Mechanical cell interactions on curved interfaces

TL;DR

This paper introduces a mathematical model for cell mechanics on curved interfaces, revealing how curvature influences cellular stress and relaxation dynamics, with implications for understanding tissue behavior in curved geometries.

Contribution

It generalizes 1D flat epithelial models to curved geometries, deriving a continuum limit where cell density follows a diffusion equation unaffected by curvature in the relaxation process.

Findings

Curved and straight spring models converge quadratically to the diffusion limit.

In the continuum limit, tissue curvature does not influence cell relaxation or tangential stress.

Normal stress on cells depends on curvature, affecting cell sensing and response.

Abstract

We propose a simple mathematical model to describe the mechanical relaxation of cells within a curved epithelial tissue layer represented by an arbitrary curve in two-dimensional space. This model generalises previous one-dimensional models of flat epithelia to investigate the influence of curvature for mechanical relaxation. We represent the mechanics of a cell body either by straight springs, or by curved springs that follow the curve's shape. To understand the collective dynamics of the cells, we devise an appropriate continuum limit in which the number of cells and the length of the substrate are constant but the number of springs tends to infinity. In this limit, cell density is governed by a diffusion equation in arc length coordinates, where diffusion may be linear or nonlinear depending on the choice of the spring restoring force law. Our results have important implications about modelling cells on curved geometries: (i) curved and straight springs can lead to different dynamics when there is a finite number of springs, but they both converge quadratically to the dynamics governed by the diffusion equation; (ii) in the continuum limit, the curvature of the tissue does not affect the mechanical relaxation of cells within the layer nor their tangential stress; (iii) a cell's normal stress depends on curvature due to surface tension induced by the tangential forces. Normal stress enables cells to sense substrate curvature at length scales much larger than their cell body, and could induce curvature dependences in experiments.