  Stochastic Chemical Reaction Networks are continuous time Markov chain models
that describe the time evolution of the molecular counts of species interacting
stochastically via discrete reactions. Such models are ubiquitous in systems
and synthetic biology, but often have a large or infinite number of states, and
thus are not amenable to computation and analysis. Due to this, approximations
that rely on the molecular counts and the volume being large are commonly used,
with the most common being the Reaction Rate Equations and the Linear Noise
Approximation. For finite time intervals, Kurtz established the validity of the
Reaction Rate Equations and Linear Noise Approximation, by proving law of large
numbers and central limit theorem results respectively. However, the analogous
question for the stationary distribution of the Markov chain model has remained
mostly unanswered, except for chemical reaction networks with special
structures or bounded molecular counts. In this work, we use Stein's Method to
obtain sufficient conditions for the stationary distribution of an
appropriately scaled Stochastic Chemical Reaction Network to converge to the
Linear Noise Approximation as the system size goes to infinity. Our results
give non asymptotic bounds on the error in the Reaction Rate Equations and in
the Linear Noise Approximation as applied to the stationary distribution. As a
special case, we give conditions under which the global exponential stability
of an equilibrium point of the Reaction Rate Equations is sufficient to obtain
our error bounds, thus permitting one to obtain conclusions about the Markov
chain by analyzing the deterministic Reaction Rate Equations.
