Multigrid discretization method for PopZ polarization in Caulobacter cell cycle

Stochastic simulation of reaction-diffusion systems presents great challenges for spatiotemporal biological models. A straightforward extension of Gillespie´s stochastic simulation algorithm (SSA) for spatially inhomogeneous systems leads to the Inhomogeneous Stochastic Simulation Algorithm (ISSA). However, the ISSA can be prohibitively expensive in computation, especially when the mesh size is too small. In this paper, a simple formula of the optimal mesh size is developed in the one dimensional case for reaction-diffusion systems. Based on our formula, we proposed a multigrid discretization method for stochastic simulation of multiscale reaction-diffusion systems. With the combination of proper discretization for species in different reaction and diffusion scales, we can greatly reduce the size of the system and achieve high efficiency. Numerical experiment with the stochastic simulation of PopZ polarization and polymerization in Caulobacter cell cycle is presented to demonstrate the accuracy and efficiency of the proposed discretization method.