Stochastic Analysis of Biochemical Systems (Mathematical Biosciences Institute Lecture Series Book 1)

This Book Focuses On Counting Processes And Continuous-time Markov Chains Motivated By Examples And Applications Drawn From Chemical Networks In Systems Biology. The Book Should Serve Well As A Supplement For Courses In Probability And Stochastic Processes. While The Material Is Presented In A Manner Most Suitable For Students Who Have Studied Stochastic Processes Up To And Including Martingales In Continuous Time, Much Of The Necessary Background Material Is Summarized In The Appendix. Students and Researchers with A Solid Understanding Of Calculus, Differential Equations, And Elementary Probability And Who are well-motivated By The Applications Will Find This Book Of Interest. David F. Anderson Is Associate Professor In The Department Of Mathematics At The University Of Wisconsin And Thomas G. Kurtz Is Emeritus Professor In The Departments Of Mathematics And Statistics At That University. Their Research Is Focused On Probability And Stochastic Processes With Applications In Biology And Other Areas Of Science And Technology. These Notes Are Based In Part On Lectures Given By Professor Anderson At The University Of Wisconsin – Madison And By Professor Kurtz At Goethe University Frankfurt. By David F. Anderson, Thomas G. Kurtz. Preface 8 Contents 10 1 Infinitesimal specification of continuous time Markov chains 12 1.1 Poisson and general counting processes 12 1.2 Modeling with intensities 16 1.3 Multivariate counting processes 18 1.4 Continuous time Markov chains 21 Problems 26 2 Models of biochemical reaction systems 29 2.1 The basic model 29 2.1.1 Example: Gene transcription and translation 32 2.1.2 Example: Virus kinetics 36 2.1.3 Example: Enzyme kinetics 38 2.2 Deterministic models of biochemical reaction systems, and first-order reaction networks 39 Problems 41 3 Stationary distributions of stochastically modeled reaction systems 42 3.1 Introduction 42 3.2 Network conditions, complex-balanced equilibria, and the deficiency zero theorem 43 3.3 Stationary distributions for complex-balanced models 45 Problems 49 4 Analytic approaches to model simplification and approximation 51 4.1 Limits under the classical scaling 51 4.2 Models with multiple time-scales 54 4.2.1 Example: Derivation of the Michaelis-Menten equation 56 4.2.2 Example: Approximation of the virus model 58 Problems 60 5 Numerical methods 62 5.1 Monte Carlo 62 5.2 Generating random variables: Transformations of uniforms 63 5.3 Exact simulation methods 65 5.3.1 Embedded discrete time Markov chains and the stochastic simulation algorithm 65 5.3.2 The next reaction method 66 5.3.2.1 Time dependent intensity functions 68 5.4 Approximate simulation with Euler's method / τ-leaping 69 5.5 Monte Carlo and multi-level Monte Carlo 70 5.5.1 Computational complexity for Monte Carlo 70 5.5.2 Multi-level Monte Carlo (MLMC) 71 Problems 74 A Notes on probability theory and stochastic processes 76 A.1 Some notation and basic concepts 76 A.1.1 Measure theoretic foundations of probability 76 A.1.2 Dominated convergence theorem 77 A.1.3 General theory of stochastic processes 78 A.2 Martingales 78 A.2.1 Doob's inequalities 79 A.3 Stochastic integrals 80 A.4 Convergence in distribution and the functional central limit theorem for Poisson processes 81 A.5 Conditioning and independence 82 A.6 Directed sets 83 A.7 Gronwall inequality 83 References 85 Index 89 Front Matter....Pages i-x Infinitesimal specification of continuous time Markov chains....Pages 1-17 Models of biochemical reaction systems....Pages 19-31 Stationary distributions of stochastically modeled reaction systems....Pages 33-41 Analytic approaches to model simplification and approximation....Pages 43-53 Numerical methods....Pages 55-68 Back Matter....Pages 69-84