Efficient simulation techniques for biochemical reaction networks

Abstract

Discrete-state, continuous-time Markov models are becoming commonplace in the modelling of biochemical processes. The mathematical formulations that such models lead to are opaque, and, due to their complexity, are often considered analytically intractable. As such, a variety of Monte Carlo simulation algorithms have been developed to explore model dynamics empirically. Whilst well-known methods, such as the Gillespie Algorithm, can be implemented to investigate a given model, the computational demands of traditional simulation techniques remain a significant barrier to modern research. In order to further develop and explore biologically relevant stochastic models, new and efficient computational methods are required. In this thesis, high-performance simulation algorithms are developed to estimate summary statistics that characterise a chosen reaction network. The algorithms make use of variance reduction techniques, which exploit statistical properties of the model dynamics, to improve performance. The multi-level method is an example of a variance reduction technique. The method estimates summary statistics of well-mixed, spatially homogeneous models by using estimates from multiple ensembles of sample paths of different accuracies. In this thesis, the multi-level method is developed in three directions: firstly, a nuanced implementation framework is described; secondly, a reformulated method is applied to stiff reaction systems; and, finally, different approaches to variance reduction are implemented and compared. The variance reduction methods that underpin the multi-level method are then re-purposed to understand how the dynamics of a spatially-extended Markov model are affected by changes in its input parameters. By exploiting the inherent dynamics of spatially-extended models, an efficient finite difference scheme is used to estimate parametric sensitivities robustly.