Intrinsic Geometry of Biological Surface Growth

Abstract The intrinsic geometry, or ‘shape’ of a growing biological surface may be regarded as a parameter of the growth process which generates that geometry. The surface geometry may then suggest an appropriate mathematical model for the underlying growth process. In axisymmetric systems (for example root tip growth in plants) the assumptions that growth is isotropic and axisymmetric determine a mathematical model for the growth process. The model in turn yields differential growth rates for a process generating the given geometry. In a surface lacking this symmetry, however, isotropy alone is insufficient to determine a mathematical model for the underlying growth process. The following methodology may be used. If S and T are compact surfaces representing two stages in the growth of a biological surface, we look at conformal (in the Riemann Surfaces sense) maps between S and T [this is equivalent to assuming isotropy]. We seek that map :S T which minimises the Dirichlet integral of log, where is the ratio of the first fundamental forms of corresponding points in S and T. represents the simplest regime of isotropic growth to generate T from S and is thus an appropriate mathematical model for the real growth process. The use of these modelling procedures will be discussed with special reference to a study on the growth and development of folding in the cerebral cortex.