Section 21.1.7.2.2. Torsion angles (dihedrals)

The conformation of the backbone of every non-terminal amino-acid residue is determined by three torsion angles, traditionally called φ (), ψ () and ω (). Owing to the peptide bond's partial double-bond character, the ω angle is restrained to values near 0° (cis-peptide) and 180° (trans-peptide). Cis-peptides are relatively rare and usually (but not always) occur if the next residue is a proline (Ramachandran & Mitra, 1976; Stewart et al., 1990). The average ω-value for trans-peptides is slightly less than 180° (MacArthur & Thornton, 1996), but surprisingly large deviations have been observed in atomic resolution structures (Sevcik et al., 1996; Merritt et al., 1998). The ω angle therefore offers little in the way of validation checks, although values in the range of ±20 to ±160° should be treated with caution in anything but very high resolution models. The φ and ψ torsion angles, on the other hand, are much less restricted, but it has been known for a long time that owing to steric hindrance there are several clearly preferred combinations of φ, ψ values (Ramakrishnan & Ramachandran, 1965). This is true even for proline and glycine residues, although their distributions are atypical (Morris et al., 1992). Also, an overwhelming majority of residues that are not in regular secondary-structure elements are found to have favourable φ, ψ torsion-angle combinations (Swindells et al., 1995). For these reasons, the Ramachandran plot (essentially a φ, ψ scatter plot) is an extremely useful indicator of model quality (Weaver et al., 1990; Laskowski, MacArthur et al., 1993; MacArthur & Thornton, 1996; Kleywegt & Jones, 1996b; Kleywegt, 1996; Hooft et al., 1997). Residues that have unusual φ, ψ torsion-angle combinations should be scrutinized by the crystallographer. If they have convincing electron density, there is probably a good structural or functional reason for the protein to tolerate the energetic strain that is associated with the unusual conformation (Herzberg & Moult, 1991). As a rule, the residue types that are most often found as outliers are serine, threonine, asparagine, aspartic acid and histidine (Gunasekaran et al., 1996; Karplus, 1996). The quality of a model's Ramachandran plot is most convincingly illustrated by a figure. Alternatively, the fraction of residues in certain predefined areas of the plot (e.g. core regions) can be quoted, but in that case it is important to indicate which definition of such areas was used. Sometimes, one may also encounter a Balasubramanian plot, which is a linear φ, ψ plot as a function of the residue number (Balasubramanian, 1977).

In protein structures, the plane of the peptide bond can have two different orientations (approximately related by a 180° rotation around the virtual C^α—C^α bond) that are both compatible with a trans configuration of the peptide (Jones et al., 1991). The correct orientation can usually be deduced from the density of the carbonyl O atom or from the geometric requirements of regular secondary-structure elements (in α-helices, all carbonyl O atoms point towards the C-terminus of the helix; in β-strands, carbonyl O atoms usually alternate their direction). In other cases, e.g. in loops with poor density, the correct orientation may be more difficult to determine and errors are easily made. By comparing the local C^α conformation to a database of well refined high-resolution structures, unusual peptide orientations can be identified and, if required, corrected (through a `peptide flip'; Jones et al., 1991; Kleywegt & Jones, 1997, 1998). Since flipping the peptide plane between residues i and i + 1 changes the ψ angle of residue i and the φ angle of residue i + 1 by ∼180°, erroneous peptide orientations may also lead to outliers in the Ramachandran plot (Kleywegt, 1996; Kleywegt & Jones, 1998).

All amino-acid residues whose side chain extends beyond the C^β atom contain one or more conformational side-chain torsion angles, termed χ₁ (N—C^α—C^β—X ^γ, where X may be carbon, sulfur or oxygen, depending on the residue type; if there are two γ atoms, the χ₁ torsion is calculated with reference to the atom with the lowest numerical identifier, e.g. O^γ1 for threonine residues), χ₂ (C^α—C^β—X ^γ—X ^δ) etc. Early on, it was found that the values that these torsion angles assume in proteins are similar to those expected on the basis of simple energy calculations and that in addition certain combinations of χ₁, χ₂ values are clearly preferred (so-called rotamer conformations; Janin et al., 1978; James & Sielecki, 1983; Ponder & Richards, 1987). Analogous to Ramachandran plots, χ₁, χ₂ scatter plots can be produced that show how well a protein's side-chain conformations conform to known preferences (Laskowski, MacArthur et al., 1993; Carson et al., 1994). Alternatively, a score can be computed for each residue that shows how similar its side-chain conformation is to that of the most similar rotamer for that residue type. This score can be calculated as an r.m.s. distance between corresponding side-chain atoms (Jones et al., 1991; Zou & Mowbray, 1994; Kleywegt & Jones, 1998) or it can be expressed as an r.m.s. deviation of side-chain torsion-angle values from those of the most similar rotamer (Noble et al., 1993).

Other torsion angles that have been used for validation purposes include the proline φ torsion (restricted to values near −65° owing to the geometry of the pyrrolidine ring; Morris et al., 1992) and the χ₃ torsion in disulfide bridges (defined by the atoms C^β—S—S′—C^β′ and restricted to values near +95 and −85°; Morris et al., 1992). In addition to the torsion-angle values of individual residues, pooled standard deviations of χ₁ and/or χ₂ torsions have been used for validation purposes (Morris et al., 1992; Laskowski, MacArthur et al., 1993).

To assess the `geometric strain' in a model on a per-residue basis, the refinement program X-PLOR (Brünger, 1992b) can produce geometric pseudo-energy plots. In such a plot, the ratio of E_geom(i)/r.m.s.(E_geom) is calculated as a function of the residue number i. The pseudo-energy term E_geom consists of the sums of the geometric and stereochemical pseudo-energy terms of the force field (E_geom = E_bonds + E_angles + E_dihedrals + E_impropers), involving only the atoms of each residue.

It has been observed that the more high-resolution protein structures become available, the more `well behaved' proteins turn out to be, i.e. the distributions of conformational torsion angles and torsion-angle combinations become even tighter than observed previously and the numerical averages tend to shift somewhat (Ponder & Richards, 1987; Kleywegt & Jones, 1998; EU 3-D Validation Network, 1998; MacArthur & Thornton, 1999; Walther & Cohen, 1999).