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. 13.2, p. 273   | 1 | 2 |

Section 13.2.5. Other rotation functions

J. Navazaa*

aLaboratoire de Génétique des Virus, CNRS-GIF, 1. Avenue de la Terrasse, 91198 Gif-sur-Yvette, France
Correspondence e-mail: jnavaza@pasteur.fr

13.2.5. Other rotation functions

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The rotation function was hitherto described in terms of self- and cross-Patterson vectors. This is perhaps inevitable in the self-rotation case, but the problem of determining the absolute orientation of the subunits when a model structure is available may be formulated in a different way. We may try to compare directly the observed and calculated intensities or structure factors by using any criterion analogous to those employed in refinement procedures, e.g., the crystallographic R factor or correlation coefficients.

When the space-group symmetry is explicitly exhibited, the structure factor corresponding to a crystal with M independent molecules in the unit cell takes the form [F({\bf h}) = {\textstyle\sum\limits_{m=1}^{M}}\ {\textstyle\sum\limits_{g=1}^{G}}\ f_{m}({\bf hM}_{g}) \exp[2 \pi i {\bf h}({\bf M}_{g}{\bf r}_{m} + {\bf t}_{g})], \eqno(13.2.5.1)] where [{\bf M}_{g}] and [{\bf t}_{g}] denote, respectively, the transformation matrix and the translation associated with the gth symmetry operation of the crystal space group. The corresponding intensity is [\eqalignno{I({\bf h}) &= {\textstyle\sum\limits_{m, \, m'=1}^{M}}\ {\textstyle\sum\limits_{g, \, g'=1}^{G}} \overline{f_{m}({\bf hM}_{g})} f_{m'}({\bf hM}_{g'}) &\cr&\quad\times\exp[2 \pi i {\bf h}({\bf M}_{g}{\bf r}_{m} + {\bf t}_{g} - {\bf M}_{g'}{\bf r}_{m'} - {\bf t}_{g'})]. &(13.2.5.2)}]

For criteria based on amplitudes, the calculated structure factor will contain only the contribution of the rotated model, [F^{\rm calc}_{\bf h}({\bf R}) = f_{m}({\bf hR}), \eqno(13.2.5.3)] i.e., the Fourier transform of a single molecule in the crystal cell, assuming P1 symmetry. For criteria based on intensities, some symmetry information may be introduced, [I^{\rm calc}_{\bf h}({\bf R}) = {\textstyle\sum\limits_{g=1}^{G}}|\ f_{m}({\bf hM}_{g}{\bf R})|^{2}. \eqno(13.2.5.4)] A criterion often considered is the correlation coefficient on intensities, [\eqalignno{{\cal R}_{\cal D} ({\bf R}) &= \langle(I^{\rm obs} - \langle I^{\rm obs}\rangle)(I^{\rm calc} - \langle I^{\rm calc}\rangle)\rangle\cr &\quad \times [\langle(I^{\rm obs} - \langle I^{\rm obs}\rangle)^{2}\rangle\langle(I^{\rm calc} - \langle I^{\rm calc}\rangle)^{2}\rangle]^{1/2}, &(13.2.5.5)}] where [\langle \ldots \rangle] means `average over reflections'. It may be calculated within reasonable computing time provided that

  • (1) the structure factors are computed by interpolation from the Fourier transform of the isolated molecule's electron density; and

  • (2) an efficient sampling set of [{\cal R}_{\cal D}] is defined.

[{\cal R}_{\cal D}] is referred to as the direct-rotation function (DeLano & Brünger, 1995[link]). A major advantage of this formulation is that the information stemming from already-positioned subunits may be taken into account, just by adding their contribution to the calculated intensities.

References

DeLano, W. L. & Brünger, A. T. (1995). The direct rotation function: rotational Patterson correlation search applied to molecular replacement. Acta Cryst. D51, 740–748.Google Scholar








































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