Editing X-ray crystallography (section)

==Methods==

===Overview===
{{See also|Resolution (electron density)}}[[File:X ray diffraction.png|thumb|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 their [[crystal structure|asymmetric 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.<ref>{{cite journal |vauthors=Jones N |date=January 2014 |title=Crystallography: Atomic secrets |journal=Nature |volume=505 |issue=7485 |pages=602–603 |bibcode=2014Natur.505..602J |doi=10.1038/505602a |pmid=24476871 |doi-access=free}}</ref>

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&nbsp;mm in all dimensions), pure in composition and regular in structure, with no significant internal [[crystal defect|imperfections]] such as cracks or [[crystal twinning|twinning]].<ref>{{cite book |last1=Ladd |first1=M. F. C. |last2=Palmer |first2=Rex A. |title=Structure Determination by X-ray Crystallography: Analysis by X-rays and Neutrons |date=2013 |publisher=Springer |location=Boston, MA |isbn=978-1-4614-3956-1 |pages=228-229 |edition=5th 2013}}</ref>

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.<ref>{{Cite web|url=https://www.imrtest.com/tests/morphology-xrd-analysis|title=Morphology XRD Analysis {{!}} IMR TEST LABS|website=www.imrtest.com|access-date=2018-04-30}}{{Dead link|date=July 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> 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.<ref>{{cite book |last1=Ladd |first1=M. F. C. |last2=Palmer |first2=Rex A. |title=Structure Determination by X-ray Crystallography: Analysis by X-rays and Neutrons |date=2013 |publisher=Springer |location=Boston, MA |isbn=978-1-4614-3956-1 |page=200 |edition=5th 2013}}</ref>

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 a ''[[crystal structure]]''—is usually stored in a public database.<ref>{{cite book |last1=Ladd |first1=Mark F. c |title=Structure determination by x-ray crystallography: analysis by x-rays and neutrons |date=2012 |publisher=Springer |location=New York |isbn=978-1-4614-3956-1 |page=417}}</ref>

===Crystallization===
{{further|Crystallization|Recrystallization (chemistry)#Single perfect crystals (for X-ray analysis)|Protein crystallization}}
[[File:Protein crystal.jpg|thumb|left|A protein crystal seen under a [[microscope]]. 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 crystal [[annealing (metallurgy)|annealing]]<ref>{{cite journal | vauthors = Harp JM, Timm DE, Bunick GJ | title = Macromolecular crystal annealing: overcoming increased mosaicity associated with cryocrystallography | journal = Acta Crystallographica. Section D, Biological Crystallography | volume = 54 | issue = Pt 4 | pages = 622–628 | date = July 1998 | pmid = 9761858 | doi = 10.1107/S0907444997019008 | bibcode = 1998AcCrD..54..622H }}</ref><ref>{{cite journal | vauthors = Harp JM, Hanson BL, Timm DE, Bunick GJ | title = Macromolecular crystal annealing: evaluation of techniques and variables | journal = Acta Crystallographica. Section D, Biological Crystallography | volume = 55 | issue = Pt 7 | pages = 1329–1334 | date = July 1999 | pmid = 10393299 | doi = 10.1107/S0907444999005442 | bibcode = 1999AcCrD..55.1329H }}</ref><ref>{{cite book | vauthors = Hanson BL, Harp JM, Bunick GJ | title = Macromolecular Crystallography, Part C | chapter = The well-tempered protein crystal: annealing macromolecular crystals | volume = 368|pages=217–35 | date = 2003 | pmid = 14674276 | doi = 10.1016/S0076-6879(03)68012-2 | isbn = 978-0-12-182271-2 | series = Methods in Enzymology }}</ref> and other methods. However, in many cases, obtaining a diffraction-quality crystal is the chief barrier to solving its atomic-resolution structure.<ref>{{cite journal | vauthors = Geerlof A, Brown J, Coutard B, Egloff MP, Enguita FJ, Fogg MJ, Gilbert RJ, Groves MR, Haouz A, Nettleship JE, Nordlund P, Owens RJ, Ruff M, Sainsbury S, Svergun DI, Wilmanns M | display-authors = 6 | title = The impact of protein characterization in structural proteomics | journal = Acta Crystallographica. Section D, Biological Crystallography | volume = 62 | issue = Pt 10 | pages = 1125–1136 | date = October 2006 | pmid = 17001090 | pmc = 7161605 | doi = 10.1107/S0907444906030307 | bibcode = 2006AcCrD..62.1125G | doi-access = free }}</ref>

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 as [[chemical vapor deposition]] and [[Recrystallization (chemistry)#Single perfect crystals (for X-ray analysis)|recrystallization]]. 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 larger [[RNA]] molecules cannot be crystallized if their tertiary structure has been [[Denaturation (biochemistry)|unfolded]]; therefore, the range of crystallization conditions is restricted to solution conditions in which such molecules remain folded.{{cn|date=July 2024}}

[[File:CrystalDrops.svg|thumb|upright|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 by ''growth'' of that crystallite, ideally to a diffraction-quality crystal.<ref>{{cite journal | vauthors = Chernov AA | title = Protein crystals and their growth | journal = Journal of Structural Biology | volume = 142 | issue = 1 | pages = 3–21 | date = April 2003 | pmid = 12718915 | doi = 10.1016/S1047-8477(03)00034-0 }}</ref><ref>{{Cite web|url=http://xray.bmc.uu.se/terese/tutorials.html|title=Protein crystallization Tutorial|vauthors=Bergfors T|date=2016|access-date=2020-01-17|archive-date=2019-12-25|archive-url=https://web.archive.org/web/20191225142449/http://xray.bmc.uu.se/terese/tutorials.html|url-status=live}}</ref> 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 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. 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.<ref name="Chayen">{{cite journal | vauthors = Chayen N |title=Limitations of crystallizing under oil |journal = Cell|year=1997 |volume=5 |issue=10 |pages=1269–1274 |doi=10.1016/s0969-2126(97)00279-7 |pmid=9351804 |doi-access=free }}</ref>

It is difficult to predict good conditions for nucleation or growth of well-ordered crystals.<ref>{{cite journal | vauthors = Rupp B, Wang J | title = Predictive models for protein crystallization | journal = Methods | volume = 34 | issue = 3 | pages = 390–407 | date = November 2004 | pmid = 15325656 | doi = 10.1016/j.ymeth.2004.03.031 }}</ref> 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.<ref>{{cite journal | vauthors = Chayen NE | title = Methods for separating nucleation and growth in protein crystallisation | journal = Progress in Biophysics and Molecular Biology | volume = 88 | issue = 3 | pages = 329–337 | date = July 2005 | pmid = 15652248 | doi = 10.1016/j.pbiomolbio.2004.07.007 | doi-access = free }}</ref> 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 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 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 100 [[nanoliter]]s 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 1 [[microliter]]).<ref>{{cite journal | vauthors = Stock D, Perisic O, Löwe J | title = Robotic nanolitre protein crystallisation at the MRC Laboratory of Molecular Biology | journal = Progress in Biophysics and Molecular Biology | volume = 88 | issue = 3 | pages = 311–327 | date = July 2005 | pmid = 15652247 | doi = 10.1016/j.pbiomolbio.2004.07.009 | doi-access = free }}</ref>

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.{{citation needed|date=March 2014}} Crystals can be marred by [[Crystal twinning|twinning]], 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.{{cn|date=July 2024}}

===Data collection===

====Mounting the crystal====
[[File:Kappa goniometer animation.ogg|thumb|left|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 (the [[mother 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 with [[liquid nitrogen]].<ref>{{cite book | vauthors = Jeruzalmi D | title = Macromolecular Crystallography Protocols, Volume 2 | chapter = First analysis of macromolecular crystals: biochemistry and x-ray diffraction | series = Methods in Molecular Biology | volume = 364|pages=43–62 | date = 2006 | pmid = 17172760 | doi = 10.1385/1-59745-266-1:43 | isbn = 1-59745-266-1 }}</ref> 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.<ref>{{cite journal | vauthors = Helliwell JR | title = Protein crystal perfection and its application | journal = Acta Crystallographica. Section D, Biological Crystallography | volume = 61 | issue = Pt 6 | pages = 793–798 | date = June 2005 | pmid = 15930642 | doi = 10.1107/S0907444905001368 | bibcode = 2005AcCrD..61..793H | author-link = John R. Helliwell | doi-access = free }}</ref> This pre-soak may itself cause the crystal to crack, ruining it for crystallography. Generally, successful cryo-conditions are identified by trial and error.{{cn|date=July 2024}}

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 ~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.{{cn|date=July 2024}}

==== Recording the reflections ====
[[File:X-ray diffraction pattern 3clpro.jpg|thumb|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 with [[photographic film]], an area detector (such as a [[hybrid pixel detector|pixel 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 of the crystal that would render it unfit for solving the structure can also be diagnosed quickly at this point.{{cn|date=July 2024}}

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 cover [[reciprocal space]], due to the curvature of the [[Ewald 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.{{cn|date=July 2024}}

Multiple data sets may be necessary for certain [[Phase problem|phasing]] 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.<ref>{{cite journal |vauthors=Ravelli RB, Garman EF |date=October 2006 |title=Radiation damage in macromolecular cryocrystallography |journal=Current Opinion in Structural Biology |volume=16 |issue=5 |pages=624–629 |doi=10.1016/j.sbi.2006.08.001 |pmid=16938450}}</ref>

=== Crystal symmetry, unit cell, and image scaling ===
{{further|Space group}}

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 with ''indexing'' 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., its ''[[space 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 an ''autoindexing'' routine.<ref>{{cite journal |vauthors=Powell HR |date=October 1999 |title=The Rossmann Fourier autoindexing algorithm in MOSFLM |journal=Acta Crystallographica. Section D, Biological Crystallography |volume=55 |issue=Pt 10 |pages=1690–1695 |bibcode=1999AcCrD..55.1690P |doi=10.1107/S0907444999009506 |pmid=10531518 |doi-access=free}}</ref> 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. 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-related [[R-factor (crystallography)|R-factor]], a reliability index based upon how similar are the measured intensities of symmetry-equivalent reflections,{{clarify|date=February 2015}} thus assessing the quality of the data.

=== Initial phasing ===
{{further|Phase problem}}

The intensity of each diffraction 'spot' is proportional to the modulus squared of the [[structure factor]]. The structure factor  is a [[complex number]] containing information relating to both the [[amplitude]] and [[Phase (waves)|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 in a variety of ways:
* '''''[[Ab initio]]'' phasing''' or '''[[Direct methods (crystallography)|direct methods]]'''&nbsp;– 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&nbsp;Å (140&nbsp;[[picometre|pm]]), [[Direct methods (crystallography)|direct methods]] can be used to obtain phase information, by exploiting known phase relationships between certain groups of reflections.<ref>{{cite journal |vauthors=Hauptman H |date=October 1997 |title=Phasing methods for protein crystallography |journal=Current Opinion in Structural Biology |volume=7 |issue=5 |pages=672–680 |doi=10.1016/S0959-440X(97)80077-2 |pmid=9345626}}</ref><ref>{{cite journal |vauthors=Usón I, Sheldrick GM |date=October 1999 |title=Advances in direct methods for protein crystallography |journal=Current Opinion in Structural Biology |volume=9 |issue=5 |pages=643–648 |doi=10.1016/S0959-440X(99)00020-2 |pmid=10508770 |doi-access=free}}</ref>
* '''[[Molecular replacement]]'''&nbsp;– 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.<ref name="Taylor">{{cite journal |vauthors=Taylor G |date=November 2003 |title=The phase problem |journal=Acta Crystallographica. Section D, Biological Crystallography |volume=59 |issue=Pt 11 |pages=1881–1890 |bibcode=2003AcCrD..59.1881T |doi=10.1107/S0907444903017815 |pmid=14573942 |doi-access=free}}</ref>
* '''[[Anomalous X-ray scattering]]''' (''[[Multi-wavelength anomalous dispersion|MAD]] or [[Single wavelength anomalous dispersion|SAD phasing]]'')&nbsp;– the X-ray wavelength may be scanned past an absorption edge{{efn|The absorption edge is originally known from [[X-ray absorption spectroscopy]]. See {{cite web |title=X-ray Anomalous Scattering |url=http://skuld.bmsc.washington.edu/scatter/ |website=skuld.bmsc.washington.edu}} for a guide to anomalous scattering.}} 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 a [[methionine]] auxotroph (a host incapable of synthesizing methionine) in a media rich in seleno-methionine, which contains [[selenium]] 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.<ref>{{cite journal |vauthors=Ealick SE |date=October 2000 |title=Advances in multiple wavelength anomalous diffraction crystallography |journal=Current Opinion in Chemical Biology |volume=4 |issue=5 |pages=495–499 |doi=10.1016/S1367-5931(00)00122-8 |pmid=11006535|doi-access=free }}</ref>
* '''Heavy atom methods''' ([[multiple isomorphous replacement]])&nbsp;– If electron-dense metal atoms can be introduced into the crystal, [[Direct methods (crystallography)|direct methods]] or [[Patterson function|Patterson-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.<ref name="Taylor" />

=== Model building and phase refinement ===
[[File:Helix electron density myoglobin 2nrl 17-32.jpg|thumb|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&nbsp;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.<ref>From PDB file 2NRL, residues 17–32.</ref>]] [[File:Fitting a model into electron density.gif|thumb|3D depiction of electron density (blue) of a ligand (orange) bound to a binding site in a protein (yellow).<ref>{{cite web |title=Garman lab: Interconversion of lysosomal enzyme specificities – Proteopedia, life in 3D |url=http://proteopedia.org/wiki/index.php/Garman_lab:_Interconversion_of_lysosomal_enzyme_specificities |access-date=2018-11-28 |website=proteopedia.org |archive-date=2018-11-28 |archive-url=https://web.archive.org/web/20181128212454/http://proteopedia.org/wiki/index.php/Garman_lab:_Interconversion_of_lysosomal_enzyme_specificities |url-status=live }}</ref> The electron density is obtained from experimental data, and the ligand is modeled into this electron density.]]
{{further|Molecular modelling{{!}}Molecular modeling}}

Having obtained initial phases, an initial model can be built. The atomic positions in the model and their respective [[Debye-Waller factor]]s (or '''B'''-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 an [[R-factor (crystallography)|''R''-factor]] defined as

:<math>R = \frac{\sum_{\text{all reflections}} \left|F_\text{obs} - F_\text{calc} \right|}{\sum_{\text{all reflections}} \left|F_\text{obs} \right|},</math>

where ''F'' is the [[structure factor]]. A similar quality criterion is ''R''<sub>free</sub>, 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''<sub>free</sub> should be approximately the resolution in angstroms divided by 10; thus, a data-set with 2&nbsp;Å resolution should yield a final ''R''<sub>free</sub> ~ 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 using ''omit maps''.<ref name="pmid25554228">{{cite journal |last1=Lamb |first1=AL |last2=Kappock |first2=TJ |last3=Silvaggi |first3=NR |date=April 2015 |title=You are lost without a map: Navigating the sea of protein structures. |journal=Biochimica et Biophysica Acta (BBA) - Proteins and Proteomics |volume=1854 |issue=4 |pages=258–68 |doi=10.1016/j.bbapap.2014.12.021 |pmc=5051661 |pmid=25554228}}</ref>

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.

=== Disorder ===
{{main|Crystallographic disorder}}

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 of [[bond stretch isomer]]ism.<ref>{{cite journal |vauthors=Parkin G |year=1993 |title=Bond-stretch isomerism in transition metal complexes: a reevaluation of crystallographic data |journal=Chem. Rev. |volume=93 |issue=3 |pages=887–911 |doi=10.1021/cr00019a003}}</ref>  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.

=== Applied computational data analysis ===
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 using [[least squares]] based minimization algorithm. Most available tools allowing phase identification and structural refinement are based on the [[Rietveld refinement|Rietveld method]],<ref>{{Cite journal |vauthors=Rietveld HM |date=1969-06-02 |title=A profile refinement method for nuclear and magnetic structures |journal=Journal of Applied Crystallography |volume=2 |issue=2 |pages=65–71 |bibcode=1969JApCr...2...65R |doi=10.1107/S0021889869006558 |doi-access=free}}</ref><ref>{{Cite book |title=The Rietveld Method |vauthors=Young RA |date=1993 |publisher=International Union of Crystallograhy |isbn=0198555776 |location=[Chester, England] |oclc=26299196}}</ref> some of them being open and free software such as FullProf Suite,<ref>{{Cite web |title=IUCr |url=https://www.iucr.org/resources/commissions/powder-diffraction/newsletter |access-date=2019-04-06 |website=www.iucr.org |archive-date=2019-04-06 |archive-url=https://web.archive.org/web/20190406103706/https://www.iucr.org/resources/commissions/powder-diffraction/newsletter |url-status=live }}</ref><ref>{{Cite web |title=Fullprof |url=https://www.ill.eu/sites/fullprof/ |access-date=2019-04-06 |website=www.ill.eu |archive-date=2019-04-02 |archive-url=https://web.archive.org/web/20190402200238/https://www.ill.eu/sites/fullprof/ |url-status=live }}</ref> Jana2006,<ref>{{Cite journal |vauthors=Petříček V, Dušek M, Palatinus L |date=2014-01-01 |title=Crystallographic Computing System JANA2006: General features |journal=Zeitschrift für Kristallographie – Crystalline Materials |volume=229 |issue=5 |pages=345–352 |doi=10.1515/zkri-2014-1737 |issn=2196-7105 |s2cid=101692863}}</ref> MAUD,<ref>{{Cite journal |vauthors=Lutterotti L |date=February 2010 |title=Total pattern fitting for the combined size–strain–stress–texture determination in thin film diffraction |journal=Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms |volume=268 |issue=3–4 |pages=334–340 |bibcode=2010NIMPB.268..334L |doi=10.1016/j.nimb.2009.09.053 |issn=0168-583X}}</ref><ref>{{Citation |title=Tenth European Powder Diffraction Conference |pages=125–130 |year=2007 |chapter=Rietveld texture analysis from diffraction images |publisher=OLDENBOURG WISSENSCHAFTSVERLAG |doi=10.1524/9783486992540-020 |isbn=9783486992540 |vauthors=Lutterotti L, Bortolotti M, Ischia G, Lonardelli I, Wenk HR}}</ref><ref>{{Cite journal |vauthors=Lutterotti L, Matthies S, Wenk HR, Schultz AS, Richardson Jr JW |date=1997-01-15 |title=Combined texture and structure analysis of deformed limestone from time-of-flight neutron diffraction spectra |journal=Journal of Applied Physics |volume=81 |issue=2 |pages=594–600 |bibcode=1997JAP....81..594L |doi=10.1063/1.364220 |issn=0021-8979}}</ref> Rietan,<ref>{{Cite web |title=Distribution Files for the RIETAN-FP-VENUS Package |url=http://fujioizumi.verse.jp/download/download_Eng.html |access-date=2019-04-06 |website=fujioizumi.verse.jp |archive-date=2019-08-10 |archive-url=https://web.archive.org/web/20190810120240/http://fujioizumi.verse.jp/download/download_Eng.html |url-status=live }}</ref> GSAS,<ref>{{Cite journal |vauthors=Toby BH, Von Dreele RB |date=2013-03-14 |title=GSAS-II: the genesis of a modern open-source all purpose crystallography software package |journal=Journal of Applied Crystallography |volume=46 |issue=2 |pages=544–549 |bibcode=2013JApCr..46..544T |doi=10.1107/s0021889813003531 |issn=0021-8898}}</ref> etc. while others are available under commercial licenses such as Diffrac.Suite TOPAS,<ref>{{Cite web |title=DIFFRAC.SUITE TOPAS - XRD Software, X-ray diffraction |url=https://www.bruker.com/products/x-ray-diffraction-and-elemental-analysis/x-ray-diffraction/xrd-software/topas.html |access-date=2019-04-06 |website=Bruker.com |archive-date=2019-04-02 |archive-url=https://web.archive.org/web/20190402174859/https://www.bruker.com/products/x-ray-diffraction-and-elemental-analysis/x-ray-diffraction/xrd-software/topas.html |url-status=live }}</ref> Match!,<ref>{{Cite web |title=Match! – Phase Identification from Powder Diffraction |url=http://www.crystalimpact.com/match/ |access-date=2019-04-06 |website=www.crystalimpact.com |archive-date=2019-04-02 |archive-url=https://web.archive.org/web/20190402175018/http://www.crystalimpact.com/match/ |url-status=live }}</ref> etc. Most of these tools also allow [[Le Bail method|Le 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,<ref>{{Cite journal |vauthors=Casas-Cabanas M, Reynaud M, Rikarte J, Horbach P, Rodríguez-Carvajal J |date=2016-12-01 |title=FAULTS: a program for refinement of structures with extended defects |journal=Journal of Applied Crystallography |volume=49 |issue=6 |pages=2259–2269 |bibcode=2016JApCr..49.2259C |doi=10.1107/S1600576716014473 |issn=1600-5767}}</ref> which allows the refinement of structures with planar defects (e.g. stacking faults, twinnings, intergrowths).

=== 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 [[Cambridge Structural Database]] (for small molecules), the [[Inorganic Crystal Structure Database (ICSD)]] (for inorganic compounds) or the [[Protein 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.