The locked translation function

Tong, L.

doi:10.1107/97809553602060000683

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. 13.3, pp. 277-278 | 1 | 2 |

Section 13.3.8. The locked translation function

L. Tong^a ^*

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

13.3.8. The locked translation function

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In the presence of noncrystallographic symmetry (NCS), locked self-rotation functions can be used to determine the orientation of the NCS elements in the crystal unit cell (Tong & Rossmann, 1990). Often, an atomic model for the monomer of the NCS assembly is available, but not the model of the entire assembly. This atomic model can be used in ordinary cross-rotation-function calculations. A more powerful technique is to use the locked cross-rotation function, which can define the orientations of all the molecules within the assembly at the same time (Tong & Rossmann, 1997). With the knowledge of the orientations, several translation searches are needed to locate the individual monomers of the assembly. For cases where the assembly has high NCS, the translation searches to locate the first few molecules may not be very successful, since the search model only represents a small portion of the diffracting power of the crystal.

A locked translation function takes into account contributions from all the monomers of the assembly at the same time (Tong, 1996b). It can determine the position of the monomer search model relative to the centre of the NCS assembly. With this knowledge, the entire assembly can be generated and can then be used in an ordinary translation search to locate the centre of this NCS assembly in the unit cell.

Given the atomic model, $[X_{j}^{0}]$ (in Cartesian coordinates), for the monomer at a starting position and the rotation, [F], that brings it into the same orientation as that of a monomer in the standard orientation, the model of the entire assembly in the standard orientation is given by $[X_{j, \, m} = [I_{m}] ([F]X_{j}^{0} + V_{0}), \eqno (13.3.8.1)]$ where $[V_{0}]$ is a translation vector and the centre of the assembly is placed at (0, 0, 0). $[[I_{m}]\ (m = 1, \ldots, M)]$ is the set of rotation matrices for the NCS point group in the standard orientation. The correct translation vector should give rise to the maximal overlap between the self vectors within the NCS assembly and the observed Patterson map. This overlap is given by the second term of equation (13.3.3.2). The locked translation function is therefore defined as $[\eqalignno{LTF(V_{0}) &= {\textstyle\sum\limits_{h}} {\textstyle\sum\limits_{m}} (F_{h}^{o})^{2}\ |\;\overline{f}_{h, \, m}|^2\cr &\quad + {\textstyle\sum\limits_{h}} {\textstyle\sum\limits_{m}} {\textstyle\sum\limits_{n \neq m}} (F_{h}^{o})^{2} \overline{f}_{h, \, m}\ \overline{f}_{h, \, n}^{*} \exp \ \{- 2\pi ih([\theta_{n}] - [\theta_{m}])V_{0}\},\cr& &(13.3.8.2)}]$ where $[\overline{f} _{h, \, m} = \textstyle\sum\limits_{j} f_{j} \exp \ (2\pi ih[\theta_{m}][F]X_{j}^{0}) \eqno (13.3.8.3)]$ and $[[\theta_{m}] = [\alpha][E][I_{m}]. \eqno (13.3.8.4)]$ [E] is the rotation matrix that brings the standard orientation to that of the assembly in the crystal and [α] is the de-orthogonalization matrix (Rossmann & Blow, 1962). Equation (13.3.8.2) can be evaluated indirectly by the FFT technique (Tong, 1996b). As with the Patterson-correlation translation function, equation (13.3.8.2) can be converted to a correlation coefficient, although the evaluation will become more time-consuming. It should be noted that equation (13.3.8.2) bears much resemblance to equation (13.3.3.2), with the interchange of the crystallographic quantities $[([T_{n}],\overline{f}_{h, \, n})]$ and the noncrystallographic quantities $[([\theta_{m}],\overline{f}_{h, \, m})]$ .

References

Rossmann, M. G. & Blow, D. M. (1962). The detection of sub-units within the crystallographic asymmetric unit. Acta Cryst. 15, 24–31.Google Scholar

Tong, L. (1996b). The locked translation function and other applications of a Patterson correlation function. Acta Cryst. A52, 476–479.Google Scholar

Tong, L. & Rossmann, M. G. (1990). The locked rotation function. Acta Cryst. A46, 783–792.Google Scholar

Tong, L. & Rossmann, M. G. (1997). Rotation function calculations with GLRF program. Methods Enzymol. 276, 594–611.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 13.3, pp. 277-278