1st Edition

Stochastic Dynamics for Systems Biology

By Christian Mazza, Michel Benaim Copyright 2014
    274 Pages 63 B/W Illustrations
    by Chapman & Hall

    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.

    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




    Christian Mazza, Michel Benaim

    "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