Editorial: special issue on stochastic modelling of reaction-diffusion processes in biology.

Reaction–diffusion equations are used to model many biological processes, ranging from intracellular signaling, metabolic processes and gene control at the cellular level, to birth–death processes and random movement at the organism and population levels. There are two fundamental approaches to the mathematical modelling of these processes: deterministic (mean-field) models which lead to partial differential equations for concentrations of biochemical species or for densities of individuals, and stochastic models in which individual events of reaction or diffusion are followed. In some cases—such as linear processes or pure reaction processes based on mass action kinetics—one can prove that the latter description converges to the former in an appropriate ‘large-number limit’, but this is still an open question in general, as some of the papers herein illustrate. Whenever the number of ‘individuals’ involved is small, stochastic effects can play an important role in the survival and spatiotemporal distribution of individuals. For instance, chemical reactions occur in discrete steps at the molecular level, the processes are inherently stochastic, and the inherent “irreproducibility” in these dynamics has been demonstrated experimentally for single-cell gene expression events (Ozbudak et al. 2002; Levsky and Singer 2003). Frequently stochastic effects simply add noise to an essentially deterministic output, but in others, such as asymmetric cell division, their role is essential.