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).

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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).

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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 ℱ_c = 0, and f_γ = 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 θ_h, 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 θ_h, 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 Ω_f, 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, ε_c, 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 = ε_f/τ_f 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 C_s 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).

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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 σ_f, τ_f, and ε_f with good accuracy (19). Using typical values for the geometry of C. crescentus [radius a_f ∼ 0.01 μm, arc length per pitch Λ ∼ 1.1 μm, total length L ∼ 6 μm, pitch angle θ_f ∼ 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).

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.