X-ray Crystallography

Two weeks ago in this blog, when reviewing Judson’s excellent book The Eighth Day of Creation, I wrote that X-ray crystallography played a central role in the development of molecular biology. But Russ Hobbie and I do not discuss X-ray crystallography in the 4th edition of Intermediate Physics for Medicine and Biology, even though it is a classic example of physics applied in the biomedical sciences. Why? I think one of the reasons for this is that Russ and I made the conscious decision to avoid molecular biophysics. In our preface we write

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 Crystallography

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_n². 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

For 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.

AAPT Summer Meeting in Portland Oregon

On Tuesday, Russ Hobbie gave a talk about “Medical Physics in the Introductory Physics Course” at the American Association of Physics Teachers Summer Meeting in Portland Oregon. His session, with over 100 people attending, focused on Reforming the Introductory Physics Courses for Life Science Majors, a topic currently of great interest and one that I have discussed before in this blog. You can find the slides that accompanied his talk at the 4th edition of Intermediate Physics for Medicine and Biology website. His talk focused on five topics that he feels are crucial for the introductory course: 1) Exponential growth and decay, 2) Diffusion and solute transport, 3) Intracellular potentials and currents, 4) Action potentials and the electrocardiogram, and 5) Fitting exponentials and power laws to data. All these topics are covered in our book. Russ and I also compiled a list of topics for the premed physics course, and cross listed them to our book, this blog, and other sources. You can find the list on the book website, or download it here.

Our book website is a source of other important information. For instance, you can download the errata, containing a list of known errors in the 4th edition of Intermediate Physics for Medicine and Biology. You will find Russ’s American Journal of Physics paper “Physics Useful to a Medical Student” (Volume 43, Pages 121–132, 1975), and Russ and my American Journal of Physics “Resource Letter MP-2: Medical Physics” (Volume 77, Pages 967–978, 2009). Other valuable items include MacDose, a computer program Russ developed to illustrate the interaction of radiation with matter, a link to a movie Russ filmed to demonstrate concepts related to the attenuation and absorption of x rays, sections from earlier editions of Intermediate Physics for Medicine and Biology that were not included in the 4th edition, and a link to the American Physical Society, Division of Biological Physics December 2006 Newsletter containing an interview with Russ upon the publication of the 4th edition of our book. You can even find a link to the Intermediate Physics for Medicine and Biology facebook group.

Russ and I hope that all this information on the book website, plus this blog, helps the reader of Intermediate Physics for Medicine and Biology keep up-to-date, and increases the usefulness of our book. If you have other suggestions about how we can make our website even more useful, please let us know. Of course, we thank all our dear readers for using our book.

The Eighth Day of Creation

The Eighth Day of Creation:
The Makers of the Revolution in Biology,
by Horace Freeland Judson.

I recently finished reading The Eighth Day of Creation, a wonderful history of molecular biology by Horace Freeland Judson. The book is divided into three parts: 1) DNA—Function and Structure: the elucidation of the structure of deoxyribonucleic acid, the genetic material, 2) RNA—The Functions of the Structure: the breaking of the genetic code, the discovery of the messenger, and 3) Protein—Structure and Function: the solution of how protein molecules work. The first part centers on the story of how Watson and Crick discovered the double-helix structure of DNA, a story also told in Watson’s book The Double Helix (required reading for any would-be scientist). I was less familiar with the RNA tale in the second part, but was fascinated by the “Good Friday” meeting in which the various roles of RNA (as both a messenger taking the genetic information from DNA to the protein, and as part of ribosomes where protein synthesis takes place) was first understood by Sydney Brenner and Francois Jacob, among others. I was somewhat familiar with Kornberg’s deciphering of the genetic code from my days at the National Institutes of Health, where Kornberg worked. The last section tells how Max Perutz used X-ray crystallography to determine the structure of hemoglobin, the first protein structure known.

New to me was the story of Jacques Monod and his study of bacteria, which led to our understanding of how protein synthesis is controlled. Last week in this blog I mentioned seeing a display about Monod at the Pasteur museum in Paris. Particularly fascinating was the story of Monod’s role as a leader of the French resistance against the Nazis during World War II, and how he continued his scientific research while participating in the resistance. Judson writes

“In the autumn of 1943, a meeting was called in Geneva of representatives of all the armed groups of the French resistance, to coordinate their military actions. Just before the meeting, Philippe Monod heard from his brother that he was the delegate of the Francs-Tireurs from Paris. In November, the Gestapo arrested a minor agent of one of the main resistance networks in France, Reseau Velites, centered on the Ecole Normale Superieur. Marchal’s identity [Marchal was an alias used by Monod] and activities were known to the agent. Monod had to go underground completely, leaving his apartment, never sleeping more than a night or two at one address, staying away from the Sorbonne. On 14 February 1944, the Gestapo caught Raymond Croland, chief of the Reseau Velites, who knew Monod.

On the run from his own laboratory, Monod was given bench space by [Andre] Lwoff. “I don’t think I was ever searched for, actually,” he said. “But the possibility existed because at least one—in fact, several men had been picked up who knew what I was doing and who knew my name and where I worked. But it was known that I lived near the Sorbonne and worked at the Sorbonne, so the Gestapo would have had no reason to hunt for me at the Pasteur Institute.” In Lwoff’s laboratory, in collaboration with Alice Audureau, a graduate student, Monod that winter began a new set of experiments…

I would rank The Eighth Day of Creation second in my list of the best scientific histories I have read, just behind Richard Rhodes’ The Making of the Atomic Bomb, and just ahead of Bruce Hunt’s The Maxwellians. Interestingly, some of the characters who appeared in The Eighth Day of Creation also played a role in The Making of the Atomic Bomb: in particular, George Gamow and Leo Szilard (Szilard was mentioned in the very first sentence of Rhodes’ book). Readers of the 4th edition of Intermediate Physics for Medicine and Biology will be interested in learning that many of the pioneers in molecular biology were trained as physicists. Judson writes “new people came into biology, and most famously the physicists: Max Delbruck, Leo Szilard, Francis Crick, Maurice Wilkins, [and] on an eccentric orbit George Gamow.” I couldn’t help but be struck by the central role of X-ray crystallography in the history of molecular biology. Under physicist William Bragg’s leadership at the Cavendish, four Nobel prizes were awarded (in the same year, 1962) for molecular structure determination: Watson and Crick for DNA, and Perutz and Kendrew for the structure of hemoglobin and myoglobin. I highly recommend the book, especially for young biology students interested in the history of their subject.

I will end with the opening paragraphs of The Eighth Day of Creation, where Judson draws parallels between the revolutions in physics in the first decades of the 20th century and the revolution in biology in the middle of the century.

The sciences in our century, to be sure, have been marked almost wherever one looks by momentous discoveries, by extraordinary people, by upheavals of understanding—by a dynamism that deserves to be called permanent revolution. Twice, especially, since 1900, scientists and their ideas have generated a transformation so broad and so deep that it touches everyone’s most intimate sense of the nature of things. The first of these transformations was in physics, the second in biology. Between the two, we are most of us spontaneously more interested in the science of life; yet until now it is the history of the transformation of physics that has been told.

The revolution in physics came earlier. It began with quantum theory and the theory of relativity, with Max Planck and Albert Einstein, at the very opening of the century; it encompassed the interior of the atom and the structure of space and time; it ran through the settling of the modern form of quantum mechanics by about 1930. Most of what has happened in physics since then, at least until recently, has been the playing out of the great discoveries—and of the great underlying shift of view—of those three decades. The decades, that shift of view, the discoveries, and the men who made them are familiar presences, at least in the background, to most of us; after all, they built the form of the world as we now take it to be. The autobiographies of the major participants, their memoirs and philosophical reflections, have been composed, their biographies written in multiple—and they remain long in print, for these were men of intelligence, originality, and, often, eccentricity. The scientific papers have been scrutinized as historical and literary objects. The letters have been catalogued and published. The collaborations have been disentangled, the conferences reconvened on paper with vivid imaginative sympathy, the encounters, the conversations, even the accidents reconstructed.

The revolution in biology stands in contrast. Beginning in the mid thirties, its first phase, called molecular biology, came to a kind of conclusion—not an end, but a pause to regroup—by about 1970. A coherent if preliminary outline of the nature of life was put together in those decades. This science appeals to us very differently from physics. It directly informs our understanding of ourselves. Its mysteries once seemed dangerous and forbidden; its consequences promise to be practical, personal, urgent. At the same time, biology has been growing accessible to the general reader as it never was before and as the modern physics never can be. Indeed, part of the plausibility of molecular biology to the scientists themselves is that it is superbly easy to visualize. The nonspecialist can understand this science, at least in outline, as it really is—as the scientist imagines it. Yet the decades of these discoveries have hardly been touched by historians before now. The Eighth Day of Creation is a historical account of the chief discoveries of molecular biology, of how they came to be made, and of their makers—for these, also, though only two or three are yet widely known, were scientists, often of intelligence, originality, even eccentricity.

Paris

I just returned from a vacation in Paris, where my wife and I celebrated our 25th wedding anniversary. Russ Hobbie was there at the same time, although conflicting schedules did not allow us to get together. My daughter Katherine posted the blog entries for the last two weeks, when I had limited computer access. Thanks, Kathy.

Although most of our time was spent doing the usual tourist activities (for example, the Arc de Triomphe, the Notre Dame Cathedral, Versailles, and, my favorite, a dinner cruise down the Seine), I did keep my eye open for those aspects of France that might be of interest to readers of the 4th edition of Intermediate Physics for Medicine and Biology. We visited the Pantheon, where we saw the tomb of Marie Curie (a unit of nuclear decay activity, the curie, was named after her and is discussed on page 489 of Intermediate Physics for Medicine and Biology). Marie Curie lies next to her husband Pierre Curie (of the Curie temperature, page 216). Also in the Pantheon is Jean Perrin, who determined Avogadro’s number (see the footnote on page 85) and Paul Langevin, of the Langevin equation (page 87). Hanging from the top of the dome is a Foucault pendulum, in the exact place where Leon Foucault publicly demonstrated the rotation of the earth in 1851. I like it when physics takes center stage like that.

Another scientific site we visited is a museum honoring Louis Pasteur at the Pasteur Institute. Pasteur chose to be buried in his home (now the museum) rather than in the Pantheon. Readers of Intermediate Physics for Medicine and Biology will find him to be an excellent example of a researcher who bridges the physical and biological sciences. His first job was as a professor of physics, although he would probably be considered more of a chemist that a physicist. His early work was on chiral molecules and how they rotated light. He later became famous for his research on the spontaneous generation of life and a vaccine for rabies. In his book Adding A Dimension, Isaac Asimov lists Pasteur as one of the ten greatest scientists of all time. The museum is enjoyable, although it is not as accessible to English speakers as some of the larger museums such as the Louvre and the delightful Musee d’Orsay. Because I speak no French, I had a difficult time following many of the Pasteur exhibits. Also at the museum was a nice display about microbiologist Jacques Monod, who I will discuss in a future entry to this blog.

A Short History of Chemistry,
by Isaac Asimov.

The only other French scientist on Asimov’s top-ten list was the chemist Antoine Lavoisier. Oddly, the French don’t seem to celebrate Lavoisier’s accomplishments as much as you might expect. (Beware, my conclusion is based on a brief 2-week vacation, and I may have missed something.) Perhaps his death by the guillotine during the French Revolution has something to do with it. We visited the Place de la Concorde, where Lavoisier was beheaded. In A Short History of Chemistry, Asimov writes

In 1794, then, this man [Lavoisier], one of the greatest chemists who ever lived, was needlessly and uselessly killed in the prime of life. “It required only a moment to sever that head, and perhaps a century will not be sufficient to produce another like it,” said Joseph Lagrange, the great mathematician. Lavoisier is universally remembered today as “the father of modern chemistry.”

I normally associate Leonardo da Vinci with Italy, but when touring the Chateau at Amboise in the Loire Valley, we stumbled unexpectedly upon his grave. He spent the last three years of his life in France. We toured an excellent museum dedicated to da Vinci, containing life-size reconstructions of some of his engineering inventions. Although da Vinci had many interests and may be best known for his paintings (yes, I saw the Mona Lisa while at the Louvre), at least some of his work might be called biomedical engineering, such as his work on an underwater breathing apparatus and on human flight.

Seventy-two famous French scientists and mathematicians are listed on the Eifel Tower, including Laplace (of the Laplacian, page 91), Ampere (of Ampere’s law, page 206, and the unit of current, page 145), Navier (of the Navier-Stokes equation, page 27), Legendre (of Legendre polynomials, page 184), Becquerel (of the unit of activity, page 489), Fresnel (of the Fresnel zone for diffraction, page 352), Coulomb (of the unit of charge and Coulomb’s law, both on page 137), Poisson (of Poisson’s ratio, page 27; the Poisson-Boltzmann equation, page 230; and the Poisson probability distribution, page 572), Clapeyron (of the Clausius-Clapeyron relation, page 78), and Fourier (of the Fourier series, page 290). I could not see all these names because the tower was partially covered for painting. Note that Lavoisier was included on the Eiffel Tower, but Poiseuille (of Poiseuille flow, page 17) was not. The view from the top of the tower is spectacular.

I admit, I am not the best of travelers and am glad to be home in Michigan. But I believe there is much in France that readers of Intermediate Physics for Medicine and Biology will find interesting.

Reynolds Number

The Reynolds number is a key concept for anyone interested in biofluid dynamics. Russ Hobbie and I discuss the Reynolds number in Section 1.18 (Turbulant Flow and the Reynolds Number) of the 4th edition of Intermediate Physics for Medicine and Biology.

The importance of turbulence (nonlarminar) flow is determined by a dimensionless number characteristic of the system called the Reynolds number N_R. It is defined by

N_R = L V ρ/η

where L is a length characteristic of the problem, V a velocity characteristic of the problem, ρ the density, and η the viscosity of the fluid. When N_R is greater than a few thousand, turbulence usually occurs…

When N_R is large, inertial effects are important. External forces accelerate the fluid. This happens when the mass is large and the viscosity is small. As the viscosity increases (for fixed L, V, and ρ) the Reynolds number decreases. When the Reynolds number is small, viscous effects are important. The fluid is not accelerated, and external forces that cause the flow are balanced by viscous forces… The low-Reynolds-number regime is so different from our everyday experience that the effects often seem counterintuitive.”

Steven Vogel, in his fascinating book Life in Moving Fluids, describes the importance of the Reynolds number more elegantly.

The peculiarly powerful Reynolds number [is] the center piece of biological (and even nonbiological) fluid mechanics. The utility of the Reynolds number extends far beyond mere problems of drag; it’s the nearest thing we have to a completely general guide to what’s likely to happen when solid and fluid move with respect to each other. For a biologist, dealing with systems that span an enormous size range, the Reynolds number is the central scaling parameter that makes order of a diverse set of physical phenomena. It plays a role comparable to that of the surface-to-volume ratio in physiology.

The Reynolds number is named after the British engineer Osborne Reynolds (1842–1912). He developed the Reynolds number as a simple way to understand the transition from laminar to turbulent flow of fluids in a pipe. Perhaps it is fitting to let Reynolds have the last word. Below he describes experiments in which he added a filament of dye to the fluid (as quoted by Vogel in Life in Moving Fluids).

When the velocities were sufficiently low, the streak of colour extended in a beautiful straight line across the tube. If the water in the tank had not quite settled to rest, as sufficiently low velocities, the streak would shift about the tube, but there was no appearance of sinuosity. As the velocity was increased by small stages, at some point in the tube, always at a considerable distance from the trumpet or intake, the colour band would all at once mix up with the surrounding water. Any increase in the velocity caused the point of break-down to approach the trumpet, but with no velocities that were tried did it reach this. On viewing the tube by the light of an electric spark, the mass of colour resolved itself into a mass of more or less distinct curls showing eddies.