Stochastic Dynamics for Systems Biology  book cover
1st Edition

Stochastic Dynamics for Systems Biology

ISBN 9781466514935
Published March 11, 2014 by Chapman and Hall/CRC
274 Pages 63 B/W Illustrations

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Book Description

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.

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



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"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