Pattern formation in icosahedral virus capsids: the papova viruses and Nudaurelia capensis beta virus

Abstract

The capsids of the spherical viruses all show underlying icosahedral symmetry, yet they differ markedly in capsomere shape and in capsomere position and orientation. The capsid patterns presented by the capsomere shapes, positions, and orientations of three viruses (papilloma, SV40, and N beta V) have been generated dynamically through a bottom-up procedure which provides a basis for understanding the patterns. A capsomere shape is represented in two-dimensional cross-section by a mass or charge density on the surface of a sphere, given by an expansion in spherical harmonics, and referred to herein as a morphological unit (MU). A capsid pattern is represented by an icosahedrally symmetrical superposition of such densities, determined by the positions and orientations of its MUs on the spherical surface. The fitness of an arrangement of MUs is measured by an interaction integral through which all capsid elements interact with each other via an arbitrary function of distance. A capsid pattern is generated by allowing the correct number of approximately shaped MUs to move dynamically on the sphere, positioning themselves until an extremum of the fitness function is attained. The resulting patterns are largely independent of the details of both the capsomere representation and the interaction function; thus the patterns produced are generic. The simplest useful fitness function is sigma 2, 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, calculated from the linearized Poisson-Boltzmann equation, is proportional to sigma 2. With disks as MUs, the model generates the coordinated lattices familiar from the quasi-equivalence theory, indexed by triangulation numbers. Using fivefold MUs, the model generates the patterns observed at different radii within the T = 7 capsid of papilloma and at the surface of SV40; threefold MUs give the T = 4 pattern of Nudaurelia capensis beta virus. In all cases examined so far, the MU orientations are correctly found.

Crystalline Order on a Sphere and the Generalized Thomson Problem

“…Although the original Thomson problem refers to the ground state of spherical shells of electrons, one can also ask for crystalline ground states of particles interacting with other potentials. Such a generalized Thomson problem arises, for example, in determining the arrangements of the protein subunits which comprise the shells of spherical viruses [5,6]. Here, the "particles" are clusters of protein subunits arranged on a shell.…”

Conserved Features in Papillomavirus and Polyomavirus Capsids

“…The nature of subunit contacts in papovaviruses has been the subject of several investigations (Rayment et al, 1982;Salunke et al, 1986Salunke et al, , 1989Baker et al, 1988Baker et al, , 1989Baker et al, , 1991Liddington et al, 1991;Marzec & Day, 1993;Stehle et al, 1994;Tarnai et al, 1995). A complete understanding of both intracapsomere and intercapsomere interactions can only come from high-resolution studies, which, to date, have only been accomplished with SV40 and murine polyomavirus (Liddington et al, 1991;Stehle et al, 1994).…”

Influence of dislocations in Thomson’s problem

“…The structures of proteins (capsomeres) in the shells (capsids) of spherical viruses are known to display icosahedral symmetry [10], and it was in this context that these compatible numbers were predicted. Each possible structure is defined by the translations along the triangular lattice vectors between each disclination (hf, kf ), which gives a "magic" number of N = 10P f + 2 particles where P = h 2 + k 2 + hk (h and k are integers without common factors, f is an integer, and P f is the triangulation number denoted by T in the literature) [11,12].…”