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Continuum mathematical models for collective cell motion normally involvereaction-diffusion equations, such as the Fisher-KPP equation, with a lineardiffusion term to describe cell motility and a logistic term to describe cellproliferation. While the Fisher-KPP equation and its generalisations arecommonplace, a significant drawback for this family of models is that they arenot able to capture the moving fronts that arise in cell invasion applicationssuch as wound healing and tumour growth. An alternative, less common, approachis to include nonlinear degenerate diffusion in the models, such as in thePorous-Fisher equation, since solutions to the corresponding equations havecompact support and therefore explicitly allow for moving fronts. We considerhere a hole-closing problem for the Porous-Fisher equation whereby there isinitially a simply connected region (the hole) with a nonzero populationoutside of the hole and a zero population inside. We outline how self-similarsolutions (of the second kind) describe both circular and non-circular frontsin the hole-closing limit. Further, we present new experimental and theoreticalevidence to support the use of nonlinear degenerate diffusion in models forcollective cell motion. Our methodology involves setting up a 2D wound healingassay that has the geometry of a hole-closing problem, with cells initiallyseeded outside of a hole that closes as cells migrate and proliferate. For aparticular class of fibroblast cells, the aspect ratio of an initiallyrectangular wound increases in time, so the wound becomes longer and thinner asit closes; our theoretical analysis shows that this behaviour is consistentwith nonlinear degenerate diffusion but is not able to be captured withcommonly used linear diffusion. This work is important because it provides aclear test for degenerate diffusion over linear diffusion in cell lines.

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