Basis extraction

Kabsch, W.

doi:10.1107/97809553602060000676

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. 11.3, p. 220 | 1 | 2 |

Section 11.3.2.6. Basis extraction

W. Kabsch^a ^*

^a Max-Planck-Institut für medizinische Forschung, Abteilung Biophysik, Jahnstrasse 29, 69120 Heidelberg, Germany
Correspondence e-mail: kabsch@mpimf-heidelberg.mpg.de

11.3.2.6. Basis extraction

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Any reciprocal-lattice vector can be written in the form $[{\bf p}_{0}^{*} = h{\bf b}_{1}^{*} + k{\bf b}_{2}^{*} + l{\bf b}_{3}^{*}]$ where h, k, l are integer numbers and $[{\bf b}_{1}^{*}, {\bf b}_{2}^{*}, {\bf b}_{3}^{*}]$ are basis vectors of the lattice. The basis vectors which describe the orientation, metric and symmetry of the crystal, as well as the reflection indices h, k, l, have to be determined from the list of strong diffraction spots $[X'_{i}, Y'_{i}, Z'_{i}\ (i = 1, \ldots, n)]$ . Ideally, each spot corresponds to a reciprocal-lattice vector $[{\bf p}_{0}^{*}]$ which satisfies the Laue equations after a crystal rotation by φ. Substituting the observed value Z′ for the unknown φ angle (see Section 11.3.2.4), $[{\bf p}_{0}^{*}]$ is found from the observed spot coordinates as $[\eqalign{{\bf p}_{0}^{*} &=D({\bf m}_{2}, -Z') ({\bf S'} - {\bf S}_{0})\cr {\bf S'} &= \left[(X'- X_{0}){\bf d}_{1} + (Y'- Y_{0}){\bf d}_{2} + F{\bf d}_{3}\right]\cr &\quad\times\left\{\lambda \cdot \left[(X'- X_{0})^{2} + (Y' - Y_{0})^{2} + F^{2}\right]^{1/2}\right\}^{-1}.}]$ Unfortunately, the reciprocal-lattice vectors $[{\bf p}^{*}_{0i}\ (i = 1, \ldots, n)]$ derived from the above list of strong diffraction spots often contain a number of `aliens' (spots arising from fluctuations of the background, from ice, or from satellite crystals) and a robust method has to be used which is still capable of recognizing the dominant lattice. One approach, suggested by Bricogne (1986) and implemented in a number of variants (Otwinowski & Minor, 1997; Steller et al., 1997), is to identify a lattice basis as the three shortest linear independent vectors $[{\bf b}_{1}, {\bf b}_{2}, {\bf b}_{3}]$ , each at a maximum of the Fourier transform $[\sum\nolimits_{i=1}^{n} \cos (2\pi {\bf b} \cdot {\bf p}^{*}_{0i})]$ . Alternatively, a reciprocal basis for the dominant lattice can be determined from short differences between the reciprocal-lattice vectors (Howard, 1986; Kabsch, 1988a). As implemented in XDS, a lattice basis is found by the following procedure.

The list of given reciprocal-lattice points $[{\bf p}^{*}_{0i}\ (i = 1, \ldots,n)]$ is first reduced to a small number m of low-resolution difference-vector clusters $[{\bf v}^{*}_{\mu}\ (\mu = 1, \ldots, m)]$ . $[f_{\mu}]$ is the population of a difference-vector cluster $[{\bf v}^{*}_{\mu}]$ , that is the number of times the difference between any two reciprocal-lattice vectors $[{\bf p}^{*}_{0i} - {\bf p}^{*}_{0j}]$ is approximately equal to $[{\bf v}^{*}_{\mu}]$ . In a second step, three linear independent vectors $[{\bf b}_{1}^{*}, {\bf b}_{2}^{*}, {\bf b}_{3}^{*}]$ are selected among all possible triplets of difference-vector clusters that maximize the function Q: $[\eqalign{Q({\bf b}_{1}^{*}, {\bf b}_{2}^{*}, {\bf b}_{3}^{*}) &={\textstyle\sum\limits_{\mu = 1}^{m}} f_{\mu} q(\xi_{1}^{\mu}, \xi_{2}^{\mu}, \xi_{3}^{\mu})\cr q(\xi_{1}^{\mu}, \xi_{2}^{\mu}, \xi_{3}^{\mu}) &= \exp\left(-2 {\textstyle\sum\limits_{k = 1}^{3}}\left\{[\max(|\xi_{k}^{\mu} - h_{k}^{\mu}| - \varepsilon,0)/\varepsilon]^{2}\right.\right.\cr&\quad\left. +\; [\max(|h_{k}^{\mu}| - \delta,0)]^{2}\right\}\bigg)\cr \xi_{k}^{\mu} &={\bf v}_{\mu}^{*} \cdot {\bf b}_{k}, \quad {\bf v}_{\mu}^{*} = {\textstyle\sum\limits_{k = 1}^{3}} \xi_{k}^{\mu}{\bf b}_{k}^{*}, \quad {\bf b}_{k} \cdot {\bf b}_{l}^{*} = {\left\{\matrix{1 \hbox{ if } k = l \hbox{;}\hfill\cr 0 \hbox{ otherwise}\cr}\right.}\cr h_{k}^{\mu} &= \hbox{ nearest integer to}\ \xi_{k}^{\mu}.}\hfill]$ The absolute maximum of Q is assumed if all difference vectors can be expressed as small integral multiples of the best triplet. Deviations from this ideal situation are quantified by the quality measure q. The value of q declines sharply if the expansion coefficients $[\xi_{k}^{\mu}]$ deviate by more than ɛ from their nearest integers $[h_{k}^{\mu}]$ or if the indices are absolutely larger than δ. The constraint on the allowed range of indices prevents the selection of a spurious triplet of very short difference vector clusters which might be present in the set. Excellent results have been obtained using $[\varepsilon = 0.05]$ and $[\delta = 5]$ . The best vector triplet thus found is refined against the observed difference-vector clusters. Finally, a reduced cell is derived from the refined reciprocal-base vector triplet as defined in IT A (2005), Chapter 9.2 .

References

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International Tables for Crystallography (2006). Vol. F. ch. 11.3, p. 220