title: A hybrid continuous-discrete method for stochastic reaction-diffusion processes.
abstract: Stochastic fluctuations in reaction-diffusion processes often have substantial effect on spatial and temporal dynamics of signal transductions in complex biological systems. One popular approach for simulating these processes is to divide the system into small spatial compartments assuming that molecules react only within the same compartment and jump between adjacent compartments driven by the diffusion. While the approach is convenient in terms of its implementation, its computational cost may become prohibitive when diffusive jumps occur significantly more frequently than reactions, as in the case of rapid diffusion. Here, we present a hybrid continuous-discrete method in which diffusion is simulated using continuous approximation while reactions are based on the Gillespie algorithm. Specifically, the diffusive jumps are approximated as continuous Gaussian random vectors with time-dependent means and covariances, allowing use of a large time step, even for rapid diffusion. By considering the correlation among diffusive jumps, the approximation is accurate for the second moment of the diffusion process. In addition, a criterion is obtained for identifying the region in which such diffusion approximation is required to enable adaptive calculations for better accuracy. Applications to a linear diffusion system and two nonlinear systems of morphogens demonstrate the effectiveness and benefits of the new hybrid method.
authors.affiliation: Department of Mathematics , City University of Hong Kong , Kowloon, Hong Kong.
authors.forename: Wing-Cheong
authors.identifier: 0000-0001-7771-5213
authors.initials: WC
authors.lastname: Lo
authors.affiliation: Department of Mathematics, University of California, Irvine, CA, USA; Center for Mathematical and Computational Biology, University of California, Irvine, CA, USA.
authors.forename: Likun
authors.initials: L
authors.lastname: Zheng
authors.affiliation: Department of Mathematics, University of California, Irvine, CA, USA; Center for Mathematical and Computational Biology, University of California, Irvine, CA, USA.
authors.forename: Qing
authors.initials: Q
authors.lastname: Nie
keywords: biological morphogen systems; hybrid method; reaction–diffusion systems; stochastic simulation