Coiled to Diffuse: Brownian Motion of a Helical Bacterium

Article pubs.acs.org/Langmuir

Coiled to Diffuse: Brownian Motion of a Helical Bacterium Alexander 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 confocal microscopy to follow the Brownian motion of a fixed helically shaped Leptospira interrogans (LI) bacterium. We extract from our measurements the translational and the rotational diffusion coefficients of this bacterium. A simple theoretical model is suggested, perfectly reproducing the experimental diffusion coefficients, with no tunable parameters. An older theoretical model, where edge effects are neglected, dramatically underestimates the observed rates of translation. Interestingly, the coiling of LI increases its rotational diffusion coefficient by a factor of 5, compared to a (hypothetical) rectified bacterium of the same contour length. Moreover, the translational diffusion coefficients would have decreased by a factor of ∼1.5, if LI were rectified. This suggests that the spiral shape of the spirochaete bacteria, in addition to being employed for their active twisting motion, may also increase the ability of these bacteria to explore the surrounding fluid by passive Brownian diffusion.

■

■

INTRODUCTION The Brownian motion, wherein small particles suspended in a fluid undergo random translations and rotations, is one of the oldest fields in modern physics. While first scientific experiments in this field date back to the early 19th century,1 active experimental2−5 and theoretical6−9 research in this field continues. In particular, even for the simplest nonspherical particles, such as rods,10−13 tetrahedra,14,15 and ellipsoids of revolution,2 the full understanding of diffusive Brownian motion has only recently been achieved. However, many of common molecules, viruses, and cells have rather complex shapes, such that both the rotational symmetry and the chiral symmetry are broken. Typically, when the diffusion rate of these complex objects is to be estimated, their shape is approximated by that of an ellipsoid,16 which is a crude approximation. The Brownian motion of chiral objects, such as a simple circular helix,17 was never quantitatively studied by a direct experimental technique.18,19 We employ direct confocal microscopy to follow the Brownian motion of a circular helix, directly, in three spatial dimensions. As a circular helix, we employ a fixed Leptospira interrogans (LI) bacterium, fluorescently labeled for confocal studies, and suspended in a density-matched solvent, to avoid any observable sedimentation on the experimental time-scales. These Gram-negative obligate aerobe spirochetes, which cause the Leptospirosis disease in humans and other mammals,20 have a helical right-handed structure, such that their chiral symmetry is broken. We use the experimental trajectories of these helical bacteria to directly measure the rotational and the translational diffusion coefficients of a circular helix; the values of these coefficients are then compared with the theoretical predictions. © 2012 American Chemical Society

EXPERIMENTAL SECTION

Shape Characterization. To measure the shape of our bacteria at high resolution, inaccessible by confocal microscopy, we obtain scanning electron microscopy (SEM) images of the LI. All the information regarding culture and harvesting of the LI is available elsewhere.21 The fixed LI were transferred, for SEM imaging, from water to ethanol, then to hexane, and deposited from hexane onto an aluminum surface. The helical right-handed shape of the bacteria is clearly visible, as shown in Figure 1a. The spirochaete bacteria, such as the LI, have their flagella running lengthwise between the bacterial inner membrane and outer membrane; these flagella, covered by the

Figure 1. (a) Scanning electron microscope image of a typical Leptospira interrogans bacterium. Note the right-helical shape of this bacterium and the typical hook-shaped ends. The image was obtained at 3 keV, with the bacterium coated by an evaporated gold layer. Some bacteria are more straight (see inset, where Ls is defined). (b) Pictorial definition of r, b, and d, which describe the shape of our LI bacterium. Received: May 20, 2012 Revised: August 12, 2012 Published: August 14, 2012 12941

dx.doi.org/10.1021/la302056j | Langmuir 2012, 28, 12941−12947

Langmuir

Article

outer membrane, cannot be observed by SEM. The flagella determine the shape of these bacteria. In particular, they are responsible for imparting a hook-shaped end to the cell during periods of directional swimming activity.20 The hook-shaped ends are clearly visible in SEM images of our fixed LI bacteria (see Figure 1a). Importantly, the confocal images of our fixed bacteria freely diffusing in the solvent indicate that they are sufficiently stiff, such that their shape does not change during Brownian motion. Clearly, the live bacteria are able to change their shape significantly, performing a twisting motion. Owing to this unique type of motion, these bacteria are able to survive and propel themselves in both highly viscous soils and low-viscosity liquids, such as water and urine,20 where the random Brownian contribution to their 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 are analyzed to yield the radius b of the bacterial cross section, the pitch of the LI helix d, the radius r of the helical central line of the LI, and the total length of the bacterium Ls, measured along the symmetry axis of the helix (see pictorial definitions in Figure 1). The resulting statistical distributions of these quantities are shown in Figure 2. We fit the

Table 1. Geometrical Characteristics of the LI Bacteria, as Obtained from the Gaussian Fits in Figure 2a b d r Ls

mean (μm)

SE (μm)

width (μm)

0.0695 0.392 0.085 9.1

0.001 0.005 0.001 0.1

0.02 0.07 0.03 4

a

The mean value, the standard error of the mean (SE), and the width of the distribution are provided.

(SSC) buffer, which is a 300 mM NaCl and 30 mM trisodium citrate (Na3C6H5O7) solution in water. The sample is left to equilibrate for about an hour on a tumbler. Then, the bacteria are sedimented by centrifugation and the supernatant is replaced by a 5 μM solution of propidium 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 and replace the supernatant by a pure 2× SSC buffer. The last step is repeated several times, to minimize the PI concentration in the free solvent. Finally, to match the density of the solvent to that of the bacteria, which minimizes sedimentation, we replace the conventional SSC buffer by a similar buffer, prepared with a carefully adjusted mixture of H2O and D2O, instead of the pure water. No measurable sedimentation or creaming of individual bacteria could be observed in this mixture over a time scale of several hours, such that the Brownian motion is undisturbed by external forces. The resulting suspension of LI bacteria in a density-matched SSC buffer is then loaded into a rectangular Vitrocom capillary 0.1 × 2 × 50 mm3 and sealed with epoxy glue. We glue the Vitrocom capillary onto a supporting glass slide, to enhance its mechanical stiffness. The walls of the capillary are 0.1 mm thick, allowing the full volume of the capillary to be imaged with a Nikon Plan Apo 100× objective, in DIC, phase contrast, or confocal mode. For confocal measurements, we employ the Nikon A1R resonant laser scanning setup, which allows 512 × 512 pixel images to be taken at a rate of 30 fps; this rate is decreased by the low signal-to-noise ratio, which makes the line-averaging necessary. Our microscope is equipped by a high-speed PIFOC piezo-z lens positioner, for rapid collection of three-dimensional image stacks through the sample. Typically, we collect one stack of ∼50 images every 4.6 s, with the voxel size being set to 0.083 × 0.083 × 0.8 μm2. The lateral digital resolution slightly oversamples beyond our optical resolution, which improves the accuracy of particle center detection.22 A finer digital resolution of 0.4 μm/slice along the z-axis, which is comparable to our optical resolution, was attempted to confirm that our results are not altered. To locate the momentary position of our bacterium, at a given time t from the beginning of the measurements, we employ a simple algorithm, based on the Crocker and Grier23 and PLuTARC22 algorithms for tracking of simple spheres; we implement this algorithm in C++, based on the PLuTARC codes. In our algorithm, the center of mass of each confocal slice through the fluorescent bacterium is determined, based on pixel brightnesses. These centers are then linked, for each individual confocal stack, under the requirement that the lateral separation between the centers of the object in consequent slices 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. For studies of Brownian motion, we choose relatively straight bacteria, where the hook-shaped ends are less pronounced (inset to Figure 1a and Supporting Information). Importantly, while our code can in general deal with more than one bacterium in a stack, such situations were avoided in real data to minimize the chances for entanglements between multiple bacteria, occurring in some of our SEM data, as also in some real-life situations.24 To obtain the approximate momentary direction of the long axis of the bacterium, we diagonalize the covariance matrix of the pixel intensities, for each of the slices. The covariance matrix is obtained from the raw confocal image of n × m pixels as

Figure 2. Statistical distributions of geometrical characteristics of our LI bacteria, as derived from scanning electron microscopy measurements. b is the radius of the bacterial cross section, d is the pitch of the helix, and r is the radius of the helix which is formed by the central line of the bacterium (see Figure 1b). Ls is the total length of the bacterium, measured along the symmetry axis of the helix. Solid symbols correspond to statistical bin heights, the horizontal separation between these symbols corresponds to the bin width. Probabilities are normalized, to have their sum equal to unity. distributions by Gaussian functions; the fitting parameters are provided in Table 1. The shape of LI does not significantly change when the bacterium is dried for SEM imaging. To test this, we have followed, by means of DIC (differential interference contrast) optical microscopy, the shape of an LI bacterium on a slide, while the solvent was allowed to evaporate. No shape changes were detected within >12 h 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-dimensional Brownian 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 12942


Langmuir ⎡ n m ⎢ ∑ ∑ Δxi2I(xi , yj ) ⎢ i j ⎢ ⎢ n m ⎢∑ ∑ ΔxiΔyj I(xi , yj ) ⎢⎣ i j

Article

⎤ ∑ ∑ ΔxiΔyj I(xi , yj )⎥ ⎥ i j ⎥ n m ⎥ 2 ∑ ∑ Δyj I(xi , yj ) ⎥ ⎥⎦ i j n

0.02 μm2/s, yielding a diffusion coefficient of Ds = s/6 = 0.152 ± 0.003 μm2/s. This result coincides, within statistical error, with the theoretical Stokes−Einstein coefficient of such sphere 0.159 ± 0.004 μm2/s, as shown by the very good match between the Stokes−Einstein theoretical ⟨|r⃗|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 semimanual detection of the (x, y, z) positions and the three-dimensional orientations of the bacteria, employing the 3D object Counter25 and the Volume Viewer plugin26 to Fiji software, which performs threedimensional reconstructions of the raw confocal data. The results do not change significantly, with either the manual or the computerized detection employed; however, clearly, only very low statistics can be achieved with manual particle tracking.

m

(1)

where I(xi, yj) is the intensity of the (i, j)-th pixel, corresponding to position (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 our bacterium, convolved with the optical resolution of our confocal microscope, are (roughly) symmetric under inversion about the length or the width of the bacterium. Thus, we find the orientation within the horizontal slice by diagonalizing the covariance matrix, such that the mixed elements ∑ni ∑mi ΔxiΔyjI(xi, yj) cancel out. In the rotated coordinate system x′−y′, where the matrix is diagonalized, the bacterium is aligned with the axes of coordinates; thus, the angle between the original x−y-coordinates and the rotated x′−y′ coordinates corresponds to the orientation of the bacterium within a given two-dimensional confocal slice. This procedure is repeated for all slices cutting through the bacterium, and the average orientation is calculated. The angle with respect to the horizontal plane is obtained from the number of confocal slices in which the bacterium is visible; a more accurate algorithm, diagonalizing the 3 × 3 covariance matrix for the 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 simple colloidal sphere, made of poly(methyl methacrylate) (PMMA). This sphere has a diameter of σ = 2.40 ± 0.05 μm. It is suspended in a 60:40 (v/v) mixture of tetrachloroethylene (Sigma-Aldrich, ≥98%) and racemic decahydronaphthalene (Sigma-Aldrich, ≥98%), which matches the density of PMMA, such that the sedimentation is completely inhibited and the sphere is separated from the capillary walls by a distance of >17 μm. Thus, we gain the full advantage of the confocal microscopy, which allows the three-dimensional path of the sphere to be reconstructed, within the bulk of the sample, where the hydrodynamic interactions with the walls of the container are completely negligible. The (dynamic) viscosity of our solvent mixture was obtained by a Cannon-Manning semimicro viscometer, yielding 1.15 ± 0.03 mPa·s. The experimental mean-squared displacement of the sphere ⟨|r⃗|2⟩ is perfectly linear in time, as shown by solid symbols in Figure 3. The slope of the best linear fit to these data is s = 0.91 ±

■

RESULTS AND DISCUSSION

To obtain the diffusion constants of the bacterium, we calculate, based on the experimental trajectories, the meansquared displacement ⟨|r|⃗ 2⟩ of the bacterium in time t. For simple spherical objects in a three-dimensional space, such as the PMMA colloids which were mentioned in the Experimental Section, ⟨|r|⃗ 2⟩ = 6Dst, where Ds = kBT/(6πrη) is the Stokes− Einstein translational diffusion coefficient of a sphere of radius r in a solvent of (dynamic) viscosity η. However, the Brownian motion of nonspherical objects, such as a cylindrical rod, is more complex.2 A rod which at time t = 0 has a certain orientation in space will diffuse faster in the direction of its long axis. At short times t, the spatial orientation of the rod does not significantly change; thus, the coefficient of diffusion, at these short t, is rotationally anisotropic. At longer t, the orientation of the rod is randomized with respect to the lab frame; thus, the diffusion becomes spherically isotropic, as for a simple sphere.2,10 The transition from an anisotropic diffusion to an isotropic diffusion occurs at the orientational decorrelation time τθ = (2Dθ)−1, which is set by the rotational diffusion coefficient2 Dθ; at times t > τθ, the directional memory is wiped out and the diffusion is isotropic. The diffusion coefficient at t > τθ depends only weakly on the shape of the particles,27 averaging over their long and short axes;2 this is the diffusion coefficient which is also typically measured in dynamic light scattering experiments, where many randomly oriented particles contribute to the signal. In our experimental system, we simplify the comparison of diffusion coefficients to theoretical predictions17,28 by measuring the translational diffusion coefficients in a coordinate system which is oriented with the long axis of the LI helix (“LIframe”). In such a coordinate system, the translational diffusion is decoupled from the rotational one.2,10 The diffusion is then anisotropic at all t, such that the contributions due to the special geometry of our bacteria are easier to separate out and test. For the lab-frame displacement vector between steps j − 1 and j being Δ⃗ rj = Δrx,jx̂ + Δry,jŷ + Δrz,jẑ, the displacement in the LI-frame Δ⃗ Xj = ΔXr,jr̂ + ΔXθ,jθ̂ + ΔXϕ,jϕ̂ is given by ⎡ ΔX r , j ⎤ ⎡ Δrx , j ⎤ ⎢ ⎥ ⎥ ⎢ ⎢ ΔXθ , j ⎥ = T ⎢ Δry , j ⎥ ⎢ ⎥ ⎥ ⎢ ⎢⎣ ΔXϕ , j ⎥⎦ ⎢⎣ Δrz , j ⎥⎦

Figure 3. Experimental mean-square displacement of a colloidal sphere (scatter) is perfectly matched by the Stokes−Einstein theoretical prediction (red dashes), confirming the accuracy of our particle tracking procedure. 12943

(2)


Langmuir ⎡ sin θj cos ϕ sin θj sin ϕ cos θj ⎤ j j ⎢ ⎥ ⎢ T = cos θj cos ϕj cos θj sin ϕj −sin θj ⎥ ⎢ ⎥ ⎢ −sin ϕ ⎥ cos ϕ 0 j j ⎣ ⎦

Article

equations,17 to obtain the mean-squared displacement of the helix as ⟨ΔX r 2(t )⟩ = 2kBT Ξθ (ΞzΞθ − r 2 Ξ2I )−1t

(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 the rotational (about the long axis of the helix) coefficients of friction,29 and ΞI measures the coupling between translations and rotations. The values of these quantities were obtained as

(3)

where x̂, ŷ, ẑ and r̂, θ̂, ϕ̂ are the unit vectors in these two different coordinate systems, and (θj, φj) is the orientation (at step j), with respect to the lab-frame.10 The mean-squared displacements along the axis of the LI helix ⟨ΔX2r (t)⟩ are obtained by summation of ΔX2r,j for j values which correspond to time-interval (τ̃, τ̃ + t), where τ̃ is an arbitrary time such that 0 < τ̃ < (T − t) and T is the duration of our experiment; an averaging is then carried out over different choices of τ̃. The resulting ⟨ΔX2r (t)⟩ are linear in time, which is typical for normal diffusion (see solid symbols in Figure 4). The

Ξz = [1 + (2πr /d)2 ]ψ1

(5)

Ξθ = 2r 2ψ1

(6)

Ξ I = −(2πr /d)ψ1

(7)

where, as above, d is the pitch of the helix and ψ1 =

4πηλd [2 + (2πr /d) ] ln[(d/2b){1 + (2π r/d)2 }]2 2

(8)

with b and λ being the radius of our LI bacteria and the number of 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 we track 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, we obtain the red dash-dotted line in Figure 4, which has a much smaller slope, compared to the experimental data. Thus, this theoretical model significantly underestimates the rate of diffusion Dr along the axis of the LI helix; tuning of our experimental input parameters, within the corresponding error bars, does not significantly improve the agreement between this theory 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 our data favorably compare with the predicted rate of diffusion of a simple cylinder approximating the dimensions of the LI helix. The length of such cylinder is the same as that of the symmetry axis of the bacterial helix, Ls. The cross-section diameter of the cylinder is taken as 2r (dashed lines in Figure 1b). This diameter is smaller than the diameter of the escribed cylinder of the LI (2r + 2b), yet larger than the diameter of the bacterial cross section 2b; thus, the resulting cylinder closely approximates the average dimensions of the bacterium. The aspect ratio of our effective cylinder is then pc = Ls/(2r) ≈ 51, such that its diffusion constant for longitudinal motion (in a coordinate system which is oriented with the cylinder) becomes28

Figure 4. Mean-square displacements, covered by the LI bacterium in the direction of the long axis of the helix in time t, in a coordinate system which is oriented with the bacterium. Note that while an elaborate 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 by a cylinder, matches the diffusion coefficient (the slope of the experimental data, divided by 2) with no free parameters (blue dashes). The slight vertical shift of the experimental data at low t may result from imperfectness of the particle tracking algorithm; the experimental data were smoothed by adjacent averaging. The green dash double-dot line shows the theoretical mean-square displacements of a rod, having the same dimensions as these of an uncoiled LI. Evidently, the coiling of LI bacteria into a helical shape increases their on-axis Brownian motility.

data are slightly up-shifted, by about (1.4 μm)2, resulting in an extrapolated nonzero value of ⟨ΔX2r (t = 0)⟩; clearly, this tiny shift is a result of our finite accuracy in location of the center of mass 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 theoretical predictions. The calculation of a diffusion coefficient of a helix is a nontrivial task, even in the LI-frame, where the orientation of the long axis of the helix is constant. Still, such a theoretical calculation was carried out several decades ago.17,29 The authors employed stokeslets approximate solution to Stokes equations,6 to obtain the friction coefficients of a helix, the end effects of which were neglected;29 in this calculation, the hydrodynamic interactions between different parts of the helix were explicitly taken into account. The resulting friction coefficients were substituted into the stochastic Langevin

Dr =

kBT ln pc 2πηLs

= 0.25 ± 0.01 μm 2/s

(9)

within statistical error from our experimental Dr = 0.24 ± 0.01 μm2/s. Indeed, the corresponding theoretical mean-squared displacements ⟨ΔX2r (t)⟩ = 2Drt are very close to the actual experimental 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 is rather remarkable, this agreement does not necessarily mean 12944


Langmuir

Article

an olive dash double-dotted line in Figure 5. Thus, the coiling of LI significantly increases the Brownian motility of this bacterium both in its axis-normal direction and in its on-axis direction. Encouraged by the good agreement of our simple cylindrical model, with no free parameters, with the translational Brownian motion of the LI bacterium, we test the performance of this theoretical model in predicting the rate of the rotational diffusion of the long axis of the LI helix. Importantly, we are not aware of any rigorous theoretical model, calculating the rotational diffusion rate of a helix; only the rotations of the helix about its long axis are discussed in the literature.17,29 We measure the direction of the unit vector u(⃗ t), along the axis of the LI helix, which yields the rotational diffusion Dθ as10

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 length would be LUR = 2πλ[r2 + (2π)−2d2]1/2 ≈ 15.8 μm, while its thickness would be equal to 2b. Thus, the rod-like shape of this uncoiled LI would have an aspect ratio of pUR ≈ 114. To obtain the on-axis diffusion of this uncoiled rod (UR), we substitute pUR and LUR into eq 9, instead of pc and Ls. The resulting meansquare displacements are shown in a green dash double dotted line in Figure 4, underestimating the experimental diffusion rate by ∼25%. To further test the observed agreement of our experimental data with the theoretical predictions for an approximating cylinder of length Ls, we measure the mean-square displacements ⟨ΔX2t (t)⟩ of our LI bacteria in-normal to the symmetry axis of the helix; as above, this quantity is measured in the coordinate system which is oriented with the bacterium. For that purpose, we sum the (1/2)[ΔX2θ,j + ΔX2ϕ,j] values (see eq 2) for j−s, which correspond to a time interval (τ̃, τ̃ + t), where τ̃ is an arbitrary time such that 0 < τ̃ < (T − t) and T is the duration of our experiment. As in the case of ⟨ΔX2r (t)⟩ (see above), an averaging is then carried out over different choices of τ̃. The resulting data, albeit quite scattered, scale roughly linearly in t, as shown in solid symbols in Figure 5; thus, half

⟨[u ⃗(t ) − u ⃗(0)]2 ⟩ = 2[1 − exp( −2Dθ t )]

the slope of these data yields the transverse diffusion coefficient of the bacterium, Dt = 0.13 ± 0.02 μm. This value is in good agreement, within the statistical accuracy, with the theoretical value for a simple cylinder30 with an aspect ratio pc and a length of Ls: kBT ln pc 4πηLs

= 0.13 ± 0.01 μm 2/s

2

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

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

Dt = Dr /2 =

(11)

The experimental ⟨Δu (t)⟩ ≡ ⟨[u(⃗ t) − u(⃗ 0)] ⟩ are nonlinear on a semilog scale, as shown by solid symbols in Figure 6a. These data are nicely matched by the theoretical scaling of eq 11 (solid curve), with the rotational diffusion coefficient fitted to Dθ = (1.62 ± 0.06) × 10−2 s−1. 2

To simplify the comparison with the theoretical predictions, we define ζ(t) ≡ −ln[1 − ⟨Δu2(t)⟩/2] = 2Dθt, which is linear in 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 the rotational diffusion of a long cylinder; however, for our shape parameters, all these expressions almost coincide numerically. The rotational diffusion coefficient is thus obtained as31

(10)

⟨ΔX2t (t)⟩

Indeed, the theoretical = 2Dtt (blue dashes in Figure 5) matches the slope of the experimental data, which are slightly up-shifted due to the finite accuracy of our particle tracking procedure. As before, the theoretical diffusion coefficient of an uncoiled LI rod (UR) of length LUR is too low by almost 50%; the corresponding ⟨ΔX2t (t)⟩ significantly underestimate the slope of the experimental values, as shown in

Dθ =

3kBT (ln pc + δc) πηLs3

(12)

(0.917pc−1)

−2

where δc = −0.662 + − (0.050pc ). With our Ls and pc (see above), this yields a theoretical estimate of Dθ = 12945


Langmuir

Article

(1.54 ± 0.04) × 10−2 s−1, which perfectly matches our experimental value. Indeed, the corresponding theoretical ζ(t) (blue dashes in Figure 6b) is almost indistinguishable from the solid black line, which is the best fit to the experimental data. Similarly, the theoretical ⟨Δu2(t)⟩ perfectly match the experimental data in Figure 6a. As in the case of translational diffusion coefficients (above), the uncoiled LI rod (UR), with its length being LUR, underestimates the rotational diffusion constant dramatically, by a factor of 4.6. The corresponding ⟨Δu2(t)⟩ and ζ(t) completely miss the experimental data, as shown in an olive dash double-dotted curve in Figure 6. Thus, the helical shape of the LI bacteria dramatically increases their rotational Brownian motility. To justify, on a hand-waving level, the very good agreement between our cylindrical model and the experimental data, we note that a length scale ξ exists, such that shape details much finer than ξ do not matter for the Brownian motion. A typical object, moving at low Reynolds numbers through a liquid of finite viscosity, disturbs the flow field in its vicinity to a distance known as the boundary layer thickness.32 With the linear dimensions of our bacterium roughly approximated by (Lsb)1/2 and the kinematic viscosity of solvent being ν ≈ 10−6 m2/s, the boundary 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 lab frame. As is well-known,5 the Brownian movement of a microparticle in a liquid consists of a series of microscopic ballistic motions (or “flights”), during which the displacement is linear in time. The velocity of a particle of mass m during such flights is33 U0 = (kBT/m)1/2, which for our LI bacteria yields U0 ≈ 2 mm/s. Thus, the boundary layer thickness in such motion is of the order of δ ≈ 20 μm. We anticipate the order of magnitude of ξ to be the same as that of δ, much larger than the size of the fine helical structure in Figure 1a. The separation between subsequent loops of the helix is much smaller than ξ, so that the fluid in between the loops is dragged together with the bacterium and the cylindrical model is valid. For helical objects with a looser coil, where the spacing between loops is comparable to δ, the cylindrical model must break down. Of course, the present hand-waving argument is only accurate to an order of magnitude; a more precise derivation must be carried out to obtain quantitative results. In conclusion, as demonstrated by both a hand-waving argument and a direct numerical calculation, while coiled structure of LI reduces their effective length by ∼70% from LUR to Ls, which dramatically increases their Brownian motility, the shape of the bacterium can 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 rotational and translational diffusion coefficients. While the longitudinal diffusion coefficient of the LI is underestimated by a detailed theory of Brownian motion of a helix, all our experimental diffusion coefficients are correctly reproduced, with no free parameters, with the shape of the LI approximated by a simple cylinder, disregarding its helical fine structure. The live LI bacteria, as also some other spirochaetes, have developed very effective mechanisms of translational motility, which make them able to move in both very viscous soils and solvents of low viscosity, such as water.20 Yet, their ability to rotate is not very impressive, limiting their ability to find the optimal direction toward an attractant or away from a repellent. We demonstrate that, at least in a low-viscosity solvent, their poor ability 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 randomize and destabilize the active directed motion of these bacteria, such that the evolutionary meaning of coiling in spirochaetes is still to be further investigated. Further studies, with different species of live spirochaetes, are necessary to confirm that the increased Brownian diffusivity is indeed used by the LI to improve their food-seeking performances.

■

ASSOCIATED CONTENT

S Supporting Information *

Confocal three-dimensional reconstruction of several LI bacteria sticking to a glass surface (movie). This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Dr. Shmuel Yitzhaki and Dr. Ada Barnea (Israel Institute for Biological Research, Ness Ziona, Israel) for providing us the LI bacteria for this research. We are grateful to Dr. Andrew B. Schofield for synthesis of PMMA colloidal spheres, to Dr. Zion Tachan for his assistance with the SEM measurements, to Dr. Manny Benish (Agentek) for his assistance with the optical microscopy, and to Dr. A. Perelman for fruitful discussions. Dr. P. J. Lu is acknowledged for sharing with us his PLuTARC codes. The Kahn foundation and the Israel Science Foundation (No. 1668/10) have generously funded some of the equipment used in this project. This research was supported by the Russell Berrie Nanotechnology Institute (Technion) and by Bar-Ilan Institute for Nanotechnology and Advanced Materials.

■

REFERENCES

(1) Brown, R. A Brief Account of Microscopical Observations Made in the Months of June, July and August, 1827, on the Particles Contained in the Pollen of Plants; and on the General Existence of Active Molecules in Organic and Inorganic Bodies. Phil. Mag. 1828, 4, 161−173. (2) Han, Y.; Alsayed, A. M.; Nobili, M.; Zhang, J.; Lubensky, T. C.; Yodh, A. G. Brownian Motion of an Ellipsoid. Science 2006, 314, 626− 630. (3) Li, T.; Kheifets, S.; Medellin, D.; Raizen, M. G. Measurement of the Instantaneous Velocity of a Brownian Particle. Science 2010, 328, 1673−1675. (4) Pusey, P. N. Brownian Motion Goes Ballistic. Science 2011, 332, 802−803. (5) Huang, R.; Chavez, I.; Taute, K. M.; Lukić, B.; Jeney, S.; Raizen, M. G.; Florin, E.-L. Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid. Nat. Phys. 2011, 7, 576. (6) Lighthill, J. Flagellar Hydrodynamics. SIAM Review 1976, 18, 161−230. (7) Carlsson, T.; Ekholm, T.; Elvingson, C. Algorithm for Generating a Brownian Motion on a Sphere. J. Phys. A: Math. Theor. 2010, 43, 505001. (8) Benyi, A.; Oh, T. Modulation Spaces, Wiener Amalgam Spaces, and Brownian Motions. Adv. Math. 2011, 228, 2943−2981. (9) Tóthová, J.; Vasziová, G.; Glod, L.; Lisý, V. Langevin Theory of Anomalous Brownian Motion Made Simple. Eur. J. Phys. 2011, 32, 645−655. 12946


Langmuir

Article

our δ scales as m−1/4, so that a factor of 2 in m is completely negligible for our order-of-magnitude type of an argument.

(10) Mukhija, D.; Solomon, M. J. Translational and Rotational Dynamics of Colloidal Rods by Direct Visualization with Confocal Microscopy. J. Colloid Interface Sci. 2007, 314, 98−106. (11) Hong, L.; Anthony, S. M.; Granick, S. Rotation in Suspension of a Rod-Shaped Colloid. Langmuir 2006, 22, 7128−7131. (12) Fung, J.; Martin, K. E.; Perry, R. W.; Katz, D. M.; McGorty, R.; Manoharan, V. N. Measuring Translational, Rotational, and Vibrational Dynamics in Colloids with Digital Holographic Microscopy. Opt. Express 2011, 19, 8051−8065. (13) Colin, R.; Yan, M.; Chevry, L.; Berret, J.-F.; Abou, B. 3D Rotational Diffusion of Micrometric Wires Using 2D Video Microscopy. Europhys. Lett. 2012, 97, 30008. (14) Hoffmann, M.; Wagner, C. S.; Harnau, L.; Wittemann, A. 3D Brownian Diffusion of Submicron-Sized Particle Clusters. ACS Nano 2009, 3, 3326−3334. (15) García de la Torre, J.; del Rio Echenique, G.; Ortega, A. Improved Calculation of Rotational Diffusion and Intrinsic Viscosity of Bead Models for Macromolecules and Nanoparticles. J. Phys. Chem. B 2007, 111, 955−961. (16) Wegener, W. A.; Koester, V. J.; Dowben, R. M. A General Ellipsoid Cannot Always Serve as a Model for the Rotational Diffusion Properties of Arbitrarily Shaped Rigid Molecules. Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 6356−6360. (17) Hoshikawa, H.; Saitô, N. Brownian Motion of Helical Flagella. Biophys. Chem. 1979, 10, 81−89. (18) Tencer, M.; Bielski, R. Mechanical Resolution of Chiral Objects in Achiral Media: Where is the Size Limit? Chirality 2011, 23, 144− 147. (19) Marcos; Fu, H. C.; Powers, T. R.; Stocker, R. Separation of Microscale Chiral Objects by Shear Flow. Phys. Rev. Lett. 2009, 102, 158103. (20) Limberger, R. J. The Periplasmic Flagellum of Spirochetes. J. Mol. Microbiol. Biotechnol. 2004, 7, 30−40. (21) Yitzhaki, S.; Barnea, A.; Keysary, A.; Zahavy, E. New approach for serological testing for leptospirosis by using detection of Leptospira agglutination by flow cytometry light scatter analysis. J. Clin. Microbiol. 2004, 42, 1680−1685. (22) Lu, P. J.; Sims, P. A.; Oki, H.; Macarthur, J. B.; Weitz, D. A. Target-Locking Acquisition with Real-Time Confocal (TARC) Microscopy. Opt. Express 2007, 15, 8702−8712. (23) Crocker, J. C.; Grier, D. G. Methods of Digital Video Microscopy for Colloidal Studies. J. Colloid Interface Sci. 1996, 179, 298−310. (24) www.youtube.com/watch?v=F9B60gsCg4w (25) Bolte, S.; Cordelières, F. P. A Guided Tour into Subcellular Colocalization Analysis in Light Microscopy. J. Microsc. 2006, 224, 213−232. (26) rsb.info.nih.gov/ij/plugins/volume-viewer.html (27) Perrin, F. Mouvement Brownien d’un Ellipsoide (I). Dispersion Diélectrique pour des Molécules Ellipsoidales. J. Phys. Radium 1934, V, 497−511. (28) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics, Oxford University Press: Oxford, 1986. (29) Hoshikawa, H.; Saitô, N. Frictional Properties of Helical Flagella. J. Phys. Soc. Jpn. 1976, 40, 877−883. (30) To the best of our knowledge, no theoretical models exist in the literature, calculating the translational diffusion of a helix, in normal to its long axis.17,29 (31) Eimer, W.; Williamson, J. R.; Boxer, S. G.; Pecora, R. Characterization of the Overall and Internal Dynamics of Short Oligonucleotides by Depolarized Dynamic Light Scattering and NMR Relaxation Measurements. Biochemistry 1990, 29, 799−811. (32) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics. In Course of Theoretical Physics; Butterworth-Heinemann: Oxford, 1987. (33) This expression, based on the equipartition theorem, must be slightly modified for longer time scales where the fluid becomes incompressible, as discussed elsewhere.5 At these longer time scales, m must be the sum of the mass of the bacterium and the mass of the displaced liquid; this increases m by roughly a factor of 2. However, 12947


Coiled to Diffuse: Brownian Motion of a Helical Bacterium

Recommend Documents