Coarse-graining active tension nets with discrete conformal geometry

TL;DR

This work develops a geometric, bottom-up coarse-graining of 2D active tissues, showing that local motor-generated tensions define a dual tension triangulation whose Voronoi dual is a macroscopically stress-free reference state. By employing discrete conformal maps, MWVTs, circle packings, and an isogonal-conformal decomposition, the authors connect microscopic tension configurations to macroscopic stress and pressure fields, revealing two soft deformation modes—isogonal (curl-free) and conformal—that govern tissue shape changes and pressure gradients. They derive a discrete-to-continuum correspondence that recovers the continuum stress–strain relations and generalizes von Neumann-type laws to generalized foams with curvature in the tension triangulation. The framework further provides a principled description of Topological T1 transitions, yielding a locally defined T1 threshold in terms of local tension-geometry (LTC) data, and connects tissue mechanics to circle-packings and Beltrami-coefficient formalisms, with potential applications to foams, granular matter, and metamaterial design.

Abstract

In contrast to inert materials, living cells use molecular motors to generate forces independently of elastic strain. How do local active forces translate into large-scale shape, and what are the mechanical properties of the resulting "living matter"? Here, we address these questions within the active tension network (ATN) model for the mechanics of 2d epithelia. We represent the configuration of active forces geometrically by a tension triangulation dual to the cell tessellation. The Voronoi dual of the tension triangulation is shown to be macroscopically stress-free -- the tensions hence define an emergent reference state for the tissue. Two soft modes, curl-free and conformal deformations, map this reference to the internally stressed, but force-balanced, physical configuration of the tissue. Conformal deformations parametrize pressure gradients, leading to a generalization of von Neumann's law for the pressure in a foam. Via finite-element-like interpolation and a notion of "discrete" conformal maps, we both construct the mechanically balanced cell tessellations exactly on the microscopic level, and pass to the continuum limit. We systematically incorporate cell rearrangement into our theory by representing cell-scale topology in terms of circle packings. The results of this bottom-up coarse-graining study match a top-down continuum analysis presented in a companion paper. Thus, the geometry of force balance bridges between micro- and macro-scales, elucidating how cell-level active forces program shape. The present formalism may be useful in the study of foams, granular matter, and metamaterials, as well as in numerical simulations.

Figures (20)

• Figure 1: Top: Laser ablations of individual junctions measure the tension $\uptau_{ij}$ on an individual cell-cell interface ($i,j$ label adjacent cells). Bottom: Laser ablations on the tissue scale measure the macroscopic stress tensor $\sigma_{ab}$ (where $a,b$ are coordinate indices).
• Figure 2: ATN geometry and notation. Red: tension triangulation with vertices ${\bm{\uptau}}_i$; Black: cell tesselation with vertices $\mathbf{r}_{ijk}$. In mechanical balance, the angles $\tilde{\gamma}_{ij}^k+\gamma_{ij}^k=\pi$. Green: Voronoi vertices are triangle circumcircle centers.
• Figure 3: Internal and boundary isogonal modes. (a) tension triangulation. (b) reference cell tessellation. (c) internal isogonal mode inflating a single cell without deforming the boundary of the cell patch. (d) global shear isogonal mode deforming the boundary. The radii of the vertex circle of the underlying decorated triangulation correspond to the values of the isogonal potential.
• Figure 4: A discrete conformal map of a polygonal cell tessellation produces a circular-arc polygonal (CAP) tiling. The discrete conformal map is parametrized by a scale factor for each cell, which is related to intracellular pressure via the Young--Laplace law.
• Figure 5: Möbius transformation of a triangle and the dual Voronoi edges (colored) and vertex $\mathbf{r}_{ijk}$. The straight Voronoi edges become circular arcs. When completed, these circles meet in $\mathbf{r}_{ijk}^*$, the image of the point at infinity. Since the three circles intersect in two points, their centers lie on a straight line.