The system of nonlinear constraint equations

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. 321 | 1 | 2 |

Section 15.1.5.1. The system of nonlinear constraint equations

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

15.1.5.1. The system of nonlinear constraint equations

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The constraints used in SQUASH/DM can be divided into three categories. The first category comprises the linear constraints, such as solvent flatness, density histogram and equal molecules. The second category comprises the nonlinear constraints, such as the local shape of electron density as expressed in Sayre's equation. The third category comprises the available structural data, such as the observed structure-factor amplitudes and the experimental phases. The first and second categories of constraints are used to solve new electron-density values. The third category of constraints is used as a means to filter the modified phases.

The modification to the density value at a grid point by a linear constraint is independent of the values at other grid points. These constraints include solvent flattening, histogram matching and molecular averaging. These density-modification methods construct an improved map directly from an initial density map as expressed by $[\rho({\bf x}) = H ({\bf x}), \eqno(15.1.5.1)]$ where $[H({\bf x})]$ is the target electron density produced by these linear constraints.

The new electron density that satisfies both the linear constraints represented by equation (15.1.5.1) and the nonlinear constraints expressed by Sayre's equation (15.1.2.31) can be obtained by solving the systems of simultaneous equations (Zhang & Main, 1990b) $[\cases{(V/N) {\textstyle\sum\limits_{\bf y}} \rho^{2} ({\bf y})\psi ({\bf x} - {\bf y}) - \rho({\bf x}) = 0\cr H({\bf x}) - \rho({\bf x}) = 0\cr}. \eqno(15.1.5.2)]$

Equation (15.1.5.2) represents a system of nonlinear simultaneous equations with as many unknowns as the number of grid points in the asymmetric unit of the map and with twice as many equations as unknowns. The functions $[H({\bf x})]$ and $[\psi({\bf x} - {\bf y})]$ are both known. The least-squares solution, using either the full matrix or the diagonal approximation, is obtained using the Newton–Raphson technique with fast Fourier transforms, as described in the next section (Main, 1990b).

References

Main, P. (1990b). The use of Sayre's equation with constraints for the direct determination of phases. Acta Cryst. A46, 372–377.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. 321