Comparing cell polarization models using local perturbation analysis

Patterns are ubiquitous in nature, but the underlying mechanisms giving rise to them are not always understood. One such pattern is the inhomogeneous domain of a cell that is established during polarization. Cell polarization is a way in which cells respond to and interact with their environment. For example, white blood cells locate and destroy bacteria, and yeast cells create buds for reproduction. Signalling proteins such as GTPases are redistributed throughout the cell and, through downstream effects, rearrangement of the actin cytoskeleton follows. This redistribution can occur in response to a stimulus, such a chemoattractant, or it may be spontaneous. Because many biological details are unknown, mathematical models are developed to recreate features of cell polarization and determine the minimal modules or characteristics for these features. Polarization models are often simple, conceptual reaction-diffusion equations for one or more signalling molecules. But comparing these models is often difficult, and there are many models in the literature for different cell types or behaviour. A new method of nonlinear stability analysis, called local perturbation analysis (LPA), was developed by Stan Maree, and later Bill Holmes, to take advantage of models with substantial diffusion disparities. This method recapitulates the dynamics of a pulse applied to a reaction-diffusion system using a system of ordinary differential equations. Bifurcation analysis of these equations is relatively easy, and LPA detects pattern formation through threshold and Turing dynamics and provides bifurcation maps of these regimes for any parameter. LPA is well-suited to cell polarization models, because the signalling proteins we model often have both fast-diffusing inactive and slow-diffusing active forms. In this thesis, I introduce LPA and its use through a wave pinning model and its extension, a model for actin waves. I then review and analyze five additional cell polarization models using combinations of LPA, simulations, and Turing analysis. In many cases, I discovered new dynamics of the models. LPA helps us to map patterning regimes and their robustness to changes in parameters, and provides a new avenue for us to compare many current models for cell polarization.Science, Faculty ofMathematics, Department ofGraduat