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The major problem in X-ray crystallography is to determine the phase angles of the X-ray reflections. In protein crystallography, this problem is solved by the application of either isomorphous replacement or molecular replacement or multiple- (MAD) or single-wavelength anomalous diffraction (SAD). In small-molecule crystallography, a completely different solution is applied. There, direct methods are the standard techniques for determining the phase angles of the structure factors. They use the principle that phase information is included in the intensities and this principle depends on the basic assumptions that the electron density is always positive [ F (0 0 0) included in the Fourier summation] and the crystal consists of discrete atoms that are sometimes even considered to be equal. Phase relations based on probability theory have been formulated and these phase relations are applied to suitably chosen clusters of reflections. Although these direct methods work perfectly well for small-molecule crystals, it has thus far not been easy to apply them for the determination of a complete protein molecular structure. However, a number of direct methods have been developed for locating the substructure atoms in a protein crystal. We present two of them: Shake-and Bake and SHELXD.

When a stationary crystal is illuminated with X-rays from a continuous range of wavelengths (polychromatic or “white” radiation), a Laue diffraction pattern is produced. The very first X-ray diffraction pictures of a crystal were in fact obtained in this way by Friedrich, Knipping, and Laue in 1912. However, since then, monochromatic beams were used nearly exclusively in X-ray crystal structure determinations. This is due to the fundamental problem that a single Laue diffraction spot can contain reflections from a set of parallel planes with different d/n , where d is the interplanar distance and n is an integer. These spots are multiples instead of singles. This is easily explained by Bragg’s law:

When the broad features of a protein molecular structure become known, structure factors calculated on the basis of this model are generally in rather poor agreement with the observed structure factors. The agreement index between calculated and observed structure factors is usually represented by an R -factor as defined in Equation (13.1). Thus an R -factor of 50% is not uncommon for the starting model, whereas for a random acentric structure it would be 59% (Wilson, 1950).

In the multiple isomorphous replacement method, the phase information from the various heavy atom derivatives and from anomalous scattering is combined by multiplying the individual phase probability curves. If the electron density map, which results from isomorphous replacement, can be fully interpreted, the crystallographer immediately starts with model refinement and the isomorphous phase information is left behind. However, if the electron density map is inadequate for complete interpretation, map improvement (i.e., phase refinement) should precede model refinement. Solvent flattening and the inclusion of molecular averaging are examples of map improvement techniques (Chapter 8). Another way to improve the existing model is by combining the isomorphous replacement phase information with phase information from the known part of the structure. It is clear that a general and convenient way of combining phase information from these various sources would be most useful. Such a method has been proposed by Hendrickson and Lattman (1970) and has been based on previous studies by Rossmann and Blow (1961). Hendrickson and Lattman proposed an exponential form for each individual probability curve of the following type:

After the molecular model of the protein structure has been refined, it might still contain errors that have creeped into the model during the interpretation of the electron density map, particularly in the regions where the electron density is weak. Some of the errors are obvious and should cause immediate suspicion; for instance, the presence of left-handed helices can almost always be ruled out. Most of the available modeling programs allow regularization of geometry, but do not guarantee overall good quality of the final model. A very qualitative impression of the accuracy of the structural model can be obtained by inspection of the electron density map:

There are several excellent books that nicely cover the topic of macromolecular crystallization and we highly recommend consulting and studying these in depth. Therefore, with this chapter we do not provide a summary or overview of these books; rather, we take the reader on a journey from sequence to crystal.