  Stochastic reaction networks are a fundamental model to describe interactions
between species where random fluctuations are relevant. The master equation
provides the evolution of the probability distribution across the discrete
state space consisting of vectors of population counts for each species.
However, since its exact solution is often elusive, several analytical
approximations have been proposed. The deterministic rate equation (DRE) gives
a macroscopic approximation as a compact system of differential equations that
estimate the average populations for each species, but it may be inaccurate in
the case of nonlinear interaction dynamics. Here we propose finite state
expansion (FSE), an analytical method mediating between the microscopic and the
macroscopic interpretations of a stochastic reaction network by coupling the
master equation dynamics of a chosen subset of the discrete state space with
the mean population dynamics of the DRE. An algorithm translates a network into
an expanded one where each discrete state is represented as a further distinct
species. This translation exactly preserves the stochastic dynamics, but the
DRE of the expanded network can be interpreted as a correction to the original
one. The effectiveness of FSE is demonstrated in models that challenge
state-of-the-art techniques due to intrinsic noise, multi-scale populations,
and multi-stability.
