Boundary geometry controls a topological defect transition that determines lumen nucleation in embryonic development

Abstract

Topological defects determine the collective properties of anisotropic materials. Nonetheless, it is not fully understood how their configurations are controlled, especially in three dimensions. In living matter, contributions of two-dimensional topological defects to biological functions have been demonstrated, but whether three-dimensional polar defects have any biological relevance is unclear. Here we report a charge-preserving transition between three-dimensional defect configurations driven by boundary geometry and independent of material parameters. Moreover, we find that three-dimensional polar defects in the mouse embryo are the sites where fluid-filled lumina form, essential structures for subsequent development. We validate these findings by experimentally perturbing embryo shape beyond the transition point, which results in the creation of additional lumen initiation sites near predicted defect locations. Overall, our results reveal how boundary geometry controls polar defects, and how embryos use this mechanism for shape-dependent lumen formation. We expect this defect-control principle to apply broadly to systems with orientational order.

The geometry of confining boundaries is shown to control three-dimensional topological defect configurations that precede formation of the central lumen in the developing mouse embryo, a key process in embryogenesis.

Main

Confinement plays a fundamental role in directing self-organization¹ in various contexts including colloids², developing tissues^3,4, liquid crystals⁵ and crowds of people⁶. Controlling confining structures and predicting their impact on the collective properties of the confined bulk can reveal organizational principles of complex systems like living matter⁷ and enable technological applications such as the design of optoelectronic⁸ and microfluidic⁹ devices. Boundary effects are especially relevant for materials composed of anisotropic particles, which possess orientational degrees of freedom and tend to align with their neighbours. In liquid crystals, for example, substrates with a particular surface topography or chemical functionality¹⁰ induce the alignment of adjacent molecules, creating an ordered layer that propagates into the bulk by elastic forces. In this way, surface-induced alignment allows one to achieve different desired configurations of the order parameter (OP) by tuning the boundary conditions, for example in nematics^11–13. However, the effects of boundary geometry on the bulk organization of anisotropic materials are not well understood, particularly for curved surfaces in three dimensions.

Confinement also influences the internal organization of living materials, for example, how multicellular systems undergo morphogenesis in biofilms¹⁴ or during embryonic development¹⁵. The interactions between anisotropic living matter and their boundaries, such as between tissues consisting of polarized cells and confining extracellular matrix (ECM) structures, involve diverse biophysical processes, including adhesion and polarity signalling^16–18, which affect ordering at the collective scale, similar to boundary-induced alignment. For instance, basement membrane layers promote specific orientations of adjacent cells for different types of polarity—parallel for epithelial planar polarity^19,20 and mesenchymal or migratory polarity^21–23, and perpendicular for apico-basal epithelial polarity²⁴. Coherent apico-basal alignment is crucial for epithelial functions such as molecule transport across the tissue, and is linked to the formation of fluid-filled cavities (lumina)²⁴. Orientational boundary effects could provide general mechanisms by which confining structures control multicellular organization.

Many collective properties of ordered materials depend on the number and spatial distribution of topological defects, that is, points in the OP field where the orientation is not defined. In the case of nematic systems confined to a surface, geometric constraints have been shown to play an important role in determining defect configurations and dynamics^25,26. In biological contexts, defects can guide diverse cellular processes²⁷ and have been shown to trigger the formation of structures such as fruiting bodies²⁸ and tentacles²⁹. However, despite their potential biological role, three-dimensional (3D) polar defects and their relation to system geometry remain unexplored, with most investigations focusing on two-dimensional (2D) domains and nematic systems, applicable in multicellular contexts where cells can be approximated as elongated units. These approaches have successfully predicted the collective dynamics of bacterial colonies^28,30,31, cell populations^25,32 and epithelial sheets^33,34. However, in contexts where cellular polarity implies distinct subcellular regions with specific functions—such as apico-basal polarity, in which the asymmetric organization of organelles and cell-surface proteins enables directed transport³⁵, or mesenchymal cell polarity, which drives active movement^36,37—the polar nature of cells is relevant for understanding collective properties.

Here we investigate how boundary effects lead to a surface–bulk coupling that permits controlling the properties of anisotropic materials through the confining geometry, focusing on boundaries capable of inducing bulk topological charge. Specifically, we analyse how the mechanical properties and geometry of the boundaries control the ordering and defect configurations of a 3D polar fluid confined by non-uniform surfaces. We identify three transitions: two in which defects in the bulk OP field appear out of a uniform field driven by changes in boundary coupling and material response, and a third charge-preserving defect transition controlled by the geometry alone.

We apply these findings to study the role of boundary geometry on the bulk organization of apico-basally polarized cells, using the mouse epiblast—the tissue that forms the embryo proper during development—as an example. The orientational order of apico-basal polarity in the epiblast is linked to the formation of a central lumen, a critical event in normal mouse development^38,39. By associating the polar OP field with the local average cellular apico-basal polarity in the tissue, we find that 3D-field defect configurations are parameter-free predictors of lumen initiation sites. Moreover, using a recently developed ex vivo culturing approach³⁸, we experimentally perturb epiblast boundary shape to induce additional lumen nucleation sites at the predicted positions.

Weak boundary-induced alignment in a 3D polar fluid

To investigate the mechanisms by which boundaries influence orientational order, we consider a polar fluid confined in a 3D space Ω with volume V₀. Such systems can be characterized by a 3D vector OP p(r) representing the average polarity of particles in a volume element defined at each point in space r (ref. ⁴⁰), which allows one to define the global degree of order

$$P\equiv {⟨|\mathbf{p}|⟩}_{\mathrm{\Omega }}=\frac{1}{V_0}{\int }_{\mathrm{\Omega }}\mathrm{d}V|\mathbf{p}|.$$ 1

When bulk and surface interactions are present and transients can be neglected, the local OP p(r) can be obtained by minimizing the effective free energy functional

$$\mathcal{F}\left[\mathbf{p}\right]={\int }_{\mathrm{\Omega }}\mathrm{d}V{f}_{\mathrm{B}}(\mathbf{p},\nabla \mathbf{p})+{\int }_{\partial \mathrm{\Omega }}\mathrm{d}S{f}_{\mathrm{S}}\left(\mathbf{p}\right),$$ 2

where f_B and f_S are the energy densities in the bulk V and on the surface S, respectively (Supplementary Note 1).

Following the Landau–de Gennes approach to the nematic–isotropic transition in liquid crystals⁴¹, we write the bulk energy density as f_B = f_R + f_E. To focus on boundary-induced order, we consider a restoring term f_R = a∣p∣²/2 with a ≳ 0 favouring the disordered state p = 0. The second term is given by the elastic contribution:

$$f_{\mathrm{E}}=\frac{k_0}{2}{(\mathrm{\nabla }\cdot \mathbf{p})}^2+\frac{k_1}{2}{[\hat{\mathbf{p}}\cdot (\mathrm{\nabla }\times \mathbf{p}\left)\right]}^2+\frac{k_2}{2}{[\hat{\mathbf{p}}\times (\mathrm{\nabla }\times \mathbf{p}\left)\right]}^2$$ 3

with $\widehat{\mathbf{p}}=\mathbf{p}/\mid \mathbf{p}\mid$, in which we do not impose the one-constant approximation for the coefficients k₀, k₁ and k₂ that penalize splay, twist and bend distortions, respectively. Note that the coupling between the magnitude and orientational degrees of freedom in equation (3) produces a regularizing cross-term absent in the classical Frank free energy (Supplementary Note 2).

The surface energy density f_S represents a weak anchoring interaction between the fluid and the confining surface ∂Ω (refs. ^42–44), where the OP at the boundary is not fixed as in the strong case^11–13,45. Deviations from the preferred value p₀ at the surface are penalized as^43,46

$$f_{\mathrm{S}}=\frac{w}{2}{(\mathbf{p}-{\mathbf{p}}_0)}^2.$$ 4

The constant w > 0 defines the anchoring length λ ≡ w/a.

Surface anchoring is known to control the defect structure in liquid crystals in the nematic phase^5,45,47. In spherical confinement, a homeotropic (that is, perpendicular) boundary condition produces a radial defect in the centre, while a longitudinal tangential alignment favours the appearance of two surface radial defects known as ‘boojums’ at the poles. For a polar OP, however, the diversity of defects is larger due to the greater variety of possible orientations at the boundary, and mixed boundary conditions—with both tangential and normal p₀ values—remain unexplored (Fig. 1a,b), although a related problem was studied for liquid crystals⁴⁵.

Fig. 1. Material parameters determine the creation of 3D polar defects.

a, Polar systems in three dimensions feature diverse defect structures, shown schematically for the radial + 1 (rad+1; black), radial − 1 (rad–1; pink), hyperbolic + 1 (hyp+1; purple) and hyperbolic − 1 (hyp–1; lilac) hedgehogs, and the vortex (empty magenta) and radial (empty blue) disclination rings (v-ring and r-ring, respectively). b, Streamlines and colour map show the local direction and magnitude of the equilibrium OP field in spherical confinement for anchoring and correlation lengths λ/R₀ = 100 and ξ/R₀ = 1 in the one-constant approximation (k₀/k₂ = 1) with different boundary orientations p₀ (black external arrows) on parallel-aligning (gray) and perpendicular-aligning (green) surfaces. c, OP fields with varying splay-to-bend ratio k₀/k₂ show how defects move from the boundary into the bulk as splay becomes the dominant mode (λ/R₀ = 100, ξ/R₀ = 1). d, The system achieves global ordering (P > 0.5; equation (1)) only if λ > R₀ (dotted line), and presents bulk defects for large splay-to-bend ratio k₀/k₂ (dashed line; stars show the parameters of the configurations in c, where ξ/R₀ = 1). e, Quantifying defect positions (filled and open points) from the OP fields reveal bulk defects entering from the boundary and changing their position and configuration as the splay-to-bend ratio increases, with boundary preference (insets) determining the type and number of defects (λ/R₀ = 100, ξ/R₀ = 1). f, In the splay-dominated regime (here, k₀/k₂ = 10), bulk defects appear when boundary anchoring dominates over the effects of bulk distortions (dashed line, ζ* ≈ 0.3R₀). The system remains disordered (P < 0.5) even in the limit λ → ∞ for small correlation length (dotted line, ξ* ≈ 0.4R₀). g, As the anchoring length λ increases, the hyp−1 bulk defect becomes a hyperbolic boojum, while the rad+1 decomposes into a hyp+1 and an r-ring (ξ/R₀ = 1, k₀/k₂ = 10). Error bars denote the spread of candidate points for the numerical detection of defect locations (Methods).

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Material parameters determine the creation of 3D polar defects

Mixed boundary conditions are relevant whenever a polar material contacts surfaces with different properties. To investigate how these non-uniform boundaries favouring distinct distortion modes affect bulk organization, we analyse the impact of different elastic coefficients^48,49. Numerical minimization of equation (2) in spherical confinement with tangential and mixed p₀ values reveals distinct field configurations and defect structures depending on the splay-to-bend ratio k₀/k₂ (Fig. 1c and Methods). Note that the twist contribution vanishes for an axially symmetric system. While there are only surface defects for k₀/k₂ ≤ 1, bulk defects form and move away from the surface as the splay-to-bend ratio increases, favouring bulk bend distortions over splay-inducing surface defects (Fig. 1c–e), similar to the transition between nematic tactoids and toroids⁴⁹. The type of bulk defects depends on the preferred orientation on the surface, with pole-to-pole tangential p₀ giving rise, in the k₀ ≫ k₂ limit, to a toroidal vortex tube centred on the symmetry axis resembling the flow lines of a passive fluid in contact with an active surface⁵⁰. By contrast, the combination of tangential and normal preferred orientations results in a configuration with a hyperbolic point defect on the symmetry axis accompanied by a radial disclination ring around it.

Within the splay-dominated regime, the correlation length of the material can be defined as $\xi \equiv \sqrt{k_0/a}$. While the development of global order is promoted by the surface anchoring (and modulated by the surface-to-volume ratio; Extended Data Fig. 1a), the correlation length determines the penetration of boundary-induced order into the bulk, with P > 0.5 achieved for ξ exceeding ξ*≈ 0.4R₀, with R₀ ≡ (3V₀/4π)^1/3 (Fig. 1f and Extended Data Fig. 1a). The extrapolation length ζ ≡ k₀/w = ξ²/λ sets the relative importance of bulk versus boundary interactions. Defects appear when surface anchoring dominates over the bulk, at ζ*≈ 0.3R₀ (Fig. 1f and Extended Data Fig. 1a), trading off the cost of bulk distortions in favour of better alignment with the preferred orientations at the boundaries, comparable to the transition in nematic tactoids⁴². We find that the number and structure of defects depend on the confinement geometry. While in both spherical and hemispherical confinement, a pair of radial (rad+1) and negative hyperbolic (hyp−1) hedgehogs nucleate at the transition, in the former, the rad+1 defect further decomposes into a positive hyperbolic (hyp+1) hedgehog and a radial disclination ring (r-ring) on increasing anchoring (Fig. 1f,g, Extended Data Fig. 1a,b and Supplementary Note 2). The topological charge for both geometries changes from neutral to +1 when the hyp−1 hedgehog moves up and becomes a hyperbolic boojum⁵¹ on the cap surface (Fig. 1g and Extended Data Fig. 1b). Note that the r-ring carries no charge⁵². In summary, we find that the competition between different distortion modes and boundary anchoring determines the global topological charge of boundary-aligned polar systems.

Extended Data Fig. 1. The degree of global order and type of defects depend on the confining geometry.

a, Global degree of order P [equation (1)] of a hemispherical system with mixed p₀. Order penetrates into the bulk more easily than in spherical confinement, obtaining P > 0.5 in the limit λ → ∞ for ξ > 0.2R₀ (dotted line). Stars denote the parameter values for example field configurations, where defects are created in order to accommodate the increasingly relevant boundary conditions (ξ/R₀ = 1, k₀/k₂ = 10). The value ζ^* ≈ 0.3R₀ (dashed line) that separates defect-free and defect-containing configurations is independent of the confining geometry. b, Increasing the anchoring length λ changes the defect height z (ξ/R₀ = 1, k₀/k₂ = 10). The hyp − 1 bulk defect moves up and becomes a hyperbolic boojum on the surface, leaving behind only the rad + 1 hedgehog. Points and error bars correspond to the mean and standard deviation of the position of the defect candidates (see Methods).

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Material parameters of the mouse epiblast

In living systems, boundary properties are often non-uniform. For example, the mouse epiblast at embryonic day 5.5 (E5.5) is confined by two distinct interfaces: an ECM layer, and a tissue–tissue interface with the extra-embryonic ectoderm (ExE; Fig. 2a,b). Epiblast cells develop apico-basal polarity, leading to directional cell–cell interactions via asymmetrically localized polarity and junctional protein complexes, which provide mechanical coupling and integration of biochemical signals^38,39,53,54. Interactions with ECM boundaries via, for example, basally localized integrin lead to the promotion of a perpendicular apico-basal orientation via polarity signalling and mechanical feedback, whereas cells orient predominantly parallel at the tissue–tissue interface^38,53,54.

Fig. 2. Orientational interactions in the mouse epiblast.

a, Apico-basally polarized cells tend to align parallel to a tissue–tissue interface (left) and perpendicular to a tissue–ECM interface (right), similar to an anchoring effect. b, Immunofluorescence image stained for an ECM component (collagen IV), actin (phalloidin), an epiblast marker (Oct3/4) and DNA (4′,6-diamidino-2-phenylindole, DAPI) shows the central cross-section of a representative mouse embryo after ex vivo culturing for 18 h from E4.5. At this stage, the epiblast consists of apico-basally polarized cells (cyan) in contact with a cup-shaped ECM layer (red) and a tissue–tissue interface with the ExE (top). Scale bar, 20 μm. c,d, Based on membrane segmentations of the individual epiblast cells from 3D images (c), cell long axes (d), representing each cell’s unsigned orientation, were extracted for n = 5 embryos. e, We obtain an estimate of the experimental director field and strength of alignment (left; lines and colour map, respectively) by computing the cross-embryo average cell orientations weighted by the corresponding cell elongation. Data also shown in ref. ⁵⁴. Using only the field magnitude, we fit the model parameters and find the polar OP that best represents the data (right; ξ/R₀ = 4.6, λ/R₀ = 10³, k₀/k₂ = 10³). Both fields present a topological defect (black point in the experimental case, with error bars denoting the s.d. of candidate point positions; Methods). f,g, Evaluating the cost function $\mathcal{C}$ (Methods) over a range of splay-to-bend ratios k₀/k₂, anchoring lengths λ and correlation lengths ξ reveals that the system is robustly represented by a splay-dominated (f), boundary-aligned (g) polar material, in the defect-containing regime (squares denote the best-fit parameters used in e). Panel b adapted from ref. ³⁸ under a Creative Commons license CC BY 4.0.

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To assess how such cell–cell and cell–boundary interactions influence the average cellular orientations, we performed a 3D-gel-embedded embryo culture, a technique that recapitulates in utero embryonic development while allowing for in toto monitoring, measurement and manipulation³⁸. Using inverted light-sheet microscopy, we imaged embryos with fluorescently labelled membranes and nuclei, observing frequent neighbour exchanges and fluid-like tissue dynamics over the course of this developmental stage (Supplementary Videos 1 and 2). Segmenting the epiblast cells, we quantified their principal axes of inertia to determine their elongation and orientation within the tissue (Fig. 2c,d; see Methods and ref. ⁵⁴). Taking advantage of the approximate axial symmetry of epiblasts at this stage, we averaged these measurements across rotational slices, and subsequently across the ensemble of five embryos, to calculate the characteristic nematic OP field (Fig. 2e). Because the underlying observables are invariant under polarity inversion, they cannot yield a direct estimate of the apico-basal polarity field: the nematic tensor captures the cell’s orientation coherence, but not the polarity direction. However, the magnitude of the nematic field can provide an unsigned read-out of the polarity magnitude, assuming that local inversions of apico-basal orientation are sufficiently rare to be neglected. Using this measured nematic magnitude, we estimated the relative strength of cell–boundary and cell–cell interaction modes by fitting the material parameters in equations (3) and (4) to the data (Methods). We find that the epiblast tissue is best represented by a correlation length ξ/R₀ = 4.6, an anchoring length λ/R₀ = 10³ and a splay-to-bend ratio k₀/k₂ = 10³, situating the embryo within the splay-dominated, defect-containing regime (Fig. 2f,g). The direction of the OP field p obtained with these values resembles well the experimental nematic director field, even though the latter was not directly included in the fitting process. In particular, both fields present a radial defect on the symmetry axis (Fig. 2e). The good agreement between confined polar fluid theory and the experimental measurements suggests that cellular configurations in the epiblast are indeed predominantly controlled by orientational cell–cell and cell–boundary interactions. In an accompanying study, we systematically analysed cell alignment patterns throughout the peri- to early post-implantation developmental stages and found an increase in the anchoring and correlation length parameters during development, in line with an observed enrichment in integrin and ECM components at the epiblast boundary⁵⁴.

Boundary shape controls charge-preserving defect transitions

Motivated by the shape of the mouse epiblast, we systematically examine the role of boundary geometry on 3D polar OP field configurations in the family of axis-symmetric ‘ideal acorn’ shapes, whose geometry is fully parametrized by ν ≡ z_α/R_β with z_α the height of the spherical-shell cap and R_β the radius of the hemispherical base (Fig. 3a and Methods). While ν = 0 corresponds to a flat tissue–tissue interface, 0 < ν ≤ 1 (respectively −1 < ν < 0) represent convex (respectively concave) interfaces, with ν = 1 the spherical case. The total volume V₀ is fixed and defines a length scale $R_0\equiv {(3V_0/4\mathrm{\pi })}^{1/3}$.

Fig. 3. Confining geometry controls charge-preserving transitions between different defect configurations.

a, We parametrize the class of axis-symmetric acorn shapes with ν ≡ z_α/R_β, where ϕ denotes the azimuthal angle around the symmetry axis z. The base S_β is defined as a hemisphere with radius R_β centred at the coordinate origin, while the spherical cap S_α has radius R_α, centre (0, c_α) on the symmetry axis and height z_α. Vectors ${\mathbf{p}}_0^{\mathrm{\text{α}}}$ and ${\mathbf{p}}_0^{\mathrm{\text{β}}}$ depict the mixed preferred orientation on the boundary. b, The OP fields for different values of the acorn parameter ν at fixed anchoring and correlation lengths λ/R₀ = 10⁶ and ξ/R₀ = 1 (that is, in the highly ordered, defect-containing regime) with splay-to-bend ratio k₀/k₂ = 10 show transitions between a radial hedgehog, and a hyp+1 defect surrounded by a radial disclination ring. c, Height z and distance to the symmetry axis x (equivalent to the radius, in the case of the r-ring) as functions of the acorn parameter ν. The transitions between defect configurations occur when the shape parameter exceeds the critical values ${\nu }_{\mathrm{c}}^{+}=0.38\pm 0.04$ and ${\nu }_{\mathrm{c}}^{-}=-0.56\pm 0.09$ (dashed lines). The transition points are invariant to changes in the material parameters, and defect positions undergo little variation (Extended Data Fig. 3). Points and error bars correspond to the mean and s.d. of the position of the defect candidates (Methods). d, The global degree of order P also depends on the acorn parameter ν, but the system remains always in the highly ordered (P > 0.5) regime (λ/R₀ = 10⁶, ξ/R₀ = 1, k₀/k₂ = 10).

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Fixing the material length scales in the splay-and-surface-dominated regime, we vary the acorn parameter ν (Fig. 3b,c) and find that the defect configuration undergoes a transition from radial hedgehog to a combination of hyp+1 point defect plus r-ring at critical values ${\nu }_{\mathrm{c}}^{+}$ and ${\nu }_{\mathrm{c}}^{-}$ (Fig. 3b,c and Extended Data Fig. 2). Notably, the surface-to-volume ratio and the global degree of order P (Fig. 3d) differ distinctly at these two points. Measuring the position of the defects (Fig. 3c) shows that their height grows approximately linearly with the acorn parameter, and that the r-ring is nucleated with a finite radius. This renders the transitions discontinuous, comparable to the first-order transition in nematics with strong homeotropic boundary conditions⁵⁵. Minimizations over intervals of the anchoring and correlation lengths and the splay-to-bend ratio $\lambda /R_0\in \left[10^2,10^6\right]$, $\xi /R_0\in \left[{\xi }^{*},\sqrt{\lambda {\zeta }^{*}}\right)/R_0$ and k₀/k₂ ∈ [10, 10²] reveal that the transition points and the position of the defects are robust to changes in material properties (Fig. 3c and Extended Data Fig. 3): the transition from the radial hedgehog to the hyp+1 and r-ring defects occurs independently of the material parameters at ${\nu }_{\mathrm{c}}^{+}=0.38\pm 0.04$ and ${\nu }_{\mathrm{c}}^{-}=-0.56\pm 0.09$ (mean ± standard deviation (s.d.) of values for different parameters), and the r-ring radius changes only twofold over an almost three orders of magnitude variation in the correlation length. In summary, we report a geometry-controlled transition in the defect structure of a boundary-aligned polar fluid, and show that defect locations are robust to changes in the material parameters.

Extended Data Fig. 2. Flipping the mixed boundary preference gives rise to the combination of hyp + 1 and r-ring for all geometries.

a, OP fields for different values of the acorn parameter ν at fixed material length scales λ/R₀ = 10⁶ and ξ/R₀ = 1 with k₀/k₂ = 10. b-d, Within the highly ordered regime, a hyp + 1 point defect and a radial disclination ring are present in the bulk for the whole range of the acorn parameter ν. Their height z and distance to the symmetry axis x varies slightly with the correlation length ξ (b) and the splay-to-bend ratio k₀/k₂ (d), but is unaffected by variations of the anchoring length λ (c). Material parameters kept constant in each case are the same as in panel a.

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Extended Data Fig. 3. The position of defects and the transitions between them are robust to parameter changes.

a, Within the highly ordered regime, the height z and distance to the symmetry axis x of the defects is not affected by a three order of magnitude variation of the anchoring length (ξ/R₀ = 1, k₀/k₂ = 10). b, By increasing the splay-to-bend ratio k₀/k₂, that is by further enforcing the splay-dominated regime, the defect positions change only slightly (λ/R₀ = 10⁶, ξ/R₀ = 1). c, The critical values of the acorn parameter where the transitions between defect configurations occur remain unchanged under variation of the mechanical parameters, giving ${\nu }_{\mathrm{c}}^{+}=0.38\pm 0.04$ and ${\nu }_{\mathrm{c}}^{-}=-0.56\pm 0.09$ (dashed lines and shaded area). Error bars show the range where the defect configuration is ambiguous.

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Polar defects determine sites of lumen nucleation

Topological defects are known to affect macroscopic properties of the systems they inhabit²⁷. In living systems, 2D nematic defects have been shown to function as organizing centres for biological processes like cell extrusion³³ or out-of-plane bending during morphogenesis⁵⁶. However, how 3D polar defects influence the multicellular organization of apico-basally polarized cells remains unexplored. Topological defects where tails of polarity arrows come together (that is, radial and hyperbolic positive defects) mark positions where contacts between multiple apical surfaces are likely. By analysing the contacts between individual epiblast cells using 3D segmentations⁵⁴, we indeed found that multicellular rosette structures cluster close to the position where we measured the topological defect in the average cellular orientation field (Fig. 4a and Methods). Epiblast cells nucleate fluid-filled cavities at such apical surface contacts^38,57, since apical surfaces feature distinct molecular compositions and functional properties, with vesicle trafficking and active osmolyte pumps localizing asymmetrically along the apico-basal axis^58–61. Actomyosin contractility drives the constriction of multicellular rosettes, while the junctional complexes seal the nascent cavity that eventually inflates^62,63. Live imaging confirmed that fluid-filled lumina nucleate from multicellular rosette structures located near the topological defect position (Fig. 4a,b, Extended Data Fig. 4a–c and Supplementary Video 1).

Fig. 4. Epiblast cells form multicellular rosettes and initiate lumen formation where their average orientation field has a topological defect.

a, Multicellular rosettes were identified from 3D segmented label images as regions where nine or more cells converge (top left; example rosette rendered in Blender). Quantified rosette positions (21 rosettes from n = 7 immunostained embryos; blue crosses, individual rosettes; blue square, mean; images acquired as part of another study⁵⁴) projected relative to embryo centroids into the y−z and x−z planes (relative positions Δx, Δy and Δz) show that these structures appear near the topological defect point of the average cellular orientation field (black point and gray dashed lines, Fig. 2e and Methods) Inset: orientation of the embryo boundaries. Estimating the positions of lumen centroids and shapes (shaded region) in the approximate x−z image plane (four lumina from n = 4 live-imaged embryos) shows lumen formation in the same region. R₀ is computed from the individual epiblast volumes V₀ via $R_0={(3V_0/4\mathrm{\pi })}^{1/3}$. Error bars, s.d. b, Central slice of a live-imaged embryo expressing membrane-targeted tdTomato (membrane, green) and histone H2B with green fluorescent protein (H2B-GFP; nuclei, magenta) shows the time points before and after lumen inflation with rosette position (white cross, quantified as membrane intensity peak; Methods) and lumen (filled circles; Extended Data Fig. 4 and Supplementary Video 1). Scale bar, 20 μm; time in hours and minutes.

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Extended Data Fig. 4. Lumina nucleate from rosette structures.

Central slice of representative live-imaged control (a-c) and ExE-devoid (d-f) embryos expressing membrane-tdTomato (membrane, green) and H2B-GFP (nuclei, magenta) at the time points before and after lumen inflation. Rosette structures were quantified using automatic detection of membrane fluorescence intensity peaks (white crosses), lumen positions were determined as the centre of a manually selected ellipsoidal region (white filled circles). Scale bars 20 μm, time in hours:minutes. Representative embryos selected from n = 17 embryos from 4 independent recovery experiments (control), and n = 9 embryos from 3 independent embryo recovery experiments (ExE-devoid).

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Continuum theory predicts how the number and spatial arrangement of defect-associated structures depends on the geometry of the confining surfaces. In particular, the geometry-driven transition we identified suggests that tissue shape controls the distribution of lumen nucleation sites, implying that externally manipulating boundary geometry could induce additional lumina: embryos with shapes $\nu \in ({\nu }_{\mathrm{c}}^{-},{\nu }_{\mathrm{c}}^{+})$ should contain a single initial lumen near the symmetry axis, whereas those outside this range should acquire additional off-axis sites at a distance comparable to the radius of the disclination ring. To test these predictions, we cultured mouse embryos for 18 h from E4.5 and performed immunostaining for an epiblast marker Oct3/4 and an apical surface marker, phosphorylated-ERM (pERM). This permitted quantifying lumen initiation sites together with the epiblast boundary geometry, for which we quantified the acorn parameter ν and a second shape parameter σ, which together capture the curvature and vertical displacement of the tissue boundary (Fig. 5, Methods).

Fig. 5. Defect positions correspond to sites of lumen nucleation in the mouse epiblast.

a, Immunostaining of a control mouse embryo cultured ex vivo for 18 h from E4.5 shows the lumen initiation site (white arrowhead) in a central cross-section, stained for actin (phalloidin), epiblast marker (Oct3/4) and either apical marker (pERM; a,g) or visceral endoderm marker (Gata4) and DNA (DAPI; d). Tissue–ECM (light green) and tissue–tissue (light gray) boundaries were segmented and fitted with the generalized acorn geometry, parameterized by $\text{σ}\equiv {\cos }^{-1}(-c_{\mathrm{\text{β}}}/R_{\mathrm{\text{β}}})$, with base centred at (0, c_β), and ν ≡ z_α/R_β. b,c, All control embryos (n = 6) have $\nu \in ({\nu }_{\mathrm{c}}^{-},{\nu }_{\mathrm{c}}^{+})$ and contain a single lumen with a centroid near the symmetry axis (black dots). The theoretically predicted lumen nucleation position (red dot) shows the defect for the average experimental shape parameters in the control condition (${\overline{\nu }}_{\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{l}}=-0.07$, ${\overline{\text{σ}}}_{\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{l}}=1.15$), using experimentally estimated material parameters (Fig. 2; R₀ computed from individual epiblast volumes V₀ as $R_0={(3V_0/4\mathrm{\pi })}^{1/3}$). Full theoretical field (c) with average and s.d. of experimental lumen positions (black dot and error bars, n = 6). d, Immunosurgical removal of ExE precursors generates a near-spherical epiblast surrounded by ECM secreted by visceral endoderm (Extended Data Figs. 4 and 5). e,f, Lumen centroid positions in immunostained ExE-devoid embryos (black dots, n = 4 cultured for 48 h after immunosurgery; gray dots, n = 7 cultured for 72 h) appear at the predicted position for uniform perpendicular anchoring (average shown in f; error bars, s.d.). Rosette and lumen positions quantified from live imaging (blue crosses and dots, n = 9) can be shown in the same coordinate system due to the rotational symmetry in this condition. g, Blocking inward folding of the ExE perturbs the tissue–tissue interface, producing shapes with $\nu >{\nu }_{\mathrm{c}}^{+}$. h,i, Shape-manipulated embryos (n = 10) display additional off-axis lumen sites. On-axis and off-axis lumina (filled and open points, respectively) nucleate on average near the predicted positions for the experimentally measured shape parameters in the shape-perturbed condition (${\overline{\nu }}_{\mathrm{m}\mathrm{a}\mathrm{n}}=0.79$, ${\overline{\text{σ}}}_{\mathrm{m}\mathrm{a}\mathrm{n}}=2.25$, n = 10; error bars, s.d.). Scale bars, 20 μm.

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We first assessed unperturbed embryos (n = 6), and found that these have shapes $\nu \in ({\nu }_{\mathrm{c}}^{-},{\nu }_{\mathrm{c}}^{+})$, that is ${\nu }_{\exp }\in [-0.14,0]$ and ${\sigma }_{\exp }\in [0.8,1.43]$. As predicted, they consistently contain a single lumen near the radial +1 defect in the theoretical OP field computed for their average shape (Fig. 5a–c).

We next analysed the behaviour in the uniform-confinement limit. For a spherical confining surface with a uniform perpendicular preferred orientation, the theory predicts the robust formation of a single lumen initiation site at the tissue centre (Fig. 1b). To test this prediction, we experimentally perturbed the boundary organization of the epiblast by immunosurgically removing precursors of the ExE tissue, thereby obtaining a near-spherical epiblast tissue uniformly surrounded by an ECM-rich boundary that imposes perpendicular anchoring (Fig. 5d). Indeed, both immunostaining (n = 11) and live-imaging (n = 9) data show that ExE-devoid epiblasts robustly form rosette structures and lumina at the tissue centre, as predicted (Fig. 5e,f, Extended Data Figs. 4d–f and 5 and Supplementary Video 4). This observation provides a mechanistic explanation for the typical behaviour of pluripotent stem cell systems, which present a single central lumen when embedded in Matrigel, a condition that mimics uniform ECM confinement^39,64,65. Conversely, upon perturbing perpendicular anchoring by genetically interfering with the deposition of the relevant ECM component laminin γ1, we observed a complete loss of rosette formation, as predicted⁵⁴ (Extended Data Fig. 6).

Extended Data Fig. 5. ExE-devoid epiblasts form a rosette structure and a subsequent lumen in the centre.

a-k, Mouse blastocysts whose trophectoderm was removed, thus not forming ExE, were cultured for 2 days (n = 4, a-d) or 3 days (n = 7, e-k). Immunostaining for Oct3/4 (epiblast marker, cyan), phalloidin (membrane, yellow), Gata4 (visceral endoderm, red), and DAPI (nucleus for all cells, white). Scale bars 20 μm. l-t, Sum projections of the central 30% slices of live-imaged ExE-devoid embryos expressing membrane-tdTomato (membrane, green) and H2B-GFP (nuclei, magenta) show central rosettes prior to lumen inflation (n = 9). Scale bars 20 μm. u, Rosette and lumen centroid coordinates in the ExE-devoid embryos measured with respect to the corresponding epiblast centroids (zoomed-in version of Fig. 5e). A spherical system with homogeneous perpendicular anchoring (Fig. 1b) predicts lumen initiation in the centre (dashed lines and red point). The volume V₀ of each individual epiblast tissue defines $R_0={\left(3V_0/4\pi \right)}^{1/3}$ for each embryo.

Source data

Extended Data Fig. 6. Inhibition of perpendicular anchoring to the ECM disrupts rosette formation.

a, Disrupting ECM deposition via genetic ablation of Laminin γ1 (Lamc1) abolishes the formation of orientational defects, as shown in Fig. 5e of ref. ⁵⁴. Immunofluorescence images with phalloidin staining show representative actin distributions in Lamc1^+/+, Lamc1^+/- and Lamc1^−/− embryos at E5.0 (images acquired as part of ref. ⁵⁴; actin rosette detection was performed as described in Methods). We also note that some wildtype and heterozygous embryos featuring an outward-curving ExE boundary prior to the invagination of this tissue (stars in panel a), display two actin rosettes instead of one, consistent with the prediction of our theory for this geometry. Scale bars 20 μm. b, Quantifying multicellular rosettes using 3D cell segmentations (see Methods) shows complete loss of rosettes in homozygous Lamc1 deleted embryos. n = 15, 17 and 10 for Lamc1^+/+, Lamc1^+/- and Lamc1^−/−, respectively. Only epiblasts with 40 or more cells were analysed. Fisher’s exact test (two-sided) with categories #rosettes = 0 and #rosettes > 0 was used to compute p-values. No adjustment was made for multiple comparisons.

Source data

Lastly, we perturbed the shape of the epiblast boundary, specifically the epiblast–ExE interface. The curvature of this interface depends on a preceding morphogenetic event, the inward folding of the adjacent ExE^38,66. Disrupting this process results in embryos with deformed interfaces between the epiblast and this extra-embryonic tissue. By blocking the inward folding of the ExE (Methods), we obtained shape-perturbed embryos (n = 10) with $\nu >{\nu }_{\mathrm{c}}^{+}$, that is ${\nu }_{\exp }\in [0.48,1.25]$, and ${\sigma }_{\exp }\in [1.42,2.82]$ (Fig. 5g and Extended Data Fig. 7). Embryos that did not undergo elongation along the distal–proximal axis or did not reach an epiblast cell number above 50 were excluded³⁸, as was one embryo in which the epiblast–ECM interface was concave and hence incompatible with our shape parameterization. As predicted by continuum theory, we identified additional off-axis lumen nucleation sites near predicted positions in the majority of these embryos (Fig. 5h,i and Extended Data Fig. 7). Notably, epiblast tissue volume remained largely unchanged by this perturbation ($V_0^{\mathrm{m}\mathrm{a}\mathrm{n}}\approx (1.8\pm 0.4)\times 10^5{\mathrm{\text{μ}}\mathrm{m}}^3$ for shape-manipulated versus $V_0^{\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{l}}\approx (2.3\pm 0.4)\times 10^5{\mathrm{\text{μ}}\mathrm{m}}^3$ for control embryos, mean ± s.d.), indicating that the observed differences in lumen position arise from altered boundary geometry rather than tissue size.

Extended Data Fig. 7. Lumina vs. generalised-acorn parameters of the epiblast samples.

Lumen positions along x and z are shown as a function of the individually fitted parameters of the generalised acorn model, ν and σ. The colour map represents the relative deviation from the ideal acorn shape, characterised by σ₀ = π/2. Coloured shaded regions show the defect position ranges for ideal acorns with λ/R₀ = 10⁶, ξ/R₀ ∈ [0.4, 100] and k₀/k₂ = 10. The volume V₀ of the epiblast tissue defines $R_0={\left(3V_0/4\pi \right)}^{1/3}$ for each embryo. Error bars are determined by residual bootstrapping of the epiblast boundaries (n = 100 bootstrapped samples from 16 embryos, see Methods). Vertical lines connect lumina from the same embryo.

Source data

Taken together, the defect positions in the theoretical OP fields quantitatively predict the lumen nucleation positions across the control and different perturbation conditions, without parameter fitting (Fig. 5). This agreement is particularly striking given the biological complexity of the system and the small number of constituent cells: despite variability in the single-cell configurations (Fig. 4a), the coarse-grained field correctly anticipates average rosette and lumen formation sites, suggesting that the underlying defect structure is a robust feature linking tissue geometry to polarity organization. In particular, we observe that external manipulation of the embryo shape produces additional nucleation sites near the position predicted by the geometry-driven defect transition we identified.

Outlook

Confinement and boundary effects impact the bulk organization of complex materials. We show how interactions with confining surfaces drive the creation of different topological defects in a polar fluid, and identify distinct roles for the confining geometry and bulk and surface mechanical parameters: while the competition between bulk distortion and surface anchoring governs the transitions between defect-free and defect-containing states, the types of defects observed and the transitions between different defect configurations depend solely on the geometry. Moreover, in the surface-dominated regime, 3D defect positions are robust against anchoring and correlation length variations. We thus report a novel, charge-preserving defect transition controlled by confining geometry. Indeed, because the preferred orientation p₀ is defined within the surface, the boundary geometry affects how this field projects into the 3D bulk. As a result, shape changes can modulate internal orientations without requiring changes in material parameters.

As the collective excitations of the orientation field, topological defects are global and quantized features, protected against local perturbations⁶⁷. As such, they act as robust organizing centres even in small or heterogeneous systems—a property broadly exploited across physical systems from liquid crystals to active matter^68–70. In living systems, moreover, the impact of topological defects on macroscopic spatial organization is not limited to distortion effects; they can also localize regulatory molecules that drive subsequent biological processes²⁷. Examples include the activation of cell-death signals leading to cell extrusion³³, or assembling organizational centres during morphogenesis^56,71. For the mouse epiblast—a tissue consisting of apico-basally polarized cells—we find that defects mark the positions where fluid-filled cavities form. Although lumina fulfil critical biological functions in tissue organization and cell fate specification^57,72,73, how their spatial arrangement within a tissue arises has been unclear. The molecular processes that drive lumen initiation in the epiblast as well as in numerous other tissues⁶¹, such as active osmolyte pumping, localize to structures at the apical side of cells^58,59. Indeed, multicellular rosettes, where several apical surfaces converge, act as canonical initiation sites of nascent lumina in diverse epithelial systems^39,72,74,75. Lumen initiation requires trans-interactions between junctional complexes at the apico-lateral interface of neighbouring cells, which seal and eventually inflate the cavity. Topological defects in the continuum field at which the tails of the polarity arrows converge mark spatial locations where contacts between the apical surfaces of multiple cells are overall most likely to form. Despite the relatively small number of cells in the mouse epiblast, and the overall variability in the specific configurations of the cells across different embryos, we indeed found that defect positions are quantitative, parameter-free predictors of lumen nucleation sites in this system. Furthermore, we induced additional lumen nucleation sites by experimentally perturbing the boundary geometry beyond the transition point, where the correspondingly confined polar fluid presents an additional defect. That the spatial distribution of early luminal structures follows the predicted geometry-dependent changes suggests that embryo shape regulates lumen formation via orientational effects. Lumen formation might therefore serve as a shape-sensing checkpoint mechanism in the embryo, determining successful amniotic cavity development. While these findings are likely relevant also to other developing systems with tissue-scale polarity organization, luminogenesis can in general proceed through diverse mechanisms, not all of which involve topological defects. For example, lumina have been reported to form via epithelial folding and closure around a cavity⁷⁶, or as networks between individual cells⁷⁷.

It will be exciting to investigate the interplay between surface-induced alignment and boundary geometry in other experimental systems to explore the generality of this mechanism for the spatial organization of defect-associated structures. Importantly, since defect configurations depend almost exclusively on the geometry, the relevant observables are directly accessible from imaging data. Quantitative predictions can be made without the fitting of material parameters, indeed without having to make detailed assumptions about specific bulk or surface properties. This robustness not only renders experimental testing feasible, but also suggests that spatial control of defect-associated processes via boundary geometry is a general phenomenon across systems with orientational degrees of freedom, independent of scale or material specifics.

For multicellular systems, our formalism offers a general and tractable framework for studying cell–ECM interactions. Using a single anchoring parameter to capture the coupling between cellular polarity and extracellular materials like basement membranes will allow the investigation of the feedback between geometry changes and boundary-induced order from active ECM remodelling by cellular processes—properties distinguishing multicellular systems from passive anisotropic materials in confinement.

In conclusion, our investigations reveal how boundary geometry controls defect configurations within polar materials, and suggest a new biological function for 3D polar defects in multicellular systems. More broadly, the fact that shape can control defect configurations independently of material specifics constitutes a general geometry-dependent organizational principle. Having demonstrated its predictive power in a system as complex and intractable as a living embryo, we expect this defect-control mechanism to be relevant in diverse orientational systems.

Methods

Ethics statement

All animal work was performed in the Institute of Laboratory Animals (ILA), with permission from the Animal Research Committee, Graduate School of Medicine (approved number, MedKyo 23065) and the Committee for Safety Control of Recombinant DNA Experiments, Kyoto University (approved number, 230029). ILA is operated according to the Regulations on Animal Experimentation at Kyoto University.

Geometrical definitions

Ideal acorn

If the total volume V₀ of the system is fixed, geometrical parameters c_α, R_α and R_β (Fig. 3a in the main text) are defined solely by ν ≡ z_α/R_β through the following system of equations:

$$\left\{\begin{array}{c}{R}_{\mathrm{\alpha }}^2=R_{\mathrm{\beta }}^2+c_{\mathrm{\alpha }}^2\hfill \\ \nu R_{\mathrm{\beta }}=c_{\mathrm{\alpha }}+\mathrm{sgn}\nu R_{\mathrm{\alpha }}\hfill \\ V_0=\frac{2\mathrm{\pi }}{3}{R}_{\mathrm{\beta }}^3+\mathrm{sgn}\nu \frac{\mathrm{\pi }}{3}{\left(R_{\mathrm{\alpha }}-\mid c_{\mathrm{\alpha }}\mid \right)}^2\left(2R_{\mathrm{\alpha }}+\mid c_{\mathrm{\alpha }}\mid \right)\hfill \end{array}\right).$$ 5

The cap S_α—representing the tissue–tissue interface—can be parameterized in spherical coordinates as $r=r_{\mathrm{\alpha }}\left(\theta \right)=c_{\mathrm{\alpha }}\cos \theta +$$\sqrt{R_{\mathrm{\alpha }}^2-c_{\mathrm{\alpha }}^2{\sin }^2\theta }$ for polar angle θ ∈ [0, π/2] (respectively θ ∈ [π/2, π]) if ν > 0 (respectively ν < 0). Note that this includes ν = 1, since in that case c_α = 0 and R_α = R_β. If ν = 0, however, S_α is given by the disc with r ∈ [0, R_β] and θ = π/2. Finally, the base S_β—corresponding to the tissue–ECM interface—is simply r = R_β and θ ∈ [π/2, π] for all ν. In all cases, $\varphi \in \left[0,2\mathrm{\pi }\right)$. With these definitions, the mixed preferred orientation in the spherical basis consists of ${\mathbf{p}}_0^{\mathrm{\beta }}=\left(1,0,0\right)\forall \nu$ on S_β, and

$${\mathbf{p}}_0^{\mathrm{\alpha }}=\left\{\begin{array}{cc}\left(1,0,0\right)\hfill & \mathrm{if}\nu =0\hfill \\ \left(\sin \theta -\mathrm{sgn}\nu g_{\mathrm{\alpha }}\left(\theta \right)\sin \theta \cos \theta \right),\hfill & \mathrm{if}0<\mid \nu \mid <1\hfill \\ \left(\cos \theta +\mathrm{sgn}\nu g_{\mathrm{\alpha }}\left(\theta \right){\sin }^2\theta ,0\right)\hfill \\ \left(0,1,0\right)\hfill & \mathrm{if}\nu =1\hfill \end{array}\right)$$ 6

with

$$g_{\mathrm{\text{α}}}\left(\theta \right)=\frac{r_{\mathrm{\text{α}}}\left(\theta \right)}{\sqrt{R_{\mathrm{\text{α}}}^2-r_{\mathrm{\text{α}}}^2\left(\theta \right){\sin }^2\theta }}$$ 7

on S_α. Note that the tissue–tissue interaction could in principle be modelled as nematic. However, we expect no notable difference, since the outward-normal ${\mathbf{p}}_0^{\mathrm{\text{β}}}$ imposed by the tissue–ECM boundary and the polar character of the constituents break this symmetry and favour the north–south orientation given by ${\mathbf{p}}_0^{\mathrm{\text{α}}}$.

Generalized acorn

In a more general scenario, the base S_β of the acorn can be centred at a point (0, c_β) different from the origin of coordinates. Such a system can be characterized by two dimensionless parameters: ν, as introduced before, and $\text{σ}\equiv {\cos }^{-1}(-c_{\mathrm{\text{β}}}/R_{\mathrm{\text{β}}})$ (Fig. 5a,g in the main text). Considering a constant volume V₀, the equations that geometrical parameters c_α, R_α, c_β and R_β must satisfy are

$$\left\{\begin{array}{cc}{R}_{\mathrm{\alpha }}^2\hfill & =R_{\mathrm{\beta }}^2+{\left(c_{\mathrm{\beta }}-c_{\mathrm{\alpha }}\right)}^2+2\left(c_{\mathrm{\beta }}-c_{\mathrm{\alpha }}\right)R_{\mathrm{\beta }}\cos \sigma \hfill \\ \nu R_{\mathrm{\beta }}\hfill & =c_{\mathrm{\alpha }}+\mathrm{sgn}\nu R_{\mathrm{\alpha }}\hfill \\ V_0\hfill & =\frac{\mathrm{\pi }}{3}{R}_{\mathrm{\beta }}^3\left(2-\cos \sigma \right){\left(1+\cos \sigma \right)}^2+\nu \frac{\mathrm{\pi }}{6}{R}_{\mathrm{\beta }}^3\left(3{\sin }^2\sigma +{\nu }^2\right)\hfill \end{array}\right).$$ 8

While S_α remains the same as in the ideal case, S_β is now too a spherical cap with $r=r_{\mathrm{\beta }}\left(\theta \right)=c_{\mathrm{\beta }}\cos \theta +\sqrt{R_{\mathrm{\beta }}^2-c_{\mathrm{\beta }}^2{\sin }^2\theta }$. By construction, the polar angle is still θ ∈ [π/2, π], and $\left(\varphi \in \left[0,2\mathrm{\pi }\right)\right)$ as always. The normal to S_β is no longer a constant and is written ${\mathbf{p}}_0^{\mathrm{\beta }}=\left(-\cos \theta +g_{\mathrm{\beta }}\left(\theta \right){\sin }^2\theta \right),$$\left(\sin \theta +g_{\mathrm{\beta }}\left(\theta \right)\sin \theta \cos \theta ,0\right)$ with g_β(θ) defined analogously to equation (7).

Numerical details

We used the FEniCSx library DOLFINx (ref. ⁷⁸) to implement the finite-element method in Python3 (ref. ⁷⁹). We minimized equation (2) in a 3D, symmetry-agnostic parametrization for selected relevant parameters and observed that no spontaneous breaking of axial symmetry occurred (Supplementary Note 2). Thereafter, the mesh (with resolution 0.05) was parameterized in terms of r and θ only, corresponding to a constant-ϕ slice. We computed the variation of $\mathcal{F}$ (equation (2)) with respect to the 3D vector OP p in the direction of a test function φ to derive its weak formulation. The resulting nonlinear problem was solved using a Newton solver with a relative tolerance of 10⁻⁶.

The number and type of defects in the OP field were determined by visual inspection of the phase and direction of p. Using the fact that ∣p∣ → 0 in the vicinity of defects due to the large local distortions they induce, we inferred the position of a defect as the central point of a mesh cell where ∣p∣ = 0, surrounded by a region with ∣p∣ > 0. The numerical uncertainty of these estimates thus depended on the size of the disordered region surrounding each defect and the resolution of the mesh. We defined a threshold p* and checked in each cell whether ∣p∣ < p*. The value of p* was fixed for each simulation, though we varied it conveniently from one to another in order to preselect the least possible number of candidate points per defect (but in all cases, p* ≤ 10⁻²). When more than one candidate point was selected, the location of the defect was calculated as the average of the candidate points. Error bars were determined as the s.d. of their spatial distribution.

Mouse work

All mice were maintained in specific pathogen-free conditions with a 14 h–10 h light–dark cycle and used for experiments at the age of 8 to 30 weeks. This study used peri-implantation and early post-implantation embryos, that is, before apparent sexual differentiation occurred.

Mouse lines and genotyping

The following mouse lines were used in this study: F1 hybrid strain between C57BL/6NCrSlc and C3H/HeSlc (B6C3F1/Slc; Japan SLC) as wild-type, mTmG (ref. ⁸⁰), H2B-GFP (ref. ⁸¹) and Lamc1^tm1Str (floxed)⁸². To generate Lamc1^+/- mice, Lamc1^{floxed/floxed} females were crossed with Zp3-Cre^tg/+ males, followed by crossing between Lamc1^floxed/+;Zp3-Cre^tg/+ females and B6C3F1 males⁸³. Standard genotyping procedures were used to genotype transgenic mice (for primers and polymerase chain reaction (PCR) product sizes; Supplementary Table 1).

Embryo recovery and culture

To obtain mouse embryos, mice were naturally mated, and the midpoint of the light period on the day when a vaginal plug was detected was defined as embryonic day 0.5 (E0.5). Recovery of embryos was performed under a stereomicroscope (Zeiss, Stemi 508) equipped with a thermo plate (Tokai Hit, TPi-STMX) heated at 37 °C. E3.5 blastocysts were recovered by flushing dissected oviducts and uteri with Global medium with HEPES buffer (LifeGlobal, LGGH-050). Blastocysts were treated with 0.5% w/v Protease (Sigma, P8811) for 2 min in an incubator with a humidified atmosphere of 5% CO₂ at 37 °C to remove zona pellucida and subsequently cultured for 24 h in 5 μl drops of Global medium (LifeGlobal, LGGG-050) under mineral oil. To culture peri-implantation embryos, a 3D-gel-embedded embryo culture (3D-geec) was performed as described in ref. ³⁸. In brief, mouse embryos at E4.5 were recovered from dissected uteri and cultured in a mixture of Matrigel (Corning, 356230, lot number 7345012) and collagen I (Corning, 354236, lot number 2055001). The mural trophectoderm (mTE) was microsurgically removed to enable polar trophectoderm (pTE) invagination in the control condition, whereas maintaining the mTE intact blocked pTE invagination and generated a boundary perturbation in 18 h of culture.

Immunosurgery

To remove the ExE precursors, the trophectoderm (TE) layer of blastocysts was removed by immunosurgery, as described previously³⁸. Briefly, cavity-expanded blastocysts after 24 h post-recovery at E3.5 were incubated for 30 min in anti-mouse serum antibody (Sigma, M5774) diluted 1:3 in Global medium at 37 °C, washed in Global medium, followed by incubation for 20 min in complement sera from guinea pig (Sigma, S1639) diluted 1:3 in Global medium at 37 °C. Lysed TE cells were removed by mouth pipetting. As the pTE is more tightly attached to the inner cell mass than the mTE, extensive mouth pipetting resulted in complete TE removal. Embryos were then allowed to recover for 6 h in IVC1 medium³⁹ before being embedded for 3D-geec as described above.

Live imaging

The 3D-geec embryos were live imaged using an inverted light-sheet microscope (Bruker, Luxendo, InVi SPIM), as described previously³⁸. Briefly, embryos were embedded in a 15 μl gel mix within the V-shaped sample holder attached with transparent FEP foil, carefully positioned so that they were in proximity but did not attach to the foil, which would disrupt morphogenesis via adhesion. After gel solidification, embryos were immersed in 75 μl IVC1 medium and further covered with 200 μl mineral oil to prevent evaporation. The sample holder was mounted in an environmentally controlled incubation box with 5% CO₂ and 5% O₂ at 37 °C. The InVi SPIM was equipped with a Nikon ×25, 1.1 numerical-aperture (NA) water-immersion detection objective and a Nikon ×10, 0.3 NA water-immersion illumination objective. The illumination plane and focal plane were aligned before the imaging session and maintained during the imaging. Images were taken every 5, 20 or 30 min depending on the experiment by a complementary metal–oxide–semiconductor (CMOS) camera (Hamamatsu, ORCA Flash4.0 V2) with line-scan mode in LuxControl (Luxendo). The imaged volume was 425.98 × 425.98 × 400 μm³ with a physical voxel size of 0.208 × 0.208 × 1.000 μm³. The lasers and filters used were 488 nm and BP525/50, and 561 nm and LP561, to image GFP and tdTomato fluorophores, respectively. Exposure time for each plane was set to 50 ms.

Immunostaining

Embryos were fixed with 4% paraformaldehyde (FUJIFILM Wako, 166-23251) in phosphate-buffered saline (PBS) for 30 min and subsequently permeabilized with 0.5% Triton X-100 (Nacalai, 12967-32) in PBS for 30 min. Embryos were incubated in 3% bovine serum albumin (BSA; Sigma, A9647) and 0.05% Triton X-100 in PBS overnight at 4 °C and then subjected to immunostaining. Primary antibodies against Oct3/4 (Santa Cruz Biotechnology, sc-5279 AF647), Gata4 (R&D Systems, AF2606), collagen IV (Millipore, AB756P), pERM (Cell Signaling, 3726) and E-cadherin (BD Biosciences, 560064) were diluted at 1:100. Donkey anti-rabbit IgG Alexa Fluor Plus 488 (Invitrogen, A32790), donkey anti-goat IgG Alexa Fluor Plus 488 (Invitrogen, A32814), DAPI (Invitrogen, D3571) and Alexa Fluor Plus 555 phalloidin (Invitrogen, A30106) were simultaneously used at the secondary antibody staining. Images were obtained by LSM880 or LSM980 Airyscan mode, equipped with a C-Apochromat ×40, 1.2 NA water-immersion objective (Zeiss).

Statistics and reproducibility

The sample size for each experiment was determined by the number of embryos available from one pregnant female, as experimental constraints. No statistical method was used to predetermine sample size. Each experiment was independently repeated three times, on different days and using embryos obtained from different female mice.

After 18 h of culture, embryos were excluded from analysis if they met either of the following criteria: (1) failure to show proximal–distal axis elongation or (2) epiblast cell number below 50. These criteria were pre-established to ensure analysis of developing embryos. In addition, one embryo was discarded in which the epiblast–ECM interface was concave and therefore incompatible with the family of shapes considered in our model. For ExE-devoid embryos, an epiblast which was not covered by visceral endoderm was excluded from analysis.

To ensure an unbiased comparison, embryos were assigned to control and experimental groups using a balanced distribution method based on their morphological ranking at embryo recovery. For example, embryos ranked from 1 to 8 according to developmental progression were distributed as (1, 4, 5, 8) and (2, 3, 6, 7), ensuring that each group spanned the full range of variability. The assignment of these balanced groups to either the control or experimental condition was then determined randomly, providing an additional level of randomization.

Blinding was not possible due to the visually distinct nature of the experimental manipulations. To minimize potential bias, quantitative analyses were performed using automated image processing whenever possible.

Image analysis

The epiblast tissue marked by Oct3/4 positive cells and the lumen inside it were manually segmented with Napari⁸⁴. Labels for the entire area were drawn every 30 slices with a 0.16 μm interval, followed by the plug-in ‘napari-label-interpolator’ to fill the entire volume. Then the label statistics function in the plug-in ‘napari-simpleitk-image-processing’ was used to count the number of voxels of the epiblast and lumen volumes^85,86. We checked that perturbation of the tissue–tissue boundary introduced only geometric changes that preserved total volume. Indeed, average epiblast tissue volume was (1.8 ± 0.4) × 10⁵ μm³ in the manipulated cohort and (2.3 ± 0.4) × 10⁵ μm³ in the control condition. For the nascent nucleated lumina, their position measurement was performed based on the space encircled with pERM signals with Fiji⁸⁷, using the middle cross-section, which was extracted by the 3D rotation function in Imaris (Bitplane).

Preprocessing for machine learning-based membrane segmentation was performed with Fiji⁸⁷. Membrane segmentation and custom-model training were performed using Cellpose^88,89 GUI or CLI using a bash script. Manual correction of segmentation and cell tracking were performed with Napari⁸⁴ (Supplementary Video 3). Segmentation to process the 3D images of the membrane signal was carried out as described in refs. ^54,90. The details of the individual steps are described below.

1. Preprocessing: Airyscan images were binned to 0.1647 × 0.1647 × 0.1600 μm³ by averaging 2 × 2 × 1 voxels, followed by isotropic transformations of voxels to a cube with a length of (0.1632 ± 0.0002) μm. To quantify spatial parameters, we set the length of the voxel to 0.1632 μm and ignored the associated error. Two channels for E-cadherin and phalloidin signals of the isotropically scaled Airyscan images were combined to generate a ubiquitous membrane channel by summing their signal intensities. The combined membrane channels were used as inputs for segmentation.

2. Developing custom segmentation models: Since the pretrained models provided by Cellpose v.2.0 required an insuperable amount of manual correction for our 3D images, we developed custom segmentation models tailored for Airyscan images. In brief, a preprocessed 3D image of an E5.0-stage embryo was resliced into three orthogonal planes (x–y, y–z and x–z), and five slices were extracted in each plane at a 50-slice interval across the entire image. The resulting slices were subjected to 2D segmentation using the Cellpose pretrained model, CPx, followed by manual corrections of all cells in the slice using Napari. Following the instructions, two embryo datasets, each consisting of 15 pairs of slices and ground truth masks, were used to train the neural network, generating AS_model_1. To further improve AS_model_1, we conducted additional training by incorporating two extra embryo datasets into the neural network, resulting in AS_model_2. While both models showed comparable performance on images that have high signal-to-noise ratios, AS_model_2 was more robust to the images with various signal-to-noise ratios. Therefore, we used AS_model_2 in this study.

3. Batch segmentation: For 3D segmentation of Airyscan images, segmentation parameters, such as flow_threshold, cellprob_threshold and stitch_threshold, were set to default values, except for the cell diameter, which was set to 60 pixels. The segmentation model AS_model_2 was chosen for this process.

4. Manual correction: Segmentation errors were manually corrected by referring to the original images, and epiblast cells were selected and counted in Napari using a custom plug-in (napari-segmentation-toolbox; ref. ⁹¹). The corrected segmentation masks were saved as .tif files for further analysis.

Multicellular rosettes were computationally identified from 3D segmented label images⁵⁴ using a custom Python pipeline. The detection method consisted of two steps: (1) generation of a multicellular contact map and (2) spatial clustering of candidate voxels.

1. Multicellular contact map: For each voxel in the label images, we computed a contact score defined as the number of distinct cells within a local neighbourhood. A cubic dilation kernel was applied to account for isotropic voxel dimensions (0.1632 μm in x, y and z). Specifically, for each cell in the label image, a binary mask was created and dilated using a maximum filter with a radius of approximately 1.0 μm in physical space (corresponding to 6 voxels). The contact score at each voxel was calculated as the sum of overlapping dilated cell masks. Background voxels (label = 0) were excluded from the analysis.

2. Rosette cluster identification: Voxels with contact scores of ≥9 (indicating proximity to nine or more cells) were identified as rosette candidates. These candidate voxels were then clustered using connected component analysis with 26-connectivity (considering all face-, edge- and corner-adjacent neighbours in three dimensions). Clusters containing fewer than 125 voxels (equivalent to a 5 × 5 × 5 cube, representing approximately 0.54 μm³ in physical volume) were excluded to eliminate noise and small-scale artefacts. Each remaining cluster was designated as a rosette structure.

In order to focus on embryos after ExE invagination, at which point they acquire an acorn-like geometry, the dataset shown in Fig. 4a contains all embryos with >75 cells. Since laminin γ1 knock-out embryos fail to develop up to later stages, epiblasts with >40 cells were considered in Extended Data Fig. 6.

To estimate the position of early lumen initiation sites in wild-type and Lamc1 mutant epiblasts, we implemented a fully automated image analysis pipeline in Fiji (ref. ⁸⁷), which detects the fluorescence intensity maxima of the actin-marker phalloidin. We performed a sum z projection of the slices corresponding to the central 30% of the imaged volume, and used the Oct3/4 channel to generate a binary mask of the epiblast tissue by performing a 10 pixel Gaussian blur, and subsequent thresholding and hole filling. Within this region of interest, we applied a 3 pixel Gaussian blur to the phalloidin channel, normalized the intensity values to a 0–255 range and detected all maxima with a prominence of 170.

A similar method was used to detect rosettes in the membrane channel of live-imaged control and ExE-devoid embryos. In the case of the former, we manually rotated the images to align the embryonic axis with the vertical direction; this was not necessary in the ExE-devoid cohort due to their rotational symmetry. After a sum z projection of the slices corresponding to the central 30% of the embryo volume, we performed a manual selection of the epiblast tissue. We then applied a 10 pixel Gaussian blur, and subsequent thresholding and hole filling to generate a binary mask and determine the epiblast centroid and characteristic size, calculated as $R_0=\sqrt{A_0/\mathrm{\pi }}$ with A₀ the area of the mask. Within this region of interest, we applied a 3 pixel Gaussian blur, normalized the intensity values to a 0–255 range and detected all maxima with a prominence of 170. To analyse the images that contained lumina, we selected only the central slice and followed the same procedure to generate the epiblast mask and determine its centroid. We then manually selected an ellipsoidal region encompassing the lumen and measured its centre, major and minor axes, and angle.

Determination of the experimental nematic field

Epiblast cell principal inertia vectors

We first convert the epiblast segmentation into a mesh representation using the Python package scikit-image (ref. ⁹²). Each segmented cell is reconstructed as an individual mesh, and a global mesh for the entire epiblast region is generated. To ensure consistency in the analysis, all meshes are post-processed to correct face orientations, and only the largest connected component is retained. The final epiblast mesh stores cell labels as vertex attributes, enabling cell-specific calculations. We then use the trimesh library⁹³ to compute various geometric and topological properties, including cell volume, surface area, centroid and principal inertia vectors. Additionally, the centroid of the entire epiblast region is determined to facilitate further spatial analyses. Details of this and the following sections are described in ref. ⁵⁴.

Distal–proximal axis

Imaging data are annotated with at least six points in Napari (ref. ⁸⁴), through which a regression plane approximately separates the boundary between the tissue–tissue and tissue–ECM boundary regions. One point is annotated to denote the distal tip of the epiblast. We define the rotation axis as a line that passes through the epiblast centroid and the annotated tip point. Two points are also annotated to indicate the transverse edge of the epiblast, the mean position of which is used to establish a plane and define angle ϕ = 0.

Weighted nematic average over rotational slices

To construct a nematic field representing average cell order in a given epiblast, we first reorient the coordinate system such that the z axis aligns with the distal–proximal axis, previously annotated. The spatial coordinates are then rescaled to normalize the epiblast volume to unity. At each spatial location r = (x, y, z), we extract the major principal inertia vector v(r) and its corresponding inertia components λ₁ ≤ λ₂ ≤ λ₃ from the occupying cell mesh. The elongation axis is then defined as $\mathbf{v}\prime \left(\mathbf{r}\right)=\eta \left(\mathbf{r}\right)\mathbf{v}\left(\mathbf{r}\right)$, where the shape anisotropy factor $\eta \left(\mathbf{r}\right)\equiv \left({\lambda }_3-\sqrt{{\lambda }_1{\lambda }_2}\right)/{\lambda }_3$ ranges from 0 for spheres up to 1 for infinitely elongated needles. Next, we define M = 36 equidistant angles ϕ_m, generating rotational slices around the z axis. For each slice, we use the rotation matrix R(ϕ_m) which maps from the rotational slice to the x–z plane to obtain elongation axes within the x–z plane as ${\mathbf{v}}_m^{″}\left(\mathbf{r}\right)=R\left({\varphi }_m\right)\mathbf{v}\prime \left(R^{\top }\left({\varphi }_m\right)\mathbf{r}\right)$. We then compute the weighted Landau–de Gennes Q-tensor, defined as

$$Q\left(\mathbf{r}\right)\equiv \frac{1}{M}\sum _{m=1}^M\mid \mid {\mathbf{v}}_m^{″}\left(\mathbf{r}\right)\mid \mid \left(\frac{d}{d-1}\frac{{\mathbf{v}}_m^{″}\left(\mathbf{r}\right)\otimes {\mathbf{v}}_m^{″}\left(\mathbf{r}\right)}{\mid \mid {\mathbf{v}}_m^{″}\left(\mathbf{r}\right)\mid {\mid }^2}-\frac{I}{d-1}\right)$$ 9

for spatial dimension d = 3.

Cross-embryo-averaged nematic field

We calculate the embryo-averaged Q-tensor $Q={\sum }_{k=1}^K{Q}_k/K$, where K is the number of embryos and Q_k denotes the previously computed Q-tensors of individual embryos (equation (9)). We then take its principal eigenvector V(r) and the corresponding eigenvalue Λ(r) as the (unsigned) director field and the strength of the nematic alignment, respectively. For visualization in two dimensions, we evaluate V(r) over a uniform grid in the x–z plane, displaying only the in-plane components (V_x, V_z), and use Λ(r) as a heat map. The latter is subsequently used as experimental data for fitting the theoretical model, as described below. The position of the defect in the director field is estimated as the average of the points where the alignment strength Λ(r) is below a threshold value of 0.05, while its type is determined by visual inspection of the direction of V(r).

Cross-embryo-averaged boundary curve

For each embryo, we compute the intersection of the epiblast mesh with the M rotational slice planes defined by angles ϕ_m. These boundary curves are converted into polar coordinates (r, θ), with the epiblast centroid as the origin. The curves are resampled at equidistant angles, and their radial distances are averaged across all slices to produce a smooth, rotationally averaged boundary. We then compute the cross-embryo average boundary by averaging with respect to all rotational slices from multiple embryos. Finally, the averaged boundary is transformed back into Cartesian coordinates for visualization in the x–z plane.

Shape model fitting

Given the collections of points ${\left\{\left(x_{\mu i}^{\exp },0,z_{\mu i}^{\exp }\right)\right\}}_{i=1,\dots ,N_{\mu }}$ corresponding to the segmented tissue–tissue (μ = α) and tissue–ECM (μ = β) contours, we identify the centroid of the epiblast and the two points b_left and b_right where the different boundaries meet. We define the axis of symmetry as the line passing through the centroid and perpendicular to the line between b_left and b_right; the point where these lines intersect defines the origin of coordinates. After referring the contour points to this origin, we transform them to spherical coordinates, ${\left\{(r_{\mu i}^{\exp },{\theta }_{\mu i}^{\exp },0)\right\}}_{i=1,\dots ,N_{\mu }}$$={\left\{{\mathbf{r}}_{\mu i}^{\exp }\right\}}_{i=1,\dots ,N_{\mu }}={\mathbf{r}}_{\mu }^{\exp }$, and minimize the cost function

$$\mathcal{H}\left({\mathbf{r}}_{\mathrm{\alpha }}^{\exp },{\mathbf{r}}_{\mathrm{\beta }}^{\exp };{\kappa }_{\mathrm{\alpha }},{\gamma }_{\mathrm{\alpha }},{\kappa }_{\mathrm{\beta }},{\gamma }_{\mathrm{\beta }}\right)=\sum _{\mu =\mathrm{\alpha },\mathrm{\beta }}\sum _{i=1}^{N_{\mu }}{\left[\mathcal{R}\left({\mathbf{r}}_{\mu i}^{\exp };{\kappa }_{\mu },{\gamma }_{\mu }\right)\right]}^2.$$ 10

Here,

$$\mathcal{R}({\mathbf{r}}_{\mu i}^{\exp };{\kappa }_{\mu },{\gamma }_{\mu })\equiv r_{\mu i}^{\exp }-r({\theta }_{\mu i}^{\exp };{\kappa }_{\mu },{\gamma }_{\mu })$$ 11

is the residual between the radius of experimental point i of boundary μ and the fitting function

$$r(\theta ;{\kappa }_{\mu },{\gamma }_{\mu })=\frac{\cos \theta }{{\gamma }_{\mu }}+\sqrt{\frac{1}{{\kappa }_{\mu }^2}-\frac{{\sin }^2\theta }{{\gamma }_{\mu }^2}}$$ 12

evaluated at that point, with κ_μ ≡ 1/R_μ (that is, the curvature of the spherical cap) and γ_μ ≡ 1/c_μ. This parametrization, albeit counterintuitive, allows us to easily fit flat caps, for which both the radius R_μ and the height c_μ of the centre would diverge. To ensure continuity of the shape profile, or in other words, to guarantee a closed surface, we impose the constraint that r(π/2; κ_α, γ_α) = r(π/2; κ_β, γ_β) at the polar angle where, by construction, the two boundaries meet. Having found the best set ${\tilde{\kappa }}_{\mu },{\tilde{\gamma }}_{\mu }$, we calculate the acorn parameter

$$\nu =(\frac{1}{{\tilde{\gamma }}_{\mathrm{\text{α}}}}+\frac{s}{{\tilde{\kappa }}_{\mathrm{\text{α}}}}){\tilde{\kappa }}_{\mathrm{\text{β}}},$$ 13

where s = 1 (s = −1) if the tissue–tissue contour is convex (concave), and we calculate the base central angle

$$\text{σ}={\cos }^{-1}(-\frac{{\tilde{\kappa }}_{\mathrm{\text{β}}}}{{\tilde{\gamma }}_{\mathrm{\text{β}}}})$$ 14

that characterize the epiblast.

Errors for the parameters are estimated by residual bootstrap. We generate a vector of residuals ${\rho }_{\mu }=\left(\mathrm{\delta }{r}_{\mu 1},\dots ,\mathrm{\delta }{r}_{\mu N_{\mu }}\right)$, where $\delta r_{\mu i}=\mathcal{R}\left({\mathbf{r}}_{\mu i}^{\exp };\right)$$\left({\tilde{\kappa }}_{\mu },{\tilde{\gamma }}_{\mu }\right)$, for each boundary μ. A residual bootstrap sample ${\rho }_{\mu }^{\prime }=\left({\mathrm{\delta }{r}^{\prime }}_{\mu 1},\dots ,\right)$$\left({\mathrm{\delta }{r}^{\prime }}_{\mu N_{\mu }}\right)$ is generated by randomly selecting N_μ elements from ρ_μ with replacement. Then, the cost function (equation (10)) is evaluated at the modified datasets ${{\mathbf{r}}^{\prime }}_{\mu }^{\exp }=\left\{\right)\left(r\left(\right){\theta }_{\mu i}^{\exp };{\tilde{\kappa }}_{\mu },{\tilde{\gamma }}_{\mu }\right)+$${\mathrm{\delta }{r}_{\mu i}^{\prime },{\theta }_{\mu i}^{\exp },0\left(\right)\left(\right\}}_{i=1,\dots ,N_{\mu }}$ and minimized to obtain the corresponding best set of parameters. We repeat this process for 100 residual bootstrap samples to generate distributions of the fitting parameters, and use their s.d. values as uncertainties $\mathrm{\Delta }{\tilde{\kappa }}_{\mu },\mathrm{\Delta }{\tilde{\gamma }}_{\mu }$. These give rise to

$$\mathrm{\Delta }\nu =\sqrt{\frac{{\tilde{\kappa }}_{\mathrm{\text{β}}}^2}{{\tilde{\gamma }}_{\mathrm{\text{α}}}^4}\mathrm{\Delta }{\tilde{\gamma }}_{\mathrm{\text{α}}}^2+\frac{{\tilde{\kappa }}_{\mathrm{\text{β}}}^2}{{\tilde{\kappa }}_{\mathrm{\text{α}}}^4}\mathrm{\Delta }{\tilde{\kappa }}_{\mathrm{\text{α}}}^2+\frac{{\nu }^2}{{\tilde{\kappa }}_{\mathrm{\text{β}}}^2}\mathrm{\Delta }{\tilde{\kappa }}_{\mathrm{\text{β}}}^2}$$ 15

and

$$\mathrm{\Delta }\text{σ}=\frac{\sqrt{\frac{{\tilde{\kappa }}_{\mathrm{\text{β}}}^2}{{\tilde{\gamma }}_{\mathrm{\text{β}}}^4}\mathrm{\Delta }{\tilde{\gamma }}_{\mathrm{\text{β}}}^2+\frac{\mathrm{\Delta }{\tilde{\kappa }}_{\mathrm{\text{β}}}^2}{{\tilde{\gamma }}_{\mathrm{\text{β}}}^2}}}{\sqrt{1-\frac{{\tilde{\kappa }}_{\mathrm{\text{β}}}^2}{{\tilde{\gamma }}_{\mathrm{\text{β}}}^2}}}.$$ 16

Fitting the material parameters to the experimental alignment field

The nematic alignment tensor captures the cell’s orientation coherence but not the polarity direction; its magnitude, however, can provide an unsigned estimation of the polarity magnitude. We define the experimental polarity magnitude $p_{\exp }\equiv \Lambda /{\Lambda }_{\max }$, where the strength of the nematic alignment Λ(r) is normalized by its maximum value ${\Lambda }_{\max }$. In order to determine the model parameters, we fit the average epiblast shape (described in ‘Shape model fitting’) and minimize the free energy functional (equation (2)) in this geometry for parameter ranges ξ/R₀ ∈ [1, 10²], λ/R₀ ∈ [10⁻¹, 10³] and k₀/k₂ ∈ [10⁻¹, 10³] to obtain the corresponding theoretical field with magnitude p. To compare the experimental and theoretical results, we define a set of N points r_i in the intersection of the theoretical and experimental embryo shapes. These points are chosen to be at a distance greater than 0.05 from the symmetry axis and greater than ℓ from the closest boundary, where ℓ is given by the mean s.d. of the experimentally determined average boundary. In this way, we exclude regions where the average experimental field has low statistics due to the different shapes of the individual embryos. Finally, we calculate the cost function

$$\mathcal{C}(\xi /R_0,\lambda /R_0,k_0/k_2)=\sum _{i=1}^N{\left[p({\mathbf{r}}_i;\xi /R_0,\lambda /R_0,k_0/k_2)-p_{\exp }\left({\mathbf{r}}_i\right)\right]}^2$$ 17

and identify the set of parameters where it attains its minimum value.

Reporting summary

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Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41563-026-02594-7.

Author contributions

P.C.G., T.I., T.H. and A.E. designed the research. P.C.G. performed the numerical work. T.I. performed the experiments. P.C.G. and T.I. analysed the data. S.P. and P.C.G. calculated the experimental nematic field. P.C.G. prepared the figures with input from A.E.; P.C.G. and A.E. wrote the paper with contributions from T.I., T.H. and S.P.

Peer review

Peer review information

Nature Materials thanks Aurélien Roux and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Funding

Open access funding provided by European Molecular Biology Laboratory (EMBL).

Data availability

Data supporting the findings of this study are available within the Article and its Supplementary Information. Raw image data are available from the corresponding authors upon request. Source data are provided with this paper.

Code availability

All the codes that support this study are available via the EMBL GitLab server at https://git.embl.de/guruciag/geometry-driven-defects or via Zenodo at 10.5281/zenodo.19248955 (ref. ⁷⁹) and via GitHub at https://github.com/SteffenPL/EpiblastOrientationMaps or via Zenodo at 10.5281/zenodo.19242875 (ref. ⁹⁰).

Competing interests

The authors declare no competing interests.

Extended data

is available for this paper at 10.1038/s41563-026-02594-7.

Supplementary information

The online version contains supplementary material available at 10.1038/s41563-026-02594-7.