Gaussian curvature as a parameter of biological surface growth.

Many biological systems may profitably be studied as surface phenomena. Green's (1965, 1969) models of tip growth, Erickson's (1966) study of leaf growth, Van Essen & Maunsell's (1980) and Todd's (1982) models of brain growth, Greenspan's (1976) tumor growth model--all have an essential surface character. Two types of surface growth may be distinguished: intrinsic surface growth, where the factors responsible for the deformation of the surface are internal to the surface itself, and extrinsic deformation, where external factors are brought into play. An example of the former type of" growth is leaf growth: here any change in shape is determined by the cellular structure of the leaf itself and not to any great extent by the nature of the surrounding medium. The growth of the surface of a tumour, on the other hand, might be expected to be influenced by growth of the underlying material and perhaps by external pressure. In this paper, we assume a model consisting of isotropic growth of a curved surface from a fiat sheet. With such a model, the Gaussian curvature of the final surface determines whether growth rate of the surface is subharmonic or superharmonic. These properties correspond to notions of convexity and concavity and thus to local excess growth and local deficiency of growth. In biological models where the major factors controlling surface growth are intrinsic to the surface, we have thus gained from geometrical study information on the differential growth undergone by the surface. We look at a sample application of the analysis to the folding of the cerebral cortex: a geometrically rather complex surface growth.