Turing Pattern Formation in Stochastic Reaction-Subdiffusion Systems
by
Jiawei Chiu
A*STAR Institute of High Performance Computing
Coauthors: K.-H. Chiam

We investigate the formation of spatial Turing patterns in stochastic reaction-diffusion systems that arise commonly in biology, such as the motion of proteins in a crowded cytoplasm, or the migration of epithelial cells driven by active biological processes. Here, we consider the case of two species of "particles, " be they proteins or cells, performing continuous random time walks specified by the probability distribution function y_i(x, t) = m(x)W_i(t), where the Laplace transform of the W_i's is ~ 1 - (h_i s)^a and a < 1 denotes subdiffusion. We seek to understand under what conditions there is Turing instability by performing linear stability analysis. We find that, if we fix the ratios between the diffusive constants, there is a critical value of a below which there is no pattern. We discuss the relevance of this critical value to several biological processes. In addition, we carry out simulations based on Gillespie's algorithm to study the effect of noise, induced by the low copy number of particles, on the conditions for Turing pattern formation. We find that Turing patterns can survive in a very noisy system.

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Date received: May 12, 2008