Inferring the plasticity of epithelial tissues from their geometry

Amorphous materials exhibit complex material properties with strongly nonlinear behaviors. At low stress they behave as plastic solids and start to yield above a yield stress. A key quantity controlling plasticity 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. 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 vertex model of epithelial tissues exhibits the phenomenology of plastic amorphous systems. We then show that, in materials where the energy functional depends on topology, x is proportional to the length L of a bond that vanishes in a plastic event and P(x) is readily measurable from geometry alone. In the developing wing epithelia of the fruit fly, we find that P(L) exhibits a power law with exponents similar to those found in the vertex model in its solid phase. This suggests that this tissues exhibit plasticity and non-linear material properties emerging from collective cell behaviors. Our approach relating topology and energetics suggests a new route to outstanding questions associated with the yielding transition.