Abstract. The reaction-diffusion master equation (RDME) has recently been used as a model for biological systems in which both noise in the chemical reaction process and diffusion in space of the reacting molecules is important. In the RDME space is partitioned by a mesh into a collection of voxels. There is an unanswered question as to how solutions depend on the mesh spacing. To have confidence in using the RDME to draw conclusions about biological systems, we would like to know that it approximates a reasonable physical model for appropriately chosen mesh spacings. This issue is investigated by studying the dependence on mesh spacing of solutions to the RDME in R 3 for the bimolecular reaction A + B → ∅, with one molecule of species A and one molecule of species B present initially. We prove that in the continuum limit the molecules never react and simply diffuse relative to each other. Nevertheless, we show that the RDME with non-zero lattice spacing yields an asymptotic approximation to a specific spatially-continuous diffusion limited reaction (SCDLR) model. We demonstrate that for realistic biological parameters it is possible to find mesh spacings such that the relative error between asymptotic approximations to the solutions of the RDME and the SCDLR models is less than one percent.

Abstract.A method is developed for incorporating diffusion of chemicals in complex geometries into stochastic chemical kinetics simulations. Systems are modeled using the reaction-diffusion master equation, with jump rates for diffusive motion between mesh cells calculated from the discretization weights of an embedded boundary method. Since diffusive jumps between cells are treated as first order reactions, individual realizations of the stochastic process can be created by the Gillespie method. Numerical convergence results for the underlying embedded boundary method, and for the stochastic reaction-diffusion method, are presented in two dimensions. A two-dimensional model of transcription, translation, and nuclear membrane transport in eukaryotic cells is presented to demonstrate the feasibility of the method in studying cell-wide biological processes.

The reaction-diffusion master equation (RDME) is a lattice stochastic reaction-diffusion model that has been used to study spatially distributed cellular processes. The RDME is often interpreted as an approximation to spatially continuous models in which molecules move by Brownian motion and react by one of several mechanisms when sufficiently close. In the limit that the lattice spacing approaches zero, in two or more dimensions, the RDME has been shown to lose bimolecular reactions. The RDME is therefore not a convergent approximation to any spatially continuous model that incorporates bimolecular reactions. In this work we derive a new convergent RDME (CRDME) by finite volume discretization of a spatially continuous stochastic reaction-diffusion model popularized by Doi. We demonstrate the numerical convergence of reaction time statistics associated with the CRDME. For sufficiently large lattice spacings or slow bimolecular reaction rates, we also show that the reaction time statistics of the CRDME may be approximated by those from the RDME. The original RDME may therefore be interpreted as an approximation to the CRDME in several asymptotic limits.

Within the nuclei of eukaryotic cells, the density of chromatin is nonuniform. We study the influence of this nonuniform density, which we derive from microscopic images [Schermelleh L, et al. (2008) Science 320:1332-1336], on the diffusion of proteins within the nucleus, under the hypothesis that chromatin density is proportional to an effective potential that tends to exclude the diffusing protein from regions of high chromatin density. The constant of proportionality, which we call the volume exclusivity of chromatin, is a model parameter that we can tune to study the influence of such volume exclusivity on the random time required for a diffusing particle to find its target. We consider randomly chosen binding sites located in regions of low (20th-30th percentile) chromatin density, and we compute the median time to find such a binding site by a protein that enters the nucleus at a randomly chosen nuclear pore. As the volume exclusivity of chromatin increases from zero, we find that the median time needed to reach the target binding site at first decreases to a minimum, and then increases again as the volume exclusivity of chromatin increases further. Random permutation of the voxel values of chromatin density abolishes the minimum, thus demonstrating that the speedup seen with increasing volume exclusivity at low to moderate volume exclusivity is dependent upon the spatial structure of chromatin within the nucleus.first passage time | gene regulation | stochastic reaction-diffusion H ow do regulatory proteins and transcription factors find specific DNA binding sites? In considering this question, it is often remarked that the rate at which proteins find specific DNA binding sites can "exceed the diffusion limit." This statement is normally interpreted to mean that the association rate for a protein to find a specific binding site is faster than the predicted rate for the protein to reach the binding site by diffusion (2). The question of whether proteins, in vivo, generally find binding sites faster than the diffusion limit is still an area of active research. One potential difficulty in addressing this problem is in understanding precisely what is meant by the term "diffusion limited" binding rate. Here we adopt the viewpoint that a diffusion limited rate refers only to the rate at which a protein undergoing pure diffusive motion in a spatially homogeneous environment finds a target binding site. This corresponds to the standard Smoluchowski diffusion limited reaction model (3). In the present paper, we consider the influence of a heterogeneous environment on the time to find a target by diffusion.A number of mechanisms that could potentially decrease the search time for a binding site, in comparison to the search time in models involving only diffusion in an empty nucleus, have been proposed and studied in experimental assays and mathematical models. For example, in ref. 2 it was discussed how the inclusion of electrostatic interactions between the protein and binding site may make predicted association r...

Tethering of the cytoplasmic tyrosine phosphatase SHP-1 to clustered receptors increases its activity 900-fold.