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Students new to the protein X-ray crystallography laboratory migt understandably be confused when colleagues discuss Fouriers and Pattersons or molecular replacement and molecular dynamics refinement. However, they understand immediately that the first requirement for protein structure determination is to grow suitable crystals. Without crystals there can be no X-ray structure determination of a protein! In this chapter, we discuss the principles of protein crystal growth, and as an exercise, we give the recipe for crystallizing the enzyme lysozyme. We will also generate an X-ray diffraction picture of a lysozyme crystal. This will provide an introduction to X-ray diffraction. The chapter concludes with a discussion of the problems encountered.

In Chapter 1 you learned how crystals of a protein can be grown and you observed a diffraction pattern. The crystalline form of a protein is required to determine the protein’s structure by X-ray diffraction, but equally necessary are the tools for recording the diffraction pattern. These will be described in this chapter on hardware. The various X-ray sources and their special properties are discussed, followed by a description of cameras and detectors for quantitative and qualitative X-ray data collection.

The beauty and regularity of crystals impressed people to such an extent that, in the past, crystals were regarded as products of nature with mysterious properties. Scientific investigation of crystals started in 1669, when Nicolaus Steno, a Dane working as a court physician in Tuscan, proposed that during crystal growth, the angles between the faces remained constant . For a given crystal form, individual crystals might differ in shape (i.e, in the development of their faces), but they always have identical angles between the same faces (Figure 3.1). The specific morphology might depend on factors such as the supply of material during growth, on the presence of certain substances in the mother liquor, or on the mother liquor itself. For a single crystal form, the angles between the faces are constant, but this is not true if the crystals belong to different crystal forms. Figure 3.2 shows four different crystal forms of deoxyhemoglobin from the sea lamprey Petromyzon marinus . Their appearance depends on the buffer and on the precipitating agent, although, occasionally, two different forms appear under the same conditions.

The best way to learn protein X-ray diffraction is by practical work in the laboratory. However, it would be very unsatisfying to perform the experiments without understanding why they have to be done in such and such a way. Moreover, at several stages in the determination of protein structures, it is necessary to decide what the next step should be. For instance, after growing suitable crystals and soaking these crystals in solutions of heavy atom reagents, applying the isomorphous replacement method, how do you obtain the positions of the heavy atoms in the unit cell and, if you do have them, how do you proceed? Questions such as these can be answered only if you have some knowledge of the theoretical background of protein X-ray crystallography. This is presented in this chapter. A slow path will be followed, and a student with a minimal background in mathematics but the desire to understand protein X-ray crystallography should be able to work through the chapter. A working knowledge of differentiation and integration is required. If you further accept that an X-ray beam can be regarded as a wave that travels as a cosine function and if you know what a vector is, you have a good start. Derivations and explanations that are not absolutely necessary to follow the text are set off within rules; these can be skipped, if you want.

A quick glance through this chapter indicates that it is short but that it is mainly of a mathematical nature. However, it is not as difficult as it seems.

In this chapter some special forms of the structure factor will be presented. It is not essential reading to understand the following chapters, but it does provide an introduction to and definitions of unitary structure factors and normalized structure factors . The chapter is put in this position in the book because the material can be easily understood using the results presented in the section on the Wilson plot (Section 5.2 of Chapter 5). However, if this is your first introduction to protein X-ray crystallography, you can skip this chapter for the time being.

As we have seen in Chapter 4, the electron density in a crystal can be obtained by calculating the Fourier summation:

After a first set of protein phases is obtained using the isomorphous replacement method, the molecular replacement method, or the single- or multiple-wavelength anomalous diffraction method and an electron density map is calculated, the next step is the interpretation of the map in terms of the polypeptide chain. If this is successful and the major part of the chain can, indeed, be followed in the electron density map, refinement of the structure can begin. However, insufficient quality of the electron density map might hamper a complete and unambiguous tracing of the polypeptide chain, increasing the risk of introducing errors in the model, which cannot be easily removed during refinement. In such a case, refinement should be preceded by a process to improve the quality of the map through improvement of the protein phase angles (Podjarny et al., 1987). During phase improvement, all available information on the structure should be used (Brünger and Nilges, 1993). This information might be in one of the following forms:

Anomalous scattering is not a new subject. It was already introduced in Chapter 7. There, you learned that anomalous scattering by an atom is due to the fact that its electrons cannot be regarded as completely free electrons. This effect depends on the wavelength, but it is, in general stronger, for the heavier atoms than for the light atoms in the periodic system.

With the isomorphous replacement method or with one of the anomalous diffraction techniques, a preliminary set of protein phase angles and a first model of the protein structure can be obtained. As we will see in Chapter 13, such a model can be refined by minimizing the difference between the observed | F | values and the | F | values calculated from the model. An easier way to obtain a first model can be followed if the structure of a protein with a homologous amino acid sequence has already been established. The structure of this homologous protein is—as it were—borrowed by the protein for which the structure must be determined and serves as a very first model that can subsequently be refined. This procedure is based on the observation that proteins, homologous in their amino acid sequence, have a very similar folding of their polypeptide chain. Also, if, for another reason, two structure can be expected to be similar, one known and the other unknown, the procedure can be applied.