Autonomously responsive pumping by a bacterial flagellar forest: A mean-field approach

This study is motivated by a microfluidic device that imparts a magnetic torque on an array of bacterial flagella. Bacterial flagella can transform their helical geometry autonomously in response to properties of the background fluid, which provides an intriguing mechanism allowing their use as an engineered element for the regulation or transport of chemicals in microscale applications. The synchronization of flagellar phase has been widely studied in biological contexts, but here we examine the synchronization of flagellar tilt, which is necessary for effective pumping. We first examine the effects of helical geometry and tilt on the pumping flows generated by a single rotating flagellum. Next, we explore a mean-field model for an array of helical flagella to understand how collective tilt arises and influences pumping. The mean-field methodology allows us to take into account possible phase differences through a time-averaging procedure and to model an infinite array of flagella. We find array separation distances, magnetic field strengths, and rotation frequencies that produce nontrivial self-consistent pumping solutions. For individual flagella, pumping is reversed when helicity or rotation is reversed; in contrast, when collective effects are included, self-consistent tilted pumping solutions become untilted nonpumping solutions when helicity or rotation is reversed.

Figure 1

The flagellar forest array within a Helmholtz coil system for 3D magnetic actuation (courtesy of M. J. Kim, BaST Lab, SMU).

Figure 2

Diagram of a single flagellum with tilt $\theta$ and rotation rate $\mathrm{\Omega }$. The vertical plane in blue is an infinite upper half-plane through which the flux $Q$ is calculated.

Figure 3

Nondimensionalized volumetric flow rate over one rotational period for the 10 helical bacterial polymorphic forms as a function of tilt angle $\theta$. Rotational direction is clockwise about the helical axis.

Figure 4

A regular $D\times D$ array of helical flagella, show here with equal tilt angle and phase. The central helix is colored black and made bold solely for emphasis in the mean-field methodology.

Figure 5

An example of the epicyclic motion of the helical axis on a unit sphere and the convergence to a stable orbit about $\{{\theta }_0,{\varphi }_0\}$ for a background shear ${\mathbf{U}}_{\infty }\left(\mathbf{x}\right)=\dot{\gamma }{x}_1\widehat{{\mathbf{x}}_{\mathbf{2}}}$. Scaled helix at final position shown in red. Inset: a highlighted epicycle to show helical dynamics.

Figure 6

The self-consistent nondimensionalized $y$ component of the shearlike velocity profile ${\mathbf{U}}_{\infty }\left(\mathbf{x}\right)$ induced on the central helix by all other helices in the array versus the normalized arc length for Mason number $\mathrm{Ma}=\mu \omega R^3/\left|\mathbf{m}\right|\left|\mathbf{B}\right|=3.32\times {10}^{-3}$ and array separation distance $d=L$.

Figure 7

An example of precessional pumping motion for a pumping angle ${\theta }^{*}$. The direction of precession is clockwise (the same direction as the rotating magnetic field $\mathbf{B}$). Inset: epicycles with circles drawn through their average helical axis about the $x_1$ axis.

Figure 8

Stable pumping angles ${\theta }^{*}$ in a phase space of scaled Mason number $\mathrm{Ma}/{\mathrm{Ma}}_{\mathrm{step}-\mathrm{out}}$ and grid separation distance $d$, normal ($n=2$) form, clockwise rotation. Patch faces are shaded by nondimensional flow rate. “X” indicates self-consistent solution is upright (nonpumping). Counterclockwise rotation always yields nonpumping behavior.

Figure 9

Stable pumping angles ${\theta }^{*}$ in a phase space of scaled Mason number $\mathrm{Ma}/{\mathrm{Ma}}_{\mathrm{step}-\mathrm{out}}$ and grid separation distance $d$, curly ($n=5$) form, counterclockwise rotation. Patch faces are shaded by nondimensional flow rate. “X” indicates self-consistent solution is upright (nonpumping). Clockwise rotation always yields nonpumping behavior.

Figure 10

A $(2D+1)\times 1$ line of helical flagella, shown here with equal tilt angle and phase. The central helix is colored black and made bold solely for emphasis in the mean-field methodology.

Figure 11

Change in tilt angle $\mathrm{\Delta }\theta$ (blue circles) and azimuthal angle $\mathrm{\Delta }\varphi$ (red squares) versus initial azimuthal angle $\varphi$ for an initial tilt $\theta =27.5^{\circ }$. We find a stable pumping configuration for $\{{\theta }^{*},{\varphi }^{*}\}=\{27.5^{\circ },60.7^{\circ }\}$.

Figure 12

Log-log plots (base e) of overall error decay rates versus array size for the (a) $x$-, (b) $y$-, and (c) $z$-component array velocities for array sizes $D=11,13,15,17$ (the solid blue lines). Decay rates from the Blakelet analysis (the blue dashed lines) are overlayed with the analytical slope from Eq. (A3).

Figure 13

Plots of (a) $x$-, (b) $y$-, and (c) $z$-component array velocities at the central helix end point for array size $D$ ranging from 3 to 39 (the solid blue curves). Resulting infinite array extrapolation (the blue dashed lines) from the velocity components for arrays of size $D=11$ and $D=17$ (circled).