Numerical Computation of Diffusion Properties in Molecular Systems on a Topology-Conforming Grid

Multiscale problems involving diffusion in molecular systems are a mainstay of computational biophysics. Given a molecular system, the local diffusion coefficient $D(\bf{r})$ as well as the equilibrium distribution function $P(\bf{r})$ that characterizes the local free energy are computed to describe the kinetics of diffusing particles at each point in space through the Smoluchowski equation (SE). An irregular grid of space-varying fineness conforming to $P(\bf{r})$ is generated via the method of topology-representing networks and a subsequent Voronoi tessellation. The discretized SE produces a rate matrix which describes the probabilities of particles hopping from point to point on the grid. We demonstrate the calculation of the rate matrix for ions diffusing through the balloon-like structure of the mechanosensitive channel of small conductance (MscS) and thence the determination of mean first-passage times that characterize conduction of ions through balloon and channel.