Direct solution of continuous densities given in the Fourier magnitudes

Abstract

In order to apply direct methods routinely to macromolecular crystals it will be necessary to generate a non-atomic theory which is applicable to continuous densities. Reformulation of the 'phase problem' in terms of deconvoluting an autocorrelation function or Patterson synthesis reduces the problem from a theoretically intractable transcendental problem to a system of simultaneous quadratic equations. These quadratic equations may, in principle, always be solved by conjugate direction search techniques. The phase problem is shown to be a class P problem, admitting a deterministic solution in polynomial time. Two algorithms are presented with running times proportional to N2points and Npoints log Npoints per step. These algorithms are a pixel-by-pixel search and a conjugate gradients search. When the data are exact and complete the Fourier magnitudes are readily inverted by them to find the image. An example with real data, from a 15mer of DNA, is also shown.

Crystal structure at 1.5-.ANG. resolution of d(CGCICICG), an octanucleotide containing inosine, and its comparison with d(CGCG) and d(CGCGCG) structures

“…The error was evaluated as the integral over the unit cell of the square of the difference between the prior map and the complete map, with the prior map truncated to reflect the previous bounds of the electron density. This is not exactly the same as the square error of Harrison (1990), but it has the same zeros and similar convergence properties. This measure is the sum of the difference between a map which has a correct Patterson (by definition as it has only observed Fourier coefficients) and one which meets the bounds on the electron density.…”

“…The Patterson deconvolution method (Harrison, 1990) was applied to give an independent solution of the packing and molecular structure. The structure was solved by a direct inversion of the Patterson function, a new method described in Harrison (1990). This approach is based on the observation that the Patterson function is a quadratic function of the electron density values.…”

Discrete Hilbert transforms and the phase problem in crystallography

“…Introduction of the ACF into the phasing process enables one to employ its physical properties (non-negativity, boundaries in space and magnitude) and thus to construct iterative ACF modification algorithms. Possibly, deconvolution of the ACF rather than the Patterson (Harrison, 1990) might give better results as the ACF is dependent on phases whereas the Patterson is not. Furthermore, DHT's applied to the intensity function in three-dimensions and the formula for in-between intensities in terms of normal ones are obtained.…”