Morphoelasticity of Large Bending Deformations of Cell Sheets during Development

TL;DR

This paper develops a new shell theory for large bending deformations in cell sheets during development, accounting for geometric anisotropy and cell constriction effects, and applies it to Volvox invagination.

Contribution

It introduces a morphoelastic shell theory valid for large bending deformations, extending classical theories to better model developmental processes.

Findings

The new theory captures large bending effects not addressed by classical shell models.

Geometric effects from the theory stabilize invagination in Volvox.

Results highlight the importance of intrinsic geometry variations in morphogenesis.

Abstract

Deformations of cell sheets during morphogenesis are driven by developmental processes such as cell division and cell shape changes. In morphoelastic shell theories of development, these processes appear as variations of the intrinsic geometry of a thin elastic shell. However, morphogenesis often involves large bending deformations that are outside the formal range of validity of these shell theories. Here, by asymptotic expansion of three-dimensional incompressible morphoelasticity in the limit of a thin shell, we derive a shell theory for large intrinsic bending deformations and emphasise the resulting geometric material anisotropy and the elastic role of cell constriction. Taking the invagination of the green alga Volvox as a model developmental event, we show how results for this theory differ from those for a classical shell theory that is not formally valid for these large bending deformations and reveal how these geometric effects stabilise invagination.