Elasticity without a reference state: continuum mechanics of active tension nets

TL;DR

This work introduces a continuum theory of active tension nets for 2D epithelia by treating cell-generated tensions as a Riemannian tension metric $oldsymbol{g}$ defined on a tension manifold. An embedding maps this manifold into physical space, and macroscopic stress is obtained from the tension metric via a matrix square root rotated by $rac{oldsymbol{\pi}}{2}$ and scaled by a reference pressure $p_0$, while intracellular pressure follows an equation of state $p=P(n)$. The statics reveal emergent elasticity with a stress-free reference state in isothermal coordinates and a biharmonic Airy-stress formulation; dynamics proceed through adiabatic Beltrami flows of the tension metric and a separate, topology-driven cell-rearrangement (T1) dynamics captured by an adjacency metric, enabling active plastic flow. The theory unifies two fundamental mechanisms—conformal (isotropic) and isogonal (deviatoric) deformations—and provides a principled framework to connect microscopic active tensions to macroscopic tissue shape, with potential applications to 2D active and granular materials.

Abstract

A constitutive relation between stress and strain relative to a reference state is the basic assumption of elasticity theory. However, in living matter, stress is governed by (motor molecule) activity rather than a constitutive law. What paradigm takes the place of elasticity in this setting? Here, we derive a continuum theory of active mechanics by taking the continuum limit of the Active Tension Network model of 2d epithelia. Instead of a reference state, we start from a prescribed active force configuration, encoded in a Riemannian "tension metric". Intuitively, one expects cells to adjust their positions to achieve force balance by rearranging local sources of active stress. More precisely, the cell positions define an embedding of the tension metric into 2d physical space, which determines the macroscopic physical stress. For free boundaries, tissue adopts a certain intrinsically defined shape, the force-balanced embedding with minimal internal stress. Boundary forces then deform this embedding. The resulting stress transformation yields an effective stress-strain relation. Key elements of elasticity hence emerge from a "stress-only" starting point, explaining how tissue shape can be adiabatically controlled by active stress during morphogenesis. Plastic behavior arises from topological cell rearrangement, which we represent by a continuous reparameterization of the tension metric, providing a principled continuum theory of emergent elasto-plastic flow. To express this physics, we use the mathematics of isothermal coordinates and quasi-conformal maps. The present theory elucidates the unconventional mechanics of living tissues and may apply to 2d active and granular materials more generally.

Figures (7)

• Figure 1: A triangulation (left) is a discrete Riemannian surface. The tension metric defines junctional tensions. An embedding of the metric into physical space is the continuum equivalent of the centroids in the cell tessellation.
• Figure 2: Left: In the continuum limit, the displacement vector between two cell centroids becomes a differential $d\mathbf{r}$. The tension metric determines the tension $d\tau$ on the interface between these cells. In isothermal coordinates, $dr= p_0^{-1} d\tau$. Right: The stress tensor $\mathbf{\sigma}$ is defined by the traction forces that act through infinitesimal virtual cuts. A cut along $d\mathbf{r}$ must yield the interfacial tension, relating tension metric and stress tensor in Eq. \ref{['eq:tension_metric_vs_stress']}.
• Figure 3: Isothermal coordinates "flatten" the tension manifold without shear. We isothermally map to the "natural domain" defined by the boundary condition $\lambda_g|_{\partial \Omega} = 1$. The quasi-conformal map $w(z,\bar{z})$ maps the reference configuration to the physical domain. It determines the macroscopic stress tensor $\bm{\sigma}$ and is subject to physical boundary conditions (gray shading illustrates a clamped boundary, red arrows illustrate a traction force along the boundary).
• Figure 4: In an hexagonal lattice, a T1 transition generates two 5-7 defect pairs. Correspondingly, a localized shear generates a curvature quadrupole in the adjacency metric $\mathbf{a}$.
• Figure 5: Illustration of the geometry of Eq. \ref{['eq:Beltrami-convective-app']}. The adjacency metric represents a triangulation where each edge has length 1, and therefore undergoes a shear deformation if adjacency is changed by an edge flip (T1 transition). The tension metric remains fixed under a T1. As a result, the tension triangle shape acquires anisotropy.