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It is easily seen from either the Sayre equation (3.52) or the tangent formula (3.60) that direct methods are likely to be much more powerful in phase extension and refinement than in ab initio phasing, since nothing can be known about the left-hand side of either the Sayre equation or the tangent formula without first putting into the right-hand side at least a small number of starting phases. One of the reasons why multi-solution procedures are so successful in practice is that they provide the possibility of having some trial sets with an initial pattern of phases able to converge to the correct point in the multi-dimensional phase space. On the other hand, if the phases of a sufficiently large number of reflections can be estimated in advance then direct methods will work even more efficiently. This gives the possibility of combining direct methods with other methods to tackle the phase problem in a number of special cases.
In the analysis of complex crystal structures, it is often the case that a fragment instead of the complete structure is first obtained. Hence fragment development plays an important role in crystal structure determination.
Fourier synthesis with partial-structure phases has been a very efficient approach to obtaining the complete structure, especially when this is associated with weighting functions (Woolfson, 1956; Sim, 1960). A reciprocal-space alternative is a phase extension and refinement procedure based on partial structure information.
Theoretical and experimental advances in the techniques available for solving crystal structures have led to the development of several powerful techniques for solving complex structures, including those of proteins. In this 1995 book, Michael Woolfson and Fan Hai-Fu describe all the available methods and how they are used. In addition to traditional methods such as the use of the Patterson function and isomorphous replacement, and the direct methods, the authors include methods that use anomalous scattering and observations from multiple-beam scattering. The fundamental physics and mathematical analyses are fully explained. Practical aspects of applying the methods are emphasised.
When von Laue and his assistants produced their first smudgy X-ray diffraction photographs in Munich in 1912 they could not have known of the developments that would follow and the impact that these would have on such a wide range of science. Structural crystallography, the ability to find the arrangement of atoms inside crystals, has advanced over the years both theoretically and experimentally. Technical advances, such as the development of computers both for control of instruments and for complex calculations, and also the advent of high power synchrotron X-ray sources have all played their part.
In this book we bring together all the methods that have been and are being used to solve crystal structures. We broadly divide these methods into two main classes, non-physical and physical methods. In the first category we place those methods that depend on a single set of diffraction data produced by the normal Thomson scattering from the individual atoms. The Patterson methods and direct methods described in chapters 2 and 3 respectively are non-physical methods. In chapter 4 the basic principles are explained for two physical methods – isomorphous replacement, which combines the data from two or more related compounds to obtain phase information, and anomalous scattering, which uses data at wavelengths for which some of the atoms scatter anomalously, i.e. with an amplitude and phase differing from that given by the Thomson process. In chapter 5 the method of isomorphous replacement is explored in much greater depth and in chapter 6 the same is done for anomalous scattering.
The process of forming an optical image is one that is very well understood and frequently occurs. In the very act of seeing what is on this page the reader is forming a retinal image of its contents which is then conveyed to the brain in the form of electrical impulses for the complex task of interpretation and comprehension. What happens in the visual cortex is poorly understood but the formation of the retinal image via the lens of the eye is straightforward and can be followed by reference to fig. 1.1. The first stage in image formation is to direct towards the object some radiation (light in this case), part of which is scattered so that each point of the object becomes a secondary source of radiation which leaves in all directions. If we look in detail at what happens at a point of the object (fig. 1.1 (a)) we see that the radiation going off in different directions is not only coherent, because it derives from the same point source, but is also all in phase. Next the scattered radiation strikes a lens (fig. 1.1(b)). Because the speed of light in the lens material is different from that in air, rays travelling by different paths to the image point have the same optical path length and so undergo constructive interference there. The amplitude, and hence intensity, of each image point is proportional to that of the corresponding object point and consequently a true image is formed.
The crystal structure analysis of a protein would be a routine procedure provided that two or more heavy-atom derivatives with good isomorphism to the native protein were available (chapter 4). Unfortunately, in practice this is often not the case. Usually there will be little difficulty in preparing one heavy-atom derivative that is isomorphous with the native protein. However, finding a second isomorphous derivative may not be straight forward so that the use of single isomorphous replacement (SIR) data is preferable if there is some way to resolve the intrinsic ambiguity of the method in the non-centrosymmetric case (§4.1.3). The double-phase method (§4.1.6) is one way of doing this but it usually gives a rather noisy map. There is now described a noise-filtering technique which can be used to develop better information in such a situation.
Resolving the SIR phase ambiguity in real space: Wang's solvent-flattening method
Protein structures are characterized by having large contiguous solvent regions surrounding other regions of somewhat higher average density within which the protein exists. The contrast between the ordered structure (protein, sometimes with some solvent molecules) and background (disordered solvent) is much less than for small-molecule structures and this is one of the reasons, additional to other factors including their size and complexity, which make protein structures difficult to solve.
The first critical step in Wang's method (Wang, 1981, 1985) is to define the molecular boundary from a noisy electron density map. Following that, the densities inside the protein envelope are raised by a constant value and then densities lower than a certain value are removed. Outside the protein region, the density is smoothed to a constant level.
In fig. 1.12 there is shown a graphical means of determining the condition under which a particular diffracted beam will be produced. If the reciprocal lattice point at position s is the only one that touches the sphere of reflection then the associated diffracted beam will be the only one to occur – except of course for the straight-through diffracted beam, corresponding to the point O, which always occurs. Such a situation is described as two-beam diffraction. It is also possible to have more than two reciprocal lattice points on the surface of the reflecting sphere and in fig. 8.1 the points O, P and Q lie on the surface corresponding to the reciprocal-lattice vectors 0, h and k. This would be a case of three-beam diffraction. Four- and more beam diffraction is also possible but here we shall be restricting our attention to the three-beam case.
The line CQ in fig. 8.1 gives the direction of the k diffracted beam. Let us suppose that a beam of radiation was incident on the crystal from that direction. The crystal would still be in the same orientation, as would be the reciprocal lattice which is rigidly attached to the crystal, and the sphere of reflection would be displaced to put the new origin point, O, on the old point Q. It is now clear that after the displacement the reciprocal lattice point h – k will now fall at the same point as P and hence that this diffracted beam would occur.
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