Miscellaneous correlation functions

Bricogne, G.

doi:10.1107/97809553602060000551

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 92 | 1 | 2 |

Section 1.3.4.4.8. Miscellaneous correlation functions

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.4.8. Miscellaneous correlation functions

| top | pdf |

Certain correlation functions can be useful to detect the presence of multiple copies of the same molecule (known or unknown) in the asymmetric unit of a crystal of unknown structure.

Suppose that a crystal contains one or several copies of a molecule $[{\scr M}]$ in its asymmetric unit. If $[\mu({\bf x})]$ is the electron density of that molecule in some reference position and orientation, then $[\rho\llap{$-\!$}^{0} = {\textstyle\sum\limits_{j \in J}} \left[{\textstyle\sum\limits_{g \in G}} S_{g}^{\#} (T_{j}^{\#} \mu)\right],]$ where $[T_{j}: {\bf x} \;\longmapsto\; {\bf C}_{j} {\bf x} + {\bf d}_{j}]$ describes the placement of the jth copy of the molecule with respect to the reference copy. It is assumed that each such copy is in a general position, so that there is no isotropy subgroup.

The methods of Section 1.3.4.2.2.9 (with $[\rho\llap{$-\!$}_{j}]$ replaced by $[C_{j}^{\#} \mu]$ , and $[{\bf x}_{j}]$ by $[{\bf d}_{j}]$ ) lead to the following expression for the auto-correlation of $[\rho\llap{$-\!$}^{0}]$ : $[\eqalign{ \breve{\rho\llap{$-\!$}}^{0} * \rho\llap{$-\!$}^{0} &= {\textstyle\sum\limits_{j_{1}}} {\textstyle\sum\limits_{j_{2}}} {\textstyle\sum\limits_{g_{1}}} {\textstyle\sum\limits_{g_{2}}} \boldtau_{{S_{g_{2}}} ({\bf d}_{j_{2}}) - s_{g_{1}} ({\bf d}_{j_{1}})}\cr &\quad \times [(R_{g_{1}}^{\#} C_{j_{1}}^{\#} \breve{\mu}) * (R_{g_{2}}^{\#} C_{j_{2}}^{\#} \mu)].}]$

If μ is unknown, consider the subfamily σ of terms with $[j_{1} = j_{2} = j]$ and $[g_{1} = g_{2} = g]$ : $[\sigma = {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{g}} \;R_{g}^{\#} C_{j}^{\#} (\breve{\mu} * \mu).]$ The scalar product $[(\sigma, R^{\#} \sigma)]$ in which R is a variable rotation will have a peak whenever $[R = (R_{g_{1}} C_{j_{1}})^{-1} (R_{g_{2}} C_{j_{2}})]$ since two copies of the `self-Patterson' $[\breve{\mu} * \mu]$ of the molecule will be brought into coincidence. If the interference from terms in the Patterson $[\pi = r * \breve{\rho\llap{$-\!$}}^{0} * \rho\llap{$-\!$}^{0}]$ other than those present in σ is not too serious, the `self-rotation function' $[(\pi, R^{\#} \pi)]$ (Rossmann & Blow, 1962; Crowther, 1972) will show the same peaks, from which the rotations $[\{C_{j}\}_{j \in J}]$ may be determined, either individually or jointly if for instance they form a group.

If μ is known, then its self-Patterson $[\breve{\mu} * \mu]$ may be calculated, and the $[C_{j}]$ may be found by examining the `cross-rotation function' $[[\pi, R^{\#} (\breve{\mu} * \mu)]]$ which will have peaks at $[R = R_{g} C_{j}, g \in G, j \in J]$ . Once the $[C_{j}]$ are known, then the various copies $[C_{j}^{\#} \mu]$ of $[{\scr M}]$ may be Fourier-analysed into structure factors: $[M_{j} ({\bf h}) = \bar{{\scr F}}[C_{j}^{\#} \mu] ({\bf h}).]$ The cross terms with $[j_{1} \neq j_{2}, g_{1} \neq g_{2}]$ in $[\breve{\rho\llap{$-\!$}}^{0} * \rho\llap{$-\!$}^{0}]$ then contain `motifs' $[(R_{g_{1}}^{\#} C_{j_{1}}^{\#} \breve{\mu}) * (R_{g_{2}}^{\#} C_{j_{2}}^{\#} \mu),]$ with Fourier coefficients $[\overline{M_{j_{1}} ({\bf R}_{g_{1}}^{T} {\bf h})} \times M_{j_{2}} ({\bf R}_{g_{2}}^{T} {\bf h}),]$ translated by $[S_{g_{2}} ({\bf d}_{j_{2}}) - S_{g_{1}} ({\bf d}_{j_{1}})]$ . Therefore the `translation functions' (Crowther & Blow, 1967) $[\eqalign{ {\scr T}_{j_{1} g_{1}, j_{2} g_{2}} ({\bf s}) &= {\textstyle\sum\limits_{{\bf h}}} |F_{{\bf h}}|^{2} \overline{M_{j_{1}} ({\bf R}_{g_{1}}^{T} {\bf h})}\cr &\quad \times M_{j_{2}} ({\bf R}_{g_{2}}^{T} {\bf h}) \exp (-2 \pi i {\bf h} \cdot {\bf s})}]$ will have peaks at $[{\bf s} = S_{g_{2}} ({\bf d}_{j_{2}}) - S_{g_{1}} ({\bf d}_{j_{1}})]$ corresponding to the detection of these motifs.

References

Crowther, R. A. (1972). The fast rotation function. In The molecular replacement method, edited by M. G. Rossmann, pp. 173–178. New York: Gordon & Breach.Google Scholar

Crowther, R. A. & Blow, D. M. (1967). A method of positioning a known molecule in an unknown crystal structure. Acta Cryst. 23, 544–548.Google Scholar

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

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 92