Sayre's equation in real and reciprocal space

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. 318 | 1 | 2 |

Section 15.1.2.5.1. Sayre's equation in real and reciprocal space

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

15.1.2.5.1. Sayre's equation in real and reciprocal space

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Sayre's equation (Sayre, 1952, 1972, 1974) expresses the constraint on structure factors when the atoms in a structure are equal and resolved, and the equation has formed the foundation of direct methods. In protein calculations, the resolution is generally too poor for atoms to be resolved, and this is reflected in the bulk of the terms required to calculate the equation for any particular missing structure factor.

For equal and resolved atoms, squaring the electron density changes only the shape of the atomic peaks and not their positions. The original density may therefore be restored by convoluting with some smoothing function, $[\psi({\bf x})]$ , which is a function of atomic shape , $[\rho({\bf x}) = (V/N) {\textstyle\sum\limits_{\bf y}} \rho^{2} ({\bf y})\psi ({\bf x} - {\bf y}), \eqno(15.1.2.31)]$ where $[\psi ({\bf x} - {\bf y}) = (1/V) {\textstyle\sum\limits_{\bf h}} \theta ({\bf h}) \exp[2\pi i{\bf h}\cdot ({\bf x} - {\bf y})]. \eqno(15.1.2.32)]$ Here, $[\theta({\bf h})]$ is the ratio of scattering factors of real, $[f({\bf h})]$ , and `squared', $[g({\bf h})]$ , atoms, and V is the unit-cell volume, i.e., $[\theta ({\bf h}) = f({\bf h})/g({\bf h}). \eqno(15.1.2.33)]$

Sayre's equation states that the convolution of the squared electron density with a shape function restores the original electron density. It can be seen from equation (15.1.2.31) that Sayre's equation puts constraints on the local shape of electron density. The local shape function is the Fourier transform of the ratio of scattering factors of the real and `squared' atoms.

Sayre's equation is more frequently expressed in reciprocal space as a system of equations relating structure factors in amplitude and phase: $[F({\bf h}) = [\theta({\bf h})/V] {\textstyle\sum\limits_{\bf k}} F({\bf k})F({\bf h} - {\bf k}). \eqno(15.1.2.34)]$ The reciprocal-space expression of Sayre's equation can be obtained directly from a Fourier transformation of both sides of equation (15.1.2.31) and the application of the convolution theorem.

References

Sayre, D. (1952). The squaring method: a new method for phase determination. Acta Cryst. 5, 60–65.Google Scholar

Sayre, D. (1972). On least-squares refinement of the phases of crystallographic structure factors. Acta Cryst. A28, 210–212.Google Scholar

Sayre, D. (1974). Least-squares phase refinement. II. High-resolution phasing of a small protein. Acta Cryst. A30, 180–184.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 15.1, p. 318