How Far Can Bacteria Coast?

Random Walks in Biology,
by Howard Berg.

In last week’s blog post, I told you about the recent death of Howard Berg, author of Random Walks in Biology. This week, I present a new homework problem based on a topic from Berg’s book. When discussing the Reynolds number, a dimensionless number from fluid dynamics that is small when viscosity dominates inertia, Berg writes

The Reynolds number of the fish is very large, that of the bacterium is very small. The fish propels itself by accelerating water, the bacterium by using viscous shear. The fish knows a great deal about inertia, the bacterium knows nothing. In short, the two live in very different hydrodynamic worlds.

To make this point clear, it is instructive to compute the distance that the bacterium can coast when it stops swimming.

Here is the new homework problem, which asks the student to compute the distance the bacterium can coast.

Section 1.20

Problem 54. When a bacterium stops swimming, it will coast to a stop. Let us calculate how long this coasting takes, and how far it will go.

(a) Write a differential equation governing the speed, v, of the bacterium. Use Newton’s second law with the force given by Stokes law. Be careful about minus signs.

(b) Solve this differential equation to determine the speed as a function of time.

(c) Write the time constant, τ, governing the decay of the speed in terms of the bacterium’s mass, m, its radius, a, and the fluid viscosity, η.

(d) Calculate the mass of the bacterium assuming it has the density of water and it is a sphere with a radius of one micron.

(e) Calculate the time constant of the decay of the speed, for swimming in water having a viscosity of 0.001 Pa s.

(f) Integrate the speed over time to determine how far the bacterium will coast, assuming its initial speed is 20 microns per second.

I won’t solve all the intermediate steps for you; after all, it’s your homework problem. However, below is what Berg has to say about the final result.

A cell moving at an initial velocity of 2 × 10^-3 cm/sec coasts 4 × 10^-10 cm = 0.04 Å, a distance small compared with the diameter of a hydrogen atom! Note that the bacterium is still subject to Brownian movement, so it does not actually stop. The drift goes to zero, not the diffusion.

Berg didn’t calculate the deceleration of the bacterium. If the speed drops from 20 microns per second to zero in one time constant, I calculate the acceleration to be be about 91 m/s², or nearly 10g. This is similar to the maximum allowed acceleration of a plane flying in the Red Bull Air Race. That poor bacterium.

Howard Berg (1934–2021)

Random Walks in Biology,
by Howard Berg.

Look up at the picture of books at the top of this blog, showing my ideal bookshelf. You see Intermediate Physics for Medicine and Biology towering in the center. The small volume two books to the right of IPMB is Random Walks in Biology, by Howard Berg.

Berg died on December 30, 2021. He was the Herchel Smith Professor of Physics in the Rowland Institute at Harvard University. He was known for his studies of flagellar motility and sensory transduction in bacteria, as described in his 2004 book E. coli in Motion.

Berg obtained his bachelor’s degree from Cal Tech, and a PhD from Harvard. He was on the faculty at the University of Colorado, then at Cal Tech, and finally at Harvard. He was a fellow of the American Physical Society and a member of the National Academy of Sciences. In 1984 he and Edward Purcell received the Max Delbrück Prize in Biological Physics from the American Physical Society “for the elucidation of complex biological phenomena, in particular chemotaxis and bacterial locomotion, through simple but penetrating physical theories and brilliant experiments.”

I had the pleasure of meeting Berg at a 2014 Gordon Research Conference about Physics Research and Education: The Complex Intersection of Biology and Physics, held at Mount Holyoke College in South Hadley, Massachusetts. He was a quiet, thoughtful, kind man. I wish I knew him better.

Purcell mentioned Berg in his influential article “Life at Low Reynolds Number” (American Journal of Physics, Volume 45, Pages 3–11, 1977).

I might say what got me into this. To introduce something that will come later, I’m going to talk partly about how microorganisms swim. That will not, however, turn out to be the only important question about them. I got into this through the work of a former colleague of mine at Harvard, Howard Berg. Berg got his Ph.D. with Norman Ramsey, working on a hydrogen maser, and then he went back into biology, which had been his early love, and into cellular physiology. He is now at the University of Colorado at Boulder, and has recently participated in what seems to me one of the most astonishing discoveries about the questions we're going to talk about. So it was partly Howard's work, tracking E. coli and finding out this strange thing about them, that got me thinking about this elementary physics stuff.

Section 4.10 of Intermediate Physics for Medicine and Biology analyzes chemotaxis, and cites Berg’s 1977 paper with Purcell “Physics of Chemoreception” (Biophysical Journal, Volume 20, Pages 119–136). Below is the abstract.

Statistical fluctuations limit the precision with which a microorganism can, in a given time T, determine the concentration of a chemoattractant in the surrounding medium. The best a cell can do is to monitor continually the state of occupation of receptors distributed over its surface. For nearly optimum performance only a small fraction of the surface need be specifically adsorbing. The probability that a molecule that has collided with the cell will find a receptor is Ns/(Ns + πa), if N receptors, each with a binding site of radius s, are evenly distributed over a cell of radius a. There is ample room for many independent systems of specific receptors. The adsorption rate for molecules of moderate size cannot be significantly enhanced by motion of the cell or by stirring of the medium by the cell. The least fractional error attainable in the determination of a concentration c is approximately (TcaD)^−1/2, where D is the diffusion constant of the attractant. The number of specific receptors needed to attain such precision is about a/s. Data on bacteriophage adsorption, bacterial chemotaxis, and chemotaxis in a cellular slime mold are evaluated. The chemotactic sensitivity of Escherichia coli approaches that of the cell of optimum design.

To learn more about Berg's life, education, and career, read his interview with Current Biology.

I will end with Berg’s introduction to his masterpiece Random Walks in Biology. If you want to learn about diffusion, start with Berg’s book.

Biology is wet and dynamic. Molecules, subcellular organelles, and cells, immersed in an aqueous environment, are in continuous riotous motion. Alive or not, everything is subject to thermal fluctuations. What is this microscopic world like? How does one describe the motile behavior of such particles? How much do they move on the average? Questions of this kind can be answered only with an intuition about statistics that very few biologists have. This book is intended to sharpen that intuition. It is meant to illuminate both the dynamics of living systems and the methods used for their study. It is not a rigorous treatment intended for the expert but rather an introduction for students who have little experience with statistical concepts.

The emphasis is on physics, not mathematics, using the kinds of calculations that one can do on the back of an envelope. Whenever practical, results are derived from first principles. No reference is made to the equations of thermodynamics. The focus is on individual particles, not moles of particles. The units are centimeters (cm), grams (g), and seconds (sec).

Topics range from the one-dimensional random walk to the motile behavior of bacteria. There are discussions of Boltzmann’s law, the importance of kT, diffusion to multiple receptors, sedimentation, electrophoresis, and chromatography. One appendix provides an introduction to the theory of probability. Another is a primer on differential equations. A third lists some constants and formulas worth committing to memory. Appendix A should be consulted while reading Chapter 1 and Appendix B while reading Chapter 2. A detailed understanding of differential equations or the methods used for their solution is not required for an appreciation of the main theme of this book.

 

Howard Berg. Marvels of Bacterial Behavior. Part 1.

Howard Berg. Marvels of Bacterial Behavior. Part 2.

The Chain of Reason vs. the Chain of Thumbs

Bully for Brontosaurus,
by Stephen Jay Gould.

I have written previously in this blog about my admiration for evolutionary biologist Stephen Jay Gould and his essays published in his monthly column “This View of Life” in the magazine Natural History. Today, I focus on one of these essays, “The Chain of Reason vs. the Chain of Thumbs,” that is related to a topic in Intermediate Physics for Medicine and Biology. You can find this essay reprinted in Gould’s book Bully for Brontosaurus.

IPMB has a chapter on biomagnetism (the production of magnetic fields by the body) and a section on the possible effects if weak external electric and magnetic fields. Much nonsense has been written about using magnetic fields to treat diseases, including to relieve pain. When did all this silliness begin? Over two hundred years ago.

Gould’s essay describes the fascinating story of the Franz Mesmer, who operated a clinic in Paris in the seventeenth century to treat various illnesses using “animal magnetism.” Gould writes

Franz Anton Mesmer was a German physician who had acquired wealth through marriage to a well endowed widow; connections by assiduous cultivation;… and renown with a bizarre, if fascinating, theory of “animal magnetism” and its role in human health.

Mesmer, insofar as one can find coherence in his ideas at all, claimed that a single (and subtle) fluid pervaded the universe, uniting and connecting all bodies. We give different names to this fluid according to its various manifestations: gravity when we consider planets in their courses; electricity when we contemplate a thunderstorm; magnetism when we navigate by compass. The fluid also flows through organisms and may be called animal magnetism. Disease results from a blockage of this flow, and cure of disease requires a reestablishment of the flux and a restoration of equilibrium.

Cure of illness requires the aid of an “adept,” a person with unusually strong magnetism who can locate the “poles” of magnetic flow on the exterior of a human body and, by massaging these areas, break the blockage within and reestablish the normal flux…

Mesmer's treatments were quite dramatic.

Within a few minutes of mesmerizing, sensitive patients would fall into the characteristic “crisis” taken by Mesmer as proof of his method. Bodies would begin to shake, arms and legs move violently and involuntarily, teeth chatter loudly. Patients would grimace, groan, babble, scream, faint, and fall unconscious.

Gould then tells the story of a Royal Commission established in 1784 by French king Louis XVI to investigate Mesmer’s claims. The commission was headed by American Benjamin Franklin, and included chemist Antoine Lavoisier and medical doctor Joseph Guillotin.

In a clever series of experiments, designed mainly by Lavoisier and carried out at Franklin’s home in Passy, the commissioners made the necessary separations and achieved a result as clear as any in the history of debunking: crises are caused by suggestion; not a shred of evidence exists for any fluid, and animal magnetism, as a physical force, must be firmly rejected.

Gould was impressed by the quality of the commission’s work.

Never in history has such an extraordinary and luminous group been gathered together in the service of rational inquiry by the methods of experimental science. For this reason alone, the Rapport des commissaires chargés par le roi de l’examen du magnétisme animal (Report of the Commissioners Charged by the King to Examine Animal Magnetism) is a key document in the history of human reason. It should be rescued from its current obscurity, translated into all languages, and reprinted by organizations dedicated to the unmasking of quackery and the defense of rational thought.

Nowadays we see a lot of ridiculous claims that magnetic fields can alleviate pain and have other health effects. Julian Whitaker and Brenda Adderly, in their book The Pain Relief Breakthrough, assert that magnets can cure backaches, arthritis, menstrual cramps, carpal tunnel syndrome, sports injuries, and more. Mexican surgeon Isaac Goiz Duran says that his “biomagnetic therapy” can cure diabetes, AIDS, cancer, and Covid-19. All these therapies are based on static magnetic fields, a type of 21st century animal magnetism.

I highly recommend all of Gould’s essays, including this one. Remember the efforts of Franklin, Lavoisier, and Guillotin before you start believing that static magnetic fields can improve your health.

Cells, Gels and the Engines of Life

Cells, Gels and the Engines of Life,
by Gerald Pollack.

I recently read Gerald Pollack’s book Cells, Gels and the Engines of Life: A New, Unifying Approach to Cell Function. I’ve always taken a simple, physicist’s view of a cell: salt water inside, salt water outside, with a membrane between; the membrane being where all the action is. Pollack’s perspective is entirely different, and challenges the standard dogma in cell biology. He focuses on the how the inside of a cell resembles a gel.

The gel-like nature of the cytoplasm forms the foundation of this book. Biologists acknowledge the cytoplasm’s gel-like character, but textbooks nevertheless build on aqueous solution behavior. A gel is quite different from an aqueous solution—it is a matrix of polymers to which water and ions cling. That’s why gelatin desserts retain water, and why a cracked egg feels gooey.

The concept of a gel-like cytoplasm turns out to be replete with power. It accounts for the characteristic partitioning of ions between the inside and outside of the cell... It also explains the cell’s electrical potential: potentials of substantial magnitude can be measured in gels as well as in demembranated cells... Thus, the gel-like character of the cytoplasm accounts for the basic features of cell biophysics.

What do I think of Pollack’s radical attitude toward biology? I’m not sold on his ideas, but his book certainly made me rethink my fundamental assumptions about biology in general, and electrophysiology in particular.

Pumps and Channels

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe the standard view of how ion gradients across the cell membrane are created by a pump (page 155).

To maintain the ion concentrations a membrane protein called the sodium-potassium pump uses metabolic energy to pump potassium into the cell and sodium out.

On page 193, we discuss ion channels.

Selective ion channels are responsible for the initiation and propagation of the action potentials.

Pollack doesn’t believe pumps and channels are important. How can this be? What about patch clamping? Section 9.7 of IPMB talks this revolutionary technique (page 251).

The next big advance was patch-clamp recording (Neher and Sakmann 1976). Micropipettes were sealed against a cell membrane that had been cleaned of connective tissue by treatment with enzymes. A very-high-resistance seal resulted [(2–3)×10⁷ Ω] that allowed one to see the opening and closing of individual channels.

Pollack says

The existence of single ion channels appeared to be confirmed by ground-breaking experiments using the patch-clamp technique… This dazzling result has so revolutionized the field of membrane electrophysiology that the originators of this technique, Erwin Neher and Bert Sakmann, were awarded the Nobel Prize. The observation of discrete events would seem to confirm beyond doubt that the ions flow through discrete channels.

Results from the laboratory of Fred Sachs, on the other hand, make one wonder. Sachs found that when the patch of membrane was replaced by a patch of silicon rubber, the discrete currents did not disappear… they remained essentially indistinguishable from those measured when the membrane was present.

Yikes! Pollack’s arguments against pumps don’t terrify me quite as much.

Pumping faces obstacles of space and energy. The membrane’s size is fixed but the number of pumps will inevitably continue to grow. At some stage the demand for space could exceed the supply, and what then? Pumping also requires energy. The Na/K pump alone is estimated to consume an appreciable fraction of the cell’s energy supply, and that pump is one of very many, including those in internal membranes. How is the cell to cope with the associated energy requirement?

 

 Membranes

Pollack goes on to renounce the importance of the cell membrane.

Continuing to move boldly, we took it upon ourselves in this chapter to reconsider the notion of the continuous ion barrier [the membrane]. If the barrier were continuous, we reasoned, violating its continuity by tearing large holes should allow ions to surge across the cell boundary and solutes to leak out, dramatically altering the cell’s makeup, shutting down cell function, and eventually killing the cell.

But that did not happen. Whether created by shoving a micropipette into the cell, plucking a patch from the membrane, riddling the membrane with an electrical barrage [electroporation], or slicing the cell into two, the wounds seemed to matter little; the cell could often continue to function as though there had been no violation. It was as though function could be sustained by the cytoplasm alone.

 

The Resting Transmembrane Potential

Russ and I explain the resting potential of a cell in the conventional way: the membrane is selectively permeable to potassium, and the potassium ion concentration is higher inside the cell than outside. The concentrations will be in equilibrium when the resting potential is negative, with a magnitude given by the Nernst equation. Pollack’s explanation is completely different, and focuses on the structuring of water near the hydrophilic surface of proteins.

Cell water excludes ions because it is structured. Exclusion is more pronounced for sodium than for potassium because sodium’s hydration shell is larger and hence more difficult to accommodate in the structured water lattice. Thus, intracellular sodium concentration remains low, whereas potassium can partition more easily into the cytoplasm.

No pumps, no channels, no membranes, no Nernst equation, and no metabolic ATP. Just water.

 

 The Action Potential

Pollack’s story may sound plausible, if not convincing, so far, but what about the action potential? This is where gels come in the forefront of the story. According to Pollack, the cytoplasm is like a gel that can undergo a polymer-gel phase transition. Normally the cytoplasmic polymers are cross linked by calcium, allowing in little water. If the calcium is not there to do the cross-linking, water gets sucked in, loosening the structure.

A plausible way in which the action potential could be initiated, then, is by replacing calcium with a monovalent [singly charged ion such as sodium or potassium]. Classically, sodium is thought to enter the cytoplasm through a localized, receptor-mediated permeability increase. In the proposed [Pollack’s] model, sodium ions flow into the peripheral cytoskeleton and begin displacing calcium. Replacement loosens the network, enabling it to adsorb water and expand. As it expands, permeability is increased, allowing for more sodium entry, further Ca displacement, additional expansion, etc.—like ripping open a zipper.

It sure doesn’t sound much like Hodgkin and Huxley.

Internal Perfusion

For a moment, I thought I had a devastating counter-example that would demolish Pollack’s theory. As mentioned before in this blog, you can squeeze the cytoplasm out of a squid giant axon and replace it by salt water, and the axon still works. But Pollack must have seen me coming; he shot down my counter-example in advance.

Lying just inside the cell membrane is a dense polymer-gel matrix known as the peripheral cytoskeleton… The presence of such a matrix had been unknown during the Hodgkin-Huxley era when experimental axons were routinely “rolled” to extrude the cytoplasm and presumably leave only the membrane. What in fact remains is the combination of membrane plus contiguous cytoskeleton.

 

Conclusion

I have only begun to cover the ideas discussed in Cells, Gels and the Engines of Life. Pollack provides us with a wonderfully written, beautifully illustrated, carefully argued, and well cited alternative view of biology. It was a joy to read, but I remain skeptical. I can think of many arguments in support of the “standard model” of cell physiology that Pollack doesn’t address. My “salt water inside, salt water outside” assumption may be too simplistic, and Pollack’s book is useful for pointing out its many limitations, but Pollack’s ideas have limitations too. Cells, Gels and the Engines of Life is an interesting read, but think long and hard before you start believing it.