Comments on the diffraction-component precision index

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. 412 | 1 | 2 |

Section 18.5.7.4. Comments on the diffraction-component precision index

D. W. J. Cruickshank^a ^*^‡

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

18.5.7.4. Comments on the diffraction-component precision index

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The DPI (18.5.6.9) or (18.5.6.10) provides a very simple formula for $[\sigma (r, B_{\rm avg})]$ . It is based on a very rough approximation to a diagonal element of the diffraction-data-only matrix. Using a diagonal element is a reasonable approximation for atomic resolution structures, but for low-resolution structures there will be significant off-diagonal terms between overlapping atoms. The effect can be simulated in the two-atom protein model of Section 18.5.3.2 by introducing positive off-diagonal elements into the diffraction-data matrix (18.5.3.3). As expected, $[\sigma_{\rm diff}^{2} (x_{i})]$ is increased. So the DPI will be an underestimate of the diffraction component in low-resolution structures.

However, the true restrained variance $[\sigma_{\rm res}^{2} (x_{i})]$ in the new counterpart of (18.5.3.12) remains less than the diagonal diffraction result (18.5.3.11) $[\sigma_{\rm diff}^{2} (x_{i}) = 1/a]$ . Thus for low-resolution structures, the DPI should be an overestimate of the true precision given by a restrained full-matrix calculation (where the restraints act to hold the overlapping atoms apart). This is confirmed by the results for the 2.1 Å study of βB2-crystallin (Tickle et al., 1998a) discussed in Section 18.5.7.3 and Table 18.5.7.3. The restrained full-matrix average for all protein atoms was $[\sigma_{\rm res}(r) = 0.15]$ Å, compared with the DPI 0.25 Å (on R) or 0.22 Å (on $[R_{\rm free}]$ ). The ratio between the unrestrained DPI and the restrained full-matrix average is consistent with a view of a low-resolution protein as a chain of effectively rigid peptide groups. The ratio no doubt gets much worse for resolutions of 3 Å and above.

The DPI estimate of $[\sigma (r, B_{\rm avg})]$ is given by a formula of `back-of-an-envelope' simplicity. $[B_{\rm avg}]$ is taken to be the average B for fully occupied sites, but the weights implicit in the averaging are not well defined in the derivation of the DPI. Thus the DPI should perhaps be regarded as simply offering an estimate of a typical $[\sigma_{\rm diff}(r)]$ for a carbon or nitrogen atom with a mid-range B. From the evidence of the tables in this section, except at low resolution, it seems to give a useful overall indication of protein precision, even in restrained refinements.

The DPI evidently provides a method for the comparative ranking of different structure determinations . In this regard it is a complement to the general use of $[d_{\min}]$ as a quick indicator of possible structural quality.

Note that (18.5.6.3) and (18.5.6.4) offer scope for making individual error estimates for atoms of different B and Z.

References

Tickle, I. J., Laskowski, R. A. & Moss, D. S. (1998a). Error estimates of protein structure coordinates and deviations from standard geometry by full-matrix refinement of γB- and βB2-crystallin. Acta Cryst. D54, 243–252.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 412