Unbuckling Mechanics of Epithelial Monolayers under Compression

When cell sheets fold during development, their apical or basal surfaces constrict and cell shapes approach the geometric singularity in which these surfaces vanish. Here, we reveal the mechanical consequences of this geometric singularity for tissue folding in a minimal vertex model of an epithelial monolayer. In simulations of the buckling of the epithelium under compression and numerical solutions of the corresponding continuum model, we discover an "unbuckling" bifurcation: at large compression, the buckling amplitude can decrease with increasing compression. By asymptotic solution of the continuum equations, we reveal that this bifurcation comes with a large stiffening of the epithelium. Our results thus provide the mechanical basis for absorption of compressive stresses by tissue folds such as the cephalic furrow during germband extension in Drosophila.