# Harmonic fields and the mechanical response of a cellular monolayer to ablation

## TL;DR

The paper introduces a mathematical framework to study how a cellular layer responds mechanically to ablation, revealing long-range effects and scaling patterns.

## Contribution

A novel discrete exterior calculus framework is developed to analyze mechanical responses in disordered cellular monolayers.

## Key findings

- Harmonic vector fields in ablated monolayers show an approximate 1/r scaling of amplitude with distance from ablation.
- Perturbation displacements exhibit long-range coherence and monopolar/quadrupolar features.
- Stress amplitudes scale approximately like 1/r², suggesting relevance to long-range mechanical signaling.

## Abstract

Multicellular tissues, such as the epithelium coating a developing embryo, often combine complex tissue shapes with heterogeneity in the spatial arrangement of individual cells. Discrete approximations, such as the cell vertex model, can accommodate these geometric features, but techniques for analysis of such models are underdeveloped. Here, we express differential operators defined on a network representing a monolayer of confluent cells in a framework inspired by discrete exterior calculus, considering scalar fields defined over cell vertices and centres and vector fields defined over cell edges. We achieve this by defining Hodge stars, wedge products and musical isomorphisms that are appropriate for a disordered monolayer for which cell edges and links between cell centres are not orthogonal, as is generic for epithelia. We use this framework to evaluate the harmonic vector field arising in an ablated planar monolayer, demonstrating an approximate 1/r scaling of the upper bound of the field’s amplitude, where r is the distance from the ablation. Using a vertex model that incorporates osmotic effects, we then calculate the mechanical response of a monolayer in a jammed state to ablation. Perturbation displacements exhibit long-range coherence, monopolar and quadrupolar features, and an approximate 1/r near-hole upper-bound scaling, implicating the harmonic field. The upper bounds on perturbation stress amplitudes scale approximately like \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$1/\textit{r}^2$$\end{document}1/r2, a feature relevant to long-range mechanical signalling.