Entropic effects in cell lineage tree packings

Abstract

Optimal packings [1, 2] of unconnected objects have been studied for centuries [3-6], 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 polyhedrons [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].

FIG. 1.

Statistics of 3D cell lineage tree (CLT) packings in D. melanogaster egg chambers. 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 20μm. b-c, Schematic of a planar CLT graph embedding (b) obtained by unfolding the 3D CLT (SI). Edges correspond to ring canals and nodes to cells labeled 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 labeled by the counter-clockwise sequence of its terminal leaves. A vertex of degree d_v permits (d_v — 1)! distinct branch permutations. Discounting mirror symmetry, the total number of possible planar embeddings is 72, see Eq. (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 gray 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 (light gray; 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 colors as in (b-d).

FIG. 2.

Entropic constraints drive departure from uniform distributions over possible cell-tree packings. 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 6 different ways on J₉₀, and hence is observed 6 times as often as tree state 48.

FIG. 3.

Energy-based models confirm non-uniform CLT distributions and capture experimentally measured cell-cell adjacencies. a, The two lowest-energy solutions T₀ and T₁ of the quadratic Thomson problem, Eq. (2), for equally sized spheres exhibit the same discrete symmetries as those of the electrostatic Thomson problem [31, 32]. 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 polyhedrons. Numerically obtained tree-state histograms for equal-size spheres (gray) approximate well the exact distributions representing the p_i-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 (cf. Fig. 1d). The simulated tree-state histogram indicates entropically favored 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 4 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 (cf. Fig. 1i) within a maximal relative entry-wise spectral error (SI) of less than 10%; corresponding maximal errors for the J₉₀ and equal-sphere Thomson models are larger by ~ 5% (SI).