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Bounds on stationary moments in stochastic chemical kinetics

In the stochastic formulation of chemical kinetics, the stationary moments of the population count of species can be described via a set of linear equations. However, except for some specific cases such as systems with linear reaction propensities, the moment equations are underdetermined as a lower order moment might depend upon a higher order moment. Here, we propose a method to find lower, and upper bounds on stationary moments of molecular counts in a chemical reaction system. The method exploits the fact that statistical moments of any positive-valued random variable must satisfy some constraints. Such constraints can be expressed as nonlinear inequalities on moments in terms of their lower order moments, and solving them in conjugation with the stationary moment equations results in bounds on the moments. Using two examples of biochemical systems, we illustrate that not only one obtains upper and lower bounds on a given stationary moment, but these bounds also improve as one uses more moment equations and utilizes the inequalities for the corresponding higher order moments. Our results provide avenues for development of moment approximations that provide explicit bounds on moment dynamics for systems whose dynamics are otherwise intractable.