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

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 two-dimensional 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, whereas standard one-dimensional experiments are unfortunately not capable of distinguishing between the two approaches.

New travelling wave solutions of the Porous–Fisher model with a moving boundary

“…For instance, the asymptotic analysis performed in this work can be extended to include higher-order terms. Another possible extension could be to consider generalizing the nonlinear degenerate diffusivity function to D(u) = u n , for some constant n > 0 [6,10,17,19,26]. We leave both extensions for future consideration.…”

Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size

“…(2) can be written as D (u ) = D u/K , and since D (0) = 0, this leads to the formation of sharp fronts [42,43], as we see in the experimental images in Figure 1. Previous studies that have compared the performance of the Porous-Fisher model to the Fisher-Kolmogorov model have often observed that the Porous-Fisher model provides a better description of various types of in vitro experiments [30,35,31].…”

“…This kind of assumption is one way to motivate the use of the Porous-Fisher model. Other arguments have been used to support the use of the Porous-Fisher model include arguments based on: (i) heuristic reasoning about the role of crowding-induced directed motion [32]; (ii) effects of cell-to-cell crowding and cell shape [34,49]; (iii) or geometric arguments relating to asymptotic observations of hole-closing near to the time of closure [31]; and (iv) heuristic arguments about the observation of sharp fronts in experimental images [27,28].…”

“…The density dependence of proliferation and diffusion in the model represents a mechanistic influence of space constraints, i.e., the availability of space for cell motion and cell division, while per-capita parameters associated with these processes represent the cell behavioural aspects. This model has been widely used to model wound healing processes in two-dimensional scratch assays [27][28][29][30][31], as well as the outward growth of initiallyconfined populations of cells [32][33][34]. Unlike other studies that connect experimental observations with the Porous-Fisher model through counting cells and constructing cell density profiles [35], here we aim to use the model in a more practical way by connecting its outputs with very simple experimental observations, such as the time to bridge.…”

Revisiting the Fisher-KPP equation to interpret the spreading-extinction dichotomy

“…Other variations include: (i) considering models with nonlinear diffusivity [28,29,30,31,32,33]; (ii) incorporating different nonlinear transport mechanisms [34,35,36]; (iii) models of multiple invading subpopulations [31,37]; and (iv) multi-dimensional models incorporating anisotropy [38]. The Fisher-KPP model gives rise to travelling wave-like solutions that do not allow the solution to go extinct.…”