Geometrical concepts in epithelial sheets

Geometrical aspects relevant to discussion of deformable epithelial cell sheets are presented. The epithelium is pictured as consisting of cells joined at their lateral surfaces and having cell heights which vary with position over the middle surface, which bisects the cell heights. The Gauss and Mean curvatures are shown to be represented by simple functions of variables S and h, where S is an angular deformation of the sheet at each point and h is the local cell height. It is argued that the Gauss curvature can often be considered as a simple quadratic function of S divided by h(2), while the Mean curvature H is defined as S/h. These two variables are in turn to be considered as monotonic functions of the morphogen(s). A closed set of coupled morphogen and geometry equations are suggested, which, however, represent an open dynamical system in the sense that there is continuous external energy input. A growing, three-dimensional entity of continuously changing shape and pattern is thus represented by this growth and form algorithm. (C) 1996 Academic Press Limited.