Crystallography phase problem

The question of determination of the phase of a field (classical or quantal, as of a wave function) from the modulus (absolute value) of the field along a real parameter (for which alone experimental determination is possible) is known as the phase problem [28]. (True also in crystallography.) The reciprocal relations derived in Section III represent a formal scheme for the determination of phase given the modulus, and vice versa. The physical basis of these singular integral relations was described in [147] and in several companion articles in that volume a more recent account can be found in [148]. Thus, the reciprocal relations in the time domain provide, under certain conditions of analyticity, solutions to the phase problem. For electromagnetic fields, these were derived in [120,149,150] and reviewed in [28,148]. Matter or Schrodinger waves were... 

Each diffracted beam, which is recorded as a spot on the film, is defined by three properties the amplitude, which we can measure from the intensity of the spot the wavelength, which is set by the x-ray source and the phase, which is lost in x-ray experiments (Figure 18.8). We need to know all three properties for all of the diffracted beams to determine the position of the atoms giving rise to the diffracted beams. How do we find the phases of the diffracted beams This is the so-called phase problem in x-ray crystallography. 

In small-molecule crystallography the phase problem was solved by so-called direct methods (recognized by the award of a Nobel Prize in chemistry to Jerome Karle, US Naval Research Laboratory, Washington, DC, and Herbert Hauptman, the Medical Foundation, Buffalo). For larger molecules, protein aystallographers have stayed at the laboratory bench using a method pioneered by Max Perutz and John Kendrew and their co-workers to circumvent the phase problem. This method, called multiple isomorphous replacement... 

When a diffracted X-ray beam hits a data collection device, only the intensity of the reflection is recorded. The other vital piece of information is the phase of the reflected X-ray beam. It is the combination of the intensity and the phase of the reflections that is needed to unravel the contributions made to the diffraction by the electrons in different parts of the molecule in the crystal. This so-called phase problem has been a challenge for theoretical crystallographers for many decades. For practical crystallography, there are four main methods for phasing the data generated from a particular crystal. 

Hauptman, H. (in press) The phase problem of X-ray crystallography. In Direct Methods for Solving Macromolecular Structures, Fortier, S. (Ed.), Kluwer, Dordrecht. 

Although the intensities of the spots can be easily measured experimentally, the phases cannot. This is the so-called phase problem of crystallography. Determining an estimate of the phases is critical to solving structures, but must be done by one of the several second-hand techniques, outlined later (O Section 2.3). 

Solving the phase problem in protein crystallography is a requirement for any structural study. The three... 

Abstract This introduction to the phase-problem in crystallography is addressed to those... 

The phase problem of X-ray crystallography may be defined as the problem of determining the phases ( ) of the normalized structure factors E when only the magnitudes E are given. Since there are many more reflections in a diffraction pattern than there are independent atoms in the corresponding crystal, the phase problem is overdetermined, and the existence of relationships among the measured magnitudes is implied. Direct methods (Hauptman and Karle, 1953) are ab initio probabilistic methods that seek to exploit these relationships, and the techniques of probability theory have identified the linear combinations of three phases whose Miller indices sum to... 

The phase problem in crystallography is on its way to being solved by completely ex novo procedures like the charge flipping method. Fast computers sweep away the problem by carrying out hundreds of thousands of stmcture factor calculations in a few hours. 

This problem is sometimes referred to as the second phase problem in crystallography. 

In crystallography, heavy atom derivatives are required to solve the phase problem before electron density maps can be obtained from the diffraction patterns. In nmr, paramagnetic probes are required to provide structural parameters from the nmr spectrum. In other forms of spectroscopy a metal atom itself is often studied. Now many proteins contain metal atoms, but even these metal atoms may not be suitable for crystallographic or spectroscopic purposes. Thus isomorphous substitution has become of major importance in the study of proteins. Isomorphous substitution refers to the replacement of a given metal atom by another metal that has more convenient properties for physical study, or to the insertion of a series of metal atoms into a protein that in its natural state does not contain a metal. In each case it is hoped that the substitution is such that the structural and/or chemical properties are not significantly perturbed. 

The "phase problem" in crystallography arises because in the usual experiment (Eq. () the magnitudes of the complex structure factors arc obtained, but not the phases. Yet in order to obtain the scattering density. 

The most demanding element of macromolecular crystallography (except, perhaps, for dealing with macromolecules that resist crystallization) is the so-called phase problem, that of determining the phase angle ahkl for each reflection. In the remainder of this chapter, I will discuss some of the common methods for overcoming this obstacle. These include the heavy-atom method (also called isomorphous replacement), anomalous scattering (also called anomalous dispersion), and molecular replacement. Each of these techniques yield only estimates of phases, which must be improved before an interpretable electron-density map can be obtained. In addition, these techniques usually yield estimates for a limited number of the phases, so phase determination must be extended to include as many reflections as possible. In Chapter 7,1 will discuss methods of phase improvement and phase extension, which ultimately result in accurate phases and an interpretable electron-density map. 

In modern crystallography virtually all structure solutions are obtained by direct methods. These procedures are based on the fact that each set of hkl planes in a crystal extends over all atomic sites. The phases of all diffraction maxima must therefore be related in a unique, but not obvious, way. Limited success towards establishing this pattern has been achieved by the use of mathematical inequalities and statistical methods to identify groups of reflections in fixed phase relationship. On incorporating these into multisolution numerical trial-and-error procedures tree structures of sufficient size to solve the complete phase problem can be constructed computationally. Software to solve even macromolecular crystal structures are now available. 

The constmction of synthetic selenocysteine-containing proteins or selenium-containing proteins attracts considerable interest at present, mainly for the reason that it can be used to solve the phase problem in X-ray crystallography. Selenomethionine incorporation has been used mostly uutil now for this purpose. There are also two reports ou uew synthetic selenocysteine-containing proteins. In one case, the active site serine of subtUisin has been converted into a selenocysteine residue by chemical means, with the result that the enzyme gains a predominant esterase instead of protease activity. In the second case, automated peptide synthesis was carried out to produce a peptide in which all seven-cysteine residues of the Neurospora crassa metallothioueiu (Cu) were replaced by selenocysteine. The replacement resulted iu au alteration of both the stoichiometry and the affinity of copper binding. ... 

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