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Research Article

Helical motion of the cell body enhances Caulobacter crescentus motility

Bin Liu, Marco Gulino, Michael Morse, Jay X. Tang, Thomas R. Powers, and Kenneth S. Breuer
  1. aSchool of Engineering, Brown University, Providence, RI 02912; and
  2. bDepartment of Physics, Brown University, Providence, RI 02912

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PNAS August 5, 2014 111 (31) 11252-11256; first published July 22, 2014; https://doi.org/10.1073/pnas.1407636111
Bin Liu
aSchool of Engineering, Brown University, Providence, RI 02912; and
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  • For correspondence: bliu27@ucmerced.edu
Marco Gulino
aSchool of Engineering, Brown University, Providence, RI 02912; and
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Michael Morse
bDepartment of Physics, Brown University, Providence, RI 02912
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Jay X. Tang
bDepartment of Physics, Brown University, Providence, RI 02912
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Thomas R. Powers
aSchool of Engineering, Brown University, Providence, RI 02912; and
bDepartment of Physics, Brown University, Providence, RI 02912
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Kenneth S. Breuer
aSchool of Engineering, Brown University, Providence, RI 02912; and
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  1. Edited by Charles S. Peskin, New York University, New York, NY, and approved June 26, 2014 (received for review April 25, 2014)

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Significance

Using a digital tracking microscope that provides both cell position and orientation, we have correlated the detailed motion of the cell body of a fast-swimming bacterium, Caulobacter crescentus, with its swimming motility. Contrary to the prevailing view that the rotating flagellum is the only means to propel the cell, we show that when the flagellum pushes the cell, the axis of the cell body precesses with a helical trajectory that enhances motility. This discovery changes our understanding regarding the role that cell shape and motion plays in bacterial motility. Furthermore, our powerful cell tracking technique enables a wide variety of studies that require extended observation of single cells, including motility, cell behavior, and aging.

Abstract

We resolve the 3D trajectory and the orientation of individual cells for extended times, using a digital tracking technique combined with 3D reconstructions. We have used this technique to study the motility of the uniflagellated bacterium Caulobacter crescentus and have found that each cell displays two distinct modes of motility, depending on the sense of rotation of the flagellar motor. In the forward mode, when the flagellum pushes the cell, the cell body is tilted with respect to the direction of motion, and it precesses, tracing out a helical trajectory. In the reverse mode, when the flagellum pulls the cell, the precession is smaller and the cell has a lower translation distance per rotation period and thus a lower motility. Using resistive force theory, we show how the helical motion of the cell body generates thrust and can explain the direction-dependent changes in swimming motility. The source of the cell body precession is believed to be associated with the flexibility of the hook that connects the flagellum to the cell body.

  • microorganisms
  • kinematics
  • fluid mechanics
  • torque
  • flicking

Motility of many flagellated bacteria, such as Escherichia coli, is achieved with the rotation of molecular motors attached to helical flagellar filaments through a flexible hook structure (1⇓–3). Because the organism must maintain zero net torque, the cell body counterrotates to balance the rotation of the flagella. A complete understanding of the role of body rotation in bacterial motility is still emerging. Qualitative studies have demonstrated that the cell shape does affect the swimming behavior of flagellated bacteria. In the extreme case of laboratory-grown cells with controlled shapes, E. coli with cell bodies having a crescent shape or a relaxed spiral were observed to swim in straight lines, whereas bacteria with tightly wound spiral-shaped cell bodies were observed to swim in circles (4, 5). In other cases, a helical cell body shape is thought to contribute to overall motility. For example, the counterrotation of the helical body shape of Spirillum is thought to contribute to the overall thrust (6). Similarly, the flagella of Spirochetes rotate between the inner and outer cell membranes, generating a helical wave that is thought to contribute to the organism’s motility (7). However, most flagellated bacteria have straight, rod-shaped cell bodies, and the effect of this body shape on motility is usually either ignored or treated as a source of passive drag (8⇓–10).

To quantitatively explore the correlation between cell kinematics and motility, we use a 3D tracking microscope—a digital implementation of the instrument first developed by Berg (11)—to follow individuals of Caulobacter crescentus for extended periods of time. A 2D microscope-stage (x–y axes) and a piezo driven-microscope objective (z axis) are moved so that the cell is kept focused in the center of the microscope’s field of view (Fig. 1A). Due to the lack of fluid inertia at these low Reynolds numbers, the motion of the cell is not affected by the moving stage, and we can thus track the (x, y, z) position of an individual cell while simultaneously recording a close-up view of the cell’s detailed motion. From the high-magnification image, we are able to measure the geometry of the cell (Fig. 1B), which we parameterize by its length, l, radius, a, and curvature, κ (all assumed to remain constant through the duration of the observation), and the instantaneous 3D orientation of the cell, characterized by the Euler angles (α, β, and γ, corresponding to yaw, pitch, and roll, respectively). Combining the cell orientation with its position yields a detailed history of the cell kinematics over an extended period (Fig. 1C).

Fig. 1.
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Fig. 1.

Extended measurements of a single motile cell are achieved using digital tracking microscopy. (A) Schematic of a 3D tracking microscope: a 3D microstage follows the motion of a motile cell so that it is kept in the center of the field of view. (B) The position (x, y, z) and orientation of the cell, denoted by the Euler angles (α, β, γ), are reconstructed from the observed image using a multiframe optimization procedure (Materials and Methods). (C) The tracking microscope allows us to follow a single cell for an extended period. Here the reconstructed cell body is rendered every eight frames and is colored according to cell speed, V. During the tracking we observe multiple transitions between pushing and pulling (e.g., red dotted circle) and pulling and pushing (e.g., blue dotted circle).

C. crescentus typically swims in a series of quasi-linear segments, reorienting its swimming direction by intermittently switching the direction of rotation of the flagellar motor (12, 13). When the right-handed flagellum rotates in a clockwise (CW) direction (viewed from outside the cell), the filament pushes the cell body forward (14), and the cell body rotates in the counterclockwise (CCW) direction to maintain torque balance. Occasionally, the flagellar motor reverses its direction, and the flagellum rotates in the CCW direction. The direction of travel changes by ∼180°, and in this reverse mode, the filament pulls the cell, which counterrotates in the CW direction. The transition from reverse back to forward motion is characterized by a distinctly different change in the direction of travel, and the recently identified flick motion (12) is thought to be associated with compressive buckling of the flexible hook protein that connects the flagellum to the cell body (13). We also observe this behavior in C. crescentus, and examples of these transitions are highlighted in Fig. 1C.

Results and Discussion

Distinct Motilities for Pushing and Pulling.

To characterize the cell kinematics, we obtain the rotation of the cell around its principal axis, γ (Fig. 1B), and thus the rotation frequency, γ˙. Perhaps not surprisingly, we find that the rotation frequency varies with time in both the pushing and pulling modes and that the cell rotation frequency is correlated with the swimming speed (typical correlation coefficient ∼0.3). However, somewhat unexpectedly, as the cell switches from a pusher to a puller, the rotation frequency of the cell increases markedly (by almost 100%) even though the average swimming speed remains constant (Fig. 2A). This difference in the frequency–speed characteristic is observed consistently over the duration of the tracking and can be characterized by two distinct nondimensional cell motilities, K+ and K−, where K=V/(γ˙l) (Fig. 2B). The difference between pushing and pulling motility, K+/K− = 1.29 ± 0.20, was found to be a general result (Fig. 2C), present for every cell tracked (n = 17) regardless of the cell length (l = 1.5–2.7 μm) and average swimming speed (V = 20−60 μm/s).

Fig. 2.
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Fig. 2.

Translation–rotation relation of a single cell during pushing (blue) and pulling (red). (A) Although the swimming speed V is almost constant, the rotation rate of the cell γ˙ increases as the flagellar motor reverses spinning direction. (B) Dependence of speed V on cell rotation rate γ˙ for a typical tracking. The dashed lines denote a linear fit with the slopes K+ and K− for pushing and pulling states, respectively. (C) The pushing motility, K+, is consistently higher than the motility during pulling, K−, over a range of swimming speeds and cell sizes.

Because low Reynolds number swimming is kinematically reversible (15), we conclude that a change in swimming kinematics underlies the difference between pushing and pulling motility. We can assess the extent of flagellar deformation due to viscous stresses from the parameter μVd2l/A, where d is the diameter of the filament and A is its bending stiffness (16). For typical values of V and l and using the elastic properties of the E. coli flagellum (16), we find that this nondimensional parameter is ∼10−3, suggesting that the flagellum can be considered to be a rigid helix at these conditions and maintains its geometry regardless of swimming direction.

Helical Motion of Cell Bodies.

Examination of the cell body kinematics reveals that the orientation of the principal axis of the cell differs from the direction of swimming and that the cell body follows a helical trajectory that is particularly pronounced during forward (pushing) motion (Fig. 3A). The average precession angle, θ, is significantly greater in the pushing state (〈θ+〉 = 0.49 ± 0.15 rad) than in the pulling state (〈θ−〉 = 0.34 ± 0.11 rad).

Fig. 3.
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Fig. 3.

Helical cell motion. (A) Three-dimensional reconstruction of the cell trajectory reveals that after the cell transitions from a puller to a pusher (denoted by the arrow) the cell body follows a helical path. The cell is color-coded by the rotation phase, γ. (B) A model cell of cylindrical shape demonstrates the thrust production. The velocity of the cell body can be decomposed to the component along the cell body V‖ and the transverse axis V⊥. Because the drag coefficients associated with these two components can be different up to a factor of 2, i.e., C⊥ ≲ 2C‖, the cell body can drift along the helical axis with the external force along the direction of rotation γ, i.e., axial force ℱc = 0, and fγ = Tc/R, where R is the radius of the helical trajectory and Tc is the external torque due to the rotating flagellum. The inset shows a view from the top. (C) The presence of high-order helical symmetry is confirmed by comparing the phase-averaged cell orientation (Euler angles α and β defined in Fig. 1C, relative to the direction of swimming), indicated by solid lines (the data range is indicated by the shaded region), with those of a perfect helical trajectory (dotted lines). For an ideal helical trajectory, both angles (α and β) have peak values equal to the precession angle θ, which differs from θh, the pitch angle of the helical trajectory.

To further quantify the helical nature of the trajectory due to cell deflection, we obtain the orientation of the principal axis of the cell as the cell rotates, which is determined by angles α and β (defined in Fig. 1B). In Fig. 3C, we show the fluctuation of these angles as a function of rotational phase of cell about its axis γ. The dashed curves show the case of a perfect helical geometry, in which the cell aligns along the tangent of the helix, with a pitch angle θ. The agreement between the experimental data and the perfect helical geometry is very good, and suggests that the cell axis precesses at the same rate as its counterrotation, γ˙, and therefore that the cell precesses about the direction of swimming.

When pushing, the tilted cell traces out a left-handed helical trajectory (Fig. 3A). Because the cell body rotates counterclockwise, the body might contribute to the propulsive force. This can be understood using a simplified physical argument in which we model the cell as an elongated rod (Fig. 3B) tilted away from the direction of travel with a precession angle θ. The axis of the rod is tangential to a cylindrical surface coaxial with the swimming direction, as shown by a view parallel to the rotation axis (inset in Fig. 3B). The hydrodynamic behavior of the rod with this oblique motion can be characterized by two drag coefficients, C‖ in the axial direction and C⊥ in the transverse direction, where C‖ ∼ 2C⊥ (17). In this simple model the rod experiences an azimuthal force from the motion of the flagellum, and due to the anisotropic nature of the drag coefficients the rod drifts persistently in the direction perpendicular to the forcing, resulting in self-propulsion with a helical trajectory. It should be noted that the wavelength of the helical trajectory, with pitch angle θh, is typically shorter than that associated with the precession angle of the cell body θ, reminiscent of the slip of a force-free swimming helix (8).

Hydrodynamic Model.

We can develop a more elaborate theory to quantitatively model this behavior. Following the general approach by Magariyama et al. (18), we describe the cell body and flagellum using a resistive force-type model (17). The cell and the flagellum both move with speed V but can rotate with different frequencies, γ˙ and Ωf, respectively. The force, F, and torque, T, acting on the cell and flagellum (subscripts c and f, respectively) are linearly related to their speed and rotation by resistance matrices:(ℱcTc)=−(σc−ϵc−ϵcτc)⋅(Vγ˙)[1]and(ℱfTf)=−(σfϵfϵfτf)⋅(VΩf).[2]Note that the off-diagonal term in the resistance matrix for the cell body, εc, represents the contribution to propulsion due to rotation of the cell, a term that is usually considered to be zero and which has a negative sign due to the fact that it rotates in the opposite direction to that of the flagellum. The cell and flagellum are coupled by the mechanical constraint that the organism swims in a force-free and torque-free state:ℱc+ℱf=0, and Tc+Tf=0.[3]Using this approach, the cell motility can be obtained asK=V/(γ˙l)=1lbτc+ϵcσc+c+bϵc,[4]where b = εf/τf and c=σf−ϵf2/τf. As discussed earlier, the cell can be modeled as a straight rod tilted away from the direction of swimming by θ (Fig. 3). The components of the resistance matrix can be readily derived asσc=l(C‖⁡cos2⁡θ+C⊥⁡sin2⁡θ),[5]τc=l[(C⊥⁡cos2⁡θ+C∥⁡sin2⁡θ)R2+Csa2⁡cos⁡θ],[6]andϵc=lR⁡cos⁡θ⁡sin⁡θ(C⊥−C∥),[7]where C‖ and C⊥ are the anisotropic translational drag coefficients of the rod (17) and Cs is its rotational drag coefficient,Cs∼4πμ.[8]R is the radius of the helical trajectory with 2R ∼ l sin θ as the projection of the cell length in the plane perpendicular to the direction of swimming (consistent with our experimental observations, as illustrated in Fig. 3A).

To demonstrate that the cell body alone can generate thrust, consider the artificial limiting case in which the flagellum generates torque but no thrust (e.g., when the length of the flagellum is much shorter than the helical pitch and the flagellum behaves as a straight rod in a spinning motion). In this case, b = c = 0, andK=12sin2⁡θ⁡cos⁡θ(C⊥−C∥)/(C∥⁡cos2⁡θ+C⊥⁡sin2⁡θ)[9](Fig. 4A, dotted line).

Fig. 4.
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Fig. 4.

Dependence of cell motility K=V/(γ˙l) on the precession angle θ for both the pushing (filled) and pulling (open) states. Symbols of identical sizes and colors represent the same individual cells. For each individual cell, the precession angle for the pushing mode (filled) is typically higher than that in the pulling mode, consistent with Fig. 2C. Cell motility, K, shows a trend of monotonic increase with θ. The dashed curve shows prediction from the resistive force-type model in the limit of vanishing flagellum length. The solid black line shows the theoretical prediction of K vs. θ achieved using the typical length and geometry of C. crescentus cell body and flagellum. The shaded area marks the range of the precession angles for the pushing case. The cell speed normalized by the external torque T vs. the precession angle (blue line, labeled on the right) shows a peak near the center of the distribution for pushers.

To apply the theory to our experimental data, we model the flagellum using slender-body theory, from which we can calculate σf, τf, and εf with good accuracy (19). Using typical values for the geometry of C. crescentus [radius af ∼ 0.01 μm, arc length per pitch Λ ∼ 1.1 μm, total length L ∼ 6 μm, pitch angle θf ∼ 0.65 rad (14)], we calculate b = 1.1 μm−1 and c = 0.65 μm × 4πμ. Using these values in Eq. 4, the predicted cell motility agrees extremely well with our observations (Fig. 4). We believe that the scatter in the data is likely due to variations in the flagellum length which we cannot measure using our current technique. At first glance, the observed pitch angle, θ, does not appear to optimize cell motility. However, plotting the swimming speed, normalized by motor torque (Fig. 4, blue line) demonstrates that the observed range of precession angles does center around the optimal speed for a fixed torque.

Selection of the Precession Angle.

The origin of the precession of the cell body almost certainly lies in the flexibility of the hook protein that connects the flagellum and cell body (13), and our observations suggest that each cell has a distinct preferred value of precession angle, θ. However, despite the success of our simple theory, it is not clear what determines this preferred angle. Potential candidates include features of the cell geometry such as length l, radius a, or curvature κ. One might also expect that at higher swimming speeds, V, the increased drag imparts a greater compressive stress on the hook and that this might lead to an increased precession angle (13). However, based on data from n = 17 individuals (each recorded moving in both the forward and backward directions), we see no statistical correlation between any of these parameters and the precession angle (Table 1).

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Table 1.

Significance tests of the effect of the different cell parameters on its motility

Natural variations in the mechanical properties of the hook could lead to changes in the preferred precession angle. The length of the hook protein has been measured (in wild-type Salmonella typhimunum) as 55 ± 6 nm (3). Because elastic bending stiffness scales with the fourth power of length (20), this variation could certainly account for the differences in the precession angle, although confirming this experimentally would be challenging. Nevertheless, our observation sheds new light on the complexity of bacterial motility and could influence prevailing ideas on the role that cell geometry plays in bacterial behavior such as chemotaxis. Lastly, another question that should be addressed is whether other bacterial species also exhibit this kind of augmented propulsion and how this mechanism might affect motility enhancement in more complex environments, for example, near solid surfaces or in non-Newtonian media such as gels, viscoelastic fluids, and mucus.

Concluding Remarks.

These data demonstrate that the motion of the cell body, usually thought of as a passive cargo propelled by the flagellum, does in fact play an important role in determining the overall cell motility. In the case of C. crescentus, the cell moves in a helical trajectory, but further experiments using the tracking microscope are needed to determine if other bacteria, with different flagellar and cell body geometries, exhibit different modes of coupled motion. For example, a cell rotating with its axis tilted from the direction of travel could follow a conical, rather than helical, motion. In this case the cell rotation rate would also decrease with increasing precession angle due to the increasing rotational drag (21). However, such rotational motion is always perpendicular to the cell axis and will not contribute to thrust production. To develop a more complete picture of the effect of cell body on its motility, the theory presented here can also be extended. Several other features, such as the shape of the cell body, the detailed motion of the bacterial flagellum (or flagella), and more general (and complex) force–torque interactions between the two will need to be considered. We expect these features may further improve the agreement between the model predictions and the experimental observations and will likely play important roles for other species that will be studied in the future.

Materials and Methods

Digital 3D Tracking Microscopy.

The bacterial samples were viewed using phase contrast on an inverted microscope (Nikon TE-200, 100× oil-immersion objective). The sample was held on a motorized 2D x–y stage (Prior ProScan II) with a travel range of 100 mm × 76 mm, controlled via USB (Phidgets bipolar controllers), whereas the objective was mounted on a piezo nano positioner (PIFOC P-721.CDQ; Physik Instrumente) with a travel range of 100 μm, used to control the position in the z direction. The images of the cells were captured using a CCD camera (Allied Vision Technology) at a rate up to 208 frames per second. From each image, the cell centroid was used to determine its x–y position, and the intensity of the cell body was used to determine its z-offset from the focal plane. Using these data, the stage and objective were moved to keep the swimming bacterium in the center of the field of view. The images were also stored for postprocessing. The target position of the stage was extrapolated using the current time stamp, and the rate of updating the stage position is thus uncorrelated with the frame rate of the camera. The real-time tracking software was written in Objective-C and ran on a Quad-Core iMac.

Reconstruction of the 3D Cell Orientation.

Making use of the crescent shape of the cell body of Caulobacter, we are able to reconstruct the 3D cell kinematics from the images captured by a single camera. We prescribe the body-centerline of the cell body with an arbitrary 3D B-spline with four equally spaced control points and fit the centerline of the cell in the raw image to the projection of the 3D string onto the x–y plane using rigid rotation. By minimizing the net error over all of the frames during the tracking, we obtain an estimate of the cell geometry (length, radius, and curvature) as well as the orientation of the cell body in every frame.

Cell Culturing and Sample Preparation.

Caulobacter strain CB15 Δpilin (YB375), which lacks pili, was grown in peptone yeast extract medium (0.2% Bacto Peptone, 0.1% yeast extract, 1.2 mM MgSO4, 0.5 mM CaCl2) for 8 h at 30 °C. The culture was then transferred to a plastic Petri dish containing fresh growth medium. The sample was incubated overnight at 30 °C and subjected to 40 rpm gyration (Lab Line Bench Top Shaker), allowing cells to differentiate into reproductive stalked cells, which attach to the bottom of the dish. The dish was then rinsed with deionized water before adding fresh growth medium and returning it to the incubator for several hours to ensure the health of attached stalked cells. To prepare a sample for observation, the dish was thoroughly rinsed with DI water and then filled with 1 mL of growth medium. The dish was allowed to sit for 5 min to allow daughter swarmer cells to separate from the attached predivisional cells. The fresh swarmer cells (mostly at the same stage of their life cycle) were then harvested in their suspension for measurement (22). A drop of sample was sealed between a glass slide and a coverslip using either vacuum grease (separation, d, ∼20 μm) or a spacer with hollow square center cut from a sheet of Parafilm M laboratory film (separation, d, ∼150 μm). We found no major difference in our results for the two geometries.

Acknowledgments

We thank H. C. Berg for helpful discussions. This work is supported by National Science Foundation Grants CBET-0854108 and CBET-1336638.

Footnotes

  • ↵1To whom correspondence should be addressed. Email: bliu27{at}ucmerced.edu.
  • Author contributions: B.L., T.R.P., and K.S.B. designed research; B.L., M.G., M.M., J.X.T., T.R.P., and K.S.B. performed research; B.L. and M.G. analyzed data; and B.L., T.R.P., and K.S.B. wrote the paper with input from other coauthors.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission.

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Helical motion of the cell body enhances Caulobacter crescentus motility
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Helical motion of the cell body enhances motility
Bin Liu, Marco Gulino, Michael Morse, Jay X. Tang, Thomas R. Powers, Kenneth S. Breuer
Proceedings of the National Academy of Sciences Aug 2014, 111 (31) 11252-11256; DOI: 10.1073/pnas.1407636111

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Helical motion of the cell body enhances motility
Bin Liu, Marco Gulino, Michael Morse, Jay X. Tang, Thomas R. Powers, Kenneth S. Breuer
Proceedings of the National Academy of Sciences Aug 2014, 111 (31) 11252-11256; DOI: 10.1073/pnas.1407636111
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  • Applied Physical Sciences
  • Biological Sciences
  • Biophysics and Computational Biology
Proceedings of the National Academy of Sciences: 111 (31)
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