Diffraction phase problem

These stmctures do not diffract as weH as smaH molecules, and as a result, there are many weak reflections and data coHection takes much longer than for smaH molecules. Also, the solution of the phase problem is more difficult and usuaHy requires the coHection of data sets with monochromatic radiation at several different wavelengths. Because of the much longer data coHection times, area detectors are almost always used. Also because of the long... 

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. 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

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. 

Application. Anomalous X-ray diffraction (AXRD), anomalous wide-angle X-ray scattering (AWAXS), and anomalous small-angle X-ray scattering (ASAXS) are scattering methods which are selective to chemical elements. The contrast of the selected element with respect to the other atoms in the material is enhanced. The phase problem of normal X-ray scattering can be resolved, and electron density maps can be computed. 

The use of blank-disc CBED patterns for solving crystal stmctures by electron diffraction (in conjunction with direct methods for the phase problem) would seem to have many advantages ... 

Similar to X-Ray and neutron diffraction analysis, electron dilFraction structure analysis consists of such main stages as the obtaining of appropriate diffraction patterns and their geometrical analysis, the precision evaluation of diffraction-reflection intensities, the use of the appropriate formulas for recalculation of the reflection intensities into the structure factors, finally the solution of the phase problem, Fourier-constructions. 

Key words phase-problem, erystallographie image processing, Patterson method, direet methods, strueture determination, eleetron diffraction... 

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... 

Data processing occurs in two stages. Initially, diffraction images are reduced to a tabulation of reflection indices and intensities or, after truncation, structure factors. The second stage involves conversion of the observed structure factors into an experimental electron density map. The choice of how to execute the latter step depends on the method used to determine the phase for each reflection. The software packages enumerated above generally focus on exploitation of anomalous signals to overcome the phase problem for a protein of unknown structure. 

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. 

In 1954, Perutz introduced the isomorphous replacement method for determining phases. In this procedure a heavy metal, such as mercury or platinum, is introduced at one or more locations in the protein molecule. A favorite procedure is to use mercury derivatives that combine with SH groups. The resulting heavy metal-containing crystals must be isomorphous with the native, i.e., the molecules must be packed the same and the dimensions of the crystal lattice must be the same. However, the presence of the heavy metal alters the intensities of the spots in the diffraction pattern and from these changes in intensity the phases can be determined. Besides the solution to the phase problem, another development that was absolutely essential was the construction of large and fast computers. It would have been impossible for Perutz to determine the structure of hemoglobin in 1937, even if he had already known how to use heavy metals to determine phases. 

Alternative methods of solving the phase problem are also used now. When a transition metal such as Fe, Co, or Ni is present in the protein, anomolous scattering of X-rays at several wavelengths (from synchrotron radiation) can be used to obtain phases. Many protein structures have been obtained using this multiple wavelength anomalous diffraction (MAD phasing) method.404 407 408 Selenocysteine is often incorporated into a protein that may be produced in... 

There are two approaches to the solution of the phase problem that have remained in favor. The first is based on the tremendously important discovery or Patterson in the 1930s ihal the Fourier summation of Eq. 3. with (he experimentally known quantities F2 (htl> replacing F(hkl) leads nol to a map of scattering density, but to a map of all interatomic vectors. The second approach involves the use of so-called direct methods developed principally by Karie and Hauptman of the U.S. Naval Research Laboratory and which led to the award of the 1985 Nobel Prize in Chemistry. Building upon earlier proposals that (he relative intensities of the spots in a diffraction pattern contain information about a crystal phase. Hauptman and Karie developed a mathematical means of extracting the information. A fundamental proposition of (heir direct method is that if thrice intense spots in the pattern have positions whose coordinates add up to zero, their relative phases will cancel out. Compulations done with many triads of spots yield probable phases for a significant number of diffracted waves and further mathematical analysis leads lo a likely solution for the structure of the molecule as a whole. 

In this case, the phase problem reduces to a question of the sign of the structure factor. Since the sign of the structure factor can be evaluated more reliably than the phase angle, the electron density in a centrosymmetric crystal can be evaluated more accurately than that in an acentric crystal (93), Equation (2) implies that total, time-averaged electron density can be determined by the diffraction method. 

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