Structure determination in "shiftless" solid state NMR of oriented protein samples

An efficient formalism for calculating protein structures from oriented-sample NMR data in the torsion-angle space is presented. Angular anisotropies of the NMR observables are treated by utilizing an irreducible spherical basis of rotations. An intermediate rotational transformation is introduced that greatly speeds up structural fitting by rendering the dependence on the torsion angles Φ and Ψ in a purely diagonal form. Back-calculation of the simulated solid-state NMR spectra of protein G involving 15N chemical shift anisotropy (CSA), and 1H-15N and 1Hα-13Cα dipolar couplings was performed by taking into account non-planarity of the peptide linkages and experimental uncertainty. Even a relatively small (to within 1 ppm) random variation in the CSA values arising from uncertainties in the tensor parameters yields the RMSD's of the back-calculated structures of more than 10 ?. Therefore, the 15N CSA has been substituted with heteronuclear dipolar couplings which are derived from the highly conserved bond lengths and bond angles associated with the amino-acid covalent geometry. Using the additional 13Cα-15N and 13C'-15N dipolar couplings makes it possible to calculate protein structures entirely from "shiftless" solid-state NMR data. With the simulated "experimental" uncertainty of 15 Hz for protein G and 120 Hz for a helical hairpin derived from bacteriorhodopsin, back-calculation of the synthetic dipolar NMR spectra yielded a converged set of solutions. The use of distance restraints dramatically improves structural convergence even if larger experimental uncertainties are assumed.