Validity conditions for stochastic chemical kinetics in diffusionlimited systems
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 "wellmixed" 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 wellmixed if and only if it is dilute. We explore the implications of these validity conditions for the reactiondiffusion (or spatially inhomogeneous) extensions of the CME and the SSA to systems whose containing volumes are not necessarily wellmixed, 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 diffusioninduced 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 reactiondiffusion 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 reactiondiffusion 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.
 Publication:

Journal of Chemical Physics
 Pub Date:
 February 2014
 DOI:
 10.1063/1.4863990
 Bibcode:
 2014JChPh.140e4111G