Dynamic Bounds on Stochastic Chemical Kinetic Systems Using Semidefinite Programming

The method of moments has been proposed as a potential means to reduce the dimensionality of the chemical master equation (CME) appearing in stochastic chemical kinetics. However, attempts to apply the method of moments to the CME usually result in the so-called closure problem. Several authors have proposed moment closure schemes, which allow them to obtain approximations of quantities of interest, such as the mean count of molecules of each species. However, these approximations have the dissatisfying feature that they come with no error bounds. Recently, a method was proposed for calculating rigorous bounds on quantities of interest for stochastic chemical kinetic systems at steady state. In particular, the method used semidefinite programming to calculate bounds on mean molecular counts, variances in these counts, and probability histograms. In this paper, we extend that idea to the associated dynamic problem – calculating rigorous time-varying bounds on means and variances.