Symmetry as a priori information: low-resolution reconstruction of viral structure from solution x-ray scattering data

The so-called “small spherical viruses” are viruses with a shell of protein (the so-called “capsid”) surrounding an inner core of nucleic acid. The capsid is “crystalline” in the sense that it is constructed from many repetitions of the same polypeptides and the entire capsid is invariant under the rotational symmetries of the icosahedron. The icosahedron, as shown in Figure 1, is constructed from 20 equilateral triangles and has 60 rotational symmetries: a 5-fold axis where 5 triangles meet, a 3-fold axis through the center of each triangle, and a 2-fold axis at the midpoint of each edge between two triangles. For the viruses discussed in this paper, the outer radius of the capsid is about 140Å.