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Stochastic Chemical Reaction Networks are continuous time Markov chain modelsthat describe the time evolution of the molecular counts of species interactingstochastically via discrete reactions. Such models are ubiquitous in systemsand synthetic biology, but often have a large or infinite number of states, andthus are not amenable to computation and analysis. Due to this, approximationsthat 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 NoiseApproximation. For finite time intervals, Kurtz established the validity of theReaction Rate Equations and Linear Noise Approximation, by proving law of largenumbers and central limit theorem results respectively. However, the analogousquestion for the stationary distribution of the Markov chain model has remainedmostly unanswered, except for chemical reaction networks with specialstructures or bounded molecular counts. In this work, we use Stein's Method toobtain sufficient conditions for the stationary distribution of anappropriately scaled Stochastic Chemical Reaction Network to converge to theLinear Noise Approximation as the system size goes to infinity. Our resultsgive non asymptotic bounds on the error in the Reaction Rate Equations and inthe Linear Noise Approximation as applied to the stationary distribution. As aspecial case, we give conditions under which the global exponential stabilityof an equilibrium point of the Reaction Rate Equations is sufficient to obtainour error bounds, thus permitting one to obtain conclusions about the Markovchain by analyzing the deterministic Reaction Rate Equations.

A Stein's Method Approach to the Linear Noise Approximation for Stationary Distributions of Chemical Reaction Networks