From discrete to continuum models of three-dimensional deformations in epithelial sheets
Résumé
Epithelial tissue, in which cells adhere tightly to each other and to the
underlying substrate, is one of the four major tissue types in adult
organisms. In embryos, epithelial sheets serve as versatile substrates
during the formation of developing organs. Some aspects of epithelial
morphogenesis can be adequately described using vertex models, in which the
two-dimensional arrangement of epithelial cells is approximated by a
polygonal lattice with an energy that has contributions reflecting the
properties of individual cells and their interactions. Previous studies
with such models have largely focused on dynamics confined to two spatial
dimensions and analyzed them numerically. We show how these models can be
extended to account for three-dimensional deformations and studied
analytically. Starting from the extended model, we derive a continuum
plate description of cell sheets, in which the effective tissue properties,
such as bending rigidity, are related explicitly to the parameters of the
vertex model. To derive the continuum plate model, we duly take into
account a microscopic shift between the two sublattices of the hexagonal
network, which has been ignored in previous work. As an application of the
continuum model, we analyze tissue buckling by a line tension applied along
a circular contour, a simplified set-up relevant to several situations in
the developmental context. The buckling thresholds predicted by the
continuum description are in good agreement with the results of direct
stability calculations based on the vertex model. Our results establish a
direct connection between discrete and continuum descriptions of cell
sheets and can be used to probe a wide range of morphogenetic processes in
epithelial tissues.
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