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Non-Local Cell Adhesion Models: Derivation, Bifurcations, and Boundary Conditions

Banff International Research Station for Mathematical Innovation and Discovery

In both normal tissue and disease states, cells interact with one another, and other tissue components using cellular adhesion proteins. These interactions are fundamental in determining tissue fates, and the outcomes of normal development, wound healing and cancer metastasis. Traditionally continuum models (PDEs) of tissues are based on purely local interactions. However, these models ignore important nonlocal effects in tissues, such as long-ranged adhesion forces between cells. For this reason, a mathematical description of cell adhesion had remained a challenge until 2006, when Armstrong et. al. proposed the use of an integro-partial differential equation (iPDE) model. The initial success of the model was the replication of the cell-sorting experiments of Steinberg (1963). Since then this approach has proven popular in applications to embryogenesis (Armstrong et. al. 2009), zebrafish development (Painter et. al. 2015), and cancer modelling (e.g. Painter et. al. 2010, Domschke et. al. 2014, Bitsouni et. al. 2018). While popular, the mathematical properties of this non-local term are not yet well understood. I will begin this talk by outlining, the first systematic derivation of non-local (iPDE) models for adhesive cell motion. The derivation relies on a framework that allows the inclusion of cell motility and the cell polarization vector in s stochastic space-jump process. The derivation's significance is that, it allows the inclusion of cell-level properties such as cell-size, cell protrusion length or adhesion molecule densities into account. In the second part, I will present the results of our study of the steady-states of a non-local adhesion model on an interval with periodic boundary conditions. The significance of the steady-states is that these are observed in experiments (e.g. cell-sorting). Combining global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the mathematical properties of the non-local term, we obtain a global bifurcation result for the branches of non-trivial solutions. Using the equationâ s symmetries the solutions of a branch are classified by the derivativeâ s number of zeros. We further show that the non-local operatorâ s properties determine whether a sub or super-critical pitchfork bifurcation occurs. Finally, I want to demonstrate how the equation's derivation from a stochastic random walk can be extended to derive different non-local adhesion operators describing cell-boundary adhesion interactions. The significance is that in the past, boundary conditions for non-local equations were avoided, because their construction is subtle. I will describe the three challenges we encountered, and their solutions.

Mathematics; Partial differential equations; Biology and other natural sciences; Mathematical biology

video/mp4

Author affiliation: University of British Columbia

2019-05-28

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70352

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/