Inferring the flow properties of epithelial tissues from their geometry

Abstract

Amorphous materials exhibit complex material properties with strongly nonlinear behaviors. Below a yield stress they behave as plastic solids, while they start to yield above a critical stress Σ _c . A key quantity controlling plasticity which is, however, hard to measure is the density P ( x ) of weak spots, where x is the additional stress required for local plastic failure. In the thermodynamic limit P ( x ) ∼ x ^θ is singular at x = 0 in the solid phase below the yield stress Σ _c . This singularity is related to the presence of system spanning avalanches of plastic events. Here we address the question if the density of weak spots and the flow properties of a material can be determined from the geometry of an amorphous structure alone. We show that a vertex model for cell packings in tissues exhibits the phenomenology of plastic amorphous systems. As the yield stress is approached from above, the strain rate vanishes and the avalanches size S and their duration τ diverge. We then show that in general, in materials where the energy functional depends on topology, the value x is proportional to the length L of a bond that vanishes in a plastic event. For this class of models P ( x ) is therefore readily measurable from geometry alone. Applying this approach to a quantification of the cell packing geometry in the developing wing epithelium of the fruit fly, we find that in this tissue P ( L ) exhibits a power law with exponents similar to those found numerically for a vertex model in its solid phase. This suggests that this tissue exhibits plasticity and non-linear material properties that emerge from collective cell behaviors and that these material properties govern developmental processes. Our approach based on the relation between topology and energetics suggests a new route to outstanding questions associated with the yielding transition.

Keywords

• tissue mechanics
• vertex model
• plasticity
• T 1 transition