## Main

Tissue morphogenesis is the emergence of increasingly complex geometry and topology out of a simple group of cells7. The geometry of a tissue characterizes its size and shape, and its topology defines how different parts are connected and characterizes the organization of cavities and passages between them2,8. Fundamental morphogenetic processes proceed as a series of size and shape changes and topological transitions3,6,9,10. For example, gastrulation involves the invagination of an epithelial cell layer of initially spherical topology that eventually transitions to a toroidal topology serving as the precursor for the passage connecting the mouth to the other end1. In the case of vasculogenesis, endothelial tissues form tubular geometries that connect and result in complex topological networks with branches and loops11. Fluid-filled cavities called lumens form the basis of transport networks such as bile canaliculi in the liver12 and ventricles in the brain13. Pathological conditions such as polycystic kidney disease are associated with altered organ topology14. Thus, topological and geometric transitions play a key role during morphogenesis and in organ function. However, the principles that guide the interplay of topology and geometry in formation of complex organ architectures remain unknown.

To address this fundamental issue, we make use of recent advances in three-dimensional (3D) organ reconstitution that provide accessible and controllable experimental systems to study how a simple assembly of cells dynamically organizes into tissues with complex architectures15,16. We differentiate mouse embryonic stem (ES) cells in vitro as free-floating aggregates that develop into neuroepithelial organoids in four days (Fig. 1a and Extended Data Fig. 1a)17,18. Staining of apical surfaces (anti-ZO1, anti-PODXL) suggests that this process involves the formation of fluid-filled lumens. The apical surface of these organoids contained many passages (Fig. 1a, top), indicating the emergence of complex tissue geometry and topology. Here, to explore organoid morphogenesis, we set out to quantify geometry and topology and use biochemical perturbations to influence geometric and topological changes. Retinoic acid (RA) is known to act as a morphogen and to instruct the size and shape of the developing neural tube19. We found that RA treatment at day 2 of differentiation had a substantial influence on the geometry and topology of organoids. Staining of apical surfaces revealed the emergence of multiple epithelial lobules by day 4 (Fig. 1a, zoomed-in view, bottom), in contrast to untreated organoids that are dominated by one large lobule (Fig. 1a, untreated).

To quantify organoid geometry and topology, we segmented the organoid architecture and constructed triangular meshes to define the outer (magenta) and apical surfaces (green) (Fig. 1b, Methods, Supplementary Videos 1 and 2, and Extended Data Fig. 1b). This allowed us to quantify key geometric and topological measures for each organoid (Extended Data Fig. 2a,b). Using the apical surface to define distinct epithelial lobules, we characterized organoid topology by the number of lobules N and their topological genus g, and the shape of each lobule by its reduced volume v and reduced curvature m (Fig. 1c and Supplementary Note). Organoid shape is then characterized by the average of the reduced volumes and reduced curvatures over all the lobules, 〈v〉 and 〈m〉, respectively (Methods and Extended Data Fig. 2c). We observed that untreated organoids contained large lobules with high genus g that increased with organoid volume (Fig. 1d and Extended Data Fig. 1d), with shape and topology that can be qualitatively captured by the wiffle ball morphology (compare with Fig. 1b,c, inset). In contrast, RA-treated organoids contained lobules of spherical topology (g = 0), with lobule number N increasing with organoid volume (Fig. 1d and Extended Data Fig. 1d).

The geometry of organoids can be represented in a shape diagram (〈v〉, 〈m〉) (refs. 20,21,22) (Fig. 1e). Untreated organoids at day 4 (blue points) had similar average reduced volumes 〈v〉 around 0.15, but varying in average reduced curvature 〈m〉 falling approximately on a vertical line (Fig. 1e, dashed line). This line corresponds to wiffle ball configurations for varying passage sizes (Supplementary Note). For RA-treated organoids at day 4 (Fig. 1e, orange points), the average reduced volume 〈v〉 decreases for increasing average reduced curvature 〈m〉. This trend falls on the line (solid) corresponding to spherocylinder shapes of varying aspect ratios, ranging from spheres to tubes (Fig. 1e and Supplementary Note). Thus, by day 4, untreated organoids develop into a morphology dominated by a large lobule of high genus resembling a wiffle ball, whereas RA-treated organoids develop into many lobules of low topological genus consisting of spherocylinders.

To address how and when these two different organoid morphologies emerge, we imaged SiR-actin-labelled organoids using light sheet microscopy over 48 h from day 2 to 4 (Fig. 2a, Extended Data Fig. 3a, and Supplementary Videos 3 and 4). At the initial stages of imaging at day 2, organoids in both conditions contained numerous small spherical lobules, which fused with each other and gave rise to elongated and tubular lobules (Supplementary Videos 5 and 6 show the surface renderings). From day 2 to 3, the organoid shapes in both conditions followed a trajectory along the spherocylinder branch in the shape diagram, starting from elongated spheres (〈v〉  1, 〈m〉  1) and increasing in aspect ratio (Fig. 2b). At about 24 h of imaging (day 3), the organoid shapes reached a region in the shape diagram where the wiffle ball and spherocylinders morphologies meet (intersection of solid and dashed lines in Fig. 1e; (〈v〉  0.15, 〈m〉  5)). After 24 h, the shape trajectories for the two conditions diverged. Untreated organoids transitioned from the spherocylinder branch to the wiffle ball branch, decreasing the average reduced curvature with time and maintaining a small average reduced volume (Fig. 2b, left). In contrast, RA-treated organoids remained on the spherocylinder branch, but returned towards larger average reduced volumes by day 4 (Fig. 2b, right). During the two days of imaging, the number of lobules N decreased monotonically in a similar manner for both conditions (Fig. 2c and Extended Data Fig. 3c). The trajectories of the total genus g in both conditions remained low until about 24 h, after which they diverged. In the untreated condition, g increased over time to large values from day 3 to 4, whereas in the RA-treated condition, g remained small even at day 4 (Fig. 2d).

The time dependence of lobule number and total genus indicated that topological transitions occur during organoid morphogenesis from day 2 to 4. A close examination of the apical surface triangulations revealed that topological transitions occurred as two distinct modes of fusion processes irrespective of geometry. One mode of fusion, which we call trans fusion, involves two separate lobules that fuse with each other (Fig. 2e, left). This results in a reduction in lobule number by one (ΔN = −1). The second mode of fusion, which we call cis fusion, involves a single lobule that fuses with itself to create a passage (Fig. 2e, right). This results in an increase in genus by one (Δg = +1). Experimentally observed examples of trans and cis fusion are shown in Fig. 2f,g, respectively, and a schematic is shown in Fig. 2e. We defined the counters for trans and cis fusion (Fig. 2e), and quantified the cumulative fusion events from day 2 to 4. In both conditions, the early time points are dominated by trans fusion events (Fig. 2h), whereas beyond the 24 h time point, cis fusion is the primary mode of fusion observed only for untreated organoids (Fig. 2i). Thus, we discovered that organoid topologies emerge from distinct modes of fusion, namely, trans and cis fusion, where only the latter leads to the creation of passages and epithelial tissues with non-spherical topology.

These findings suggest that the topology and shape of the emerging tissue are determined by the trans and cis modes of topological transitions that are generated during morphogenesis. This raises the question of which epithelial properties determine the dominant mode of fusion and how they are controlled. One possibility is that tissue mechanics23,24,25 governs the fusion events and favours one mode of fusion over the other. An example of a mechanical property that distinguishes between the trans and cis modes of fusion is the bending energy of a fluid surface21,26,27,28,29

$${E}_{\mathrm{b}}=\int [\kappa {H}^{2}+\bar{\kappa }K]\mathrm{d}A,$$
(1)

where H denotes the local mean curvature of the surface, K denotes the Gaussian curvature and dA is the area element. For fluidized epithelial surfaces, the bending rigidities can be estimated from cell-based physical models of tissue mechanics30,31,32,33,34 (Supplementary Note). The bending rigidity κ and Gaussian rigidity $$\bar{\kappa }$$ are elastic moduli that describe the resistance of the shape to bending and saddle-splay deformations, respectively. Note that ∫KdA = 4π(N − g) only depends on the indices N and g counting the lobules and passages, respectively. Therefore, Gaussian rigidity $$\bar{\kappa }$$ describes the resistance to topological changes. In addition, bending rigidity κ not only governs changes in shape, but also changes in lobule number N. We discuss how $$\bar{\kappa }$$ and κ could affect the mode of fusion by comparing bending energy Eb before and after fusion (Fig. 3a and Supplementary Note). Briefly, the change in Eb associated with trans fusion of two lobules (Fig. 3a, left) is $${{\Delta }}{E}_{\mathrm{b}}\simeq -4\uppi (\kappa +\bar{\kappa })$$; thus, trans fusion is energetically favoured when $$\kappa +\bar{\kappa } > 0$$. The change in Eb for cis fusion is $${{\Delta }}{E}_{\mathrm{b}}\simeq -4\uppi \bar{\kappa }$$; thus, passage formation via cis fusion in a lobule is energetically favoured when $$\bar{\kappa } > 0$$ (Fig. 3a, right). If $$\bar{\kappa } < -\kappa$$, neither trans or cis fusion is energetically favoured. These criteria can be summarized in a state diagram as a function of the ratio $$\bar{\kappa }/\kappa$$ of the two elastic moduli, which is the reduced Gaussian rigidity. We can distinguish three parameter regions, where morphologies evolve differently when starting from an initial state of N spherical lobules (Fig. 3b). In region I, $$\bar{\kappa }/\kappa < -1$$ and fusion is energetically disfavoured. The system remains in a configuration with many spherical lobules (large N, g = 0). In region II, $$-1 < \bar{\kappa }/\kappa < 0$$ and only trans fusion is energetically favoured. In this case, lobules will tend to fuse, resulting in fewer lobules of spherical topology (small N, g = 0). In region III, $$0 < \bar{\kappa }/\kappa$$, both trans and cis fusion are energetically favoured. As a result, large lobules with many passages exemplified by wiffle ball morphology emerge (small N, large g). Our results suggest that RA treatment shifts the behaviour of the system from region III to region II. To test this idea, we estimate the scaled bending energy Eb/κ of lobule geometries as a function of time (Supplementary Videos 7, 8, 9 and 10) considering different values of reduced Gaussian rigidity $$\bar{\kappa }/\kappa$$ (Methods, Fig. 3c (left) and Extended Data Fig. 4). Only for positive $$\bar{\kappa }/\kappa$$ corresponding to region III, scaled bending energy Eb/κ decreases with time for untreated organoids, consistent with the idea that untreated organoids operate in region III. After RA treatment, a decrease in scaled bending energy Eb/κ with time requires a shift in $$\bar{\kappa }/\kappa$$ to negative values (Fig. 3c, right). This suggests that after RA treatment, the mechanical properties of the epithelia is modified, possibly leading to a reduction in $$\bar{\kappa }/\kappa$$ below zero at later times.

Thus, the primary effect of RA treatment on topological morphogenesis is consistent with a decrease in the reduced Gaussian rigidity $$\bar{\kappa }/\kappa$$. This raises the question whether we can tune the relative rates of trans and cis fusion via further molecular perturbations and control the topological morphogenesis. To identify such a molecular pathway, we performed RNA sequencing (RNAseq) on organoids treated with or without RA (Methods). RA treatment led to a broad change in the expression of developmental genes (Supplementary Table 1 and Extended Data Fig. 5) consistent with previous work35,36. We found that RA treatment downregulated the expression of Enpp2, which encodes the cellular enzyme that produces lysophosphatidic acid (LPA)37. LPA is known to enhance the folding of developing mouse brains38 and increase the apical membrane area in human39 and gorilla40 neuroepithelial cells. To test the role of LPA synthesis on organoid morphogenesis, we treated day 2 organoids with varying concentrations of HA130, a small molecule inhibitor of LPA synthesis41, and analysed their morphology at day 4 (Fig. 3d (top) and Extended Data Fig. 6a,b). At low concentrations of HA130 (31–125 nM), the organoids primarily had a single lobule with many passages, resembling wiffle ball morphologies, similar to the untreated case and consistent with region III (Fig. 3e,f). Increasing the concentration of HA130 resulted in morphologies with fewer passages until at the highest concentration of HA130 used (2000 nM), the organoids resembled a set of elongated spheres, consistent with the gradual shift from region III to region II (Fig. 3d (top right) and Supplementary Video 11). This perturbation shows that the inhibition of LPA synthesis mimics the effect of RA treatment in organoid morphogenesis. Our theory based on reduced Gaussian rigidity predicts that further reduction in $$\bar{\kappa }/\kappa$$ can result in a transition from region II to region I, where both trans and cis fusion are disfavoured. This behaviour was not originally observed in the presence or absence of RA. To test this prediction, we applied varying concentrations of HA130 to RA-treated organoids, thus combining the effect of RA treatment and inhibition of LPA synthesis (Fig. 3d (bottom) and Extended Data Fig. 6a,c). For increasing concentrations of HA130 (31–2,000 nM), RA-treated organoids exhibited increasing numbers of lobules (Fig. 3f), consistent with the gradual suppression of trans fusion. At the highest concentrations of HA130 (500 and 2,000 nM), RA-treated organoids resulted in an ensemble of spherical lobules with lobule number N at day 4 comparable with day 2 organoids (Fig. 3d (bottom right) and Supplementary Video 12). This is consistent with the suppression of trans and cis fusion and a shift to region I in the state diagram (Fig. 3b). To follow the changes in topology for these conditions, we performed live imaging experiments and found that 2,000 nM HA130 greatly reduced the rates of trans and cis fusion (Extended Data Fig. 6d,e). By combining the effects of RA treatment with HA130, we achieved an attenuation of topological transitions corresponding to a shift in organoid morphogenesis from region III via region II to region I. The corresponding organoid shapes followed a trajectory in the (〈v〉, 〈m〉) diagram starting from the wiffle ball branch (Fig. 3e, dashed) at low 〈m〉, moving upwards and shifting towards region II. The trajectory transitions to the spherocylinder branch (Fig. 3e, solid) when passages are no longer observed corresponding to region II. The trajectory follows this branch towards large 〈v〉 and small 〈m〉 as the system shifts to region I. At the cellular level, the treatment with RA and HA130 lead to the reduced production of LPA, which is known to increase cortical contractility via the Rho/ROCK/myosin pathway42. To directly test the role of cell contractility on topological morphogenesis, we treated organoids with the ROCK kinase inhibitor Y-27632, which led to organoid topology and shape consistent with the prediction of reduced $$\bar{\kappa }/\kappa$$ and attenuation of fusion (Extended Data Fig. 7 and Supplementary Note). Note, however, that changes in cell mechanics may have consequences beyond Gaussian and bending rigidity. These pharmacological modulations allowed us to tune the topological morphogenesis by controlling the modes of fusion and their rates, giving rise to a broad range of morphologies with varying shapes and topologies.

In summary, by combining quantitative four-dimensional microscopy, theory and pharmacological perturbations, we have discovered how topological transitions drive neuroepithelial morphogenesis through trans and cis epithelial fusion. The morphological space can be captured by a single control parameter of epithelial fusion rates. We propose that this parameter is governed by the reduced Gaussian rigidity $$\bar{\kappa }/\kappa$$ of an epithelial surface. The physical principle of topological morphogenesis could apply to organ development in diverse contexts43,44. Topological morphogenesis may explain the normal development and pathology of the posterior neural tube, in which loops45,46 and multiple ectopic tubes47,48 have been reported. In engineering applications, we envision that the control of tissue topology will guide new strategies to design synthetic tissues.

## Methods

### Mouse ES cell and neuroepithelial organoid culture

Mouse ES cells were passaged in N2B27 medium with 2i/LIF following standard protocols49. To generate neuroepithelial organoids, we modified a published protocol for optic cup organoids17. On day 0, the ES cells were dissociated into single cells and re-aggregated in N2B27 neural induction medium (typically 1,000 cells per 50 μl per ml per well) in 96-well U-bottom low adhesion plates (ThermoFisher, 174925). On day 1, Matrigel (BD Biosciences, 354234) dissolved in 50 μl N2B27 medium was added to each well for a final concentration of 2.5% (v/v). On day 2, 50 μl N2B27 with or without all-trans RA (Sigma R2625) was added to each well. The final concentration of RA was 250 nM. In some experiments, organoids were treated with varying concentrations of HA130 (Echelon Biosciences, B-0701) to inhibit LPA synthesis, or with 20 μM Y-27632 (Tocris, #1254) to inhibit ROCK kinase. All the 3D morphological data in this study were collected from neuroepithelial organoids made from E14 mouse ES cells (a gift from M. Shahbazi, MRC Laboratory of Molecular Biology). The major morphological phenotypes induced by the addition of RA and HA130 were confirmed by experiments using R1 wild-type mouse ES cells (a gift from R. Naumann, MPI-CBG).

### Immunofluorescence and optical clearing of organoids

The organoid samples were washed in phosphate-buffered saline (PBS), fixed with ice cold 4% paraformaldehyde solution for 30 min, rinsed and stored in PBS at 4 °C. The organoids were permeabilized and blocked in a blocking solution (PBS + 0.5% (w/v) bovine serum albumin + 0.3% (v/v) Triton X-100) for 1 h at room temperature. Primary antibodies were diluted in 50 μl blocking solution and applied to organoids overnight with occasional mixing. After removal of the primary antibody solution, organoids were washed with 0.5 ml PBST (PBS + 0.3% Triton X-100) for 2 h and three times with occasional mixing. Secondary antibodies and DNA stains were diluted in 50 μl blocking solution overnight with occasional mixing. After removal of the secondary antibody solution, the organoids were washed with 0.5 ml PBST (for 2 h and three times) and finally stored in PBS. The primary antibodies used were rat anti-PODXL (R&D, MAB1556) at 1:40 dilution and mouse anti-ZO1 (Invitrogen, #33-9100) at 1:40 dilution. The secondary antibodies used were Alexa 568 anti-rat at 1:100 dilution and Alexa 647 anti-mouse at 1:100 dilution. The DNA stain was SYTOX Green (Invitrogen, S7020) used at 84 nM. Immunostained organoid samples were optically cleared using the second-generation ethyl cinnamate clearing protocol50.

### Fixed-sample imaging and 3D segmentation

Optically cleared organoids were transferred to an ethyl-cinnamate-resistant 96-well plate (Ibidi, #89621) for high-throughput imaging. The imaging was performed with an automated spinning-disc microscope (Yokogawa, Cell Voyager 7000S) equipped with a CSU-W1 spinning disc and a complementary metal–oxide–semiconductor camera (1,280 pixels × 1,080 pixels). To cover an entire well from a 96-well plate, four fields of view were acquired with a ×4 objective lens. These overview images were processed on the fly via a custom ImageJ macro executed by the SearchFirst module in the Wako Software Suite 2.0.20 that identified the coordinates of individual organoids. These coordinates were subsequently revisited with a ×20 air (NA = 0.75) Olympus objective. For each position, 300 planes with 0.8 μm spacing were acquired. Due to the ‘fish-bowl’ effect of ethyl cinnamate’s high refractive index (RI = 1.56), the effective z steps were 1.25 μm and the entire z stack encompassed 374 μm in sample depth. We used 2 × 2 binning on the camera, which resulted in images at 0.648 μm per pixel. For image analysis, we further binned the images in the xy plane so that the 3D voxel dimensions were comparable (1.30 μm × 1.30 μm × 1.25 μm in the x, y and z directions). To segment the organoid outer boundary, we applied Otsu thresholding to the z stacks from the SYTOX Green channel. To segment the apical surface, we applied Otsu thresholding to the z stacks from the PODXL channel. The cell height of the epithelia was calculated for each organoid from identifying the boundaries from projections: cell height $$h=\sqrt{{A}_{\mathrm{outer}}/(4\uppi )}-\sqrt{{A}_{\mathrm{inner}}/(4\uppi )}$$, where Aouter and Ainner are the areas enclosed by the outer and inner (apical) convex hull boundaries, respectively. All the reported measurements have accounted for the effect of tissue shrinkage (factor of 0.603 in linear dimensions, experimentally determined) of the ethyl cinnamate clearing protocol.

### Live imaging and 3D segmentation

Neuroepithelial organoids were cultured as described above with the addition of 100 nM SiR-actin (Spirochrome) in the medium from day 1. Immediately after RA treatment on day 2, the organoids were transferred to custom multiwell chambers and imaged on a light sheet microscope (Viventis, LS1) equipped with a scientific complementary metal–oxide–semiconductor camera and 638 nm laser line. Every 30 min, z stacks were acquired at 3 µm intervals, covering a sample depth of 200 µm. For image analysis, we binned the images in the xy plane so the voxel dimensions were nearly isotropic (2.76 μm × 2.76 μm × 3.00 μm in the x, y and z directions). To segment the organoid outer boundary, we applied multi-Otsu thresholding (classes = 3 using the lowest threshold) to the SiR-actin z stacks. To segment the apical surface, we applied multi-Otsu thresholding (classes = 4 using the highest threshold) to the SiR-actin z stacks.

### Surface construction and morphological analysis

The marching cubes algorithm was used to extract triangulated meshes from the segmented 3D images that represent the organoid outer boundary and apical surfaces. To reduce the mesh complexity, a combination of PyMesh functions were used to collapse the short edges (tolerance = 2 × the minimum voxel length of the input image), remove duplicated vertices, remove duplicated faces and remove degenerated/obtuse triangles. For each closed surface, we calculated volume V, surface area A, integral mean curvature M and Euler characteristic χ using the functions implemented in PyMesh and our custom code (Supplementary Note). These quantities were used to calculate the reduced volume $$v=3\sqrt{4\uppi }\,V/{A}^{3/2}$$, reduced curvature $$m=M/\sqrt{4\uppi A\,}$$, and topological genus g = 1 − χ/2 of epithelial lobules. Epithelial lobules with lumen volume V < 100 μm3 were excluded from the analysis. Organoid-level quantities were defined as total quantities g = ∑igi or as weighted averages 〈v〉 = ∑iviVi/∑iVi and 〈m〉 = ∑imiVi/∑iVi, where index i enumerates all the N epithelial lobules of an organoid. All the computational analyses and visualizations in this study were performed using Python 3.7 with the libraries NumPy51, SciPy52, Pandas53, Matplotlib54, seaborn55, scikit-image56, PyMesh57, PyVista58 and Polyscope59.

### RNAseq and data processing

Organoids were grown in 35 mm MatTek dishes by seeding 3,000 5,000 R1 mouse ES cells in 50 μl Matrigel and cultured in 2 ml N2B27 medium at 37 °C, 5% CO2 via clonal growth as described in other work60,61. At day 2 (48 h post-seeding), the organoids were treated with or without 250 nM all-trans RA. The organoids were extracted from the Matrigel with a cell recovery solution (Corning 354253) at day 2 or at 8, 18, 30, 42, 54, 66 and 78 h after day 2. Three biological replicates were prepared for each cell population, yielding 45 independent samples for RNA extraction. For each sample, a minimum of 50,000 cells were collected for RNA extraction using the QIAGEN RNeasy Micro Kit. RNAseq libraries were prepared using the NEBNext Poly(A) mRNA Magnetic Isolation Module (E7490) and the NEBNext Ultra Directional RNA Library Prep Kit for Illumina (E7420) according to the manufacturer’s instructions. For ligation, custom adaptors were used (Adaptor-Oligo 1: 5’-ACA CTC TTT CCC TAC ACG ACG CTC TTC CGA TCT-3’; Adaptor-Oligo 2: 5’-P-GAT CGG AAG AGC ACA CGT CTG AAC TCC AGT CAC-3’). The libraries were sequenced as 75 bp single end on an Illumina NextSeq500 system. RNAseq reads were trimmed using Trim Galore v0.5.0, filtered to remove abundant sequences using Bowtie 2 v2.3.4.1, aligned to the GRCm38 genome (Ensembl release 94) using STAR v2.6.0c and summarized per gene with featureCounts (subread v1.6.2). Further analysis was performed using DESeq2 v1.18.1. To identify significantly differentially regulated genes between untreated versus RA-treated cell populations, we focused on the 18 and 30 h post-day 2 time points, and selected genes with log2 fold change less than –1 or greater than 1 and false discovery rate of 0.01 (based on adjusted p values) in both time points, identifying 569 upregulated and 250 downregulated genes.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.