Scaling the observed structure-factor amplitudes according to the ideal density histogram

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. 316 | 1 | 2 |

Section 15.1.2.2.4. Scaling the observed structure-factor amplitudes according to the ideal density histogram

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

15.1.2.2.4. Scaling the observed structure-factor amplitudes according to the ideal density histogram

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In the process of density modification, electron density or structure factors from different sources are compared and combined. It is, therefore, crucial to ensure that all the structure factors and maps are on the same scale. The observed structure factors can be put on the absolute scale by Wilson statistics (Wilson, 1949) using a scale and an overall temperature factor. This is accurate when atomic or near atomic resolution data are available. The scale and overall temperature factor obtained from Wilson statistics are less accurate when only medium- to low-resolution data are available. A more robust method of scaling non-atomic resolution data is through the density histogram (Cowtan & Main, 1993; Zhang, 1993).

The ideal density histogram defines the mean and variance of an electron density, as shown in equations (15.1.2.15) and (15.1.2.16). We can scale the observed structure-factor amplitudes to be consistent with the target histogram using the following formula, obtained from the structure-factor equation and Parseval's theorem. The mean density and the density variance of the observed map can be calculated as $[\eqalignno{\overline{\rho}' &= (1/V)F(000), &(15.1.2.19)\cr \sigma '(\rho) &= (1/V) \left[{\textstyle\sum\limits_{\bf h}} | F({\bf h})|^{2}\right]^{1/2}. &(15.1.2.20)}%(15.1.2.20)]$

The mean and variance of the electron-density map at the desired resolution are calculated using the target histogram, the mean value of the solvent density, $[\overline{\rho}_{\rm solv}]$ , and the solvent volume of the cell, $[V_{\rm solv}]$ . The F(000) term can then be evaluated from equations (15.1.2.15) and (15.1.2.19): $[{F(000) = (V - V_{\rm solv})\overline{\rho} + V_{\rm solv} \overline{\rho}_{\rm solv}.} \eqno(15.1.2.21)]$ The scale of the observed amplitudes can be obtained from equations (15.1.2.16) and (15.1.2.20), $[F'({\bf h}) = KF({\bf h}), \eqno(15.1.2.22)]$ where $[K = \left[(\overline{\rho^{2}} - \overline{\rho}^{2})\right]^{1/2}\bigg/\bigg\{(1/V) \left[{\textstyle\sum\limits_{\bf h}} | F({\bf h})|^{2}\right]^{1/2}\bigg\}. \eqno(15.1.2.23)]$ This method is adequate for scaling observed structure factors at any resolution.

References

Cowtan, K. D. & Main, P. (1993). Improvement of macromolecular electron-density maps by the simultaneous application of real and reciprocal space constraints. Acta Cryst. D49, 148–157.Google Scholar

Wilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Cryst. 2, 318–321.Google Scholar

Zhang, K. Y. J. (1993). SQUASH – combining constraints for macromolecular phase refinement and extension. Acta Cryst. D49, 213–222.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 15.1, p. 316