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.