Intracellular coupling modulates biflagellar synchrony

Abstract

Beating flagella exhibit a variety of synchronization modes. This synchrony has long been attributed to hydrodynamic coupling between the flagella. However, recent work with flagellated algae indicates that a mechanism internal to the cell, through the contractile fibres connecting the flagella basal bodies, must be at play to actively modulate flagellar synchrony. Exactly how basal coupling mediates flagellar coordination remains unclear. Here, we examine the role of basal coupling in the synchronization of the model biflagellate Chlamydomonas reinhardtii using a series of mathematical models of decreasing levels of complexity. We report that basal coupling is sufficient to achieve inphase, antiphase and bistable synchrony, even in the absence of hydrodynamic coupling and flagellar compliance. These modes can be reached by modulating the activity level of the individual flagella or the strength of the basal coupling. We observe a slip mode when allowing for differential flagellar activity, just as in experiments with live cells. We introduce a dimensionless ratio of flagellar activity to basal coupling that is predictive of the mode of synchrony. This ratio allows us to query biological parameters which are not yet directly measurable experimentally. Our work shows a concrete route for cells to actively control the synchronization of their flagella.

Keywords: Chlamydomonas reinhardtii; cilia; coupled oscillators; cytoskeleton; elastic filaments; microhydrodynamics.

Figure 1.

(a) Schematic of the biflagellate alga Chlamydomonas reinhardtii (CR), held in place using a micropipette as in [19], with cis and trans flagella connected via basal fibres (BF); the flagella are modelled as a pair of filaments coupled at their base via elastic springs. The filaments are driven into oscillations by an active moment M at the filament base that switches direction when the basal angle θ reaches pre-defined locations ±Θ relative to the average basal angle $\overline{\mathit{\Theta }}$, akin to the geometric switch model [–16]. (b) Experimental data: snapshots of the flagellar waveforms during one oscillation cycle and time evolution of basal angles for breaststroke, freestyle and slip motions. Data reproduced from [, fig. 3]. Five consecutive beats are shown with time being colour-coded. (c) Filament pair model: snapshots of filament waveforms during one beating period and time evolution of basal angles as well as the respective basal spring force showing inphase (Θ = 0.2π), antiphase (Θ = 0.1π) and phase-slip. The bases of the flagella are fixed in the snapshots for aesthetic purposes. Model parameters and filament simulations are provided in the electronic supplementary material.

Figure 2.

Time evolutions of the filament model and the minimal model. (a,b) Snapshots of filament waveforms and time evolutions of the basal angles θ⁽¹⁾ and θ⁽²⁾ and basal spring force $-K(x_{\mathrm{b}}^{(1)}-x_{\mathrm{b}}^{(2)})$ showing (a) inphase synchrony (breaststroke) for Θ = 0.175π and (b) antiphase synchrony (freestyle) for Θ = 0.125π. Other parameter values are set to K = 50, K_c = 10, M = 2. The reference curvature of the filament is set to be zero. (c,d) Snapshots of dumbbell configuration and time evolution of basal angles and basal spring force showing (c) inphase synchrony for Θ = 0.25π and (d) antiphase synchrony for Θ = 0.1π. Other parameter values are set to K = 20, M = 1.

Figure 3.

Synchronization order parameter as a function of the filament activity parameters Θ, M and basal coupling strength K, colour-coded by the initial condition: simulations with nearly inphase or antiphase initial conditions are depicted in thick red lines or thin blue lines, respectively. The values of the parameters held constant are set to (a) M = 3, K = 50, (b) K = 30, Θ = 0.2π, (c) M = 3, Θ = 0.25π. Bistable regions are highlighted by shaded boxes.

Figure 4.

Long-term dynamics of the filament model and the minimal model. (a–c) Three slices of the parameter space for the filament model at (a) K = 50, (b) Θ = 0.2π, (c) M = 3 respectively. (d–f) Three slices of the parameter space for the minimal model at (d) K = 20, (e) Θ = 0.4π, (f) M = 1 respectively. Each data point comprises the long-term dynamics of two different initial conditions, distinguished by the marker edge and face colours. Flagella synchronized into breaststroke or freestyle are in magenta or cyan colours, respectively.

Figure 5.

(a) Synchronization modes of dumbbell model plotted in the condensed two-dimensional parameter space (Θ, M/Kℓ²). Dashed lines delineate boundaries between different regions by their stable synchronization modes. (b) Synchronization modes of filament model plotted in the same parameter space. The dashed lines in (b) have the same slopes as those in (a).