# Role of Hydrodynamics in the Synchronization of Chlamydomonas Flagella

Luc Zorrilla, Antoine Allard, Krish Desai, Marco Polin

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Luc Zorrilla, Antoine Allard, Krish Desai, Marco Polin. Role of Hydrodynamics in the Synchronization of Chlamydomonas Flagella. Physical Review Letters, 2025, 135 (21), pp.218402. 10.1103/zjr8-3tyj. hal-05382984

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# The role of hydrodynamics in the synchronisation of *Chlamydomonas* flagella

Luc Zorrilla,¹ Antoine Allard,² Krish Desai,³,\* and Marco Polin¹,†

¹Mediterranean Institute for Advanced Studies, IMEDEA, UIB-CSIC, Esporles, 07190, Spain

²Univ. Bordeaux, CNRS, LOMA, UMR 5798, F-33400 Talence, France

³Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK

(Dated: April 4, 2025)

While hydrodynamic coupling has long been considered essential for synchronisation of eukaryotic flagella, recent experiments on the unicellular biflagellate model organism *Chlamydomonas* demonstrate that -at the single cell level- intracellular mechanical coupling is necessary for coordination. It is therefore unclear what role, if any, hydrodynamic forces actually play in the synchronisation of multiple flagella within individual cells, arguably the building block of large scale coordination. Here we address this question experimentally by transiently blocking hydrodynamic coupling between the two flagella of single *Chlamydomonas*. Our results reveal that in wild type cells intracellularly-mediated forces are necessary and sufficient for flagellar synchronisation, with hydrodynamic coupling causing minimal changes in flagellar dynamics. However, fluid-mediated ciliary coupling is responsible for the extended periods of anti-phase synchronisation observed in a mutant with weaker intracellular coupling. At the single-cell level, therefore, flagellar coordination depends on a subtle balance between intracellular and extracellular forces.

Cilia and flagella, henceforth used interchangeably, are slender organelles ubiquitous in the eukaryotic world, where they support tasks ranging from mechanochemical sensing to regulate cell proliferation in renal ducts [1], to pumping of cerebrospinal flow by ependymal cells [2] and establishment of planar cell polarity [3, 4]. Motile cilia combine thousands of molecular motors within a microtubule-based passive scaffold, the axoneme, rooted in an intracellular basal body [5]. Their regular beating results from the motors' active stresses, the mechanical properties of the axoneme and the viscoelastic properties of the extracellular fluid [6].

In groups of motile cilia, whether belonging to the same cell or to different ones, coordinated motion is a ubiquitous feature [7]. The metachronal waves observed at a large scale can increase transport efficiency in ciliated epithelia in a noise-dependent manner [8] and improve swimming efficiency of ciliated microorganisms [9]. At the single-cell level, active modulation of flagellar synchrony can help microorganisms explore space [10, 11] and respond to environmental stimuli [12]. Fluid-mediated interactions between beating flagella have commonly been considered the cornerstone of flagellar synchronisation, usually coupled with either a modulation of the ciliary driving force [13], waveform compliance [14, 15] or a combination of the two [16]. Direct experiments with pairs of flagellated cells support this hypothesis: synchronisation develops spontaneously below a critical cell-to-cell separation as a result of hydrodynamic interactions [17]. Ciliary beating can also be entrained by an externally imposed flow, if sufficiently strong [18].

The focus on hydrodynamic coupling between flagella has been more recently upended by experiments on single cells. Studies on *Xenopus* multiciliated cells [19] and *Paramecium* [20] suggested that intracellular mechanical coupling between flagellar basal bodies are critical for flagellar coordination. This was confirmed by experiments on the *vfl3* mutant strain of the biflagellate unicellular green alga *Chlamydomonas reinhardtii* (CR), a major model system to study eukaryotic flagella [21, 22]. These mutants lack the intracellular fibres that, in the wild type, connect the two flagellar basal bodies [23]. In their absence, flagella do not synchronise, showing that intracellular mechanical coupling is necessary for the coordination of CR flagella. The importance of direct basal body connections has also been shown in *Tetrahymena*, where cortical actin is required for correct basal body placement and ciliary coordination [24]. The mechanism leading to flagellar coordination within a single cell has been explored theoretically through minimal models combining basal coupling and hydrodynamic interactions between flagella [25-28]. These models provide useful insights on the different synchronisation regimes possible through the interplay of extracellular and intracellular coupling, but are all based on experiments that have probed only the effect of removing the latter. Understanding the mechanism leading to flagellar coordination within a single cell requires us to be able to selectively remove hydrodynamic coupling between the flagella. So far this has proved challenging.

Here we use a tip-less cantilever to separate the flow fields generated by the two flagella of individual CR cells, allowing us to compare their behaviour with and without hydrodynamic coupling. While the standard in-phase (IP) synchrony is maintained in wild type (*wt*) regardless of the cantilever, the prolonged periods of anti-phase (AP) coordination displayed by the flagellar dominance mutant *ptx1* require hydrodynamic coupling. Our results provide novel critical insights into the role of hydrodynamics on the collective behaviour of eukaryotic flagella.

*Chlamydomonas reinhardtii* strains CC125 (*wt*, wild type), CC2894 (*ptx1*, flagellar dominance mutant) and CC1926 (*uni1*, uniflagellated) (*chlamycollection*.org) were grown in Tris-Acetate-Phosphate medium (TAP; [29]) within
<figure><img src="image_2.png" /><figcaption>FIG. 1. **Flow field for uniflagellated mutants.** a) Signal to noise ratio of the flagellar flow field without (left) and with the cantilever (right). b) Time-dependent flow field averaged across the highlighted area. Peaks and valleys are used to define the Poincaré sections of the phase cycle. c) Autocorrelation of the normalised instantaneous periods between beats.</figcaption></figure>

a diurnal growth chamber (20° C; 14/10 day/night; 80 µmol/m²s PAR). A small aliquot of exponentially-growing cells was then injected into a 4 mm-thick poly(dimethylsiloxane) observation chamber filled with a suspension of 1 µm carboxylate polystyrene tracer particles (Polysciences Europe, Germany) in TAP (0.026% solids). Individual cells (13 *uni1* ; 8 *wt*; 14 *ptx1*) were captured on the tip of a glass micropipette hosted on a micromanipulator (PatchStar, Scientifica, UK) and realigned to beat their flagella along the focal plane of a 63× water immersion objective (NA 1.0; Zeiss, Germany). The support section of a tip-less silicon nitride AFM cantilever (Quest R 20, NuNano, UK; 0.65 × 30 × 200 µm) was previously inserted into the top side of the chamber, allowing us to place the cantilever sideways in front of the cell (Fig. 1a). Pre-treating the chamber with a 0.5% w/v pluronic solution prevented bead adhesion to the cantilever. The flagellar flow field was imaged in brightfield through a 610 nm long-pass filter and recorded at 500 fps (*wt*,*ptx1*; 1000 fps *uni1*) (EoSens Cube6/mini1, Mikrotron, Germany). The local instantaneous flow field around each cell with/without the cantilever was estimated *via* Particle Image Velocimetry [30].

Figure 1a characterises the local strength of the periodic flow from the single flagellum of *uni1*, as the normalised ratio between the excess spectral power around the beating frequency and the baseline power level from the noise [31]. We see that an aptly placed cantilever confines flagellar flow to one side of the cell (configuration *ctl*; see also Movie M1), confirming that this configuration can block inter-flagellar hydrodynamic coupling in biflagellate cells. The local oscillatory flow (Fig. 1b) can also be used to estimate the beating phase *φ(t)* of each flagellum through Poincaré sectioning of the dynamics, as done previously [17, 32, 33]. These phases can be used to compare, for each cell, characteristics of flagellar beating with and without the cantilever. As seen in Fig. 1c the cantilever does not alter the autocorrelation of the instantaneous beating periods, which resembles that of a slightly underdamped oscillator as previously reported [34]. Having established that the cantilever prevents the flow generated by one flagellum from crossing over to the site of the other, we now turn to *wt* biflagellate cells and use the cantilever to prevent hydrodynamic coupling. Movies M2,M3 show that, when placing the cantilever, flagellar synchronisation in *wt* cells not only is preserved, but it is also recovered after a photoshock, induced here with a flash of blue light (470 nm). Figure 2 shows that the presence of the cantilever does not significantly alter the waveform (Fig. 2a), the longitudinal flow field close to the flagellum *u(t)* (Fig. 2b), or the mean around which the inter-flagellar phase difference *Δφ(t)* fluctuates (Fig. 2c). This proves that flagellar synchronisation in *wt* does not *per se* require hydrodynamic coupling: intracellular coupling is both necessary *and sufficient* to establish and maintain the synchronised motion that is usually observed. Yet, one would be naive to conclude that hydrodynamic forces have no measurable consequences in this system. Figure 3a shows that all of the strains we tested display a minute increase in beating frequency (~5%) without the cantilever (a configuration henceforth referred to as *bulk*), probably due to a slightly lower flagellar drag coefficient (for a table collating all numerical values see [31]). A more pronounced difference, however, can be found
<figure><img src="image_3.png" /><figcaption>FIG. 2. Flagellar synchronisation: bulk vs. cantilever (wt). a) Stroboscopic traces of experimental flagellar waveforms without/with the cantilever. b) Longitudinal flagellar flow fields of top and bottom without (left) and with (right) the cantilever. c) Inter-flagellar phase difference without/with the cantilever.</figcaption></figure>

in the fluctuations of the inter-flagellar phase difference $\Delta\phi$ during synchrony. Its variance $\mathbb{V}(\Delta\phi)$ increases by a factor of $2.10 \pm 0.23$ when the cantilever is placed between the flagella ($1.56 \pm 0.15$ for *ptx1*). The change is not due to a difference in the dynamics of inter-flagellar phase fluctuations. Figure 3b shows that both with and without the cantilever, the power spectrum $S_{\Delta\phi}$ of $\Delta\phi(t)$ follows the same curve. This is compatible with an inverse power law $\sim 1/f^\alpha$ with $\alpha_{\text{bulk}}^{\text{wt}} = 0.37 \pm 0.02$ and $\alpha_{\text{ctl}}^{\text{wt}} = 0.37 \pm 0.03$. Scale-free power spectra indicate long-range fractal-like temporal correlations in the fluctuations, and are a common phenomenon in nature, where they have been proposed to signal systems working at criticality [35]. In CC125, temporal fluctuations in the flagellar beating period, as opposed to $\Delta\phi$, have also been reported to display scale-free fluctuations [34]. In line with past works [8, 14, 15] it is useful to conceptualise fluctuations in $\Delta\phi$ as a stochastic process confined within a one-dimensional potential $\mathbb{U}(\Delta\phi)$. This implies that the force $-\partial\mathbb{U}(\Delta\phi)$ that the system feels at $\Delta\phi$ can be estimated as the expectation value $\mathbb{E}[\Delta\phi(t + \delta t) - \Delta\phi(t)|\Delta\phi(t) = \Delta\phi]/\delta t$ and then integrated to give the effective potential. Figure 3c shows that $\mathbb{U}(\Delta\phi) \simeq \epsilon\Delta\phi^2/2$ both with and without the cantilever. The effective coupling constants $\epsilon_{\text{bulk}}^{\text{wt}} = 0.434 \pm 0.04$ and $\epsilon_{\text{ctl}}^{\text{wt}} = 0.402 \pm 0.06$ are not significantly different, in line with the fact that the cell maintains its in-phase (IP) beating regardless of the configuration. There is, however, a sizeable difference in the magnitude of the fluctuations within these potentials. This can be estimated by calculating the probability distribution function $\mathbb{P}(\Delta\phi)$ of the excursions of the phase difference with respect to the plateau value during IP, and comparing $\mathbb{U}(\Delta\phi)$ to $-\log \mathbb{P}$. Figure 3d) shows that the two align well, implying that -despite their memory- fluctuations in $\Delta\phi$ are ergodic with respect to a simple Boltzmann-like measure. The slope of the linear fit can then be interpreted as the inverse of an effective temperature $T_{\text{eff}}$, characterising the noise magnitude. We obtain $T_{\text{eff, bulk}}^{\text{wt}} = (3.46 \pm 0.11) \times 10^{-4}$, $T_{\text{eff,ctl}}^{\text{wt}} = (5.56 \pm 0.29) \times 10^{-4}$. The change in variance in Fig. 3a can then be understood as a change in the ratio $\epsilon/T_{\text{eff}}$, where the main contribution comes from an increase in the magnitude of intrinsic fluctuations $((\epsilon_{\text{ctl}}^{\text{wt}}T_{\text{eff, bulk}}^{\text{wt}})/(\epsilon_{\text{bulk}}^{\text{wt}}T_{\text{eff,ctl}}^{\text{wt}})) = 0.64 \pm 0.3)$.

Flagella from *wt* cells, termed *cis* and *trans* depending on the position with respect to the cell's eyespot, are not identical organelles. They differ in structure [36], intrinsic frequency ($\sim 30\%$ [37]) and coupling [18], causing the *cis* flagellum to dominate the dynamics of the pair. A conceptually simpler system is provided by *ptx1*, a mutant strain which putatively possesses two identical flagella of *trans* type [38]. We previously reported that, differently from *wt*, pipette-held *ptx1* cells can display extended periods of either in-phase (IP) or anti-phase (AP) beating [32], the latter being the state favoured by hydrodynamic interactions [17]. Figure 4 compares flagellar behaviour in *ptx1* with and without the cantilever. It shows that, while cells fluctuate between AP and IP periods in bulk, AP disappears when the cantilever is in place (fig. 4a). Instead, IP synchrony is punctuated by sporadic short periods of large $\Delta\phi$ deviations accompanied by the beating frequency increase characteristic of AP ($\bar{\nu} = (77.0 \pm 4)$ Hz vs $\bar{\nu}_{AP} = (82.2 \pm 5)$ Hz and $\bar{\nu}_{IP} = (58.4 \pm 2)$ Hz in bulk) but never lasting more than 200 ms. AP synchronisation, therefore, requires inter-flagellar
<figure><img src="image_4.png" /><figcaption>FIG. 3. Inter-flagellar phase difference $\Delta\phi$: bulk vs. cantilever (wt). a) Variance of $\Delta\phi$ ($V(\Delta\phi)$) and mean beating frequency ($\nu$): ratio between bulk and cantilever values (subscripts b and c respectively). b) Power spectrum $S_{\Delta\phi}$ of phase difference fluctuations during synchronous periods with (orange dashed) and without (purple dashed) the cantilever. Solid line: fit of the bulk case to the inverse power law $A/f^\alpha$. c) Effective potential $U(\Delta\phi)$ with (orange dot-dashed) and without (purple dashed) the cantilever, and quadratic fits (pink and blue solid lines respectively). d) Scatter plot of $\mathbb{P}(\Delta\phi)/\mathbb{P}(0)$ vs. $U(\Delta\phi)$ and fit to $\exp(-U/T_{\text{eff}})$ with (orange squares and dashed line) and without (purple disks and solid line) the cantilever. Shaded regions indicate the ranges of data used for the fits.</figcaption></figure>

hydrodynamic coupling. Comparing the IP periods with and without the cantilever, we recover a similar behaviour as observed in wt. Cells in bulk have a slightly higher beating frequency ($12 \pm 3\%$) and lower variance $V(\Delta\phi)$ ($64 \pm 6\%$) (fig. 3a). Their $\Delta\phi$ fluctuations are characterised by compatible inverse-power-law-type power spectra $S_{\Delta\phi}(f) \sim 1/f^\alpha$ with $\alpha_{\text{bulk}}^{\text{ptx1}} = 0.49 \pm 0.03$ and $\alpha_{\text{ctl}}^{\text{ptx1}} = 0.48 \pm 0.02$. These are slightly larger than for wt, signalling a process with a somewhat shorter memory. As it is the case for wt, the deterministic part of the fluctuations dynamics of $\Delta\phi$ in ptx1 is well captured by a quadratic effective potential, with coupling constants $\epsilon_{\text{bulk}}^{\text{ptx1}} = 0.275 \pm 0.020$ and $\epsilon_{\text{ctl}}^{\text{ptx1}} = 0.299 \pm 0.024$ (fig. 4c). These are not significantly different between them, but are only $\sim 69\%$ of the corresponding values in wt, suggesting a weaker intracellular coupling between the two identical flagella of ptx1. Comparison between $U(\Delta\phi)$ and $\mathbb{P}(\Delta\phi)$ shows that the fluctuations in $\Delta\phi$ are again compatible with a simple Boltzmann-like measure (fig. 4d). This can be characterised by effective temperatures $T_{\text{eff, bulk}}^{\text{ptx1}} = (5.56 \pm 0.29) \times 10^{-4}$, $T_{\text{eff,ctl}}^{\text{ptx1}} = (8.75 \pm 0.16) \times 10^{-4}$, which are significantly larger ($\sim 167\%$) than for the wt case. Once again, the change in variance in fig. 3a can be understood as a change in the ratio $\epsilon/T_{\text{eff}}$, driven by a variation in $T_{\text{eff}}$ $((\epsilon_{\text{ctl}}^{\text{ptx1}} T_{\text{eff, bulk}}^{\text{ptx1}})/(\epsilon_{\text{bulk}}^{\text{ptx1}} T_{\text{eff,ctl}}^{\text{ptx1}}) = 0.69 \pm 0.3)$.

As mentioned earlier, the IP state in ptx1 is accompanied -in bulk- by periods of AP synchronisation. Although the lack of AP states with the cantilever prevents a comparative analysis as done for IP, it is still instructive to analyse AP synchrony by itself. This reveals both quantitative and qualitative differences between AP and IP. Figure 4c shows that the effective potential $U(\Delta\phi)$ for AP is significantly weaker than for the IP of either ptx1 or wt. The effective inter-flagellar coupling $\epsilon_{\text{bulk,AP}}^{\text{ptx1}} = 0.140 \pm 0.034$ has a magnitude which is only $\sim 51\%$ of the corresponding IP values in ptx1. Note that the coupling constants for IP and AP states appear with opposite signs within the standard stochastic Adler-model used frequently in the past to rationalise the dynamics of $\Delta\phi$ [26]. Furthermore, $\Delta\phi$ fluctuations within this effective potential follow a power spectral density (fig. 4b) which -surprisingly- is compatible with the Lorentzian decay predicted for a standard Ornstein-Uhlenbeck process, rather than the scale-free power law observed for IP in both ptx1 and wt (see [31] for the corresponding autocorrelation functions).

The dynamic of the inter-flagellar phase difference is a window into the mechanism responsible for synchronisation. Altogether, the differences in dynamics between IP (wt, ptx1) and AP (ptx1) suggest that these states stem from fundamentally different processes. On the one hand, the IP results presented here show unequivocally that this mode of flagellar synchronisation is independent of hydrodynamic coupling. Building upon an existing body of evidence [19–22, 24] we can therefore conclude that, in CR, IP stems exclusively from intracellular coupling, most likely realised by the fibrous connections between the flagellar basal bodies. The fluctuation dynamics observed in IP states (wt, ptx1) suggest a non-trivial stochastic process with a long memory. This might then be mediated by changes in the
<figure><img src="image_5.png" /><figcaption>FIG. 4. Inter-flagellar phase difference: bulk vs. cantilever (ptx1). a) Representative dynamics of $\Delta\phi$ without (purple solid line) and with (dot-dashed orange line) the cantilever. Integer/half integer values correspond to IP and AP synchronisation respectively. b) Power spectrum $S_{\Delta\phi}$ of phase difference fluctuations during synchronous periods with (IP: orange dashed) and without (IP: purple dotted; AP: pink dot-dashed) the cantilever. Solid lines: fit of bulk IP to an inverse power law $A/f^\alpha$ (blue) and of AP to a Lorentzian (pink). c) Effective potentials $U(\Delta\phi)$: IP with (orange dashed) and without (purple dotted) the cantilever; AP (pink dot-dashed). Solid lines: corresponding quadratic fits (orange, blue and pink respectively). d) Scatter plot of $P(\Delta\phi)/P(0)$ vs. $U(\Delta\phi)$ and fit to $\exp(-U/T_{\text{eff}})$ with (orange squares and dashed line) and without (purple disks and solid line) the cantilever. Shaded regions indicate the ranges of data used for the fits.</figcaption></figure>

fibres' stiffness, possibly caused by local [Ca$^{2+}$] fluctuations [39, 40]. On the other hand, we see that AP requires hydrodynamic interactions, is associated with an effective coupling of reduced magnitude -and of opposite sign than IP- as well as a qualitatively different fluctuation dynamic which is conceptually simpler than in the IP case. We propose that these observations support the hypothesis of a fundamentally hydrodynamic origin of AP synchrony in *ptx1*, in line with reports on synchrony in pairs of nearby cells [15, 22]. Previous theoretical investigations demonstrated the possibility to realise both AP and IP states with only hydrodynamic coupling [27], or only through basal connections [26, 28], or combining both [25, 26]. In these models, areas of stable IP and AP in parameter space are separated by transition regions characterised by bi-stability and hysteresis, similarly to what is encountered in a subcritical pitchfork bifurcation [41]. Taken together with the observation of short large-magnitude lapses in IP synchrony when the cantilever is present, this suggests a picture where fluctuations of intracellular coupling in *ptx1* cause it to move outside the region of stable IP synchrony. In bulk, hydrodynamics can then stabilise the system to AP. In order to revert to IP synchrony, then, the intracellular coupling will need to recover enough to overcome the hysteresis intrinsic within the bistable region, leading to prolonged periods of AP. In turn, the dependence of AP on hydrodynamic coupling suggests that the presence of the cantilever should modify the stability profile in parameter space, drastically reducing both the AP region and the width of the AP-IP transition. This would then preclude the AP-like fluctuations observed in *ptx1* with the cantilever to extend into fully-fledged intervals of AP synchrony, as observed in experiments. At the same time, the existence of these fluctuations -lasting up to $\sim 10$ beats- hints at the potential for a weak metastability induced solely by intracellular coupling in *ptx1*, as predicted by some models [26, 28]. It would be interesting in the future to extend these studies by modulating the level of hydrodynamic interaction through partial insertion of the cantilever. In the *wt* case, even without considering the effect of coupling asymmetries [18], the lack of AP synchronisation suggests that the intracellular coupling is large enough to guarantee that the system will fluctuate well within the stable IP phase. Indeed, this is supported by our estimate of the effective coupling strength, which is $\sim 1.5 \times$ the IP value of *ptx1*. Besides its effects on the synchronisation state, the cantilever influences also the spread of phase difference fluctuations. Measured as an increase in $T_{\text{eff}}$, this suggests some level of flagellar sensing of the cantilever. *Chlamydomonas* flagella are decorated by mastigonemes, tiny transversal filaments $\sim 0.8~\mu\text{m}$ long [42]. They do not play a direct role in motility [43], but are tethered to mechanosensitive proteins [44]. We therefore surmise that the increase in $T_{\text{eff}}$ is the result of mastigoneme-mediated mechanosensitivity, an effect that would widen our current understanding of the link between mechanosensitivity and flagellar dynamics [45]. This will
be probed further with targeted experiments using mutants lacking mastigonemes and/or mechanosensitive channels.

We thank Douglas Brumley for insightful discussions. The authors acknowledge support of the European Training Network PHYMOT (Horizon 2020 Marie Skłodowska-Curie grant agreement No 955910; L.Z. and M.P.); the Leverhulme Trust (grant RPG-2018-345; A.A. and M.P.); MICINN (grant PID2019-104232B-I00/AEI; M.P.). M.P. acknowledges the fact that IMEDEA is an accredited 'María de Maeztu Excellence Unit' (grant CEX2021-001198, funded by MCIN/AEI/10.13039/501100011033). A.A. acknowledge support from the SMR Department and CNRS Tremplin@INP.

* Current affiliation: Department of Metabolism, Digestion and Reproduction, Imperial College London, UK.
† email: mpolin@imedea.uib.csic

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## SUPPLEMENTARY PLOTS

<figure><img src="image_6.png" /><figcaption>FIG. 5. Autocorrelation of the phase difference for different mutants and types of synchronization, with fits made on the whole visible range of data. A homographic function is fitted for In-Phase synchrony, and only the fit to the bulk case is shown. An exponential function is fitted to Anti-Phase synchrony.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 6. Probability distribution function against potential, for *ptx1* cells in the AP synchronized regime, without cantilever. A fit to a Boltzmann distribution is made.</figcaption></figure>

## SUPPLEMENTARY MOVIES

* Movie S1. Micropipette-held uniflagellated cell (CC1926) with/without the cantilever (same cell). Movie frame rate: 1000fps.

* Movie S2. Micropipette-held wild type cell (CC125) with/without the cantilever (same cell). Movie frame rate: 500fps.

* Movie S3. Photo-shock and recovery of a micropipette-held wild type cell (CC125) with the cantilever. The photo-shock was elicited with a sudden pulse from a 470 nm blue LED (M470L2, Thorlabs). Movie frame rate: 500fps.
<figure><img src="image_8.png" /><figcaption>FIG. 7. Constructing a Signal-to-Noise Ratio (SNR). (Left) An example time-averaged flow field retrieved with Particle-Image Velocimetry performed on a uniflagellated cell. (Right) Fourier transforms of the temporal flow at flagellum position (up) and far from the cell (down). The red color denotes the area under the curve used for SNR calculations.</figcaption></figure>

## EVALUATION OF THE STRENGTH OF THE PERIODIC FLOW

We use PIV to estimate the instantaneous flow field $(u(\mathbf{x}, t), v(\mathbf{x}, t))$ on a $32 \times 32$ pxl grid ($\sim 5.632 \times 5.632$ µm). At each point $\mathbf{x}$ along the grid we calculate the Fourier transform $\tilde{u}(\mathbf{x}, \nu)$ of the $y$-component of the flow field $u(\mathbf{x}, t)$. When present, the signature of flagellar beating is clearly visible as a peak in the power spectrum $|\tilde{u}(\mathbf{x}, \nu)|^2$ at the beating frequency $\nu_0$ (see Fig. 7). We then calculate the signal $S(\mathbf{x})$ as the integral

$$S_u(\mathbf{x}) = \int_{\nu_0 - \Delta\nu_0}^{\nu_0 + \Delta\nu_0} |\tilde{u}(\mathbf{x}, \nu)|^2 d\nu,$$

where $\nu_0$ and $\Delta\nu_0$ are respectively the mean and the standard deviation of the instantaneous beating frequency (calculated from the time series of peak-to-peak intervals, see fig.1b of the main manuscript). We then perform a linear fit $P_{\text{lin}}(\mathbf{x}, \nu)$ of the part of the power spectrum outside of the interval $[\nu_0 - \Delta\nu_0, \nu_0 + \Delta\nu_0]$ and use it to calculate

$$S_{\text{lin}}(\mathbf{x}) = \int_{\nu_0 - \Delta\nu_0}^{\nu_0 + \Delta\nu_0} P_{\text{lin}}(\mathbf{x}, \nu) d\nu.$$

This estimates the power that would have been in $[\nu_0 - \Delta\nu_0, \nu_0 + \Delta\nu_0]$ in absence of a beating flagellum. The signal-to-noise ratio

$$SNR(\mathbf{x}) := \frac{S_u(\mathbf{x})}{S_{\text{lin}}(\mathbf{x}) \max_{\mathbf{x}}(S_u(\mathbf{x})/S_{\text{lin}}(\mathbf{x}))}.$$

For sake of clarity, in fig.1 the signal $SNR(\mathbf{x})$ is binned in 6 bins for which the lowest corresponds to the noise level.

## COLLATED NUMERICAL RESULTS
<table><thead><tr><th></th><th>CC1926 (uni1)</th><th>CC125 (wt, IP)</th><th>CC2894 (ptx1, IP)</th><th>CC2894 (ptx1, AP)</th></tr></thead><tbody><tr><td>ν<sub>b</sub></td><td>50.0 ± 3.4</td><td>52.8 ± 2.9</td><td>58.4 ± 1.6</td><td>82.2 ± 4.9</td></tr><tr><td>ν<sub>c</sub></td><td>50.1 ± 2.1</td><td>50.4 ± 2.9</td><td>51.7 ± 2.5</td><td>77.0 ± 4.0</td></tr><tr><td>⟨ν<sub>b</sub>/ν<sub>c</sub>⟩</td><td>1.054 ± 0.018</td><td>1.048 ± 0.012</td><td>1.124 ± 0.033</td><td>1.093 ± 0.075</td></tr><tr><td>V(Δφ)<sub>b</sub></td><td>N/A</td><td>(0.66 ± 0.10) × 10<sup>-3</sup></td><td>(1.86 ± 0.22) × 10<sup>-3</sup></td><td>(95.4 ± 37.4) × 10<sup>-3</sup></td></tr><tr><td>V(Δφ)<sub>c</sub></td><td>N/A</td><td>(1.55 ± 0.36) × 10<sup>-3</sup></td><td>(2.91 ± 0.22) × 10<sup>-3</sup></td><td>N/A</td></tr><tr><td>⟨V(Δφ)<sub>b</sub>/V(Δφ)<sub>c</sub>⟩</td><td>N/A</td><td>0.476 ± 0.053</td><td>0.642 ± 0.060</td><td>N/A</td></tr><tr><td>α<sub>bulk</sub></td><td>N/A</td><td>0.37 ± 0.02</td><td>0.51 ± 0.03</td><td>N/A</td></tr><tr><td>α<sub>ctl</sub></td><td>N/A</td><td>0.37 ± 0.03</td><td>0.48 ± 0.02</td><td>N/A</td></tr><tr><td>τ<sup>ac</sup><sub>bulk</sub></td><td>N/A</td><td>0.98 ± 0.04</td><td>1.62 ± 0.07</td><td>5.32 ± 0.12<sup>a</sup>, 2.89 ± 0.03<sup>b</sup></td></tr><tr><td>τ<sup>ac</sup><sub>ctl</sub></td><td>N/A</td><td>1.00 ± 0.09</td><td>1.73 ± 0.06</td><td>N/A</td></tr><tr><td>ε<sub>bulk</sub></td><td>N/A</td><td>0.434 ± 0.037</td><td>0.275 ± 0.020</td><td>0.141 ± 0.034</td></tr><tr><td>ε<sub>ctl</sub></td><td>N/A</td><td>0.402 ± 0.058</td><td>0.299 ± 0.024</td><td>N/A</td></tr><tr><td>T<sub>eff,bulk</sub></td><td>N/A</td><td>(3.46 ± 0.11) × 10<sup>-4</sup></td><td>(5.56 ± 0.29) × 10<sup>-4</sup></td><td>(15.45 ± 6.36) × 10<sup>-4</sup></td></tr><tr><td>T<sub>eff,ctl</sub></td><td>N/A</td><td>(5.03 ± 0.19) × 10<sup>-4</sup></td><td>(8.75 ± 0.16) × 10<sup>-4</sup></td><td>N/A</td></tr></tbody></table>

a Autocorrelation function is fitted to an exponential function, whereas the rest is fitted to a homographic function.

b Value coming from a fit of the power spectrum to a Lorentzian function

<caption>TABLE I. Synopsis of numerical values of the different observables. Notice that the average values of the ratios $\langle \nu_b/\nu_c \rangle$ and $\langle V(\Delta\phi)_b/V(\Delta\phi)_c \rangle$ are calculated directly as averages of the ratios and are therefore different from the ratio of the averages.</caption>