Icosahedral lattice

Polyoma- and papillomaviruses share a common capsid structure that is assembled from 72 pentameric capsomeres arranged on a T = 7 icosahedral lattice. Assembly studies of polyomaviruses were first initiated by expression of the major capsid protein of mouse polyomavirus, VPl, in... 

Ciyo-EM studies of the CLPs demonstrated that the organization of VPS and VP7 is essentially identical to that seen in native cores (Hewat et al, 1992). VPS forms a thin 120 suhunit-containing shell that has T = 1 icosahedral symmetry. This shell is surrounded by a layer of 260 VP7 trimers arranged on a T= 13 icosahedral lattice. In the CLPs, the second layer composed of VP7 tended to be incomplete, "with trimers missing at the fivefold axes, a feature not observed in native cores. 

The capsid of STMV, 170 A in diameter, with 60 monomeric subunits, is organized on a T=1 icosahedral lattice. The capsid encloses an ssRNA genome of 1059 bases. The 159-amino acid-long capsid protein exhibits a canonical antiparallel 8-stranded /3-sandwich fold, with the N-terminal 36 residues in highly extended conformation facing the interior of capsid. About 45% of the genome is seen in the electron density map along with... 

Perturbation theory. The Hubbard model defined on the truncated icosahedral lattice of Cao is a many-electron problem of considerable complexity consequently, a direct numerical assault on this problem is likely to be unsuccessful. Faced with this dilemma we adopt the textbook approach of second-order perturbation theory in the Hubbard-F 13). 

These vectors are linearly independent on the rationals. The set of all their integral linear combinations defines a Z-module of dimension 3 and rank 6 called an icosahedral lattice A o, in a way similar to the axially symmetric case. A lattice point P has position vector rp with integer indices n,... 

The combination of these rescaled dodecahedral points with the vertices of the icosahedron indicated above yields another indexed polyhedron with icosahedral symmetry the triacontahedron discovered by Kepler in 1611. The triacontahe-dron has 32 vertices, 12 icosahedral and 20 dodecahedral ones and 30 rhombic faces, or 60 triangular ones in the latter case it is then denoted ico-dodecahedron. The triacontahedron is the projection in space of a six-dimensional hypercube and has two points [200000] and [11 111 I] as generators with integral indices (Figure 11-7). Further polyhedra with icosahedral symmetry and vertices at icosahedral lattice points are obtainable from one or more generators with integer indices [28]. 

Figure 11-7. Indexed ico-dodecahedron (triacontahedron) viewed along the fivefold, twofold and threefold axes, respectively. It has 12 icosahedral vertices (empty circles) and 20 dodecahedral vertices (filled circles) all belonging to the same icosahedral lattice (adapted from [27], courtesy lUCr)...

Accordingly, all vertices of the molecular form encapsulating the capsid of the rhinovirus are at points of the same icosahedral lattice, proving that the icosahedral lattice is indeed the form lattice for the viral capsid, which has its envelope and hole related by a three-dimensional crystallographic scaling. This property provides an answer to the first question. 

Figure 11-9. The cluster with 222 symmetry of four VPl coat proteins taken from the human rhinovirus is enclosed in molecular forms with vertices at points of an integral tetragonal lattice with lattice parameters a = ra( /9, b = c = tuq/S, in which Uq is the icosahedral lattice parameter and t the golden ratio (adapted from [27], courtesy lUCr)...

It may therefore be concluded that the asymmetric structure unit of the picornaviral capsid is composed of three polypeptides and conforms to the prediction of Finch and Klug. It has a molec]jlar mass of approximately 86,000 daltons, a diameter of about 68 A (35) and is repeated 60 times in the virus capsid. This corresponds to the simplest icosahedral lattice, with a triangulation number of 1 (57) Considering the I4S pentamers as capsomeres, there would be 12 capsomeres per virion. The location of the 6 (7P4) polypeptides in the capsid has not yet been established, but it is possible that they are distributed over the internal surface and in direct contact with the virion RNA (5Q, 39I see below). 

---