Section 21.1.7.3. Model quality, temperature factors

In crystallographic refinement, atomic displacement parameters (ADPs; often referred to as temperature factors or B factors) model the effects of static and dynamic disorder. Except at high resolution (typically better than 1.5 Å), where there are sufficient observations to warrant refinement of anisotropic temperature factors, ADPs are usually constrained to be isotropic. The isotropic temperature factor B of an atom is related to the atom's mean-square displacement 〈Δr ²〉 according to B = 8π²〈Δr ²〉/3. Compared with the atomic coordinates, there are usually comparatively few restraints on temperature factors during refinement. Therefore, particularly at low resolution, temperature factors often function as `error sinks' (Read, 1990). They absorb not only the effects of static and dynamic disorder, but also of various kinds of model errors.

Compared with the wealth of statistics that can be used to check and validate coordinates, there are relatively few methods available to assess how reasonable a model's temperature factors are. One obvious check is to see how well the average temperature factor of the model matches the value calculated from the data, using either a Wilson plot (Wilson, 1949) or the Patterson origin peak (Vaguine et al., 1999). Since the average temperature factor of a model is usually not restrained, this is a useful check that has been used on several occasions to justify high average B factors. One should keep in mind that a low average B factor, per se, is not necessarily an indication of high model quality. For instance, a backwards-traced protein model can have a considerably lower average B factor than a correct model at a similar resolution (Kleywegt & Jones, 1995b). Average (and minimum and maximum) temperature-factor values can also be listed separately for various groups of atoms (e.g. individual protein or nucleic acid molecules, ligands, solvent molecules). A simple plot of residue-averaged temperature factors as a function of residue number may reveal regions of the molecule that have consistently high B factors, which may be a consequence of problems in the model (Kleywegt et al., 1996).

Other statistics pertain to the r.m.s. differences in B factors between atoms that are somehow related, for example through a chemical bond (r.m.s. ΔB_bonded), through a 1–3 interaction or through noncrystallographic symmetry (possibly after correcting for any differences between the average B factors of the NCS-related molecules). Sometimes these statistics are calculated separately for main-chain and side-chain atoms. If the B factors of such related atoms have been restrained to be similar during refinement, these checks do not provide a convincing indication of the quality of the model. On the other hand, the B factors of atoms that have non-bonded interactions are usually not restrained to be similar, which renders the r.m.s. B-factor difference between such atoms (r.m.s. ΔB_non-bonded) slightly more informative.

Since proteins tend to consist of a tightly packed core with more flexible regions at the surface, a radial B-factor plot (i.e. a plot of the average B factor of all atoms in a certain distance range from the centre of the molecule as a function of the distance) is expected to be shaped roughly like a half-parabola. Kuriyan & Weis (1991) used a ten-parameter isotropic rigid-molecule model of the mean-square atomic displacement (Schomaker & Trueblood, 1968). After obtaining values for the ten parameters (either by refinement against the structure-factor data or by fitting to the refined B factors of the model), the B factor of any atom can be calculated and depends only on its coordinates. They found that regions with large discrepancies between the refined and fitted B factors tend to be associated with errors or problems in a model.

Validation of anisotropic ADPs (Merritt, 1999), non-unit occupancies and H atoms, all of which are usually associated with high-resolution data, is still in its infancy. The validity of modelling anisotropic ADPs can be assessed by comparing the reduction of the conventional and free R values. If occupancies are used for multiple conformations of, for example, a side chain, the sum of the occupancies should be unity.