# Accurate reaction rates for the reaction-diffusion master equation

**Stefan Hellander
**

Department of Computer Science

University of California, Santa Barbara

Santa Barbara, California, USA

### Abstract:

Some reaction networks in cells can only be fully understood via stochastic models. The cellular environment is spatially heterogeneous, and therefore we need to explicitly model both the diffusion of molecules as well as discrete stochastic reaction events. A common approach is to divide space into voxels, each considered spatially homogeneous, and then model diffusion as discrete jumps between the voxels. Reactions between molecules occur with some probability when molecules occupy the same voxel. This particular model is described mathematically by the reaction-diffusion master equation (RDME). The RDME has the unfortunate numerical property of not converging as the voxel size decreases. A challenge is also to select correct reaction rates for a given voxel size. However, given that the reaction rates are accurate, it is a useful model for a range of voxel size. In this talk I will present a method of computing accurate reaction rates on Cartesian as well as on unstructured meshes. From this approach follows lower bounds on the voxel size, where the RDME can be shown to get increasingly accurate down to this lower bound, but less accurate beyond the lower bound. This methodology also provides a way to estimate how small the voxels need to be for the system to be well resolved.