Hydrodynamic interactions are key in thrust-generation of hairy flagella

The important role of unicellular flagellated micro-organisms in aquatic food webs is mediated by their flagella, which enable them to swim and generate feeding currents. The flagellum in many predatory flagellates is equipped with hairs (mastigonemes) that reverse the direction of thrust compared to the thrust due to a smooth flagellum. Conventionally, the mechanism of such reversal has been attributed to drag-based thrust of individual hairs, neglecting their hydrodynamic interactions. However, at natural densities of hairs, hydrodynamic interactions are important. In fact, using fully resolved three-dimensional computational fluid dynamics, we show here that hydrodynamic interactions are key to thrust-generation and reversal in hairy flagellates, making their hydrodynamics fundamentally different from the slender-body theory governing smooth flagella. We reveal the significant role of the curvature of the flagellum, and using model analysis we demonstrate that strongly curved flagellar waveforms are optimal for thrust-generation. Our results form a basis for understanding the diverse flagellar architectures and feeding modes of predatory flagellates.

Figure 1

Appearance and function of flagellar hairs. (a) Transmission electron micrograph of Paraphysomonas sp with tubular hairs (short arrow, red) on the flagellum (long arrow, green). Image courtesy of Helge A. Thomsen. (b) Light micrograph of a tethered individual of Pteridomonas danica. The hairy flagellum drives a feeding current (dashed arrows) towards the cell in the opposite direction of the propagating flagellar wave (blue arrow) [16], and the flow goes past the prey-intercepting tentacles extending from the cell (solid black arrows). The hairs on the flagellum are too thin to be visible in light microscopy. Image courtesy of Sei Suzuki-Tellier. Scale bars in (a) and (b): $2\mu \mathrm{m}$.

Figure 2

Model morphology of Pteridomonas danica and the computational mesh. (a) The model of P. danica has a spherical cell (blue) of diameter $5.0\mu \mathrm{m}$, a flagellum (green) of length $L_f=12.5\mu \mathrm{m}$ and diameter $D_f=0.20\mu \mathrm{m}$, and hairs (red) each of length $l=1.50\mu \mathrm{m}$ and diameter $D_h=20\mathrm{nm}$ [3, 27]. (b) A snapshot of the finite volume polyhedral cells in the beat plane around the base of the flagellum.

Figure 3

Overall hydrodynamics of Pteridomonas danica. (a) CFD results of averaged flow field over the beat cycle at $l=1.50\mu \mathrm{m}$. (b) Swimming speed at different values of ${\lambda }_{\varphi }$: in the observed range of ${\lambda }_{\varphi }$ (dotted lines), there is a good agreement between CFD prediction and observation of the swimming speed. For the case with amplitude modulation factor $\delta =1.0\mu \mathrm{m}$ (${\mathrm{\Phi }}_1$), the averaged generated thrust by the flagellum $\overline{F_y}=6.8\mathrm{pN}$, consistent with indirect estimates based on observed flow fields [21], $\sim 7\mathrm{pN}$. ${\mathrm{\Phi }}_1=$ Eq. (1) with $\delta =1.0\mu \mathrm{m}$, and ${\mathrm{\Phi }}_2=$ Eq. (1) with $\delta =2.0\mu \mathrm{m}$.

Figure 4

Effect of the density of hairs on flow and thrust. The kinematic parameters in Eq. (1) are chosen based on the beat analysis of Pteridomonas danica, i.e., $f=50\mathrm{Hz}$, ${\lambda }_{\varphi }=10\mu \mathrm{m}$, $A_{\varphi }=2\pi /5$, $\delta =2.0\mu \mathrm{m}$, and $L_f=12.5\mu \mathrm{m}$. The hairs are positioned in the beat plane, and the length of the hairs is reduced from the observed length $l=1.50$ to $0.90\mu \mathrm{m}$ to avoid physical interference during the beat. Snapshots of the velocity field at (a) low density of hairs ($N=1\mu {\mathrm{m}}^{-1}$), and (b) high (observed) density of hairs ($N=7\mu {\mathrm{m}}^{-1}$). (c) Time-averaged thrust as a function of density of hairs (data points), and for a flexible, plane sheet ($N=\infty$) with the same thickness as that of the diameter of the hairs, and with the same beat kinematics. At high densities, as for P. danica, the induced flow by neighboring hairs on an individual hair becomes completely dominant over the local velocity of hairs, and hydrodynamic interactions dominate.

Figure 5

Hydrodynamic interactions among parallel hairs in sideways and lengthwise motion. Both the length of the hairs and the sideways length of the arrays are $l=2.0\mu \mathrm{m}$, and the diameter of the hairs is $20\mathrm{nm}$. (a) The induced flow by one hair on the other one, i.e., the hydrodynamic interactions between two hairs ($N=0.5\mu {\mathrm{m}}^{-1}$), is stronger when they move sideways than when they move lengthwise. (b) By adding more hairs and increasing their density, $N=7.5\mu {\mathrm{m}}^{-1}$, the flows become similar for the two directions of motion. (c) The drag on the array in sideways, $F_n$, and lengthwise, $F_t$, motion approaches the asymptotic value of an equivalent square sheet (top), and the drag anisotropy of the system vanishes (bottom). The theoretical prediction for a single hair (red symbols) is $F_n/F_t=1.6$, in agreement with the CFD results, and the theoretical analysis of two hairs in Eq. (8) predicts $F_n^{\left(2\right)}/F_t^{\left(2\right)}=1.4$, in agreement with the CFD simulations for two hairs, i.e., $N=0.5\mu {\mathrm{m}}^{-1}$.

Figure 6

Location of thrust-generation on flagella with sparse and dense hairs. We apply the kinematics in Eq. (9), and in (a)–(e) we use $f=27.5\mathrm{Hz}$, $\lambda =12\mu \mathrm{m}$, $A=2.0\mu \mathrm{m}$, $0<y<3\mu \mathrm{m}$, and $l=0.90\mu \mathrm{m}$. The dashed arrows show the overall motion of the hairs, and the red arrows show the thrust due to the hairs, ${\overline{F}}_{y,\text{hairs}}$. (a), (b) In the sparse system with $N=0.8\mu {\mathrm{m}}^{-1}$, ${\overline{F}}_{y,\text{hairs}}$ is produced at the straight segment of the flagellum. (c), (d) In contrast, in the dense system with $N=6.8\mu {\mathrm{m}}^{-1}$, ${\overline{F}}_{y,\text{hairs}}$ is produced at the curved part. (e) The thrust of the hairs and (central) flagellum in the sparse and the dense system during half of the beat period. Snapshots in (a), (c) and (b), (d) correspond to $t/T=0.14$ and 0.38, respectively. (f) The averaged thrust (over the beat cycle) due to the hairs as a function of the wavelength and with all other parameters as in (a)–(e). The thrust in the sparse system depends only weakly on the wavelength and is described well by the resistive force theory (RFT) model in Eq. (10), whereas the thrust in the dense system decreases strongly with increasing wavelength (decreasing curvature), and it is significantly overestimated by resistive force theory.

Figure 7

Parameter exploration and analytical thrust model. (a) Snapshot of the velocity field for a hairy flagellum with the kinematics in Eq. (1) and the parameters $f=50\mathrm{Hz}$, ${\lambda }_{\varphi }=10\mu \mathrm{m}$, $A_{\varphi }=2\pi /5$, $\delta =2.0\mu \mathrm{m}$, $L_f=12.5\mu \mathrm{m}$, $l=0.90\mu \mathrm{m}$, and $N=7\mu {\mathrm{m}}^{-1}$. (b) Snapshot of the velocity field produced by an equivalent plane sheet. (c) Basic thrust unit assuming that the dense system of hairs functions as a sheet (red) on either side of the flagellum (green). (d)–(f) Dimensionless thrust as a function of the dimensionless parameters $l/{\lambda }_{\varphi }$, $A_{\varphi }$, and $L_f/{\lambda }_{\varphi }$. (d) Parameters: ${\lambda }_{\varphi }=10\mu \mathrm{m}$, $A_{\varphi }=2\pi /5$, and $L_f=12.5\mu \mathrm{m}$. (e) Parameters: ${\lambda }_{\varphi }=9.0\mu \mathrm{m}$, $L_f=12.5\mu \mathrm{m}$, and $l=1.50\mu \mathrm{m}$. (f) Parameters: ${\lambda }_{\varphi }=10\mu \mathrm{m}$, $A_{\varphi }=2\pi /5$, and $l=0.90\mu \mathrm{m}$. The diamond refers to the case with $l=1.50\mu \mathrm{m}$, where hairs are slightly off the beat plane to avoid physical interference [Fig. 3]. In all CFD simulations, we used $f=50.0\mathrm{Hz}$ and $N=8\mu {\mathrm{m}}^{-1}$. CFD results (red points) and analytical model in Eq. (14) with $\alpha =3.80$ (blue curves).