  Turing patterns can be observed in reaction-diffusion systems where chemical
species have different diffusion constants. In recent years, several studies
investigated the effects of noise on Turing patterns and showed that the
parameter regimes, for which stochastic Turing patterns are observed, can be
larger than the parameter regimes predicted by deterministic models, which are
written in terms of partial differential equations for species concentrations.
A common stochastic reaction-diffusion approach is written in terms of
compartment-based (lattice-based) models, where the domain of interest is
divided into artificial compartments and the number of molecules in each
compartment is simulated. In this paper, the dependence of stochastic Turing
patterns on the compartment size is investigated. It has previously been shown
(for relatively simpler systems) that a modeller should not choose compartment
sizes which are too small or too large, and that the optimal compartment size
depends on the diffusion constant. Taking these results into account, we
propose and study a compartment-based model of Turing patterns where each
chemical species is described using a different set of compartments. It is
shown that the parameter regions where spatial patterns form are different from
the regions obtained by classical deterministic PDE-based models, but they are
also different from the results obtained for the stochastic reaction-diffusion
models which use a single set of compartments for all chemical species. In
particular, it is argued that some previously reported results on the effect of
noise on Turing patterns in biological systems need to be reinterpreted.
