Introduction

Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000556

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 258-259 | 1 | 2 |

Section 2.3.7.1. Introduction

M. G. Rossmann^a ^* and E. Arnold^b

^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: mgr@indiana.bio.purdue.edu

2.3.7.1. Introduction

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The problem of determining the position of a noncrystallographic symmetry element in space, or the position of a molecule of known orientation in a unit cell, has been reviewed by Rossmann (1972), Colman et al. (1976), Karle (1976), Argos & Rossmann (1980), Harada et al. (1981) and Beurskens (1981). All methods depend on the prior knowledge of the object's orientation implied by the rotation matrix [C]. The various translation functions, T, derived below, can only be computed given this information.

The general translation function can be defined as $[T({\bf S}_{x}, {\bf S}_{x'}) = {\textstyle\int\limits_{U}} \rho_{1} ({\bf x}) \cdot \rho_{2} ({\bf x}')\;\hbox{d}{\bf x},]$ where T is a six-variable function given by each of the three components that define $[{\bf S}_{x}]$ and $[{\bf S}_{x'}]$ . Here $[{\bf S}_{x}]$ and $[{\bf S}_{x'}]$ are equivalent reference positions of the objects, whose densities are $[\rho_{1}({\bf x})]$ and $[\rho_{2}({\bf x}')]$ . The translation function searches for the optimal overlap of the two objects after they have been similarly oriented. Following the same procedure used for the rotation-function derivation, Fourier summations are substituted for $[\rho_{1}({\bf x})]$ and $[\rho_{2}({\bf x}')]$ . It can then be shown that $[\eqalign{ T({\bf S}_{x}, {\bf S}_{x'}) &= \int\limits_{U} \left\{{1 \over V_{\bf h}} {\sum\limits_{\bf h}} |{\bf F}_{\bf h}| \exp [i(\alpha_{\bf h} - 2\pi {\bf h} \cdot {\bf x})]\right\}\cr &\quad \times \left\{{1 \over V_{\bf p}} {\sum\limits_{\bf p}} |{\bf F}_{\bf p}| \exp [i(\alpha_{\bf p} - 2\pi {\bf p} \cdot {\bf x}')]\right\}\;\hbox{d}{\bf x}.}]$

Using the substitution $[{\bf x}' = [{\bi C}]{\bf x} + {\bf d}]$ and simplifying leads to $[\eqalign{ T({\bf S}_{x}, {\bf S}_{x}') &= {1 \over V_{\bf h} V_{\bf p}} {\sum\limits_{\bf h}} {\sum\limits_{\bf p}} |{\bf F}_{\bf h}| |{\bf F}_{\bf p}|\cr &\quad \times \exp [i(\alpha_{\bf h} + \alpha_{\bf p} - 2\pi {\bf p} \cdot {\bf d})]\cr &\quad \times {\int\limits_{U}} \exp \{-2\pi i({\bf h} + [{\bi C}]^T{\bf p}) \cdot {\bf x}\}\;\hbox{d}{\bf x}.}]$ The integral is the diffraction function $[G_{\bf hp}]$ (2.3.6.4). If the integration is taken over the volume U, centred at $[{\bf S}_{x}]$ and $[{\bf S}_{x'}]$ , it follows that $[\eqalignno{ T({\bf S}_{x}, {\bf S}_{x'}) &= {2 \over V_{\bf h} V_{\bf p}} {\sum\limits_{\bf h}} {\sum\limits_{\bf p}} |{\bf F}_{\bf h}| |{\bf F}_{\bf p}| G_{\bf hp}\cr &\quad \times \cos [\alpha_{\bf h} + \alpha_{\bf p} - 2\pi ({\bf h}\cdot {\bf S}_{x} + {\bf p}\cdot {\bf S}_{x'})]. &(2.3.7.1)}]$

References

Argos, P. & Rossmann, M. G. (1980). Molecular replacement methods. In Theory and practice of direct methods in crystallography, edited by M. F. C. Ladd & R. A. Palmer, pp. 361–417. New York: Plenum.Google Scholar

Beurskens, P. T. (1981). A statistical interpretation of rotation and translation functions in reciprocal space. Acta Cryst. A37, 426–430.Google Scholar

Colman, P. M. & Fehlhammer, H. (1976). Appendix: the use of rotation and translation functions in the interpretation of low resolution electron density maps. J. Mol. Biol. 100, 278–282.Google Scholar

Harada, Y., Lifchitz, A., Berthou, J. & Jolles, P. (1981). A translation function combining packing and diffraction information: an application to lysozyme (high-temperature form). Acta Cryst. A37, 398–406.Google Scholar

Karle, J. (1976). Partial structures and use of the tangent formula and translation functions. In Crystallographic computing techniques, edited by F. R. Ahmed, K. Huml & B. Sedlacek, pp. 155–164. Copenhagen: Munksgaard.Google Scholar

Rossmann, M. G. (1972). The molecular replacement method. New York: Gordon & Breach.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 258-259