Strongly correlated structure of axial-symmetric proteins. II. Pentagonal, heptagonal, octagonal, nonagonal and ondecagonal symmetries.

The investigation of the geometry of the molecular envelope and channel in the proteins discussed in part I [Janner (2005a), Acta Cryst. D61, 247-255] is extended to axial-symmetric proteins with orders of rotation N = 5, 7, 8, 9 and 11, non-crystallographic in dimension 3. In these cases also, the vertices of the molecular form which encapsulate the C(alpha) backbone have integral coordinates (indices) in a symmetry-adapted basis which generates a polygonal lattice. As in the crystallographic case of part I, a characteristic rational axial ratio squared is observed that reduces to one the number of free lattice parameters and enhances the symmetry. Furthermore, there is a crystallographic scaling relation between the envelope and the channel which depends on the order of the axial symmetry and is expressible in terms of star polygons. Possible biological implications are suggested within a more general context.