A stochastic cellular automaton simulation of chemical oscillations in small systems

We propose a stochastic cellular automaton method to simulate chemical reactions in small systems. Unlike the standard Gillespie method, which simulates chemical reactions with a few thousand molecules reacting with each other but without spatial considerations, our systems are divided into independent cells, each containing only a few molecules. Our simulation of the Brusselator produces chemical oscillations that agree extremely well with solutions to deterministic rate equations, and we can see strong oscillations in systems with as few as 10 cells. We are able to study several factors that affect the robustness of these small chemical oscillators: system size, spatial distribution, and correlation of molecules. We have found that non-Poisson particle distributions can greatly suppress chemical oscillations and that chemical reactions can induce correlation between the spatial distributions of particles of different species and create large-scale inhomogeneity in particle concentrations. In addition, incomplete oscillations (misfirings) can appear among strong, regular oscillations when the system size is smaller than a certain threshold, and these misfirings are triggered by random events, with a probability that is related to the system size. Since these effects, resulting from several different physical causes, are difficult to accurately model by adding generic noise factors to deterministic rate equations, as is frequently done in theoretical studies, we argue that our stochastic cellular automaton method is a useful addition to the existing tools for studying small, inhomogeneous, and non-equilibrium reaction-diffusion systems, especially those of biological nature.