Lattice models of pattern formation in bacterial dynamics
Thompson, Alasdair Graham
In this thesis I study a model of self propelled particles exhibiting run-and tumble dynamics on lattice. This non-Brownian diffusion is characterised by a random walk with a finite persistence length between changes of direction, and is inspired by the motion of bacteria such as Escherichia coli. By defining a class of models with multiple species of particle and transmutation between species we can recreate such dynamics. These models admit exact analytical results whilst also forming a counterpart to previous continuum models of run-and- tumble dynamics. I solve the externally driven non-interacting and zero-range versions of the model exactly and utilise a field theoretic approach to derive the continuum fluctuating hydrodynamics for more general interactions. I make contact with prior approaches to run-and-tumble dynamics of lattice and determine the steady state and linear stability for a class of crowding interactions, where the jump rate decreases as density increases. In addition to its interest from the perspective of nonequilibrium statistical mechanics, this lattice model constitutes an efficient tool to simulate a class of interacting run-and-tumble models relevant to bacterial motion. Pattern formation in bacterial colonies is confirmed to be able to stem solely from the interplay between a diffusivity that depends on the local bacterial density and regulated division of the cells, in particular without the need for any explicit chemotaxis. This simple and generic mechanism thus provides a null hypothesis for pattern formation in bacterial colonies which has to be falsified before appealing to more elaborate alternatives. Most of the literature on bacterial motility relies on models with instantaneous tumbles. As I show, however, the finite tumble duration can play a major role in the patterning process. Finally a connection is made to some real experimental results and the population ecology of multiple species of bacteria competing for the same resources is considered.
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