(39g) 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