Abstract
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.
Highlights
Oscillatory systems are ubiquitous in the natural world, ranging from simple physical systems, such as pendulums, to complicated biological systems, such as cellular cycles.[1,2] What these oscillators have in common is a restoring mechanism that can pull the system back to a certain configuration—a fixed point—when the system moves away from it
Chemical oscillations can occur in systems with numerous chemical species that undergo multistep reactions
Because the laws of thermodynamics dictate that a closed system eventually reaches a state of equilibrium, where the free energy is a minimum, an oscillatory chemical system is obviously not at equilibrium
Summary
Oscillatory systems are ubiquitous in the natural world, ranging from simple physical systems, such as pendulums, to complicated biological systems, such as cellular cycles.[1,2] What these oscillators have in common is a restoring mechanism that can pull the system back to a certain configuration—a fixed point—when the system moves away from it. Despite the enormous success of the Gillespie algorithm and several subsequent modifications that are computationally more efficient,[10,11,12,13,14] the issues of how to treat spatial inhomogeneity and take into account diffusion of molecules remain an ongoing research effort.[15] Methods of simulating reaction-diffusion in heterogeneous systems usually involve dividing systems into many subvolumes and allowing chemical reactions to occur in the subvolumes with diffusion of molecules between subvolumes.[16] For example, in the binomial tau-leap spatial stochastic simulation algorithm,[17] at each time step a chosen subvolume can undergo either a chemical reaction using a Gillespie-based method, or molecules in the subvolume can diffuse into neighboring subvolumes, depending on various parameters specified by the algorithm. Even though the scope of the current study is limited principally to homogeneous systems, we hope to further develop this method for application to spatially inhomogeneous reaction-diffusion systems — and to networks of biochemical systems — where fluctuations due to small system size can play critical roles
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