Averaging data derived from images of helical structures with different symmetries

There are many examples of macromolecules that form helical tubes or crystals, which are useful for structure determination by electron microscopy and image processing. Helical crystals can be thought of as two-dimensional crystals that have been rolled into a cylinder such that two lattice points are superimposed. In many real cases, helical crystals of a particular macromolecule derive from an identical two-dimensional lattice but have different lattice points superimposed, thus producing different helical symmetries which cannot be simply averaged in Fourier-space. When confronted with this situation, one can select images corresponding to one of the observed symmetries at the expense of reducing the number of images that can be used for data collection and averaging, or one can calculate separate density maps from each symmetry, then align and average them together in real-space. Here, we present a third alternative, which is based on averaging of the Fourier-Bessel coefficients, gn,l(r), and which allows the inclusion of data from all symmetry groups derived from a common two-dimensional lattice. The method is straightforward and simple in practice and is shown, through a specific example with real data, to give results comparable to real-space averaging.