# What it feels like for a sperm, or how to get around when you are really, really small

We don’t usually learn about the physics of squishy things. Physics textbooks are filled with solid objects such as incompressible blocks, inclined planes and inelastic strings. This is the rigid world that obeys Newton’s laws of motion. Here, squishiness is an exception and drag is routinely ignored. The only elastic object around is a spring, and it is perfectly elastic. It will never bend too far and lose its shape. But any child who has played vigorously with a Slinky has stretched past the limits of this Newtonian world.

Whereas the rigid universe is notable for its strict adherence to a few basic principles, the squishy universe is a different beast altogether.

I was recently out paddling, and noticed that as you move the paddle through water, tiny whirlpools begin to develop along its sides. The whirlpools grow in size, become self-sustaining, and break off and float away. Eventually they die out, as they lose their energy to the fluid around them.

You could also watch the spirals and vortices created by rising smoke. Or notice the strange shapes made by the wind as it sweeps through the clouds. It’s as if fluids have a life of their own, often wondrous and beautiful, and other times surprising and counter-intuitive.

But the motion of fluids is notoriously hard to predict. It’s so difficult that if you can solve the equations of fluid flow, there are people willing to offer you a million dollars. The difficulty comes from a mathematical property of the equations known as non-linearity. Simply put, a non-linear system is one where a small change can lead to a large effect. The same thing that makes these equations difficult to solve is also what makes fluids surprising and interesting. It’s why the weather is so hard to predict – tiny changes in local temperatures and pressures can have a large effect.

At this point, most reasonable people would throw their arms up in despair. But physicists are an unreasonably persistent bunch, and when faced with an equation that they can’t solve, they try to get some insight by looking at what happens at extremes. For example, thick and syrupy fluids like glycerine behave in a surprisingly orderly fashion. Take a look at this video (watch through to the end, it’s worth it).

I bet you’ve never seen a fluid do that before. So what’s going on here? And what does this have to do with swimming sperm?

Let’s take a step back. Picture a flowing river. If there is an obstruction to the water’s path, like a rock jutting out of the surface, the water will move around it and swirl back upstream. Behind the rock, the water remains relatively calm. What you get is a spot on a moving river where the water is remarkably still. These calm spots are called eddies, and kayakers treat them as parking spots on the river.

But fluids don’t always behave like this. If you replace all the water in a river with a viscous fluid like glycerine, there won’t be any eddies. The syrup will simply follow the contours of the rock and smoothly flow around it.

In one case we have smooth, orderly flow, and in the other case we have eddies and turbulent flow. The question arises, is there any way to know what kind of flow will result in a given situation? This question was answered by the physicist Osborne Reynolds in 1883, and he answered it in style.

Here is Reynolds’ elegant experiment. He sent fluid flowing through a thin pipe (analogous to the river), and injected colored dye in a small section of the flow. He watched the dye flow down the tube, and could plainly see whether the flow was smooth or disorderly. By tweaking the parameters in this experiment, he was able to discover the conditions that ensure an orderly flow.

What he found is that there is one simple, magic number that can predict what is going to happen. It neatly ties together all the different physical quantities involved. It’s been named Reynolds number (Re for short), and is given by

$Re = \dfrac{\textrm{density}\times\textrm{speed}\times\textrm{length}}{\textrm{viscosity}}$

These are all quantities that you can directly measure. The viscosity of a fluid is a measure of how slowly it flows. Thick and syrupy fluids like honey and corn syrup have a high viscosity, gases like air have a very low viscosity, and water is somewhere in between. The length in the above equation is a length that describes the object that you are studying (say the width of the rock). Reynolds used the diameter of the pipe. And the speed is that of the fluid.

The Reynolds number has the nice property of being dimensionless, meaning that the number is the same in whatever system of units you choose to measure the above quantities (dimension-full quantities are things like speed, which you could measure in km/h or mph). What Reynolds found is that as this number exceeds 2000, you suddenly get turbulent flow. In fact, this week’s issue of Science magazine mentions a new experiment that verifies this surprising result, and puts the turning point at Re = 2040. (The specifics of this number has to do with a fluid moving through a cylindrical tube with smooth walls. In a different situations, the number will change, but the principle is the same. There is a sudden jump from order to turbulence.)

The above figure gives you an idea of what happens as you increase Reynolds number. Here’s an analogy. The low Reynolds number world is like a collectivist ideal, where water moves along uniformly like soldiers marching in step. The high Reynolds number world is the individualist nightmare, where everyone looks out for themselves. Think of a march versus a mob.

We can arrive at this number from another route. There are two fundamentally different type of forces that act on an object immersed in a fluid. The first kind are inertial forces. This is like the push you give to the water when you take a stroke while swimming. Inertia is what allows water particles to keep moving undisturbed. On the other hand, you have viscous forces which measure the tendency for the fluid to smooth out any irregularities. To use the above analogy, inertial forces reflect the individuality of bits of fluid, and viscous forces are like a communist government enforcing conformity. And when you take the ratio of these forces, you get back the Reynolds number.

$Re = \dfrac{\textrm{inertial forces}}{\textrm{viscous forces}}$

This number is of immense importance to aeronautical engineers and to biologists interested in locomotion.

Let’s say you want to simulate the effect of wind on a new wing design. You build a scale model in the lab that is one tenth the size of the actual wing.

But remember how the Reynolds number is defined.

$Re = \dfrac{\textrm{density}\times\textrm{speed}\times\textrm{length}}{\textrm{viscosity}}$

If you shrink the size of the wing by a factor of 10, you have to increase the windspeed by the same amount in order to keep the number fixed. The key point is that systems with the same Reynolds number have essentially the same nature of flow. If you didn’t account for this, your wing would be quite a disaster.

How would a biologist use this idea? Well, nature presents us with organisms that cover an incredible range of sizes, from the tiniest microbes to the blue whales. Here is a table of Reynolds numbers across this range.

The list covers 14 orders of magnitude. A whale swims at a huge Reynolds number. This means that inertial forces completely dominate. If it flaps its tail once, it can coast ahead for an incredible distance. Bacteria live at the other extreme. In a delightful paper entitled Life at low Reynolds number, the physicist Edward Purcell calculated that if you a push a bacteria and then let go, it will coast for a distance equal to one tenth the diameter of a hydrogen atom before coming to a stop. And it will do this in 3 millionths of a second. Bacteria clearly inhabit a world where inertia is utterly irrelevant.

Eels and sperms may look similar, but their method of moving is very different, as their Reynolds numbers are far apart. In fact, we can now answer the question, what would it feel like to swim like a sperm or a bacteria? To do this, you have to somehow get down to their Reynolds number. We can’t change our size, but we can shrink our Reynolds number by swimming in a very viscous fluid. Purcell estimated that you would have to submerge yourself in a swimming pool full of molasses, and move your arms at the speed of the hands of a clock. (Don’t try this at home. Swimming in molasses is not a good idea.) Under these conditions, if you managed to cover a few meters in a few weeks, then you qualify as a low Reynolds number swimmer.

This clearly isn’t a hospitable environment for denizens of our Middle World. But yet this is the scale of the task that microbes face simply to get around.

Except, it’s even harder. Remember the youtube video of the colored dye swirling in the glycerine? The reason that the colors come back to where they start is because at low Reynolds number, flow is reversible. Because inertial forces are so small, certain terms drop out of the complicated fluid flow equations. The equations simplify considerably, and not only are they now solvable, they don’t depend on time any more. If you took the youtube video and played it backwards, you wouldn’t be able to tell the difference.

But this reversibility has a surprising consequence. It means that anything that swims using a repeating flapping motion can’t get anywhere. If it moves forward in one stroke, the other stroke will bring it right back to where it started. Scallops swim by opening their jaws and snapping it shut. In low Reynolds number, scallops can’t get anywhere.

Don’t believe me? See it for yourself. Here’s a rubber band powered toy that paddles forward when in water.

Woohoo! Look at it go. Now, take the same toy and place it in a vat of viscous corn syrup.

The reversibility of the flow ensures that the boat can’t make any progress.

So how, then, do microbes manage to get anywhere? Well, many don’t bother swimming at all, they just let the food drift to them. This is somewhat like a lazy cow that waits for the grass under its mouth to to grow back. But many microbes do swim, and they make use of remarkable adaptations to get around in an environment that is entirely alien to us.

One trick they can use is to deform the shape of their paddle. By cleverly contorting the paddle create more drag on the power stroke than on the recovery stroke, single cell organisms like paramecia break the symmetry of their stroke and thus elude the scallop conundrum. Indeed, this is how the flapping structures known as cilia thrust a cell forward: they flex.

There is an even more ingenious solution that has been hit upon by bacteria, sperm and other cells. Rather than having a cilia, which is essentially a flexible paddle, these cells adopt a different strategy: they use a corkscrew for a propeller. Just as a corkscrew used on a wine bottle converts winding motion into motion along its axis, these organisms spin their helical tails (flagellum) to push themselves forward.

But don’t expect to see human swimmers doing ‘the corkscrew’ anytime soon. This strategy works only at low Reynolds number, where water ‘feels’ as thick as cork, so you can push against it effectively.

And here’s proof. Whereas our rubber band powered stiff paddle couldn’t make any headway in the corn syrup, take a look at what happens if you instead have a helical propeller.

It winds its way into the fluid and inches forwards.

Motion in this viscous world is counter-intuitive and puzzling. By applying science, we can imagine what it must feel like to be very small. And we can work out how to build tiny ships in such a world. But evolution has beaten us to the punchline, and microorganisms have evolved intricate and wonderful structures that pulsate rhythmically and take advantage of the quirks of physics at this scale.

References

Purcell, E. (1977). Life at low Reynolds number American Journal of Physics, 45 (1) DOI: 10.1119/1.10903

Avila K, Moxey D, de Lozar A, Avila M, Barkley D, & Hof B (2011). The onset of turbulence in pipe flow. Science (New York, N.Y.), 333 (6039), 192-6 PMID: 21737736

Reynolds, O. (1883). An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels. Proceedings of the Royal Society of London, 35 (224-226), 84-99 DOI: 10.1098/rspl.1883.0018

In addition to the above papers, I learnt a lot about this subject from the following excellent book, from which many of the figures in this post are taken:
Life in moving fluids: the physical biology of flow by Steven Vogel (1996)

The theme of this post came from reading a following wonderful out-of-print book that I discovered in the basement of Strand bookstore in NYC:
On Size and Life (Scientific American Library) (1983)

Image Credits

Figures from the cited papers or from Life in moving fluids by Steven Vogel are attributed in place.

Cartoon of eddies was lifted from Whitewater kayaking: the ultimate guide by Ken Whiting & Kevin Varette

Difference of beating pattern of flagellum and cilia courtesy Wikimedia Commons

Nice post I’d really like to see more research being done on how different sperm cells swim, and why there are so many different strategies. Maybe the viscosity is different depending on the female or other environment they’re in? By the way, if you want to see some videos of Drosophila sperm swimming, there are some neat ones here (but you can only see the heads moving; the tails are too long): http://pitnicklab.syr.edu/Pitnick/proposal_movies.html

• http://www.empiricalzeal.com Aatish

Hi Brooke. There really are a lot of cool strategies. My favorite is that of the deer mice sperm, where the sperm somehow sniffs out its relatives, and they clump together to swim faster. Video

The videos you linked to are quite cool. The last one reminded me of the final scene in Star Wars where Luke blows up the death star!

Really awesome post, this. I, for one, never realised that movement for a bacterium would be this challenging. Which is a pity. Those small “wonders” would have got many more kids interested in biology at high school.

• James Byrne

I’d never heard of a Reynolds Number before and it explain so much. Phenomenal post

• Marci

Fascinating. I love all the videos and images.  Makes me wish I had taken more physics.  Though the physics textbooks I had were never as interesting as this.

• Marci

Fascinating. I love all the videos and images.  Makes me wish I had taken more physics.  Though the physics textbooks I had were never as interesting as this.

• sinisa

Very nice post (again) Aatish!

• Kshitij

Awesome science and science blogging – very cool post!
Btw I have always had a desire for visiting a microbe zoo where instead of animals you have to look through a microscope at microbes. Does anyone know of anything similar?

• http://www.empiricalzeal.com Aatish

Cool idea. NSF grants often have a requirement for science outreach, and this would be an awesome way for a bio lab to do it. The natural history museum in London (my mecca) has this incredible structure called the cocoon, where you can chat with researchers at the museum. And you can do stuff like hand them some dirt or bugs or whatever, and they’ll stick it under a microscope and show it to you on the screen.

• http://www.empiricalzeal.com Aatish

Thanks all. Very happy that you liked it. Now that the physics is out of the way, I’m thinking of doing a follow up which focuses more on strategies and adaptations in organisms.

• Nikos Svoronos

Have you ever read about the structure of the proteins that allow cells to twirl flagelli? It’s unusual to see rotating wheels in biology. You should read about the enzyme ATP synthase, which has a similar structure to the motor driving a flagellum.

• http://www.empiricalzeal.com Aatish

Thanks for the tip Nikos. Will do.

• RK

Fascinating post — best piece of science blogging I’ve read in ages!

• http://www.empiricalzeal.com Aatish

Thanks Raghu. Very glad you think so.

• Sitapriya Moorthi

Hey, that was a very interesting post. Congrats for the award. But I have a few questions. I know physics/maths helps to explain this complex science of motion. But as a physics person, what would be an ideal motion through viscous liquid? If there is such a thing? Minimum energy maximum propulsion?  And why hasnt evolution diversified into it ?

• http://www.empiricalzeal.com Aatish

Hey Sita. That’s a really insightful question, and not at all easy to answer. I’ll try to take a stab at it. After some googling around, I found that there’s a whole literature out there on the energy efficiency of microbes and low Reynolds number swimmers (both theoretical calculations and experimental measurements). Purcell himself was one of the first people to ask what is the efficiency of a bacteria with a rotating flagellum. Normally, it would be really hard to figure this out, but thanks to the fortunate physics simplifications that happen at very low Reynolds number (time disappearing from the equation, and the equations becoming linear), it can be done. In fact, in many idealized models, people have figured out what is the optimal stroke pattern.

What does it mean to be optimal? One interesting thing that happens at low Reynolds number is that two surprisingly different optimization problems – trying to maximize speed, and trying to maximize efficiency – turn out to amount to the same thing.  In other words, the sprinters are also the endurance runners! [1]

So, Purcell figured out the efficiency of propelling yourself like a sperm or a bacteria (in the same paper linked in the post, but also elaborated in [2]). He modeled this as a spherical bead connected to a rigid helical coil, and found the efficiency to be about 1% (this number turns out to match the measured efficiency of E Coli, which is 2% [3]).

This really bothered Purcell.. why would nature select an organism that is so inefficient? He worried about it for a while, but then concluded that “efficiency is not the primary problem of an animal’s motion.” What he found was that if you look at the energy expenditure *per unit mass*, it’s not so bad, even with such a low efficiency. It turns out be be about 0.5 watts / kg, which is a small fraction of its energy budget. It’s also 30-40 times lower energy consumption per unit mass than our own! The idea is that they can afford the cost of low efficiency. In Purcell’s words, these creatures are “driving a Datsun in Saudi Arabia.”

On the other hand, while this may be true for a bacteria, later studies showed that paramecium use half their energy budget on flapping their cilia. I found a paper that theoretically models the efficiency of swimming with cilia. [4] They then compare these numbers to measurements for a paramecium, and they find that “the experimentally estimated efficiency of Paramecium is surprisingly close to the theoretically possible optimum.” So you may be right about evolution optimizing efficiency after all!

References:
[1] http://boulder.research.yale.edu/Boulder-2011/Lectures/Hosoi/Boulder1.pdf[2] http://www.pnas.org/content/94/21/11307.full[3] http://www.pnas.org/content/103/37/13712.short[4] http://www.pnas.org/content/108/38/15727.full

• Sitapriya Moorthi

Thanks for that in-depth answer. But I think you just opened doors to a whole array of questions now brewing in my head. Sorry!

Okay, as much as I could gather from  your explanation, improvisation/optimization does occur. I am assuming that this optimization is species limited ( But is it?). And as per your explanation this means as long as a species can afford a sub-optimal  expenditure of energy on movement ( which I am assuming is very critical for survival…probably the most relevant phenotypic aspect) it does so ( the 2% in the case of bacteria), is their any evidence that this compromises on other functions ? Would it make sense to compare species that have lost their cilia and see gain in functions/metabolic advantages?

Next if the paramecium does come close to the theoretical value of energy optimization of movement is there any evidence of reduced loco-motor development… ( being a firm believer in evolution this points to a crazy idea of no more evolution!) ?

• http://www.empiricalzeal.com Aatish

Hey Sita, I have no idea if these questions have been studied at all (I’m not familiar with the literature). But they’re definitely intriguing questions and have got me wondering.

• david luong

this is pretty rad!  i just finished a yearlong fellowship (NSF GK12) all about science in the classroom, and you would be blow these students minds.

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