Two drags, or nothing swims

Taylor’s waving sheet ends its own list of limitations with the case it cannot treat: the sheet is planar, most flagella are helical, and a plane wave produces no torque while a helical one does. That is stated there as a shortcoming of the model. It is better read as a different mechanism.

A bacterium does not wave anything. It has a rotary motor in its cell wall, the motor turns a rigid helical filament at about a hundred revolutions a second, and the cell moves forward — a displacement that, like every other one at this scale, is the area its shape traces out rather than a consequence of how fast it is traced. Nothing about that arrangement is obviously propulsive. A corkscrew in a nut goes forward because the nut’s thread holds it; a corkscrew in water has nothing to push against, the water has no inertia to leave behind, and turning something about its own axis is an operation with no preferred direction along that axis at all.

The thing that supplies the direction is not the helix. It is a property of the filament that the helix merely makes use of.

One filament, two drags #

Pull a thin straight rod through a viscous fluid end-on and broadside at the same speed, and the broadside case resists more. That is unsurprising; what matters is by how much, and the answer has a peculiar form.

A slender body of length much greater than its radius has drag coefficients per unit length

$$\zeta_\perp = \frac{4\pi\mu}{\ln(2q/r) + \tfrac12}, \qquad \zeta_\parallel = \frac{2\pi\mu}{\ln(2q/r) - \tfrac12},$$

where $r$ is the filament’s radius and $q$ the length over which its direction changes — for a helical filament, its wavelength. The logarithms are there for the same reason logarithms appear all over creeping flow: a line force in three dimensions makes a disturbance that decays as $1/\text{distance}$, its integral diverges at both ends, and what cuts it off is the filament’s own radius at one end and its own curvature at the other. The ratio of those two lengths appears, and it appears inside a logarithm because that is what the integral gives.

For a bacterial flagellum — a filament twenty nanometres across on a helix of 2.3 micrometres pitch — the logarithm is 5.44, and the two coefficients are $2.12\times10^{-3}$ and $1.27\times10^{-3}$ pascal seconds. Their ratio is 1.663.

Three things about those expressions repay reading rather than using. The first is that the coefficients are not properties of the filament alone: $q$ is a length belonging to the shape the filament has been bent into, so the same wire has different drag coefficients when it is wound into a tight helix and a loose one. That is not a defect of the theory but a statement about what a local drag coefficient can mean — the flow at a point on a curved filament depends on where the rest of the filament is, and the logarithm is the crudest possible way of admitting it.

The second is that the two coefficients are not independent constants of a material but both built from one logarithm, which is why their ratio is so tightly constrained. There is no filament with a large $\zeta_\perp$ and a small $\zeta_\parallel$; making the filament thinner raises both, in a fixed proportion set by the numerators 4π and 2π and shifted only slightly by the halves.

The third is the halves themselves. They are the difference between the modern coefficients and the ones Gray and Hancock published, and they matter more than their size suggests: it is the difference of the two coefficients that does all the work below, and a half in each denominator moves that difference by more than it moves either coefficient. A theory accurate to twenty per cent in $\zeta_\perp$ and $\zeta_\parallel$ separately is accurate to rather less than that in the quantity the organism actually uses.

The coupling term, and what it is made of #

Turning a helix about its axis is a rigid motion, and in Stokes flow the force follows from the instantaneous geometry alone, so the velocity of each point of the filament is known: an axial translation $U$ plus a rotation $\Omega$ about the same axis. Resolving that velocity onto the local tangent and perpendicular, multiplying by the two coefficients, and integrating along the filament gives the force and the torque in the form

$$F = -(A\,U + B\,\Omega), \qquad T = -(B\,U + C\,\Omega),$$

with, for a helix whose tangent makes an angle $\alpha$ with the axis,

$$A = \Lambda\!\left(\zeta_\parallel\cos^2\!\alpha + \zeta_\perp\sin^2\!\alpha\right), \qquad B = \Lambda R \sin\alpha\cos\alpha\left(\zeta_\parallel - \zeta_\perp\right), \qquad C = \Lambda R^2\!\left(\zeta_\parallel\sin^2\!\alpha + \zeta_\perp\cos^2\!\alpha\right).$$

$A$ is the drag of a helix towed along its axis and $C$ is its resistance to being spun; both are positive sums of positive things and neither is interesting. Everything the organism gets is in $B$, which is the term that says a rotation produces an axial force, and it is proportional to $\zeta_\parallel - \zeta_\perp$ and to nothing else that can vanish.

The mechanism, said without algebra: each short piece of the filament is moving sideways, in a circle about the axis. Because it is tilted, that sideways motion is partly along its own length and partly across it. The fluid resists the across-part harder than the along-part, so the reaction on the filament is not antiparallel to its motion — it is tilted away from it, towards the filament’s own normal. Summed round the helix, the components in the plane of rotation cancel and the axial components do not.

If the two resistances were equal the reaction would be exactly antiparallel to the motion, there would be no tilt, and the axial components would be zero everywhere rather than cancelling. Computed that way — setting $\zeta_\perp = \zeta_\parallel$ and evaluating — the coupling term is identically zero, and so is the swimming speed of the whole organism.

It is worth being clear about what that thought experiment is and is not. An isotropic drag law is not a fluid anybody can make; it is the statement that the filament’s resistance tensor is a multiple of the identity, which for a slender body it is not and for a sphere it is. A helix made of beads that were each a sphere would not swim, and this is a real prediction rather than a rhetorical one: the beads would have to be spaced far enough apart not to see one another, and a chain of independent spheres turned about an axis produces no net axial force at all.

The other half of the same statement is that the coupling is largest when the tilt is largest, which is not at the extremes of pitch. $B$ carries a factor $\sin\alpha\cos\alpha$, so it vanishes for a straight filament along the axis — no radius to make a circle with — and for a ring perpendicular to it — every element moving exactly along its own length — and it is largest at forty-five degrees, where the two are balanced. The geometry contributes a factor that peaks in the middle; the material contributes a factor that cannot be changed. Everything a designer or an organism can do is contained in the first.

A ratio with a ceiling it cannot approach #

Since everything depends on the anisotropy, the natural question is how large it can be made, and the answer is a good example of a bound that is useless.

As $r \to 0$ the logarithm dominates both denominators and the ratio tends to $2$. That limit is quoted everywhere, and it is quoted as though it were nearly achieved. It is not:

$$\gamma = \frac{\zeta_\perp}{\zeta_\parallel} = \frac{2\left(\ln(2q/r) - \tfrac12\right)}{\ln(2q/r) + \tfrac12},$$

and inverting it for a stated $\gamma$ gives the logarithm, and thence the radius, in closed form. A ratio of 1.7 needs a radius of $2\times10^{-11}$ metres, which is a fifth of an atom. A ratio of 1.8 needs $2.6\times10^{-14}$, which is smaller than a proton. A ratio of 1.9 needs $5\times10^{-23}$ metres, which is not a length anything has.

So the honest statement is that the anisotropy is fixed at about five-thirds for every filament that exists, and it is not a parameter an organism can select. The design freedom in flagellar propulsion is entirely in the geometry — the radius, the pitch, the length — and not at all in the property that makes propulsion possible in the first place.

The matrix is symmetric, and that is not a convenience #

The same $B$ appears in the force equation and in the torque equation. It was not put there twice.

Computing the matrix by summing the drag on twenty thousand short elements of the helix — one calculation for a unit translation, a second for a unit rotation, with the axial force and the axial torque read off each — gives the off-diagonal entry twice, by two routes that share no arithmetic. They agree exactly: the difference is zero to the last bit, and the diagonal entries match the closed forms to three parts in $10^{13}$.

That symmetry is the reciprocal theorem in the form it takes for a rigid body. A Stokes flow’s resistance matrix is symmetric for the same reason a conservative system’s stiffness matrix is: it is the second derivative of a single quadratic functional, the dissipation. The physical statement is worth having because it is not obvious. The axial force produced by a unit of rotation is numerically equal to the axial torque produced by a unit of translation — so a helix dragged through a fluid spins at exactly the rate needed to make those two statements the same number, and a helix held against rotation and pushed feels a torque a measurement of the first experiment would have predicted.

It also guarantees, together with positivity, that the determinant $AC - B^2$ is positive at every pitch, which is what stops the arithmetic below producing a swimmer that extracts work from the fluid.

The pitch that is best, and the one bacteria use #

With the matrix in hand, a free swimmer is three lines. Attach a sphere of radius $r_b$ as a cell body, let the motor turn the filament at $\omega$ relative to it, and impose no net force and no net torque on the pair. The result is

$$U = \frac{-B\,C_b\,\omega}{(A + A_b)(C + C_b) - B^2},$$

with $A_b = 6\pi\mu r_b$ and $C_b = 8\pi\mu r_b^3$ the body’s own two resistances.

The speed peaks at fifty degrees and the efficiency at forty, and a bacterial flagellum’s tangent sits at about twenty-nine. That it lies below both optima is not a puzzle about efficiency. The sweep above holds the contour length fixed, which means a steeper helix is a shorter one along the axis; an organism that instead holds its body length fixed, or that must fit its filament past a cell wall, is optimising a different thing. What the curve does establish is that the optimum is broad — the speed is within ten per cent of its maximum over a range of twenty degrees — so a flagellum does not have to be precisely made.

One bacterium, end to end #

What the motor has to be able to do #

The same three lines that give the swimming speed also say what load the motor is working against, and the answer is not the one an engineer would guess.

Torque balance on the cell body alone says that the motor’s torque equals the body’s own rotational drag: $T = C_b\,|\Omega_{\text{body}}|$. So the motor is not primarily fighting the filament. It is fighting the body, through the filament, and the division of the motor’s rotation between the two is set by the ratio of their rotational resistances — the same kind of statement as twice as slippery along as across, read for rotation instead of translation. With the numbers above, the filament turns at 86 per cent of the motor’s rate and the body counter-rotates at 14 per cent, and the torque is whatever the body’s drag demands at that rate.

The effective rotational load the motor sees — its torque divided by its own rate — is $4.5\times10^{-22}$ newton-metre-seconds, which is smaller than either the body’s $C_b$ or the filament’s $C$ taken alone. That is the same sharing that makes two springs in series softer than either: the motor’s shaft is free to turn either end, and it turns whichever end is easier.

The consequence for the organism is a genuine design statement. A bacterium with a larger body has a larger $C_b$, turns its filament closer to the full motor rate, and swims faster for the same motor — up to the point where the body’s translational drag $A_b$ starts to dominate the denominator of the swimming speed. There is therefore a body size that swims fastest for a given filament and a given motor, and it is not the smallest one. The usual intuition that a microswimmer should minimise its drag gets the sign of one of the two terms wrong.

The computed swimming speed is 13.6 micrometres a second against a measured 25 to 30. That is a factor of two, and it is the honest accuracy of resistive force theory applied to a real cell: the coefficients assume an isolated filament in unbounded fluid, and a bacterium’s flagella are gathered into a bundle a few filament diameters apart, immediately behind a body several times their own thickness. Both corrections go the same way — a bundle experiences less drag per filament than an isolated one, and the body shields part of the flow — and both are known to be tens of per cent.

Three other numbers come out of the same three lines and are worth more than the speed, because nothing was tuned to produce them. The body counter-rotates at seventeen per cent of the flagellum’s rate, against a measured twenty or so; that number is a ratio of two resistances, $C_b$ against $C$, and it contains none of the filament’s drag coefficients at all. The motor torque is 283 piconewton-nanometres, which is the right order for a flagellar motor running near a hundred hertz. And the power delivered into the fluid is 0.18 femtowatts.

That last number is the one that decides how to read the efficiency. Two per cent sounds like a failure; 0.18 femtowatts is about one part in $10^5$ of a bacterium’s total metabolic rate. Propulsion is not what a bacterium spends its energy on, so the efficiency of its propeller is not a quantity evolution had much reason to optimise — which is consistent with a flagellum sitting twenty degrees off the best pitch and not caring.

The two per cent also has a floor underneath it that has nothing to do with the helix. At this Reynolds number thrust and drag are produced by the same mechanism — every part of the filament that pushes on the fluid is also being dragged by it — so there is no arrangement that separates them, and the Lighthill efficiency of any low-Reynolds swimmer is a few per cent. A fish reaches tens of per cent by leaving a jet behind, which requires inertia to carry the jet away, and inertia is exactly what is absent here. The propeller is not badly designed; the regime does not permit a good one, and the same statement in different words is the second-order smallness that makes a waving sheet inefficient too.

The sheet is the cleaner of the two comparisons, because it isolates what the anisotropy is doing. A plane transverse wave has no preferred sense of rotation, so nothing about it needs a counter-rotating body; in the helix the thrust and the torque are two entries of one matrix and cannot be had separately. That is the whole of why a bacterium spends seventeen per cent of its motor rate turning its own cell body backwards and a waving sheet spends none — and it is also why the sheet’s efficiency, small as it is, is not limited by anything a helix could borrow.

Three assumptions doing more work than the arithmetic #

Three approximations carry more of this calculation than any of its numbers do, and the first is the condition under which the drag law itself holds. Stokes’ law is the leading term of an expansion in the Reynolds number, and the correction to it is not quadratic but linear, so it dies away far more slowly than a first reading suggests — which is worth knowing before quoting a drag coefficient for anything that moves at a Reynolds number near one.

Every element of the filament is treated as though the rest of the filament were not there. Resistive force theory replaces the filament by a line of local drag coefficients and ignores the flow each element makes at the others. That is the largest error in the calculation, it is what the logarithm is a crude stand-in for, and slender-body theory — which keeps the interaction and is an integral equation rather than a formula — changes the coefficients by ten to twenty per cent.

There is one filament and a real cell has several. E. coli carries four to six, which coalesce into a rotating bundle when the motors turn the same way and fly apart when one reverses. A bundle is not a thicker filament and it is not several independent ones; the correct treatment needs the interaction the previous paragraph discards.

The body is a sphere and the filament is attached to nothing. A real cell is a rod, the hook joining motor to filament is flexible, and the flexibility matters: it is what lets a bundle form at all.

The filament is rigid. It is elastic, and at high enough motor torque it buckles — polymorphic transitions, in which the helix switches to a different pitch and handedness, are how the organism changes direction.

And nothing here is near a wall. A swimmer within a body length of a surface has an image system, swims in circles rather than straight lines, and is attracted to the surface — which is why bacteria accumulate on glass and why a measurement made in a thin chamber is not a measurement of free swimming.

Who found it, and when #

Gray and Hancock gave the resistive coefficients and the first calculation of flagellar propulsion in 1955, working on sea-urchin spermatozoa. Lighthill corrected the coefficients in 1976 — the logarithm’s argument and the half in the denominator are his — and gave the slender-body treatment that supersedes them. Purcell wrote the propulsion matrix in the form used here in 1997, in a short paper whose title, The efficiency of propulsion by a rotating flagellum, understates what it contains.

The surprising connection is with a result about a completely different kind of object, and it is the same anisotropy. A long rigid rod settling under gravity in a viscous fluid does not fall vertically unless it happens to be vertical or horizontal: its drag coefficients across and along differ, so the drag is not antiparallel to the velocity, and the rod drifts sideways as it sinks. The angle of that drift is set by the same ratio $\gamma$ and vanishes when $\gamma = 1$. A settling rod and a swimming bacterium are the same tilted reaction, once with gravity supplying the motion and once with a motor, and neither would happen in a fluid that resisted a thin filament equally in both directions.

Still open: what the bundle is worth #

The factor of two between the computed speed and the measured one is not a mystery, and it is not currently a number either.

Two corrections stand between them and they have opposite signs in the places it matters. A bundle of four filaments turning together, each a few diameters from the next, has a lower drag per filament than an isolated one, because each sits in the flow the others make — which raises the swimming speed for a given torque. The cell body immediately ahead of the bundle shields part of that flow and adds a drag the sphere model already counts once, which lowers it. Neither is small, and their sum has been computed in full only by numerical methods that give a number rather than a dependence.

The calculation that would say something general is the propulsion matrix of a bundle as a function of filament spacing, with the interaction kept at the level of the Oseen tensor and nothing else — which is the same approximation the three-sphere swimmer is solved with, applied to a different geometry. The two questions it would answer are how tightly a bundle has to be wound before it behaves as a single thicker filament, and whether the spacing bacteria actually maintain is near the optimum of that curve or merely near whatever the hooks allow.