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Icosahedron construction

Geometric principles for generating icosahedral quasi-equivalent surface lattices. These constructions show the relation between icosahedral symmetry axes and quasi-equivalent symmetry axes. The latter are symmetry elements that hold only in a local environment.

Hexamers are initially considered planar (an array of hexamers forms a flat sheet as shown in a) and pentamers are considered convex, introducing curvature in the sheet of hexamers when they are inserted. The closed icosahedral shell, composed of hexamers and pentamers, is generated by inserting 12 pentamers at appropriate positions in the hexamer net. To construct a model of a particular quasi-equivalent lattice, one face of an icosahedron is generated in the hexagon net. The origin is replaced with a pentamer and the (h,k) hexamers is replaced by a pentamer. The third replaced hexamer is identified by 3-fold symmetry (i.e. complete the equilateral triangle of the face). The icosahedral face for a T=3 surface lattice is defined by the triangle with blue edges (h=1, k=1). Two possible T=7 lattice choices are also marked with green and yellow lines (h=2, k=1 or h=1, k=2, these being mirror images of each other), and require knowledge of the arrangement of hexamers and pentamers and the enantiomorph of the lattice for a complete lattice definition.

In figure b), seven hexamer units (bold outlines in a) defined by the T=3 lattice choice are shown and the T=3 icosahedral face defined in (a) has been shaded. A three dimensional model of the lattice can be generated by arranging 20 identical faces of the icosahedron as shown, and folded into a quasi-equivalent icosahedron.

Johnson, J.E. & Speir, J.A. (1997) Quasi-equivalent Viruses: A Paradigm for Protein Assemblies JMB, 269, 665-675
Caspar DLD, Klug A. Physical principles in the construction of regular viruses. Cold Spring Harbor Symp Quant Biol 1962;27:1.

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