International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

International Tables for Crystallography (2006). Vol. F. ch. 21.1, p. 505   | 1 | 2 |

Section 21.1.7.4.3. Coordinate error estimates

G. J. Kleywegta*

aDepartment of Cell and Molecular Biology, Uppsala University, Biomedical Centre, Box 596, SE-751 24 Uppsala, Sweden
Correspondence e-mail: gerard@xray.bmc.uu.se

21.1.7.4.3. Coordinate error estimates

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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]; Ten Eyck, 1996[link]; Cruickshank, 1999[link]). 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[link]) 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[link]). A Luzzati plot is a plot of R value versus [2\sin\theta/\lambda] , 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[link]), Kleywegt et al. (1994[link]) 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[link]; Brünger, 1997[link]). Read (1986[link], 1990[link]) estimated coordinate error from σ A plots; the cross-validated modification of this method also yields reasonable error estimates (Brünger, 1997[link]).

Cruickshank, almost 50 years after his work on the precision of small-molecule crystal structures (Cruickshank, 1949[link]), introduced the diffraction-component precision index (DPI; Dodson et al., 1996[link]; Cruickshank, 1999[link]) 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[link]) 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

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