## Optimal packing of linked biomatter

-- This work is published in Nature Physics --

What is the optimal packing of a group of objects, say a case of oranges? Solution to the problem of sphere packing is given by Kepler’s conjecture that dates back to 17th century. A very similar way of thinking allows physicists to describe crystal structures, which is just an example of how pure mathematical modeling has various applications in physics. While today the question of optimal packing of unlinked objects is still keeping many minds busy, a recent study establishes the principles of packing of linked biological objects. Jasmin Imran Alsous from Princeton University and coworkers investigate egg development in Drosophila melanogaster (fruit fly) egg chambers by mathematical modelling of 3D images.

Biological processes are robust and incredibly faultless while occurring in countless times in biomatter through complex reactions. Optimal packing is essential in many of these processes such as DNA and protein folding. In order to understand the mechanism of how packing controls bio-reactions researchers address geometrically frustrated tree packing problem (TPP) in D. melanogaster egg chambers during oogenesis. TPP follows from topological tree-constrains leading to a cell positioning in the form of hierarchical lineage tree (evolutionary tree is an example to this kind of lineage). An egg chamber is generated by incomplete cytokinesis of germline cells such that the cells remain connected via ring canals. In simple words one can view the egg chambers as consisting of germline cells represented by spherical nodes connected via thin cylinders (ring canals) in an orange like combined geometrical shape. Individual cells and ring canals can be captured on confrontal images using fluorescent markers such that the authors are able to map the 3D images to 2D graph maps and show that each embedding of the egg chamber is uniquely determined by the terminal nodes of the tree, while the total number of planar arrangements that a cell linage tree can form is given by $$\Pi_{\rm v=1}^{\rm V}(d_{\rm v}-1)!$$ where $$d$$ is degree of the node. In the paper the 16-vertex lineage tree is investigated, for which there are 144 (72 after taking mirror symmetry into account) distinct embeddings present following from the above formula but the methodology can also be applied to any lineage tree. Mathematical and statistical analysis of graph automorphisms of 100 000 computational realizations of 72 embeddings show that certain tree states occurs more than the expected average occurrences. This nonuniformity in distributions, as observed in the measurements as well, is quantified by the authors using entropic and energetic effects by developing two models considering experimental adjacency (whether the nearest neighboring nodes are linked via ring canals). In a purely geometrical framework the tree embedding are considered on a type of convex equilateral polyhedra called Johnson solid by placing the cells (nodes) on the vertices. In fact only one Johnson solid, $$J_{90}$$, is shown to agree with the experimental adjacency. The reason of more frequent observation of some tree states is that for those tree states there are more than one way of placing the cells at the vertices of $$J_{90}$$ as calculated from statistical analysis of 72 embeddings mapped on $$J_{90}$$. The energetic arguments are introduced by the electrostatic interactions present between the spheres while the ring canals modelled as harmonic oscilators. The energy term is minimized by a Monte Carlo algorithm giving the necessary parameters.

Investigation of biological processes purely based on mathematical and biophyiscal principles provides deep understanding of living matter. The developed models in this work based on topology and cell positioning can be extended to study tree packing problems seen in other germline cells including of humans.