General considerations

Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000684

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. 13.4, p. 282 | 1 | 2 |

Section 13.4.5.1. General considerations

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

^aDepartment of Biological Sciences, Purdue University, West Lafayette, IN 47907-1392, USA, and ^bBiomolecular Crystallography Laboratory, CABM & Rutgers University, 679 Hoes Lane, Piscataway, NJ 08854-5638, USA
Correspondence e-mail: mgr@indiana.bio.purdue.edu

13.4.5.1. General considerations

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Let Y and X be position vectors in a Cartesian coordinate system whose components have dimensions of length, in the p- and h-cells, which utilize the same origin as the fractional coordinates, y and x, respectively. Let $[[\beta_{p}]]$ and $[[\alpha_{h}]]$ be `orthogonalization' and `de-orthogonalization' matrices in the p- and h-cells, respectively (Rossmann & Blow, 1962). Then $[\eqalign{{\bf Y} &= [\beta_{p}]{\bf y} \qquad \quad \hbox{and}\qquad \quad {\bf x} = [\alpha_{h}]{\bf X},\cr [\alpha_{p}] &= [\beta_{p}]^{-1}\quad \quad \;\hbox{ and}\quad \quad [\alpha_{h}] = [\beta_{h}]^{-1}.} \eqno(13.4.5.1)]$ Thus, for instance, $[[\alpha_{h}]]$ denotes a matrix that transforms a Cartesian set of unit vectors to fractional distances along the unit-cell vectors $[{\bf a}_{h}, {\bf b}_{h}, {\bf c}_{h}]$ .

Let the Cartesian coordinates Y and X be related by the rotation matrix [ω] and the translation vector D such that $[{\bf X} = [\omega]{\bf Y} + {\bf D}. \eqno(13.4.5.2)]$ If the molecules are to be averaged among different unit cells, then each p-cell must be related to the standard h-cell orientation by a different [ω] and D. Then, from (13.4.5.1) and (13.4.5.2) $[{\bf X} = [\omega][\beta_{p}]{\bf y} + {\bf D}. \eqno(13.4.5.3)]$

Now, if [ω] represents the rotational relationship between the `reference' molecule, [m = 1] , in the p-cell with respect to the h-cell, then from (13.4.5.3) $[{\bf X} = [\omega][\beta_{p}]{\bf y}_{m = 1} + {\bf D},]$ where $[{\bf y}_{m}]$ refers to the fractional coordinates of the mth molecule in the p-cell.

Assuming there is only one molecule per asymmetric unit in the p-cell, let the mth molecule in the p-cell be related to the reference molecule by the crystallographic rotation $[[\hbox{T}_{m}]]$ and translational operators $[{\bf t}_{m}]$ , such that $[{\bf y}_{m} = [\hbox{T}_{m}]{\bf y}_{m=1} + {\bf t}_{m}. \eqno(13.4.5.4)]$ For convenience, all translational components will initially be neglected in the further derivations below, but they will be reintroduced in the final stages. Hence, from (13.4.5.3) and (13.4.5.4) $[{\bf X} = \{[\omega][\beta_{p}][\hbox{T}_{m}^{-1}]\}{\bf y}_{m}. \eqno(13.4.5.5)]$ Further, if $[{\bf X}_{n}]$ refers to the nth subunit within the molecule in the h-cell, and similarly if $[{\bf y}_{m,\, n}]$ refers to the nth subunit within the mth molecule of the p-cell, then from (13.4.5.5) $[{\bf X}_{n} = \{[\omega][\beta_{p}][\hbox{T}_{m}^{-1}]\}{\bf y}_{m,\, n}. \eqno(13.4.5.6)]$ Finally, the rotation matrix $[[\hbox{R}_{n}]]$ is used to define the relationship among the N ( [N = 2] for a dimer, 4 for a 222 tetramer, 60 for an icosahedral virus etc.) noncrystallographic asymmetric units of the molecule within the h-cell. Then $[{\bf X}_{n} = [\hbox{R}_{n}]{\bf X}_{n=1}. \eqno(13.4.5.7)]$

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

International Tables for Crystallography (2006). Vol. F. ch. 13.4, p. 282