Jean Perrin and Avogadro’s Number

Regular readers of this blog may recall that last summer I visited Paris for my 25th wedding anniversary, which was followed by a string of blog entries about famous French scientists. During this trip, my wife and I toured the Pantheon, where we saw the burial site of French scientist Jean Baptiste Perrin (1870–1942). Russ Hobbie and I mention Perrin in a footnote on page 85 of the 4th edition of Intermediate Physics for Medicine and Biology.

The Boltzmann factor provided Jean Perrin with the first means to determine Avogadro’s number [N_A]. The density of particles in the atmosphere is proportional to exp(−mgy/k_BT), where mgy is the gravitational potential energy of the particles. Using particles for which m was known, Perrin was able to determine [Boltzmann’s constant] k_B for the first time. Since the gas constant R was already known, Avogadro’s number was determined from the relationship R = N_Ak_B.

This brief footnote does not do justice to Perrin’s extensive accomplishments. He played a key role in establishing that matter is not a continuum, but rather is made out of atoms. He performed experiments not only on the exponential distribution of particles (described above, and also known as sedimentation equilibrium), but also on Brownian motion. Russ and I describe this phenomenon in Chapter 4:

This movement of microscopic-sized particles, resulting from bombardment by much smaller invisible atoms, was first observed by English botanist Robert Brown in 1827 and is called Brownian motion.

Molecular Reality:
A Perspective on the
Scientific Work of Jean Perrin,
by Mary Jo Nye.

One can learn more about Perrin in the book Molecular Reality: A Perspective on the Scientific Work of Jean Perrin, by Mary Jo Nye. I would not rank this book with the best histories of science I have read (my top three would be The Making of the Atomic Bomb, The Eighth Day of Creation, and The Maxwellians), or among the best scientific biographies (such as Subtle is the Lord: The Science and Life of Albert Einstein). However, it did provide some valuable insight into Perrin’s achievements. Ney states in her introduction that

What has struck me in a perusal of the literature on these topics [discoveries in physics during the early 20th century] is the tendency to assume what so many of the physical scientists of this pivotal period did not for one minute assume—the discontinuity of the matter which underlies visible reality. In looking back upon the discoveries and theories of particles, one perhaps fails to realize that the focus was not simply upon the nature of the molecules, ions and atoms, but upon the very fact of their existence…

In analyzing the role of Jean Perrin in the eventual acceptance of this assumption among the outspoken majority of the scientific community, I have concentrated upon the period of experimental, theoretical, philosophical and popular science which climaxed with the Solvay conference of 1911 and with the publication of Perrin’s book Les Atomes [read an online English translation here] in 1913…

In conclusion, I have discussed the reception of Perrin’s scientific experimentation and propagandisation on the subject of molecular reality, especially his work on Brownian movement, which climaxed in 1913 with the completion of a number of national and international conferences and the publication of Les Atomes. Though Perrin himself did not view his task as completed at that time, the question was no longer central to the basic working assumptions of scientists, and polemics on this question were no longer an impediment or impetus to the progress of general scientific conceptualization. That Perrin’s role was historically essential to this denouement cannot, in my opinion, be doubted.

Nye’s first chapter on 19th-century background contains a little too much philosophy of science for my taste. But her historical review does indicate that, despite what our footnote says, Perrin did not provide the first estimate for Avogadro’s number, but rather provided a definitive early measurement of that value. Her second chapter about Young Perrin: Initial Investigations was better, and the book really captured my attention in the third chapter on The Essential Debate.

The exponential law which Perrin announced in 1908, describing the vertical distribution of a colloid at equilibrium, was the fruit of laborious experiments on Brownian movement after several years of apprenticeship in the study of colloids. Included in his first 1908 paper on Brownian movement was a successful application of the concepts of osmotic pressure and mean kinetic energy to the visible Brownian particles, as well as a convincing calculation of Avogadro’s number. These endevours were but the prelude to a five-year drama devoted to the erection of an unassailable edifice to house the dictum of molecular reality, a structure buttressed at its most vulnerable point of criticism by the observed laws of visible Brownian movement.

I was particularly fascinated by how Perrin knew the mass of the particles he studied.

In order to find m, Perrin utilized Stoke’s law [see Section 4.5 of Intermediate Physics for Medicine and Biology], applying it to a column of the emulsion in a vertical capillary tube, and observing the fact that when the emulsion is very far from equilibrium, the Brownian granules in the upper layers of the column fall as if they were droplets of a cloud. Using Stokes’ formula relating the velocity of a spherical droplet, its radius, and the viscosity of the medium, Perrin found the radius of the granules [on the order of a micron].

Then from the known density, he could determine the mass. Perrin had to go to great lengths to obtain particles with a uniform distribution of radii, starting with 1200 grams of particles and, after repeated centrifugation, ending with less than a gram of uniform particles.

In 1926, Jean Perrin won the 1926 Nobel Prize in physics “for his work on the discontinuous structure of matter, and especially for his discovery of sedimentation equilibrium.”

e, The Story of a Number

On page 33 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce the constant e.

The number e is approximately equal to 2.71828… and is called the “base of the natural logarithms.” Like π (3.14159…) e has a long history [Maor (1994)].

e: The Story of a Number,
by Eli Maor.

The citation is to the delightful book e: The Story of a Number, by Eli Maor. In his preface, Maor explains why he wrote the book.

My goal is to tell the story of e on a level accessible to readers with only a modest background in mathematics. I have minimized the use of mathematics in the text itself, delegating several proofs and derivations to the appendixes. Also, I have allowed myself to digress from the main subject on occasion to explore some side issues of historical interest. These include biographical sketches of the many figures who played a role in the history of e, some of whom are rarely mentioned in textbooks. Above all, I want to show the great variety of phenomena—from physics and biology to art and music—that are related to the exponential function e^x, making it a subject of interest in fields well beyond mathematics.

Our Chapter 2, about exponential growth, centers on the exponential and logarithm functions, and our Appendix C lists many of the properties of these functions. Maor explores all sorts of interesting facts about e. For instance, 878/323 is a very good rational approximation to this irrational number. You can recall the first ten digits of e by remembering 2.7 (Andrew Jackson)² [Jackson was elected president in 1828]. In his Chapter 13, Maor presents some beautiful continued fractions for e that I will not attempt to reproduce here using html.

When developing the Fourier series in Chapter 11 of Intermediate Physics for Medicine and Biology, Russ and I note that “the remarkable property of imaginary numbers that make them useful in this context is that e^iθ = cosθ + i sinθ.” (Here, i is the square root of minus one.) Maor sets θ = π to obtain an equation studied by the Swiss mathematician Leonhard Euler

e^iπ = −1 ,

and claims

it must surely rank among the most beautiful formulas in all of mathematics. Indeed, by rewriting it as e^πi + 1 = 0, we obtain a formula that connects the five most important constants of mathematics (and also the three most important mathematical operation—addition, multiplication, and exponentiation). These five constants symbolize the four major branches of classical mathematics: arithmetic, represented by 0 and 1; algebra, by i; geometry, by π; and analysis, by e.

Taking a less aesthetic view, Russ and I downplay the use of complex exponentials in Intermediate Physics for Medicine and Biology.

The Fourier transform is usually written in terms of complex exponentials. We have avoided using complex exponentials. They are not necessary for anything done in this book. The sole advantage of complex exponentials is to simplify the notation. The actual calculations must be done with real numbers.

Another reason I often steer clear of complex exponentials is that I place great importance on being able to visualize physically what a mathematical expression is saying, and I find trigonometric functions far easier to envision than complex exponentials. So, while I concede the abstract beauty of the formula e^iπ = −1, I don’t find it so useful when thinking about physics.

While educating his readers about e, Maor also introduces them to many famous mathematicians, including Archimedes, Napier, Newton, Gauss, the Bernoullis, and above all Euler, who is apparently one of Maor’s favorites.

Leonhard Euler (1707–1783) is unquestionably the Mozart of mathematics, a man whose immense output--not yet published in full—is estimated to fill at least seventy volumes. Euler left hardly an area of mathematics untouched, putting his mark on such diverse fields as analysis, number theory, mechanics and hydrodynamics, cartography, topology, and the theory of lunar motion.

Maor discusses the uses of logarithms and exponentials in biology. He talks about the logarithmic spiral and its role in growth, for instance, of a nautilus shell. He also makes an interesting comparison between the ear and the eye.

The remarkable sensitivity of the human ear to frequency changes is matched by its audibile range—from about 20 cycles per second to about 20,000 (the exact limits vary somewhat with age). In terms of pitch, this corresponds to about ten octaves (an orchestra rarely uses more then seven). By comparison, the eye is sensitive to a wavelength range from 4,000 to 7,000 angstroms (10⁻⁸ cm)—a range of less than two “octaves.” [Doesn’t Maor mean: less than one “octave”?]

I’m particularly fond of Maor’s recreation of a meeting between Bach and one of the Bernoulli’s

Let us imagine a meeting between Johann Bernoulli (Johann I, that is) and Johann Sebastian Bach. The year is 1740. Each is at the peak of his fame. Bach, at the age of fifty-five, is organist, composer, and Kapellmeister (musical director) at St. Thomas’s Church in Leipzig. Bernoulli, at seventy three, is the most distinguished professor of the University of Basel.

The resulting imagined conversation is fascinating and amusing. Musicians interested in the “equal tempered scale” will enjoy this section.

I will close this blog entry the same way Maor ended his book, by letting e take a final bow. Here is e to one hundred decimal places:

2.7182818284590452353

60287471352662497757

24709369995957496696

76277240766303535475

94571382178525166427

Non-Newtonian Fluids and the Rheology of Blood

In Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I explain the difference between a Newtonian fluid and a non-Newtonian fluid.

A fluid can support a viscous shear stress if the shear strain is changing. One way to create such a situation is to immerse two parallel plates, each of area S, in the fluid, and to move one parallel to the other … The variation of velocity between the plates gives rise to a velocity gradient dv_x/dy …

In order to keep the top plate moving and the bottom plate stationary, it is necessary to exert a force of magnitude F on each plate: to the right on the upper plate and to the left on the lower plate. The resulting shear stress or force per unit area is in many cases proportional to the velocity gradient:

F/S = η dv_x/dy .   (1.33)

The constant η is called the coefficient of viscosity … Fluids that are described by Eq. 1.33 are called Newtonian fluids. Many fluids are not Newtonian.

At the end of the chapter, we give an example of a biologically important non-Newtonian fluid.

Blood is not a Newtonian fluid. The viscosity depends strongly on the fraction of volume occupied by red cells (the hematocrit).

An excellent review of blood’s fluid behavior can be found in the article “Rheology of Blood” by Edward Merrill (Physiological Reviews, Volume 49, Pages 863–888, 1969). Rheology is the part of fluid mechanics that deals with non-Newtonian fluids. Merrill explains clearly the difference between a Newtonian fluid with a high viscosity and a Non-Newtonian fluid.

A Newtonian liquid is one in which the viscosity, at fixed temperature and pressure, is independent of the shear stress. Thus, a non-Newtonian liquid is one in which the viscosity depends on shear stress. Water and honey are Newtonian, but many aqueous suspensions of fine particulate matter such as water-base paint, plaster, and oil emulsions are non-Newtonian. The distinction is qualitatively obvious if one imagines two spoons, one in a pot of honey (Newtonian) and the other in a pot of mayonnaise (non-Newtonian emulsion). The honey is harder to stir (has a higher viscosity) than the mayonnaise, but when the spoons are removed and held above the pots, the honey continues to drizzle off its spoon, whereas the mayonnaise coating the other spoon clings indefinitely to it without flow, thus exhibiting “infinite” viscosity.

An important concept when discussing the rheology of blood is yield stress. Merrill explains

Blood … exhibits a “yield stress.” This means that, if …one increases from zero the stress, but keeps it less than a critical value, the response will be elastic … On removal of the stress, the shape of the blood film will be unaltered, i.e., no flow will have occurred. However, if the yield stress is exceeded, irreversible deformation will occur.

In other words, it acts like a solid at low stress, and a fluid at high stress. Merrill concludes by discussing the physiological significance of the non-Newtonian nature of blood.

In summary, the relevance of blood rheology to physiological fluid mechanics is to make stopping of flows easier, starting of flows more difficult, and slow flows more energy consuming than would be expected if blood were a simple, cell-less, micromolecular fluid of equal viscosity—and these effects are increasingly emphasized with increase of hematocrit and fibrinogen concentration.

Besides blood, another dramatic example of a non-Newtonian fluid is a mixture of corn starch and water. My Oakland University colleague Alberto Rojo (whose office is next door to mine) has made a fun video demonstrating how you can “walk on water” by taking advantage of this mixture’s non-Newtonian properties. The effect is fascinating.

Alberto Rojo walks on a mixture of corn starch and water.

https://www.youtube.com/watch?v=5BdRLQCumXA

Drawing Figures

Two of my favorite figures in the 4th edition of Intermediate Physics for Medicine and Biology are Fig. 7.13 (the extracellular potential produced by an action potential along a nerve axon) and Fig. 8.14 (the magnetic field produced by the same axon). John Wikswo and I prepared these figures when I was in graduate school at Vanderbilt University.

Fig. 7.13. The exterior potential calculated using the method of Clark and Plonsey.
From Intermediate Physics for Medicine and Biology, 4th edition.

Fig. 8.14. A three-dimensional plot of the magnetic field around the crayfish axon.
From Intermediate Physics for Medicine and Biology, 4th edition.

Soon after entering graduate school in 1982, I took a class taught by John based on Russ Hobbie’s first edition of Intermediate Physics for Medicine and Biology. Clearly, the book had a significant influence on my subsequent career. (I remember the bright yellow cover of the first edition: my office is probably one of the few places outside of Minnesota where all four editions of the book sit proudly, side-by-side, on a bookshelf.) When preparing the second edition, Russ added a chapter on biomagnetism, and asked John to contribute a figure showing the magnetic field produced by an axon. Of course, this is just the sort of work graduate students are good for, and I was given the task of preparing the figure (actually two figures, as we decided to make a similar figure for the extracellular potential). This was not a big job, because I already had access to the computer code that my friend Jim Woosley had written for his master’s thesis, and which I used when preparing our paper “The Magnetic Field of a Single Axon: A Volume Conductor Model,” (Woosley, Roth, and Wikswo, 1985, Mathematical Biosciences, Volume 76, Pages 1–36).

In the mid-1980s, three-dimensional graphics programs were not as common as they are now, but we had one and I was able to create the figure. What we didn’t have was a publication-quality printer or software to prepare and manipulate figures. Therefore, once I had the plots created, they went to the drafting room to be finished. John usually had one or more undergraduates hired for the sole task of preparing figures. I don’t remember exactly who worked on the two figures for the 2nd edition, but it may have been David Barach, son of Vanderbilt physics professor John Barach. The daftsman’s job was to retrace the figure, thereby providing a higher quality appearance than a dot-matrix printer could provide. As I recall, his job was also to remove hidden lines (I don’t think that our 3-d graphics program was “smart” enough to remove hidden lines on its own). He also labeled all the axes using some really neat rub-on letters that John was able to purchase in both Roman and Greek fonts (note the “μ” in μV in Fig. 7.13). I remember David working on figures at a large, slanted drafting table, using very high quality, vellum-like paper. He had rulers, triangles, and “French curves” of all types. First the drawing was done in pencil, and then traced with black ink. Once finished, additional copies were made using photography by a center in the Vanderbilt Medical School dedicated to such work. Before Photoshop, Powerpoint, and other such programs, that is the way figures were prepared. John had a policy that all graduate students had to get some experience at the drafting table, which I didn’t mind at all. At the risk of sounding like a Luddite who is nostalgic for the days of buggy whips, I think those figures have a little more personality and visual appeal than computer-generated figures drawn today.

The figures appeared in the second edition of Russ’s book, and have continued on through subsequent editions (including the 4th edition, on which I have the high honor of becoming a coauthor). Figures like that required much time and expense to prepare, and are difficult to edit. But my, it was more fun to really “draw” those figures than it is to churn out figures using graphics software.

Central Slice Theorem and Ronald Bracewell

Chapter 12 of the 4th edition of Intermediate Physics for Medicine and Biology deals with images and tomography. One of the key ideas in tomography is the “central slice theorem.” Russ Hobbie and I write in Section 12.4 that

The Fourier transform of the projection at angle θ is equal to the two-dimensional Fourier transform of the object, evaluated in the direction θ in Fourier transform space. This result is known as the projection theorem or the central slice theorem (Problem 17). The transforms of a set of projections at many different angles provide values of C and S [the cosine and sine parts of the 2-d Fourier transform] throughout the k_xk_y plane [frequency space] that can be used in Eq. 12.9a [the definition of the 2-d Fourier transform] to calculate f(x,y).

I consider the central slice theorem to be one of the most important concepts in medical imaging. How was this fundamental idea first developed? The answer to that question provides a fascinating example of how physics and engineering can contribute to medicine.

Ronald Bracewell first developed the central slice theorem while working in the field of radio astronomy. His 2007 New York Times obituary states

Ronald N. Bracewell, an astronomer and engineer who used radio telescopes to make early images of the Sun’s surface, in work that also led to advances in medical imaging, died on Aug. 12 at his home in Stanford, Calif. He was 86…

With his colleagues at Stanford University in the 1950s, Dr. Bracewell designed a specialized radio telescope, called a spectroheliograph, to receive and evaluate microwaves emitted by the Sun…

Later, in the 1970s, the techniques and a formula devised by Dr. Bracewell were applied by other scientists in developing X-ray imaging of tumors, called tomography, and other forms of medical imaging that scan electromagnetic and radio waves. Dr. Bracewell advised researchers at Stanford and other institutions, but did not conduct laboratory research in the field.

The Fourier Transform
and Its Applications,
by Ronald Bracewell.

Russ and I cite Bracewell’s 1990 paper “Numerical Transforms” (Science, Volume 248, Pages 697–704). The central slice theorem was published in 1956 in the Australian Journal of Physics (Volume 9, Pages 198–217). Early in this career Bracewell published a lot in that journal, which is now defunct but maintains a website with free access to all the papers. Bracewell also wrote a marvelous book: The Fourier Transform and Its Applications (originally published in 1965, the revised 2nd edition is published by McGraw-Hill, New York, 1986). When writing this blog entry, I checked this book out of Kresge Library here at Oakland University. Once I opened it, I realized it is an old friend. I am sure I read this book in graduate school. It contains many pictures that allow the student to gain an intuition about the Fourier transform; an extraordinarily valuable skill to develop. The introduction states

The present work began as a pictorial guide to Fourier transforms to complement the standard lists of pairs of transforms expressed mathematically. It quickly became apparent that the commentary would far outweigh the pictorial list in value, but the pictorial dictionary of transforms is nevertheless important, for a study of the entries reinforces the intuition, and many valuable and common types of function are included which, because of their awkwardness when expressed algebraically, do not occur in other lists.

The text also does a fine job describing convolutions.

Convolution is used a lot here. Experience shows that it is a fairly tricky concept when it is presented bluntly under its integral definition, but it becomes easy if the concept of a functional is first understood.

Many of the ideas that Russ and I present in Chapter 11 of Intermediate Physics for Medicine and Biology are examined in more detail in Bracewell’s book. I recommend it as a reference to keep at your side as your plow through the mathematics of Fourier analysis.

Finally, Bracewell’s view of homework problems, as stated in his Preface to the second edition, mirrors my own.

A good problem assigned at the right stage can be extremely valuable for the student, but a good problem is hard to compose. Among the collection of supplementary problems now included at the end of the book are several that go beyond being mathematical exercises by inclusion of technical background or by asking for opinions.

Bursting

Last week in this blog I talked briefly about bursting in pancreatic beta cells. A bursting cell fires several action potential spikes consecutively, followed by an extended quiescent period, followed again by another burst of action potentials, and so on. One of the first and best-known models for bursting was developed by James Hindmarsh and Malcolm Rose (“A Model of Neuronal Bursting Using Three Coupled First Order Differential Equations,” Proceedings of the Royal Society of London, B, Volume 221, Pages 87–102, 1984). Their analysis was an extension of the FitzHugh-Nagumo model, with an additional variable governed by a very slow time constant. Their system of equations is

dx/dt = y – x³ + 3 x² – z + I

dy/dt = 1 – 5 x² – y

dz/dt = 0.001 [ 4(x + 1.6) – z]

where x is the membrane potential (appropriately made dimensionless), y is a recovery variable (like a sodium channel inactivation gate), z is the slow bursting variable, and I is an external stimulus current. For some values of I, this model predicts bursting behavior.

Bursting: The Genesis of
Rhythm in the Nervous System,
by Stephen Coombes and Paul Bressloff.

There is an entire book dedicated to this topic: Bursting--The Genesis of Rhythm in the Nervous System, by Stephen Coombes and Paul Bressloff (World Sci. Pub. Co., 2005). The first chapter, co-written by Hindmarsh, provides a little of the history behind the Hindmarsh-Rose model:

The collaboration that led to the Hindmarsh-Rose model began in 1979 shortly after Malcolm Rose joined Cardiff University. The particular project was to model the synchronization of firing of two snail neurons in a relatively simple way that did not use the full Hodgkin-Huxley equations... A natural choice at the time was to use equations of the FitzHugh [type]…

A problem with this choice was that these equations do not provide a very realistic description of the rapid firing of the neuron compared to the relatively long interval between firing. Attempts were made to achieve a more realistic description by making the time constants … voltage dependent. In particular so the rates of change of x and y were much smaller in the subthreshold or recovery phase. These were not convincing and it was not until Malcolm raised the question about whether the FitzHugh equations could account for “tail current reversal” that progress was made.

The modification of the FitzHugh equations to account for tail current reversal was crucial for the development of the Hindmarsh-Rose model.

For those not familiar with the FitzHugh-Nagumo model, see Problem 33 in Chapter 10 of the 4th edition of Intermediate Physics for Medicine and Biology, or see the Scholarpedia article by FitzHugh himself, written before he died in 2007. If you want to see some bursting patterns, check out this youtube video. It is not great, but you will get the drift of what the model predicts.

My friend Artie Sherman also had a chapter in the bursting book, titled “Beyond Synchronization: Modulatory and Emergent Effects of Coupling in Square-Wave Bursting.” He has been working on bursting in pancreatic beta cells for years, as a member (and now chief) of the Laboratory of Biological Modeling in the Mathematical Research Branch, the National Institute of Diabetes, Digestive and Kidney Diseases, part of the National Institutes of Health. His work is the best I am aware of for modeling bursting.

Effects of Rapid Buffers on Ca2+ Diffusion and Ca2+ Oscillations

I enjoy taking a scientific paper and reducing it to a homework problem. For example, one of the new homework problems in the 4th edition of Intermediate Physics for Medicine and Biology is Problem 23 of Chapter 4 (Transport in an Infinite Medium), based on a paper by John Wagner and Joel Keizer.

Problem 23 Calcium ions diffuse inside cells. Their concentration is also controlled by a buffer:

Ca + B ⇐⇒ CaB.

The concentrations of free calcium, unbound buffer, and bound buffer ([Ca], [B], and [CaB]) are governed, assuming the buffer is immobile, by the differential equations

∂[Ca]/∂t= D∇²[Ca] − k⁺[Ca][B] + k⁻[CaB],
∂[B]/∂t= −k⁺[Ca][B] + k⁻[CaB],
∂[CaB]/∂t= k⁺[Ca][B] − k⁻[CaB].

(a) What are the dimensions (units) of k⁺ and k⁻ if the concentrations are measured in mole l⁻¹ and time in s?

(b) Derive differential equations governing the total calcium and buffer concentrations, [Ca]_T = [Ca]+[CaB] and [B]_T= [B] + [CaB] . Show that [B]_T is independent of time.

(c) Assume the calcium and buffer interact so rapidly that they are always in equilibrium:

[Ca][B]/[CaB]= K,

where K = k⁻/k⁺.Write [Ca]_T in terms of [Ca] , [B]_T , and K (eliminate [B] and [CaB]).

(d) Differentiate your expression in (c) with respect to time and use it in the differential equation for [Ca]_T found in (b). Show that [Ca] obeys a diffusion equation with an “effective” diffusion constant that depends on [Ca]:

D_eff = D/(1 + K [B]_T/(K+[Ca])²) .

(e) If [Ca] < < K and [B]_T = 100K (typical for the endoplasmic reticulum), determine D_eff/D.

For more about diffusion with buffers, see Wagner and Keizer (1994).

The reference and abstract of the paper is given below.

John Wagner and Joel Keizer (1994) “Effects of Rapid Buffers on Ca2+ Diffusion and Ca2+ Oscillations,” Biophysical Journal, Volume 67, Pages 447–456.

Based on realistic mechanisms of Ca²⁺ buffering that include both stationary and mobile buffers, we derive and investigate models of Ca²⁺ diffusion in the presence of rapid buffers. We obtain a single transport equation for Ca²⁺ that contains the effects caused by both stationary and mobile buffers. For stationary buffers alone, we obtain an expression for the effective diffusion constant of Ca²⁺ that depends on local Ca²⁺ concentrations. Mobile buffers, such as fura^-2, BAPTA, or small endogenous proteins, give rise to a transport equation that is no longer strictly diffusive. Calculations are presented to show that these effects can modify greatly the manner and rate at which Ca²⁺ diffuses in cells, and we compare these results with recent measurements by Allbritton et al. (1992). As a prelude to work on Ca²⁺ waves, we use a simplified version of our model of the activation and inhibition of the IP3 receptor Ca²⁺ channel in the ER membrane to illustrate the way in which Ca²⁺ buffering can affect both the amplitude and existence of Ca²⁺ oscillations.

John Wagner is currently with the Functional Genomics and Systems Biology Group of the IBM T. J. Watson Research Center. In the mid 1990s he was a research assistant with Joel Keizer.

Joel Keizer was a long-time member of the University of California at Davis. A UC Davis website states

Joel’s scientific legacy encompassed several fields. Joel originally trained as a chemist at the University of Oregon under Terrell Hill, where he received his doctorate in theoretical physical chemistry, and did postdoctoral work in chemical physics at the Battelle Institute in Columbus, Ohio. He began his career in 1971 at the University of California, Davis, as an assistant professor of chemistry. He pioneered an approach to the thermodynamics of non-equilibrium steady states, which culminated in the monograph, Statistical Thermodynamics of Nonequilibrium Processes in 1987. By this time, he had over 60 journal publications to his credit.

In the 1980s, Joel gradually shifted his research program and focused his powerful intellect on problems within the biological sciences, first on mathematical models of insulin secretion, and later on intracellular calcium oscillations and diffusion. He subsequently transferred his appointment to the Division of Biological Sciences, where both theoreticians and empiricists respected and admired Joel for his strong modeling work and his insightful collaborations with experimental biologists.

I never met Joel Keizer, but I did know a couple of his collaborators, John Rinzel and Arthur Sherman, both at NIH when I was there in the early 1990s. They worked on bursting in pancreatic beta-cells, and published some influential papers with Keizer (for example, see: Sherman, Rinzel, and Keizer (1988) “Emergence of Organized Bursting in Clusters of Pencreatic Beta-Cells by Channel Sharing,” Biophysical Journal, Volume 54, Pages 411–425).

Finally, the paper by Allbritton et al. cited in the Wagner and Keizer paper is:

Allbritton, Meyer, Stryer (1992) “Range of Messenger Action of Calcium-Ion and Inositol 1,4,5-Trisphosphate,” Science, Volume 258, Pages 1812–1815.

Superconductivity

This month marks the hundredth anniversary of the discovery of superconductivity. An article in the magazine IEEE Spectrum states

On April 8, 1911, physicist Heike Kamerlingh Onnes of Leiden University used an intricate glass cryostat to cool mercury down to just a few degrees above absolute zero. Then he scribbled down three words that ultimately marked the discovery of an entirely new physical phenomenon.

The phrase, jotted more than halfway down the page of a messy lab notebook, didn’t really match the occasion. What Kamerlingh Onnes wrote was “Mercury practically zero,” or, according to a more literal translation, “Quick[silver] near-enough dull.” But what he saw was the first evidence of superconductivity, the ability of some substances to conduct electricity with no resistance at all.

You can learn more about this landmark event in a Physics Today September 2010 article “The Discovery of Superconductivity,” by Dirk van Delft and Peter Kes, and the article “Superconductivity’s Smorgasbord of Insights: A Movable Feast,” in the April 8, 2011 issue of Science by Adrian Cho. Also, see the biography of Onnes on Nobelprize.org.

The Quest for Absolute Zero,
by Kurt Mendelssohn.

One of my favorite books is The Quest for Absolute Zero, by Kurt Mendelssohn. He starts his tale in 1877 with the liquefaction of oxygen and then tells the subsequent history of low temperature physics, including the fascinating story of how Onnes liquefied helium and his early superconductivity studies. According to Mendelssohn, the reason mercury was used for the first experiment is because it could be purified:

There was one other metal which might be obtained in an even purer state than gold, and that was mercury. Being a liquid at room temperatures, it can be distilled and re-distilled again and again until an extreme degree of purity is reached. The results were communicated to the Netherlands Royal Academy on the 28th April 1911, when Onnes reported that mercury, as well as a sample of very pure gold, had, at helium temperature, reached resistivities so low that his instruments had failed to detect them. He was particularly intrigued with the behavior of the mercury sample because it had still a fairly high resistance at liquid hydrogen temperatures and could also be recorded at the boiling point of liquid helium but then vanished at lower temperatures.

Russ Hobbie and I discuss superconductivity in Section 8.9 (Detection of Weak Magnetic Fields) of the 4th edition of Intermediate Physics for Medicine and Biology.

The [magnetic] signals from the body are weaker, and their measurement requires higher sensitivity and often special techniques to reduce noise. Hämäläinen et al. (1993) present a detailed discussion of the instrumentation problems. Sensitive detectors are constructed from superconducting materials. Some compounds, when cooled below a certain critical temperature, undergo a sudden transition and their electrical resistance falls to zero. A current in a loop of superconducting wire persists for as long as the wire is maintained in the superconducting state. The reason there is a superconducting state is a well-understood quantum-mechanical effect that we cannot go into here. It is due to the cooperative motion of many electrons in the superconductor [Eisberg and Resnick (1985), Sec. 14.1; Clarke (1994)].

We then go on to discuss superconducting quantum interference device (SQUID) magnetometers, which are often used to measure the small magnetic fields produced by the brain or the heart. Although not discussed in our book, superconductivity is also used in many MRI machines to produce the strong static magnetic field without losses due to heating of a copper coil.

The citations in the quote from our book are to:

Clarke, J. (1994) “SQUIDS,” Scientific American, Volume 1994, Pages 46–53.

Eisberg, R., and R. Resnick (1985) Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, 2nd ed. New York, Wiley.

Hämäläinen, M., R. Harri, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounasmaa (1993) “Magnetoencephalography—Theory, Instrumentation, and Applications to Noninvasive Studies of the Working Human Brain.” Reviews of Modern Physics, Volume 65, Pages 413–497.

Point/Counterpoint Revisited

In one of my first entries in this blog, I introduced readers to the Point/Counterpoint articles in the journal Medical Physics. I enjoy these articles immensely, and they provide valuable insight into important and controversial questions in the medical physics field. Each issue of Medical Physics contains one Point/Counterpoint, in which a proposition is stated and two prominent medical physicists debate it, one for and one against. They each have “opening statements” and then provide a “rebuttle” to their opponent’s claims. The articles make for more lively reading than a typical scientific paper full of jargon and technical content.

The September 2010 issue of Medical Physics contains a Point/Counterpoint article titled “Ultrasonography is Soon Likely to Become a Viable Alternative to X-Ray Mammography for Breast Cancer Screening.” Arguing for the proposition is Carri Glide-Hurst, a Senior Associate Physicist at Henry Ford Hospital in Detroit. Her opponent is Andrew Maidment, Associate Professor of Radiology at the University of Pennsylvania in Philadelphia. I am in favor of the proposition, for the silly reason that I always root for the home team. (Oakland University, where I work, is about 20 miles north of Detroit, and our medical physics program has several adjunct faculty at Henry Ford Hospital.)

The 4th edition of Intermediate Physics for Medicine and Biology provides much of the scientific background needed to understand this debate. Russ Hobbie and I added a new chapter to the 4th edition that describes ultrasound and its applications to medical imaging (Chapter 13, Sound and Ultrasound). After introducing the wave equation, we describe the decibel intensity scale, attenuation, medical uses of ultrasound, and the Doppler effect. Two chapters are dedicated to understanding x rays and x-ray imaging. In Chapter 15 (Interaction of Photons and Charged Particles with Matter) we analyze the basic mechanisms by which an x-ray photon affects tissue, including the photoelectric effect, Compton scattering, and pair production. Chapter 16 (Medical Use of X Rays) focuses on applications, including a section dedicated entirely to mammography.

Magnetic resonance imaging is another modality that produces no ionizing radiation. It is described in Chapter 18 of Intermediate Physics for Medicine and Biology. However, MRI is expensive, takes a long time, and cannot be used in some patients, such as those with surgical clips. Therefore the American Cancer Society recommends MRI only for a small group of patients.

Which side wins the debate? It’s always hard to say. I’m sure the moderator, Colin Orton, chooses only those questions that do not have obvious answers. Glide-Hurst concludes her opening statement by arguing

Ultrasound poses a practical and affordable solution for screening younger women with dense breasts, pregnant females, and those who do not meet the risk level requirements of breast MRI screening. Overall, whole-breast ultrasound is advantageous because it is volumetric, noninvasive, and nonionizing, and the current literature supports the routine implementation for breast cancer screening, particularly for women with dense breasts.

Maidment ends his opening statement by stating

Since ultrasound can distinguish solid tumors from fluidfilled cysts, it has a clear clinical role as a diagnostic tool in breast imaging. However, ultrasound does not appear useful for routine screening because of lower sensitivity and specificity compared to mammography, the suboptimal imaging of microcalcifications with ultrasonography, and the projected costs.

All things considered, I do know who wins the debate. The winner is the reader, who witnesses two experts carefully weighing the evidence, analyzing the physics, and predicting future trends. I encourage any student reading Intermediate Physics for Medicine and Biology to also browse through recent issues of Medical Physics. If, like me, you’re often short on time, skip the articles and just read Point/Counterpoint. You won’t regret it.

Fukushima Nuclear Reactors

Because of the scary events at the Fukushima nuclear reactors in Japan, the health hazards of radiation is in the news a lot. One place I turn to for authoritative information is the Health Physics Society. Here is what their website says:

As you are well aware, the Japanese experienced the worst earthquake in their history, followed by a devastating tsunami. These natural disasters have had a serious impact on several Japanese nuclear reactors, principally those at the Fukushima Daiichi site. The Health Physics Society is concerned about radiation exposures associated with these reactor problems and desires to keep our members and the concerned public advised on current events associated with the Japanese nuclear plants.

For information on the potential for radiation from the Japanese Nuclear Plants reaching the United States, see this Health Physics Society Ask the Experts FAQ. For information on radiation particle effects on food, read this Bloomberg FAQ.

Details of the status of the reactors at Fukushima are available in a document issued by the Japan Atomic Industrial Forum that is provided here. We will be updating this news item periodically to provide current information.

The Health Physics Society links to an interesting youtube video: an interview with John Boice of Vanderbilt University. He says “the fear is out of proportion to the risk,” and claims this event is no where near the situation in the Chernobyl diasater. (Warning: The interview was on March 20, and events seem to change daily.)

The Health Physics Society website also links to the following statement:

RADIATION RISKS TO HEALTH
A Joint Statement from the American Association of Clinical Endocrinologists, the American Thyroid Association, The Endocrine Society, and the Society of Nuclear Medicine
March 18, 2011

The recent nuclear reactor accident in Japan due to the earthquake and tsunami has raised fears of radiation exposure to populations in North America from the potential plume of radioactivity crossing the Pacific Ocean. The principal radiation source of concern is radioactive iodine including iodine-131, a radioactive isotope that presents a special risk to health because iodine is concentrated in the thyroid gland and exposure of the thyroid to high levels of radioactive iodine may lead to development of thyroid nodules and thyroid cancer years later. During the Chernobyl nuclear plant accident in 1986, people in the surrounding region were exposed to radioactive iodine principally from intake of food and milk from contaminated farmlands. As demonstrated by the Chernobyl experience, pregnant women, fetuses, infants and children are at the highest risk for developing thyroid cancer whereas adults over age 20 are at negligible risk.

Radioiodine uptake by the thyroid can be blocked by taking potassium iodide (KI) pills or solution, most importantly in these sensitive populations. However, KI should not be taken in the absence of a clear risk of exposure to a potentially dangerous level of radioactive iodine because potassium iodide can cause allergic reactions, skin rashes, salivary gland inflammation, hyperthyroidism or hypothyroidism in a small percentage of people. Since radioactive iodine decays rapidly, current estimates indicate there will not be a hazardous level of radiation reaching the United States from this accident. When an exposure does warrant KI to be taken, it should be taken as directed by physicians or public health authorities until the risk for significant exposure to radioactive iodine dissipates, but probably for no more than 1-2 weeks. With radiation accidents, the greatest risk is to populations close to the radiation source. While some radiation may be detected in the United States and its territories in the Pacific as a result of this accident, current estimates indicate that radiation amounts will be little above baseline atmospheric levels and will not be harmful to the thyroid gland or general health.

We discourage individuals needlessly purchasing or hoarding of KI in the United States. Moreover, since there is not a radiation emergency in the United States or its territories, we do not support the ingestion of KI prophylaxis at this time. Our professional societies will continue to monitor potential risks to health from this accident and will issue amended advisories as warranted.

News sources have been reporting that higher-than-normal radiation levels were detected in the United States. These observations say more about our ability to detect small amounts of radiation than about any risk to Americans. People living in the United States are at no risk of health hazards from radiation exposure caused by the Fukushima reactors.

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the risk of radiation in Section 13 of Chapter 16 (Medical Use of X Rays). We introduce the unit of the sievert (Sv), one of the most important units used when discussing radiation risk.

Both the sievert and the gray are J kg^-1. Different names are used to emphasize the fact that they are quite different quantities. One is physical, and the other includes biological effects. An older unit … is the rem. 100 rem = 1 Sv.

We then analyze the natural background dose, which is about 3 mSv per year, and which arises from several sources, including cosmic radiation, terrestrial rocks, and inhalation of radon gas.

If you prefer learning from a video, watch Understanding the Reactor Meltdown in Fukushima, Japan from a Physics Perspective on Youtube.

Time will tell if this event turns into a full-scale disaster. At the moment, it is a serious situation, but does not appear to be a serious health hazard, except perhaps for the workers trying to repair the power plants.