Flexible single molecule simulation of reaction–diffusion processes

Abstract

An algorithm is developed for simulation of the motion and reactions of single molecules at a microscopic level. The molecules diffuse in a solvent and react with each other or a polymer and molecules can dissociate. Such simulations are of interest e.g. in molecular biology. The algorithm is similar to the Green's function reaction dynamics (GFRD) algorithm by van Zon and ten Wolde where longer time steps can be taken by computing the probability density functions (PDFs) and then sample from its distribution function. Our computation of the PDFs is much less complicated than GFRD and more flexible. The solution of the partial differential equation for the PDF is split into two steps to simplify the calculations. The sampling is without splitting error in two of the coordinate directions for a pair of molecules and a molecule-polymer interaction and is approximate in the third direction. The PDF is obtained either from an analytical solution or a numerical discretization. The errors due to the operator splitting, the partitioning of the system, and the numerical approximations are analyzed. The method is applied to three different systems involving up to four reactions. Comparisons with other mesoscopic and macroscopic models show excellent agreement.

Coupled Mesoscopic and Microscopic Simulation of Stochastic Reaction-Diffusion Processes in Mixed Dimensions

“…In a microscopic model, the paths and reactions of individual molecules are tracked requiring much more computing time for the same number of molecules. If the mesh is kept fixed, the cost of simulating a trajectory on the mesoscale scales linearly with the number of molecules in a fixed domain while it scales quadratically for the microscopic method [29].…”

“…Then the time for the next reaction in the system is sampled with arbitrary precision. We show that by using the techniques developed in [29] also in two dimensions we obtain an accurate and efficient algorithm for simulating diffusion and reactions on surfaces. The accuracy of the method is analyzed in the special case when a sphere is approximated by planar facets.…”

“…In the microscopic model, individual molecules move by Brownian motion and when they are close they can react with each other. The molecules diffuse in the simulation either by taking small time steps in a solution of a Langevin equation [5,32] or by sampling a probability distribution for the new position [29,40,52]. Such microscopic models are much more computationally demanding than a corresponding mesoscopic simulation, at least for reasonable mesh resolutions.…”

“…Our contribution differs from these two methods in the following respects: Firstly, the mesoscopic mesh is unstructured as in [22] where a voxel shares faces with a variable number of other voxels. Secondly, assuming Brownian dynamics the microscopic part is based on Smoluchowski theory for diffusion and reactions [45] and is simulated by an efficient version from [29] of the GFRD algorithm [52] allowing longer microscopic time steps. Thirdly, the simulation in time is split in a time step such that the meso level is treated first with a frozen micro level and then the micro level is advanced with a frozen meso level.…”

“…The method we propose is made sufficiently general to simulate complex biochemical models in mixed dimensions by extending the method in [29] to handle an arbitrary 2D boundary. Our microscopic method simulates Brownian motion and reactions on any smooth 2D surface embedded in 3D by a locally planar approximation determined by the triangulation of the surface.…”