Accurate and Efficient Solution of the Smoluchowski Equation

Abstract: The probability density function (PDF) of the relative position of molecules diffusing independently in three dimensional space according to Brownian motion and reacting with a certain probability when any two of them collide is given by the Smoluchowski equation. The PDF is used to sample particle positions in simulations of reaction-diffusion processes by particle-based simulation methods, like Green's Function Reaction Dynamics (GFRD) proposed by van Zon and ten Wolde. The GFRD algorithm is an event-driven algorithm, allowing the use of longer time steps, which is particularly efficient for simulating chemical reactions at low concentration in molecular biology. This study is based on the improved version of the GFRD algorithm developed by S. Hellander and P. Lötstedt, where the applicability of the algorithm is increased and computing the PDFs is simplified by using an operator splitting approach. The main idea is to split the spatial differential operator of the Smoluchowski equation into a radial part and an angular part, resulting in two one-dimensional time-dependent partial differential equations (PDEs) to be solved independently and sequentially. These equations can be solved analytically but the solutions are complicated and computationally expensive to evaluate. This thesis intends to compare the accuracy and efficiency of sampling the radial distance between two molecules and the relative angular position of the two molecules from directly evaluated exact PDFs or their finite difference approximations and interpolating the positions from precomputed tabulated data.