Operator representation and definitions

Furey, W.

doi:10.1107/97809553602060000724

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. 25.2, p. 702

Section 25.2.1.10.1. Operator representation and definitions

W. Furey^a ^*

25.2.1.10.1. Operator representation and definitions

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NC symmetry operators are specified in terms of the parameters φ, ψ, χ, $[O_{x}]$ , $[O_{y}]$ , $[O_{z}]$ and t, which refer to a Cartesian coordinate system in Å, obtained by orthogonalization of the unit cell as in the Protein Data Bank (Bernstein et al., 1977). The angles φ and ψ determine the direction of the NC rotation axis, while χ determines the amount of rotation about it. $[O_{x}]$ , $[O_{y}]$ and $[O_{z}]$ are coordinates of a point through which the rotation axis passes, and t is a post-rotation translation parallel to the rotation axis. The relationships between the angles, orthogonal reference axes X, Y, Z and the unit cell are given in Fig. 25.2.1.2. Coordinates for a pair of points related by NC symmetry are then expressed in the orthogonal system by $[P_{2} = R_{\varphi,\, \psi,\, \chi} (P_{1} - O) + O + tD_{\varphi,\, \psi}, \eqno(25.2.1.32)]$ where $[P_{1}]$ and $[P_{2}]$ are three-element column vectors containing coordinates for the related points, $[R_{\varphi,\, \psi,\, \chi}]$ is a 3 × 3 rotation matrix derived from the angles, O is a three-element column vector containing coordinates for a point through which the rotation axis passes, t is the post-rotation translation scalar in Å and $[D_{\varphi,\, \psi}]$ is a three-element column vector containing direction cosines of the rotation axis. This type of parameterization simplifies transfer of information from self-rotation functions, which are usually calculated in spherical polar angles anyway, and also makes obvious pseudo-space-group symmetry type operations such as pseudo-screw axes. For convenience, a program O_TO_SP is provided to convert from a 3 × 3 rotation matrix and 1 × 3 column vector representation of the NC symmetry operation, as used in some programs, to the parameters described here.

Figure 25.2.1.2| top | pdf |

Relationships between noncrystallographic symmetry rotation axis direction, orthogonal reference system axes X, Y, Z and crystallographic axes. The X axis is aligned with the crystal a. The Y axis is parallel to $[{\bf a} \times {\bf c}^{*}]$ . The Z axis is parallel to $[\bf{X} \times \bf{Y}]$ , i.e. $[{\bf c}^{*}]$ . ψ is the angle between the NC rotation and +Y axes. φ is the angle between the projection of the NC rotation axis in the XZ plane and the +X axis, with +φ counterclockwise when viewed from +Y toward the origin. χ is the amount of rotation about the directed axis, with +χ clockwise when viewed from the axis toward the origin.

References

Bernstein, F. C., Koetzle, T. F., Williams, G. J. B., Meyer, E. F. Jr, Brice, M. D., Rodgers, J. R., Kennard, O., Shimanouchi, T. & Tasumi, M. (1977). The Protein Data Bank: a computer-based archival file for macromolecular structures. J. Mol. Biol. 112, 535–542.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 25.2, p. 702