Rotational dynamics

Rotational dynamics of a superhelix towed in a Stokes fluidSunghwan JungApplied Mathematics Laboratory, Courant Institute of Mathematical Sciences, New York University,251 Mercer Street, New York, New York 10012, USA

Kathleen MareckApplied Mathematics Laboratory, Courant Institute of Mathematical Sciences, New York University,251 Mercer Street, New York, New York 10012, USA and Department of Mathematics, Tulane University,6823 St. Charles Avenue, New Orleans, Louisiana 70118, USA

Lisa FauciDepartment of Mathematics, Tulane University, 6823 St. Charles Avenue,New Orleans, Louisiana 70118, USA

Michael J. ShelleyApplied Mathematics Laboratory, Courant Institute of Mathematical Sciences, New York University,251 Mercer Street, New York, New York 10012, USA

Received 24 June 2007; accepted 24 September 2007; published online 26 October 2007

Motivated by the intriguing motility of spirochetes helically shaped bacteria that screw throughviscous fluids due to the action of internal periplasmic flagella, we examine the fundamental fluiddynamics of superhelices translating and rotating in a Stokes fluid. A superhelical structure may bethought of as a helix whose axial centerline is not straight, but also a helix. We examine theparticular case in which these two superimposed helices have different handedness, and employ acombination of experimental, analytic, and computational methods to determine the rotationalvelocity of superhelical bodies being towed through a very viscous fluid. We find that the directionand rate of the rotation of the body is a result of competition between the two superimposed helices;for small axial helix amplitude, the body dynamics is controlled by the short-pitched helix, whilethere is a crossover at larger amplitude to control by the axial helix. We find far better, and excellent,agreement of our experimental results with numerical computations based upon the method ofRegularized Stokeslets than upon the predictions of classical resistive force theory. 2007 American Institute of Physics. DOI: 10.1063/1.2800287

I. INTRODUCTION

The study of swimming micro-organisms, including bac-teria, has long been of scientific interest.13 Bacteria swim bythe action of rotating, helical flagella driven by reversiblerotary motors embedded in the cell wall.2 Typically, theseflagella visibly emanate from the cell body. The external fla-gella of rod-shaped bacteria, such as E. coli, form a coherenthelical bundle when rotating counterclockwise, causing for-ward swimming. When these flagella rotate in the oppositedirection, the flagellar bundle unravels, causing the cell totumble. This run and tumble mechanism allows a bacteriumto swim up a chemoattractant gradient as it senses temporalchanges in concentration.4,5 Many studies have focused onthe fundamental fluid mechanics surrounding this locomo-tion affected by a simple helical flagellum attached to andextruded from the cell body.3,6 Recently, there have beenadditional studies that investigate the hydrodynamics offlagellar bundling.7,8

In contrast, swimming bacteria with more complicatedbody-flagella arrangements are less studied. Spirochetes aresuch a group of bacteria. They have a helically shaped cellbody,9,10 and although they also swim due to the action ofrotating flagella, these do not visibly project outward fromtheir cell body. Instead, the cell body is surrounded by an

outer sheath, and it is within this periplasmic space that ro-tation of periplasmic flagella PFs occurs. These helicalperiplasmic flagella emanate from each end of the cell body,but rather than extend outwards, they wrap back around thehelical cell body. In the case of Leptospiracaeae, there aretwo PFs, one emerging from each end of the cell body, whichdo not overlap in the center of the cell. Rotation of eachflagellum is achieved by a rotary motor embedded in the cellbody. The shapes of both ends of the helical cell body arethen determined by the intrinsic helical structure of the peri-plasmic flagella, as well as their direction of rotation. Duringforward swimming, L. illini exhibit an anterior region that issuperhelical, due to this interplay of helical cell body andhelical flagellum.10 In fact, the handedness of these two he-lical structures are opposite, with the flagellum axial helixexhibiting a much larger pitch than the cell body helix. Fig-ure 1 shows a photograph of the spirochete L. illini with ananterior superhelical region at the left.

The overall swimming dynamics of spirochetes involvesnonsteady coupling of the complex geometry of the cellbody, the flexible outer sheath, and the counter-rotation ofthe cell body with the internal flagella. However, a naturalquestion is how the effectiveness of spirochete locomotiondepends upon the detailed superhelical geometry of the an-terior region of the bacterium. With this as motivation, we

PHYSICS OF FLUIDS 19, 103105 2007

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present here a careful study of the fundamental fluid mechan-ics of superhelical bodies translating and rotating throughhighly viscous fluids. We extend the classical analytic andexperimental results of Purcell6 and the numerical results ofCortez et al.11 performed for regular helices. In addition, weoffer coordinated laboratory and computational experimentsas validation of the method of Regularized Stokeslets forzero Reynolds number flow coupled with an immersed, geo-metrically complex body. This method uses modified expres-sions for the Stokeslet in which the singularity has been mol-lified. The regularized expression is derived as the exactsolution to the Stokes equations consistent with forces givenby regularized delta functions.

We focus on a typical body that is a short-pitched helixwhose axis is itself shaped as a helix of larger pitch andopposite handedness. In the following sections, we describethe experimental setup as well as the construction of thesesuperhelical bodies. We experimentally measure the rota-tional velocities of the bodies as they are towed with a con-stant translational velocity through a very viscous fluid. Notethat rotational and translational velocities should be propor-tional, with the constant of proportionality ratio of resis-tance coefficients dependent upon the body geometry. Therotational velocities corresponding to translational velocitiesare also predicted analytically using resistive force theory, aswell as using the method of Regularized Stokeslets.11,12 Wefind compatible behavior between experiments and the resis-

tive force theory, but excellent quantitative agreement be-tween experiments and the method of Regularized Stokes-lets.

II. SUPERHELIX CONSTRUCTION

A superhelix is formed from a copper wire chosen to besufficiently malleable to deform into a desired shape, butrigid enough not to deform as it moves through the viscousfluid. The superhelix is made in two steps see Fig. 2b.First, a copper wire of diameter 0.55 mm is wound tightly ina clockwise direction up a rod of diameter 3.15 mm, forminga tight coil. After removing the coil from the rod, we stretchit out into a smaller radius, larger pitch helix, simply bypulling the ends of the coil away from each other. The axialhelix is made in the same manner, but we use lead wire of athicker diameter 3.15 mm and a larger rod 4.7 mm diam-eter. The most important difference between the two helicesis handedness; the axial helix is wound counterclockwise upthe rod, whereas the small helix is wound clockwise. Oncethe parameters of the small and axial helices are measured,the axial helix is threaded through the small helix, forming asuperhelix; i.e., the small helix is placed back on a rod thathas been distorted into a helical shape. The last step is toremove the axial helix. This is done by simply rotating theaxial helix while keeping the superhelix fixed.

The defining geometric parameters of the superhelix arethe radius r and pitch p of the small helix, and the radius Rand pitch P of the axial helix see Fig. 2. At the extremevalues of pitch for the small helix p=0 and p=, the su-perhelix reduces to a regular helix. Similarly, for R=0 orP=, the superhelix reduces to a regular helix. The super-helix construction described above requires the removal of athin helical wire from the larger lead wire. This procedurepresents difficulties for large values of r and small values ofp. For this reason, we limit our experiments to two differentsets of small helices. The corresponding geometric param-eters of these small helices are the pitch 5.580.25 mmfor set I and 5.040.36 mm for set II and radius

FIG. 1. Photograph of L. illini. Note the superhelical anterior region at theleft. We thank Professor S. Goldstein, Dept. of Genetics, Cell Biology andDevelopment, University of Minnesota for providing this image.

FIG. 2. a Schematic of experimental setup. A motorpulls a rigid body through silicon oil, a highly viscousStokes fluid =10 000 cSt. b Procedure for makingsuperhelix.

103105-2 Jung et al. Phys. Fluids 19, 103105 2007


1.910.14 mm for set I and 1.750.21 mm for set II. Thesmall less than 12% variations of pitch and radius are pre-sumably due to mechanical relaxation of material when it ispulled off the axial helix. Seven different axial helices areprepared from the same initial coil see Fig. 3.

We now construct a mathematical representation ofthe superhelix. The coordinates of an axial helix areX= R cosKz ,R sinKz ,z, where K=2 / P. The distancemeasured along this helix is linearly proportional to theaxial distance z=s. The unit vector tangential to theaxial helix is tA=X /s. The principal normal vector isnA= cosKz ,sinKz ,0 and the binormal is bA= tA nA=sinKz ,cosKz ,RK. Since tA is a unit vector, we set as

2R2K2 + 1 = 1. 1

The coordinates of the one-dimensional curve describingthe superhelix are

Rs = Rx,Ry,Rz = Xs + r cosksnA + r sinksbA. 2

Recall that the actual superhelices have nonzero thicknessthe diameter of the copper wire, and hence are true three-dimensional structures.

III. EXPERIMENT

The classical experiments of Purcell, elaborated on inRef. 6, examined the relationship between angular and trans-lational velocities of helical objects at very low Reynoldsnumbers. Here we extend these experiments to the superhe-lical objects described above. The experimental setup wasoriginally designed for sedimentation experiments13 see Fig.2a. A tall transparent container is filled with silicone oilwith large viscosity =104 cS, =0.98 g/cm3. The oil be-haves as a Newtonian fluid in the regime of interest here.Rather than allowing the superhelical object to descend bygravity, our experiment is designed to measure its rotationalspeed as it is towed up through the viscous column of fluid at

a specified translational speed. To drag the superhelix, asmall hook 2 mm is used to attach the superhelix to athread from a motor. Note that the dimensions of this hookare quite small compared to the superhelix length 4 cm.By experimentally testing with an axisymmetric bodysphere, we found that this towing system the thread plusthe motor, does not produce any torque on the body.

The superhelix is initially positioned near the bottom ofthe container, and then is dragged upwards by the motorClifton Precision-North at constant speed. In the interme-diate region in the container, steady state motion constanttranslational velocity, rotational velocity, and drag force isassumed. The superhelix positions, orientations, and veloci-ties are measured from a 30 frames per second video streamof the camcorder. The translational velocity in our experi-ments varies by changing power input to the motor. We havechosen a velocity range of 310 cm/s. Below 3 cm/s, thestep motor produces nonuniform pulsed axle rotations, whichlead to irregular translational velocity. The Reynolds numberbased upon the towing velocity and radius of the superhelicalstructure 1 cm is at most

Re =UR

0.1. 3

We assume therefore that the steady Stokes equations governthe fluid mechanics of the translating superhelix. Within thistranslational velocity range, a linear relationship between ro-tational velocity and translational velocity U is observedsee Fig. 4.

A translating helix in a viscous solution rotates in thedirection in which it screws. Following this rule, the smallstraight helix in our experiments would rotate clockwiseand the axial straight helix would rotate counterclockwisewhen viewed from above. In Purcells work,6 the jointedstructure built by connecting two helices of opposite hand-edness, otherwise identical, showed no rotation during itssedimentation. The superhelix of interest here is the super-position of two helices with opposite handedness. The inher-ent rotational directions of these superimposed helices are incompetition. For very small values of the nondimensionalparameter RK of the axial helix, the superhelical structure

FIG. 3. Seven superhelices with increasing axial helix radius from left toright. The upper panel is the side view and the lower panel is the axial view.Radius R and wavenumber K of the axial helix. As the initial coil ispulled apart, the radius and wavenumber of the axial helix increase. Thisrelation can be predicted by the simple scaling relation with inextensibilityof a wire.

FIG. 4. A linear relationship between the translational velocity and rota-tional frequency of a superhelix r=0.89 mm, R=4.62 mm, p=5.5 mm, andP=19.4 mm. Triangles are from experimental observations. Dashed line isa least-squares fit of experimental data.

103105-3 Rotational dynamics of a superhelix Phys. Fluids 19, 103105 2007


reverts to the straight small helix, and would rotate clock-wise. One expects that for larger values of the parameter RK,the axial helix would be dominant, and the superhelicalstructure would rotate counterclockwise. For some criticalvalue of RK, we would expect a transition in direction, andhence, a structure that would show no rotation as it is towedthrough the fluid. We performed experiments that systemati-cally varied RK, and observed this expected change in rota-tional direction. Figure 5 shows the ratio of angular velocityto translational velocity as a function of RK, for the twodifferent sets of superhelices sets I and II. Positive rota-tional rate is clockwise, and negative is counterclockwise. Ineach set of experiments, the measured ratio is depicted bytriangles. Note that each of these data points is arrived at byaveraging the results of about ten realizations of the towingexperiment for each superhelix. The experimental error,based upon the standard deviation, is at most five percent. Inthe next sections, we describe mathematical formulationsthat model these observations.

IV. NUMERICAL RESULTSA. Regularized Stokeslets

We assume that the superhelix is a rigid body moving ina Stokes fluid. The governing equations of motion are

p + 2u = 0, u = 0. 4

The total hydrodynamic force and torque exerted by the su-perhelix with surface D on the surrounding fluid is

F = xD

fxdx , 5

L = xD

x x0 fxdx , 6

where f is the surface traction.A solution to the Stokes equations in three dimensions

3D with a point force centered at x0 is the classicalStokeslet.14 Due to the linearity of the Stokes equations, su-perposition of these fundamental solutions allows the con-struction of the velocity field induced by a distribution ofpoint forces. The method of Regularized Stokeslets eases theevaluation of integrals with singular kernels by replacing thedelta distribution of forces by a smooth, localizeddistribution.11,12 The force f= fx0xx0 is replaced byf= fx0xx0, where is a cutoff, or blob, function withintegral 1. This blob function is an approximation to the 3DDirac delta function, with a small parameter. FollowingRef. 11, we choose

x x0 =154

8x x02 + 27/2. 7

For N regularized point forces distributed on the surface of abody in rigid rotation and translation, the fluid velocity atany point x is evaluated as

8uix = jn=1

N

Sij x,xnf jxn . 8

For the given cutoff function, the kernel S is

Sij x,xn = ij

r2 + 22

r2 + 23/2+

xi xn,ixj xn,jr2 + 23/2

, 9

where r= xxn.Note that evaluating Eq. 8 at each of the N points of

the superhelix surface gives us a linear relation between thevelocities and the forces exerted at these points. The matrixSij for a given cutoff parameter depends only upon the

geometry of the superhelix.For a rigid body moving in a Stokes flow, there is a

linear relationship between the total hydrodynamic force andtorque and the translational and rotational velocity of thebody.6 Following Refs. 6 and 11, we focus on the z compo-nents of total hydrodynamic force F and torque L, along withthe z component of translational velocity U, and rotationalvelocity about the z axis . These are related by resistanceor propulsion coefficients

FL = A BB D U . 10Here, A, B, and D depend only upon the geometry of theobject.

In order to compute these coefficients, we describe thesuperhelix by a discrete set of points. The discrete points ofthe superhelix lie on its surface, and not along the centerline.The diameter of the superhelical wire is a free parameter ofthis model. Here, each circular cross section of the copperwire is approximated by a hexagon, with six azimuthal gridpoints. We choose a cutoff parameter on the order of thedistance between discrete points see Ref. 11 for details. At

FIG. 5. Ratio of angular velocity to translational towing velocity. Trianglesare values measured experimentally. Circles connected by lines are valuespredicted using the method of Regularized Stokeslets. Squares are valuespredicted using resistive force theory. a Superhelices of set I. Positiverotational rate is clockwise and negative rate is counterclockwise. Rotationaldirection changes around 0.7 for RK. b Superhelices for set II. Same tran-sition also occurs around 0.7 for RK.



each point on this discretized superhelix, we impose a unittranslational velocity and zero rotational velocity in the zdirection. We use the Regularized Stokeslet linear relation8 to solve for the forces on the superhelix that producedthis velocity. We then evaluate the integrals for total force Fand total torque L in Eqs. 5 and 6 above. Using the linearequations 10, we compute the resistance coefficientsA and B. Similarly, we can compute D by imposing a unitrotational velocity and zero translational velocity.

These resistance coefficients allow us to predict the ratioof rotational velocity to translational velocity in our torque-free experiments described above, as

U=

BD

. 11

Figure 5 shows the Regularized Stokelet predictions circlesof these ratios for both sets of superhelices. The agreementwith experimental data is excellent. Indeed, the transitionfrom clockwise to counterclockwise rotation is captured veryprecisely.

B. Resistive force theory

Resistive force theory15,16 is widely used to give an ap-proximate description of a slender body moving in a viscousfluid. However, the nonlocal interactions of stress along thebody are not taken into account. To see how important thisnonlocal interaction is, we estimate analytically the ratio ofangular to translational velocities in this section, and com-pare the predictions with experiment and the numerical cal-culations using Regularized Stokeslets.

The Stokes drag force is proportional to its velocity asf=Ctu tt+Cnu nn, where t and n are tangential and nor-mal directions, respectively, and Ct and Cn are drag coeffi-cients. The normal direction is arbitrary, but is uniquely de-termined if the motion is given.

First, we consider pure body-rotation about the z axis.The body velocity is u=R=Ry ,Rx ,0=Ru,where R= Rx ,Ry ,0, R= R, u=R z, and =z. Theangle between the direction of motion and the tangentialof the body is expressed by the superhelix coordinates asu t=cos = RysRxRxsRy /R, where t=Rs. Similarly,u n=sin .

Using the vector relation AB C=AC B, thez-component torques associated with forces are

Lzs = R f z = Ctu tR t z + Cnu nR n z

= Ctu tR z t Cnu nR z n

= R2 Ct cos2 + Cn sin2 , 12

and the total torque due to its rotational motion is

LzRotation

= Lzsds= R2 Ct cos2 + Cn sin2 ds . 13

Similarly, the total torque associated with the pure transla-tion, i.e., u= 0,0 ,U=Uz, is

LzTranslation

= U Cn CtsRzR cos ds . 14Decoupling the body motion into a pure rotation and a puretranslation, the z component of total torque on the body isexpressed as

Lz = LzTranslation + Lz

Rotation. 15

In our experiments, we do not apply any external torque.Balancing two torques gives an expression for the ratio ofangular velocity to translational velocity as

LzTranslation

= LzRotation

U

=

Cn CtsRzR cos ds R

2 Ct cos2 + Cn sin2 ds. 16

Note that this ratio of velocities depends only upon the ratioof drag coefficients Cn /Ct. Evaluating these integrals nu-merically, and using R and Cn=2Ct the leading order resultof slender-body theory, /U is plotted as squares in Fig. 5.

The slope B /D in relation 11 is measured in experi-ments by using a least-squares fit of the data see Fig. 4.Although the resistive force theory predictions of /U showthe same general trend as the experimental measurements,the quantitative agreement is quite poor, and the transitionfrom clockwise to counterclockwise rotation is not captured.This discrepancy is likely due to the omission of nonlocalinteractions in the resistive force theory formulation. Thesenonlocal interactions become more important for larger val-ues of RK.

V. CONCLUSION

In conclusion, we have studied the rotational dynamicsof superhelices built out of superimposed helices of oppositehandedness in a viscous fluid. As the radius of the axial helixincreases, we have observed the transition of rotational di-rection to the natural direction of rotation of the axial helix.The rotational velocities corresponding to translational ve-locities are also predicted semi-analytically using resistiveforce theory, as well as numerically, using the method ofRegularized Stokeslets.11,12 We see that although there isqualitative agreement between experiments and resistiveforce theory, for larger radii of the axial helix, the predictedratios of rotation to translational velocities differ from ex-perimental results by more than 100%, and show the incor-rect direction of rotation. In contrast, the easily implementedcomputational framework of Regularized Stokeslets demon-strates excellent quantitative agreement with experiments.

103105-5 Rotational dynamics of a superhelix Phys. Fluids 19, 103105 2007


Our numerical results based upon Regularized Stokesletsassumed that the superhelix was towed through an un-bounded three-dimensional fluid. Since there is such closeagreement between our laboratory experiments and the com-putations, the effects of the container walls seem to be neg-ligible. In previous sedimenting experiments, in which theratio of body size to container width was also small, a similarconclusion was reached.13 However, the motility of micro-organisms in confined geometries is of biological interest,and we plan to investigate wall effects by performing thesetowing experiments in smaller containers.

While this study is motivated by the fascinating geom-etry of spirochetes, we recognize that the rotation of the an-terior superhelix of a spirochete is not due to an imposedrotation about a vertical axis, but due to the counter-rotationof the periplasmic flagellum which determines the axial he-lix, against the cell body the small helix. The fluid me-chanic implications of this counter-rotation that governs spi-rochete motility will be the subject of future work.

ACKNOWLEDGMENTS

We thank J. Zhang, E. Kim, A. Medovikov, R. Cortez,and S. Goldstein for helpful discussions. This work waspartially supported by the DOE Grant No. DE-FG02-88ER25053, and by NSF Grant Nos. DMS 0201063 andDMS-0412203.

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Rotational dynamics