title: From discrete to continuum models of three-dimensional deformations in epithelial sheets.
abstract: 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 contexts. The buckling thresholds predicted by the continuum description are in good agreement with the results of 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.
authors.affiliation: Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey.
authors.forename: Nebojsa
authors.initials: N
authors.lastname: Murisic
authors.affiliation: CNRS & Laboratoire de Physique Statistique, Ecole Normale Supérieure, Paris, France.
authors.forename: Vincent
authors.initials: V
authors.lastname: Hakim
authors.affiliation: Chemical and Biological Engineering, Princeton University, Princeton, New Jersey.
authors.forename: Ioannis G
authors.initials: IG
authors.lastname: Kevrekidis
authors.affiliation: Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey.
authors.forename: Stanislav Y
authors.initials: SY
authors.lastname: Shvartsman
authors.affiliation: Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190 Institut Jean Le Rond d'Alembert, Paris, France. Electronic address: audoly@lmm.jussieu.fr.
authors.forename: Basile
authors.initials: B
authors.lastname: Audoly
mesh_terms: D003198:Computer Simulation; D004548:Elasticity; D004847:Epithelial Cells / Q000502:physiology; D004848:Epithelium / Q000502:physiology*; D008954:Models, Biological