An Accelerated Algorithm for Stochastic Simulation of Reaction-Diffusion Systems using Gradient-Based Diffusion and Unified Tau-Leaping

Abstract

Stochastic simulation of reaction-diffusion systems enables the investigation of stochastic events arising from the small numbers and heterogeneous distribution of molecular species in biological cells. Stochastic variations in intracellular microdomains and in diffusional gradients that are especially prominent in neurites with elongated morphology play a significant part in the spatiotemporal activity and behavior of cells. Although an exact stochastic simulation that simulates every individual reaction and diffusion event occurring in the system gives a most accurate trajectory of the system's state over time, it can be too slow for many practical applications. We present an accelerated algorithm for discrete stochastic simulation of reaction-diffusion systems designed to improve the speed of simulation by reducing the number of time steps required to complete a simulation run. Our method is unique in that it employs two strategies that have not been incorporated in existing spatial stochastic simulation algorithms. First, we treat diffusive transfers between neighboring subvolumes based on concentration gradients. Our treatment necessitates sampling of only the net or observed diffusion events from higher to lower concentration gradients rather than sampling all diffusion events regardless of local concentration gradients. Second, we extend the non-negative Poisson tau-leaping method that was originally developed for speeding up non-spatial or homogeneous stochastic simulation algorithms. Our method calculates each leap time in a unified step for both reaction and diffusion processes while satisfying the leap condition that the propensities do not change appreciably during the leap and ensuring that leaping does not cause molecular populations to become negative. We also present numerical results that illustrate the improvement in simulation speed achieved by incorporating these two new strategies.