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. 

A Problem with Testing

Chance in Biology,
by Mark Denny and Steven Gaines.

With a pandemic raging, I hear a lot about testing. One problem is we don’t have an adequate supply of test kits to screen for Covid-19. But another problem would arise even if we had enough kits to test everyone. To explain, I’ll describe an example presented in Chance in Biology: Using Probability to Explore Nature, by Mark Denny and Steven Gaines. Their analysis was based on testing for HIV, but I’ll recast the story in terms of the coronavirus.

Suppose we have an accurate test for Covid-19. No test is perfect, so let’s assume it’s correct 99.5% of the time. In other words, its error rate is 0.5% (one error for every two hundred tests). We’ll assume this error rate is the same for false positives and false negatives. Furthermore, we’ll assume Covid-19 is not prevalent, infecting only 0.1% of the population. I doubt this is a good assumption right now, when the virus seems to be infecting everyone, but I can imagine a time not too far in the future (a few weeks maybe, a few months probably) when the fraction of people having the virus is small.

In a population of a million people, 1000 would have Covid-19 and 999,000 would not. First, consider what happens when you test the thousand that are infected. The test would come back positive for Covid-19 in 99.5% of the cases, so it would produce 995 true positives. The test would be in error and give a negative result 0.5% of the time, giving 5 false negatives (the test would say you don’t have the disease when in fact you do). Next, consider the results from testing the 999,000 people who are not infected. Again, the test accuracy is 99.5%, so you would get a negative result (true negatives) for 994,005 people (0.995 times 999,000). You’d make a mistake 0.5% of the time, so you get false positives in 4995 cases (the test would say you have the disease when in fact you don’t). Let’s summarize:

True Positives	995
False Positives	4,995
True Negatives	994,005
False Negatives	5

Now, suppose Michigan Governor Gretchen Whitmer (I’m a big fan of the governor) decides that—to prevent the virus from flaring up again—everyone will be tested; anyone who tests positive for Covid-19 must be quarantined, and anyone who tests negative is free to go wherever they please (restaurants, sporting events, movies... oh how I miss them!). Out of a million people, 5990 will test positive (995 + 4995). Of those, 4995 are mistakes (false positives). In other words, 83% of the people who are forced into quarantine are false positives; they don’t have the disease, but the blasted test said they do and they must suffer for it. Only 17% of the quarantined people are infected.

Is this acceptable? Maybe. We might decide as a state that it’s worth it; we insist on quarantines, even if five out of six people forced into isolation are actually healthy. The needs of the many outweigh the needs of the few. Or, we might decide this is too high a price to pay; people are innocent until proven guilty, so to speak. This decision is not simple. Whitmer will have people mad at her regardless of what she does. But we must decide based on the facts. If the incidence of Covid-19 is 0.1%, and the accuracy of the test is 99.5%, then five out of six quarantined people will be false positives. That’s how the math works. Denny and Gaines summarize it this way:

Although individual tests have a low chance of error, most individuals who are tested are not infected with [Covid-19]. Therefore, we are multiplying a small probability of false positives by a large number of uninfected individuals. Even a minute probability of false positives for individual tests can in this circumstance produce many more false positives than true positives. As long as the disease is rare, even a very accurate test of infection will not be able to accurately identify infected individuals in a random test.

I don’t know the prevalence of Covid-19 or the accuracy of Covid-19 tests. The numbers for the coronavirus may be different than what I used in this example. My point is that even an accurate test can produce many false positives for a rare disease. That’s an important insight, whether or not the numbers are accurate for our current plague.

Chance in Biology is full of examples like this one. It’s a good book (although I like Denny’s Air and Water better). It’s a useful supplement to Intermediate Physics for Medicine and Biology, providing all the probability you need to understand biological physics.

No-Slip Boundary Condition

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the no-slip boundary condition

The velocity of the fluid immediately adjacent to a solid is the same as the velocity of the solid itself.

Life in Moving Fluids,
by Steven Vogel.

This seemingly simple condition is not obvious. To learn more, let’s consult Steven Vogel’s masterpiece Life in Moving Fluids: The Physical Biology of Flow.

The No-Slip Condition

The properly skeptical reader may have detected a peculiar assumption in our demonstration of viscosity: the fluid had to stick to the walls… in order to shear rather than simply slide along the walls. Now fluid certainly does stick to itself. If one tiny portion of a fluid moves, it tends to carry other bits of fluid with it—the magnitude of that tendency is precisely what viscosity is about. Less obviously, fluids stick to solids quite as well as they stick to themselves. As nearly as we can tell from the very best measurements, the velocity of a fluid at the interface with a solid is always just the same as that of the solid. This last statement expresses something called the “no-slip condition”—fluids do not slip with respect to adjacent solids. It is the first of quite a few counterintuitive concepts we’ll encounter in this world of fluid mechanics; indeed, the dubious may be comforted to know that the reality and universality of the no-slip condition was heatedly debated through most of the nineteenth century. Goldstein (1938) devotes a special section at the end of his book to the controversy. The only significant exception to the condition seems to occur in very rarefied gases, where molecules encounter one another too rarely for viscosity to mean much.

The reference to a book by Sydney Goldstein

Goldstein, S. (1938) Modern Developments in Fluid Dynamics. Reprint. New York: Dover Publications, 1965.

The no-slip boundary condition is important not only a low Reynolds number but also (and more surprisingly) at high Reynolds number. When discussing a solid sphere moving through a fluid, Russ and I say

At very high Reynolds number, viscosity is small but still plays a role because of the no-slip boundary condition at the sphere surface. A thin layer of fluid, called the boundary layer, sticks to the solid surface, causing a large velocity gradient and therefore significant viscous drag.

Vogel also addresses this point

Most often the region near a solid surface in which the velocity gradient is appreciable is a fairly thin one, measured in micrometers or, at most, millimeters. Still, its existence requires the convention that when we speak of velocity we mean velocity far enough from a surface so the combined effect of the no-slip condition and viscosity, this velocity gradient, doesn’t confuse matters. Where ambiguity is possible, we’ll use the term “free stream velocity” to be properly explicit.

Many fluid problems in IPMB occur at low Reynolds number, where thin boundary layers are not relevant. However, at high Reynolds number the no-slip condition causes a host of interesting behavior. Russ and I write

A thin layer of fluid, called the boundary layer, sticks to the solid surface, causing a large velocity gradient… At extremely high Reynolds number, the flow undergoes separation, where eddies and turbulent flow occur downstream from the sphere.

Turbulence! That’s another story.


See y’all next week for more coronavirus bonus posts.

Electrophoresis

Random Walks in Biology,
by Howard Berg.

Electrophoresis is used widely in biomedical research. It’s an example of physics applied to biology that is not discussed in Intermediate Physics for Medicine and Biology. In Random Walks in Biology, Howard Berg describes electrophoresis.

If a particle carries an electric charge, then one can exert a force on it with an electric field. An ion carrying a charge q (esu) in an electric field of intensity E (statvolts/cm) experiences a force in the direction of the field Eq (dynes). [Sorry about the cgs units.] Unfortunately, q is not easy to define. Particles of biological interest contain a variety of ionizable groups whose charges depend strongly on pH. These charges are shielded by counter-ions attracted from the medium in which the particles are suspended. The effectiveness of the shielding depends on the ionic strength. So you do not hear much about particles that have specified electrophoretic drift rates per unit field (as you do, for example, about 30, 50, or 70 S ribosomes [S is a parameter commonly used in centrifuge work, corresponding to velocity per unit acceleration and measured in svedbergs, 1 Sv = 10⁻¹³ seconds]). Nevertheless, electrophoretic methods of separating and characterizing biological materials are extremely useful. In practice, they are remarkably simple.

I like the comparison of electrophoresis to centrifugation. Both are examples of diffusion with drift; the physics is nearly the same. Berg continues

…One layers a mixture of particles at the top of a medium designed to suppress convective stirring and passes an electrical current through that, generating patterns analogous to those shown at the bottom of Fig. 4.7… The relative displacement of the … [bands] increases linearly with time, while the spreading increases as the square-root of time; so the separation improves as the square-root of time….

The bottom of Berg’s Fig. 4.7 looks like this:

As noted earlier, convective stirring is suppressed in the ultracentrifuge by the use of density gradients, e.g., of sucrose or CsCl. In an electrophoresis experiment, it is more convenient to use a gel, e.g., polyacrylamide or agarose. At the end of the experiment the bands can be precipitated into the gel and/or stained, e.g., with colored or fluorescent dyes, or the gel can be dried down and exposed to X-ray film to reveal components that are radioactive…

Electrophoretic gels are a workhorse of molecular biology.

Gels not only suppress convective stirring, they act as molecular sieves. The rate of migration of a particle through the gel is strongly dependent on size. A particle that is small compared with the pores in the gel can diffuse through it, almost as if the gel were not there. Particles of intermediate size get through with varying degrees of difficulty. Particles that move through a dilute aqueous medium at roughly the same rate move through the gel at rates that decrease exponentially with size; as a result, an estimate of size (or mass) can be made from a measurement of the logarithm of the displacement. Pieces of DNA and RNA are routinely sorted in this way, as are proteins dissolved in ionic detergents, such as sodium dodecyl sulfate. It is easy to distinguish gels of this kind, because the faster moving bands always are broader; the molecules that drift more rapidly are smaller and have larger diffusion constants.

In a later chapter, Berg develops the analogy between electrophoresis and centrifugation further.

An analogous situation [to density-gradient sedimentation] arises in electrophoresis when the experiment is run in a pH gradient. At equilibrium, a protein will form a band centered at the pH at which it is electrically neutral, i.e., at its isoelectric point. A particle at a more acid pH is positively charged and moves toward the cathode; a particle at a more basic pH is negatively changed and moves toward the anode. Thus, the pH gradient must be acidic near the anode and basic near the cathode.

The Localization of Sound

In Chapter 13 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I include a homework problem about the localization of sound.

Problem 20. People use many cues to estimate the direction a sound came from. One is the time delay between sound arriving at the left and right ears. Estimate the maximum time delay. Ignore any diffraction effects caused by the head.

Air and Water,
by Mark Denny.

What other ways do people use to sense the direction to the source of a sound? In Air and Water, Mark Denny discusses four.

Sense the Direction that Fluid Moves in a Sound Wave

Some animals can sense the direction that fluid moves in a sounds wave. This requires an ear that responds to motion (a vector) instead of pressure (a scalar). Apparently this ability is common in fish, but not in terrestrial animals.

Comparing the Intensity at Each Ear

Most animals have two ears. If one is closer to the source of a sound than the other, it hears a louder sound. Also, the presence of the body may attenuate or diffract the sound that reaches the far ear, changing its intensity. Denny notes two problems with this mechanism. First, attenuation in air occurs over large distances (70 dB per kilometer), so we would not expect much difference of intensity because the ears are, say, 20 cm apart. Second, the perceived direction is ambiguous. The sound from a source in front of us produces the same intensity at each ear, but so does sound from a source behind us. One way to resolve the ambiguity is to tilt your head as you listen, providing two data point: before and after the tilt.

Comparing the Delay at Each Ear

The homework problem Russ and I wrote is based on the time difference of sound arriving at each ear. This mechanism shares the problem mentioned earlier of direction ambiguity. The biggest problem, however, is that the arrival time at each ear differs by only a small amount: less than a millisecond. Nevertheless, bats appear to make use of this mechanism. For smaller animals (such as hummingbirds) the delays may be too short to be perceptible. Moreover, the speed of sound in water is more than four times the speed of sound in air, so this mechanism is unreliable for aquatic animals. SCUBA divers have trouble localizing sound.

Detecting the Phase Shift Between Each Ear

This mechanism is similar to comparing arrival times, except instead of sensing the delay you sense the phase difference. The method suffers from the same ambiguities discussed earlier, plus another unique to the detection of phase. If the phase shifts by an entire wavelength, it sounds the same as if it had no phase shift at all. So, you don’t want large phase shifts (greater than 2π), but you don’t want small phase shifts that are lost in the noise. Some small animals (such as crickets) have their ears connected by an air-filled tube, so they only detect sound when the two ears are out of phase. Because the speed of sound changes with temperature, any mechanism based on the speed of sound might function differently on a cold day than on a hot one.

Denny concludes

Despite the inherent problems of determining direction, animals combine the methods described above and thereby perform admirably. For example, bats and owls have been shown to localize sounds with 1° to 2°, and dolphins have similar directional acuity. Humans, cats, and opossums can localize sounds within 1° to 6° (Lewis 1983). These abilities are a tribute to the ability of the nervous system to assimilate complex data.

I don’t mind being beaten out by a cat, but we humans need to up our game if we want to do better than those possums.

Time-Dependent Solutions to Fick’s Equations

Solving the diffusion equation (also known as Fick’s second law) can require fancy mathematics. After discussing a few solutions in Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

Many other solutions to the diffusion equation and techniques for solving it are known. See Crank (1975) and Carslaw and Jaeger (1959).

Random Walks in Biology,
by Howard Berg.

In Random Walks in Biology, Howard Berg addresses the same point.

Time-dependent solutions to Fick’s equations

One way to find solutions to Fick’s equations is to look them up! An excellent source is Carslaw and Jaeger (1959), a book dealing with the conduction of heat in solids. The heat equation has the same form as the diffusion equation [see Chapter 4, Homework Problem 19 in IPMB]. In the notation of Carslaw and Jaeger,

where ν is the temperature and κ is the thermal diffusivity. So, take their results and read C for ν and D for κ. Sources that do not require such translation include Crank (1975) and Jost (1960). But this strategy requires luck. If you happen to find a discussion of just the problem that you are trying to solve, well and good. If not, you will soon be lost in a morass of complex equations.

After reading this, my first thought was: I know Crank, and I know Carslaw and Jaeger, but who’s Jost? The reference is to

Jost, W. 1960. Diffusion in Solids, Liquids, Gases. Academic.

The Oakland University library does not own this book, but even if it did the book might as well be on the moon for all the access I have to it. I can still download journal articles through the library website, but the library building itself is locked up tight because of the coronavirus.

If Jost’s book is anything like Crank’s or Carslaw and Jaeger’s, it’s overflowing with mathematics (just the way I like it).

I was able to find an obituary for Wilhelm Jost, written by Hartweil Calcote.

F. Wilhelm Jost

1903-1988

Professor Dr. sc. nat. Dres. h. c. W. Jost died on September 25, 1988 in G6ttingen. W. Jost was a preeminent scientist who pioneered research and development in combustion, reaction-kinetics in gases and solids, diffusion in solids and phase separation. He authored several monographs that have become classics in the field, among them “Explosions- und Verbrennungsvorg/inge in Gasen” with English and Russian editions. Several Academies of Science elected him as a member and he served on the boards of many scientific societies.

W. Jost was a director of the International Combustion Institute for decades and he founded and directed its German section. His many distinguished awards include the Institute's Sir Alfred Egerton Gold Medal.

F. W. Jost was born in Friedberg in Hessen. He studied chemistry in Halle and Munich and received his Dr. sc. nat. degree in chemistry from the University of Halle in 1926. He then joined M. Bodenstein in Berlin where his work on gas kinetics and hydrocarbon oxidation started. In 1929 Jost became Privatdozent in Hannover. From 1932 to 1933 he spent a year at MIT in Cambridge, Massachusetts, where he founded the basis for the understanding of disorder energies. In 1937 W. lost became a Professor of Physical Chemistry in Leipzig, in 1943 at the University of Marburg and 1951 in Darmstadt, in 1952 he accepted a call to the chair of W. Nernst at the University in G6ttingen.

On the occasion of the honorary promotion in Cambridge it was stated: "This man has always shown himself zealous for liberty, careful of truth and thoroughly civilized".

The members of the International Combustion Institute express their deep sympathy to his wife and his family.

Solving a differential equation by looking the solution up is an odd way to do math, but for the diffusion equation it often works. I will add Jost’s book to my list of post-pandemic reading.