# Topological and geometrical quantities in active cellular structures

## TL;DR

This study investigates whether equilibrium topological laws apply to active cellular structures, using large-scale simulations, and finds that these laws are valid in certain active tissue states, with differences depending on collective motion.

## Contribution

It demonstrates that well-known equilibrium topological laws also describe active cellular structures, bridging passive and active tissue mechanics.

## Key findings

- Active structures follow Lewis's and Aboav-Weaire's laws.
- Quantitative agreement with passive systems in collective motion states.
- Essential cellular properties like T1 transitions are naturally captured.

## Abstract

Topological and geometrical properties and the associated topological defects find a rapidly growing interest in studying the interplay between mechanics and the collective behavior of cells on the tissue level. We here test if well studied equilibrium laws for polydisperse passive systems such as the Lewis's and the Aboav-Weaire's law are applicable also for active cellular structures. Large scale simulations, which are based on a multi phase field active polar gel model, indicate that these active cellular structures follow these laws. If the system is in a state of collective motion also quantitative agreement with typical values for passive systems is observed. If this state has not developed quantitative differences can be found. We further compare the model with discrete modeling approaches for cellular structures and show that essential properties, such as T1 transitions and rosettes are naturally fulfilled.