Phase problem in X-ray crystallography

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. 

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. ... 

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

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

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... 

In early 1948 I thought that there was an experimental solution of the phase problem of X-ray crystallography. The idea was to use a double reflection hj followed by I12 which diffracts in the direction of I13 = hi + I12. If hi is set on the sphere of reflection so that it diffracts for any orientation of the crystal about a suitably chosen rotation axis, then hi and I12 should show an interference effect. This idea, beautiful in principle, was defeated by the mosaic character of crystals and possibly also crystal boimdary effects. Our experiment in which hi is 040 of a glycine crystal failed, although some reflections which were forbidden as single diffractions were observed. Shortly thereafter (1951) Bijvoet published his experimental solution to the phase problem by multiple isomorphous replacement methods, and I thought then that his discovery opened the way to solve protein structures. However, I did not start work in this direction until about 1958, and pursued it seriously beginning in 1961. 

Notice that while F is a complex number, only the absolute square, F, appears in eq. (9). This gives rise to the well-known phase problem of X-ray crystallography. The exponential in eq. (9), the Debye-Waller factor, involves the terms hor and u, which are the mean-square atomic displacements in the horizontal and vertical directions, respectively. The factor 7 (ap) describes interference of X-rays diffracted upwards with X-rays diffracted down and subsequently reflected back up by the interface. The factor ITCap) equals unity except near ap= ttc where it peaks sharply (the Yoneda-Vineyard peak) [65, 66]. It is convenient for deducing the zero-point of the f scale, which often covers a range of 0 to 10 , but otherwise... 

This relationship indicates the origin of the phase problem (Section 3.1.1) whose solution is one of the important tasks in X-ray crystallography there exist an infinite number of functions p(r) which give rise to the same function /(S). If p(r) is given, we can always calculate the corresponding function G(S). The passage from I G(S) to p(r), i.e. the solution of the phase problem, is only possible on the basis of models the most important will be developed in Sections 3.3.3 and 3.4.1. 

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. 

Gas-phase electron diffraction is the technique of choice for many special problems of molecular structure determination. However, it has not become a mass-producing technique like X-ray crystallography or the quantum chemical calculations. With the proliferation of quantum chemical calculations some of the problems, namely, the accurate determination of relatively simple organic molecules that used to be solved by gas-phase electron diffraction have moved to the realm of these calculations. There are a wealth of other problems, mainly in inorganic chemistry, that still necessitate the application of this rather demanding but instructive and amazing approach. 

Starting around 1948, I learned about X-ray crystallography and the phase problem, which, incidentally, had not been of any interest to Jerry at that time. His work was concerned exclusively with electron diffraction. Jerry and Isabella were not at all involved in X-ray diffraction. I had a period of free time for a year or so with not much else to do, but learn about X-ray diffraction and the phase problem. Jerry knew something about this, he had some background in it, but it was not an area in which he was planning to do any research or in which he had done any research. 

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