Experimental and theoretical studies of elastic instability in growing yeast colonies and thin sheets

Abstract

The thesis gives a comprehensive study of elastic instability in growing yeast colonies and thin sheets. The differential adhesion between cells is believed to be the major driving force behind the formation of tissues. The idea is that an aggregate of cells minimizes the overall adhesive energy between cell surfaces. We demonstrate in a model experimental system that there exist conditions where a slowly growing tissue does not minimize this adhesive energy. A mathematical model demonstrates that the instability of a spherical shape is caused by the competition between elastic and surface energies. The mechanism is similar to the Asaro-Tiller instability in prestressed solids. We also study the buckling of a highly constrained thin elastic plate under edge compression. The plate is clamped lengthwise on two edges and constrained by foam pieces along one of the shorter edges. The remaining edge is free. Applying uniform compression along the clamped edges generates a cascade of parabolic singularities. We apply the theories pioneered by Pogorelov, who showed that any zero gaussian curvature surfaces are solutions of the von Karman equations. When two such surfaces intersect, the adjoint surfaces remains a solution everywhere except at the boundary of intersection. However, for small plate thickness and the asymptotic limit, it is possible to construct a solution for the boundary. The total energy of the solution is then given as the sum of the energy of individual surfaces and the boundary energy. We demonstrate that by intersecting a cone and a cylinder the deformation of a parabolic singularity is entirely determined.