International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. F. ch. 15.1, p. 318   | 1 | 2 |

Section 15.1.2.5.2. The application of Sayre's equation to macromolecules at non-atomic resolution – the θ([{\bf h}]) curve

K. Y. J. Zhang,a K. D. Cowtanb* and P. Mainc

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

15.1.2.5.2. The application of Sayre's equation to macromolecules at non-atomic resolution – the θ([{\bf h}]) curve

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Sayre's equation is exact for an equal-atom structure at atomic resolution. The reciprocal-space shape function, [\theta({\bf h})], can be calculated analytically from the ratio of the scattering factors of real and `squared' atoms, which can both be represented by a Gaussian function. At infinite resolution, we expect [\theta({\bf h})] to be a spherically symmetric function that decreases smoothly with increased h. However, for data at non-atomic resolution, the [\theta({\bf h})] curve will behave differently because atomic overlap changes the peak shapes. Therefore, a spherical-averaging method is adopted to obtain an estimate of the shape function empirically from the ratio of the observed structure factors and the structure factors from the squared electron density using the formula [\theta (s) = V\left\langle F\left({\bf h}\right)\Big/{\textstyle\sum\limits_{\bf k}} F\left({\bf k}\right)F\left({\bf h} - {\bf k}\right)\right\rangle _{|{\bf h}|}, \eqno(15.1.2.35)] where the averaging is carried out over ranges of [|{\bf h}|], i.e., over spherical shells, each covering a narrow resolution range. Here, s represents the modulus of h.

The empirically derived shape function only extends to the resolution of the experimentally observed phases. This is sufficient for phase refinement. However, there are no experimentally observed phases to give the empirical [\theta(s)] for phase extension. Therefore, a Gaussian function of the form [\theta(s) = K\exp (- Bs^{2}) \eqno(15.1.2.36)] is fitted to the available values of [\theta(s)], and the parameters K and B are obtained using a least-squares method. The shape function [\theta(s)] for the resolution beyond that of the observed phases is extrapolated using the fitted Gaussian function. The derivation of the shape function [\theta(s)] from a combination of spherical averaging and Gaussian extrapolation is the key to the successful application of Sayre's equation for phase improvement at non-atomic resolution (Zhang & Main, 1990b[link]).

References

First citation Zhang, K. Y. J. & Main, P. (1990b). The use of Sayre's equation with solvent flattening and histogram matching for phase extension and refinement of protein structures. Acta Cryst. A46, 377–381.Google Scholar








































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