Stochastic simulation with diffusion in molecular biology

Abstract: 

Stochastic models are needed to model the diffusing and reacting molecules in biological cells. The reason is that the number of molecules of each chemical species is low and the reactions occur with a certain probability. The diffusion and the chemical processes can be simulated in space and time with a Monte Carlo method called the Stochastic Simulation Algorithm which was invented by Gillespie. The equation for the mean value of the concentration of the species is the diffusion (or heat) equation. Space is discretized by a mesh with voxels (or computational cells). The number of molecules in each voxel defines the state of the system and evolves in time. If the mesh is Cartesian or unstructured of good quality, the procedure is well defined but on an unstructured mesh of poor quality different options are possible. These can be based on modified finite element discretizations, finite volume discretizations, the mean first exit time, or optimization. The methods can be interpreted as solving a modified diffusion equation. Examples of simulations of different biochemical systems will be presented. This problem is related to the problem of satisfying the discrete maximum principle for finite element discretizations of the diffusion equation.