On surface geometry coupled to morphogen

An expression is derived for both the Gauss and the Mean curvature of a surface, in terms of three simple cell parameters. The surface is thought of as composed of a single-cell thick sheet of cells joined laterally. The three cellular parameters involved are the ratios of (linear) basal to apical dimension in two orthogonal directions, S1 and S2, and the cell thickness "h". These three parameters may be envisioned as functions of a morphogen or morphogens which vary from point to point over the (middle) surface. As an example, the "reaction-diffusion" equations which are often used to describe pattern-formation in early development can be seen as possible candidates for these morphogens, when the resultant surface deformations are given when the dependence of the three cellular parameters are specified as a function of morphogen concentration. The coupling back of the surface deformations to the set of reaction-diffusion equations is simply given, and is through the dependence on geometry of the Laplacian operator which enters these equations.