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

TL;DR

This paper introduces a one-dimensional discrete model for cell movement that includes adhesion and chemotaxis, analyzes its continuum limit, and explores solution behaviors including oscillations, plateaus, and singular problems.

Contribution

It presents a novel discrete model incorporating cell adhesion and chemotaxis, analyzes its continuum limit, and addresses ill-posed cases with a Stefan-problem formulation.

Findings

Continuum limit leads to a nonlinear Keller-Segel type equation with potential negative diffusivity.

Numerical solutions show spatial oscillations and plateaus due to ill-posedness.

Stefan-problem approach effectively models high-adhesion plateaus and matches discrete model results.

Abstract

We develop and analyse a discrete, one-dimensional model of cell motility which incorporates the effects of volume filling, cell-to-cell adhesion and chemotaxis. The formal continuum limit of the model is a nonlinear generalisation of the parabolic-elliptic Keller-Segel equations, with a diffusivity which can become negative if the adhesion coefficient is large. The consequent ill-posedness results in the appearance of spatial oscillations and the development of plateaus in numerical solutions of the underlying discrete model. A global-existence result is obtained for the continuum equations in the case of favourable parameter values and data, and a steady-state analysis which, amongst other things, accounts for high-adhesion plateaus is carried out. For ill-posed cases, a singular Stefan-problem formulation of the continuum limit is written down and solved numerically, and the numerical solutions are compared with those of the original discrete model