Transitions in synchronization states of model cilia through basal-connection coupling

Abstract

Despite evidence for a hydrodynamic origin of flagellar synchronization between different eukaryotic cells, recent experiments have shown that in single multi-flagellated organisms, coordination hinges instead on direct basal body connections. The mechanism by which these connections lead to coordination, however, is currently not understood. Here, we focus on the model biflagellate Chlamydomonas reinhardtii, and propose a minimal model for the synchronization of its two flagella as a result of both hydrodynamic and direct mechanical coupling. A spectrum of different types of coordination can be selected, depending on small changes in the stiffness of intracellular couplings. These include prolonged in-phase and anti-phase synchronization, as well as a range of multi-stable states induced by spontaneous symmetry breaking of the system. Linking synchrony to intracellular stiffness could lead to the use of flagellar dynamics as a probe for the mechanical state of the cell.

Keywords: Chlamydomonas reinhardtii; cilia; cytoskeleton; flagella; microhydrodynamics; oscillators; synchronization.

Figure 1.

Physical configuration. (a) Two external rotors (blue), moving in the fluid, mimic the beating motion of flagellar filaments. Internal rotors (red) represent the flagellar basal bodies and are coupled through an anisotropic spring. (b) Anti-phase (AP) and (c) in-phase (IP) states. (Online version in colour.)

Figure 2.

Synchronization dynamics for pairs of model cilia in the absence of hydrodynamic interactions. Phase sum, , for (a) , (b) and (c) . In each case (starting with ), IP (blue) and AP (red) synchronized states are obtained for and , respectively, over a timescale inversely proportional to . (d) Values of measured from numerical simulations (circles) compare favourably with equation (3.2) for sufficiently soft internal springs. Other model parameters as given in table 1. (Online version in colour.)

Figure 3.

Steady state solutions when radial dynamics are fast compared to the phase dynamics. This is achieved by setting . In each simulation . Other parameters are shown in table 1. (a–c) Phase sum and (d–f) external rotor radii are shown as functions of time for various values of k_y. (g) The mean value of phase sum and (h) external rotor radius are computed from numerical solutions (green) and compared with analytical predictions of equation (4.1) (dashed) and equations (4.2), (4.3) (dotted). (Online version in colour.)

Figure 4.

Transitions between synchronization states mediated by basal body coupling. Radial dynamics are again fast compared to phase dynamics (). (a) Steady state as a function of k_y for various ciliary spacings (, 20, 25, 35, 50, ). Numerical solutions (smooth, coloured) are shown alongside analytical predictions to leading order (dashed, black) and second order in (dashed, coloured). The transition zone boundaries are quantified by (red crosses) and (blue crosses), respectively. (b) These are shown as a function of l, together with leading order (black) and second order in (coloured) analytical predictions. (Online version in colour.)

Figure 5.

Basal body coupling and hydrodynamic interactions. (a) Steady state phase sum and (b) external radii and . Results are shown for , 0.006, 0.007, 0.017, with and . For intermediate values of k_y, the external rotors display a permanent difference in their average radii despite being identical. Electronic supplementary material movies 1–5 show the dynamics for (i)–(v), respectively. (c) Value of as a function of k_y for various values of l (results shown for , 20, 25, 35, 50, ). The transition zone boundaries are quantified by (red crosses) and (blue crosses) respectively; dashed black lines show far-field analytical predictions to leading order (from equation (4.1)). (d) Measured boundaries compared with leading order far-field analytical results. (e) The variance of the time-dependent phase sum, , for given values of k_y and l, reveals large excursions in the phase sum prior to the bifurcation (see (a(ii)) for the raw time-dependent signal). (Online version in colour.)

Figure 6.

The transition between AP and IP states is mediated by internal spring constants. Mean and standard deviation of as functions of k_x and k_y for external rotor separations (a) , (b) , and (c) . White dotted lines correspond to the blue bifurcation plot in figure 5c. For small k_x, the system is capable of supporting intermediate phase-locked states, with . However, for larger k_x, an abrupt transition between AP and IP occurs. (Online version in colour.)