Lotka-Volterra equations

Russ Hobbie and I don’t study population dynamics much in the 4th edition of Intermediate Physics for Medicine and Biology. To me, it’s more of a mathematical biology topic rather than an application of physics to biology. However, we do discuss one well-known model for population dynamics, the Lotka-Volterra equations, in a Homework Problem in Chapter 2.

Problem 34 Consider a classic predator-prey problem. Let the number rabbits be R and the number of foxes be F. The rabbits eat grass, which is plentiful. The foxes eat only rabbits. The number of rabbits and foxes can be modeled by the Lotka-Volterra equations

dR/dt = a R – b R F

dF/dt = - c F + d R F .

(a) Describe the physical meaning of each term on the right-hand side of each equation. What does each of the constants a, b, c, and d denote?

(b) Solve for the steady-state values of R and F.

These differential equations are difficult to solve because they are nonlinear (see Chapter 10). Typically, R and F oscillate about the steady-state solutions found in part (b). For more information, see Murray (2001).

There are two steady-state solutions. One is the trivial R = F = 0. The most interesting aspect of this solution is that it is not stable. If R and F are both small, the nonlinear terms in the Lotka-Volterra equations are negligible, and the number of foxes falls exponentially (they is no prey to eat) but the number of rabbits rises exponentially (there is no predator to eat them).

The other steady-state solution is (spoiler alert!) R = c/d and F = a/b. We claim in the problem that these equations are difficult to solve, and that is true in general, at least when searching for analytical solutions. However, if we focus on small deviations from this steady-state, we can solve the equations. Let  

R = c/d + r 

F = a/b + f ,

where r and f are small (much less than the steady state solutions). Plug these into the original differential equations, and ignore any terms containing r times f (these “doubly small” terms are negligible). The new equations for r and f are

dr/dt = - b (c/d) f 

df/dt =   d (a/b) r .

Now let’s use my favorite technique for solving differential equations: guess and check. I will guess

r = A sin(ωt)

f = B cos(ωt) .

If we plug these expressions into the differential equations, we get a solution only if ω² = ac. In that case, B = -(d/b) √(a/c) A. You can’t get A in this way; it depends on the initial conditions.

A plot of the solution shows two oscillating populations, with the rabbits lagging 90 degrees behind the foxes. In words, suppose you start with foxes at their equilibrium value, but a surplus of rabbits above their equilibrium. In this case, there are lots of rabbits for the foxes to eat, so the foxes gorge themselves and their population grows. However, as the number of foxes rises, the number of rabbits starts to fall (they are being ravaged by all those foxes). After a while, the number of rabbits declines back to its equilibrium value, but by then the number of foxes has surged above its steady-state value. Foxes continue to devour rabbits, reducing the rabbit population below equilibrium. Now there are too many foxes competing for too few rabbits, so the fox population starts to shrink as some inevitably go hungry. During this difficult time, both populations are plummeting as a large but decreasing number of ravenous foxes hunt the rare and frightened rabbits. When the foxes finally fall back to their equilibrium value there is a shortage of rabbits, so the foxes continue to starve and their number keeps falling. With less foxes, the rabbits breed like…um…rabbits and begin to make a comeback. Once they climb to their equilibrium value, there are still relatively few foxes, so the rabbits prosper all the more. With the rabbit population surging, there is plenty of food for the foxes, and the fox population begins to increase. During these happy days, both populations thrive. Eventually, the foxes return to their equilibrium value, but by this time the rabbits are plentiful. But this is just where we started, so the process repeats, over and over again. I needed a lot of words to explain about those foxes and rabbits. I think you can begin to see the virtue of a succinct mathematical analysis, rather than a verbose nonmathematical description.

For larger oscillations, the nonlinear nature of the model becomes important. The populations still oscillate, but not sinusoidally. For some parameters, one population may rise slowly and then suddenly drop precipitously, only to gradually rise again. You can see some of those results here and here.

The Lotka-Volterra model is rather elementary. For instance, there is no damping; the oscillations never decay away but instead continue forever. Moreover, the oscillations do not approach some fixed amplitude (a limit cycle behavior). Instead, the amplitude depends entirely on the initial conditions. Many more realistic models have a threshold, above which oscillations occur but below which the systems returns to its steady state.

Mathematical Biology,
by James Murray.

Population dynamics is a large field, of which we only scratch the surface. One place to learn more is James Murray’s book that we cite at the end of the homework problem:

Murray, J. D. (2001) Mathematical Biology. New York, Springer-Verlag.

The most recent (3rd) edition of Murray’s book is actually in two volumes:

Murray, J. D. (2002) Mathematical Biology: I. An Introduction. New York, Springer-Verlag. 

Murray, J. D. (2002) Mathematical Biology: II. Spatial Models and Biomedical Applications. New York, Springer-Verlag.

Hear Murray talk about his research here. 

Listen to James Murray talk about Mathematical Biology.
https://www.youtube.com/watch?v=6Yj5Nyb_VyU

Alfred Lotka (1880–1949) was an American scientist. In 1925 he published a book, Elements of Physical Biology, that is in some ways a precursor to Intermediate Physics for Medicine and Biology, or perhaps an early version of Murray’s Mathematical Biology. You can download a copy of the book here.

Life’s Ratchet

Life’s Ratchet: How Molecular Machines
Extract Order from Chaos,
by Peter Hoffmann.

This week I finished reading Life’s Ratchet: How Molecular Machines Extract Order from Chaos, by Peter Hoffmann. This book is mostly about molecular biophysics, which Russ Hobbie and I purposely avoid in the 4th edition of Intermediate Physics for Medicine and Biology. But the workings of tiny molecular motors is closely related to thermal motion (Hoffmann calls it the “molecular storm”) and the second law of thermodynamics, topics that Russ and I do address. One fascinating topic I want to focus on is a discussion of Feynman’s ratchet.

Let us begin with Richard Feynman’s discussion in Chapter 16 of Volume 1 of The Feynman Lectures on Physics. I recall reading The Feynman Lectures the summer between graduating from the University of Kansas and starting graduate school at Vanderbilt University. All physics students should find time to read these great lectures. Feynman writes

Let us try to invent a device which will violate the Second Law of Thermodynamics, that is, a gadget which will generate work from a heat reservoir with everything at the same temperature. Let us say we have a box of gas at a certain temperature and inside there is an axle with vanes in it… Because of the bombardments of gas molecules on the vane, the vane oscillates and jiggles. All we have to do is to hook onto the other end of the axle a wheel which can turn only one way—the ratchet and pawl. Then when the shaft tries to jiggle one way, it will not turn, and when it jiggles the other, it will turn… If we just look at it, wee see, prima facie, that it seems quite possible. So we must look more closely. Indeed, if we look at the ratchet and pawl, we see a number of complications.

First, our idealized ratchet is as simple as possible, but even so, there is a pawl, and there must be a spring in the pawl. The pawl must return after coming off a tooth, so the spring is necessary…

Feynman goes on to explore this device in detail. He concludes that, as we would expect, the device does not violate the second law. He explains

It is necessary to work against the spring in order to lift the pawl to the top of a tooth. Let us call this energy ε… The chance that the system can accumulate enough energy ε to get the pawl over the top of the tooth is e^−ε/kT [T is the absolute temperature, and k is Boltzmann’s constant]. But the probability that the pawl will accidentally be up is also e^−ε/kT. So the number of times that the pawl is up the wheel can turn backwards freely is equal to the number of times that we have enough energy to turn it forward when the pawl is down. We thus get a “balance,” and the wheel will not go around.

Hoffmann explains that a lot of molecular machines important in biology operate analogously to Feynman’s ratchet and pawl. He writes

What kind of molecular device could channel random molecular motion into oriented activity? Such a device would need to allow certain directions of motion, while rejecting others. A ratchet, that is, a wheel with asymmetric teeth blocked by a spring-loaded pawl, could do the job... Maybe nature has made molecular-size ratchets that allow favorable pushes from the molecular storm in one direction, while rejecting unfavorable pushes from the opposite direction…

For the ratchet-and-pawl machine to extract energy from the molecular storm, it has to be easy to push the pawl over one of the teeth of the ratchet. The pawl spring must be very weak to allow the ratchet to move at all. Otherwise, a few water molecules hitting the ratchet would not be strong enough to force the pawl over one of the teeth. Just like the ratchet wheel, the pawl is continuously bombarded by water molecules. Its weak spring allows the pawl to bounce up and down randomly, opening from time to time, allowing the ratchet to slip backward… Worse, because the spring is most relaxed when the pawl is at the lowest point between two teeth [the compressed spring pushes the pawl down against the ratchet], the pawl spends most of its time touching the steep edge of one of the teeth. When an unfavorable hit pushes the ratchet backward just as the pawl has opened, it does not need to go far to end up on the incline of the next tooth—rotating the ratchet backward!... The ratchet will move, bobbing back and forth, but it will not make any net headway.

How then do molecular machines work? They require in input of energy, which eventually gets dissipated into heat. Hoffmann concludes

We could, in fact, make Feynman’s ratchet work, if from time to time, we injected energy to loosen and then retighten the pawl’s spring. On loosening the spring, the wheel would rotate freely, with a slightly higher probability of rotating one way rather than the other. Tightening the pawl’s spring would push the wheel further in the direction we want. On average, the wheel would move forward and do work. In fact, it can be shown that any molecular machine that operates on an asymmetric energy landscape and incorporates and irreversible, energy-degrading step can extract useful work from the molecular storm.

This may all seem abstract, but Hoffmann brings it down to specifics. The molecular machine could be myosin moving along actin (as in muscles) or kinesin moving along a microtubule (as in separating chromosomes during mitosis). The energy source for the irreversible step is ATP. This step allows the motor to extract energy from the “molecular storm” of thermal energy that is constantly bombarding it.

I WENT TO PARIS AND I MISSED THE VERY BEST THING!

Three summers ago, my wife and I visited Paris for our 25th wedding anniversary. We carefully planned our trip so we could see all the most famous sites—the Eifel Tower, the Arc de Triomphe, the Notre Dame Cathedral, the Palace of Versailles, the Pantheon, the Louvre, and the Musee d’Orsay—but somehow WE MISSED THE MOST IMPORTANT THING! Apparently there is a giant painting in the Musee d’Art Moderne by Raoul Dufy, depicting many scientists who have contributed to the study of electricity. What more could a physicist like me ask for? I first learned about this painting in a book I am now reading, The Spark of Life: Electricity and the Human Body, by Frances Ashcroft. I’ll have more on that book in a future post. Here is what she writes about the painting:

An unusual tribute to the scientists and philosophers who contributed to the discovery of electricity hangs in Musee d’Art Moderne in Paris. A giant canvas known as “La Fee Electricite,” which measures 10 metres high and 60 metres long, it was commissioned by a Paris electricity company to decorate its Hall of Light at the 1937 world exhibition in Paris. It is the work of French Fauvist painter Raoul Dufy, better known for his wonderful colourful depictions of boats, and it took him and two assistants four months to complete. The Electricity Fairy sails through the sky at the far left of the painting above some of the world’s most famous landmarks, the Eiffel Tower, Big Ben and St Peter’s Basilica in Rome among them. Behind her follow some 110 people connected with the development of electricity, from Ancient Greece to modern times. As time and the canvas progress, the landscape changes from scenes of rural idyll to steam trains, furnaces, the trappings of the industrial revolution and finally the giant pylons that support the power lines carrying electricity to the planet.

Short of going to see the painting in Paris, the next best thing is to view it in sections at the Electricity Online website of the University of Leeds. I won’t list all the scientists depicted in it, but let me note those Russ Hobbie and I mention in the 4th edition of Intermediate Physics for Medicine and Biology (roughly in chronological order): Newton, Bernoulli, Laplace, Poisson, Gauss, Ohm, Oersted, Clausius, Clapeyron, Fourier, Savart, Fresnel, Biot, Ampere, Faraday, Gibbs, Helmholtz, Maxwell, Poincare, Moseley, Lorentz, and Pierre Curie. A few were only present in IPMB because they have a unit named after them: Pascal, Watt, Joule, Kelvin, Roentgen, Becquerel, Hertz, and Marie Curie. Galvani is shown with a frog, Faraday with a coil and galvanometer, Pierre Curie (mentioned in IPMB through the Curie temperature) is standing next to his wife Marie Curie (only mentioned in IPMB in association with her unit, and the only female scientist in the painting), and Edison is next to his light bulbs.

I’m still not sure how I never knew about this magnificent painting. I guess we need to take another trip to Paris. Honey, start packing!

Resource Letter BSSMF-1: Biological Sensing of Static Magnetic Fields

In the October 2012 issue of the American Journal of Physics, physicist Leonard Finegold published “Resource Letter BSSMF-1: Biological Sensing of Static Magnetic Fields” (Volume 80, Pages 851–861). Finegold recommends that a good starting point for mastering the topic of magnetoreception is Kenneth Lohmann’s News and Views article in Nature.

35. “Magnetic-field perception: News and Views Q and A,” K. J. Lohmann, Nature, 464, 1140–1142 (2010). (E) 

I looked it up, and it does indeed provide a well-written summary of the field in a reader-friendly question-and-answer format.

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss magnetotactic bacteria. We write that

Bacteria in the northern hemisphere have been shown to seek the north pole. Because of the tilt of the earth’s field, they burrow deeper into the environment in which they live. Similar bacteria in the southern hemisphere burrow down by seeking the south pole.

Finegold also reviews this topic. The excerpt reproduced below serves both as an up-date to IPMB and as a sample of the style of an American Journal of Physics resource letter.

Certain bacteria move in response to the earth’s magnetic field (Ref. 35), swimming along the field lines, and have been excellently reviewed (Ref. 36). The “sensing” element is magnetite (an iron oxide) or greigite (an iron sulfide) (Ref. 37). The bacteria would swim toward the boundary between oxygenated and oxygen-poor regions. Until recently, there was the comforting idea that there are two groups of bacteria with opposite sensors, depending on which of the earth’s hemispheres they reside. Alas, both groups have now been found in the same place; it appears that their polarity is correlated with the local redox potential (Ref. 38 and 39). In addition, some bacteria use only the axial property of the field (i.e., they swim both with or against the field direction), whereas others use the vector property (i.e., they swim either with or against the field direction). Details of the behavior have been elucidated by applying magnetic fields to bacteria in a spectrophotometer cuvette, with genetic analysis (Ref. 39).

35. “South-seeking magnetotactic bacteria in the Southern Hemisphere,” R. P. Blakemore, R. B. Frankel, and Ad. J. Kalmijn, Nature 286, 384–385 (1980). (A)

36. “Bacteria that synthesize nano-sized compasses to navigate using Earth’s geomagnetic field,” L. Chen, D. A. Bazylinski, and B. H. Lower, Nature Education Knowledge 1(10), 14 (2010). (I)

37. “The identification and biogeochemical interpretation of fossil magnetotactic bacteria,” R. E. Kopp and J. L. Kirschvink, Earth-Sci. Rev. 86, 42–61 (2008). (A)

38. “South-seeking magnetotactic bacteria in the northern hemisphere,” S. L. Simmons, D. A. Bazylinski, and K. J. Edwards, Science 311, 371–374 (2006). (A)

39. “Characterization of bacterial magnetotactic behaviors by using a magnetospectrophotometry assay,” C. T. Lefevre, T. Song, J. P. Yonnet, and L. F. Wu, Appl. Environ. Microbiol. 75, 3835–3841 (2009). (A)”

Magnetoreception is a field that often stirs debate. Russ and I outline one such debate in IPMB

Kirschvink (1992) proposed a model whereby a magnetosome in a field of 10⁻⁴–10⁻³ T could rotate to open a membrane channel. As an example of the debate that continues in this area, Adair (1991, 1992, 1993, 1994) argued that a magnetic interaction cannot overcome thermal noise in a 60-Hz field of 5 × 10⁻⁶ T. However, Polk (1994) argues that more biologically realistic parameters, including a large number of magnetosomes in a cell, could allow an interaction at 2 × 10⁻⁶ T.

The key citations in the debate are

Adair, R. (1991) “Constraints on biological effects of weak extremely-low-frequency electromagnetic fields,” Phys. Rev. A, Volume 43, Pages 1039–1048.

Kirschvink, J. L. (1992) “Comment on “Constraints on biological effects of weak extremely-low-frequency electromagnetic fields,” Phys. Rev. A, Volume 46, Pages 2178–2184.

Adair, R. (1992) “Reply to “Comment on ‘Constraints on biological effects of weak extremely-low-frequency electromagnetic fields’,” Phys. Rev. A, Volume 46, Pages 2185–2187.

For those of you who like this sort of thing, here is another example from Finegold’s resource letter. The debate is about, of all things, if cows align themselves in magnetic fields!

A surprising finding is that cattle and deer seem to align themselves in an approximate north-south (geomagnetic) direction. The evidence is from world-wide satellite photographs from Google Earth, supported by ground observations of more than 10,000 animals, and is hard to rebut. The satellite photographs do not have enough resolution to show the direction (north or south) in which the animals face.

72. “Magnetic alignment in grazing and resting cattle and deer,” S. Begall, J. Cerveny, J. Neef, O. Vojtech, and H. Burda, Proc. Natl. Acad. Sci. U.S.A. 105, 13453–13455 (2008). (I)

As Usherwood asks, why on Earth should cattle and deer prefer this alignment? Possible interpretations are that the satellite photographs are made close to noon, so there may be physiological reasons (heating, cooling) for animals to align or to view predators better.

73. “Cattle and deer align north (-north-east),” J. Usherwood, J. Exp. Biol. 212, iv (2009). (E)

Partly to rule out sun compass effects, Burda et al. investigated ruminant alignment under high-voltage (and hence high-current, low-frequency) power lines and found that the geomagnetic north-south alignment was disturbed; the disturbance was correlated with the alternating fields. Such disturbance might instead be because the animals felt protected by (or preferring) the overhead lines or pylons or because of the audible (to humans at least) corona discharge. A good control for this would be to look at ruminants under power lines being repaired, carrying no current; this is difficult to do. The authors ingeniously compared the nonalignment under N-S and E-W trending power lines and found that the nonalignment followed the resultant total magnetic field. Their conclusions have been challenged (Ref. 75), and they have a lively rebuttal (Ref. 76), to which the challengers have replied (Ref. 77). Hence, the initially persuasive evidence, that cattle and deer detect magnetic fields, may need re-examination.

74. “Extremely low-frequency electromagnetic fields disrupt magnetic alignment of ruminants,” H. Burda, S. Begall, J. Cerven, J. Neef, and P. Nemec, Proc. Natl. Acad. Sci. U.S.A. 106, 5708–5713 (2009). (I)

75. “No alignment of cattle along geomagnetic field lines found,” J. Hert, L. Jelinek, L. Pekarek, and A. Pavlicek, J. Comp. Physiol., A 197, 677–682 (2011). (I)

76. “Further support for the alignment of cattle along magnetic field lines: Reply to Hert et al.,” S. Begall, H. Burda, J. Cerveny, O. Gerter, J. Neef-Weisse, and P. Nemec, J. Comp. Physiol. [A] 197, 1127–1133 (2011). (I)

77. “Authors’ Response,” J. Hert, L. Jelinek, L. Pekarek, and A. Pavlicek, J. Comp. Physiol. [A] 197(12), 1135– 1136 (2011). (I) 

Finegold also discusses magnet therapy, a topic I am extremely skeptical about, and that I have discussed before in this blog. He cites his own editorial with Flamm

“Magnet therapy,” L. Finegold and B. L. Flamm, Br. Med. J. 332, 4 (2006) (E) 

which concludes

Extraordinary claims demand extraordinary evidence. If there is any healing effect of magnets, it is apparently small since published research, both theoretical and experimental, is weighted heavily against any therapeutic benefit. Patients should be advised that magnet therapy has no proved benefits. If they insist on using a magnetic device they could be advised to buy the cheapest—this will at least alleviate the pain in their wallet.

Rounding Off the Cow

In the October 2012 issue of the American Journal of Physics, Dawn Meredith and Jessica Bolker published an article about “Rounding Off the Cow: Challenges and Successes in an Interdisciplinary Physics Course for Life Science Students” (Volume 80, Pages 913–922). The article is interesting, and much of the motivation for their work is nearly identical to that of Russ Hobbie and I in writing the 4th edition of Intermediate Physics for Medicine and Biology. They focus on an introductory physics class, whereas Russ and I wrote an intermediate level textbook. Nevertheless, many of the ideas and challenges are the same. Here, I want to focus on their Table 1, in which they list topics that are emphasized and deemphasized compared to standard introductory classes.

Table I. Changes in topic emphasis compared to standard course

	Semester 1	Semester 2
Included/stressed	Kinematics	Heat transfer
	Dynamics	Kinetic theory of gases
	Static torque	Entropy
	Energy	Diffusion, convection, conduction
	Stress/strain and fracture	Simple harmonic motion
	Fluids (far more)	Waves (sound, optics)
Omitted/de-emphasized	Projectile motion	Heat engines
	Relative motion	Magnetism (less)
	Rotational motion	Induction (qualitatively)
	Statics	Atomic physics (instrumentation)
	Collisions	Relativity
	Newton’s law of gravitation
	Kepler’s laws

How does this list compare with the content of IPMB? We don’t stress kinematics and dynamics much. In fact, most of our mechanics discussion centers on static equilibrium. Interestingly, Meredith and Bolker emphasize static torque, which is absolutely central to our analysis of biomechanics in Chapter 1. Rotational equilibrium and torque is what explains why bones, muscles and tendons often experience forces far larger than the weight of the body. It also underlies our rather extensive discussion of the role of a cane. We discuss mechanical energy in Chapter 1, but energy doesn’t become an essential topic until our Chapter 3 on thermodynamics. We agree completely with Meredith and Bolker’s listing of “stress/strain and fracture” and “Fluids (far more)”, and I second the “far more”. Our Chapter 1 contains a lot of fluid dynamics, including the biologically-important concept of buoyancy, the idea of high and low Reynolds number, and applications of fluid dynamics to the circulatory system.

The time allotted to an introductory physics class is limited, so something must get deemphasized to free up time for topics like fluids. Meredith and Bolker mention projectile motion (we agree, it is nowhere in IPMB), relative motion (not crucial if not covering relativity), and rotational motion (we don’t emphasize this either, except when analyzing the centrifuge). I don’t really understand the omission of statics, because as I said earlier static mechanical equilibrium is crucial for biomechanics. They deemphasize collisions, and so do we, although we do discuss the collision of an electron with a photon when analyzing Compton Scattering in Chapter 15. Newton’s law of gravity and Kepler’s laws of planetary motion are absent from both our book and their class.

In the second semester, Meredith and Bolker stress heat transfer (convection and conduction), the kinetic theory of gases, and entropy. Russ and I discuss all these topics in our Chapter 3. Diffusion is a topic they emphasize, and rightly so. It is a topic that is typically absent from an introductory physics class, but is crucial for biology. We discuss it in detail in Chapter 4 of IPMB. Meredith and Bolker list simple harmonic motion among the topics they stress. We talk about harmonic motion in Chapter 10, but mainly as a springboard for the study of nonlinear dynamics. Much of the analysis of linear harmonic motion is found in IPMB in an appendix. Finally, they stress waves (sound and optics). We do too, mainly in our Chapter 13 about sound and ultrasound; a new chapter in the 4th edition.

Topics they omit or deemphasize in the second semester include heat engines. We barely mention heat engines at the end of Chapter 3, and the well-known Carnot heat engine is never analyzed in our book. Meredith and Bolker deemphasize magnetism and magnetic induction. As a researcher in biomagnetism, I would hate to see these topics go. Russ and I analyze biomagnetism in Chapter 8. However, I could see how one might be tempted to deemphasize these topics; biomagnetic fields are very weak and do not play a large role in either biology or medicine. I personally would keep them in, and they remain an important part of IPMB. They do not stress “Atomic Physics (Instrumentation)” and I am not sure exactly what they mean, especially with their parenthetical comment about instruments. We talk a lot about atomic physics in Chapter 14 on Atoms and Light. Finally, Meredith and Bolker omit relativity, and so do Russ and I, except as needed to understand photons. We never discuss the more traditional phenomena of relativity, such as the Lorentz contraction, time dilation, or simultaneity.

Some topics should get about the same amount of attention as in a traditional class, but with slight changes in emphasis. For instance, I would cover geometrical optics, including lenses (when discussing the eye and eyeglasses) but I would skip mirrors. I would cover nuclear physics, but I would skip fission and fusion, and focus on radioactive decay, including positron decay (positron emission tomography).

I think that Meredith and Bolker provide some useful guidance on how to construct an introductory physics class for students interested in the life sciences. Russ and I aim at an intermediate class for students who have taken a traditional introductory class and want to explore applications to biology and medicine in more detail. Our book is clearly at a higher mathematical level: we use calculus while most introductory physics classes for life science majors are algebra based. But for the most part, we agree with Meredith and Bolker about what physics topics are central for biology majors and pre-med students.

Eleanor Adair (1926-2013)

Eleanor Adair, who studied the health risks of microwave radiation, died on April 20 in Hamden, Connecticut at the age of 86. A 2001 interview with Adair, published in the New York Times, began

Eleanor R. Adair wants to tell the world what she sees as the truth about microwave radiation.

New widely reported studies have failed to find that cellular phones, which use microwaves to transmit signals, cause cancer. And most academic scientists say the microwave radiation that people are exposed to with devices like cell phones is harmless. But still, Dr. Adair knows that many people deeply fear these invisible rays.

She knows that many people hear the word “radiation” and assume that all radiation is dangerous, equating microwaves to the very different X-rays.

Microwaves, she points out, are at the other end of the electromagnetic spectrum from high energy radiation like X-rays and gamma rays. And unlike gamma rays and X-rays, which can break chemical bonds and injure cells, even causing cancer, microwaves, she says, can only heat cells. Of course, if cells get hot enough, they can die, but the heat level has to be closer to that in an oven than the extremely low level from cell phones.

The interview ends with this exchange:

Q. If I were to say to people, “Hey there’s this really cool idea: Why heat your whole house when you could use microwaves to heat yourself?” they would say: “You’ve got to be kidding. Don’t you know that microwaves are dangerous? They can even cause cancer.” What do you say to people who respond like that?

A. I try to educate them in exactly what these fields are. That they are part of the full electromagnetic spectrum that goes all the way from the radio frequency and microwave bands, through infrared, ultraviolet, the gamma rays and all that.

And the difference between the ionizing X-ray, gamma ray region and the microwave frequency is in the quantum energy. The lower you get in frequency the lower you get in quantum energy and the less it can do to the cells in your body.

If you have a really high quantum energy such as your X-rays and ionizing-radiation region of the spectrum, this energy is high enough that it can bump electrons out of the orbit in your cells and it can create serious changes in the cells of your body such that they can turn into cancers and various other things that are not good for you.

But down where we are working, in the microwave band, you are millions of times lower in frequency and there the quantum energy is so low that they can’t do any damage to the cells whatsoever. And most people don’t realize this.

Somehow, something is missing in their basic science education, which is something I keep trying to push. Learn the spectrum. Learn that you’re in far worse shape if you lie out on the beach in the middle of summer and you soak up that ultraviolet radiation than you are if you use your cell phone.

Q. Some people say that with the ever-increasing exposure of the population to microwaves—cell phones have really taken off in the past few years—we need to redouble our research efforts to look for dangerous effects of microwaves on cells and human tissues. Do you agree?

A. No. All the emphasis that we need more research on power line fields, cell phones, police radar—this involves billions of dollars that could be much better spent on other health problems. Because there is really nothing there.

We don’t cite Adair’s research in the 4th edition of Intermediate Physics for Medicine and Biology, but we do cover the interaction of electromagnetic fields with tissue in Chapter 9. Much of our discussion is about powerline (60 Hz) fields, but many of the same considerations apply to microwaves. In our discussion, we do cite Robert Adair, Eleanor’s husband and an emeritus professor of physics at Yale, who shares his wife’s interest in the health effect of microwave radiation.

Adair won the d’Arsonval Award, presented by the Bioelectromagnetics Society, to recognize her accomplishments in the field of bioelectromagnetics. In an editorial announcing the award, Ben Greenebaum writes (Bioelectromagnetics, Volume 29, Page 585, 2008)

It gives me great pleasure to introduce Dr. Eleanor R. Adair, the recipient of the Bioelectromagnetics Society’s 2007 D’Arsonval Award, as she presents her Award Lecture (Fig. 1). Dr. Adair is being honored by the Society for her body of work investigating physiological thermoregulatory responses to radio frequency and microwave fields. Her bioelectromagnetic career began with extensive experimental studies of electromagnetic radiation-induced thermophysiological responses in monkeys and concluded with experiments that accomplished the critical extrapolation of the earlier findings to humans. I believe that this body of work constitutes a majority of the literature on the latter topic.

She spent most of her career as a research scientist at the John B. Pierce Foundation Laboratory at Yale University, but finished it as a scientist at the US Air Force’s Brooks City Base in San Antonio, Texas. As she notes in her D’Arsonval address [Adair, 2008], she took her undergraduate degree at Mount Holyoke College in 1948 and her doctorate in psychology at the University of Wisconsin-Madison in 1955. Interspersed among her academic accomplishments in Madison were others—marriage to Robert Adair and children. We should not forget that combining a research career and family at that time was much rarer and required overcoming greater difficulties than those still encountered today. Those of us who have interacted with Dr. Adair over the years know that she has determination in plenty.

Dr. Adair was a charter member of the Society and was its Secretary-Treasurer (1983–1986) during a difficult time, when the Society decided to replace its first Executive Director with Bill Wisecup. She has also been active outside the Society, both with groups concerned with research into bioelectromagnetic effects and with groups concerned with the implications of these results.

However, it is for her overall scientific contributions to bioelectromagnetics that she is being presented the D’Arsonval Award. The criteria for the Award state that “. . . the D’Arsonval Medal is to recognize outstanding achievement in research in the field of Bioelectromagnetics.” And that is the topic that she will address today in her presentation entitled, “Reminiscences of a Journeyman Scientist.”

For those who want to read Adair's own words, you can find her presentation at:

Adair. E. R. (2008) “Reminiscences of a journeyman scientist: Studies of thermoregulation in non-human primates and humans,”  Bioelectromagnetics  Volume 29, Pages 586–597.

The Lorenz equations and chaos

Fifty years ago, Edward Lorenz (1917–2008) published an analysis of Rayleigh-Benard convection that began the study of a field of mathematics called chaos theory. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce chaos by analyzing the logistic map, which

is an example of chaotic behavior or deterministic chaos. Deterministic chaos has four important characteristics: 1. The system is deterministic, governed by a set of equations that define the evolution of the system. 2. The behavior is bounded. It does not go off to infinity. 3. The behavior of the variables is aperiodic in the chaotic regime. The values never repeat. 4. The behavior depends very sensitively on the initial conditions.

The sensitivity to initial conditions is sometimes called the “butterfly effect,” a term coined by Lorenz. His model is a simplified description of the atmosphere, and has implications for weather prediction.

The mathematical model that Lorenz analyzed consists of three first-order coupled nonlinear ordinary differential equations. Because of their historical importance, I have written a new homework problem that introduces Lorenz’s equations. These particular equations don’t have any biological applications, but the general idea of chaos and nonlinear dynamics certainly does (see, for example, Glass and Mackey’s book From Clock’s to Chaos.

Section 10.7

Problem 33 1/2. Edward Lorenz (1963) published a simple, three-variable (x, y, z) model of Rayleigh-Benard convection

dx/dt = σ (y – x)

dy/dt = x (ρ – z) – y

dz/dt = xy – β z

where σ=10, ρ=28, and β=8/3.

(a) Which terms are nonlinear?

(b) Find the three equilibrium points for this system of equations.

(c) Write a simple program to solve these equations on the computer (see Sec. 6.14 for some guidance on how to solve differential equations numerically). Calculate and plot x(t) as a function of t for different initial conditions. Consider two initial equations that are very similar, and compute how the solutions diverge as time goes by.

(d) Plot z(t) versus x(t), with t acting as a parameter of the curve.

Lorenz, E. N. (1963) “Deterministic nonperiodic flow,” Journal of the Atmospheric Sciences, Volume 20, Pages 130–141.

If you want to examine chaos in more detail, see Steven Strogatz’s excellent book Nonlinear Dynamics and Chaos. He has an entire chapter (his Chapter 9) dedicated to the Lorenz equations.

The story of how Lorenz stumbled upon the sensitivity of initial conditions is a fascinating tale. Here is one version in a National Academy of Sciences Biographical Memoir about Lorenz written by Kerry Emanuel.

At one point, in 1961, Ed had wanted to examine one of the solutions [to a preliminary version of his model that contained 12 equations] in greater detail, so he stopped the computer and typed in the 12 numbers from a row that the computer had printed earlier in the integration. He started the machine again and stepped out for a cup of coffee. When he returned about an hour later, he found that the new solution did not agree with the original one. At first he suspected trouble with the machine, a common occurrence, but on closer examination of the output, he noticed that the new solution was the same as the original for the first few time steps, but then gradually diverged until ultimately the two solutions differed by as much as any two randomly chosen states of the system. He saw that the divergence originated in the fact that he had printed the output to three decimal places, whereas the internal numbers were accurate to six decimal places. His typed-in new initial conditions were inaccurate to less than one part in a thousand.

“At this point, I became rather excited,” Ed relates. He realized at once that if the atmosphere behaved the same way, long-range weather prediction would be impossible owing to extreme sensitivity to initial conditions. During the following months, he persuaded himself that this sensitivity to initial conditions and the nonperiodic nature of the solutions were somehow related, and was eventually able to prove this under fairly general conditions. Thus was born the modern theory of chaos.

To learn more about the life of Edward Lorenz, see his obituary here and here. I have not read Chaos: Making a New Science by James Gleick, but I understand that he tells Lorenz’s story there.

Graduation

Today, my wife Shirley and I will attend the graduation of our daughter Kathy from Vanderbilt University. She is getting her undergraduate degree, with a double major in biology and history.

Kathy spent part of her time in college working with John Wikswo in the Department of Physics. As regular readers of this blog may know, Wikswo was my PhD advisor when I was a graduate student at Vanderbilt in the 1980s. Russ Hobbie and I often cite Wikswo’s work in the 4th edition of Intermediate Physics for Medicine and Biology, for his contributions to both cardiac electrophysiology and biomagnetism. You can see a picture of Kathy and John in an article about undergraduate research in Arts and Science, the magazine of Vanderbilt University’s College of Arts and Science. Interestingly, there are now publications out there with “Roth and Wikswo” among the authors that I had nothing to do with; for example, see poster A53 at the 6th q-bio conference (Santa Fe, New Mexico, 2012). You can watch and listen to Wikswo give his TEDxNashville talk here. Kathy also worked with Todd Graham of the Department of Cell and Developmental Biology at Vanderbilt. This fall, she plans to attend graduate school studying biology at Michigan State University.

Biodamage Via Shock Waves Initiated by Irradiation With Ions

Just two doors down the hall from my office in Hannah Hall of Science at Oakland University is the office of my friend Gene Surdutovich. Gene teaches physics at OU, and is an international expert on the interaction of heavy ions with biological tissue. Recently he and his collaborators have published a review describing their work, titled “Biodamage Via Shock Waves Initiated by Irradiation With Ions,” in the journal Scientific Reports (Volume 3, Article Number 1289). Their abstract states

Radiation damage following the ionising radiation of tissue has different scenarios and mechanisms depending on the projectiles or radiation modality. We investigate the radiation damage effects due to shock waves produced by ions. We analyse the strength of the shock wave capable of directly producing DNA strand breaks and, depending on the ion’s linear energy transfer, estimate the radius from the ion’s path, within which DNA damage by the shock wave mechanism is dominant. At much smaller values of linear energy transfer, the shock waves turn out to be instrumental in propagating reactive species formed close to the ion’s path to large distances, successfully competing with diffusion.

Except for the British spelling, I enjoyed reading this paper very much.

Russ Hobbie and I discuss the interaction of ions with tissue in the 4th edition of Intermediate Physics for Medicine and Biology, in the context of proton therapy.

Protons are also used to treat tumors. Their advantage is the increase of stopping power at low energies. It is possible to make them come to rest in the tissue to be destroyed, with an enhanced dose relative to intervening tissue and almost no dose distally (“downstream”) as shown by the Bragg peak in Fig. 16.51.

Surdutovich and his colleagues note that the Bragg peak occurs for heavier ions too, such as carbon. What is really fascinating about their work is one of the mechanisms they propose. They note that

while the Bragg peak location answers a question of where most of the damage occurs, we are raising a question of how the damage happens… We investigate the effects that stem from a large inhomogeneity of the dose distribution in the vicinity of the Bragg peak on biological damage.

Their hypothesis is that when a carbon ion interacts with tissue, energy is abruptly deposited

within a cylinder of about one nm radius, which is so small that the temperature within this cylinder increases by over 1000 K by 10⁻¹³ s (we will refer to it as the ‘‘hot cylinder’’). This increase of temperature brings about a rapid increase of pressure (up to 1 GPa) compared to the atmospheric pressure outside the cylinder. Such circumstances cause the onset of a cylindrical shock wave described by the strong explosion scenario. The pressure rapidly increases on the wave front and then decreases in the wake.

I find this mechanism to be fascinating. The theory is noteworthy for its multiscale approach: analyzing events spanning a wide range of time, space, and energy scales. Interestingly, their theory also spans multiple chapters in Intermediate Physics for Medicine and Biology: Chapter 1 about pressure, Chapter 3 on temperature and heat, Chapter 4 on diffusion, Chapter 13 on sound, Chapter 15 about the interaction of radiation with tissue, and Chapter 16 about the medical use of radiation. They conclude

The notion of thermomechanical effects represents a paradigm shift in our understanding of radiation damage due to ions and requires re-evaluation of relative biological effectiveness because of collective transport effects for all ions and direct covalent bond breaking by shock waves for ions heavier than argon. These effects will also have to be considered for high-density ion beams, irradiation with intensive laser fields, and other conditions prone to causing high gradients of temperature and pressure on a nanometre scale.

This paper appears in an open access online journal, so you don’t need a subscription to read it. Be sure to look at the paper’s supplementary information, especially the movie showing a molecular dynamics simulation of the shock wave distorting a DNA molecule. And if you read very closely, you will find this nugget appearing in the acknowledgments: “We are grateful to … Bradley Roth who critically read the manuscript.”

By the way, this fall Gene is scheduled to teach PHY 325, Biological Physics, using the textbook….you guessed it….Intermediate Physics for Medicine and Biology.

Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid

Sixty years ago yesterday the journal Nature published a letter by James Watson and Francis Crick titled “Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid” (Volume 171, Pages 737–738), announcing the discovery of the double helix structure of DNA.

The 4th edition of Intermediate Physics for Medicine and Biology doesn’t discuss the structure of DNA much. As Russ Hobbie and I say in the preface, “Molecular biophysics has been almost completely ignored: excellent texts already exist, and this is not our area of expertise.” Yet, we do mention DNA occasionally. In the first section of our book, about distances and sizes, we say

Genetic information is stored in long, helical strands of deoxyribonucleic acid (DNA). DNA is about 2.5 nm wide, and the helix completes a turn every 3.4 nm along its length.

Problem 2 in the first chapter analyzes the volume of DNA

Problem 2 Our genetic information or genome is stored in the parts of the DNA molecule called base pairs. Our genome contains about 3 billion (3×10⁹) base pairs, and there are two copies in each cell. Along the DNA molecule, there is one base pair every one-third of a nanometer. How long would the DNA helix from one cell be if it were stretched out in a line? If the entire DNA molecule were wrapped up into a sphere, what would be the diameter of that sphere?

A problem in Chapter 3 considers errors in DNA in the context of the Boltzmann factor.

Problem 25 The DNA molecule consists of two intertwined linear chains. Sticking out from each monomer (link in the chain) is one of four bases: adenine (A), guanine (G), thymine (T), or cytosine (C). In the double helix, each base from one strand bonds to a base in the other strand. The correct matches, A–T and G–C, are more tightly bound than are the improper matches. The chain looks something like this, where the last bond shown is an “error.”

A  T  G  C  G

T  A  C  G  A (error)

The probability of an error at 300 K is about 10⁻⁹ per base pair. Assume that this probability is determined by a Boltzmann factor e^−U/k_BT, where U is the additional energy required for a mismatch.

(a) Estimate this excess energy.

(b) If such mismatches are the sole cause of mutations in an organism, what would the mutation rate be if the temperature were raised 20° C?

We discuss DNA again in Chapter 16, when considering radiation damage to tissue.

Cellular DNA is organized into chromosomes. In order to understand radiation damage to DNA, we must recognize that there are four phases in the cycle of cell division …

Figure 16.33 shows, at different magnifications, a strand of DNA, various intermediate structures which we will not discuss, and a chromosome as seen during the M phase of the cell cycle. The size goes from 2 nm for the DNA double helix to 1400 nm for the chromosome. In addition to cell survival curves one can directly measure chromosome damage. There is strong evidence that radiation, directly or indirectly, breaks a DNA strand. If only one strand is broken, there are efficient mechanisms that repair it over the course of a few hours using the other strand as a template. If both strands are broken, permanent damage results, and the next cell division produces an abnormal chromosome.¹⁹ Several forms of abnormal chromosomes are known, depending on where along the strand the damage occurred and how the damaged pieces connected or failed to connect to other chromosome fragments. Many of these chromosomal abnormalities are lethal: the cell either fails to complete its next mitosis, or it fails within the next few divisions. Other abnormalities allow the cell to continue to divide, but they may contribute to a multistep process that sometimes leads to cancer many cell generations later.

The Double Helix,
by James Watson.

The story of how the structure of DNA was discovered is nearly as fascinating as the structure itself. James Watson provides a first-person account in his book The Double Helix. It begins

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 him. 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.

The Eighth Day of Creation,
by Horace Freeland Judson.

The Double Helix is one of those iconic books that everyone should read for the insights it provides into how science is done, and for what is simply a fascinating story. The tale is also told from a more unbiased perspective in Horace Freeland Judson’s book The Eighth Day of Creation: The Makers of the Revolution in Biology. Let us end with Judson’s discussion of Watson and Crick’s now 60-year-old letter.

The letter to Nature appeared in the April 25th issue. To those of its readers who were close to the questions, and who had not already heard the news, the letter must have gone off like a string of depth charges in a calm sea. “We wish to suggest a structure for the salt of deoxyribose nucleic acid (D.N.A.). This structure has novel features which are of considerable biological interest,” the letter began; at the end, “It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material.” The last sentence has been called one of the most coy statements in the literature of science.