Reaction-diffusion master equation in the microscopic limit

Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice reaction-diffusion master equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. Here we give a general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model and lets us quantify this limit in two and three dimensions. In this light we review and discuss recent work in which the RDME has been modified in different ways in order to better agree with the microscale model for very small voxel sizes.

Figure 1

The expected time until the molecules are in the same voxel for the first time, ${\tau }_D$, is computed with the RDME on a Cartesian grid on a square (a) and a cube (b) with reflective boundary conditions. To the right of the vertical line we have ${\tau }_D<{\tau }_{\mathrm{micro}}$. In that region the mesoscopic reaction rate can be corrected so that the expected time until the molecules react matches the expected time in the microscale model. To the left of the vertical line we have ${\tau }_D>{\tau }_{\mathrm{micro}}$, and no such correction is possible. We have used the parameters $\rho =2\times {10}^{-9}$ m, $D={10}^{-12}{\text{m}}^2{\text{s}}^{-1}$ (3D), and $D={10}^{-14}{\text{m}}^2{\text{s}}^{-1}$ (2D).

Figure 2

Schematic representation of the RDME's behavior as a function of $h$. For $h<h^{*}$, no local correction to the conventional mesoscopic reaction rates exists for the simple problem of diffusion to a target.

Figure 3

The average mesoscopic association time using different reaction rates, compared to the average microscopic association time. The expressions for the discretization-dependent mesocopic reaction rates from [17, 18] and those obtained here all depend on the ansatz used to derive them. All expressions produce more accurate results than the conventional expression for our model problem for $h>h^{*}$.