THE MORPHOGENESIS OF HIGH SYMMETRY: THE WARPING THEOREM

Abstract

A theorem is presented concerning the morphogenesis of high-symmetry structures made of three-dimensional morphological units (MU's) free to move in three dimensions or constrained to a surface. All parts of each MU interact non-specifically with the rest of the structure, via an isotropic function of distance. Summing all interactions gives a net figure of merit, ℐ, that depends upon MU positions and orientations. A structure evolves via gradient dynamics, each MU moving down the local gradient of ℐ. The analysis is represented with generality in Fourier space. A "warping" from a configuration of MU's is a set of MU displacements and/or rotations that slightly perturb nearest neighbor relations; deviations can accrue across the structure, producing large global distortions. A warping behaves qualitatively like a small perturbation, so a warping from a stable equilibrium decays under gradient dynamics. Connection to the Symmetrization Theorem greatly extends the basin of attraction of stable symmetrical configurations. Warped configurations are equivalent as precursors of structure, which helps to understand assembly by accretion. Numerical illustrations are given in cylindrical geometry, for application to phyllotaxis; and in spherical geometry, for virus capsid structure. For animations of numerical evolutions that find high symmetry via unwarping, see