# Hole-closing model reveals exponents for nonlinear degenerate diffusivity functions in cell biology

## TL;DR

This paper investigates nonlinear degenerate diffusion models, specifically the Porous-Fisher equation, to better describe cell invasion and wound healing, demonstrating their ability to capture moving fronts and complex wound geometries.

## Contribution

It introduces a hole-closing model using the Porous-Fisher equation, providing experimental and theoretical evidence supporting nonlinear degenerate diffusion in cell migration.

## Key findings

- Nonlinear degenerate diffusion captures moving fronts in cell invasion.
- Wound aspect ratio changes are consistent with degenerate diffusion models.
- Linear diffusion fails to explain the elongation of wounds during closure.

## Abstract

Continuum mathematical models for collective cell motion normally involve reaction-diffusion equations, such as the Fisher-KPP equation, with a linear diffusion term to describe cell motility and a logistic term to describe cell proliferation. While the Fisher-KPP equation and its generalisations are commonplace, a significant drawback for this family of models is that they are not able to capture the moving fronts that arise in cell invasion applications such as wound healing and tumour growth. An alternative, less common, approach is to include nonlinear degenerate diffusion in the models, such as in the Porous-Fisher equation, since solutions to the corresponding equations have compact support and therefore explicitly allow for moving fronts. We consider here a hole-closing problem for the Porous-Fisher equation whereby there is initially a simply connected region (the hole) with a nonzero population outside of the hole and a zero population inside. We outline how self-similar solutions (of the second kind) describe both circular and non-circular fronts in the hole-closing limit. Further, we present new experimental and theoretical evidence to support the use of nonlinear degenerate diffusion in models for collective cell motion. Our methodology involves setting up a 2D wound healing assay that has the geometry of a hole-closing problem, with cells initially seeded outside of a hole that closes as cells migrate and proliferate. For a particular class of fibroblast cells, the aspect ratio of an initially rectangular wound increases in time, so the wound becomes longer and thinner as it closes; our theoretical analysis shows that this behaviour is consistent with nonlinear degenerate diffusion but is not able to be captured with commonly used linear diffusion. This work is important because it provides a clear test for degenerate diffusion over linear diffusion in cell lines.