Multi-phase Stefan problems for a non-linear one-dimensional model of cell-to-cell adhesion and diffusion

Abstract

We consider a family of multi-phase Stefan problems for a certain 1-d model of cell-to-cell adhesion and diffusion, which takes the form of a nonlinear forward-backward parabolic equation. In each material phase the cell density stays either high or low, and phases are connected by jumps across an 'unstable' interval. We develop an existence theory for such problems which allows for the annihilation of phases and the subsequent continuation of solutions. Stability results for the long-time behaviour of solutions are also obtained, and, where necessary, the analysis is complemented by numerical simulations. *

A one-dimensional model for the interaction between cell-to-cell adhesion and chemotactic signalling

“…Next, pick an initial datum ρ 0 (x) such that avg(ρ 0 ) := 1 L L 0 ρ 0 (x)dx =ρ and max ρ 0 (x) 6 ρ δ 1 , and introduce a smooth and modified diffusivity D * (ρ), which is equal to D(ρ) for ρ 6 ρ δ 1 , and is greater than 1 2 δ 1 for ρ > ρ δ 1 . Next, pick an initial datum ρ 0 (x) such that avg(ρ 0 ) := 1 L L 0 ρ 0 (x)dx =ρ and max ρ 0 (x) 6 ρ δ 1 , and introduce a smooth and modified diffusivity D * (ρ), which is equal to D(ρ) for ρ 6 ρ δ 1 , and is greater than 1 2 δ 1 for ρ > ρ δ 1 .…”

“…Following the approach of [1], we solve the ρ-equation in each given phase by rescaling the spatial coordinate so as to fix the relevant moving boundary (or boundaries). Also, the solution of Figure 4 unfortunately gains a little mass during the course of the simulation.…”

“…Our intention in this paper is to extend the modelling and analysis of [1,2] in a rather obvious way, namely, by factoring in the directed response of cells to an extracellular chemical gradient (i.e. chemotaxis), and then examining the resulting interaction between such long-range signalling and the short-range signalling of cell-to-cell adhesion.…”

“…Finally, and in analogy with [1], we consider the idea that the Stefan-problem-type framework, in which the density is allowed to jump across the unstable region I α , might be an appropriate way of treating (1.3)-(1.6) as the limit of (1.1) in ill-posed cases. Although such problems seem to be analytically intractable at present, there is a sense in which solving them numerically may nevertheless be more efficient than discretising the Neumann problem for (1.3)-(1.6) directly, since this (the direct approach) requires a very fine mesh to properly resolve the singular behaviour observed near I α ; simulations obtained using both of these methods are compared and contrasted in Section 5.…”

Continuum and lattice models analysis of the aggregation diffusion cell movement

“…The second one is the continuous model where all variables are considered to be defined at every point in space and time changes continuously. The last one is the hybrid which is a mixture of the previous two [1], [3]. Each of these models have advantages and disadvantages depending on the phenomenon under consideration, and on the length scale over which we wish to investigate the phenomenon.…”