Terminal Speed of Microorganisms

A Paramecium aurelia seen through an optical microscope.
Source: Wikipedia (http://en.wikipedia.org/wiki/Image:Paramecium.jpg)

Homework Problem 28 at the end of Chapter 2 in Intermediate Physics for Medicine and Biology asks the reader to calculate the terminal speed of an animal falling in air. Although this problem provides insight, it includes a questionable assumption. Russ and I tell the student to “assume that the frictional force is proportional to the surface area of the animal.” If, however, the animal falls at low Reynolds number, this assumption is not valid. Instead, the drag force is given by Stokes’ law, which is proportional to the radius, not the surface area (radius squared). The new homework problem given below asks the reader to calculate the terminal speed for a microorganism falling through water at low Reynolds number.

Section 2.8

Problem 28 ½. Calculate the terminal speed, V, of a paramecium sinking in water. Assume that the organism is spherical with radius R, and that the Reynolds number is small so that the drag force is given by Stokes’ law. Include the effect of buoyancy. Let the paramecium’s radius be 100 microns and its specific gravity be 1.05. Verify that its Reynolds number is small.

The reader will first need to get the density ρ and viscosity η of water, which are ρ = 1000 kg/m³ and η = 0.001 kg/(m s). The specific gravity is not defined in IPMB, but it’s the density divided by the density of water, implying that the density of the paramecium is 1050 kg/m³. Finally, Stokes’ law is given in IPMB as Eq. 4.17, F_drag = –6πRηV.

I’ll let you do your own calculation. I calculate the terminal speed to be about 1 mm/s, so it takes about a fifth of a second to sink one body diameter. The Reynolds number is 0.1, which is small, but not exceptionally small.

You should find that the terminal speed increases as the radius squared, in contrast to a drag force proportional to the surface area for which the terminal speed increases in proportion to the radius. Bigger organisms sink faster. The dependence of terminal speed on size is even more dramatic for aquatic microorganisms than for mammals falling in air. To paraphrase Haldane’s quip, “a bacterium is killed, a diatom is broken, a paramecium splashes,” except the speeds are small enough that none of the “wee little beasties” are really killed (the terminal speed is not terminal...get it?) and splashing is a high Reynolds number phenomenon.

Buoyancy is not negligible for aquatic animals. The effective density of a paramecium in air would be about 1000 kg/m³, but in water its effective density drops to a mere 50 kg/m³. Microorganisms are made mostly of water, so they are almost neutrally buoyant. In this homework problem, the effect of gravity is reduced to only five percent of what it would be if buoyancy were ignored.

A paramecium is a good enough swimmer that it can swim upward against gravity if it wants to. Its surface is covered with cilia that beat together like a Roman galley to produce the swimming motion (ramming speed!).

Whenever discussing terminal speed, one should remember that we assume the fluid is initially at rest. In fact, almost any volume of water will have currents moving at speeds greater than 1 mm/s, caused by tides, gravity, thermal convection, wind driven waves, or the wake of a fish swimming by. A paramecium would drift along with these currents. To observe the motion described in this new homework problem, one must be careful to avoid any bulk movement of water.

If you watched a paramecium sink in still water, would you notice any Brownian motion? You can calculate the root-mean-squared thermal speed with Eq. 4.12 in IPMB, using the mass of the paramecium as four micrograms and a temperature of 20° C. You get approximately 0.002 mm/s. That is less than 1% of the terminal speed, so you wouldn’t notice any random Brownian motion unless you measured extraordinarily carefully.

Breathless

Breathless,
by David Quammen.

Whenever David Quammen has a new book, I put it on my “to read” list. Recently I finished his latest: Breathless: The Scientific Race to Defeat a Deadly Virus. Here’s the opening paragraph:

To some people it wasn’t surprising, the advent of this pandemic, merely shocking in the way a grim inevitability can shock. Those unsurprised people were infectious disease scientists. They had for decades seen such an event coming, like a small, dark dot on the horizon of western Nebraska, rumbling toward us at indeterminable speed and with indeterminable force, like a runaway chicken truck or an eighteen-wheeler loaded with rolled steel. The agent of the next catastrophe, they knew, would almost certainly be a virus. Not a bacterium as with bubonic plague, not some brain-eating fungus, not an elaborate protozoan of the sort that cause malaria. No, a virus—and, more specifically, it would be a “novel” virus, meaning not new to the world but newly recognized as infecting humans.

Quammen—a national treasure—is writing about covid (or, to use its official name, SARS-CoV-2). The coronavirus pandemic did not startle him; he almost predicted it in his earlier book Spillover. Quammen’s book Breathless is to tracing the origins and variants of covid as Walter Isaacson’s book The Code Breaker is to developing a vaccine for covid: required reading to understand what we’ve all been through the last three years. (And what I went through last month with my first case of covid, but I’m healthy now and feeling fine.)

Breathless describes the scientists who developed amazing software to analyze the virus’s genome, such as Áine O’Toole’s genomic pipeline PANGOLIN. Intermediate Physics for Medicine and Biology doesn’t discuss computational genomics, but at the heart of IPMB is the idea that you can combine a hard science like computer programming with a biological science like genomics to gain more information about, and insight into, biology and medicine. Quammen interviewed O’Toole about her experience writing the PANGOLIN program (“O’Toole stayed up late one night, and the next morning, there it was.”). But he didn’t interview just her. He talked to 96 heroic scientists and medical doctors who sought to understand covid, from those I’ve never heard of such as O’Toole to those we all are familiar with such as the brilliant Anthony Fauci. These interviews give the book credibility, especially given all the covid conspiracy theories and anti-vaccine nonsense that floats around the internet these days.

For anyone who may doubt the reality of evolution, I challenge you to try making sense of covid variants without it. Quammen takes us through the list: Alpha, Beta, Gamma, and the frightening Delta.

And after Delta, we knew, would come something else. The Greek alphabet contains twenty-four letters; at that point, the WHO [World Health Organization] list of variants only went up to mu. A virus will always and continually mutate, as I’ve noted, and the more individuals it infects, the more mutations it will produce. The more mutations, the more chances to improve its Darwinian success. Natural selection will act on it, eliminating waste, eliminating ineptitude, carving variation like a block of Carrara marble at the hands of Michelangelo, finding beautiful shapes, preserving the fittest. Evolution will happen. That’s not a variable, it’s a constant.

The latest variant, Omicron, seems to have appeared just as Quammen was finishing his book.

Omicron’s panoply of mutations reflects a period of active, extensive evolution—because the mutations not only occurred but they were preserved, within the lineage, suggesting they offered adaptive value.

One of the most interesting questions addressed in Breathless is the source of covid. Was it a lab accident, a spillover from an animal host (called a zoonotic event), or a malevolent attempt at biological warfare? Quammen doesn’t provide a definitive answer, but he favors the conclusions reached in a review article written by a group of prominent virologists led by Eddie Holmes.

Yes, Holmes and his coauthors agreed, the possibility of a lab accident can’t be entirely dismissed. Furthermore, that hypothesis may be nearly impossible to disprove. But it’s “highly unlikely,” they judged, “relative to the numerous and repeated human-animal contacts that occur routinely in the wildlife trade.” Failure to investigate that zoonotic dimension, with collaborative studies, crossing borders between countries and boundaries between species, would leave this pandemic festering and the world still very vulnerable to the next one.

Run, do not walk, to your library or bookstore and get Breathless. You need to read this book. Take special heed of Quammen’s alarming, disturbing, terrifying last sentence.

There are many more fearsome viruses where SARS-CoV-2 came from, wherever that was.

 A conversation with author and journalist David Quammen.

https://www.youtube.com/watch?v=jsREp6XI0nY

The Unscientific King: Charles III’s History Promoting Homeopathy

King Charles III of England was crowned this week. What’s that got to do with Intermediate Physics for Medicine and Biology? Well, the king is a big supporter of alternative medicine and one goal of IPMB is to highlight science-based medicine. If you believe in science, you don’t believe in alternative medicine. If science shows that some treatment works, it becomes part of medicine; there is nothing “alternative” about it. If science doesn’t show that some treatment works, then advocating for that treatment as “alternative medicine” is silly and foolish. In the realm of medicine, the king is a snake oil salesman.

Voodoo Science
by Robert Park.

Particularly worrisome is the king’s support for homeopathy. For those not familiar with homeopathic medicine, it works like this: a drug is repeatedly diluted, first by a 10:1 ratio of water to active ingredient (1X), then again a 10:1 dilution so the total dilution is by a factor of 100 (2X), then again a 10:1 dilution (3X), and so on. In Voodo Science, Bob Park described it this way:

The dilution limit is reached when a single molecule of the medicine remains. Beyond that point, there is nothing left to dilute. In over-the-counter homeopathic remedies, for example, a dilution of 30X is fairly standard. The notation 30X means the substance was diluted one part in ten and shaken, and then this was repeated sequentially thirty times. The final dilution would be one part medicine to 1,000,000,000,000,000,000,000,000,000,000 parts of water. That would be far beyond the dilution limit. To be precise, at a dilution of 30X you would have to drink 7,874 gallons of the solution to expect to get just one molecule of the medicine.

The supporters of homeopathy would have us believe that the water “remembers” the presence of the active ingredient.

King Charles’s support of alternative medicine was discussed in a recent article in The Scientist by Sophie Fessl, titled “The Unscientific King: Charles III’s History Promoting Homeopathy.” The first paragraph is reproduced below.

King Charles III has been conferred many new titles following the recent death of his mother, Queen Elizabeth II, but one existing title that remains is “Royal Patron of the Faculty of Homeopathy,” an organization of healthcare practitioners who also practice the pseudoscientific form of medicine. And the new king’s ties with alternative medicine go beyond this patronship and a dalliance with alternative medicine: In several instances, then-Prince Charles appears to have lobbied for homeopathy and other fields of alternative medicine. As King Charles ascends the throne, experts are reflecting on his influence on medical science in the UK as Prince of Wales, and how he might affect alternative medicine in the UK going forward as monarch. 

One book I have not read yet but is on my to-read list is Charles, The Alternative King, by Edzard Ernst, an advocate for evidence-based medicine and one of my heroes. In his preface, Ernst writes

This book chronicles Charles’s track record in promoting pseudo- and anti-science in the realm of alternative medicine. The new edition includes an additional final chapter with a summary of some of the scientific evidence that has emerged since this biography [originally titled Charles, The Alternative Prince] was first published. It demonstrates that the concerns about the safety and efficacy of the treatments in question are becoming even more disquieting. Whether such data will tame the alternative bee under the royal bonnet seems, however, doubtful.

This is the man who now sits on the thrown of England. We Americans owe George Washington so much.

Plans for a 6th Edition of Intermediate Physics for Medicine and Biology

Thank you for your interest in the 5th edition of Intermediate Physics for Medicine and Biology. We’re considering a 6th edition and we would like to get your opinions and input. Some of you may have heard that about a year ago the senior author Russ Hobbie passed away. He was a joy to work with and will be missed. The 6th edition will have three authors: Hobbie, me, and a new coauthor Gene Surdutovich.

Gene and I would appreciate any feedback you have about our plans for the 6th edition. If you have time, please answer the questions below and email your responses to me at roth@oakland.edu. It would help us a lot to hear from you. I know that I listed many questions. Don’t worry if you have answers to only a few (or even just one). 

Thank you. 

1. Was there any topic in the textbook that you think could be removed? 
2. Was there any topic NOT in the textbook that you would like to see added in the 6th edition? 
3. Regarding the homework problems, were they too easy? Too difficult? Too many? Too few? 
4. How useful was the list of symbols at the end of each chapter? 
5. Regarding the references, were there too many? Too few?
   
6. Were the figures useful? Not useful? Easy to understand? Difficult to understand? 
7. A few chapters had computer code. Was it useful? Was it unnecessary? 
8. Are the Appendices useful? Unnecessary? Too many? Too few? 
9. Do you have any general input or advice? 
10. Do you have any specific issues with the 5th edition that you would like us to address?