Maximum entropy prediction of non-equilibrium stationary distributions for stochastic reaction networks with oscillatory dynamics

Abstract

Many chemical reaction networks in biological systems present complex oscillatory dynamics. In systems such as regulatory gene networks, cell cycle, and enzymatic processes, the number of molecules involved is often far from the thermodynamic limit. Although stochastic models based on the probabilistic approach of the Chemical Master Equation (CME) have been proposed, studies in the literature have been limited by the challenges of solving the CME and the lack of computational power to perform large-scale stochastic simulations. In this paper, we show that the infinite set of stationary moment equations describing the stochastic Brusselator and Schnakenberg oscillatory reactions networks can be truncated and solved using maximization of the entropy of the distributions. The results from our numerical experiments compare with the distributions obtained from well-established kinetic Monte Carlo methods and suggest that the accuracy of the prediction increases exponentially with the closure order chosen for the system. We conclude that maximum entropy models can be used as an efficient closure scheme alternative for moment equations to predict the non-equilibrium stationary distributions of stochastic chemical reactions with oscillatory dynamics. This prediction is accomplished without any prior knowledge of the system dynamics and without imposing any biased assumptions on the mathematical relations among species involved.

Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers

“…Oscillatory behaviors are particularly challenging to capture with stochastic models [ 46 ]. We have reported earlier how methods that can capture multistable behaviors fail to appropriately capture oscillatory ones [ 39 ].…”

“…For more information on how the lower-order moments affects probability distributions the reader is directed to the literature [ 29 ]. It has been reported that the Brusselator requires more than fourth-order moments for accurate results [ 46 ].…”

On Differences between Deterministic and Stochastic Models of Chemical Reactions: Schlögl Solved with ZI-Closure

“…In this section, we briefly discuss the elements of the ZI-closure scheme. More details on the method can be found in [ 14 , 35 , 36 ].…”

“…The dependence of the lower-order vector on the higher-order one is evident in this equation. This is the closure scheme challenge, which we have previously solved by developing the ZI-closure scheme [ 14 , 35 , 36 ].…”

Data-driven, variational model reduction of high-dimensional reaction networks

“…The most widely applied reduction methods can be roughly classified as timescale exploitation approaches [23,8,78,49,77,73,83,63,95,106,88], reduction methods based on sensitivity analysis [61,21,2,93,92,61,65,41], optimization methods [65,66,35,62,1,75], and lumping-based methods [20,24,50,86]. Finally, an important class of model reduction methods are based on maximum entropy techniques for the closure of the equation that determines the time evolution of the probabilistic description of the stochastic reaction network (see for instance [16,58,31,34]). In particular, our proposed method can be considered as a combination of sensitivity and optimization-based methods, which uses information theory approaches for both; in Section 7 we discuss the relations between our method and the current literature.…”