Stochastic Dynamics for Systems Biology: 1st Edition (Hardback) book cover

Stochastic Dynamics for Systems Biology

1st Edition

By Christian Mazza, Michel Benaim

Chapman and Hall/CRC

274 pages | 63 B/W Illus.

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pub: 2014-03-11
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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.


"This book is the ideal media for introducing many stochastic models from systems biology and the biological meaning of some biological notions, like Hill functions and binding curves, to mathematicians, and likewise providing the biologists with a mathematical framework of simulating and theoretically studying the biological processes. … The book also presents an original and up-to-date treatment of cooperativity …"

Zentralblatt MATH 1305

Table of Contents

Dynamics of Reaction Networks: Markov Processes

Reaction Networks: Introduction

Introduction to modelling: a self-regulated gene

Birth and death processes to model basic chemical reactions

Some results on the self-regulated gene

Continuous-Time Markov Chains


General time-continuous Markov chains

Some important Markov chains

Two-time-scale stochastic simulations

Illustrations from Systems Biology

First-Order Chemical Reaction Networks

Reaction networks

Linear first-order reaction networks

Statistical descriptors for linear rate functions

Open and closed conversion systems

Illustration: Intrinsic noise in gene regulatory networks

Biochemical Pathways

Stochastic fluctuations in metabolic pathways

Signalling networks

Binding Processes and Transcription Rates

Positive and negative control

Binding probabilities

Gibbs-Boltzmann distributions

Local Hill coefficients

Cooperativity in the microstate

The sigmoidal nature of the binding curve

Cooperativity in the Hill sense

ηH(v) as an indicator of cooperativity

The cooperativity index

Macroscopic cooperativity

The case N = 3

Transcription rates for basic models

A genetic switch: regulation by λ phage repressor

Kinetics of Binding Processes

A mathematical model of eukaryotic gene activation

Steady state distribution of more general binding processes

Transcription Factor Binding at Nucleosomal DNA

Competition between nucleosomes and TF

Nucleosome-mediated cooperativity between TF

Signalling Switches

Ordered phosphorylation

Unordered phosphorylation

A Short Course on Dynamical Systems

Differential Equations, Flows, and Vector Fields

Some examples

Vector fields and differential equations

Existence and uniqueness theorems

Higher order and nonautonomous equations

Flow and phase portrait

Equilibria, Periodic Orbits and Limit Cycles

Equilibria, periodic orbits and invariant sets

Alpha and omega limit sets

The Poincaré-Bendixson theorem


Lyapunov functions


Stability in autonomous systems

Application to Lotka-Volterra equations


Linear differential equations

Linearization and stable manifolds

Linear Noise Approximation

Density-Dependent Population Processes and the Linear Noise Approximation

A law of large numbers

Illustration: bistable behaviour of self-regulated genes

Epigenetics and multistationarity

Gaussian approximation

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

Mass Action Kinetics

Deterministic mass action kinetics and the deficiency zero theorem

Stochastic mass action kinetics

Extension to more general dynamics


Self-Regulated Genes


Transcription with fast dimerisation

Asymptotic Behaviour of the Solutions to Time-Continuous Lyapunov Equations

Time-continuous Lyapunov equations

Asymptotically autonomous dynamical systems



About the Series

Chapman & Hall/CRC Mathematical and Computational Biology

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General
MEDICAL / Biotechnology