Scutoids are a geometrical solution to three-dimensional packing of epithelia

Abstract

As animals develop, tissue bending contributes to shape the organs into complex three-dimensional structures. However, the architecture and packing of curved epithelia remains largely unknown. Here we show by means of mathematical modelling that cells in bent epithelia can undergo intercalations along the apico-basal axis. This phenomenon forces cells to have different neighbours in their basal and apical surfaces. As a consequence, epithelial cells adopt a novel shape that we term "scutoid". The detailed analysis of diverse tissues confirms that generation of apico-basal intercalations between cells is a common feature during morphogenesis. Using biophysical arguments, we propose that scutoids make possible the minimization of the tissue energy and stabilize three-dimensional packing. Hence, we conclude that scutoids are one of nature's solutions to achieve epithelial bending. Our findings pave the way to understand the three-dimensional organization of epithelial organs.

Fig. 1

A mathematical model for curved epithelia uncovers a novel geometrical solid. a Scheme representing planar columnar/cubic monolayer epithelia. Cells are simplified as prisms. b Scheme illustrating an invagination or fold in a columnar/cubic monolayer epithelium. Cells adopt the called “bottle shape” that would be simplified as frusta. c Mathematical model for an epithelial tube. A Voronoi diagram is drawn on the surface of a cylinder (representing the apical surface of the epithelial tube). The seeds of each Voronoi cells are projected in an outer cylinder (representing the basal surface of the epithelial tube). This can induce a topological change, a cell intercalation. Yellow and blue cells are neighbours in the apical surface but not in the basal surface. The reciprocal occurs for red and green cells. R_a, radius from the centre of the cylinders to the apical surface. R_b, radius from the centre of the cylinders to the basal surface. d Modelling clay figures illustrating two scutoids participating in a transition and two schemes for scutoids solids. Scutoids are characterized by having at least a vertex in a different plane to the two bases and present curved surfaces. e A dorsal view of a Protaetia speciose beetle of the Cetoniidae family. The white lines highlight the resemblance of its scutum, scutellum and wings with the shape of the scutoids. Illustration from Dr. Nicolas Gompel, with permission. f 3D reconstruction of the cells forming a tube with $R_b∕R_a=2.5$. The four-cell motif (green, yellow, blue, and red cells) shows an apico-basal cell intercalation. g Detail of the apico-basal transition, showing how the blue and yellow cells contact in the apical part, but not in the basal part. The figure also shows that scutoids present concave surfaces

Fig. 2

3D tissue packing of curved epithelia. a Example of Drosophila salivary gland and its processed images. Scale bar = 100 μm. b Confocal images showing the apico-basal cell intercalation of epithelial cells marked with green, yellow, red, and blue pseudo-colours. The green cell participates in two apico-basal transitions. c 3D reconstruction of the same cells labelled in b using the same colour code. The image confirms the presence of concave surfaces predicted by the mathematical model

Fig. 3

Apico-basal transitions are favoured by geometrical factors. a Examples of the 40, 200, and 800 cells models resulting from basal expansions of two and five times the apical surface. Data are represented as mean ± SEM. b Percentage of cells involved in transitions (scutoids) in relation to the increase of the surface ratio for five conditions (40, 80, 200, 400, and 800 cells). c Scheme showing how are measured the edge angle with respect the transverse axis (θ) and the edge length in a four-cells motif of the basal surface of the salivary gland. Scale bar = 100 μm. d Polar scatter showing the length and the angle of the contacting edge in basal four-cell motifs from the salivary gland epithelium. Light blue points stand for motifs that exchange neighbours; orange points stand for motifs that do not intercalate. e Polar scatter showing the length and the angle of the contacting edge in basal four-cell motifs from the tubular model with the 1/0.6 surface ratio. Light blue points stand for 200 motifs that exchange neighbours; orange points stand for 200 motifs that do not intercalate

Fig. 4

Energy minimization on tubular epithelia. a Packing configurations are characterized by four-cells motifs. Our theoretical argument to explain packing relies on line-tension energy minimization. Two idealized, stable, packing configurations are possible along the apico-basal axis as represented by l_w and l_w. We decompose the experimental measured value of the line-tension energy into these fundamental modes. b Density plot of the energy profile (dimensionless) as a function of the aspect ratio, $ϵ=h∕w$, and l (in units of w) as obtained by the theoretical model. If $ϵ<1\mathrm{∕}\sqrt{3}$ the only stable configuration is l_w, if $ϵ>\sqrt{3}$ the only stable configuration is l_h, and if $ϵ\in \left(1\mathrm{∕}\sqrt{3},\sqrt{3}\right)$ both configurations are stable. Cells adopt the “scutoidal” shape when there is a $l_w\leftrightarrow l_h$ transition. The dashed magenta line indicates the location of the unstable configuration $l_w=l_h=0$. The potential wells (top) indicate schematically the shape of the energy profile within each zone: the green-shaded regions indicate the stable energy attractors and the green dotted line the absolute energy minimum (see d, e). c Scheme showing how the experimental values for h, w, and l were measured in the apical and basal surfaces. Four-cell motifs were identified and we characterized the aspect ratio $⟨h⟩\mathrm{∕}⟨w⟩$ by measuring $⟨w⟩=\left(w_1+w_2\right)\mathrm{∕}2$ and $⟨h⟩=\left(h_1+h_2\right)\mathrm{∕}2$. d–e Decomposition of the tensile energies into the fundamental modes ${Ê}_w$ and ${Ê}_h$ (Methods) for apico-basal events in salivary glands (d) and 1/0.6 Voronoi tubes (e): “no transition” (left) “transition” (right) events. Individual packing configuration are represented by connected dots that account for the energy at the basal (grey) and apical (blue) surfaces. The magenta and green dotted lines indicate the theoretical energy levels of the unstable fourfold configuration and the absolute energy minimum respectively. The stable energy attractors are located within the green-shaded region. Insets: polar histogram accounting for the directionality of the trajectories from basal to apical

Fig. 5

Graphic summary of 3D packing in epithelia. a Section of a stage 4 egg chamber with nuclei (blue) and cell membranes (green) labelled. The green staining decorates the contours of the follicular cells (FC), the nurse cells (NC) and the oocyte (Oo). Scale bar = 20 μm. b Section of a stage 8 egg chamber with nuclei (blue) and cell membranes (green) labelled. The green staining decorates the contours of the follicular cells (FC), the nurse cells (NC) and the oocyte (Oo). Scale bar = 20 μm. c Close up of the surface of a stage 4 egg chamber (a), showing two four-cell motifs that are involved in an apico-basal transition. The numbers indicate the confocal plane, increasing from basal to apical regions of the follicular cells. In this case, the sections were taken every 0.4 μm. Scale bar = 10 μm. d Polar scatters showing the length and the angle of the contacting edge in basal four-cell motifs from the stage 4 and stage 8 egg chambers. Light blue points stand for motifs that exchange neighbours; orange points stand for motifs that do not intercalate. e Voronoi spheroid model with the same dimensions that the average egg chamber 4. f Voronoi spheroid model with the same dimensions that the average egg chamber 8. g–h Linear decomposition of the experimental energies into the fundamental modes ${Ê}_w$ and ${Ê}_h$. Colour codes as in Fig. 4

Fig. 6

Graphic summary of 3D packing in epithelia. a The graph represents the curvature ratio (surface ratio) depending of the dimensionless values of ${\widehat{R}}_a$ and ${\widehat{R}}_b$. The curvature ratio is defined by the ratio ${\widehat{R}}_a∕{\widehat{R}}_b$ in the case of basal reduction and ${\widehat{R}}_b∕{\widehat{R}}_a$ in the case of apical reduction. The bottom triangle (blue shaded region) covers the region where the epithelial bend generating a reduction of the apical surface ${\widehat{R}}_b>{\widehat{R}}_a$. The top triangle (grey shaded region) shows the region where a reduction of the basal surface occurs ${\widehat{R}}_a>{\widehat{R}}_b$. In all the geometrical solids and epithelial representations basal is at top (grey) and apical is at bottom (blue). The black dashed arrow in the main diagonal indicates the epithelium without curvature (ratio = 1). This planar epithelium is formed by prisms. The packing configuration space is compartmentalized into regions depending on the relative values of the curvature radii (or equivalently the aspect ratio) of epithelia. The curved arrows indicate the types of solids that are more favourable in the epithelia depending of the curvature: close to the diagonal the frusta shapes of the cells are preferred. On the contrary, in the top-right and bottom-left corners all the cells tend to be scutoids that take part in several apico-basal transitions. There is also an intermediate situation where the epithelium is packed using both scutoids and frusta shapes. b The model can be generalized to describe tissues with two main axes of curvature (h, transversal, and w, longitudinal), where R represents the Cauchy radius, e.g. $R_b^h$ accounts for the Cauchy radius along the h axis in the basal, b, surface. Here we assume that the apical surface lies in the inner part of the tissue. As a function of the Cauchy radii it is possible to assess the surface ratio anisotropy for different geometries as measured by the relative change of the aspect ratio from apical to basal (colour code): $\left(R_b^h\mathrm{∕}{R}_a^h\right)\mathrm{∕}\left(R_b^w\mathrm{∕}{R}_a^w\right)-1$. The increment of the surface ratio anisotropy positively correlates with the percentage of scutoids