Cell bulging and extrusion in a three-dimensional bubbly vertex model for curved epithelial sheets

Cell extrusion is an essential mechanism for controlling cell density in epithelial tissues. Another essential element of epithelia is curvature, which is required to achieve complex shapes, like in the lung or intestine. Here we introduce a three-dimensional bubbly vertex model to study the interplay between extrusion and curvature. We find a generic cellular bulging instability at topological defects which is much stronger than for standard vertex models. Analyzing cell shapes in three-dimensional imaging data of spherical mouse colon organoids, we infer that pentagonal cells have an increased basal interfacial tension, suggesting that cells at topological defects react to the different force conditions. Using the bubbly vertex model, we show that such basal tensions stabilize against the predicted instability and result in better cell shape control than tissue-scale mechanisms such as lumen pressure and spontaneous curvature. Our theory suggests that epithelial curvature naturally leads to bulged and extrusion-like cell shapes because the interfacial curvature of individual cells at the defects strongly amplifies buckling effected by tissue-scale topological defects in elastic sheets. Our results highlight the complex interplay of forces across scales in three-dimensional tissue organization.

💡 Research Summary

This paper introduces a three‑dimensional “bubbly vertex model” (BVM) to investigate how curvature and topological defects together influence cell bulging and extrusion in epithelial sheets. Traditional vertex models (VM) treat cell interfaces as flat facets, which limits their ability to capture curvature‑driven phenomena that are intrinsic to curved tissues such as intestinal villi or lung alveoli. In the BVM each cell is bounded by a curved apical surface, a curved basal surface, and curved lateral faces. All three interfaces carry uniform surface tensions (Γ_a, Γ_b, Γ_l) and the total mechanical energy is the sum of the corresponding area contributions, with cell volume constrained to a constant value. By allowing curvature, the model naturally incorporates the geometric coupling between tissue‑scale curvature and the local curvature of individual cell interfaces.

The authors first generate spherical epithelial shells by random sequential adsorption of cell centers on a sphere, constructing a Voronoi tessellation, and minimizing the VM energy to obtain a low‑energy configuration. This configuration, with a mixture of pentagonal, hexagonal and heptagonal cells (the latter appearing as defect pairs that screen the curvature), is then used as the initial condition for BVM energy minimization performed in Surface Evolver. The comparison reveals a striking “bulging instability” in the BVM: pentagonal cells (the inevitable 5‑fold disclinations required by Euler’s theorem) develop markedly larger half‑opening angles than in the VM, and some even bend outward opposite to the lumen. The distribution of opening angles in the BVM is far broader, indicating that curved interfaces dramatically amplify the buckling tendency at topological defects.

To test whether such an instability occurs in real tissue, the authors image mouse colon organoids—spherical monolayers derived from single stem cells—using 3‑D confocal microscopy. They segment the organoids, assign each cell its number of neighbors, and compute the half‑opening angle δ/2 from the lateral interfaces. Contrary to the BVM prediction, most organoids show pentagonal cells with smaller opening angles than hexagons, suggesting that the tissue actively suppresses the instability. By applying a force‑inference pipeline they estimate the effective tensions on apical, basal and lateral interfaces for each cell. Pentagonal cells exhibit a statistically significant increase in basal tension relative to the average, implying that cells at disclinations reinforce their basal surface to counteract curvature‑driven buckling.

The stabilizing role of the actomyosin cortex is probed pharmacologically. Treatment with Latrunculin A (LatA), which depolymerizes actin, leaves overall organoid morphology intact but reveals a pronounced increase in pentagonal opening angles, reproducing the BVM‑predicted bulging. This demonstrates that the actin‑myosin network provides a compensatory basal tension that normally keeps the tissue in a regular shape.

A continuum mean‑field description of a defect cell embedded in an elastic sheet is then derived. By integrating the BVM energy over a spherical shell with icosahedral symmetry, the authors show that curved interfaces lower the effective saddle‑splay modulus locally, concentrating Gaussian curvature at the defect. Reducing apical‑basal tension or weakening a supracellular actin ring can trigger a transition from a partially bulged geometry to full extrusion, and the energy barrier for extrusion is lowest at pentagonal defects.

Overall, the study uncovers a multiscale mechanical picture: curvature inevitably creates topological defects that, through interface curvature, amplify a buckling instability; however, living epithelia counterbalance this tendency by locally increasing basal tension via the actomyosin cortex. The BVM thus provides a powerful theoretical framework for predicting when and where extrusion will occur in curved tissues, with implications for developmental biology, tissue engineering, and disease contexts where extrusion is dysregulated.