### Swimming driven by a Helical Flagellum

The simulation codes used to create the numerical data described below may be downloaded here. The same files are also available from the Mathworks File Exchange by searching for the “Helical Swimming Simulator.”

The swimming of a bacterium or a biomimetic nanobot driven by a rotating helical flagellum is often interpreted using the resistive force theory developed by Gray and Hancock (1955) and Lighthill (1976), but this theory has not been tested for a range of physically relevant parameters. We test resistive force theory in experiments on macroscopic swimmers in a fluid that is highly viscous so the Reynolds number is small compared to unity, just as for swimming microorganisms. The measurements are made for the range of helical wavelengths $\lambda$, radii $R$, and lengths $L$ relevant to bacterial flagella. The experiments determine thrust, torque, and drag, thus providing a complete description of swimming driven by a rotating helix at low Reynolds number. Complementary numerical simulations are conducted using the resistive force theories of Gray and Hancock (1955) and Lighthill (1976), the slender body theories of Lighthill (1976) and Johnson (1980), and the regularized Stokeslet method of Cortez et al. (2005).

## Bacterial Flagella

Movie frame showing swimming Rhodobacter Spheroides Rowland Institute at Harvard (http://www.rowland.harvard.edu/labs/bacteria).

Many microorganisms use a rotating helical flagellum for propulsion such as the Rhodobacter Spheroides shown in the image above. Because of their small size, microorganisms swim in a very low Reynolds number $\left ( Re \equiv \frac{UL}{\nu}<<1\right )$ environment, where $U$ is a typical velocity, $L$ is a typical length scale and $\nu$ is the kinematic viscosity of the fluid. The fluid dynamics at low Reynolds number are very different from those we experience in our everyday lives. Gray and Hancock (1955) and Lighthill (1976) developed resistive force theory to describe swimming at low Reynolds number because numerical techniques to calculate the forces and torques on a swimming microorganism were not feasible at that time.

Resistive force theory continues to be used in a variety of contexts. The theory is used to interpret propulsion by a planar wave in sperm (Friedrich (2010)), small worms (Sznitman (2010)), Chlamydomonas reinhardtii (Bayly (2011)), and swimmers in a granular material (Maladen et al. (2011)); however, rather than using the drag coefficients given by Gray and Hancock or Lighthill, the coefficients are usually adjusted to fit the observations. Using drag coefficients as free parameters to fit experimental data to resistive force theory has never been proven to be a valid experimental technique. In fact, our analysis shows that whether using drag coefficients from fitting the experimental data or analytic expressions for the drag coefficients resistive force theory fails to properly describe the dynamics of a rotating helical flagellum. This failure results from a fundamental assumption in resistive force theory that long range hydrodynamic interactions can be ignored as we show below.

## Flagellum Schematic

We model a bacterial flagellum as a rigid helix rotating about the helical axis. We are therefore ignoring the effects of the body of a bacterium and any flexibility in the flagellum. However, experiments have shown that a bacterial flagellum has essentially a fixed geometry when used for propulsive motion (Darnton et al. (2007)) though the flagellum may change its shape when the bacterium reverses its motor direction or due to environmental conditions.

Schematic of a helical flagellum (axis in the $\hat{x}$ direction) with radius $R$, pitch $\lambda$, axial length $L$, filament radius $a$, contour length $\Lambda=L/\cos\theta$, and pitch angle $\theta$, where $\tan \theta=2\pi R/\lambda$. The inset shows a filament segment of length $ds = dx/\cos \theta$. The segment’s tangential unit vector is $\hat{t}(x)= \left[\cos \theta,\,-\sin \theta\sin\phi,\, \sin \theta\cos\phi\right]$, where $\phi = 2\pi x/ \lambda$ is the phase angle of the helix.

## Swimming with a Flagellum at Low Reynolds Number

At low Reynolds number a rotating flagellum exerts an axial thrust $\mathcal{F}$ and torque $\tau$ related to the flagellum’s axial velocity $U$ and rotation rate $\Omega$ by (Happel and Brenner (1965), Kim and Karilla (1991)):

$\left(\begin{array}{ccc}\mathcal{F}\\ \tau\\ \end{array} \right) = \left(\begin{array}{ccc}A_{11} & A_{12} \\ A_{12} & A_{22}\\\end{array} \right) \cdot \left(\begin{array}{ccc}U \\ \Omega \\ \end{array} \right).$

The symmetric 2 X 2 in the above matrix depends only on the geometry of the flagellum. The elements of the propulsive matrix can be determined by measuring the axial thrust $F$ and torque $T$ for a rotating non-translating flagellum, and the axial drag $D$ on a translating non-rotating flagellum: $D=A_{11}U, \,F=A_{12}\Omega, \& T=A_{22}\Omega$. We determine these matrix elements using experiments and numerical simulations and compare them to resistive force theory predictions for these matrix elements

## Resistive Force Theory

Resistive force theory obtains the total force and torque for motion of a flagellum by integrating the local forces on each small segment (see inset of Schematic above). The local forces are calculated using drag coefficients per unit length in the directions normal and tangential to the segment, $C_n$ and $C_t$, respectively. Resistive force theory predicts that the thrust, torque, and drag on a flagellum are given by
\begin{aligned} F&=\left(\Omega R\right)\left(C_{n}-C_{t}\right)\sin\theta\cos\theta\frac{L}{\cos\theta}\\T&=\left(\Omega R^{2}\right)\left(C_{n}\cos^{2}\theta+C_{t}\sin^{2}\theta\right)\frac{L}{\cos\theta}\\D&=U\left(C_{n}\sin^{2}\theta+C_{t}\cos^{2}\theta\right)\frac{L}{\cos\theta} \end{aligned}

Two sets of drag coefficients are commonly used in the literature, those by Gray and Hancock (1955) and those by Lighthill (1976); both are based on slender body theory, assuming that the effect of each small filament segment is only locally important. Gray and Hancock’s drag coefficients are

$C_{t}=\frac{2\pi\mu}{\ln\frac{2\lambda}{a}-1/2} \text{ }\text { and } \text{ } C_{n}=\frac{4\pi\mu}{\ln\frac{2\lambda}{a}+1/2}$

and Lighthill’s are

$C_{t}=\frac{2\pi\mu}{\ln\frac{0.18\lambda}{a\cos\theta}} \text{ }\text{ and } \text{ } C_{n}=\frac{4\pi\mu}{\ln\frac{0.18\lambda}{a\cos\theta}+1/2}.$

These drag coefficients are used in resistive force theory equations for the matrix elements for comparison with experiments.

## Experiments

The image shows a mechanical bacterium made with a wire helix and immersed in silicone oil with viscosity 100,000 times the viscosity of water so that it is Reynolds number-matched to bacteria. $Re\sim 10^{-2}$

Experiments were performed in an 80-liter tank (520 mm X 495 mm X 330 mm high) filled with silicone oil (Clearco\textsuperscript{\textregistered}) of density $970\, kg/m^3$ and dynamic viscosity (at $25^\circ \,C$) $\mu = 1.0 \times 10^{2}\, kg/(m\cdot s)$, about \$100,000 times that of water. Model flagella were constructed from initially straight pieces of type 304 stainless steel welding wire with radius 0.397 mm. The wire was wrapped around aluminum mandrels with helical V-shaped grooves of varying pitches on rods with radius 6.4 mm and lengths of either 152.5 or 305 mm. Measurements were made with each flagellum’s axis about 250 mm from the tank walls to minimize boundary effects. The ends of a flagellum were kept at least 100 mm from the wall boundaries and from the free surface. To examine possible changes of the radius and wavelength of the wire helices under load, we analyzed videos for different rotation rates and for different rotation directions, and the change in wavelength and radius was smaller than the measurement uncertainty.

## Simulations

We compute the propulsive matrix elements using a technique developed by Cortez et al. (2005), who used a “regularized” Stokeslet, which is an approximate point force, $\mathbf{f\phi_{\varepsilon}\left(r\right)}$, where the radial cutoff function $\mathbf{\phi_{\varepsilon}}$ avoids singular, non-integrable kernels in numerical simulations. We also compute the propulsive matrix using the slender body theory approximation to describe flagellar swimming, as developed by Lighthill (1976) and Johnson (1980). Slender body theory uses Lorentz’s (Lorentz (1976)) result that the far field fluid response to a moving sphere can be represented by a Stokeslet and a source dipole (doublet) of the same strength at the center of the sphere. In this approach Stokeslets and doublets are arranged along the axis of the flagellum, and the force per unit length on a flagellum is obtained by inverting the integral equation relating the velocity of the flagellum’s center line and the strength of the Stokeslets and doublets. The difference between the two methods is shown graphically above.

## Comparison of Theory, Simulation, and Data

The experimental results differ qualitatively and quantitatively from the predictions of resistive force theory. The difference is especially large for $\lambda < 6R$ and/or $L > 3\lambda$, parameter ranges common for bacteria. In contrast, the predictions of Stokeslet and slender body analyses agree with the laboratory measurements within the experimental uncertainty (a few percent) for all $\lambda$, $R$, and $L$. Our code implementing the slender body, regularized Stokeslet, and resistive force theories will be made available on the MATLAB file exchange; thus readers can readily compute force, torque and drag for any bacterium or nanobot driven by a rotating helical flagellum.

## Discussion

Our experiments reveal that resistive force theory fails to provide an accurate description of low Reynolds number swimmers driven by a rotating helical flagellum for helices with $\lambda 3\lambda$, which is the range relevant to bacteria. The measured values of thrust, torque, and drag differ significantly from the predictions of resistive force theory with either Gray and Hancock’s or Lighthill’s drag coefficients. The measurements were made for values of helical pitch $\lambda$ and length $L$ that include the ranges relevant to bacteria.

Hancock (1953) pioneered the analysis of swimming microorganisms more than a half century ago. His “slender body model” yielded predictions in terms of integrals that could not be numerically evaluated at that time. Then Gray and Hancock (1955) developed “resistive force theory,” which yielded expressions for thrust, torque, and drag in terms of the normal and tangential resistance coefficients.

Resistive force theory was subsequently reanalyzed by Lighthill (1976). He computed zero-thrust free swimming speed (Eq. 57 in his paper) for “an infinitely long flagellum” from his slender-body theory. He then used this computed zero-thrust swimming speed and expressions for the force distribution as “experimental data” to calibrate resistive force theory and got “improved” expressions for drag coefficients. Thus one can use Lighthill’s resistive force theory coefficients to predict the free speed for a helical flagellum without a cell body, but the theory fails when the cell body produces significant drag (non-zero thrust) (Chattopadhyay (2008)).

The primary advantage of resistive force theory has been its computational simplicity; the results are given in terms of algebraic expressions without the necessity of numerical integration. But numerical integration is now straightforward on desktop computers. We have developed numerical implementations of regularized Stokeslet theory and two versions of slender body theory, which were used to produce the curves in the figure above. These algorithms (available here) can be used by a reader to compute thrust, torque, and drag for any rotating helix with parameters $R$, $\lambda$, $L$, and $a$.

In conclusion, we have shown that slender body theory and regularized Stokeslet theory predictions are in good accord with measurements on low Reynolds number swimmers driven by a rotating helix. This work indicates that it should also be straightforward to apply slender body theory to microscopic undulating swimmers, whose motion is often interpreted using resistive force theory.

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