Stochastic dynamics for systems biology / Christian Mazza, Michel Benaïm.

Format:

Book

Author/Creator:

Mazza, Christian.

Contributor:

Benaim, Michel.

Series:

Chapman and Hall/CRC mathematical & computational biology series

Chapman & Hall/CRC Mathematical and Computational Biology Series

Language:

English

Subjects (All):

Systems biology--Mathematical models.

Systems biology.

Mathematical models.

Physical Description:

xii, 260 pages : illustrations ; 24 cm.

Place of Publication:

Boca Raton, FL : CRC Press, Taylor & Francis Group, 2014.

Summary:

Stochastic Dynamics for Systems Biology is one of the first books to provide a systematic study of the many stochastic models used in systems biology. The book shows how the mathematical models are used as technical tools for simulating biological processes and how the models lead to conceptual insights on the functioning of the cellular processing system. Most of the text should be accessible to scientists with basic knowledge in calculus and probability theory. The authors illustrate the relevant Markov chain theory using realistic models from systems biology, including signaling and metabolic pathways, phosphorylation processes, genetic switches, and transcription. A central part of the book presents an original and up-to-date treatment of cooperativity. The book defines classical indexes, such as the Hill coefficient, using notions from statistical mechanics. It explains why binding curves often have S-shapes and why cooperative behaviors can lead to ultrasensitive genetic switches. These notions are then used to model transcription rates. Examples cover the phage lambda genetic switch and eukaryotic gene expression. The book then presents a short course on dynamical systems and describes stochastic aspects of linear noise approximation. This mathematical framework enables the simplification of complex stochastic dynamics using Gaussian processes and nonlinear ODEs. Simple examples illustrate the technique in noise propagation in gene networks and the effects of network structures on multistability and gene expression noise levels. The last chapter provides up-to-date results on stochastic and deterministic mass action kinetics with applications to enzymatic biochemical reactions and metabolic pathways. Book jacket.

Contents:

I Dynamics of reaction networks: Markov processes 1

1 Reaction networks: introduction 3

1.1 Introduction to modelling: a self-regulated gene 4

1.2 Birth, and death processes to model basic chemical reactions 9

1.2.1 Degradation 9

1.2.2 A basic transcriptional unit 11

1.2.3 Conversion 17

1.3 Some results on the self-regulated gene 20

1.3.1 The case of constant g and k 22

2 Continuous-time Markov chains 27

2.1 Introduction 27

2.1.1 Birth and death processes 28

2.1.2 The Kolmogorov equation associated with birth and death processes 28

2.1.3 The Poisson process 31

2.2 General time-continuous Markov chains 32

2.2.1 Spectral properties* 35

2.2.2 Jump chain and holding times 38

2.2.3 Convergence to equilibrium 40

2.3 Some important Markov chains 43

2.3.1 The Metropolis-Hastings chain 43

2.3.2 A Metropolis chain on the d-cube* 45

2.4 Two-time-scale stochastic simulations* 46

II Illustrations from systems biology 51

3 First-order chemical reaction networks 53

3.1 Reaction networks 53

3.2 Linear first-order reaction networks 54

3.3 Statistical descriptors for linear rate functions 57

3.4 Open and closed conversion systems 59

3.5 Illustration: Intrinsic noise in gene regulatory networks 60

4 Biochemical pathways 65

4.1 Stochastic fluctuations in metabolic pathways 65

4.2 Signalling networks 69

5 Binding processes and transcription rates 77

5.1 Positive and negative control 78

5.2 Binding probabilities 79

5.3 Gibbs-Boltzmann distributions 83

5.4 Site-specific Hill coefficients 85

5.5 Cooperativity in the microstate* 86

5.6 The sigmoidal nature of the binding curve* 87

5.7 Cooperativity in the Hill sense 89

5.8 ηH(V) as an indicator of cooperativity 92

5.9 The cooperativity index 93

5.10 Macroscopic cooperativity 95

5.11 The case N = 3* 96

5.12 Transcription rates for basic models 99

5.13 A genetic switch: regulation by λ phage repressor 102

6 Kinetics of binding processes 109

6.1 A mathematical model .of eukaryotic gene activation 110

6.2 Steady state distribution of more general binding processes 117

7 Transcription factor binding at nucleosomal DNA 119

7.1 Competition between nucleosomes and TF 119

7.2 Nucleosome-mediated cooperativity between TF 120

8 Signalling switches 131

8.1 Ordered phosphorylation 132

8.2 Unordered phosphorylation 133

III A short course on dynamical systems 137

9 Differential equations, flows and vector fields 139

9.1 Some examples 139

9.1.1 Malthus and Verhulst equations 139

9.1.2 Predators-Preys systems: The Lotka-Volterra model 140

9.2 Vector fields and differential equations 142

9.3 Existence and uniqueness theorems 142

9.3.1 A global existence criterion 144

9.4 Higher order and nonautonomous equations 145

9.5 Flow and phase portrait 146

9.5.1 Phase portrait 148

9.5.2 The variational equation 150

9.5.2.1 Liouville formula 150

10 Equilibria, periodic orbits and limit cycles 153

10.1 Equilibria, periodic orbits and invariant sets 153

10.2 Alpha and omega limit sets 154

10.2.1 Limit cycles 155

10.2.2 Heteroclinic cycle: The May and Leonard example 155

10.3 The Poincaré-Bendixson theorem 159

10.4 Chaos 159

10.4.1 Lotka-Volterra and chaos 161

10.5 Lyapunov functions 162

10.6 Attractors 163

10.7 Stability in autonomous systems 164

10.8 Application to Lotka-Volterra equations 165

10.8.1 Lotka-Volterra with limited growth 165

10.8.2 Lotka-Volterra in dimension n 166

11 Linearisation 169

11.1 Linear- differential equations 169

11.1.1 Hyperbolic matrices, sources, sinks and saddles 171

11.1.2 Two dimensional linear systems 173

11.2 Linearization and stable manifolds 176

11.2.1 Nonlinear sinks 177

11.2.2 The stable manifold theorem 178

11.2.3 The Hartmann-Grobman linearization theorem* 181

11.2.4 The May and Leonard model (the end) 181

IV Linear noise approximation 183

12 Density dependent population processes and the linear noise approximation 185

12.1 A law of large numbers 185

12.2 Illustration: bistable behaviour of self-regulated genes 190

12.3 Epigenetics and multistationarity 191

12.4 Gaussian approximation 193

12.4.1 Steady state approximations 197

12.5 Illustration: attenuation of noise using negative feedback loops in prokaryotic transcription 199

13 Mass action kinetics 205

13.1 Deterministic mass action kinetics and the deficiency zero theorem* 207

13.2 Stochastic mass action kinetics 212

13.3 Extension to more general dynamics 216

V Appendix 219.

Notes:

Includes bibliographical references (pages 243-255) and index.

ISBN:

9781466514935

1466514930

OCLC:

769420425

Publisher Number:

99958550411