Antiphase Synchronization in a Flagellar-Dominance Mutant of Chlamydomonas

Groups of beating flagella or cilia often synchronize so that neighboring filaments have identical frequencies and phases. A prime example is provided by the unicellular biflagellate Chlamydomonas reinhardtii, which typically displays synchronous in-phase beating in a low-Reynolds number version of breaststroke swimming. We report the discovery that ptx1, a flagellar-dominance mutant of C. reinhardtii, can exhibit synchronization in precise antiphase, as in the freestyle swimming stroke. High-speed imaging shows that ptx1 flagella switch stochastically between in-phase and antiphase states, and that the latter has a distinct waveform and significantly higher frequency, both of which are strikingly similar to those found during phase slips that stochastically interrupt in-phase beating of the wild-type. Possible mechanisms underlying these observations are discussed.

Figure 1

Waveforms of C. reinhardtii. Logarithmically scaled residence time plots averaged over $\mathcal{O}({10}^2)$ beats overlaid by waveforms, color coded in time. The $wt$ displays IP breaststroke beating (a) stochastically interrupted by phase slips (b) in which one flagellum (here, trans) beats faster with an attenuated waveform. ptx1 displays an IP state (c) nearly identical to the wild-type (a) and a high-frequency AP state (d). Large and small ovals indicate cell body and eyespot, respectively.

Figure 2

Beating dynamics. (a) Phase difference $\Delta =({\psi }_{\mathrm{trans}}-{\psi }_{\mathrm{cis}})/2\pi$ showing half-integer jumps between IP and AP states. Insets show waveforms in the two states. (b) Instantaneous frequencies of AP and IP states. (c) Distribution of instantaneous frequencies during IP beating and of the faster flagellum during slips, across all sampled $wt$ cells. (d) IP and AP instantaneous frequency distributions, across all sampled ptx1 cells.

Figure 3

Synchronization dynamics. Phase plane of polar angles ${\theta }_{\mathrm{cis},\mathrm{trans}}$ reveals the IP (green) and AP (red) synchronization of a single cell (a), and (b) the average over six cells, averaged over $\mathcal{O}({10}^3)$ beats and resampled at 15 points, equally spaced in time. Shaded regions in (a) indicate 1 standard deviation of fluctuations. (c),(d) Sample time series for evolution of ${\theta }_{\mathrm{cis},\mathrm{trans}}$ during a transition event. (e) Phase difference dynamics during $\mathrm{AP}\to \mathrm{IP}$ (orange) and $\mathrm{IP}\to \mathrm{AP}$ (blue) transitions for 60 events, with means (solid lines) and standard deviations (shaded), vertically aligned by plotting difference modulo 1. Dashed lines are fits to data.

Figure 4

Synchronization mechanisms. (a,b) Top row: Motion of sphere 1 at two possible phases ${\varphi }_1$ (solid and open circles) induces flows (blue arrows) which alter the trajectory of sphere 2, either speeding it up (green), or slowing it down (red). (a,b) Bottom row: Converse perspective. (c) Elastic coupling between flagella can induce either IP or AP modes.