Fluid mechanics of swimming bacteria with multiple flagella

It is known that some kinds of bacteria swim by forming a bundle of their multiple flagella. However, the details of flagella synchronization as well as the swimming efficiency of such bacteria have not been fully understood. In this study, swimming of multiflagellated bacteria is investigated numerically by the boundary element method. We assume that the cell body is a rigid ellipsoid and the flagella are rigid helices suspended on flexible hooks. Motors apply constant torque to the hooks, rotating the flagella either clockwise or counterclockwise. Rotating all flagella clockwise, bundling of all flagella is observed in every simulated case. It is demonstrated that the counter rotation of the body speeds up the bundling process. During this procedure the flagella synchronize due to hydrodynamic interactions. Moreover, the results illustrated that during running the multiflagellated bacterium shows higher propulsive efficiency (distance traveled per one flagellar rotation) over a bacterium with a single thick helix. With an increasing number of flagella the propulsive efficiency increases, whereas the energetic efficiency decreases, which indicates that efficiency is something multiflagellated bacteria are assigning less priority to than to motility. These findings form a fundamental basis in understanding bacterial physiology and metabolism.

Figure 1

Bundling behavior and trajectory of body center motion for a bacterium with three flagella and a random initial configuration.

Figure 2

Time course of swimming behavior during the bundling process: (a) bundling index $\gamma$; (b) energy rate ${\dot{e}}_{\mathrm{in}}$; (c) velocity component in the final swimming direction $v$; (d) the angular velocities of body ${\omega }_{\mathrm{B}}$ and on average for flagella ${\overline{\omega }}_{\mathrm{F}}$ in observation space. The angular velocities are divided by the angular velocity of an isolated rotating flagellum ${\omega }_0$.

Figure 3

Bundling index $\gamma$ over time of a bacterium with a total of three flagella for clockwise rotation of 1, 2, and all flagella.

Figure 4

Trajectory of a bacterium tumbling by two flagella driven clockwise.

Figure 5

Synchronization over time of two flagella for different initial phase separations (gray scales).

Figure 6

Bundling and synchronization of a bacterium with five flagella. (a) Image of flagellar configuration at $t=0$ (left), $t=750$ (center), and $t=1500$ (right); (b) synchronization over time.

Figure 7

Comparison between the simulation results of a bacterium with multiple flagella (flagellar bundle) and the analytical results of an ellipsoid with single helix with filament radius ${\epsilon }_{\mathrm{helix}}=\epsilon \sqrt{N_{\mathrm{F}}}$ (single helix). (a) Averaged swimming velocity with respect to number of flagella on logarithmic axis; (b) efficiency of flagellar propulsion with respect to number of flagella.

Figure 8

Energy efficiency of a bacterium with multiple flagella with respect to number of flagella on logarithmic axis.

Figure 9

Effect of flagella number on the bundling time. (a) Bundling times with body $t_{\mathrm{bun}}$ and without body $t_{\mathrm{bun}}^{*}$ over number of flagella on double logarithmic axis; (b) ratio of both times $t_{\mathrm{bun}}/t_{\mathrm{bun}}^{*}$.