Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.3, p. 259   | 1 | 2 |

Section Position of a noncrystallographic element relating two unknown structures

M. G. Rossmanna* and E. Arnoldb

aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and  bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA
Correspondence e-mail: Position of a noncrystallographic element relating two unknown structures

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The function ([link] is quite general. For instance, the rotation function corresponds to a comparison of Patterson functions [P_{1}] and [P_{2}] at their origins. That is, the coefficients are [F^{2}], phases are zero and [{\bf S}_{x} = {\bf S}_{x'} = 0]. However, the determination of the translation between two objects requires the comparison of cross-vectors away from the origin.

Consider, for instance, the determination of the precise translation vector parallel to a rotation axis between two identical molecules of unknown structure. For simplicity, let the noncrystallographic axis be a dyad (Fig.[link]. Fig.[link] shows the corresponding Patterson of the hypothetical point-atom structure. Opposite sets of cross-Patterson vectors in Fig.[link] are related by a twofold rotation and a translation equal to twice the precise vector in the original structure. A suitable translation function would then compare a Patterson at S with the rotated Patterson at [-{\bf S}]. Hence, substituting [{\bf S}_{x} = {\bf S}] and [{\bf S}_{x'} = - {\bf S}] in ([link], [T({\bf S}) = {2 \over V^{2}} {\sum\limits_{\bf h}} {\sum\limits_{\bf p}} |{\bf F}_{\bf h}|^{2} |{\bf F}_{\bf p}|^{2} G_{\bf hp} \cos [2\pi ({\bf h} - {\bf p})\cdot {\bf S}]. \eqno(]


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Crosses represent atoms in a two-dimensional model structure. The triangles are the points chosen as approximate centres of molecules A and B. [\Delta^{AB}] has components t and s parallel and perpendicular, respectively, to the screw rotation axis. [Reprinted from Rossmann et al. (1964)[link].]


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Vectors arising from the structure in Fig.[link]. The self-vectors of molecules A and B are represented by + and ·; the cross-vectors from molecules A to B and B to A by × and ○. Triangles mark the position of [+\Delta^{AB}] and [-\Delta^{AB}]. [Reprinted from Rossmann et al. (1964)[link].]

The opposite cross-vectors can be superimposed only if an evenfold rotation between the unknown molecules exists. The translation function ([link] is thus applicable only in this special situation. There is no published translation method to determine the interrelation of two unknown structures in a crystallographic asymmetric unit or in two different crystal forms. However, another special situation exists if a molecular evenfold axis is parallel to a crystallographic evenfold axis. In this case, the position of the noncrystallographic symmetry element can be easily determined from the large peak in the corresponding Harker section of the Patterson.

In general, it is difficult or impossible to determine the positions of noncrystallographic axes (or their intersection at a molecular centre). However, the position of heavy atoms in isomorphous derivatives, which usually obey the noncrystallographic symmetry, can often determine this information.

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