Bounding Transient Moments of Stochastic Chemical Reactions

The predictive ability of stochastic chemical reactions is currently limited by the lack of closed form solutions to the governing chemical master equation. To overcome this limitation, this letter proposes a computational method capable of predicting mathematically rigorous upper and lower bounds of transient moments for reactions governed by the law of mass action. We first derive an equation that transient moments must satisfy based on the moment equation. Although this equation is underdetermined, we introduce a set of semidefinite constraints known as moment condition to narrow the feasible set of the variables in the equation. Using these conditions, we formulate a semidefinite program that efficiently and rigorously computes the bounds of transient moment dynamics. The proposed method is demonstrated with illustrative numerical examples and is compared with related works to discuss advantages and limitations.