  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.
