Extension for low-resolution structures and use of Rfree

Cruickshank, D. W. J.

doi:10.1107/97809553602060000697

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

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International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 410 | 1 | 2 |

Section 18.5.6.3. Extension for low-resolution structures and use of R_free

D. W. J. Cruickshank^a ^*^‡

^a Chemistry Department, UMIST, Manchester M60 1QD, England
Correspondence e-mail: dwj_cruickshank@email.msn.com

18.5.6.3. Extension for low-resolution structures and use of R_free

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For low-resolution structures, the number of parameters may exceed the number of diffraction data. In (18.5.6.3) and (18.5.6.5), $[p = n_{\rm obs} - n_{\rm params}]$ is then negative, so that $[\sigma (x)]$ is imaginary. This difficulty can be circumvented empirically by replacing p with $[n_{\rm obs}]$ and R with $[R_{\rm free}]$ (Brünger, 1992). The counterpart of the DPI (18.5.6.5) is then $[\sigma (x, B_{\rm avg}) = 1.0 (N_{i} / n_{\rm obs})^{1/2} C^{-1/3} R_{\rm free}d_{\min}. \eqno(18.5.6.6)]$ Here $[n_{\rm obs}]$ is the number of reflections included in the refinement, not the number in the $[R_{\rm free}]$ set.

It may be asked: how can there be any estimate for the precision of a coordinate from the diffraction data only when there are insufficient diffraction data to determine the structure? By following the line of argument of Cruickshank's (1960) analysis, (18.5.6.6) is a rough estimate of the square root of the reciprocal of one diagonal element of the diffraction-only least-squares matrix. All the other parameters can be regarded as having been determined from a diffraction-plus-restraints matrix.

Clearly, (18.5.6.6) can also be used as a general alternative to (18.5.6.5) as a DPI, irrespective of whether the number of degrees of freedom $[p = n_{\rm obs} - n_{\rm params}]$ is positive or negative.

Comment . When p is positive, (18.5.6.6) would be exactly equivalent to (18.5.6.5) only if $[R_{\rm free} = R[n_{\rm obs}/(n_{\rm obs} - n_{\rm params})]^{1/2}]$ . Tickle et al. (1998b) have shown that the expected relationship in a restrained refinement is actually $[{R_{\rm free} = R\{[n_{\rm obs} + (n_{\rm params} - h)] / [n_{\rm obs} - (n_{\rm params} - h)]\}^{1/2},} \eqno(18.5.6.7)]$ where $[h = n_{\rm restraints} - \sum w_{\rm geom}(\Delta Q)^{2}]$ , the latter term, as in (18.5.3.1), being the weighted sum of the squares of the restraint residuals.

References

Brünger, A. T. (1992). Free R-value: a novel statistical quantity for assessing the accuracy of crystal structures. Nature (London), 355, 472–475.Google Scholar

Cruickshank, D. W. J. (1960). The required precision of intensity measurements for single-crystal analysis. Acta Cryst. 13, 774–777.Google Scholar

Tickle, I. J., Laskowski, R. A. & Moss, D. S. (1998b). R_free and the R_free ratio. I. Derivation of expected values of cross-validation residuals used in macromolecular least-squares refinement. Acta Cryst. D54, 547–557.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 410

Section 18.5.6.3. Extension for low-resolution structures and use of Rfree

18.5.6.3. Extension for low-resolution structures and use of Rfree

References

Section 18.5.6.3. Extension for low-resolution structures and use of R_free

18.5.6.3. Extension for low-resolution structures and use of R_free