Approximation and Model Reduction for the Stochastic Kinetics of Reaction Networks

Abstract

The mathematical modeling of the dynamics of cellular processes is a central part of systems biology. It has been realized that noise plays an important role in the behavior of these processes. This includes not only intrinsic noise, due to random molecular events within the cell, but also extrinsic noise, due to the varying environment of a cellular (sub-)system. These environmental effects and their influence on the system of interest have to be taken into account in a mathematical model. The thesis at hand deals with the (exact or approximate) reduced or marginal description of cellular subsystems when the environment of the subsystem is of no interest, and also with the approximate solution of the forward problem for biomolecular reaction networks in general. These topics are investigated across the hierarchy of possible models for reaction networks, from continuous-time Markov chains to stochastic differential equations to ordinary differential equation models. The first contribution is the derivation of moment closure approximations via a variational approach. The resulting viewpoint sheds light on the problems usually associated with moment closure, and allows one to correct some of them. The full probability distributions obtained from the variational approach are used to find approximate descriptions of heterogeneous rate equations with log-normally distributed extrinsic noise. The variational method is also extended to the approximation of multi-time joint distributions. Finally, the general form of moment equations and cumulant equations for mass-action kinetics is derived in the form of a diagrammatic technique. The second contribution is the investigation of the use of the Nakajima-Zwanzig-Mori projection operator formalism for the treatment of heterogeneous kinetics. Cumulant expansions in terms of partial cumulants are used to obtain approximate convolutional forward equations for the process of interest, with the heterogeneous reaction rates or the environment marginalized out. The performance of the approximation is investigated numerically for simple linear networks. Finally, extending previous work, a marginal description of the subsystem of interest on the process level, for fully bi-directionally coupled reaction networks, is obtained by means of stochastic filtering equations in combination with entropic matching. The resulting approximation is interpreted as an orthogonal projection of the full joint master equation, making it conceptually similar to the projection operator formalism. For mass-action kinetics, a product-Poisson ansatz for the filtering distribution leads to the simplest possible marginal process description, which is investigated analytically and numerically.