Analytical Framework for Protein Structure Determination by Solid-State NMR of Aligned Samples

Abstract

A compact formulation of the experimental solid-state NMR observables in terms of the irreducible basis of rotations is presented. Quadratic-form representations are derived for the 15N and 1H chemical shift anisotropy, as well as 1H-15N and 1Hα-13Cα dipolar interactions. Peptide plane geometries together with the torsion angles Φ and Ψ are incorporated into the corresponding Wigner rotation matrices, which allow one to establish the mapping of the protein structure onto its multidimensional solid-state NMR spectra in analytically semi-closed form. The structural fitting of experimental two-dimensional NMR spectra of Pf1 bacteriophage is presented. Examples of protein structure calculations from simulated three-dimensional solid-state NMR spectra for protein G, two-helical hairpin fragment of bacteriorhodopsin, and a loop region from KcsA are also presented. Moreover, analytical expression is obtained for the periodicity of secondary structures as an explicit function of Φ and Ψ.