Transition to bound states for bacteria swimming near surfaces

It is well known that flagellated bacteria swim in circles near surfaces. However, recent experiments have shown that a sulfide-oxidizing bacterium named Thiovulum majus can transition from swimming in circles to a surface bound state where it stops swimming while remaining free to move laterally along the surface. In this bound state, the cell rotates perpendicular to the surface with its flagella pointing away from it. Using numerical simulations and theoretical analysis, we demonstrate the existence of a fluid-structure interaction instability that causes cells with relatively short flagella to become surface bound.

Figure 1

Scanning electron microscope image of a Thiovulum majus bacterium with a slightly shrunken spherical cell body of radius $a\sim 4\mu \mathrm{m}$ and flagella $L\sim 2\mu \mathrm{m}$ long, reproduced with permission from Ref. [9] (see similar pictures in Ref. [10]).

Figure 2

Schematic diagram of a flagellated bacterium swimming near a plane surface. The cell body is modeled as a rigid sphere and the flagellar filament as either a rigid right-handed helix rotating along its axis for the numerical simulations in Sec. 3 or a rod placed along the helix axis with an active force ${\mathit{f}}_a$ acting along it for the theoretical model in Sec. 4. The cell body undergoes rigid body translation and rotation with velocities $\mathit{U}$ and $\mathit{\Omega }$, respectively. The flagellum tilt angle is denoted by $\theta$ (see Sec. 3 for description of other notations). The two possible final steady states of a bacterium near a surface are shown in the insets, namely, the surface-bound and circular-swimming states.

Figure 3

Temporal evolution plot of (a) the minimum distance between the spherical cell body surface and the rigid wall ${\delta }^{*}$, showing attraction of the bacterium toward the wall for various flagellum axial lengths, $L_{\lambda }^{*}=1–6$ (time $t$ is scaled using ${\mathrm{\Omega }}_f^{-1}$), and (b) the tilt angle $\theta /\pi$. (c) Pitchfork bifurcation of the tilt angle, $\theta /\pi$, plotted against the flagellum axial length $L_{\lambda }^{*}$ for the numerical simulations (red dashed lines/circles for sim-I and green dash-dotted lines/triangles for sim-II) and the theoretical model (blue solid lines/squares). The critical flagellum axial length below which bound state is possible, obtained by Eq. (14), is shown in purple dashed line. Bound and swimming states are represented by $\theta =\pi /2$ and $\theta \ne \pi /2$, respectively. (d) Pitchfork bifurcation of the circular radius $R$ traced by the bacteria when it swims parallel to the wall (numerical simulations).

Figure 4

Relative errors of the elements of the resistance matrix of a sphere near a wall computed by boundary element method for a uniform and non-uniform mesh when compared with exact solutions using bispherical coordinates.

Figure 5

Relative errors of the elements of the resistance matrix of a sphere near a wall obtained using lubrication and far-field theory when compared with exact solutions using bispherical coordinates.

Figure 6

Hydrodynamic torque $T_y$ acting on a rotating helix measured about its tapered end for two different heights (a) $d=1.005$, and (b) $d=0.25$ above the wall plotted against various tilt angles $\theta$ for various flagellar axial lengths $L_{\lambda }$. See the schematic diagram in Fig. 2 for an illustration of geometrical parameters and coordinate axis definition.