Section 21.1.7.2. Model quality, coordinates

The covalent geometry of a model can be assessed by comparing bond lengths and angles to a library of `ideal' values. In the past, every refinement and modelling program had its own set of `ideal' values. This even made it possible to detect (with 95% accuracy) with which program a model had been refined, simply by inspecting its covalent geometry (Laskowski, Moss & Thornton, 1993). Nowadays, standard sets of ideal bond lengths and bond angles derived from an analysis of small-molecule crystal structures from the CSD (Allen et al., 1979) are available for proteins (Engh & Huber, 1991; Priestle, 1994) and nucleic acids (Parkinson et al., 1996). For other entities, typical bond lengths and bond angles can be taken from tables of standard values (Allen et al., 1987) or derived by other means (Kleywegt & Jones, 1998; Greaves et al., 1999).

For bond lengths, the r.m.s. deviation from ideal values is invariably quoted. Deviations from ideality of bond angles can be expressed directly as an angular r.m.s. deviation or in terms of angle distances (i.e. the angle ∠ABC is measured by the 1–3 distance |AC|; note that this distance is also implicitly dependent on the bond lengths |AB| and |BC|). There are some indications that protein geometry cannot always be captured by assuming unimodal distributions (i.e. geometric features with only a single `ideal' value). For example, Karplus (1996) found that the main-chain bond angle τ₃ (∠N—C^α—C) varies as a function of the main-chain torsion angles φ and ψ.

Chirality is another important criterion in the case of biomacromolecules: most amino-acid residues will have the L configuration for their C^α atom. Also, the C^β atoms of threonine and isoleucine residues are chiral centres (IUPAC–IUB Commission on Biochemical Nomenclature, 1970; Morris et al., 1992). Chirality can be assessed in terms of improper torsion angles or chiral volumes. For example, to check if the C^α atom of any residue other than glycine has the L configuration, the improper (or virtual) torsion angle C^α—N—C—C^β should have a value of about +34° (a value near −34° would indicate a D-amino acid). The torsion angle is called improper or virtual because it measures a torsion around something other than a covalent bond, in this case the N—C `virtual bond'. The chiral volume is defined as the triple scalar product of the vectors from a central atom to three attached atoms (Hendrickson, 1985). For instance, the chiral volume of a C^α atom is defined as where r_X is the position vector of atom X. It should be noted that the chiral volume also implicitly depends on the bond lengths and angles involving the four atoms.

Another issue to consider is that of moieties that are necessarily planar (e.g. carboxylate groups, phenyl rings; Hooft et al., 1996a). Again, planarity can be assessed in two different ways: by inspecting a set of (possibly improper) torsion angles and calculating their r.m.s. deviation from ideal values (e.g. all ring torsions in a perfectly flat phenyl ring should be 0°) or by fitting a least-squares plane through each set of atoms and calculating the r.m.s. distance of the atoms to that plane. Note that for double bonds, cis and trans configurations cannot be distinguished by deviations from a least-squares plane, but they can be distinguished by an appropriately defined torsion angle.

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).

Validation of C^α-only models may be necessary if such a model is retrieved from the PDB to be used in molecular replacement or homology modelling exercises; however, not many validation tools can handle such models (Kleywegt, 1997). The C^α backbone can be characterized by C^α—C^α distances (∼2.9 Å for a cis-peptide and ∼3.8 Å for a trans-peptide), C^α—C^α—C^α pseudo-angles and C^α—C^α—C^α—C^α pseudo-torsion angles (Kleywegt, 1997). The pseudo-angles and torsion angles turn out to assume certain preferred value combinations (Oldfield & Hubbard, 1994), much like the backbone φ and ψ torsions, and this can be employed for the validation of C^α-only models (Kleywegt, 1997). In addition to these straightforward methods, the mean-field approach of Sippl (1993) is also applicable to C^α-only models.

Hydrophobic, electrostatic and hydrogen-bonding interactions are the main stabilizing forces of protein structure. This leads to packing arrangements where hydrophobic residues tend to interact with each other, where charged residues tend to be involved in salt links and where hydrophilic residues prefer to interact with each other or to point out into the bulk solvent. Serious model errors will often lead to violations of such simple rules of thumb and introduce non-physical interactions (e.g. a charged arginine residue located inside a hydrophobic pocket; Kleywegt et al., 1996) that serve as good indicators of model errors. Directional atomic contact analysis (Vriend & Sander, 1993) is a method in which these empirical notions have been formalized through database analysis. For every group of atoms in a protein, it yields a score which in essence expresses how `comfortable' that group is in its environment in the model under scrutiny (compared with the expectations derived from the database). If a region in a model (or the entire model) has consistently low scores, this is a very strong indication of model errors. The ERRAT program is based on the same principle, but it is less specific in that it assesses only six types of non-bonded interactions (CC, CN, CO, NN, NO and OO; Colovos & Yeates, 1993).

Hydrogen-bonding analysis can often be used to determine the correct orientation of asparagine, glutamine and histidine residues (McDonald & Thornton, 1995). Similarly, an investigation of unsatisfied hydrogen-bonding potential can be used for validation purposes (Hooft et al., 1996b), as can calculation of hydrogen-bonding energies (Morris et al., 1992; Laskowski, MacArthur et al., 1993).

Finally, a model should not contain unusually short non-bonded contacts. Although most refinement programs will restrain atoms from approaching one another too closely, if any serious violations remain they are worth investigating, since they may signal an underlying problem (e.g. erroneous omission of a disulfide restraint or incorrect side-chain assignment).

Molecules that are related by noncrystallographic symmetry exist in very similar, but not identical, physical environments. This implies that their structures are expected to be quite similar, although different relative domain orientations and local variations may occur (e.g. owing to different crystal-packing interactions; Kleywegt, 1996). Many criteria have been developed to quantify the differences between (NCS) related models. Some, such as the r.m.s. distance (e.g. on all atoms, backbone atoms or C^α atoms) are based on distances between equivalent atoms, measured after a (to some extent arbitrary; Kleywegt, 1996) structural superpositioning operation has been performed. Others are based on a comparison of torsion angles, be it of main-chain φ, ψ angles [e.g. Δφ, Δψ plot (Korn & Rose, 1994); multiple-model Ramachandran plot (Kleywegt, 1996); σ(φ), σ(ψ) plot (Kleywegt, 1996); circular variance (Allen & Johnson, 1991) plots of φ and ψ (G. J. Kleywegt, unpublished results); Euclidian φ, ψ distances (Carson et al., 1994) or pseudo-energy values (Carson et al., 1994)] or side-chain χ₁, χ₂ angles [e.g. multiple-model χ₁, χ₂ plot (Kleywegt, 1996); σ(χ₁), σ(χ₂) plots (Kleywegt, 1996); circular variance (Allen & Johnson, 1991) plots of χ₁ and χ₂ (G. J. Kleywegt, unpublished results); Euclidian χ₁, χ₂ distances (Carson et al., 1994) or pseudo-energy values (Carson et al., 1994)]. Still other methods are based on analysing differences in contact-surface areas (Abagyan & Totrov, 1997), temperature factors (Kleywegt, 1996) or the geometry of the C^α backbone alone (Flocco & Mowbray, 1995; Kleywegt, 1996). Many of these methods can also be used to compare the structures of related molecules in different crystals or crystal forms (e.g. complexes, mutants).

Solvent molecules provide an excellent means of `absorbing' problems in both the experimental data and the atomic model. Neither their position nor their temperature factor are usually restrained (other than by the data and restraints that prevent close contacts) and sometimes even their occupancy is refined. At a resolution of ∼2 Å, crystallographers tend to model roughly one water molecule for every amino-acid residue and at 1.0 Å resolution this number increases to ∼1.6 (Carugo & Bordo, 1999). When waters are placed, it should be ascertained that they can actually form hydrogen bonds, be it to protein atoms or to other water molecules. Considering that several ions that are isoelectronic with water (Na⁺, ) are often used in crystallization solutions, one should keep in mind the possibility that some entities that have been modelled as water molecules could be something else (Kleywegt & Jones, 1997). A method to check if water molecules could actually be sodium ions, based on the surrounding atoms, has been published (Nayal & Di Cera, 1996).

Many other coordinate-based methods for assessing the validity or correctness of protein models have been developed. These include the profile method of Eisenberg and co-workers (Bowie et al., 1991; Lüthy et al., 1992), the inspection of atomic volumes (Pontius et al., 1996), and the use of threading and other potentials (Sippl, 1993; Melo & Feytmans, 1998; Maiorov & Abagyan, 1998). Some of these methods are described in more detail elsewhere in this volume. The program WHAT IF (Vriend, 1990) contains a large array of quality checks, many of which are not available in other programs, that span the spectrum from administrative checks to global quality indicators (Hooft et al., 1996). During the refinement process, coordinate shifts can be used as a rough indication of `quality' or, rather, convergence (Carson et al., 1994; Kleywegt & Jones, 1996a). Crude models tend to undergo much larger changes during refinement than models that are essentially correct and complete. Also at the residue level, large coordinate shifts indicate residues that are worth a closer look.

Laskowski et al. (1994) have formulated single-number geometrical quality criteria, which they dubbed `G factors' in analogy to crystallographic R values. These G factors combine the results of a number of quality checks (covalent geometry, main-chain and side-chain torsion angles etc.) in a single number.