The stochastic dynamics of systems in switching environments

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Abstract

No physical or biological system can fully be decoupled from its surroundings; the effect of the environment on such a system therefore needs to be understood. In this thesis, we present a number of methods to understand the dynamics of stochastic systems coupled to a time-varying environment. We focus primarily on the case of an environment which randomly switches between conditions, however deterministic environments are also considered. For all our methods and applications, the starting point is an individual-based model: a microscopic description of the system. The first model-reduction method we develop considers an approximation to the dynamics in the limit of a large system. As the system's size approaches infinity, its dynamics can be approximated by a piecewise-deterministic Markov process (PDMP), where the dynamics are characterised by deterministic motion in between random switches of the environment; this approximation neglects the effects of demographic noise. We go beyond this approximation and explicitly include the effects of demographic stochasticity, resulting in a description of the system as a stochastic differential equation with switching. We derive an expression for the stationary distribution for certain cases, and show how this method leads to strong agreement with simulations. The second method considers the dynamics of a system in an environment when there is a large separation between systemic and environmental time scales. In the limit in which environment's time scale is infinitely faster than the system's---the adiabatic limit---the environmental dynamics can be eliminated. For fast, but finite, environments we show how reduced master equations can be derived beyond this adiabatic limit. These are characterised by bursting events not found in the original master equation. The above two methods can be combined to consider scenarios with both a large system and fast environmental switching. This results in a range of different approximation schemes valid in different situations. New methods are also developed for the approximation of first-passage times, subject to a deterministic environment. Applications of these methods to models in biology, medicine, and otherwise, are explored. One focus is on bet-hedging strategies in populations of cells: in the face of uncertain environmental conditions, cells are understood to switch between phenotypes with different growth properties. We present a microscopic model of such a system, and use the PDMP to derive analytical expressions for the mean instantaneous growth rates and thereby study bet-hedging. Another study considers a large genetic network, representing an embryonic stem cell. We derive a simplified model, where the system is a high-dimensional population of gene-products, and we show how the PDMP can be used as an efficient method of simulation. We also consider applications to normal tissue complications, which are caused by damage to normal cells in the radiotherapy of neighbouring cancer cells. Our approaches provide an approximation to the first and second moments of normal tissue complication probabilities in the limit of a large, but finite, population of cells. Other applications considered include a bi-stable genetic circuit and crack propagation.

Keyword(s)

Model reduction methods; NTCP; PDMP; bet hedging; cancer; demographic noise; extrinsic noise; first-passage time; gene switching; genetic networks; intrinsic noise; linear noise approximation; nonlinear dynamics; phenotypic switching; statistical physics; switching environments; switching noise; system-size expansion

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