|
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. 21.1, p. 505
Section 21.1.7.4.3. Coordinate error estimates
aDepartment of Cell and Molecular Biology, Uppsala University, Biomedical Centre, Box 596, SE-751 24 Uppsala, Sweden |
Since a measurement without an error estimate is not a measurement, crystallographers are keen to assess the estimated errors in the atomic coordinates and, by extension, in the atomic positions, bond lengths etc. In principle, upon convergence of a least-squares refinement, the variances and covariances of the model parameters (coordinates, ADPs and occupancies) may be obtained through inversion of the least-squares full matrix (Sheldrick, 1996![[link]](/graphics/greenarr.gif)
; Ten Eyck, 1996; Cruickshank, 1999). In practice, however, this is seldom performed as the matrix inversion requires enormous computational resources. Therefore, one of a battery of (sometimes quasi-empirical) approximations is usually employed.
For a long time, the elegant method of Luzzati (1952) has been used for a different purpose (namely, to estimate average coordinate errors of macromolecular models) than that for which it was developed (namely, to estimate the positional changes required to reach a zero R value, using several assumptions that are not valid for macromolecules; Cruickshank, 1999). A Luzzati plot is a plot of R value versus ![[2\sin\theta/\lambda]](/teximages/cbch7o4/cbch7o4fi43.svg)
, and a comparison with theoretical curves is used to estimate the average positional error. Considering the problems with conventional R values (discussed in Section 21.1.7.4.1), Kleywegt et al. (1994) instead plotted free R values to obtain a cross-validated error estimate. This intuitive modification turned out to yield fairly reasonable values in practice (Kleywegt & Brünger, 1996; Brünger, 1997). Read (1986, 1990) estimated coordinate error from σ A plots; the cross-validated modification of this method also yields reasonable error estimates (Brünger, 1997).
Cruickshank, almost 50 years after his work on the precision of small-molecule crystal structures (Cruickshank, 1949), introduced the diffraction-component precision index (DPI; Dodson et al., 1996; Cruickshank, 1999) to estimate the coordinate or positional error of an atom with a B factor equal to the average B factor of the whole structure. In several cases for which full-matrix error estimates are available, the DPI gives quantitatively similar results. SFCHECK (Vaguine et al., 1999) calculates both the DPI and Cruickshank's 1949 statistic (now termed the `expected maximal error') based on the slope and the curvature of the electron-density map.
References
Brünger, A. T. (1997). The free R value: a more objective statistic for crystallography. Methods Enzymol. 277, 366–396.Google Scholar
Cruickshank, D. W. J. (1949). The accuracy of electron-density maps in X-ray analysis with special reference to dibenzyl. Acta Cryst. 2, 65–82.Google Scholar
Cruickshank, D. W. J. (1999). Remarks about protein structure precision. Acta Cryst. D55, 583–601.Google Scholar
Dodson, E., Kleywegt, G. J. & Wilson, K. S. (1996). Report of a workshop on the use of statistical validators in protein X-ray crystallography. Acta Cryst. D52, 228–234.Google Scholar
Kleywegt, G. J., Bergfors, T., Senn, H., Le Motte, P., Gsell, B., Shudo, K. & Jones, T. A. (1994). Crystal structures of cellular retinoic acid binding proteins I and II in complex with all-trans-retinoic acid and a synthetic retinoid. Structure, 2, 1241–1258.Google Scholar
Kleywegt, G. J. & Brünger, A. T. (1996). Checking your imagination: applications of the free R value. Structure, 4, 897–904.Google Scholar
Luzzati, V. (1952). Traitement statistique des erreurs dans la determination des structures crystallines. Acta Cryst. 5, 802–810.Google Scholar
Read, R. J. (1986). Improved Fourier coefficients for maps using phases from partial structures with errors. Acta Cryst. A42, 140–149.Google Scholar
Read, R. J. (1990). Structure-factor probabilities for related structures. Acta Cryst. A46, 900–912.Google Scholar
Sheldrick, G. M. (1996). Least-squares refinement of macromolecules: estimated standard deviations, NCS restraints and factors affecting convergence. In Proceedings of the CCP4 study weekend. Macromolecular refinement, edited by E. Dodson, M. Moore, A. Ralph & S. Bailey, pp. 47–58. Warrington: Daresbury Laboratory.Google Scholar
Ten Eyck, L. F. (1996). Full matrix least squares. In Proceedings of the CCP4 study weekend. Macromolecular refinement, edited by E. Dodson, M. Moore, A. Ralph & S. Bailey, pp. 37–45. Warrington: Daresbury Laboratory.Google Scholar
Vaguine, A. A., Richelle, J. & Wodak, S. J. (1999). SFCHECK: a unified set of procedures for evaluating the quality of macromolecular structure-factor data and their agreement with the atomic model. Acta Cryst. D55, 191–205.Google Scholar
