Entropic effects in cell lineage tree packings.
Optimal packings [1, 2] of unconnected objects have been studied for centuries [3-6], but the packing principles of linked objects, such as topologically complex polymers [7, 8] or cell lineages [9, 10], are yet to be fully explored. Here, we identify and investigate a generic class of geometrically frustrated tree packing problems, arising during the initial stages of animal development when interconnected cells assemble within a convex enclosure . Using a combination of 3D imaging, computational image analysis, and mathematical modelling, we study the tree packing problem in Drosophila egg chambers, where 16 germline cells are linked by cytoplasmic bridges to form a branched tree. Our imaging data reveal non-uniformly distributed tree packings, in agreement with predictions from energy-based computations. This departure from uniformity is entropic and affects cell organization during the first stages of the animal's development. Considering mathematical models of increasing complexity, we investigate spherically confined tree packing problems on convex polyhedrons  that generalize Platonic and Archimedean solids. Our experimental and theoretical results provide a basis for understanding the principles that govern positional ordering in linked multicellular structures, with implications for tissue organization and dynamics [12, 13].