TOOLS OF BIOCHEMISTRY 4A
AN INTRODUCTION TO X-RAY DIFFRACTION
Only a few decades ago, virtually nothing was known about the three-dimensional structures of nucleic acids, proteins, and poly-saccharides. Today, largely as a result of the technique of X-ray diffraction, many of these molecules are understood at the atomic level—detail that would have astounded the biochemists of 1950. The method is complicated and it is possible to give only a brief introduction here, describing what is measured and what can be obtained.
When radiation of any kind passes through a regular, repeat-ing structure, diffraction is observed. This means that radiation scattered by the repeating elements in the structure shows rein-forcement of the scattered waves in certain specific directions and weakening of the waves in other directions. A simple example is given in Figure 4A.1, which shows radiation being scattered from a row of equally spaced atoms. Only in certain directions will the scattered waves be in phase and therefore constructively interfere with (reinforce) one another. In all other directions they will be out of phase and destructively interfere with one another. Thus, a diffraction pattern is generated. For the diffraction pattern to be sharp, it is essential that the wavelength of the radiation used be somewhat shorter than the regular spacing between the elements of the structure. This is why X-rays are used in studying mole-cules, for X-rays typically have a wavelength of only a few tenths of a nanometer. If the regular spacing in the object being studied is large (as in a window screen), we can observe exactly the same phenomenon with visible light, which has a wavelength thou-sands of times longer than X-rays. We will find that a point source, seen through a window screen, gives a rectangular diffrac-tion pattern of spots.
The rule relating the periodic spacings in object and diffraction pattern is simple: Short spacings in the periodic structure corre-spond to large spacings in the diffraction pattern, and vice versa. In addition, by determining the relative intensities of different spots, we can tell how matter is distributed within each repeat of the structure.
Fiber Diffraction
We consider first the diffraction from helical molecules, aligned approximately parallel to the axis of a stretched fiber. A helical molecule, like the one shown schematically in Figure 4A.2, is characterized by certain parameters:
The repeat (c) of the helix is the distance parallel to the axis in which the structure exactly repeats itself. The repeat contains some integral number (m) of polymer residues. In Figure 4A.2,
The pitch (p) of the helix is the distance parallel to the helix axis in which the helix makes one turn. If there is an integral num-ber of residues per turn (as here), the pitch and repeat are equal.
The rise (h) of the helix is the distance parallel to the axis from the level of one residue to the next, so If we think of a spiral staircase as an example of a helix, the rise is the height of each step and the pitch is the distance from where one is standing to the corresponding spot directly overhead.
Suppose we wish to investigate a polymer with the helical structure shown in Figure 4A.2. A fiber is pulled from a concen-trated solution of the polymer. Stretching the fiber further will produce approximate alignment of the long helical molecules
h = c/m.
m = 4.
Distance between atoms
Scattered waves are in phase at this angle Scattered
radiation Incoming
radiation
One row of regularly spaced atoms or molecules
Path difference equals one wavelength at this angle
FIGURE 4A.1
Diffraction from a very simple structure—a row of atoms or molecules.
Helix axis
Repeat, c
Pitch, p
Rise, h
FIGURE 4A.2
A simple helical molecule.
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CHAPTER 4 NUCLEIC ACIDSa b
c
FIGURE 4A.4
Schematic drawing of a molecular crystal.
Fiber axis Direction of
fiber axis
Layer line number
8 7 6 5 4 3 2 1 0
−1
−2
−3
−4
−5
−6
−7
−8 Inversely
proportional to rise of helix Inversely proportional to repeat distance Film
L4
L3
L2
L1
L0
L−1
L−2 Incoming
radiation
Fiber
(a) (b)
FIGURE 4A.3
Diffraction from fibers. (a) A fiber in an X-ray beam. (b) The diffraction pattern.
with the fiber axis. The fiber is then placed in an X-ray beam, and a photographic film or comparable detection medium is posi-tioned behind it, as shown in Figure 4A.3a. The diffraction pat-tern, which consists of spots or short arcs, will look like that in Figure 4A.3b. It can be read as follows: According to the mathe-matics of diffraction theory, a helix always gives rise to this kind of cross-shaped pattern. Therefore, we know we are dealing with a helical structure. The spots all lie on lines perpendicular to the fiber axis; these are called layer lines. The spacing between these lines is inversely proportional to the repeat of the helix, c, which in this case equals the pitch. Note that the cross pattern repeats itself on every fourth layer line. This repetition pattern tells us that there are exactly 4.0 residues per turn in the helix. Thus, the rise in the helix is c/4. This is the kind of evidence telling Watson and Crick that B-DNA was a helix with 10 residues per turn.
The information above is given directly by the pattern. To find out exactly how all of the atoms in each residue are arranged in each repeat, a more detailed analysis is necessary. Usually, a model is made using the correct repeat, pitch, and rise. Model making is simplified because we know approximate bond lengths and the angles between many chemical bonds. The model must also be inspected to see that no two atoms approach closer than their van der Waals radii. From such a model, the intensities of the various spots can be predicted. These predictions are compared with the observed intensities, and the model is readjusted until a best fit is obtained. The initial determination of the structure of DNA was done in just this way. As you can see from Figure 4.9, real fiber dif-fraction patterns are not as neat as the idealized example, mainly because of incomplete alignment of the molecules.
Crystal Diffraction
To study molecular crystals such as those formed by small oligonucleotides, molecules like tRNA, and globular proteins, the experimenter faces a rather different problem from the study of helical fibers and proceeds in a quite different way. A schematic drawing of such a crystal is shown in Figure 4A.4.
AN INTRODUCTION TO X-RAY DIFFRACTION
127
FIGURE 4A.5
Diffraction pattern produced by a molecular crystal of a small DNA.
P. S. Ho, Colorado State University.
The repeating unit is now the unit cell, which may contain one, two, or more molecules. The unit cell may be thought of as the basic building block of the crystal. Repetition of the unit cell in three dimensions (marked by arrows on the figure) creates the whole crystal. A simple two-dimensional analog of the crystal unit cell is the repeating pattern in wallpaper. No matter how random a wallpaper pattern may seem, if you stare at it long enough, you can always find a unit that, by repetition, fills the entire wall.
Just as in fiber diffraction, passing an X-ray beam through a molecular crystal produces a diffraction pattern. The pattern shown in Figure 4A.5 was obtained from a crystal of a small DNA. Again, the spacing of the spots allows us to determine the repeating distances in the periodic structure—in this case the x, y, and z dimensions of the unit cell labeled a, b, and c in Figure 4A.4.
But the important information in crystal diffraction studies is just how the atoms are arranged within each unit cell, for that arrangement describes the molecule. Again, this information is contained in the relative intensities of the diffraction spots in a pattern like that shown in Figure 4A.5. But in crystal diffraction, more exact information can be extracted than from a fiber dif-fraction pattern because the corresponding molecules in each unit cell of the crystal are of the same shape and are oriented in the same way. In fiber diffraction, the helical molecules may all have their long axes pointed in the same direction, but they are rotated randomly about these axes. This difference in exactness of arrangement can be appreciated by comparing the sharpness of the crystal diffraction pattern shown in Figure 4A.5 with the fiber pattern depicted in Figure 4.9.
After obtaining the diffraction pattern from a molecular crys-tal, the experimenter measures the intensities of a large number of the spots. If the molecule being studied is a small one, it is pos-sible to proceed in much the same manner as with fiber patterns.
A structure is guessed at, and expected intensities are calculated and compared with the observed intensities. The structure is refined until the relative intensities of all spots are correctly
pre-dicted. However, such a procedure won’t work with a molecule as complex as the tRNA shown in Figure 4.20—there is simply no way to guess such a structure.
Why not proceed directly from spot intensities to the structure? The difficulty is that some of the information con-tained in the spot intensities is hidden. To greatly simplify a complex problem, we may say that it is as if the quantities that the experimenter needed in order to deduce the structure (which are called structure factors) were the square roots of the intensities.* If the intensity has a value of, say, 25, the investigator knows that the number needed is or But which? This sort of quandary is the essence of the phase prob-lem, which prevented progress in large-molecule crystallogra-phy for many years. One way of solving the problem was dis-covered in the early 1950s. Suppose a heavy metal atom, such as mercury, can be introduced into some point in the molecule in such a way that the molecule and crystal are otherwise unchanged. This process is called an isomorphous replace-ment. Now suppose the heavy metal contributes a value of to the structure factor for the spot we were just discussing. If the original value was its new value is and its square is 49. If the original value was the value now becomes 3 and its square is 9. The investigator takes a diffraction photograph of the crystal with the heavy metal inserted. If the new crystal has an intensity of 9 for this spot, the original structure factor must have been not Although an oversimplification, this example gives the essence of the method. Usually multiple isomorphous replacements are necessary to determine the phases of the structure factors.
Given structure factors for all of the spots, the investigator can calculate the positions of all atoms in the unit cell. What is actually calculated is an electron density distribution (Figure 4A.6), but this amounts to the same thing, for regions of high electron den-sity are where the atoms are. In the particular view shown in Figure 4A.6, we are looking at a two-dimensional “slice” through the three-dimensional electron density distribution.
It is now appropriate to review the steps that must be taken to determine the three-dimensional structure of a macromolecule from crystal diffraction studies:
1. Obtain satisfactory crystals. This step is often the hardest part of the procedure, for the crystals must be of good quality and at least a few tenths of a millimeter in minimum dimension.
Crystals that are too small will not give sharp diffraction pat-terns. Getting macromolecules to crystallize well is still more of an art than a science.
2. Record the diffraction pattern from the crystal, and measure the intensities of many of the spots.
3. Find some way to make isomorphous replacements in the molecule. Usually two or more replacements are required.
+5.
-5,
-5, +7
+5,
+2 -5.
+5
*For the more mathematically sophisticated reader, we note that the structure factors are usually complex numbers and can thus be represented as vectors in the complex plane. What are determined from the intensities are their amplitudes, but what is not known are their phases.
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CHAPTER 4 NUCLEIC ACIDS4. Repeat steps 1 and 2 for each isomorphous derivative.
5. Calculate structure factors and, from them, the electron den-sity distribution. These calculations are usually done on a large computer.
In most cases, the investigator will first carry through this analysis with a relatively small number of spots. This procedure will give a low-resolution structure. If all is going well, more spots will be measured and the calculations refined to give higher resolution. With the best crystals, it is now possible to obtain res-olutions of about 1Å. This resolution is sufficient to identify individual groups and even some atoms and to show how they interact with one another. The detail in the phenolic ring of a protein side chain revealed at different resolutions is shown in Figure 4A.7.
Most of the detailed three-dimensional structures of biologi-cal macromolecules shown in this book have been determined by X-ray diffraction studies of crystals. At present, tens of thousands of such structures are known. This knowledge represents an enormous amount of labor in many laboratories, but the results allow us to understand macromolecular function at a level that would have been unbelievable only a short time ago.
References
van Holde, K. E., W. C. Johnson, and P. S. Ho (2006) Principles of Physical Biochemistry (2nd ed., Chapter 6). Pearson/Prentice Hall, Upper Saddle River, N.J. A more detailed treatment of X-ray diffraction of biopolymers.
FIGURE 4A.6
Part of an electron density map derived from the DNA crystal diffraction pattern in Figure 4A.5.
P. S. Ho, Colorado State University.
C C
C C C
C C
C C
C C C
C C
O
C
H H
H
0.12 nm resolution 0.15 nm resolution 0.20 nm resolution
H C
C C C
C C
O O
FIGURE 4A.7
Effect of increased resolution on molecular detail observed by X-ray diffraction. The amino acid shown in this illustration is tyrosine.
Reprinted from Journal of Molecular Biology
138:615–633, K. D. Watenpaugh, L. K. Sieker, and L. H.
Jensen, Crystallographic refinement of rubredoxin at 1·2 Å resolution. © 1980, with permission from Elsevier.