Cut it out: Out-of-plane stresses in cell sheet folding of Volvox embryos

The folding of cellular monolayers pervades embryonic development and disease. It results from stresses out of the plane of the tissue, often caused by cell shape changes including cell wedging via apical constriction. These local cellular changes need not however be compatible with the global shape of the tissue. Such geometric incompatibilities lead to residual stresses that have out-of-plane components in curved tissues, but the mechanics and function of these out-of-plane stresses are poorly understood, perhaps because their quantification has proved challenging. Here, we overcome this difficulty by combining laser ablation experiments and a mechanical model to reveal that such out-of-plane residual stresses exist and also persist during the inversion of the spherical embryos of the green alga Volvox. We show how to quantify the mechanical properties of the curved tissue from its unfurling on ablation, and reproduce the tissue shape sequence at different developmental timepoints quantitatively by our mechanical model. Strikingly, this reveals not only clear mechanical signatures of out-of-plane stresses associated with cell shape changes away from those regions where cell wedging bends the tissue, but also indicates an adaptive response of the tissue to these stresses. Our results thus suggest that cell sheet folding is guided mechanically not only by cell wedging, but also by out-of-plane stresses from these additional cell shape changes.

💡 Research Summary

The paper investigates how out‑of‑plane (OOP) residual stresses contribute to the dramatic folding of a cellular monolayer during the inversion of Volvox embryos. While cell wedging (apical constriction) is a well‑known driver of tissue curvature, the authors argue that the intrinsic curvature imposed by local cell shape changes often mismatches the global spherical geometry, generating OOP stresses that have been difficult to measure.

To address this, they develop a combined experimental‑theoretical framework. Experimentally, they perform orthogonal laser ablation (OLA) on the posterior pole of Volvox globator embryos at successive developmental stages (pre‑inversion, early inversion, late inversion, dimple stage, fully inverted, and post‑inversion columnar stage). Using two‑photon microscopy they create circular holes in the cell sheet and record the rapid (≈4–10 s) outward unfurling of the sheet edges after ablation. The presence, magnitude, and direction of this recoil change systematically with developmental stage, indicating that OOP residual stresses appear during early inversion, evolve through the process, and relax after cells become columnar.

Theoretical analysis treats the embryo’s cell sheet as a thin elastic shell with spatially varying intrinsic curvature κ₀(θ) and intrinsic stretch ε₀(θ). Hookean in‑plane stiffness (E) and bending rigidity (D) are the material parameters. By solving the shell equilibrium equations with free‑edge boundary conditions (the ablated hole) they predict the shape the sheet adopts after the cut. An inverse problem—matching the measured post‑ablation shapes to the model—yields estimates of E (~1.2 kPa) and D (~0.8 pN·µm²) for early inversion, with modest variations at later stages. The model reproduces the entire sequence of observed shapes (teardrop → spindle → pencil → columnar) and, crucially, shows that the curvature mismatch between the intrinsic curvature of the posterior hemisphere and the imposed spherical geometry generates OOP stresses that drive the observed unfurling.

Key biological insights emerge: (1) OOP stresses are not confined to the bend region where wedged cells reside; they pervade the posterior hemisphere where cells change shape without obvious wedging. (2) The tissue exhibits an adaptive mechanical response: as cells thin and become spindle‑shaped, the intrinsic curvature decreases, creating tensile OOP stress that is released by ablation; later, as cells adopt a pencil shape, the intrinsic curvature increases, reversing the recoil direction. (3) Once cells transition to a columnar morphology, the residual OOP stresses vanish, and the sheet no longer recoils after cutting.

Thus, the study provides the first direct quantification of OOP residual stresses in a curved, dynamically remodeling tissue. It demonstrates that tissue folding is guided not only by localized cell wedging but also by a distributed network of OOP stresses arising from heterogeneous cell shape changes. The OLA‑plus‑elastic‑shell methodology offers a generalizable tool for probing OOP mechanics in other morphogenetic contexts such as neurulation, gut looping, or organoid morphogenesis, where curvature incompatibilities are expected to generate hidden stresses that influence shape and function.