Model reduction, mechanistic modelling and transience in models of stochastic chemical kinetics

Abstract

Now, it is long known that gene expression and chemical kinetics are subject to random fluctuations. These lead to deviations from deterministic models that do not account for the random nature of biochemical kinetics. Successfully incorporating these stochastic dynamics is of great interest so that one can better model, and more closely understand, the intricate phenomena inherent in biological mechanisms. Many previous studies have been conducted in modelling such processes stochastically, for instance processes such as genetic autoregulation, Michaelis-Menten enzyme action and ant recruitment models. However, the majority of these studies explore only the steady state solutions of such processes while assuming mass-action kinetics, without considering: (1) extrinsic noise, (2) transience from an initial condition, or even (3) the finite, non-continuous nature of molecule or agent numbers.

This thesis focuses on the aforementioned complex systems, with an emphasis on how to use toy models in responsible and informed ways. Responsible refers to a knowledge of how good our approximations of microscopic dynamics are and their limitations: Do we understand the assumptions that commonly employed approximations rely on? Informed refers to whether a model we design is sufficiently minimal or complex to represent the underlying biochemical (or economical) kinetics: Can we use alternative models of similar simplicity (possibly mechanistically informed) to more properly capture the dynamics of the system we are attempting to model? Further issues pursued in this thesis are whether common approximative methods can be extended to effectively include details of more complex underlying dynamics, or whether we can move beyond typical steady state solutions and explore transience from an initial condition.

There are several main findings from our studies. We find that for non mass-action Hill-type propensities, often used in biochemical kinetics, that typically only assume time scale separation as the basis of approximation, that finite molecule number effects can greatly perturb their accuracy. Then, we show that the addition of non-Gaussian colored noise to biochemical rate parameters can capture intricate characteristics of gene expression that are not explicitly modelled. For common two-state gene models, we explore why they seem to be so effective at approximating gene expression, where it is known that several key rate limiting steps are ignored. Finally, we develop transient solutions to master equations describing Michaelis-Menten enzyme kinetics and ant recruitment, and we show how to extend the solutions therein to more general forms.