Donnan Equilibrium

Russ Hobbie and I analyze Donnan equilibrium in Chapter 9 of Intermediate Physics for Medicine and Biology.

Section 9.1 discusses Donnan equilibrium, in which the presence of an impermeant ion on one side of a membrane, along with other ions that can pass through, causes a potential difference to build up across the membrane. This potential difference exists even though the bulk solution on each side of the membrane is electrically neutral.

Today I present two new homework problems based on one of Donnan’s original papers.

Donnan, F. G. (1924) “The Theory of Membrane Equilibria.” Chemical Reviews, Volume 1, Pages 73-90.

Here’s the first problem.

Section 9.1

Problem 2 ½. Suppose you have two equal volumes of solution separated by a semipermeable membrane that can pass small ions like sodium and potassium but not large anions like A⁻. Initially, on the left is 1 mole of Na⁺ and 1 mole of A⁻, and on the right is 10 moles of K⁺ and 10 moles of A⁻. What is the equilibrium amount of Na⁺, K⁺, and A⁻ on each side of the membrane?

Stop and solve the problem using the methods described in IPMB. Then come back and compare your solution with mine (and Donnan’s).

In equilibrium, x moles of sodium will cross the membrane from left to right. To preserve electroneutrality, x moles of potassium will cross from right to left. So on the left you have 1 – x moles of Na⁺, x moles of K⁺, and 1 mole of A⁻. On the right you have x moles of Na⁺, 10 – x moles of K⁺, and 10 moles of A⁻.

Both sodium and potassium are distributed by the same Boltzmann factor, implying that

           [Na⁺]_left/[Na⁺]_right = [K⁺]_left/[K⁺]_right = exp(−eV/kT)            (Eq. 9.4)

where e is the elementary charge, V is the voltage across the membrane, k is Boltzmann’s constant, and T is the absolute temperature. Therefore

           (1 – x)/x = x/(10 – x)

or x = 10/11 = 0.91. The equilibrium amounts (in moles) are

                          left       right
           Na⁺        0.09      0.91
           K⁺          0.91      9.09
           A⁻          1.00    10.00

The voltage across the membrane is

           V = kT/e ln([Na⁺]_right/[Na⁺]_left) = (26.7 mV) ln(10.1) = 62 mV .

Donnan writes

In other words, 9.1 per cent of the potassium ions originally present [on the right] diffuse to [the left], while 90.9 per cent of the sodium ions originally present [on the left] diffuse to [the right]. Thus the fall of a relatively small percentage of the potassium ions down a concentration gradient is sufficient in this case to pull a very high percentage of the sodium ions up a concentration gradient. The equilibrium state represents the simplest possible case of two electrically interlocked and balanced diffusion-gradients.

Like this problem? Here’s another. Repeat the last problem, but instead of initially having 10 moles of K⁺ on the right, assume you have 10 moles of Ca⁺⁺. Calcium is divalent; how will that change the problem?

Problem 3 ½. Suppose you have two solutions of equal volume separated by a semi-impermeable membrane that can pass small ions like sodium and calcium but not large anions like A⁻ and B⁻⁻.  Initially, on the left is 1 mole of Na⁺ and 1 mole of A⁻, and on the right is 10 moles of Ca⁺⁺ and 10 moles of B⁻⁻. What is the equilibrium amount of Na⁺, Ca⁺⁺, A⁻ and B⁻⁻ on each side of the membrane?

Again, stop, solve the problem, and then come back to compare solutions.

Suppose 2x moles of Na⁺ cross the membrane from left to right. To preserve electroneutrality, x moles of Ca⁺⁺ move from right to left. Both cations are distributed by a Boltzmann factor (Eq. 9.4)

           [Na⁺]_left/[Na⁺]_right = exp(−eV/kT)

           [Ca⁺⁺]_left/[Ca⁺⁺]_right  = exp(−2eV/kT) .

However,

          exp(−2eV/kT) = [ exp(−eV/kT) ]²

so

      { [Na⁺]_left/[Na⁺]_right }² = [Ca⁺⁺]_left/[Ca⁺⁺]_right

or
        [ (1 –2 x)/(2x) ]² = x/(10 – x)

This is a cubic equation that I can’t solve analytically. Some trial-and-error numerical work suggests x = 0.414. The equilibrium amounts are therefore

                          left       right
           Na⁺        0.172    0.828
           Ca⁺⁺      0.414    9.586
           A⁻          1           0
           B⁻⁻        0          10 

The voltage across the membrane is

           V = kT/e ln([Na⁺]_right/[Na⁺]_left) = (26.7 mV) ln(4.814) = 42 mV .

I think this is correct; Donnan didn’t give the answer in this case, so I’m flying solo.

Frederick Donnan.
From an article in the Journal of Chemical Education,
Volume 4(7), page 819.

Who was Donnan? Frederick Donnan (1870 – 1956) was an Irish physical chemist. He obtained his PhD at the University of Leipzig under Wilhelm Ostwald, and then worked for Henry van’t Hoff. Most of his career was spent at the University College London. He was elected a fellow of the Royal Society and won the Davy Medal in 1928 “for his contributions to physical chemistry and particularly for his theory of membrane equilibrium.”

The Effects of Spiral Anisotropy on the Electric Potential and the Magnetic Field at the Apex of the Heart

Readers of Intermediate Physics for Medicine and Biology may enjoy this story about some of my research as a graduate student, working for John Wikswo at Vanderbilt University. My goal was to determine if the biomagnetic field contains new information that cannot be obtained from the electrical potential.

In 1988, Wikswo, fellow grad student Wei-Qiang Guo, and I published an article in Mathematical Biosciences (Volume 88, Pages 191-221) about the magnetic field at the apex of the heart.

The Effects of Spiral Anisotropy on the Electric Potential and the Magnetic Field at the Apex of the Heart.

B. J. Roth, W.-Q. Guo, and J. P. Wikswo, Jr. 

Living State Physics Group, Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235

This paper describes a volume-conductor model of the apex of the heart that accounts for the spiraling tissue geometry. Analytic expressions are derived for the potential and magnetic field produced by a cardiac action potential propagating outward from the apex. The model predicts the existence of new information in the magnetic field that is not present in the electrical potential.

The analysis was motivated by the unique fiber geometry in the heart, as shown in the figure below, from an article by Franklin Mall. It shows how the cardiac fibers spiral outward from a central spot: the apex (or to use Mall’s word, the vortex).

The apex of the heart.
From Mall, F. P. (1911) “On the Muscular Architecture of the Ventricles of the Human Heart.” American Journal of Anatomy, Volume 11, Pages 211-266.

Our model was an idealization of this complicated geometry. We modeled the fibers as making Archimedean spirals throughout a slab of tissue representing the heart wall, perfused by saline on the top and bottom.

The geometry of a slab of cardiac tissue. The thickness of the tissue is l, the conductivity of the saline bath is σ_e, and the conductivity tensors of the intracellular and interstitial volumes are σ̃_i and σ̃_o. The variables ρ, θ, and z are the cylindrical coordinates, and the red curves represent the fiber direction. Based on Fig. 2 of Roth et al. (1988).

Cardiac tissue is anisotropic; the electrical conductivity is higher parallel to the fibers than perpendicular to them. This is taken into account by using conductivity tensors. Because the fibers spiral and make a constant angle with the radial direction, the tensors have off-diagonal terms when expressed in cylindrical coordinates.

Consider a cardiac wavefront propagating outward, as if stimulated at the apex. Two behaviors occur. First, ignore the spiral geometry. A wavefront produces intracellular current propagating radially outward and extracellular current forming closed loops in the bath (blue). This current produces a magnetic field above and below the slab (green).

The current (blue) and magnetic field (green) created by an action potential propagating outward from the apex of the heart if no off-diagonal terms are present in the conductivity tensors. Based on Fig. 5a of Roth et al. (1988).

Second, ignore the bath but include the spiral fiber geometry. Although the wavefront propagates radially outward, the anisotropy and fiber geometry create an intracellular current that has a component in the θ direction (blue). This current produces its own magnetic field (green).

The azimuthal component of the current (blue) and the electrically silent components of the magnetic field (green) produced by off-diagonal terms in the conductivity tensor, with σ_e = 0. Based on Fig. 5b of Roth et al. (1988).

Of course, both of these mechanisms operate simultaneously, so the total magnetic field distribution looks something like that shown below.

The total magnetic field at the apex of the heart. This figure is only qualitatively correct; the field lines may not be quantitatively accurate. Based on Fig. 5e of Roth et al. (1988).

The original versions of these beautiful figures were prepared by a draftsman in Wikswo’s laboratory. I can’t remember who, but it might have been undergraduate David Barach, who prepared many of our illustrations by hand at the drafting desk. I added color for this blog post.

The main conclusion of this study is that there exists new information about the tissue in the magnetic field that is not present from measuring the electrical potential. The ρ and z components of the magnetic field are electrically silent; the spiraling fiber geometry has no influence on the electrical potential.

Is this mathematical model real, or just the musings of a crazy physics grad student? Two decades after we published our model, Krista McBride—another of Wikswo’s grad students, making her my academic sister—performed an experiment to test our prediction, and obtained results consistent with our calculations.

I’m always amazed when one of my predictions turns out to be correct.

Consequences of the Inverse Viscosity-Temperature Relationship

In Homework Problem 50 of Chapter 1 in Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

Section 1.19

Problem 50. The viscosity of water (and therefore of blood) is a rapidly decreasing function of temperature. Water at 5° C is twice as viscous as water at 35° C. Speculate on the implications of this extreme temperature dependence for the circulatory system of cold-blooded animals. (For a further discussion, see Vogel 1994, pp. 27–31.)

Life in Moving Fluids,
by Steven Vogel.

Let’s see what Steven Vogel discusses. The citation is to Life in Moving Fluids: The Physical Biology of Flow.

CONSEQUENCES OF THE INVERSE VISCOSITY-TEMPERATURE RELATIONSHIP

At 5° C water is about twice as viscous (dynamically or kinematically) as at 35° C; organisms live at both temperatures and, indeed, at ones still higher and lower. Some experience an extreme range within their lifetimes—seasonally, diurnally, or even in different parts of the body simultaneously. Does the consequent variation in viscosity ever have biological implications?...

Consider the body temperature of animals. At elevated temperatures less power ought to be required to keep blood circulating if the viscosity of blood follows the normal behavior of liquids. And, in our case, it does behave in the ordinary way—human blood viscosity (ignoring blood’s minor non-Newtonianism) is 50% higher at 20° than at 37° C… Is this a fringe benefit of having a high body temperature? Probably the saving in power is not especially significant—circulation costs only about 6% of basal metabolic rate. More interesting is the possibility of compensatory adjustments in the bloods and circulatory systems of animals that tolerate a wide range of internal temperatures. The red blood cells of cold-blooded vertebrates, and therefore presumably their capillary diameters, are typically larger than either the nucleated cells of birds or the nonnucleated ones of mammals… The shear rate of blood is greatest in the capillaries; must these be larger in order to permit circulation at adequate rates without excessive cost in a cold body?...

Is the severe temperature dependence of viscosity perhaps a serendipitous advantage on occasion? A marine iguana of the Galapagos basks on warm rocks, heating rapidly, and then jumps into the cold Humboldt current to graze on algae, cooling only slowly. Circulatory adjustments as the animal takes the plunge have been postulated…, but no one seems to have looked at whether part of the circulatory reduction in cold water is just a passive consequence of an increase in viscosity. A variety of large, rapid, pelagic fish have circulatory arrangements that permit locomotory muscles to get quite hot when they’re in use…; blood flow ought to increase automatically at just the appropriate time.

A less speculative case is that of Antarctic mammals and birds… [They] must commonly contend with cold appendages, since full insulation of feet and flippers would be quite incompatible with their normal functions. The circulation of such an appendage often includes a [countercurrent] heat exchanger at the base of the limb so that, in effect, a cold-blooded appendage and a warm-blooded body can be run on the same circulatory system without huge loses of heat… Changes in blood viscosity will reduce flow to appendages when they get cold quite without active adjustments within the circulatory systems.

Vogel goes on for another couple pages. I love the way he uses comparative physiology to illustrate physics. He also covers a lot of ground, ranging from the Galapagos islands to Antarctica. Such discussions are typical of Life in Moving Fluids.

The Rayleigh-Einstein-Jeans law

In Chapter 14 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss blackbody radiation. Max Planck’s blackbody radiation formula is given in Eq. 14.33

where λ is the wavelength and T is the absolute temperature. This equation, derived in December 1900, is the first formula that contained Planck’s constant, h.

Often you can recover a classical (non-quantum) result by taking the limit as Planck’s constant goes to zero. Here’s a new homework problem to find the classical limit of the blackbody radiation formula.

Section 14.8

Problem 26 ½. Take the limit of Planck’s blackbody radiation formula, Eq. 14.33, when Planck’s constant goes to zero. Your result should should be the classical Rayleigh-Jeans formula. Discuss how it behaves as λ goes to zero. Small wavelengths correspond to the ultraviolet and x-ray part of the electromagnetic spectrum. Why do you think this behavior is known as the “ultraviolet catastrophe”?

Subtle is the Lord,
by Abraham Pais.

I always thought that the Rayleigh-Jeans formula was a Victorian result that Planck knew about when he derived Eq. 14.33. However, when thumbing through Subtle is the Lord: The Science and the Life of Albert Einstein, by Abraham Pais, I learned that the Rayleigh-Jeans formula is not much older than Planck’s formula. Lord Rayleigh derived a preliminary version of it in June 1900, just months before Planck derived Eq. 14.33. He published a more complete version in 1905, except he was off by a factor of eight. James Jeans caught Rayleigh’s mistake, corrected it, and thereby got his name attached to the Rayleigh-Jeans formula. I am amazed that Planck’s blackbody formula predates the definitive version of the Rayleigh-Jeans formula.

Einstein rederived Rayleigh’s formula from basic thermodynamics principles in, you guessed it, his annus mirabilis, 1905. Pais concludes “it follows from this chronology (not that it matters much) that the Rayleigh-Jeans law ought properly to be called the Rayleigh-Einstein-Jeans law.”

The Double Helix

The Double Helix,
by James Watson.

So you’re stuck at home because of the coronavirus pandemic, and you want to know what one book you should you read to get an idea what science is like? I recommend The Double Helix, by James Watson. It’s a lively and controversial story about how Watson and Francis Crick determined the structure of DNA, launching a revolution in molecular biology.

The very first few paragraphs of The Double Helix describe the influence of physics on biology. Readers of Intermediate Physics for Medicine and Biology will enjoy seeing how the two disciplines interact.

To whet your appetite, below are the opening paragraphs of The Double Helix. Enjoy!

I have never seen Francis Crick in a modest mood. Perhaps in other company he is that way, but I have never had reason so to judge hm. It has nothing to do with his present fame. Already he is much talked about, usually with reverence, and someday he may be considered in the category of Rutherford or Bohr. But this was not true when, in the fall of 1951, I came to the Cavendish Laboratory of Cambridge University to join a small group of physicists and chemists working on the three-dimensional structures of proteins. At that time he was thirty-five, yet almost totally unknown. Although some of his closest colleagues realized the value of his quick, penetrating mind and frequently sought his advice, he was often not appreciated, and most people thought he talked too much.

Leading the unit to which Francis belonged was Max Perutz, and Austrian-born chemist who came to England in 1936. He had been collecting X-ray diffraction data from hemoglobin crystals for over ten years and was just beginning to get somewhere. Helping him was Sir Lawrence Bragg, the director of the Cavendish. For almost forty years Bragg, a Nobel Prize winner and one of the founders of crystallography, and been watching X-ray diffraction methods solve structures of ever-increasing difficulty. The more complex the molecule, the happier Bragg became when a new method allowed its elucidation. Thus in the immediate postwar years he was especially keen about the possibility of solving the structures of proteins, the most complicated of all molecules. Often, when administrative duties permitted, he visited Perutz’ office to discuss recently accumulated X-ray data. Then he would return home to see if he could interpret them.

Somewhere between Bragg the theorist and Perutz the experimentalist was Francis, who occasionally did experiments but more often was immersed in the theories for solving protein structures. Often he came up with something novel, would become enormously excited, and immediately would tell it to anyone who would listen. A day or so later he would often realize that his theory did not work and return to experiments, until boredom generated a new attack on theory.

Biological Physics/Physics of Living Systems: A Decadal Survey

I want you to complete the
Biological Physics/Physics of Living Systems
decadal survey.

Hey readers of Intermediate Physics for Medicine and Biology! I’ve got a job for you. The National Academies is performing a decadal survey of biological physics, and they want your input.

The National Academies has appointed a committee to carry out the first decadal survey on biological physics. The survey aims to help federal agencies, policymakers, and academic leadership understand the importance of biophysics research and make informed decisions about funding, workforce, and research directions. This study is sponsored by the National Science Foundation.

Anyone who reads a blog like mine probably has plenty to say about biological physics. Here’s their request:

We invite you to share your thoughts on the future of biophysical science with the study committee and read the input already given to the committee. Input will be accepted throughout the study but will only receive maximum consideration if submitted by April 30, 2020.

Below is some more detail about what they’re looking for.

Description The committee will be charged with producing a comprehensive report on the status and future directions of physics of the living world. The committee’s report shall:

1. Review the field of Biological Physics/Physics of Living Systems (BPPLS) to date, emphasize recent developments and accomplishments, and identify new opportunities and compelling unanswered scientific questions as well as any major scientific gaps. The focus will be on how the approaches and tools of physics can be used to advance understanding of crucial questions about living systems.

2. Use selected, non-prioritized examples from BPPLS as case studies of the impact this field has had on biology and biomedicine as well as on subfields of physical and engineering science (e.g., soft condensed-matter physics, materials science, computer science). What opportunities and challenges arise from the inherently interdisciplinary nature of this interface?

3. Identify the impacts that BPPLS research is currently making and is anticipated to make in the near future to meet broader national needs and scientific initiatives.

4. Identify future educational, workforce, and societal needs for BPPLS. How should students at the undergraduate and graduate levels be educated to best prepare them for careers in this field and to enable both life and physical science students to take advantage of the advances produced by BPPLS. The range of employment opportunities in this area, including academic and industry positions, will be surveyed generally.

5. Make recommendations on how the U.S. research enterprise might realize the full potential of BPPLS, specifically focusing on how funding agencies might overcome traditional boundaries to nurture this area. In carrying out its charge, the committee should consider issues such as the state of the BPPLS community and institutional and programmatic barriers.

I’ve already submitted my comments. Now it’s your turn. The deadline is April 30.

Murray Eden

In 1992, when I was working at the National Institutes of Health, I wrote a review article about magnetic stimulation with my boss’s boss, Murray Eden. We submitted it to IEEE Potentials, a magazine aimed at engineering students. I liked our review, but somehow we never heard back from the journal. I pestered them a few times, and finally gave up and focused on other projects. I hate to waste anything, however, so I give the manuscript to you, dear readers (click here). It’s well written (thanks to our editor Barry Bowman, who improved many of my papers from that era) and describes the technique clearly. You can use it to augment the discussion in Section 8.7 (Magnetic Stimulation) in Intermediate Physics for Medicine and Biology. Unfortunately the article is out of date by almost thirty years.

I reproduce the title page and abstract below.

 

Eden was our fearless leader in the Biomedical Engineering and Instrumentation Program. He was an interesting character. You can learn more about him in an oral history available at the Engineering and Technology History Wiki. In our program, Eden was known for his contribution to barcodes. He was on the committee to design the ubiquitous barcode that you find on almost everything you buy nowadays. Just when the design was almost complete, Eden piped up and said they should include written numbers at the bottom of the barcode, just in case the barcode reader was down. There they have been, ever since (thank goodness!). I didn’t work too closely with Eden; I generally interacted with him through my boss, Seth Goldstein (inventor of the everting catheter). But Eden suggested we write the article, and I was a young nobody at NIH, so of course I said yes.

In Eden’s oral history interview, you can read about the unfortunate end of his tenure leading BEIP.

The world changed and I got a new director in the division, a woman who had been Director of Boston City Hospital’s Clinical Research Center. She and I battled a good deal and I just didn’t like it. By this time I was well over seventy and I said, “Okay, the hell with it. I’m going to retire.” I retired in the spring of ’94. It’s a very sad thing; I don’t like to talk about it very much. My program was essentially destroyed. A few years thereafter NIH administration took my program out of her control. They are currently trying to build the program up again, but most of the good people left.

I was one of the people who left. That woman who became the division director (I still can’t bring myself to utter her name) made it clear that all of us untenured people would not have our positions renewed, which is why I returned to acedemia after seven wonderful years at NIH. I shouldn’t complain. I’ve had an excellent time here at Oakland University and have no regrets, but 1994–1995 was a frustrating time for me.

After I left NIH, I stopped working on magnetic stimulation. I was incredibly lucky to be at NIH at a time when medical doctors were just starting to use the technique and needed a physicist to help. Even now, my most highly cited paper is from my time at NIH working on magnetic stimulation.

Announcement of Murray Eden's retirement in the NIH Record, March 15, 1994.

NMR Imaging of Action Currents

Vanderbilt Notebook 11,
Page 69, dated April 3, 1985

In graduate school, I kept detailed notes about my research. My PhD advisor, John Wikswo, insisted on it, and he provided me with sturdy, high-quality notebooks that are still in good shape today. I encourage my students to keep a notebook, but most prefer to record “virtual” notes on their computer, which is too newfangled for my taste.

My Vanderbilt Notebook 11 covers January 28 to April 25, 1985 (I was 24 years old). On page 69, in an entry dated April 3, I taped in a list of abstracts from the Sixth Annual Conference of the IEEE Engineering in Medicine and Biology Society, held September 15–17, 1984 in Los Angeles. A preview of the abstracts were published in the IEEE Transactions on Biomedical Engineering (Volume 31, Page 569, August, 1984). I marked one as particularly important:

NMR Imaging of Action Currents 

J. H. Nagel

The magnetic field that is generated by action currents is used as a gradient field in NMR imaging. Thus, the bioelectric sources turn out to be accessible inside the human body while using only externally fitted induction coils. Two- or three-dimensional pictures of the body’s state of excitation can be displayed.

That’s all I had: a three sentence abstract by an author with no contact information. I didn’t even know his first name. Along the margin I wrote (in blue ink):

I can’t find J H Nagel in Science Citation Index, except for 3 references to this abstract and 2 others at the same meeting (p. 575, 577 of same Journal [issue]). His address is not given in IEEE 1984 Author index. Goal: find out who he is and write him for a reprint.

How quaint; I wanted to send him a little postcard requesting reprints of any articles he had published on this topic (no pdfs back then, nor email attachments). I added in black ink:

3–25–88 checked biological abstracts 1984–March 1, 1988. None

Finally, in red ink was the mysterious note

See ROTH21 p. 1

In Notebook 21 (April 11, 1988 to December 1, 1989) I found a schedule of talks at the Sixth Annual Conference. I wrote “No Nagel in Session 14!” Apparently he didn’t attend the meeting.

Why tell you this story? Over the years I’ve wondered about using magnetic resonance imaging to detect action currents. I’ve published about it:

Wijesinghe, R. and B. J. Roth, 2009, Detection of peripheral nerve and skeletal muscle action currents using magnetic resonance imaging. Ann. Biomed. Eng., 37:2402-2406.

Jay, W. I., R. S. Wijesinghe, B. D. Dolasinski and B. J. Roth, 2012, Is it possible to detect dendrite currents using presently available magnetic resonance imaging techniques? Med. & Biol. Eng. & Comput., 50:651-657.

Xu, D. and B. J. Roth, 2017, The magnetic field produced by the heart and its influence on MRI. Mathematical Problems in Engineering, 2017:3035479.

I’ve written about it in this blog (click here and here). Russ Hobbie and I have speculated about it in Intermediate Physics for Medicine and Biology:

Much recent research has focused on using MRI to image neural activity directly, rather than through changes in blood flow (Bandettini et al. 2005). Two methods have been proposed to do this. In one, the biomagnetic field produced by neural activity (Chap. 8) acts as the contrast agent, perturbing the magnetic resonance signal. Images with and without the biomagnetic field present provide information about the distribution of neural action currents. In an alternative method, the Lorentz force (Eq. 8.2) acting on the action currents in the presence of a magnetic field causes the nerve to move slightly. If a magnetic field gradient is also present, the nerve may move into a region having a different Larmor frequency. Again, images taken with and without the action currents present provide information about neural activity. Unfortunately, both the biomagnetic field and the displacement caused by the Lorentz force are tiny, and neither of these methods has yet proved useful for neural imaging. However, if these methods could be developed, they would provide information about brain activity similar to that from the magnetoencephalogram, but without requiring the solution of an ill-posed inverse problem that makes the MEG so difficult to interpret.

Notebook 11.

Apparently all this activity began with my reading of Nagel’s abstract in 1985. Yet, I was never able to identify or contact him. Recent research indicates that the magnetic fields in the brain are tiny, and they produce effects that are barely measurable with modern technology. Could Nagel really have detected action currents with nuclear magnetic resonance three decades ago? I doubt it. But there is one thing I would like to know: who is J. H. Nagel? If you can answer this question, please tell me. I’ve been waiting 35 years!

Life in Moving Fluids (continued)

Life in Moving Fluids,
by Steven Vogel.

Yesterday, I quoted excerpts from Steven Vogel’s book Life in Moving Fluids about the Reynolds number. Today, I’ll provide additional quotes from Vogel’s Chapter 15, Flow at Very Low Reynolds Number.

[Low Reynolds number] is the world, as Howard Berg puts it, of a person swimming in asphalt on a summer afternoon—a world ruled by viscosity. It’s the world of a glacier of particles, the world of flowing glass, of laboriously mixing cold molasses (treacle) and corn (maise) syrup. Of more immediate relevance, it’s the everyday world of every microscopic organism that lives in a fluid medium, of fog droplets, of the particulate matter called “marine snow”… “Creeping flow” is the common term in the physical literature; for living systems small size rather than (or as well as) low speed is the more common entry ticket. And it’s a counterintuitive—which is to say unfamiliar—world.

Vogel then lists properties of low Reynolds number flow.

At very low Reynolds number, flows are typically reversible: a curious temporal symmetry sets in, and the flow may move matter around but in doing so doesn’t leave much disorder in its wake. Concomitantly, mixing is exceedingly difficulty…

Inertia is negligible compared to drag: when propulsion ceases, motion ceases…

Separation behind bluff bodies is unknown…

Boundary layers are thick because velocity gradients are gentle, and the formal definition of a boundary layer has little or no utility…

Nor can one create appreciable circulation around an airfoil… Turbulence, of course, is unimaginable…

While this queer and counterintuitive range is of some technological interest, its biological importance is enormous… since the vast majority of organisms are tiny, they live in this world of low Reynolds number. Flow at very low Reynolds number may seem bizarre to us, but the range of flow phenomena that we commonly contend would undoubtedly seem even stranger to someone whose whole experience was at Reynolds number well below unity.

Ha! Try explaining turbulence to Covid-19.

Vogel then discusses Edward Purcell’s classic paper “Life at Low Reynolds Number.” He notes

But while these slow, small-scale flows may seem peculiar, they’re orderly (Purcell calls them “majestic”) and far more amenable to theoretical treatment than the flows we’ve previously considered.

You can find an example of the theoretical analysis of low-Reynolds number flow in Homework Problem 46 in Chapter 1 of Intermediate Physics for Medicine and Biology, which discusses creeping flow around a sphere.

As you can probably tell, Vogel is a master writer. If you are suffering from boredom during this coronavirus pandemic, order a copy of Life in Moving Fluids from Amazon. I own the Second Edition, Revised and Expanded. It's the perfect read for anyone interested in biological fluid dynamics.

Life in Moving Fluids

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the central concept of fluid dynamics: The Reynolds number.

The importance of turbulence (nonlaminar 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 ,            (1.62)

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.

Life in Moving Fluids,
by Steven Vogel.

To provide a more in-depth analysis of Reynolds number, I will quote some excerpts from Life in Moving Fluids by Steven Vogel. I chose this book in part because of its insights into fluid dynamics, and in part because it is written so clearly. I use Vogel’s writing as a model for how to explain complicated concepts using vivid and simple language, metaphors, and analogies. He begins by analyzing the drag force on an object immersed in a moving fluid, and then introduces the “peculiarly powerful” Reynolds number, that “centerpiece of biological... fluid dynamics.”

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 surface-to-volume ratio in physiology….

  I love the analogy to surface-to-volume ratio. Vogel continues

One of the marvelous gifts of nature is that this index proves to be so simple—a combination of four variables [L, V, ρ, and η], each with an exponent of unity. It has, however, a few features worth some comment. First, the Reynolds number is dimensionless… so its value is independent of the system of units in which the variables are expressed. Second, in it reappears the kinematic viscosity… What matters isn’t the dynamic viscosity, μ [Russ and I use the symbol η], and the density, ρ, so much as their ratio… Finally, a bit about L, commonly called the “characteristic length.” For a circular pipe, the diameter is used; choosing the diameter rather than the radius is entirely a matter of convention… The value of the Reynolds number is rarely worth worrying about to better than one or at most two significant figures. Still, that’s not trivial when biologically interesting flows span at least fourteen orders of magnitude[!]…

Of greatest importance in the Reynolds number is the product of size and speed, telling us that the two work in concert, not counteractively. For living systems “small” almost always mean slow, and “large” almost always implies fast. That’s why the range of Reynolds numbers so far exceed the eight or so orders of magnitude over which the lengths of organisms vary…

Russ and I explain how the Reynolds number arises from the ratio of two forces, but I don't think we are as clear as Vogel.

What distinguishes regimes of flow is the relative importance of inertial and viscous forces. The former keeps things going; the latter makes them stop. High inertial forces favor turbulence… High viscous forces should prevent sustained turbulence and favor laminar flow by damping incipient eddies…

Another point should be made emphatically. If, for example, the Reynolds number is low, the situation is highly viscous. The flow will be dominated by viscous forces, vortices will be either nonexistent or nonsustained, and velocity gradients will be very gentle… If, in nature, small means slow and large means fast, then small creatures will live in a world dominated by viscous phenomena and large ones by inertial phenomena—this, even though the bacterium swims in the same water as the whale.

 The bacterium-whale comparison is just the sort of insight that Vogel excels at.

Tomorrow, I’ll provide a few more excerpts from Live in Moving Fluids, in which Vogel studies low Reynolds number flow in more detail.