Multiclass maximum-likelihood symmetry determination and motif reconstruction of 3-D helical objects from projection images for electron microscopy.

Many micro- to nano-scale 3-D biological objects have a helical symmetry. Cryo electron microscopy provides 2-D projection images where, however, the images have low SNR and unknown projection directions. The object is described as a helical array of identical motifs, where both the parameters of the helical symmetry and the motif are unknown. Using a detailed image formation model, a maximum-likelihood estimator for the parameters of the symmetry and the 3-D motif based on images of many objects and algorithms for computing the estimate are described. The possibility that the objects are not identical but rather come from a small set of homogeneous classes is included. The first example is based on 316 128 × 100 pixel experimental images of Tobacco Mosaic Virus, has one class, and achieves 12.40-Å spatial resolution in the reconstruction. The second example is based on 400 128 × 128 pixel synthetic images of helical objects constructed from NaK ion channel pore macromolecular complexes, has two classes differing in helical symmetry, and achieves 7.84- and 7.90-Å spatial resolution in the reconstructions for the two classes.