The refinement of noncrystallographic symmetry

Zhang, K. Y. J.; Cowtan, K. D.; Main, P.

doi:10.1107/97809553602060000687

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. 15.1, p. 317 | 1 | 2 |

Section 15.1.2.3.3. The refinement of noncrystallographic symmetry

K. Y. J. Zhang,^a K. D. Cowtan^b ^* and P. Main^c

^a Division of Basic Sciences, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave N., Seattle, WA 90109, USA,^bDepartment of Chemistry, University of York, York YO1 5DD, England, and ^cDepartment of Physics, University of York, York YO1 5DD, England
Correspondence e-mail: cowtan+email@ysbl.york.ac.uk

15.1.2.3.3. The refinement of noncrystallographic symmetry

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The initial NCS operation obtained from rotation and translation functions or heavy-atom positions can be fine-tuned by a density-space R-factor search in the six-dimensional rotation and translation space. The density-space R factor is defined as $[R = {\textstyle\sum\limits_{\bf r}} | \rho({\bf r}) - \rho({\bf r}') |\big/{\textstyle\sum\limits_{\bf r}} | \rho({\bf r}) + \rho({\bf r}')|, \eqno(15.1.2.24)]$ where $[{\bf r} = \{xyz\}]$ is the set of Cartesian coordinates, $[{\bf r}' = \Omega{\bf r}]$ is the NCS-related set of coordinates of r and Ω represents the NCS operator.

The six-dimensional search is very time-consuming. The search rate can be increased by using only a representative subset of grid points. The NCS operation is systematically altered to find the lowest density-space R factor for the selected subset of grid points.

The solution of the NCS operation from the six-dimensional search can be further refined by the following least-squares procedure. If $[\rho({\bf r})]$ is related to $[\rho({\bf r}')]$ by the NCS operation, Ω, $[\rho({\bf r}') = \rho(\Omega{\bf r}). \eqno(15.1.2.25)]$ Here, Ω is a function of $[\omega, \Omega = f(\omega)]$ , where $[\omega = \{\alpha, \beta, \gamma, t_{x},t_{y},t_{z}\}]$ represents the rotation and translation components of the NCS operation. The solution to the NCS parameters, ω, can be obtained by minimizing the density residual between the NCS-related molecules, $[\varepsilon ({\bf r}) = \rho({\bf r}) - \rho(\Omega{\bf r}), \eqno(15.1.2.26)]$ using a least-squares formula of the form $[\left({\partial \rho \over \partial \omega}\right)^{T} \left({\partial \rho \over \partial \omega}\right)\Delta \omega = \left({\partial \rho \over \partial \omega}\right)^{T} \varepsilon ({\bf r}), \eqno(15.1.2.27)]$ where Δω is the shift to the NCS parameters. Here, $[{\partial \rho \over \partial \omega} = {\partial \rho \over \partial {\bf r}} {\partial {\bf r} \over \partial \omega}. \eqno(15.1.2.28)]$ The partial derivatives, $[\partial \rho/\partial {\bf r} = \{\partial \rho/\partial x, \ \partial \rho/\partial y, \ \partial \rho/\partial z\}]$ , can be calculated by Fourier transforms, $[\eqalign{{\partial \rho \over \partial x} &= - {2\pi i \over V} {\sum\limits_{hkl}} hF_{hkl} \exp [- 2\pi i(hx + ky + lz)]\cr {\partial \rho \over \partial y} &= - {2\pi i \over V} {\sum\limits_{hkl}} kF_{hkl} \exp [- 2\pi i(hx + ky + lz)]\cr {\partial \rho \over \partial z} &= - {2\pi i \over V} {\sum\limits_{hkl}} lF_{hkl} \exp [- 2\pi i(hx + ky + lz)],} \eqno(5.1.2.29)]$ or more efficiently with a single Fourier transform by the use of spectral B-splines (Cowtan & Main, 1998). $[\partial {\bf r}/\partial \omega]$ is derived analytically based on the relationship between the Cartesian coordinates, r, and the rotational and translational coordinates of the NCS operation, ω, $[\left(\matrix{x'\cr y'\cr z'\cr}\right) = \left(\matrix{\cos \alpha \cos \beta \cos \gamma - \sin \alpha \sin \gamma &- \cos \alpha \cos \beta \sin \gamma - \sin \alpha \sin \gamma &\cos \alpha \sin \beta\cr \sin \alpha \cos \beta \cos \gamma + \cos \alpha \sin \gamma &- \sin \alpha \cos \beta \sin \gamma + \cos \alpha \cos \gamma &\sin \alpha \sin \beta\cr - \sin \beta \cos \gamma &\sin \beta \sin \gamma &\cos \beta\cr}\right) \left(\matrix{x\cr y\cr z\cr}\right) + \left(\matrix{t_{x}\cr t_{y}\cr t_{z}}\right). \eqno(15.1.2.30)]$

References

Cowtan, K. D. & Main, P. (1998). Miscellaneous algorithms for density modification. Acta Cryst. D54, 487–493.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 15.1, p. 317