Anomalous scattering problem

The problems surrounding Hamilton s test (vide supra), as well as some misconceptions and misuses encountered in the literature, led Rogers to propose an alternative and more reliable method for determining the absolute structure58. A factor >7 is introduced which multiplies the imaginary component Aff of the anomalous scattering terms of the atomic scattering factors of all atoms (equation 11, which replaces equation 9, see Section 4.2.2,1.1), and which is treated as a variable in the least-squares refinement. 

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. 

An alternative procedure, called molecular replacement, uses information about known structures that are believed to be similar to that of the species being investigated. The known structure is used to estimate the electron density of the unknown structure, which is then refined and improved. Another method of dealing with the phase problem is to introduce atoms which absorb radiation in the region of the incident X-rays, leading to a process called anomalous scattering . For proteins, a popular method is to replace S by Se by using selenomethionine in place of methionine. For nucleic acids, iodouracil or iodocytosine can be used in place of thymine and cytosine respectively. 

Okaya, Y., and Pepinsky, R. New formulation and solution of the phase problem in X-ray analysis of noncentric crystals containing anomalous scatterers. Phys. Rev. 103, 1645-1647 (1956). 

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. 

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 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 anomalous scattering is extensively used in modem macromolecular crystallography to solve the phase problem. To understand how this is done, we need to return to the simple picture of X-rays reflecting from Bragg planes, where it makes no difference which side of the plane is the reflecting surface . This leads to two structure factors Fhki and F h differing only in the sign of their phase. The phase — a complex number - drops out because we measure intensities (/= F2 see above) and I k,i and are equal. 

As mentioned at the beginning of this section, anomalous scattering based methods (SAD and MAD) have become very important. This is due to two reasons. First, unlike in the isomorphous replacement methods, the methods do not require multiple crystals, one with and one without a heavy atom. All the measurements are made from one crystal, so there is no problem with isomorphicity,- a crystal is by definition isomorphous with itself. As long as the protein has a 3rd transition row or heavier element bound, MAD and SAD experiments can be performed. Second, modern techniques have made the method extremely easy to use in most cases, it is not even necessary to introduce a heavy atom into the protein after crystallization. Selenium has a K-edge at 12.6578 keV, or 0.9795 A, a very good energy for data collection at a synchrotron and cells can be grown on... 

Figure 12. Solid curve the refractive index of water showing a simple, monotonic dispersion curve. Dotted curve the contribution of the 189 nm band of iV-methylacetamide at a 1 M concentration added on to the water curve using equation 19. The point to be made is that there is no way to match the background curve of a good solvent to the anomalous dispersion of a chromophoric system under study. Such matching is commonly attempted to remove light scattering problems which depend on the difference in refractive index of the particle, n, with that of the solvent, n, i.e., (tip—nf). The dashed curve adds the second dispersion term in equation 25. Also included is the calculated refractive index of particulate poly-L-glutamic acid (PGA) (see section 4(cKii))-...

The first X-ray photographs of a protein crystal were described 50 years ago by Bernal and Crowfoot [1], These remarkable photographs indicated that a wealth of structural information was available for protein molecules once methods for the solution of the patterns had been developed. At that time the determination of atomic positions even in the crystals of small molecules was a difficult task. In 1954, Perutz and his colleagues [2] showed that the technique of heavy atom isomorphous replacement could be used to solve the phase problem. The method was put on a sound systematic basis by Blow and Crick [3] and extended to include the use of anomalous scattering [4,5]. Until recently, these methods provided the basis for all protein structure determinations. They have been remarkably effective (as illustrated below) and new developments have both increased the size of the problem solvable and provided greater insights. 

Resolution of the Difficulty Some years earlier we4 had added anomalous scattering terms to the usual least squares program. In 1966 the polar dispersion error was discovered.5 When the above problem came to our attention, we felt that the difficulty might be in an incorrect space group determination. Ensuing calculations using our modified least squares program quickly revealed that the difficulty arose from the... 

One potential problem of introducing a relatively large number of anomalously scattering atoms into a protein is that of determining their... 

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