Emergent Elasticity and Quasiconformal Flow in Active Solids

Abstract

A constitutive relation between stress and strain relative to a reference state is the basic assumption of elasticity theory. However, in living matter, force generation is governed by motor molecule activity, which does not depend on deformation relative to a reference. A different approach is needed to describe how cells sculpt tissues through local active forces. We develop a theory of two-dimensional continuum mechanics where the active stress configuration, rather than a reference shape, is the fundamental input. Motivated by the Active Tension Network model for epithelia, we encode motor-driven forces between cells in a Riemannian tension metric. We derive a stress-metric relation for the macroscopic stress that results from embedding the tension manifold into physical space (defining cell positions). Despite the absence of constitutive laws, a stress-free reference state and an effective stress-strain relation arise from the tension metric, making the system an effectively elastic active solid. Moving from statics to dynamics, our framework describes how an active solid can morph its shape through adiabatic dynamics of active stress. To capture large, plastic deformations through cell rearrangement, we introduce a second metric that geometrizes the cell network topology. Topological rearrangement appears as a continuous reparametrization of the tension manifold. This mathematical framework, based on Riemannian geometry, isothermal coordinates, and quasi-conformal flows, quantitatively predicts how local contractile activity determines large-scale shape and provides a principled continuum description of active plasticity. A companion paper validates the continuum analysis through coarse-graining of discrete cell networks. Our theory identifies a geometric origin of emergent elasticity and plasticity in living matter and, more broadly, in active and granular materials.