“Biophysics is a very broad subject. Nearly every branch of physics has something to contribute, and the boundaries between physics and engineering are blurred. Each chapter could be much longer; we have attempted to provide the essential physical tools. Molecular biophysics has been almost completely ignored: excellent texts already exist, and this is not our area of expertise. This book has become long enough.”Nevertheless, sometimes--to amuse myself--I play a little game. I say to myself “Brad, suppose someone pointed a gun to your head and demanded that you MUST include X-ray crystallography in the next edition of Intermediate Physics for Medicine and Biology. Where would you put it?”

My first inclination would be to choose Chapters 15 and 16, about how X-rays interact with tissue and their use in medicine, which seems a natural place because crystallography involves X-rays. Yet, these two chapters deal mainly with the particle properties of X-rays, whereas crystallography arises from their wave properties. Also, Chapters 15 and 16 make a coherent, self-contained story about X-rays in medical physics for imaging and therapy, and a digression on crystallography would be out of place. An alternative is Chapter 14 about Atoms and Light. This is a better choice, but the chapter is already long, and it does not discuss electromagnetic waves with wavelengths shorter than those in the ultraviolet part of the spectrum. Chapter 12 on Images is another possibility, as crystallography uses X-rays to produce an image at the molecular level based on a complicated mathematical algorithm, much like tomography uses X-rays to predict an image at the level of the whole body. Nevertheless, if that frightening gun were held to my head, I believe I would put the section on X-ray crystallography in Chapter 11, which discusses Fourier analysis. It would look something like this:

11.6 ½ X-ray CrystallographyFor more information on X-ray crystallography, see http://www.ruppweb.org/Xray/101index.html or http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html.

One application of the Fourier series and power spectrum is in X-ray crystallography, where the goal is to determine the structure of a molecule. The method begins by forming a crystal of the molecule, with the crystal lattice providing the periodicity required for the Fourier series. DNA and some proteins form nice crystals, and their structures were determined decades ago.* Other proteins, such as those that are incorporated into the cell membrane, are harder to crystallize, and have been studied only more recently, if at all (for instance, see the discussion of the potassium ion channel in Sec. 9.7).

X-rays have a short wavelength (on the order of Angstroms), but not short enough to form an image of a molecule directly, like one would obtain using a light microscope to image a cell. Instead, the image is formed by diffraction. X-rays are an electromagnetic wave consisting of oscillating electric and magnetic fields (see Chapter 14). When an X-ray beam is incident on a crystal, some of these oscillations add in phase, and the resulting constructive interference produces high amplitude X-rays that are emitted (diffracted) in some discrete directions but not others. This diffraction pattern (sometimes called the structure factor, F) depends on the wavelength of the X-ray and the direction (see Prob. 19 2/3). One useful result from electromagnetic theory is that the structure factor is related to the Fourier series of the electron density of the molecule: F is just the a_{n}and b_{n}coefficients introduced in the previous three sections, extended to account for three dimensions. Therefore, the electron density (and thus the molecular structure) can be determined if the structure factor is known.

A fundamental limitation of X-ray crystallography is that the crystallographer does not measure F, but instead detects the intensity |F|^{2}. To understand this better, recall that the Fourier series consists of a sum of both cosines (the a_{n}coefficients) and sines (b_{n}). You can always write the sum of a sine and cosine having the same frequency as a single sine with an amplitude c_{n}and phase d_{n}(See Prob. 19 1/3)

a_{n}cos(ω_{n}t) + b_{n}sin(ω_{n}t) = c_{n}sin(ω_{n}t + d_{n}) . (1)

The measured intensity is then c_{n}^{2}. In other words, an X-ray crystallography experiment allows you to determine c_{n}, but not d_{n}. Put in still another way, the experiment measures the power spectrum only, not the phase. Yet, in order to do the Fourier reconstruction, phase information is required. How to obtain this information is known as the “phase problem,” and is at the heart of crystallographic methods. One way to solve the phase problem is to measure the diffraction pattern with and without a heavy atom (such as mercury) attached to the molecule: some phase information can be obtained from the difference of the two patterns (Campbell and Dwek (1984)). In order for this method to work, the molecule must have the same shape with and without the attached heavy atom present.

* for a fascinating history of these developments, see Judson (1979)

Problem 19 1/3 Use the trigonometric identity sin(A+B) = sinA cosB + cosA sinB to relate a_{n}and b_{n}in Eq. (1) to c_{n}and d_{n}.

Problem 19 2/3 Bragg’s law can be found by assuming that the incident X-rays (having wavelength λ) reflect off a plane formed by the regular array of points in the lattice. Assume that two adjacent planes are separated by a distance d, and that the incident X-ray bean falls on this plane at an angle θ with respect to the surface. The condition for constructive interference is that the path difference between reflections from the two planes is an integral multiple of λ. Derive Bragg’s law relating θ, λ and d.

Campbell, I. D., and R. A. Dwek (1984) Biological Spectroscopy. Menlo Park, CA, Benjamin/Cummings.

Judson, H. F. (1979) The Eighth Day of Creation. Touchstone Books