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.
Bibliographical noteFunding Information:
This work was supported by a grant from the National Science Foundation (CBET-1412283) and a grant from the National Institutes of Health (GM111358). This work utilized the high-performance computational resources of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575. Support from the University of Minnesota Digital Technology Center and from the Minnesota Supercomputing Institute is gratefully acknowledged. This paper was written in part while YNK was a Visiting Scholar at the Isaac Newton Institute of Mathematical Sciences at the University of Cambridge. Computational support from the Minnesota Supercomputing Institute (MSI) is gratefully acknowledged. Support from the University of Minnesota Digital Technology Center, the University of Minnesota Biotechnology Institute and CAPES ? Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior ? Brazil is also acknowledged.
- Chemical Master equation
- Maximum entropy
- Moment closure
- Oscillatory reactions