  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.
