On the stochastic modelling of reaction-diffusion processes - Twomey, A (University of Oxford;Mathematical Institute, 2007)
Traditionally, the law of mass action has been used to deterministically model chemical reactions. There are, however, fundamental limitations to this approach when the number of molecules involved in the reactions is small. This is often the case in biological cells, and many authors have reported that deterministic models do not adequately describe processes such as gene expression. This dissertation shows circumstances in which stochastic simulation describes non-linear reactions more accurately than deterministic models. Exact stochastic algorithms due to Gillespie and Gibson-Bruck, based on the chemical master equation, are introduced. A compartment based model is described and analysed for reaction-diffusion systems with low copy numbers of molecules. Finally, this approach is used to produce Turing patterns that may be useful in describing pattern formation in developmental biology.