Anomalous elasticity of cellular tissue vertex model

Abstract

Vertex Models, as used to describe cellular tissue, have an energy controlled by deviations from both a target area and a target perimeter. The constrained nonlinear relation between area and perimeter, as well as subtleties in selecting the appropriate reference state, lead to a host of interesting mechanical responses. Here we provide a mean-field treatment of a highly simplified model: a network of regular polygons with no topological rearrangements. Since all polygons deform in the same way we need only analyze the ground states and the response to deformations of a single polygon (cell). The model exhibits the known transition between a fluid/compatible state, where the cell can accommodate both target area and perimeter, and a rigid/incompatible state. The fluid state has a manifold of degenerate zero energy states. The rigid state has a single gapped ground state. We show that linear elasticity fails to describe the response of the vertex model to even infinitesimal deformations in both regimes and that the response depends on the precise deformation protocol. We give a pictorial representation in configuration space that reveals that the complex elastic response of the Vertex Model arises from the presence of two distinct sets of reference configurations (associated with target area and target perimeter) that may be either compatible or incompatible, as well as from the underconstrained nature of the model energy. An important result of our work is that the elasticity of the Vertex Model cannot be captured by a Taylor expansion of the energy for small strains, as Hessian and higher order gradients are ill-defined.

Shear-Driven Solidification and Nonlinear Elasticity in Epithelial Tissues

“…We find that nonlinear elasticity becomes increasingly dominant closer to the critical point, where the mechanical response is completely nonlinear. This intrinsic critical nonlinearity was also demonstrated in recent work on a vertex models of regular polygons, where it was shown to arise from purely geometric constraints [66]. While Ref.…”

“…[66] focused on the response to infinitesimal perturbations, demonstrating the failure of linear elasticity, here we examine the nonlinear response in the presence of topological rearrangements that mediate plasticity. We additionally extend the mean-field (MF) formulation of [66] to account for the emergence of shear-induced rigidity in the liquid state. The MF predicts exactly the nonlinear response and stress-stiffening exponents observed in the simulations.…”

“…Mean-field model of a sheared tissue.-To gain a theoretical understanding of the strain-driven rigidity and emergence of nonlinear elasticity, we examine a mean-field theory (MFT) formulation of the vertex model [66,86,100]. Neglecting cell-cell correlations, we consider the shear deformation of a single n-sided polygonal cell.…”

Substrate disorder promotes cell motility in confluent tissues

“…Despite the broad range of physico-chemical processes, which in many cases are system dependent [2], there are convincing arguments that simple, mechanistic models can provide valuable insight into the dynamics of living systems [30][31][32][33]. Several models have been proposed to understand their collective behavior, from single particle descriptions to density field models [30,31,[33][34][35][36][37][38]. The Self-Propelled Voronoi model (SPV) has been one of the models of choice to study confluent tissues [39,40].…”

Yield Stress and Compliance in Active Cell Monolayers

“…An important question is whether the solidto-liquid transition has an impact in tissue function in health and disease. Theory and experiments have begun to address this issue by exploring the rheological and mechanical properties of biological tissues, which have key consequences to their macroscopic biophysical behavior [13,[22][23][24][25][26][27][28][29]. A mechanistic and quantitative understanding of the impact of cell surface tension and cell motility on the rheology and transport properties of biophysical tissues is, however, still lacking.…”