Moment Closure by Entropy Maximization for Stochastic Oscillators and Sparse Control of Modular Networks

Abstract

Chemical reactions in regulatory gene networks and other biological processes often present complex nonlinear dynamics involving a small number of molecules. Population balances based on stochastic theory, such as the Chemical Master Equation (CME), along with kinetic Monte Carlo simulations constitute the most accurate approach to modeling such systems. These models, however, remain limited by the challenges of solving the CME and handling the computational cost to perform large-scale stochastic simulations. In the first part of this thesis, we will demonstrate how the maximum information entropy principle can be used as an efficient moment-based method to predict product distributions for stochastic reaction networks with oscillatory dynamics. The numerical results compare well with kinetic Monte Carlo and reach solutions significantly faster. We also perform a stability analysis of the internally disturbed Brusselator network, a benchmark oscillatory model in kinetics. Particularly, we discuss the effects of kinetic and size parameters on the dynamics of this stochastic oscillatory system with intrinsic noise. Our numerical experiments reveal that changes in kinetic parameters lead to phenomenological and dynamical Hopf bifurcations, while reduced system sizes in the oscillatory region can reverse the stochastic Hopf dynamical bifurcations at the ensemble level. The second part of the thesis will employ a sparsity-promoting optimal control framework to address the question of the evolutionary origin of modularity and organization in complex biological networks. We show that the cost of feedback channels selects organized topological structures such as modular ones in undirected networks with Laplacian dynamics as the easiest option to control. Employing genetic algorithms of network populations that use the total control cost as the fitness function for natural selection, we also show that purely blind random mutations do not create modular networks. However, mutation schemes combining up to 80% of random mutations and 20% of biased mutations to maximize the diffusion of biological information increases the average modularity of the population. We conclude that control efficiency can be an important driver of modularity in biological networks if the evolutionary process is not entirely random and gradual, as a growing number of evolutionary development biologists have recently argued.