Not all the diffracted photons from a crystal end up in the Bragg reflections from specified (hkl) planes. Indeed, for quite a large number of macromolecular crystals the non-Bragg diffraction or diffuse scattering is strong in intensity. The diffuse scattering is due to a breakdown in the periodicity of the crystal and carries information on the mobility and flexibility of the molecules in the crystal. There are text-books describing diffuse scattering from small molecule crystals such as Amorós and Amorós (1968) and Wooster (1962).

At the synchrotron the long exposure times used in the measurement of the high resolution Bragg data on film or IP also automatically give the diffuse scattering and reveal a diversity of diffuse background patterns from different crystals. These observations have stimulated considerable interest in trying to understand and interpret these features. Of great interest to the molecular biologist is the relationship between macromolecular structure and function. Recent years have shown that besides the static/time-averaged structural information, appreciation of the molecular flexibility and dynamics is essential. Usually this information has been derived from the crystallographic atomic thermal parameters and also from molecular dynamics simulations (see, e.g., McCammon (1984)) which yield individual atomic trajectories. A characteristic feature of macromolecular crystals compared to small molecule crystals, however, is that their diffraction patterns extend to quite limited resolution even employing SR.

This appendix is based, with permission, on my sections in the new International Tables for Crystallography, Volume C (editor A. J. C. Wilson, 1991).

There are text-books which concentrate on almost every diffraction geometry. References to these books are given in the respective sections in the following pages. However, in addition, there are several books which contain details of diffraction geometry. Blundell and Johnson (1976) described the use of the various diffraction geometries with the examples taken from protein crystallography. There is an extensive discussion and many practical details to be found in the text-books of Stout and Jensen (1968, 1989), Woolfson (1970), Glusker and Trueblood (1971, 1985), Vainshtein (1981) and McKie and McKie (1986), for example. A collection of early papers on the diffraction of X-rays by crystals involving, inter alia, experimental techniques and diffraction geometry, can be found in Bijvoet, Burgers and Hägg (1969, 1972). A collection of recent papers on primarily protein and virus crystal data collection via the rotation film method and diffractometry can be found in Wycoff, Hirs and Timasheff (1985); detailed references are also made to this volume later.

In this appendix which deals with monochromatic methods, the convention is adopted that the Ewald sphere takes a radius of unity and the magnitude of the reciprocal lattice vector is λ/d.

EARLY HISTORY AND GENERAL INTRODUCTION

The first discussions in the literature concerning the applications of SR in protein crystallography were given by Harrison (1973), Wyckoff (1973) and Holmes (1974). The first experimental tests were made on SPEAR by Webb et al (1976, 1977) and reported by Phillips et al (1976, 1977); precession photographs of protein crystals were obtained with a 60-fold reduction in exposure times over a home laboratory X-ray source (in this case a conventional fine focus Cu Kα tube running at 1200W) and test data were collected about the iron K edge for rubredoxin and the copper K edge for azurin. The azurin crystal suffered much less from radiation damage in the intense beam than during a longer equivalent exposure on a conventional source. This was the first indication that radiation damage to a protein crystal was less with a more intense X-ray source (figure 10.1). The anomalous dispersion effects using the Fe K edge enabled phases to be determined for rubredoxin with a mean figure of merit of 0.5 (mean phase error of 60°). The anomalous dispersion effects using the Cu K edge were used to confirm the copper sites in azurin utilising phases determined from conventional source data (Adman, Stenkemp, Sieker and Jensen 1978).

X-rays are used to probe the atomic or molecular structure of matter because the wavelength of the radiation is of approximately the same dimension as an atom. Similarly longer wavelength visible light is appropriate for studying larger structures, e.g. cell organelles. However, since there is no known X-ray lens the equivalent function of a glass lens for visible light in a conventional microscope has to be performed by computational transformation of X-ray diffraction patterns.

The basic steps in a macromolecular crystal structure analysis involve:

1. (i) crystallisation;

2. (ii) space group and cell parameter determination;

3. (iii) data collection;

4. (iv) phase determination;

5. (v) electron density map interpretation;

6. (vi) refinement of the molecular model.

Figure 2.1 (a)—(f) illustrates some of these steps showing, as an example, the structure determination of human erythrocyte purine nucleoside phosphorylase (PNP) (Ealick et al (1990)). A list of general texts on crystallography is given in the bibliography, section 1.

CRYSTALLISATION, CRYSTALS AND CRYSTAL PERFECTION, SYMMETRY

Crystallisation is a process involving precipitation of the dissolved protein from solution. This is achieved by decreasing the protein solubility, decreasing any repulsive forces between individual protein molecules and/or increasing the attractive forces. The crystals that might be produced need to be of ‘X-ray diffraction quality’.

The use of focussed, monochromatised radiation at the synchrotron has so far yielded the most results in terms of biological molecular structure compared with the other methods being developed. This is readily explained because of the ease with which the monochromatic diffraction data measured at the synchrotron have been processed with existing computer programs for data from monochromatic, emission line, laboratory X-ray sources. In contrast, the Laue method, although it is being very actively developed at the synchrotron (chapter 7), had been abandoned in the home laboratory. Hence, the monochromatic method is covered first in this book. In appendix 1 details are given of the various monochromatic diffraction geometries. These geometries are:

1. (a) monochromatic still exposure;

2. (b) rotation/oscillation geometry;

3. (c) Weissenberg geometry;

4. (d) precession geometry;

5. (e) diffractometry.

Quantitative X-ray crystal structure analysis usually involved methods (b), (c) and (e) although (d) has certainly been used. Photographic film is being replaced by use of electronic area detectors or, even more recently, the IP.

At the various synchrotrons all these geometries have been exploited for macromolecular crystal data collection as they have also on conventional X-ray sources. Once the polychromatic synchrotron X-ray beam has been rendered monochromatic the single crystal data can be measured and processed as for a conventional X-ray source.

The systems to which thermodynamics have been applied have become more and more complex. The analysis and understanding of these systems requires a knowledge and understanding of the methods of applying thermodynamics to multiphase, multicomponent systems. This book is an attempt to fill the need for a monograph in this area.

The concept for this book was developed during several years of teaching a one-year advanced graduate course in chemical thermodynamics at the Illinois Institute of Technology. Students who took the course were studying chemistry, chemical engineering, gas technology, or biochemistry. During those years we came to believe that the major difficulty that students have is not with the numerical solution of a problem; the difficulty is with the development of the pertinent relations in terms of experimentally determinable quantities. Moreover, during the initial writing of the book, it became evident that chemical thermodynamics was being applied in many new fields and to systems having more than two or three components. These new fields are so numerous that any attempt to illustrate the application of thermodynamics to each of them would make this book much too long. Therefore, the aim of the book is to develop in a general way the concepts and relations that are pertinent to the solution of many thermodynamic problems encountered in multiphase, multicomponent systems. The emphasis is on obtaining exact expressions in terms of experimentally determinable quantities. Simplifying assumptions can be made as necessary after the exact expression has been obtained.

The conditions of equilibrium expressed by Equations (5.25)–(5.29) and (5.46) involve the temperature, pressure, and chemical potentials of the components or species. The chemical potentials are functions of the temperature, pressure or volume, and composition, according to Equations (5.54) and (5.56). In order to study the equilibrium properties of systems in terms of these experimentally observable variables, expressions for the chemical potentials in terms of these variables must be obtained. This problem is considered in this chapter and in Chapter 8.

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature–pressure–volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities: the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty.