Flagellar Synchronization Independent of Hydrodynamic Interactions

Inspired by the coordinated beating of the flagellar pair of the green algae Chlamydomonas, we study theoretically a simple, mirror-symmetric swimmer, which propels itself at low Reynolds number by a revolving motion of a pair of spheres. We show that perfect synchronization between these two driven spheres can occur due to the motion of the swimmer and local hydrodynamic friction forces. Hydrodynamic interactions, though crucial for net propulsion, contribute little to synchronization for this free-moving swimmer.

Figure 1

(a) Simplified flagellar beat of Chlamydomonas showing flagellar shapes at equidistant times representing a full beat cycle ($T\approx 15\mathrm{ms}$), adapted from 34. The flagellar bending waves are approximately planar and mirror-symmetric. Each point on a flagellum moves on a periodic orbit with respect to a material frame of the cell body. (b) The idealized model swimmer consists of three equal spheres connected by a frictionless scaffold. The first and second sphere (located at ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$) can move along a circular trajectory as indicated, being driven by internally generated active torques. (The arrows correspond to the case ${\omega }_0>0$).

Figure 2

(a) Phase synchronization behavior of our model swimmer (here for $m_1=-m_2<0$) can be read off from a Poincaré return map $\Lambda =\delta (T)-\delta (0)$ that tracks the change of the phase sum $\delta ={\phi }_1+{\phi }_2$ after a full revolution of the first sphere as function of initial $\delta =\delta (t=0)$. Fixed points $\Lambda =0$ correspond to limit cycles of the (${\phi }_1$, ${\phi }_2$)-phase dynamics; see Fig. 3(b1). For the dashed and solid curve, hydrodynamic interactions were neglected or accounted for, respectively. (b) Constraining translation and rotation of the swimmer by clamping the third sphere changes its synchronization behavior completely. (c) In the presence of fluctuations, the phase dynamics of the free swimmer exhibits stochastic phase slips that occur at a frequency $(2\pi {)}^{-2}d⟨{\delta }^2⟩/dt$; shown is $d⟨{\delta }^2⟩/dt$ normalized by $D{\omega }_0$ (solid curve), as well as the analytical result [25] $d⟨{\delta }^2⟩/dt\approx 4D{\omega }_0{I}_0^{-2}[\lambda /(4\pi D)]$ for the approximate dynamics given by Eq. (6), (dotted curve). Parameters: $a/l=0.1$, $R/a=5$, $\kappa =\eta l^3$.

Figure 3

(a) Depending on the ratio of the active driving torques $m_1$ and $m_2$, phase locking of the phase variables ${\phi }_1$ and ${\phi }_2$ describing our model swimmer can occur giving rise to a distinct pattern of Arnold tongues. (b) As specific example, a limit cycle with $\delta ={\phi }_1+{\phi }_2=0$ is globally attractive for opposite torques with $m_1=-m_2<0$ (b1), but repulsive for $m_1=-m_2>0$ (b2). (c) For equal torques $m_1=m_2$, the (${\phi }_1$, ${\phi }_2$) phase space is foliated by neutrally stable orbits. Parameters: $a/l=0.1$, $R/a=5$, $\kappa =\eta l^3$.

Figure 4

Flagellar synchronization of Chlamydomonas computed for the beat pattern from Fig. 1a using Eq. (7) and resistive force theory. (a) The Poincaré return map $\Lambda (\delta )$ [defined analogous to that of Fig. 2a] shows that in-phase flagellar beating with $\delta =0$ is stable with respect to perturbations. (b),(c) The synchronization parameter $\lambda =-d\Lambda /d{\delta }_{|\delta =0}$ depends on the ratio of hydrodynamic dissipation and internal dissipation (b) as well as on the size of the cell body (c). Parameters (unless indicated otherwise): flagellar length, $L=12\mu \mathrm{m}$; semiaxes of spheroidal cell body $3.7\mu \mathrm{m}$, $5\mu \mathrm{m}$ [34]; resistive force coefficients, ${\zeta }_{\parallel }=2\pi \eta /[\ln (2L/r)-3/2]$, ${\zeta }_{\perp }=4\pi \eta /[\ln (2L/r)-3/2]$ [29] with flagellar radius $r=0.1\mu \mathrm{m}$ [2], $\eta L^3/\kappa =100$.