X-ray crystallography is the experimental science of determining the atomic and molecular structure of acrystal, in which the crystalline structure causes a beam of incidentX-rays todiffract in specific directions. By measuring the angles and intensities of theX-ray diffraction, acrystallographer can produce a three-dimensional picture of the density ofelectrons within the crystal and the positions of the atoms, as well as theirchemical bonds,crystallographic disorder, and other information.
X-ray crystallography has been fundamental in the development of many scientific fields. In its first decades of use, this method determined the size ofatoms, the lengths and types of chemical bonds, and the atomic-scale differences between various materials, especially minerals andalloys. The method has also revealed the structure and function of many biological molecules, includingvitamins, drugs,proteins andnucleic acids such asDNA. X-ray crystallography is still the primary method for characterizing the atomic structure of materials and in differentiating materials that appear similar in other experiments. X-raycrystal structures can also help explain unusualelectronic orelastic properties of a material, shed light on chemical interactions and processes, or serve as the basis fordesigning pharmaceuticals against diseases.
Modern work involves a number of steps all of which are important. The preliminary steps include preparing good quality samples, careful recording of the diffracted intensities, and processing of the data to remove artifacts. A variety of different methods are then used to obtain an estimate of the atomic structure, generically called direct methods. With an initial estimate further computational techniques such as those involving difference maps are used to complete the structure. The final step is a numerical refinement of the atomic positions against the experimental data, sometimes assisted byab-initio calculations. In almost all cases new structures are deposited in databases available to the international community.
Drawing of square (A) and hexagonal (B) packing fromKepler's work,Strena seu de Nive Sexangula.
The hexagonal symmetry of snowflakes results from thetetrahedral arrangement ofhydrogen bonds about each water molecule.
Crystals, though long admired for their regularity andsymmetry, were not investigated scientifically until the 17th century.Johannes Kepler hypothesized in his workStrena seu de Nive Sexangula (A New Year's Gift of Hexagonal Snow) (1611) that the hexagonal symmetry ofsnowflake crystals was due to a regular packing of spherical water particles.[1] The Danish scientistNicolas Steno (1669) pioneered experimental investigations of crystal symmetry. Steno showed that the angles between the faces are the same in every exemplar of a particular type of crystal (law of constancy of interfacial angles).[2]René Just Haüy (1784) discovered that every face of a crystal can be described by simple stacking patterns of blocks of the same shape and size (law of decrements). Hence,William Hallowes Miller in 1839 was able to give each face a unique label of three small integers, theMiller indices which remain in use for identifying crystal faces. Haüy's study led to the idea that crystals are a regular three-dimensional array (aBravais lattice) of atoms andmolecules; a singleunit cell is repeated indefinitely along three principal directions. In the 19th century, a complete catalog of the possible symmetries of a crystal was worked out byJohan Hessel,[3]Auguste Bravais,[4]Evgraf Fedorov,[5]Arthur Schönflies[6] and (belatedly)William Barlow (1894). Barlow proposed several crystal structures in the 1880s that were validated later by X-ray crystallography;[7] however, the available data were too scarce in the 1880s to accept his models as conclusive.
Model of the arrangement of water molecules in ice, revealing thehydrogen bonds (1) that hold the solid together.
Wilhelm Röntgen discovered X-rays in 1895.[8] Physicists were uncertain of the nature of X-rays, but suspected that they were waves ofelectromagnetic radiation. TheMaxwell theory ofelectromagnetic radiation was well accepted, and experiments byCharles Glover Barkla showed that X-rays exhibited phenomena associated with electromagnetic waves, including transversepolarization andspectral lines akin to those observed in the visible wavelengths. Barkla created the x-ray notation for sharp spectral lines, noting in 1909 two separate energies, at first naming them "A" and "B" and then supposing that there may be lines prior to "A", he started an alphabet numbering beginning with "K."[9][10] Single-slit experiments in the laboratory ofArnold Sommerfeld suggested that X-rays had awavelength of about 1angstrom.[11] X-rays are not only waves but also have particle properties causing Sommerfeld to coin the nameBremsstrahlung for the continuous spectra when they were formed when electrons bombarded a material.[10]Albert Einstein introduced the photon concept in 1905,[12] but it was not broadly accepted until 1922,[13][14] whenArthur Compton confirmed it by the scattering of X-rays from electrons.[15] The particle-like properties of X-rays, such as their ionization of gases, had promptedWilliam Henry Bragg to argue in 1907 that X-rays werenot electromagnetic radiation.[16][17][18][19] Bragg's view proved unpopular and the observation ofX-ray diffraction byMax von Laue in 1912[20] confirmed that X-rays are a form of electromagnetic radiation.
One of the copper sulfate X-ray interference patterns published in Von Laue's 1912 paper[20].
The idea that crystals could be used as adiffraction grating forX-rays arose in 1912 in a conversation betweenPaul Peter Ewald andMax von Laue in theEnglish Garden in Munich. Ewald had proposed a resonator model of crystals for his thesis, but this model could not be validated usingvisible light, since the wavelength was much larger than the spacing between the resonators. Von Laue realized that electromagnetic radiation of a shorter wavelength was needed, 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 acopper sulfate crystal and record its diffraction on aphotographic 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. The results were presented to theBavarian Academy of Sciences and Humanities in June 1912 as "Interferenz-Erscheinungen bei Röntgenstrahlen" (Interference phenomena in X-rays).[20][21] 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 theNobel Prize in Physics in 1914.[22]
Although diamonds (top left) andgraphite (top right) are identical in chemical composition—being both purecarbon—X-ray crystallography revealed the arrangement of their atoms (bottom). In diamond, the carbon atoms are arrangedtetrahedrally and held together by singlecovalent bonds. By contrast, graphite is composed of stacked sheets. Within the sheet, the bonding is covalent and has hexagonal symmetry, but there are no covalent bonds between the sheets.
After Von Laue's pioneering research, the field developed rapidly, most notably by physicistsWilliam Lawrence Bragg and his fatherWilliam Henry Bragg. In 1912–1913, the younger Bragg developedBragg's law, which connects the scattering with evenly spaced planes within a crystal.[8][23][24][25] The Braggs, father and son, shared the 1915 Nobel Prize in Physics for their work in crystallography. The earliest structures were generally simple; as computational and experimental methods improved over the next decades, it became feasible to deduce reliable atomic positions for more complicated arrangements of atoms.
The earliest structures were simple inorganic crystals and minerals, but even these revealed fundamental laws of physics and chemistry. The first atomic-resolution structure to be "solved" (i.e., determined) in 1914 was that oftable salt.[26][27][28] The distribution of electrons in the table-salt structure showed that crystals are not necessarily composed ofcovalently bonded molecules, and proved the existence ofionic compounds.[29] The structure of diamond was solved in the same year,[30][31] proving the tetrahedral arrangement of its chemical bonds and showing that the length of C–C single bond was about 1.52 angstroms. Other early structures included copper,[32]calcium fluoride (CaF2, also known asfluorite),calcite (CaCO3) andpyrite (FeS2)[33] in 1914;spinel (MgAl2O4) in 1915;[34][35] therutile andanatase forms oftitanium dioxide (TiO2) in 1916;[36]pyrochroite (Mn(OH)2) and, by extension,brucite (Mg(OH)2) in 1919.[37][38] Also in 1919,sodium nitrate (NaNO3) and caesium dichloroiodide (CsICl2) were determined byRalph Walter Graystone Wyckoff, and thewurtzite (hexagonal ZnS) structure was determined in 1920.[39]
The structure ofgraphite was solved in 1916[40] by the related method ofpowder diffraction,[41] which was developed byPeter Debye andPaul Scherrer and, independently, byAlbert Hull in 1917.[42] The structure of graphite was determined from single-crystal diffraction in 1924 by two groups independently.[43][44] Hull also used the powder method to determine the structures of various metals, such as iron[45] and magnesium.[46]
X-ray crystallography has led to a better understanding ofchemical bonds andnon-covalent interactions. The initial studies revealed the typical radii of atoms, and confirmed many theoretical models of chemical bonding, such as the tetrahedral bonding of carbon in the diamond structure,[30] the octahedral bonding of metals observed in ammonium hexachloroplatinate (IV),[47] and the resonance observed in the planar carbonate group[33] and in aromatic molecules.[48]Kathleen Lonsdale's 1928 structure ofhexamethylbenzene[49] established the hexagonal symmetry ofbenzene 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 ofresonance between chemical bonds, which had profound consequences for the development of chemistry.[50] Her conclusions were anticipated byWilliam Henry Bragg, who published models ofnaphthalene andanthracene in 1921 based on other molecules, an early form ofmolecular replacement.[48][51]
The application of X-ray crystallography tomineralogy began with the structure ofgarnet, which was determined in 1924 by Menzer. A systematic X-ray crystallographic study of thesilicates was undertaken in the 1920s. This study showed that, as theSi/O ratio is altered, the silicate crystals exhibit significant changes in their atomic arrangements. Machatschki extended these insights to minerals in which aluminium substitutes for thesilicon atoms of the silicates. The first application of X-ray crystallography tometallurgy also occurred in the mid-1920s.[78][79][80][81][82][83] Most notably,Linus Pauling's structure of the alloy Mg2Sn[84] led to his theory of the stability and structure of complex ionic crystals.[85] Many complicatedinorganic andorganometallic systems have been analyzed using single-crystal methods, such asfullerenes,metalloporphyrins, and other complicated compounds. Single-crystal diffraction is also used in thepharmaceutical industry. TheCambridge Structural Database contains over 1,000,000 structures as of June 2019; most of these structures were determined by X-ray crystallography.[86]
The three-dimensional structure ofpenicillin, solved byDorothy Crowfoot Hodgkin in 1945. The green, red, yellow and blue spheres represent atoms ofcarbon,oxygen,sulfur andnitrogen, respectively. The white spheres representhydrogen, which were determined mathematically rather than by the X-ray analysis.
Ribbon diagram of the structure ofmyoglobin, showingalpha helices. Suchproteins are long, linearmolecules with thousands of atoms; yet the relative position of each atom has been determined with sub-atomic resolution by X-ray crystallography. Since it is difficult to visualize all the atoms at once, the ribbon shows the rough path of the protein'sbackbone from its N-terminus to its C-terminus.
Crystal structures of proteins (which are irregular and hundreds of times larger than cholesterol) began to be solved in the late 1950s, beginning with the structure ofsperm whalemyoglobin bySir John Cowdery Kendrew,[88] for which he shared theNobel Prize in Chemistry withMax Perutz in 1962.[89] Since that success, 190,000 X-ray crystal structures of proteins, nucleic acids and other biological molecules have been determined.[90] The nearest competing method in number of structures analyzed isnuclear magnetic resonance (NMR) spectroscopy, which has resolved less than one tenth as many.[91] Crystallography can solve structures of arbitrarily large molecules, whereas solution-state NMR is restricted to relatively small ones (less than70 kDa). X-ray crystallography is used routinely to determine how a pharmaceutical drug interacts with its protein target and what changes might improve it.[92] However, intrinsicmembrane proteins remain challenging to crystallize because they require detergents or otherdenaturants to solubilize them in isolation, and such detergents often interfere with crystallization. Membrane proteins are a large component of thegenome, and include many proteins of great physiological importance, such asion channels andreceptors.[93][94]Helium cryogenics are used to prevent radiation damage in protein crystals.[95]
Workflow for solving the structure of a molecule by X-ray crystallography.
Two limiting cases of X-ray crystallography—"small-molecule" (which includes continuous inorganic solids) and "macromolecular" crystallography—are often used. Small-molecule crystallography typically involves crystals with fewer than 100 atoms in theirasymmetric unit; such crystal structures are usually so well resolved that the atoms can be discerned as isolated "blobs" of electron density. In contrast, macromolecular crystallography often involves tens of thousands of atoms in the unit cell. Such crystal structures are generally less well-resolved; 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 and proteins with hundreds of thousands of atoms, through improved crystallographic imaging and technology.[96]
The technique of single-crystal 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 (typically larger than 0.1 mm in all dimensions), pure in composition and regular in structure, with no significant internalimperfections such as cracks ortwinning.[97]
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 pattern of reflections. The angles and intensities of diffracted X-rays are measured, with each compound having a unique diffraction pattern.[98] As the crystal is gradually rotated, previous reflections disappear and new ones appear; the intensity of every spot is recorded 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 typically containing tens of thousands of reflections.[99]
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 of the atomic arrangement—now called acrystal structure—is usually stored in a public database.[100]
A protein crystal seen under amicroscope. Crystals used in X-ray crystallography may be smaller than a millimeter across.
Although crystallography can be used to characterize the disorder in an impure or irregular crystal, crystallography generally requires a pure crystal of high regularity to solve the structure of a complicated arrangement of atoms. Pure, regular crystals can sometimes be obtained from natural or synthetic materials, such as samples of metals, minerals or other macroscopic materials. The regularity of such crystals can sometimes be improved with macromolecular crystalannealing[101][102][103] and other methods. However, in many cases, obtaining a diffraction-quality crystal is the chief barrier to solving its atomic-resolution structure.[104]
Small-molecule and macromolecular crystallography differ in the range of possible techniques used to produce diffraction-quality crystals. Small molecules generally have few degrees of conformational freedom, and may be crystallized by a wide range of methods, such aschemical vapor deposition andrecrystallization. By contrast, macromolecules generally have many degrees of freedom and their crystallization must be carried out so as to maintain a stable structure. For example, proteins and largerRNA molecules cannot be crystallized if their tertiary structure has beenunfolded; therefore, the range of crystallization conditions is restricted to solution conditions in which such molecules remain folded.[citation needed]
Three methods of preparing crystals, A: Hanging drop. B: Sitting drop. C: Microdialysis
Protein crystals are almost always grown in solution. The most common approach is to lower the solubility of its component molecules very gradually; if this is 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 bygrowth of that crystallite, ideally to a diffraction-quality crystal.[105][106] The solution conditions that favor the first step (nucleation) are not always the same conditions that favor the second step (subsequent growth). The solution conditions shoulddisfavor the first step (nucleation) butfavor 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. Other approaches involve crystallizing proteins under oil, where aqueous protein solutions are dispensed under liquid oil, and water evaporates through the layer of oil. Different oils have different evaporation permeabilities, therefore yielding changes in concentration rates from different percipient/protein mixture.[107]
It is difficult to predict good conditions for nucleation or growth of well-ordered crystals.[108] In practice, favorable conditions are identified byscreening; a very large batch of the molecules is prepared, and a wide variety of crystallization solutions are tested.[109] Hundreds, even thousands, of solution conditions are generally tried before finding the successful one. 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 theHofmeister series or chemicals that lower the dielectric constant of the solution, and still others contain large polymers such aspolyethylene 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 difficulty in obtaining such large quantities (milligrams) of crystallization-grade protein, robots have been developed that are capable of accurately dispensing crystallization trial drops that are in the order of 100nanoliters in volume. This means that 10-fold less protein is used per experiment when compared to crystallization trials set up by hand (in the order of 1microliter).[110]
Several factors are known to inhibit 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. Molecules that tend to self-assemble into regular helices are often unwilling to assemble into crystals. Crystals can be marred bytwinning, which can occur when a unit cell can pack equally favorably in multiple orientations; although recent advances in computational methods may allow solving the structure of some twinned crystals. 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.[citation needed]
For proteins, impurities have been shown to sometimes enhance and sometimes disrupt crystal growth. Oscillation with audible sound sometimes works. These issues may be related to thephase separation characteristic of protein crystalation.[111]
Animation showing the five motions possible with a four-circle kappa goniometer. The rotations about each of the four angles φ, κ, ω and 2θ leave the crystal within the X-ray beam, but change the crystal orientation. The detector (red box) can be slid closer or further away from the crystal, allowing higher resolution data to be taken (if closer) or better discernment of the Bragg peaks (if further away).
The crystal is mounted for measurements so that it may be held in the X-ray beam and rotated. There are several methods of mounting. In the past, crystals were loaded into glass capillaries with the crystallization solution (themother liquor). Crystals of small molecules are typically attached with oil or glue to a glass fiber or a loop, which is made of nylon or plastic and attached to a solid rod. Protein crystals are scooped up by a loop, then flash-frozen withliquid nitrogen.[112] This freezing reduces the radiation damage of the X-rays, as well as thermal motion (the Debye-Waller effect). However, untreated protein crystals often crack if flash-frozen; therefore, they are generally pre-soaked in a cryoprotectant solution before freezing.[113] This pre-soak may itself cause the crystal to crack, ruining it for crystallography. Generally, successful cryo-conditions are identified by trial and error.[citation needed]
The capillary or loop is mounted on agoniometer, 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 ~25 micrometers 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 perpendicular to the beam; the κ angle, about an axis at ~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.[citation needed]
An X-ray diffraction pattern of a crystallized enzyme. The pattern of spots (reflections) and the relative strength of each spot (intensities) can be used to determine the structure of the enzyme.
The relative intensities of the reflections provides information to determine the arrangement of molecules within the crystal in atomic detail. The intensities of these reflections may be recorded withphotographic film, an area detector (such as apixel detector) or with acharge-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 themosaicity of the crystal and its overall disorder, as observed in the peak widths. Some pathologies of the crystal that would render it unfit for solving the structure can also be diagnosed quickly at this point.[citation needed]
One set of spots is insufficient to reconstruct the whole crystal; it represents only a small slice of the full three dimensional set. To collect all the necessary information, the crystal must be rotated step-by-step through 180°, with an image recorded at every step; actually, slightly more than 180° is required to coverreciprocal space, due to the curvature of theEwald sphere. However, if the crystal has a higher symmetry, a smaller angular range such as 90° or 45° may be recorded. The rotation axis should 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.[citation needed]
Multiple data sets may be necessary for certainphasing methods. For example,multi-wavelength anomalous dispersion phasing requires that the scattering be recorded 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.[114]
The recorded series of two-dimensional diffraction patterns, each corresponding to a different crystal orientation, is converted into a three-dimensional set. Data processing begins withindexing the reflections. 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., itsspace group. Some space groups can be eliminated from the beginning. For example, reflection symmetries cannot be observed in chiral molecules; thus, only 65 space groups of 230 possible are allowed for protein molecules which are almost always chiral. Indexing is generally accomplished using anautoindexing routine.[115] Having assigned symmetry, the data is thenintegrated. This converts the hundreds of images containing the thousands of reflections into a single file, consisting of (at the very least) records of theMiller 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. These have to be merged and scaled using peaks that appear in two or more images (merging) and scaling so there is a consistent intensity scale. Optimizing the intensity scale is critical because the relative intensity of the peaks is the key information from which the structure is determined. The repetitive technique of crystallographic data collection and the often high symmetry of crystalline materials cause the diffractometer to record many symmetry-equivalent reflections multiple times. This allows calculating the symmetry-relatedR-factor, a reliability index based upon how similar are the measured intensities of symmetry-equivalent reflections,[clarification needed][116] thus assessing the quality of the data.
The intensity of each diffraction 'spot' is proportional to the modulus squared of thestructure factor. The structure factor is acomplex number containing information relating to both theamplitude andphase of awave. In order to obtain an interpretableelectron 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 thephase problem. Initial phase estimates can be obtained in a variety of ways:
Ab initio phasing ordirect 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.[117][118]
Molecular replacement – if a related structure is known, 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.[119]
Anomalous X-ray scattering (MAD orSAD phasing) – the X-ray wavelength may be scanned past an absorption edge[a] 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 hence the structure of the whole molecule. The most popular method of incorporating anomalous scattering atoms into proteins is to express the protein in amethionine auxotroph (a host incapable of synthesizing methionine) in a media rich in seleno-methionine, which containsselenium atoms. A multi-wavelength anomalous dispersion (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.[120]
Heavy atom methods (multiple isomorphous replacement) – If electron-dense metal atoms can be introduced into the crystal,direct methods orPatterson-space methods can be used to determine their location and to obtain initial phases. Such heavy atoms can be introduced 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 multi-wavelength anomalous dispersion 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 multi-wavelength anomalous dispersion phasing with selenomethionine.[119]
Structure of a protein alpha helix, with stick-figures for the covalent bonding within electron density for the crystal structure at ultra-high-resolution (0.91 Å). The density contours are in gray, the helix backbone in white, sidechains in cyan, O atoms in red, N atoms in blue, and hydrogen bonds as green dotted lines.[121]3D depiction of electron density (blue) of a ligand (orange) bound to a binding site in a protein (yellow).[122] The electron density is obtained from experimental data, and the ligand is modeled into this electron density.
Having obtained initial phases, an initial model can be built. The atomic positions in the model and their respectiveDebye-Waller factors (orB-factors, accounting for the thermal motion of the atom) 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 successive rounds of refinement are carried out. This iterative process continues until the correlation between the diffraction data and the model is maximized. The agreement is measured by anR-factor defined as
whereF is thestructure factor. A similar quality criterion isRfree, which is calculated from a subset (~10%) of reflections that were not included in the structure refinement. BothR factors depend on the resolution of the data. As a rule of thumb,Rfree should be approximately the resolution in angstroms divided by 10; thus, a data-set with 2 Å resolution should yield a finalRfree ~ 0.2. Chemical bonding features such as stereochemistry, hydrogen bonding and distribution of bond lengths and angles are complementary measures of the model quality. In iterative model building, it is common to encounter phase bias or model bias: because phase estimations come from the model, each round of calculated map tends to show density wherever the model has density, regardless of whether there truly is a density. This problem can be mitigated by maximum-likelihood weighting and checking usingomit maps.[123]
It may not be possible to observe every atom in the asymmetric unit. In many cases,crystallographic disorder smears 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 post-translational modifications that were not detected prior to the crystallization.
A common challenge in refinement of crystal structures results from crystallographic disorder. Disorder can take many forms but in general involves the coexistence of two or more species or conformations. Failure to recognize disorder results in flawed interpretation. Pitfalls from improper modeling of disorder are illustrated by the discounted hypothesis ofbond stretch isomerism.[124] Disorder is modelled with respect to the relative population of the components, often only two, and their identity. In structures of large molecules and ions, solvent and counterions are often disordered.
The use of computational methods for the powder X-ray diffraction data analysis is now generalized. It typically compares the experimental data to the simulated diffractogram of a model structure, taking into account the instrumental parameters, and refines the structural or microstructural parameters of the model usingleast squares based minimization algorithm. Most available tools allowing phase identification and structural refinement are based on theRietveld method,[125][126] some of them being open and free software such as FullProf Suite,[127][128] Jana2006,[129] MAUD,[130][131][132] Rietan,[133] GSAS,[134] etc. while others are available under commercial licenses such as Diffrac.Suite TOPAS,[135] Match!,[136] etc. Most of these tools also allowLe Bail refinement (also referred to as profile matching), that is, refinement of the cell parameters based on the Bragg peaks positions and peak profiles, without taking into account the crystallographic structure by itself. More recent tools allow the refinement of both structural and microstructural data, such as the FAULTS program included in the FullProf Suite,[137] which allows the refinement of structures with planar defects (e.g. stacking faults, twinnings, intergrowths).
Once the model of a molecule's structure has been finalized, it is often deposited in acrystallographic database such as theCambridge Structural Database (for small molecules), theInorganic Crystal Structure Database (ICSD) (for inorganic compounds) or theProtein Data Bank (for protein and sometimes nucleic acids). Many structures obtained in private commercial ventures to crystallize medicinally relevant proteins are not deposited in public crystallographic databases.
A number of women were pioneers in X-ray crystallography at a time when they were excluded from most other branches of physical science.[138]
Kathleen Lonsdale was a research student ofWilliam Henry Bragg, who had 11 women research students out of a total of 18. She is known for both her experimental and theoretical work. Lonsdale joined his crystallography research team at theRoyal Institution in London in 1923, and after getting married and having children, went back to work with Bragg as a researcher. She confirmed the structure of the benzene ring, carried out studies of diamond, was one of the first two women to be elected to theRoyal Society in 1945, and in 1949 was appointed the first female tenured professor of chemistry and head of the Department of crystallography atUniversity College London.[139] Lonsdale always advocated greater participation of women in science and said in 1970: "Any country that wants to make full use of all its potential scientists and technologists could do so, but it must not expect to get the women quite so simply as it gets the men.... It is utopian, then, to suggest that any country that really wants married women to return to a scientific career, when her children no longer need her physical presence, should make special arrangements to encourage her to do so?".[140] During this period, Lonsdale began a collaboration with William T. Astbury on a set of 230 space group tables which was published in 1924 and became an essential tool for crystallographers.
Molecular model of penicillin by Dorothy Hodgkin, 1945
In 1932Dorothy Hodgkin joined the laboratory of the physicist John Desmond Bernal, who was a former student of Bragg, in Cambridge, UK. She and Bernal took the first X-ray photographs of crystalline proteins. Hodgkin also played a role in the foundation of theInternational Union of Crystallography. She was awarded the Nobel Prize in Chemistry in 1964 for her work using X-ray techniques to study the structures of penicillin, insulin and vitamin B12. Her work on penicillin began in 1942 during the war and on vitamin B12 in 1948. While her group slowly grew, their predominant focus was on the X-ray analysis of natural products. She is the onlyBritish woman ever to have won a Nobel Prize in a science subject.
Photograph of DNA (photo 51), Rosalind Franklin, 1952
Rosalind Franklin worked on the X-ray photography of a DNA fibre that proved key toJames Watson andFrancis Crick's discovery of the double helix, for which they both won the Nobel Prize for Physiology or Medicine in 1962. Watson revealed in his autobiographic account of the discovery of the structure of DNA,The Double Helix,[141] that he had used Franklin's X-ray photograph without her permission. Franklin died of cancer in her 30s, before Watson received the Nobel Prize. Franklin also carried out important structural studies of carbon in coal and graphite, and of plant and animal viruses.
Isabella Karle of the United States Naval Research Laboratory developed an experimental approach to the mathematical theory of crystallography. Her work improved the speed and accuracy of chemical and biomedical analysis. Yet only her husband Jerome shared the 1985 Nobel Prize in Chemistry with Herbert Hauptman, "for outstanding achievements in the development of direct methods for the determination of crystal structures". Other prize-giving bodies have showered Isabella with awards in her own right.
Women have written many textbooks and research papers in the field of X-ray crystallography. For many years Lonsdale edited theInternational Tables for Crystallography, which provide information on crystal lattices, symmetry, and space groups, as well as mathematical, physical and chemical data on structures.Olga Kennard of theUniversity of Cambridge, founded and ran theCambridge Crystallographic Data Centre, an internationally recognized source of structural data on small molecules, from 1965 until 1997.Jenny Pickworth Glusker, a British scientist, co-authoredCrystal Structure Analysis: A Primer,[142] first published in 1971 and as of 2010 in its third edition.Eleanor Dodson, an Australian-born biologist, who began as Dorothy Hodgkin's technician, was the main instigator behindCCP4, the collaborative computing project that currently shares more than 250 software tools with protein crystallographers worldwide.
"For their contribution to the understanding of the connection between chemical structure and catalytic activity of the active centre of theribonuclease molecule"[148]
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