Asymptotic analysis for stationary distributions of scaled reaction networks

Abstract: We study stationary distributions in the context of stochastic reaction networks. In particular, we are interested in complex balanced reaction networks and reduction of such networks by assuming a set of species (called non-interacting species) are degraded fast (and therefore essentially absent in the network), implying some reaction rates are large compared to others. Technically, we assume these reaction rates are scaled by a common parameter $N$ and let $N\to\infty$. The limiting stationary distribution as $N\to\infty$ is compared to the stationary distribution of the reduced reaction network obtained by algebraic elimination of the non-interacting species. In general, the limiting stationary distribution might differ from the stationary distribution of the reduced reaction network. We identify various sufficient conditions for when these two distributions are the same, including when the reaction network is detailed balanced and when the set of non-interacting species consists of intermediate species. In the latter case, the limiting stationary distribution essentially retains the form of the complex balanced distribution. This finding is particularly surprising given that the reduced reaction network might be non-weakly reversible and exhibit unconventional kinetics.