2.1. Icosahedral Symmetry

The chiral icosahedral rotation symmetry group (I) is a set of 60 rotations consisting of 2-fold, 3-fold and 5-fold rotations on a sphere, and is the largest discrete finite point group without reflections (modulo spatial inversions). Reflections are not considered due to the chirality of the amino acids which make up proteins. Spherical capsids may be subdivided into 60 equivalent sections, known as the asymmetric unit (AU), similar to a unit cell in crystallography (Figure 1). Due to the reducibility of the full capsid to the AU, we will often express our results in terms of it. The group of 60 icosahedral rotation matrices can be generated by successive combinations of the 2-fold rotation a and the 3-fold rotation b which border the asymmetric unit (Figure 1), as

These rotation matrices and all our polyhedral vertices are given in the Appendix A.

We begin with the vertices of the three standard (base) representations of icosahedral symmetry the icosahedron (ICO), Dodecahedron (DOD) and Icosadodecahedon (IDD) which are representations of the 5, 3 and 2-fold symmetries, respectively (Figure 3). The vertices of these polyhedra will also serve as our translation vectors for the affine extensions below. We align all of our structures with the Viper Database orientation [21] of the spherical volume with a 2-fold axis aligned with the +z direction and a 5-fold aligned with the vector (0,1,ϕ), where ϕ=1+52≈1.618 is the golden ratio. The vertices of each of the three respective polyhedra are all equidistant from the origin and therefore constitute only a single radial level.

The three standard polyhedra with icosahedral symmetry and the affine extension translation vectors. (a) The 12 vertices of the icosahedron are on the six 5-fold axes, and the structure is generated by applying all 60 icosahedral rotations on a single point [0,1,ϕ], which also serves as the translation vector T→5. (b) The 20 vertices of the dodecahedron are on the ten 3-fold axes, and the structure is generated by applying the full icosahedral group to the point [−ϕ′,0,ϕ], which also serves as the T→3 translation vector. (c) The 30 vertices of the icosadodecahedron are on the fifteen 2-fold axes, and the structure is generated by applying the full icosahedral group to the point 12[−1,ϕ2,ϕ], which serves as the T→2 translation vector.

Throughout this work we will show images of point arrays layered atop virus capsids. We have introduced a standard color scheme, inspired by the icosahedral building kits made by Zometool. The symmetry axes have primary colors 5-fold (red), 3-fold (yellow) and 2-fold (blue). The points along the great circles connecting neighboring symmetry axes (Figure 1) are based on the paint color addition of those two colors, see Table 1.

The color scheme used for point array elements throughout this paper. Each color specifies the type of geometric location, e.g., 5-fold points are red, 2-fold points are blue, and purple are the points on the great circle between the two (Figure 1).