Samuel A. Isaacson

Incorporating Diffusion in Complex Geometries into Stochastic Chemical Kinetics Simulations

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 as an Asymptotic Approximation of Diffusion to a Small Target

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

A convergent reaction-diffusion master equation

\mathbb{R}^{3}

The influence of volume exclusion by chromatin on the time required to find specific DNA binding sites by diffusion

for the bimolecular reaction

Imaging and characterizing cells using tomography

\mathrm{A}+\mathrm{B}\to\varnothing

A Comparison of Bimolecular Reaction Models for Stochastic Reaction–Diffusion Systems

, with one molecule of species

A First-Passage Kinetic Monte Carlo method for reaction-drift-diffusion processes

\mathrm{A}

The influence of spatial variation in chromatin density determined by X-ray tomograms on the time to find DNA binding sites.

and one molecule of species

Numerical solution of linear Volterra integral equations of the second kind with sharp gradients

\mathrm{B}

Uniform asymptotic approximation of diffusion to a small target.

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 nonzero lattice spacin...