  The chemical master equation (CME) is frequently used in systems biology to
quantify the effects of stochastic fluctuations that arise due to biomolecular
species with low copy numbers. The CME is a system of ordinary differential
equations that describes the evolution of probability density for each
population vector in the state-space of the stochastic reaction dynamics. For
many examples of interest, this state-space is infinite, making it difficult to
obtain exact solutions of the CME. To deal with this problem, the Finite State
Projection (FSP) algorithm was developed by Munsky and Khammash (Jour. Chem.
Phys. 2006), to provide approximate solutions to the CME by truncating the
state-space. The FSP works well for finite time-periods but it cannot be used
for estimating the stationary solutions of CMEs, which are often of interest in
systems biology. The aim of this paper is to develop a version of FSP which we
refer to as the stationary FSP (sFSP) that allows one to obtain accurate
approximations of the stationary solutions of a CME by solving a finite
linear-algebraic system that yields the stationary distribution of a
continuous-time Markov chain over the truncated state-space. We derive bounds
for the approximation error incurred by sFSP and we establish that under
certain stability conditions, these errors can be made arbitrarily small by
appropriately expanding the truncated state-space. We provide several examples
to illustrate our sFSP method and demonstrate its efficiency in estimating the
stationary distributions. In particular, we show that using a quantised tensor
train (QTT) implementation of our sFSP method, problems admitting more than 100
million states can be efficiently solved.
