An iterative Fourier-Bessel algorithm for reconstruction of helical structures with severe Bessel overlap

Classical Fourier-Bessel methodology fails when used to reconstruct helical structures with severe Bessel overlap on the layer lines. In the reconstruction of a peculiar type of double-layered helical tube of GDP-tubulin, we face the problem of Bessel overlap on all the layer lines due to the superposition of the Fourier components from the inner and outer layers of the tube. In order to decompose the Fourier terms of the inner and outer layers more than one image of the tubes must be combined and the orientations of their inner and outer layer helices must be determined. While there is no direct analytical method to determine these orientational parameters, we have devised an iterative Fourier-Bessel algorithm to calculate the correct orientations and thus allow us to obtain a reconstruction from multiple images of the double-layered tubes. The algorithm successfully works for the reconstruction of computer-modeled double-layered helical tubes as well as with real images obtained by cryo-electron microscopy. The algorithm has also been applied with very satisfactory results to the reconstruction of 13-protofilament microtubules, which is another helical structure that suffer Bessel overlap, suggesting its generality.