Finding the steady-state solution of the chemical master equation

In Systems Biology, the chemical master equation (CME) is often used to understand the effects of randomness that is generated by reactions involving biomolecular species with low copy-numbers. The CME is a system of ordinary differential equations (ODEs) describing the dynamics of the probability distribution of the underlying Markov chain. For many examples, the state-space of this Markov chain is infinite, and hence the CME cannot be directly solved. In such cases, approximate solutions of the CME can be found by truncating the state-space and using the Finite State Projection (FSP) algorithm, developed by Munsky and Khammash (Jour. Chem. Phys. 2006). The FSP is only applicable for finite time-periods and it cannot estimate the stationary solutions of CME which are often of interest in biological applications. In this paper we present a version of FSP which enables accurate estimation of the stationary CME solution. We illustrate our approach using a simple example.