Validity conditions for stochastic chemical kinetics in diffusion-limited systems

• Type of Material: Article
• Authors: Daniel T. Gillespie, Dan T Gillespie Consulting; Linda R. Petzold, University of California Santa Barbara; Effrosyni Seitaridou, Emory University
• Language: English
• Date: 2014-02-07
• Publisher: AIP
• Publication Version: Final Published Version
• Copyright Statement: © 2014 AIP Publishing LLC
• Final Published Version (URL): http://dx.doi.org/10.1063/1.4863990
• Title of Journal or Parent Work: The Journal of Chemical Physics
• Volume: 140
• Issue: 5
• Start Page: 054111
• End Page: 054111
• Grant/Funding Information: D.T.G. was funded by the University of California, Santa Barbara under professional services agreement 130401A40, pursuant to NIH award R01-EB014877-01.; L.R.P. was funded by NSF award DMS-1001012, ICB award W911NF-09-0001 from the U.S. Army Research Office, NIBIB of the NIH under award R01-EB014877-01, and U.S. DOE award DE-SC0008975.
• Keywords: biomolecular reaction; dilute; well-mixed; stochastic simulation algorithm; chemical master equation
• Research Categories: Biology, Molecular

Abstract

The chemical master equation (CME) and the mathematically equivalent stochastic simulation algorithm (SSA) assume that the reactant molecules in a chemically reacting system are “dilute” and “well-mixed” throughout the containing volume. Here we clarify what those two conditions mean, and we show why their satisfaction is necessary in order for bimolecular reactions to physically occur in the manner assumed by the CME and the SSA. We prove that these conditions are closely connected, in that a system will stay well-mixed if and only if it is dilute. We explore the implications of these validity conditions for the reaction-diffusion (or spatially inhomogeneous) extensions of the CME and the SSA to systems whose containing volumes are not necessarily well-mixed, but can be partitioned into cubical subvolumes (voxels) that are. We show that the validity conditions, together with an additional condition that is needed to ensure the physical validity of the diffusion-induced jump probability rates of molecules between voxels, require the voxel edge length to have a strictly positive lower bound. We prove that if the voxel edge length is steadily decreased in a way that respects that lower bound, the average rate at which bimolecular reactions occur in the reaction-diffusion CME and SSA will remain constant, while the average rate of diffusive transfer reactions will increase as the inverse square of the voxel edge length. We conclude that even though the reaction-diffusion CME and SSA are inherently approximate, and cannot be made exact by shrinking the voxel size to zero, they should nevertheless be useful in many practical situations.