UBC Math Bio Seminar: Sofie Verhees

As part of cell signalling, cells sense and respond to mechanical stimuli. This process is called mechanotransduction and is important in many functions of the cell, such as cell migration and tissue homeostasis. Although the biochemistry of cell signalling has been extensively studied, unraveling its connection to mechanical properties remains challenging. In this talk, I will explore the coupling between the mechanical properties of the cell and the chemical signalling.

I propose a model of the RhoA signalling pathway with a novel two-way coupling between reaction-diffusion equations and the equations of linear elasticity. The signalling molecules involved in the RhoA signalling pathway are essential in mechanotransduction as they, for example, sense the stiffness of the extracellular matrix and influence the mechanical properties of the cell. Using a numerical method based on the bulk-surface finite element method, I show simulations of the model for different couplings and cell shapes. Results suggest that cell shape and substrate stiffness have a significant effect on the dynamics of the signalling molecules, and the two-way coupling between the biochemistry and mechanical properties can give rise to mechanical homeostasis.

Secondly, I will discuss the existence of a unique solution to the proposed model and convergence of the numerical scheme. I use concepts from classical PDE analysis such as the Galerkin method, a priori estimates, and Banach's fixed point theorem to prove well-posedness. The convergence proof makes use of stability estimates, proven with a similar approach as the a priori estimates, and a decomposition of the error into the geometric error, the error due to the approximated finite element spaces and the error due to the time discretization.