InternationalCrystallography of biological macromoleculesTables for Crystallography Volume F 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 |

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; 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* , 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.

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