|
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. 18.5, p. 409
Section 18.5.5.3. Application of the modified Fourier method
a
Chemistry Department, UMIST, Manchester M60 1QD, England |
An extreme example of an apparently successful gross approximation to protein precision is represented by Daopin et al.'s (1994)![[link]](/graphics/greenarr.gif)
treatment of two independent determinations (at 1.8 and 1.95 Å) of the structure of TGF-β2. They reported that the modified Fourier-map formulae given in Section 18.5.5.2 yielded a quite good description of the B dependence of the positional differences between the two independent determinations. However, there is a formal difficulty about this application. Equation (18.5.5.1) derives from a diffraction-data-only approach, whereas the two structures were determined from restrained refinements. Even though the TNT restraint parameters and weights may have been the same in both refinements, it is slightly surprising that (18.5.5.1) should have worked well.
Equation (18.5.2.1) requires the summation of various series over all (hkl) observations; such calculations are not customarily provided in protein programs. However, due to the fundamental similarities between Fourier and least-squares methods demonstrated by Cochran (1948), Cruickshank (1949b, 1952, 1959), and Cruickshank & Robertson (1953), closely similar estimates of the precision of individual atoms can be obtained from the reciprocal of the diagonal elements of the diffraction-data-only least-squares matrix. These elements will often have been calculated already within the protein refinement programs, but possibly never output. Such estimates could be routinely available.
Between approximations using largish blocks and those using only the reciprocals of diagonal terms, a whole variety of intermediate approximations involving some off-diagonal terms could be envisaged.
Whatever method is used to estimate uncertainties, it is essential to distinguish between coordinate uncertainty, e.g., ![[\sigma (x)]](/teximages/fach18o5/fach18o5fi109.svg)
, and position uncertainty ![[\sigma (r) = [\sigma^{2} (x) + \sigma^{2} (y) + \sigma^{2} (z)]^{1/2}]](/teximages/fach18o5/fach18o5fi184.svg)
.
The remainder of this chapter discusses two rough-and-ready indicators of structure precision: the diffraction-component precision index (DPI) and Luzzati plots.
References
Cochran, W. (1948). The Fourier method of crystal-structure analysis. Acta Cryst. 1, 138–142.Google Scholar
Cruickshank, D. W. J. (1949b). The accuracy of atomic coordinates derived by least-squares or Fourier methods. Acta Cryst. 2, 154–157.Google Scholar
Cruickshank, D. W. J. (1952). On the relations between Fourier and least-squares methods of structure determination. Acta Cryst. 5, 511–518.Google Scholar
Cruickshank, D. W. J. (1959). Statistics. In International tables for X-ray crystallography, Vol. 2, edited by J. S. Kasper & K. Lonsdale, pp. 84–98. Birmingham: Kynoch Press.Google Scholar
Cruickshank, D. W. J. & Robertson, A. P. (1953). The comparison of theoretical and experimental determinations of molecular structures, with applications to naphthalene and anthracene. Acta Cryst. 6, 698–705.Google Scholar
Daopin, S., Davies, D. R., Schlunegger, M. P. & Grütter, M. G. (1994). Comparison of two crystal structures of TGF-β2: the accuracy of refined protein structures. Acta Cryst. D50, 85–92.Google Scholar
