The Technique of X-ray Crystallography

A special case of microscopy where x-rays are needed to get information about interatomic dimensions, and crystals are used to hold and orient the object (molecule).

The Technique of X-ray Crystallography

A special case of microscopy where x-rays are needed to get information about interatomic dimensions, and crystals are used to hold and orient the object (molecule).

The goal in macromolecular crystallography is to obtain a three-dimensional picture of a molecule. The general method, and the source of some of the main difficulties involved in applying it, can be described by an analogy with an ordinary light microscope.

The light microscope uses a lens to recombine the light scattered by the subject into an image. But it is necessary to use light of a wavelength not much greater than the size of the features we want to see in the images. for molecular structures this means using x-rays.

The characteristic x-ray emission of copper, which is often used for protein crystallography, has a wavelength of 1.54 Å. There is no known way of making a lens that will directly refract x-rays to make an image, so we are forced to measure the scattered x-rays at the place where the lens should have been, and use a computational method of recombining them into an image. This would not be too difficult a procedure if it were not for two very serious technical problems that arise: the size problem and the phase problem.

(Yes, a simple lens makes an inverted image. For our objects of interest, we need more magnification and the most comprehensible image from which to construct a model, so complex "lens" systems for any wavelength!)

Match of wavelength to features we want to see:

x-ray: ~1 Å, atom: ~3Å, protein molecule: ~50Å, (bonded atoms: 1.5Å apart)

Green light: 5000 Å, scale to 5meters = 5000 mm , scaled protein: 50mm = 5cm, (bonded atoms: 1.5 mm)

→ Draw 1 wavelength across both boards (e.g. of 147 Nanaline Duke bldg.),

(Note change of phase along wave)

look at two 5 cm patchs "molecules": (or two neighboring 3mm patches "atoms"), myoglobin would be about 5cm in diameter.

even if they would scatter this light, could we tell we had two of them?

5 meters scaled green light wave with scaled molecule

expanded to

32 cycles of 1.5Å across the diameter of myoglobin:

Show here Myoglobin 0.9Å structure with electron density

→ 2NRL.kin with 2nrl.omap in KiNG

myoglobin 2NRL.kin(3.2MB)

myoglobin 2nrl.omap(2.2MB)

myoglobin-withXrayWave.kin shown in KiNG Note that the atoms are almost resolved, and that the wave length is approximately the length of an atom-atom bond. This structure had data to 0.91 Å resolution. X-rays of 1.5 Å wavelength can possible resolve things spaced as closely as 1.5/2 = .75 Å.

Macromolecular Crystallography, The Experiment

General Issues

What is the information content of the experiment?

distance & direction

How does one get information out of the experiment and into the model?

measure, calculate, make image, interpret...

What can we know about the reliability of the model, both in general and in detail?

validation and feedback

Specific Toe-catchers

All atoms contribute to all Data, all Data contribute to all parts of the Map.

Map shows SUM of all conformations.

There are TWO kinds of waves:

real x-rays

computed waves to construct the image

The Size Problem ---> CRYSTALS

A single molecule (even a large protein molecule) is so small that it does not by itself scatter a sufficient amount of x-rays in a reasonable time to accumulate enough information to form a detailed picture. Also, x-rays of a wavelength short enough to resolve molecular details are energetic enough that in usual practice the molecule would be destroyed long before its image could be produced.

The solution to this problem is to use a very great number of identical molecules packed together in a very orderly three dimensional array: a crystal.

This gives scattering power.

But one will need to understand how x-rays interact with crystals. This is the general subject of x-ray diffraction.

One will also need to grow suitable crystals (of dimensions on the order of a few tenths of a millimeter,) which is a very tricky art indeed.

Unit Cell & possible symmetry in a crystal lattice

CRYSTAL Packing

A unit cell fills all space by translations along its edges.

Such a unit-cell translation relates a point in space to an identical point. That is, the space surrounding one point looks exactly the same from any other point related by unit-cell translations.

n-fold axes and unit cell translations.

2-fold axes:

3-fold axes (also 6-fold):

4-fold axes:

What about 5-fold axes?

Objects in a crystal

1: Only unit cell translations, and there are no restrictions on the angles relating the three directions (axes) that describe the crystal.

All three angles free leads to the term **“triclinic”** for this kind of crystal.

Here, the whole “unit cell” (that box that fills all space by translations along three axes) is the “asymmetric unit” (that volume whose contents are all the unique parts that exactly repeat through 3D space to make this crystal).

Note that this allows further “Non-Crystallographic Symmetry” within the asymmetric unit, (e.g. 2 arms, 2legs, etc.) but these relationships are not constrained to be exact.

However, for a single bear per cell, each unique feature, e.g. the earring, is exactly related to all other identical features by unit-cell translations.

2-fold axes constrain 2 of the angles relating crystal axes to be 90°, but the third is not constrained. So this type of crystal is termed **“monoclinic”** for its one free angle.

2-fold axes and unit cell translations. The 2-fold axes run through the whole crystal and relate not only the contents of the “asymmetric unit” with its pair within the unit cell, but also the asymmetric units throughout all 3-D space.

Note that this allows further “Non-Crystallographic Symmetry” within the asymmetric unit, (e.g. 2 arms, 2legs, etc.) but these relationships are not constrained to be exact.

Dancing-Bears.kin shown in KiNG

Download:
Dancing-Bears.kin(20KB)
(show with downloaded KiNG)

2-fold screw axes and unit cell translations.
The 2-fold screw axes run through the whole crystal and relate not only the contents of the “asymmetric unit” with its pair within the unit cell, but also the asymmetric units throughout all 3-D space.

Here the bears are individually lined up and embedded where lysozyme molecules are in a common crystal form, P2_{1}2_{1}2_{1}.

Download: Lysozyme-bears-P212121.kin(290KB) (show with downloaded KiNG, but works better in Mage)

Additional 2-fold axes perpendicular to the first will constrain all angles to be 90°.
Hence **“orthorhombic”**.

A 4-fold axis constrains 2 of the axes to be equal, thus **“tetragonal”**

A 3-fold at least constrains the three axes to be equal, with **“rhombohedral”** or at least **“trigonal”** shape.

A 6-fold leads to **“hexagonal”**.

Combinations of axes can lead to a **“cubic”** overall shape.

2D patterns:

Imagine Unit Cell translations in the plane of the screen.

"unit cells" are parallelograms whose corners are all on identical points displaced by the length of the edges, and the parallelogram by sliding one unit cell in the directions of its edges would eventually fill all of space (in 2D the plane).

Imagine 2-fold axes perpendicular to the screen.

The 2-fold axes relate both the contents of the
“asymmetric unit” with its pair within the unit cell,
and also the asymmetric units throughout all this 2-D space.

Squirrels

**Lizards arranged around 4-fold axes.**

Image from MAGE utility:

Mage menu: MAGE-HELP/Make kinemage.../Practice Docking with tetramers

Download: PS-1-5fold-UnitCells_mod.pdf(140KB) Problem set for this section

bottom 1, Introduction to X-ray Crystallography