Combining crystallographic and noncrystallographic symmetry

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, pp. 282-283 | 1 | 2 |

Section 13.4.5. Combining crystallographic and noncrystallographic symmetry

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. Combining crystallographic and noncrystallographic symmetry

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Transformations will now be described which relate noncrystallographically related positions distributed among several fragmented copies of the molecule in the asymmetric unit of the p-cell and between the p-cell and the h-cell.

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 $[{\bf X}_{n} = [\hbox{R}_{n}]{\bf X}_{n=1}. \eqno(13.4.5.7)]$

13.4.5.2. Averaging with the p-cell

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Consider averaging the density at N noncrystallographically related points in the p-cell and replacing that density into the p-cell. By substituting for $[{\bf X}_{n}]$ and $[{\bf X}_{n=1}]$ in (13.4.5.7) and using (13.4.5.6), $[\{[\omega][\beta_{p}][\hbox{T}_{m}^{-1}]\}{\bf y}_{m,\, n} = [\hbox{R}_{n}]\{[\omega][\beta_{p}][\hbox{T}_{m}^{-1}]\}{\bf y}_{m,\, n=1},]$ or $[{\bf y}_{m,\, n} = \{[\omega][\beta_{p}][\hbox{T}_{m}^{-1}]\}^{-1} \times [\hbox{R}_{n}]\{[\omega][\beta_{p}][\hbox{T}_{m}^{-1}]\}{\bf y}_{m,\, n=1}. \eqno(13.4.5.8)]$ Now set $[\eqalignno{[{\rm E}_{m,\, n}] &= \{[\omega][\beta_{p}][\hbox{T}_{m}^{-1}]\}^{-1}[\hbox{R}_{n}]\{[\omega][\beta_{p}][\hbox{T}_{m}^{-1}]\}\cr &= [\hbox{T}_{m}][\alpha_{p}][\omega^{-1}][\hbox{R}_{n}][\omega][\beta_{p}][\hbox{T}_{m}^{-1}], &(13.4.5.9)}]$ giving $[{\bf y}_{m,\, n} = [\hbox{E}_{m,\, n}]{\bf y}_{m,\, n=1} + {\bf e}_{m,\, n}, \eqno(13.4.5.10)]$ where $[{\bf e}_{m,\, n}]$ is the corresponding translational element. Note that multiplication by $[[\hbox{E}_{m,\, n}]]$ thus corresponds to the following sequence of transformations: (1) placing all the crystallographically related subunits into the reference orientation with $[[\hbox{T}_{m}^{-1}]]$ ; (2) `orthogonalizing' the coordinates with $[[\beta_{p}]]$ ; (3) rotating the coordinates into the h-cell with [ω]; (4) rotating from the reference subunit of the molecule of the h-cell with $[[\hbox{R}_{n}]]$ ; (5) rotating these back into the p-cell with $[[\omega^{-1}]]$ ; (6) `de-orthogonalizing' in the p-cell with $[[\alpha_{p}]]$ ; and (7) placing these back into each of the M crystallographic asymmetric units of the p-cell with $[[\hbox{T}_{m}]]$ .

The translational elements, $[{\bf e}_{m,\, n}]$ , can now be evaluated. Let $[{\bf s}_{p,\, m}]$ be the fractional coordinates of the centre (or some arbitrary position) of the mth molecule in the p-cell; hence, $[{\bf s}_{p,\, m=1}]$ denotes the molecular centre position of the reference molecule in the p-cell. If $[{\bf s}_{p,\, m}]$ is at the intersection of the molecular rotation axes, then it will be the same for all n molecular asymmetric units. Therefore, it follows from (13.4.5.10) that $[{\bf e}_{m,\, n} = {\bf s}_{p,\, m} - [\hbox{E}_{m,\, n}]{\bf s}_{p,\, m=1}, \eqno(13.4.5.11a)]$ or $[{\bf y}_{m,\, n} = [\hbox{E}_{m,\, n}]{\bf y}_{m,\, n=1} + ({\bf s}_{p,\, m} - [\hbox{E}_{m,\, n}]{\bf s}_{p,\, m=1}). \eqno(13.4.5.11b)]$ Equation (13.4.5.11b) can be used to find all the N noncrystallographic asymmetric units within the crystallographic asymmetric unit of the p-cell. Thus, this is the essential equation for averaging the density in the p-cell and replacing it into the p-cell.

13.4.5.3. Averaging the p-cell and placing the results into the h-cell

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Consider averaging the density at N noncrystallographically related points in the p-cell and placing that result into the h-cell. From (13.4.5.7), multiplying by $[[\alpha_{h}]]$ , $[[\alpha_{h}] [{\bf X}_{n=1}] = [\alpha_{h}][\hbox{R}_{n}^{-1}]{\bf X}_{n}.]$ From (13.4.5.1) and (13.4.5.6), $[{\bf x}_{n=1} = [\alpha_{h}][\hbox{R}_{n}^{-1}]\{[\omega][\beta_{p}][\hbox{T}_{m}^{-1}]\}{\bf y}_{m,\, n}. \eqno(13.4.5.12)]$ Since it is only necessary to place the reference molecule of the p-cell into the h-cell, it is sufficient to consider the case when [m = 1] , in which case $[[\hbox{T}_{m}^{-1}]]$ is the identity matrix [I]. It then follows, by inversion, that $[\eqalign{{\bf y}_{m=1,\, n} &= \{[\omega][\beta_{p}]\}^{-1}[\hbox{R}_{n}][\beta_{h}]{\bf x}_{n=1}\cr &= [\alpha_{p}][\omega^{-1}][\hbox{R}_{n}][\beta_{h}]{\bf x}_{n=1},}]$ which corresponds to: (1) `orthogonalizing' the h-cell fractional coordinates with $[[\beta_{h}]]$ ; (2) rotating into the nth noncrystallographic unit within the molecule using $[[\hbox{R}_{n}]]$ ; (3) rotating into the p-cell with $[[\omega^{-1}]]$ ; and (4) `de-orthogonalizing' into fractional p-cell coordinates with $[[\alpha_{p}]]$ .

Now, if $[{\bf s}_{h}]$ is the molecular centre in the h-cell (usually $[{1 \over 2}, {1 \over 2}, {1 \over 2}]$ ), then $[{\bf y}_{m=1,\, n} = [\hbox{E}_{m=1,\, n}']{\bf x} + ({\bf s}_{p,\, m=1} - [\hbox{E}_{m=1,\, n}']{\bf s}_{h}),]$ and $[[\hbox{E}_{m=1,\, n}'] = [\alpha_{p}][\omega^{-1}][\hbox{R}_{n}][\beta_{h}]. \eqno(13.4.5.13)]$ Equation (13.4.5.13) determines the position of the N noncrystallographically related points $[{\bf y}_{m=1,\, n}]$ in the p-cell whose average value is to be placed at x in the h-cell.

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, pp. 282-283