Chapter 18.3. Structure quality and target parameters

One set of restraints arises from the expectation that the macromolecular structure should be chemically reasonable, that is, it should reflect the geometries required by chemical physics. This restricts especially values of bond lengths, bond angles, planarity and improper dihedral angles. Additional restriction comes from the expectation that these values should most closely conform to structures determined by the same experimental method and corresponding model assumptions. There are now two sources of structural information of sufficient quality for use as restraints in structure refinement: chemical fragments from the Cambridge Structural Database (CSD) (Chapter 23.4 ) and the very high resolution protein structures that can be refined without recourse to restraining parameters (Longhi et al., 1998; Dauter et al., 1997). Restraints derived from the CSD have come into widespread use for protein (Engh & Huber, 1991; Brünger, 1993; Priestle, 1994) and nucleic acid (Parkinson et al., 1996) refinement; since their derivation, the database has grown from some 80 000 structures to around 200 000. The number of very high resolution protein structures solved has also increased to include dozens of structures (or thousands of chemical fragments). Chemical-fragment geometries can be studied using either or both sets of structures.

The optimal choice of restraints to use depends on the character and expected use of the refined structure. For example, the quality of comparative studies (Kleywegt & Jones, 1998) is enhanced by standardized refinement conditions, removing at least one source of systematic variation (e.g. Laskowski, Moss & Thornton, 1993), although effects from individual crystals and data collection and processing remain. On the other hand, a structure with an unusual chemical environment might require a tailored parameterization scheme to study the effects of this environment. Such cases might occur, for example, at enzyme active sites, where bound ligands may be distorted toward intermediate geometries (Bürgi & Dubler-Steudle, 1988).

As an alternative to implementation as restraints, statistical information from known structures provides an independent check of structural integrity. However, the more information which is used for restraints, the less is available for such cross validation. For example, the use of a force field in protein refinement that strictly enforces a physical distribution of the protein backbone φ–ψ angles of the Ramachandran plot would transfer error-induced strain into other degrees of freedom while eliminating a Ramachandran analysis as a tool for judging protein quality (Hooft et al., 1997). There is an additional risk associated with the intentional introduction of bias into protein refinement: Restraining a model to conform to known structures is in error if the structure is unique in some way and therefore should not conform to these structures, or if there is erroneous bias in the set of structures from which the restraints were derived. Furthermore, the bias may be exaggerated, especially if the biased feature is so probable that deviations are considered suspect. This can be seen, for example, in the increasing frequency of cis-prolines as protein-structure resolution increases (Stewart et al., 1990): when there is uncertainty in lower-resolution structures, the more probable trans-prolines are preferentially chosen in model building, exaggerating their frequency.

Standard refinement procedures usually include the use of harmonic restraints of bond lengths, bond angles, planarity and `improper' dihedrals, which, along with nuclear repulsions, comprise the `hardest' restraints. Other geometric properties, such as dihedral angles, electrostatic interactions, Ramachandran energies etc., can contribute further information to the refinement; these softer restraints can either distort the structure if weighted too strongly, or not contribute significantly to the refinement if weighted too weakly. Further, the softer restraints may be more useful as statistical parameters for judging the quality of a structure than as refinement parameters themselves. Particular care should be taken when refining with significant electrostatic energies: the forces have significant effects over long ranges, are usually based on simplified polarizability models and usually arise from assumptions of standard intrinsic pK_a values.

For a given distribution of parameter values, we take the average as the target value. If the choice of fragment geometries gives a distribution that corresponds to the varieties in the refinement problems, restraints should be applied to allow a similar range of freedom. In general, Gaussian distributions are assumed (with some obvious checks to avoid bimodal distributions) and are not corrected for non-orthogonality when parameters are introduced. If the refinement program uses a least-squares minimization method, the statistical mean and variance values can be used as tabulated. If an energy-function target is used, the variance values may be converted to force constants k defined by by equating the Boltzmann probability, , with the Gaussian distribution function, , such that , where b is the Boltzmann constant. (Note that this force constant k is one-half the magnitude of the force constant defined by Newton's law, .) This treatment resembles the generation of a mean field potential from the structure (e.g. Sippl, 1995), but is simplified for bond and angle parameters in several respects. Firstly, a single temperature is assumed; temperature dependence of equilibrium constants would require consideration of the temperatures of individual crystal structure determinations. Secondly, a normal-mode analysis would be required to eliminate redundancies arising from coupling between the parameters. Finally, the values of the variances are in many cases uncertain, due to poor statistical representation, non-Gaussian effects, or other causes (see below). Note that this simplified treatment of parameter uncertainty can be implemented more rigorously in maximum-entropy refinement methods (Pannu & Read, 1996; Bricogne, 1993).

Planarity and improper dihedral restraints, being `hard' restraints, are amenable to the same kind of parameterization described above. Physically realistic deviations should be allowed. A survey of several planar atoms, such as CG of aromatic residues, the inter-ring carbons (CD2, CE2) of tryptophan and CZ of tyrosine, showed standard deviations about strict planarity of 1–2°. Statistically significant deviations of average values from perfect planarity might also be expected, particularly as a function of the protein fold environment. For example, an average nonzero planarity of the ω angle of the peptide bond has been noted (Marquart et al., 1983) and attributed at least in part to secondary structure; such effects may be misrepresented in statistics from small molecules.

Since torsion angles are generally more adequately determined by protein structures of typical resolutions, there is less need to derive parameters from the CSD for refinement purposes. Further, distributions derived from small molecules may not be representative of torsion angles among proteins, since typical fragments lack `typical' protein secondary structure. This kind of environmental dependence will affect softer parameters such as dihedral angles more than bonds or angles. However, since the torsion-angle distribution is a function not only of potential interactions of the peripheral groups but of the electronic character of the bond itself, some questions may be best examined by the study of chemical fragments. One such feature might be the distributions for aromatic residues. Statistical distributions of side-chain orientations show the apparent stability of an eclipsed conformation, particularly for ( conformer, Fig. 18.3.2.3). If the relative dearth of secondary structure leads to atypical torsion-angle distributions in proteins, it must be conversely expected that torsion-angle statistics derived from the set of all proteins will be less applicable, for example, to helical proteins than statistics derived from mostly helical proteins. This illustrates the dilemma of selecting the ideal statistical database for protein refinement.

Figure 18.3.2.3| top | pdf | Tryptophan dihedral angle distribution. The 36 tryptophan fragments in the CSD show several apparent minima, including an eclipsed Cα—Cβ—Cγ—Cδ1 dihedral conformation. This is apparent in , tryptophan distribution plots from protein structures as well (Laskowski, MacArthur et al., 1993).

Like the potential parameterization of torsion-angle statistics with the CSD, parameterization of non-bonded interactions, typically into terms representing packing (an empirical mix of London dispersion forces and solvent effects), electrostatics and hydrogen bonding, is probably more strongly influenced by protein environment than are bond and angle terms. Specific chemical questions are likely to be best addressed with fragment structure databases, however, for improved parameterization or structure-quality evaluation. Possible examples include hydrogen bonds, salt bridges (see, e.g., the discussion on charge delocalization of Glu and Asp above) etc. These are particularly relevant for structures generally atypical among proteins, such as at enzyme reactive sites.

While many CSD fragments include hydrogen-atom positions, their accuracy is necessarily the most limited. Evaluation of CSD statistics without hydrogens and the subsequent addition of parameters to refine hydrogens adds an additional artifactual coupling between parameters involving non-hydrogen atoms as well. This artifactual coupling might theoretically be of some concern, but is a second-order effect and presumably introduces effects smaller than the artifactual coupling between non-hydrogen parameters, for example.

Most crystallographers will experience neither the need nor desire to derive their own parameterization for general protein structure refinement; many, however, need new parameters for ligands or other entities that are not amino-acid residues. The accuracy required for their application will determine the appropriate effort. If the purpose of the structure is to determine the general orientation of an inhibitor for molecular modelling studies with a still lower effective accuracy, it may be that no refinement (and no parameterization) is necessary at all. As the accuracy requirements increase, so does the need for good parameterization and a way of estimating when the density is incompatible with known structures. Such incompatibility may be decisive in identifying the stereochemistry of ligands selected from racemic mixtures, or the occurrence of a chemical reaction, or even falsely characterized substances. For such applications, small-molecule structural databases will remain the only choice for parameter derivation, which can be done exactly as for amino-acid fragments.

When specific structural effects are observed which are not otherwise parameterized, new parameters might be desirable to encode this information. In the case of new statistical minima or variances of parameters already encoded by the refinement program, it is for most programs a simple matter to introduce new atom types, residues, or fragment names to the topology and parameter libraries. Examples include new parameterization for charged states or for cis- and trans-proline. If the reason for the new statistics is structural, the structure must be appropriately monitored during refinement to ensure that the conditions continue to hold, particularly in the case of simulated-annealing refinement steps.

Primarily in cases of new structures, such as small-ligand- or metal-binding proteins, the refinement may indicate that the expected geometries and applied restraints seem incompatible with evidence from the electron density. Several sources for such discrepancies must be considered for an evaluation of the true geometries or the confidence level of such an evaluation. The quality of the experimental information, such as data resolution and reduction parameters, must be considered. Physical phenomena possibly ignored by the refinement model might include anisotropies of motion and/or electron distribution, or disorder in the crystal. These might lead to systematic deviations in the refined structure that mimic alternate parameterizations. Finally, newly derived parameters should be examined to decide whether the fragments and chemical environments were inappropriate for the refinement problem, or whether errors in fragment structures artifactually distorted the parameterization.