Single and two-cells shape analysis from energy functionals for three-dimensional vertex models

TL;DR

This paper investigates the shapes of single and two-cell configurations in 3D vertex models, analyzing how energy functional parameters influence cellular geometry and relating findings to capillarity theory.

Contribution

It provides analytical and numerical analysis of cell shape dependence on energy parameters in 3D vertex models, extending understanding beyond 2D models.

Findings

Analytical expressions for cell radius and contractility limits in 3D.

Aspect ratio evolution in two-cell systems with different energy terms.

Distinct configurations for quadratic surface terms compared to linear ones.

Abstract

Vertex models have been extensively used for simulating the evolution of multicellular systems, and have given rise to important global properties concerning their macroscopic rheology or jamming transitions. These models are based on the definition of an energy functional, which fully determines the cellular response and conclusions. While two-dimensional vertex models have been widely employed, three-dimensional models are far more scarce, mainly due to the large amount of configurations that they may adopt and the complex geometrical transitions they undergo. We here investigate the shape of single and two-cells configurations as a function of the energy terms, and we study the dependence of the final shape on the model parameters: namely the exponent of the term penalising cell-cell adhesion and surface contractility. In single cell analysis, we deduce analytically the radius and limit values of the contractility for linear and quadratic surface energy terms, in 2D and 3D. In two-cells systems, we deduce the evolution of the aspect ratio. While in functionals with linear surface terms yield the same aspect ratio in 2D and 3D, the configurations when using quadratic surface terms are distinct. We relate our results with well-known solutions from capillarity theory, and verify our analytical findings with a three-dimensional vertex model.