Theory of post-jamming rigidity in feedback-regulated cellular packings

Abstract

Budding-cell packings jam before all buds are mechanically constrained, so the post-jamming state is set not by the pressure $P$ alone but also by the fraction $u$ of buds that remain unconstrained. We develop a mean-field theory in these two variables for this regime. Stress feedback suppresses growth on the loaded buds, so continued growth is redirected onto the remaining free ones. As those buds are completed they add contacts that raise the excess coordination without a comparable rise in prestress. We introduce a modified Maxwell count, a bud-depletion relation, and a flux-partition argument to predict the post-jamming coordination, the density at which the reservoir of initially free buds is exhausted, and how strong feedback can stiffen the packing while generating little internal pressure. Because the added contacts raise the rigidity while the prestress stays low, we conclude that feedback-regulated growth provides a distinct mechanism of self-rigidification.