Experimental Design, Quantitative Analysis, and the Cartography of Crystal Growth

This chapter is about practical uses of mathematical models to simplify the task of finding the best conditions under which to crystallize a macromolecule. The models describe a system’s response to changes in the independent variables under experimental control. Such a mathematical description is a surface, whose two-dimensional projections can be plotted, so it is usually called a ‘response surface’. Various methods have been described for navigating an unknown surface. They share important characteristics: experiments performed at different levels of the independent variables are scored quantitatively, and fitted implicitly or explicitly, to some model for system behaviour. Initially, one examines behaviour on a coarse grid, seeking approximate indications for multiple crystal forms and identifying important experimental variables. Later, individual locations on the surface are mapped in greater detail to optimize conditions. Finding ‘winning combinations’ for crystal growth can be approached successively with increasingly well-defined protocols and with greater confidence. Whether it is used explicitly or more intuitively, the idea of a response surface underlies the experimental investigation of all multivariate processes, like crystal growth, where one hopes to find a ‘best’ set of conditions. The optimization process is illustrated schematically in Figure 1. In general, there are three stages to this quantitative approach: (a) Design. One must first induce variation in some desired experimental result by changing the experimental conditions. Experiments are performed according to a plan or design. Decisions must be made concerning the experimental variables and how to sample them. (b) Experiments and scores. Each experiment provides an estimate for how the system behaves at the corresponding point in the experimental space. When these estimates are examined together as a group, patterns often appear. For example, a crystal polymorphism may occur only in restricted regions of the variable space explored by the experiment. (c) Fitting and testing models. Imposing a mathematical model onto such patterns provides a way to predict how the system will behave at points where there were no experiments. The better the predictions, the better the model. Adequate models provide accurate interpolation within the range of experimental variables originally sampled; occasionally a very good model will correctly predict behaviour outside it (1).

Molecular Biology for Structural Biology

The number of published 3D structures has increased exponentially in the last decade and the resulting mass of structural data has contributed significantly to the understanding of mechanisms underlying the biology of living cells. However, these mechanisms are so complex that structural biologists face still greater challenges, such as the study of higher-order functional complexes. As an example, we can mention the protein complexes that assemble around activated growth factor receptors to allow the transduction of extracellular signals through the membrane and inside the cell (1). Because of their diverse intrinsic properties, proteins exhibit variable difficulty for structural biology studies. Before the rise of recombinant expression methods, only a minority of protein structures were determined, representing mainly favourable cases: proteins of high abundance in their natural source which could be purified and crystallized, in contrast to rare proteins that were often refractory to crystallization. The advent of methods for recombinant protein overexpression was a breakthrough in this area. It was followed by an increasing number of publications describing the crystallization of proteins, not under their native form, but in modified versions after sequence engineering. First we will consider the classical use of molecular biology applied to optimize the expression system for a recombinant protein for structural biology, without modification of its sequence. In the second part, we will deal with molecular biology procedures aimed at engineering the properties of a protein through sequence modifications in order to make its crystallization possible. In the last part we will give an example where molecular biology can help solve a crystallographic problem, namely that of phase determination by introducing anomalous scatterers (e.g. selenium atoms) into the protein of interest. Whenever extraction of a protein from its natural source appears unsuitable for structural studies, molecular biology resources can be brought in, initially aiming at choosing and setting up an appropriate expression system. This initial approach could involve comparing various expression hosts and vectors and deciding if the protein is to be produced as a fusion to facilitate its purification.

X-ray analysis

This chapter covers the preliminary characterization of the crystals in order to determine if they are suitable for a full structure determination. Probably more frustrating than failure to produce crystals at all, is the growth of beautiful crystals which do not diffract, which have very large unit cell dimensions, or which decay very rapidly in the X-ray beam, though this last problem has been largely overcome by freezing the sample. It is impossible in one brief chapter to give more than a flavour of what the X-ray crystallographic technique entails and it is assumed that the protein chemist growing the crystals will have contact with a protein crystallographer, who will carry out the actual structure determination and in whose laboratory state-of-the-art facilities exist. However, preliminary characterization can often be carried out with little more than the equipment which is widely available in Chemistry and Physics Departments and so the crystal grower remote from a protein crystallography laboratory can monitor the success of their experiments. The reader should refer to the first edition for protocols useful for photographic characterization but such techniques are seldom used nowadays. It must be remembered, in any case, that X-rays are dangerous and the inexperienced should not try to X-ray protein crystals without help. It is necessary to provide an overview of X-ray crystallography, to put the preliminary characterization in context. For a general description of the technique the reader should refer to Glusker et al. (1) or Stout and Jensen (2). For protein crystallography in particular, the books by McRee (3) and Drenth (4) describe many of the advances since the seminal work of Blundell and Johnson (5). Amongst many excellent introductory articles, those by Bragg (6), published years ago, and Glusker (7) are particularly recommended. The scattering or diffraction of X-rays is an interference phenomenon and the interference between the X-rays scattered from the atoms in the structure produces significant changes in the observed diffraction in different directions. This variation in intensity with direction arises because the path differences taken by the scattered X-ray beams are of the same magnitude as the separation of the atoms in the molecule.

Crystallization of Membrane Proteins

Crystallization of membrane proteins is one of the most recent developments in protein crystal growth; in 1980, for the first time, two membrane proteins were successfully crystallized, bacteriorhodopsin (1) and porin (2). Since then, a number of membrane proteins (about 30) yielded three-dimensional crystals. In several cases, the quality of the crystals was sufficient for X-ray diffraction studies. The first atomic structure of a membrane protein, a photosynthetic bacterial reaction centre, was described in 1985 (3), followed by the structure of about ten other membrane protein families. Crystallization of membrane proteins is now an actively growing field, and has been discussed in several recent reviews (4-8). The major difficulty in the study of membrane proteins, which for years hampered their crystallization, comes from their peculiar solubility properties. These originate from their tight association with other membrane components, particularly lipids. Indeed integral membrane proteins contain hydrophobic surface regions buried in the lipid bilayer core, as well as hydrophilic regions with charged or polar residues more or less exposed at the external faces of the membrane. Disruption of the bilayer for isolating a membrane protein can be done in various ways: extraction with organic solvents, use of chaotropic agents, or solubilization by a detergent. The last method is the most frequently used, since it maintains the biological activity of the protein if a suitable detergent is found. This chapter will be restricted to specific aspects of three-dimensional crystallizations done in micellar solutions of detergent. In some cases, it is possible to separate soluble domains from the membrane protein either by limited proteolysis or by genetic engineering. Such protein fragments can then be treated as soluble proteins and so will not be discussed further in this chapter. We refer to Chapter 12 and the review by Kühlbrandt (9) for the methodology of two-dimensional crystallization used for electron diffraction. The general principles discussed in this book for the crystallization of soluble biological macromolecules apply for membrane proteins; the protein solution must be brought to supersaturation by modifying its physical parameters (concentrations of constituents, ionic strength, and so on), so that nucleation may occur.

Two-Dimensional Crystallization of Soluble Proteins on Planar Lipid Films

Electron crystallography of protein two-dimensional (2D) crystals constitutes a fast-expanding method for determining the structure of macromolecules at near-atomic resolution (1, 2). The main limitation in the application and generalization of this approach remains in obtaining highly ordered 2D crystals, as is the case of 3D crystals in X-ray crystallography. Several methods of 2D crystallization are available which can be classified into two families, depending on the type of proteins under investigation, either membrane proteins (3, 4) or soluble proteins (5, 6). In both cases, 2D crystallization is a self-organization process which spontaneously occurs between macromolecules which are restricted to diffusing by translation and rotation in a 2D space, with a fixed orientation along the normal to this plane. The scope of this chapter is restricted to the 2D crystallization of soluble proteins on planar lipid films, by the so-called ‘lipid monlayer crystallization method’ (5). Our aim is to present a step-by-step description of the experimental procedures involved in the application of this method. The method of protein 2D crystallization on planar lipid films was introduced about 15 years ago (5) and has since been successfully applied to about 30 proteins. Its principle is based on the specific interaction between soluble proteins and lipid ligands inserted in a lipid monolayer, at an air-water interface. In practice, a lipid monolayer is formed by spreading lipids dissolved in an organic solvent on a water surface. Proteins present in the aqueous subphase bind to their ligand of lipidic nature and spontaneously form 2D domains and, in favourable cases, 2D crystals. The process of 2D crystal formation relies on three successive steps: (a) Molecular recognition between a protein and its ligand. (b) Diffusion and concentration of the protein-lipid complexes in the plane of the lipid film. (c) Self-organization of the proteins into 2D crystals. As indicated in Table 1, three different types of systems can be distinguished, depending on the nature of the lipid ligand: • natural lipids • synthetic lipids made of a protein ligand coupled to a lipid molecule • charged lipids.

Diagnostic of Pre-Nucleation and Nucleation By Spectroscopic Methods and Background on the Physics of Crystal Growth

Unlike the crystallization of small inorganic molecules, the problem of protein crystallization was first approached by trial and error methods without any theoretical background. A physico-chemical approach was chosen because crystallographers and biochemists needed criteria to rationally select crystallization conditions. In fact, the problem of the production of homogeneous and structurally perfect protein crystals is set the same as the production of high-quality crystals for opto-electronic applications, because, in both cases, the crystal growth mechanisms are the same. Biological macromolecules and small organic molecules follow the same rules concerning crystallization even if each material exhibits specific characteristics. This chapter introduces the fundamentals of crystallization: supersaturation, nucleation, and crystal growth mechanisms. Phase diagrams are presented in Chapter 10. Special attention will be paid to the behaviour of the macromolecules in solution and to the techniques used for their analysis: light scattering (LS), small angle X-ray scattering (SAXS), small angle neutron scattering (SANS), and osmotic pressure (OP). Before obtaining any nucleation or growth, it is necessary to dissolve the biological macromolecules under consideration in some good solvent. However, it may immediately be asked whether a good solvent is a solvent in which the material is highly soluble, or in which nucleation is easily controlled, or in which growth is fast, or solvent in which the crystals exhibit the appropriate morphology. In practice, the choice of the solvent often depends on the nature of the material to be dissolved, taking into account the well known rule which says that ‘like dissolves like’. This means that, for dissolution to occur, it is necessary that the solute and the solvent exchange bonds: between an ion and a dipole, a dipole and another dipole, hydrogen bonds, and/or Van der Waals bonds. Therefore, the nature of the bonds depends on both the nature of the solute and the solvent which can be dipolar protic, dipolar aprotic, or completely apolar. Once the material has dissolved, the solution must be supersaturated in order to observe nucleation or growth. The solution is supersaturated when the solute concentration exceeds its solubility. There are several ways to achieve supersaturation.

From Solution to Crystals With a Physico-Chemical Aspect

Biological macromolecules follow the same thermodynamic rules as inorganic or organic small molecules concerning supersaturation, nucleation, and crystal growth (1). Nevertheless macromolecules present particularities, because the intramolecular interactions responsible of their tertiary structure, the intermolecular interactions involved in the crystal contacts, and the interactions necessary to solubilize them in a solvent are similar. Therefore these different interactions may become competitive with each other. In addition, the biological properties of biological macromolecules may be conserved although the physico-chemical properties, such as the net charge, may change depending on the crystallization conditions (pH, ionic strength, etc.). A charged biological macromolecule requires counterions to maintain the electroneutrality of the solution; therefore it should be considered as a protein (or nucleic acid) salt with its own physico-chemical properties, depending on the nature of the counterions. To crystallize a biological macromolecule, its solution must have reached supersaturation which is the driving force for crystal growth. The understanding of the influence of the crystallization parameters on protein solubility of model proteins is necessary to guide the preparation of crystals of new proteins and their manipulation. Only the practical issues are developed in this chapter, and the reader should refer to recent reviews (2-4) for a description of the fundamental physical chemistry underlying crystallogenesis. The solubilization of a solute (e.g. a biological macromolecule) in an efficient solvent requires solvent-solute interactions, which must be similar to the solvent-solvent interactions and to the solute-solute interactions of the compound to be dissolved. All of the compounds of a protein solution (protein, water, buffer, crystallizing agents, and others) interact with each other via various, often weak, types of interactions: monopole-monopole, monopole-dipole, dipole-dipole, Van der Waals hydrophobic interactions, and hydrogen bonds. Solubility is defined as the amount of solute dissolved in a solution in equilibrium with its crystal form at a given temperature. For example, crystalline ammonium sulfate dissolves at 25°C until its concentration reaches 4.1 moles per litre of water, the excess remaining non-dissolved. More salt can be dissolved when raising the temperature, but if the temperature is brought back to 25°C, the solution becomes supersaturated, and the excess of salt crystallizes until its concentration reaches again its solubility value at 25°C (4.1 moles per litre of water).

Nucleic Acids and Their Complexes

At first glance crystallizing nucleic acids poses the same problems as crystallizing proteins since most of the variables to investigate are alike. It is thus astonishing that crystallization data banks (1) that describe so many successful protein crystallizations are so poor in information on nucleic acids. This relies on the physico-chemical and biochemical characteristics of nucleic acids distinguishing them from proteins. The aim of this chapter is to underline features explaining the difficulties often encountered in nucleic acid crystallization and to discuss strategies that could help to crystallize them more readily, either as free molecules or as complexes with proteins. Other general principles, in particular for RNA crystallization, are discussed in ref. 2. Among natural nucleic acids only the smaller ones provide good candidates for successful crystallizations. Large DNAs or RNAs can a priori be excluded because of their flexibility that generates conformational heterogeneity not compatible with crystallization. Thus the smaller RNAs with more compact structures (with 75-120 nt), especially transfer RNAs (tRNAs), but also 5S RNA, were the first natural nucleic acids to be crystallized (3, 4). At present attempts are being made with other RNA systems, such as ribozymes and introns, fragments of mRNA, viroids, viral and other tRNA-like RNAs, SELEX-evolved RNAs, and crystallization successes leading to X-ray structure determinations were reported for RNA domains of up to 160 nt long, with the resolution of the P4-P6 domain of the self-splicing Tetrahymena intron (5). The recent excitement in nucleic acid crystallography, and particularly in RNA crystallography, have partly been due to technological improvements in the preparation methods of the molecules. Advances in oligonucleotide chemical synthesis provide opportunity for making large amounts of pure desoxyribo- and more recently of ribo-oligomers of any desired sequence. This led to the crystallization of a number of DNA and RNA fragments and was followed by the co-crystallization of complexes between proteins and such synthetic fragments. Transcription methods of RNAs from synthetic DNA templates were also essential for rejuvenating the structural biology of RNAs. In the case of complexes of proteins with RNAs, the main difficulty was to purify large quantities of homogeneous biological material with well defined physico-chemical properties.

Seeding Techniques

A seed provides a template for the assembly of molecules to form a crystal with the same characteristics as the crystal from which it originated. Seeding has often been used as a method of last resort, rather than a standard practice. Recently, these techniques have gained popularity, in particular, macroseeding, used to enlarge the size of crystals. Seeding has many more applications, and the use of seeding in crystallization can simplify the task of the crystallographer even when crystals can be obtained without it. We will explore the various seeding techniques, and their applications, in the growth of large single crystals and the methods by which we may attempt to obtain crystals that diffract to higher resolution. Crystallogenesis can be divided into two separate phases. The first being the screening of crystallization conditions to obtain the first crystals, the second consisting of the optimization of these conditions to improve crystal size and quality. Seeding can be used advantageously in both these situations. The first stage in crystallogenesis consists of the discovery of initial crystals, crystalline aggregates, or microcrystalline precipitate. This may result from a standardized screening method (1, 2), a systematic method (3), an incomplete factorial search (see Chapter 4 and refs 4 and 5), or by extensive screening of many conditions. This may be bypassed by starting with seeds from crystals of a related molecule that has been previously crystallized. Molecules that have been obtained by genetic or molecular engineering of a previously crystallized macromolecule fall in this category. This method is termed cross-seeding. It has been used to obtain crystals of pig aspartate aminotransferase starting with crystal from the chicken enzyme (6) and between native and complexed Fab molecules (7). Whatever the method used to obtain the initial crystals, seeding may provide a fast and effective way to facilitate the optimization of growth conditions without the uncertainty which is intrinsic in the process of spontaneous nucleation. The streak seeding technique can be used to carry out a search quickly and efficiently over a wide range of growth conditions.

Soaking Techniques

Once crystals of a macromolecule are obtained there are many circumstances where it is necessary to change the environment in which the macromolecule is bathed. Such changes include the addition of inhibitors, activators, substrates, products, cryo-protectants, and heavy atoms to the bathing solution to achieve their binding to the macromolecule, which may have sufficient freedom to undergo some conformational changes in response to these effectors. In fact, macromolecular crystals have typically a high solvent content which ranges from 27-95% (1, 2). Although, part of this solvent, ‘bound solvent’ (typically 10%) is tightly associated with the protein matrix consisting of both water molecules and other ions that occupy well defined positions in refined crystal structure it can be replaced in soaking experiments, at a slower rate compared to the ‘free solvent’. In this chapter we will consider the relative merits of various methods for modifying crystals, the restraints that the lattice may impose on the macromolecule, and the relative merits of soaking compared to co-crystallization. The size and configuration of the channels within the lattice of macromolecular crystals will determine the maximum size of the solute molecules that may diffuse in. The solvent channels are sufficiently large to allow for the diffusion of most small molecules to any part of the surface of the macromolecule accessible in solution except for the regions involved in crystal contacts, although in some cases lattice forces may hinder conformational changes or rearrangements of the macromolecule in crystal. In other cases, the forces that drive the conformational changes can be sufficient to overcome the constraints imposed by the crystalline lattice leading to the disruption of intermolecular and crystal contacts resulting in the cracking and dissolution of the crystals. Some lattices may be more flexible and capable of accommodating conformational changes, and while crystals may crack initially, they may subsequently anneal into a new rearrangement and occasionally improve their crystallinity. In general small changes are easily accommodated and many macromolecules maintain their activity in the crystalline state. This is exploited in time-resolved crystallography to obtain structural information of transition states of enzymes.