A Graph-Based Approach for the Approximate Solution of the Chemical Master Equation

The chemical master equation (CME) represents the accepted stochastic description of chemical reaction kinetics in mesoscopic systems. As its exact solution—which gives the corresponding probability density function—is possible only in very simple cases; there is a clear need for approximation techniques. Here, we propose a novel perturbative three-step approach, which draws heavily on graph theory: (i) we expand the eigenvalues of the transition state matrix in the CME as a series in a nondimensional parameter that depends on the reaction rates and the reaction volume; (ii) we derive an analogous series for the corresponding eigenvectors via a graph-based algorithm; (iii) we combine the resulting expansions into an approximate solution to the CME. We illustrate our approach by applying it to a reversible dimerization reaction; then we formulate a set of conditions, which ensure its applicability to more general reaction networks, and we verify those conditions for two common catalytic mechanisms. Comparing our results with the linear-noise approximation (LNA), we find that our methodology is consistently more accurate for sufficiently small values of the nondimensional parameter. This superior accuracy is particularly evident in scenarios characterized by small molecule numbers, which are typical of conditions inside biological cells.