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. This paper considers chemical reactions that start from a set of base molecular species and can allow the composition of viable compounds from these species. The reactions are assumed to result either between molecules of any two compounds or by transformation of some compounds into another one. Such reactions are of particular interest in biochemistry. Within this context, we derive a chemical master equation (CME), and then obtain its steady-state solution under the physically motivated assumption that the overall arrival rate of molecules into the system is identical to the rate at which molecules are removed from the system. This steady-state solution has the remarkable property that the joint probability distribution of the number of molecules is the product of the marginal distributions for the number of molecules of each species. The marginal probability distribution for the number of molecules of each compound is shown to follow a Poisson law.