3D reconstruction problems for cryo electron microscopy of viruses

In this paper we describe several signal reconstruction problems that arise during the determination of the 3D structure of so-called spherical viruses. Spherical viruses are viruses with a shell of protein (the 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 some rotational symmetry, often the rotational symmetry of the icosahedron which is the case we focus on. The icosahedron 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. A typical outer radius of the capsid is in the range 102-103Å.