Biopolymer dynamics driven by helical flagella

Microbial flagellates typically inhabit complex suspensions of polymeric material which can impact the swimming speed of motile microbes, filter feeding of sessile cells, and the generation of biofilms. There is currently a need to better understand how the fundamental dynamics of polymers near active cells or flagella impacts these various phenomena, in particular, the hydrodynamic and steric influence of a rotating helical filament on suspended polymers. Our Stokesian dynamics simulations show that as a stationary rotating helix pumps fluid along its long axis, polymers migrate radially inward while being elongated. We observe that the actuation of the helix tends to increase the probability of finding polymeric material within its pervaded volume. This accumulation of polymers within the vicinity of the helix is stronger for longer polymers. We further analyze the stochastic work performed by the helix on the polymers and show that this quantity is positive on average and increases with polymer contour length.

Figure 1

Snapshots from a single simulation of a rotating helix and an advected polymer taken at times $t_0<t_1<t_2$. A torque $\mathbf{\tau }$ is distributed uniformly on the helix and, as it rotates, it generates a pumping flow. The polymer is drawn in radially and axially toward the helix, stretching in the process due to the induced shear flow. Once captured, it winds around the helix and is pumped axially in the direction of the average flow.

Figure 2

[(a), (b)] Cross sections of the instantaneous flow field generated by a rotating helix, decomposed into components $v_z(\mathit{r},t)$ and $v_{\perp }(\mathit{r},t)$ respectively. (c) Streamlines of the full flow field $\mathit{v}(\mathit{r},t)$ show both the net flow along $\widehat{\mathit{z}}$ and the chiral winding caused by the rotation of the helix. (d) time-averaged fields as a function of radial distance from helix center line, ${\overline{v}}_z(r,t)$ and ${\overline{v}}_{\perp }(r,t)$. Within the volume of the helical filament $\left(r<R\right)$, the fluid rotates like a rigid body, $v_{\perp }\propto r$, and flows axially at a constant rate $v_r=\text{const}$. (e) Slice in the $xy$ plane of the mean rotational flow. (f) Slice in the $xz$ plane of the mean axial flow.

Figure 3

The three components of the helix friction tensor as a function of pitch length $2\pi /\kappa$. The data (black dots) are determined by measuring $f_z$, and ${\tau }_z$ separately as functions of $v$ and $\omega$ for different $\kappa$ in order to determine the matrix components ${\mu }_{tt},{\mu }_{tr}$, and ${\mu }_{rr}$. The solid lines are calculated using analytical results from slender body theory (Eqs. (44) from Ref. [16]). Increasing the pitch length decreases the isotropic components of the friction tensor, while increasing the coupling between rotation and translation. Inset: The helix shapes for the extremal choices of pitch.

Figure 4

[(a)–(c)] Comparison of the time-averaged flow fields generated by (a) MCPD fluid simulations of a cell attached to a wall (b) SD simulations of a helix rotating in a bulk fluid computed in the $xy$ plane. These two measured fields differ only by Gaussian fluctuations inherent in the MPCD simulation as shown in (c). [(d)–(f)] Same as panels (a)–(c) except in $xz$ plane. Panel (f) shows that there are systematic differences in the flow fields in this plane, mainly due to the presence of the wall. The contour-enclosed region in the lower quarter of the image represents where the flow field differs by $|{\overline{v}}_z^{\mathrm{MPCD}}-{\overline{v}}_z^{\mathrm{SD}}|>0.25$. Everywhere else, the fields behave similarly.

Figure 5

[(a)–(d)] Polymer distributions averaged over all simulations and over all time show the net behavior of polymers being pumped in the positive-$z$ direction, given an initial distribution of polymers on a disk of radius $r=4R$, located at $z\approx -10R$ in cylindrical coordinates $(r,z)$. Each distribution is averaged over 200 simulations for polymers of size (a) $N=1$ (colloidal tracers), (b) $N=10$, (c) $N=30$, and (d) $N=50$. The net behavior is a drift toward the right of each image, due to the helix (not shown) pumping the fluid. [(e)–(h)] The same data as in panels (a)–(c) respectively, but plotted as distributions over $r$ only, with each curve representing contiguous quarter intervals of the simulation time. In each case, the initial quarter is the shaded region, and we observe that for increasing polymer size, a strong tendency for the polymers to migrate inward is observed. For larger polymers, this tendency is stronger and occurs faster.

Figure 6

[(a)–(c)] Stochastic excess work performed by a rotating helix in transporting polymers of size $N=(10,20,40)$ respectively in 200 experiments performed for each case. The markers correspond to the point at which the center of mass of the polymer exits the negative end of the helix. Once this occurs, the polymers collapse and advect with the fluid; no more work is performed on them. The markers are color coded by the point of closest approach to the central long axis of the helix for the polymer in that experiment, ${\text{min}}_t\left(r\right)$. More work on average was performed to transport polymers that migrated nearer (magenta) the central axis than on polymers which failed to become captured (cyan). [(d)–(f)] Mean excess power per revolution, averaged over 200 simulations. For each time step, $d{w}_{ex}$ was calculated and smoothed with a window size of $\sim 1/4$ revolution. The shaded region corresponds to the standard error on the mean. [(g)–(i)] Mean increase in polymer energy calculated using the potentials $H_{\text{repel}}$ and $H_{\text{bond}}$ with the same smoothing and averaging procedure employed in panels (d)–(f).