Sayre's equation

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. Sayre's equation

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. Sayre's equation

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Sayre's equation constrains the local shape of electron density. It provides a link between all structure-factor amplitudes and phases. It is an exact equation at atomic resolution in an equal-atom system. It is, therefore, very powerful for phase refinement and extension for small molecules at atomic resolution (Sayre, 1952, 1972, 1974). However, its power diminishes as resolution decreases. It can still be an effective tool for macromolecular phase refinement and extension if the shape function can be modified to accommodate the overlap of atoms at non-atomic resolution (Zhang & Main, 1990b).

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.

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).

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

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

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

Section 15.1.2.5. Sayre's equation

15.1.2.5. Sayre's equation

15.1.2.5.1. Sayre's equation in real and reciprocal space

15.1.2.5.2. The application of Sayre's equation to macromolecules at non-atomic resolution – the θ() curve

References

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