Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models

NONLINEAR SCIENCE THEORY AND APPLICATIONS Numerical experiments over the last thirty years have revealed that simple no...

Author: Péter Érdi | János Tóth

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NONLINEAR SCIENCE THEORY AND APPLICATIONS Numerical experiments over the last thirty years have revealed that simple nonlinear systems can have surprising and complicated behaviours. Nonlinear phenomena include waves that behave as particles, deterministic equations having irregular, unpredictable solutions, and the formation of spatial structures from an isotropic medium. The applied mathematics of nonlinear phenomena has provided metaphors and models for a variety of physical processes: solitons have been described in biological macromolecules as well as in hydrodynamic systems; irregular activity that has been identified with chaos has been observed in continuously stirred chemical flow reactors as well as in convecting fluids: nonlinear reaction diffusion systems have been used to account for the formation of spatial patterns in homogeneous chemical systems as well as biological morphogenesis; and discrete-time and discrete-space nonlinear systems (cellular automata) provide metaphors for processes ranging from the microworld of particle physics to patterned activity in computing neural and self-replicating genetic systems. Nonlinear Science: Theory and Applications will deal with all areas of nonlinear science - its mathematics, methods and applications in the biological, chemical, engineering and physical sciences.

Nonlinear science: theory and applications Series editor: Arun V. Holden, Reader in General Physiology, Centre for Nonlinear Studies, The University, Leeds LS2 9NQ, UK Editors: S. I. Amari (Tokyo), P. L. Christiansen (Lyngby), D. G. Crighton (Cambridge), R. H. G. Heileman (Houston), D. Rand (Warwick), J. C. Roux (Bordeaux)

Chaos A. V. Holden (Editor) Control and optimization J. E. Rubio Automata networks in computer science F. Fogelman Soulie, Y. Robert and M. Tchuente (Editors) Oscillatory evolution processes I. Gumowski Introduction to the theory of algebraic invariants of differential equations K. S. Sibirsky Simulation of wave processes in excitable media V. Zykov (Edited by A. T. Winfree and P. Nandapurkar) Almost periodic operators and related nonlinear integrable systems V. A. Chulaevsky Other volumes are in preparation

Mathematical models of '· chemical reaction~_/ Theory and applications of deterministic and stochastic models P. Erdi and J. T6th Central Research Institute for Physics, Hungarian Academy of Sciences and Computer and Automation Institute, Hungarian Academy of Sciences

c

Manchester University Press

Copyright© P. Erdi and J. Toth 1989 Published by Manchester University Press Oxford Road, Manchester MI3 9PL, UK British Library cataloguing in publication data Erdi, P. Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models.- (Nonlinear science) I. Chemical reactions - Mathematical models I. Title II. Toth, J. III. Series 541.3'9'0724 QD501

ISBN 0 7190 2208 8 hardback Typeset in Times 10/12 pt by Graphicraft Typesetters Ltd, Hong Kong Printed in Great Britain by Biddies Ltd., Guildford and King's Lynn

Contents

Preface and acknowledgements Symbols used in the text

XI

xm

1 Chemical kinetics: a prototype of nonlinear science 1.1 Mass action kinetics: macroscopic and microscopic approach 1.2 Physical models of chemical reactions 1.3 Deterministic and stochastic models 1.4 Regular and exotic behaviour 1.5 Chemical kinetics as a metalanguage

I 4 6 II 12

2 The structure of kinetic models 2.1 Temporal processes 2.2 Properties of process-time 2.2.1 Discrete versus continuous 2.2.2 Time, thermodynamics, chemical kinetics 2.3 Structure of state-space 2.3.1 Discrete versus continuous 2.3.2 State and site 2.4 Nature of determination 2.5 X YZ models

14 14 14 15 15 16 16 17 18 19

3 Stoichiometry: the algebraic structure of complex chemical reactions 3.1 Conventional stoichiometry 3.2 Atom-free stoichiometry 3.3 Retrospective and prospective remarks. Suggested further reading 3.4 Exercises

21 21 26 28 29

Contents

vi 3.5 Problems 3.6 Open problems 4 Mass action kinetic deterministic models 4.1 Kinetic equations: their structure and properties 4.1.1 Introduction 4.1.2 Introduction reconsidered 4.1.3 Exercises 4.1.4 Problems 4.1.5 Open problems 4.2 Verifications and falsifications of traditional beliefs 4.2.1 The zero deficiency theorem 4.2.2 Vol'pert's theorem 4.2.3 Remarks on related literature 4.2.4 Exercises 4.2.5 Problems 4.2.6 Open problems 4.3 Exotic reactions: general remarks 4.4 Multistationarity 4.4.1 Multistability 4.4.2 Multistationarity in kinetic experiments 4.4.3 Multistationarity in kinetic models of continuous flow stirred tank reactors 4.4.4 Exercises 4.4.5 Problems 4.4.6 Open problems 4.5 Oscillatory reactions: some exact results 4.5.1 Periodicity in kinetic experiments 4.5.2 Excluding periodicity in differential equations 4.5.3 Excluding periodicity in reactions 4.5.4 Sufficient conditions of periodicity in differential equations 4.5.5 Sufficient conditions of periodicity in reactions 4.5.6 Designing oscillatory reactions 4.5.7 Overshoot-undershoot kinetics 4.5.8 Exercises 4.5.9 Problems 4.5.10 Open problems 4.6 Chaotic phenomena in chemical kinetics 4.6.1 Chaos in general 4.6.2 Chaos in kinetic experiments 4.6.3 Chaos in kinetic models

29 32 33 33 33

35 39 39 40 40 42 45

46

47 48 48 49 49 49 50 50 51

52 52 54 54 54

55 55 56 56

57 57 58 59 59 59

60 61

Contents

4.6.4 On the structural characterisation of chaotic chemical reactions 4.6.5 Problems 4.6.6 Open problems 4.7 The inverse problems of reaction kinetics 4.7.1 Polynomial differential equations, kinetic differential equations, kinetic initial value problems 4.7.1.1 Polynomial and kinetic differential equations 4.7.1.2 Further problems 4.7.1.3 The density of kinetic differential equations 4.7.1.4 Uniqueness questions 4.7.1.5 A sufficient condition for the existence of an inducing reaction of deficiency zero 4.7.1.6 On the inverse problem of generalised compartmental systems 4.7.2 The classical problem of parameter estimation 4.7.3 Exercises 4.7.4 Problems 4.7.5 Open problems 4.8 Selected addenda 4.8.1 Lumping 4.8.1.1 Lumping in general 4.8.1.2 Lumping in reaction kinetics 4.8.1.3 Possible further directions 4.8.2 Continuous components 4.8.3 Kinetic gradient systems 4.8.4 Structural identifiability 4.8.5 Parameter sensitivity 4.8.6 Symmetries 4.8.7 Principle of quasistationarity 4.8.8 Exercises 4.8.9 Problems 4.8.10 Open problem 5 Continuous time discrete state stochastic models

5.1 On the nature and role of fluctuations: general remarks 5.1.1 The logical status of stochastic reaction kinetics 5.1.2 Fluctuation phenomena in physics and chemistry: an introduction 5.1.2.1 Stochastic thermostatics, stochastic thermodynamics 5.1.3 Stochastic processes: concepts

VII

62 62 63 63 64 64 65 67 67 69 72 74 74 75 75 75 75 75 76 77 78 80 82 83 84 88 89 89 90 91 91 91 93 93 96

viii

5.2

5.3

5.4

5.5

5.6

Contents

5.1.3.1 Introductory remarks 5.1.3.2 Continuous state-space processes 5.1.3.3 Discrete state-space processes 5.1.4 Operator semigroup approach: advantages coming from the use of more sophisticated mathematics Stochasticity due to internal fluctuations: alternative models 5.2.1 Some historical remarks 5.2.2 Models On the solutions of the CDS models 5.3.1 General remarks 5.3.2 Chemical reaction X !... Y 5.3.3 Compartmental systems 5.3.4 Bicomponential reactions: general remarks 5.3.5 Chemical reaction X + Y ~ Z 5.3.5.1 The master equation 5.3.5.2 Use of Laplace transformation 5.3.5.3 Determination of expectation 5.3.5.4 The behaviour of the reaction during the initial period of the processes 5.3.5.5 Determination ofstationary distribution 5.3.6 General equation for the generating function 5.3.7 Approximations 5.3.8 Simulation methods The fluctuation-dissipation theorem of chemical kinetics 5.4.1 Stochastic reaction kinetics: 'nonequilibrium thermodynamics of state-space'? 5.4.2 Fluctuation-dissipation theorem of linear nonequilibrium thermodynamics 5.4.3 Determination of rate constants from equilibrium fluctuations: methods of calculation Small systems 5.5.1 Enzyme kinetics 5.5.2 Ligand migration in biomolecules 5.5.3 Membrane noise 5.5.4 Kinetic examinations of fast reactions Fluctuations near instability points 5.6.1 An example of the importance of fluctuations 5.6.2 Stochastic Lotka-Volterra model 5.6.3 Stochastic Brusselator model 5.6.4 The Schlogl model of second-order phase transition 5.6.5 The Schlogl model of first-order phase transition 5.6.6 Stochastic theory of bistable reactions

96 97 99 99 101 I0 I 102 I05 I 05 I 06 107 107 I 08 108 108 I08 109 109 109 110 112 115 115 116 117 119 119 121 123 125 128 128 129 130 131 134 135

Contents

5. 7 Stationary distributions: uni- versus multimodality 5.7.1 The scope and limits of the Poisson distribution in the stochastic models of chemical reactions: motivations 5.7.2 Sufficient conditions of unimodality 5.7.3 Sufficient condition for a Poissonian stationary distribution 5.7.4 Multistationarity and multimodality 5.7.5 Transient bimodality 5.8 Stochasticity due to external fluctuations 5.8.1 Motivations 5.8.2 Stochastic differential equations: some concepts and comments 5.8.3 Noise-induced transition: an example for white noise idealisation 5.8.4 Noise-induced transition: the effect of coloured noise 5.8.5 On the effects of external noise on oscillations 5.8.6 Internal and external fluctuations: a unified approach 5.8.7 Estimation of reaction rate constants using stochastic differential equations 5.8.8 Exercises 5.8.9 Problems 5.8.10 Open problem 5.9 Connections between the models 5.9.1 Similarities and differences: some remarks 5.9.2 Blowing up 5.9.3 Kurtz's results: consistency in the thermodynamic limit 5.9.4 Exercise 5.9.5 Problems

6 Chemical reaction accompanied by diffusion 6.1 What kinds of models are relevant? 6.2 Continuous time, continuous !>tate-space deterministic models 6.3 Stochastic models: difficulties and possibilities 6.3.1 Introductory remarks 6.3.2 Two-cell stochastic models 6.3.3 Cellular model 6.3.4 Other models

tx

138

138 140 142 143 144 146 146 147 149 151 153 156 157 I 58 159 159 159 159 159 160 160 161

162 162 163 167 167 168 169 171

Contents

x

6.4 Spatial structures 6.5 Pattern formation and morphogenesis 7 Applications 7.1 Introductory remarks 7.2 Biochemical control theory

7.3 Fluctuation and oscillation phenomena in neurochemistry 7.4 Population genetics 7.5 Ecodynamics 7.5.1 The theory of interacting populations 7.5.1.1 Boulding ecodynamics 7.5.1.2 Compartmental ecokinetics 7.5.1.3 Generalised Lotka-Volterra models 7.5.1.4 The advantages of stochastic models: illustrations 7.5.2 An ecological case study 7.5.2.1 Arguments for a stochastic model 7.5.2.2 A common description of the deterministic and stochastic models 7.5.2.3 Exercise 7.6 Aggregation, polymerisation, cluster formation 7.7 Chemical circuits 7.8 Kinetic theories of selection 7.8.1 Prebiological evolution 7.8.1.1 Introductory remarks 7.8.1.2 The hypercycle: The basic model 7.8.2 The origin of asymmetry of biomolecules

172 174 177 177 177 185 192 194 194 194 195 196 199 202 202 204 207 207 210 213 213 213 214 216

References

220

Index

252

Preface and acknowledgements

Chemical kinetics may be considered as a prototype of nonlinear science, since the velocity of reactions is generally a nonlinear function of the quantities of the reacting chemical components. Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and 'exotic' begins to become 'common'. Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics; second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points. In addition to the direct utility of mathematical models in the analysis of\ complex chemical systems a unified conceptual framework is offered to the \ mathematical treatment of problems of chemical kinetics and related areas in ! biomathematics. Biochemical control processes, oscillation and fluctuation / p~enomena in neurochemical systems, coexistence and extinction in popul- j at10ns, prebiological evolution and certain ecological problems of Lake 1 Balaton can be treated in terms of this framework. Though the main body of/ the book deals with spatially homogeneous systems, spatial structures in chemical systems, pattern formation and morphogenesis related to reaction-diffusion models are also mentioned briefly. The material that the book contains has until now been scattered in journals and proceedings. In spite of the undoubted penetration of modern mathematical techniques into the kingdom of chemical kinetics, the gap between the practical necessity of the chemist and the theorems of the

xii

Preface and acknowledgements

mathematician is obvious. This book is the result of a ·quasicontinuous discussion between a chemist (P.E.) and a mathematician (J.T.) about mathematics and the natural sciences, and we certainly do not want it to be either a 'user's manual' for kineticists or a zoo of 'pseudo-applications' of , mathematical problems. What we hope to emphasise is that chemical 1 kinetics, a beautiful discipline in its own right, is really a pro,totype of nonlinear science. We are deeply indebted to many Hungarian scientists working in Buda and Pest, Debrecen and Veszprem, with whom we have had many discussions, informal and formal, about mathematical techniques and theoretical concepts of chemistry. We would like to express particular gratitude to our former colleague, Vera Hars with whom we have discussed many topics studied and not studied in this book. Many problems on stochastic kinetics have been debated with Professor Michel Moreau (Paris). One of us (P.E.) enjoyed his hospitality in June 1985; the main part of Chapter 5 was written during this time. Some short visits to Bordeaux were useful for both of us. Very special thanks to Dr Holden for inviting us to contribute this monograph to the series. A large part of the figures are reprinted with permission of the authors (as cited at the appropriate place) and of the copyright holders, as follows: Acta Biochimica, Elsevier Scientific Publishers Ireland Ltd, North-Holland Physics Publishing, Pergamon Journals Ltd, Plenum Publishing Corp., Publishing House of the Hungarian Academy of Sciences.

l

Budapest, February 1986

Peter Erdi, Janos T6th

Symbols used in the text

Chapter 1 A A( solid)

notation for an abstract chemical component (or species) in the elementary reaction aA + bB -+ cC + dD reactant component in the solid phase reaction A(solid)

A a

a•.•

8 B(solid)

A(solid)

b

c

c

-+ B(solid)

+ C(gas)

matrix of infinitesimal transition probabilities (infinitesimal operator) of the stochastic model of reactions stoichiometric coefficient (or molecularity) of the chemical component A in the elementary reaction aA + bB -+ cC + dD element of the matrix of infinitesimal transition probabilities (infinitesimal operator) of the stochastic model of reactions element of the matrix of infinitesimal transition probabilities (infinitesimal operator) of the stochastic model of reactions notation for an abstract chemical component (or species) in the elementary reaction aA + bB -+ cC + dD product component in the solid phase reaction -+ B(solid)

+ C(gas)

stoichiometric coefficient (or molecularity) of the chemical component B in the elementary reaction aA + bB -+ cC + dD notation for an abstract chemical component (or species) in the elementary reaction aA + bB -+ cC + dD product component in lhe solid phase reaction A(solid)

-+ B(solid)

+ C(gas)

stoichiometric coefficient (or molecularity) of the chemical component C in the elementary reaction aA + bB -+ cC + dD initial vector of concentrations


xiv

vector of concentrations of chemical components at time t (the unit of a component of the concentration vector is to be understood as kmol/m 3 here and everywhere below) time derivative of the concentration vector versus time c(t) function at time t (the unit of a component of the time derivative of the concentration vector is to be understood as kmol/m 3 s) here and everywhere below) concentration of the component A at time m the elementary reaction aA + bB-+ cC + dD concentration of the component B at time in the elementary reaction aA + bB-+ cC + dD concentration of the component C at time in the elementary reaction aA + bB-+ cC + dD concentration of the component D at time m the elementary reaction aA + bB -+ cC + dD c 1 (t), c 2 (t), c 3 (t) components of the vector of concentrations at time t notation for an abstract chemical component (or species) D in the elementary reaction aA + bB-+ cC + dD stoichiometric coefficient (or molecularity) of the chemd ical component D m the elementary reaction c(t)

aA f

/;,jj ks M N

NM n(t) n, n'

~M

r(t)

-S-ST t

v

+ bB-+ cC + dD

right-hand side of the kinetic differential equation scalar valued functions of the state variables in eqn ( 1.6) Boltzmann's constant (1.38 x 10- 23 J/K) number of chemical components in the investigated complex chemical reaction (perhaps after lumping) number of chemical components of a chemical system (if this number is very large) space of M-dimensional vectors of nonnegative integers number of components at time t possible number of components in the stochastic model probability of the event written in parenthesis is the absolute distribution of the number of components at time t space of M-dimensional vectors of real numbers of the elementary reaction reaction rate aA + bB-+ cC + dD (kmol/(m 3 s)) covalent bond between sulphuric atoms absolute temperature (K) time (s) volume of the chemical system (m- 3 ) scalar valued functions of time in eqn (1.6) a vector valued stochastic process describing the time evolution of the number of components at time t


XV

this sign is used throughout to mean 'is defined by' or 'is by definition'

Chapter 2 state-space of a dynamic system state of a thermodynamic system at time t function giving the state of a thermodynamic system at time t as a Jf function of site and time site of a thermodynamic system at time t history of the site of a thermodynamic system up to time t the set of positive integers the set of real numbers euclidean space of M-dimensional real vectors number of constitutive quantities point of the time domain T of a dynamic system time domain of a dynamic system point of the time domain T of a dynamic system state of a dynamic system X initial state of a dynamic system Xo a function from T x A into A describing the time evolution of a dynamic system for different initial states the set of integers the set (or lattice) of ordered M-tuples of integers motion of a dynamic system starting from state x 0 x. 0,

therefore p(m, r) < cx(m, r)

should hold. But, because ofO ~ p(m, r) and cx(m, r) ~ I, the product above surely does contain xm among its factors. The consequence of the theorem as applied to the Lorenz equation is that no reaction can induce the Lorenz equation, and so this equation cannot be considered as the kinetic differential equation of a complex chemical reaction. It is an astonishing fact that the converse of the theorem holds as well. If the right-hand side of a differential equation is an (M, M)-polynomial without negative cross-effects then it may be considered as the induced kinetic differential equation of a reaction, or, in other words, if there is no negative cross-effect in the right-hand side then there exists a reaction with the given equation as its deterministic model. A constructive proof of this theorem is stated as a problem: one of the inducing reactions can easily be constructed. Polynomial differential equations without negative cross-effects will usually be called kinetic differential equations from now on. 4. 7.1.2 Further problems Two examples will be shown here proving that even the mechanism corresponding to a given kinetic differential equation is not unique. Any reaction can be enlarged by adding the reaction I

2X~X+

I

Y--.2Y

(4.21)

without any change of the original kinetic differential equation. (It is interesting, however, that the simpler reaction I

I

2X ~X--. 0, where Y is considered to be an external component, is easy to distinguish from the empty reaction, the complex chemical reaction without elementary reactions and components, if one considers the usual stochastic model. In this case the variance of the quantity of X tends to infinity, see Chapter 5.) The canonic reaction corresponding to a kinetic differential equation

66

Mathematical models of chemical reactions

constructed as described by Hius & T6th (1981) is: 2X+ 2V

~

2X+ V-+X+ V

~

2X+ Y+ V

P/3Y + V

2

/

3P

3Y+ 2V-+2Y+ 2V ""'- 4P

"-..'3Y+ Z 1

+ 2V

/'4Z + V

/2y

4Z-+X+ 4Z.

~3Z On the other hand, the same equation is the induced kinetic differential equation of the reaction 3

2X + V-+ 3 Y

' 1

+ 4V

,/

4Z p

as well. This reaction is weakly reversible, conservative, and of deficiency zero. This example shows why the two statements above are so important for a pure mathematician: no general theorem other than the zero deficiency theorem is known to provide statements so simply about the qualitative behaviour of the solutions of the (complicated nonlinear) kinetic differential equation induced by the above reactions. This example also shows that the canonic mechanism is not the simplest one and it is not minimal in any sense (although it is unique). Its major advantage is that it can quickly be constructed and by an algorithm. Several groups of problems arise here: (I) Is the lack of negative cross-effects not too strong a restriction in the sense that a 'randomly selected' polynomial differential equation is usually nonkinetic. How 'dense' is the set of kinetic differential equations within the set of polynomial ones? (2) For the sake of easy manipulation it would be useful to find an inducing reaction with the minimal number of complexes, elementary reactions, linkage classes etc. What kind of reasonable assumptions assure the

Deterministic models

67

existence or the uniqueness of such a mechanism? (3) The reaction may be looked for within a class of reactions with a given, chemically relevant property. Such a property may be conservativity or subconservativity (the canonic reaction is never conservative!), reversibility, weak reversibility or acyclicity, small (zero or one) deficiency. When does an inducing reaction exist within a given class to a given differential equation, when is it unique, or if it is not unique then how far is it nonunique? The chemically relevant property may also relate qualitative properties of solutions of induced kinetic differential equations, such as, presence or absence of multistationarity, oscillation and chaos. Representatives of the 'kinetic logic' approach such as Thomas, King and Glass have given a solution to these kinds of problems. Thomas (1981) has shown by an example how to derive a set of differential equations with a given type of solutions, such as, for example, having three stable steady states. The only problem of this method at present seems to be that the emerging equations are not of the polynomial type. Therefore the ideas are to be tailored according to the needs of formal reaction kinetics. (4) Similar but more complex problems will be obtained if only the 'essential' part of a differential equation is considered. (5) Is it possible to obtain a kinetic differential equation from a nonkinetic one with transformation of a given type? Let us formulate some of the questions more precisely in order either to answer them or to set them as a target. Before doing so we mention that question (5) will not be treated here separately, as a question of this type has been treated in Subsection 4.6.4, and other similar ones will be treated in Subsection 4.8.6 (on symmetries). 4.7.1.3 The density of kinetic differential equations According to several different real situations different definitions have been given of the random event that a polynomial differential equation is kinetic (Toth, 1981 b, pp. 44-8). The results can be summarised as follows. If one selects a polynomial differential equation with fixed coefficients and the random selection only concerns the exponents than the probability of getting a kinetic differential equation is I. If the exponents are fixed and the coefficients are randomly chosen then the probability of getting a kinetic differential equation is 0. Finally, as a consequence of the statements above, if both the coefficients and the exponents are randomly selected then the probability of getting a kinetic differential equation is again 0. 4.7.1.4 Uniqueness questions A general theorem can be stated saying that for any given kinetic differential

68

Mathematical models of' chemical reactions

equation there exists an inducing reaction with the minimal number of complexes and elementary reactions. If a certain inequality holds for the coefficients then it is also true that the number of complexes equals the different exponent vectors on the right-hand side of the equation. This theorem is rather awkward to formulate, and is given as Problem I of Subsection 4.7.5. The right-hand side of the induced kinetic differential equation of reaction (4.21) is the (2, 2)-polynomial 0. Such a case cannot occur with reversible, or even with weakly reversible reactions. Neither can it occur with acyclic reactions. The right-hand side of the induced kinetic differential equation

c =Poe of a weakly reversible reaction cannot be the polynomial 0. The proof of this statement is based on the fact that in this case span(9Pp)

= §,

(Feinberg & Horn, 1977, p. 90), where§ is the stoichiometric subspace. But dimS~ I, thus span(9Pp) cannot be the single 0 vector. On the other hand, it is easy to construct two essentially different conservative, reversible reactions with the same induced kinetic differential equation (see Exercise 4). Such an example shows that it is not true that a kinetic differential equation is induced by a unique reaction even within the class of reversible and conservative reactions. The addition of other, chemically relevant properties may be enough to ensure uniqueness. The right-hand side of the induced kinetic differential equation of an acyclic reaction cannot be the polynomial 0. As the reaction is acyclic it should contain at least one point without arrows pointing towards it. If one of these points corresponds to a component then the derivative of this component surely contains a (negative) term that cannot be counterbalanced as a result of other elementary reactions as no other point points toward this point. If all these points correspond to elementary reactions, than the components formed in this elementary reactions have a constant inflow that cannot be counterbalanced. Summarising the result of the previous two paragraphs: the right-hand side of an induced kinetic differential equation can only be zero, if the reaction is cyclic but not weakly reversible. This class of reactions has again proved to be the most complicated and interesting. The design of periodic reactions may also be treated as a uniqueness problem. In Subsection 4.5.6 we sketched how is it possible to use the methods of mathematical programming to design periodic reactions, or, to use the present terminology, to investigate the uniqueness of reactions with a prescribed linearised part.


69

4.7.1.5 A sufficient condition for the existence of an inducing reaction of deficiency zero As is known from an earlier exercise (Problem 4(ii) of Section 3.5) a generalised compartmental system is a reaction consisting of elementary reactions of three types k,m

.

y"' X(m) ~ y'X(i) y"' X(m) 0 ~ y"' X(m),

;;o

(4.22a) (4.22b) (4.22c)

where MeN;

={1, 2, ... , M}; y"', /eN for all i, meJI =Jt' u {0} (i ¥- m). In other words,

m, ieJt'

for all i, meJt', k;mE ~R; all the components (here: compartments) are contained in a single complex and all the complexes, except the empty one, contain a single component. The example 2X~O~X

shows that it is not true that a reaction only cons1stmg of elementary reactions of the type above is a generalised compartmental system, because in the example component X is contained in both the complexes X and 2X. Let us repeat that a generalised compartmental system is closed if it only contains elementary reactions of the type (4.22a), while it is strictly half open or strictly open according to whether it contains elementary reactions of the type (4.22b) or (4.22c) too. These definitions express that a generalised compartmental system is closed if and only if no matter enters it from and leaves it for the outside world. It is strictly half open if and only if no matter enters it but matter does leave it. Finally, it is strictly open if and only if matter does enter it and may leave it. These formulations are in accord with the results of Problem, Section 3.5. As we know from Problem 2 of Subsection 4.2.5, the deficiency of a generalised compartmental system is zero. It is also known that there exists a directed route from one component into the another in the V-graph of a generalised compartmental system if and only if there is a directed route from the first given component into the second one in the FHJ-graph of the reaction. Earlier it was shown that if a reaction is acyclic then it cannot be weakly reversible (Exercise I of Subsection 4.2.4). This statement shows that in the case of generalised compartmental systems the connection between the two graphs is even stronger: there exists not just a one-to-one correspondence between cycles and closed directed routes; the two graphs are essentially identical. (Here the empty complex may not be excluded.)


70

These statements together with the results of Feinberg, Horn & Jackson, and Vol'pert imply that the dynamic behaviour of a generalised compartmental system is completely characterised in two extreme cases: if the graph (either the V- or the FHJ-graph) is weakly reversible, or, if the graph is acyclic then the reaction behaves regularly in the classical sense. The zero deficiency theorem throws light on intermediate cases too: in the cyclic but not weakly reversible case, and in the acyclic and not weakly reversible case, some of the co-ordinates of the equilibrium point of the kinetic differential equation must be zero. The induced kinetic differential equations of generalised compartmental systems of the three types are as follows: (I) in the case of a closed generalised compartmental system X;= ( -

~kj)/(xY! + /~k;;(xj)·
0 (only if abc :F 0

has previously been assumed) and the vector in the definition of conservativity may be chosen p = (lfa, lfb, lfc). Hint. Start from the definitions. 2. The property that the vector on the right-hand side of a differential equation points into the interior of the first orthant is obviously a necessary condition for negative cross-effects to be absent. Give a polynomial example showing that it is not sufficient. 4.7.5 Open problems 1. Try to formulate and give a (constructive) proof of the uniqueness theorem on the inducing reaction to a given polynomial differential equation mentioned in Subsection 4. 7.1.4. 2. Does there exist a kinetic differential equation for which all the conditions (4.24) hold except the inequalities and which is still induced by a deficiency zero mechanism (although surely not by a generalised compartmental system)? Hint. Cf. the example at the end of Subsection 4. 7.1.5. 4.8 Selected Addenda 4.8.1 Lumping 4.8.1.1 Lumping in general Let us start from outside chemical reaction kinetics, from the theory and practice of modelling in general. Let us suppose that the phenomenon to be described can bt: characterised by a vector-valued function x of N coordi-


7~

n~es,

where N is an enormously large number. One should like to reduce the niiillWr of coordinates using, for example, a linear transformation, that is by 1i1ear lumping:

i=Mx,

(4.27)

Vl\ere M is an M x N matrix, and M is much smaller than N. Such a lumping is only meaningful if the resultant process behaves s~ilarily, with respect to some of its properties. For example, if the original poce~s obeys a differential equation

i=Kx tkn ;1 requirement may be that the resultant process obeys a differential ~ 11 ation of the same form:

i= tx \lith ;tn M x M matrix t. In this case the lumping is said to be exact. [t jS an easy exercise (Exercise 1 of Subsection 4.8.9) to show that the lllatriX M in eqn (4.27) realises an exact lumping only if there exists an },f x M matrix K for which

KM=Mi.

(4.28)

In this case K may even be determined from (4.28), although not necessarily ~niq 11ely.

It may also be interesting whether the lumping is proper or not. Linear lumping can be written in the following way: M

Xm =

L mmjxj

(m = l, 2, ... , M).

(4.29)

j=l

lhe tum ping (4.29) is said to be proper, if no xj is contained in two different

.r s ~sa summand with a nonzero coefficient mmj· This is equivalent to saying th"atno column of M contains more than one nonzero element, or, to put it another way, all the columns of M are either zero vectors, or lie in the direition of the coordinate axes.

4.8.1.2 Lumping in reaction kinetics An~e statement related the application oflumping in reaction kinetics is due to russ & Hutchinson (1971): A cpsed compartmental system is transformed into another closed compartmental systc111 by proper exact lumping.

Thii statement means that the transformed equation may be considered as theinduced kinetic differential equation of another closed compartmental sysr-m with newly introduced quasicomponents.


77

In the same paper cited above, Luss & Hutchinson consider the reaction MA(i)-+B

(i=1,2, ... ,N)

where M is a nonnegative real number. This system of equations may be lumped exactly into a scalar equation only if the trivial case M = I holds. In the case when M may be different from I upper and lower bounds may be obtained for x. The authors succeeded in giving these bounds in such a way that they only depend on the initial concentration of the lumped component. Based upon the difference between the two bounds they suggest how to lump the components, which are to be lumped into a new quasicomponent. Another interesting result of theirs is that they are able to estimate the sensitivity of the reaction rate of the lumped component with respect to changes in the original reaction rates and concentrations without knowing original reaction rate constants and initial concentrations, using nothing other than easily measurable data. Finally, let us mention that lumping is especially important in the presence (in the model) of continuous components (Aris, 1969; Bailey, 1972). It may be worth citing Aris (1969) here: 'When infinitely many components are involved, a first order system may be lumped into virtually any kinetic law, and probably an approximate form of such lumping accounts for the success of some of the less enlightening empirical rate expressions.' 4.8.1.3 Possible further directions The number of open questions is much larger than the number of solved ones in this area. We cannot even formulate them in a form suitable for the 'Open problems' subsections. (I) It is obvious that one can also define nonlinear lumping. (This is the case e.g. if one replaces mole numbers by mole fractions or weighted mole fractions in a nonconservative system.) (2) The same, or almost the same definitions apply to other types of models as well (inside and outside reaction kinetics). The mathematical problems arising are quite different. This may be the appropriate place to mention that lumping is widely used in econometrics (see, for example the basic reference text by Malinvaud (1969), or the book edited by Los & Los (1974). In this context lumping is called aggregation. Another name for the same notion is collapsing the states, and this expression is used in the theory of stochastic processes. In this area some of the questions are as follows: - Is the lumped process Markovian, if the original one was? -Does it fulfil at least the Chapman-Kolmogorov equations? (This is a less severe requirement than the previous one.) It is very hard even to formulate questions in this field. Starting points for the interested

78


reader may be the works by Erickson (1970), Heller (1965) and Rosenblatt (1971 ). (3) The basic requirement may be other than conservation of the form of the constitutive equation; other meaningful requirements in reaction kinetics include conservation of conservativity, deficiency, acyclicity, and controllability. (4) From the point of view of practice, by far the most important question is the following one. What can be said about the original system when some of the properties of the lumped system are known? This area seems to be unjustifiably neglected. Still, two recent works may worth mentioning, the one by Balakotaoiah and Luss (1982) connecting this area with multistationarity using the method of singularity theory, and the papers by Pismen (1984, 1985) on the dynamics of lumped reactions near singular bifurcation points. It may be the case that reducing the number of components does not help to solve a kinetic problem. In this case increasing the number of components may be a solution. How, and why, is the topic of the next subsection.

4.8.2 Continuous components The state of the system we investigate in the present book is almost exclusively characterised by a finite dimensional vector. Sometimes it may prove insufficient. The usual stochastic model can be conceived of as one where the state is a random variable, i.e. an element of an infinite dimensional linear space. If one wants to describe spatial effects (in a deterministic model) then a possible way to do so is to characterise the state by a function (by the mass density) and to write down a differential equation for the time versus mass density function. This will be a differential equation for a function with values in an infinite dimensional state space. (Memory effects can also be taken into consideration using an infinite dimensional state space. Cf. Atlan & Weisbuch, 1973.) In the present section we should like to present a model in which the infinite dimensional state space arises in the most natural way: in the case of continuous components. There are two problems leading to the notion of continuous components. At first, the number of species or chemical components may reach a huge number in such areas as oil chemistry, polymerisation or biochemistry. In these cases the number of the variables in the induced kinetic differential equation is so large that this system is difficult to treat; it may prove more promising to have a continuous manifold of chemical components. Theoretical, as opposed to practical, considerations may also lead to the introduction of the notion of continuous components. Let us consider the case of the ethane molecule. This molecule can in the first approximation


79

H H

H

H

(a)

(b)

Fig. 4.4 Eclipsed (a) and staggered (b) conformations of ethane.

have two basic conformations: the eclipsed one and the staggered one depending on the angle between the two methyl groups that can take the value ofO" or 180". Actually, this angle may take any value between o· and 180" (see Fig. 4.4), and the C-C bond energy of these species is different. Therefore a more subtle analysis should take into consideration ethane species characterised by an angle anywhere between o· and 180". The same kind of problem arises when considering transition states in the transition state theory. Let us consider the isomerisation of ethane in a more detailed way. Let the species characterised by the angle m be X(m), then one has the following elementary reactions: X(m)

k(p.m)

-+ X(p)

(p, me .A)

where .A is the set of components. If .A is a finite set then one has the induced kinetic differential equation Xm

=

L k(p, m) + L k(m, p)xp

-xP

peJI

(me .A).

(4.30)

peJI

If .A is an infinite set, say the interval [0, 180), then it is natural to replace the sums in (4.30) with integrals. In order to do so a measure Jl is to be introduced on .A to express the importance of the component X(m). Thus the generalised form of eqn (4.30) is xnr = -xP

f k(p, m)dJ.l(p) + f k(m, p)xpdJl(p)

(me .A).

(4.30)

./{

./{

In the more general case the basic equation R

Xm =

L (~(m, r) r= I

M

cx(m, r))k(r)

fl

x~ 0) .X"= 0 i =X, _i· = J' .X"= X

Hamiltonian

First integral

div f= 0?

xy

xy

+ + +

X

xjy

mentioned. However the examples describe nonconservative reactions, and so the analogy is rather strained. To avoid the disturbing effects of analogies perhaps the best procedure


87

would be to abandon their advantages as well. The precisely defined induced kinetic differential equations of chemical reactions can be investigated by the aid of pure mathematical concepts. The structure and role of 'concentration space' is distinct from those of real space, and so there is not too much to be done (if anything) with co-ordinate transformations leading to 'relativistic reaction kinetics'. However, other, not-too-sophisticated transformations may be relevant from the practical point of view of the chemist. Let us consider the differential equation i =fox

(4.34)

with an everywhere-defined, continuously differentiable right-hand side. A (state) transformation is an everywhere-defined, continuously differentiable function «p. Such a transformation transforms (4.34) into

:X= «p'o«p-•ox-fo«p-•ox = («p'·f)o«p-•ox.

(4.35)

The transformation of eqn (4.34) is called linear, if «pEGL(M) i.e. if «pis an invertible linear transformation. The transformation is said to be a change of scale, if

pr 1 0«pOpr 1 «p=

prMO«pOprM and [}l~m

C

IR+,

where

If a transformation D is linear and it is also a change of scale, then D is a change of the unit of measure. This is equivalent to saying that Dis a positive definite diagonal transformation. (Time) transformation of equation (4.34) is defined by a diffeomorphism

by which the new function X:=xoT is introduced. The transformed equation of (4.34) is: X=fOX·t. Change of time scale and change of time unit can be defined analogously to the previous case.

Mathematical models of chemical reactions 1he class of differential equations for which (~ + )M is an invariant set will (~+)M is an invariant set if and only if the vector f(x) shoWs inward from every point x of (~+)M. If (~+)M is an invariant set then l!l~)M has the same property. According to a theorem due to Vol'pert (~+)M is an invariant set of the induced kinetic differential equations of reactions. Of course, polynomial and nonkinetic differential equations, or even nonpolynomial ones may have this property. l3ased upon the definitions above the following statements can be made.

\e interesting for us.

II) Every change of scale (including every change of unit of measure)

preserves the character of polynomial equations in the sense that kinetic differential equations remain kinetic, while nonkinetic ones remain nonkinetic. (2) Every change of time scale (including every change of time unit) preserves the character of polynomial differential equations in the sense above. (3) If(~+)M is an invariant set of a differential equation then the change of scale and the change of time scale preserves this property. Group analysis is the general mathematical method for searching symmetries of differential equations. Group analysis of differential equations was initiated by the Norwegian mathematician Sophus Lie at the end of the ni11eteenth century. The first problem is to determine the general form of differential equations that admit a given group as a symmetry group. Loosely speaking G is a symmetry group for a differential equation if for all g EGg o p is asolution to the differential equation if p itself was a solution. The inverse to this problem is, how to find the Lie-group of symmetries of a given differential equation. (The fundamental books on the subject are: OvsjaJlnikov (1962, 1982), Bluman & Cole (1974) and Sibirskii (1982).) In a series of papers Steeb (1978, 1979; Schwarz & Steeb, 1983) pointed out (cf. w~lfman, 1979) that if an autonomous differential equation (for which all the solutions are defined on the whole real line) has a limit cycle solution, then the limit cycle trajectory is an invariant function of a one-parameter group admitted by the equation.

4.8-7 Principle of quasistationarity frOm time to time a paper emerges in the literature aimed at emphasising the n~essity of the foundation of the principle of quasistationarity, pseudosteady state hypothesis, or Bodenstein(-Semenov) method. The essence of the method seems to be an absolutely crazy idea- from the mathematical point ofview. In a system of differential equations let us consider the variables that ta~e on 'small' values to be constant. So: if a function is small, so is its derivative! It turns out that among the conditions that occur in chemical re~ction kinetics it does work well.


89

According to our knowledge, after Bodenstein (1913) it was Semenov ( 1939), Frank-Kamenetskii ( 1940) and Hirschfelder ( 1957) who treated the problem from the chemists' side. Sayasov & Vasil'eva (1955) and Heineken eta/. (1967) were the first to cite and use the seminal paper by Tikhonov ( 1952) on singular perturbation theory that is the adequate tool to treat such problems. Relevant contributions have since been made by Aris and coworkers (Aris, 1972; Georgakis & Aris, 1975; Viswanathan & Aris, 1975) and by Vasil'ev eta/. (1973). It was Noyes (1978) who from the chemical side was to reconsider things again. And now we have the (by the way, very useful) paper by Klonowski (1983) aimed at the mathematical foundation of simplifying principles for reaction kinetics. The paper is based upon the mathematical results of the Russian school, but seems to acknowledge nobody from among the other authors mentioned above. In spite of this, we would recommend just this paper the reader as it is self-contained; and this is very important as Tikhonov's theory is not an easy one for the nonmathematician. 4.8.8 Exercises I. Show the sufficiency of condition (4.28) for exact lumping. 2. Investigate exact lumping in the case when M = I. 3. Formulate a continuous component model for xylene isomerisation. (Xylene is known to have three distinct isomers: ortho-, meta- and paraxylene and the transition between them is usually described by the triangle reaction, cf. Aly eta/. (1965).) 4. Verify that (i) a generalised compartmental system is weakly realistic, (ii) a second order reaction (in which all the reactant complex vectors have a length smaller than three) is weakly realistic, (iii) the reaction rates of all the elementary reactions of type (ii) in a weakly realistic mechanism are linear functions of all the concentrations separately. 4.8.9 Problems Show that condition (4.28) for exact lumping is necessary as well. Hint. Start from the definition. 2. Give a necessary and sufficient condition for the solvability of (4.28) inK. What about the case when M is of full rank? 3. State and prove existence and uniqueness theorems on the continuous component model. Hint. The Picard-Lindelof theorem for differential equations in Banach spaces may be used; see, for example, Lang ( 1962). 4. Reformulate the abstract differential equation (4.33) into a partial I.


cJO

differential equation in the case when ..It = [0, I), q{ = [0, I) and both J.1 and A. are the Lebesgue measure. ?· Prove the statement of Subsection 4.8.3. Hint. Cf. Rudin, 1978, Remark 10.35. 6. Solve the induced kinetic differential equation of the reaction 1

2

X+-0+-X+Y 1! 1! 1j 2X2. Y.!.-2Y

Hint. The induced kinetic differential equation for x and y is a Cauchy-Riemann- (or Erugin-) system, therefore for z x + iy an easily solvable (separable) differential equation can be written down.

=

4.8.10 Open problem

I.

Find a physicochemical meaning of the measures J.1 and A. and devise a method (theoretically) to determine them.

5

Continuous time discrete state stochastic models

5.1 On the nature and role of fluctuations: general remarks 5.1.1 The logical status of stochastic reaction kinetics

A new approach has been adopted in the last two decades in the theory of chemical reactions that considers the chemical reaction as a stochastic process. In this chapter we deal with stochastic models of chemical reaction. More specifically, our intention is (I) (2) (3) (4)

to to to to

discuss the theoretical foundation, set up the model, investigate its properties, give some hints on the applications.

Arguments for the application of stochastic models for chemical reactions might come at least from three directions, since the model (a) takes into consideration - the discrete character of the quantity of components - the inherently random character of the phenomena· (b) is in accordance (more or less) with the theories of - thermodynamics - stochastic processes (c) is appropriate to describe - 'small systems' - instability phenomena. We have to make clear that the formulation of the theory of stochastic


92

kinetics does not reduce the importance of deterministic kinetics, since for great classes of phenomena the stochastic model is only slightly 'better' than the deterministic approach, while the mathematics of the stochastic model is much more complicated. Practically speaking, the set of stochastic models for which we can find an analytical solution is much more restricted than that of deterministic models. The subject of this chapter is organised as shown in Table 5.1.

Table 5.1 The logical status and mathematical framework of stochastic models of macroscopic physicochemical systems 5.1

Theoretical background

Stochastic models of chemical reactions 5.2, 5.8

The fluctuation-dissipation theorem Examination '- of chemical kinetics: of properties 1 theory and applications 5.4 ,(;)

.s'I

J

~--------~~v~-------,

Methods of solutions: exact results approximations qualitative properties simulation methods 5.3

QJ,..~,

~.s'f.

.

/c V~

&~ ;so

I'"

Stationary probability distributions 5.7

Small systems 5.5

Unstable systems 5.6

Stochastic models

93

5.1.2 Fluctuation phenomena in physics and chemistry: an introduction More than !50 years ago the Scottish botanist Robert Brown discovered the existence of fluctuations when he had studied microscopic living phenomena. However, the physical nature of the motion, which was named after its discoverer, was not known for a long time. As Darwin wrote in 1876: 'I called on him [Brown] two or three times before the voyage of the Beagle (1831 ), and on one occasion he asked me to look through a microscope and describe what I saw. This I did, and believe now that it was the marvelous currents of protoplasm in some vegetable cell. I then asked him what I had seen; but he answered me, "That is my little secret".' The knowledge that Brownian motion could be detected particularly well in colloidal solutions dates from the eighteen-seventies. The mass of the literally microscopic - Brownian particle is much greater than the mass of the solvent molecules, and the observable motion is the result of the individual motions of the small molecules. (For the early history of Brownian motion up to 1900 see Kerker (1974), and particularly Brush (1976, Chapter 15). A rather sophisticated explanation of Brownian motion was given by Einstein (1905) and Smoluchowski (1916), who calculated the temporal change of the expectation of the square of the displacement of the Brownian particle, and the connection between the mobility of the particle and the macroscopic - diffusion constant. The experimental verification of the heterogeneous nature of colloid solutions, and of the EinsteinSmoluchowski theory resulted in three Nobel prizes for colloid chemists in 1925-26 (Zsigmondy, Svedberg, Perrin). A time-dependent theory of fluctuations has been formulated in connection with Brownian motion (Uhlenbeck & Ornstein, 1930; Wang & Uhlenbeck; 1945). However, the theory of Brownian motion is not closed, and it is still open for physical and mathematical researches (Levy, 1948; Kac, 1959; Ito & McKean, 1965). The other class of fluctuation phenomena, well-known since the work of Gibbs (done in 1902, see Gibbs (1948)) and Einstein (1910), is equilibrium fluctuation. The theory of equilibrium (thermostatic) fluctuations considers the equilibrium state as a stationary stochastic process (see, for example, Tisza & Quay (1963) and Tisza (1966)). By thermostatic fluctuation theory the statistical character (e.g. the distribution functions and moments derived from it) can be computed.

5.1.2.1 Stochastic thermostatics, stochastic thermodynamics It was ·logical to extend the results of the thermostatic theory to temporal changes. Stochastic thermostatics adopts an intermediate level between statistical and phenomenological thermodynamics. Analogously, in principle the stochastic treatment of thermodynamics processes has an intermediate character between nonequilibrium statistical mechanics and phenomenolog-

94


ical (deterministic) thermodynamics. While statistical thermodynamics deals with the motion of microparticles, conventional thermodynamics describes the temporal change of macroscopic quantities which used to be interpreted as expectations. Taking into consideration the variance and the higher moments a more detailed description of irreversible processes might be given. Since the stochastic treatment of thermodynamics adopts an intermediate level between the strict microscopic and strict macroscopic levels, it might be called mesoscopic, according to the terminology of van Kampen (1976). A typical question of thermostatic fluctuation theory is the following: a system with volume V consists of n particles. What can we tell about the number of particles in a volume ~V ,

keN: re.Jik- 1

Pt(t) = L-., ~ Pk(t) ~ k(r)ka.(·,r) 1 1 L.. keN(,

(5.36)

reJI1- k

- Pit)

L

k(r) /"(·.r>,

The Kolmogorov-like equations for the absolute probability distribution function can be derived by using assumption (4): Pj(t)

=

4 P (t) L

k(r)f"(·.r)

1

/eN(,

re.Jfj-/

L

-Pit)

(5.37)

k(r)j"(·.r>(jeN~)

reA,_ j

Equations similar to (5.36) and (5.37) can be derived using assumption (4)* instead of (4): in this case we might call the equations the combinatorial model. Generally these equations are called the master equations. The structure of (5.37) is clear. Two types of elementary reactions are taken into consideration. The effect of the first class of reaction is that the state j is available from I (I can denote different possible states). The second class describes all of the possible transitions from the state j. Therefore we can write: Pj(t) =

L transition to state j

from /- transition from state j to /.

I

Sometimes the expressions 'gain term' and 'loss term' are used. Unfortunately the stochastic model appears more 'complex' than the associated deterministic model. Even in the case of the reaction X~ Y for Pit) we get Pj(t)

= &.(j +

l)Pj+ l(t)- &Pit) Pj(O) = oii.•

(5.38)

where j 0 is the quantity of X at the time 0. The dimension of the system of differential equation is j 0 + I. It is often remarked that stochastic models of chemical reactions can be easily extended to 'birth and death' type phenomena that take place in other populations of entities. Although we agree with this approach in principle, we have to remark that from the mathematical point of view the relationship between the three categories, namely stochastic models of reactions, simple birth and death processes (Karlin & MacGregor, 1957) and Markov population processes (Kingman, 1969) is not simple and is illustrated in Figure 5.1.

Stochastic models

105

Stochastic models of reactions

Population Markov processes

Simple birth and death processes

Fig. 5.1 Logical relationships among population Markov processes, stochastic models of chemical reactions and simple birth and death processes.

5.3 On the solutions of the CDS models 5.3.1 General remarks

The solution of the master equation (5.37) contains all the information about the system that is required in practice. Unfortunately, closed form solutions cannot be obtained even for the large class of reactions that are not important in practice. We just mention some methods that have been used in simple special cases: -

successive integration and induction (Sole 1972), determination of eigenvalues and eigenvectors (in cases when the statespace is finite and the system size is small), application of Laplace transformation (for the case X+ Y~Z (see Renyi, 1953).

The most important technique is the generating function method that transforms the system of(ordinary) differential-difference equations into one partial differential equation. Examples will be given for illustrating the scope (and limit) of this method. Not being able to solve the master equation in the more general cases we are often satisfied by the determination of the first and second moments. Furthermore, different techniques can be applied to approximate the 'jump processes' by continuous processes, which are more easily solvable. The clear structure of the stochastic model of chemical reactions allows the possibility of simulating the reaction. By simulation procedures realisations of the processes can. be obtained. The methods for obtaining solutions will be illustrated by discussing particular examples.

106


5.3.2 Chemical reaction X~ Y The reaction consists of one elementary reaction step with the rate constant k. For the absolute distribution Pj(t) = &'(l;(t) = j) the form of the master equation is:

Here j 0 is the quantity of X at t = 0. (Again the initial condition is deterministic.) Equation (5.39) is a system of linear differential equations with constant coefficients and, in principle, its solution can be given by the eigenvalues and eigenvectors of the coefficient matrix. However, because of the large number of the unknown functions this method cannot be used in practice. To solve the equation the generating function can be introduced:

=L Pj(t)zj 00

F(z, t)

(zeC; lzl ~ I)

(5.40)

j=O

Without giving the details of the calculations the partial differential equation for F is: oF(z, t)fot = k(l - z)oF(z, t)foz;

F(z, 0) = Zj'

(5.41)

(5.41) is a first-order linear partial differential equation with the known solution: F(z, t) =[I+ (z- l)exp(-kt}F•.

(5.42)

Pit) can be obtained by using the formula Pj(t) = (1/j!)o{F(O, t)

(5.43)

and so Pj(t) =

~J) exp(- jkt)(l -

exp(- kt))j-j•.

(5.44)

The moments are also directly obtained from the generating function: E[l;(t)] = o 1 F(l, t) = Dexp( -kt) D 2 [!;(t)] = oyF(l, t)

+ o 1 F(l,

(5.45)

t)- [F(l, t)F = j 0 exp(-kt)(l- exp(-kt)). (5.46)

In words, at every time point t -F 0 the quantity of component X can be given by a binomial distribution with expectation j 0 exp(- kt) and variance j 0 exp(- kt) (I - exp(- kt) ). As can be seen, the expectation agrees with the well-known deterministic solution. The two models are said to be 'consistent in the mean' (Bartholomay, 1957).

Stochastic models

107

5.3.3 Compartmental systems Compartmental systems might be considered -from a formal point of view - to be the analogues of isomerisation reactions. They are also quite often used in the 'biomathematical' literature. It is very important that stochastic models of compartmental systems are completely solved, since by the aid of the method of generating functions the absolute probability distribution can be expressed for every time instant as a function of the rate constants and the initial conditions. In particular, the state of the system is characterised by the sum of random variables having independent multinomial distributions, which can always be determined exactly. General compartmental systems have been solved by Siegert (1949) and simplified by Krieger & Gans (1960) and Gans (1960). Their model was interpreted in terms of transport among levels of internal energy. Darvey & Staff ( 1966) interpreted the earlier results in terms of chemical reactions. The master equation for the probability distribution function is n

dPl1h···ln · · (t)fdt =

n

~ ~ i...J L.

I= lm= I

-

k 1m (J"1 + I)P.l1h···ln . . (t) (5.47)

n

n

~

~

~

~

k,m }·,pl1h···ln. (t) •

I= lm= I

The multivariate generating function is defined by :c

F(z, t) = F(ziz2 ... z., t)

=L

oo

:c

L ... L

j, = 0 j, = 0

nz{'. m

Pj,j, ... j.(t)

j0 = 0

(5.48)

I= I

The differential equation for F(z, t) is aF(z, t)jat

=

n

n

L L klm(ZI -

zm)aF(z, t)jaz

(5.49)

I= II= I

(with F(z, 0) = Zmj'). Equation (5.49) can be solved, and it is possible to show that F(z, t) is the generating function of the multinomial distribution: (5.50) where w;(t) are time-dependent functions (0:::; w;(t) :::; 1). The consistency in the mean can also be proved. 5.3.4 Bicomponential reactions: general remarks The mathematical treatment of stochastic models of bicomponential reactions is rather difficult. The reactions X + Y-+ Z and X + Y ¢ Z were investigated by Renyi ( 1953) using Laplace transformation. The method of the generating function does not operate very well in the general case, since it leads to higher-order partial differential equations. In principle chemical


108

reactions can be classified according to whether the equation for the generating function can be solved exactly or not. Methods of solution have been developed in the last decade, but not too much can be expected from the practical point of view in the near future. Information obtainable by elementary methods and approximations will be illustrated by the example of the reaction X + Y ¢ Z. 5.3.5 Chemical reaction X+ Y¢Z 5.3.5.1 The master equation The master equation will be given. Let ~(t) be a random variable denoting the quantity of X; D 1 , and D 3 are the (deterministic) initial conditions for X, Y and Z; and 1.1 and A. are the 'forward' and 'backward' rate constants. Then for Pj(t) &'(~(t) = j)

=

dPif)/dt

= A.[(j + l)(D 2 - D 1 + j + l)Pj+ 1(t)- j(D 2 - D 1 + j)Pit)] + Jl[(D 3 + D 1 - j + l)Pj_ 1(t)- (D 3 + D 1 - j)Pit))

Pj(O) = Ojo,·

(5.51)

5.3.5.2 Use of Laplace transformation Renyi (1953) gave a recursive formula for the Laplace transform of the distribution, 00

~(s) = f e-" Pj(t) dt,

(5.52)

0

but he did not calculate the distribution Pj(t) itself. 5.3.5.3 Determination of expectation It is sometimes sufficient in practice just to determine the expectation and variance. Multiplying eqn (5.51) by j and summing over all possible j the

equation for the expectation can be obtained: d/dt E(~(t))

=

A.E[(D 2

-

~(t)]E[D 3 -

~(t)]

-

1.1E(~(t))

+ A.D 2 (~(t)). (5.53)

We see that in the equation there is a term containing the variance D 2 [~(t)]

=

L (j- E[~(t)])2 Pj(t) j

= L F Pit) =

{E[~(t)]}2

(5.54)

E[~(t) 2 )- {E[~(t)]}2.

In general the treatment of bicomponential reactions is rather difficult

Stochastic models

109

because the expression for the derivatives of the moments contain the higher moments, and so a closed system of equations cannot be set up. Sometimes the approximation E[!;(t) 2 ] = (E[!;(t)])l is adopted. This assumption is equivalent to the condition that D 2 [!;(t)] = 0 and so it corresponds to the omission of the stochastic character. If D 1 -+ oo (i.e. for large systems), then A.D 2 [!;(t)]-+ 0 and with TJ(t) = D 2 - !;(t) and l;(t) = D 3 - l;(t), (ll(t) is the quantity of Y, l;(t) is that of Z) we obtain dfdt E[!;(t)]

= A.E[TJ(t)]E[!;(t)] -

~E[I;(t)].

(5.55)

This equation is analogous to that for the deterministic model. Similar kinds of equations can be derived for more general cases. 5.3.5.4 The behaviour of the reaction during the initial period of the processes Pj(t) can be approximated using Maclaurin series expansion: Pj(t)

=

t" L Pj"l(O)nt. I, s < 0 or s > s 0 - I + e, where s 0 is the initial number of substrate molecules S: s(O) = s0 . The evolution equation for the generating function would be of the second order. However, introducing the marginal generating function s0

F.(z, t)

=

-

I+ e

L

z' P.,(t)

(e

= 0,

(5.85)

I)

s=O

we get

0

0

ofot F.(z, t) = ex(e + I) ot F.+ I (z, t) - exez ot F.(z, t)

-

(~

+

(~

+ y)(l -

(5.86)

e)F.(z, t) + y)z(2- e)F._ 1(z, t) (e = 0, 1).

The solution of this equation is (Aninyi, 1976):

F0 (z, t) =

rexp[-~/ex(z-

+ +

r

:z:

f.L., fL.,

i= In =0

F 1 (z, t) = r -

l)exp(-yt)

~exp(-(ex + y)t) rt) i _ J..(•J I

(5.88) '

1

nwhere A.; =I= Y; and _ (A.\">) 2 + (ex+ ~ + y)A.\">) + exy q. = ex(y + J..~•l) r, rand

(i = I, 2).

r can be determined from the

F.(l, t) + F 2 (l, t) =I;

F0 (z, 0) = 0;

F 1 (z, 0) =

z0 (Az)

boundary and initial conditions. Assuming that the solutions F.(z, t) are true generating functions, i.e. they are polynomials of finite degree in z, it can be shown that the summation, contains a finite number of terms only, and r = = 0. The q.s are integers,

r

Stochastic models 0

~

q. < S 0

-

121

1, (q. = n), and the A.s are the roots of the equations

+ [~ + cx(n + 1) + y]A. + cxy(n + 1) = 0;

1.. 2

(n

= 0, 1, ... S 0

-

1).

All the A.s are different negative real numbers. The absolute probabilities can be calculated from the generating function, and they appear as finite sums of exponential functions. The CDS and CCD models of Subsection 4.1.1 have been compared. The deterministic value always evolves above the expectation, and their differences can be 20-30% of the former (see Fig. 5.3). E, S

(a)

1.0 (b)

E, S

1.0

\ 0.5

\

\ ·,

' ·,·,E(~) ...... 0

-0.2~

--·-

1'-.2345678

~

E(t;)- D(s)

0123456789

Time Fig. 5.3 Time course of the Michaelis-Menten reaction. k 1 = k_r= k 2 = I. Deterministic solution ([£], [S]) and stochastic expected values (£@, E(t;>) are compared (a). The stochastic expected value £(~) is plotted together with its variance (b).

5.5.2 Ligand migration in hiomolecules The binding of ligands to proteins can be typically treated by CDS models, see, for example, Alberding eta/. ( 1978). In particular consider the binding of carbon monoxide to myoglobin where the ligand CO may encounter four potential barriers. The random variable ~L(t) denotes the number of ligands

122


in well L in a given biomolecule with L = I, 2, ... Lmax· (in this example Lmax = 4). The transition from stage L to K is proportional to the number XL. (XL is the value of ~L(t)). The binding is so tight that well I acts as a trap, and so transitions from well I can be neglected. The first ligand to occupy well I blocks further transitions. Adopting a Markovian approach to ligand migration the system is characterised by a distribution function P(~L(t) =XL; A) (L = I, ... , Lmax). where A denotes such parameters as temperature, pressure, pH and ligand concentration. Using the abbreviation X= (Xi, ... , XLm.J the stochastic evolution equation is iJP(X,t) _

Lmax Lmax

- I I

a

L=2K=2

f

+

(XL+ I)yKLP(XL

Lm••

L

(XL+ l)ysLP(XL L=2 L#K

+

+ I, X')

Lmax

+

Lmax Lmax

L PLP(XL- I, X')- L L XLyKLP(X)

L=2

- L

L=2

L=2K=2 L#K

Lm.u

Lmax

+

,

1, xK- 1, x)

XLYsLP(X)-

L

L=2

PLP(X)

Lmax

I

L=2

(5.89)

(XL+ I)y,Lox,.,P(XL

+ 1,

xi- 1, X')

Lmax

- L

L=2

XLYlLOx,.oP(X).

Here X' is the reduced vector derived from X by removal of the explicitly written components. YKv YsL and PL are rate parameters, S refers to the solvent, t and A are omitted for the sake of brevity. Using again the multivariate generating function (5.48), F is in the form: oo

I

F(z,, ...•

ZLmax;

oo

L L ... L

t, A):=

X 1 =OX2 =0

n z[L.

Lmu

P(X; t, A)

XLmax""o

(5.90)

L= I

The evolution equation for F is Lmax

iJF(z, t; A)jot =

L PL(zLL=2

l)F +

Lmax

L YsL(l L=2

Lmax Lmax

+ L"J;2K"J;2 YKL(zK+ L ... X1

oF - zL)"JZ L

oF zL) iJzL

Lmax

L L y,d(XL + l)Ox,.lP(XL + I, xi XLmax L =

I, X')

2

n z{

Lmax

- XLOx,.oP(X)]

1•

I= I

(5.91)

Stochastic models

123

Before solving the equation we have to mention an experimental observation done by optical absorption measurements, which measures an average over a very large number of biomolecules: (5.92) From (5.90) and (5.92) N•• p(t, A) = F(O, I, ... , I; t, A),

(5.93)

To solve (5.90) analytically for F(O, I, ... , I; t, A) with the initial distribution P(X, t = 0, A) we need the 'phenomenological coefficient' matrix: (5.94) Lmu

MKK

= YsK-

L

YLK((~L(t))),

(5.95)

L,.l

L,.K

where the (~K(t)) bracket denotes the average occupation number in well K at time t. The calculated value of N.,P(t, A) is Ncxp(l, A) =

exp { -llbll- 1 :X:

*

:~: ~L(A>[:~>K 1 BKL(exp llKt- l)bK

)I {I -

+ B 1Lb11

r]}

:X:

L ... L X.z = I

1

XLmax

P(O,

x2, ... , XLmax; t = 0,

A)

(5.96)

=1

llbll- 1

:~~ BKLbK expll~rL 1

Here bK = (bK 1 , ••• , bKL ) are the eigenvectors of M with corresponding eigenvalue IlK• B is the c~"factor matrix of eigenvectors, and II b II denotes the determinant of b. Experiments suggested that the deterministic and stochastic approaches give different results when the probability of finding more than one ligand bound to a given biomolecule cannot be neglected, even if we accept that continuous deterministic models might be defined for small systems. 5.5.3 Membrane noise Electrical noise in biological membrane systems is explained in terms of opening and closing of ionic channels (for two useful review see De Felice ( 1981) and Frehland (1982)). The qualitative behaviour of membrane activity strongly depends on the density ofmembrane channels (Holden, 1981; Holden & Yoda 1981). Integrating numerically the celebrated HodgkinHuxley equations (Hodgkin & Huxley, 1952) it was demonstrated that the number of channels is a bifurcation parameter. The fact that channel

124


numbers can be less than 104 and channel density can be as low as I J.lm- 2 implies the adequacy of using stochastic methods for describing membrane kinetics. The ionic transport system is thought to consist of a number NP of identical channels. According to the basic assumption the measured current is proportional to the number of channels being in conducting 'open' state. The measured current J(t) as a random variable is given: J(t) := Npj 5 P 1 (t)

= j 5 Nt(t)

(5.97)

where } is the current through a single open channel, P 1 (t) is the probability that one channel is in the open state. Current fluctuations occur because of the fluctuation in the number of open channels N 1 (t) The variance crf is given by 5

crf = (} 5 ) 2NpPf(I - Pf), where

Pf =

(5.98)

lim P 1 (t). r~x

The kinetic model adopted by Hill & Chen ( 1972), (but see Chen, 1978), assumed that the channels are independent and that each one consists of x independent subunits. Each subunit can be in either one of two configurations. The state of the channel is given by the number of subunits in a particular configuration. A channel is thought to be nonconducting unless all x subunits are in the appropriate configuration. For the particular case of x = 2; the kinetic scheme is rm~rn~rn

P0

P1

(5.99)

P2

where [1] denotes a channel in which j of the subunits are in adequate configuration, pj is the fraction of M such channels in that state. At equilibrium N]. =

Mp~

= M[cx/(cx +

~)]2

+

~)]2

N 0 = Mp"•

= M[~/(cx

Nr =

= M[2cx~/(cx

Mp;

+

=Mn~ =M(l -

~)2]

(5.100)

nx) 2

=M2nx(l -

nx).

Often the current spectrum is measured. Theoretically, the spectrum of 'component' fluctuation is S(ro)

=

8Mn~(l- nx>[l +n~ 2 , 2 + 4 1+-ro~~ 2 l

(5.101)

The current spectrum S1(w) can be associated to S(ro) by the relation (5.102) where gK is the conductivity of an open (potassium) channel, Vis the actual potential, EK is the equilibrium potential for K+. More complex models exist to describe different types of membrane noise.

Stochastic models

125

Noise analysis is a method of differentiating between concurrent transport mechanisms (Fig. 5.4). A particular problem, transmitter-induced membrane noise, will be studied in Chapter 7.

Formal chemical reaction

Assumed transport mechanism

Kolmogorov equations

Correlation function

Calculated noise spectrum

no

Noise analysis cannot provide sufficient information to reject the mechanism assumed; if you have another assumption. repeat the calculation

yes

Fig. 5.4 Scheme of an algorithm to differentiate among mechanisms of membrane transport processes.

5.5.4 Kinetic examinations of fast reactions It was seen earlier (Subsection 5.4.3) that at least the 'spirit' of the fluctuation-dissipation theorem can be utilised in chemical kinetics, since rate constants can be evaluated from equilibrium fluctuation measurements. According to the classical approach of experimental reaction kinetics, the reagents have to be mixed, then the temporal changes of the components have to be followed. The classical methods are not appropriate for the study of fast reactions, since the reaction is much more rapid than the mixing.


126

The relaxation method is an important technique for obtaining rapid kinetic measurements in solution (e.g. Eigen & De Mayer, 1963). According to this technique the chemical and physical equilibrium is disturbed by perturbing the system with a jump in an intensive variable (such as temperature, pressure or concentration). Certain functions of the rate constants can be determined by the process relaxing to a new equilibrium. The relaxation technique is very useful, though its theoretical foundations are not completely well-established. First, the perturbation disturbs the physical as well as the chemical equilibrium; the common treatment of physical and chemical relaxation processes - mostly with deterministic models- is difficult (see Czerlinski, 1966; Pecht & Riegler, 1977). Though it is ingenious, the method offered by Hayman (1971), according to which the volume and the entropy are interpreted as 'chemical components', and their change is considered as a formal chemical reaction, it cannot qualify as the final solution of the problems. Second, there are opposing requirements for the size of the concentration change due to the perturbation: it has to be small enough to allow the use of the linear equation, but it also has to be large enough to justify the neglect of fluctuations. From the theoretical point of view measurements based on fluctuation phenomena are better, since the equilibrium fluctuation may be interpreted as spontaneous perturbation and relaxation. Consequently, kinetic information is available without perturbing the system externally. In practice, chemical fluctuation measurements refer to small systems, therefore the problem is discussed in this section. Fluctuation measurements can be classified as follows (Romine 1976): indirect measurements, and direct measurements, which may be based on the time representation of fluctuations, or the frequency representations of fluctuations. The indirect measurements are derived from static not kinetic measurements, where the effect of chemical fluctuations appears as a disturbance. It is well-known (e.g. Gordon, 1969) that the connection between the function describing the shape of the spectrum line and the associated physical quantity is

I X

/(ro)

= l/27t

exp(iro•)C('t)d't,

(5.1 03)

-x

where the physical quantity is considered as a stationary stochastic process ex(t) with a correlation function c('t). ex can be the magnetic momentum of the spin, or the dipole moment of a molecule. ex might be a function of temperature T, pressure, P, and chemical composition, c; (the function is denoted by &). The effect of fluctuation of these intensive variables and of time: dex

= (iJ&jiJp) dp + (iJ&jiJT) dT + ;~1 (~~)de; + (iJ&jiJt) dt.

(5.1 04)

Stochastic models

127

& is considered as a stationary stochastic process, and so the derivatives are taken in some stochastic sense. It is a problem to separate the effect of the component fluctuation from those resulting from temperature and pressure fluctuations. Another, finer problem is the decomposition of the effect of component fluctuation on 'diffusion' and 'reaction' terms. Extensively used measurement procedures are based on the line broadening of nuclear magnetic resonance peaks (e.g. Bradley, 1975); however, the area of application is limited. Another indirect method is based on the light-scattering analysis of chemically reactive solutions (when Cl is taken as the relative permittivity). The fluctuation of the relative permittivity reflects the fluctuation of the quantities of the components (at least when the polarisability of the reagent and product molecules differs). Sometimes a rough approach is adopted, according to which temperature and pressure fluctuations can be neglected (Berne, eta/., 1968). The broadening of the Rayleigh component of scattered light has been analysed by the methods of conventional linear nonequilibrium thermodynamics (Blum & Salsburg, 1968, 1969). They pointed out that in the limit of zero scattering angle the (spatial gradient-dependent) transport processes do not contribute to the line broadening. They considered at how small an angle the measurements have to be obtained. Other applications are by Yeh & Keeler (1969, 1970), who determined certain functions of rate constants of dissociation reactions taking place in 0.1-1 M electrolytes and of conformation reactions of macromolecules. The conformation change of DNA has been investigated by Simon (1971). The results of these measurements sometimes gave relaxation time with a 10- 8 s order of magnitude. The modern theory· of light scattering, and its kinetic and biological applications are surveyed in the book of Berne & Pecora (1976). The direct measurement of fluctuations seems to be feasible, given current and anticipated measurement techniques. The measurement of optically active systems is particularly favourable, since laser-based absorbance measurements, fluorescence spectroscopy, circular dichroism, rotary dispersion and Raman spectroscopy are all avariable. Conductance measurements have also been used in electrolytes for determination of composition fluctuation. Most of the techniques mentioned above are appropriate to investigate both the time, and spectral, representation of fluctuations. The scope, and limits of, the experimental techniques for determining the composition fluctuation was analysed by Magde ( 1977). The representation of fluctuations connected with fluorescence correlation spectroscopy has been adopted for the study of the reversible binding of ethidium bromide to DNA. The reaction has biological significance in connection with the discovery of the transcriptive mechanism of the genetic code, since ethidium bromide inhibits nucleic acid synthesis. The DNAethidium bromide complex is strongly fluorescent, its fluorescent quantum


128

yield is 20 times larger than that of pure ethidium bromide. A single-step reversible bimolecular reaction scheme can be associated with a lumped model of the reaction. The rate constants have been determined from the time correlation function and the expectation of the quantity of DNA. An example of direct measurements based on the frequency representation of fluctuation is the study of the association-dissociation reaction of BeS04 in a 0.03 M solution, with conductance measurements (Feher & Weismann, 1973). It is particularly interesting that they could increase the ratio of the reaction noise to Johnson noise of the circuit, since the former is a quadratic function of the applied direct voltage, while the latter is independent of the voltage. Though modern experimental techniques are appropriate for fluctuation measurements, the specific examination of the fluctuation phenomena due to chemical reaction is still rather difficult. 5.6 Fluctuations near instability points 5.6.1 An example of the importance offluctuations

Let us consider the reaction ).

A+ X -+2X 11 X-+0,

(5.105)

where A is the external and X the internal component, and 0 is the zero complex. This reaction can be associated with a simple birth-death process. The deterministic model is the following: dx(t)/dt

= (A.- jl}x(t); x(O) =

(5.106)

x0 •

(Here A.= A.[A].) The solution is x(t) = x 0 exp(A. - jl}t.

(5.107)

If A. > jl, i.e. the birth rate constant is greater than the death rate constant, x is an exponentially increasing function of the time. If A. < jl, x decreases exponentially. For the case of A.= ll x(t)

=

(5.108)

x0 .

The stochastic model of the reaction is dPk(t)/dt = -k(A.

+ !l)Pk(t) + A.(k- 1)Pk_ 1(t) + !l(k Pk(O) = okx.; k = I, 2, ... N.

+

I)Pk+ 1(1)

(5.109)

There are two consequences of the model: (1) the expectation coincides with the process coming from the deterministic

theory, i.e.

129

Stochastic models E[~(t)]

= x 0 exp(A. - ll)t,

(5.ll0)

which in case of A. = ll reduces to the form E[~(t)]

= X0.

(5.lll)

(2) the variance of the process is D 2 [~(t)]

= (A.+

ll)t.

(5.ll2)

For the case of A.= ll D 2 [~(t)] = 2A.t,

(5.113)

i.e. progressing with time, larger and larger fluctuations around the expectation occur (Fig. 5.5). E[l;(t)]

±

D[l;(t)]:_. _ _ _ _ _ E[l;(t)]:------

...-· ,..,..... .,.,.

--·-·

..--·---·

-·-

Fig. 5.5 Amplification of fluctuations might imply instability.

It is quite obvious that in this situation it is very important to take the fluctuations into consideration. Such kinds of formal reactions are used to describe the chain reactions in nuclear reactors (e.g. Williams, 1974). In this context it is clear that the fluctuations have to be limited, since they could imply undesirable instability phenomena. 5.6.2 Stochastic Lotka- Volterra model Let us consider the (irreversible) Lotka-Volterra reaction:

130

Mathematical models of chemical reactions k,

A+ X-+2X k,

(5.114)

X+ Y-+2Y k,

y -+0. Here X, Yare internal and A external components, and 0 is the zero complex. The deterministic kinetic equation is dx(t)/dt dy(t)fdt

= k 1 ax(t)- k 2 x(t)y(t)

= k 2 x(t)y(t) -

The equation (5.115) has one trivial stationary solution:

(x~1

k 3 y(t).

(5.ll5)

= y~1 = 0) and one nontrivial (5.116)

According to linear stability analysis the trivial stationary point is an unstable saddle point, while the nontrivial stationary point is a marginally stable centre. The Lotka-Volterra system exhibits undamped oscillation, and the amplitude of the oscillation is determined by the initial values (and not by the structure of the system). The equation of the trajectory is x

+y

- x 0 In x - y 0 In y - H

= 0,

(5.117)

where H is the integration constant that is determined by the initial value. The stochastic version of the Lotka-Volterra model leads to qualitatively different results from the deterministic approach. The master equation is dP,,.(t)/dt

= - (k 1 xa + k 2 xy + k 3 y)Pxy(t) + k 1 (x - I )aPx _ 1.,.(t)

+ k 2 (x + l)(y- l)Pn 1.y- 1(t) + k 3 (y + l)P,_,.+ 1(t). (5.118) It can be shown (Reddy, 1975) that P~~.:= &'(~ 1

= 0,

~2

= 0) =

I,

(5.119)

i.e. the absorbing state x = 0, y = 0 is the only stationary state. This result is different from that obtained from the deterministic model. Similar results were given by Keizer (1976). However, he also demonstrated that the fluctuations might be bounded, if fluctuations for the external components (including the zero complex) are also allowed. Possible behaviours of the realisations were demonstrated by simulation methods (Fig. 5.2). 5.6.3 Stochastic Brusselator model The Brusselator model (Prigogine & Lefever, 1968) is a formal reaction scheme:

Stochastic models

131 k,

A-+X k,

B+X-+Y+C

(5.120)

k,

2X+ Y-+3X k.

X-+E,

that has a deterministic kinetic equation: dx(t)fdt = k 1 a- k 2 hx(t) + k 3 x 2 (t)y(t)- k 4 x(t) dy(t)/dt = k 2 hx(t) - k 3 x 2 (t)y(t).

(5.121)

Introducing the dimension-free variables

- yk;.x, {k;

X=

y- {k; =

yk;.y,

and the parameters A= (kta)/k 4 jk 3 /k 4 ,

B

= (k 2 /k 4 )h,

the following system of differential equations is obtained: dXfd• =A- BX+ X 2 Y- X dYfd• = BX- X 2 Y.

(5.122)

Equation (5.122) has one stationary solution: X 0 = A,

Y 0 = B/A.

(5.123)

The necessary and sufficient condition of the stability of the stationary point is B < I + A 2 • It can be shown for this system that an unstable stationary point implies a limit cycle. ·stationary and time-dependent solutions of the master equation of the Brusselator have been approximately established by Turner ( 1979). The main point of his procedure is that the stationary solution of the master equation can be seen as a time average over a period of the limit cycle of Pn(t), the time-dependent solution. Some illustrative results of the symmetry-breaking bifurcations in the stochastic Brusselator model are shown in Fig. 5.6 (after Nicolis, 1984). 5.6.4 The Sch/ogl model of second-order phase transition Chemical reaction models of 'phase transition-like' phenomena have extensively been studied. Schlogl's model (1972) of the second-order phase transition is A+

k~

X~2X k~

(5.124)


132

log P

c)

(b)

y

(a) X

y

X

"'(X,, Y,)

Fig. 5.6 Transition to a limit cycle in a two-variables stochastic model: (a) For A. < A.,, there is only one stationary distribution of probability P, peaking on the stable steady state (x,, y,); For A. > A.c, there is a family of time-periodic distributions (differing by a phase), peaking on the stable limit cycle solution (x,(t), y,(t)).

Here A, B and C are external components, and X is the single internal component. The deterministic model is dx(t)Jdt

=

=-

x2

+ (l

= =

- ~)x

+ y,

=

setting knA1 I; kl I; ~ k;[B]; y k2[C]. The stationary value of x will be denoted by X 00 • The two 'phases' are represented by the cases whether x differs from 0 or not. y > 0 implies X 00 > 0, therefore 'phase transition' does not occur. For

y=O

_{I - ~.

Xoo -

0,

if ~
1.

It means that for ~ < I the stationary concentration is finite. A change in ~ could lead to a 'phase-transition'. The analogy with the most frequently investigated second-order phase transition phenomena, namely to the behaviour of ferromagnetic materials, is trivially demonstrated by making the following correspondences: Xx

+-+M;

y+-+

H;

~+-+ TJTc

where M is the magnetisation, H is the magnetic field, T is the absolute temperature and T,. is the critical temperature. What follows from the stochastic analysis of the system? In the case of y = 0 the only stationary state, analogously to the Lotka-Volterra case, is the absorbing state 0, i.e. P,(oo)

=&I(~=

0) = I.

(5.125)

This result is different from that obtained based on the deterministic model. Oppenheim et a/. (1977) attempted to resolve this contradiction (but see Horsthemke & Brenig, 1971 ). They restricted the state-space by making

Stochastic models

133

y

0.75

0.5

0.25

0.25

0.5

0.75

n

P=1~T=Tc

Fig. 5.7 Second-order transition in the deterministic (Schlogl) model.

x = 1 a reflecting state and introduced the concept of quasistationary state. The starting points of their analysis are:

(a) in the case ofmultistationarity in a deterministic model it depends on the initial value which stationary state will be realised; (b) the stationary distribution of the master equation does not depend on the initial distribution; (c) the temporal change of the distribution depends on the initial distribution; (d) the time course of the reaction can be characterised by two separable time scales: the quasistationary states which are associated with the deterministic solution in the thermodynamic limit are realised much more rapidly than the states of the stationary distribution. Though (a)-(d) are not generally fulfilled, e.g. the range of validity of(b) is definitely narrow ('ergodicity') in the case of y = 0 they hold. The temporal


134

change of the shape of the distribution and the relaxation times are visualised in Fig. 5.8.

T = O(e•)

Initial distribution T = O(N") Quasistationary t = 0 distribution

P.:.:U_ p:.:.u (1 - ~)

0

p:lL

(1 - ~)

0

n

Equilibrium distribution

(1- ~)

0

n

n

Fig. 5.8 Relaxation of distribution function Pn(t) from initial state to final state via quasistationary state.

5.6.5 The Schlogl model offirst-order phase transition

The Schlogl model of the first-order phase transition is given by the reaction A+

k:

2X~3X

(5.126)

A, Band Care external components, X is the only internal component. With the notation

a= (kt/k!)[A];

k

=(k;/k!)[B]; b =(k2/k!)[C],

the deterministic model is dx(t)/dt = x 3

-

ax 2

+ kx -

b

=R(x).

Under the circumstances a > 0, k > 0 and b > 0 and the polynomial R(x) has three real, positive roots if R

(a-

(a 2

-

3

3k) 112)

~

0 and R

(a -

(a 2

-

3

3k) 112)

~

0

R has a triple root, if k =am and b = a 3 /27. The phases are represented by the stationary points. The triple root might be associated with the 'critical point'. Since the constitutive equation of the ':'an der Waals gases is also a third order polynomial, R(x) can be associated with the equation

p _ RT _ a 1

-y

V2

+

a2

V3

Stochastic models

135

by making the following correspondences: x+-+

v-•;

k+-+RT;

b+-+p,

where V is the volume, p is the pressure and T is the temperature. Stochastic analysis of this model has been reported many times in different contexts. Janssen (1974) calculated the stationary distribution: _ I b Tin (i- l)(i- 2)a + b pn- p 0 fifki= 2 (i- l)(i- 2) + k

(5.127)

This is not the only example dealing with the role of fluctuations in general, and in the vicinity of bifurcation points, specifically. Based on the approximative calculations of Nitzan et a/. (l974a) it was suggested that the stationary distribution can be multimodal, and the location of the maxima can be associated with the stable stationary points of the deterministic model, and the minimum corresponds to the unstable stationary point. This result does not hold for small systems; and the question ofmultimodality will be discussed in Subsection 5.7.4. Matheson et a/. (1975) obtained a qualitatively similar result, they also estimated the relaxation time of the process from the quasistationary state. Nicolis & Turner (1977) calculated the variance at the critical point and in its vicinity. Under and beyond the critical point the variance is a linear function of the volume, but at the critical point it shows a stronger volume-dependence: lim

cr 2 ( oo) = const. V 312 + o( V).

v-oo N-oo N/V-consl.

This result shows that the formulae (5.1) and (5.2), which suggest that the fluctuations are insignificant for large systems, are not always valid. Further remarks

Gillespie (1979, 1980) has emphasised the possibility of using stochastic simulation methods for identifying the quasistable stationary states and calculating the associated mean transition times, mean fluctuation periods, and effective fluctuation ranges. It was argued (Horsthemke & Brenig, 1971; Blomberg, 1981; Haenggi et a/., 1984) that the nonlinear Fokker-Planck equations (derived in a slightly different way) also operate correctly in the critical point. Related questions in connection with stochastic bistable systems are discussed in Subsection 5.6.6. 5.6.6 Stochastic theory of bistable reactions I. The motivation for studying bistable reaction systems came from two different directions. First, a model of a Brownian particle moving in a


136

potential well was adopted by Kramers (1940) to reformulate the diffusion model of the chemical reactions at the microscopic level. Microscopic models of chemical reactions are beyond the scope of this book. From a more general point of view it is interesting that Fokker-Pianck equations in a double well potential are widely used to describe phase transition phenomena (see van Kampen, 1981, VIII. 7 and X1.6). Exact solutions for diffusion in bistable potentials in the case of particular potential functions have been given (Hongler, 1979a, 1982; Hongler & Zheng, 1982, 1983). Second, multistationarity has been demonstrated experimentally in continuous stirred tank reactors, as was mentioned earlier in Subsection 4.4.2. 2. The estimation of relaxation times from the Fokker-Planck equations was discussed by Matsuno eta/. (1978). A Fokker-Pianck equation can be written as a atP(x, t)

=-

a ax[A(x)P(x, t)]

az

+ (I/2V) axz [B(x)P(x,

t)].

(5.128)

The stationary distribution is P 51 (x) = (K/B(x))exp[- VU(x)], where K is a normalisation constant and X

U(x) = -2 f[A(x')/B(x')]dx', 0

In the physical literature, adopting the Landau-Ginzburg picture, U is often referred to as the 'free energy'. In the chemical context it is better to use the term 'free-energy-like'. The stationary states of the system occur at the extrema of U(x). Metastable states can be identified with local minima. The relaxation time of the process leading from metastable to stable state is estimated as .\'11

'"' =

X

2 V f exp[VU(x)] dx f 1/B( y){exp[- VU( y)]} dy

(5.129)

-"" where x. and xm are neighbouring unstable and metastable states. It can be seen that the relaxation time is exponentially dependent on the volume. By evaluating the integrals the relaxation time is estimated as 'tm ~

const.exp[V(U(x.)- U(xm))].

This formula shows not only the exponential dependence of the relaxation time on the system size, but also the effect of the height of the potential barrier ( U(x.) - U(xm)). 3. Estimation of relaxation times from the master equation using spectral analysis of the coefficients matrix was used by Dambrine & Moreau (1981a, b) for a chemical system having a finite state-space. The reaction

Stochastic models

137 X+

Y~2X,

X~Y

(5.130)

obeys the master equation with the following transition probabilities: ~

=

o,

if IJ- II

2,

;?;

Pli+ 1 = ct)(N- J) k • Pj-l.j- C2)

p'ff = (c 1 j(N- j) + c2j). Here, c 1 and c 2 are constants determined by the rate constants, N is the total number of molecules. Exact bounds for the first two nontrivial eigenvalues of the coefficient matrix were given as: IA2I- 2

In thermodynamic limit 't 1

"'

= 't2

4i IAtl-'

= 'tt

(D(exp(CN)) and (D(l) < 't 2 < (D(ln N).

Remark The method has also been extended to infinite state-space models by Borgis & Moreau (l984a) who studied the reaction A -+ X, A + X-+ 2X, A + 2X-+ 3X. The coefficient matrix of the master equation is a tridiagonal infinite matrix. The eigenvalues and eigenvectors of this matrix are the limits of eigenvalues and eigenvectors of truncated matrices obtained by introducing a reflexing state for X= N where N is the number of molecules at which the system is truncated. 4. A scaling theory of transient phenomena near an instability point has been formulated by Suzuki (1976a, b, 1980, 1981, 1984). The theory is based on a generalised scale transformation of time and equivalently on a nonlinear transformation of stochastic variables. The whole range of time is divided into three regions, namely the initial, scaling and final regimes (Fig. 5.9). He

Fig. 5.9 Illustration of macroscopic enhancement of fluctuation from the initially microscopic one. Fluctuations in the initial and final regime can be well described by Gaussian approximation. In the transient regime fluctuation enhances macroscopically, as can be calculated based on a generalised scale transformation of time. (a) Initial regime. (b) Scaling regime. (c) Final regime.

138


,,

o•[J(t)J

/ ' /(t)

--

t Fig. 5.10 Anomalous fluctuation near the instability point: J(t) denotes the most probable path or deterministic motion y(t) and D 2 [y(t)) is the variance of the order parameter.

calculated that the fluctuation becomes anomalously large in the scaling regime (Fig. 5.10). 5. Stochastic bifurcation problems in chemical systems have been analysed by Lemarchand ( 1980). He proposed a systematic expansion of the freeenergy-like quantity ('stochastic potential') in powers of v- 1 : (5.131) to obtain the stationary distribution as well as the time-dependent evolution near the maximum of the probability distribution. (Some further readings are: Lemarchand & Fraikin, 1984; Fraikin, 1984; Walgraef eta/., 1982). The expansion of around an extremum is expressed in the representation provided by the eigenfunctions of the linear stability operator. General expressions of the successive derivatives are obtained independently of the nature of the bifurcation. The method has been extended to describe the effect of inhomogeneous fluctuations as well as in reaction diffusion systems (Fraikin & Lemarchand, 1985). 5. 7 Stationary distributions: uni- versus bimodality 5. 7.1 The scope and limits of the Poisson distribution in the stochastic models of chemical reactions: motivations I. In the literature of stochastic reaction kinetics it was often assumed that the stationary distributions of chemical reactions were generally Poissonian (Prigogine, 1978). The statement is really true for systems containing only first-order elementary reactions, even when inflow and outflow are taken into account (i.e. for open compartmental systems; see Gans, 1960, p. 692). If the model of open compartmental systems is considered as an approximation of an arbitrary chemical reaction 'near' equilibrium, then in this approximation the statement is true.

Stochastic models

139

2. Thermodynamic fluctuation theory characterises equilibrium fluctuation by the so-called Einstein relation connecting the probability density function g with the (appropriately defined) entropy function S: g'1(x)

= C exp(- S(x)fk 8 ),

(5.132)

where k 8 is the Boltzmann constant, C is a normalisation constant. Expanding S around x*, which is considered a thermodynamic equilibrium point, i.e. a stationary point of S, and stopping at the second term in the expansion, the density function is approximated by g'1(x)

= [ det ;;~x*) JM12 exp[(- I /2k 8)(x - x*)T S" (x*)(x - x*)]. (5.133)

From a mathematical point of view the approximation by a Gaussian process is appropriate, in the sense that for every stochastic process having first and second moment there exists a Gaussian process with the same first and second moment. Fluctuations might also be expressed without utilising the entropy function: g'1 (x)

=

[21t mt x! JM/2exp( -~(x- x*)T(dgx*)- (x- x*)) 1

(5. 134)

(x* is ~ector variable, dg x* is the diagonal matrix formed from x*). In small systems, i.e. for relatively large fluctuations (5. 134), approximation can be by a Poisson distribution: P(N)

= (x

*N

) e

N!

-N

Il (x!)Nmexp(- I Nm)

= m= 1

m= 1

M

0Nm! m= I

This relationship can be interpreted as an inverse procedure of the usual approximation by a normal distribution. 3. Some historical remarks. The physical assumption adopted by van Kampen (1976) is that the grand-canonical distribution of the particle number of an ideal mixture is Poissonian. Based on this - strongly restrictive - assumption, and utilising the conservation of the total number of atoms the stationary distribution can be obtained. This stationary distribution can be identified with the stationary solution of the master equation, and it is not Poissonian in general, even for large systems. Hanusse (1976, pp. 86-8) claimed that in the case when not only monomolecular reactions are in the system, the stationary distribution can be different from the Poissonian. Zurek & Schieve (1980) have simulated a particular cheq~ical reaction exhibiting a non-Poisson distribution. The Poisson distribution seems to have particular role, since it is sometimes considered as a reference distribution. For nonlinear systems the


140

cr 2 variance of the stationary distribution is expressed as the function of the expectation m, and 11: cr 2 = m(l - f.l) where ll (the parameter of the Poisson distribution) is qualified as a parameter characterising the reaction (e.g. Malek-Mansour & Nicolis, 1975). A sufficient condition for having Poissonian stationary distribution in a certain class of birth and death processes was given by Whittle (1968). However, his assumptions are slightly different from those of chemical reactions, therefore the search for precisely defined assumptions or for certain classes of reactions is necessary. The fundamental works of P. Medgyessy on the decomposition of superpositions of density functions and discrete distributions (Medgyessy 1977) has been applied to set up conditions for determining the modality (i.e. the number of maxima) of density functions and distributions. 5.7.2 Sufficient conditions of unimodality The following definition is applied: the {Pj:j E N 0 } distribution is unimodal, if in the series Pi - P 0 , P 2 - Pi, P 3 - P 2 , ••• there is precisely one change of sign. Let us consider the reaction having one internal component: k(r)

cx(r)X-+ ~(r)X

(5.136)

For the generating function F of the stationary distribution we can write the following ordinary differential equation: R

L k(r) (z~(r)

-

z"(r))~"(r) F(z)

=0

(5.137)

r= I

where ~denotes the differential operator. A theorem of Medgyessy (1977): Let us assume that the generating function F of the {P.} (P 0 > 0) discrete distribution satisfies the differential equation M

L

(Amz'"

+ Bmz'"+ 1 )~mF(z) = 0,

(0 < lzl ~ 1),

(5.138)

m=O

where MEN, Am, Bm E IR are constants. Furthermore let

m~/m(n:l)m!#O,

ifnEN

(5.139)

and (5.140)

Stochastic models

141

and we assume that in the series (5.141) there is at most one change of sign. Then {Pn} is unimodal. Applying this theorem for the reaction (5.136) we get the existence of m such that ~(r) = CI(r) = m + I. Additionally k(r) = Am = - Bm. Therefore

m~O Am M

(n + )) m

m!

(n 1)

R + = r~l k(r) ~(r) (~(r))!

and

,t.

m~o Bm(~)m! = k(r)(~~r))(~(r))!. Equations (5.139) and (5.140) hold precisely if r exists such that ~(r) = 0 (and then CI(r) = 1). From the chemical point of view it means that the reaction (5.136) contains an elementary reaction X-+ 0 (first-order decay or outflow). Based on another theorem we may state that a birth and death-type chemical reaction with one internal component leads to unimodal stationary distribution. More precisely, let us assume that a chemical reaction can be identified with a birth and death-type process with the birth \j1 and death J.1 rate: I

\j!(j)

=\j!o, and JJ(j) =L J.li i=l

(jei\1 0 ;

\j! 0 EIR+;

/eN)

Then the stationary distribution of the stochastic model of the reaction exists, and it is unimodal. The proof of this proposition is based on a slight generalisation of a theorem of Medgyessy. Let us assume that are functions such that y(j)

+ o(j)

~

I

(jE 1\1 0 )

holds, and suppose that for the distribution {Pi} pj ~ y(j)Pj+ I+ O(j)Pj-1

holds. Then Pi is unimodal. Going back to the statement on birth and death-type reactions, existence and uniqueness of the stationary distribution follows from the KarlinMcGregor (1957) condition. For the stationary distribution of the reaction we can write

Mathematical models of' chemical reactions

142 (J.t{j) + \jl(j))Pj

= J.l{j + l)Pj+ I + \jl(j- l)Pj- I

(5.142)

Utilising the particular form of \jl, the monotonicity of J.l can be written as: -

pj

J.l{j+ l)

\j/(j-1)

= pj +I J.l{j) + \jl(j) + pj- I J.l{j) + \jl(j)

o/o

+P

>-: P Jl{j) ...- j+ I J.l{j) + \jl(j)

j-1

J.l{j) + \jl(j)

Applying Medgyessy's above mentioned theorem with Y( 1.) _

J.l{j) . - J.l{j) + \jl(j),

~( ·) _

o/o

u 1 - J.l{j) + \jl(j)

~~., ) 1 E '~o

(·

the statement is proven. From the chemical point of view these kinds of result suggest that the stationary distribution of the reaction kM

MX-+(M- l)X-+ ... (M > 0, k_ 1 > 0, kM > 0; km

~

k2

k1

-+X~O k_,

(5.143)

0, m = l, 2 ... M- l) is unimodal.

5. 7.3 Sufficient condition for a Poisson ian stationary distribution Theorem. The necessary and sufficient condition for a simple birth and deathtype process to have a Poissonian stationary distribution (if it has a nondegenerate stationary distribution at all) is that it be a linear one (Erdi & T6th, 1979). Sketch of the proof If a simple birth and death-type process has a unique stationary distribution then its form is: Pi

rzi

= const. - M , - - - - - - - - (jE 1\J~)

TI

(5.144)

0. Some basic problems in connection with (6.1 ):

n.

=

-

the initial boundary value problem: to find a continuous function p: Qr--+ ~·. bounded in x satisfying the initial condition p(x, 0) = p(x), and the boundary conditions on Q x (0, T). Various types of boundary conditions can be given: (a) Individuals are imported or exported at a given rate: ~ vk(x)H;k = a;(x, t); (the Neumann condition). (b) the efflux is a linear function of p: ~ vk(x)H;k = a;(x, t) + b;(x, t)p: (the Robin condition). Here v(x) is the unit outward normal at x, and h;> 0. (c) P;(x, t) = c;(x, t) (the Dirichlet condition). - initial value problems: similar to the previous problems, with Q = ~m. and without boundary conditions. - boundary value problems: to find a bounded function p: fiX (- 00, 00)--+ ~·, satisfying one of the three boundary COnditions. - stationary problems: to find a time-independent solution of the boundary value problem. - periodic problem: to find a solution of a boundary value problem which is periodic in time under certain conditions. The asymptotic state is defined as a collection of solutions, all approaching the same solution as t --+ oo. In the case of the occurrence of symmetries in reaction-diffusion systems, a family of solutions can be generated from any given solution by using an appropriate group of transformations. The application of the Lie theory of transformation groups of reaction-diffusion equations (e.g. Steeb & Strampp, 1981) can be a very efficient method to study the symmetry properties of the solutions.

Chemical reaction and diffusion

165

=

Let us consider the particular case L; D;llp; + F;(p), where D; are nonnegative constants, l1 is the Laplacian operator, and F is independent of (x, t).

(6.2) Let the group of transformations G be generated by rigid motions and reflections in x and translations in t. Since the solutions of (6.2) are invariant under these transformations, the solutions p of the original equation are equivalent with the solutions Tp, for TeG. More precisely, two solutions p 1 , p2 are asymptotically equivalent for some Te G, if lim supjp 1 (x, t)- Tp 2 (x, t)l r-:n

= 0.

x

From a practical point of view it is the stable asymptotic states that are important. One of the main questions in the theory of reaction-diffusion systems is to determine which are the stable asymptotic states for a given reaction-diffusion system. A solution p(x, t) is called permanent if it is defined, and is a solution, for all real t, negative as well as positive. Familiar examples of solutions of the permanent type (the classes are not disjoint) are -

time periodic solutions

# x-independent oscillatory solutions # wave trains

# target patterns: p(x, t) = U(lxl, t), U is periodic in t # rotating spiral patterns: n = 2, x = r(cos 9, sin 9), p(x, t) = U(r, -

9 - ct), U is periodic in the second argument. travelling wave solutions: the form of the solutions: P(x, t) = p(x - ct), c is a velocity vector

# stationary solutions (c = 0, p is time-independent) # plane waves: p(x, t) = U(x - ct)v. vis a unit vector in the direction of c +wave trains: U is periodic +wave fronts: U is monotone and bounded +pulses: U(- oo) = U( oo ), U is not constant.

Remarks

1. For the one-component case the reaction-diffusion system can be considered as a gradient system, therefore the results of elementary catastrophe theory can be applied. Ebeling & Malchow (1979) analysed bifurcations in (pseudo)-one-component systems by this technique. They showed that stable homogeneous stationary states are also stable in reaction-diffusion systems. 2. The two-component model plays an important role in the analysis of

166


wave propagation in chemical systems. Many special systems have been studied by linear stability analysis. Diffusion has been taken into account, (i) as transport between two homogeneous compartments; (ii) as diffusion in a continuous system. The occurrence of nonuniform steady states, as well of chemical waves in the Brusselator model, was demonstrated by linear stability analyisis (see Nicolis & Prigogine, 1977). 3. A class of two-component models with one diffusion coefficient much smaller than the other was studied by Fife ( 1976). The concentrations of u and v of the components U and V satisfy the equation oufot - E 2 V 2 u - f(u, v) = 0 ovfot- V 2 v- g(u, v) = 0.

(6.3a) (6.3b)

The boundary condition is expressed as E2 0ujon

= Uo - u; OVjOn = Vo - V.

The determination of the time-independent functions u(x) and v(x) is easier for the special case when/= 0 is expressible as an S-shaped curve of v versus u. Fife demonstrated that in the first stage of the evolution sharply differentiated subregions, bounded by layers within which one of the concentrations has a large spatial gradient, are formed. At this time the initial distributions determine the shape of the subregions. The final distribution that is achieved with a much smaller time scale depends on the stoichiometry of the reaction, the larger diffusion coefficient, and the reference concentrations on the boundary, but is relatively independent of the initial distribution. 4. Bifurcations. In many situations the uniform solution of the reaction-diffusion equation exists, but is stable only for certain regimes of the parameters. Near to the transition zone between stability and instability of this uniform solution, other nonuniform, small-amplitude, solutions exist as well. If these solutions are stable, their appearance can be considered as bifurcation phenomena. The equation oufot

=

Do 2 ujox2 + (A + A.B)u + g(u)

is an example for illustrating diffusion-induced instability. The stationary solution of the kinetic equation (i.e. when D = 0) is destabilised by a nonuniform perturbation, but preserves its stability in the case of an xindependent perturbation (Segel & Jackson, 1972). 5. Solitary waves were found in a model of a biochemical reaction system catalysed by an allosteric protein, as a result of a threshold phenomenon and diffusion (Anan & Go, 1979). They studied the simplified equation: osjot = -s- cp + V 2 s opfot = s + hp- p 3 + a + dV 2 p In the spatially homogeneous system for certain parameter values of a, h and


167

c one stable stationary point, one unstable stationary point and one saddle point can be found. A separatrix (i.e. a trajectory flowing into the saddle point) goes very close to the stable stationary point. In response to a perturbation it could occur that the trajectory leaves the stable point and goes around the unstable steady state. Inclusion of diffusion can imply the occurrence of solitary waves. 6.3 Stochastic models: difficulties and possibilities 6.3.1 Introductory remarks

The fact, that it is necessary to take into account spatial fluctuations arose in consequence of a strange debate. Kuramoto (1973) criticised the statement of Nicolis & Prigogine ( 1971) that the stochastic kinetic model is not in accordance with the microscopic description based on the Boltzmann equation. In their comprehensive answer to Kuramoto's short paper Nicolis et a/. ( 1974) stated that the assumptions of Kuramoto were not natural, and the stochastic kinetic model is not appropriate to describe certain microscopic fluctuations. In fact, stochastic kinetics represents an intermediate level between 'microscopic reaction kinetics' operating with elementary collisions and 'phenomenological reaction kinetics' working with macroscopic quantities. In the course of this debate the view that chemical fluctuations lead 'locally' to a Poissonian distribution, but deviate from it 'globally' has emerged (Malek-Mansour & Nicolis, 1975). The debate seems to be incoherent, since three distinct models were compared. Not only a microscopic and a global stochastic model were involved, but also a-not well-defined-local stochastic model. Global and local kinetics can be compared only partially. In the case of deterministic models an ordinary differential equation would not be blamed for not being able to describe spatial changes if the elements of the domain of the unknown function were interpreted as points of time. The usual stochastic approach tends to associate a stochastic process with the partial differential equations of the deterministic model, and not a random field. However, three directions in the theory of random fields seem to be able to cope with such complexity: the theory of random measures (Prekopa, 1956, 1957a, b; Kendall, 1975), the theory of stochastic partial differential equations relating to trajectories (see Skorohod, 1978) and the theory of Hilbert space valued stochastic processes (Ichikawa, Arnold 1982; eta/., 1980 Dawson 1972, 1975). At present, this last direction seems to be the most fruitful: it seems to be broad enough to provide the framework for the generalisation of (l) nonequilibrium thermodynamics to include randomness,

(2) stochastic reaction kinetics to include spatial effects (such as diffusion) correctly, and

168


(3) stochastic theories of diffusion to include sources and sinks (such as chemical reactions). Numerous efforts have been made in the last decade to evaluate spatial correlations of fluctuations. The examination of specific models was given preference over the formulation of general models. Different approaches to stochastic modelling of reaction-diffusion systems will be shortly reviewed in this chapter. 6.3.2 Two-cell stochastic models The simplest extension of stochastic models of homogeneous kinetic systems for describing spatial effects is the introduction of the two-cell model. According to this approach reactions take place within the homogeneous cells, while transport of matter between the cells is taken into account as a random exchange between cells. The two-cell stochastic model of the Schlogl reaction describing 'firstorder phase transition' has been carefully studied (Borgis & Moreau, 1984b; Moreau & Borgis, 1984). Let P(n 1 , n 2 ; t) denote the absolute distribution function. Accordingly, at a fixed time t, the number of molecules in cell 1 is n 1 , and in cell 2 is n 2 • Denoting with R; the operator of the reaction operating on n;, which describes the effect of chemical reactions the master equation can be described: ojotP(n~>

n 2 ; t) = (R 1

+ D(n 1 + l)P(n 1 + + D(n 2 + l)P(n 1 -

+ R 2 )(P(n 1 ,

n 2 ; t)) I, n 2 - 1; t)- D(n 1 1, n 2 + 1; t)

+ n 2 )P(n 1 ,

n 2 ; t)

(6.4)

Approximations or simulation experiments are necessary to find even the stationary solution. The behaviour of the system is particularly interesting for small values of D, since large D implies a homogeneous solution. The simulation technique used earlier (Frankowicz & Gudowska-Nowak, 1982) was adopted by Borgis & Moreau ( 1984b) to calculate the stationary distribution. Its shape is shown by Fig. 6.1. Let (X and y denote the stable steady states in single cells. The stationary distribution presents two peaks on the homogeneous steady states ((X, (X), (y, y) and two peaks on the inhomogeneous states ((X', y') and (y', (X'). (For D = 0, (X' = (X andy' = y, otherwise there is a shift toy and (X repectively.) In the plane (n 1 , n 2 ) there are only four regions around these peaks, where the stationary distribution differs significantly from zero. An approximate master equation has been derived (Borgis & Moreau, 1984) based on the quasistationary assumption. Accordingly, exchanges between the regions ((X) and (y) are much slower than the relaxation inside these regions. This assumption led to a simple approximate equation for the


169

Fig. 6.1 Qualitative shape of the stationary distribution for small D.

probability P*(x, y; t) to find cell I in region (x) and cell2 in region (y). The definition of P*(x, y; t) is: P*(x, y; t)

=L

L

P(n 1 , n 2 ; t).

n, E(x) n,E(y)

The approximate solution of the evolution equation for P*(x, y; t) based on the hypothesis of low coupling between cells shows that the transition between two homogeneous stationary states occurs via an inhomogeneous transient state. Transient inhomogeneous structures may be also responsible for the stabilisation of an unstable state. Such kinds of phenomena are reminiscent of nucleation processes (Blanche, 1981; Frankowicz, 1984) and might be considered as a nonequilibrium analogue of an equilibrium phenomenon. 6.3.3 Cellular model According to the basic assumption of this model, not only is the component space discrete, but the real space is also subdivided into mesoscopic cells. The meaning of the term 'mesoscopic' here is that the size of cells is larger than the size of the constituent molecules, but much smaller than the characteristic scale of the total system. While from a heuristic point of view the discrete state-space description of chemical reactions seems to be natural, the discretisation of the space can be qualified as a more or less forced technical procedure. Subdividing the total system into JVd cells of equal size, the state of the


170

whole system can always be characterised by a finite-dimensional vector (dis the dimension). Let X;, denote the number of particles of species i in cell r (r is the central point of the cell), and P({X;,}. t) is the joint probability distribution of all X;,. Since the chemical reaction is modelled as in the homogeneous case, and diffusion is considered as a random walk between adjacent spatial cells, a Markovian process can be constructed in the space combined from the component space and the real space. The probability distribution P({X;,}, t) is modified due to chemical reactions and diffusion:

iJjiJt P({X;,}, t)Jt' =''reactions within cell r' + 'diffusion among cells' While the first term is described according to the rules of homogeneous stochastic kinetics, the second term is as follows (Nicolis & Malek-Mansour, 1980): LDJ2d[(X;, + J)P(X;,r+l- J, X;,+ J, {X;,}, t)- X;,P({X;,}, f)] s, i

where s denotes the first neighbours of cell r. Remarks 1. Many difficulties occur in connection with the discretisation of the space. In practice a large set of rate constants and diffusion constants are necessary to evaluate the model. In general there is little chance of obtaining a solution of the model. Approximations, as by the mean-field approach (which neglects the spatial fluctuations), cumulant expansions and perturbative methods (see, for example, Nicolis, 1984) can be applied. 2. Kurtz (1981) introduced a different, better founded model. It was mentioned (Subsection 5.1.4) that reaction-diffusion systems can be derived by the semigroup operator approach giving the generator. For sake of simplicity Kurtz considered the reaction A + B -+C. The state of the system is described by the vector (k, /, X~o x 2 , ... , xko y 1 , y 2 , ... , y 1), where k is the number of molecules of A and X~o x 2 , ... , xk are their locations, I is the number of molecules B with coordinates y 1 , y 2 , ••• , y 1• According to the physical assumptions, the molecules undergo Brownian motion with normal reflection at the boundary, the probability of a reaction between a molecule A at x, and B at y in a time interval!l.t is C(x - y)M + o(M). Associating to the vector x e Rk the vector tt;E Rk- 1 by dropping X; the generator A of the process has the form k

Af(k, /,

X,

y) =

I

L d./l.J(k, I, X, y) + L d,AyJ(k, I, X, y) + L L c(x;- y;)(f(k- I,/- 1, tt;. qj)- f(k,

i= I k

I

i= lj= I

where fl. denotes the discrete Laplacian.

j= I

/, x, y)),


171

3. The connection between the stochastic cell model and the usual deterministic model of reaction-diffusion systems were given by Kurtz for the homogeneous case in the same spirit. To give the relationship among a Markovian jump process and the solution p(r, t) of the deterministic model a law of large numbers and a centra/limit theorem hold (Arnold & Theodosopulu, 1980; Arnold 1980). Roughly speaking the law of large numbers is (at least for finite time in the thermodynamic limit):

limsupP(IIx(t)- p*(t)IIL > t) = 0,

t

> 0, 0

~

t

~

T.

Problem Rewrite the central limit theorem given by Kurtz for the homogeneous case Subsection 5.9.3 for inhomogeneous systems. 6.3.4 Other models To avoid the drawbacks of the discretisation of space a particle density function u(r) can be introduced by the relation u(r)

X- (t) =~·-o lim -"-. Av

The joint distribution function P({X;,}, t) is converted formally into a probability in function space, P([u(r)], t), which is a functional of u(r); (see van Kampen, 1981, pp. 346-7). From a mathematical point of view this procedure is badly founded, since the probability density in function space is not defined. The explicit use of probability in function space can be avoided by the method of compounding moments. For the reaction A -+X,

2X-+ B

evolution equations can be derived for the first and second moment. For the first moment: o,(u*(r, t)) = 1 - (u*(r, t) 2)

+ DV 2(u*(r,

t)).

(6.5)

Equations can also be derived for the factorial cumulants defined by the relation o,[xlx2]

=<xlx2>- (xl) (x2)- 012(nl);

o,[u*(rl, t)u*(r2, t)] = -2{(u*(r 1 , t)) + (u*(r 2, t))}[u*(r 1 , t)u*(r 2, t)] - {r 1 - r 2)(u*(r 1 , t)) 2 + D(Vi + VD[u*(r 1 , t)u*(r 2, t)].

(6.6)

(6.7)

These kinds of equations are appropriate to describe the spatio-temporal

172


fluctuations in reaction-diffusion systems. As in the case of homogeneous systems, there are two kinds of stochastic descriptions for reaction-diffusion systems as well: the 'master equation' approach and the stochastic differential equation method. Until now we have dealt with the first approach; however, stochastic partial differential equations are also used extensively. Most often partial differential equations are supplemented with a term describing fluctuations. In particular, timedependent Ginzburg-Landau equations describe the behaviour of the system in the vicinity of critical points (Haken, 1977; Nitzan, 1978; Suzuki, 1984). A usual formulation of the equation is: ap(r, t)jat

= ap(r,

t)

+ DV 2 p(r,

t) - bp 3 (r, t)

+ )J dt y

Bx- x 2 y

0

has been studied (Tomita, 1982). The undriven system shows limit cycle behaviour in a certain region of the parameter space. It was clearly demonstrated that four different types of region might be recognised on the (a-co) plane (Fig. 7.4), namely entrainment (phase locking), quasiperiodic oscillation, periodic doubling cascade, and chaos have been found. Although elaborate mathematical techniques exist to analyse bifurcations from and to the regions just mentioned (e.g. Ioos & Joseph, 1980), numerical methods often offer a technically simpler, 'down-to-earth' treatment of dynamic problems. The analysis of systems, when the unperturbed system is higher than three-dimensional is particularly difficult (but not hopeless-see, for example, Rossler & Hudson, 1985). Periodically perturbed chemical systems were reviewed recently by Rehmus & Ross (1984). IX

0.16

0.12

0.08

0.04

0 0

0.2: 1/3 ~ 2/3: 1/2 3/4

'0.6 3/2 4n

:o.a 2

1.0

1.2 3

(J)

wlw0

Fig. 7.4 Phase diagram. The numbers indicate the harmonic periods appearing in the respective regions in the unit of forced period. A limit cycle of nonintegral period appears in the shaded region Q, and a chaotic response is found in the region indicated by J(.

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185

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