Spatial heterogeneity and the stability of reaction states in autocatalysis

The impact of stochasticity and spatial heterogeneity on the quadratic autocatalytic system is studied. In a nonspatial setting the reactive state of the system is found to be unstable in small volumes where internal fluctuations drive the system to the unreactive state. This phenomena is of potential importance to the stability of reactions in biological cells. A simple spatial model is constructed by linking N nonspatial models via migration of reactants controlled by a mixing rate lambda. Simulation of this stochastic process demonstrates the importance of such mixing in controlling the impact of internal fluctuations on the stability of the autocatalytic reaction. For high mixing rate the mean reactant levels in equilibrium correspond to the well-mixed deterministic system, although a significant degree of spatial heterogeneity remains. For intermediate mixing rates, mean reactant levels vary continuously with lambda, where the interaction of internal fluctuations with limited spatial mixing modifies the reactive states of the deterministic system. However, there is a threshold below which mixing is unable to control internal fluctuations which drive the system into the unreactive state. Thus a critical minimum level of communication between the cells is required to stabilize the reaction across the entire system. Approximate analytic results, obtained using moment-closure techniques, support these findings and demonstrate the relationship between the spatial stochastic and nonspatial deterministic models.