Aspects of mathematical biology : from self-organisation of the cytoskeleton to transport of migratory species

Abstract

This thesis spans scales of mathematical biology, from single molecules to groups of organisms. We explore questions regarding the self-organisation of the cytoskeleton and the long distance migration of animals. Though disparate at first glance, both topics revolve around transport and self-organisation of biological particles. We first model the microtubule cytoskeleton: a self-organising dynamic scaffolding along which cellular components, e.g. proteins, are transported. Its organisation is crucial for correct cellular functions; for example, maintaining the correct distribution of E-cadherin (the epithelial cell adhesion protein) along the cell boundary to ensure tissue integrity. Using stochastic simulations, genetic manipulations of the Drosophila epithelial cells and a probabilistic model we show that microtubule cytoskeleton selforganisation principally depends on cell geometry and microtubule seed density and is robust at the tissue scale. We then extend this work. Specifically, we build and explore an analytical model and perform stochastic simulations to explain microtubule self-organisation in crowded cytoplasm, i.e. containing various highly anisotropic barriers. We consider Drosophila follicular epithelium cells, which contain actin cables throughout. We find that anisotropy in the cell interior leads to a significant increase in the number of microtubules pointing in the direction of the anisotropy. This allows us to deduce the type of interaction between microtubules and actin cables. We introduce a new measure of self-organisation of microtubules, the bundling factor, and use it to explore the persistent direction of transport created by microtubule bundles. A second research topic is subsequently discussed. Many animals navigate long distances for purposes including foraging or nesting. While often mysterious, various lines of research support the idea that navigation is aided by a combination of cues whose magnitudes change with distance from the target. Motivated by agent-based simulations from a study of green sea turtle migration, we construct an abstract model for taxis-based animal navigation. We investigate the key properties of various navigating cues and their impact on animal migration, and discuss how the starting location can affect the mean first passage time of a migratory journey.