Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models (Nonlinear science)

Contents......Page 6 Preface and acknowledgements......Page 12 Symbols used in the text......Page 14 1.1 Mass action kinetics: macroscopic and microscopic approach......Page 26 1.2 Physical models of chemical reactions......Page 29 1.3 Deterministic and stochastic models......Page 31 1.4 Regular and exotic behaviour......Page 36 1.5 Chemical kinetics as a metalanguage......Page 37 2.2 Properties of process-time......Page 39 2.2.1 Discrete versus continuous......Page 40 7.6 Aggregation......Page 0 2.3.1 Discrete versus continuous......Page 41 2.3.2 State and site......Page 42 2.4 Nature of determination......Page 43 2.5 X YZ models......Page 44 3.1 Conventional stoichiometry......Page 46 3.2 Atom-free stoichiometry......Page 51 3.3 Retrospective and prospective remarks. Suggested further reading......Page 53 3.5 Problems......Page 54 3.6 Open problems......Page 57 4.1.1 Introduction......Page 58 4.1.2 Introduction reconsidered......Page 60 4.1.4 Problems......Page 64 4.2 Verifications and falsifications of traditional beliefs......Page 65 4.2.1 The zero deficiency theorem......Page 67 4.2.2 Vol'pert's theorem......Page 70 4.2.3 Remarks on related literature......Page 71 4.2.4 Exercises......Page 72 4.2.6 Open problems......Page 73 4.4.1 Multistability......Page 74 4.4.3 Multistationarity in kinetic models of continuous flow stirred tank reactors......Page 75 4.4.4 Exercises......Page 76 4.4.6 Open problems......Page 77 4.5.2 Excluding periodicity in differential equations......Page 79 4.5.4 Sufficient conditions of periodicity in differential equations......Page 80 4.5.6 Designing oscillatory reactions......Page 81 4.5.8 Exercises......Page 82 4.5.9 Problems......Page 83 4.6.1 Chaos in general......Page 84 4.6.2 Chaos in kinetic experiments......Page 85 4.6.3 Chaos in kinetic models......Page 86 4.6.5 Problems......Page 87 4.7 The inverse problems of reaction kinetics......Page 88 4.7.1.1 Polynomial and kinetic differential equations......Page 89 4.7.1.2 Further problems......Page 90 4.7.1.4 Uniqueness questions......Page 92 4.7.1.5 A sufficient condition for the existence of an inducing reaction of deficiency zero......Page 94 4.7.1.6 On the inverse problem of generalised compartmental systems......Page 97 4.7.3 Exercises......Page 99 4.8.1.1 Lumping in general......Page 100 4.8.1.2 Lumping in reaction kinetics......Page 101 4.8.1.3 Possible further directions......Page 102 4.8.2 Continuous components......Page 103 4.8.3 Kinetic gradient systems......Page 105 4.8.4 Structural identifiability......Page 107 4.8.5 Parameter sensitivity......Page 108 4.8.6 Symmetries......Page 109 4.8.7 Principle of quasistationarity......Page 113 4.8.9 Problems......Page 114 4.8.10 Open problem......Page 115 5.1.1 The logical status of stochastic reaction kinetics......Page 116 5.1.2 Fluctuation phenomena in physics and chemistry: an introduction......Page 118 5.1.3.1 Introductory remarks......Page 121 5.1.3.2 Continuous state-space processes......Page 122 5.1.4 Operator semigroup approach: advantages coming from the use of more sophisticated mathematics......Page 124 5.2.1 Some historical remarks I 0......Page 126 5.2.2 Models......Page 127 5.3 On the solutions of the CDS models I......Page 130 5.3.2 Chemical reaction X !... Y I......Page 131 5.3.4 Bicomponential reactions: general remarks......Page 132 5.3.5.3 Determination of expectation I......Page 133 5.3.6 General equation for the generating function......Page 134 5.3.7 Approximations......Page 135 5.3.8 Simulation methods......Page 137 5.4.1 Stochastic reaction kinetics: 'nonequilibrium thermodynamics of state-space'?......Page 140 5.4.2 Fluctuation-dissipation theorem of linear nonequilibrium thermodynamics......Page 141 5.4.3 Determination of rate constants from equilibrium fluctuations: methods of calculation......Page 142 5.5.1 Enzyme kinetics......Page 144 5.5.2 Ligand migration in biomolecules......Page 146 5.5.3 Membrane noise......Page 148 5.5.4 Kinetic examinations of fast reactions......Page 150 5.6.1 An example of the importance of fluctuations......Page 153 5.6.2 Stochastic Lotka-Volterra model......Page 154 5.6.3 Stochastic Brusselator model......Page 155 5.6.4 The Schlogl model of second-order phase transition......Page 156 5.6.5 The Schlogl model of first-order phase transition......Page 159 5.6.6 Stochastic theory of bistable reactions......Page 160 5.7.1 The scope and limits of the Poisson distribution in the stochastic models of chemical reactions: motivations......Page 163 5.7.2 Sufficient conditions of unimodality......Page 165 5.7.3 Sufficient condition for a Poissonian stationary distribution......Page 167 5.7.4 Multistationarity and multimodality......Page 168 5.7.5 Transient bimodality......Page 169 5.8.1 Motivations......Page 171 5.8.2 Stochastic differential equations: some concepts and comments......Page 172 5.8.3 Noise-induced transition: an example for white noise idealisation......Page 174 5.8.4 Noise-induced transition: the effect of coloured noise......Page 176 5.8.5 On the effects of external noise on oscillations......Page 178 5.8.6 Internal and external fluctuations: a unified approach......Page 181 5.8.7 Estimation of reaction rate constants using stochastic differential equations......Page 182 5.8.8 Exercises I......Page 183 5.9.2 Blowing up......Page 184 5.9.4 Exercise......Page 185 5.9.5 Problems......Page 186 6.1 What kinds of models are relevant?......Page 187 6.3.1 Introductory remarks......Page 192 6.3.2 Two-cell stochastic models......Page 193 6.3.3 Cellular model......Page 194 6.3.4 Other models......Page 196 6.4 Spatial structures......Page 197 6.5 Pattern formation and morphogenesis......Page 199 7.2 Biochemical control theory......Page 202 7.3 Fluctuation and oscillation phenomena in neurochemistry......Page 210 7.4 Population genetics......Page 217 7.5.1.1 Boulding ecodynamics......Page 219 7.5.1.2 Compartmental ecokinetics......Page 220 7.5.1.3 Generalised Lotka-Volterra models......Page 221 7.5.1.4 The advantages of stochastic models: illustrations......Page 224 7.5.2.1 Arguments for a stochastic model......Page 227 7.5.2.2 A common description of the deterministic and stochastic models......Page 229 7.5.2.3 Exercise......Page 232 7.7 Chemical circuits......Page 235 7.8.1.1 Introductory remarks......Page 238 7.8.1.2 The hypercycle: The basic model......Page 239 7.8.2 The origin of asymmetry of biomolecules......Page 241 References......Page 245 Index......Page 277 Contents 6 Preface and acknowledgements 12 Symbols used in the text 14 1 Chemical kinetics: a prototype of nonlinear science 26 1.1 Mass action kinetics: macroscopic and microscopic approach 26 1.2 Physical models of chemical reactions 29 1.3 Deterministic and stochastic models 31 1.4 Regular and exotic behaviour 36 1.5 Chemical kinetics as a metalanguage 37 2 The structure of kinetic models 39 2.1 Temporal processes 39 2.2 Properties of process-time 39 2.2.1 Discrete versus continuous 40 2.2.2 Time -1 2.3 Structure of state-space 41 2.3.1 Discrete versus continuous 41 2.3.2 State and site 42 2.4 Nature of determination 43 2.5 X YZ models 44 3 Stoichiometry: the algebraic structure of complex chemical reactions 46 3.1 Conventional stoichiometry 46 3.2 Atom-free stoichiometry 51 3.3 Retrospective and prospective remarks. Suggested further reading 53 3.4 Exercises 54 3.5 Problems 54 3.6 Open problems 57 4 Mass action kinetic deterministic models 58 4.1 Kinetic equations: their structure and properties 58 4.1.1 Introduction 58 4.1.2 Introduction reconsidered 60 4.1.3 Exercises 64 4.1.4 Problems 64 4.1.5 Open problems 65 4.2 Verifications and falsifications of traditional beliefs 65 4.2.1 The zero deficiency theorem 67 4.2.2 Vol'pert's theorem 70 4.2.3 Remarks on related literature 71 4.2.4 Exercises 72 4.2.5 Problems 73 4.2.6 Open problems 73 4.3 Exotic reactions: general remarks 74 4.4 Multistationarity 74 4.4.1 Multistability 74 4.4.2 Multistationarity in kinetic experiments 75 4.4.3 Multistationarity in kinetic models of continuous flow stirred tank reactors 75 4.4.4 Exercises 76 4.4.5 Problems 77 4.4.6 Open problems 77 4.5 Oscillatory reactions: some exact results 79 4.5.1 Periodicity in kinetic experiments 79 4.5.2 Excluding periodicity in differential equations 79 4.5.3 Excluding periodicity in reactions 80 4.5.4 Sufficient conditions of periodicity in differential equations 80 4.5.5 Sufficient conditions of periodicity in reactions 81 4.5.6 Designing oscillatory reactions 81 4.5.7 Overshoot-undershoot kinetics 82 4.5.8 Exercises 82 4.5.9 Problems 83 4.5.10 Open problems 84 4.6 Chaotic phenomena in chemical kinetics 84 4.6.1 Chaos in general 84 4.6.2 Chaos in kinetic experiments 85 4.6.3 Chaos in kinetic models 86 4.6.4 On the structural characterisation of chaotic chemical reactions 87 4.6.5 Problems 87 4.6.6 Open problems 88 4.7 The inverse problems of reaction kinetics 88 4.7.1 Polynomial differential equations -1 4.7.1.1 Polynomial and kinetic differential equations 89 4.7.1.2 Further problems 90 4.7.1.3 The density of kinetic differential equations 92 4.7.1.4 Uniqueness questions 92 4.7.1.5 A sufficient condition for the existence of an inducing reaction of deficiency zero 94 4.7.1.6 On the inverse problem of generalised compartmental systems 97 4.7.2 The classical problem of parameter estimation 99 4.7.3 Exercises 99 4.7.4 Problems 100 4.7.5 Open problems 100 4.8 Selected addenda 100 4.8.1 Lumping 100 4.8.1.1 Lumping in general 100 4.8.1.2 Lumping in reaction kinetics 101 4.8.1.3 Possible further directions 102 4.8.2 Continuous components 103 4.8.3 Kinetic gradient systems 105 4.8.4 Structural identifiability 107 4.8.5 Parameter sensitivity 108 4.8.6 Symmetries 109 4.8.7 Principle of quasistationarity 113 4.8.8 Exercises 114 4.8.9 Problems 114 4.8.10 Open problem 115 5 Continuous time discrete state stochastic models 116 5.1 On the nature and role of fluctuations: general remarks 116 5.1.1 The logical status of stochastic reaction kinetics 116 5.1.2 Fluctuation phenomena in physics and chemistry: an introduction 118 5.1.2.1 Stochastic thermostatics -1 5.1.3 Stochastic processes: concepts 121 5.1.3.1 Introductory remarks 121 5.1.3.2 Continuous state-space processes 122 5.1.3.3 Discrete state-space processes 124 5.1.4 Operator semigroup approach: advantages coming from the use of more sophisticated mathematics 124 5.2 Stochasticity due to internal fluctuations: alternative models 126 5.2.1 Some historical remarks I 0 126 5.2.2 Models 127 5.3 On the solutions of the CDS models I 130 5.3.1 General remarks I 131 5.3.2 Chemical reaction X !... Y I 131 5.3.3 Compartmental systems 132 5.3.4 Bicomponential reactions: general remarks 132 5.3.5 Chemical reaction X + Y ~ Z I 133 5.3.5.1 The master equation 133 5.3.5.2 Use of Laplace transformation 133 5.3.5.3 Determination of expectation I 133 5.3.5.4 The behaviour of the reaction during the initial period of the processes 134 5.3.5.5 Determination ofstationary distribution 134 5.3.6 General equation for the generating function 134 5.3.7 Approximations 135 5.3.8 Simulation methods 137 5.4 The fluctuation-dissipation theorem of chemical kinetics 140 5.4.1 Stochastic reaction kinetics: 'nonequilibrium thermodynamics of state-space'? 140 5.4.2 Fluctuation-dissipation theorem of linear nonequilibrium thermodynamics 141 5.4.3 Determination of rate constants from equilibrium fluctuations: methods of calculation 142 5.5 Small systems 144 5.5.1 Enzyme kinetics 144 5.5.2 Ligand migration in biomolecules 146 5.5.3 Membrane noise 148 5.5.4 Kinetic examinations of fast reactions 150 5.6 Fluctuations near instability points 153 5.6.1 An example of the importance of fluctuations 153 5.6.2 Stochastic Lotka-Volterra model 154 5.6.3 Stochastic Brusselator model 155 5.6.4 The Schlogl model of second-order phase transition 156 5.6.5 The Schlogl model of first-order phase transition 159 5.6.6 Stochastic theory of bistable reactions 160 5.7 Stationary distributions: uni- versus multimodality 163 5.7.1 The scope and limits of the Poisson distribution in the stochastic models of chemical reactions: motivations 163 5.7.2 Sufficient conditions of unimodality 165 5.7.3 Sufficient condition for a Poissonian stationary distribution 167 5.7.4 Multistationarity and multimodality 168 5.7.5 Transient bimodality 169 5.8 Stochasticity due to external fluctuations 171 5.8.1 Motivations 171 5.8.2 Stochastic differential equations: some concepts and comments 172 5.8.3 Noise-induced transition: an example for white noise idealisation 174 5.8.4 Noise-induced transition: the effect of coloured noise 176 5.8.5 On the effects of external noise on oscillations 178 5.8.6 Internal and external fluctuations: a unified approach 181 5.8.7 Estimation of reaction rate constants using stochastic differential equations 182 5.8.8 Exercises I 183 5.8.9 Problems 184 5.8.10 Open problem 184 5.9 Connections between the models 184 5.9.1 Similarities and differences: some remarks 184 5.9.2 Blowing up 184 5.9.3 Kurtz's results: consistency in the thermodynamic limit 185 5.9.4 Exercise 185 5.9.5 Problems 186 6 Chemical reaction accompanied by diffusion 187 6.1 What kinds of models are relevant? 187 6.2 Continuous time -1 6.3 Stochastic models: difficulties and possibilities 192 6.3.1 Introductory remarks 192 6.3.2 Two-cell stochastic models 193 6.3.3 Cellular model 194 6.3.4 Other models 196 6.4 Spatial structures 197 6.5 Pattern formation and morphogenesis 199 7 Applications 202 7.1 Introductory remarks 202 7.2 Biochemical control theory 202 7.3 Fluctuation and oscillation phenomena in neurochemistry 210 7.4 Population genetics 217 7.5 Ecodynamics 219 7.5.1 The theory of interacting populations 219 7.5.1.1 Boulding ecodynamics 219 7.5.1.2 Compartmental ecokinetics 220 7.5.1.3 Generalised Lotka-Volterra models 221 7.5.1.4 The advantages of stochastic models: illustrations 224 7.5.2 An ecological case study 227 7.5.2.1 Arguments for a stochastic model 227 7.5.2.2 A common description of the deterministic and stochastic models 229 7.5.2.3 Exercise 232 7.6 Aggregation -1 7.7 Chemical circuits 235 7.8 Kinetic theories of selection 238 7.8.1 Prebiological evolution 238 7.8.1.1 Introductory remarks 238 7.8.1.2 The hypercycle: The basic model 239 7.8.2 The origin of asymmetry of biomolecules 241 References 245 Index 277 0719022088,9780719022081 Manchester University Press,1989