X-ray crystallography

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X-ray crystallography is the science of determining the arrangement of atoms within a crystal from the manner in which a beam of X-rays is deflected by the crystal. The method produces a three-dimensional picture of the atoms within the crystal, from which their chemical bonds and other information can be derived. By definition, a crystal is a solid in which a particular arrangement of atoms (its unit cell) is repeated endlessly in all directions. Crystals can be composed of any solid material, such as ionic salts, metals, minerals, semiconductors, as well as various inorganic, organic and biological molecules.

The oldest and most precise method of X-ray crystallography is single-crystal X-ray diffraction, in which a beam of X-rays is reflected from evenly spaced planes of a single crystal, producing a diffraction pattern of spots called reflections. The arrangement of atoms within the crystal is determined from the position and brightness of the various reflections observed as the crystal is gradually rotated in the X-ray beam, together with supplementary data. An analogous diffraction pattern may be observed by shining a laser pointer on a compact disc or DVD; the periodic spacing of the CD tracks correspond to the periodic arrangement of atoms in a crystal. For single crystals of sufficient purity and regularity, X-ray diffraction data can determine the positions of most atoms in a crystal structure to within a few tenths of an Ångström. Such crystal structures are useful in characterizing new materials and in discerning materials that appear similar by all other means. However, such structures have a vast array of other scientific and medical applications; for example, they can account for unusual electronic or elastic properties of a material, shed light on chemical interactions and processes, or serve as the basis for designing pharmaceuticals against diseases.

The technique of X-ray crystallography has three basic steps. The first — and often most difficult — step is to obtain an adequate crystal of the material under study. The crystal should be sufficiently large, pure in composition and regular in structure, with no large internal imperfections such as cracks or twinning. In the second step, the crystal is placed in an intense beam of X-rays, usually of a single wavelength (monochromatic X-rays), producing the regular array of reflections. As the crystal is gradually rotated, previous reflections disappear and new ones appear; the intensity of every spot is recorded meticulously at every orientation of the crystal. Multiple data sets may have to be collected, with each set covering slightly more than half a full rotation of the crystal and containing tens of thousands of reflection intensities. In the third step, these data are combined computationally with complementary chemical information to produce and refine a model of the arrangement of atoms within the crystal. The final, refined model — now called a crystal structure — is usually stored in a public database.

As the crystal's unit cell becomes larger, the atomic-level picture provided by X-ray crystallography becomes less well-resolved (more "fuzzy") for a given number of observed reflections. Two limiting cases of X-ray crystallography are often discerned, "small-molecule" and "macromolecular" crystallography. Small-molecule crystallography typically involves crystals with fewer than 100 atoms in their asymmetric unit; such crystal structures are usually so well resolved that its atoms can be discerned as isolated "blobs" of electron density. By contrast, macromolecular crystallography often involves tens of thousands of atoms in the unit cell. Such crystal structures are generally less well-resolved (more "smeared out"); the atoms and chemical bonds appear as tubes of electron density, rather than as isolated atoms. In general, small molecules are also easier to crystallize than macromolecules; however, X-ray crystallography has proven possible even for viruses with hundreds of thousands of atoms.

The term "X-ray crystallography" is also sometimes applied to methods that involve X-ray diffraction from polycrystalline materials, such as powders of small crystals studied by X-ray powder diffraction. X-ray diffraction is just one type of crystal diffraction; other types, such as electron diffraction and neutron diffraction, give similar information but have had technical limitations that, until recently, made their use more restricted than X-ray diffraction. Similarly, X-ray crystallography belongs to a large family of X-ray scattering techniques, such as fiber diffraction and inelastic scattering (X-ray spectroscopic) methods; these latter techniques may be applied when there is no periodicity, e.g., Raman scattering, or periodicity only along one direction, as in fiber diffraction.

1 Definition
2 History
- 2.1 Pre-history of crystallography and X-rays
- 2.2 The idea of combining crystals and X-rays
- 2.3 Development through 1930
- 2.4 Early organic and small biological molecules
- 2.5 Protein crystallography
- 2.6 Applications in chemistry and material science
3 Scattering techniques
- 3.1 Elastic vs. inelastic scattering
- 3.2 Other types of X-ray scattering
- 3.3 Electron and neutron diffraction
- 3.4 Advantages of a crystal
4 Methods
- 4.1 Crystallization
- 4.2 Data collection
  - 4.2.1 Mounting the crystal
  - 4.2.2 X-ray sources
  - 4.2.3 Recording the reflections
- 4.3 Data analysis
  - 4.3.1 Crystal symmetry, unit cell, and image scaling
  - 4.3.2 Initial phasing
  - 4.3.3 Model building and phase refinement
- 4.4 Deposition of the structure
5 Diffraction theory
- 5.1 Scattering as a Fourier transform
- 5.2 Ewald's sphere
- 5.3 Patterson function
6 See also
7 Bibliography
- 7.1 International Tables for Crystallography
- 7.2 Bound collections of articles
- 7.3 Textbooks
- 7.4 Historical
8 References
9 External links
- 9.1 Tutorials
- 9.2 Primary databases
- 9.3 Derivative databases
- 9.4 Structural validation

[edit] Definition

An X-ray diffraction image for the protein myoglobin. The spots (reflections) are clearly visible.

X-ray crystallography is a technique used to determine the arrangement of atoms within a crystal. Strictly speaking, X-ray crystallography measures only the density of electrons within the crystal, from which the atomic positions can be inferred. The crystal may be of any material composition, which makes X-ray crystallography a favorite technique of mineralogists, metallurgists and other material scientists who wish to characterize a new crystalline material at the atomic level. For example, the technique allows one to determine the exact differences between different crystals having the same chemical composition but different properties, such as different forms of silica or the different allotropes of tin that lead to the tin pest infecting organ pipes. In another illustrative example, crystallography has helped physicists to understand molecular forces and how the arrangement of atoms can produce the observed material properties of crystals.

X-ray diffraction involves the scattering of X-rays of a single wavelength ("monochromatic X-rays") from a single, pure crystal.^[1] This scattering produces a diffraction pattern, a set of intense spots (also called reflections) on a screen behind it.

The spots can be related to the density of electrons in the crystal through a mathematical operation called a Fourier transform. This technique decomposes the electron density into its spatial frequency components, just as the human ear can discern different musical notes in a chord. Each spot observed on the screen corresponds to a spatial oscillation of the electron density along a particular direction within the crystal. By combining these independent oscillations in density, the electron distribution can be reconstructed, just as chord can be played on a piano once its individual musical notes are known.^[1]

The reflections vary in intensity, and by gradually rotating the crystal and recording the intensities of the spots, one may determine the magnitude of the Fourier transform of the density of electrons within the crystal. By using data on related molecules, or by recording several sets of data with specific changes in the scattering, the phases corresponding to these magnitudes may be computed. Combining the phases and magnitudes yields the full Fourier transform of the electron density, which may be inverted to obtain the electron density in terms of position within the crystal. Complementary chemical data on the crystal allows the electron density to be converted into a model of the position of every atom of the molecule(s) within the crystal.

In principle, the continuous X-ray scattering from a single molecule, if measured with sufficient accuracy, would suffice to determine its structure. However, such single-molecule scattering is weak and difficult to measure accurately. Moreover, it may not be possible to extract a molecule from its environment without changing its structure. The coherent scattering from a crystal strengthens the signal quadratically with the number of scatterers, which can be quite large even for very small crystals.

[edit] History

[edit] Pre-history of crystallography and X-rays

Crystals have long been admired for their regularity and symmetry, but they were not investigated scientifically until the 17th century, when Johannes Kepler hypothesized (1611) that the hexagonal symmmetry of snowflake crystals was due to a regular packing of spherical water particles.^[2] Crystal symmetry was investigated experimentally by Nicolas Steno (1669), who showed that the angles between the faces are the same in every exemplar of a single type of crystal,^[3] and by René Just Haüy (1784), who discovered that every face of a crystal can be described by three small integers, the so-called Miller indices. These studies led Haüy to the correct idea that crystals are a regular three-dimensional array (a Bravais lattice) of atoms and molecules; a single unit cell is repeated indefinitely along three principal directions that are not necessarily perpendicular. In the 19th century, a complete catalog of the possible symmetries of a crystal was worked out by Hessel,^[4] Bravais,^[5] Fedorov,^[6] and Schönflies.^[7]

X-rays were discovered by Wilhelm Conrad Röntgen in 1895, just as the studies of crystal symmetry were being concluded. However, physicists were initially uncertain of the nature of X-rays, although it was soon suspected (correctly) that they were waves of electromagnetic radiation, in other words, another form of light. At that time, the wave model of light — specifically, the Maxwell theory of electromagnetic radiation — was well accepted among scientists, and experiments by Charles Glover Barkla showed that X-rays exhibited phenomena associated with electromagnetic waves, including transverse polarization and spectral lines akin to those observed in the visible wavelengths. Single-slit experiments in the laboratory of Arnold Sommerfeld suggested the wavelength of X-rays was roughly 1 Angström, one ten millionth of a millimetre. Being composed of photons, X-rays also exhibit particle-like properties, e.g., in the ionization of gases; these properties led William Henry Bragg to suggest in 1907 that X-rays were not electromagnetic radiation, since the concept of the photon was relatively new (1905) and not generally accepted. However, Bragg's view was itself not broadly accepted and the observation of X-ray diffraction in 1912 confirmed for most scientists that X-rays were a form of electromagnetic radiation. Any remaining doubt was resolved in 1922, when Arthur Compton confirmed the photon model by studying the scattering of X-rays from electrons.

[edit] The idea of combining crystals and X-rays

To summarize, crystals are regular arrays of atoms and X-rays are waves of electromagnetic radiation. Atoms scatter X-rays, primarily through their electrons; just as an ocean wave striking a lighthouse will produce secondary circular waves emanating from the lighthouse, so an X-ray striking an atom will produce secondary spherical waves emanating from the atom. This phenomenon is known as scattering, and the atom (or lighthouse) is known as the scatterer.

Any wave impinging on a regular array of scatterers produces diffraction, as predicted first by Francesco Maria Grimaldi in 1665. To produce significant diffraction, the spacing between the scatterers and the wavelength of the impinging wave should be roughly similar in size. James Gregory observed the optical diffraction of sunlight through a bird's feather in the later 17th century. The first man-made diffraction grating was constructed by David Rittenhouse in 1787, and improved by Joseph von Fraunhofer in 1821. Similarly, X-ray diffraction results from an electromagnetic wave (the X-ray) impinging on a regular array of scatterers (the crystal). By good fortune, the wavelength of X-rays and the spacing between unit cells in crystals are roughly of the same size (1-100 Ångströms); prior to X-ray diffraction, however, the spacings between unit cells in a crystal were unknown.

The idea that crystals could be used as a diffraction grating for X-rays arose in 1912 in a conversation between Paul Peter Ewald and Max von Laue in the English Garden in Munich. Ewald had proposed a resonator model of crystals for his thesis, but this model could not be validated using visible light, since the wavelength was much larger than the spacing between the resonators. von Laue realized that electromagetic radiation of a shorter wavelength was needed to observe such small spacings, and suggested that X-rays might have a wavelength comparable to the unit-cell spacing in crystals. von Laue worked with two technicians, Walter Friedrich and his assistant Paul Knipping, to shine a beam of X-rays through a sphalerite crystal and record its diffraction on a photographic plate. After being developed, the plate showed a large number of well-defined spots arranged in a pattern of intersecting circles around the spot produced by the central beam.^[8] von Laue developed a law that connects the scattering angles and the size and orientation of the unit-cell spacings in the crystal, for which he was awarded the Nobel Prize in Physics in 1914.^[9]

[edit] Development through 1930

After von Laue's pioneering research, the field developed rapidly, most notably by physicists William Lawrence Bragg and his father William Henry Bragg. In 1913, the younger Bragg developed Bragg's law, which connects the periodicities in the crystal with the scattering observed. The earliest structures were generally simple and marked by one-dimensional symmetry; as the field progressed over the next decades, the structures of two- and three-dimensional arrangements of atoms in the unit-cell became feasible.

The potential of X-ray crystallography for determining the structure of molecules and minerals — then only known vaguely from chemical and hydrodynamic experiments — was realized immediately. The earliest structures were generally simple inorganic crystals and minerals. The first atomic-resolution structure to be solved (in 1913) was that of table salt,^[10] which proved the existence of ionic compounds and that crystals are not necessarily comprised of molecules. The structure of diamond was solved in the same year,^[11] proving the tetrahedral arrangement of its chemical bonds and showing that the C-C single bond was 1.52 Ångströms. Other early structures included copper, calcium fluoride (Ca₂F) and pyrite (FeS₂) (all in 1914); spinel and calcite (in 1915); rutile and anatase (all in 1916); magnesium hydroxide (in 1919); and wurtzite (1920). The structure of graphite was solved in 1916 by the related method of powder diffraction, which was developed by Peter Debye and Paul Scherrer and, independently, by Albert Hull in 1917.

The 1920s saw many advances in X-ray crystallography. The first structure of an organic compound, hexamethylenetetramine, was solved in 1923.^[12] In mineralogy, a systematic study of the silicates was undertaken, beginning with the structure of garnet in 1924 by Menzer. As the Si/O ratio is altered, the resulting crystals exhibited significant changes in their internal arrangements. Machatschki extended these insights to minerals in which aluminum substitutes for silicon in the silicates. During the same decade, Victor Moritz Goldschmidt and later Linus Pauling developed rules for eliminating chemically unlikely structures and for determining the relative sizes of atoms. These rules led to the structure of brookite (1928) and an understanding of the relative stability of the rutile, brookite and anatase forms of titanium oxide. Also in 1928, Kathleen Lonsdale solved the structure of hexamethylbenzene,^[13] which established the hexagonal symmetry of benzene and showed a clear difference in bond length between the aliphatic C-C bonds and aromatic C-C bonds; this finding led to the idea of resonance between chemical bonds, which had profound consequences for the development of chemistry.^[14]

X-ray crystallography made significant contributions to metallurgy. Linus Pauling determined the structure of the first alloy, Mg₂Sn,^[15] which led to his theory governing the stability and structure of complex ionic crystals.^[16]

Another early pioneer was John Desmond Bernal.

[edit] Early organic and small biological molecules

The three-dimensional structure of penicillin, for which Dorothy Crowfoot Hodgkin was awarded the Nobel Prize in Chemistry in 1964.

In the 1930s, the structures of much larger molecules with two-dimensional complexity began to be solved. A significant advance was the structure of phthalocyanine,^[17] a large planar molecule that has an approximate four-fold symmetry and resembles porphyrins found in nature, such as heme, corrin and chlorophyll.

X-ray crystallography of biological molecules took off with Dorothy Crowfoot Hodgkin, who solved the structures of cholesterol (1937), vitamin B12 (1945) and penicillin (1954), for which she was awarded the Nobel Prize in Chemistry in 1964.

A representation of the 3D structure of myoglobin, showing colored alpha helices.

[edit] Protein crystallography

Crystal structures of proteins (which are irregular and hundreds of times larger than cholesterol) began to be solved in the late 1950's, beginning with the structure of sperm whale myoglobin by Max Perutz and Sir John Cowdery Kendrew, for which they were awarded the Nobel Prize in Chemistry in 1962.^[18] Since that success, over 36000 X-ray crystal structures of proteins, nucleic acids and other biological molecules have been determined. For comparison, the nearest competing method, NMR spectroscopy has produced roughly 6000 structures.^[19] Moreover, crystallography can solve structures of arbitrarily large molecules, whereas solution-state NMR is restricted to relatively small molecules (less than 70 kDa). X-ray crystallography is now used routinely by scientists to determine how a phamaceutical interacts with its protein target and what changes might be advisable to improve it.^[20].

The chief obstacle in the X-ray crystallography of biological molecules is growing a crystal of sufficient size and regularity. The crystallization process is poorly understood and growing adequate crystals is often a matter of luck and exhaustively trying out solution conditions that have worked in the past. Robots are now being employed to facilitate this process and ensure reproducibility. Intrinsic membrane proteins are especially challenging to crystallize because they require detergents or other means to solubilize them in isolation, and such detergents often interfere with crystallization. Such membrane proteins are a large component of the genome and include many proteins of great physiological importance, such as ion channels and receptors.^[21]^[22]

[edit] Applications in chemistry and material science

X-ray crystallography gives the positions of atoms very precisely, which has led to a better understanding of chemical bonds and non-covalent interactions, such as van der Waals forces. The distance between two covalently bonded atoms is a sensitive measure of the bond strength and its bond order; thus, X-ray crystallographic studies have led to the discovery of novel types of bonding in inorganic chemistry, such as metal-metal double bonds,^[23] metal-metal quadruple bonds,^[24] and three-center, two-electron bonds.^[25] In the field of organometallic chemistry, the X-ray structure of ferrocene initiated scientific studies of sandwich compounds,^[26] while that of Zeise's salt stimulated research into "back bonding" and metal-pi complexes in general.^[27] Finally, X-ray crystallography had a pioneering role in the development of supramolecular chemistry, particularly in clarifying the structures of the crown ethers and the principles of host-guest chemistry.

In material sciences, many complicated inorganic and organometallic systems have been analyzed using single-crystal methods, such as fullerenes, metalloporphyrins, and other complicated compounds. Single-crystal diffraction is also used in the pharmaceutical industry, due to recent problems with polymorphs. The major factors affecting the quality of single-crystal structures is crystal's size and regularity; recrystallization is a commonly used technique to improve these factors in small-molecule crystals. The Cambridge Structural Database contains over 400,000 structures; over 99% of these structures were determined by X-ray diffraction.

[edit] Scattering techniques

Further information: X-ray scattering techniques

[edit] Elastic vs. inelastic scattering

X-ray crystallography is a form of elastic scattering; the outgoing X-rays have the same energy as the incoming X-rays, only with altered direction. Since the energy of a photon is inversely proportional to its wavelength, elastic scattering means that the outgoing photons have the same wavelength as the incoming photons. By contrast, inelastic scattering occurs when energy is transferred from the incoming X-ray to the crystal, e.g., by exciting an inner-shell electron to a higher energy level. Such inelastic scattering changes the wavelength of the outgoing beam, making it longer and less energetic. Inelastic scattering is useful for probing such excitations of matter, but are not as useful in determining the distribution of scatterers within the matter, which is the goal of X-ray crystallography.

The wavelength of an X-ray is roughly 1 Å (0.1 nm = 10^-10 m), which is on the scale of a single atom. Longer wavelength photons (such as ultraviolet radiation) would not have sufficient resolution to determine the atomic positions. At the other extreme, shorter wavelength photons such as gamma rays are difficult to produce in large numbers, difficult to focus, and interact too strongly with matter, producing particle-antiparticle pairs. Therefore, X-rays are the "sweetspot" for wavelength when determining the structure of molecules using electromagnetic radiation.

[edit] Other types of X-ray scattering

X-ray diffraction involves the scattering of X-rays from a single crystal. Other forms of elastic X-ray scattering include powder diffraction, SAXS and several types of X-ray fiber diffraction, which was used by Rosalind Franklin in determining the double-helix structure of DNA. In general, X-ray diffraction produces isolated spots ("reflections"), while the other methods produce smooth, continuous scattering. X-ray diffraction, if possible, offers more structural information than these other techniques; however, it requires a sufficiently large and regular crystal, which is not always possible to obtain.

All of these scattering methods generally use monochromatic X-rays, X-rays that are restricted to a single wavelength with minor deviations. A broad spectrum of X-rays (that is, a blend of X-rays with different wavelengths) can also be used to carry out X-ray diffraction, a technique known as the Laue method. This is the method used in the original discovery of X-ray diffraction. Laue scattering provides much structural information with only a short exposure to the X-ray beam, and is therefore used in structural studies of very rapid events (time-resolved X-ray crystallography). However, it is not as well-suited as monochromatic scattering for determining the full atomic structure of a crystal. It is better suited to crystals with relatively simple atomic arrangements, such as minerals.

The Laue back reflection mode records X-rays scattered backwards also from a broad spectrum source. This is useful if the sample is too thick or bulky for X-rays to transmit through it. The diffracting planes in the crystal are determined by knowing that the normal to the diffracting plane bisects the angle between the incident beam and the diffracted beam. A Greninger chart can be used ^[28] to interpret the back reflection Laue photograph. The X-calibre RTXDB and MWL 110 are commercial systems for Laue back reflection pattern recording. This technique can be used in materials analysis or non destructive inspection.

Crystals can produce tens of thousands of well-resolved reflections, each of which represents an independent piece of data about the structure. Full sets of reflections are typically collected under different conditions, either by scattering with multiple wavelengths of X-rays or with small metallic additives that help in solving the structure. These hundreds of thousands of data are assembled by powerful computers into the atomic-resolution model of the electron density.

No other structural technique offers so many independent data on the structure; for example, protein NMR typically collects a hundredfold fewer data. Thanks to its enormous amount of independent data, X-ray crystallography is the best technique for determining a static atomic-resolution structure of any molecule. However, X-ray crystallography requires a crystal, and obtaining a crystal of sufficient quality is generally the key stumbling block to determining the structure. Some crystallographers have devoted over ten years of their lives to obtaining a single crystal, e.g., the viral protein gp120 used by HIV to enter human cells.

[edit] Electron and neutron diffraction

As derived below, elastic scattering can be represented as a Fourier transform of the density of the objects doing the scattering (the "scatterers"), as long as the scattering is weak; the scattered beams should be much less intense than the incoming beam. When the scattering is weak, the scattered beams do not produce re-scattered beams of their own, at least not with any significant amplitude. Re-scattered waves are called "secondary scattering". This is not a problem for X-ray diffraction, since the X-rays interact relatively weakly with the electrons. However, this may be a problem for diffraction with other types of beams. For example, in electron diffraction, the electrons in the incoming beam interact strongly with those of the crystal, resulting in significant secondary scattering even for the smallest samples typically used in X-ray crystallography (roughly, 100 microns). The solution is to use extremely thin samples, roughly 100 nanometers or less; the primary scattered electron beams leave the sample before they have a chance to undergo secondary scattering. Since this thickness corresponds roughly to the diameter of many viruses, a promising direction is the electron diffraction of isolated macromolecular assemblies, such as viral capsids and molecular machines, which may be carried out with a cryo-electron microscope.

Neutron diffraction is an excellent method for structure determination, although it has been difficult to obtain intense, monochromatic beams of neutrons in sufficient quantities. Traditionally, nuclear reactors have been used, although the new Spallation Neutron Source holds much promise in the near future. Being uncharged, neutrons scatter much more readily from the atomic nuclei rather than from the electrons. Therefore, neutron scattering is very useful for observing the positions of light atoms with few electrons, especially hydrogen, which is essentially invisible in the X-ray diffraction of larger molecules. Neutron scattering also has the remarkable property that the solvent can be made invisible by adjusting the ratio of normal water, H₂O, and heavy water, D₂O.

[edit] Advantages of a crystal

A crystalline sample is by definition periodic; a crystal is composed of many unit cells repeated over and over in all three independent directions. Such periodic systems have a Fourier transform that is concentrated at periodically repeating points in reciprocal space known as Bragg peaks; the Bragg peaks correspond to the reflection spots observed in the diffraction image. Since the amplitude at these reflections grows linearly with the number N of scatterers, the observed intensity of these spots should grow quadratically, like N². In other words, using a crystal concentrates the weak scattering of the individual unit cells into a much more powerful, coherent reflection that can be observed above the noise. This is an example of constructive interference.

In a non-crystalline sample, molecules within that sample would be in random orientations and therefore would have a continuous Fourier spectrum that spreads its amplitude more uniformly and with a much reduced intensity, as is observed in SAXS. More importantly, the orientational information is lost. In the crystal, the molecules are all held at precisely the same orientation within the crystal, whereas in a liquid, powder or amorphous state, the observed signal is averaged over all possible orientations of the molecules. It has proven nearly impossible to obtain atomic-resolution structures from such rotationally averaged scattering data. An intermediate case is fiber diffraction in which the subunits are arranged periodically in at least one dimension, if not in three dimensions.

[edit] Methods

[edit] Crystallization

A protein crystal seen under a microscope. Most crystals used in X-ray crystallography are less than a millimeter across.

Further information: Crystallization

Crystallography requires a pure crystal of high regularity. In some cases, such crystals can be obtained readily, such as samples of metals, minerals or other macroscopic materials. The regularity of such crystals can sometimes be improved with annealing and other methods. However, in many cases, obtaining a diffraction-quality crystal is the chief barrier to solving its atomic-resolution structure.^[29]

One of the major differences between small molecule and macromolecular crystallography is the methods used in the production of diffraction quality crystals. Small molecule crystals are generally grown using chemical vapor deposition and recrystallisation techniques. However, these methods are generally unsuitable for macromolecular crystals as the conditions are likely to unfold the protein. Protein crystals are formed more commonly by growing them in solution; the most common method is to lower the solubility of its component molecules very gradually. If done too quickly, the molecules will precipitate from solution, forming a useless dust or amorphous gel on the bottom of the container. Crystal growth in solution is characterized by two steps: nucleation of a microscopic crystallite (possibly having only 100 molecules), followed by growth of that crystallite to a useful crystal.^[30] The solution conditions that favor the first step (nucleation) are not always the same conditions that favor the second step (its subsequent growth). The crystallographer's goal is to identify solution conditions that favor the development of a single, large crystal, since larger crystals offer improved resolution of the molecule. Consequently, the solution conditions should disfavor the first step (nucleation) but favor the second (growth), so that only one large crystal forms per droplet. If nucleation is favored too much, a shower of small crystallites will form in the droplet, rather than one large crystal; if favored too little, no crystal will form whatsoever.

In some cases, the crystallographer can identify good solution conditions for growing very small crystals that do not continue to grow and which are too small for crystallography. In such cases, the tiny crystals can be transferred to new solution conditions that favor growth more strongly; the small crystals act as pre-nucleated seeds for subsequent growth. In an alternative approach, a larger but poor-quality crystal may be crushed, and the pieces used as seed crystals to obtain higher quality crystals.

It is extremely difficult to predict good conditions for nucleation or growth of well-ordered crystals.^[31] In practice, favorable conditions are identified by screening; a very large batch of the molecules is prepared, and a wide variety of crystallization solutions are tested.^[32] Hundreds, even thousands, of solution conditions are generally tried before finding one that succeeds in crystallizing the molecules. The various conditions can use one or more physical mechanisms to lower the solubility of the molecule; for example, some may change the pH, some contain salts of the Hofmeister series or chemicals that lower the dielectric constant of the solution, and still others contain large polymers such as polyethylene glycol that drive the molecule out of solution by entropic effects. It is also common to try several temperatures for encouraging crystallization, or to gradually lower the temperature so that the solution becomes supersaturated. These methods require large amounts of the target molecule, as they use high concentration of the molecule(s) to be crystallized. Due to the difficult in obtaining such large quantities (milligrams) of crystallisation grade protein, dispensing robots have been developed that are capable of accurately dispensing crystallisation trial drops that are of the order on 100 nanoliters in volume. This means that roughly 10-fold less protein is used per-experiment when compared to crystallisation trials setup by hand (on the order on 1 microliters) ^[33].

Several factors are known to inhibit or mar crystallization. The growing crystals are generally held at a constant temperature and protected from shocks or vibrations that might disturb their crystallization. Impurities in the molecules or in the crystallization solutions are often inimical to crystallization. Conformational flexibility in the molecule also tends to make crystallization less likely, due to entropy. Ironically, molecules that tend to self-assemble into regular helices are often unwilling to assemble into crystals. Crystals can be marred by twinning, which can occur when a unit cell can pack equally favorably in multiple orientations; although recent advances in computational methods have begun to allow the structures of twinned crystals to be solved, it is still very difficult. Having failed to crystallize a target molecule, a crystallographer may try again with a slightly modified version of the molecule; even small changes in molecular properties can lead to large differences in crystallization behavior.

[edit] Data collection

[edit] Mounting the crystal

Once they are full-grown, the crystals are mounted so that they may be held in the X-ray beam and rotated. There are several methods of mounting. Although crystals were once loaded into glass capillaries with the crystallization solution (the mother liquor), a more modern approach is to scoop the crystal up in a tiny loop, made of nylon or plastic and attached to a solid rod, that is then flash-frozen with liquid nitrogen.^[34] This freezing reduces the radiation damage of the X-rays, as well as the noise in the Bragg peaks due to thermal motion (the Debye-Waller effect). However, untreated crystals often crack if flash-frozen; therefore, they are generally pre-soaked in a cryoprotectant solution before freezing.^[35] Unfortunately, this pre-soak may itself cause the crystal to crack, ruining it for crystallography. Generally, successful cryo-conditions are identified by trial and error.

The capillary or loop is mounted on a goniometer, which allows it to be positioned accurately within the X-ray beam and rotated. Since both the crystal and the beam are often very small, the crystal must be centered within the beam to within roughly 25 microns accuracy, which is aided by a camera focused on the crystal. The most common type of goniometer is the "kappa goniometer", which offers three angles of rotation: the ω angle, which rotates about an axis roughly perpendicular to the beam; the κ angle, about an axis at roughly 50° to the ω axis; and, finally, the φ angle about the loop/capillary axis. When the κ angle is zero, the ω and φ axes are aligned. The κ rotation allows for convenient mounting of the crystal, since the arm in which the crystal is mounted may be swung out towards the crystallographer. The oscillations carried out during data collection (mentioned below) involve the ω axis only. An older type of goniometer is the four-circle goniometer, and its relatives such as the six-circle goniometer.

[edit] X-ray sources

Further information: Diffractometer and Synchrotron

The mounted crystal is then irradiated with a beam of monochromatic X-rays. The brightest and most useful X-ray sources are synchrotrons; their much higher luminosity allows for better resolution. They also make it convenient to tune the wavelength of the radiation, which is useful for multi-wavelength anomalous dispersion (MAD) phasing, described below. Synchrotrons are generally national facilities, each with several dedicated beamlines where data is collected around the clock, seven days a week. Crystallographers apply for a slot of time, which they must use whenever it is granted, even at 3am on a national holiday. Crystallographers will sometimes stay awake for days, collecting data continuously until their allotted time runs out.

A diffractometer

Smaller, weaker X-ray sources are often used in laboratories to check the quality of crystals before bringing them to a synchrotron and sometimes to solve a crystal structure. In such systems, electrons are boiled off of a cathode and accelerated through a strong electric potential of roughly 50 kV; having reached a high speed, the electrons collide with a metal plate, emitting bremsstrahlung and some strong spectral lines corresponding to the excitation of inner-shell electrons of the metal. The most common metal used is copper, which can be kept cool easily, due to its high thermal conductivity, and which produces strong K_α and K_β lines. The K_β line is sometimes suppressed with a thin layer (0.0005 in. thick) of nickel foil. The simplest and cheapest variety of sealed X-ray tube has a stationary anode and produces circa 2 kW of X-ray radiation. The more expensive variety has a rotating-anode type source that produces circa 14 kW of X-ray radiation.

X-rays are generally filtered to a single wavelength (made monochromatic) and collimated to a single direction before they are allowed to strike the crystal. The filtering not only simplifies the data analysis, but also removes radiation that degrades the crystal without contributing useful information. Collimation is done either with a collimator (basically, a long tube) or with a clever arrangement of gently curved mirrors. Mirror systems are preferred for small crystals (under 0.3 mm) or with large unit cells (over 150 Å).

[edit] Recording the reflections

When a crystal is mounted and exposed to an intense beam of X-rays, it scatters the X-rays into a pattern of spots or reflections that can be observed on a screen behind the crystal. A similar pattern may be seen by shining a laser pointer at a compact disc. The relative intensities of these spots provide the information to determine the arrangement of molecules within the crystal in atomic detail. The intensities of these reflections may be recorded with photographic film, an area detector or with a charge-coupled device (CCD) image sensor. The peaks at small angles correspond to low-resolution data, whereas those at high angles represent high-resolution data; thus, an upper limit on the eventual resolution of the structure can be determined from the first few images. Some measures of diffraction quality can be determined at this point, such as the mosaicity of the crystal and its overall disorder, as observed in the peak widths. Some pathologies can be quickly diagnosed as well, such as twinning or a prominent ice ring.

One image of spots is insufficient to reconstruct the whole crystal; it represents only a small slice of the full Fourier transform. To collect all the necessary information, the crystal must be rotated step-by-step through 180°, with an image recorded at every step; acrually, slightly more than 180° is required to cover reciprocal space, due to the curvature of the Ewald sphere. However, if the crystal has a higher symmetry, a smaller angle such as 90° or 45° may be recorded. The axis of the rotation should generally be changed at least once, to avoid developing a "blind spot" in reciprocal space close to the rotation axis. It is customary to rock the crystal slightly (by 0.5-2°) to catch a broader region of reciprocal space.

Multiple data sets may be necessary for certain phasing methods. For example, MAD phasing requires that the scattering be recorded at at least three (and usually four, for redundancy) wavelengths of the incoming X-ray radiation. A single crystal may degrade too much during the collection of one data set, owing to radiation damage; in such cases, data sets on multiple crystals must be taken.^[36]

[edit] Data analysis

[edit] Crystal symmetry, unit cell, and image scaling

Further information: Space group

In order to process the data, a crystallographer must first index the reflections within the multiple images recorded. This means identifying the dimensions of the unit cell and which image peak corresponds to which position in reciprocal space. A byproduct of indexing is to determine the symmetry of the crystal, i.e., its space group. Some space groups can be eliminated from the beginning, since they require symmetries known to be absent in the molecule itself. For example, symmetries with reflection symmetries cannot be observed in chiral molecules; thus, only 65 space groups of 243 possible are allowed for protein molecules which are almost always chiral. Indexing is generally accomplished using an autoindexing routine^[37]. Having assigned symmetry, the data is then integrated. This converts the hundreds of images containing the thousands of reflections into a single file, consisting of (at the very least) records of the Miller index of each reflection, and an intensity for each reflection (at this state the file often also includes error estimates and measures of partiality (what part of a given reflection was recorded on that image).

A full data set may consist of hundreds of separate images taken at different orientations of the crystal. The first step is to merge and scale these various images, that is, to identify which peaks appear in two or more images (merging) and to scale the relative images so that they have a consistent intensity scale. This is important, since the relative intensities of the peaks is the key information from which the structure is determined. The technique of crystallographic data collection and the often high symmetry of crystalline materials, means that many symmetry-equivalent reflections are recorded multiple times - this allows a merging or symmetry related R-factor to be calculated, based upon how similar the measured intensities of symmetry equivalent reflections are, thus giving a score to assess the quality of the data.

[edit] Initial phasing

Further information: Phase problem

The data collected from a diffraction experiment is a reciprocal space representation of the crystal lattice. The position of each diffraction 'spot' is governed by the size and shape of the unit cell, and the inherent symmetry within the crystal. The intensity of each diffraction 'spot' is recorded, and is proportional to the square root of the structure factor amplitude. The structure factor is a complex number containing information relating to both the amplitude and phase of a wave. In order to obtain an interpretable electron density map, both amplitude and phase must be known (an electron density map allows a crystallographer to build a starting model of the molecule). The phase cannot be directly recorded during a diffraction experiment: this is known as the phase problem. Initial phase estimates can be obtained be in a variety of ways:

Ab Initio phasing aka Direct Methods - This is usually the method of choice for small molecules (<1000 non-hydrogen atoms), and has been used successfully to solve the phase problems for small proteins. If the resolution of the data is better than 1.4 Å (140 pm), direct methods can be used to obtain phase information, by exploiting known phase relationships between certain groups of reflections^[38] ^[39]

Further information: direct methods

Molecular replacement - if a structure exists of a related crystal structure, it can be used as a search model in molecular replacement to determine the orientation and position of the molecules within the unit cell. The phases obtained this way can be used to generate electron density maps.^[40]

Anomalous scattering (MAD or SAD phasing) - the X-ray wavelength may be scanned past an absorption edge of an atom, which changes the scattering in a known way. By recording full sets of reflections at three different wavelengths (far below, far above and in the middle of the absorption edge) one can solve for the substructure of the anomalously diffracting atoms and thence the structure of the whole molecule. The most popular method of incorporating anomalous scattering atoms into proteins is to express the protein in a methionine auxotroph (a host incapable of synthesising methionine) in a media rich in Seleno-methionine, which contains Selenium atoms. A MAD experiment can then be conducted around the absorption edge, which should then yield the position of any methionine residues within the protein, providing initial phases.^[41]

Heavy atom methods - If electron-dense metal atoms can be introduced into the crystal , direct methods or Patterson-space methods can be used to determine their location and to obtain initial phases. Typically, a crystallographer can dope a crystal with heavy atoms either by soaking the crystal in a heavy atom-containing solution, or by co-crystallization (growing the crystals in the presence of a heavy atom). As in MAD phasing, the changes in the scattering amplitudes can be interpreted to yield the phases. Although this is the original method by which protein crystal structures were solved, it has largely been superseded by MAD phasing with selenomethionine.^[40]

Whilst all four of the above methods are used to solve the phase problem for protein crystallography, small molecule crystallography generally yields data suitable for structure solution using Direct methods/ab initio phasing.

[edit] Model building and phase refinement

An example of electron density for a protein crystal structure at 2.7 Å resolution. A disulphide bond, an Arginine residue and a Tyrosine residue are highlighted

Further information: Molecular modeling

Having obtained initial phases, an initial model can be built. This model can be used to refine the phases, leading to an improved model, and so on. Given a model of some atomic positions, these positions and their respective Debye-Waller factors (accounting for the thermal motion of the atom - aka B-factors) can be refined to fit the observed diffraction data, ideally yielding a better set of phases. A new model can then be fit to the new electron density map and a further round of refinement is carried out. This continues until the correlation between the diffraction data and the model is maximized. The agreement is measured by an R-factor defined as

$R = \frac{\sum_{\mathrm{all\ reflections}} \left| F_{o} - F_{c} \right|}{\sum_{\mathrm{all\ reflections}} \left| F_{o} \right|}$

A similar quality criterion is R_free, which is calculated from a subset (~10%) of reflections that were not included in the structure refinement. Both R factors depend on the resolution of the data. As a rule of thumb, R_free should be approximately the resolution in Ångströms divided by 10; thus, a data-set with 2 Å resolution should yield a final R_free of roughly 0.2. Chemical bonding features such as stereochemistry, hydrogen bonding and distribution of bond lengths and angles are complementary measures of the model quality. Phase bias is a serious problem in such iterative model building. Omit maps are a common technique used to check for this.

It may not be possible to observe every atom of the crystallized molecule - it must be remembered that the resulting electron density is an average of all the molecules within the crystal. In some cases, there is too much residual disorder in those atoms, and the resulting electron density for atoms existing in many conformations is smeared to such an extent that it is no longer detectable in the electron density map.. Weakly scattering atoms such as hydrogen are routinely invisible. It is also possible for a single atom to appear multiple times in an electron density map, e.g., if a protein sidechain has multiple (<4) allowed conformations. In still other cases, the crystallographer may detect that the covalent structure deduced for the molecule was incorrect, or changed. For example, proteins may be cleaved or undergo posttranslational modifications that were not detected prior to the crystallization.

[edit] Deposition of the structure

Once the model of a molecule's structure has been finalized, it is often deposited in a crystallographic database such as the Protein Data Bank (for protein structures) or the Cambridge Structural Database (for small molecules). Many structures obtained in private commercial ventures to crystallize medicinally relevant proteins, are not deposited in public crystallographic databases.

[edit] Diffraction theory

Further information: Dynamical theory of diffraction and Bragg diffraction

The main goal of X-ray crystallography is to determine the density of electrons f(r) throughout the crystal. To do this, X-ray scattering is used to collect data about its Fourier transform F(q), which is then inverted mathematically to obtain the density defined in real space, using the formula

$f(\mathbf{r}) = \int \frac{d\mathbf{q}}{\left(2\pi\right)^{3}} F(\mathbf{q}) e^{i\mathbf{q}\cdot\mathbf{r}}$

The corresponding formula for a Fourier transform is

$F(\mathbf{q}) = \int d\mathbf{r} f(\mathbf{r}) e^{-i\mathbf{q}\cdot\mathbf{r}}$

which will be used below. The Fourier transform F(q) is generally a complex number, and therefore has a magnitude |F(q)| and a phase φ(q) related by the equation

$F(\mathbf{q}) = \left| F(\mathbf{q}) \right| e^{i\phi(\mathbf{q})}$

The intensities of the reflections observed in X-ray diffraction give us the magnitudes |F(q)| but not the phases φ(q). To obtain the phases, full sets of reflections are collected with known alterations to the scattering, either by modulating the wavelength past a certain absorption edge or by adding strongly scattering (i.e., electron-dense) metal atoms such as mercury. Combining the magnitudes and phases yields the full Fourier transform F(q), which may be inverted to obtain the electron density f(r).

[edit] Scattering as a Fourier transform

The incoming X-ray beam has a polarization and should be represented as a vector wave; however, for simplicity, let it be represented here as a scalar wave. We also ignore the complication of the time dependence of the wave and just focus on the wave's spatial dependence. Plane waves can be represented by a wave vector k_in, and so the strength of the incoming wave at time t=0 is given by

$A e^{i\mathbf{k}_{in} \cdot \mathbf{r}}$

At position r within the sample, let there be a density of scatterers f(r); these scatterers should produce a scattered spherical wave of amplitude proportional to the local amplitude of the incoming wave times the number of scatterers in a small volume dV about r

$\mathrm{amplitude\ of\ scattered\ wave} = A e^{i\mathbf{k} \cdot \mathbf{r}} S f(\mathbf{r}) dV$

where S is the proportionality constant.

Let's consider the fraction of scattered waves that leave with an outgoing wave-vector of k_out and strike the screen at r_screen. Since no energy is lost (elastic, not inelastic scattering), the wavelengths are the same as are the magnitudes of the wave-vectors |k_in| = |k_out|. From the time that the photon is scattered at r until it is absorbed at r_screen, the photon undergoes a change in phase

$e^{i \mathbf{k}_{out} \cdot \left( \mathbf{r}_{\mathrm{screen}} - \mathbf{r} \right)}$

The net radiation arriving at r_screen is the sum of all the scattered waves throughout the crystal

$A S \int d\mathbf{r} f(\mathbf{r}) e^{i \mathbf{k}_{in} \cdot \mathbf{r}} e^{i \mathbf{k}_{out} \cdot \left( \mathbf{r}_{\mathrm{screen}} - \mathbf{r} \right)} = A S e^{i \mathbf{k}_{out} \cdot \mathbf{r}_{\mathrm{screen}}} \int d\mathbf{r} f(\mathbf{r}) e^{i \left( \mathbf{k}_{in} - \mathbf{k}_{out} \right) \cdot \mathbf{r}}$

which may be written as a Fourier transform

$A S e^{i \mathbf{k}_{out} \cdot \mathbf{r}_{\mathrm{screen}}} \int d\mathbf{r} f(\mathbf{r}) e^{-i \mathbf{q} \cdot \mathbf{r}} = A S e^{i \mathbf{k}_{out} \cdot \mathbf{r}_{\mathrm{screen}}} F(\mathbf{q})$

where q = k_out - k_in. The measured intensity of the reflection will be square of this amplitude

$A^{2} S^{2} \left| F(\mathbf{q}) \right|^{2}$

The electron density f(r) is a real function, which imposes a constraint on its Fourier transform. Specifically, the Fourier transform of a negative frequency must have the same magnitude as the corresponding positive frequency, but opposite phase

$F(-\mathbf{q}) = \left| F(-\mathbf{q}) \right| e^{i\phi(-\mathbf{q})} = F^{*}(-\mathbf{q}) = \left| F(\mathbf{q}) \right| e^{-i\phi(\mathbf{q})}$

In other words, the Fourier transforms of the negative and positive frequency vectors are complex conjugates of one another; in X-ray crystallography, these corresponding reflections are called Friedel mates. This allows one to measure the full Fourier transform from only half the reciprocal space, e.g., by slightly more than a 180° rotation (see next section). In symmetric crystals, other reflections may have the same intensity (Bijvoet mates); in such cases, one can measure even less of the reciprocal space, e.g., slightly more than 90°.

[edit] Ewald's sphere

Further information: Ewald's sphere

Each X-ray diffraction image represents only a slice, a spherical slice of reciprocal space, as may be seen by the Ewald sphere construction. Both k_out and k_in have the same length, due to the elastic scattering, since the wavelength has not changed. Therefore, they may be represented as two radial vectors in a sphere in reciprocal space, which shows the values of q that are sampled in a given diffraction image. Since there is a slight spread in the incoming wavelengths of the incoming X-ray beam, the values of |F(q)| can be measured only for q vectors located between the two spheres corresponding to those radii. Therefore, to obtain a full set of Fourier transform data, it is necessary to rotate the crystal through slightly more than 180°, or sometimes less if sufficient symmetry is present. A full 360° rotation is not needed because of a symmetry intrinsic to the Fourier transforms of real functions (such as the electron density), but "slightly more" than 180° is needed to cover all of reciprocal space within a given resolution because of the curvature of the Ewald sphere (add Figure to illustrate this). In practice, the crystal is rocked by a small amount (0.25-1°) to incorporate reflections near the boundaries of the spherical Ewald shells.

[edit] Patterson function

Further information: Patterson function

A well-known result of Fourier transforms is the autocorrelation theorem, which states that the autocorrelation c(r) of a function f(r)

$c(\mathbf{r}) = \int d\mathbf{x} f(\mathbf{x}) f(\mathbf{x} + \mathbf{r}) = \int \frac{d\mathbf{q}}{\left(2\pi\right)^{3}} C(\mathbf{q}) e^{i\mathbf{q}\cdot\mathbf{r}}$

has a Fourier transform C(q)that is the squared magnitude of F(q)

$C(\mathbf{q}) = \left| F(\mathbf{q}) \right|^{2}$

Therefore, the autocorrelation function c(r) of the electron density (also known as the Patterson function) can be computed directly from the reflection intensities, without computing the phases. In principle, this could be used to determined the crystal structure directly; however, it is difficult to realize in practice. The autocorrelation function corresponds to the distribution of vectors between atoms in the crystal; thus, a crystal of N atoms in its unit cell may have N(N-1) peaks in its Patterson function. Given the inevitable errors in measuring the intensities, and the mathematical difficulties of reconstructing atomic positions from the interatomic vectors, this technique is rarely used to solve structures, except for the simplest crystals.