# X-ray Crystallography

translations: Romanian Danish

Crystals are three dimensional ordered structures than can be described as a repetition of identical unit cells. The unit cell is made up of the smallest possible volume that when repeated, is representative of the entire crystal. The dimensions of a unit cell can be described with 3 edge lengths (a,b,c) and 3 angles (alpha, beta, gamma). The 3D location of atoms within a unit cell can be listed as their x, y, z Cartesian Coordinates. Space groups describe the symmetry of a unit cell, of which there are 230 variations. In the molecular origami program, by clicking on Use Crystallographic Info NOW, you can experiment with all the different types of space groups.

## Why are X-rays used ?

Using visible light, it will never be possible to see atoms under even the most powerful of microscopes. In order for an object to be seen, its size needs to be at least half the wavelength of the light being used to see it. Since visible light has a wavelength much longer that the distance between atoms it is useless to see molecules. In order to see molecules it is necessary to use a form of electromagnetic radiation with a wavelength on the order of bond lengths, such as X-rays.

## Why X-ray diffraction ?

Unfortunately, unlike with visible light, there is no known way to focus x-rays with a lens. This causes an x-ray microscope to be unfeasible unless someone finds a way of focusing x-rays. Until then it is necessary to use crystals to diffract x-rays and create a diffraction pattern which can be interpreted mathematically by a computer. This turns the computer into a virtual lens, so it on a monitor we can look at the structure of a molecule. Crystals are important because by definition they have a repeated unit cell within them. The x-ray diffraction from one unit cell would not be significant. Fortunately, the repetition of unit cells within a crystal amplifies the diffraction enough to give results that computers can turn into a picture.

## Growing Crystals

To perform x-ray crystallography, it is necessary to grow crystals with edges around 0.1-0.3 mm. Crystals are formed as the conditions in a supersaturated solution slowly change. There are three degrees of saturation in solution, and crystallographers take advantage of these when growing crystals:
• Unsaturated - where no crystals will form or grow.
• Low supersaturated - where crystals will grow but no new ones will form.
• High supersaturated - where crystals will both form and grow.
One theory of crystal growth is to start by getting a few crystals to grow in the highly supersaturated solution. Then the crystals are exposed to a less saturated solution so they can grow. This is done either by moving the crystals or changing the saturation of the solution.
For small molecules, growing large enough crystals is relatively simple. By taking a supersaturated solution of solution and gradually changing the conditions, crystals will begin to grow. If left undisturbed for a few days ideally a few large crystals will grow.
Proteins are difficult to crystallize because of their complexity and the fact that protein scientists are usually working with small amounts of protein.
There are various methods of growing protein crystals:
Vapor Diffusion -(Hanging Drop Method)
This is probably the most common ways of crystal growth. A drop of protein solution is suspended over a reservoir containing buffer and precipitant. Water diffuses from the drop to the solution leaving the drop with optimal crystal growth conditions.
Batch crystallization
A saturated protein solution left in a sealed container to let the crystals grow.
Microbatch crystallization
A drop of protein solution is put in inert oil and left to grow. Here there probably is some diffusion of proteins into the oil, lowering the saturation over time.
Macroseeding
A crystal is grown in a highly saturated solution and placed in a less saturated one where only growth of the crystal will occur.
Microseeding
A few crystals are grown, then crushed, and put into a final solution that combines them into a few nice crystals. This involves quite a bit of experimentation with solutions' concentrations to get the desired number of crystals.
Free interface diffusion
A container has levels of varying saturation. Crystals form initially in the highly saturated part, but as the solution mixes, it eventually only supports crystal growth.
Dialysis
Similar to the previous, but with a semipermeable membrane separating the layers.

Proteins are crystallized on such a small scale that it is difficult to reproduce concentrations. This makes crystallizing proteins almost more of an art than a science, and sometimes multiple methods are tried before crystals of the required size are grown.

## X-ray diffraction

When X-rays are beamed at the crystal, electrons diffract the X-rays, which causes a diffraction pattern. Using the mathematical Fourier transform these patterns can converted into electron density maps. These maps show contour lines of electron density. Since electrons more or less surround atoms uniformly, it is possible to determine where atoms are located. Unfortunately since hydrogen has only one electron, it is difficult to map hydrogens. To get a three dimensional picture, the crystal is rotated while a computerized detector produces two dimensional electron density maps for each angle of rotation. The third dimension comes from comparing the rotation of the crystal with the series of images. Computer programs use this method to come up with three dimensional spatial coordinates.

For further reference visit Crystallography 101
Or check out:
Carter, Charles W. Jr. and Robert M. Sweet eds. Methods in Enzymology. 276, [2] (1997).