 0.366, 1.085)T . (30) For comparison, we attempt to use standard mathematical tools, in particular via Matlab and Mathematica. We hence use the functions “solve” (Matlab function for both analytical and numerical techniques) and ”ﬁndroot” (Mathematica function for numerical technique) which ﬁnd only one solution. This happens because the equations in (12) are neither polynomial nor rational in this case, which means that no analytical technique exist for ﬁnding the solutions in this case. Existing tools therefore apply numerical techniques which allow to ﬁnd a local solution starting from an initial point, but the other solutions are lost.

278

4.2

G. Chesi

Genetic Regulatory Network in SUM Form with Hill Function

In this example we consider the genetic regulatory network in SUM form with ⎧ m ˙ 1 (t) = −2.0m1 (t) + 0.9(1 − f (p2 )) + 0.5f (p3) ⎪ ⎪ ⎪ ⎪ m ˙ 2 (t) = −2.2m2 (t) + 0.9(1 − f (p3 )) + 0.5f (p4) ⎪ ⎪ ⎪ ⎪ m ˙ 3 (t) = −2.4m3 (t) + 0.9(1 − f (p4 )) + 0.5f (p5) ⎪ ⎪ ⎪ ⎨ .. . (31) ⎪ m ˙ (t) = −3.4m8 (t) + 0.9(1 − f (p9 )) + 0.5f (p10 ) ⎪ 8 ⎪ ⎪ ⎪ m ˙ 9 (t) = −3.6m9 (t) + 0.9(1 − f (p10 )) + 0.5f (p1 ) ⎪ ⎪ ⎪ ⎪ m ˙ (t) = −3.8m10 (t) + 0.9(1 − f (p1 )) + 0.5f (p2 ) ⎪ ⎪ ⎩ 10 p˙i (t) = −pi (t) + mi (t) ∀i = 1, . . . , 10 where the saturation function is chosen as the Hill function f (pi (t)) =

1 . 1 + pi (t)6

(32)

This genetic regulatory network is characterized by the cyclic structure where gene i has TF i + 1 as regressor and TF i + 2 as activator. By using the algorithm proposed in Theorem 3 we have that the positive octant R10 + shrinks to the set E = (0.449, 0.408, 0.375, 0.346, 0.321, 0.300, 0.281, 0.267, 0.251, 0.236)T , (33) hence implying that there is one equilibrium point only in this genetic regulatory network. Also in this case we attempt to use standard mathematical tools as done in the previous example. However, by using analytical techniques (which can be used since the equations in (12) are rational for this example) we do not obtain any solution. This happens because the degree of the one-variable polynomial that the analytical techniques allow one to ﬁnd is prohibitive in this case since the equations in (12) have degree 12 (the degree of b(p)) and 10 variables (the p-components of the state), therefore there can be up to 1210 solutions. Also, we attempt to use numerical techniques, and ﬁnd that they return the sought equilibrium point. Unfortunately, these techniques are not able to establish whether this solution is unique or not. 4.3

Repressilator Model in E. Coli

Here we consider the repressilator investigated in Escherichia coli [24]: ⎧ ˙ i (t) = −mi (t) + αrep (1 − f (pj (t))) ⎨m p˙ i (t) = −β rep (pi (t) − mi (t)) ⎩ i = lacl, tetR, cl; j = cl, lacl, tetR where the saturation function is the Hill function

(34)

Computing Equilibrium Points of Genetic Regulatory Networks

279

8

p3

6 4 2 0 0

2

4

6

p2

8

8

7

6

5

4

3

2

0

1

p1

Fig. 2. Found equilibrium point and limit cycle in the example of Section 4.3

f (pi (t)) =

1 1 + pi (t)2

(35)

and αrep , β rep ∈ R+ are positive constants. Let us select the plausible values αrep = 10 and β rep = 1. By using the algorithm proposed in Theorem 3 we ﬁnd that there is a unique equilibrium point, in particular E = (2, 2, 2)T .

(36)

For this example it is interesting to observe that, in addition to the found equilibrium point, there exists a limit cycle that the proposed algorithm can help to ﬁnd. Indeed, as explained in Remark 4, at the ﬁrst recursion of the proposed algorithm one obtains the rectangle B(R3+ ), which is equal to [0.1010, 9.899]3. Then, the limit cycle is revealed by simply computing the trajectory of the system starting at the vertices of this rectangle. Figure 2 shows the projection on the plane p1 -p2 of the found limit cycle.

280

4.4

G. Chesi

Genetic Regulatory Network in SUM Form with Non-Hill Function

As last example, we consider the genetic regulatory network in SUM form described by ⎧ ˙ 1 (t) = −2m1 (t) + 0.5f (p5 ) ⎪ ⎪m ⎪ ⎪ m ˙ 2 (t) = −m2 (t) + 0.1(1 − f (p2 )) + 0.4(1 − f (p4 )) ⎪ ⎪ ⎨ m ˙ 3 (t) = −0.6m3(t) + 0.2f (p1) + 1.1(1 − f (p4 )) (37) m ˙ 4 (t) = −m4 (t) + 0.5(1 − f (p3 )) + 1.5f (p4 ) ⎪ ⎪ ⎪ ⎪ m ˙ 5 (t) = −2m5 (t) + 0.3f (p2 ) + 0.3(1 − f (p5 )) ⎪ ⎪ ⎩ p˙ i (t) = −pi (t) + mi (t) ∀i = 1, . . . , 5 and the saturation function f (pi (t)) =

2 arctan(pi (t)2 ). π

(38)

This genetic regulatory network is characterized by the fact that TF 1 is an activator of gene 3, TF 2 is an activator of gene 5 and a regressor of gene 2, TF 3 is a regressor of gene 4, TF 4 is a regressor of genes 2 and 3 and an activator of gene 4, and TF 5 is an activator of gene 1 and a regressor of gene 5. By using the algorithm proposed in Theorem 3 as done in the previous examples we conclude that this system has three equilibrium points, in particular the set E in (17) is given by E = (0.0037, 0.1961, 0.4518, 1.566, 0.1515)T , (0.0039, 0.3130,1.003, 0.9278, 0.1570)T , (0.0046, 0.4827, 1.821, 0.1035, 0.1691)T .

(39)

However, by using standard mathematical tools, we obtain only one solution similarly to the example in Section 4.1.

5

Conclusion

We have proposed an algorithm which allows one to ﬁnd the equilibrium points of genetic regulatory networks described by diﬀerential equation models and which include both SUM form and PROD form with saturation functions of any type. The proposed algorithm is guaranteed to ﬁnd all sought equilibrium points, moreover as shown by some numerical examples the computation is reasonably fast also in cases where standard mathematical tools for solving systems of nonlinear equations may fail. It is hence expected that the proposed algorithm represents a useful tool for researchers working in the area of genetic regulatory networks. In particular, the proposed algorithm can allow one to investigate issues such as stability, disturbance rejection, and robustness, for which the knowledge of the equilibrium points is required, see for instance [25,26,27,28].

Computing Equilibrium Points of Genetic Regulatory Networks

281

Acknowledgement The author would like to thank the Editor and the Reviewers for their time and useful comments.

References 1. D’haeseleer, P., Wen, X., Fuhrman, S., Somogyi, R.: Mining the gene expression matrix: Inferring gene relationships from large scale gene expression data. In: Paton, R.C., Holcombe, M. (eds.) Information Processing in Cells and Tissues. Plenum Publishing, New York (1998) 2. Davidson, E.H.: The Regulatory Genome: Gene Regulatory Networks In Development And Evolution. Academic Press, London (2006) 3. D’haeseleer, P.: Reconstructing Gene Networks from Large Scale Gene Expression Data. PhD thesis, University of New Mexico (2000) 4. D’haeseleer, P., Liang, S., Somogyi, R.: Genetic network inference: From coexpression clustering to reverse engineering. Bioinformatics 16(8), 707–726 (2000) 5. Li, C., Chen, L., Aihara, K.: A systems biology perspective on signal processing in genetic network motifs. IEEE Signal Processing Magazine 221(3), 136–142 (2007) 6. Yuh, C.H., Bolouri, H., Davidson, E.H.: Genomic cis-regulatory logic: Experimental and computational analysis of a sea urchin gene. Science 279, 1896–1902 (1998) 7. Tsai, H.K., Yang, J.M., Tsai, Y.F., Kao, C.Y.: An evolutionary approach for gene expression patterns. IEEE Transactions on Information Technology in Biomedicine 8(2), 69–78 (2004) 8. Maraziotis, I.A., Dragomir, A., Bezerianos, A.: Gene networks reconstruction and time-series prediction from microarray data using recurrent neural fuzzy networks. IET Systems and Biology 1(1), 41–50 (2007) 9. Smolen, P., Baxter, D.A., Byrne, J.H.: Mathematical modeling of gene networks. Neuron 26(3), 567–580 (2000) 10. Bower, J.M., Bolouri, H. (eds.): Computational Modeling of Genetic and Biochemical Networks. Computational Molecular Biology. MIT Press, Cambridge (2001) 11. Jong, H.D.: Modeling and simulation of genetic regulatory systems: A literature review. Journal of Computation Biology 9, 67–103 (2002) 12. D’haeseleer, P., Liang, S., Somogyi, R.: Gene expression data analysis and modeling. In: Proc. Paciﬁc Symposium on Biocomputing, Hawaii, USA (1999) 13. Aracena, J., Lamine, S.B., Mermet, M.A., Cohen, O., Demongeot, J.: Mathematical modeling in genetic networks: Relationships between the genetic expression and both chromosomic breakage and positive circuits. IEEE Transactions on Systems, Man, and Cybernetics–Part b: Cybernetics 33(5), 825–834 (2003) 14. Bintu, L., Buchler, N.E., Garcia, H.G., Gerland, U., Hwa, T., Kondev, J., Phillips, R.: Transcriptional regulation by the numbers: models. Current Opinion in Genetics and Development 15(2), 116–124 (2005) 15. Li, C., Chen, L., Aihara, K.: Stability of genetic networks with sum regulatory logic: Lure system and lmi approach. IEEE Trans. on Circuits and Systems I 53(11), 2451–2458 (2006) 16. Li, C., Chen, L., Aihara, K.: Stochastic stability of genetic networks with disturbance attenuation. IEEE Transactions on Circuits and Systems II 54(10), 892–896 (2007)

282

G. Chesi

17. Chesi, G., Hung, Y.S.: Stability analysis of uncertain genetic SUM regulatory networks. Automatica 44(9), 2298–2305 (2008) 18. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach. Int. Journal of Robust and Nonlinear Control 13(13), 1239–1257 (2003) 19. Mora, T.: Solving Polynomial Equation Systems II. Cambridge University Press, Cambridge (2005) 20. Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, Heidelberg (2006) 21. Chesi, G.: Optimal representation matrices for solving polynomial systems via LMI. Int. Journal of Pure and Applied Mathematics 45(3), 397–412 (2008) 22. Stetter, H.J.: Numerical Polynomial Algebra. SIAM, Philadelphia (2004) 23. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia (1987) 24. Elowitz, M.B., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403, 335–338 (2000) 25. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Englewood Cliﬀs (2001) 26. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Homogeneous Lyapunov functions for systems with structured uncertainties. Automatica 39(6), 1027–1035 (2003) 27. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Solving quadratic distance problems: an LMI-based approach. IEEE Trans. on Automatic Control 48(2), 200–212 (2003) 28. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems. Lecture Notes in Control and Information Sciences, vol. 390. Springer, London (2009)

Code, Context, and Epigenetic Catalysis in Gene Expression Rodrick Wallace1 and Deborah Wallace2 1

The New York State Psychiatric Institute, 549 W. 123 St., Suite 16F, New York, NY, 10027. Tel.: (212) 865-4766 [email protected] 2 Consumers Union [email protected]

Abstract. We examine a class of probability models describing how epigenetic context aﬀects gene expression and organismal development, using the asymptotic limit theorems of information theory in a highly formal manner. Taking classic results on spontaneous symmetry breaking abducted from statistical physics in groupoid, rather than group, circumstances, the work suggests that epigenetic information sources act as analogs to a tunable catalyst, directing development into diﬀerent characteristic pathways according to the structure of external signals. The results have signiﬁcant implications for epigenetic epidemiology, in particular for understanding how environmental stressors, in a large sense, can induce a broad spectrum of developmental disorders in humans.

1 1.1

Introduction Toward New Tools

Researchers have begun to explore a de-facto cognitive paradigm for gene expression in which contextual factors determine the behavior of what Cohen calls a ‘reactive system’, not at all a deterministic, or even stochastic, mechanical process (e.g., [18, 19, 74]). The diﬀerent approaches, while highly formal, are nonetheless much in the spirit of the pioneering eﬀorts of Maturana and Varela [53, 54] who foresaw the essential role that cognitive process must play in a vast realm of biological phenomena. O’Nuallain [57] has recently placed gene expression ﬁrmly in the realm of complex linguistic behavior, for which context imposes meaning, claiming that the analogy between gene expression and language production is useful both as a fruitful research paradigm and also, given the relative lack of success of natural language processing (nlp) by computer, as a cautionary tale for molecular biology. First O’Nuallain argues that, at the orthographic or phonological level, depending on whether the language is written or spoken, we can map from phonetic elements to nucleotide sequence. His second claim is that Nature has designed highly ambiguous codes in both cases, and left disambiguation to the context. C. Priami et al. (Eds.): Trans. on Comput. Syst. Biol. XI, LNBI 5750, pp. 283–334, 2009. c Springer-Verlag Berlin Heidelberg 2009

284

R. Wallace and D. Wallace

He notes that, given our concern with the Human Genome Project (HGP) and its implications for human health, only 2% of diseases can be traced back to a straightforward genetic cause. As a consequence the HGP will have to be redone for a variety of metabolic contexts in order to establish a sound technology of genetic engineering [58]. Here we investigate a broad class of probability models based on the asymptotic limit theorems of information theory that instantiate this perspective, ﬁnding a ‘natural’ means by which epigenetic context ‘farms’ gene expression in an inherently punctuated manner via a kind of tunable catalysis. The models will be used to study how normal developmental modes can be driven by external context into pathological trajectories often expressed, in humans, as comorbid psychiatric and physical disorders, expanding recent work [71]. It appears possible to convert such models to powerful tools for data analysis, much as those based on the Central Limit Theorem can be converted to parametric statistics. A more formal version of the underlying mathematics can be found in [34]. We will begin with a summary of the biological context, then examine the popular spinglass model of development taken from neural network studies that we will ultimately generalize using a cognitive paradigm. The expanded approach permits import of tools and methods from statistical physics via the homology between information source uncertainty and free energy density, and this leads directly to the idea of epigenetic catalysis. It is worth keeping in mind throughout the formal mathematics that Feynman’s basic measure of information is simply the free energy needed to erase it [31]. 1.2

Epigenetic Epidemiology

What we attempt is itself embedded in a large and lively intellectual context. Jablonka and Lamb [41, 42] have long argued that information can be transmitted from one generation to the next in ways other than through the base sequence of DNA. It can be transmitted through cultural and behavioral means in higher animals, and by epigenetic means in cell lineages. All of these transmission systems allow the inheritance of environmentally induced variation. Such Epigenetic Inheritance Systems are the memory structures that enable somatic cells of diﬀerent phenotypes but identical genotypes to transmit their phenotypes to their descendants, even when the stimuli that originally induced these phenotypes are no longer present. In chromatin-marking systems information is carried from one cell generation to the next because it rides with DNA as binding proteins or additional chemical groups that are attached to DNA and inﬂuence its activity. When DNA is replicated, so are the chromatin marks. One type of mark is the methylation pattern a gene carries. The same DNA sequence can have several diﬀerent methylation patterns, each reﬂecting a diﬀerent functional state. These alternative patterns can be stably inherited through many cell divisions. Epigenetic inheritance systems are very diﬀerent from the genetic system. Many variations are directed and predictable outcomes of environmental changes.

Code, Context, and Epigenetic Catalysis in Gene Expression

285

Epigenetic variants are, in the view of [41, 42], often, although not necessarily, adaptive. The frequency with which variants arise and their rate of reversion varies widely and epigenetic variations induced by environmental changes may be produced coordinatedly at several loci. Parenthetically, some authors, e.g., [39], disagree with the assumption of adaptiveness, inferring that input responsible for methylation eﬀects simply produces a phenotypic variability then subject to selection. The matter remains open. Jablonka and Lamb [42] conclude that epigenetic systems may therefore produce rapid, reversible, co-ordinated, heritable changes. However such systems can also underlie non-induced changes, changes that are induced but non-adaptive, and changes that are very stable. What is needed, they feel, is a concept of epigenetic heritability comparable to the classical concept of heritability, and a model similar to those used for measuring the eﬀects of cultural inheritance on human behavior in populations. Following a furious decade of research and debate, this perspective received much empirical conﬁrmation. Backdahl et al. [6], for example, write that epigenetic regulation of gene expression primarily works through modifying the secondary and tertiary structures of DNA (chromatin), making it more or less accessible to transcription. The sum and interaction of epigenetic modiﬁcations has been proposed to constitute an ‘epigenetic code’ which organizes the chromatin structure on diﬀerent hierarchical levels [67]. Modiﬁcations of histones include acetylation, methylation, phosphorylation, ubiquitination, and sumoylation, but also other modiﬁcations have been observed. Some such modiﬁcations are quite stable and play an important part in epigenetic memory although DNA methylation is the only epigenetic modiﬁcation that has maintenance machinery which preserves the marks through mitosis. This argues for DNA methylation to function as a form of epigenetic memory for the epigenome. Codes and memory, of course, are inherent to any cognitive paradigm. Jaenish and Bird [45] argue that cells of a multicellular organism are genetically homogeneous but structurally and functionally heterogeneous owing to the diﬀerential expression of genes. Many of these diﬀerences in gene expression arise during development and are subsequently retained through mitosis. External inﬂuences on epigenetic processes are seen in the eﬀects of diet on long-term diseases such as cancer. Thus, epigenetic mechanisms seem to allow an organism to respond to the environment through changes in gene expression. Epigenetic modiﬁcations of the genome provide a mechanism that allows the stable propagation of gene activity states from one generation of cells to the next. Because epigenetic states are reversible they can be modiﬁed by environmental factors, which may contribute to the development of abnormal responses. What needs to be explained, from their perspective, is the variety of stimuli that can bring about epigenetic changes, ranging from developmental progression and aging to viral infection and diet. Jaenish and Bird conclude that the future will see intense study of the chains of signaling that are responsible for epigenetic programming. As a result, we will

286

R. Wallace and D. Wallace

be able to understand, and perhaps manipulate, the ways in which the genome learns from experience. Indeed, our central interest precisely regards the manner in which the asymptotic limit theorems of information theory constrain such chains of signaling, in the same sense that the Central Limit Theorem constrains sums of stochastic variates. Crews et al. [21, 22] provide a broad overview of induced epigenetic change in phenotype, as do Guerrero-Bosagna et al. [39], who focus particularly on early development. They propose that changes arising because of alterations in early development processes, in some cases environmentally induced, can appear whether or not such changes could become ﬁxed and prosper in a population. They recognize two ways for this to occur, ﬁrst by dramatically modifying DNA aspects in the germ line with transgenerational consequences – mutations or persistent epigenetic modiﬁcations of the genome – or by inducing ontogenetical variation in every generation, although not inheritance via the germ line. From their perspective inductive environmental forces can act to create, through these means, new conformations of organisms which also implies new possibilities within the surrounding environment. Foley et al. [32] take a very general perspective on the prospects for epigenetic epidemiology. They argue that epimutation is estimated to be 100 times more frequent than genetic mutation and may occur randomly or in response to the environment. Periods of rapid cell division and epigenetic remodeling are likely to be most sensitive to stochastic or environmentally mediated epimutation. Disruption of epigenetic proﬁle is a feature of most cancers and is speculated to play a role in the etiology of other complex diseases including asthma, allergy, obesity, type 2 diabetes, coronary heart disease, autism spectrum disorders, bipolar disorders, and schizophrenia. They ﬁnd evidence that a small change in the level of DNA methylation, especially in the lower range in an animal model, can dramatically alter expression for some genes. The timing of nutritional insuﬃciency or other environmental exposures may also be critical. In particular low-level maternal care was associated with developmental dysfunction and altered stress response in the young. Foley et al. emphasize the potential implications of such ﬁndings, given how widely stress is implicated in disease onset and relapse. They especially note that when epigenetic status or change in status over time is the outcome, then models for either threshold-based dichotomies or proportional data will be required. Threshold models, deﬁned by a given level or pattern of methylation or a degree of change in methylation over time, will, in their view, beneﬁt from relevant functional data to identify meaningful thresholds. A special contribution of the approach taken here is that just such threshold behavior leads ‘naturally’ to a language-like ‘dual information source’ constrained by the necessary conditions imposed by information theory’s asymptotic limit theorems, allowing development of statistical models of complicated cognitive phenomena, including but not limited to cognitive gene expression.

Code, Context, and Epigenetic Catalysis in Gene Expression

287

A recent review by Weaver [77] focuses speciﬁcally on the epigenetic eﬀects of glucocorticoids – stress hormones. In mammals, Weaver argues, the closeness or degree of positive attachment in parent-infant bonding and parental investment during early life has long-term consequences on development of interindividual diﬀerences in cognitive and emotional development in the oﬀspring. The long-term eﬀects of the early social experience, he continues, particularly of the mother-oﬀspring interaction, have been widely investigated. The nature of that interaction inﬂuences gene expression and the development of behavioral responses in the oﬀspring that remain stable from early development to the later stages of life. Although enhancing the oﬀspring’s ability to respond according to environmental clues early in life can have immediate adaptive value, the cost, Weaver says, is that these adaptations serve as predictors of ill health in later life. He concludes that maternal inﬂuences on the development of neuroendocrine systems that underlie hypothalamic-pituitary-adrenal (HPA) axis and behavioral responses to stress mediate the relation between early environment and health in the adult oﬀspring. In particular, he argues, exposure of the mother to environmental adversity alters the nature of mother-oﬀspring interaction, which, in turn, inﬂuences the development of defensive responses to threat and reproductive strategies in the progeny. In an updated review of epigenetic epidemiology, Jablonka [43] ﬁnds it clear that the health and general physiology of animals and people can be aﬀected not only by the interplay of their own genes and conditions of life, but also by the inherited eﬀects of the interplay of genes and environment in their ancestors. These ancestral inﬂuences on health, Jablonka says, depend neither on inheriting particular genes, nor on the persistence of the ancestral environment. Signiﬁcantly, Bossdorf et al. [11] invoke ‘contexts’ much like Baars’ model of consciousness [68], and infer a need to expand the concept of variation and evolution in natural populations, taking into account several likely interacting ecologically relevant inheritance systems. Potentially, this may result in a signiﬁcant expansion, though by all means not a negation, of the Modern Evolutionary Synthesis as well as in more conceptual and empirical integration between ecology and evolution. More formally, Scherrer and Jost [62, 63] use information theory arguments to extend the deﬁnition of the gene to include the local epigenetic machinery, something they characterize as the ‘genon’. Their central point is that coding information is not simply contained in the coded sequence, but is, in their terms, provided by the genon that accompanies it on the expression pathway and controls in which peptide it will end up. In their view the information that counts is not about the identity of a nucleotide or an amino acid derived from it, but about the relative frequency of the transcription and generation of a particular type of coding sequence that then contributes to the determination of the types and numbers of functional products derived from the DNA coding region under consideration. From our perspective the formal tools for understanding such phenomena involve asymptotic limit theorems aﬀecting information sources – active systems

288

R. Wallace and D. Wallace

that generate or ‘provide’ information – and these are respectively the Rate Distortion Theorem and its zero error limit, the Shannon-McMillan Theorem, described in the Mathematical Appendix. We begin with a reconsideration of the current de-facto standard systems biology neural network-analog model of development, and proceed to its generalization.

2 2.1

Models of Development The Spinglass Model

Ciliberti et al.[16, 17], culminating a long series of papers, apply the spinglass model from statistical physics to organisimal development in an evolutionary context. We summarize their formalism and look at some of the less obvious topological implications – in particular the mapping of disjoint directed homotopy classes of phenotype paths into interaction matrix space. We then extend the approach by applying a cognitive paradigm for gene expression ﬁrst developed in [74]. Analogs to phase transition arguments in physical systems generate punctuated equilibrium evolutionary transitions in a ‘highly natural’ manner, even for the spinglass treatment, and a hierarchical extension permits incorporation of epigenetic eﬀects as a kind of tunable catalysis. The spinglass model of development assumes that N transcriptional regulators are represented by their expression patterns S(t) = [S1 (t), ..., SN (t)] at some time t during a developmental or cell-biological process and in one cell or domain of an embryo. The transcriptional regulators inﬂuence each other’s expression through cross-regulatory and autoregulatory interactions described by a matrix w = (wij ). For nonzero elements, if wij > 0 the interaction is activating, if wij < 0 it is repressing. w represents, in this model, the regulatory genotype of the system, while the expression state S(t) is the phenotype. These regulatory interactions change the expression of the network S(t) as time progresses according to a diﬀerence equation N Si (t + Δt) = σ[ wij Sj (t)],

(1)

j=1

where Δt is a constant and σ a sigmodial function whose value lies in the interval (−1, 1). In the spinglass limit σ is the sign function, taking only the values ±1. The networks of interest in the spinglass model are those whose expression state begins from a prespeciﬁed initial state S(0) at time t = 0 and converge to a prespeciﬁed stable equilibrium state S∞ . Such a network is termed viable, for obvious reasons. After an elaborate and very diﬃcult simulation exercise, a particular series of results emerges. Reference [16] ﬁnds that viable networks comprise a tiny

Code, Context, and Epigenetic Catalysis in Gene Expression

289

fraction of possible ones. They could be widely scattered in the space of all possible networks and occupy disconnected islands in this space. However, direct computation indicates precisely the opposite. The metagraph of viable networks has one ‘giant’ connected component that comprises most or all viable networks. Any two networks in this component can be reached from one another through gradual changes of one regulatory interaction at a time, changes that never leave the space of viable networks, for this calculation. In general, within the giant component, randomly chosen pairs of networks with the same phenotype will have vastly diﬀerent organization, in terms of the matrix (wij ). Deﬁne 0 ≤ d ≤ 1 as the the fraction of genes that diﬀer in their expression state between S0 and S∞ . A typical result is that for N = 5 genes, 6 ≤ M ≤ 7 total regulatory interactions, and d = 0.4, full enumeration ﬁnds a total of only 37,338 viable networks out of 6.3 × 107 possible ones [16]. Long random walks through the space of viable networks, however, visit all but a very small fraction of the nodes of the metagraph, and this missing fraction decreases as N increases. Large N require elaborate Monte Carlo sampling for simulation, a diﬃcult and computationally intensive enterprise. In w-space [16, 17] deﬁne a metric characterizing the distance between two network topologies as D(w, w ) =

1 |sign(wij ) − sign(wij )|, 2M+ i,j

where M+ is the maximum number of regulatory interactions, and sign(x)=±1 depends on the sign of x, and is 0 for x = 0. Several observations emerge directly. 1. This approach is formally similar to spinglass neural network models of learning by selection, e.g., as proposed by Toulouse et al. [66] nearly a generation ago. Subsequent work [4, 5], summarized in [23], suggests that such models are simply not suﬃcient to the task of understanding high level cognitive function, and these have been largely supplanted by complicated ‘global workspace’ concepts whose mathematical characterization is highly nontrivial [3]. 2. What [16, 17] observe, in another idiom, is that in phenotype space, in S-space, the set of all paths associated with viable networks forms an equivalence class, closely analogous to the directed homotopy equivalence classes in the sense of [36, 37]. Directed homotopy diﬀers from simple homotopy (e.g., [50]) in that one uses paths from one point to another rather than loops, and seeks continuous deformations between them. See [74] for discussion in a biological context. Thus there is, in this spinglass model, a mapping from S-space into (wij ) space, characterized by the metric D, that associates a unique simply connected component with each dihomotopy-like equivalence class of paths connecting two particular phenotype points. Indeed, the w-space component might well be treated according to standard homotopy arguments, i.e., using loops.

290

R. Wallace and D. Wallace

3. What one does with homotopically simply connected components is patch them together to build larger, and more interesting, topological structures, using the Seifert-Van Kampen Theorem (SVKT) (e.g., [50], Ch. 10). If paths within Sspace are not continuously transformable into one another, (if there are ‘holes’), then several distinct dihomotopy classes will exist, e.g., as in ﬁgures 1 and 2 of [74], explored further below in terms of developmental critical periods and their ‘shadows’. The obvious conjecture is that, under such a circumstance, very complex topological objects may lurk in w-space, not just the simply connected component discovered by by [16, 17]. These may, according to the SVKT, intersect as well as exist as isolated and disconnected sets. In particular, if there are dihomotopy ‘holes’ in S-space, consequently reﬂected in disconnected patches in w-space, then punctuated transition events of various sorts may well become an evolutionary norm, as in [38], even for the spinglass model. 4. A large and increasing body of work surrounding coupled cell networks invokes groupoids, a natural generalization of symmetry groups. As [25] remarks, until recently the abstract theory of coupled cell systems has mainly focused on the eﬀects of symmetry in the network and the consequent formation of spatial and spatiotemporal patterns. The formal setting for this theory centers upon the symmetry group of the network. Reference [25] concludes that analysis of robust patterns of synchrony in general coupled cell systems – that is, dynamics in which sets of cells behave identically as a consequence of the network topology – leads to the fruitful notion of the ‘symmetry groupoid’ of a coupled cell network. A groupoid is a generalization of a group, in which products of elements are not always deﬁned. The symmetry groupoid of a coupled cell network is a natural algebraic formalization of the ‘local symmetries’ that relate subsets of the network to each other. In particular ‘admissible’ vector ﬁelds – those speciﬁed by the network topology – are precisely those that are equivariant under the action of the symmetry groupoid. The Appendix provides a summary of standard material on groupoids that will be of later use. 5. Both of these – analogous – approaches can apparently be coarse-grained into a symbolic dynamics associated with (simple) information sources having particular grammar and syntax. The method is straightforward (e.g., [7, 55]). One could, thus, probably translate the spinglass results of Ciliberti et al. into symbolic dynamics, using groupoid methods to study the underlying topological objects. 6. The spinglass model of development is abstracted from longstanding (if ultimately unsucessful) attempts at similar treatments of neural networks involved in high level cognition (e.g., [44, 56, 61, 64]). Thus and consequently [16, 17] are invoking an implicit cognitive paradigm for gene expression (e.g., [18, 19, 74]). Cognitive process, as the philosopher Fred Dretske eloquently argues (e.g., [26]), is constrained by the necessary conditions imposed by the asymptotic limit theorems of information theory. A little work produces a very general cognitive gene expression metanetwork structure recognizably similar to that found

Code, Context, and Epigenetic Catalysis in Gene Expression

291

in [16, 17]. The massively parallel computations are hidden, somewhat, in the required empirical ﬁtting of regression model analogs based on the asymptotic limit theorems of information theory rather than on the central limit theorem. 7. A salient characteristic of high level cognitive process is precisely its inherent punctuation (e.g., [4, 5, 68]), and this emerges directly using an information theory approach via the famous homology between information and free energy (e.g., [31]). ‘Simple’ neural network analogs will inevitably have more diﬃculty replicating such behavior, but as discussed, the mapping of disjoint dihomotopy equivalence classes from phenotype sequence space to disjoint sets in interaction matrix space provides a straightforward example for spinglass models. The next sections use information theory methods to make the transition from crossectional w-space into that of serially correlated sequences of phenotypes, expanding on the results of [74]. 2.2

Shifting Perspective: Cognition as an Information Source

Atlan and Cohen [2], in the context of a study of the immune system, argue that the essence of cognition is the comparison of a perceived signal with an internal, learned picture of the world, and then choice of a single response from a large repertoire of possible responses. Such choice inherently involves information and information transmission since it always generates a reduction in uncertainty, as explained in [1] (p. 21). More formally, a pattern of incoming input – like the S(t) of equation (1) – is mixed in a systematic algorithmic manner with a pattern of internal ongoing activity – like the (wij ) according to equation (1) – to create a path of combined signals x = (a0 , a1 , ..., an , ...) – analogous to the sequence of S(t+Δt) of equation (1), with, say, n = t/Δt. Each ak thus represents some functional composition of internal and external signals. This path is fed into a highly nonlinear decision oscillator, h, a ‘sudden threshold machine’, in a sense, that generates an output h(x) that is an element of one of two disjoint sets B0 and B1 of possible system responses. Let us deﬁne the sets Bk as B0 = {b0 , ..., bk }, B1 = {bk+1 , ..., bm }. Assume a graded response, supposing that if h(x) ∈ B0 , the pattern is not recognized, and if h(x) ∈ B1 , the pattern has been recognized, and some action bj , k + 1 ≤ j ≤ m takes place.

292

R. Wallace and D. Wallace

The principal objects of formal interest are paths x triggering pattern recognition-and-response. That is, given a ﬁxed initial state a0 , examine all possible subsequent paths x beginning with a0 and leading to the event h(x) ∈ B1 . Thus h(a0 , ..., aj ) ∈ B0 for all 0 < j < m, but h(a0 , ..., am ) ∈ B1 . For each positive integer n, let N (n) be the number of high probability grammatical and syntactical paths of length n which begin with some particular a0 and lead to the condition h(x) ∈ B1 . Call such paths ‘meaningful’, assuming, not unreasonably, that N (n) will be considerably less than the number of all possible paths of length n leading from a0 to the condition h(x) ∈ B1 . While the combining algorithm, the form of the nonlinear oscillator, and the details of grammar and syntax are all unspeciﬁed in this model, the critical assumption which permits inference of the necessary conditions constrained by the asymptotic limit theorems of information theory is that the ﬁnite limit H = lim

n→∞

log[N (n)] n

(2)

both exists and is independent of the path x. Deﬁne such a pattern recognition-and-response cognitive process as ergodic. Not all cognitive processes are likely to be ergodic in this sense, implying that H, if it indeed exists at all, is path dependent, although extension to nearly ergodic processes seems possible [73]. Invoking the spirit of the Shannon-McMillan Theorem, whose content is described in more detail in the Appendix, as choice involves an inherent reduction in uncertainty, it is then possible to deﬁne an adiabatically, piecewise stationary, ergodic (APSE) information source X associated with stochastic variates Xj having joint and conditional probabilities P (a0 , ..., an ) and P (an |a0 , ..., an−1 ) such that appropriate conditional and joint Shannon uncertainties satisfy the classic relations log[N (n)] = n→∞ n

H[X] = lim

lim H(Xn |X0 , ..., Xn−1 ) =

n→∞

lim

n→∞

H(X0 , ..., Xn ) . n+1

(3)

See the Mathematical Appendix for a summary of basic information theory results. This information source is deﬁned as dual to the underlying ergodic cognitive process. Adiabatic means that the source has been parametized according to some scheme, and that, over a certain range, along a particular piece, as the parameters vary, the source remains as close to stationary and ergodic as needed for information theory’s central theorems to apply. Stationary means that the system’s probabilities do not change in time, and ergodic, roughly, that the cross

Code, Context, and Epigenetic Catalysis in Gene Expression

293

sectional means approximate long-time averages. Between pieces it is necessary to invoke various kinds of phase transition formalisms, as described more fully in [68, 74]. Using the developmental vernacular of [16, 17], we now examine paths in phenotype space that begins at some S0 and converges n = t/Δt → ∞ to some other S∞ . Suppose the system is conceived at S0 , and h represents (for example) reproduction when phenotype S∞ is reached. Thus h(x) can have two values, i.e., B0 not able to reproduce, and B1 , mature enough to reproduce. Then x = (S0 , SΔt , ..., SnΔt , ...) until h(x) = B1 . Structure is now subsumed within the sequential grammar and syntax of the dual information source rather than within the cross sectional internals of (wij )space, a simplifying shift in perspective. This transformation carries heavy computational burdens, as well as providing deeper mathematical insight. First, the fact that viable networks comprise a tiny fraction of all those possible emerges easily from the spinglass formulation simply because of the ‘mechanical’ limit that the number of paths from S0 to S∞ will always be far smaller than the total number of possible paths, most of which simply do not end on the target conﬁguration. From the information source perspective, which inherently subsumes a far larger set of dynamical structures than possible in a spinglass model – not simply those of symbolic dynamics – the result is what [47] characterizes as the ‘Eproperty’ of a stationary, ergodic information source. This property is that, in the limit of inﬁnitely long output, the classiﬁcation of output strings into two sets: 1. A very large collection of gibberish which does not conform to underlying (sequential) rules of grammar and syntax, in a large sense, and which has nearzero probability, and 2. A relatively small ‘meaningful’ set, in conformity with underlying structural rules, having very high probability. The essential content of the Shannon-McMillan Theorem is that, if N (n) is the number of meaningful strings of length n, then the uncertainty of an information source X can be deﬁned as H[X] = limn→∞ log[N (n)]/n, that can be expressed in terms of joint and conditional probabilities as in equation (3) above. Proving these results for general stationary, ergodic information sources requires considerable mathematical machinery [20, 24, 47]. Second, information source uncertainty has an important heuristic interpretation that [1] describes as follows: ...[W]e may regard a portion of text in a particular language as being produced by an information source. The probabilities P [Xn = an |X0 = a0 , ...Xn−1 = an−1 ] may be estimated from the available data about the language; in this way we can estimate the uncertainty associated with the language. A large uncertainty means, by the [Shannon-McMillan Theorem], a large number of ‘meaningful’ sequences. Thus given two languages with uncertainties H1 and H2 respectively, if H1 > H2 , then in

294

R. Wallace and D. Wallace

the absence of noise it is easier to communicate in the ﬁrst language; more can be said in the same amount of time. On the other hand, it will be easier to reconstruct a scrambled portion of text in the second language, since fewer of the possible sequences of length n are meaningful. This will prove important below. Third, information source uncertainty is homologous with free energy density in a physical system, a matter having implications across a broad class of dynamical behaviors. The free energy density of a physical system having volume V and partition function Z(K) derived from the system’s Hamiltonian – the energy function – at inverse temperature K is (e.g., [49]) F [K] = lim − V →∞

1 log[Z(K, V )] = K V

ˆ log[Z(K, V )] , V →∞ V lim

(4)

where Zˆ = Z −1/K . The partition function for a physical system is the normalizing sum in an equation having the form exp[−Ei /kT ] P [Ei ] = j exp[−Ej /kT ] where Ei is the energy of state i, k a constant, and T the system temperature, and P [Ei ] is the probability of state i. Feynman [31], following the classic arguments of [9] that present idealized machines using information to do work, concludes the information contained in a message is most simply measured by the free energy needed to erase it. The arguments of [9] are clever indeed, and the Feynman treatment of them in [31] is well worth reading. Thus, according to this argument, source uncertainty is homologous to free energy density as deﬁned above, i.e., from the similarity with the relation H = limn→∞ log[N (n)]/n. Ash’s comment above then has an important corollary: If, for a biological system, H1 > H2 , source 1 will require more metabolic free energy than source 2.

3

Symmetry Arguments

A formal equivalence class algebra, in the sense of the groupoid section of the Appendix, can now be constructed by choosing diﬀerent origin and end points S0 , S∞ and deﬁning equivalence of two states by the existence of a high probability meaningful path connecting them with the same origin and end. Disjoint

Code, Context, and Epigenetic Catalysis in Gene Expression

295

partition by equivalence class, analogous to orbit equivalence classes for dynamical systems, deﬁnes the vertices of the proposed network of cognitive dual languages, much enlarged beyond the spinglass example. We thus envision a network of metanetworks, in the sense of [16]. Each vertex then represents a different equivalence class of information sources dual to a cognitive process. This is an abstract set of metanetwork ‘languages’ dual to the cognitive processes of gene expression and development. This structure generates a groupoid, in the sense of [78]. States aj , ak in a set A are related by the groupoid morphism if and only if there exists a high probability grammatical path connecting them to the same base and end points, and tuning across the various possible ways in which that can happen – the diﬀerent cognitive languages – parametizes the set of equivalence relations and creates the (very large) groupoid. There is an implicit hierarchy. First, there is structure within the system having the same base and end points, as in [16]. Second, there is a complicated groupoid structure deﬁned by sets of dual information sources surrounding the variation of base and end points. We do not need to know what that structure is in any detail, but can show that its existence has profound implications. We begin with the simple case, the set of dual information sources associated with a ﬁxed pair of beginning and end states. 3.1

The First Level

The spinglass model of [16, 17] produced a simply connected, but otherwise undiﬀerentiated, metanetwork of gene expression dynamics that could be traversed continuously by single-gene transitions in the highly parallel w-space. Taking the serial grammar/syntax model above, we ﬁnd that not all high probability meaningful paths from S0 to S∞ are actually the same. They are structured by the uncertainty of the associated dual information source, and that has a homological relation with free energy density. Let us index possible dual information sources connecting base and end points by some set A = ∪α. Argument by abduction from statistical physics is direct: Given metabolic energy density available at a rate M , and an allowed development time τ , let K = 1/κM τ for some appropriate scaling constant κ, so that M τ is total developmental free energy. Then the probability of a particular Hα will be determined by the standard expression (e.g., [49]), exp[−Hβ K] P [Hβ ] = , α exp[−Hα K]

(5)

where the sum may, in fact, be a complicated abstract integral. This is just a version of the fundamental probability relation from statistical mechanics, as above. The sum in the denominator, the partition function in statistical physics, is a crucial normalizing factor that allows the deﬁnition of of P [Hβ ] as a probability. A basic requirement, then, is that the sum/integral always converges. K is the inverse product of a scaling factor, a metabolic energy density rate term, and

296

R. Wallace and D. Wallace

a characteristic development time τ . The developmental energy might be raised to some power, e.g., K = 1/(κ(M τ )b ), suggesting the possibility of allometric scaling. Thus, in this formulation, there must be structure within a (cross sectional) connected component in the w-space of [16, 17], determined in no small measure by available energy. Some dual information sources will be ‘richer’/smarter than others, but, conversely, must use more metabolic energy for their completion. 3.2

The Second Level

The next generalization is crucial: While we might simply impose an equivalence class structure based on equal levels of energy/source uncertainty, producing a groupoid in the sense of the Appendix (and possibly allowing a Morse Theory approach in the sense of [52, 59]), we can do more by now allowing both source and end points to vary, as well as by imposing energy-level equivalence. This produces a far more highly structured groupoid that we now investigate. Equivalence classes deﬁne groupoids, by standard mechanisms [13, 35, 78]. The basic equivalence classes – here involving both information source uncertainty level and the variation of S0 and S∞ , will deﬁne transitive groupoids, and higher order systems can be constructed by the union of transitive groupoids, having larger alphabets that allow more complicated statements in the sense of Ash above. Again, given an appropriately scaled, dimensionless, ﬁxed, inverse available metabolic energy density rate and development time, so that K = 1/κM τ , we propose that the metabolic-energy-constrained probability of an information source representing equivalence class Di , HDi , will again be given by exp[−HDi K] P [HDi ] = , j exp[−HDj K]

(6)

where the sum/integral is over all possible elements of the largest available symmetry groupoid. By the arguments of Ash above, compound sources, formed by the union of underlying transitive groupoids, being more complex, generally having richer alphabets, as it were, will all have higher free-energy-densityequivalents than those of the base (transitive) groupoids. Let ZD = exp[−HDj K]. (7) j

We now deﬁne the Groupoid free energy of the system, FD , at inverse normalized metabolic energy density K, as 1 FD [K] = − log[ZD [K]], (8) K again following the standard arguments from statistical physics [31, 49]. The groupoid free energy construct permits introduction of important ideas from statistical physics.

Code, Context, and Epigenetic Catalysis in Gene Expression

3.3

297

Spontaneous Symmetry Breaking

We have expressed the probability of an information source in terms of its relation to a ﬁxed, scaled, available (inverse) metabolic free energy, seen as a kind of equivalent (inverse) system temperature. This gives a statistical thermodynamic path leading to deﬁnition of a ‘higher’ free energy construct – FD [K] – to which we now apply Landau’s fundamental heuristic phase transition argument [49, 59, 65]. The essence of Landau’s insight was that certain phase transitions were usually in the context of a signiﬁcant symmetry change in the physical states of a system, with one phase being far more symmetric than the other. A symmetry is lost in the transition, a phenomenon called spontaneous symmetry breaking. The greatest possible set of symmetries in a physical system is that of the Hamiltonian describing its energy states. Usually states accessible at lower temperatures will lack the symmetries available at higher temperatures, so that the lower temperature phase is less symmetric: The randomization of higher temperatures – in this case limited by available metabolic free energy – ensures that higher symmetry/energy states – mixed transitive groupoid structures – will then be accessible to the system. Absent high metabolic free energy, however, only the simplest transitive groupoid structures can be manifest. A full treatment from this perspective requires invocation of groupoid representations, no small matter (e.g., [10, 14]). Somewhat more rigorously, the biological renormalization schemes of the Appendix to [74] may now be imposed on FD [K] itself, leading to a spectrum of highly punctuated transitions in the overall system of developmental information sources. Most deeply, however, an extended version of Pettini’s Morse-Theory-based topological hypothesis [59] can now be invoked, i.e., that changes in underlying groupoid structure are a necessary (but not suﬃcient) consequence of phase changes in FD [K]. Necessity, but not suﬃciency, is important, as it, in theory, allows mixed groupoid symmetries. The essential insight is that the single simply connected giant component of [16, 17] is unlikely to be the full story, and that more complete models will likely be plagued – or graced – by highly punctuated dynamics. Several matters are worth noting. First, Landau’s spontaneous symmetry breaking arguments are perhaps the simplest approach possible here. The formal mathematical development requires invoking holonomy groups and groupoids, as in [34]. Second, one need not be restricted to terms of the form exp[−Hj K], as any f (Hj , K) such that the sum over j converges will serve, although the resulting ‘thermodynamic’ relations between variates of central interest may then be less elegant. Third, there may be some allometric scaling tradeoﬀ between metabolic energy rate and development time determined by a relation of the form K ∝ (τ M )α .

298

4

R. Wallace and D. Wallace

Tunable Epigenetic Catalysis

Incorporating the inﬂuence of embedding contexts – epigenetic eﬀects – is most elegantly done by invoking the Joint Asymptotic Equipartition Theorem (JAEPT) [20]. For example, given an embedding contextual information source, say Z, that aﬀects development, then the dual cognitive source uncertainty HDi is replaced by a joint uncertainty H(XDi , Z). The objects of interest then become the jointly typical dual sequences y n = (xn , z n ), where x is associated with cognitive gene expression and z with the embedding context. Restricting consideration of x and z to those sequences that are in fact jointly typical allows use of the information transmitted from Z to X as the splitting criterion. One important inference is that, from the information theory ‘chain rule’ [20], H(X, Y ) = H(X) + H(Y |X) ≤ H(X) + H(Y ), while there are approximately exp[nH(X)] typical X sequences, and exp[nH(Z)] typical Z sequences, and hence exp[n(H(x) + H(Y ))] independent joint sequences, there are only about exp[nH(X, Z)] ≤ exp[n(H(X) + H(Y ))] jointly typical sequences, so that the eﬀect of the embedding context, in this model, is to lower the relative free energy of a particular developmental channel. Thus the eﬀect of epigenetic regulation is to channel development into pathways that might otherwise be inhibited by an energy barrier. Hence the epigenetic information source Z acts as a tunable catalyst, a kind of second order cognitive enzyme, to enable and direct developmental pathways. This result permits hierarchical models similar to those of higher order cognitive neural function that incorporate Baars’ contexts in a natural way [73, 74]. It is worth emphasizing that this is indeed a relative energy argument, since, metabolically, two systems must now be supported, i.e., that of the ‘reaction’ itself and that of its catalytic regulator. ‘Programming’ and stabilizing inevitably intertwined, as it were. This elaboration allows a spectrum of possible ‘ﬁnal’ phenotypes, what [33] calls developmental or phenotype plasticity. Thus gene expression is seen as, in part, responding to environmental or other, internal, developmental signals. West-Eberhard [79] argues that any new input, whether it comes from the genome, like a mutation, or from the external environment, like a temperature change, a pathogen, or a parental opinion, has a developmental eﬀect only if the preexisting phenotype is responsive to it. A new input causes a reorganization of the phenotype, or ‘developmental recombination.’ In developmental recombination, phenotypic traits are expressed in new or distinctive combinations during ontogeny, or undergo correlated quantitative change in dimensions. Developmental recombination can result in evolutionary divergence at all levels of organization. Individual development can be visualized as a series of branching pathways. Each branch point, according to [79], is a developmental decision, or switch point, governed by some regulatory apparatus, and each switch point deﬁnes a modular trait. Developmental recombination implies the origin or deletion of a

Code, Context, and Epigenetic Catalysis in Gene Expression

299

branch and a new or lost modular trait. It is important to realize that the novel regulatory response and the novel trait originate simultaneously. Their origins are, in fact, inseparable events. There cannot, [79] concludes, be a change in the phenotype, a novel phenotypic state, without an altered developmental pathway. These mechanisms are accomplished in our formulation by allowing the set B1 in section 2.2 to span a distribution of possible ‘ﬁnal’ states S∞ . Then the groupoid arguments merely expand to permit traverse of both initial states and possible ﬁnal sets, recognizing that there can now be a possible overlap in the latter, and the epigenetic eﬀects are realized through the joint uncertainties H(XDi , Z), so that the epigenetic information source Z serves to direct as well the possible ﬁnal states of XDi . Again, [62, 63] use information theory arguments to suggest something similar to epigenetic catalysis, ﬁnding the information in a sequence is not contained in the sequence but has been provided by the machinery that accompanies it on the expression pathway. That work does not, however, invoke a cognitive paradigm, its attendant groupoid symmetries, or the homology between information source uncertainty and free energy density that drives dynamics. The mechanics of channeling can be made more precise as follows.

5

Rate Distortion Dynamics

Real time problems, like the crosstalk between epigenetic and genetic structures, are inherently rate distortion problems, and the interaction between biological structures can be restated in communication theory terms. Suppose a sequence of signals is generated by a biological information source Y having output y n = y1 , y2 , .... This is ‘digitized’ in terms of the observed behavior of the system with which it communicates, say a sequence of observed behaviors bn = b1 , b2 , .... The bi happen in real time. Assume each bn is then deterministically retranslated back into a reproduction of the original biological signal, bn → yˆn = yˆ1 , yˆ2 , .... Here the information source Y is the epigenetic Z, and B is XDi , but the terminology used here is more standard [20]. Deﬁne a distortion measure d(y, yˆ) which compares the original to the retranslated path. Many distortion measures are possible, as described in the Mathematical Appendix. The distortion between paths y n and yˆn is deﬁned as 1 d(yj , yˆj ). n j=1 n

d(y n , yˆn ) =

A remarkable fact of the Rate Distortion Theorem is that the basic result is independent of the exact distortion measure chosen [20, 24].

300

R. Wallace and D. Wallace

Suppose that with each path y n and bn -path retranslation into the y-language, denoted yˆn , there are associated individual, joint, and conditional probability distributions p(y n ), p(ˆ y n ), p(y n , yˆn ), p(y n |ˆ y n ). The average distortion is deﬁned as D= p(y n )d(y n , yˆn ).

(9)

yn

It is possible, using the distributions given above, to deﬁne the information transmitted from the Y to the Yˆ process using the Shannon source uncertainty of the strings: I(Y, Yˆ ) = H(Y ) − H(Y |Yˆ ) = H(Y ) + H(Yˆ ) − H(Y, Yˆ ),

(10)

where H(..., ...) is the joint and H(...|...) the conditional uncertainty [1, 20]. If there is no uncertainty in Y given the retranslation Yˆ , then no information is lost, and the systems are in perfect synchrony. In general, of course, this will not be true. The rate distortion function R(D) for a source Y with a distortion measure d(y, yˆ) is deﬁned as R(D) =

p(y,ˆ y );

min

(y,y) ˆ

p(y)p(y|ˆ y )d(y,ˆ y)≤D

I(Y, Yˆ ).

(11)

The minimization is over all conditional distributions p(y|ˆ y) for which the joint distribution p(y, yˆ) = p(y)p(y|ˆ y) satisﬁes the average distortion constraint (i.e., average distortion ≤ D). The Rate Distortion Theorem states that R(D) is the minimum necessary rate of information transmission which ensures communication does not exceed average distortion D. Thus R(D) deﬁnes a minimum necessary channel capacity. References [20, 24] provide details. The rate distortion function has been explicitly calculated for a number of simple systems. Recall, now, the relation between information source uncertainty and channel capacity [1, 20]: H[X] ≤ C,

(12)

where H is the uncertainty of the source X and C the channel capacity, deﬁned according to the relation [1, 20] C = max I(X|Y ).

(13)

P (X)

X is the message, Y the channel, and the probability distribution P (X) is chosen so as to maximize the rate of information transmission along a Y .

Code, Context, and Epigenetic Catalysis in Gene Expression

301

Finally, recall the analogous deﬁnition of the rate distortion function above, again an extremum over a probability distribution. Recall, again, equations (4-8), i.e., that the free energy of a physical system at a normalized inverse temperature-analog K = 1/κT is deﬁned as F (K) = − log[Z(K)]/K where Z(K) the partition function deﬁned by the system Hamiltonian. More precisely, if the possible energy states of the system are a set Ei , i = 1, 2, ... then, at normalized inverse temperature K, the probability of a state Ei is determined by the relation P [Ei ] = exp[−Ei K]/ j exp[−Ej K]. The partition function is simply the normalizing factor. Applying this formalism, it is possible to extend the rate distortion model by describing a probability distribution for D across an ensemble of possible rate distortion functions in terms of available free metabolic energy, K = 1/κM τ . The key is to take the R(D) as representing energy as a function of the average distortion. Assume a ﬁxed K, so that the probability density function of an average distortion D, given a ﬁxed K, is then exp[−R(D)K] P [D, K] = Dmax . Dmin exp[−R(D)K]dD

(14)

Thus lowering K in this model rapidly raises the possibility of low distortion communication between linked systems. We deﬁne the rate distortion partition function as just the normalizing factor in this equation:

Dmax

ZR [K] =

exp[−R(D)K]dD,

(15)

Dmin

again taking K = 1/κM τ . We now deﬁne a new free energy-analog, the rate distortion free-energy, as FR [K] = −

1 log[ZR [K]], K

(16)

and apply Landau’s spontaneous symmetry breaking argument to generate punctuated changes in the linkage between the genetic information source XDi and the embedding epigenetic information source Z. Recall that Landau’s insight was that certain phase transitions were usually in the context of a signiﬁcant symmetry change in the physical states of a system. Again, the biological renormalization schemes of the Appendix to [74] may now be imposed on FR [K] itself, leading to a spectrum of highly punctuated transitions in the overall system of interacting biological substructures. Since 1/K is proportional to the embedding metabolic free energy, we assert that 1. The greatest possible set of symmetries will be realized for high developmental metabolic free energies, and 2. Phase transitions, related to total available developmental metabolic free energy, will be accompanied by fundamental changes in the ﬁnal topology of the

302

R. Wallace and D. Wallace

system of interest – phenotype changes – recognizing that evolutionary selection acts on phenotypes, not genotypes. The relation 1/K = κM τ suggests the possibility of evolutionary tradeoﬀs between development time and the rate of available metabolic free energy.

6

More Topology

It seems possible to extend this treatment using standard topological arguments. Taking T = 1/K in equations (6) and (14) as a product of eigenvalues, we can deﬁne it as the determinant of a Hessian matrix representing a Morse Function, f , on some underlying, background, manifold, M, characterized in terms of (as yet unspeciﬁed) variables X = (x1 , ..., xn ), so that 1/K = det(Hi,j ), Hi,j = ∂ 2 f /∂xi ∂xj .

(17)

Again, see the Appendix for a brief outline of Morse Theory. Thus κ, M , and the development time τ are seen as eigenvalues of H on the manifold M in an abstract space deﬁned by some set of variables X . By construction H has everywhere only nonzero, and indeed, positive, eigenvalues, whose product thereby deﬁnes T as a generalized volume. Thus, and accordingly, all critical points of f have index zero, that is, no eigenvalues of H are ever negative at any point, and hence at any critical point Xc where df (Xc ) = 0. This deﬁnes a particularly simple topological structure for M: If the interval [a, b] contains a critical value of f with a single critical point Xc , then the topology of the set Mb deﬁned above diﬀers from that of Ma in a manner determined by the index i of the critical point. Mb is then homeomorphic to the manifold obtained from attaching to Ma an i-handle, the direct product of an i-disk and an (m − i)-disk. One obtains, in this case, since i = 0, the two halves of a sphere with critical points at the top and bottom [52, 59]. This is, as in [16], a simply connected object. What one does then is to invoke the Seifert-Van Kampen Theorem (SVKT, [50]) and patch together the various simply connected subcomponents to construct the larger, complicated, topological object representing the full range of possibilities. The physical natures of κ, M , and τ thus impose constraints on the possible complexity of this system, in the sense of the SVKT.

7

Inherited Epigenetic Memory

The cognitive paradigm for gene expression invoked here requires an internal picture of the world against which incoming signals are compared – algorithmically

Code, Context, and Epigenetic Catalysis in Gene Expression

303

combined according to the rules of Section 2.2 – and then fed into a sharply stepwise decision oscillator that chooses one (or a few) action(s) from a much large repertoire of possibilities. Memory is inherent, and much recent work, as described in the introduction, suggests that epigenetic memory is indeed heritable. The abduction of spinglass and other models from neural network studies to the analysis of development and its evolution carries with it the possibility of more than one system of memory. What Baars called ‘contexts’ channeling high level animal cognition may often be the inﬂuence of cultural inheritance, in a large sense. Our formalism suggests a class of statistical models that indeed greatly generalize those used for measuring the eﬀects of cultural inheritance on human behavior in populations. Epigenetic machinery, as a dual information source to a cognitive process, serves as a heritable system, intermediate between (relatively) hard-wired classical genetics, and a (usually) highly Larmarckian embedding cultural context. In particular, the three heritable systems interact, in our model, through a crosstalk in which the epigenetic machinery acts as a kind of intelligent catalyst for gene expression.

8

Multiple Processes

The argument to this point has, in large measure, been directly abducted from recent formal studies of high level cognition – consciousness – based on a Dretskestyle information theoretic treatment of Bernard Baars’ global workspace model [3, 68]. A deﬁning and grossly simplifying characteristic of that phenomenon is its rapidity: typically the global broadcasts of consciousness occur in a matter of a few hundred milliseconds, limiting the number of processes that can operate simultaneously. Slower cognitive dynamics can, therefore, be far more complex than individual consciousness. One well known example is institutional distributed cognition that encompasses both individual and group cognition in a hierarchical structure typically operating on timescales ranging from a few seconds or minutes in combat or hunting groups, to years at the level of major governmental structures, commercial enterprises, religious organizations, or other analogous large scale cultural artifacts. Reference [73] provides the ﬁrst formal mathematical analysis of institutional distributed cognition. Clearly cognitive gene expression is not generally limited to a few hundred milliseconds, and something much like the distributed cognition analysis may be applied here as well. Extending the analysis requires recognizing an individual cognitive actor can participate in more than one ‘task’, synchronously, asynchronously, or strictly sequentially. Again, the analogy is with institutional function whereby many individuals often work together on several distinct projects: Envision a multiplicity of possible cognitive gene expression dual ‘languages’ that themselves form a higher order network linked by crosstalk. Next, describe crosstalk measures linking diﬀerent dual languages on that meta-meta (MM) network by some characteristic magnitude ω, and deﬁne a topology on the MM network by renormalizing the network structure to zero if

304

R. Wallace and D. Wallace

the crosstalk is less than ω and set it equal to one if greater or equal to it. A particular ω, of suﬃcient magnitude, deﬁnes a giant component of network elements linked by mutual information greater or equal to it, in the sense of [29], as more fully described in [73] (Section 3.4). The fundamental trick is, in the Morse Theory sense [52], to invert the argument so that a given topology for the giant component will, in turn, deﬁne some critical value, ωC , so that network elements interacting by mutual information less than that value will be unable to participate, will be locked out and not active. ω becomes an epigenetically syntactically-dependent detection limit, and depends critically on the instantaneous topology of the giant component deﬁning the interaction between possible gene interaction MM networks. Suppose, now, that a set of such giant components exists at some generalized k system ‘time’ k and is characterized by a set of parameters Ωk = ω1k , ..., ωm . Fixed parameter values deﬁne a particular giant component set having a particular set of topological structures. Suppose that, over a sequence of times the set of giant components can be characterized by a possibly coarse-grained path γn = Ω0 , Ω1 , ..., Ωn−1 having signiﬁcant serial correlations that, in fact, permit deﬁnition of an adiabatically, piecewise stationary, ergodic (APSE) information source Γ . Suppose that a set of (external or internal) epigenetic signals impinging on the set of such giant components can also be characterized by another APSE information source Z that interacts not only with the system of interest globally, but with the tuning parameters of the set of giant components characterized by Γ . Pair the paths (γn , zn ) and apply the joint information argument above, generating a splitting criterion between high and low probability sets of pairs of paths. We now have a multiple workspace cognitive genetic expression structure driven by epigenetic catalysis.

9

‘Coevolutionary’ Development

The model can be applied to multiple interacting information sources representing simultaneous gene expression processes, for example across a spatially diﬀerentiating organism as it develops. This is, in a broad sense, a ‘coevolutionary’ phenomenon in that the development of one segment may aﬀect that of others. Most generally we assume that diﬀerent cognitive developmental subprocesses of gene expression characterized by information sources Hm interact through chemical or other signals and assume that diﬀerent processes become each other’s principal environments, a broadly coevolutionary phenomenon. We write Hm = Hm (K1 ...Ks , ...Hj ...),

(18)

where the Ks represent other relevant parameters and j = m. The dynamics of such a system is driven by a recursive network of stochastic diﬀerential equations, similar to those used to study many other highly parallel dynamic structures (e.g., [83]).

Code, Context, and Epigenetic Catalysis in Gene Expression

305

Letting the Kj and Hm all be represented as parameters Qj , (with the caveat that Hm not depend on itself), one can deﬁne, according to the generalized Onsager development of the Appendix, S m = Hm −

Qi ∂Hm /∂Qi

i

to obtain a complicated recursive system of phenomenological ‘Onsager relations’ stochastic diﬀerential equations, dQjt =

[Lj,i (t, ...∂S m /∂Qi ...)dt + σj,i (t, ...∂S m /∂Qi ...)dBti ],

(19)

i

where, again, for notational simplicity only, we have expressed both the Hj and the external K’s in terms of the same symbols Qj . m ranges over the Hm and we could allow diﬀerent kinds of ‘noise’ dBti , having particular forms of quadratic variation that may, in fact, represent a projection of environmental factors under something like a rate distortion manifold [73, 74]. As usual for such systems, there will be multiple quasi-stable points within a given system’s Hm , representing a class of generalized resilience modes accessible via punctuation. Second, however, there may well be analogs to fragmentation when the system exceeds the critical values of Kc according to the approach of [74]. That is, the K-parameter structure will represent full-scale fragmentation of the entire structure, and not just punctuation within it. We thus infer two classes of punctuation possible for this kind of structure. There are other possible patterns: 1. Setting equation (19) equal to zero and solving for stationary points again gives attractor states since the noise terms preclude unstable equilibria. 2. This system may converge to limit cycle or ‘strange attractor’ behaviors in which the system seems to chase its tail endlessly, e.g., the cycle of climate-driven phenotype changes in persistent temperate region plants. 3. What is converged to in both cases is not a simple state or limit cycle of states. Rather it is an equivalence class, or set of them, of highly dynamic information sources coupled by mutual interaction through crosstalk. Thus ‘stability’ in this extended model represents particular patterns of ongoing dynamics rather than some identiﬁable ‘state’, although such dynamics may be indexed by a ‘stable’ set of phenotypes. Here we become enmeshed in a system of highly recursive phenomenological stochastic diﬀerential equations, but at a deeper level than the standard stochastic chemical reaction model (e.g., [84]), and in a dynamic rather than static manner: the objects of this system are equivalence classes of information sources and their crosstalk, rather than simple ﬁnal states of a chemical system.

306

10

R. Wallace and D. Wallace

Multiple Models

Recent work [75] argues that consciousness may have undergone the characteristic branching and pruning of evolutionary development, particularly in view of the rapidity of currently surviving conscious mechanisms. According to that study, evolution is littered with polyphyletic parallelisms: many roads lead to functional Romes, and consciousness, as a particular form of high order cognitive process operating in real time, embodies one such example, represented by an equivalence class structure that factors the broad realm of necessary conditions information theoretic realizations of Baars’ global workspace model. Many diﬀerent physiological systems, then, can support rapidly shifting, highly tunable, and even simultaneous assemblages of interacting unconscious cognitive modules. Thus [75] concludes the variety of possibilities suggests minds today may be only a small surviving fraction of ancient evolutionary radiations – bush phylogenies of consciousness pruned by selection and chance extinction. Even in the realms of rapid global broadcast inherent to real time cognition, [75] speculates, following a long tradition, that ancient backbrain structures instantiate rapid emotional responses, while the newer forebrain harbors rapid ‘reasoned’ responses in animal consciousness. The cooperation and competition of these two rapid phenomena produces, of course, a plethora of systematic behaviors. Since consciousness is necessarily restricted to realms of a few hundred milliseconds, evolutionary pruning may well have resulted in only a small surviving fraction of previous evolutionary radiations. Processes operating on longer timescales may well be spared such draconian evolutionary selection. That is, the vast spectrum of mathematical models of cognitive gene expression inherent to our analysis here, in the context of development times much longer than a few hundred milliseconds, implies current organisms may simultaneously harbor several, possibly many, quite diﬀerent cognitive gene expression mechanisms. It seems likely, then, that, with some generality, slow phenomena, ranging from institutional distributed cognition to cognitive gene expression, permit the operation of very many quite diﬀerent cognitive processes simultaneously or in rapid succession. One inference is, then, that gene expression and its epigenetic regulation are, for even very simple organisms, far more complex than individual human consciousness, currently regarded as one of the ‘really big’ unsolved scientiﬁc problems. Neural network models adapted or abducted from inadequate cognitive studies of a generation ago are unlikely to cleave the Gordian Knot of scientiﬁc inference surrounding gene expression.

11

Epigenetic Focus

The Tuning Theorem analysis of the Appendix permits an inattentional blindness/concentrated focus perspective on the famous computational ‘no free lunch’ theorem of [81, 82]. Following closely the arguments of [28], [81, 82] have established that there exists no generally superior function optimizer. There is no ‘free

Code, Context, and Epigenetic Catalysis in Gene Expression

307

lunch’ in the sense that an optimizer ‘pays’ for superior performance on some functions with inferior performance on others. If the distribution of functions is uniform, then gains and losses balance precisely, and all optimizers have identical average performance. The formal demonstration depends primarily upon a theorem that describes how information is conserved in optimization. This Conservation Lemma states that when an optimizer evaluates points, the posterior joint distribution of values for those points is exactly the prior joint distribution. Put simply, observing the values of a randomly selected function does not change the distribution: An optimizer has to ‘pay’ for its superiority on one subset of functions with inferiority on the complementary subset. As [28] describes, anyone slightly familiar with the evolutionary computing literature recognizes the paper template ‘Algorithm X was treated with modiﬁcation Y to obtain the best known results for problems P1 and P2 .’ Anyone who has tried to ﬁnd subsequent reports on ‘promising’ algorithms knows that they are extremely rare. Why should this be? A claim that an algorithm is the very best for two functions is a claim that it is the very worst, on average, for all but two functions. It is due to the diversity of the benchmark set of test problems that the ‘promise’ is rarely realized. Boosting performance for one subset of the problems usually detracts from performance for the complement. Reference [28] argues that hammers contain information about the distribution of nail-driving problems. Screwdrivers contain information about the distribution of screw-driving problems. Swiss army knives contain information about a broad distribution of survival problems. Swiss army knives do many jobs, but none particularly well. When the many jobs must be done under primitive conditions, Swiss army knives are ideal. Thus, according to [28], the tool literally carries information about the task optimizers are literally tools-an algorithm implemented by a computing device is a physical entity. Another way of looking at this is to recognize that a computed solution is simply the product of the information processing of a problem, and, by a very famous argument, information can never be gained simply by processing. Thus a problem X is transmitted as a message by an information processing channel, Y , a computing device, and recoded as an answer. By the Tuning Theorem argument of the Appendix there will be a channel coding of Y which, when properly tuned, is most eﬃciently transmitted by the problem. In general, then, the most eﬃcient coding of the transmission channel, that is, the best algorithm turning a problem into a solution, will necessarily be highly problem-speciﬁc. Thus there can be no best algorithm for all equivalence classes of problems, although there may well be an optimal algorithm for any given class. The tuning theorem form of the No Free Lunch theorem will apply quite generally to cognitive biological and social structures, as well as to massively parallel machines. Rate distortion, however, occurs when the problem is collapsed into a smaller, simpliﬁed, version and then solved. Then there must be a tradeoﬀ between allowed average distortion and the rate of solution: the retina eﬀect. In a very fundamental

308

R. Wallace and D. Wallace

sense – particularly for real time systems – rate distortion manifolds present a generalization of the converse of the no free lunch arguments. The neural corollary is known as inattentional blindness [69]. We are led to suggest that there may well be a considerable set of no free lunch-like conundrums confronting highly parallel real-time structures, including epigenetic control of gene expression, and that they may interact in distinctly complicated ways.

12 12.1

Developmental Disorders Network Information Theory

Let U be an information source representing a systematic embedding environmental ‘program’ interacting with the process of cognitive gene expression, here deﬁned as a complicated information set of sources having source joint uncertainty H(Z1 , ..., Zn ) that guides the system into a particular equivalence class of desired developmental behaviors and trajectories. To model the eﬀect of U on development one can, most simply, invoke results from network information theory, ([20], p. 388). Given three interacting information sources, say Y1 , Y2 , Z, the splitting criterion between high and low probability sets of states, taking Z as the external context, is given by I(Y1 , Y2 |Z) = H(Z) + H(Y1 |Z) + H(Y2 |Z) − H(Y1 , Y2 , Z), where, again, H(...|...) and H(..., ..., ...) represent conditional and joint uncertainties. This generalizes to the relation I(Y1 , ..., Yn |Z) = H(Z) +

n

H(Yj |Z) − H(Y1 , ..., Yn , Z).

j=1

Thus the fundamental splitting criterion between low and high probability sets of joint developmental paths becomes I(Z1 , ..., Zn |U ) = H(U ) +

n

H(Zj |U ) − H(Z1 , ..., Zn , U ).

(20)

j=1

Again, the Zi represent internal information sources and U that of the embedding environmental context. The central point is that a one step extension of that system via the results of network information theory [20] allows incorporating the eﬀect of an external environmental ‘farmer’ in guiding cognitive developmental gene expression. 12.2

Embedding Ecosystems as Information Sources

The principal farmer for a developing organism is the ecosystem in which it is embedded, in a large sense. Summarizing brieﬂy the arguments of [74], ecosystems, under appropriate coarse graining, often have reconizable grammar and

Code, Context, and Epigenetic Catalysis in Gene Expression

309

syntax. For example, the turn-of-the-seasons in a temperate climate, for most natural communities, is remarkably similar from year to year in the sense that the ice melts, migrating birds return, trees bud, ﬂowers and grass grow, plants and animals reproduce, the foliage turns, birds migrate, frost, snow, the rivers freeze, and so on in a predictable manner from year to year. Suppose, then, that we can coarse grain an ecosystem at time t according to some appropriate partition of the phase space in which each division Aj represents a particular range of numbers for each possible species in the ecosystem, along with associated parameters such as temperature, rainfall, humidity, insolation, and so on. We examine longitudinal paths, statements of the form x(n) = A0 , A1 , ..., An deﬁned in terms of some ‘natural’ time unit characteristic of the system. Then n corresponds to a time unit T , so that t = T, 2T, ..., nT . Our interest is in the serial correlation along paths. If N (n) is the number of possible paths of length n that are consistent with the underlying grammar and syntax of the appropriately coarse grained ecosystem, for example, spring leads to summer, autumn, winter, back to spring, etc., but never spring to autumn to summer to winter in a temperate climate. The essential assumption is that, for appropriate coarse graining, N (n), the number of possible grammatical paths, is much smaller than the total conceivable number of paths, and that, in the limit of large n, log[N (n)] n→∞ n

H = lim

both exists and is independent of path. Not all possible ecosystem coarse grainings are likely to lead to this result, as is sometimes the case with Markov models. Reference [40] in particular emphasizes that mesoscale ecosystem processes are most likely to entrain dynamics at larger and smaller scales, a process [74] characterizes as mesoscale resonance, a generalization of the Baldwin eﬀect. See that reference for details, broadly based on the Tuning Theorem. 12.3

Ecosystems Farm Organismal Development

The environmental and ecosystem farming of development may not always be benign. Suppose we can operationalize and quantify degrees of both overfocus or inattentional blindness (IAB) and of overall structure or environment distortion (D) in the actions of a highly parallel cognitive epigenetic regulatory system. The essential assumption is that the (internal) dual information source of a cognitive structure that has low levels of both IAB overfocus and structure/environment distortion will tend to be richer than that of one having greater levels. This is shown in ﬁgure 1a, where H is the source uncertainty dual to internal cognitive process, X = IAB, and Y = D. Regions of low X, Y , near the origin,

310

R. Wallace and D. Wallace

have greater source uncertainty than those nearby, so H(X, Y ) shows a (relatively gentle) peak at the origin, taken here as simply the product of two error functions. We are, then, particularly interested in the internal cognitive capacity of the structure itself, as paramatized by degree of overfocus and by the (large scale) distortion between implementation and impact. That capacity, a purely internal quantity, need not be convex in the parameter D, which is taken to characterize interaction with an external environment, and thus becomes a context for internal measures. Such measures need not themselves be convex in D. The generalized Onsager argument, based on the homology between information source uncertainty and free energy, as explained more fully in the Appendix, is shown in ﬁgure 1b. S = H(X, Y )− XdH/dX − Y dH/dY , the ‘disorder’ analog to entropy in a physical system, is graphed on the Z axis against the X − Y plane, assuming a gentle peak in H at the origin. Peaks in S, according to theory, constitute repulsive system barriers, which must be overcome by external forces. In ﬁgure 1b there are three quasi-stable topological resilience modes, in the sense of [71], marked as A, B, and C. The A region is locked in to low levels of both overfocus and distortion, as it sits in a pocket. Forcing the system in either direction, that is, increasing either IAB or D, will, initially, be met by homeostatic attempts to return to the resilience state A, according to this model. If overall distortion becomes severe in spite of homeostatic developmental mechanisms, the system will then jump to the quasi-stable state B, a second pocket. According to the model, however, once that transition takes place, there will be a tendency for the system to remain in a condition of high distortion. That is, the system will become locked-in to a structure with high distortion in the match between structure implementation and structure impact, but one having lower overall cognitive capacity, i.e., a lower value of H in ﬁgure 1a. The third pocket, marked C, is a broad plain in which both IAB and D remain high, a highly overfocused, poorly linked pattern of behavior which will require signiﬁcant intervention to alter once it reaches such a quasi-stable resilience mode. The structure’s cognitive capacity, measured by H in ﬁgure 1a, is the lowest of all for this condition of pathological resilience, and attempts to correct the problem – to return to condition A, will be met with very high barriers in S, according to ﬁgure 1b. That is, mode C is very highly resilient, although pathologically so, much like the eutrophication of a pure lake by sewage outﬂow. See [70, 71] for discussions of ecological resilience and literature references. We can argue that the three quasi-equilibrium conﬁgurations of ﬁgure 1b represent diﬀerent dynamical states of the system, and that the possibility of transition between them represents the breaking of the associated symmetry groupoid by external forcing mechanisms. That is, three manifolds representing three diﬀerent kinds of system dynamics have been patched together to create a more complicated topological structure. For cognitive phenomena, such behavior is likely to be the rule rather than the exception. ‘Pure’ groupoids are abstractions, and the fundamental questions will involve linkages which break the underlying symmetry.

Code, Context, and Epigenetic Catalysis in Gene Expression

311

Fig. 1. a. Source uncertainty, H, of the dual information source of epigenetic cognition, as parametized by degrees of focus, X = IAB and distortion, Y = D, between implementation and actual impact. Note the relatively gentle peak at low values of X, Y . Here H is generated as the product of two error functions. b. Generalized Onsager treatment of ﬁgure 1a. S = H(X, Y ) − XdH/dX − Y dH/dY . The regions marked A, B, and C represent realms of resilient quasi-stability, divided by barriers deﬁned by the relative peaks in S. Transition among them requires a forcing mechanism. From another perspective, limiting energy or other resources, or imposing stress from the outside, driving down H in ﬁgure 1a, would force the system into the lower plain of C, in which the system would then become trapped in states having high levels of distortion and inattentional blindness/overfocus.

In all of this, as in equation (19), system convergence is not to some ﬁxed state, limit cycle, or pseudorandom strange attractor, but rather to some appropriate set of highly dynamic information sources, i.e., behavior patterns constituting, here, developmental trajectories, rather than to some ﬁxed ‘answer to a computing problem’ [72]. What this model suggests is that suﬃciently strong external perturbation can force a highly parallel real-time cognitive epigenetic structure from a normal, almost homeostatic, developmental path into one involving a widespread, comorbid, developmental disorder. This is a well studied pattern for humans and their institutions, reviewed at some length elsewhere [71, 73]. Indeed, this argument provides the foundation of a fairly comprehensive model of chronic developmental dysfunction across a broad class of cognitive systems, including, but not limited to, cognitive epigenetic control of gene expression. One approach might be as follows: A developmental process can be viewed as involving a sequence of surfaces like ﬁgure 1, having, for example, ‘critical periods’ when the barriers between

312

R. Wallace and D. Wallace

the normal state A and the pathological states B and C are relatively low. This might particularly occur under circumstances of rapid growth or long-term energy demand, since the peaks of ﬁgure 1 are inherently energy maxima by the duality between information source uncertainty and free energy density. During such a time the peaks of ﬁgure 1 might be relatively suppressed, and the system would become highly sensitive to perturbation, and to the onset of a subsequent pathological developmental trajectory. To reiterate, then, during times of rapid growth, embryonic de- and re- methylation, and/or other high system demand, metabolic energy limitation imposes the need to focus via something like a rate distortion manifold. Cognitive process requires energy through the homologies with free energy density, and more focus at one end necessarily implies less at some other. In a distributed zero sum developmental game, as it were, some cognitive or metabolic processes must receive more free energy than others, and these may then be more easily aﬀected by external chemical, biological, or social stressors, or by simple stochastic variation. Something much like this has indeed become a standard perspective (e.g., [76]). A structure trapped in region C might be said to suﬀer something much like what [80] describes as the loss of gradient problem, in which one part of a multiple population coevolutionary system comes to dominate the others, creating an impossible situation in which the other participants do not have enough information from which to learn. That is, the cliﬀ just becomes too steep to climb. Reference [80] also characterizes focusing problems in which a two-population coevolutionary process becomes overspecialized on the opponent’s weaknesses, eﬀectively a kind of inattentional blindness. Thus there seems some consonance between our asymptotic analysis of cognitive structural function and current studies of pathologies aﬀecting coevolutionary algorithms (e.g. [30, 72]). In particular the possibility of historic trajectory, of path dependence, in producing individualized failure modes, suggests there can be no one-size-ﬁts-all amelioration strategy. Equation (20) basically enables a kind of environmental catalysis to cognitive gene expression, in a sense closely similar to the arguments of Section 4. This is analogous to, but more general than, the ‘mesoscale resonance’ invoked by [74]: during critical periods, according to these models, environmental signals can have vast impact on developmental trajectory. 12.4

A Simple Probability Argument

Again, critical periods of rapid growth require energy, and by the homology between free energy density and cognitive information source uncertainty, that energy requirement may be in the context of a zero-sum game so that the barriers of ﬁgure 1 may be lowered by metabolic energy constraints or high energy demand. In particular the groupoid structure of equation (5) changes progressively as the organism develops, with new equivalence classes being added to A = ∪α. If metabolic energy remains capped, then exp[−Hβ K] P [Hβ ] = α exp[−Hα K]

Code, Context, and Epigenetic Catalysis in Gene Expression

313

must decrease with increase in α, i.e., with increase in the cardinality of A. Thus, for restricted K, barriers between diﬀerent developmental paths must fall as the system becomes more complicated. A precis of these results can be more formally captured using methods closely similar to recent algebraic geometry approaches to concurrent, i.e., highly parallel, computing [26, 37, 60].

13

Reconsidering Directed Homotopy: Shadows

Here we reconsider directed homotopy in a developmental context, as shadowed by critical developmental periods. First, we restrict the analysis to a two dimensional phenotype space, and begin development at some S0 as in ﬁgure 2. If one requires temporal path dependence – no reverse development – then ﬁgure 2 shows two possible ﬁnal states, S1 and S2 , separated by a critical point C that casts a path-dependent developmental shadow in time. There are, consequently, two separate ‘ways’ of reaching a ﬁnal state in this model. The Si thus represent (relatively) static phenotypic expressions of the solutions to equation (19) that are, of themselves, highly dynamic information sources. Elements of each ‘way’ can be transformed into each other by continuous deformation without crossing the impenetrable shadow cast by the critical period C. These ways are the equivalence classes deﬁning the system’s topological structure, a groupoid analogous to the fundamental homotopy group in spaces that admit of loops [50] rather than time-driven, one-way paths. That is, the closed loops needed for classical homotopy theory are impossible for this kind of system because of the ‘ﬂow of time’ deﬁning the output of an information source – one goes from S0 to some ﬁnal state. The theory is thus one of directed homotopy, dihomotopy, and the central question revolves around the continuous deformation of paths in development space into one another, without crossing the shadow cast by the critical period C. Reference [36] provides another introduction to the formalism. Thus the external signals U of equation (20), as a catalytic mechanism, can deﬁne quite diﬀerent developmental dihomotopies. Such considerations suggest that a multitasking developmental process that becomes trapped in a particular pattern cannot, in general, expect to emerge from it in the absence of external forcing mechanisms or the stochastic resonance/mutational action of ‘noise’. Emerging from such a trap involves largescale topological changes, and this is the functional equivalent of a ﬁrst order phase transition in a physical systems and requires energy. The fundamental topological insight is that environmental context – the U in equation (20) – can be imposed on the ‘natural’ groupoids underlying massively parallel gene expression. This sort of behavior is, as noted in [71], central to ecosystem resilience theory. Apparently the set of developmental manifolds, and its subsets of directed homotopy equivalence classes, formally classiﬁes quasi-equilibrium states, and

314

R. Wallace and D. Wallace

Fig. 2. Given an initial developmental state S0 and a critical period C casting a pathdependent developmental shadow, there are two directed homotopy equivalence classes of deformable paths leading, respectively, to ﬁnal phenotype states S1 and S2 that are expressions of the highly dynamic information source solutions to equation (19). These equivalence classes deﬁne a topological groupoid on the developmental system.

thus characterizes the diﬀerent possible developmental resilience modes. Some of these may be highly pathological. Shifts between markedly diﬀerent topological modes appear to be necessary effects of phase transitions, involving analogs to phase changes in physical systems. It seems clear that both ‘normal development’ and possible pathological states can be represented as topological resilience/phase modes in this model, suggesting a real equivalence between diﬃculties in carrying out gene expression and its stabilization. This mirrors recent results on the relation between programming diﬃculty and system stability in highly parallel computing devices [70].

14

Epigenetic Programming of Artificial Systems for Biotechnology

Reference [72] examines how highly parallel ‘Self-X’ computing machines – selfprogramming, protecting, repairing, etc. – are inevitably coevolutionary in the sense of Section 9 above, since elements of a dynamic structural hierarchy always interact, an eﬀect that will asymptotically dominate system behavior at great

Code, Context, and Epigenetic Catalysis in Gene Expression

315

scale. The ‘farming’ paradigm provides a model for programming such devices, that, while broadly similar to the liquid state machines of [51], diﬀers in that convergence is to an information source, a systematic dynamic behavior pattern, rather than to a computed ﬁxed ‘answer’. As the farming metaphor suggests, stabilizing complex coevolutionary mechanisms appears as diﬃcult as programming them. Suﬃciently large networks of even the most dimly cognitive modules will become emergently coevolutionary, suggesting the necessity of ‘second order’ evolutionary programming that generalizes the conventional Nix/Vose models. Although we cannot pursue the argument in detail here, very clearly such an approach to programming highly parallel coevolutionary machines – equivalent to deliberate epigenetic farming – should be applicable to a broad class of artiﬁcial biological systems/machines for which some particular ongoing behavior is to be required, rather than some ﬁnal state ‘answer’. Examples might include the manufacture, in a large sense, of a dynamic product, e.g., a chemical substance, anti-cancer or artiﬁcial immune search-and-destroy strategy, biological signal detection/transduction process, and so on. Tunable epigenetic catalysis lowers an ‘eﬀective energy’ associated with the convergence of a highly coevolutionary cognitive system to a ﬁnal dynamic behavioral strategy. Given a particular ‘farming’ information source acting as the program, the behavior of the ﬁnal state of interest will become associated with the lowest value of the free energy-analog, possibly calculable by optimization methods. If the retina-like rate distortion manifold has been properly implemented, a kind of converse to the no free lunch theorem, then this optimization procedure should converge to an appropriate solution, ﬁxed or dynamic. Thus we invoke a synergism between the focusing theorem and a ‘tunable epigenetic catalysis theorem’ to raise the probability of an acceptable solution, particularly for a real-time system whose dynamics will be dominated by rate distortion theorem constraints. The degree of catalysis needed for convergence in a real time system would seem critically dependent on the rate distortion function R(D) or on its product with an acceptable reaction time, τ , that is, on there being suﬃcient bandwidth in the communication between a cognitive biological ‘machine’ and its embedding environment. If that bandwidth is too limited, or the available reaction time too short, then the system will inevitably freeze out into what amounts to a highly dysfunctional ‘ground state’. The essential point would seem to be a convergence between emerging needs in biotechnology and general strategies for programming coevolutionary computing devices.

15

Discussion and Conclusions

We have hidden the kind of massive calculations made explicit in [16, 17], burying them as ‘ﬁtting regression-model analogs to data’, possibly at a second order epigenetic hierarchical level. In the real world such calculations would be quite diﬃcult, particularly given the introduction of punctuated transitions that must

316

R. Wallace and D. Wallace

be ﬁtted using elaborate renormalization calculations, typically requiring such exotic objects as Lambert W-functions (e.g., [68, 73, 74]). Analogies with neural network studies suggest, however, intractable conceptual diﬃculties for spinglass-type models of gene expression and development dynamics, much as claimed by [57]. In spite of nearly a century of sophisticated neural network model studies – including elegant treatments like [66] – Atmanspacher [3] claims that to formulate a serious, clear-cut and transparent formal framework for cognitive neuroscience is a challenge comparable to the early stage of physics four centuries ago. Only a very few contemporary approaches, including that of [68], are worth mentioning, in his view. Furthermore, [48] has identiﬁed what might well be described as the suﬃciency failing of neural network models, that is, neural networks can be constructed as Turing machines that can replicate any known dynamic behavior in the same sense that the Ptolemaic Theory of planetary motion, as a Fourier expansion in epicycles, can, to suﬃcient order, mimic any observed orbit. Keplerian central motion provides an essential reduction. The particular characterization of [48] is that ‘neural possibility is not neural plausibility’. Likewise, [8] concludes that neural-centered explanations of high order mental function commit the mereological fallacy, that is, the fundamental logical error of attributing what is in fact a property of an entirety to a limited part of the whole system. ‘The brain’ does not exist in isolation, but as part of a complete biological individual who is most often deeply embedded in social and cultural contexts. Neural network-like models of gene expression and development applied to complex living things inherently commit both errors, particularly in a social, cultural, or environmental milieu. This suggests a particular necessity for the formal inclusion of the eﬀects of embedding contexts – the epigenetic Z and the environmental U – in the sense of [4, 5]. That is, gene expression and development are conditioned by signals from embedding physiological, social, and for humans, cultural, environments. As described above, our formulation can include such inﬂuences in a highly natural manner, as they inﬂuence epigenetic catalysis. In addition, multiple, and quite diﬀerent, cognitive gene expression mechanisms may operate simultaneously, or in appropriate sequence, given suﬃcient development time. Although epigenetic catalysis, as we have explored it here, might seem worthy of special focus, this would be a kind of intellectual optical illusion akin to inattentional blindness. Epigenetic catalysis is only one aspect of a general cognitive paradigm for gene expression, and this larger, and very complicated ‘perceptual ﬁeld’ should remain the center of intellectual attention, rather than any single element of that ﬁeld. This is to take, perhaps, an ‘East Asian’ rather than ‘Western’ perspective on the matter [69]. Developmental disorders, in a broad sense that must include comorbid mental and physical characteristics, emerge as pathological ‘resilience’ modes, in the sense of [71], a viewpoint from ecosystem theory quite similar to that of epigenetic epidemiology [32, 76]. Environmental farming through an embedding

Code, Context, and Epigenetic Catalysis in Gene Expression

317

information source aﬀecting internal epigenetic regulation of gene expression, can, as a kind of programming of a highly parallel cognitive system, place the organism into a quasi-stable pathological developmental pattern converging on a dysfunctional phenotype. The probability models of cognitive process presented here will lead, most fundamentally, to statistical tools based on the asymptotic limit theorems of information theory, in the same sense that the usual parametric statistics are based on the Central Limit Theorem. We have not, then, given ‘a’ model of development and its disorders in cognitive gene expression, but, rather, outlined a possible general strategy for ﬁtting empirically-determined statistical models to real data, in precisely the sense that one would ﬁt the usual parametric statistical models to normally distributed data. The ﬁtting of statistical models does not, of itself, perform scientiﬁc inference. That is done by comparing ﬁtted models for similar systems under diﬀerent, or diﬀerent systems under similar, conditions, and by examining the structure of residuals. One implication of this work, then, is that understanding complicated processes of gene expression and development – and their pathologies – will require construction of data analysis tools considerably more sophisticated than now available, including the present crop of simple models abducted from neural network studies or stochastic chemical reaction theory. Most centrally, however, currently popular (and fundable) reductionist approaches to understanding gene expression must eventually exhaust themselves in the same desert of sand-grain hyperparticularity that appears to have driven James Crick from molecular biology into consciousness studies, a ﬁeld now mature enough to provide tools for use in the other direction.

Acknowledgments The author thanks Dr. C. Guerrero-Bosagna and two anonymous reviewers for comments useful in revision.

References 1. Ash, R.: Information Theory. Dover Publications, New York (1990) 2. Atlan, H., Cohen, I.: Immune information, self-organization, and meaning. International Immunology 10, 711–717 (1998) 3. Atmanspacher, H.: Toward an information theoretical implementation of contextual conditions for consciousness. Acta Biotheoretica 54, 157–160 (2006) 4. Baars, B.: A Cognitive Theory of Consciousness. Cambridge University Press, New York (1988) 5. Baars, B.: Global workspace theory of consciousness: toward a cognitive neuroscience of human experience. Progress in Brain Research 150, 45–53 (2005) 6. Backdahl, L., Bushell, A., Beck, S.: Inﬂammatory signalling as mediator of epigenetic modulation in tissue-speciﬁc chronic inﬂammation. The International Journal of Biochemistry and Cell Biology (2009), doi:10.1016/j.biocel.2008.08.023

318

R. Wallace and D. Wallace

7. Beck, C., Schlogl, F.: Thermodynamics of Chaotic Systems. Cambridge University Press, Cambridge (1995) 8. Bennett, M., Hacker, P.: Philosophical Foundations of Neuroscience. Blackwell Publishing, Malden (2003) 9. Bennett, C.: Logical depth and physical complexity. In: Herkin, R. (ed.) The Universal Turing Machine: A Half-Century Survey, pp. 227–257. Oxford University Press, Oxford (1988) 10. Bos, R.: Continuous representations of groupoids. arXiv:math/0612639 (2007) 11. Bossdorf, O., Richards, C., Pigliucci, M.: Epigenetics for ecologists. Ecology Letters 11, 106–115 (2008) 12. Britten, R., Davidson, E.: Gene regulation for higher cells: a theory. Science 165, 349–357 (1969) 13. Brown, R.: From groups to groupoids: a brief survey. Bulletin of the London Mathematical Society 19, 113–134 (1987) 14. Buneci, M.: Representare de Groupoizi. Editura Mirton, Timisoara (2003) 15. Cannas Da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras. American Mathematical Society, RI (1999) 16. Ciliberti, S., Martin, O., Wagner, A.: Robustness can evolve gradually in complex regulatory networks with varying topology. PLoS Computational Biology 3(2), e15 (2007) 17. Ciliberti, S., Martin, O., Wagner, A.: Innovation and robustness in complex regulatory gene networks. Proceedings of the National Academy of Sciences 104, 13591–13596 (2007) 18. Cohen, I.: Immune system computation and the immunological homunculus. In: Nierstrasz, O., Whittle, J., Harel, D., Reggio, G. (eds.) MoDELS 2006. LNCS, vol. 4199, pp. 499–512. Springer, Heidelberg (2006) 19. Cohen, I., Harel, D.: Explaining a complex living system: dynamics, multi-scaling, and emergence. Journal of the Royal Society: Interface 4, 175–182 (2007) 20. Cover, T., Thomas, J.: Elements of Information Theory. John Wiley and Sons, New York (1991) 21. Crews, D., McLachlan, J.A.: Epigenetics, evolution, endocrine disruption, health, and disease. Endocrinology 147, S4–S10 (2006) 22. Crews, D., Gore, A., Hsu, T., Dangleben, N., Spinetta, M., Schallert, T., Anway, M., Skinner, M.: Transgenerational epigenetic imprints on mate preference. Proceedings of the National Academy of Sciences 104, 5942–5946 (2007) 23. Dehaene, S., Naccache, L.: Towards a cognitive neuroscience of consciousness: basic evidence and a workspace framework. Cognition 79, 1–37 (2001) 24. Dembo, A., Zeitouni, O.: Large Deviations: Techniques and Applications, 2nd edn. Springer, New York (1998) 25. Dias, A., Stewart, I.: Symmetry groupoids and admissible vector ﬁelds for coupled cell networks. Journal of the London Mathematical Society 69, 707–736 (2004) 26. Dretske, F.: The explanatory role of information. Philosophical Transactions of the Royal Society A 349, 59–70 (1994) 27. Emery, M.: Stochastic Calculus on Manifolds. Springer, New York (1989) 28. English, T.: Evaluation of evolutionary and genetic optimizers: no free lunch. In: Fogel, L., Angeline, P., Back, T. (eds.) Evolutionary Programming V: Proceedings of the Fifth Annual Conference on Evolutionary Programming, pp. 163–169. MIT Press, Cambridge (1996) 29. Erdos, P., Renyi, A.: On the evolution of random graphs (1960); reprinted in The Art of Counting, pp. 574–618 (1973), and in Selected Papers of Alfred Renyi, pp. 482–525 (1976)

Code, Context, and Epigenetic Catalysis in Gene Expression

319

30. Ficici, S., Milnik, O., Pollak, J.: A game-theoretic and dynamical systems analysis of selection methods in coevolution. IEEE Transactions on Evolutionary Computation 9, 580–602 (2005) 31. Feynman, R.: Lectures on Computation. Westview Press, New York (2000) 32. Foley, D., Craid, J., Morley, R., Olsson, C., Dwyer, T., Smith, K., Saﬀery, R.: Prospects for epigenetic epidemiology. American Journal of Epidemiology 169, 389–400 (2009) 33. Gilbert, S.: Mechanisms for the environmental regulation of gene expression: ecological aspects of animal development. Journal of Bioscience 30, 65–74 (2001) 34. Glazebrook, J.F., Wallace, R.: Small worlds and red queens in the global workspace: an information-theoretic approach. Cognitive Systems Reserch (2009), doi:10.1016/j.cogsys.2009.01.002 35. Golubitsky, M., Stewart, I.: Nonlinear dynamics and networks: the groupoid formalism. Bulletin of the American Mathematical Society 43, 305–364 (2006) 36. Goubault, E., Raussen, M.: Dihomotopy as a tool in state space analysis. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 16–37. Springer, Heidelberg (2002) 37. Goubault, E.: Some geometric perspectives on concurrency theory. Homology, Homotopy, and Applications 5, 95–136 (2003) 38. Gould, S.: The Structure of Evolutionary Theory. Harvard University Press, Cambridge (2002) 39. Guerrero-Bosagna, C., Sabat, P., Valladares, L.: Environmental signaling and evolutionary change: can exposure of pregnant mammals to environmental estrogens lead to epigenetically induced evolutionary changes in embryos? Evolution and Development 7, 341–350 (2005) 40. Holling, C.: Cross-scale morphology, geometry and dynamicsl of ecosystems. Ecological Monographs 41, 1–50 (1992) 41. Jablonka, E., Lamb, M.: Epigenetic Inheritance and Evolution: The Lamarckian Dimension. Oxford University Press, Oxford (1995) 42. Jablonka, E., Lamb, M.J.: Epigenetic inheritance in evolution. Journal of Evolutionary Biology 11, 159–183 (1998) 43. Jablonka, E.: Epigenetic epidemiology. International Journal of Epidemiology 33, 929–935 (2004) 44. Jaeger, J., Surkova, S., Blagov, M., Janssens, H., Kosman, D., Kozlov, K., Manu, M., Myasnikova, E., Vanario-Alonso, C., Samsonova, M., Sharp, D., Reintiz, J.: Dynamic control of positional information in the early Drosophila embryo. Nature 430, 368–371 (2004) 45. Jaenisch, R., Bird, A.: Epigenetic regulation of gene expression: how the genome integrates intrinsic and environmental signals. Nature Genetics Supplement 33, 245–254 (2003) 46. Kastner, M.: Phase transitions and conﬁguration space topology. ArXiv condmat/0703401 (2006) 47. Khinchin, A.: Mathematical Foundations of Information Theory. Dover, New York (1957) 48. Krebs, P.: Models of cognition: neurological possibility does not indicate neurological plausibility. In: Bara, B., Barsalou, L., Bucciarelli, M. (eds.) Proceedings of CogSci 2005, Stresa, Italy, pp. 1184–1189 (2005), http://cogprints.org/4498/ 49. Landau, L., Lifshitz, E.: Statistical Physics, Part I, 3rd edn., Part I. Elsevier, New York (2007) 50. Lee, J.: Introduction to topological manifolds. Springer, New York (2000)

320

R. Wallace and D. Wallace

51. Maas, W., Natschlager, T., Markram, H.: Real-time computing without stable states: a new framework for neural computation based on perturbations. Neural Computation 14, 2531–2560 (2002) 52. Matsumoto, Y.: An Introduction to Morse Theory. American Mathematical Society, Providence (2002) 53. Maturana, H.R., Varela, F.J.: Autopoiesis and Cognition. Reidel Publishing Company, Dordrecht (1980) 54. Maturana, H.R., Varela, F.J.: The Tree of Knowledge. Shambhala Publications, Boston (1992) 55. McCauly, J.: Chaos, Dynamics, and Fractals. Cambridge Nonlinear Science Series, Cambridge, UK (1994) 56. Mjolsness, E., Sharp, D., Reinitz, J.: A connectionist model of development. Journal of Theoretical Biology 152, 429–458 (1991) 57. O’Nuallain, S.: Code and context in gene expression, cognition, and consciousness. In: Barbiere, M. (ed.) The Codes of Life: The Rules of Macroevolution, ch. 15, pp. 347–356. Springer, New York (2008) 58. O’Nuallain, S., Strohman, R.: Genome and natural language: how far can the analogy be extended? In: Witzany, G. (ed.) Proceedings of Biosemiotics. Tartu University Press, Umweb (2007) 59. Pettini, M.: Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics. Springer, New York (2007) 60. Pratt, V.: Modeling concurrency with geometry. In: Proceedings of the 18th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 311–322 (1991) 61. Reinitz, J., Sharp, D.: Mechanisms of even stripe formation. Mechanics of Development 49, 133–158 (1995) 62. Scherrer, K., Jost, J.: The gene and the genon concept: a functional and information-theoretic analysis. Molecular Systems Biology 3, 87–95 (2007) 63. Scherrer, K., Jost, J.: Gene and genon concept: coding versus regulation. Theory in Bioscience 126, 65–113 (2007) 64. Sharp, D., Reinitz, J.: Prediction of mutant expression patterns using gene circuits. BioSystems 47, 79–90 (1998) 65. Skierski, M., Grundland, A., Tuszynski, J.: Analysis of the three-dimensional timedependent Landau-Ginzburg equation and its solutions. Journal of Physics A (Math. Gen.) 22, 3789–3808 (1989) 66. Toulouse, G., Dehaene, S., Changeux, J.: Spin glass model of learning by selection. Proceedings of the National Academy of Sciences 83, 1695–1698 (1986) 67. Turner, B.: Histone acetylation and an epigeneticv code. Bioessays 22, 836–845 (2000) 68. Wallace, R.: Consciousness: A Mathematical Treatment of the Global Neuronal Workspace Model. Springer, New York (2005) 69. Wallace, R.: Culture and inattentional blindness. Journal of Theoretical Biology 245, 378–390 (2007) 70. Wallace, R.: Toward formal models of biologically inspired, highly parallel machine cognition. International Journal of Parallel, Emergent, and Distributed Systems 23, 367–408 (2008) 71. Wallace, R.: Developmental disorders as pathological resilience domains. Ecology and Society 13(1), 29 (2008), http://www.ecologyandsociety.org/vol13/iss1/art29/ 72. Wallace, R.: Programming coevolutionary machines: the emerging conundrum. International Journal of Parallel, Emergent, and Distributed Systems (in press, 2009)

Code, Context, and Epigenetic Catalysis in Gene Expression

321

73. Wallace, R., Fullilove, M.: Collective Consciousness and its Discontents: Institutional Distributed Cognition, Racial Policy, and Public Health in the United States. Springer, New York (2008) 74. Wallace, R., Wallace, D.: Punctuated equilibrium in statistical models of generalized coevolutionary resilience: how sudden ecosystem transitions can entrain both phenotype expression and Darwinian selection. In: Priami, C. (ed.) Transactions on Computational Systems Biology IX. LNCS (LNBI), vol. 5121, pp. 23–85. Springer, Heidelberg (2008) 75. Wallace, R.G., Wallace, R.: Evolutionary radiation and the spectrum of consciousness. Consciousness and Cognition (2009), doi:10.1016/j.concog.2008.12.002 76. Waterland, R., Michels, K.: Epigenetic epidemiology of the developmental origins hypothesis. Annual Reviews of Nutrition 27, 363–388 (2007) 77. Weaver, I.: Epigenetic eﬀects of glucocorticoids. Seminars in Fetal and Neonatal Medicine (2009), doi:10.1016/j.siny.2008.12.002 78. Weinstein, A.: Groupoids: unifying internal and external symmetry. Notices of the American Mathematical Association 43, 744–752 (1996) 79. West-Eberhard, M.: Developmental plasticity and the origin of species diﬀerences. Proceedings of the National Academy of Sciences 102, 6543–6549 (2005) 80. Wiegand, R.: An analysis of cooperative coevolutionary algorithms. PhD Thesis, George Mason University (2003) 81. Wolpert, D., Macready, W.: No free lunch theorems for search. Santa Fe Institute, SFI-TR-02-010 (1995) 82. Wolpert, D., Macready, W.: No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation 1, 67–82 (1997) 83. Wymer, C.R.: Structural nonlinear continuous-time models in econometrics. Macroeconomic Dynamics 1, 518–548 (1997) 84. Zhu, R., Rebirio, A., Salahub, D., Kaufmann, S.: Studying genetic regulatory networks at the molecular level: delayed reaction stochastic models. Journal of Theoretical Biology 246, 725–745 (2007)

16 16.1

Mathematical Appendix The Shannon-McMillan Theorem

According to the structure of the underlying language of which a message is a particular expression, some messages are more ‘meaningful’ than others, that is, are in accord with the grammar and syntax of the language. The ShannonMcMillan or Asymptotic Equipartition Theorem, describes how messages themselves are to be classiﬁed. Suppose a long sequence of symbols is chosen, using the output of a random variable X, so that an output sequence of length n, with the form xn = (α0 , α1 , ..., αn−1 ) has joint and conditional probabilities P (X0 = α0 , X1 = α1 , ..., Xn−1 = αn−1 ) P (Xn = αn |X0 = α0 , ..., Xn−1 = αn−1 ).

322

R. Wallace and D. Wallace

Using these probabilities we may calculate the conditional uncertainty H(Xn |X0 , X1 , ..., Xn−1 ). The uncertainty of the information source, H[X], is deﬁned as H[X] = lim H(Xn |X0 , X1 , ..., Xn−1 ). n→∞

In general

(21)

H(Xn |X0 , X1 , ..., Xn−1 ) ≤ H(Xn ).

Only if the random variables Xj are all stochastically independent does equality hold. If there is a maximum n such that, for all m > 0 H(Xn+m |X0 , ..., Xn+m−1 ) = H(Xn |X0 , ..., Xn−1 ), then the source is said to be of order n. It is easy to show that H[X] = lim

n→∞

H(X0 , ...Xn ) . n+1

In general the outputs of the Xj , j = 0, 1, ..., n are dependent. That is, the output of the communication process at step n depends on previous steps. Such serial correlation, in fact, is the very structure which enables most of what is done in this paper. Here, however, the processes are all assumed statble in time, that is, the probabilities and serial correlations do not change in time, and the system is stationary. A very broad class of such self-correlated, stationary, information sources, the so-called ergodic sources for which the long-run relative frequency of a sequence converges stochastically to the probability assigned to it, have a particularly interesting property: It is possible, in the limit of large n, to divide all sequences of outputs of an ergodic information source into two distinct sets, S1 and S2 , having, respectively, very high and very low probabilities of occurrence, with the source uncertainty providing the splitting criterion. In particular the Shannon-McMillan Theorem states that, for a (long) sequence having n (serially correlated) elements, the number of ‘meaningful’ sequences, N (n) – those belonging to set S1 – will satisfy the relation log[N (n)] ≈ H[X]. n More formally, lim

n→∞

log[N (n)] = H[X] n

= lim H(Xn |X0 , ..., Xn−1 ) n→∞

(22)

Code, Context, and Epigenetic Catalysis in Gene Expression

= lim

n→∞

H(X0 , ..., Xn ) . n+1

323

(23)

Using the internal structures of the information source permits limiting attention only to high probability ‘meaningful’ sequences of symbols. 16.2

The Rate Distortion Theorem

The Shannon-McMillan Theorem can be expressed as the ‘zero error limit’ of the Rate Distortion Theorem [20, 24] which deﬁnes a splitting criterion that identiﬁes high probability pairs of sequences. We follow closely the treatment of [20]. The origin of the problem is the question of representing one information source by a simpler one in such a way that the least information is lost. For example we might have a continuous variate between 0 and 100, and wish to represent it in terms of a small set of integers in a way that minimizes the inevitable distortion that process creates. Typically, for example, an analog audio signal will be replaced by a ‘digital’ one. The problem is to do this in a way which least distorts the reconstructed audio waveform. Suppose the original stationary, ergodic information source Y with output from a particular alphabet generates sequences of the form y n = y1 , ..., yn . These are ‘digitized,’ in some sense, producing a chain of ‘digitized values’ bn = b1 , ..., bn , where the b-alphabet is much more restricted than the y-alphabet. bn is, in turn, deterministically retranslated into a reproduction of the original signal y n . That is, each bm is mapped on to a unique n-length y-sequence in the alphabet of the information source Y : bm → yˆn = yˆ1 , ..., yˆn . Note, however, that many y n sequences may be mapped onto the same retranslation sequence yˆn , so that information will, in general, be lost. The central problem is to explicitly minimize that loss. The retranslation process deﬁnes a new stationary, ergodic information source, Yˆ . The next step is to deﬁne a distortion measure, d(y, yˆ), which compares the original to the retranslated path. For example the Hamming distortion is d(y, yˆ) = 1, y = yˆ d(y, yˆ) = 0, y = yˆ.

(24)

324

R. Wallace and D. Wallace

For continuous variates the Squared error distortion is d(y, yˆ) = (y − yˆ)2 .

(25)

There are many possibilities. The distortion between paths y n and yˆn is deﬁned as 1 d(yj , yˆj ). n j=1 n

d(y n , yˆn ) =

(26)

Suppose that with each path y n and bn -path retranslation into the y-language and denoted y n , there are associated individual, joint, and conditional probability distributions p(y n ), p(ˆ y n ), p(y n |ˆ y n ). The average distortion is deﬁned as D= p(y n )d(y n , yˆn ).

(27)

yn

It is possible, using the distributions given above, to deﬁne the information transmitted from the incoming Y to the outgoing Yˆ process in the usual manner, using the Shannon source uncertainty of the strings: I(Y, Yˆ ) = H(Y ) − H(Y |Yˆ ) = H(Y ) + H(Yˆ ) − H(Y, Yˆ ). If there is no uncertainty in Y given the retranslation Yˆ , then no information is lost. In general, this will not be true. The information rate distortion function R(D) for a source Y with a distortion measure d(y, yˆ) is deﬁned as R(D) = p(y,ˆ y);

min

(y,y) ˆ

I(Y, Yˆ ).

(28)

p(y)p(y|ˆ y)d(y,ˆ y)≤D

The minimization is over all conditional distributions p(y|ˆ y) for which the joint distribution p(y, yˆ) = p(y)p(y|ˆ y) satisﬁes the average distortion constraint (i.e., average distortion ≤ D). The Rate Distortion Theorem states that R(D) is the maximum achievable rate of information transmission which does not exceed the distortion D. See [20, 24] details. More to the point, however, is the following: Pairs of sequences (y n , yˆn ) can be deﬁned as distortion typical ; that is, for a given average distortion D, deﬁned in terms of a particular measure, pairs of sequences can be divided into two sets, a high probability one containing a relatively small number of (matched) pairs with d(y n , yˆn ) ≤ D, and a low probability one containing most pairs. As n → ∞, the smaller set approaches unit probability, and, for those pairs, p(y n ) ≥ p(ˆ y n |y n ) exp[−nI(Y, Yˆ )].

(29)

Code, Context, and Epigenetic Catalysis in Gene Expression

325

Thus, roughly speaking, I(Y, Yˆ ) embodies the splitting criterion between high and low probability pairs of paths. For the theory of interacting information sources, then, I(Y, Yˆ ) can play the role of H in the dynamic treatment above. The rate distortion function can actually be calculated in many cases by using a Lagrange multiplier method – see Section 13.7 of [20]. 16.3

Groupoids

Basic ideas. Following [78] closely, a groupoid, G, is deﬁned by a base set A upon which some mapping – a morphism – can be deﬁned. Note that not all possible pairs of states (aj , ak ) in the base set A can be connected by such a morphism. Those that can deﬁne the groupoid element, a morphism g = (aj , ak ) having the natural inverse g −1 = (ak , aj ). Given such a pairing, it is possible to deﬁne ‘natural’ end-point maps α(g) = aj , β(g) = ak from the set of morphisms G into A, and a formally associative product in the groupoid g1 g2 provided α(g1 g2 ) = α(g1 ), β(g1 g2 ) = β(g2 ), and β(g1 ) = α(g2 ). Then the product is deﬁned, and associative, (g1 g2 )g3 = g1 (g2 g3 ). In addition, there are natural left and right identity elements λg , ρg such that λg g = g = gρg [78]. An orbit of the groupoid G over A is an equivalence class for the relation aj ∼ Gak if and only if there is a groupoid element g with α(g) = aj and β(g) = ak . Following [15], we note that a groupoid is called transitive if it has just one orbit. The transitive groupoids are the building blocks of groupoids in that there is a natural decomposition of the base space of a general groupoid into orbits. Over each orbit there is a transitive groupoid, and the disjoint union of these transitive groupoids is the original groupoid. Conversely, the disjoint union of groupoids is itself a groupoid. The isotropy group of a ∈ X consists of those g in G with α(g) = a = β(g). These groups prove fundamental to classifying groupoids. If G is any groupoid over A, the map (α, β) : G → A × A is a morphism from G to the pair groupoid of A. The image of (α, β) is the orbit equivalence relation ∼ G, and the functional kernel is the union of the isotropy groups. If f : X → Y is a function, then the kernel of f , ker(f ) = [(x1 , x2 ) ∈ X × X : f (x1 ) = f (x2 )] deﬁnes an equivalence relation. Groupoids may have additional structure. As [78] explains, a groupoid G is a topological groupoid over a base space X if G and X are topological spaces and α, β and multiplication are continuous maps. A criticism sometimes applied to groupoid theory is that their classiﬁcation up to isomorphism is nothing other than the classiﬁcation of equivalence relations via the orbit equivalence relation and groups via the isotropy groups. The imposition of a compatible topological structure produces a nontrivial interaction between the two structures. It is possible to introduce a metric structure on manifolds of related information sources, producing such interaction.

326

R. Wallace and D. Wallace

In essence, a groupoid is a category in which all morphisms have an inverse, here deﬁned in terms of connection to a base point by a meaningful path of an information source dual to a cognitive process. As [78] points out, the morphism (α, β) suggests another way of looking at groupoids. A groupoid over A identiﬁes not only which elements of A are equivalent to one another (isomorphic), but it also parametizes the diﬀerent ways (isomorphisms) in which two elements can be equivalent, i.e., all possible information sources dual to some cognitive process. Given the information theoretic characterization of cognition presented above, this produces a full modular cognitive network in a highly natural manner. Brown [13] describes the fundamental structure as follows: A groupoid should be thought of as a group with many objects, or with many identities... A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, ﬂexibility and range of applications... EXAMPLE 1. A disjoint union [of groups] G = ∪λ Gλ , λ ∈ Λ, is a groupoid: the product ab is deﬁned if and only if a, b belong to the same Gλ , and ab is then just the product in the group Gλ . There is an identity 1λ for each λ ∈ Λ. The maps α, β coincide and map Gλ to λ, λ ∈ Λ. EXAMPLE 2. An equivalence relation R on [a set] X becomes a groupoid with α, β : R → X the two projections, and product (x, y)(y, z) = (x, z) whenever (x, y), (y, z) ∈ R. There is an identity, namely (x, x), for each x ∈ X... [78] makes the following fundamental point: Almost every interesting equivalence relation on a space B arises in a natural way as the orbit equivalence relation of some groupoid G over B. Instead of dealing directly with the orbit space B/G as an object in the category Smap of sets and mappings, one should consider instead the groupoid G itself as an object in the category Ghtp of groupoids and homotopy classes of morphisms. The groupoid approach has become quite popular in the study of networks of coupled dynamical systems which can be deﬁned by diﬀerential equation models, [35]. Global and local symmetry groupoids. Here we follow [78] fairly closely, using the example of a ﬁnite tiling. Consider a tiling of the euclidean plane R2 by identical 2 by 1 rectangles, speciﬁed by the set X (one dimensional) where the grout between tiles is X = H ∪V , having H = R×Z and V = 2Z ×R, where R is the set of real numbers and Z the integers. Call each connected component of R2 \X, that is, the complement of the two dimensional real plane intersecting X, a tile. Let Γ be the group of those rigid motions of R2 which leave X invariant, i.e., the normal subgroup of translations by elements of the lattice Λ = H ∩ V =

Code, Context, and Epigenetic Catalysis in Gene Expression

327

2Z × Z (corresponding to corner points of the tiles), together with reﬂections through each of the points 1/2Λ = Z × 1/2Z, and across the horizontal and vertical lines through those points. As noted in [78], much is lost in this coarsegraining, in particular the same symmetry group would arise if we replaced X entirely by the lattice Λ of corner points. Γ retains no information about the local structure of the tiled plane. In the case of a real tiling, restricted to the ﬁnite set B = [0, 2m] × [0, n] the symmetry group shrinks drastically: The subgroup leaving X ∩ B invariant contains just four elements even though a repetitive pattern is clearly visible. A two-stage groupoid approach recovers the lost structure. We deﬁne the transformation groupoid of the action of Γ on R2 to be the set G(Γ, R2 ) = {(x, γ, y|x ∈ R2 , y ∈ R2 , γ ∈ Γ, x = γy}, with the partially deﬁned binary operation (x, γ, y)(y, ν, z) = (x, γν, z). Here α(x, γ, y) = x, and β(x, γ, y) = y, and the inverses are natural. We can form the restriction of G to B (or any other subset of R2 ) by deﬁning G(Γ, R2 )|B = {g ∈ G(Γ, R2 )|α(g), β(g) ∈ B} 1. An orbit of the groupoid G over B is an equivalence class for the relation x ∼G y if and only if there is a groupoid element g with α(g) = x and β(g) = y. Two points are in the same orbit if they are similarly placed within their tiles or within the grout pattern. 2. The isotropy group of x ∈ B consists of those g in G with α(g) = x = β(g). It is trivial for every point except those in 1/2Λ ∩ B, for which it is Z2 × Z2 , the direct product of integers modulo two with itself. By contrast, embedding the tiled structure within a larger context permits definition of a much richer structure, i.e., the identiﬁcation of local symmetries. We construct a second groupoid as follows. Consider the plane R2 as being decomposed as the disjoint union of P1 = B ∩ X (the grout), P2 = B\P1 (the complement of P1 in B, which is the tiles), and P3 = R2 \B (the exterior of the tiled room). Let E be the group of all euclidean motions of the plane, and deﬁne the local symmetry groupoid Gloc as the set of triples (x, γ, y) in B × E × B for which x = γy, and for which y has a neighborhood U in R2 such that γ(U ∩ Pi ) ⊆ Pi for i = 1, 2, 3. The composition is given by the same formula as for G(Γ, R2 ). For this groupoid-in-context there are only a ﬁnite number of orbits: O1 O2 O3 O4

= = = =

interior points of the tiles. interior edges of the tiles. interior crossing points of the grout. exterior boundary edge points of the tile grout.

328

R. Wallace and D. Wallace

O5 = boundary ‘T’ points. O6 = boundary corner points. The isotropy group structure is, however, now very rich indeed: The isotropy group of a point in O1 is now isomorphic to the entire rotation group O2 . It is Z2 × Z2 for O2 . For O3 it is the eight-element dihedral group D4 . For O4 , O5 and O6 it is simply Z2 . These are the ‘local symmetries’ of the tile-in-context. 16.4

Morse Theory

Morse theory examines relations between analytic behavior of a function – the location and character of its critical points – and the underlying topology of the manifold on which the function is deﬁned. We are interested in a number of such functions, for example information source uncertainty on a parameter space and ‘second order’ iterations involving parameter manifolds determining critical behavior, for example sudden onset of a giant component in the mean number model [74], and universality class tuning in the mean ﬁeld model of the next section. These can be reformulated from a Morse theory perspective. Here we follow closely the elegant treatments of [46, 59]. The essential idea of Morse theory is to examine an n-dimensional manifold M as decomposed into level sets of some function f : M → R where R is the set of real numbers. The a-level set of f is deﬁned as f −1 (a) = {x ∈ M : f (x) = a}, the set of all points in M with f (x) = a. If M is compact, then the whole manifold can be decomposed into such slices in a canonical fashion between two limits, deﬁned by the minimum and maximum of f on M . Let the part of M below a be deﬁned as Ma = f −1 (−∞, a] = {x ∈ M : f (x) ≤ a}. These sets describe the whole manifold as a varies between the minimum and maximum of f . Morse functions are deﬁned as a particular set of smooth functions f : M → R as follows. Suppose a function f has a critical point xc , so that the derivative df (xc ) = 0, with critical value f (xc ). Then f is a Morse function if its critical points are nondegenerate in the sense that the Hessian matrix of second derivatives at xc , whose elements, in terms of local coordinates are Hi,j = ∂ 2 f /∂xi ∂xj , has rank n, which means that it has only nonzero eigenvalues, so that there are no lines or surfaces of critical points and, ultimately, critical points are isolated.

Code, Context, and Epigenetic Catalysis in Gene Expression

329

The index of the critical point is the number of negative eigenvalues of H at xc . A level set f −1 (a) of f is called a critical level if a is a critical value of f , that is, if there is at least one critical point xc ∈ f −1 (a). Again following [59], the essential results of Morse theory are: 1. If an interval [a, b] contains no critical values of f , then the topology of f −1 [a, v] does not change for any v ∈ (a, b]. Importantly, the result is valid even if f is not a Morse function, but only a smooth function. 2. If the interval [a, b] contains critical values, the topology of f −1 [a, v] changes in a manner determined by the properties of the matrix H at the critical points. 3. If f : M → R is a Morse function, the set of all the critical points of f is a discrete subset of M , i.e., critical points are isolated. This is Sard’s Theorem. 4. If f : M → R is a Morse function, with M compact, then on a ﬁnite interval [a, b] ⊂ R, there is only a ﬁnite number of critical points p of f such that f (p) ∈ [a, b]. The set of critical values of f is a discrete set of R. 5. For any diﬀerentiable manifold M , the set of Morse functions on M is an open dense set in the set of real functions of M of diﬀerentiability class r for 0 ≤ r ≤ ∞. 6. Some topological invariants of M , that is, quantities that are the same for all the manifolds that have the same topology as M , can be estimated and sometimes computed exactly once all the critical points of f are known: Let the Morse numbers μi (i = 0, ..., m) of a function f on M be the number of critical points of f of index i, (the number of negative eigenvalues of H). The Euler characteristic of the complicated manifold M can be expressed as the alternating sum of the Morse numbers of any Morse function on M , χ=

m

(−1)i μi .

i=1

The Euler characteristic reduces, in the case of a simple polyhedron, to χ=V −E+F where V, E, and F are the numbers of vertices, edges, and faces in the polyhedron. 7. Another important theorem states that, if the interval [a, b] contains a critical value of f with a single critical point xc , then the topology of the set Mb deﬁned above diﬀers from that of Ma in a way which is determined by the index, i, of the critical point. Then Mb is homeomorphic to the manifold obtained from attaching to Ma an i-handle, i.e., the direct product of an i-disk and an (m − i)-disk. Again, see [52, 59] for details. 16.5

Generalized Onsager Theory

Understanding the time dynamics of groupoid-driven information systems away from phase transition critical points requires a phenomenology similar to the

330

R. Wallace and D. Wallace

Onsager relations of nonequilibrium thermodynamics. This also leads to a general theory involving large-scale topological changes in the sense of Morse theory. If the Groupoid Free Energy (GFE) of a biological process is parametized by some vector of quantities K = (K1 , ..., Km ), then, in analogy with nonequilibrium thermodynamics, gradients in the Kj of the disorder, deﬁned as SG = FG (K) −

m

Kj ∂FG /∂Kj

(30)

j=1

become of central interest. Equation (30) is similar to the deﬁnition of entropy in terms of the free energy of a physical system. Pursuing the homology further, the generalized Onsager relations deﬁning temporal dynamics of systems having a GFE become dKj /dt = Lj,i ∂SG /∂Ki , (31) i

where the Lj,i are, in ﬁrst order, constants reﬂecting the nature of the underlying cognitive phenomena. The L-matrix is to be viewed empirically, in the same spirit as the slope and intercept of a regression model, and may have structure far diﬀerent than familiar from more simple chemical or physical processes. The ∂SG /∂K are analogous to thermodynamic forces in a chemical system, and may be subject to override by external physiological or other driving mechanisms: biological and cognitive phenomena, unlike simple physical systems, can make choices as to resource allocation. That is, an essential contrast with simple physical systems driven by (say) entropy maximization is that complex biological or cognitive structures can make decisions about resource allocation, to the extent resources are available. Thus resource availability is a context, not a determinant, of behavior. Equations (30) and (31) can be derived in a simple parameter-free covariant manner which relies on the underlying topology of the information source space implicit to the development [74]. We will not pursue that development here. The dynamics, as we have presented them so far, have been noiseless, while biological systems are always very noisy. Equation (31) might be rewritten as dKj /dt = Lj,i ∂SG /∂Ki + σW (t) i

where σ is a constant and W (t) represents white noise. This leads directly to a family of classic stochastic diﬀerential equations having the form dKtj = Lj (t, K)dt + σ j (t, K)dBt ,

(32)

where the Lj and σ j are appropriately regular functions of t and K, and dBt represents the noise structure, and we have readjusted the indices. Further progress in this direction requires introduction of methods from stochastic diﬀerential geometry and related topics in the sense of [27]. The obvious inference is that noise – not necessarily ‘white’ – can serve as a tool to

Code, Context, and Epigenetic Catalysis in Gene Expression

331

shift the system between various topological modes, as a kind of crosstalk and the source of a generalized stochastic resonance. Eﬀectively, topological shifts between and within dynamic manifolds constitute another theory of phase transitions [59], and this phenomenological Onsager treatment would likely be much enriched by explicit adoption of a Morse theory perspective. 16.6

The Tuning Theorem

Messages from an information source, seen as symbols xj from some alphabet, each having probabilities Pj associated with a random variable X, are ‘encoded’ into the language of a ‘transmission channel’, a random variable Y with symbols yk , having probabilities Pk , possibly with error. Someone receiving the symbol yk then retranslates it (without error) into some xk , which may or may not be the same as the xj that was sent. More formally, the message sent along the channel is characterized by a random variable X having the distribution P (X = xj ) = Pj , j = 1, ..., M. The channel through which the message is sent is characterized by a second random variable Y having the distribution P (Y = yk ) = Pk , k = 1, ..., L. Let the joint probability distribution of X and Y be deﬁned as P (X = xj , Y = yk ) = P (xj , yk ) = Pj,k and the conditional probability of Y given X as P (Y = yk |X = xj ) = P (yk |xj ). Then the Shannon uncertainty of X and Y independently and the joint uncertainty of X and Y together are deﬁned respectively as H(X) = −

M

Pj log(Pj )

j=1

H(Y ) = −

L

Pk log(Pk )

k=1

H(X, Y ) = −

M L j=1 k=1

Pj,k log(Pj,k ).

(33)

332

R. Wallace and D. Wallace

The conditional uncertainty of Y given X is deﬁned as H(Y |X) = −

M L

Pj,k log[P (yk |xj )].

(34)

j=1 k=1

For any two stochastic variates X and Y , H(Y ) ≥ H(Y |X), as knowledge of X generally gives some knowledge of Y . Equality occurs only in the case of stochastic independence. Since P (xj , yk ) = P (xj )P (yk |xj ), we have H(X|Y ) = H(X, Y ) − H(Y ). The information transmitted by translating the variable X into the channel transmission variable Y – possibly with error – and then retranslating without error the transmitted Y back into X is deﬁned as I(X|Y ) = H(X)− H(X|Y ) = H(X)+ H(Y )− H(X, Y )

(35)

See, for example, [1, 20, 47] for details. The essential point is that if there is no uncertainty in X given the channel Y , then there is no loss of information through transmission. In general this will not be true, and herein lies the essence of the theory. Given a ﬁxed vocabulary for the transmitted variable X, and a ﬁxed vocabulary and probability distribution for the channel Y , we may vary the probability distribution of X in such a way as to maximize the information sent. The capacity of the channel is deﬁned as C = max I(X|Y ) P (X)

(36)

subject to the subsidiary condition that P (X) = 1. The critical trick of the Shannon Coding Theorem for sending a message with arbitrarily small error along the channel Y at any rate R < C is to encode it in longer and longer ‘typical’ sequences of the variable X; that is, those sequences whose distribution of symbols approximates the probability distribution P (X) above which maximizes C. If S(n) is the number of such ‘typical’ sequences of length n, then log[S(n)] ≈ nH(X), where H(X) is the uncertainty of the stochastic variable deﬁned above. Some consideration shows that S(n) is much less than the total number of possible messages of length n. Thus, as n → ∞, only a vanishingly small fraction of all possible messages is meaningful in this sense. This observation, after some considerable development, is what allows the Coding Theorem to work so well. In sum, the prescription is to encode messages in typical sequences, which are sent at very nearly the capacity of the channel. As the encoded messages become

Code, Context, and Epigenetic Catalysis in Gene Expression

333

longer and longer, their maximum possible rate of transmission without error approaches channel capacity as a limit. Again, [1, 20, 47] provide details. This approach can be, in a sense, inverted to give a tuning theorem which parsimoniously describes the essence of the Rate Distortion Manifold. Telephone lines, optical wave, guides and the tenuous plasma through which a planetary probe transmits data to earth may all be viewed in traditional information-theoretic terms as a noisy channel around which we must structure a message so as to attain an optimal error-free transmission rate. Telephone lines, wave guides, and interplanetary plasmas are, relatively speaking, ﬁxed on the timescale of most messages, as are most other signaling networks. Indeed, the capacity of a channel, is deﬁned by varying the probability distribution of the ‘message’ process X so as to maximize I(X|Y ). Suppose there is some message X so critical that its probability distribution must remain ﬁxed. The trick is to ﬁx the distribution P (x) but modify the channel – i.e., tune it – so as to maximize I(X|Y ). The dual channel capacity C ∗ can be deﬁned as C∗ =

max

I(X|Y ).

max

I(Y |X)

P (Y ),P (Y |X)

(37)

But C∗ =

P (Y ),P (Y |X)

since I(X|Y ) = H(X) + H(Y ) − H(X, Y ) = I(Y |X). Thus, in a purely formal mathematical sense, the message transmits the channel, and there will indeed be, according to the Coding Theorem, a channel distribution P (Y ) which maximizes C ∗ . One may do better than this, however, by modifying the channel matrix P (Y |X). Since M P (yj ) = P (xi )P (yj |xi ), i=1

P (Y ) is entirely deﬁned by the channel matrix P (Y |X) for ﬁxed P (X) and C∗ =

max

P (Y ),P (Y |X)

I(Y |X) = max I(Y |X). P (Y |X)

Calculating C ∗ requires maximizing the complicated expression I(X|Y ) = H(X) + H(Y ) − H(X, Y ), that contains products of terms and their logs, subject to constraints that the sums of probabilities are 1 and each probability is itself between 0 and 1. Maximization is done by varying the channel matrix terms P (yj |xi ) within the constraints. This is a diﬃcult problem in nonlinear optimization. However, for the special case M = L, C ∗ may be found by inspection:

334

R. Wallace and D. Wallace

If M = L, then choose

P (yj |xi ) = δj,i ,

where δi,j is 1 if i = j and 0 otherwise. For this special case C ∗ = H(X), with P (yk ) = P (xk ) for all k. Information is thus transmitted without error when the channel becomes ‘typical’ with respect to the ﬁxed message distribution P (X). If M < L, matters reduce to this case, but for L < M information must be lost, leading to Rate Distortion limitations. Thus modifying the channel may be a far more eﬃcient means of ensuring transmission of an important message than encoding that message in a ‘natural’ language which maximizes the rate of transmission of information on a ﬁxed channel. We have examined the two limits in which either the distributions of P (Y ) or of P (X) are kept ﬁxed. The ﬁrst provides the usual Shannon Coding Theorem, and the second a tuning theorem variant, a tunable retina-like Rate Distortion Manifold. It seems likely, however, than for many important systems P (X) and P (Y ) will interpenetrate, to use Richard Levins’ terminology. That is, P (X) and P (Y ) will aﬀect each other in characteristic ways, so that some form of mutual tuning may be the most eﬀective strategy.

Author Index

Aman, Bogdan

26

Bantang, Johnrob Y. 164 Bortolussi, Luca 216 Boˇsnaˇcki, D. 69

Harmer, Russ 116 Heiner, Monika 138 Hillston, Jane 1 Jack, John

200

Calder, Muﬀy 1 Chesi, Graziano 268 Ciobanu, Gabriel 26 Ciocchetta, Federica 45

Krivine, Jean

Danos, Vincent 116 Dassow, J¨ urgen 187 David, Maria Pamela C. de Vink, E.P. 69

Marwan, Wolfgang 138 Mendoza, Eduardo R. 164 Mitrana, Victor 187

Lehrack, Sebastian

164

138

P˘ aun, Andrei 200 Policriti, Alberto 216 Pronk, T.E. 69

Feret, J´erˆ ome 116 Fontana, Walter 116 Gilbert, David 138 Guerriero, Maria Luisa

116

90

Wallace, Deborah Wallace, Rodrick

283 283

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close