Capturing Brownian dynamics with an on-lattice model of hard-sphere diffusion

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.