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Introduction to Macromolecular Crystallography

John Wiley & Sons

Chapter One

The only technique that allows direct visualization of protein structure at the atomic or near-atomic level is X-ray diffraction analysis as applied to single crystals of pure proteins. Such techniques have been applied to conventional small molecules now for more than 70 years with extraordinary success, and very few chemical structures of less than 100 atoms, obtainable in the crystalline state, have proven refractory to this technique. The successful application of X-ray diffraction to protein structure is relatively new, the first protein structure, that of myoglobin, having been solved only in 1960 (Dickerson, 1991; Dickerson et al., 1961). Since that time, nearly 10,000 additional protein structures have been added to our database, but even this collection represents only a very small fraction of the hundreds of thousands of different protein molecules that play some role in living processes. Thus, the determination of protein structure by X-ray crystallography occupies the energy of several hundred laboratories in the world, and this number is ever growing, as the need for more and increasingly precise structural information expands in step with the molecular biological revolution.

What Do We Mean by the Structure of Something?

In common language, when we ask, "What is its structure?" wemean by that, how are the various components or elements that make up the object disposed or placed with respect to one another in three-dimensional space. More simply, "What does it look like?"

Although seemingly a straightforward question, it is one that has perplexed scientists, philosophers, and poets for centuries. Answers have been formulated by homology (e.g., "a rock, a craig, nay a peninsula"); by describing the physical qualities of the object (e.g., "it was a one-eyed, one-horned, flyin', purple people eater"); by analytical expressions (e.g., r = a]theta]); by visual illustration and, undoubtedly, by other means as well.

However, in proper scientific terms, there is only one way to precisely describe the structure of an object, be it simple or intricate and complex: by specifying, as in Figure 1.1, the coordinates in three-dimensional space of each point within the object, each with respect to some defined and agreed upon system of axes in space, that is, a coordinate system. Generally, this is chosen to be an orthogonal, Cartesian coordinate system, but it need not be. It may be a nonorthogonal, cylindrical, or spherical system, or one of any number of other systems.

The object's inherent structure, being fixed, remains the same no matter how the spatial coordinate system is chosen, or where its origin is taken to be. Because the structure is invariant, even if its constituent points are transformed from one coordinate system to another, the relative positions of the points within the object remain the same. The structure is not dependent on the coordinate system we choose. Thus, if the structure of a molecule is defined by specifying the coordinates in space [x.sub.j], [y.sub.j], [z.sub.j] of each atom j in the molecule, atom 5 (or 7 or 18 or whatever) maintains the same relationship in space to atom 14 (or 3 or whatever), no matter what the coordinate system.

To define the structure of a molecule in a precise manner then, we must create a list (the order is not important) of atomic positions [x.sub.j], [y.sub.j], [z.sub.j] (and here the order is important). A molecular structure becomes a set of ordered triples [x.sub.j], [y.sub.j], [z.sub.j], one for every atom. This is imminently suitable not only for translation into a visual representation on a graph, but for manipulation and analysis in a computer, and presentation on the screen of a computer graphics workstation in any number of manifestations.

When we solve the structure of a molecule, any kind of molecule, including proteins, nucleic acids, or even large assemblies such as viruses or ribosomes, in the end we are identifying and specifying the [x.sub.j], [y.sub.j], [z.sub.j] coordinates of every atom in the molecule. The form, the shape, the image must always, first, be defined in these simple numerical terms, as ordered triplets. Only from these can we faithfully reproduce the precise structure of the molecule in more familiar visual terms, as pictures or images.

An Analogy

Let us assume that we want to determine the structure, as defined above, of some object that is invisible. It has the supernatural property that it is nonresponsive to any electromagnetic radiation such as light. But, to make the example more concrete, let's assume it is an invisible, yellow, 1963 Volkswagen Beetle, like that shown in Figure 1.2. If we have never seen such a glorious object before, how can we learn of its structure? How can we visualize it?

One way we might approach this problem is to take advantage of the fact that, although the Volkswagen is impervious to light, it retains all of its other physical properties. We might, for example, take a basketball and throw it at the invisible object from some direction [[bar.k].sub.o] (a vector, which has direction, is defined by some character or symbol with a line over the top), and note which direction [bar.k] it bounces off the object. More informatively, we might throw 100 or 1,000 balls at the invisible Volkswagen and note how many balls bounce in all directions [bar.k]. Some will hit the fender, others the hood, others the windshield, etc., and, depending on the orientation of the car with respect to the balls thrown along [[bar.k].sub.o], some directions [bar.k] for the reflected balls will be much favored over others. If the direction [[bar.k].sub.o] corresponds to one aiming directly at the front of the car, for example, balls bouncing off the hood and windshield will be strongly favored.

Lets assume, however, that we can walk around the invisible Volkswagen and throw the basketballs from many, in fact all, possible directions [[bar.k].sub.o], and, each time, we note carefully how many balls bounce in which direction. Then, ultimately, we will know for every direction of our incoming beam of basketballs, [[bar.k].sub.o], how many are reflected in every direction [bar.k].

This assembly of observations, [[bar.k].sub.o], [bar.k], and the number or intensity of balls, I, in the direction [bar.k] contains information about the structure of the invisible object, the directions of its various external planes (doors, windows, hoods, fenders, etc.) from which the balls bounce. Now, the question is, can we, from these observations, synthesize the shape of the object that gave rise to the pattern of reflected basketballs? The answer is, of course, yes. Mathematical procedures do indeed exist for extracting the shape of a 1963 Volkswagen Beetle (see Fig. 1.2) from a scattering pattern of basketballs. We might even invent some analogue device that we could place in a manner that it accumulated automatically the reflected balls and somehow translated the pattern into an image of the object. We would call such a device a lens.

Now, basketballs are rather large objects (probes), and when they bounce from a surface plane, they are rather insensitive to its finer details such as windshield wipers, door handles, bolt heads, etc. We could, however, make our investigation more sensitive by using, instead of basketballs, tennis balls, and even more sensitive still by using ping-pong balls, or even marbles (no, let's not use marbles, they would damage the paint job). Then, the direction in which our probe bounced would very closely reflect the undulations of the hood and the presence of door handles. That is, we would obtain a more refined, higher-resolution image.

The approach illustrated here is not exactly what is done in X-ray diffraction, but it is similar. For example, we don't learn anything about the shape of the engine, because our various balls cannot penetrate the interior of the car, whereas x-radiation can penetrate and reflect from the internal atoms of molecules. But in many other ways it is quite the same.

Let us alter our analogy a bit and now assert that the reason our 1963 Volkswagen Beetle is invisible is because it is too small to see. It is smaller than the wavelength of visible light. We could, in principle, however, carry out the same experiment of walking around the minute Volkswagen and directing a probe at it from all directions [[bar.k].sub.o] and noting in every case what intensity of reflected probes I was observed for all directions [bar.k]. If the probe we used sometimes penetrated into the interior parts of the object (and struck the transmission, for example) so much the better, for, although our pattern of diffracted probes would be considerably more complex, we would then learn about the structures of things inside the car as well. As long as the size of the probe was comparable to the sizes of the molecular features we wished to see (a very important point), then we could do as we did with the basketballs or ping-pong balls.

In organic molecules, the distances between bonded atoms are usually 1-2 Å, hence the size of our probe must be comparable. The wavelength, [lambda], of X rays used in diffraction experiments is usually between 1 and 2 Å. Cu[K.sub.[alpha]] radiation produced by most conventional laboratory sources, for example, is 1.54 Å wavelength. [lambda] is exactly analogous to the probe size, the shorter [lambda], the smaller the diameter of the ball we are using.

Now, a single, molecule-sized Volkswagen, impressive though it might otherwise be, would nonetheless be very, very small and, in practice, it would be impossible to hit it with enough balls (or probes or waves) from a particular direction [[bar.k].sub.o] and measure the intensity of reflected waves, I, in a particular direction, [bar.k]. How could we amplify the effect so that we could measure it?

Consider an enormous parking lot full of identical 1963 Volkswagen Beetles, all perfectly parked by their drivers so that every car is identically oriented and placed in exact order. That is, they form a vast periodic array of cars. If we now direct millions of basketballs at this Volkswagen array from the same direction [[Bar.k].sub.o], then every car, having identically the same disposition, would reflect the balls exactly the same. The signal, or reflected pattern of probes, would be amplified by the number of cars in the parking lot, and the end result, which we call the signal, would be far more easily detected because of its strength. In our diffraction experiment, which is what we are really doing here, the Volkswagen Beetles are the molecules, the basketballs analogous to x-radiation, and the numbers of basketballs scattered in each direction are the intensities of the diffracted waves. Instead of an automobile parking lot, we have a molecular parking lot, a crystal.

A single voice in a coliseum, though shouting, cannot be heard at a distance. Even a stadium full of voices cannot be heard far away if each individual is shouting a different cheer at random times. But if every voice in the stadium (or at least those favoring one particular team) is united in time in a single, mighty cheer, the sound echoes far and wide:

Then from five thousand throats and more there rose a lusty yell; It rumbled through the valley, it rattled in the dell; It pounded through on the mountain and recoiled upon the flat; For Casey, mighty Casey, was advancing to the bat.

It is this cooperative effort of many individuals united in space and time, as occurs in a crystal, that makes a molecular diffraction experiment possible.

Now, clearly, in this analogy, we have simplified things a bit to get at the essentials, but the details will come later. It is important to emphasize at this point, however, and it will become very evident later, that in carrying out our experiments on vast, ordered arrays of structurally unknown objects, we do sacrifice some information that might have been obtained from a single individual (i.e., there is no free lunch). In addition, because our probes in X-ray crystallography are not particles, or balls, but are waves, an additional complication is introduced. This is because waves add together, or interfere with one another, in a manner quite different from single-particle probes. Thus, we ultimately must consider the diffraction pattern from our molecular array, or crystal, as sums of waves. Later, this sacrifice of information will emerge as what is known as "the phase problem" in X-ray crystallography-a problem we will address in due course.

A Lens

In our common experience, we rarely even think of waves and how they add together, even though we depend on light (wavelength [lambda] = 3500-6000 Å) for visualizing nearly everything. We use our eyes, microscopes, telescopes, cameras, and other optical devices that depend on waves of light, yet we never, it seems, have to deal directly with waves. The reason is that we have lenses that gather the light waves scattered by objects together, and focus them into an image of the original object. The lens of our eye focuses light waves scattered by an object at a distance into an image of that object on our retinas. The lens of a microscope focuses the light scattered by a minute object in the path of a light beam into an image of the object, and magnifies it for us at the same time.

Figure 1.3 illustrates the essential features of image formation by a lens using a simple ray diagram. There are two unique planes where the rays emitted by the light-scattering object intersect after passage through the lens. One plane is twice the focal length of the lens (2 f). There, an inverted image of the object is formed by the summation of rays from discrete points on the object converging at corresponding points on the plane. The rays converge in a different manner, however, on a second plane at a distance f between the lens and the image plane. In that plane, rays intersect that do not originate at the same point on the scattering object, but that have the same direction (defined by the angle [gamma]) in leaving the object. The convergence of the various sets of rays, each having a different direction parameter [gamma], forms in this plane a second kind of image, which is called the diffraction pattern of the object. The diffraction pattern is known mathematically as the Fourier transform of the object.

What does the diffraction pattern of an object look like? We can visualize the diffraction pattern, the Fourier transform, of an object by making a mask about the object and then passing a collimated beam of light through the mask and onto a lens.

Continues...

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Excerpted from Introduction to Macromolecular Crystallography by Alexander McPherson Excerpted by permission.
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