Maximum likelihood three-dimensional virus reconstruction from projections of unknown orientation and cryo electron microscopy application

Biological spherical viruses are nanometer (10–100nm) objects of roughly spherical shape where in many species each example of the species is identical. Cryo electron microscopy of such viruses provides images that are essentially 2-D tomographical projections of the 3-D virus scattering intensity. Three key problems are that the projection angles are not known, the projections are modified by a contrast transfer function with zeros, and the signal to noise ratio is less than 1. The goal is to compute a 3-D reconstruction of the virus scattering intensity. Motivated by the low signal to noise ratio, we developed statistical approaches. In particular, we developed a statistical model of the image formation process which forms the basis for a maximum likelihood estimation problem. We developed expectation maximization algorithms to solve the maximum likelihood problem. These algorithms are unusual because the maximization step is easy but the expectation step is hard because multi-dimensional numerical integrations are required. To incorporate the uncertainty in the image origin offset, 5-D integration rules are necessary and have been introduced. Several examples of 3D virus reconstruction from image data, which include classification and reconstruction using synthetic mixed particle data and single particle reconstruction using experimental data, successfully demonstrate the use of the proposed algorithms