User:WillowW/X-ray crystallography

This page is for snippets to improve X-ray crystallography, which is the May 2007 Science collaboration of the month.

Awadewit's suggestions from October 2007

• In the image at the top right, the x-rays kind of look like they are emanating from the crystal rather than being passed through the crystal.
• The lead is a bit long.
• Perhaps a few images in the "History" section?
• From Bragg on to the end of that paragraph in "Scientific pre-history", I got a little lost about what is actually the case.
• I found the history in the second paragraph of "The idea of combining crystals and X-rays" distracting as I was trying to understand diffraction.

I agree with all these comments and will try to address them.

• "The earliest structures were generally simple and marked by one-dimensional symmetry; as the field progressed over the next decades, the structures of two- and three-dimensional arrangements of atoms in the unit-cell became feasible." - Do you mean it became feasible to see/record them?

No, in this case, I think I mean "As computational and experimental methods improved, it became to feasible to deduce reliable atomic positions for more complicated 2D and 2D molecules."

That was what I was trying to say - sorry I wasn't clearer. It was the way "feasible" was being used that seemed odd - as if the atoms themselves somehow became feasible. Awadewit | talk 09:23, 18 October 2007 (UTC)

• "Contributions to chemistry" became a little jargony, but that is probably inevitable.

It is pretty dense, no? Pth is not always a good thing.

• "Contributions to mineralogy" is short, isn't it?

Yes, I'm keenly aware of that, thanks especially to KP. More on that and other subjects will be added shortly; they're in my library notes.

• Is "solubilize" a word?

Ummm, in some circles. ;)

Just like in other circles, "problematize" is a word. :) Awadewit | talk 09:23, 18 October 2007 (UTC)

• "As derived below, elastic scattering can be represented as a Fourier transform of the density of the scatterers" - it's not "transformation"?

The "transformation" is the process, whereas the "transform" is the result of that process.

Ah, I see. Awadewit | talk 09:23, 18 October 2007 (UTC)

• "Most crystals used in X-ray crystallography are less than a millimeter across." - Shouldn't this be "fewer than one millimeter"?

How about "Crystals used in X-ray crystallography are generally smaller than a millimeter across"? Willow 09:55, 15 October 2007 (UTC)

Better. Awadewit | talk 09:23, 18 October 2007 (UTC)

• Why is it so difficult to obtain a suitable sample for crystallography? Perhaps this could be addressed under "Methods".

• I think a little more linking might be a good idea (pH, supersaturated, etc.)

• It was not clear to me until the "Methods" section that the crystals have to be produced in some way. I, stupidly, thought you "mined" them or something - that they would be readily available in nature, in other words.

• Crystallographers apply for a slot of time, which they must use whenever it is granted, even at 3am on a national holiday. - This sentence broke the tone of the article. It's funny, but perhaps not encyclopedic.

• Some pathologies can be quickly diagnosed as well, such as twinning or a prominent ice ring. - Why would there be ice? What did I miss?

• I felt that I was only getting bits and pieces of "Crystal symmetry, unit cell, and image scaling". For a reader such as myself, several intervening steps seemed to missing in the explanations.

• Typically, a crystallographer can dope a crystal with heavy atoms either by soaking the crystal in a heavy atom-containing solution, or by co-crystallization (growing the crystals in the presence of a heavy atom). - I assume "dope" is a technical term? It sounds funny. :)

• I didn't read the "Diffraction theory" section. It is just way beyond me, I'm afraid. The rest of the article, however, I thought was quite accessible. In fact, I was surprised how much I understood.

• Just sort of thinking how I, as a lay reader, would have liked the information arranged:

• Summary explanation of technique
• History (but I am particularly interested in history)
• Detailed explanation of technique
• [Insert math here for other readers, perhaps?]
• Application of technique

• If you add much more, I would suggest starting to cut some other things. It is starting to get long. Awadewit | talk 09:23, 18 October 2007 (UTC)

Reminders of what yet to add

• something about the people: the International Union, the congresses, the profession of being a crystallographer
• look at mineralogy books for clues to "unbalanced"
• add small-molecule structure with anisotropic ellipsoids for disorder
• discussion of heavy vs. hydrogen atoms

Theoretical treatment of scattering

Context of all types of scattering

Xray crystallography is a form of elastic scattering; the outgoing waves have the same energy as the incoming waves, and have only altered their direction. Since the energy of a photon is inversely proportional to its wavelength, elastic scattering means that the outgoing photons have the same wavelength as the incoming photons. For X-rays, this wavelength is roughly 1 Å (0.1 nm = 10^-10 m), which is on the scale of a single atom.

By contrast, inelastic scattering occurs when energy is transferred from the incoming beam to the crystal, e.g., exciting an inner-shell electron. Such inelastic scattering changes the wavelength of the outgoing beam, making it longer. In elastic scattering is useful for probing such excitations of matter, but are not as useful in determining the distribution of scatterers within the matter.

The elastic scattering of radiation from anything can be represented as a Fourier transform of the density of the objects doing the scattering, as long as the scattering is weak; the scattered waves should be much less intense than the incoming wave. If this is so, the scattered waves do not produce re-scattered waves of their own, at least not significantly. Re-scattered waves are called "secondary scattering". Another trick for reducing secondary scattering is to use very thin samples; the primary scattered waves leave the sample before they have a chance to do secondary scattering.

Within most crystals, the electrons are scattering the incoming X-rays; hence, the scattered radiation describes the density of electrons within the crystal. Diffraction patterns can also be produced by incoming beams of electrons (electron diffraction) or neutrons (neutron diffraction); in the former case, electrons interact a million-fold more strongly with other electrons, so that samples must be very thin to avoid secondary scattering. In the latter case, it is much harder to get intense, monochromatic beams of neutrons, although the new Spallation source holds much promise in the near future. Being uncharged, neutrons scatter much more readily from the atomic nuclei, rather than from the electrons. Neutron scattering is very useful for several reasons, especially for seeing hydrogen atoms in larger crystals; such light atoms have few electrons and, therefore, are often not discernible. Neutron scattering also has the cool property that the solvent can be made invisible by adjusting the ratio of H₂O and heavy water, D₂O.

X-ray crystallography involves the scattering of X-rays from a crystal. There are other forms of X-ray scattering, such as SAXS and various forms of X-ray fiber diffraction, which was so useful in determining the double-helix structure of DNA. The technique of powder diffraction amounts to rotationally averaging the scattering of zillions of little crystals. None of these techniques offer as much structural information as X-ray diffraction. In contrast to these other methods which produce smooth scattering plots, crystal diffraction produces spots known as Bragg peaks or reflections. A typical protein crystal might produce 30,000 reflections, each of which represents an independent piece of data about the structure. The crystal is also typically scattered with multiple wavelengths of X-rays o with small metallic additives that help in solving the structure. These hundreds of thousands of data are assembled by powerful computers into an atomic-resolution model of the electron density. Using the known chemical bonds of the molecule(s) in the crystal, the positions of the atomic nuclei can be inferred, producing a crystal strcture. Thanks to its enormous amount of independent data, X-ray crystallography is the best technique for determining the atomic-resolution structure of any molecule. However, a crystal is needed, and that often proves to be the key stumbling block to determining the structure.

Scattering as a Fourier transform

Let the incoming wave be represented as a scalar. We ignore for now the complications of the polarization and the time dependence of the wave and just focus on its spatial dependence. Plane waves can be represented by a wave vector k_in, and so the strength of the incoming wave at time t=0 is given by

$Ae^{i\mathbf {k} \cdot \mathbf {r} }$

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 number of scatterers in a small volume dV about r

$Ae^{i\mathbf {k} \cdot \mathbf {r} }f(\mathbf {r} )dV$

Let's consider the fraction of scattered waves that leave with a outgoing wave-vector of k_out and strike the screen at r′. Since no energy is lost (elastic, not inelastic scattering), the wavelengths are the same as are the magnitudes of the wave-vectors |k_in| = |k_out|.

Outgoing propagation from scattering point r to screen [amplitude of weak wave scattered at r] * eik′∙(r′ - r) = [Aeik∙r f(r)] * eik′∙(r′ - r) Total amplitude scattered into k′ (q = k′ - k) eik′∙r′ A ∫ dr f(r) eir∙(k - k′) = eik′∙r′ A ∫ dr f(r) e-ir∙q = eik′∙r′ A F(q)

Fourier transforms are generally complex F(q) = |F(q)| eif(q) We don’t measure amplitude, but rather intensity α |F(q)|2, which gives |F(q)|

Each X-ray image represents only a slice, a spherical slice of reciprocal space; Ewald sphere construction

k and k′ both lie on a sphere, i.e., have the same magnitude, because they have the same energy. It’s elastic, not inelastic scattering

To measure a full set of |F(q)| for all q, you need to collect a set of such spherical slices by rotating the crystal; typically with a 0.5-1° rock (oscillation)

Get the phases f(q), then inverse Fourier transform to get the density f(r)

Methods of phasing

Direct methods

MAD phasing and selenomethionine

isomorphous replacement with heavy metals mercuric compounds are cool, but dangerous

molecular replacement

refinement, modern software

Sources of crystals

the Black Art

Obtaining diffraction-quality crystals is the rate-determining step; the process of crystallization is haphazard and poorly understood

Crystals may assemble but not diffract well

Basic idea is to lower the solubility, induce nucleation without precipitation, encourage subsequent growth

problem of twinning

ironically, molecules that interact tightly don't crystallize well, e.g., fibers

Crystal screening

modern crystallization robots

kits, e.g., from Hampton's and Emerald

lists of good solutions, e.g., Peter Kwong's list

sitting-drop, hanging drop, droplet in oil, at interface, zillion other methods

limitation of having enough pure protein to check everything; typically 50-100 mg of pure protein are needed

Purity is essential to have a prayer of getting crystallization — so they say ;)

Structural genomics centers (maybe put into Modern Developments section?)

Sources of X-rays

synchrotron sources vs. in-house sources higher resolution often observed in purer, brighter beams

Sources of error

Thermal motions, Debye-Waller factor lower the temperature

intrinsic disorder, mosaicity

diffuse scattering

high luminosity, radiation damage to the crystal, chemical origin, minimize with ultracold frozen crystals, need for a cryoprotectant

assessment of errors, R and R_free give definitions, motivations; typical good R values are below 0.25 (also called 25%)

Definition section

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

X-ray diffraction involves the scattering of X-rays of a single wavelength ("monochromatic X-rays") from a single, pure crystal. This scattering produces a diffraction pattern, a set of intense spots (also called reflections) on a screen behind it. The spots can be related to the density of electrons in the crystal through a mathematical operation called a Fourier transform. This technique decomposes the electron density into its spatial frequency components, just as the human ear can discern different musical notes in a chord. Each spot observed on the screen corresponds to a spatial oscillation of the electron density along a particular direction within the crystal. By combining these independent oscillations in density, the electron distribution can be reconstructed, just as chord can be played on a piano once its individual musical notes are known.

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

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

Gallery

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