Inferring the flow properties of epithelial tissues from their geometry

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 sigma(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) similar to x(theta) is singular at x = 0 in the solid phase below the yield stress sigma(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 tau 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.