Flagellar synchronization through direct hydrodynamic interactions

Abstract

Flows generated by ensembles of flagella are crucial to development, motility and sensing, but the mechanisms behind this striking coordination remain unclear. We present novel experiments in which two micropipette-held somatic cells of Volvox carteri, with distinct intrinsic beating frequencies, are studied by high-speed imaging as a function of their separation and orientation. Analysis of time series shows that the interflagellar coupling, constrained by lack of connections between cells to be hydrodynamical, exhibits a spatial dependence consistent with theory. At close spacings it produces robust synchrony for thousands of beats, while at increasing separations synchrony is degraded by stochastic processes. Manipulation of the relative flagellar orientation reveals in-phase and antiphase states, consistent with dynamical theories. Flagellar tracking with exquisite precision reveals waveform changes that result from hydrodynamic coupling. This study proves unequivocally that flagella coupled solely through a fluid can achieve robust synchrony despite differences in their intrinsic properties.DOI: http://dx.doi.org/10.7554/eLife.02750.001.

Keywords: Volvox; flagella; synchronization.

Figure 1.. Synchronized pairs of beating flagella.

(A) Experimental apparatus and (B) cell configuration. (C) Extracted phase difference $\text{Δ}=\left({\varphi }_1-{\varphi }_2\right)/2\pi$ at four different interflagellar spacings, as indicated. These separations correspond to scaled spacing L = d/l of 0.85, 1.22, 1.69, and 2.27. (D) fluctuations during phase-locked periods around the average phase lag, Δ₀, and (E) the fluctuations’ probability distribution functions (PDFs), each cast in terms of the rescaled separation-specific variable (Δ − Δ₀)/√L. Solid lines represent Gaussian fits. Further details of the phase extraction procedure can be found in Figure 1—figure supplement 1. Samples of the four processed videos corresponding to the cells in Figure 1C are shown in Video 1. DOI: http://dx.doi.org/10.7554/eLife.02750.003

Figure 1—figure supplement 1.. Phase extraction.

(A–C) Snapshots from three different processed videos showing the same cells at different interflagellar spacings (d = 14.2, 20.4, 28.2 μm). (D–F) The signal extracted from the interrogation regions is used to reconstruct the flagellar phases. For each row, the frame on the left corresponds to the time indicated by the dashed line. DOI: http://dx.doi.org/10.7554/eLife.02750.004

Figure 2.. Measured flagellar flow field.

(A) Time-dependent flow field for an individual cell measured using particle image velocimetry. Results are shown for the first half of the beating cycle. (B) Time-averaged flow field ${\langle u\rangle }_t=(1/\tau ){{\displaystyle \int }}_0^{\tau }\left|\mathit{u}\left(\mathit{x},t\right)\right|dt$ (averaged across 4 cells with τ ∼ 1000 beats for each). The velocity magnitude (colour) and streamlines (white) are shown. (C) Velocity magnitude upstream (red) and downstream (blue) of the origin (black dot in B). DOI: http://dx.doi.org/10.7554/eLife.02750.005

Figure 3.. Force amplitude of flagellum.

(A) Fitted instantaneous velocity field at various stages during the first half of one representative flagellar beat. (B) The fitted Stokeslet is shown at evenly-spaced times throughout the average flagellar beat cycle. The red dots indicate the Stokeslet position extracted from every frame. (C) Amplitude of the fitted point force as a function of time throughout the flagellar beat period T. DOI: http://dx.doi.org/10.7554/eLife.02750.006

Figure 3—figure supplement 1.. Time-dependent flow fields.

Instantaneous fluid velocity corresponding to various stages during one representative flagellar beat. Also shown are the fitted flow fields for each frame, corresponding to the application of a point force on the fluid. Very good qualitative agreement can be seen. DOI: http://dx.doi.org/10.7554/eLife.02750.007

Figure 4.. Resistive force theory analysis.

(A) Instantaneous velocity distribution along the flagellum during one complete beat cycle (indexed by frame number, imaged at 1000 fps). (B) Components of integrated force density produced by a flagellum executing characteristic power and recovery strokes, as a function of arclength along the flagellum measured from the basal to the distal end. (C) Integrated vector forces F(t) shown localised at centre-of-mass coordinates x(t) (red: per frame, black: averaged over O(10³) frames), evolve cyclically around an average trajectory. The average value is |F|/8πμ ∼ 1910 μm²/s. DOI: http://dx.doi.org/10.7554/eLife.02750.008

Figure 5.. Coupling strength.

(A) Dimensionless interflagellar coupling strength $\kappa =\mathit{ϵ}/\overline{\omega }$ as a function of the scaled spacing L = d/l (log–log scale). The dotted lines represent fits of the form $\left|\kappa \right|=k\times L^{-1}$ with k = 0.016 (in-phase) and k = 0.014 (antiphase). (B) Measured beat frequency $\omega /{\overline{\omega }}_{\text{cell}}$ of each flagellum, nondimensionalised by the average value for that cell across several videos. (C) Measured frequency difference $\delta \omega /\delta {\omega }_{\text{far}}$ as a function of spacing L. The curves represent the predictions based on the average extracted model parameters in the absence (orange) and presence of noise (green). Symbols represent different pairs of cells, with the in-phase (blue) and antiphase (red) configurations shown. DOI: http://dx.doi.org/10.7554/eLife.02750.010

Figure 6.. Waveform characteristics.

(A) Logarithmically-scaled residence time plots of the entire flagella. The displayed waveforms correspond to 1 ms time intervals over several successive flagellar beats. (B) Angles x_a, x_b, x_c (in radians) measured and (C) their characteristic 3D trajectories. Results are shown for the right flagellum, corresponding to three different interflagellar spacings. As the spacing d is increased, the flagellar waveform exhibits a systematic change. DOI: http://dx.doi.org/10.7554/eLife.02750.011

Figure 6—figure supplement 1.. Flagellar filaments are tracked for cells in the (A) antiphase state, as well as (B) the situation in which one of the cells does not possess a flagellum (dummy cell).

For each configuration, the waveform of the left cell is analysed at three different cell–cell separations. DOI: http://dx.doi.org/10.7554/eLife.02750.012

Figure 6—figure supplement 2.. Additional waveform data collected for 5 different cells in various geometric configurations.

The colours RGB correspond to increasing inflagellar spacing respectively, with distances (in micrometres) given by (A) {7.3, 10.9, 72.2}, (B) {6.5, 11.0, 27.1}, (C) {9.2, 14.2, 34.7}, (D) {10.3, 31.0, 84.0}, (E) {12.1, 16.2, 67.3}. DOI: http://dx.doi.org/10.7554/eLife.02750.013

Figure 7.. Model parameters.

Two of the experimental observables (A) C₀ and (B) ${\tau }_{\text{sync}}$, and the two additional model parameters (C) $T_{\text{eff}}/\overline{\omega }$ and (D) $\delta \nu /\overline{\omega }$ are shown as functions of interflagellar spacing for all experiments conducted. DOI: http://dx.doi.org/10.7554/eLife.02750.015

Figure 8.. Effect of nearby pipette.

The time-averaged flow field associated with one captured cell is measured as a second pipette slowly approaches. This demonstrates that the precise angle from which the cell is held by the micropipette has very little effect on the resultant flow field. DOI: http://dx.doi.org/10.7554/eLife.02750.016

Figure 9.. Effect of force modulation.

Evolution of the phase difference $\delta ={\varphi }_1-{\varphi }_2$ among two identical model oscillators, each composed of a sphere driven around a circular trajectory by a tangential driving force. The trajectories each possess a radial stiffness λ. Smaller values of λ yield rapid convergence towards synchrony (δ = 0), in a manner essentially independent of the functional form of the driving force. Parameters used are given by a = 0.75 μm, r₀ = 8 μm, d = 20 μm, A₀ = 1076 μm²/s and A₁ = 0.56. DOI: http://dx.doi.org/10.7554/eLife.02750.017

Figure 10.. Effect of force modulation.

Re-run of the simulations in Figure 9 with properties inspired by real flagella. DOI: http://dx.doi.org/10.7554/eLife.02750.018