UBC Theses and Dissertations
From signalling to cell behaviour : modelling multi-scale organization in single and collective cellular systems Zmurchok, Cole
Individually and collectively, cells are organized systems with many interacting parts. Mathematical models allow us to infer behaviour at one level of organization from information at another level. In this thesis, I explore two biological questions that are answered through the development of new mathematical approaches and novel models. (1) Molecular motors are responsible for transporting material along molecular tracks (microtubules) in cells. Typically, transport is described by a system of reaction-advection-diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the PDE system. To understand how nonlinear reactions affect the overall transport process at the cellular level, I extend the QSS approach to certain nonlinear reaction models, reducing the full PDE system to a single nonlinear PDE. I find that the approximating PDE is a conservation law for the total density of motors within the cell, with effective diffusion and velocity that depend nonlinearly on the motor densities and model parameters. Cell-scale predictions about the organization and distribution of motors can be drawn from these effective parameters. (2) Rho GTPases are a family of protein regulators that modulate cell shape and forces exerted by cells. Meanwhile, cells sense forces such as tension. The implications of this two-way feedback on cell behaviour is of interest to biologists. I explore this question by developing a simple mathematical model for GTPase signalling and cell mechanics. The model explains a spectrum of behaviours, including relaxed or contracted cells and cells that oscillate between these extremes. Through bifurcation analysis, I find that changes in single cell behaviour can be explained by the strength of feedback from tension to signalling. When such model cells are connected to one another in a row or in a 2D sheet, waves of contraction/relaxation propagate through the tissue. Model predictions are qualitatively consistent with developmental-biology observations such as the volume fluctuations in a cellular monolayer. The model suggests a mechanism for the organization of tissue-scale behaviours from signalling and mechanics, which could be extended to specific experimental systems.
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