PIMS- UBC Math Distinguished Colloquium: Cole Zmurchok

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 talk, I will share 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). To understand how the behaviour of many molecular motors, various model parameters, and nonlinear interactions affect the overall transport process at the cellular level, I develop an asymptotic quasi-steady-state approach, 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.

(2) Protein regulators (GTPases) 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.

This is joint work with Dhananjay Bhaskar, Leah Edelstein-Keshet, Tim Small, and Michael Ward.