Controlling tissue size by active fracture

Groups of cells, including clusters of cancerous cells, multicellular organisms, and developing organs, may both grow and break apart. What physical factors control these fractures? In these processes, what sets the eventual size of clusters? We first develop a one-dimensional framework for understanding cell clusters that can fragment due to cell motility using an active particle model. We compute analytically how the break rate of cell-cell junctions depends on cell speed, cell persistence, and cell-cell junction properties. Next, we find the cluster size distributions, which differ depending on whether all cells can divide or only the cells on the edge of the cluster divide. Cluster size distributions depend solely on the ratio of the break rate to the growth rate - allowing us to predict how cluster size and variability depend on cell motility and cell-cell mechanics. Our results suggest that organisms can achieve better size control when cell division is restricted to the cluster boundaries or when fracture can be localized to the cluster center. Additionally, we derive a universal survival probability for an intact cluster $S(t)=\mathrm{e}^{-k_d t}$ at steady state if all cells can divide, which is independent of the rupture kinetics and depends solely on the cell division rate $k_d$. Finally, we further corroborate the one-dimensional analytics with two-dimensional simulations, finding quantitative agreement with some - but not all - elements of the theory across a wide range of cell motility. Our results link the general physics problem of a collective active escape over a barrier to size control, providing a quantitative measure of how motility can regulate organ or organism size.