Coiled to Diffuse: Brownian Motion of a Helical BacteriumAlexander V. Butenko,† Emma Mogilko,† Lee Amitai,‡ Boaz Pokroy,*,‡ and Eli Sloutskin*,†

†Physics Department and Institute for Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel‡Department of Materials Engineering and the Russell Berrie Nanotechnology Institute, Technion Israel Institute of Technology,Haifa, Israel

*S Supporting Information

ABSTRACT: We employ real-time three-dimensional con-focal microscopy to follow the Brownian motion of a fixedhelically shaped Leptospira interrogans (LI) bacterium. Weextract from our measurements the translational and therotational diffusion coefficients of this bacterium. A simpletheoretical model is suggested, perfectly reproducing theexperimental diffusion coefficients, with no tunable parame-ters. An older theoretical model, where edge effects areneglected, dramatically underestimates the observed rates of translation. Interestingly, the coiling of LI increases its rotationaldiffusion coefficient by a factor of 5, compared to a (hypothetical) rectified bacterium of the same contour length. Moreover, thetranslational diffusion coefficients would have decreased by a factor of ∼1.5, if LI were rectified. This suggests that the spiralshape of the spirochaete bacteria, in addition to being employed for their active twisting motion, may also increase the ability ofthese bacteria to explore the surrounding fluid by passive Brownian diffusion.

■ INTRODUCTIONThe Brownian motion, wherein small particles suspended in afluid undergo random translations and rotations, is one of theoldest fields in modern physics. While first scientific experi-ments in this field date back to the early 19th century,1 activeexperimental2−5 and theoretical6−9 research in this fieldcontinues. In particular, even for the simplest nonsphericalparticles, such as rods,10−13 tetrahedra,14,15 and ellipsoids ofrevolution,2 the full understanding of diffusive Brownianmotion has only recently been achieved. However, many ofcommon molecules, viruses, and cells have rather complexshapes, such that both the rotational symmetry and the chiralsymmetry are broken. Typically, when the diffusion rate ofthese complex objects is to be estimated, their shape isapproximated by that of an ellipsoid,16 which is a crudeapproximation. The Brownian motion of chiral objects, such asa simple circular helix,17 was never quantitatively studied by adirect experimental technique.18,19

We employ direct confocal microscopy to follow theBrownian motion of a circular helix, directly, in three spatialdimensions. As a circular helix, we employ a fixed Leptospirainterrogans (LI) bacterium, fluorescently labeled for confocalstudies, and suspended in a density-matched solvent, to avoidany observable sedimentation on the experimental time-scales.These Gram-negative obligate aerobe spirochetes, which causethe Leptospirosis disease in humans and other mammals,20

have a helical right-handed structure, such that their chiralsymmetry is broken. We use the experimental trajectories ofthese helical bacteria to directly measure the rotational and thetranslational diffusion coefficients of a circular helix; the valuesof these coefficients are then compared with the theoreticalpredictions.

■ EXPERIMENTAL SECTIONShape Characterization. To measure the shape of our bacteria at

high resolution, inaccessible by confocal microscopy, we obtainscanning electron microscopy (SEM) images of the LI. All theinformation regarding culture and harvesting of the LI is availableelsewhere.21 The fixed LI were transferred, for SEM imaging, fromwater to ethanol, then to hexane, and deposited from hexane onto analuminum surface. The helical right-handed shape of the bacteria isclearly visible, as shown in Figure 1a. The spirochaete bacteria, such asthe LI, have their flagella running lengthwise between the bacterialinner membrane and outer membrane; these flagella, covered by the

Received: May 20, 2012Revised: August 12, 2012Published: August 14, 2012

Figure 1. (a) Scanning electron microscope image of a typicalLeptospira interrogans bacterium. Note the right-helical shape of thisbacterium and the typical hook-shaped ends. The image was obtainedat 3 keV, with the bacterium coated by an evaporated gold layer. Somebacteria are more straight (see inset, where Ls is defined). (b) Pictorialdefinition of r, b, and d, which describe the shape of our LI bacterium.

Article

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© 2012 American Chemical Society 12941 dx.doi.org/10.1021/la302056j | Langmuir 2012, 28, 12941−12947

outer membrane, cannot be observed by SEM. The flagella determinethe shape of these bacteria. In particular, they are responsible forimparting a hook-shaped end to the cell during periods of directionalswimming activity.20 The hook-shaped ends are clearly visible in SEMimages of our fixed LI bacteria (see Figure 1a). Importantly, theconfocal images of our fixed bacteria freely diffusing in the solventindicate that they are sufficiently stiff, such that their shape does notchange during Brownian motion. Clearly, the live bacteria are able tochange their shape significantly, performing a twisting motion. Owingto this unique type of motion, these bacteria are able to survive andpropel themselves in both highly viscous soils and low-viscosity liquids,such as water and urine,20 where the random Brownian contribution totheir motility is significant.To quantify the shape of LI bacteria, we obtain several dozen SEM

images of individual bacteria, such as in Figure 1a. The images areanalyzed to yield the radius b of the bacterial cross section, the pitch ofthe LI helix d, the radius r of the helical central line of the LI, and thetotal length of the bacterium Ls, measured along the symmetry axis ofthe helix (see pictorial definitions in Figure 1). The resulting statisticaldistributions of these quantities are shown in Figure 2. We fit the

distributions by Gaussian functions; the fitting parameters areprovided in Table 1. The shape of LI does not significantly changewhen the bacterium is dried for SEM imaging. To test this, we havefollowed, by means of DIC (differential interference contrast) opticalmicroscopy, the shape of an LI bacterium on a slide, while the solventwas allowed to evaporate. No shape changes were detected within >12h after the complete drying of the solvent.While the high-resolution shape details of LI are best observed by

SEM, confocal microscopy allows us to follow the three-dimensionalBrownian motion of these bacteria in a free solvent.Confocal Measurements. To fluorescently stain the bacteria for

confocal imaging, we transfer them into a 2× saline-sodium citrate

(SSC) buffer, which is a 300 mM NaCl and 30 mM trisodium citrate(Na3C6H5O7) solution in water. The sample is left to equilibrate forabout an hour on a tumbler. Then, the bacteria are sedimented bycentrifugation and the supernatant is replaced by a 5 μM solution ofpropidium iodide (PI, C27H34I2N4) fluorescent dye in SSC buffer.After ∼1 h of incubation time, we wash out the excess fluorescent dye;for that purpose, we sediment the bacteria by centrifugation andreplace the supernatant by a pure 2× SSC buffer. The last step isrepeated several times, to minimize the PI concentration in the freesolvent. Finally, to match the density of the solvent to that of thebacteria, which minimizes sedimentation, we replace the conventionalSSC buffer by a similar buffer, prepared with a carefully adjustedmixture of H2O and D2O, instead of the pure water. No measurablesedimentation or creaming of individual bacteria could be observed inthis mixture over a time scale of several hours, such that the Brownianmotion is undisturbed by external forces. The resulting suspension ofLI bacteria in a density-matched SSC buffer is then loaded into arectangular Vitrocom capillary 0.1 × 2 × 50 mm3 and sealed with epoxyglue. We glue the Vitrocom capillary onto a supporting glass slide, toenhance its mechanical stiffness. The walls of the capillary are 0.1 mmthick, allowing the full volume of the capillary to be imaged with aNikon Plan Apo 100× objective, in DIC, phase contrast, or confocalmode.

For confocal measurements, we employ the Nikon A1R resonantlaser scanning setup, which allows 512 × 512 pixel images to be takenat a rate of 30 fps; this rate is decreased by the low signal-to-noiseratio, which makes the line-averaging necessary. Our microscope isequipped by a high-speed PIFOC piezo-z lens positioner, for rapidcollection of three-dimensional image stacks through the sample.Typically, we collect one stack of ∼50 images every 4.6 s, with thevoxel size being set to 0.083 × 0.083 × 0.8 μm2. The lateral digitalresolution slightly oversamples beyond our optical resolution, whichimproves the accuracy of particle center detection.22 A finer digitalresolution of 0.4 μm/slice along the z-axis, which is comparable to ouroptical resolution, was attempted to confirm that our results are notaltered.

To locate the momentary position of our bacterium, at a given timet from the beginning of the measurements, we employ a simplealgorithm, based on the Crocker and Grier23 and PLuTARC22

algorithms for tracking of simple spheres; we implement this algorithmin C++, based on the PLuTARC codes. In our algorithm, the center ofmass of each confocal slice through the fluorescent bacterium isdetermined, based on pixel brightnesses. These centers are then linked,for each individual confocal stack, under the requirement that thelateral separation between the centers of the object in consequentslices is below a certain limiting value. This yields the momentary (x, y,z) positions of the centers of mass of the bacteria, at a given time t. Forstudies of Brownian motion, we choose relatively straight bacteria,where the hook-shaped ends are less pronounced (inset to Figure 1aand Supporting Information). Importantly, while our code can ingeneral deal with more than one bacterium in a stack, such situationswere avoided in real data to minimize the chances for entanglementsbetween multiple bacteria, occurring in some of our SEM data, as alsoin some real-life situations.24

To obtain the approximate momentary direction of the long axis ofthe bacterium, we diagonalize the covariance matrix of the pixelintensities, for each of the slices. The covariance matrix is obtainedfrom the raw confocal image of n × m pixels as

Figure 2. Statistical distributions of geometrical characteristics of ourLI bacteria, as derived from scanning electron microscopy measure-ments. b is the radius of the bacterial cross section, d is the pitch of thehelix, and r is the radius of the helix which is formed by the central lineof the bacterium (see Figure 1b). Ls is the total length of thebacterium, measured along the symmetry axis of the helix. Solidsymbols correspond to statistical bin heights, the horizontal separationbetween these symbols corresponds to the bin width. Probabilities arenormalized, to have their sum equal to unity.

Table 1. Geometrical Characteristics of the LI Bacteria, asObtained from the Gaussian Fits in Figure 2a

mean (μm) SE (μm) width (μm)

b 0.0695 0.001 0.02d 0.392 0.005 0.07r 0.085 0.001 0.03Ls 9.1 0.1 4

aThe mean value, the standard error of the mean (SE), and the widthof the distribution are provided.

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∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

Δ Δ Δ

Δ Δ Δ

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

x I x y x y I x y

x y I x y y I x y

( , ) ( , )

( , ) ( , )

i

n

j

m

i i ji

n

j

m

i j i j

i

n

j

m

i j i ji

n

j

m

j i j

2

2

(1)

where I(xi, yj) is the intensity of the (i, j)-th pixel, corresponding toposition (xi, yj) of the image, Δxi = xi − ⟨x⟩, Δyj = yj − ⟨y⟩, and (⟨x⟩,⟨y⟩) is the center of mass of the slice. The slices through ourbacterium, convolved with the optical resolution of our confocalmicroscope, are (roughly) symmetric under inversion about the lengthor the width of the bacterium. Thus, we find the orientation within thehorizontal slice by diagonalizing the covariance matrix, such that themixed elements ∑i

n∑imΔxiΔyjI(xi, yj) cancel out. In the rotated

coordinate system x′−y′, where the matrix is diagonalized, thebacterium is aligned with the axes of coordinates; thus, the anglebetween the original x−y-coordinates and the rotated x′−y′coordinates corresponds to the orientation of the bacterium within agiven two-dimensional confocal slice. This procedure is repeated for allslices cutting through the bacterium, and the average orientation iscalculated. The angle with respect to the horizontal plane is obtainedfrom the number of confocal slices in which the bacterium is visible; amore accurate algorithm, diagonalizing the 3 × 3 covariance matrix forthe full three-dimensional stack, is currently under construction.Finally, to obtain the trajectory of our helical bacterium we link the (x,y, z) positions of its center of mass obtained at different time steps,subject to a limiting spatial separation between them.To test our technique for tracking the position of the center of

mass, we measure the translational diffusion coefficient of a simplecolloidal sphere, made of poly(methyl methacrylate) (PMMA). Thissphere has a diameter of σ = 2.40 ± 0.05 μm. It is suspended in a60:40 (v/v) mixture of tetrachloroethylene (Sigma-Aldrich, ≥98%)and racemic decahydronaphthalene (Sigma-Aldrich, ≥98%), whichmatches the density of PMMA, such that the sedimentation iscompletely inhibited and the sphere is separated from the capillarywalls by a distance of >17 μm. Thus, we gain the full advantage of theconfocal microscopy, which allows the three-dimensional path of thesphere to be reconstructed, within the bulk of the sample, where thehydrodynamic interactions with the walls of the container arecompletely negligible. The (dynamic) viscosity of our solvent mixturewas obtained by a Cannon-Manning semimicro viscometer, yielding1.15 ± 0.03 mPa·s. The experimental mean-squared displacement ofthe sphere ⟨|r|2⟩ is perfectly linear in time, as shown by solid symbolsin Figure 3. The slope of the best linear fit to these data is s = 0.91 ±

0.02 μm2/s, yielding a diffusion coefficient of Ds = s/6 = 0.152 ± 0.003μm2/s. This result coincides, within statistical error, with thetheoretical Stokes−Einstein coefficient of such sphere 0.159 ± 0.004μm2/s, as shown by the very good match between the Stokes−Einsteintheoretical ⟨|r|2⟩ (red dashes in Figure 3) and our experimental data.This confirms the accuracy of our particle tracking algorithm.

In addition, we test the results of our algorithm by semimanualdetection of the (x, y, z) positions and the three-dimensionalorientations of the bacteria, employing the 3D object Counter25 and theVolume Viewer plugin26 to Fiji software, which performs three-dimensional reconstructions of the raw confocal data. The results donot change significantly, with either the manual or the computerizeddetection employed; however, clearly, only very low statistics can beachieved with manual particle tracking.

■ RESULTS AND DISCUSSION

To obtain the diffusion constants of the bacterium, wecalculate, based on the experimental trajectories, the mean-squared displacement ⟨|r|2⟩ of the bacterium in time t. Forsimple spherical objects in a three-dimensional space, such asthe PMMA colloids which were mentioned in the ExperimentalSection, ⟨|r|2⟩ = 6Dst, where Ds = kBT/(6πrη) is the Stokes−Einstein translational diffusion coefficient of a sphere of radius rin a solvent of (dynamic) viscosity η. However, the Brownianmotion of nonspherical objects, such as a cylindrical rod, ismore complex.2 A rod which at time t = 0 has a certainorientation in space will diffuse faster in the direction of its longaxis. At short times t, the spatial orientation of the rod does notsignificantly change; thus, the coefficient of diffusion, at theseshort t, is rotationally anisotropic. At longer t, the orientation ofthe rod is randomized with respect to the lab frame; thus, thediffusion becomes spherically isotropic, as for a simplesphere.2,10 The transition from an anisotropic diffusion to anisotropic diffusion occurs at the orientational decorrelation timeτθ = (2Dθ)

−1, which is set by the rotational diffusion coefficient2

Dθ; at times t > τθ, the directional memory is wiped out and thediffusion is isotropic. The diffusion coefficient at t > τθ dependsonly weakly on the shape of the particles,27 averaging over theirlong and short axes;2 this is the diffusion coefficient which isalso typically measured in dynamic light scattering experiments,where many randomly oriented particles contribute to thesignal.In our experimental system, we simplify the comparison of

diffusion coefficients to theoretical predictions17,28 by measur-ing the translational diffusion coefficients in a coordinatesystem which is oriented with the long axis of the LI helix (“LI-frame”). In such a coordinate system, the translational diffusionis decoupled from the rotational one.2,10 The diffusion is thenanisotropic at all t, such that the contributions due to thespecial geometry of our bacteria are easier to separate out andtest. For the lab-frame displacement vector between steps j − 1and j being Δrj = Δrx,jx + Δry,jy + Δrz,jz, the displacement inthe LI-frame ΔXj = ΔXr,jr + ΔXθ,jθ + ΔXϕ,jϕ is given by

Δ

Δ

Δ

=

Δ

Δ

Δθ

ϕ

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

X

X

X

T

r

r

r

r j

j

j

x j

y j

z j

,

,

,

,

,

, (2)

Figure 3. Experimental mean-square displacement of a colloidalsphere (scatter) is perfectly matched by the Stokes−Einsteintheoretical prediction (red dashes), confirming the accuracy of ourparticle tracking procedure.

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θ ϕ θ ϕ θ

θ ϕ θ ϕ θ

ϕ ϕ

= −

−

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥T

sin cos sin sin cos

cos cos cos sin sin

sin cos 0

j j j j j

j j j j j

j j (3)

where x , y, z and r, θ, ϕ are the unit vectors in these twodifferent coordinate systems, and (θj, φj) is the orientation (atstep j), with respect to the lab-frame.10

The mean-squared displacements along the axis of the LIhelix ⟨ΔXr

2(t)⟩ are obtained by summation of ΔXr,j2 for j values

which correspond to time-interval (τ, τ + t), where τ is anarbitrary time such that 0 < τ < (T − t) and T is the duration ofour experiment; an averaging is then carried out over differentchoices of τ. The resulting ⟨ΔXr

2(t)⟩ are linear in time, which istypical for normal diffusion (see solid symbols in Figure 4). The

data are slightly up-shifted, by about (1.4 μm)2, resulting in anextrapolated nonzero value of ⟨ΔXr

2(t = 0)⟩; clearly, this tinyshift is a result of our finite accuracy in location of the center ofmass of the bacteria. The slope of the linear fit (black solid line)to our data, divided by two, yields a diffusion coefficient of Dr =0.24 ± 0.01 μm2/s, which we compare to various theoreticalpredictions.The calculation of a diffusion coefficient of a helix is a

nontrivial task, even in the LI-frame, where the orientation ofthe long axis of the helix is constant. Still, such a theoreticalcalculation was carried out several decades ago.17,29 Theauthors employed stokeslets approximate solution to Stokesequations,6 to obtain the friction coefficients of a helix, the endeffects of which were neglected;29 in this calculation, thehydrodynamic interactions between different parts of the helixwere explicitly taken into account. The resulting frictioncoefficients were substituted into the stochastic Langevin

equations,17 to obtain the mean-squared displacement of thehelix as

⟨Δ ⟩ = Ξ Ξ Ξ − Ξθ θ−X t k T r t( ) 2 ( )z Ir

2B

2 2 1(4)

where r is the radius of the helix. In this expression, Ξz and Ξθ

are the translational (along the axis of the helix) and therotational (about the long axis of the helix) coefficients offriction,29 and ΞI measures the coupling between translationsand rotations. The values of these quantities were obtained as

π ψΞ = + r d[1 (2 / ) ]z2

1 (5)

ψΞ =θ r2 21 (6)

π ψΞ = − r d(2 / )I 1 (7)

where, as above, d is the pitch of the helix and

ψ πηλπ π

=+ +

dr d

4[2 (2 / ) ] ln[(d/2b){1 (2 r/d) }]1 2 2 2 (8)

with b and λ being the radius of our LI bacteria and the numberof pitches in its full length.To compare the experimental rate of diffusion to the

theoretical predictions, we employ the SEM-derived values of b,d, r, and Ls, listed in Table 1. The lengths of our bacteria (Ls)are large, well beyond the resolution of confocal microscopy.Thus, we confirm that the lengths of those bacteria which wetrack by confocal microscopy coincide with the peak of the(rather wide) Ls distribution. Consequently, we obtain λ = Ls/d≈ 23, as observed directly in our SEM images. Unfortunately,on substituting all these measured parameters into eqs 4−8, weobtain the red dash-dotted line in Figure 4, which has a muchsmaller slope, compared to the experimental data. Thus, thistheoretical model significantly underestimates the rate ofdiffusion Dr along the axis of the LI helix; tuning of ourexperimental input parameters, within the corresponding errorbars, does not significantly improve the agreement between thistheory and our experimental data.Strikingly, in view of the poor agreement of our data with the

rather complex theory of a Brownian motion of a helix,17,29 ourdata favorably compare with the predicted rate of diffusion of asimple cylinder approximating the dimensions of the LI helix.The length of such cylinder is the same as that of the symmetryaxis of the bacterial helix, Ls. The cross-section diameter of thecylinder is taken as 2r (dashed lines in Figure 1b). Thisdiameter is smaller than the diameter of the escribed cylinder ofthe LI (2r + 2b), yet larger than the diameter of the bacterialcross section 2b; thus, the resulting cylinder closelyapproximates the average dimensions of the bacterium. Theaspect ratio of our effective cylinder is then pc = Ls/(2r) ≈ 51,such that its diffusion constant for longitudinal motion (in acoordinate system which is oriented with the cylinder)becomes28

πημ= = ±D

k T

L

ln p

20.25 0.01 m /sr

B c

s

2

(9)

within statistical error from our experimental Dr = 0.24 ± 0.01μm2/s. Indeed, the corresponding theoretical mean-squareddisplacements ⟨ΔXr

2(t)⟩ = 2Drt are very close to the actualexperimental data, as shown in a dashed blue line in Figure 4.Importantly, while the agreement between the diffusion

coefficient of the LI and that of the approximating cylinder israther remarkable, this agreement does not necessarily mean

Figure 4. Mean-square displacements, covered by the LI bacterium inthe direction of the long axis of the helix in time t, in a coordinatesystem which is oriented with the bacterium. Note that while anelaborate theory for a diffusion of a helix17 (dash-dotted red line)misses the experimental (solid symbols) rate of diffusion dramatically,a simpler model, where we approximate the shape of this bacterium bya cylinder, matches the diffusion coefficient (the slope of theexperimental data, divided by 2) with no free parameters (bluedashes). The slight vertical shift of the experimental data at low t mayresult from imperfectness of the particle tracking algorithm; theexperimental data were smoothed by adjacent averaging. The greendash double-dot line shows the theoretical mean-square displacementsof a rod, having the same dimensions as these of an uncoiled LI.Evidently, the coiling of LI bacteria into a helical shape increases theiron-axis Brownian motility.

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that the helical shape of the LI plays no role in its diffusion.Indeed, if the LI were not coiled into a helix, its extended lengthwould be LUR = 2πλ[r2 + (2π)−2d2]1/2 ≈ 15.8 μm, while itsthickness would be equal to 2b. Thus, the rod-like shape of thisuncoiled LI would have an aspect ratio of pUR ≈ 114. To obtainthe on-axis diffusion of this uncoiled rod (UR), we substitutepUR and LUR into eq 9, instead of pc and Ls. The resulting mean-square displacements are shown in a green dash double dottedline in Figure 4, underestimating the experimental diffusion rateby ∼25%.To further test the observed agreement of our experimental

data with the theoretical predictions for an approximatingcylinder of length Ls, we measure the mean-square displace-ments ⟨ΔXt

2(t)⟩ of our LI bacteria in-normal to the symmetryaxis of the helix; as above, this quantity is measured in thecoordinate system which is oriented with the bacterium. Forthat purpose, we sum the (1/2)[ΔXθ,j

2 + ΔXϕ,j2 ] values (see eq

2) for j−s, which correspond to a time interval (τ, τ + t), whereτ is an arbitrary time such that 0 < τ < (T − t) and T is theduration of our experiment. As in the case of ⟨ΔXr

2(t)⟩ (seeabove), an averaging is then carried out over different choicesof τ. The resulting data, albeit quite scattered, scale roughlylinearly in t, as shown in solid symbols in Figure 5; thus, half

the slope of these data yields the transverse diffusion coefficientof the bacterium, Dt = 0.13 ± 0.02 μm. This value is in goodagreement, within the statistical accuracy, with the theoreticalvalue for a simple cylinder30 with an aspect ratio pc and a lengthof Ls:

πημ= = = ±D D

k T

L/2

ln p

40.13 0.01 m /st r

B c

s

2

(10)

Indeed, the theoretical ⟨ΔXt2(t)⟩ = 2Dtt (blue dashes in

Figure 5) matches the slope of the experimental data, which areslightly up-shifted due to the finite accuracy of our particletracking procedure. As before, the theoretical diffusioncoefficient of an uncoiled LI rod (UR) of length LUR is toolow by almost 50%; the corresponding ⟨ΔXt

2(t)⟩ significantlyunderestimate the slope of the experimental values, as shown in

an olive dash double-dotted line in Figure 5. Thus, the coilingof LI significantly increases the Brownian motility of thisbacterium both in its axis-normal direction and in its on-axisdirection.Encouraged by the good agreement of our simple cylindrical

model, with no free parameters, with the translational Brownianmotion of the LI bacterium, we test the performance of thistheoretical model in predicting the rate of the rotationaldiffusion of the long axis of the LI helix. Importantly, we are notaware of any rigorous theoretical model, calculating therotational diffusion rate of a helix; only the rotations of thehelix about its long axis are discussed in the literature.17,29 Wemeasure the direction of the unit vector u (t), along the axis ofthe LI helix, which yields the rotational diffusion Dθ as

10

⟨ − ⟩ = − − θu t u D t[ ( ) (0)] 2[1 exp( 2 )]2(11)

The experimental ⟨Δu2(t)⟩ ≡ ⟨[u (t) − u (0)]2⟩ are nonlinearon a semilog scale, as shown by solid symbols in Figure 6a.These data are nicely matched by the theoretical scaling of eq11 (solid curve), with the rotational diffusion coefficient fittedto Dθ = (1.62 ± 0.06) × 10−2 s−1.

To simplify the comparison with the theoretical predictions,we define ζ(t) ≡ −ln[1 − ⟨Δu2(t)⟩/2] = 2Dθt, which is linearin t; the experimental ζ(t) appear in solid symbols in Figure 6b,together with the fitted ζ(t) = 2 × 1.62 × 10−2t (solid line).Several different expressions appear in the literature28,31 for therotational diffusion of a long cylinder; however, for our shapeparameters, all these expressions almost coincide numerically.The rotational diffusion coefficient is thus obtained as31

δ

πη=

+θD

k T p

L

3 (ln )B c c

s3

(12)

where δc = −0.662 + (0.917pc−1) − (0.050pc

−2). With our Lsand pc (see above), this yields a theoretical estimate of Dθ =

Figure 5. Mean-square displacements, covered by the LI bacterium innormal to the long axis of its helix in time t, in a coordinate systemwhich is oriented with the bacterial symmetry axis. Note that thesimple model (blue dashes), where the shape of this bacterium isapproximated by a cylinder of length Ls, matches the experimentaldiffusion coefficient (the slope of the experimental data, divided by 2)with no free parameters. The uncoiled bacterial rod (UR) is subject toa much higher friction, significantly slowing its Brownian motion(olive dash double dotted line).

Figure 6. Rotational diffusion of the LI bacterium. (a) Theexperimental (symbols) and the theoretical, based on the cylindricalmodel (blue dashes), mean-square displacements of the unit vector,which is oriented with the long axis of the bacterium. Note the perfectagreement between theory and experiment, obtained with no tunableparameters. (b) ζ(t) = −ln(1 − ⟨u2(t)⟩/2) = 2Dθt, obtained from thedata in section (a); the slope of ζ(t) yields the rotational diffusioncoefficient. As before, the agreement with the cylindrical model (bluedashes) is perfect. The theoretical mean-square displacements and ζ(t)of an uncoiled bacterium are lower, by a factor of 4.6, compared to theactual experimental data, as demonstrated by olive dash double dottedlines in sections (a) and (b).

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(1.54 ± 0.04) × 10−2 s−1, which perfectly matches ourexperimental value. Indeed, the corresponding theoretical ζ(t)(blue dashes in Figure 6b) is almost indistinguishable from thesolid black line, which is the best fit to the experimental data.Similarly, the theoretical ⟨Δu2(t)⟩ perfectly match theexperimental data in Figure 6a. As in the case of translationaldiffusion coefficients (above), the uncoiled LI rod (UR), withits length being LUR, underestimates the rotational diffusionconstant dramatically, by a factor of 4.6. The corresponding⟨Δu2(t)⟩ and ζ(t) completely miss the experimental data, asshown in an olive dash double-dotted curve in Figure 6. Thus,the helical shape of the LI bacteria dramatically increases theirrotational Brownian motility.To justify, on a hand-waving level, the very good agreement

between our cylindrical model and the experimental data, wenote that a length scale ξ exists, such that shape details muchfiner than ξ do not matter for the Brownian motion. A typicalobject, moving at low Reynolds numbers through a liquid offinite viscosity, disturbs the flow field in its vicinity to a distanceknown as the boundary layer thickness.32 With the lineardimensions of our bacterium roughly approximated by (Lsb)

1/2

and the kinematic viscosity of solvent being ν ≈ 10−6 m2/s, theboundary layer thickness is obtained as δ = v1/2U0

−1/2(Lsb)1/4,

where U0 is the velocity of the bacterium with respect to the labframe. As is well-known,5 the Brownian movement of amicroparticle in a liquid consists of a series of microscopicballistic motions (or “flights”), during which the displacement islinear in time. The velocity of a particle of mass m during suchflights is33 U0 = (kBT/m)

1/2, which for our LI bacteria yields U0≈ 2 mm/s. Thus, the boundary layer thickness in such motionis of the order of δ ≈ 20 μm. We anticipate the order ofmagnitude of ξ to be the same as that of δ, much larger than thesize of the fine helical structure in Figure 1a. The separationbetween subsequent loops of the helix is much smaller than ξ,so that the fluid in between the loops is dragged together withthe bacterium and the cylindrical model is valid. For helicalobjects with a looser coil, where the spacing between loops iscomparable to δ, the cylindrical model must break down. Ofcourse, the present hand-waving argument is only accurate toan order of magnitude; a more precise derivation must becarried out to obtain quantitative results. In conclusion, asdemonstrated by both a hand-waving argument and a directnumerical calculation, while coiled structure of LI reduces theireffective length by ∼70% from LUR to Ls, which dramaticallyincreases their Brownian motility, the shape of the bacteriumcan still be very well approximated by that of a simple cylinder.We have followed the three-dimensional Brownian motion of

an LI helix by confocal microscopy and measured its rotationaland translational diffusion coefficients. While the longitudinaldiffusion coefficient of the LI is underestimated by a detailedtheory of Brownian motion of a helix, all our experimentaldiffusion coefficients are correctly reproduced, with no freeparameters, with the shape of the LI approximated by a simplecylinder, disregarding its helical fine structure. The live LIbacteria, as also some other spirochaetes, have developed veryeffective mechanisms of translational motility, which makethem able to move in both very viscous soils and solvents oflow viscosity, such as water.20 Yet, their ability to rotate is notvery impressive, limiting their ability to find the optimaldirection toward an attractant or away from a repellent. Wedemonstrate that, at least in a low-viscosity solvent, their poorability to rotate is compensated, to some extent, by their(almost) 5-fold increased Brownian rotational diffusivity. At the

same time, this extended Brownian motility must randomizeand destabilize the active directed motion of these bacteria,such that the evolutionary meaning of coiling in spirochaetes isstill to be further investigated. Further studies, with differentspecies of live spirochaetes, are necessary to confirm that theincreased Brownian diffusivity is indeed used by the LI toimprove their food-seeking performances.

■ ASSOCIATED CONTENT*S Supporting InformationConfocal three-dimensional reconstruction of several LIbacteria sticking to a glass surface (movie). This material isavailable free of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: bpokroy@tx.technion.ac.il; eli.sloutskin@biu.ac.il.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe thank Dr. Shmuel Yitzhaki and Dr. Ada Barnea (IsraelInstitute for Biological Research, Ness Ziona, Israel) forproviding us the LI bacteria for this research. We are gratefulto Dr. Andrew B. Schofield for synthesis of PMMA colloidalspheres, to Dr. Zion Tachan for his assistance with the SEMmeasurements, to Dr. Manny Benish (Agentek) for hisassistance with the optical microscopy, and to Dr. A. Perelmanfor fruitful discussions. Dr. P. J. Lu is acknowledged for sharingwith us his PLuTARC codes. The Kahn foundation and theIsrael Science Foundation (No. 1668/10) have generouslyfunded some of the equipment used in this project. Thisresearch was supported by the Russell Berrie NanotechnologyInstitute (Technion) and by Bar-Ilan Institute for Nano-technology and Advanced Materials.

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