Breakdown of the reaction-diffusion master equation with nonelementary rates

The chemical master equation (CME) is the exact mathematical formulation of chemical reactions occurring in a dilute and well-mixed volume. The reaction-diffusion master equation (RDME) is a stochastic description of reaction-diffusion processes on a spatial lattice, assuming well mixing only on the length scale of the lattice. It is clear that, for the sake of consistency, the solution of the RDME of a chemical system should converge to the solution of the CME of the same system in the limit of fast diffusion: Indeed, this has been tacitly assumed in most literature concerning the RDME. We show that, in the limit of fast diffusion, the RDME indeed converges to $a$ master equation but not necessarily the CME. We introduce a class of propensity functions, such that if the RDME has propensities exclusively of this class, then the RDME converges to the CME of the same system, whereas if the RDME has propensities not in this class, then convergence is not guaranteed. These are revealed to be elementary and nonelementary propensities, respectively. We also show that independent of the type of propensity, the RDME converges to the CME in the simultaneous limit of fast diffusion and large volumes. We illustrate our results with some simple example systems and argue that the RDME cannot generally be an accurate description of systems with nonelementary rates.

Figure 1

Steady-state solution of the CME for system (25) (blue histogram) compared with the steady-state solutions of the global molecule numbers calculated using the RDME for very slow diffusion (green dashed line) and for very fast diffusion (red dash-dotted line), all obtained with the SSA. Note that the RDME converges to the CME in the limit of fast diffusion, as predicted by our theory. The parameter values are $K=0.03,M=10$. Note that $D_0$ is varied through $D$ while all other parameters are constant.

Figure 2

(a) Steady-state solution of the CME for system (29) (blue histogram) compared with the RDME solutions for slow diffusion (green dashed line) and for fast diffusion (red dash-dotted line), all obtained with the SSA, and the exact RDME solution for infinite diffusion given by Eq. (34) (blue circles). (b) Steady-state exact solution of the CME given by Eq. (33) compared with the steady-state exact solution of the RDME for infinite diffusion given by Eq. (34) and the LNA for system (29). Parameter values are $K_1=1.15,M=10$. The nondimensional volume is $K_2=0.01$ in (a) and $K_2=200$ in (b). $D_0$ is varied through the diffusion rate $D$. Note that while the RDME does not agree with the CME in the fast diffusion limit for small volumes (a), it does agree with the CME for fast diffusion in large-enough volumes (b).