Section 18.1.7. Models

Atomic resolution models are generally straightforward. A reasonably well phased diffraction pattern at atomic resolution shows the location of each atom. The primary problem (which can be substantial) is deciding how to model any disorder that may be present. Structural chemistry is derived from the model. Macromolecular models generally have most of the structural chemistry built in as part of the model. This approach is required as a direct consequence of having too little data at too limited a resolution to determine the positions of all of the atoms without using this additional information.

There are two procedures for building structural chemistry into a model. The first is to use known molecular geometry to reduce the number of variables. For example, if the distance between two atoms is held constant, the locus of possible positions for the second atom is the surface of a sphere centred on the first atom. This means that the position of the second atom can be specified given the position of the first atom and two variables to locate the point on the sphere – a total of five variables instead of six. Every non-redundant constraint reduces the number of degrees of freedom in the model by one. If the second atom in this example were replaced by a group of atoms with known geometry (e.g. a phenyl group containing six atoms), the number of positional parameters could be reduced from 21 to eight. Constrained refinement is discussed extensively in IT C Chapter 8.3.

The second procedure is to treat the additional information as additional observations. A bond length is assumed to be an observation, based on other crystal structures, which has a mean value and a variance. This observation is added to the data instead of being used to reduce the number of parameters in the model.

The two approaches have different consequences on the ratio of observations to parameters. If we have observations, parameters and non-redundant geometric features to add to the problem, we have either and or and , where C is the ratio of observations to parameters. The former are parameter constraints and the latter are parameter restraints. Constraints are more effective at increasing the ratio of observations to parameters, but since these features are built into the model, it is difficult to evaluate how appropriate they actually are for the problem at hand. Restraints provide an automatic evaluation of the appropriateness of the assumed geometry to the current data, because the deviations from the assumed values can be tested for statistical significance.

The most common constraints and restraints applied to macromolecular crystal structures are those which preserve or reinforce the molecular geometry of the amino acid or nucleotide residues (Chapter 18.3 ). Expected values for the geometry of these structural fragments are available from the small-molecule crystallographic literature and databases. A further step, which reduces the parameter count substantially, is to treat parts of the molecule as a set of linked rigid groups. This is particularly appropriate for aromatic fragments such as the side chains of phenylalanine, tyrosine, tryptophan and histidine, but can also be appropriate for small groups like valine and threonine. The extreme form of this approach is torsion-angle dynamics (Rice & Brünger, 1994), in which the only variables are torsion angles about bonds, and the position and orientation of the whole molecule. This description of the model works well with the right kind of optimization procedure.

Positional restraints can be parameterized in a variety of ways. For example, the geometry of three atoms can be treated as the three distances involved or as two distances and the angle between them. Several of the more popular restrained refinement programs treat the parameters for bond distances, bond angles and planarity as distances with a set of standard deviations. Others treat them as bond distances, bond angles and torsion angles weighted by the energy terms derived from experimental conditions. Different methods of parameterization and weighing have different effects on the refinement process, but to date these differences are not well characterized. The primary effects should be on the approach to convergence, as all of these formulations are normally satisfied by correct structures.

Additional criteria can be added to the model besides simple geometry. Preservation of bond lengths is usually done by adding terms to the objective function, where is the distance between atoms i and j, is the ideal bond length, and is the weight applied to the bond. This is formally equivalent to treating bond stretching as a spring. Additional energy parameters can be added, such as electrostatic energy terms. Whatever vision of reality is applied to the objective function becomes part of the model.

The atomic displacement factors (B factors) present a different set of problems from the coordinates. The behaviour of these parameters is strongly affected by coordinate errors, and in fact large atomic displacement parameters are frequently used to determine which parts of a structure are likely to contain errors. The B factors are strongly related to the rate at which the diffraction pattern diminishes with resolution and thus cannot be accurately determined unless the diffraction pattern has been measured over a sufficiently wide range of resolution to determine this rate. As a practical matter, it is not feasible to refine individual atomic displacement parameters at resolution less than about 2 Å, and they frequently present problems even in atomic resolution small-molecule structures. In high-resolution small-molecule structures, B factors are frequently represented as anisotropic ellipsoids described by six parameters per atom. In spite of the larger displacements found in macromolecules relative to small molecules, it is rarely possible to support the number of parameters required to refine a structure with independent anisotropic displacement factors. Nevertheless, the B factors of the atoms are essential parts of the crystallographic model. Several methods for reducing the number of independent B factors have been developed. The simplest is group B factors, in which one parameter is refined for all atoms in a particular group of atoms. Another method is to apply a simple model to the change in displacement parameter within a group of atoms. In this treatment, a B factor is refined for one atom, say the atom of an amino-acid residue, and the remainder of the atoms in the residue are assigned displacement parameters that depend on their distance from the atom (Konnert & Hendrickson, 1980). A third method is to enforce similarity of displacement parameters based on the correlation coefficients between pairs of displacement parameters in highly refined high-resolution structures (Tronrud, 1996).

Small-molecule refinement programs also apply restraints to the displacement parameters. The SIMU command of SHELX restrains the axes of the anisotropic displacement parameters of bonded atoms to be similar. This approach has been applied to a number of very high resolution macromolecular refinements.

Large B factors do not represent large thermal motions of the atom but rather a distribution of positions occupied by the atom over time or in different unit cells of the crystal. The line between describing atoms with large B factors as distributed about a single point or several points (disordered atoms) is sometimes blurred. At some point, the disorder can become resolved into alternative positions or the atoms disappear from the observable electron density. There are two kinds of disorder that can be easily modelled if data are available to sufficient resolution:

At high resolution, it is sometimes possible to model the correlated motion of atoms in rigid groups by a single tensor that describes translation, libration and screw. This is rarely done for macromolecules at present, but may be an extremely accurate way to model the behaviour of the molecules. The recent development of efficient anisotropic refinement methods for macromolecules by Murshudov et al. (1999) will undoubtedly produce a great deal more information about the modelling of dynamic disorder and anisotropy in macromolecular structures.

Macromolecular crystals contain between 30 and 70% solvent, mostly amorphous. The diffraction is not accurately modelled unless this solvent is included (Tronrud, 1997). The bulk solvent is generally modelled as a continuum of electron density with a high atomic displacement parameter. The high displacement parameter blurs the edges, so that the contribution of the bulk solvent to the scattering is primarily at low resolution. Nevertheless, it is important to include this in the model for two reasons. First, unless the bulk solvent is modelled, the low-resolution structure factors cannot be used in the refinement. This has the unfortunate effect of rendering the refinement of all of the atomic displacement parameters ill-determined. Second, omission or inaccurate phasing of the low-resolution reflections tends to produce long-wavelength variations in the electron-density maps, rendering them more difficult to interpret. In some regions, the maps can become overconnected, and in others they can become fragmented.