Determination of the Backbone Dihedral Angles φ in Human Ubiquitin from Reparametrized Empirical Karplus Equations

The backbone dihedral angle φ in polypeptides is characterized by four different J couplings:  ³J_H^NH^α, ³J_H^NC‘, ³J_H^NC^β, and ³J_H^αC‘. E.COSY and quantitative J correlation techniques have been used to measure these couplings in the protein human ubiquitin, uniformly enriched in ¹³C and ¹⁵N. Assuming that the dihedral backbone angles in solution are identical to those in the X-ray structure of this protein and that H^N is located in the C‘−N−C^α plane, Karplus relations for ³J_H^NH^α, ³J_H^αC‘, and ³J_H^NC^β, have been reparametrized. The root-mean-square (rms) difference between measured values of ³J_H^NH^α, ³J_H^αC‘, ³J_H^NC^β, and ³J_H^NC‘ and their corresponding Karplus curves are 0.53, 0.25, 0.24, and 0.36 Hz, respectively, whereas the precision of these measurements is considerably better. For any given residue, the differences between the four measured J couplings and values predicted by their Karplus curves on the basis of the X-ray structure-derived φ angle are highly correlated with one another. On average, a root-mean-square change of 5.7° in the X-ray derived φ angles is needed to obtain optimal agreement with all four measured J couplings. There is no clear correlation between the φ angle correction needed and the out-of-plane position of the amide proton predicted by ab initio calculations. The small differences in φ angles therefore presumably result from small uncertainties in the atomic positions of the 1.8 Å X-ray structure. However, they may also be caused by genuine differences between the structure of the protein in solution and in the crystalline state or contain a contribution resulting from deviations from the assumption that the H^N−N−C^α−H^α dihedral angle equals φ − 60°.