Hydrodynamics Versus Intracellular Coupling in the Synchronization of Eukaryotic Flagella

The influence of hydrodynamic forces on eukaryotic flagella synchronization is investigated by triggering phase locking between a controlled external flow and the flagella of C. reinhardtii. Hydrodynamic forces required for synchronization are over an order of magnitude larger than hydrodynamic forces experienced in physiological conditions. Our results suggest that synchronization is due instead to coupling through cell internal fibers connecting the flagella. This conclusion is confirmed by observations of the vfl3 mutant, with impaired mechanical connection between the flagella.

Figure 1

(a) Experimental setup. (b) Snapshots representing the stroke patterns of C. reinhardtii, for a cell with $f_0=54.3\mathrm{Hz}$ and an external flow with $f_F=52.7\mathrm{Hz}$ and $U_F=\pm 527\mu \mathrm{m}.{\mathrm{s}}^{-1}$. Time between snapshots: 6 ms. Arrows represent the direction of the background flow. In this sequence, the flagella beat with the flow and the flagellar tip motion has the same direction as the external flow. (c) Snapshots of same cell as in (b) 0.33 s later. The flagella beat against the flow.

Figure 2

Experimental data for one data set. The same cell is subject to an external flow $A_F=5\mu \mathrm{m}$ and $f_F=49.9–60.3\mathrm{Hz}$. (a) Time variations of the phase difference between the flagella and the external flow for separate recordings. Inset for $f_F=57.5\mathrm{Hz}$ shows a stepwise decay of $\mathrm{\Delta }(t)$. Inset for $f_F=51.2\mathrm{Hz}$ shows fluctuations during phase locking. (b) Spectrograms of flagellar motion for different $f_F$. Black (white) correspond to a low (high) amplitude in the frequency spectrum. From top: no synchrony ($f_F=60.3\mathrm{Hz}$), partial locking ($f_F=55.8\mathrm{Hz}$), and complete phase locking ($f_F=55.0\mathrm{Hz}$). (c) $f_F-f$ as a function of $f_F-f_0$: Symbols, experimental data; solid line, $f$ is computed from Eq. (4) with the fitted values for $f_0$, ${\epsilon }_0$, and $T_{\text{eff}}$; dotted line, $f$ is computed from Eq. (4) in the limit of zero noise.

Figure 3

Histograms showing the distributions of phase difference $\mathrm{\Delta }(t)$ in separate recordings of the same cell, with external flows at $A_F=5\mu \mathrm{m}$ and different $f_F$. Red line: $\widehat{P}(\mathrm{\Delta })$ computed with the fitted parameters $f_0=53.6\mathrm{Hz}$, ${\epsilon }_0/f_0=0.042$, and $T_{\text{eff}}/f_0=0.0006$.

Figure 4

(a) Synchronization region in $(f_F,U_F)$ domain. Each marker represents a separate recording. $U_F$ ranges from $86\mu \mathrm{m}·{\mathrm{s}}^{-1}$ to $1300\mu \mathrm{m}·{\mathrm{s}}^{-1}$ and is normalized with $U_0$. Color map represents the time fraction when flagellar beating is phase locked with external flow. It is equal to 1 (black) when phase locking is observed for the entire 30 s recording, while it is 0 (white) when phase locking is never observed. (b) Coupling strength $\epsilon$ as a function of external flow $U_F/U_0$.