Fast rotational matching

A computationally efficient method is presented - `fast rotational matching' or FRM -  that significantly accelerates the search of the three rotational degrees of freedom (DOF) in biomolecular matching problems. This method uses a suitable parametrization of the three-dimensional rotation group along with spherical harmonics, which allows efficient computation of the Fourier Transform of the rotational correlation function. Previous methods have used Fourier techniques only for two of the rotational DOFs, leaving the remaining angle to be determined by an exhaustive search. Here for the first time a formulation is presented that makes it possible to Fourier transform all three rotational DOFs, resulting in notable improvements in speed. Applications to the docking of atomic structures into electron-microscopy maps and the molecular-replacement problem in X-ray crystallography are considered.