Approximation methods for stochastic systems biology

Abstract

Biochemical reactions involved in complex cellular mechanisms are driven by inherently stochastic molecular interactions. Although the intrinsic noise is often negligible in the macroscopic world, it has been established experimentally that intracellular processes can be subject to substantial stochasticity due to a low number of molecules present. Therefore, modelling the dynamics of such biological systems necessitates the use of stochastic rather than deterministic methods.

The Chemical Master Equation (CME) gives an accurate mathematical description of stochastic chemical reaction kinetics in well-mixed conditions. However, analytical solutions to the CME are available only for a handful of biologically relevant systems and its exact stochastic simulation with Monte Carlo methods can be prohibitively computationally expensive. This in turn motivates the development of approximation methods that provide more e cient ways of investigating the system behaviour. The study and the development of novel analytical and computational approximations to the CME is the focus of this thesis.

First, we develop an approximate time-dependent closed-form solution to the CME describing the Michaelis-Menten reaction mechanism of enzyme catalysis. The derivation is based on a time scale separation technique called averaging, allowing us to treat the Markovian dynamics on the slower time scale as a one-dimensional master equation that can be solved exactly in time using methods from linear algebra and complex analysis.

Second, we introduce MomentClosure.jl, a Julia package for automated derivation of the moment equations applicable to any biochemical system. As the moment expansion of the CME can lead to an in nite hierarchy of coupled moment equations, MomentClosure implements a wide array of moment closure methods that truncate the moment hierarchy and provide a closed set of equations describing approximate moment dynamics. The package integrates seamlessly with other Julia libraries and makes moment closure approximations more accessible to the broader scienti c community.

Lastly, we propose a surrogate modelling framework that allows us to approximate the solution of the CME by training neural networks on stochastic simulation data. We showcase our approach on several models of gene expression, nding that relatively simple neural networks can learn to approximate highly complex distributions of molecule numbers over time and parameter space, and hence greatly accelerate otherwise computationally expensive parameter exploration and inference studies.