Abstract

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.

Main

Packing problems1 are part and parcel of life. From DNA folding in the nucleus14 and cell assembly during embryogenesis15 to crystal growth16,17, densification of granular materials4 and masonry18, 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 packings3, a rich body of experimental and theoretical work19 has helped clarify the physical20 and mathematical1,2 principles underlying optimal embeddings of unconnected objects5,6,21. By contrast, much less is known about the packings of topologically linked structures in confined spaces, a setting relevant to the understanding of chromosome condensation8, protein folding22, multi-scaffold DNA-origami structures7, synthetic multicellular self-assembly23 and the positioning of cell lineage-trees (CLTs) in embryonic cysts10,24.

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 organisms10, 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 objects1,2, implying that topological constraints can provide an efficient biophysical mechanism for controlling cell positioning during early oogenesis10 and embryogenesis13.

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 studied25,26, 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 molluscs27, and similar topological linkages play an important role in the colony formation of choanoflagellates, the closest living relatives of animals9. Although the structure and function of cytoplasmic bridges have long been studied in the context of intercellular biochemical transport10,28,29, 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 cysts10, 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 algae28,30, 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 embryos13. 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.

Fig. 1: Statistics of 3D CLT packings in D. melanogaster egg chambers.
Fig. 1

a, Volume rendering of a 3D confocal image of two egg chambers, each comprising a germline cyst of 16 cells that is enveloped by an epithelial layer of cells. In each egg chamber, the oocyte lies at the most posterior (P) pole, with the 15 nurse cells lying at its relative anterior (A). Incomplete cytokinesis leaves sibling cells connected by ring canals (red) resulting in a hierarchical CLT. Scale bar, 20 μm. b,c, Schematic of a planar CLT graph embedding (b) obtained by unfolding the 3D CLT (Supplementary Information). Edges correspond to ring canals and nodes to cells labelled according to their order of appearance in the division sequence that gives rise to the 16-cell germline cyst (c). Each planar CLT embedding is uniquely labelled by the counter-clockwise sequence of its terminal leaves. A vertex of degree dv permits (dv − 1)! distinct branch permutations. Discounting mirror symmetry, the total number of possible planar embeddings is 72, see equation (1). d, Example of an annotated membrane-based 3D cell volume reconstruction for an egg chamber from our confocal microscopy data, showing front and back views and demonstrating layered cell arrangements. e, CLT reconstructed from d with red edges indicating physical ring canal connections between cells and grey edges additional contact adjacencies (top), and its corresponding planar unfolding (bottom). f, Experimentally measured frequency histogram over the 72 topologically different tree configurations, suggesting a non-uniform distribution. g,h, Large-deviations statistics support the non-uniformity hypothesis. Simulated histograms (grey; sample size 105) showing the expected probability of observing >4 counts (g) and a given spread of counts (h) when n = 121 samples are drawn from a uniform distribution over P′ = 72 distinct tree states. Red bars indicate the expected probabilities for observing the experimental outcome in f. i, Experimentally measured adjacency probability distribution, with rows and columns ordered according to a cell’s CLT distance from the oocyte, barcoded using the same colours as in bd.

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 dv, basic combinatorial considerations (Fig. 1b) show that there exist

$$P = \mathop {\prod}\limits_{v = 1}^V \left( {d_v - 1} \right)!$$
(1)

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 dv {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 en=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 fc = 4 times (Fig. 1g) and the largest highest count (Fig. 1h), with the frequency cut-off fc 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 polyhedra31 (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 Thomson32 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.

Fig. 2: Entropic constraints drive departure from uniform distributions over possible cell-tree packings.
Fig. 2

a, The 4 convex equilateral Johnson solids with 16 vertices. b, The tree-state histogram calculated from all possible 5,184 embeddings of the 16-cell lineage tree on the disphenocingulum J90 shows strong non-uniformity, reflecting that certain tree (macro)states e = [16, a, b, c, d, e, f, g] can be realized by a larger number of spatial embeddings (microstates). c, Discounting trivial rotations and reflections, tree state 48 with leaf sequence e48 = [16, 12, 15, 11, 13, 9, 14, 10] permits only one possible embedding on J90. d, Tree state 13 with leaf sequence e13 = [16, 12, 11, 15, 9, 13, 10, 14] can be embedded in six different ways on J90, and hence is observed six times as often as tree state 48.

Fig. 3: Energy-based models confirm non-uniform CLT distributions and capture experimentally measured cell–cell adjacencies.
Fig. 3

a, The two lowest-energy solutions T0 and T1 of the quadratic Thomson problem, equation (2), for equally sized spheres exhibit the same discrete symmetries as those of the electrostatic Thomson problem35,37. For equal-size spheres, our MC simulation runs (n = 104) converged to either T0 or T1 with probability p0 = 0.86 and p1 = 0.14, in good agreement with the theoretical predictions 24/(24 + 4) ≈ 0.857 and 4/(24 + 4) ≈ 0.143 based on the automorphisms of the two polyhedra. Numerically obtained tree-state histograms for equal-size spheres (grey) approximate well the exact distributions representing the pi-weighted average over all 62,256 CLT embeddings on T0 and 94,344 embeddings on T1 (green). These results also support an entropically driven departure from a uniform distribution over cell-tree configurations. b, A typical realization of the n = 4,000 simulated CLT packings with sphere volumes matched to the experimental average values for the corresponding cells, showing layered cell arrangements consistent with experiments (see Fig. 1d). The simulated tree-state histogram indicates entropically favoured configurations. c, Experimental adjacency-graph characteristics are best reproduced with volume-matched spheres (empty circles), whereas the equal-sphere model (blue) underestimates the vertex degree. Among the four Johnson solids (red), the disphenocingulum J90 is closest to the experimental data. d,e, Adjacency probability matrices for the volume-matched energy model (d) agree with the experimental data (see Fig. 1i) within a maximal relative entry-wise spectral error (Supplementary Information) of less than 10%; corresponding maximal errors for the J90 and equal-sphere Thomson models are larger by ~5% (Supplementary Information).

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 solids31, 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 solids11. 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 J90 is approximately realized in some of the cysts, whereas the other three (J28, J29 and J85) show a larger or smaller number of degree-4-vertices than observed in our experiments (Fig. 3c). Analysis of the J90 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 J90 and determined the frequencies of the 72 embedding states en (Supplementary Information). The tree-state histogram shows that certain tree configurations can be mapped onto J90 in more ways than others (Fig. 3b). For example, the state e48 = [16, 12, 15, 11, 13, 9, 14, 10] has exactly one embedding on J90 (Fig. 2c), whereas the state e13 = [16, 12, 11, 15, 9, 13, 10, 14] can be embedded in six different ways on J90 (Fig. 2d). In thermodynamic terminology, this means that certain macrostates en 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 sphere32. The model represents the cell v as a soft sphere of radius rv at position xv, 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 kR. Similarly, adhesion to the epithelium and steric repulsion between adjacent spheres without ring canal connections are described by quadratic potentials, yielding the energy

$$\begin{array}{*{20}{l}} E \hfill & = \hfill & {\frac{{k_R}}{2}\mathop {\sum}\limits_{\langle v,w\rangle } \left[ {\left| {{\bf x}_v - {\bf x}_w} \right| - \left( {r_v + r_w} \right)} \right]^2 + } \hfill \cr {} \hfill & {} \hfill & {\frac{{k_{\mathrm{S}}}}{2}\mathop {\sum}\limits_{{\mathrm{adj}}(v,w)} \left[ {\left| {{\bf x}_v - {\bf x}_w} \right| - \left( {r_v + r_w} \right)} \right]^2 + } \hfill \cr {} \hfill & {} \hfill & {\frac{{k_{\mathrm{E}}}}{2}\mathop {\sum}\limits_{v = 1}^{16} \left[ {\left| {{\bf x}_v} \right| - \left( {R - r_v} \right)} \right]^2} \hfill \end{array}$$
(2)

The kS term represents steric repulsion between unconnected adjacent spheres with \(\left| {{\bf{x}}_v - {\bf{x}}_w} \right| < r_v + r_w\), and the kE term epithelial adhesion. Results presented below use kR = 1, kS = 0.2 and kE = 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) algorithm33,34 with 3D random initial conditions (Supplementary Information).

The basic model assumes equal cell radii, rv = r, in equation (2). In this case, our 10,000 MC runs always converged to one of the two polyhedra T0 and T1 shown in Fig. 3a. T0 and T1 are energetically equivalent within 0.01% and have the same symmetries as the two lowest-energy solution of the classical electrostatic Thomson problem32,35,36. Their different occurrence probabilities p0 = 0.86 and p1 = 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 T0 and T1, respectively (green in Fig. 3a; Supplementary Information), providing further support for the hypothesis that certain CLT embeddings are entropically favoured.

The Johnson solid J90 and the Thomson graphs T0 and T1 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 rv 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 rv 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, molluscs27, birds, humans and other mammals10, suggesting that tree-packing problems play a fundamental role at the onset of oogenesis and embryogenesis13 in a wide range of organisms. Entropic constraints can favour particular cell-packing configurations, similar to conformational entropy barriers in protein folding22. Since oocytes receive essential biochemical signals from adjacent nurse cells during development29, 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 aggregates12, 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 origami7 and recently developed molecular linker toolboxes for multicellular self-assembly23.

Methods

Tree state combinatorics

To derive equation (1), we start from cell v = 1 in Fig. 1b that has vertex degree d1 = 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 (d1 − 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 d2 = 4 and the three emanating edges {(2, 10), (2, 6), (2, 4)} can be permuted in (d2 − 1)! = 3! possible ways relative to the incoming edge (1, 2). Similarly, the two edges emanating from cell 5 can be permuted in (d5 − 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

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

Affiliations

  1. Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ, USA

    • Jasmin Imran Alsous
    •  & Stanislav Y. Shvartsman
  2. The Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, USA

    • Jasmin Imran Alsous
    • , Paul Villoutreix
    •  & Stanislav Y. Shvartsman
  3. Department of Molecular Genetics, Weizmann Institute of Science, Rehovot, Israel

    • Paul Villoutreix
  4. 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

  1. Supplementary Information

    Supplementary Figures S1–S5, Supplementary References 1–14

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DOI

https://doi.org/10.1038/s41567-018-0202-0