Anomalous Scattering Approaches

Excellent and detailed treatments of the use of anomalous dispersion data in the deduction of phase information can be found elsewhere (Smith et al., 2001), and no attempt will be made to duplicate them here. The methodology and underlying principles are not unlike those for conventional isomorphous replacement based on heavy atom substitution. Here, however, the anomalous scatterers may be an integral part of the macromolecule sulfurs (or selenium atoms incorporated in place of sulfurs), the iron in heme groups, Ca++, Zn++, and so on. Anomalous scatterers can also be incorporated by diffusion into the crystals or by chemical means. With anomalous dispersion techniques, however, all data necessary for phase determination are collected from a single crystal (but at different wavelengths) hence non-isomorphism is less of a problem. 

As with the isomorphous replacement method, the locations x, y, z in the unit cell of the anomalous scatterers must first be determined by Patterson techniques or by direct methods. Patterson maps are computed in this case using the anomalous differences Fi,u — F-h-k-i-Constructions similar to the Harker diagram can again be utilized, though probability-based mathematical equivalents are generally used in their stead. 

For more than 50 years it has been known that the barely measurable differences between Fhki and F-n-k-t contained useful phase information. For macromolecular crystals lacking anomalous scattering atoms, this phase information was impossible to extract and use because it was below the measurement error of reflections. Anomalous dispersion was, however, sometimes useful in conjunction with isomorphous replacement where the heavy atom substitutent provided a significant anomalous signal. The difference between F ki and F-h-k-i was, for example, employed to resolve the phase ambiguity when only a single isomorphous derivative could be obtained (known as single isomorphous replacement, or SIR) or used to improve phases in MIR analyses. 

In the past 15 years a number of technical advances have made it possible to maximize and precisely measure anomalous dispersion differences, and more powerful mathematical approaches have been devised to optimize its use for phase determination. The experimental problem was always to amplify the difference between and and obtain its 

Currently, use of selenium methionine, recombinant protein crystals, or crystals of wild type protein into which have been introduced heavy atom anomalous scatterers, has made single and multiple wavelength anomalous scattering the method of choice for phase determination. In addition, the ways it can be used, and the increasing opportunities for its application are dramatically expanding its popularity. 

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Direct methods are likely to increase in importance and have more widespread application for several reasons. As experimental tools become more powerful, and we learn to grow better crystals, atomic resolution data for proteins of increasing size will become more common. As the use of anomalous scattering approaches expands, the opportunities for their application to deduce the constellations and substmctures will increase as well. Finally, as the algorithms strengthen and computing methods become even more powerful, the direct methods themselves will become more effective. 

Although the physical processes responsible for X-ray and neutron resonance scattering are vastly different a unified approach of resonance (or anomalous) scattering can be given on the basis of the famous optical theorem ... 

The Patterson synthesis (Patterson, 1935), or Patterson map as it is more commonly known, will be discussed in detail in the next chapter. It is important in conjunction with all of the methods above, except perhaps direct methods, but in theory it also offers a means of deducing a molecular structure directly from the intensity data alone. In practice, however, Patterson techniques can be used to solve an entire structure only if the structure contains very few atoms, three or four at most, though sometimes more, up to a dozen or so if the atoms are arranged in a unique motif such as a planar ring structure. Direct deconvolution of the Patterson map to solve even a very small macromolecule is impossible, and it provides no useful approach. Substructures within macromolecular crystals, such as heavy atom constellations (in isomorphous replacement) or constellations of anomalous scattered, however, are amenable to direct Patterson interpretation. These substructures may then be used to solve the phase problem by one of the other techniques described below. 

As with the isomorphous replacement technique it is necessary to identify the positions, the x, y, z coordinates of the anomalous scatterers. This can be done by anomalous difference Patterson maps, which are Patterson syntheses that use the anomalous differences Fhki — F—h—k—i as coefficients (Blow and Rossmann, 1961). These maps are interpreted identically to isomorphous difference Patterson maps (see Chapter 9). Rapidly surpassing Patterson approaches, particularly for selenomethionine problems and others where the number of anomalous scatterers tends to be large, are direct methods (see below). These are strictly mathematical methods that have proved to be surprisingly effective in revealing the constellation of anomalous scatterers in a unit cell. 

The phenomenon of diffraction and its description as a Fourier transform (FT) is explained. The measured intensity of the diffracted X-rays related to the FT of the electron density, and the electron density - seen as an electron density map - is related to the (inverse) Fourier sum of the intensity of the diffracted X-rays. As we can only measure their intensity, we do not know the phases of the diffracted X-rays we have to determine them to solve the structure. Therefore, three principal methods are used, two experimental approaches (isomorphous replacement and anomalous scattering) and one based on known structures (molecular replacement). 

For example, if we have another heavy atom isomorphous derivative available with heavy atom sites different from those found in the first derivative, when the preceding process is repeated, we will get two solutions, one true and one false for each reflection from the second derivative as well. The true solutions should be consistent between the two derivatives while the false solution should show a random variation. Thus, by comparing the solutions obtained from these two calculations, one (the computer) can establish which solution represents the true phase angle. This is the principle of the MIR method. One can also utilize the anomalous scattering (AS) data of the first derivative to resolve the phase ambiguity. In this case, the technique is called the SIRAS approach. If two derivatives and anomalous data are used, then it is called the MIRAS approach. 

This approach is known as single isomorphous replacement with anomalous scattering (SIRAS). Trigonometric equations can be derived... 

Other theoretical approaches (e.g. section 9.3) have treated separately the anomalously scattering atom. This was done largely because the mathematics paralleled the chemical procedure of heavy atom derivatisation. 

But absolute configurations can be obtained from an analysis of small differences in diffraction intensities by a method developed by J.M. Bijvoet. The method makes use of extra phase shifts that occur when the frequency of the X-rays approaches an absorption frequency of atoms in the compound. The phase shifts are called anomalous scattering and result in different intensities in the diffraction patterns of different enantiomers. See Section 2.3.7(b) of the 7th edition of this text for an explanation of the origin of this anomalous phase shift. The incorporation of heavy atoms into the compound makes the observation of the extra phase shift easier to observe, but with very seasitive modern diffractometers this is no longer strictly neces.sary. 

S. Zumer, Fight scattering from nematic droplets anomalous-drffiaction approach, Phys. Rev. A, 37, 4006 (1988). 

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