International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 403
Section 18.5.1.1. Background^{a}Chemistry Department, UMIST, Manchester M60 1QD, England |
Even in 1967 when the first few protein structures had been solved, it would have been hard to imagine a time when the best protein structures would be determined with a precision approaching that of small molecules. That time was reached during the 1990s. Consequently, the methods for the assessment of the precision of small molecules can be extended to good-quality protein structures.
The key idea is simply stated. At the conclusion and full convergence of a least-squares or equivalent refinement, the estimated variances and covariances of the parameters may be obtained through the inversion of the least-squares full matrix.
The inversion of the full matrix for a large protein is a gigantic computational task, but it is being accomplished in a rising number of cases. Alternatively, approximations may be sought. Often these can be no more than rough order-of-magnitude estimates. Some of these approximations are considered below.
Caveat. Quite apart from their large numbers of atoms, protein structures show features differing from those of well ordered small-molecule structures. Protein crystals contain large amounts of solvent, much of it not well ordered. Parts of the protein chain may be floppy or disordered. All natural protein crystals are noncentrosymmetric, hence the simplifications of error assessment for centrosymmetric structures are inapplicable. The effects of incomplete modelling of disorder on phase angles, and thus on parameter errors, are not addressed explicitly in the following analysis. Nor does this analysis address the quite different problem of possible gross errors or misplacements in a structure, other than by their indication through high B values or high coordinate standard uncertainties. These various difficulties are, of course, reflected in the values of used in the precision estimates.
On the problems of structure validation see Part 21 of this volume and Dodson (1998).
Some structure determinations do make a first-order correction for the effects of disordered solvent on phase angles by application of Babinet's principle of complementarity (Langridge et al., 1960; Moews & Kretsinger, 1975; Tronrud, 1997). Babinet's principle follows from the fact that if is constant throughout the cell, then , except for F(0). Consequently, if the cell is divided into two regions C and D, . Thus if D is a region of disordered solvent, can be estimated from . A first approximation to a disordered model may be obtained by placing negative point-atoms with very high Debye B values at all the ordered sites in region C. This procedure provides some correction for very low resolution planes. Alternatively, corrections are sometimes made by a mask bulk solvent model (Jiang & Brünger, 1994).
The application of restraints in protein refinement does not affect the key idea about the method of error estimation. A simple model for restrained refinement is analysed in Section 18.5.3, and the effect of restraints is discussed in Section 18.5.4 and later.
Much of the material in this chapter is drawn from a Topical Review published in Acta Crystallographica, Section D (Cruickshank, 1999).
Protein structures exhibiting noncrystallographic symmetry are not considered in this chapter.
References
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