## Abstract

Optimal packings^{1,2} of unconnected objects have been studied for centuries^{3,4,5,6}, 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^{10}. 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 polyhedra^{11} 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}.

## Main

Packing problems^{1} are part and parcel of life. From DNA folding in the nucleus^{14} and cell assembly during embryogenesis^{15} to crystal growth^{16,17}, densification of granular materials^{4} and masonry^{18}, the spatial packings of fundamental building blocks play an essential role in determining the structure and function of physical and biological matter. Since Kepler’s seventeenth century conjecture on dense sphere packings^{3}, a rich body of experimental and theoretical work^{19} has helped clarify the physical^{20} and mathematical^{1,2} principles underlying optimal embeddings of unconnected objects^{5,6,21}. By contrast, much less is known about the packings of topologically linked structures in confined spaces, a setting relevant to the understanding of chromosome condensation^{8}, protein folding^{22}, multi-scaffold DNA-origami structures^{7}, synthetic multicellular self-assembly^{23} and the positioning of cell lineage-trees (CLTs) in embryonic cysts^{10,24}.

Here, we combine 3D imaging of fruit fly egg chambers with mathematical modelling to investigate a generic yet previously unexplored class of geometrically frustrated tree packing problems (TPPs). This class of problems arises during oogenesis in mammals, birds, insects and other multicellular organisms^{10}, when interconnected germline cells assemble within an enclosed domain. Our analysis shows that topological and symmetry-based entropic constraints inherent to TPPs severely restrict and bias the observed packing distributions. This finding is in stark contrast to the conventional packings of unconnected objects^{1,2}, implying that topological constraints can provide an efficient biophysical mechanism for controlling cell positioning during early oogenesis^{10} and embryogenesis^{13}.

Higher life forms develop through successive symmetry breaking transitions that transform a cell cluster with little obvious structure into a highly compartmentalized functional organism. While gene expression and chemically induced morphogenesis have long been studied^{25,26}, the role that topological and geometric constraints play in determining the positioning of cells during the initial stages of development is not yet well understood. Furthermore, physical linkages between germline cells through tubular cytoplasmic bridges have been confirmed in germline tissues of various mammals, insects, amphibians (Table 1 in ref. ^{10}) and molluscs^{27}, and similar topological linkages play an important role in the colony formation of choanoflagellates, the closest living relatives of animals^{9}. Although the structure and function of cytoplasmic bridges have long been studied in the context of intercellular biochemical transport^{10,28,29}, their role for physical cell positioning, and tissue organization and dynamics has yet to be explored.

Cytoplasmic bridges often originate from incomplete cytokinesis, forming a hierarchical CLT network that encodes the history of cell divisions. In several species, including the fruit fly *Drosophila melanogaster*, these lineage trees are encapsulated into approximately spherical cysts^{10}, giving rise to a geometrically constrained TPP. A closely related packing problem with directly visible dynamical implications is encountered in the embryos of colonial *Volvox* algae^{28,30}, which are among the simplest multicellular organisms. In *Volvox*, the spatial embedding of the cytoplasmic bridge network along the surface of a fluid-filled vesicle determines the position of the phialopore, an opening at the anterior pole that defines the initiation point for the inversion of the embryos^{13}. This example demonstrates that tree embeddings can provide a robust physical symmetry breaking mechanism. It is therefore desirable to develop a better empirical and theoretical understanding of TPPs in convex enclosures. As we shall show below, the associated mathematical challenge involves solving the subgraph isomorphism problem of identifying all topology-preserving embeddings of a given planar tree in a convex polyhedron.

To investigate TPPs systematically in a biologically relevant model organism, we acquired 3D confocal images of *D. melanogaster* egg chambers, the precursors of mature oocytes (Fig. 1a). An egg chamber is a cluster of 16 germline cells (1 oocyte and 15 supporting nurse cells), generated by four synchronous divisions of a founder cell. The divisions occur with incomplete cytokinesis, so that the cells remain connected through membranous bridges called ring canals and form a hierarchical lineage tree (Fig. 1a–c). In the rounded 3D egg chamber, each of the 16 germline cells is additionally attached to an outer epithelium, a single-layered sheet of smaller somatic cells that provides a convex hull (Fig. 1a). Using 3D confocal images with fluorescently labelled nuclei, membranes and ring canals, we identified the individual cells and ring canals in *n* = 121 rounded cysts (Fig. 1d). The cells and ring canals define the nodes and edges of the CLT, respectively (Fig. 1b,c). The nodes of the lineage trees can be unambiguously labelled by a cell’s position in the division history (Fig. 1c; Supplementary Information). This convention is adopted throughout, with the oocyte cell *v* = 1 denoting the root of the tree and nurse cells *v* = 2, 3, …, 16 (Fig. 1b,c). Armed with this unique tree topology, we next focus on the question of how the lineage tree is spatially embedded in 3D.

To this end, we first generated 3D membrane-based reconstructions of individual egg chambers from our imaging data (Fig. 1d). The 3D spatial organization of the cell-tree packing resembles that of an orange, in that each germline cell contacts the outer epithelium; the germline cells correspond to the segments, and the epithelium to the outer peel. Our 3D reconstructions reveal that the topological tree-constraints impose a highly conserved layered organization of the 15 nurse cells relative to the oocyte: nurse cells separated from the oocyte by the same number of ring canals, regardless of generation, typically reside within the same layers (Fig. 1c), but a cell’s relative position within a layer is found to vary between different egg chambers. By considering the 2D planar unfolding of the 3D cell-tree packing (Supplementary Information), tree states can be distinguished and uniquely labelled by their leaf sequences (Fig. 1e). Generally, for an unfolded CLT with *v* = 1, …, *V* vertices of degree *d*_{v}, basic combinatorial considerations (Fig. 1b) show that there exist

planar embeddings up to 2D rotational symmetry, where *n*! = *n*(*n* − 1) …1 and 0! = 1. For the 16-vertex lineage tree of the egg chamber, we have *d*_{v} ∈ {4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, yielding *P* = 144 distinct embeddings. Each of these embeddings is uniquely determined by the order in which the eight leaves (terminal nodes) of the tree, given by the nodes *v* = 9, …, 16, are arranged. This allows us to assign a unique tree-state vector *e* to each embedding by starting with *v* = 16 and then listing the leaf indices in counter-clockwise order; for example, the sequence *e* = [16, 14, 9, 13, 15, 11, 10, 12] uniquely identifies the embedding realized in Fig. 1e. For each tree state *e* = [16, *a*, *b*, *c*, *d*, *e*, *f*, *g*] there exists a corresponding mirror-symmetric embedding \(\bar e\) = [16, *g*, *f*, *e*, *d*, *c*, *b*, *a*]. When embedded in 3D, *e* and \(\bar e\) can give rise to packings of opposite handedness, reflecting the fact that pure rotations and reflections belong to different branches of the special orthogonal group. In general, it is not clear a priori that *e* and \(\bar e\) are biologically equivalent. However, to reduce the complexity of the TPP, we will treat mirror-symmetric packings as mathematically equivalent in the remainder, distinguishing only the *P*′ = 144/2 = 72 embeddings *e*_{n=1,…,72} that are not related by rotations or reflections.

We first explored how frequently specific tree states *e* are realized in the egg chambers. Our *n* = 121 experimental samples suggest that certain tree configurations appear more frequently than others (Fig. 1f). To test the likelihood of observing our experimentally measured statistics under a uniform null-hypothesis, we numerically computed 100,000 realizations of 121 samples from a uniform distribution over the 72 distinct tree states. For each instance, we recorded the number of configurations observed more than *f*_{c} = 4 times (Fig. 1g) and the largest highest count (Fig. 1h), with the frequency cut-off *f*_{c} chosen to be more than twice the expected frequency \(\bar f\) = 121/72 under a uniform distribution. The resulting histograms for these two large-deviations statistics indicate that the experimentally observed data are highly atypical under the uniformity hypothesis (red bars in Fig. 1g,h). As we show below, mathematical analysis of graph automorphisms and a detailed statistical mechanics model both strongly favour the non-uniformity hypothesis.

To quantify metric properties of the 3D tree embeddings, we estimated the cell–cell adjacency probabilities from the 121 rounded cysts (Supplementary Information). Adjacent cells can be connected by ring canals (red edges in Fig. 1e) or not (grey edges in Fig. 1e). The adjacency probabilities extracted from the data confirm quantitatively the highly conserved layered structure of the 3D embedding and also reveal cell permutations within a layer across different spatial embeddings (Fig. 1i). The measured adjacency probabilities and the underlying contact graph topologies provide benchmarks for testing theoretical models.

To rationalize the experimental observations and provide a mathematical framework for convex TPPs, we analysed and compared conceptually related models of increasing complexity: first, we consider tree embeddings on convex equilateral polyhedra^{31} (Fig. 2). This purely geometric framework, which neglects energetic and cell-size effects, is well suited for clarifying the role of entropic effects. Subsequently, we generalize to an energy-based model by considering CLT embeddings on graphs arising from generalized Thomson^{32} packing problems for equal and unequal spheres on spherical surfaces (Fig. 3a,b). As we shall see, the energy-based approach reproduces many qualitative characteristics of the polyhedral model but yields better agreement for the adjacency statistics. Throughout, we focus on 16-vertex models as relevant to the experiments, although the underlying ideas generalize to arbitrary vertex numbers.

A minimal geometric model accounting for basic aspects of the CLT embeddings in rounded cysts identifies the cell positions with the vertices of a convex equilateral polyhedron. Convexity is required as the cells adhere to the epithelium, which forms a convex enclosure (Fig. 1a). Equilaterality assumes approximately equal cell diameters, a simplification that will be dropped later. The most famous polyhedra of this type are the 5 Platonic and 13 Archimedean solids^{31}, which satisfy, however, additional symmetry requirements that prohibit 16-vertex realizations. We therefore consider here the more general class of convex equilateral polyhedra with non-uniform faces, known as Johnson solids^{11}. There exist 92 different Johnson solids in total, with vertex numbers ranging from 5 to 75, but only four of these have exactly 16 vertices (Fig. 2a). By comparing with topological features of the experimentally measured adjacency networks, we find that, among those four, the packing structure of the disphenocingulum *J*_{90} is approximately realized in some of the cysts, whereas the other three (*J*_{28}, *J*_{29} and *J*_{85}) show a larger or smaller number of degree-4-vertices than observed in our experiments (Fig. 3c). Analysis of the *J*_{90} solid helps explain why certain tree packings occur more frequently than others. To illustrate the underlying entropic argument, we computed all 5,184 possible CLT embeddings on *J*_{90} and determined the frequencies of the 72 embedding states *e*_{n} (Supplementary Information). The tree-state histogram shows that certain tree configurations can be mapped onto *J*_{90} in more ways than others (Fig. 3b). For example, the state *e*_{48} = [16, 12, 15, 11, 13, 9, 14, 10] has exactly one embedding on *J*_{90} (Fig. 2c), whereas the state *e*_{13} = [16, 12, 11, 15, 9, 13, 10, 14] can be embedded in six different ways on *J*_{90} (Fig. 2d). In thermodynamic terminology, this means that certain macrostates *e*_{n} can be realized by a larger number of microstates and hence have higher entropy; such highly degenerate macrostates are more frequently observed than low-entropy states. Analogous arguments apply to the refined energy-based models discussed next.

To obtain a more accurate description of the experimentally observed adjacencies, we introduce an energy-based model that generalizes the classic Thomson problem of arranging particles with electrostatic Coulomb repulsion on a sphere^{32}. The model represents the cell *v* as a soft sphere of radius *r*_{v} at position **x**_{v}, and the epithelium as a spherical container of radius *R*. Ring canals \(\left\langle {v,w} \right\rangle\) connecting cells *v* and *w* are modelled as harmonic springs of strength *k*_{R}. Similarly, adhesion to the epithelium and steric repulsion between adjacent spheres without ring canal connections are described by quadratic potentials, yielding the energy

The *k*_{S} term represents steric repulsion between unconnected adjacent spheres with \(\left| {{\bf{x}}_v - {\bf{x}}_w} \right| < r_v + r_w\), and the *k*_{E} term epithelial adhesion. Results presented below use *k*_{R} = 1, *k*_{S} = 0.2 and *k*_{E} = 0.3, reflecting the relative strength of the interaction forces, and were found to be robust against parameter variations. To identify tree embeddings and their statistics, we minimized equation (2) numerically, employing an annealed Metropolis–Hastings Monte Carlo (MC) algorithm^{33,34} with 3D random initial conditions (Supplementary Information).

The basic model assumes equal cell radii, *r*_{v} = *r*, in equation (2). In this case, our 10,000 MC runs always converged to one of the two polyhedra *T*_{0} and *T*_{1} shown in Fig. 3a. *T*_{0} and *T*_{1} are energetically equivalent within 0.01% and have the same symmetries as the two lowest-energy solution of the classical electrostatic Thomson problem^{32,35,36}. Their different occurrence probabilities *p*_{0} = 0.86 and *p*_{1} = 0.14 in the MC simulations reflect the cardinalities 24 and 4 of their graph automorphism sets; that is, the number of ways in which vertices can be relabelled without changing adjacencies. Intuitively, the automorphism sets determine the effective degeneracies of the two low-energy solutions, after factoring out trivial global rotations and translations. The tree-state histogram obtained from the MC simulations (grey in Fig. 3a) agrees well with the exact histogram obtained by determining all 62,256 and 94,344 CLT embeddings on *T*_{0} and *T*_{1}, respectively (green in Fig. 3a; Supplementary Information), providing further support for the hypothesis that certain CLT embeddings are entropically favoured.

The Johnson solid *J*_{90} and the Thomson graphs *T*_{0} and *T*_{1} do not yet have the correct degree distribution to reproduce the experimentally measured adjacency probabilities (Fig. 3c) and, in particular, the highly conserved layered structure observed in the egg chambers (Fig. 1d). The discrepancy is resolved by adapting the sphere radii *r*_{v} in equation (2) to match the experimentally measured average volumes of the corresponding cells in each layer (Supplementary Information). As for equal radii, the tree-state distribution remains non-uniform (Fig. 3b); this non-uniformity now arises due to a combination of entropic and energetic effects as differences in the cell radii *r*_{v} lift the degeneracies of the minima of equation (2). The associated adjacency probability matrix (Fig. 3d) agrees with the experimental data (Fig. 1i) within 8% relative error (Fig. 3e; Supplementary Information), corroborating that energy models of the type (2) define a useful theoretical framework for studying biologically relevant TPPs under convexity constraints.

To conclude, although our investigation focused on the insect model organism *Drosophila*, encapsulated CLTs have also been reported in the germlines of amphibians, molluscs^{27}, birds, humans and other mammals^{10}, suggesting that tree-packing problems play a fundamental role at the onset of oogenesis and embryogenesis^{13} in a wide range of organisms. Entropic constraints can favour particular cell-packing configurations, similar to conformational entropy barriers in protein folding^{22}. Since oocytes receive essential biochemical signals from adjacent nurse cells during development^{29}, topologically supported relative cell localization may be important for reproducible oogenesis and morphogenesis. In particular, the presence of topological links arising from incomplete cytokinesis suggests that cell ordering and rearrangement processes can deviate strongly from the soap bubble paradigm used to describe patterns of organization of simple cell aggregates^{12}, with potentially profound implications for tissue-scale organization and dynamics. More broadly, the insights from this study can provide useful guidance for the controlled self-assembly of topologically linked microstructures in confined geometries, as realizable with modern DNA origami^{7} and recently developed molecular linker toolboxes for multicellular self-assembly^{23}.

## Methods

### Tree state combinatorics

To derive equation (1), we start from cell *v* = 1 in Fig. 1b that has vertex degree *d*_{1} = 4 and fix the edge (1, 2) as a reference axis. The remaining three edges {(1, 3), (1, 5), (1, 9)} emanating from cell 1 can then be arranged in (*d*_{1} − 1)! = 3! different ways relative to edge (1, 2). Next consider the four cells *v* = 2, 3, 5, 9 that are topologically connected to cell 1 and coloured in blue in Fig. 1b,c. Cell 2 has vertex degree *d*_{2} = 4 and the three emanating edges {(2, 10), (2, 6), (2, 4)} can be permuted in (*d*_{2} − 1)! = 3! possible ways relative to the incoming edge (1, 2). Similarly, the two edges emanating from cell 5 can be permuted in (*d*_{5} − 1)! = 2! possible ways relative to the incoming edge (1, 5), and so on. Repeating this procedure for each tree layer and multiplying all the permutations yields equation (1).

### Code availability

All used codes can be downloaded from https://github.com/stoopn/CLTPackings.

### Data availability

The experimental data are publicly accessible at https://github.com/stoopn/CLTPackings.

## Additional information

**Publisher’s note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Change history

### 08 August 2018

In the version of this Letter originally published, the citations to equation (1) in Fig. 3 caption and the main text were incorrect; they should have been to equation (2). This has now been corrected.

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## Acknowledgements

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.

## Author information

### Author notes

These authors contributed equally: Jasmin Imran Alsous, Paul Villoutreix, Norbert Stoop.

### Affiliations

#### Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ, USA

- Jasmin Imran Alsous
- & Stanislav Y. Shvartsman

#### The Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, USA

- Jasmin Imran Alsous
- , Paul Villoutreix
- & Stanislav Y. Shvartsman

#### Department of Molecular Genetics, Weizmann Institute of Science, Rehovot, Israel

- Paul Villoutreix

#### Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

- Norbert Stoop
- & Jörn Dunkel

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### Contributions

All authors designed the research. J.I.A. performed the experiments. J.I.A., P.V. and N.S. analysed the data. N.S. and J.D. developed the theory. N.S. performed the simulations. All authors wrote the paper.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Jörn Dunkel.

## Supplementary information

### Supplementary Information

Supplementary Figures S1–S5, Supplementary References 1–14

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