Tracking phase synchronization between flagella in the time-frequency domain resolves photophobic response

TL;DR

The study tackles how two-flagellate coordination in Chlamydomonas reinhardtii reorganizes after a photoshock stimulus. It develops a time-frequency phase synchronization framework based on the continuous wavelet transform to track transient coupling via a complex Phase Synchronization Index ($PSI$), enabling simultaneous tracking of instantaneous frequency and phase across time scales. The key contributions include identifying three swimming stages (pre-stimulus breaststroke, post-stimulus high-frequency backward beating, and resynchronization), revealing persistent harmonic components that modulate beating, and proposing a spectral reserve mechanism that supports robust adaptation. The approach provides a general tool for resolving nonstationary synchronization in biological oscillators and could be extended to mutant strains or direct flagellar imaging to further dissect coupling pathways and amplitude dynamics.

Abstract

The unicellular microalga Chlamydomonas reinhardtii (CR) is well known for its bi-flagellated swimming in response to light stimuli. This work aims to study the resynchronization of CR flagella after a high light intensity stimulus, known as photoshock. The synchronization is estimated thanks to a quantity defined as the Phase Synchronization Index (PSI). The originality of this approach is to perform a time-frequency computation of a complex PSI based on continuous wavelet transform. Thanks to this analysis, we distinguish three swimming stages involving different frequency bands and phase shifts: normal breaststroke, escaping, and resynchronization. This approach also reveals the presence of signal harmonics that set the photoshock response, independently of cell variability. Our results suggest that CR modulates the balance between fundamental and harmonic beating modes, providing a mechanism for robust adaptation to sudden environmental stresses.

Figures (11)

• Figure 1: Experimental setup. (a) C. reinhardtii cell trapped in a micropipette and bathed with tracer beads. Dashed squares indicate the regions of interest where velocity signals are extracted ($Y=0, \pm 1$). Schematic lines represent a screenshot of the flagella before photoshock (black, -PS) and after photoshock (gray, +PS). Scale bar: 10. (b-c) Velocity components along the $x$-axis (b) and $y$-axis (c), averaged in front of the cell ($Y=0$), before photoshock (black) and after photoshock (gray).
• Figure 2: Simulations of two coupled oscillators with Gaussian noise, modeled by Eq. (\ref{['eq:S_delta_noise']}). (a) Strong negative coupling ($B/2\pi=\qty{-20}{\hertz}<0$, $|2B|>\Delta \omega$), with $\varphi_1 (0) = 0$, $\varphi_2 (0) = \pi$: leads to stable in-phase synchronization. (b) Weak negative coupling ($B/2\pi=\qty{-3}{\hertz}<0$, $|2B|<\Delta \omega$), with same initial phases: leads to desynchronization with transient coordination. (c) Strong positive coupling ($B/2\pi=\qty{20}{\hertz}>0$, $|2B|>\Delta \omega$) with $\varphi_1 (0) =\varphi_2 (0) = 0$: leads to stable anti-phase synchronization. Left panels: $S_1=\cos(\varphi_1)$ (gray) and $S_2=\cos(\varphi_2)$ (black). Right panels: $\delta=\varphi_1 - \varphi_2$. Dashed and solid lines represent anti-phase and in-phase locking, respectively.
• Figure 3: System of two coupled oscillators with linearly increasing frequency. (a) Simulated signals generated from Eq. (\ref{['eq:S_delta_noise']}). Gray: $\cos(\varphi_1)$; black: $\cos(\varphi_2)$. Frequencies: $\omega_1/2\pi=5Hz$, $\omega_2/2\pi=4Hz$. Coupling strength: $B_1/2\pi=B_2/2\pi=0.5Hz$ (b) $|W_1(t,f)|$ for the first oscillator (gray line in (a)) using Morlet wavelet with $n_0=6$. The gray line indicates the local frequency maxima $f_1^*(t)$. (c) $|W_2(t,f)|$ for the second oscillator (black line in (a)) using Morlet wavelet with $n_0=6$. The black line indicates the local frequency maxima $f_2^*(t)$. (d) $\Upsilon_\Psi(t,f)$ computed with an adaptive time window $\Delta T_\Psi \propto n_0/f^*$ ($n_0=6$). Frequency ridges $f_1^*(t)$ (gray) and $f_2^*(t)$ (black) are overlaid. (e) $\Upsilon_\Psi \left(t,f^*_{12}(t)\right)$ (black), computed along the frequency trajectory $f^*_{12}(t) = (f_1^*(t) + f_2^*(t))/2$. The derivative of the phase difference between $S_1$ and $S_2$, $\dot\delta_W (t)$ (gray) was computed using a Gaussian derivative wavelet to filter out the signal noise (Supplementary Fig. \ref{['fig:wavelet_derivative']}) mallat_wavelet_2008.
• Figure 4: Flagellar resynchronization of CR following a photoshock. (a) Averaged fluid velocity signals along the $x$-axis (left) and $y$-axis (right), extracted from regions on each side of the alga: $Y=-1$ (gray) and $Y=+1$ (black), as shown in Fig. \ref{['fig:exp_setup']}. (b-c) Time-frequency representation of the CWT signals, $\log |W_Y(t,f)|$ (Morlet wavelet with $n_0=12$), for $Y=-1$ (b) and $Y=+1$ (c). Frequency ridges $f_{-1}^*(t)$ (gray) and $f_{+1}^*(t)$ (black) are also plotted. (d) Phase Synchronization Index, $|\tilde{\Upsilon_\Psi}| (t, f^*_{\pm 1}(t))$, computed at the mean instantaneous frequency $f^*_{\pm1}(t) = [f_{-1}^*(t)+f_{+1}^*(t)]/2$. (e) Phase $\Phi(t)$ obtained from the complex PSI $\tilde{\Upsilon}_\Psi$. In all panels, the shaded gray region marks the duration of the photoshock.
• Figure 5: Flagellar synchronization and frequency dynamics around photoshock. (a-b) PSI averaged over 44 photoshocks (13 algae) computed using the two signals $Y=-1$ and $Y=1$, along the $x$-axis (a) and the $y$-axis (b). Black line: Median PSI. Gray shade: First quartile (Q1) and Third quartile (Q3). (c-d) The red lines correspond to the averaged wavelet frequency maxima of each band accross all the algae, computed using signal in $Y=0$. The gray shades are the standard deviations. The color map is linked to the amplitude and represents the average wavelet transform squared modulus of the corresponding frequency maxima, in log scale. Frequencies between $t=-\qty{50}{\milli\second}$ and $t=\qty{100}{\milli\second}$ are not shown because they are not robustly detected using the wavelet transform.