Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers

Abstract

The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the time evolution of stochastic reaction networks. Based on the assumption that the entropy of the reaction network is maximum, Lagrange multipliers are introduced. The proposed method derives equations that model the time derivatives of these Lagrange multipliers. We present detailed steps to transform moment equations to Lagrange multiplier equations. In order to demonstrate the method, we present examples of non-linear stochastic reaction networks of varying degrees of complexity, including multistable and oscillatory systems. We find that the new approach is as accurate and significantly more efficient than Gillespie’s original exact algorithm for systems with small number of interacting species. This work is a step towards solving stochastic reaction networks accurately and efficiently.

On the Properties of the Reaction Counts Chemical Master Equation

“…Even though the dynamics of the derivative are fairly simple, solving the CME is a hard problem [5][6][7][8][9][10][11][12]. A special case of the CME which is truly unyielding to approximation is when a system is close to its population boundary (for example, close to zero).…”