Symmetry properties of the rotation function

Navaza, J.

doi:10.1107/97809553602060000682

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. 13.2, p. 272 | 1 | 2 |

Section 13.2.3.4. Symmetry properties of the rotation function

J. Navaza^a ^*

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

13.2.3.4. Symmetry properties of the rotation function

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The overlap integral that defines $[{\cal R}]$ may be calculated by rotating $[P_{t}]$ instead of $[P_{s}]$ , but with the inverse rotation, $[\eqalignno{{\cal R} ({\bf R}) &= (1/v) {\textstyle\int\limits_{\Omega}} P_{t}({\bf r}) P_{s}({\bf R}^{-1}{\bf r})\ \hbox{d}^{3} {\bf r}\cr &= (1/v) {\textstyle\int\limits_{\Omega}} P_{t}({\bf Rr}) P_{s}({\bf r})\ \hbox{d}^{3} {\bf r}.&(13.2.3.18)}]$ This property enables the analysis of the consequence of the symmetries of the Patterson functions upon the rotation function (Tollin et al., 1966; Moss, 1985). For example, when the target and search functions are the same, a trivial symmetry of the self-rotation results: its values at R and $[{\bf R}^{-1}]$ are the same or, in Euler angles, at (α, β, γ) and $[(180 - \gamma, \beta, 180 - \alpha)]$ . More generally, if $[P_{t}]$ is invariant under the rotation T, i.e., $[P_{t}({\bf T}^{-1}{\bf r}) = P_{t}({\bf r})]$ [and similarly for the search function, $[P_{s}({\bf S}^{-1}{\bf r}) = P_{s}({\bf r})]$ ], then $[\eqalignno{{\cal R}({\bf R}) &= (1/v) {\textstyle\int\limits_{\Omega}} P_{t}({\bf r}) P_{s}({\bf R}^{-1}{\bf r})\ \hbox{d}^{3} {\bf r}\cr &= (1/v) {\textstyle\int\limits_{\Omega}} P_{t}({\bf T}^{-1}{\bf r}) P_{s}({\bf S}^{-1}{\bf R}^{-1}{\bf r}) \hbox{ d}^{3} {\bf r}\cr &= (1/v) {\textstyle\int\limits_{\Omega}} P_{t}({\bf r}) P_{s}({\bf S}^{-1}{\bf R}^{-1}{\bf T} \ {\bf r}) \hbox{ d}^{3} {\bf r} = {\cal R}({\bf T}^{-1}{\bf RS}). \cr &&(13.2.3.19)}]$ However, the actual symmetry displayed by $[{\cal R}]$ depends on the parameterization of the rotations and on the orientation of the orthonormal axes with respect to the crystal ones. Within the Euler parameterization, if the target function has an n-fold rotation axis parallel to the orthonormal z axis, then, according to equation (13.2.2.3), $[{\cal R}]$ will have a periodicity of [360/n] along α. Similarly, a rotation axis of order n along z of $[P_{s}]$ gives rise to a periodicity of [360/n] along γ. Therefore, the amount of calculation is reduced by choosing z along the Patterson functions' highest rotational symmetry axes [see equation (13.2.3.11)].

References

Moss, D. S. (1985). The symmetry of the rotation function. Acta Cryst. A41, 470–475.Google Scholar

Tollin, P., Main, P. & Rossmann, M. G. (1966). The symmetry of the rotation function. Acta Cryst. 20, 404–417.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 13.2, p. 272