Network of interacting synthetic molecules in steady state

In this paper, we study the steady-state behaviour of a reaction network of interacting molecules using the chemical master equation (CME). The model considers a set of base species from which further compounds are created via binary reactions, as well as by monomolecular and dissipation reactions. The model includes external arrivals of molecules into the reaction volume and assumes that the reaction rates are proportional to the number of molecules of the reactants that are present. We obtain an explicit expression for the solution of the CME in equilibrium under the assumption that the system obeys a mass conservation law for the overall rate of incoming and outgoing molecules. This closed-form solution shows that the joint probability distribution of the number of molecules of each species is in ‘product form’, i.e. it is the product of the marginal distributions for the number of molecules of each species. We also show that the steady-state distribution of the number of molecules of each base and synthesized species follows a Poisson distribution.