Rapid automated superposition of shapes and macromolecular models using spherical harmonics

Abstract

A rapid algorithm to superimpose macromolecular models in Fourier space is proposed and implemented (SUPALM). The method uses a normalized integrated cross-term of the scattering amplitudes as a proximity measure between two three-dimensional objects. The reciprocal-space algorithm allows for direct matching of heterogeneous objects including high- and low-resolution models represented by atomic coordinates, beads or dummy residue chains as well as electron microscopy density maps and inhomogeneous multi-phase models (e.g. of protein-nucleic acid complexes). Using spherical harmonics for the computation of the amplitudes, the method is up to an order of magnitude faster than the real-space algorithm implemented in SUPCOMB by Kozin & Svergun [J. Appl. Cryst. (2001 ▸), 34, 33-41]. The utility of the new method is demonstrated in a number of test cases and compared with the results of SUPCOMB. The spherical harmonics algorithm is best suited for low-resolution shape models, e.g. those provided by solution scattering experiments, but also facilitates a rapid cross-validation against structural models obtained by other methods.

Keywords: SUPALM; model superposition; protein structure; scattering amplitudes; small-angle scattering.

Figure 1

The crystal structures of several proteins (green Cα traces) with superimposed ab initio shapes obtained by SUPALM (red spheres) and by SUPCOMB (blue spheres). (a) Z1Z2 protein (PDB code 2a38; Marino et al., 2006 ▸), (b) G protein (PDB code 1got; Lambright et al., 1996 ▸), (c) internalin (PDB code 1o6v; Schubert et al., 2002 ▸), (d) lumazine synthase (PDB code 1hqk; Zhang et al., 2001 ▸). The right view is rotated counterclockwise by 90° around the vertical axis.

Figure 2

A hybrid high-resolution model of the human gamma-secretase derived from cryoEM (PDB code 4uis; Sun et al., 2015 ▸) (green Cα traces) and the bead models from the electron-density map (ID EMD-2974.map) superimposed onto it by SUPALM (red spheres) and by SUPCOMB (blue spheres). The right view is rotated counterclockwise by 90° around the vertical axis.

Figure 3

The crystal structure of the 70S ribosome (PDB code 4v4w; Mitra et al., 2006 ▸) (green Cα traces correspond to the protein parts, the RNA parts are shown in yellow) with superimposed two-phase ab initio shapes obtained by SUPALM (red spheres) and by SUPCOMB (blue spheres). (a) The complete 70S ribosome models, (b) the protein parts, (c) the RNA parts. The right view is rotated counterclockwise by 90° around the vertical axis.

Figure 4

Two-dimensional contour plots of NSD (left column) and 1/NCC (right column) in the vicinity of the SUPALM solution for G protein (see Fig. 1 ▸ b). The SUPALM solution corresponds to the origin of the coordinates for all plots (marked with a cross symbol). (a), (b) The contour plots versus the rotations around x/y and x/z axes, respectively. (c), (d) The contour plots versus the shifts along x/y and x/z axes, respectively. The ‘true’ SUPCOMB solution (the minimum of the two-dimensional contour plot) is positioned close to the SUPALM solution (RMSD = 3.3 Å).

Figure 5

Different conformers of UVI31+ protein as determined by NMR (PDB code 2ma0) (a) and SUPALM alignments of these conformations onto the reference model (b). The reference NMR model is shown with a bold blue Cα trace in both panels.