Chemical Master Equations for Non-linear Stochastic Reaction Networks: Closure Schemes and Implications for Discovery in the Biological Sciences

Abstract

With the development of genome-wide sequencing, DNA synthesis technologies and the continued growth of supercomputing resources, biology has become a new focus of engineering research across the globe. Our ability to analyze gene function and develop novel synthetic biological systems has made engineered biological constructs a reality. Despite the advancement of computational resources and numerical methods biological research, however, remains the domain of experimental scientists. Novel simulation methods and theories for biological simulation are sorely needed in order to bridge the gap between the experimental and computational sides of biological engineering.One major issue facing biological simulations is that these systems experience random fluctuations that can strongly influence and drive overarching function. The importance of these random fluctuations to the accuracy of the simulation requires the use of stochastic mathematics. Instead of describing and simulating a single deterministic trajectory through time for a chemical system, a probabilistic distribution of possible states must be determined. In such systems the master equation describes, in full detail, the underlying dynamics. In practice, however, such a solution for non-linear systems has been elusive for over 50 years. From a statistical perspective what is missing is a relationship between complex sets of statistics that has remained unresolved for decades called the \textit{moment closure problem}. Solving this problem would allow for a new way to analyze and optimize stochastic simulations using deterministic numerical methods.The work presented herein focuses on the full development of a numerical solution to the moment closure problem using maximum-entropy distributions. The intentions of my work were: (1) To develop an algorithm to quickly produce moment equations that fully describe the dynamics of the chemical master equation deterministically; (2) Develop a novel moment closure method using maximum-entropy optimization to solve the master equation; (3) Demonstrate the potential of this method for performing non-linear analysis, power spectral density determination and model reduction on stochastic systems. I will demonstrate in this initial study a new method for the simulation of biological systems (and other systems with a random nature) that is entirely separate from the methods that currently dominate stochastic biological simulation.