Using Semidefinite Programming to Calculate Bounds on Stochastic Chemical Kinetic Systems

Using Semidefinite Programming to Calculate Bounds on Stochastic Chemical Kinetic Systems Garrett R. Dowdy and Paul I. Barton Process Systems Engineering Laboratory, MIT

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. 

This talk will describe a fundamentally different approach to the closure problem in stochastic chemical kinetics. Instead of making an approximation to compute a single number for the quantity of interest, we calculate mathematically rigorous bounds on the quantity by solving convex optimization problems called semidefinite programs (SDPs).  These bounds provide a check on the validity of the moment closure approximations, and are in some cases so tight that they effectively provide the desired quantity. 

The talk will outline methods for calculating bounds on each of the following, for both steady state and transient conditions:

·         The mean count of molecules of each species

·         The variance in this count

·         The probability that the count lies within a specified interval