Phase problem, in crystallography

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

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. 

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. 

Although the structure factor modulus (structure amplitude) can be obtained from the intensity data (eqn [8]), the corresponding phase cannot be measured directly experimentally. The amplitude values alone cannot be used to determine directly the atomic positions. Incorrect phase values will lead to an incorrect calculated electron density and an erroneous final structure. This is referred to as the phase problem in crystallography and is the central problem in X-ray structure analysis. 

However, there is a major difficulty in carrying out this inversion process. The electric field vector E h,k/) of the scattered x-rays is directly proportional to Sc(hM> at the present there is no method for directly measuring this E-field. For x-rays, we are limited to the intensity I as the observable, which is given by I E / The E is a complex function involving both amplitude and phase, but the phase irrformation is lost unless one uses a coherent source to record the phase information as is done in holography. Since we do not yet have x-ray lasers suitable for this purpose, this is known as the phase problem in crystallography (see McPherson, 1999). 

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. 

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

Because Fhkl is a periodic function, it possesses amplitude, frequency, and phase. It is a diffracted X ray, so its frequency is that of the X-ray source. The amplitude of Fhkl is proportional to the square root of the reflection intensity lhkl, so structure amplitudes are directly obtainable from measured reflection intensities. But the phase of Fhkl is not directly obtainable from a single measurement of the reflection intensity. In order to compute p(x,y,z) from the structure factors, we must obtain, in addition to the intensity of each reflection, the phase of each diffracted ray. In Chapter 6,1 will present an expression for p(x,y,z) as a Fourier series in which the phases are explicit, and I will discuss means of obtaining phases. This is one of the most difficult problems in crystallography. For now, on the assumption that the phases can be obtained, and thus that complete structure factors are obtainable, I will consider further the implications of Eqs. (5.15) (structure factors F expressed in terms of atoms), (5.16) [structure factors in terms of p(x,y,z)], and (5.18) [p(x,y,z) in terms of structure factors]. 

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. ... 

V is the volume of the unit cell and should not be confused with V(r) = V(xyz). From the fact that only the absolute square of F can be measured, the phase of the complex value of F is lost. This is the phase problem of crystallography, which found its solution by the introduction of the method of isomorphous replacement in the fifties... 

Hauptman, H. A. The phase problem in X-ray crystallography. Physics Today November, 24-29 (1989). 

Colella, R. Multiple diffraction of X-rays and the phase problem. In P. P. Ewatd and his Dynamical Theory of X-ray Diffraction. A memorial Volume for Paul P. Ewald. S3 January 1888 — SS August 1985. (Eds., Cruickshank, D. W.. 1 Juretschke, H. J., and Kato, N.) International Union of Crystallography/OxforJ University Press Oxford (1992). 

The central problem in crystallography lies in obtaining the phase for every observed structure factor amplitude. We judge how correct a given set of phases is by the result does the electron density map make chemical sense For small molecules, very accurate data is usually available to high resolution (1 A or better), which allows the use of direct methods 9 to obtain the phases rapidly and correctly. The approach uses statistical relationships between the phases of certain reflections. Unfortunately, direct methods are not easily... 

The classical method for solving the phase problem in macromolecular crystal structures, known as isomorphous replacement, dates back to the earliest days of protein crystallography.10,16 The concept is simple enough we introduce into the protein crystal an atom or atoms heavy enough to affect the diffraction pattern measurably. We aim to figure out first where those atoms are (the heavy atom substructure) by subtracting away the protein component, and then bootstrap — use the phases based on the heavy atom substructure to solve — the structure of the protein. 

Typical X-ray diffraction experiments provide structure factor moduli, while the relative phases are lost. Recovery of the phase information is crucial for crystal structure solution and is referred to in crystallography as the phase problem. In single-crystal diffraction this problem is solved by different approaches ... 

However, when the intensities of the X-rays are recorded in this manner all information of the phase is lost. Thus, the fundamental problem in a structure determination is the phase problem. Until recently, the phase problem in protein crystallography has been solved by the heavy atom isomorphous replacement method (sections 2(d) and (e)), but other methods are also available (sections 2(e) and (f)). 

The maximum entropy method achieves a remarkable universality and unification based on common sense reduced to calculation . It has been applied to information theory, statistical mechanics, image processing in radio astronomy, and now to X-ray crystallography. The prospects for a computational solution to the phase problem in protein crystallography appear promising and developments in the field are awaited eagerly. 

Heavy atoms are often essential for solving the phase problem in the X-ray crystallography of biomacromolecules. In many cases, crystals are soaked with a solution of the compound, and it is hoped that the heavy atoms will occupy well-defined sites in the crystal with a high occupancy factor. Organomercury compounds are frequently used for this purpose. 

As the measured intensities provide (after the necessary corrections for background, instrumental resolution, etc.) the moduli of the structure factor, but not the phase factor, the Fourier synthesis (inverse Fourier transformation) cannot be applied to the determination of the nuclei in the unit cell this is the well-known phase problem of crystallography. 

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