International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 13.3, p. 349   | 1 | 2 |

Section 13.3.6. The unique region of a translation function (the Cheshire group)

L. Tonga*

aDepartment of Biological Sciences, Columbia University, New York, NY 10027, USA
Correspondence e-mail: tong@como.bio.columbia.edu

13.3.6. The unique region of a translation function (the Cheshire group)

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The region of the unit cell that should be covered during a translation search does not generally correspond to the asymmetric unit of the space group. Since the search model has a defined orientation, it can only reside in one of the asymmetric units in the unit cell. Lacking knowledge as to which asymmetric unit the model occupies, the entire unit cell would need to be searched. However, most space groups possess alternative origins, which means the position of a molecule in the unit cell can only be determined to within certain sets of translations. For example, in space group [P2_{1}2_{1}2_{1}], there are eight alternative origins at [(0, 0, 0)], [({1 \over 2}, 0, 0)], [(0, {1 \over 2}, 0)], [(0, 0, {1 \over 2})], [({1 \over 2}, {1 \over 2}, 0)], [({1 \over 2}, 0, {1 \over 2})], [(0, {1 \over 2}, {1 \over 2})] and [({1 \over 2}, {1 \over 2}, {1 \over 2})]. This implies that the region that should be searched to locate a molecule need only be [{1 \over 8}] of the volume of the unit cell [for example, [({1 \over 2}, {1 \over 2}, {1 \over 2})]]. In addition, for polar space groups, the position of the molecule along the polar axis is arbitrary. The symmetry, as defined by these unique regions, is also known as the Cheshire group (Hirshfeld, 1968[link]), and has been defined for all the 230 space groups.

Once the first molecule is positioned, the origin of the unit cell is fixed as well. The search for subsequent molecules will need to cover the entire unit cell, unless the crystal lattice is centred.

References

Hirshfeld, F. L. (1968). Symmetry in the generation of trial structures. Acta Cryst. A24, 301–311.Google Scholar








































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