Bounds on stochastic chemical kinetic systems at steady state

Abstract

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 molecular count for each species. However, these approximations have the dissatisfying feature that they come with no error bounds. This paper presents 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 this quantity by solving semidefinite programs. 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. In this paper, the bounded quantities of interest are the mean molecular count for each species, the variance in this count, and the probability that the count lies in an arbitrary interval. At present, we consider only steady-state probability distributions, intending to discuss the dynamic problem in a future publication.

Dynamic bounds on stochastic chemical kinetic systems using semidefinite programming

“…In particular, we have formulated SDPs for calculating time-varying bounds on the mean molecular count of each species in the system and the variances in these counts. This idea is an extension of the method described by several authors [8][9][10][11][12] for calculating bounds on the steady-state (i.e., stationary) distribution of a stochastic chemical kinetic system. As a proof of concept, we have demonstrated the bounding method for a simple stochastic chemical system for which analytical means and variances are available.…”

“…This last analogy may not be so obvious, but it becomes clearer if we expand Equation (33) using Equation (9). Doing so, we see that for any j ∈ NN ,…”

Bounding Transient Moments of Stochastic Chemical Reactions

“…In this section, we present a mathematical optimization problem for rigorously bounding the transient moments. Our derivation is based on the recently proposed method for computing the steady state moments [20]- [25].…”

“…In [18], a recursive algorithm was proposed to obtain bounds of moments based on concentration inequalities. More recently, a semidefinite program (SDP) [19] was formulated to compute guaranteed upper/lower bounds of steady state moments based on the moment equation [20]- [25]. These works also used the truncated moment equation used in the moment closure, but they compensated for the truncated moments based on a relaxation that a moment matrix, a matrix defined by a product of monomial vectors, is positive semidefinite.…”

“…The use of the semidefinite relaxation was motivated by its close connection with a so-called moment condition, a necessary and sufficient condition for a given sequence of real numbers to be moments of some non-negative measure (probability distribution) under some assumptions [26]. Although complete understanding the underlying mechanism requires further study, the previous works demonstrated that this approach could give surprisingly tight bounds of steady state moments only with a small number of moments [20]- [25].…”