(589d) Zero-Information Moment Closure: Novel Stochastic Analysis for Chemical Simulation

Using mathematical modeling to design biological systems with a desired function remains a challenging goal of biological engineering. One major hurdle is that as the molecular count of chemical species in biological systems can be in the single digits, stochastic simulation is necessary to capture system dynamics that are dominated by “noise”. Traditionally, stochastic simulations are solved using kinetic Monte Carlo (kMC) algorithms and have become more feasible in the recent past with the development of high-performance algorithms, but issues remain. It is still practically impossible to quickly determine steady-state solutions or perform a priori analysis, valuable tools in non-linear deterministic models.

Deterministic analysis of stochastic systems may be possible if one considers the moments (the mean, variance, etc.) of the underlying distribution. While individual trajectories of the kMC simulations are not deterministic, the distribution moments are deterministic. The equations that describe the dynamics of the moments are simple, even for complex reaction networks. Unfortunately, for any reaction network containing 2^nd-order reactions the moment equations are open, the lower-order moment dynamics depends on higher-order moments. The moment-closure problem has prevented the widespread use of moment-closure techniques for decades.

Our work concerns a novel closure scheme that we call Zero-Information Closure. The goal of a closure scheme is to find a relation (analytical or numerical approximation) between the unknown higher-order moments and lower-order moments such that the moment equations are closed and can be solved. In Zero-Information Closure this relationship is based on uncertainty maximization through Shannon’s entropy. Zero-Information Closure assumes all higher-order moments beyond a certain point contain no relevant information about the underlying distribution. Zero-Information closure demonstrates universality and accuracy for arbitrary chemical systems.

To show a proof of concept for Zero-Information Closure, results for three systems are presented. The three systems cover both single and multi-species chemical systems along with demonstrating increasing accuracy at progressively higher closure-order. Even for complex (bimodal) distributions the closure scheme remains valid. Dynamic trajectories and de novo steady-state results are provided to show Zero-Information closure’s potential power in analyzing chemical networks.

Moment-closure opens up a variety of new analytical tools as stochastic systems are able to become a non-linear, but deterministic, set of equations. Zero-Information Closure represents perhaps the first closure scheme to remain theoretically applicable at any closure-order for arbitrary chemical networks.