### Talk by Badal Joshi (Cal State San Marcos)

Title: Detailed balance, Complex balance, and Product-Form Stationary Distribution in Stochastic Models of Reaction Networks

Abstract:
Biochemical reaction networks when considered with stochastic mass-action kinetics give rise to continuous-time Markov chains. We characterize the stationary distribution of such Markov chains when the reaction network has some special symmetry properties. Anderson et al. showed that if the reaction network is complex balanced then the stationary distribution is product-form Poisson. Recently Cappelletti et al. showed that if the stationary distribution of a reaction network is product-form Poisson and the reaction network is essential then the reaction network must be stochastically (as well as deterministically) complex balanced. We show that detailed balanced reaction networks (which are a subset of complex balanced reaction networks) possess a stationary distribution that is detailed balanced. Furthermore, if the stationary distribution is detailed balanced and product-form Poisson and the reaction network is essential then the reaction network must be stochastically (as well as deterministically) detailed balanced. We provide a complete equivalence between the symmetry properties of the stationary distribution and the symmetry properties of the underlying reaction network.