Synopsis for C6.4b: Stochastic Modelling of Biological Processes

Level: M-level Method of Assessment: Written examination.
Weight: Half-unit, (OSS paper codes tbc)

Recommended Prerequisites

A basic understanding of probability is sufficient. The course is designed in such a way that a Part C student should be able to understand it without taking special stochastic or biological classes.

Overview

This course provides an overview of stochastic methods which are used for modelling biological systems. The course starts with stochastic modelling of chemical reactions, introducing stochastic simulation algorithms and mathematical methods which can be used for analysis of stochastic models (chemical master equation). Systems with increasing level of complexity are used to illustrate the theory. Then stochastic differential equations are introduced (from the computational point of view), explaining their connections with modelling chemical systems and the Fokker-Planck equation. Different models of molecular diffusion (on-lattice and off-lattice models, velocity jump processes) and their properties are studied, before moving to stochastic reaction-diffusion models. Compartment-based and molecular-based approaches to stochastic reaction-diffusion modelling (Brownian dynamics) are discussed together with properties of stochastic spatially-distributed models (pattern formation). The final lectures include discussion of bacterial chemotaxis, Metropolis-Hastings algorithm and multiscale modelling.

Learning Outcomes

The student will learn:
(i) about biological systems which are often described in terms of stochastic models;
(ii) mathematical techniques which are used for the analysis of stochastic models;
(iii) how the models can be efficiently simulated using a computer;
(iv) connections and differences between different stochastic methods, and between stochastic and deterministic modelling.

Synopses

Stochastic simulation of chemical reactions: well-stirred systems, Gillespie algorithm, chemical master equation, analysis of simple systems, deterministic vs. stochastic modelling, systems with multiple favourable states, stochastic resonance, stochastic focusing.

Stochastic differential equations: numerical methods, Fokker-Planck equation, first exit time, backward Kolmogorov equation, chemical Fokker-Planck equation.

Diffusion: Brownian motion, on-lattice and off-lattice models, compartment-based approach, velocity jump processes, Einstein-Smoluchowski relation, diffusion to adsorbing surfaces, reactive boundary conditions.

Stochastic reaction-diffusion models: compartment-based reaction-diffusion algorithm, reaction-diffusion master equation, pattern formation, morphogen gradients, Turing patterns, molecular-based approaches to reaction-diffusion modelling, Brownian dynamics, reaction radius.

Bacterial chemotaxis: reaction-diffusion-advection processes, velocity jump processes with internal dynamics, agent-based modelling.

Metropolis-Hastings algorithm: Markov chain Monte Carlo methods.

Multiscale modelling: efficient stochastic modelling of chemical reactions, multiscale SSA with partial equilibrium assumption, hybrid modelling approaches.

Reading

1. R. Erban, J. Chapman and P. Maini: "A practical guide to stochastic simulations of reaction-diffusion processes, available as http://arxiv.org/abs/0704.1908, 2007 (course lecture notes extend this material)

Further Reading

1. H. Berg: "Random Walks in Biology", new, expanded edition, Princeton University Press, 1993,
2. D. Gillespie: "Markov Processes, an Introduction for Physical Scientists", Academic Press, Inc.,1992
3. A.Chorin and O.Hald: "Stochastic Tools for Mathematics and Science", 2nd Edition, Springer, 2009