A New Theory and Algorithm for Reconstructing Helical Structures with a Seam

Conventional helical reconstruction is a general method to obtain three-dimensional structures of many filamentous biological macromolecules. The method assumes helical symmetry, and generates the three-dimensional structures from two-dimensional projection images. However, the theory is inadequate to describe filamentous structures discontinuities, which are called seams in the case of microtubules or perturbations in the case of tobacco mosaic virus or the bacterial flagellar filament. To study such structures, a new theory and algorithm are required. To this aim, we developed a new algorithm, namely, asymmetric helical reconstruction, which is based on our new theory that describes a “helical” object with a seam. In the theory, “helical” objects with a seam are indexed with a non-integral order of ν. Like the conventional helical reconstruction, the layer-line data are extracted from the Fourier transform of the images. We show that the Fourier–Bessel transform using the Bessel functions of fractional order can, to good approximation, reconstruct the three-dimensional structure of the object. To test the new algorithm, we reconstructed three-dimensional structures of a kinesin–microtubule complex with a seam from both model data and experimental data from cryo-electron microscopic images. The reconstructed structures are almost identical with those reconstructed from conventional helical reconstruction demonstrating the validity of the algorithm. The algorithm enables the analysis of various “helical” specimens with seams and also significantly improves the throughput and the resolution of kinesin–microtubule complexes.