Stochastic Dynamics for Systems Biology

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
Introduction
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
Chaos
Lyapunov functions
Attractors
Stability in autonomous systems
Application to Lotka-Volterra equations

Linearisation
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

Appendix
Self-Regulated Genes
Dimerisation
Transcription with fast dimerisation

Asymptotic Behaviour of the Solutions to Time-Continuous Lyapunov Equations
Time-continuous Lyapunov equations
Asymptotically autonomous dynamical systems

Bibliography

Index