Section 5.1.1. Crystal morphology and optical properties

Morphology is the general study of the overall shape of a crystal, that is, the arrangement of faces of a crystal. It can often provide useful information about the internal symmetry of the arrangement of atoms within the crystal (Mighell et al., 1993). The periodicity of the arrangement of molecules or ions in a crystal can be represented by three non-collinear vectors, a, b and c, which give a unit cell in the form of a parallelepiped with axial edges a, b and c, and interaxial angles α, β and γ (α between b and c, etc.). The vectors a, b and c from the chosen origin of the unit cell are, by convention, selected in a right-handed system. Since there may be several possible choices of unit cell, the simplest, with the smallest possible repeats and with interaxial angles nearest to 90°, is the best choice. One method used to highlight the periodicity of the atomic arrangement within a crystal is to replace each unit cell by a point; this mathematical construction gives the crystal lattice. The entire crystal structure is the convolution of the unit-cell contents with the crystal lattice.

Biological macromolecules are, in general, chiral and can only crystallize in those space groups that do not contain symmetry operations that would convert a left-handed molecule into a right-handed one (improper symmetry operations). Proper symmetry operations involve translations, rotation axes and screw axes. These maintain the chirality of the molecule and hence are appropriate for crystals of biological macromolecules. The number of possible space groups is therefore reduced by this constraint on the types of symmetry operations allowed from the usual 230 for molecules in general down to 65 for chiral molecules.

The appearance of a crystal that has grown under a particular set of experimental conditions is called its habit. It is a result of the different relative growth rates of various crystal faces, and these rates, in turn, depend on the nature of the interactions between the molecules in the crystal, the degree of supersaturation of the solution and the presence of any impurities which may affect the growth rates of specific crystal faces. The term `habit' is only used to describe the various appearances of crystals that are composed of identical material and maintain the same unit-cell dimensions and space group. The faces that have developed on these crystals are various subsets of those implied in the overall morphological description of the crystal. Any change in the experimental conditions under which a crystal is grown may alter its habit; a judicious selection of experimental conditions may permit formation of crystals with a chunky habit that are more suitable for X-ray diffraction analysis than thin plates or needle-like crystals. Examples of the crystalline forms of haemoglobins are provided by Reichert & Brown (1909).

Various descriptions of crystal habits appear in the literature. These include terms such as `tabular', `platy' or `acicular' crystals, `hexagonal rods' and `truncated tetragonal bipyramids', among others. Some crystal habits are not very appropriate for X-ray diffraction analyses; these include spherulites, which are polycrystalline aggregates of fine needles with an approximately radial symmetry, and dendrites, which have a tree-like structure. The habit of a crystal can sometimes give information on the molecular arrangement within it. For example, flat molecules that stack readily upon each other produce long crystalline needles, because interactions in the stacking direction are stronger than those in other directions.

A crystal is bounded by those faces that have grown most slowly. Fast-growing faces quickly disappear as more and more molecules are deposited on them, constrained by surrounding faces that are growing more slowly. Any factor that changes the relative rates of growth of crystal faces, such as impurities in the crystallizing solution, will affect the overall habit. Different faces of protein crystals have different arrangements of side chains on their surfaces; thus, an impurity may bind to certain faces rather than others. Adsorption of an impurity on a particular face of a crystal may retard the growth of that face, causing it to become more prominent than normal in the growing crystal.

Protein and nucleic acid crystals contain a high proportion of water in each unit cell and are therefore fragile. The proportion of solvent to macromolecule in the crystal can be expressed, as described by Matthews (1968), as V_m in ų Da⁻¹ for the asymmetric unit. Values in the range 1.7 to 4.0 are usual for proteins, but nucleic acid crystals generally have a higher water content. Crystal fragility due to water content may be used to determine whether or not a crystal contains protein or buffer salt. Pressure with a fine probe will settle this question because a protein crystal will shatter, while a salt crystal, which is much sturdier, will generally withstand such treatment. If crystals have grown into one another, or appear as clumps, it is sometimes possible to split off a single crystal by prodding the clump gently at the junction point between the crystals with a scalpel or a glass fibre.

Intermolecular contacts between protein molecules in the crystalline state determine the mechanical stability of the crystal. If the conditions used for crystallization vary, the number and identity of these contacts may be changed, and polymorphs will result. Polymorphism is the existence of two or more crystalline forms of a given material. Polymorphs have different unit-cell dimensions and hence different molecular arrangements within them. This property is common for biological macromolecules and can be used to select the best crystalline form for X-ray diffraction studies. Different polymorphs of a particular material are often prepared by varying the crystallization conditions. They may also develop in the same crystallizing drop because supersaturation conditions may change while a crystal is growing. Examples of polymorphs are provided by hen egg-white lysozyme, which can form tetragonal, triclinic, monoclinic or orthorhombic crystals, depending on the pH, temperature and nature of added salts in the crystallization setup (Steinrauf, 1959; Ducruix & Giegé, 1992: Oki et al., 1999).

A regular surface that offers a charge distribution pattern that is complementary to a possible protein layer in a crystal can sometimes be useful in producing a starting point for the nucleation of new protein crystals. Epitaxy is the oriented growth of one material on a crystal of an entirely different material. Regularities on the surface of the first crystal can act as a nucleus for the oriented growth of the second material. Generally, there should be similar, but not necessarily exactly matching, repeat distances in the two crystals. Epitaxy has been used with considerable success for the growth of protein crystals on selected mineral surfaces (McPherson & Shlichta, 1988). For example, lysozyme crystals grow well on the surface of the mineral apophyllite. Crystals of related macromolecules can also be used as nucleation sources for protein crystallization. Epitaxy can, however, sometimes be a nuisance rather than a benefit if the crystallization setup contains surfaces with unwanted regularity.

A change in the environment around a protein crystal may also cause a change in unit-cell dimensions, and possibly even in space group. For example, the transference of a RuBisCO crystal from a high-salt, low-pH mother liquor to a low-salt, high-pH synthetic mother liquor produced a more densely packed polymorph. The overall unit-cell dimensions were smaller in the latter (V_m changed from 3.16 to 2.74 ų Da⁻¹ ) (Zhang & Eisenberg, 1994).

Twinning is a phenomenon that can cause much grief in X-ray diffraction data measurements. It has been described as `a crystal growth anomaly in which the specimen is composed of separate crystal domains whose orientations differ in a specific way … some or all of the lattice directions in the separate domains are parallel' (Yeates, 1997). Thus, a twin consists of two (or more) distinct but coalescent crystals. This effect has been described in terms of their diffraction patterns as follows: `Some crystals show splitting of diffraction spots owing to the different tilts of the two lattices. Others pretend to be single crystals with no split spots, and their symmetries of intensity distribution vary with every data set. These latter have been called hemihedral, and in them the unique axes of the two crystals are exactly reversed parallel with each other.' (Igarashi et al., 1997). Perfect hemihedral twinning (when there are equal proportions of each twin member) can be detected from the value of for the acentric data; it is near 2 for untwinned data and 1.5 for twinned data (Yeates, 1997).

Twinning may sometimes be prevented by introducing variations into the crystallization setup. Such changes could involve the pH or the nature of the buffer. Variations in the seeding technique used, or the introduction of additional agents, such as metal ions or salts, detergents, or certain amino acids, can also be tried. One method for estimating the degree of twinning is based on the fact that each measured X-ray diffraction intensity is the sum of the intensities from the two (or more) crystal lattices (suitably weighted according to the proportion in which each lattice alignment occurs in the crystal). The relative proportion of each component can therefore be estimated. Detwinned intensities obtained by this method should only be positive or zero (not negative) within experimental error (Stanley, 1972; Britton, 1972; Rees, 1980). Crystal structures of hemihedrally twinned crystals are now being determined (Gomis-Rüth et al., 1995; Breyer et al., 1999).

Crystal faces are described by three numbers, the Miller indices, that define the relative positions at which the three axes of the unit cell are intercepted by the crystal face (Blundell & Johnson, 1976; Phillips, 1957). The crystal face that is designated hkl makes intercepts , and with the three axes (a, b and c) of the unit cell. When the Miller indices are negative, because the crystal face intercepts an axis in a negative direction, they are designated or . Most faces of a crystal are, as required by the law of rational indices, represented by small integers. For symmetry reasons, planes in hexagonal crystals are conveniently described by four axes, three in a plane at 120° to each other. This leads to four indices, hkil, where . Crystal faces are designated by round brackets, e.g. (100), as shown in two simple examples in Fig. 5.1.1.1. A set of faces then defines a crystal form. All sets of planes hkl related by symmetry, such as the main faces of an octahedron (111), (), (), (), (), (), () and (), may be represented by the use of curly brackets, e.g. . Square brackets, e.g. [111], are used to indicate a direction in a crystal, [001] being the c axial direction. Thus, information about several faces of a crystal is contained in information about one face if the relevant symmetry of the crystal is known. If the atomic structure of the crystal is known, it is possible to determine which aspects of the macromolecule lie on the various crystal faces.

Figure 5.1.1.1 | top | pdf | Crystal faces. (a) Cube and (b) octahedron. (c) Unit-cell axes.

The habit of a crystal may be described in detail by measurements, particularly of the angles between adjacent faces. This can be done with an optical goniometer, in which a collimated visible beam of light is reflected from different faces as the crystal is rotated, and the angles between positions of high light intensity are noted. Such studies can be combined with X-ray diffraction measurements of the crystal orientation of the unit cell with respect to the various crystal faces (Oki et al., 1999).

Most protein crystals are birefringent and are brightly coloured in polarized light. In order to view these effects, crystals (in their mother liquor) are set on a microscope stage with a Nicol prism (polarizing material) between the light source and the microscope slide (the polarizer). Another Nicol prism is set between the crystal and the eyepiece (the analyser). The crystal should not be in a plastic container, since this would produce too many colours. If the vibration plane of the analyser is set perpendicular to that of the polarizer (to give `crossed Nicols'), no light will pass through in the absence of crystals, and the background will be dark. If the crystal is isotropic, the image will remain dark as the crossed Nicol prisms are rotated. If, however, the crystal is birefringent (with two refractive indices), the crystal will appear coloured except at four rotation positions (90° apart) of the crossed Nicol prisms, where the crystal and background will be dark (extinguished). At these positions, the vibration directions of the Nicol prisms coincide with those of the crystal. If one is looking exactly down a symmetry axis of a crystal that is centrosymmetric in projection (such as a tetragonal or hexagonal crystal), the crystal will not appear birefringent, but dark. By noting the external morphology of the crystal with respect to its angle of rotation, one can often deduce the directions of the unit-cell axes in the crystal (Hartshorne & Stuart, 1960). Examination of a crystal under crossed Nicol prisms can also provide information on crystal quality. For example, sometimes the components of a twinned crystal extinguish plane-polarized light independently. Other methods of examining crystals include Raman spectroscopy (Kudryavtsev et al., 1998).

The refractive index of a crystal can be measured by immersing it in a mixture of liquids of a known refractive index in which the crystal is insoluble. The liquid composition is then varied until the crystal appears invisible. At this point, the refractive indices of the crystal and the liquid are the same. If the refractive index is the same in all directions, the crystal is optically isotropic, but most protein crystals are optically anisotropic and have more than one refractive index. For example, tetragonal crystals have different refractive indices for light vibrating parallel to the fourfold axis and for light vibrating perpendicular to it. These refractive indices are measured by the use of plane-polarized light.