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 emphasize 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 stabilize invagination.
Highlights
Cell division, cell shape changes, and related processes can drive deformations of cell sheets during animal and plant development [1,2,3,4,5,6]
Just as classical thin shell theories arise from an asymptotic expansion of bulk elasticity in the small thickness of the shell [14,15,16], these morphoelastic shell theories should be asymptotic limits of a bulk theory
Even in a constitutively isotropic material, this biological scaling limit of large bending deformations induces, in the thin shell limit, a geometric anisotropy absent from classical shell theories: different deformation directions exhibit different deformation responses
Summary
Cell shape changes, and related processes can drive deformations of cell sheets during animal and plant development [1,2,3,4,5,6]. Other studies [9,10] took a more geometric approach, mirroring geometric derivations of classical shell theories [25] based on the so-called Kirchhoff “hypothesis” This is the asymptotic result [15] that the normals of the midsurface of the undeformed shell remain, at leading order, normal to the deformed midsurface. There is, one more serious limitation of these models: Tissues in development undergo large bending deformations (Fig. 1) that are outside the formal range of validity of the underlying thin shell theories, which assume that the thickness of the shell is much smaller than all length scales of the midsurface of the shell [15,25,26]. Invagination is stabilized by the geometry of large bending deformations
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