Complex balanced distributions for chemical reaction networks

Reaction networks have become a major modelling framework in the biological sciences from epidemiology and population biology to genetics and cellular biology. In recent years, much progress has been made on stochastic reaction networks (SRNs) that in particular are applicable when copy numbers are low (as in cellular systems) or when noise and drift are ubiquitous (as in population genetic models). Often a main interest is the long term behaviour of a system: does the system settle towards stochastic equilibrium in some sense? In that context, we are concerned with SRNs, modelled as continuous time Markov chains (CTMCs) and their stationary distributions. In particular, we are interested in complex balanced stationary distributions, where the probability flow out of a complex equals the flow into the complex. We characterise the existence and the form of complex balanced distributions of SRNs with arbitrary transition functions through conditions on the cycles of a corresponding reaction graph (a digraph). Furthermore, we give a sufficient condition for the existence of a complex balanced distribution and give precise conditions for when it is also necessary. The sufficient condition is also necessary for mass-action kinetics (and certain generalisations of that) or if the connected components of the digraph are cycles. Moreover, we state a deficiency theorem, a generalisation of the deficiency theorem for stochastic mass-action kinetics to arbitrary stochastic genetics. The theorem gives the co-dimension of the parameter space for which a complex balanced distribution exists.
Comment: 30 pages, 2 figures