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

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