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

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. 283 | 1 | 2 |

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

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

13.4.5.3. 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

International Tables for Crystallography (2006). Vol. F. ch. 13.4, p. 283