Abstract
Conventional master equation approaches approximate the diffusion of molecules in continuum space by the process of particles hopping on a spatial lattice. The hopping probability from one voxel (spatial lattice point) to its neighbor is usually considered to be constant throughout space. Such an assumption is only consistent with pointlike molecules and thus neglects volume-exclusion effects due to finite particle size. A few studies have attempted to introduce volume-exclusion effects by choosing the hopping probability from one voxel to a neighboring one to be a linear function of the number density. Here, we formulate an alternative master equation in which the hopping probability is equal to the fraction of available space in the neighboring voxel as estimated using scaled particle theory. This leads to the hopping probability being a nonlinear function of the number density. A mean-field approximation (mfa) leads to a partial differential equation of the advection-diffusion type. We show that the time evolution of the particle number density sampled using the stochastic simulation algorithm associated with the new master equation and the number density obtained by numerical integration of the mfa are in good agreement with each other. They are also distinctly different than the time evolution predicted by the conventional master equation and those with hopping probabilities which are linear functions of the number density. The results from the new lattice description are also shown to be in very good agreement with the lattice-free method of Brownian dynamics, even for highly crowded scenarios.
Highlights
Reaction-diffusion processes have a long history of being modeled using partial differential equations [1]
We presented an effective rescaling of the propensities of the reaction-diffusion master equation (RDME) such that we obtain a stochastic spatial description which takes into account volume exclusion due to the finite size of particles and which is applicable to a mixture of particles of different sizes
This is in contrast to the conventional RDME which considers only point particles and to recent modified versions of the RDME [15,17,18,28] which consider volume exclusion but implicitly assume all particles are of the same size
Summary
Reaction-diffusion processes have a long history of being modeled using partial differential equations [1] This deterministic approach is fine when the standard deviation of intrinsic noise is small compared to the mean molecule numbers, a condition which is typically satisfied in the limit of large molecule numbers. The assumption of point particles is a problem for describing reaction-diffusion processes in crowded conditions where clearly the size of particles has a considerable impact on the dynamics Such is the case inside cells where it is estimated that up to about 40% of the cell’s volume is occupied by various macromolecules, a condition termed macromolecular crowding [10,11].
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