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Modelling spatio-angular patterns in cell biology

University of British Columbia

In this thesis I investigate the formation of spatio-angular order in populations of interacting individuals. Mathematical models are used to describe distributions of orientation, and how these evolve under different assumptions about the turning behaviour of individuals. Three types of models are introduced: in one, turning is described by a gradual direction shift (rather than abrupt transition). In two other variants, interactions between individuals cause abrupt turning and mutual displacements. Linear stability analysis and weakly non-linear (synergetics) analysis of interacting modes are used to understand the nature and stability properties of the steady state solutions. I investigate how nonhomogeneous pattern evolves close to the bifurcation point and find that individuals tend to cluster together in one direction of alignment. A special limiting case of nearly complete alignment occurs when the rotational diffusion of individual objects becomes very slow. In this case, motion of the objects is essentially deterministic, and individuals tend to gather in clusters at various orientations. To understand analytically the behaviour of the deterministic models, I represent the angular distribution as a number of 8-\ike peaks. For weak but nonzero angular diffusion, the peaks are smoothed out. The analysis of this case leads to the study of a singular perturbation problem. Next, I generalize the models to examine the dynamic behaviour of ensembles of individuals whose interactions depend on angular orientations as well as spatial positions. I show how processes such as mutual alignment, pattern formation, and aggregation can be described by sets of partial differential equations containing convolution terms. Such models appear to contain a rich diversity of possible behaviour and dynamics, depending on the details of the convolution kernels involved. The analysis of the equations, and predictions in several test cases are presented. Finally, I introduce a partial integro-differential model of aggregation phenomena. The model can be derived from a system of chemotactic partial differential equations (PDE). A linear stability analysis reveals that under certain conditions, periodic patterns may emerge. A weakly non-linear analysis predicts stripe patterns. In the limit of slow diffusion I use a Lagrangian approach to derive and analyse the gradient system and obtain periodic peak-like patterns. The possibility of existence of other types of patterns is discussed. I consider the application of the model to a phenomenon known as rippling in a swarm of social bacteria observed in Myxobacteria.

Thesis/Dissertation

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2009-04-24

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http://hdl.handle.net/2429/7546

Mathematics

University of British Columbia

UBCV

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