Stochastic reaction-diffusion methods for modeling gene expression and spatially distributed chemical kinetics

In order to model fundamental cell biological processes including the transcription, translation, and nuclear membrane transport of biological molecules within a eukaryotic cell it is necessary to be able to approximate the stochastic reaction and diffusion of a small number of molecules in the complex three dimensional geometry of a cell. For this reason a method is developed that incorporates diffusion and active transport of chemicals in complex geometries into stochastic chemical kinetics simulations. Systems undergoing stochastic reaction and diffusion are modeled using a discrete state master equation. It is shown how the jump rates for spatial movement between mesh cells can be derived from the discretization weights of embedded boundary methods. Spatial motion is modeled as first order reactions between neighboring mesh cells using the predetermined jump rates. Individual realizations of the master equation can be created by the Gillespie Method, allowing numerical simulation of the underlying stochastic process. We investigate the numerical convergence properties of both the underlying embedded boundary methods, and the reaction-diffusion master equation model. Several continuum limits of the reaction-diffusion master equation are investigated. In addition, a proposed model for the problem of locating a point binding site by diffusive motion is studied. This problem is an idealization of one proposed mechanism by which gene regulatory proteins may bind to specific sites on DNA. A model of transcription, translation, and nuclear membrane transport in eukaryotic cells is presented and solved using this stochastic reaction-diffusion master equation approach in order to demonstrate the feasibility of the method in studying fundamental spatially distributed cell biological processes.