Andreas Buttenschoen

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, 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. 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'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 outline some mathematical results for the non-local adhesion model. In particular, I will highlight our result on steady-states of the non-local 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).

Finally, I will discuss 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.

At the end of the talk, I will discuss possible extensions of our work, to models of cancer and evolution.