Life at Low Reynolds Number
E.M. Purcell, Lyman Laboratory, Harvard University, Cambridge, Mass 02138
June 1976
American Journal of Physics vol 45, pages 3-11, 1977.
Editor's note: This is a reprint of a (slightly edited) paper of
the same title that appeared in the book Physics and Our World: A Symposium
in Honor of Victor F. Weiskopf, published by the American Journal of Physics
(1976). The personal tone of the original talk has been preserved in the
paper, which was itself a slightly edited transcript of a tape. The figures
reproduce transparencies used in the talk. The demonstration involved a
tall rectangular transparent vessel of corn syrup, projected by an overhead
projector turned on its side. Some essential hand waving could not be reproduced.
This is a talk that I would not, I'm afraid, have the nerve to give
under any other circumstances. It's a story I've been saving up to tell
Viki. Like so many of you here, I've enjoyed from time to time the wonderful
experience of exploring with Viki some part of physics, or anything to
which we can apply physics. We wander around strictly as amateurs equipped
only with some elementary physics even if we don't throw much light on
the other subjects. Now this is that kind of a subject, but I have still
another reason for wanting to, as it were, needle Viki with it, because
I'm going to talk for a while about viscosity. Viscosity in a liquid will
be the dominant theme here and you know Viki's program of explaining everything
including the height of mountains, with the elementary constants. The viscosity
of a liquid is a very tough nut to crack, as he well knows, because when
the stuff is cooled by merely 40 degrees, its viscosity can change by a
factor of a million. I was really amazed by fluid viscosity in the early
days of NMR, when it turned out that glycerin was just what we needed to
explore the behavior of spin relaxation. And yet if you were a little bug
inside the glycerin looking around, you wouldn't see much change in your
glycerin as it cooled. Viki will say that he can at least predict the logarithm
of the viscosity. And that, of course, is correct because the reason
viscosity changes is that it's got one of these activation energy things
and what he can predict is the order of magnitude of the exponent. But
it's more mysterious than that, Viki, because if you look at the Chemical
Rubber Handbook table you will find that there is almost no liquid with
viscosity much lower than that of water. The viscosities have a big range
but they stop at the same place. I don't understand that. That's
what I'm leaving for him.
I'm going to talk about a world which, as physicists, we almost never think
about. The physicist hears about viscosity in high school when he's repeating
Millikan's oil drop experiment and he never hears about it again, at least
not in what I teach. And Reynolds's number, of course, is something for
the engineers. And the low Reynolds's number regime most engineers
aren't even interested in--except possibly chemical engineers, in connection
with fluidized beds, a fascinating topic I heard about from a chemical
engineering friend at MIT. But I want to take you into the world of very
low Reynolds number--a world which is inhabited by the overwhelming majority
of the organisms in this room. This world is quite different form the one
that we have developed our intuitions in.
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.
Now,
Well here we go. In Fig. 1., you see an object which is moving through
a fluid with velocity v . It has dimension a. In Stoke's
law, the object is a sphere, but here it's anything;
and
are the viscosity and density of the fluid. The ratio of the inertial forces
to the viscous forces, as Osborne
Reynolds pointed out slightly less than a hundred years ago, is given by
av / or av/ , where is called the
kinematic viscosity. It's easier to remember its dimensions for water:
10-2 cm2/sec. The ratio is called the Reynolds number
and when that number is small the viscous forces dominate. Now there is
an easy way, which I didn't realize at first, to see who should be interested
in small Reynolds numbers. If you take the viscosity, and square
it and divide by by the density, you get a force (Fig. 2).
No other dimensions
come in at all, 2/ is a force. For water, since
10-2 and 1, 2/
10-4 dynes. That is a force that will tow anything, large or small,
with a Reynolds number of order of magnitude 1. In other words, if you
want to tow a submarine with Reynolds number 1 (or strictly
speaking 1/6
if it's a spherical submarine) tow it with 10-4 dyne in water. So
it's clear in this case that you're interested in small Reynolds number
if you're interested in small forces in an absolute sense. The only
other people who are interested in low Reynolds numbers, although they
usually don't have to invoke it, are the geophysicists. The Earth's mantle
is supposed to have a viscosity of 1021 P. If you now work out
2/, the force is 1041 dyne.
That is more than 109 times the gravitational
force that half the Earth exerts on the other half! So the conclusion is,
of course, that in the flow of the mantle of the Earth the Reynolds number
is very small indeed.
Now consider things that move through a liquid (Fig. 3). The Reynolds number
for a man swimming in a liquid might be 104, if we put in reasonable dimensions,
for a goldfish or a tiny guppy it might get down to 102. For the animals
that we're going to be talking about, as we'll see in a moment it's about
10-4 or 10-5 For these animals inertia is totally irrelevant. We
know that F=ma, but they could scarcely care less. I'll show you a picture
of the real animals in a bit but we are going to be taking about objects
which are the order of a micron in size (Fig. 4).
That's a micron scale,
not a suture, in the animal in Fig. 4. In water where the kinematic viscosity
is 10-2 cm/sec these things move around with a typical speed of 30 micron/sec.
If I have to push that animal to move it, and suddenly I stop pushing,
how far will it coast before it slows down? The answer is, about 0.1 angstrom.
And
it takes it about 0.6 microsec to slow down. I think this makes it clear
what low Reynolds number means. Inertial plays no role whatsoever. If you
are at very low Reynolds number, what you are doing at the moment is entirely
determined by the forces that are exerted on you at that moment, and by
nothing in the past.
It helps to imagine under what conditions a man would be swimming at,
say, the same Reynolds number as his own sperm. Well you put him in a swimming
pool that is full of molasses, and the you forbid him to move any pare
of his body faster than 1 cm/min. Now imagine yourself in that condition;
you're under the swimming pool in molasses, and now you can only move like
the hands of a clock. If under those ground rules you are able to move
a few meters in a couple of weeks, you may qualify as a low Reynolds number
swimmer.
I want to talk
about swimming at low Reynolds number in a very general way. What does
it mean to swim? Well, it means simply that you are in some liquid and
are allowed to deform your body in some manner. That's all you can do.
Move it around and move it back. Of course, you choose some kind of cyclic
deformation because you want to keep swimming, and it doesn't do any good
to use a motion that goes to zero asymptotically. You have to keep moving.
So, in general, we are interested n cyclic deformations of a body
on which there are no external torques or forces except those exerted by
the surrounding fluid. In Fig. 5, there is an object which has a shape
shown by the solid line; it changes its shape to the dashed contour and
then it changes back. When it finally gets back to its original shape,
the dotted contour, it has moved over and rotated a little. It has been
swimming. When it executed the cycle, a displacement resulted. If it repeats
the cycle, it will, of course, effect the same displacement, and in tow
dimensions we'd see it progressing around a circle. In three dimensions
its most general trajectory is a helix consisting of little kinks, each
of which is the result of one cycle of shape change.
There is a very funny thing about motion at low Reynolds number, which is the
following. One special kind of swimming motion is what I cal a reciprocal
motion. That is to say, I change my body into a certain shape and then
I go back to the original shape by going through the sequence in reverse.
At low Reynolds number, everything reverses just fine. Time, in fact, makes
no difference---only configuration. If I change quickly or slowly, the
pattern of motion is exactly the same. If you take the Navier-Stokes equation
and throw away the inertia terms, all you have left is
2v = p / , where p is the pressure (Fig. 6). So,
if the animal tries to swim by a reciprocal motion, it can't go anywhere.
Fast or slow, it exactly retraces its trajectory and it's back where it
started. A good example of that is a scallop. You know, a scallop opens
its shell slowly and closes its shell fast, squirting out water. The moral
of this is that the scallop at low Reynolds number is no good.
It can't swim because it only has one hinge, and if you have only one degree
of freedom in configuration space, you are bound to make a reciprocal motion.
There is nothing else you can do. The simplest animal that can swim that
way is an animal two hinges. I don't know whether one exists but Fig. 7
shows a hypothetical one. This animal is like a boat with a rudder at both
front and back, and nothing else. This animal can swim. All it has to do
is go through the sequence to configurations shown, returning to the original
one at S_5. Its configuration space, of course, is two dimensional with
coordinates
1,
2.
The animal is going around a loop in that
configuration space, and that enables it to swim. In fact, I worked this
one out just for fun and you can prove from symmetry that it goes along
the direction shown in the figure. As an exercise for the student, what
is it that distinguishes that direction?
You can invent
other animals that have no trouble swimming. We had better be able to invent
them, since we know they exist. One you might think of first as a physicist,
is a torus. I don't know whether there is a toroidal animal, but whatever
other physiological problems it might face, it clearly could swim at low
Reynolds number (Fig. 8). Another animal might consist of two cells which
were stuck together and were able to roll on one another by having some
kind of attraction here while releasing there. That thing will "roll" along.
I described it once a s a combination caterpillar tractor and bicycle built
for two, but that isn't the way ti really works. In the animal kingdom,
there are at least two other more common solutions to the problem of swimming
at low Reynolds number (Fig. 9). One might be called the flexible oar.
You see, you can't row a boat at low Reynolds number in molasses--if you
are submerged--because the stiff oars are just reciprocating things. But
if the oar is flexible, that's not true, because then the car bends one
way during the first half of the stroke and the other during the second
half. That's sufficient to elude the theorem that got the scallop. Another
method, and the one we'll mainly be talking about, is what I call a corkscrew.
If you keep turning it, that, of course, is not a reciprocal change in
configuration space and that will propel you. At this point, I wish I could
persuade you that the direction in which this helical drive will move in
not obvious.
Put
yourself back in that swimming pool under molasses and move around very,
very slowly. Your intuitions about pushing water backwards are irrelevant.
That's not what counts. Now, unfortunately, it turns out that the thing
does move the way your naive, untutored, and actually incorrect argument
would indicate, but that's just a pedagogical misfortune we are always
running into.
Well, let's look at some real animals (Fig. 10). This figure I've taken
from a paper of Howard Berg that he sent me. Here are three real swimmers.
The one we're going to be talking about most is the famous animal, Escherichia
coli, at A, which is a very tiny thing. Then there are two larger animals.
I've copied down their Latin names and the may be old friends to some of
you here. This thing (S. volutans) swims by waving its body as well
as its tail and roughly speaking, a spiral wave runs down that tail. The
bacterium E. coli on the left is about 2 micron long. The tail is
the part that we are interested in. That's the flagellum. Some E. coli
cells have them coming out the sides; and they may have several, but
when they have several they tend to bundle together. Some cells are nonmotile
and don't have flagella. They
live perfectly well, so swimming is not an absolute necessity for this
particular animal, but the one in the figure does swim. The flagellum is
only about 130 angstrom in diameter. It is much thinner than the cilium
which is another very important kind of propulsive machinery. There is
a beautiful article on cilia in the October 1974 issue of Scientific American.
Cilia are about 2000 angstrom in diameter, with a rather elaborate apparatus
inside. There's not room for such apparatus inside this flagellum.
For a long time there has been interest in how the flagellum works.
Classic work in this field was done around 1951, as I'm sure some of you
will remember, by Sir Geoffrey Taylor, the famous fluid dynamicist of Cambridge.
One time I heard him give a fascinating lecture at the National Academy.
Out of his pocket at the lecture he pulled his working model, a cylindrical
body with a helical tail driven by a rubber-band motor inside the body.
He had tested it in glycerin. In order to make the tail he hadn't just
done the simple thing of having a turning corkscrew, because at that time
nearly everyone had persuaded themselves that the tail doesn't rotate,
it waves. Because, after all, to rotate you'd have to have a rotary joint
back at the animal. So he had sheathed the turning helix with rubber tubing
anchored to the body. The body had a keel. I remember Sir Geoffrey Taylor
saying in his lecture that he was embarrassed that he hadn't put the keel
on it first and he'd had to find out that he needed it. There haas since
been a vast literature on this subject, only a small part of which I'm
familiar with. But at that time G. I. Taylor's paper in the Proceedings
of the Royal Society could conclude with just three references: H.
Lamb, Hydrodynamics ; G. I. Taylor (his previous paper); G. N. Watson,
Bessel Functions . That is called getting in on the ground floor.
To come now
to modern times, I want to show a picture of these animals swimming or
tracking. This is the work of Howard Berg, and I'll first describe what
he did. He started building the apparatus when he was at Harvard. He was
interested in studying not the actual mechanics of swimming at all but
a much more interesting question, namely, why these things swim and where
they swim. In particular, he wanted to study chemotaxis in E. coli --seeing
how they behave in gradients of nutrients and things like that. So he built
a little machine which would track a single bacterium in x,y,z coordinates--just
lock onto it optically and track it. He was able then to track one of these
bacteria while it was behaving in its normal manner, possibly subject to
the influence of gradients of one thing or another. A great advantage of
working with a thing like E. coli is that there are so many mutant
strains that have been well studied that you can use different mutants
for different things. The next picture (Fig. 11) is one of his tracks.
It shows a projection on a plane of the track of one bacterium. The little
dots are 0.1 sec apart so that it was actually running along one of the
legs for a second or two and the speed is typically 20--40 microns/sec.
Notice that it swims for a while and then stops and goes off in another
direction. We'll see later what that might suggest. A year ago, Howard
Berg went out on a limb and wrote a paper in Nature in which he
argued that, on the basis of available evidence, E. coli must swim
by rotating their flagella, not by waving them.
Within
the year a very elegant, crucial experiment by Silverman and Simon at UC-San
Diego showed that this fact is the case. Their experiment involved a mutant
strain of E. coli bacteria which don't make flagella at all but
only make something called the proximal hook to which the flagella would
have been attached. They found that whit antihook antibodies they could
cause these things to glue together. And once in a while one of the bacteria
would have its hook glued to the microscope slide, in which case the whole
body rotated at constant angular velocity. And when two hooks glued together,
the two bodies counter-rotated, as you would expect. It's a beautiful technique.
Howard was ready with his tracker and the next picture (Fig. 12) shows
his tracker following the end of one of these tethered E. coli cells
which is stuck to the microscope slide by antibody at the place where the
flagellum should have been. Plotted here are the two velocity components
V_x and V_y. The two velocity components are 90 degrees out of phase. The
point being tracked is going in a circle. In the middle of the figure,
you see a 90 degrees phase change in one component, a reversal of rotation.
They can rotate hundreds of revolutions at constant speed and then turn
around and rotate the other way. Evidently the animal actually has a rotary
joint, and has a motor inside that's able to drive a flagellum in one direction
or the other, a most remarkable piece of machinery.
I got interested
in the way a rotating corkscrew can propel something. Let's consider propulsion
in one direction only, parallel to the axis of the helix. The helix can
translate and it can rotate; you can apply a force to it and a torque.
It has a velocity v and an angular velocity
. And
now remember, at low Reynolds number everything is linear. When everything
is linear, you expect to see matrices come in. Force and torque must be
related by matrices with constant coefficients, to linear and angular velocity.
I call this little 2x2 matrix the propulsion matrix (Fig. 13). If I knew
its elements A, B, C, D, I could then find out how good this rotating
helix is for propelling anything.
Well, let's try to go on by making some assumptions. If two corkscrews
or other devices on the same shaft are far enough from one another so that
their velocity patterns don't interact, their propulsive matrices just
add. If you allow me that assumption, then there is a very nice way, which
I don't have time to explain, of proving that the propulsion matrix must
be symmetrical (Fig. 14).
So actually the motion is described by only three
constants, not four, and they are very easily measured. All you have to
do is make a model of this thing and drop in a fluid that you are interested
in or not, because these constants are independent of that. And so I did
that and that's my one demonstration. I thought this series of talks ought
to have one experiment and there it is. We're looking through a tank not
of glycerin but of corn syrup, which is cheaper, quite uniform, and has
a viscosity of about 50 P or 5000 times the viscosity of water. The nice
part of this is you can just lick the experimental material off your fingers.
Motion at low
Reynolds number is very majestic, slow, and regular. You'll notice that
the model is actually rotating but rather little. If that were a corkscrew
moving through a cork of course, the pattern in projection wouldn't change.
It's very very far from that, it's slipping, so that it sinks by
several wavelengths while it's turning around once. If the matrix were
diagonal, the thing would not rotate at all. So all you have to do is just
see how much it turns as it sinks and you have got a handle on the off-diagonal
element. A nice way to determine the other elements is to run two of these
animals, one of which is a spiral and the other is two spirals, in series,
of opposite handedness. The matrices add and with two spirals of opposite
handedness, the propulsions matrix must be diagonal (Fig. 14). That's not
going to rotate; it better not.
The propulsive efficiency is more or less proportional to the square
of the off-diagonal element of the matrix. The off-diagonal element depends
on the difference between the drag on a wire moving perpendicular to its
length and the drag on a wire moving parallel to its length (Fig. 15).
These are supposed to differ in a certain limit by a factor of 2. But for
the models I've tested that factor is more like 1.5. Since it's that factor
minus 1 that counts, that's very bad for efficiency. We thought that if
you want something to rotate more while sinking, it would be better not
to use a round wire. Something like a slinky ought to be better. I made
one and measure its off diagonal elements. Surprise, surprise, it was no
better at all!
I don't really understand that, because the fluid mechanics of these two
situations is not at all simple. In each case there is a logarithmic divergence
that you have to worry about, and the two are somewhat different in character.
So that theoretical ratio of two I referred to is probably not even right.
When you put all this in and calculate the efficiency, you find that
it's really rather low even when the various parameters of the model are
optimized. For a sphere which is driven by one of these helical propellers
(Fig. 16), I will define the efficiency as the ratio of the work that I
would have to do just to putt that thing along to what the man inside it
turning the crank has to do. And that turns out to be about 1%. I worried
about that result for a while and tried to get Howard interested in it.
He didn't pay much attention to it, and he shouldn't have, because it turns
out that efficiency is really not the primary problem of the animal's motion.
We'll see that when we look at the energy requirement. How much power does
it take to run one of these things with a 1% efficient propulsion system,
at this speed in these conditions? We can work it out very easily.
Going 30 micron/sec, at 1% efficiency will cost us about 2x10-8 ergs/sec
at the motor. On a per weight basis, that's a 0.5 W/kg, which is really
not very much. Just moving things around in out transportation system,
we use energy at 30 or 40 times that rate. This bug runs 24 hours a day
and only uses 0.5 W/kg. That's a small fraction of its metabolism and energy
budget. Unlike us, they do not squander their energy budget just moving
themselves around. So they don't care whether they have a 1% efficient
flagellum or a 2% efficient flagellum. It doesn't really make that much
difference. They're driving a Datsun in Saudi Arabia.
So the interesting question is not how they swim. Turn anything--if
it isn't perfectly symmetrical, you'll swim. If the efficiency is only
1%, who cares. A better way to say it is that the bug can collect, by diffusion
through the surrounding medium, enough energetic molecules to keep moving
when the concentration of those molecules is 10-9 M. I've now introduced
the word diffusion. Diffusion is important because of another very peculiar
feature of the world at low Reynolds number, and that is, stirring isn't
any good. The bug's problem is not its energy supply; its problem is its
environment. At low Reynolds number you can't shake off your environment.
If you move, you take it along; it only gradually falls behind. we can
use elementary physics to look at this in a very simple way.
The time for transporting anything a distance l by stirring, is about
l divided by the stirring speed v. Whereas, for transport
by diffusion, it's l2 divided by D, the diffusion constant.
The ratio of those two times is a measure of the effectiveness of stirring
versus that of diffusion for any given distance and diffusion constant.
I'm sure this ratio has someone's name but I don't know the literature
and I don't know whose number that's called. Call it S for stirring
number, it's just lv / D . You'll notice by the way that the
Reynolds number was lv / ,
is the kinematic viscosity in
cm2/sec, and D is the diffusion constant in cm2/sec,
for whatever
it is that we are interested in following--let us say a nutrient molecule
in water. Now, in every reasonably sized molecules, something like 10-5
cm2/sec. In the size domain that we're interested in, of micron distances,
we find that the stirring number S is 10-2 for the velocities
that we are talking about (Fig. 18).
In other words, this bug can't do anything by stirring its local surroundings.
It might as well wait for things to diffuse, either in or out. The transport
of wastes away from the animal and food to the animal is entirely controlled
locally by diffusion. You can thrash around a lot, but the fellow who just
sits there quietly waiting for stuff to diffuse will collect just as much.
At one time I thought that the reason the thing swims is that if it
swims it can get more stuff, because the medium is full of molecules the
bug would like to have. All my instincts as a physicist say you should
move if you want to scoop that stuff up. You can easily solve the problem
of diffusing in the velocity field represented by the Stokes flow around
a sphere---for instance, by a relaxation method. I did so and found out
how fast the cell would have to go to increase--its food supply. The food
supply if it just sits there is 4 a ND ,
where a is the
cell's radius (Fig. 19) and N is the concentration of nutrient molecules.
To increase its food supply by 10% it would have to move at a speed of
700 microns/sec, which is 20 times as fast as it can swim. The increased
intake varies like the square root of the bug's velocity so the swimming
does no good at all in that respect. But what it can do is find places
where the food is better or more abundant. that is, it does not move like
a cow that is grazing a pasture--it moves to find greener pastures.
And how far does it have to move? Well, it has to move far enough to outrun
diffusion. We said before that stirring wouldn't do any good locally, compared
to diffusion. But suppose it wants to run over there to see whether there
is more over there. Then it must outrun diffusion, and ho do you do that?
Well, you go that magic distance, D/v. So the rule is then, to out
swim diffusion you have to go a distance which is equal to or greater than
this number we had in our S constant. For typical D and v,
you have to go about
30 microns and that's just about what the swimming
bacteria were doing. If you don't swim that far, you haven't gone anywhere,
because it's only on that scale that you could find a difference in your
environment with respect to molecules of diffusion constant D (Fig. 20).
Let's go back
and look at one of those sections from Berg's track (Fig. 11). You'll see
that there are some little trips, but otherwise you might ask why did it
go clear over here and stop. Why did it go back? Well, my suggestions is,
and I'd like to put this forward very tentatively, that the reason it does
is because it's trying to outrun diffusion. Otherwise, it might as well
sit still, as indeed do the mutants who don't have flagella. Now there
is still another thing that I put forward with even more hesitation because
I haven't tried tried this out on Howard yet. When he did his chemotaxis
experiments, he found a very interesting behavior. If these things are
put in a medium where there is a gradient of something that they like,
they gradually work their way upstream. But if you look at how they do
it and ask what rules they are using, what the algorithm is to use the
current language, for finding your way upstream, it turns out that it's
very simple. The algorithm is: if things are getting better don't stop
so soon. If, in other words, you plot, as Berg has done in some of his
papers, the distribution of path lengths between runs and the little stops
that he calls "twiddles," the distribution of path lengths if they are
going up the gradient gets longer. That's a very simple rule for
working your way to where things are better. If they're going down the
gradient, though, they don't get shorter. And that seems a little puzzling.
Why, if things are getting worse, don't they change sooner? My suggestion
is that there is no point in stopping sooner. there is a sort of bedrock
length which outruns diffusion and is useful for sampling the medium. shorter
paths would be a ridiculous way to sample. It may be something like that,
but as I say, I don't know. the residue of education that I got from this
is partly this stuff about simple fluid mechanics, partly the realization
that the mechanism of propulsion is really not very important except, of
course, for the physiology of that very mysterious motor, which physicists
aren't competent even to conjecture about.
I come back for a moment to Osborne Reynolds. That was a very great
man. He was a professor of engineering, actually. He was the one who not
only invented Reynolds number, but he was also the one who showed what
turbulence amounts to and that there is an instability in flow, and all
that. He is also the one who solved the problem for how you lubricate a
bearing, which is a very subtle problem that I recommend to anyone who
hasn't looked into it. But I discovered just recently in reading in his
collected works that toward the end of his life, in 1903, he published
a very long paper on the details of the sub mechanical universe,
and he had a complete theory which involved small particles of diameter
10-18 cm. It gets very nutty from there on. It's a mechanical model,
the particles interact with one another and fill all space. But I thought
that, incongruous as it may have seemed to put this kind of stuff in between
our studies of the sub mechanical universe today, I believe that Osborne
Reynolds would not have found that incongruous, and I'm quite positive
that Viki doesn't.
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