An Introduction to X‐Ray Crystallography

Abstract

X‐ray crystallography is the prime technique for determining the 3D structure of chemical and biochemical compounds, providing the conformation of the molecules in the crystal in addition to their configuration. The use of X‐rays with wavelengths around 1 Å means that the positions of individual atoms in the molecule can be defined. However, the technique needs a crystal to amplify the signal from a single molecule, and crystal growth requires the compound under study to be homogenous and that all the molecules take up a single stable conformation in the crystal, which restricts crystallography to the study of long‐lived states. In small molecule structures, the molecules are closely packed with few solvent molecules present. In contrast, crystals of macromolecules contain on average roughly 50% aqueous solvent and as result are less well ordered, with the resolution of the experimental X‐ray data being less than atomic. Thus, while for small molecule structures there are sufficient experimental observations to provide accurate atomic positions, the more limited resolution of macromolecular structures means that the X‐ray observations must be supplemented by stereochemical restraints.

Key Concepts

  • It is vital to visualise molecules in 3D at the atomic level to fully understand their function.
  • X‐rays allow us to see such atomic structures, but require crystals to produce an amplified view of the structure.
  • The crystal works as an amplifier, producing an averaged view of many millions of similarly oriented molecules.
  • It is the electrons around the atoms which scatter X‐rays and thus provide an image of the electron density.
  • Crystallography reveals not only the configurations but in addition the 3D conformations of molecules.

Keywords: X‐ray crystallography; microscopy; visualising atoms; the nature of crystals; atomic models

Figure 1. The human eye. Highly schematic view of the human eye. Incident light rays are focussed by the lens onto the retina, while the rods and cones in the retina respond to different parts of the spectrum. The complex image including both spatial and spectral (colour) information is transmitted via the optic nerve to the brain.
Figure 10. Protein crystals. (a) The electron density map for a protein crystal (PDB code 4cfy). This illustrates the problems faced by the protein crystallographer. There is well‐resolved high density for the protein molecules with low‐level flat density between them corresponding to the disordered solvent channels. (b) Ligand complexes. The structure of a complex of the cellulase Cel5A (PDB code 4a3h) is shown with the protein surface in white, and the ligand shown as spheres.
Figure 11. Crystals and goniostats. (a) A protein crystal mounted in its vitrified mother liquor in a loop. (b) A four‐circle kappa goniostat with the crystal mounted on its goniometer at the centre of the rotation axes. Three of the rotations (φ, ω, γ) are required for orientation of the crystal, and the fourth allows movement of the detector.
Figure 12. A diffraction image from the protein Savinase recorded with a Pilatus 6‐m detector on beam line I04 at the Diamond Light Source. The data extend to 1.4‐Å resolution. The dark ring corresponds to the scattering from the disordered vitrified solvent, and the white lines result from the fact that the detector is made up of a number of independent segments.
Figure 13. The effect of limited resolution. The density around Phe30 in rubredoxin (PDB: 1rb9) using data to the resolution limit of 0.92Å, and then truncated to 1.5, 2.0 and 3.0 Å. Figure produced using CCP4mg (McNicholas et al., ).
Figure 14. The wwPDB quality indicator for 1svn.
Figure 2. Spectroscopy, scattering and microscopes. (a) Schematic view of a spectroscopy experiment. The absorption of the radiation is measured as a function of wavelength comparing the intensity of the incident beam, I0 with the transmitted intensity, I. They are related by the Beer Lambert law, with the absorption depending on the concentration, c, of the absorbent in the sample, the path length, L and the wavelength‐dependent extinction coefficient, ϵ. (b) A scattering experiment. While many of the incident rays pass straight through the sample, others are scattered at a variety of angles. It is these inelastically scattered rays which reveal the structures within the sample and form the basis of the family of microscopy, scattering and diffraction experiments. The experiment shown is the scattering from a set of randomly oriented particles in solution. (c,d) Schematics of three microscopy/diffraction instruments. (c) An optical microscope. The object scatters the incident visible light, and the scattered rays are focussed by the physical lens system onto the eye. (d) An electron microscope. The incident radiation is a beam of electrons, which are scattered by the object. Here, the lens system is a set of focussing electromagnets, which focus the image onto a photographic plate or electronic detector. (e) X‐ray crystallography. There is no physical lens system available for X‐rays, and the diffraction image of the scattered X‐rays is recorded in reciprocal space. Only half of the information can be recorded, namely the amplitudes, with the phases being lost. After the phases have been recovered, the crystallographer uses sophisticated software to compute a magnified image, a process entirely analogous to that of the lens system in the other two methods.
Figure 3. Waves and interference. (a) Three parameters are required to describe a wave. The wave repeats every 360° along the x‐axis. The wavelength is the distance between the successive peaks and is inversely related to the frequency. The amplitude (F in crystallographic terminology) is the height of the wave and corresponds to the square root of the intensity. The phase (α) defines the distance of the first peak from the origin. α can vary between 0° and 360°; in this example, the phase is 90°. (b) Interference of waves. Two waves with the same amplitude can interfere constructively to give a wave with twice the amplitude if they have the same phase, destructively to give zero amplitude if they are out of phase or to give an intermediate resultant amplitude if they are partially out of phase. (c) A simple magnifying lens. The rays scattered by each of the three objects (atoms) A, B and C are focused by the lens onto the eye (or film) on the right, where they combine with the correct relative amplitude and phase to create a magnified image of the three objects. (b) Reprinted from Glusker JP and Trueblood KN (2010) Crystal Structure Analysis: A Primer (International Union of Crystallography Texts on Crystallography). OUP Oxford; 3rd edition by permission from Oxford University Press.
Figure 4. Isotropic (a) and anisotropic (b) atomic models. The structure shown is phenylalanine 30 from Desulfovibrio vulgaris rubredoxin (PDB 1rb9) at 0.92 Å resolution, exceptionally high for a protein. At this resolution, there are sufficient parameters to refine an anisotropic atomic model. The size of each atom represents the Gaussian atom at rest expanded to cover its atomic displacement due to a combination of thermal motion and static disorder. The isotropic model (a) assumes that the displacement is spherically symmetric, and the more realistic anisotropic model (b) allows differential displacement along the three axes and is described by six parameters. From (b), the phenylalanine is seen to vibrate in the plane of the ring, the vibration increasing towards the end of the side chain. This and other structural figures were made with CCP4mg (McNicholas et al., ).
Figure 5. Scattering from a single atom. (a) The rays scattered by different regions of the electron cloud interfere constructively at the centre of the scattered pattern and fall off smoothly with increasing scattering angle. (b) X‐ray form factors for four atom types. The value at zero scattering angle corresponds to the number of electrons and falls off smoothly with increasing scattering angle. (c) Extension of the scattering in (a) to 2D. The atom is spherical to a first approximation, and so the scattering pattern is radially symmetric, with the highest intensity and the centre, and falling off smoothly with increased scattering angle. Real systems have a spherical fall off in three dimensions.
Figure 6. Real and Reciprocal Space. (a) The duck: an optical analogue. Using a small cut‐out drawing of a duck as the object in an optical microscope leads to a scattered image – a reciprocal duck (b) if the lens is replaced by a photographic plate. The relation between (a) and (b) is not immediately obvious. Single features in the scatter pattern are not simply related in a 1:1 manner with features such as the eye or wing in the duck. The lens system refocuses the scattered (reciprocal) image to give a magnified image of the duck in real space (c). In the right‐hand panel, the duck and its image are rotated by 90° in the real experiment so that they lie perpendicular to the paper. The term reciprocal is used as the features at the centre of the scattered image relate to the broad features of the structure (its overall ‘duckiness’), those towards the edge of the image provide data on the details such as the eye, wing and tail. (d) Scattering from a 2Dmolecule’. On the left is a representation of a 2D single molecule (real space); on the right the resulting scattering pattern (reciprocal space). To provide an atomic resolution image of the molecule, it is necessary to record all the scattered data to the edge of the image. If the outer data are excluded, the magnified image will be less than atomic – as is usually the case for protein structures – see main text. (b,d) Courtesy of Kevin Cowtan.
Figure 7. The unit cell. (A) The unit cell is characterised by six parameters, the cell edges a, b and c and the angles between them, α, β and γ. (B) A 2D array of simple unit cells containing a single molecule, showing how the complete array is generated by a simple translation of the individual cells. (C) The diffraction pattern from this 2D crystal. (B, C) Courtesy of Kevin Cowtan.
Figure 8. Fourier series and unit cells. (a) Schematic view of a Fourier summation for a very small crystal, with only four unit cells in 1D. The top row shows the real electron density for a structure with two identical atoms. These lie in identical positions in the four unit cells, their positions being related by a translation along the horizontal axis. Below lie the first four Fourier components with correct amplitude (height) and phase. At the bottom is shown the result of the summation of these four Fourier terms, providing a reasonable approximation to the two‐atom structure. Addition of more terms (higher resolution data in crystallographic terminology) would improve the model. Figure based on the example used by Glusker and Trueblood . (b) The electron density for a two‐dimensional slice through a 2D unit cell. The cell is divided into a grid with points about every 1/3 Å. Equation is applied to calculate the density at the top left hand point, summing over all the reflections. This is repeated for the next point along the row, followed by row two and so on. The numbers at the grid points represent the electron density at that point. Drawing contour levels highlights the higher density regions and reveals the atomic positions. Left hand side: atomic resolution; right hand side: simulated lower resolution. (b) Courtesy of Kevin Cowtan.
Figure 9. X‐ray sources. (a) A conventional X‐ray tube. The electrons are ripped off the cathode filament by the high voltage and hit the metal anode (usually Cu or Mo). The anode then emits X‐rays at the characteristic wavelength of the metal. The cathode and anode must be maintained under high vacuum and the anode cooled, conventionally by circulating cold water. (b) A synchrotron ring. The electrons are accelerated to relativistic speeds, and as the magnets bend their trajectory they give off an intense beam of radiation, from the UV to X‐rays, in the forward direction. This provides a powerful source of X‐rays many orders of magnitude greater than that available in a conventional home laboratory, which is important for small crystals or crystals of large molecules which provide a weak diffraction pattern. Wigglers/undulators are series of dipoles producing a higher bending angle and a more intense and focussed beam.
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References

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Cowtan KD (2001) Phase problem in X‐ray crystallography, and its solution. In: eLS. Chichester: John Wiley & Sons, Ltd. www.els.net.

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Glusker JP and Trueblood KN (2010) Crystal Structure Analysis: A Primer (International Union of Crystallography Texts on Crystallography), 3rd edn. Oxford, UK: OUP.

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Further Reading

The textbook by Glusker and Trueblood, there is an excellent web resource from CSIC in Spain (http://www.xtal.iqfr.csic.es/Cristalografia/index‐en.html).

Jaeger J (2004) Macromolecular structure determination by X‐ray crystallography. In: eLS. Chichester: John Wiley & Sons, Ltd. www.els.net.

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Wilson, Keith S(May 2018) An Introduction to X‐Ray Crystallography. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1002/9780470015902.a0025432]