PATTERN-FORMATION IN ICOSAHEDRAL VIRUS CAPSIDS - THE PAPOVA VIRUSES AND NUDAURELIA CAPENSIS BETA-VIRUS

The capsids of the spherical viruses all show underlying icosahedral s ymmetry, yet they differ markedly in capsomere shape and in capsomere position and orientation. The capsid patterns presented by the capsome re shapes, positions, and orientations of three viruses (papilloma, SV 40, and NbetaV) have been generated dynamically through a bottom-up pr ocedure which provides a basis for understanding the patterns. A capso mere shape is represented in two-dimensional cross-section by a mass o r charge density on the surface of a sphere, given by an expansion in spherical harmonics, and referred to herein as a morphological unit (M U). A capsid pattern is represented by an icosahedrally symmetrical su perposition of such densities, determined by the positions and orienta tions of its MUs on the spherical surface. The fitness of an arrangeme nt of MUs is measured by an interaction integral through which all cap sid elements interact with each other via an arbitrary function of dis tance. A capsid pattern is generated by allowing the correct number of appropriately shaped MUs to move dynamically on the sphere, positioni ng themselves until an extremum of the fitness function is attained. T he resulting patterns are largely independent of the details of both t he capsomere representation and the interaction function; thus the pat terns produced are generic. The simplest useful fitness function is SI GMA2, the average square of the mass (or charge) density, a minimum of which corresponds to a ''uniformly spaced'' MU distribution; to good approximation, the electrostatic free energy of charged capsomeres, ca lculated from the linearized Poisson-Boltzmann equation, is proportion al to SIGMA2. With disks as MUs, the model generates the coordinated l attices familiar from the quasi-equivalence theory, indexed by triangu lation numbers. Using fivefold MUs, the model generates the patterns o bserved at different radii within the T = 7 capsid of papilloma and at the surface of SV40; threefold MUs give the T = 4 pattern of Nudaurel ia capensis beta virus. In all cases examined so far, the MU orientati ons are correctly found.