(191dk) Maximum Entropy Prediction of Distributions for Stochastic Biochemical Reaction Networks with Oscillatory Dynamics

Ever since the Belousov-Zhabotinsky discovery, and with the molecular revolution in biology, an increasing number of experimental papers have been published on a variety of biological systems that exhibit oscillatory behavior (Novák and Tyson, 2008). In systems such as regulatory gene networks, cell cycle, and enzymatic processes, the number of molecules involved is often far from the thermodynamic limit. Therefore, the internal noise or the inherent stochasticity for such systems has to be taken into account while modeling these reactions (Gillespie, 2007). 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.

An alternative approach through a series of mathematical transformations is to rewrite the CME in terms of probability moments (Smadbeck and Kaznessis, 2013). Since the dynamics of the lower order moments explicitly depends on the higher ones, the set of moment equations for nonlinear systems is underspecified. In this presentation, we show that the infinite set of stationary moment equations describing stochastic reactions networks with oscillatory behavior can be truncated and solved using maximization of the entropy of the distributions. 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 (Constantino et al., 2016).

We explore the Brusselator and Schnakenberg models as study cases. These are hypothetical sets of chemical reactions that can produce limit cycle oscillations and provide a qualitative description of biochemical oscillators. The results from our numerical experiments compare with the distributions obtained from well-established kinetic Monte Carlo methods, such as the Stochastic Simulations Algorithm (SSA). They suggest that the accuracy of the prediction increases exponentially with the closure order chosen for the system. Hence, 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.

References

Constantino, P.H., Vlysidis, M., Smadbeck, P., Kaznessis, Y.N. (2016) Modeling stochasticity in biochemical reaction networks. J. Phys. D: Appl. Phys. 49 093001; DOI: 10.1088/0022-3727/49/9/093001

Gillespie, D. T. (2007) Stochastic Simulation of Chemical Kinetics, Ann. Rev. Phys. Chem. 58:35-55; DOI: 10.1146/annurev.physchem.58.032806.104637

Novák, B., Tyson, J.J. (2008) Design Principles of Biochemical Oscillators. Nature Reviews Molecular Cell Biology, 9, 981-991; DOI: 10.1038/nrm2530

Smadbeck, P., Kaznessis, Y. N. (2013) A closure scheme for chemical master equations. PNAS, 110, 35:14261-65; DOI: 10.1073/pnas.1306481110