Protein imperfections: separating intrinsic from extrinsic variation of torsion angles

In this paper, the variation of the values of dihedral angles in proteins is divided into two categories by analyzing distributions in a database of structures determined at a resolution of 1.8 Å or better [Lovell et al. (2003), Proteins Struct. Funct. Genet. 50, 437–450]. The first analysis uses the torsion angle for the C^α—C^β bond (χ₁) of all Gln, Glu, Arg and Lys residues (`unbranched set'). Plateaued values at low B values imply a root-mean-square deviation (RMSD) of just 9° for χ₁ related to intrinsic structural differences between proteins. Extrapolation to high resolution gives a value of 11°, while over the entire database the RMSD is 13.4°. The assumption that the deviations arise from independent intrinsic and extrinsic sources gives ∼10° as the RMSD for χ₁ of these unbranched side chains arising from all disorder and error over the entire set. It is also found that the decrease in χ₁ deviation that is correlated with higher resolution structures is almost entirely a consequence of the higher percentage of low-B-value side chains in those structures and furthermore that the crystal temperature at which diffraction data are collected has a negligible effect on intrinsic deviation. Those intrinsic aspects of the distributions not related to statistical or other errors, data incompleteness or disorder correlate with energies of model compounds computed with high-level quantum mechanics. Mean side-chain torsion angles for specific rotamers correlate well with local energy minima of Ace-Leu-Nme, Ace-Ile-Nme and Ace-Met-Nme. Intrinsic RMSD values in examples with B ≤ 20 Ų correlate inversely with calculated values for the relevant rotational energy barriers: from a low of 6.5° for χ₁ of some rotamers of Ile to a high of 14° for some Met χ₃ for fully tetrahedral angles and much higher for χ angles around bonds that are tetrahedral at one end and planar at the other (e.g. 30° for χ₂ of the gauche⁻ rotamer of Phe). For the lower barrier Met χ₃ rotations there are relatively more well validated cases near eclipsed values and calculated torques from the rest of the protein structure either confine or force the C^[epsilon] atom into the strained position. These results can be used to evaluate the variability and accuracy of χ angles in crystal structures and also to decide whether to restrain side-chain angles in refinement as a function of the resolution and atomic B values, depending on whether one aims for a realistic distribution of values or a spread that is statistically suitable to the probable data-set errors.