Nonlinear surface deformation mechanism coupled to the morphogen field

The surface deformation from a preexisting spherical surface of a multi-cellular monolayer is explained by a spherical symmetry breaking mechanism in the case of axial symmetry. A multi-cellular sheet consists of the cells linked together laterally. The cellular sheet is presumed to be a regular and closed surface so that differential geometry and the Gauss-Bonnet theorem can be applied. The Gaussian curvature is considered as a function of the morphogen field to derive the morphogen field equation coupled to the middle surface geometry. The morphogen variation over the middle surface is coupled to the curvature change, and the metric of the surface g(u,v) is determined simultaneously with the morphogen field and the Gaussian curvature with respect to u and v in the conformal coordinates. In the case of axial symmetry, two principal curvatures are not independent of each other. A differentiation between two principal curvatures appears in the deformed regions from their sphere. The surface determined by the model presents a scale-invariance: a size-invariant pattern formation.;Numerical solutions of the coupled non-linear equations are given as a computer simulation for a surface deformation pattern.