# The younger flagellum coordinates the beating in C. reinhardtii

Da Wei1,3,Greta Quaranta2, Marie-Eve Aubin-Tam${}^{1}\dagger$, Daniel S.W.
Tam${}^{2}\ast$

1Department of Bionanoscience, Delft University of Technology,
2628CJ Delft, Netherlands.
2Laboratory for Aero and Hydrodynamics, Delft University of Technology,
2628CD Delft, Netherlands.
3Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics,
Chinese Academy of Sciences; Beijing 100190, China.

$\dagger$Corresponding author. Email<EMAIL_ADDRESS>
$\ast$Corresponding author. Email<EMAIL_ADDRESS>
###### Abstract

Eukaryotes swim with coordinated flagellar (ciliary) beating and steer by
fine-tuning the coordination. The model organism for studying flagellate
motility, _C. reinhardtii_ (CR), employs synchronous, breast-stroke-like
flagellar beating to swim, and it modulates the beating amplitudes
differentially to steer. This strategy hinges on both inherent flagellar
asymmetries (e.g. different response to chemical messengers) and such
asymmetries being effectively coordinated in the synchronous beating. In CR,
the synchrony of beating is known to be supported by a mechanical connection
between flagella, however, how flagellar asymmetries persist in the synchrony
remains elusive. For example, it has been speculated for decades that one
flagellum leads the beating, as its dynamic properties (i.e. frequency,
waveform, etc.) appear to be copied by the other one. In this study, we
combine experiments, computations, and modeling efforts to elucidate the roles
played by each flagellum in synchronous beating. With a non-invasive technique
to selectively load each flagellum, we show that the coordinated beating
essentially responds to only load exerted on the cis flagellum; and that such
asymmetry in response derives from a unilateral coupling between the two
flagella. Our results highlight a distinct role for each flagellum in
coordination and have implication for biflagellates’ tactic behaviors.

One-Sentence Summary: The younger flagellum of _C. reinhardtii_ coordinates
the synchronous beating and couples to external forces.

## Introduction

The ability to swim towards desirable environments and away from hazardous
ones is fundamental to the survival of many microorganisms. These so-called
tactic behaviors are exhibited by many motile microorganisms ranging from
bacteria (?, ?) to larger flagellates and ciliates (?, ?, ?). Different
microorganisms have developed specific strategies for steering, depending on
the tactic behavior and on their specific sensory and motility repertoire. For
example, bacteria modulate the tumbling rate (?) while flagellates and
ciliates modulate the waveform (?, ?, ?, ?), amplitude (?, ?) and frequency of
their flagellar/ciliary (?, ?) beating. The goal of these active modulations
of the motility is to achieve a spatially asymmetric generation of propulsive
force to steer the organism.

C. reinhardtii (CR), the model organism for studies of flagellar motility,
achieves tactic navigation by a fine-tuned differential modulation on its two
flagella. Studying this organism offers great opportunities to look into how
flagella coordinate with each other and how such coordination helps facilitate
targeted steering. CR has a symmetric cell body and two near-identical
flagella inherited from the common ancestors of land plants and animals (?).
It swims by beating its two flagella synchronously and is capable of photo-
and chemotaxis (?, ?). For this biflagellated organism, effective steering
hinges on both flagellar asymmetry and flagellar coordination. On the one
hand, the two flagella must be asymmetric to respond differentially to stimuli
(?, ?); on the other hand, the differential responses must be coordinated by
the cell such that the beating would remain synchronized to guarantee
effective swimming. Understanding this remarkable feat requires knowledge
about both flagellar asymmetry and coordination.

The two flagella are known to be asymmetric in several, possibly associated,
aspects. First of all, they differ in developmental age (?, ?). The flagellum
closer to the eyespot, the cis(-eyespot) flagellum, is always younger than the
other one, the trans(-eyespot) flagellum. This is because the cis is organized
by a basal body (BB) that develops from a pre-matured one in the mother cell;
and this younger BB also organizes the flagellar root (D4 rootlet) that
dictates the eyespot formation (?). Second, the two flagella have asymmetric
protein composition (?, ?, ?). For example, the trans flagellum is richer in
CAH6, a protein possibly involved in CO2 sensing (?, ?). Finally, the flagella
have different dynamic properties (?, ?, ?). Their beating is modulated
differentially by second messengers such as calcium (?, ?) and cAMP (?). When
beating alone, the trans beats at a frequency 30%-40% higher than the cis (?,
?, ?, ?); the trans also displays an attenuated waveform (?) and a much
stronger noise (?, ?).

Remarkably, despite these inherent asymmetries, CR cells establish robust
synchronization between the flagella. Such coordination enables efficient
swimming and steering of the cells and takes basis on the fibrous connections
between flagellar bases (?, ?). Intriguingly, in the coordinated beating, both
flagella display dynamic properties, i.e., flagellar waveform, beating
frequency ($\sim$50 Hz), and frequency fluctuation, that are more similar to
those of the cis flagellum (?, ?, ?, ?, ?). This has led to a long-standing
hypothesis that ”the cis somehow tunes the trans flagellum” (?). This implies
that the symmetric flagellar beating (”breast-stroke”) observed is the result
of interactions between two flagella playing differential roles in
coordination. How does the basal coupling make this possible? Recent
theoretical efforts show that the basal coupling can give rise to different
synchronization modes (?, ?, ?); and that flagellar dynamics, such as beating
frequency, may simply emerge from the interplay between mechanics of basal
coupling and bio-activity (?). Yet, most theoretical efforts examining
flagellar synchronization have assumed two identical flagella, limiting the
results’ implication for the realistic case. Moreover, little experiments
directly probe the flagella’s differential roles during synchronous beating
(?). Therefore, flagellar coordination in this model organism remains unclear.
To clarify the picture experimentally, one needs to selectively force each
flagellum, and characterize the dynamics of the flagellar response.

In this study, we address this challenge and devise a non-invasive approach to
apply external forces selectively on the cis\- or the trans-flagella.
Oscillatory background flows are imposed along an angle with respect to the
cell’s symmetry axis. Such flows result in controlled hydrodynamic forces,
which are markedly different on the two flagella. With experiments,
hydrodynamic computations, and modeling, we show definitively that the two
flagella are unilaterally coupled, such that the younger flagellum (cis)
coordinates the beating, whereas the elder one simply copies the dynamic
properties of the younger. This also means that only external forces on the
cis may mechanically fine-tune the coordination. We also study the effect of
calcium in the cis’ leading role as calcium is deeply involved in flagellar
asymmetry and hence phototactic steering. In addition, a well-known mutant
that lacks flagellar dominance (ptx1) (?, ?) is examined. Results show that
the coordinating role of cis does not need environmental free calcium, whereas
it does require the genes lost in ptx1. Our results discern the differential
roles of CR’s flagella, highlight an advanced function of the inter-flagellar
mechanical coupling, and have implications for biflagellates’ tactic motility.

## Experimental scheme for selective loading

We set out to establish a non-invasive experimental technique that exerts
differential loads on the flagella of CR. Following the study by Quaranta et
al. (?), we induce oscillatory background flows to exert hydrodynamic forcing
to flagella of captured cells. With programmed oscillations of the
piezoelectric stage, the amplitude, frequency, and direction of the background
flows are all controlled, enabling selective loading.

To quantitatively estimate the selectivity of the flows along different angles
($\theta$), we compute the flagellar loads under the flows along
$\theta=-45\degree$, $0\degree$, and $45\degree$, see Fig. 1A. Computations
based on boundary element methods (BEM) and slender-body theory (SBT) give the
real-time drag force F on each flagellum and the power $P$ exerted by the
viscous forces on each flagellum. For given realistic flagellar shapes, we
compare computed loads with and without external flows. From these we isolate
the loads from the induced flows $\textbf{F}_{\rm Flow}$ and $P_{\rm Flow}$
(Methods).

Loads on each flagellum under flows of $\theta=0\degree,\ -45\degree,\
45\degree$ are presented in Fig. 2. Upper panels display the magnitude of the
drag force $F_{\rm Flow}=|\textbf{F}_{\rm Flow}|$; while lower panels show
viscous power $P_{\rm Flow}$. Force magnitudes are scaled by $F_{0}=6\pi\mu
RU_{0}$ = 9.9 pN; while the powers by $P_{0}=F_{0}U_{0}=1.1$ fW. $F_{0}$ is
the Stokes drag on a typical free-swimming cell (radius $R=5\ \mu$m, speed
$U_{0}=110\ \mu$m/s, water viscosity $\mu=0.95$ mPa$\cdot$s).

Evidently, along $\theta=0\degree$, flows load the flagella equally (Fig. 2A).
However, at $\theta=-45\degree$, flows load the cis flagellum $\sim 2$ times
larger than the trans (Fig. 2B, $F^{\rm c}_{\rm Flow}\approx 2F^{\rm t}_{\rm
Flow}$); whereas flows at $\theta=45\degree$ do the opposite (Fig. 2C). The
selectivity also manifests in (the absolute values of) $P_{\rm Flow}$. We do
notice that flows along $\theta=+45\degree$ are able to synchronize the
flagella with $P_{\rm Flow}<0$, meaning that the flagella are working against
the flows, and this shall be discussed in later sections.

Hereon forward, we refer to $\theta_{\rm c}$-flows, flows for which
$\theta=-45\degree$ and the cis-flagellum is selectively loaded. Likewise,
$\theta_{\rm t}$-flows denote flows on $\theta=+45\degree$ that selectively
load the trans. $\theta_{\rm a}$-flows denote the axial flow along
$\theta=0\degree$. We next introduce how we quantify the flows’ effective
forcing strength ($\varepsilon$) on the cell.

Phase dynamics of flagellar beating is extracted from videography (?, ?, ?).
Recordings are masked and thresholded to highlight the flagella (Fig. 1B-C).
Then the mean pixel values over time within two sampling windows (Fig. 1D) are
converted to observable-invariant flagellar phases (?), Fig. 1E. Throughout
this study, as cis and trans always beat synchronously (Fig. 1E inset), their
phases $\varphi_{\rm c,t}$ are used interchangeably as the flagellar phase
$\varphi$. The flagellar phase dynamics under external periodic forcing is
described by Adler equation (?, ?, ?):

$\displaystyle\frac{d\Delta\varphi}{dt}=-2\pi\nu-2\pi\varepsilon\sin(\Delta\varphi)+\zeta(t).$
(1)

$\Delta\varphi=\varphi-2\pi f_{\rm f}t$ is the phase difference between the
beating and the forcing, with $f_{\rm f}$ the forcing frequency, and
$\varepsilon$ the forcing strength. The detuning $\nu=f_{\rm f}-f_{0}$ is the
frequency mismatch between the beating ($f_{0}$) and forcing. $\zeta(t)$
represents a white noise that satisfies
$\langle\zeta(\tau+t)\zeta(\tau)\rangle=2T_{\rm eff}\delta(t)$, with $T_{\rm
eff}$ an effective temperature and $\delta(t)$ the Dirac delta function.

When the forcing strength outweighs the detuning ($\varepsilon>|\nu|$),
synchronization with the flow ($d\Delta\varphi/dt=0$) emerges, see the
plateaus marked black in Fig. 1F. We characterize synchronization with
$\tau=t_{\rm sync}$/$t_{\rm tot}$, where $t_{\rm sync}$ is the total time of
flow synchronization and $t_{\rm tot}$ the flow duration. Fig. 1F presents the
phase dynamics which are representative and range from: no synchronization
($\tau$=0, i), unstable synchronization ($0<\tau<1$, ii-iii), and stable
synchronization ($\tau$=1, iv). In this study, the frequency range in $\nu$
for which $\tau\geq 0.5$ is used to measure $\varepsilon$ (see Fig. 1F inset).
This method is equivalent to previous fitting-based methods (?, ?), see SM.
Sec.S1.

## Asymmetric susceptibility to flow synchronization

Now we examine cell responses to flows of various amplitudes and along
different directions. First we explore flow synchronization over a broad range
of amplitudes and frequencies. $\theta_{\rm a}$-flows with frequencies $f_{\rm
f}\in[40,\,75]$ Hz and amplitudes $U\in[390,2340]\ \mu$m/s are imposed. The
scanned range covers reported intrinsic frequencies of both the cis and trans
flagellum (?, ?, ?, ?); while the amplitude reaches the maximum instantaneous
speed of a beating flagellum ($\sim 2000\ \mu$m/s). Fig. 3A displays the
resultant flow-synchronized time fractions $\tau$. Up until the strongest flow
amplitude, the large forces cannot disrupt the synchronized flagellar beating.
In addition, synchronization is never established around frequencies other
than $f_{0}$. This shows that the inter-flagella coupling is much stronger
than the maximum amplitude of forcing.

Next we examine the synchronization with the $\theta_{\rm c}$-flows and
$\theta_{\rm t}$-flows. Flows of a fixed amplitude ($\sim 7U_{0}$) but varying
frequencies around $f_{0}$ are applied to each captured cell (see Methods).
With these, the flow-synchronized time fraction $\tau$ as a function of the
detuning ($\nu$) and flow direction ($\theta_{\rm c,a,t}$) is recorded and
helps quantify the flows’ effective forcing $\varepsilon(\theta)$.

Comparing $\tau(\nu;\theta_{\rm c})$ to $\tau(\nu;\theta_{\rm t})$, with
$\tau(\nu;\theta_{\rm a})$ as reference, we find that $\theta_{\rm c}$-flows
are the most effective in synchronizing the beating (Fig. 3B). We illustrate
this point with the profiles of an exemplary cell (Fig. 3B inset). First,
although both the $\theta_{\rm c}$-flow (red) and the $\theta_{\rm t}$-flow
(blue) can synchronize the cell at small detunings ($|\nu|<$0.5Hz), the
$\theta_{\rm c}$-flow maintains the synchronization for the whole time
($\,\tau(\theta_{\rm c})~{}$=1), while the $\theta_{\rm t}$-flow for a
slightly smaller time fraction ($\,\tau(\theta_{\rm c})~{}$$\approx$0.85).
This is due to phase-slips (step-like changes in $\Delta\varphi(t)$ in Fig.
1F) between flagella and the flow, and means that the $\theta_{\rm t}$-flow
synchronization is less stable. Additionally, for intermediate detuning (0.5
Hz$<|\nu|<$4 Hz), $\,\tau(\theta_{\rm c})~{}$is always larger than
$\,\tau(\theta_{\rm t})~{}$. In some cases, the $\theta_{\rm c}$-flow
synchronizes the cell fully whereas the $\theta_{\rm t}$-flow fails completely
(e.g., at $\nu=-2$ Hz). Together, these results imply that a flow of given
amplitude synchronizes flagellar beating more effectively if it selectively
loads the cis.

We repeat the experiments with cells from multiple cultures, captured on
different pipettes, and with different eyespot orientations ($\sim$50% heading
rightward in the imaging plane) to rule out possible influence from the setup.
$\tau(\nu;\theta)$ of N=11 wt cells tested in the TRIS-minimal medium (pH=7.0)
are displayed in Fig. 3B (labeled as ”TRIS”). On average,
$\,\varepsilon(\theta_{\rm c})~{}$= 2.9 Hz and is 70% larger than
$\,\varepsilon(\theta_{\rm t})~{}$= 1.7 Hz. It bears emphasis that
$\,\varepsilon(\theta_{\rm c})~{}$$>$$\,\varepsilon(\theta_{\rm t})~{}$holds
true for every single cell tested (11/11). In Fig. 3C, we show this by
representing each cell as a point in the $\,\varepsilon(\theta_{\rm
c})~{}$-$\,\varepsilon(\theta_{\rm t})~{}$plane. A point being below the first
bisector line ($\,\varepsilon(\theta_{\rm c})~{}$=$\,\varepsilon(\theta_{\rm
t})~{}$) indicates that $\,\varepsilon(\theta_{\rm
c})~{}$$>$$\,\varepsilon(\theta_{\rm t})~{}$for this cell. All cells cluster
clearly below the line. This asymmetry manifest equivalently through $\tau$.
In Fig. 3D, each point represents the time fractions of the same cell
synchronized by the $\theta_{\rm c}$-flow and the $\theta_{\rm t}$-flow at the
same frequency. Most points ($>$90%) are below the first bisector line,
meaning that $\,\tau(\theta_{\rm c})~{}$$>$ $\,\tau(\theta_{\rm t})~{}$.
Altogether, all results show that selectively loading the cis flagellum
establishes synchronization with the flow more effectively, pointing to cis
and trans playing differential roles in the coordinated beating.

We next study whether this newly observed cis-trans asymmetry is affected by
calcium depletion. Calcium is a critical second messenger for modulating
flagellates motility and is deeply involved in phototaxis (?). The depletion
of the free environmental calcium is known to degrade flagellar
synchronization and exacerbate flagellar asymmetry (?). Here we focus on
whether calcium depletion affects the asymmetry $\,\varepsilon(\theta_{\rm
c})~{}$$>$$\,\varepsilon(\theta_{\rm t})~{}$. We deplete environmental calcium
by EGTA-chelation, following the protocol in Ref. (?). Similar to previous
reports (?, ?), the number of freely swimming cells drops significantly in
EGTA-containing medium. However, the remaining cells beat synchronously for
hours after capture. For these beating cells, calcium depletion is first
confirmed by characterizing their deflagellation behavior. Indeed, calcium
depletion is reported to inhibit deflagellation (?, ?). In experiments with
standard calcium concentration, all cells deflagellated under pipette suction
(20/20). For experiments conducted in calcium depleting EGTA-containing
medium, we observe deflagellation to occur in none but one cell (1/19).

After confirming the calcium depletion, we perform the same sets of flow
synchronization experiments. The dashed lines in Fig. 3B show the median
synchronization profiles $\tau(\nu;\theta)$ (N=6 cells, labeled as ”EGTA”).
The flagellar asymmetry is unaffected, see also Fig. 3E. Note that
$\,\varepsilon(\theta_{\rm c})~{}$$>$$\,\varepsilon(\theta_{\rm t})~{}$again
applies for every single cell tested. The mean values of $\varepsilon$ drop
slightly. However, the different effectiveness between $\theta_{\rm c}$-flows
and $\theta_{\rm t}$-flows, $\,\varepsilon(\theta_{\rm
c})~{}$$-$$\,\varepsilon(\theta_{\rm t})~{}$, is not affected, see Fig. 3E
inset.

Finally, we determine how the forcing strength of the flow depends on the
hydrodynamic forces exerted by the flow on the flagella. We compute the
hydrodynamic beat-averaged loads, $\overline{F}_{\rm
Flow}=\int_{0}^{2\pi}F_{\rm Flow}d\varphi/2\pi$, $\overline{P}_{\rm
Flow}=\int_{0}^{2\pi}{P}_{\rm Flow}d\varphi/2\pi$, induced by the flow on the
trans and on the cis flagella, see the horizontal lines in Fig. 2. These loads
are computed for the $\theta_{\rm c}$-flow, $\theta_{\rm t}$-flow,
$\theta_{\rm a}$-flow and we also include experiments and computations
performed with flows along $\theta=90\degree$ (circles), see SM. Sec.S2. Fig.
3F and G represent $\varepsilon$ as a function of the loads on the cis and
trans flagellum respectively, with each symbol representing one of the four
different flow directions, see the drawings. We find that the effective
forcing strength scales with the time-averaged drag on the cis,
$\varepsilon\sim\overline{F}^{\rm c}_{\rm Flow}$, while we find no such
correlation between $\varepsilon$ and $\overline{F}^{\rm t}_{\rm Flow}$. The
linear relation between $\varepsilon$ and $\overline{F}^{\rm c}_{\rm Flow}$
has an intercept near zero ($\varepsilon|_{\overline{F}^{\rm c}_{\rm
Flow}=0}\approx 0$). Given the total forces on both flagella
($\overline{F}^{\rm c}_{\rm Flow}+\overline{F}^{\rm t}_{\rm Flow}$) for these
flows remains almost constant (0.74-0.79$F_{0}$), the zero-intercept implies
that for a hypothetical flow that exerts no load on the cis but solely forces
the trans, it will not be able to synchronize the cell at all. This suggests a
negligible contribution of the forcing on the trans in establishing
synchronization with flows.

## The asymmetry is lost in _ptx1_ mutants

Furthermore, we examine the flagellar dominance mutant ptx1. In this mutant,
both flagella respond similarly to changes of calcium concentrations (?) and
have similar beating frequencies when demembranated and reactivated (?).

Ptx1 mutants have two modes of coordinated beating, namely, the in-phase (IP)
synchronization and the anti-phase (AP) synchronization (?, ?). First, we
apply $\theta_{\rm a}$-flow in the same frequency and amplitude ranges as for
wt cells. We find that the IP mode around $f_{0}\approx 50$ Hz is the only
mode that can be synchronized by external flows. We focus on this mode and
report $\tau$ as $\tau=t_{\rm sync}/t_{\rm IP}$ for this mutant, where $t_{\rm
IP}$ is the total time of IP-beating under the applied flows, see Fig. 4A.
Synchronization profiles $\tau(\nu;\theta)$ of ptx1 are shown in Fig. 4B. The
median profiles are of similar width and height, indistinguishable from each
other, and hence indicate a loss of asymmetric susceptibility to flow
synchronization. The loss is further confirmed by the extracted
$\varepsilon(\theta)$ (?) and $\tau(\theta)$ (Fig. 4C-D). Cells and
synchronization attempts are distributed evenly across the first bisector
lines (7/14 cells are below $\,\varepsilon(\theta_{\rm
c})~{}$=$\,\varepsilon(\theta_{\rm t})~{}$in Fig. 4C, and $\sim$50% points are
below $\,\tau(\theta_{\rm c})~{}$$=$$\,\tau(\theta_{\rm t})~{}$in Fig. 4D).
Altogether, all results show consistently that the asymmetry is lost in ptx1.

## Modeling

### Framework

To investigate the implications of our experimental results on the coupling
between flagella and their dynamics, we develop a model for the system (SM.
Sec.S3), representing flagella and external flows as oscillators with
directional couplings:

$\begin{cases}\dot{\varphi_{\rm f}}=2\pi f_{\rm f}\\\ \dot{\varphi_{\rm
c}}=2\pi[f_{\rm c}-\lambda_{\rm t}\sin(\varphi_{\rm c}\text{-}\varphi_{\rm
t})-\varepsilon_{\rm c}\sin(\varphi_{\rm c}\text{-}\varphi_{\rm
f})]+\zeta_{\rm c}(t)\\\ \dot{\varphi_{\rm t}}=2\pi[f_{\rm t}-\lambda_{\rm
c}\sin(\varphi_{\rm t}\text{-}\varphi_{\rm c})-\varepsilon_{\rm
t}\sin(\varphi_{\rm t}\text{-}\varphi_{\rm f})]+\zeta_{\rm t}(t).\end{cases}$
(2)

$\varphi_{\rm f,c,t}(t)$ respectively represent the phase of the flow, the
cis, and the trans flagellum. $f_{\rm f,c,t}$ represents the inherent
frequency of the forcing (flow), the cis, and the trans respectively. The
phase dynamics of each flagellum is modulated by its interactions with the
other flagellum as well as the background flow. Take the cis
($\dot{\varphi}_{\rm c}$) for example, the effect of the trans and the forcing
on the cis are respectively accounted for by the $\lambda_{\rm t}$-term and
the $\varepsilon_{\rm c}$-term, see Eq. (2). In other words, $\lambda_{\rm t}$
and $\varepsilon_{\rm c}$ measure the sensitivity of the actual cis-frequency
to the phase differences between oscillators ($\varphi_{\rm c}-\varphi_{\rm
t,f}$), see the arrows in Fig. 5A. Lastly, $\zeta_{\rm c,t}$ represent the
white noise of the cis and trans flagellum respectively. In the following
parts, without loss of generality, the noise are assumed equally strong and
uncorrelated ($\langle\zeta^{2}_{\rm c}\rangle=\langle\zeta^{2}_{\rm
t}\rangle$, or $T^{\rm c}_{\rm eff}=T^{\rm t}_{\rm eff}$). Nuanced phase
dynamics under differential noise levels can be found in SM. Sec.S4.

Eq. (2) can be readily reduced to Eq. (1), which allows us to write the
experimentally measured values ($f_{0},\ \varepsilon(\theta),\ T_{\rm eff}$)
analytically with $\varepsilon_{\rm c,t}$, $\lambda_{\rm c,t}$, and
$\zeta_{\rm c,t}$. The asymptotic behavior of the model under the condition
$\varphi_{c}\approx\varphi_{t}\approx\varphi_{f}$ are (SM. Sec.S3):

$\begin{cases}f_{0}&=\alpha f_{\rm c}+(1-\alpha)f_{\rm t},\\\ T_{\rm
eff}&=\alpha^{2}T_{\rm eff}^{\rm c}+(1-\alpha)^{2}T_{\rm eff}^{\rm t},\\\
\varepsilon(\theta)&=\alpha\varepsilon_{\rm
c}(\theta)+(1-\alpha)\varepsilon_{\rm t}(\theta),\end{cases}$ (3)

with $\alpha=\lambda_{\rm c}/(\lambda_{\rm c}+\lambda_{\rm t})$ representing
the dominance of cis. It is then clear that when $\alpha\approx 1$, the
coordinated beating will display dynamic properties of the cis flagellum.

Fig. 5A illustrates an exemplary modeling scheme describing flagellar beating
subjected to $\theta_{\rm c}$-flows. The direction and thickness of arrows
represent coupling direction and strength respectively. The selective loading
on the cis is represented by $\varepsilon_{\rm c}>\varepsilon_{\rm t}$; while
$\lambda_{\rm c}>\lambda_{\rm t}$ reflects that the cis has a more dominant
role in the coordinated beating. We run Monte-Carlo simulation with Eq. (2)
using customized matlab scripts.

### Coordinated beating under symmetric forcing

We first model the flow synchronization induced by $\theta_{\rm a}$-flow
(symmetric flagellar loads). In this case,
$\varepsilon(\theta)=\alpha\varepsilon_{\rm
c}(\theta)+(1-\alpha)\varepsilon_{\rm t}(\theta)$ (Eq. (3)) reduces to
$\varepsilon=\varepsilon_{\rm c,t}$ and is independent of $\alpha$. We set
$\varepsilon_{\rm c,t}$ as 2.4 Hz to match the measured
$\,\varepsilon(\theta_{\rm a})~{}$=2.4 Hz (Fig. 3B).

At similar detunings as in the experimental results in Fig. 1F, our Monte-
Carlo simulations reproduces the phase dynamics with: (i) no flow
synchronization, (ii-iii) unstable synchronization, and (iv) stable
synchronization (Fig. 5B). Repeating the simulations for varying forcing
strength $\varepsilon$ ($=\varepsilon_{\rm c,t}$) and frequency $f_{\rm f}$
yields Arnold tongue diagrams in agreement with those reported from our
experiments. The Arnold Tongue for wt in Fig. 3A and ptx1 in Fig. 4A are
reproduced with simulations shown in Fig. 5C and D respectively. The only
parameter value changed between Fig. 5C and D is the level of noise ($T^{\rm
c,t}_{\rm eff}$), which is increased by an order of magnitude. The differences
in phase dynamics between wt and ptx1, when subjected to symmetric external
loading, are therefore accounted by solely varying the noise.

### Coordinated beating under selective loading

We next model flow synchronization by the $\theta_{\rm c}$-flows and the
$\theta_{\rm t}$-flows. The selective forcing ($\varepsilon_{\rm
c}\neq\varepsilon_{\rm t}$) allows the effect of flagellar dominance
($\lambda_{\rm c}\neq\lambda_{\rm t}$) to manifest in the effective forcing
strength $\varepsilon(\theta)$ and hence in the synchronization profiles
$\tau(\nu;\theta)$, Fig. 5E. Similar to our experimental observations,
$\theta_{\rm c}$-flow synchronizes the coordinated beating over the broadest
range of $\nu$ (i.e. largest $\varepsilon$). This is directly attributed to
the dominance $\lambda_{\rm c}>\lambda_{\rm t}$: by setting $\lambda_{\rm
c}=\lambda_{\rm t}$, the differences between $\,\tau(\theta_{\rm c})~{}$and
$\,\tau(\theta_{\rm t})~{}$disappear even under selective loading (Fig. 5E
inset). Fig. 5F details how the asymmetry of inter-flagellar coupling
($\lambda_{\rm c}/\lambda_{\rm t}$) affects the asymmetry between
$\,\tau(\theta_{\rm c})~{}$and $\,\tau(\theta_{\rm t})~{}$. The open symbols
represent $\varepsilon(\theta)$ measured from modeled $\tau(\nu;\theta)$ and
the lines represent Eq. (3). The difference between $\,\varepsilon(\theta_{\rm
c})~{}$and $\,\varepsilon(\theta_{\rm t})~{}$increases with $\lambda_{\rm
c}/\lambda_{\rm t}$, and they each saturates to reflect only the forcing on
the cis ($\varepsilon_{\rm c}$, the grey dashed lines). With $f_{c}=45$ Hz,
$f_{t}=65$ Hz (?, ?), and $f_{0}\approx 50$ Hz, we deduce from Eq. (3) that
$\lambda_{\rm c}=4\lambda_{\rm t}$ for wt cells. For wt cells under calcium
depletion, experimental results are reproduced with a lower total forcing
strength (Fig. 5G). $\varepsilon_{\rm c}+\varepsilon_{\rm t}$ is set to 4.08
Hz (15% lower) to reflect the $7\%-20\%$ decrease in $\varepsilon(\theta)$
induced by calcium depletion.

The ptx1 results are reproduced with a stronger noise ($T^{\rm c,t}_{\rm
eff}=9.42\ {\rm rad}^{2}/{\rm s}$) and a symmetric inter-flagellar coupling
$\lambda_{\rm c}/\lambda_{\rm t}=1$, see Fig. 5H and Table. 1. Both changes
are necessary for reproducing the synchronization profiles of ptx1 in Fig. 5H:
while the stronger noise lowers the maximal values of $\tau(\theta,\nu)$,
setting $\lambda_{\rm c}/\lambda_{\rm t}=4$ would still result in
$\,\tau(\theta_{\rm c})~{}$$>$$\,\tau(\theta_{\rm t})~{}$in the central range
($|\nu|\lesssim 2.4$ Hz). Finally, it is noteworthy that the noise in ptx1
increases not only because a higher noise value for individual flagella, but
also because the cis-trans coupling has become symmetric. As shown by Eq. (3),
the unilateral coupling promotes not only the cis-frequency in the synchrony
but also the cis-noise. Given $T^{\rm c}_{\rm eff}\ll T^{\rm t}_{\rm eff}$ and
$\lambda_{\rm c}=4\lambda_{\rm t}$, we confirm with simulations that the cis
stabilizes the beating frequency of the trans and decreases its beating noise.
The simulations are in good agreement with experimental noise measurements,
see SM. Sec.S4 for details.

## Discussion

The two flagella of C. reinhardtii have long been known to have inherently
different dynamic properties such as frequency, waveform, level of active
noise, and responses to second messengers (?, ?, ?, ?, ?). Intriguingly, when
connected by basal fibers and beating synchronously, they both adopt the
kinematics of the cis-(eyespot) flagellum, which led to the assumption that
the flagella may have differential roles in coordination. In this work, we
test this hypothesis by employing oscillatory flows applied from an angle with
respect to the cells’ symmetry axis and thus exert biased loads on one
flagellum.

Without an exception, in wt cells, $\theta_{\rm c}$-flows, the ones that
selectively load the cis flagellum, are always more effective in synchronizing
the flagellar beating than the $\theta_{\rm t}$-flows. This is shown by the
larger effective forcing strengths ($\,\varepsilon(\theta_{\rm
c})~{}$$>$$\,\varepsilon(\theta_{\rm t})~{}$, Fig. 3B-C) and larger
synchronized time fractions ($\,\tau(\theta_{\rm c})~{}$$>$$\,\tau(\theta_{\rm
t})~{}$, Fig. 3D). Mapping the measured forcing strength $\varepsilon(\theta)$
as a function of the loads, we find empirically that
$\varepsilon\propto\overline{F}_{\rm Flow}^{\rm c}$ (Fig. 3F) and that trans-
loads appear to matter negligibly. These observations all indicate that the
cis-loads determine whether an external forcing can synchronize the cell.
Moreover, this point is further highlighted by an unexpected finding: when
$\theta_{\rm t}$-flows are applied, the trans flagellum always beats against
the external flow ($P_{\rm Flow}^{\rm t}<0$) and the only stabilizing factor
for flow synchronization is the cis flagellum working along with the flow
during the recovery stroke (Fig. 2C lower panel). These observations
definitively prove that the two flagella have differential roles in the
coordination and interestingly imply that flagella are coupled to external
flow only through the cis.

To have a mechanistic understanding of this finding, we model the system with
Eq. (2). In the model, selective hydrodynamic loading and flagellar dominance
in the coordinated beating are respectively represented by $\varepsilon_{\rm
c}\neq\varepsilon_{\rm t}$ and $\lambda_{\rm c}\neq\lambda_{\rm t}$. Setting
out from the model, we obtain closed-form expressions for observables such as
$f_{0}$ and $\varepsilon$ (Eq. (3)), which illustrate how flagellar dominance
and selective loading affect the coordinated flagellar beating. Moreover, with
Monte-Carlo simulation, we clarified the interplay between flows and flagella
(SM. Sec.S3), and reproduces all experimental observations.

With the model, we show that a ”dominance” of the cis ($\lambda_{\rm
c}>\lambda_{\rm t}$) is sufficient to explain why the coordinated flagellar
beating bears the frequency and the noise level of the cis flagellum. In the
model, such dominance means that the cis-phase is much less sensitive to the
trans-phase than the other way around. We then reproduce the phase dynamics of
flow synchronization at varying detunings (Fig. 5B), amplitudes (Fig. 5C), and
noise (Fig. 5D). Exploiting the observation that the coordination between
flagella cannot be broken by external flows up to the strongest ones tested
($\varepsilon^{\rm max}\sim 10$ Hz, Fig. 3A), we quantify the lower limit of
the total basal coupling, $\lambda_{\rm c}+\lambda_{\rm t}$, to be
approximately 40 Hz (deduced in SM. Sec.S3), which is an order magnitude
larger than the hydrodynamic inter-flagellar coupling (?, ?, ?, ?).

The modulation of flagellar dominance mediates tactic behaviors (?, ?, ?, ?).
Calcium is hypothesized to be underlying the modulation of dominance, as it
causes the connecting fiber between flagella to contract (?), modulates the
cis- and trans activity (e.g. beating amplitude) differentially (?), and
calcium influx comprises the initial step of CR’s photo- (?) and
mechanoresponses (?). We therefore investigate flagellar coupling in the
context of tactic steering by depleting the environmental free calcium and
hence inhibiting signals of calcium influxes. Cells are first acclimated to
calcium depletion, and then tested with the directional flows. Our results
show that the cis dominance does not require the involvement of free
environmental calcium. Calcium depletion merely induces an overall drop in the
forcing strength perceived by the cell $\varepsilon(\theta)$ ($7\%-20\%$),
which is captured by reducing $\varepsilon_{\rm c}+\varepsilon_{\rm t}$ for
15% (mean drop) in the model (Fig. 5G). Together, our results indicate that
the leading role of cis, is an inherent property, that does not require active
influx of external calcium, and possibly reflects an intrinsic mechanical
asymmetry of the cellular mesh that anchors the two flagella into the cell
body.

In ptx1 cells, a lack of flagellar dominance ($\lambda_{\rm c}=\lambda_{\rm
t}$) and a stronger noise level help reproduce our experimental observations.
Previous studies suggested that both flagella of ptx1 are similar to the
wildtype trans (?), and that the noise levels of this mutant’s synchronous
beating are much greater than those of wt (?) (see also SM. Sec.S4). If both
flagella and their anchoring roots indeed have the composition of the wildtype
trans, such symmetry would predict $\lambda_{\rm c}=\lambda_{\rm t}$. This
symmetric coupling renders the noise of ptx1 $T_{\rm eff}=T_{\rm eff}^{\rm t}$
(Eq. (3)), which is about an order of magnitude larger than the noise of wt
$T_{\rm eff}\approx T_{\rm eff}^{\rm c}$.

The comparison between ptx1 and wt highlights an intriguing advantage of the
observed unilateral coupling ($\lambda_{\rm c}\gg\lambda_{\rm t}$); that is,
it strongly suppresses the high noise of the trans. Considering that the trans
is richer in CAH6 protein and this protein’s possible role in inorganic carbon
sensing (?, ?), the potential sensing role of the trans is worth noticing.
Assuming the strong noise present in the trans originates from the biochemical
processes related to sensing, then the unilateral coupling effectively
prevents such noise from perturbing the cell’s synchronous beating and
effective swimming. In this way, the asymmetric coupling may combine the
benefit of having a stable cis as the driver while equipping a noisy trans as
a sensor.

## Material and methods

### Cell culture

CR wildtype (wt) strain cc125 (mt+) and flagellar dominance mutant ptx1 cc2894
(mt+) are cultured in TRIS-minimal medium (pH=7.0) with sterile air bubbling,
in a 14h/10h day-night cycle. Experiments are performed on the 4th day after
inoculating the liquid culture, when the culture is still in the exponential
growth phase and has a concentration of $\sim 2\times 10^{5}$ cells/ml. Before
experiments, cells are collected and resuspended in fresh TRIS-minimal
(pH=7.0).

### Calcium depletion

In calcium depletion assays, cells are cultured in the same fashion as
mentioned above but washed and resuspended in fresh TRIS-minimal medium + 0.5
mM EGTA (pH=7.0). Free calcium concentration is estimated to drop from 0.33 mM
in the TRIS-minimal medium, to 0.01 µM in the altered medium (?). Experiments
start at least one hour after the resuspension in order to acclimate the
cells.

### Experimental setup

Single cells of CR are studied following a protocol similar to the one
described in (?). Cell suspensions are filled into a customized flow chamber
with an opening on one side. The air-water interface on that side is pinned on
all edges and is sealed with silicone oil. A micropipette held by
micromanipulator (SYS-HS6, WPI) enters the chamber and captures single cells
by aspiration. The manipulator and the captured cell remain stationary in the
lab frame of reference, while the flow chamber and the fluid therein are
oscillated by a piezoelectric stage (Nano-Drive, Mad City Labs), such that
external flows are applied to the cell. Frequencies and amplitudes of the
oscillations are individually calibrated by tracking micro-beads in the
chamber. Bright field microscopy is performed on an inverted microscope (Nikon
Eclipse Ti-U, 60$\times$ water immersion objective). Videos are recorded with
a sCMOS camera (LaVision PCO.edge) at 600-1000 Hz.

### Measurement scheme

The flagellar beating of each tested cell is recorded before, during, and
after the application of the flows. We measure the cell’s average beating
frequency $f_{0}$ over 2 s ($\sim$100 beats). For ptx1 cells, $f_{0}$ is
reported for the in-phase (IP) synchronous beating. Unless otherwise stated,
directional flows ($\theta=0,\pm 45\degree$) are of the same amplitude
(780$\pm$50 $\mu$m/s, mean$\pm$std), similar to those used in Ref. (?). Flow
frequencies $f_{\rm f}$ are scanned over [$f_{0}-7$, $f_{0}+7$] Hz for each
group of directional flows.

### Computation of the flagellar loads

To quantify the hydrodynamic forces on the flagella, we first track realistic
flagellar deformation from videos wherein background flows are applied. Then
we employ a hybrid method combining boundary element method (BEM) and slender-
body theory (?, ?) to compute the drag forces exerted on each flagellum and
the forces’ rates of work. In this approach, each flagellum is represented as
a slender-body (?) with 26 discrete points along its centerline and the time-
dependent velocity of each of the 26 points is calculated by its displacement
across frames. The cell body and the pipette used to capture the cell are
represented as one entity with a completed double layer boundary integral
equation (?). Stresslet are distributed on cell-pipette’s surface; while
stokeslet and rotlet of the completion flow are distributed along cell-
pipette’s centerline (?). The no-slip boundary condition on the cell-pipette
surface is satisfied at collocation points. Lastly, stokeslets are distributed
along the centerlines of the flagella, so that no-slip boundary conditions are
met on their surfaces. Integrating the distribution of stokeslets
$\textbf{f}(s)$ over a flagellar shape, one obtains the total drag force
$\textbf{F}=\int\textbf{f}(s)ds$ is obtained. Similarly, the force’s rate of
work is computed as $P=\int\textbf{f}(s)\cdot\textbf{U}(s)ds$, where
$\textbf{U}(s)$ is the velocity of the flagellum at the position $s$ along the
centerline.

The computations shown in this study are based on videos of a representative
cell which originally beats at $\sim$50 Hz. The cell is fully synchronized by
flows along different directions ($\theta=0\degree,\ \pm 45\degree$ and
$90\degree$) at 49.2 Hz. In the computations, the applied flows are set to
have an amplitude of 780 $\mu$m/s to reflect the experiments. Computations
begin with the onset of the background flows (notified experimentally by a
flashlight event), and last for $\sim$30 beats (500 frames sampled at 801
fps). Additionally, we confirm the results of $\theta_{\rm t}$-flow-
synchronization, that both flagella spend large fractions of time beating
against the flows, with other cells and with $\theta_{\rm t}$-flows at other
frequencies.

### Isolate loads of external flows

The total loads (F and $P$) computed consist of two parts, one from the flow
created by the two flagella themselves and the other from the applied flow. In
the low Reynolds number regime, the loads of the two parts add up directly
(linearity): $\textbf{F}=\textbf{F}_{\rm Self}+\textbf{F}_{\rm Flow}$, and
$P=P_{\rm Self}+P_{\rm Flow}$. To isolate $\textbf{F}_{\rm Flow}$ and $P_{\rm
Flow}$, we compute $\textbf{F}^{\prime}=\textbf{F}_{\rm Self}$ and
$P^{\prime}=P_{\rm Self}$ by running the computation again but without the
external flows, and obtain $\textbf{F}_{\rm
Flow}=\textbf{F}-\textbf{F}^{\prime}$ and $P_{\rm Flow}=P-P^{\prime}$.

### Modeling parameters

We assume the flagellar intrinsic frequencies $f_{c}$ and $f_{t}$ to be 45 Hz
and 65 Hz respectively (?, ?, ?). On this basis, $\lambda_{\rm c}:\lambda_{\rm
t}$ is assumed to be 4:1 to account for the observed $f_{0}$ ($\sim 50$ Hz).
$\varepsilon_{\rm c}:\varepsilon_{\rm t}$ is set as 2:1, 1:1, and 1:2 for the
$\theta_{\rm c}$-flows, the $\theta_{\rm a}$-flows, and the $\theta_{\rm
t}$-flows respectively, see Fig. 2A-C. Additionally, $\varepsilon_{\rm
c}+\varepsilon_{\rm t}$ is assumed to be constant to reflect the fact that
$\overline{F}_{\rm Flow}^{\rm c}+\overline{F}_{\rm Flow}^{\rm t}$
approximately does not vary with flow directions. We take a typical value of
$T^{\rm c,t}_{\rm eff}=1.57$ rad2/s (?). The sum of inter-flagellar coupling
$\lambda_{\rm tot}=\lambda_{\rm c}+\lambda_{\rm t}$ is set to be large enough,
i.e., $\lambda_{\rm tot}=3\nu_{\rm ct}$ with $\nu_{\rm ct}=|f_{\rm t}-f_{\rm
c}|$, to account for the fact that: 1) the coordinated beating is approximated
in-phase, and 2) up until the strongest flow applied, the coordinated beating
cannot be broken (quantitative evaluation is detailed in SM. Sec.S3). To model
wt cells under calcium depletion, we decrease $\varepsilon_{\rm
c}+\varepsilon_{\rm t}$ by 15% - which is the mean decrease in the observed
$\,\varepsilon(\theta_{\rm c})~{}$,$\,\varepsilon(\theta_{\rm a})~{}$, and
$\,\varepsilon(\theta_{\rm t})~{}$(Fig. 3E). For ptx1 cells, we assume a
symmetric inter-flagellar coupling ($\lambda_{\rm c}=\lambda_{\rm t}$) and a
stronger noise level (SM. Sec.S4). The parameters are summarized in Table. 1.

Table 1: Modeling parameters

variable | symbol (unit) | TRIS | EGTA | ptx1
---|---|---|---|---
Intrinsic freq. (?, ?) |  $f_{c},\ f_{t}$ (Hz) | 45,65 | 45,65 | 45,65
Basal coupling∗ |  $\lambda_{c}+\lambda_{t}$ (Hz) | 60 | 60 | 60
cis dominance (?, ?) |  $\lambda_{c}:\lambda_{t}$ (-) | 4:1 | 4:1 | 1:1
Flow detuning |  $\nu$ (Hz) | [-10,10] | [-10,10] | [-10,10]
Total forcing (?) |  $\varepsilon_{\rm c}+\varepsilon_{\rm t}$ (Hz) | 4.8 | 4.08 | 4.8
Noise∗ (?) |  $T^{\rm c,t}_{\rm eff}$ (rad2/s) | 1.57 | 1.57 | 9.42

$*$ detailed in SM. Sec.S3

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## Acknowledgments

The authors thank Roland Kieffer for technical support. D.W. thanks Ritsu
Kamiya for helpful discussions. The authors acknowledge support by the
European Research Council (ERC starting grants no. 716712 and no. 101042612).

## Author Contributions

D.W. performed experiments, computations, designed the model, and drafted the
manuscript. G.Q. performed early experiments and obtained preliminary results.
M.A. and D.T. conceived the study, supervised the project and critically
revised the manuscript.

## Competing interests

Authors declare that they have no competing interests.

## Supplementary materials

Supplementary Text
Figs. S1 to S5
References (?, ?, ?, ?, ?, ?, ?, ?)

Figure 1: Experimental workflow. (A) Captured CR cells are subjected to
sinusoidal flows of frequency $f_{\rm f}$ along given angles ($\theta$) in the
$xy$-plane. Flows along $\theta=-45\degree,0\degree,45\degree$ of same
amplitude (780$\pm$50 $\mu$m/s, mean$\pm$std.) are used and termed as shown.
(B-E) Extracting flagellar phase $\varphi_{c}$ and $\varphi_{t}$ by image
processing. Raw images (B) are thresholded and contrast-adjusted to highlight
the flagella (C). Mean pixel values within the user-defined interrogation
windows (red and blue circles) capture the raw phases of beating (D), which
are then converted to observable-independent phases (E). Inset: phase
difference $\varphi_{c}-\varphi_{t}$. (F) Flagella-flow phase dynamics at
decreasing detuning $\nu=f_{\rm f}-f_{\rm 0}$ with $f_{\rm 0}$ the cell’s
beating frequency without external flow. Traces i to iv are taken at detunings
marked in the inset. Plateaus marked black represent flow synchronization,
whose time fractions $\tau=t_{\rm sync}$/$t_{\rm tot}$ are noted. $t_{\rm
tot}$ is the total time of recording. Inset: the flow synchronization profile,
$\tau(\nu)$, reports the effective forcing strength 2$\varepsilon$ by its
width.

Figure 2: External flagellar loads when beating is synchronized. Force
magnitude (upper panels) and power (lower panels) exerted by external flows of
$\theta=0\degree$ (A, $\theta_{\rm a}$-flow), $-45\degree$ (B, $\theta_{\rm
c}$-flow), and $+45\degree$ (C, $\theta_{\rm t}$-flow). The medians (solid
lines) and interquartile ranges (shadings) are computed over $\sim$20
synchronized beats. Dashed horizontal lines: loads averaged over a
synchronized beat. Force magnitudes and powers are scaled by $F_{0}$=9.9 pN
and $P_{0}$=1.1 fW respectively. Flagellar phase corresponds to the displayed
shapes in the middle $x$-axis.

Figure 3: Flow synchronization of wt cells. (A) Arnold tongue of a
representative cell tested with $\theta_{\rm a}$-flow. The contour is
interpolated from N=132 measurements (6 equidistant amplitudes $\times$ 22
equidistant frequencies), and color-coded by the entrained time fraction
$\tau$. (B) The synchronization profiles $\tau(\nu;\theta)$ of a
representative wt cell (inset), the median profile of the TRIS group wt cells
(N=11, solid lines) and the EGTA group (N=6, dashed lines), with either
$\theta_{\rm c}$-flows (red), $\theta_{\rm a}$-flows (yellow) or $\theta_{\rm
t}$-flows (blue). Shaded areas are the interquartile ranges for the TRIS
group. (C) Tested wt cells represented on the $\,\varepsilon(\theta_{\rm
c})~{}$$-$$\,\varepsilon(\theta_{\rm t})~{}$plane (TRIS group). Solid line:
the first bisector line ($y=x$). (D) Comparing $\tau(\nu;\theta_{\rm c})$ and
$\tau(\nu;\theta_{\rm t})$ for each cell at each applied frequency. N=132
pairs of experiments are represented on the $\,\tau(\theta_{\rm
c})~{}$$-$$\,\tau(\theta_{\rm t})~{}$plane. More than 90% of them are below
the first bisector line. (E) The coupling strengths $\varepsilon(\theta)$ of
the TRIS group (black) and the EGTA group (gray). Bars and error bars: mean
and 1 std., respectively. Inset:
$\delta\varepsilon=$$\,\varepsilon(\theta_{\rm
c})~{}$$-$$\,\varepsilon(\theta_{\rm t})~{}$. NS: not significant, p$>$0.05,
Kruskal-Wallis test, One-Way ANOVA. Relations between the forcing strength
$\varepsilon$ and the loads on the cis (F) and the trans flagellum (G).
Markers represent different flow angles, see the drawings.

Figure 4: The asymmetric susceptibility to flow synchronization is lost in the
flagellar dominance mutant _ptx1_. (A) Arnold tongue of a representative ptx1
cell tested with $\theta_{\rm a}$-flow. The contour is interpolated from N=132
measurements (6 equidistant amplitudes $\times$ 22 equidistant frequencies).
Color bar: the entrained time fraction $\tau=t_{\rm sync}/t_{\rm IP}$. (B)
Flow synchronization profiles $\tau(\nu;\theta)$ of N=14 ptx1 cells, tested
with $\theta_{\rm c}$-flows (red), $\theta_{\rm a}$-flows (yellow) and
$\theta_{\rm t}$-flows (blue). (C) $\,\varepsilon(\theta_{\rm c})~{}$and
$\,\varepsilon(\theta_{\rm t})~{}$of the tested cells. The first bisector line
(solid): $y=x$. (D) $\tau(\nu;\theta_{\rm c,t})$ for each cell at each applied
frequency. N=154 points are present.

Figure 5: Modeling the asymmetric flow synchronization. (A) Modeling scheme
describing a cell beating under directional flow ($\theta_{\rm c}$-flow as an
example). Arrows represent the directional coupling coefficients with line
thickness representing the relative strength. For example, $\lambda_{c}$
points from cis to trans, representing how the latter ($\varphi_{c}$) is
sensitive to the former ($\varphi_{t}$); meanwhile, the arrow of $\lambda_{c}$
being thicker than $\lambda_{t}$ means that $\varphi_{t}$ is much more
sensitive to $\varphi_{c}$ than the other way around. (B) Modeled phase
dynamics of flow synchronization under $\theta_{\rm a}$-flows, analogous to
Fig. 1F. Reproducing the Arnold tongue diagrams at the noise level of wt (C)
and ptx1 (D), analogous to Fig. 3A and Fig. 4A respectively. (E) Flow
synchronization profiles $\tau(\nu;\theta)$ obtained experimentally (upper
panel) and by modeling (lower panel). Inset: the modeling results with
symmetric inter-flagellar coupling. (F) Effective forcing strength
$\varepsilon(\theta)$ as a function of the inter-flagellar coupling asymmetry
$\lambda_{\rm c}/\lambda_{\rm t}$. Points: measured from simulation; lines:
analytical approximation (Eq. (3)); dashed lines: $\varepsilon_{\rm c}$
respectively for the $\theta_{\rm c}$-flow, $\theta_{\rm a}$-flow, and
$\theta_{\rm t}$-flow (from top to bottom). (G) Reproducing the flow
synchronization of wt cells under calcium depletion (H) Reproducing results of
ptx1. See Table. 1 for the modeling parameters.