Contents 1 History 1.1 Early scientific history of crystals and X-rays
1.2
2 Contributions to chemistry and material science 2.1
3 Relationship to other scattering techniques 3.1 Elastic vs. inelastic scattering
3.2 Other
4 Methods 4.1 Overview of single-crystal
4.1.1 Procedure 4.1.2 Limitations 4.2 Crystallization 4.3 Data collection 4.3.1 Mounting the crystal
4.3.2
4.3.2.1 Rotating anode
4.3.2.2
4.3.3 Recording the reflections 4.4 Data analysis 4.4.1
4.5 Deposition of the structure 5
5.1 Intuitive understanding by Bragg's law 5.2 Scattering as a Fourier transform 5.3 Friedel and Bijvoet mates 5.4 Ewald's sphere 5.5 Patterson function 5.6 Advantages of a crystal 6 Nobel Prizes involving
7.1
8 See also 9 References 10 Further reading 10.1 International Tables for Crystallography 10.2 Bound collections of articles 10.3 Textbooks 10.4 Applied computational data analysis 10.5 Historical 11 External links 11.1 Tutorials 11.2 Primary databases 11.3 Derivative databases 11.4 Structural validation History[edit] Early scientific history of crystals and X-rays[edit] Drawing of square (Figure A, above) and hexagonal (Figure B, below) packing from Kepler's work, Strena seu de Nive Sexangula. Crystals, though long admired for their regularity and symmetry, were not investigated scientifically until the 17th century. Johannes Kepler hypothesized in his work Strena seu de Nive Sexangula (A New Year's Gift of Hexagonal Snow) (1611) that the hexagonal symmetry of snowflake crystals was due to a regular packing of spherical water particles.[1] As shown by
The Danish scientist
The incoming beam (coming from upper left) causes each scatterer to re-radiate a small portion of its intensity as a spherical wave. If scatterers are arranged symmetrically with a separation d, these spherical waves will be in sync (add constructively) only in directions where their path-length difference 2d sin θ equals an integer multiple of the wavelength λ. In that case, part of the incoming beam is deflected by an angle 2θ, producing a reflection spot in the diffraction pattern. Crystals are regular arrays of atoms, and
2 d sin θ = n λ displaystyle 2dsin theta =nlambda Here d is the spacing between diffracting planes, θ displaystyle theta is the incident angle, n is any integer, and λ is the wavelength of
the beam. These specific directions appear as spots on the diffraction
pattern called reflections. Thus,
I o = I e ( q 4 m 2 c 4 ) 1 + cos 2 2 θ 2 = I e 7.94.10 − 26 1 + cos 2 2 θ 2 = I e f displaystyle I_ o =I_ e left( frac q^ 4 m^ 2 c^ 4 right) frac 1+cos ^ 2 2theta 2 =I_ e 7.94.10^ -26 frac 1+cos ^ 2 2theta 2 =I_ e f Hence the atomic nuclei, which are much heavier than an electron, contribute negligibly to the scattered X-rays. Development from 1912 to 1920[edit] Although diamonds (top left) and graphite (top right) are identical in
chemical composition—being both pure carbon—
After Von Laue's pioneering research, the field developed rapidly,
most notably by physicists
First
Since the 1920s,
The three-dimensional structure of penicillin, solved by Dorothy Crowfoot Hodgkin in 1945. The green, white, red, yellow and blue spheres represent atoms of carbon, hydrogen, oxygen, sulfur and nitrogen, respectively. The first structure of an organic compound, hexamethylenetetramine,
was solved in 1923.[74] This was followed by several studies of
long-chain fatty acids, which are an important component of biological
membranes.[75][76][77][78][79][80][81][82][83] 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,[84] a large planar molecule that is closely related to
porphyrin molecules important in biology, such as heme, corrin and
chlorophyll.
Biological macromolecular crystallography[edit]
Workflow for solving the structure of a molecule by X-ray crystallography. The oldest and most precise method of
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[99][100][101] and other methods. However, in many cases,
obtaining a diffraction-quality crystal is the chief barrier to
solving its atomic-resolution structure.[102]
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. 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
Three methods of preparing crystals, A: Hanging drop. B: Sitting drop. C: Microdialysis
Play media 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
The crystal is mounted for measurements so that it may be held in the
An
When a crystal is mounted and exposed to an intense beam of X-rays, it
scatters the
Model building and phase refinement[edit] A protein crystal structure at 2.7 Å resolution. The mesh encloses the region in which the electron density exceeds a given threshold. The straight segments represent chemical bonds between the non-hydrogen atoms of an arginine (upper left), a tyrosine (lower left), a disulfide bond (upper right, in yellow), and some peptide groups (running left-right in the middle). The two curved green tubes represent spline fits to the polypeptide backbone. 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 (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 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 = ∑ a l l r e f l e c t i o n s
F o − F c
∑ a l l r e f l e c t i o n s
F o
displaystyle R= frac sum _ mathrm all reflections leftF_ o -F_ c right sum _ mathrm all reflections leftF_ o right where F is the structure factor. A similar quality criterion is Rfree,
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, Rfree should be
approximately the resolution in angstroms divided by 10; thus, a
data-set with 2 Å resolution should yield a final Rfree ~ 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.[clarification needed]
It may not be possible to observe every atom in the asymmetric unit.
In many cases, 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[edit]
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 isomerism.[121] 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.
Deposition of the structure[edit]
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
f ( r ) = 1 ( 2 π ) 3 ∫ F ( q ) e i q ⋅ r d q displaystyle f(mathbf r )= frac 1 left(2pi right)^ 3 int F(mathbf q )e^ mathrm i mathbf q cdot mathbf r mathrm d mathbf q where the integral is taken over all values of q. The three-dimensional real vector q represents a point in reciprocal space, that is, to a particular oscillation in the electron density as one moves in the direction in which q points. The length of q corresponds to 2 π displaystyle pi divided by the wavelength of the oscillation. The corresponding
formula for a
F ( q ) = ∫ f ( r ) e − i q ⋅ r d r displaystyle F(mathbf q )=int f(mathbf r )mathrm e ^ -mathrm i mathbf q cdot mathbf r mathrm d mathbf r where the integral is summed over all possible values of the position
vector r within the crystal.
The
F ( q ) =
F ( q )
e i ϕ ( q ) displaystyle F(mathbf q )=leftF(mathbf q )rightmathrm e ^ mathrm i phi (mathbf q ) The intensities of the reflections observed in
2 d sin θ = n λ displaystyle 2dsin theta =nlambda , A reflection is said to be indexed when its Miller indices (or, more
correctly, its reciprocal lattice vector components) have been
identified from the known wavelength and the scattering angle 2θ.
Such indexing gives the unit-cell parameters, the lengths and angles
of the unit-cell, as well as its space group. Since
A e i k i n ⋅ r displaystyle Amathrm e ^ mathrm i mathbf k _ mathrm 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 a m p l i t u d e o f s c a t t e r e d w a v e = A e i k ⋅ r S f ( r ) d V displaystyle mathrm amplitude of scattered wave =Amathrm e ^ mathrm i mathbf k cdot mathbf r Sf(mathbf r )mathrm d V where S is the proportionality constant. Let's consider the fraction of scattered waves that leave with an outgoing wave-vector of kout and strike the screen at rscreen. Since no energy is lost (elastic, not inelastic scattering), the wavelengths are the same as are the magnitudes of the wave-vectors kin=kout. From the time that the photon is scattered at r until it is absorbed at rscreen, the photon undergoes a change in phase e i k o u t ⋅ ( r s c r e e n − r ) displaystyle e^ imathbf k _ out cdot left(mathbf r _ mathrm screen -mathbf r right) The net radiation arriving at rscreen is the sum of all the scattered waves throughout the crystal A S ∫ d r f ( r ) e i k i n ⋅ r e i k o u t ⋅ ( r s c r e e n − r ) = A S e i k o u t ⋅ r s c r e e n ∫ d r f ( r ) e i ( k i n − k o u t ) ⋅ r displaystyle ASint mathrm d mathbf r f(mathbf r )mathrm e ^ mathrm i mathbf k _ in cdot mathbf r e^ imathbf k _ out cdot left(mathbf r _ mathrm screen -mathbf r right) =ASe^ imathbf k _ out cdot mathbf r _ mathrm screen int mathrm d mathbf r f(mathbf r )mathrm e ^ mathrm i left(mathbf k _ in -mathbf k _ out right)cdot mathbf r which may be written as a Fourier transform A S e i k o u t ⋅ r s c r e e n ∫ d r f ( r ) e − i q ⋅ r = A S e i k o u t ⋅ r s c r e e n F ( q ) displaystyle ASmathrm e ^ mathrm i mathbf k _ out cdot mathbf r _ mathrm screen int dmathbf r f(mathbf r )mathrm e ^ -mathrm i mathbf q cdot mathbf r =ASmathrm e ^ mathrm i mathbf k _ out cdot mathbf r _ mathrm screen F(mathbf q ) where q = kout – kin. The measured intensity of the reflection will be square of this amplitude A 2 S 2
F ( q )
2 displaystyle A^ 2 S^ 2 leftF(mathbf q )right^ 2 Friedel and Bijvoet mates[edit]
For every reflection corresponding to a point q in the reciprocal
space, there is another reflection of the same intensity at the
opposite point -q. This opposite reflection is known as the Friedel
mate of the original reflection. This symmetry results from the
mathematical fact that the density of electrons f(r) at a position r
is always a real number. As noted above, f(r) is the inverse transform
of its
F ( − q ) =
F ( − q )
e i ϕ ( − q ) = F ∗ ( q ) =
F ( q )
e − i ϕ ( q ) displaystyle F(-mathbf q )=leftF(-mathbf q )rightmathrm e ^ mathrm i phi (-mathbf q ) =F^ * (mathbf q )=leftF(mathbf q )rightmathrm e ^ -mathrm i phi (mathbf q ) The equality of their magnitudes ensures that the Friedel mates have
the same intensity F2. This symmetry allows one to measure the full
f ( r ) = ∫ d q ( 2 π ) 3 F ( q ) e i q ⋅ r = ∫ d q ( 2 π ) 3
F ( q )
e i ϕ ( q ) e i q ⋅ r displaystyle f(mathbf r )=int frac dmathbf q left(2pi right)^ 3 F(mathbf q )mathrm e ^ mathrm i mathbf q cdot mathbf r =int frac dmathbf q left(2pi right)^ 3 leftF(mathbf q )rightmathrm e ^ mathrm i phi (mathbf q ) mathrm e ^ mathrm i mathbf q cdot mathbf r Since
f ( r ) = ∫ d q ( 2 π ) 3
F ( q )
e i ( ϕ + q ⋅ r ) = ∫ d q ( 2 π ) 3
F ( q )
cos ( ϕ + q ⋅ r ) + i ∫ d q ( 2 π ) 3
F ( q )
sin ( ϕ + q ⋅ r ) = I c o s + i I s i n displaystyle f(mathbf r )=int frac dmathbf q left(2pi right)^ 3 leftF(mathbf q )rightmathrm e ^ mathrm i left(phi +mathbf q cdot mathbf r right) =int frac dmathbf q left(2pi right)^ 3 leftF(mathbf q )rightcos left(phi +mathbf q cdot mathbf r right)+iint frac dmathbf q left(2pi right)^ 3 leftF(mathbf q )rightsin left(phi +mathbf q cdot mathbf r right)=I_ mathrm cos +iI_ mathrm sin The function f(r) is real if and only if the second integral Isin is zero for all values of r. In turn, this is true if and only if the above constraint is satisfied I s i n = ∫ d q ( 2 π ) 3
F ( q )
sin ( ϕ + q ⋅ r ) = ∫ d q ( 2 π ) 3
F ( − q )
sin ( − ϕ − q ⋅ r ) = − I s i n displaystyle I_ mathrm sin =int frac dmathbf q left(2pi right)^ 3 leftF(mathbf q )rightsin left(phi +mathbf q cdot mathbf r right)=int frac dmathbf q left(2pi right)^ 3 leftF(mathbf -q )rightsin left(-phi -mathbf q cdot mathbf r right)=-I_ mathrm sin since Isin = −Isin implies that Isin=0.
Ewald's sphere[edit]
Further information: Ewald's sphere
Each
c ( r ) = ∫ d x f ( x ) f ( x + r ) = ∫ d q ( 2 π ) 3 C ( q ) e i q ⋅ r displaystyle c(mathbf r )=int dmathbf x f(mathbf x )f(mathbf x +mathbf r )=int frac dmathbf q left(2pi right)^ 3 C(mathbf q )e^ imathbf q cdot mathbf r has a
C ( q ) =
F ( q )
2 displaystyle C(mathbf q )=leftF(mathbf q )right^ 2 Therefore, the autocorrelation function c(r) of the electron density
(also known as the Patterson function[122]) can be computed directly
from the reflection intensities, without computing the phases. In
principle, this could be used to determine 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.
Advantages of a crystal[edit]
In principle, an atomic structure could be determined from applying
Year Laureate Prize Rationale 1914
Max von Laue
Physics
"For his discovery of the diffraction of
1915 William Henry Bragg Physics "For their services in the analysis of crystal structure by means of X-rays",[124] 1915 William Lawrence Bragg Physics "For their services in the analysis of crystal structure by means of X-rays",[124] 1962 Max F. Perutz Chemistry "for their studies of the structures of globular proteins"[125] 1962 John C. Kendrew Chemistry "for their studies of the structures of globular proteins"[125] 1962 James Dewey Watson Medicine "For their discoveries concerning the molecular structure of nucleic acids and its significance for information transfer in living material"[126] 1962 Francis Harry Compton Crick Medicine "For their discoveries concerning the molecular structure of nucleic acids and its significance for information transfer in living material"[126] 1962 Maurice Hugh Frederick Wilkins Medicine "For their discoveries concerning the molecular structure of nucleic acids and its significance for information transfer in living material"[126] 1964
Dorothy Hodgkin
Chemistry
"For her determinations by
1972 Stanford Moore Chemistry "For their contribution to the understanding of the connection between chemical structure and catalytic activity of the active centre of the ribonuclease molecule"[128] 1972 William H. Stein Chemistry "For their contribution to the understanding of the connection between chemical structure and catalytic activity of the active centre of the ribonuclease molecule"[128] 1976 William N. Lipscomb Chemistry "For his studies on the structure of boranes illuminating problems of chemical bonding"[129] 1985 Jerome Karle Chemistry "For their outstanding achievements in developing direct methods for the determination of crystal structures"[130] 1985 Herbert A. Hauptman Chemistry "For their outstanding achievements in developing direct methods for the determination of crystal structures"[130] 1988 Johann Deisenhofer Chemistry "For their determination of the three-dimensional structure of a photosynthetic reaction centre"[131] 1988 Hartmut Michel Chemistry "For their determination of the three-dimensional structure of a photosynthetic reaction centre"[131] 1988 Robert Huber Chemistry "For their determination of the three-dimensional structure of a photosynthetic reaction centre"[131] 1997 John E. Walker Chemistry "For their elucidation of the enzymatic mechanism underlying the synthesis of adenosine triphosphate (ATP)"[132] 2003 Roderick MacKinnon Chemistry "For discoveries concerning channels in cell membranes [...] for structural and mechanistic studies of ion channels"[133] 2003 Peter Agre Chemistry "For discoveries concerning channels in cell membranes [...] for the discovery of water channels"[133] 2006 Roger D. Kornberg Chemistry "For his studies of the molecular basis of eukaryotic transcription"[134] 2009 Ada E. Yonath Chemistry "For studies of the structure and function of the ribosome"[135] 2009 Thomas A. Steitz Chemistry "For studies of the structure and function of the ribosome"[135] 2009 Venkatraman Ramakrishnan Chemistry "For studies of the structure and function of the ribosome"[135] 2012 Brian Kobilka Chemistry "For studies of G-protein-coupled receptors"[136] Applications of
Beevers–Lipson strip
Bragg diffraction
Crystallographic database
Crystallographic point groups
Difference density map
References[edit] ^ Kepler J (1611). Strena seu de Nive Sexangula. Frankfurt: G.
Tampach. ISBN 3-321-00021-0.
^ Steno N (1669). De solido intra solidum naturaliter contento
dissertationis prodromus. Florentiae.
^ Hessel JFC (1831). Kristallometrie oder Kristallonomie und
Kristallographie. Leipzig.
^ Bravais A (1850). "Mémoire sur les systèmes formés par des points
distribués regulièrement sur un plan ou dans l'espace". Journal de
l'Ecole Polytechnique. 19: 1.
^ Shafranovskii I I & Belov N V (1962). Paul Ewald, ed. "E. S.
Fedorov" (PDF). 50 Years of X-Ray Diffraction. Springer: 351.
ISBN 90-277-9029-9.
^ Schönflies A (1891). Kristallsysteme und Kristallstruktur.
Leipzig.
^ Barlow W (1883). "Probable nature of the internal symmetry of
crystals". Nature. 29 (738): 186. Bibcode:1883Natur..29..186B.
doi:10.1038/029186a0. See also Barlow, William (1883). "Probable
Nature of the Internal Symmetry of Crystals". Nature. 29 (739): 205.
Bibcode:1883Natur..29..205B. doi:10.1038/029205a0. Sohncke, L.
(1884). "Probable Nature of the Internal Symmetry of Crystals".
Nature. 29 (747): 383. Bibcode:1884Natur..29..383S.
doi:10.1038/029383a0. Barlow, WM. (1884). "Probable Nature of
the Internal Symmetry of Crystals". Nature. 29 (748): 404.
Bibcode:1884Natur..29..404B. doi:10.1038/029404b0.
^ Einstein A (1905). "Über einen die Erzeugung und Verwandlung des
Lichtes betreffenden heuristischen Gesichtspunkt" [A Heuristic Model
of the Creation and Transformation of Light].
Further reading[edit] International Tables for Crystallography[edit] Theo Hahn, ed. (2002). International Tables for Crystallography.
Volume A, Space-group Symmetry (5th ed.). Dordrecht: Kluwer Academic
Publishers, for the International Union of Crystallography.
ISBN 0-7923-6590-9.
Michael G. Rossmann; Eddy Arnold, eds. (2001). International Tables
for Crystallography. Volume F,
Bound collections of articles[edit] Charles W. Carter; Robert M. Sweet., eds. (1997). Macromolecular
Crystallography, Part A (Methods in Enzymology, v. 276). San Diego:
Academic Press. ISBN 0-12-182177-3.
Charles W. Carter Jr.; Robert M. Sweet., eds. (1997). Macromolecular
Crystallography, Part B (Methods in Enzymology, v. 277). San Diego:
Academic Press. ISBN 0-12-182178-1.
A. Ducruix; R. Giegé, eds. (1999).
Textbooks[edit] B.E. Warren (1969).
Applied computational data analysis[edit] Young, R.A., ed. (1993). The Rietveld Method. Oxford: Oxford University Press & International Union of Crystallography. ISBN 0-19-855577-6. Historical[edit] Bijvoet JM, Burgers WG, Hägg G, eds. (1969). Early Papers on
External links[edit] Library resources about
Resources in your library Resources in other libraries Wikibooks has a book on the topic of: Xray Crystallography Tutorials[edit] Learning Crystallography
Simple, non technical introduction
The
Primary databases[edit]
Derivative databases[edit] PDBsum
Proteopedia – the collaborative, 3D encyclopedia of proteins
and other molecules
RNABase
HIC-Up database of PDB ligands
Structural validation[edit] MolProbity structural validation suite ProSA-web NQ-Flipper (check for unfavorable rotamers of Asn and Gln residues) DALI server (identifies proteins similar to a given protein) v t e
High resolution Cryo-electron microscopy
Medium resolution Fiber diffraction Mass spectrometry SAXS Spectroscopic NMR
Circular dichroism
Dual-polarization interferometry
Absorbance
Fluorescence
Translational Diffusion Analytical ultracentrifugation
Size exclusion chromatography
Rotational Diffusion
Chemical Hydrogen-deuterium exchange Site-directed mutagenesis Chemical modification Thermodynamic Equilibrium unfolding Computational
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