Coarse-graining Active Tension Nets using discrete conformal geometry

TL;DR

This paper develops a geometric theory linking microscopic tensions in epithelial tissues to macroscopic shape and stress, using discrete conformal geometry and Thurston's Circle Packing.

Contribution

It introduces a novel geometric framework for understanding tissue mechanics that accounts for active tensions and cell rearrangements without fixed reference shapes.

Findings

Identifies a link between microscopic tensions and macroscopic stress in tissues.

Provides a geometric method to predict tissue shape changes from tension variations.

Unifies elasticity and plasticity in epithelial tissue modeling.

Abstract

Connecting cell behavior to tissue shape and mechanics is a fundamental challenge in the physics of morphogenesis. In epithelia, cell-cell interfaces are under internal active tension, and cytoskeletal turnover precludes a fixed "reference shape" that could anchor conventional elastic theory. This "tension-first" setting calls for a different approach. Here, we develop a geometric theory for epithelia in quasi-static force balance. Cell interfaces under prescribed tension act as force dipoles whose embedding in physical space is constrained by force balance. To solve this constraint problem geometrically, we represent the tensions as a triangulation dual to the cell tiling. For a given tension triangulation, there is a manifold of balanced states with different macroscopic shapes and stresses, parametrized by two types of geometric modes. Curl-free modes determine macroscopic stress, whereas conformal modes determine intracellular pressure. This identifies a non-trivial link between microscopic tensions and macroscopic stress with experimentally testable consequences. Balancing pressure and tensile stress, together with boundary conditions, selects the physically realized configuration and gives rise to an emergent elastic response. Adiabatic changes of tensions cause changes of the physical configuration, explaining how local forces under cellular control define tissue shape. Our geometric framework builds on the mathematics of discrete differential geometry. Remarkably, representing the adjacency relations of cells via Thurston's Circle Packing accounts for cell rearrangements within the same mathematical framework. Thus, one arrives at a unified description of emergent elasticity of epithelial tissues and their plastic morphing due to adiabatic tension dynamics and cell rearrangements.