Modeling polymorphic transformation of rotating bacterial flagella in a viscous fluid

The helical flagella that are attached to the cell body of bacteria such as Escherichia coli and Salmonella typhimurium allow the cell to swim in a fluid environment. These flagella are capable of polymorphic transformation in that they take on various helical shapes that differ in helical pitch, radius, and chirality. We present a mathematical model of a single flagellum described by Kirchhoff rod theory that is immersed in a fluid governed by Stokes equations. We perform numerical simulations to demonstrate two mechanisms by which polymorphic transformation can occur, as observed in experiments. First, we consider a flagellar filament attached to a rotary motor in which transformations are triggered by a reversal of the direction of motor rotation [L. Turner et al., J. Bacteriol. 182, 2793 (2000)]. We then consider a filament that is fixed on one end and immersed in an external fluid flow [H. Hotani, J. Mol. Biol. 156, 791 (1982)]. The detailed dynamics of the helical flagellum interacting with a viscous fluid is discussed and comparisons with experimental and theoretical results are provided.

Figure 1

A schematic showing the physiology of an E. coli cell. The flagellar filaments are randomly distributed along the cell body. Each filament is attached to a rotary motor via a flexible hook.

Figure 2

An illustration of the parametrization of the flagellar filament. The centerline of the flagellum is parameterized by $\mathit{X}(s,t)$ and its orientation is determined by an orthonormal triad $\{{\mathit{D}}^1(s,t),{\mathit{D}}^2(s,t),{\mathit{D}}^3(s,t)\}$.

Figure 3

The regularized $\delta$ function for various smoothing parameter values of $c$.

Figure 4

Snapshots of a bistable helix with ${\tau }_1=-2.1472\mu {\mathrm{m}}^{-1}$ and ${\tau }_2=1.4310\mu {\mathrm{m}}^{-1}$, rotating at 100 Hz. The shade of the rod indicates the torsion value where gray is negative (left-handed) and black is positive (right-handed). The horizontal planes show the vertical velocity of the fluid and the markers shown in magenta are passive fluid tracers.

Figure 5

A schematic diagram of the block angle $\alpha$ formed by a kink in the helical filament (left) and estimated block angles from our simulations for various helical shapes (right). Simulation results and experimental data from Hotani [52] are in good agreement.

Figure 6

Torsion along the filament over time (left) and kink propagation speed plotted against motor rotation frequency (middle) and fluid viscosity (right). The intrinsic twist values are given as ${\tau }_1=-2.1472\mu {\mathrm{m}}^{-1}$ and ${\tau }_2=2.1472\mu {\mathrm{m}}^{-1}$. The left panel illustrates how torsion changes over time at the rotating frequency of 100 Hz. The negative value (blue) of torsion corresponds to the left-handed helix and the positive value (red) corresponds to the right-handed helix. The kink speed is estimated as the slope of the curve where the filament is torsion-free. The middle and right panels show that the kink speed is approximately proportional to the motor frequency and decreases logarithmically with increasing fluid viscosity, respectively.

Figure 7

Mean fluid flux driven by a helical flagellum rotating at 100 Hz for various pitches and radii. The markers on the left panel denote the mean flux for four helical shapes commonly observed in nature [11, 17], which are shown on the right.

Figure 8

Simulation of an experiment by Hotani [17] using intrinsic torsion values ${\tau }_1=-2.1472\mu {\mathrm{m}}^{-1}$ and ${\tau }_2=1.4310\mu {\mathrm{m}}^{-1}$. Fluid (water) is flowing upward at 2000 $\mu \mathrm{m}/\mathrm{s}$. The shade of the rod indicates the torsion value where gray is negative (left-handed) and black is positive (right-handed). The arrows indicate CW (red) or CCW (blue) rotation and their length is proportional to the motion over the previous 0.008 s.

Figure 9

Values of torsion ${\mathrm{\Omega }}_3$ from simulations of Hotani's experiment plotted against time $t$ and filament position $s$. The negative value (blue) of torsion corresponds to the left-handed helix and the positive value (red) corresponds to the right-handed helix. The intrinsic torsion values used are ${\tau }_1=-2.1472\mu {\mathrm{m}}^{-1},{\tau }_2=2.1472\mu {\mathrm{m}}^{-1}$ (top) and ${\tau }_1=-2.1472\mu {\mathrm{m}}^{-1},{\tau }_2=1.4310\mu {\mathrm{m}}^{-1}$ (bottom). The fluid viscosity is $0.01\times {10}^{-4}\mathrm{g}/(\mu \mathrm{m}·\mathrm{s})$. The background flow speed is set to 2000 $\mu \mathrm{m}/\mathrm{s}$.

Figure 10

(Left) Observed handedness change for various background flow velocity and fluid viscosity. The vertical dashed line and the dashed-dot line denote the viscosity of water and methylcellulose, respectively. Both axes are plotted on a log scale. (Right) Time evolution of $n_3$ and $f_3$ at the motor for flow speed 2000 $\mu \mathrm{m}/\mathrm{s}$. In both plots, the intrinsic torsion is set to ${\tau }_1=-2.1472\mu {\mathrm{m}}^{-1}$ and ${\tau }_2=1.4310\mu {\mathrm{m}}^{-1}$.

Figure 11

Background flow velocity plotted against the energy barrier $H=\frac{a_3}{4}{\tau }^4$ for the symmetric twist energy $\tau =-{\tau }_1={\tau }_2$. The result by Coombs et al. (solid curve) was scaled to fit the data. Both axes are plotted on a log scale.

Figure 12

Kink propagation speed plotted against motor rotation frequency (left) and fluid viscosity (right) for various regularization parameters $c=2\mathrm{\Delta }s,...,6\mathrm{\Delta }s$. Here, $\mathrm{\Delta }s=0.03\mu \text{m}$, and intrinsic twist values are set to ${\tau }_1=-2.1472\mu {\mathrm{m}}^{-1}$ and ${\tau }_2=2.1472\mu {\mathrm{m}}^{-1}$. The dotted lines on the left panel denote the lines of best fit, and the dotted curves on the right panel denote a logarithmic best fit curves. The solid curve in each panel estimates the result for $c=0.01\mu \mathrm{m}$ based on the best fit parameters.