Amplified effect of Brownian motion in bacterial near-surface swimming

Results and Discussion

Characteristic Swimming Trajectories Observed by Dark-Field Microscopy.

We start off with observation of the swimming trajectories of C. crescentus swarmer cells near a glass surface by dark-field microscopy. C. crescentus is a Gram-negative bacterium that, at the developmental stage called the swarmer phase, has a single polar flagellum (9, 10). We use a mutant strain that does not express pilus, thus decoupling specific adhesion from near-surface swimming. The mutant swarmer cells swim back and forth like the wild type, but they display circular trajectories only when swimming backward near a surface. A typical circular trajectory observed is shown in Fig. 1B, which is similar to those observed for E. coli (11, 12) and Vibrio alginolyticus (13, 14). These circular trajectories can be explained based on the hydrodynamic property that a viscous fluid closer to a stationary boundary exerts more drag on an moving object than one further away (7). Specifically, swimming flagellated bacteria are propelled by their rotating helical flagellar filament, and the cell body rotates in the opposite direction to counterbalance it. According to the hydrodynamic property stated above, the rotating cell body and flagellar filament each experience a net lateral drag force near the glass surface simply because more drag force per unit area is exerted on the side facing the glass surface. Because the cell body and the flagellar filament rotate in opposite directions, the lateral forces acting on them also point in opposite directions (Fig. 1A), which give rise to a net torque. This torque constantly alters the bacterial swimming direction, resulting in a circular trajectory. Note that the observed trajectory is not a perfect circle with a fixed radius. Instead, deviations from the circular trajectory can be characterized by a locally varying radius of curvature along the swimming path, as shown in Fig. 1C.

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Fig. 1.

Circular swimming trajectory of a Caulobacter swarmer cell. (A) A schematic drawing illustrating the backward swimming of a cell above a surface. The 2 solid curved arrows indicate the rotation directions of the cell body and flagellar filament, and the 2 straight arrows indicate the net lateral drag forces on them. The dashed arrow depicts the circular trajectory. (B) An overlay of consecutive frames taken at 10 frames per second showing a typical swimming trajectory observed by dark-field microscopy with a 10× objective. The arrows indicate the swimming direction. The cell followed a circular trajectory while swimming backward. (C) Local radius of curvature along the circular trajectory. For each position of interest, we selected it along with the positions 0.1 s before and after to define a circle whose radius was taken as the local radius of curvature.

The deviation from a circular trajectory can be readily attributed to Brownian motion, which changes local radius of curvature in 2 ways. First, rotational Brownian motion directly alters the orientation of cell body and thus the swimming direction (2, 15). Second, Brownian motion changes the distance between the cell and the surface, thereby varying the surface–cell hydrodynamic interaction. Because the net torque on the bacterium caused by this hydrodynamic interaction decreases as the distance between the cell and the surface increases, a larger distance leads to a more gradual turning of the swimming direction and thus a larger radius of curvature, and vice versa.

TIRF Microscopy Reveals Rapid Changes of Distance from the Surface in Near-Surface Swimming.

To confirm the variation in distance between swimming Caulobacter cells and the surface, we observed swimming trajectories of fluorescently labeled cells with TIRF microscopy. TIRF microscopy uses the evanescent wave of total reflection to illuminate fluorescently-labeled cells (Fig. 2A). The intensity of evanescent light decreases exponentially as the distance from the glass surface increases (16). Only cells very close to the surface can be imaged, and the distance of the cell to the surface can be calculated from the intensity of its fluorescence image. The TIRF setup used for this study is capable of detecting the cell body only when it is within 300 nm from the surface. We use it to observe swimming trajectories of fluorescently-labeled Caulobacter swarmer cells. Instead of continuous trajectories as seen by dark-field microscopy, only segments of circular trajectories are observed (Fig. 2B). This observation indicates that a cell in a circular trajectory as shown in Fig. 1B wanders >300 nm away from the surface. Another feature of the trajectories taken by TIRF microscopy is that the fluorescence intensity of a cell varies rapidly along its trajectory, suggesting that the distance of a swimming cell from the surface varies rapidly. We calculated the distance at each position along trajectories from the TIRF images, assuming that the closest distance the cell can reach is 10 nm from the surface. Fig. 2C shows a typical segment plotted in 3D (red) and its projection (blue) on the glass surface. The variation in distance is large and random, up to >100 nm per 0.05-s interval. While swimming near the surface, the cells were actually >100 nm away over half of the instances when the images were taken.

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Fig. 2.

TIRF observation of swimming trajectories of Caulobacter swarmer cells near a glass surface. (A) A schematic drawing showing the principle of TIRF observation. The cell is labeled with NanoOrange and illuminated with blue light through a FITC filter. Arrows indicate incident light totally reflected at the glass–water interface. (B) An overlay of 71 consecutive frames taken at 20 frames per second showing segments of circular trajectories. (C) A 3D plot (red) of a segment of circular trajectory and its projection on the glass surface (blue). The distance is measured from the intensity of TIRF images with an estimated penetration depth of 80 nm [see supporting information (SI) Fig. S1].

A number of factors were considered but ruled out as possible causes of the large variation of distance. There was no flow in our experiments. The glass surface of commercial microscope coverslip is known to be locally flat. Because the large variation is found in any particular cell over time, it cannot be explained by differences among individual cells. We also considered the wobble of a swimming bacterium caused by asymmetric shape of the cell body around its long axis. The Caulobacter cell body rotates at ≈50 Hz, and the filament rotates even faster, at ≈300 Hz. From Fig. 2C, we see that the cell in 3 consecutive positions spanning 0.15 s is obviously closer than in the next 3 consecutive positions. The cell has turned ≈7.5 turns in 0.15 s, and the effect of wobble would have been largely averaged off. The fast rotation also rules out the influence of nonuniform distribution of fluorescent dyes on the cell body. Note that the variation is random in nature, and similar variation has been observed for colloidal particles of sizes close to bacteria near a surface because of Brownian motion (17). Therefore, it is reasonable to assume that the dominant source of the variation we observed is Brownian motion as well.

Dependence of Trajectory Curvature and Swimming Speed on Cell–Surface Distance.

Results of the TIRF measurements confirm the dependence of curvature of trajectory and swimming speed on the cell–surface distance. Hydrodynamic theory predicts that the cell experiences a smaller lateral force when the distance between the cell and the surface is larger (7). Therefore, the curvature of circular trajectory decreases as the distance increases. This dependence is discernible by comparing the distance of a cell from the surface and the local radius of curvature along its trajectory observed by TIRF microscopy. To better demonstrate this relationship, we measured positions of 33 trajectories and plotted the local curvature as a function of the distance. As shown in Fig. 3A, the average curvature (solid circles) clearly decreases as the distance increases, although the data are highly scattered. Similarly, we measured the dependence of swimming speed on distance. When a bacterium swims near a surface, it experiences a larger drag than when in bulk fluid. The closer the bacterium is to the surface, the larger the drag. Therefore, it swims slower when closer to the surface. The results shown in Fig. 3B support this prediction, although the data are also highly scattered. Hydrodynamics also predicts that the dependence of speed on distance when the cell swims parallel to the surface is not as strong as when the cell approaches the surface. It has been shown that there is no observable difference in swimming speed parallel to the surface between cells < 10 μm and >10 μm from the surface, whereas the approaching speed is heavily influenced by the distance (11, 18). Our observation indicates that even within a couple of hundred nanometers the distance dependence of swimming speed parallel to the surface is modest.

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Fig. 3.

Dependence of local curvature (A) and swimming speed (B) on the distance. The curvature at a position is the inverse of its local radius of curvature and its corresponding distance averaged over the 3 positions used for determining the curvature. The speed is determined by the displacement between 2 consecutive positions, and the corresponding distance from the surface is an average of that of the 2 positions. An open circle represents a local data point in a trajectory. The data were collected from 33 trajectories. Each solid circle is an average of 20 data points binned by the distance. The dashed line is a linear fit, serving as a guide to the eye for the average curvature and speed.

To explain the experimental observation, we performed a theoretical analysis of the bacterial swimming near a surface by considering 3 physical factors of relevance. Besides the well-studied hydrodynamic interaction (7, 8, 14, 19), the role of van der Waals and electrostatic interaction, collectively described by the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory, has also been extensively discussed in the literature (11, 12, 20). We describe below simulations that include the treatments of hydrodynamic interaction and the DLVO theory. Because we have observed the influence of Brownian motion with TIRF microscopy, we also include in the simulations random forces and torques responsible for translational and rotational Brownian motion.

Simulation Without Brownian Motion.

We first examined the swimming trajectories without Brownian motion, in which case only hydrodynamic and DLVO interactions were considered. The cell was approximated as a sphere connected rigidly to a helical filament (Fig. 4A). We chose the spherical radius to be 0.4 μm, which matches a Caulobacter swarmer cell body in terms of the viscous drag in the swimming direction (7). For a bacterium swimming near a surface, there is a hydrodynamic effect that tends to bring it toward the surface (7). However, the cell body does not come into close contact with the surface because of electrostatic repulsion since the opposing surfaces are both negatively charged. After accounting for the van der Waals interaction, which is attractive, there is a secondary minimum in the combined DLVO energy curve, which defines the most preferable distance of the cell from the surface. Using the typical parameters for a bacterium near a glass surface listed in Fig. 4, the secondary minimum is found at distance h = 17.5 nm away from the surface (see Fig. S3). The simulation considering DLVO and hydrodynamics shows that a backward-swimming cell reaches an equilibrium spacing of 17.1 nm from the surface while swimming in a perfect circle of radius 7.9 μm (magenta in Fig. 4C), maintaining a constant tilt angle of 0.04. Changing physical parameters such as the Hamaker constants and the surface potentials in the simulation would result in different predicted values of radius, distance, and tilt angle. Whatever values we obtain, this oversimplified treatment does not agree with the microscopy observation, which manifests the strong influence of Brownian motion.

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Fig. 4.

Model and simulations of near-surface swimming. (A) A simple model bacterium (see also Fig. S2) with its position specified by the closest distance h of its spherical head to a surface and angle θ of its helical tail with respect to the surface. We chose Cartesian coordinates originating at the sphere center, with X₁ parallel to the flat surface and perpendicular to the axis of the helical filament, X₂ along the helical axis, and X₃ perpendicular to both X₁ and X₂. The sphere is pulled backward by a rotating helix joined by a rotary motor. (B) An example of simulated angle (blue) and distance (red) of a trajectory as functions of time. (C) The projection of a simulated trajectory on the surface. The cell starts at h = 20 nm and θ = 0. Arrows indicate the swimming direction. The magenta represents the trajectory when Brownian motion is omitted from the simulation. Parameters for the simulation are: sphere radius, 0.4 μm; flagellar filament radius, 7 nm; helix radius, 0.14 μm; helix pitch length, 1.08 μm; filament length, 6 μm (27); Hamaker constant, 1 × 10⁻²¹ J; zeta potential, −20 mV (20); Debye length, 2 nm; and time interval Δt = 10⁻⁶ s. (D) A 3D plot (red) of the last 1.6 s of the trajectory. Blue line is its projection on the coverslip. (E and F) Dependence of local curvature (E) and the swimming speed (F) on the cell body to surface distance. The data were acquired from 200 simulated trajectories. Red dots are values each averaged over 100 points binned by distance.

Simulation with Brownian Motion.

By inclusion of Brownian motion, the simulation generates trajectories comparable with experimental observation. Fig. 4B gives an example of simulated angle and distance over time. The distance fluctuates quickly with time at amplitudes up to a few hundred nanometers, in agreement with the TIRF microscopy observations. Fig. 4C shows an example of a 2D trajectory (black), which is an imperfect circle, much like what is observed with dark-field microscopy (Fig. 1B). A 3D plot of the last 1.6 s of the trajectory is shown in Fig. 4D, by the end of which the cell swam away from the surface. Local radius of curvature along the trajectory also displays a large variation, in agreement with the microscopy observation. To show the dependence of curvature on distance, we plot a large set of data acquired from >200 simulated trajectories. The results are shown in Fig. 4E. The curvatures at a given distance are highly scattered, but the average curvature decreases as the distance increases. Both features are consistent with the measured data as shown in Fig. 3A. Similar comparison is made between the dependence of simulated (Fig. 4F) and measured (Fig. 3B) swimming speed on the distance. These simulation results indicate that, by including Brownian motion, the main features observed with dark-field and TIRF microscopy are reproduced.

Coupling Between Brownian Motion and Hydrodynamic Interaction.

Further analysis suggests that the coupling between Brownian motion and hydrodynamic interaction is the primary cause of the large variation in curvature in Fig. 3A. From hydrodynamic theory, the curvature depends not only on the distance of cell body to the surface h, but also on that of flagellar filament, which is described by the tilt angle θ (Fig. 4A), as discussed by Goto et al. (8). To show these dependencies, we calculated the curvature as a function of the distance and the angle, in the absence of Brownian motion. As shown in Fig. 5, the curvature decreases when the distance increases or when the helix tilts away from the surface. These dependencies are primarily the result of hydrodynamic interaction. DLVO interaction is perpendicular to the surface and does not affect the curvature. From Fig. 5 we can understand the large variation in curvature in Figs. 3A and 4E when Brownian motion is considered. Brownian motion causes fluctuations in both distance and angle (see Fig. 4B, for example). At a given distance, the model bacterium samples through various angles caused by Brownian motion, and accordingly different values of curvature result. Fig. 5 shows a range of curvatures as a function of distance of the cell body from the surface and the tilt angle of the flagellar filament calculated for the model bacterium. Additional variations are the direct consequence of rotational Brownian motion around the axis perpendicular to the surface, which accounts for a variance in curvature of ≈0.01 μm⁻¹. This value is smaller than the one caused by the coupling between Brownian motion and hydrodynamic interaction. Note that the fluctuations in distance h and angle θ are not directly seen on the trajectory, which consists of images of the cell body projected onto the surface. The fluctuations in distance and angle alter the curvature only through coupling with the hydrodynamic interaction, and the coupling magnifies the influence of Brownian motion.

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Fig. 5.

Dependence of curvature on angle and distance without Brownian motion. Color bar represents curvature in the unit of μm⁻¹. Schematic drawings at the 4 corners depict the respective positions and orientations of the model bacterium to the surface.

Evident by both microscopy observations and computer simulations, Brownian motion significantly influences bacterial swimming trajectory near a surface, which subsequently affects swimming-related behaviors, such as food uptake and adhesion. Brownian motion increases the range of food scavenging near surfaces, which is illustrated in Fig. 4C. Without Brownian motion a bacterium can only search a small area repeatedly, whereas with Brownian motion the cell wanders through a much larger area.

Brownian motion is also an important factor in the consideration of adhesion and biofilm formation. On one hand, because of Brownian motion, the average distance of the cell body to the surface has increased dramatically from 17 to >100 nm, rendering the adhesion process less sensitive to ionic conditions than expected based on solution electrostatics (20–22). However, the random collisions caused by Brownian motion also provide occasional close contact, which might facilitate adhesion by allowing for some specific interaction at molecular proximity to take place.

The coupling between Brownian motion and hydrodynamics occurs not only to motile bacteria as studied in this article, but also to nonmotile microbes and cells, and colloidal particles near the surface. For example, platelets undergo constant Brownian motion while flowing in blood vessels (23, 24). The moving speed of a platelet depends on its distance to the wall, which is constantly altered by Brownian motion. Accordingly, Brownian motion is coupled to the shear flow and thereby affects platelet transport along the vessel. Another relevant situation occurs during the transport of microorganisms through porous media such as sand or soil (25), where the speed of a cell also depends on its position in the pores. Therefore, illuminating the intricate coupling between Brownian motion and hydrodynamics extends our understanding of their broader roles in a wide range of biological functions and phenomena.