Optimal packings1,2 of unconnected objects have been studied for centuries3,4,5,6, but the packing principles of linked objects, such as topologically complex polymers7,8 or cell lineages9,10, 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 enclosure10. 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 polyhedra11 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 dynamics12,13.
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We thank G. Laevsky for expert help with imaging, and E. Gavis, A. Spradling, T. Orr-Weaver, F. Nijhout, L. Manning, H. Mattingly, Y. H. Song, T. Stern, S. Okuda and C. Doherty for helpful discussions. This work was supported by the National Science Foundation Science and Technology Center for Emergent Behaviors of Integrated Cellular Systems CBET-0939511 (J.I.A. and S.Y.S.), the NIH R01GM107103 (S.Y.S.), the WIN programme between Princeton University and the Weizmann Institute (P.V.), a James S. McDonnell Foundation Complex Systems Scholar Award (J.D.) and an Edmund F. Kelly Research Award from the MIT Department of Mathematics (J.D.). This research was partially supported by the Allen Discovery Center programme through The Paul G. Allen Frontiers Group.
The authors declare no competing interests.
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Imran Alsous, J., Villoutreix, P., Stoop, N. et al. Entropic effects in cell lineage tree packings. Nature Phys 14, 1016–1021 (2018). https://doi.org/10.1038/s41567-018-0202-0