Lag, lock, sync, slip: the many “phases” of coupled flagellaKirsty Y. Wan, Kyriacos C. Leptos, Raymond E. Goldstein

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences,University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

In a multitude of life’s processes, cilia and flagella are found indispensable. Recently, the biflagellated chlorophyte algaChlamydomonas has become a model organism for the study of ciliary coordination and synchronization. Here, we usehigh-speed imaging of single pipette-held cells to quantify the rich dynamics exhibited by their flagella. Underlyingthis variabiltiy in behaviour, are biological dissimilarities between the two flagella – termed cis and trans, withrespect to a unique eyespot. With emphasis on the wildtype, we use digital tracking with sub-beat-cycle resolutionto obtain limit cycles and phases for self-sustained flagellar oscillations. Characterizing the phase-synchrony of acoupled pair, we find that during the canonical swimming breaststroke the cis flagellum is consistently phase-laggedrelative to, whilst remaining robustly phase-locked with, the trans flagellum. Transient loss of synchrony, or phase-slippage, may be triggered stochastically, in which the trans flagellum transitions to a second mode of beating withattenuated beat-envelope and increased frequency. Further, exploiting this alga’s ability for flagellar regeneration,we mechanically induced removal of one or the other flagellum of the same cell to reveal a striking disparity betweenthe beating of the cis vs trans flagellum, in isolation. This raises further questions regarding the synchronizationmechanism of Chlamydomonas flagella.

1 Introduction

Periodicity permeates Nature and its myriadlifeforms. Oscillatory motions lie at the heartof many important biological and physiologicalprocesses, spanning a vast dynamic range of spatialand temporal scales. These oscillations seldom occurin isolation; from the pumping of the humanheart, to the pulsating electrical signals in ournervous systems, from the locomotive gaits ofa quadruped, to cell-cycles and circadian clocks,these different oscillators couple to, entrain, or areentrained by each other and by their surroundings.Uncovering the mechanisms and consequences ofthese entrainments provides vital insight intobiological function. Often, it is to the aid ofquantitative mathematical tools that we must turnfor revealing analyses of these intricate physiologicalinteractions.The striking, periodic flagellar beats of one

particular organism shall dictate the followingdiscussion: Chlamydomonas reinhardtii is aunicellular alga whose twin flagella undergo bilateralbeating to elicit breaststroke swimming. For thesemicron-sized cells, their motile appendages, termedflagella, are active filaments that are actuated byinternal molecular motor proteins. Each full beatcycle comprises a power stroke which generatesforward propulsion, and a recovery stroke in whichthe flagella undergo greater curvature excursions,thereby overcoming reversibility of Stokes flows [1].A single eyespot breaks cell bilateral symmetry,distinguishing the cis flagellum (closer to theeyespot), from the trans flagellum (figure 1).Subject to internal modulation by the cell, andbiochemical fluctuations in the environs, the two

flagella undergo a rich variety of tactic behaviours.For its ease of cultivation and well-studiedgenotype, Chlamydomonas has become a modelorganism for biological studies of flagella/cilia-related mutations. For its simplistic cell-flagellaconfiguration, Chlamydomonas has also emergedas an idealised system onto which more physicalmodels of flagellar dynamics and synchronizationcan be mapped [2–5]. With this versatility inmind, the present article has two goals. First, weproffer a detailed exposition of Chlamydomonasflagella motion as captured experimentally byhigh-speed imaging of live cells; second, we developa quantitative framework for interpreting thesecomplex nonlinear motions.In the light of previous work, we have found the

motion of Chlamydomonas flagella to be sufficientlyregular to warrant a low-dimensional phase-reduceddescription [2, 3, 6, 7]. Single flagellum limit cyclesare derived from real timeseries, and are associatedwith a phase (§3.b). For each cell, dynamics ofthe flagella pair can thus be formulated in termsof mutually coupled phase oscillators (§3.c), whosepairwise interactions can be determined to sub-beat-cycle resolution. Just as marching soldiersand Olympic hurdlers alike can have preferentialfootedness, we find that Chlamydomonas is ofno exception; resolving within each cycle ofits characteristic breaststroke gait we see thatone flagellum is consistently phase-lagged withrespect to the other. These transient episodes,previously termed slips [2], are to be identifiedwith phase slips that occur when noisy fluctuationsdegrade the phase-locked synchronization of twoweakly coupled oscillators of differing intrinsicfrequencies [8]. For each cell, sampled here overthousands of breaststroke cycles, and supported bymulti-cell statistics, we clarify the non-constancy

Journal of the Royal Society Interface 1–12; doi: 10.1098/rspa.00000000December 12, 2013

This journal is c© 2013 The Royal Society

2

(a) (b)

C1 C2

AB radial

spoke

outerdynein

inner dynein

B2

cis

B1

trans

M4

M2D2

D4

Figure 1. (a) Asymmetric cytoskeletal organization thatunderlies beating differences between cis and trans flagella.During development, eyespot positioning delineates betweenthe two flagella; that closer to the eyespot is the cis flagellum,and the farther one is the trans flagellum. (b) Inside theaxoneme: a peripheral arrangement of microtubule doubletsencircles a central pair, and specialized dynein motors initiateinterdoublet sliding and beat generation.

of synchrony observed over a typical cycle. Inparticular, the two flagella are found to be mostrobustly synchronized in the power stroke, and leastsynchronized at the transition to the succeedingrecovery stroke. This trend appears to be universalto all cells of the wildtype strain. However, thetendency for the two flagella of a given cellto experience phase slips exhibits much greatervariation across the population (figure 6). Thiswe take as further indication that Chlamydomonascells are highly sensitive to fluctuating biochemicalcues. Sampled across multiple cells, phase slipexcursions between synchronized states can bevisualized by a dimension-reduced Poincaré return-map of interflagellar phase difference, showing thesynchronized state itself to be globally stable.Examining each flagellar phase slip in detail weshow further that the trans flagellum reproduciblytransitions to a well-defined transient state withhigher beat frequency and attenuated waveform.This evidences an alternative mode of beating- which we conjecture to exist as one of adiscrete spectrum of modes on the eukaryoticflagellum. This second mode can also be sustainedfor longer durations, and in both flagella of aparticular phototaxis mutant of Chlamydomonas,as detailed elsewhere [7]. Taken together, figure 3encapsulates in a single diagram the three possiblebiflagellate “gaits”, their differences and similarities,highlighting the need for a quantitative formulationsuch as that we present in this article.Intrinsic differences between the two

Chlamydomonas flagella, their constructionand actuation, underlie this rich assemblage ofbiflagellate phenomenology. Past experiments haveshown such differences to exist, for example inreactivated cell models of Chlamydomonas [9],in which the trans flagellum has a faster beatfrequency than the cis. In contrast, we considerhere the in vivo case (§3.d); by mechanicallyinducing deflagellation of either trans or cisflagellum, we render live wildtype cells uniflagellate.

This allowed us to compare the intrinsic beatingdynamics of cis vs trans flagella. We found thatwhilst cis- uniflagellated cells tend to beat withthe canonical breastroke-like mode (BS-mode),trans- uniflagellated cells can instead sustain thefaster mode of beating associated with the phaseslip (aBS-mode) (figure 10a). Yet this cis-transspecialization is lost once the cell is allowed toregrow the lost flagellum to full-length, by whichtime both flagella have recovered the BS-mode.Flagellar tracking has enabled us to acquire true

spatial localization of the flagellum throughout itsdynamic rhythmicity, complementing recent effortsaiming in this direction [4, 10] The need to knowprecise waveforms has long been an ambition ofhistoric works, in which manual light-table tracingsof Chlamydomonas flagella were used to elucidatebehaviour of the wildtype [11, 12], and latterlyalso of flagellar mutants [13]. We hope that thefindings and methodologies herein presented shall beof broad interest to physicists and biologists alike.

2 Background

a The enigmatic flagellum beat

At a more fundamental level, how is beating of asingle flagellum or cilium generated, and moreover,how can multi-ciliary arrays spontaneouslysynchronize? For each of us, or at least, forthe hair-like appendages lining the epithelial cells ofour respiratory tracts, this is indeed an importantquestion. Beating periodically, synchronously, andmoreover metachronously, multitudes of thesecilia drive extracellular fluid flows which mediatemucociliary clearance. These motile cilia, and theirnon-motile counterparts, are regulated by complexbiochemical networks to perform highly-specificfunctions [14, 15]. Mutations and defects in theseorganelles have been increasingly implicated inmany human disorders including blindness, obesity,cystic kidney and liver disease, hydrocephalus, aswell as laterality defects such as situs inversus totalis[16, 17]. Mice experiments in which nodal flows areartificially disrupted directly link mechanical flowsto positioning of morphogens, which in turn triggerlaterality-signalling cascades [18].Across the eukaryotic phylogeny these slender,

propulsion-generating appendages possess aremarkably conserved ultrastructure [19]. In recentdecades, causality from structure to functionwithin eukaryotic ciliary/flagellar axonemes hasbeen established using sophisticated moleculargenetics tools. For the Chlamydomonas inparticular, rapid freezing of specimens has madepossible the capture of axonemal componentsin near-physiological states, at molecular-levelresolution [20]. Chlamydomonas flagella have awell-characterized 9 + 2 structure of microtubuledoublets (MTD), along which are rows of themolecular motor dynein (figure 1). These directional

3

motors generate forces parallel to filamentous MTDtracks, which slide neighbouring MTDs past eachother. Anchored to the A-tubule of each MTDby their tail domains, these dyneins detach andreattach onto the next B-tubule, consuming ATPin the stepping process. Different dynein speciescoexist within the flagellar axoneme, with force-generation principally provided by outer dyneins,and modulation of flagellar waveform by the innerdyneins. The central pair, is thought to transducesignals via the radial spokes [21]. Approximatelyevery 96 nm this precise arrangement of dyneins,radial spokes, and linkers repeats itself [22]. Periodicelastic linkages between neighbouring MTDs callednexins provide the coupling by which dynein-drivenfilament sliding produces macroscopic bending ofthe flagellum, which in turn propels the cell throughthe fluid. Treatments of axonemes which disruptdynein domains have shown these nexin linkages tofunction in a non-Hookean manner to provide theelastic restoring force that resists internal filamentshear [23]. More recently, detailed 3D-tomographicanalysis revealed that nexins are in fact one andthe same with the dynein regulatory complex,collectively termed the NDRC [24]. The importanceof the NDRC within the functioning axoneme,from those of algae to that of humans, has beenhighlighted in a recent review [25].With regard to the flagellum-cycling mechanism,

consensus has been lacking. Timing-control appearsto be absent, yet much experimental evidencepoints to selective, periodic, dynein-activation [26].Both power and recovery strokes of the beat cycleinvolve active force generation by differentiallyactuated dyneins. Rhythmic beating of the flagellummay arise from dynamical instabilities of dyneinoscillations [27, 28], and may be closely coupled tothe intrinsic geometry of the axoneme [29, 30]. Withthese unanswered questions in mind, there is thusmuch incentive to analyze the flagellum beat in vivo,as the present study seeks to demonstrate.

b Molecular origins of cis-trans difference

As with many species of flagellated algae, motilityis essential not just for swimming, but also forcell taxis. For Chlamydomonas, its two equal-length flagella, each 10− 15 µm long, emergefrom an ellipsoidal cell body ∼ 5 µm in radius.Cell bilateral symmetry is broken by the presenceof an eyespot, which functions as a primitivephotosensor. Perceived directional light is thenconverted downstream via secondary messenger ionsinto differential flagellar response and hence a turnof the cell [22, 31, 32]. The two anterior basal bodiesfrom which the two flagella protrude are connectedto each other via distal striated fibres [33]. A systemof four acetylated microtubule rootlet-bundles liesbeneath the cell membrane, and extends towardsthe cell posterior [32]. The eyespot is assembled denovo after each cell division, breaking cell bilateralsymmetry by its association with the daughter

four-membered rootlet (figure 1a, D4). The transflagellum, nucleated by the mother basal body(B1), has been shown in reactivated cell models tobeat at an intrinsic frequency that is ∼ 30− 40%higher than that of the cis [9]. This frequencymismatch is discernible in vivo in wildtype cellsthat we rendered uniflagellated through mechanicaldeflagellation (§3.d), additionally with a discrepancyin beating waveform. Differential phosphorylation ofan axonemal docking complex has been suggestedto underlie the distinctive cis-trans beat [34].In particular, differential cis-trans sensitivity tosubmicromolar Ca2+ in cell models and in isolatedaxonemes [9, 36] is consistent with the opposingflagellar modulation necessary for cells to performphototactic turns [13].Yet, despite these intrinsic differences, the two

flagella maintain good synchrony during freeswimming [4, 10, 11], as well as when held stationaryby a micropipette (here). Interflagellar couplingmay be provided by the motion of the fluidmedium [35, 37], by rocking of the cell-body [5], orfurther modulated internally via elastic componentsthrough physical connections in the basal region[33]. However, stochastically-induced flagellar phaseslips can appear in otherwise synchronized beatingflagella in a distinctive, reproducible manner (figure3). We find that the propensity to undergo thesetransient slips [13] can vary significantly evenbetween cells of a clonal population (figure 6).

3 Results

a Three gaits of biflagellate locomotion

The breaststroke In their native habitat,Chlamydomonas cells swim in water (kinematicviscosity ν = 10−6 m2 s−1), at speeds on the orderof 100 µm/s, up to a maximum of 200 µm/sdepending on strain and culture growth conditions[38]. Oscillatory flows set up by these flagellahave a frequency f = ω/2π' 60 Hz during thebreaststroke. Stroke lengths of L= 10 µm thusproduce a tip velocity scale of U =Lω∼ 4 mm/s.An (oscillatory) Reynolds number Re = ωL2/νgauges the viscous and inertial force contributionsto the resulting flow. Here, Re≈ 0.001, and cellpropulsion thus relies on viscous resistance toshearing of the fluid by the flagellar motion.To overcome the reversibility of such flows, abreaking of spatial symmetry is essential duringthe swimming breaststroke. The rhythmic sweepingmotion of each flagellum can be partitioned intodistinct power and recovery strokes: during thepower stroke, the flagella are extended to maximizeinteraction with the fluid, but undergo greaterbending excursions during the recovery stroke(figure 3a). Net swimming progress results fromthe drag anisotropy of slender filaments and thefolding of flagella much closer to the cell body duringthe recovery stroke. Interestingly, a qualitatively

4slip

breaststroke

antiphase cis

breastroke

cistrans

trans cis

cistrans

Figure 2. Three Chlamydomonas breaststroke swimming gaits recorded at 3000 fps and shown at intervals of 5 frames (i.e. 1.7ms intervals). Orange dot marks cell eyespot location, red curves: cis flagellum, blue curves trans flagellum. Shown in order,in-phase synchronized breaststroke (both flagella in BS-mode), a phase slip in the same cell (trans flagellum in aBS-mode), andantiphase synchronization in the phototaxis mutant ptx1 (both flagella in aBS-mode). Scale bar is 5µm.

objective

micropipette

breaststroke

slip

antiphase

(a)

(b)

(c)

θ (

rad)

θ (

rad)

θ (

rad)

three gaits

cell axis

-1

3

-1

3

-1

θ

3

-1

3

trans cis

φH

cis

trans

θ

(e)

(f)

(d)

Figure 3. Overlaid sequences of tracked flagella showing (a) normal breaststroke (BS) (5 consecutive beats), (b) a stochastic slipevent (4 consecutive slips), in which the trans flagellum transiently enters a different mode (aBS), and (c) in which both flagellaof ptx1 sustain the aBS mode (5 consecutive beats). For each flagellum, progression through the beat cycle can be trackedusing an angle θ defined relative to the cell bilateral axis (d); sample timeseries for θ are shown for each gait. The aBS-modecan clearly be seen to have an attenuated beat amplitude and a faster beat frequency. (e)-(f): Differences between cis and translimit cycles are clarified in phase-space coordinates (θ,H(θ)), where H(.) denotes the Hilbert transform (see §3.b), and thecolor-intensity is obtained by logarithmically scaling the probability of recurrence.

5

similar bilateral stroke can emerge from a theoreticaloptimisation performed on swimming gaits ofbiflagellate microorganisms [39] and on singleflagella near surfaces [40].

The phase slip Early microscopic analyses ofChlamydomonas flagella suggested that the normalbreaststroke synchrony “may be disturbed forbrief periods” [33]. These interruptions to ‘normal’beating, subsequently detailed in manual waveformtracings by Rüffer and Nultsch, were shown tooccur in the absence of obvious stimulation, inboth free-swimming cells [11] and cells affixedto micropipettes [12]. Crucially, these transientasynchronies do not significantly alter the trajectoryof swimming (unpublished observation); instead,during each such episode the cell body is seen torock back and forth slightly from frame to framewithout altering its prior course. These asynchroniesare termed slips [2, 3] by analogy with an identicalphenomenon in weakly-coupled phase oscillators.Physically, phase slips are manifest in these coupledflagella in a strikingly reproducible manner. Underour experimental conditions (detailed in §5), beatingof the trans flagellum transitions during a slip toa distinct waveform, concurrently with a ∼ 30%higher frequency [7], whilst at the same time thecis-flagellum maintains its original mode of beatingthroughout, apparently unaffected (figure 3b). Wefind also that the faster, attenuated breaststrokemode (aBS) is sustained by the trans flagellum foran integer number of full beat cycles, after whichnormal synchronized breaststroke (BS) resumes.

The antiphase The aBS waveform assumedby the trans-flagellum during a slip turns outto be markedly similar to that identified in ananti-synchronous gait displayed by a particularphototaxis mutant of Chlamydomonas called ptx1[41]. In recent, related work, we make thesecomparisons more concrete, and show that this gait(figure 3c) involves actuation of both flagella in aBS-mode, and in precise antiphase with each other [7].Although the precise nature of the ptx1 mutationremains unclear, it is thought that emergence of thisnovel gait in the mutant is closely associated withloss of calcium-mediated flagellar dominance.

b Phase-dynamics of a single flagellum

Many biological oscillators are spatially extendedand are therefore fundamentally high-dimensionaldynamical objects, but adopting a phase-reductionapproach facilitates quantitative analyses. In suchcases, stable self-sustained oscillations can berepresented by dynamics on a limit cycle, for whichmonotonically increasing candidate phases ϕ maybe extracted.The natural or interpolated phase of an oscillator

is defined to increment linearly between successivecrossings of a Poincaré section, and as such isstrongly dependent on the precise choice of the

(c)600

200 π 2π0

1

23

4

5

Poincaré section

0.5 1.0 1.5

-0.5

0

0.5

0

π

2π

(a) 1

2

3

4

5

(b)

Figure 4. (a) Hilbert embedding for the cis-flagellum ofa sample cell, recorded over thousands of beat cycles,and coloured according to the equivalent transformedphase φ (via equation 3.6). Points of equal phase lie onisochrones, highlighted here at equi-phase intervals. Therate of phase rotation varies systematically throughout thebeat cycle, as indicated by the variable inter-isochronespacing. Inset: successive zero-crossings of φ− 〈φ〉 mapout one Poincaré-section. (b) Snapshots 1− 5 show typicalpositions of the flagellum at phases corresponding to fiverepresentative isochrones. (c) Phase-velocity Γ(ϕ) = dϕH/dtis approximated by a truncated Fourier series. Shaded regionsshow one standard deviation of fluctuations in the raw data.

section. For discrete marker events {tn}, and tn ≤t≤ tn+1,

ϕP (t) = 2π

(n+

t− tntn+1 − tn

). (3.1)

This method was used in earlier work [2, 3],to extract flagellar phases φcis,trans, by samplingpixel intensity variations over pre-defined regionsof interest, on individual frames of recordedvideo. Specifically, phase values are interpolatedbetween successive peaks in intensity. However asbeating of the flagellum corresponds to smoothdynamics, by Poincaré sectioning the dynamicsin this way, sub-beat-cycle information is lost.In the present work, we take a more continuousapproach by incorporating enhanced resolution toelucidate within-beat-cycle dynamics, and make useof flagellum tracking to define 2D-projections offlagellum oscillations.For this we choose an embedding via the Hilbert

transform. This technique derives from the analyticsignal concept [42], and is used to unambiguouslydefine an instantaneous phase (and amplitude) from

6

0 10 20

time (s)

-10

10

30

11.9 12.0 12.1 12.2

phas

e de

viat

ion

(rad

)

2

4

Figure 5. Phase-deviation over long-timescales obtained by subtracting from unwrapped phases the linear component thatscales with oscillator frequency. General trends are preserved by the phase transformation (equation 3.4), but within-cyclefluctuations are removed

scalar signals with slowly-varying frequency. Froma periodic scalar timeseries x(t) we construct itscomplex extension ζ(t) = x(t) + ix(t), where x(t) isgiven by

H(x) = x(t) =1

πPV

∫∞−∞

x(τ)

t− τdτ . (3.2)

Here, the integral to be taken in the sense of theCauchy principal value. Polar angle rotation in thex−x phase-plane gives the Hilbert phase,

ϕH(t) = tan−1

(x− x0

x− x0

), (3.3)

which, when unwrapped, serves as a monotonically-increasing candidate phase on our limit cycleprojection. The origin (x0, x0) is chosen to bestrictly interior of the limit cycle; here, x0 = 〈x〉,x0 = 〈x〉.Candidate phases such as the Hilbert phase do not

uniformly rotate and are sensitive to cycle geometryand nonlinearity. That is, ϕ= Γ(ϕ) is in generalsome non-constant but 2π-periodic function. Forany given limit cycle, there is however a unique truephase φ for which ω= dφ/dt is constant, the rate ofrotation of φ is equal to the autonomous oscillatorfrequency. The desired transformation ϕ→ φ is

φ= ω

∫ϕ(Γ(ϕ))−1 dϕ . (3.4)

Whilst φ is unique, ambiguity remains in the choiceof limit cycle that best characterizes the originalphase-space. Furthermore for noisy timeseries, limitcycle trajectories do not repeat themselves exactly,so that the phase transformation (3.4) can only beperformed in a statistical sense. Accuracy of phaseestimation is improved with longer observation time.To derive an approximation for dϕ/dt as a

function of ϕ, we first sort data pairs 〈ϕ, dϕ〉and then average over all ensemble realizations ofϕ (figure 4). Direct numerical approximations fordt/dϕ are sensitive to noise, due to the heavy-tailednature of ratio distributions, to remedy this, wefollow the approach of Revzen et al [43] and begin

by finding an N-th order truncated Fourier seriesapproximation Γ =FN [Γ], to Γ. Next, we find asimilar Fourier approximation to 1/Γ,

dt

dϕ(ϕ)≈FN [1/Γ] =

N∑k=−N

fkeikϕ . (3.5)

The zeroth coefficient f0 = 12π

∫T0

(dtdϕ

)dϕ= ω−1

where ω is the intrinsic frequency of the oscillator, sothat to lowest order, ϕ and φ coincide. Substituting(3.5) into (3.4) gives

φ=ϕ+ 2ω

N∑k=1

Im(fkk

(eikϕ − 1)

). (3.6)

For flagellum oscillations, we choose scalartimeseries x= θcis/trans (Fig. 3). For each flagellum,pairs of values x= (x, x) define a noisy limit cycle inphase space. Using equation. 3.6, we can associatea phase at each x, consistent with the notion ofasymptotic phase defined in the attracting regionaround a limit cycle. For different flagella, thefunction Γ(ϕ) takes a characteristic form (4c).Points of equal phase lie on isochrones, which foliatethe attracting domain (figure 4a).Perturbations that are 2π periodic functions of ϕ

are eliminated by the transformation 3.6. Within-period oscillations are averaged out, whilst long-timescale dynamics are preserved, by virtue of theinvertibility of the transformation. This can beseen over long-time recordings, in which the twomeasures of phase deviation: Dφ = φ− ωt and Dϕ =ϕH − ωt coincide (figure 5), but the periodicity of∆ϕ has been smoothed out (inset).

c Phase-dynamics of coupled flagella pair

For microorganisms that rely on multiple flagellafor swimming motility, precision of coordinationis essential to elicit high swimming efficacy.The bilateral geometric disposition of the twoChlamydomonas flagella facilitates extraction ofphases for an individual flagellum’s oscillations, and

7

0 0.4 0.80

0.4

0.8

(b)

(c)

τavg=0.0524

lag1time1(τ/T)

Λ(τ

)

power

1

reco

very

1

recurrentnot1

recurrent0

2π

0 2π

(a) cis trans

lag

θ tra

ns1(

rad)

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20

iii)

ii)

i)

no1slips

few1slips

many1slips1

0 10

50

40

30

20

10

0

0 2π

2π

Figure 6. (a) Lag synchronization in bivariate timeseries of flagella beats, shown for a typical cell. Inset: cis flagellum begins itsrecovery stroke fractionally ahead of trans. (b) Multi-cell similarity functions show a similar trend. (c) Noisy flagellar dynamicswithin a population of 60 cells, as represented by the tendency of each cell to undergo slips. Most cells remain synchronized formore than 20 s, whilst some exhibit frequent asynchronies. Red to blue: from high to low probability of recurrence.

in turn, derivation of phase synchrony relationsbetween coupled pairs of flagella. However, atransformation function similar to equation 3.4 thatis bivariate in the two phases cannot be derivedfrom observations of the synchronized state alone;therefore in the following, we make use of the Hilbertphase (equation 3.3).

Phase difference derivation To monitorbiflagellar synchrony, the phase differenceϕtrans − ϕcis is of particular interest. Moregenerically, for coupled noisy phase oscillatorsi and j, the dynamics of each is perturbed bythe motion of the other, as well as by stochasticcontributions:

ϕi = f(ϕi) + εi g(ϕi, ϕj) + ξi(t) , (i= 1, 2) ,(3.7)

where g(ϕi, ϕj) is the coupling function, and ξ is anoise term. When ε1,2 = 0 we recover the intrinsicmotion of a single flagellum. The two oscillatorsare said to be n :m phase-locked when their cyclicrelative phase given by ∆n,m = nϕ1 −mϕ2, satisfies∆n,m = Const.. For noisy or chaotic systems thiscondition may be relaxed to ∆n,m <Const. Here,we define the trans-cis phase difference from therespective angle signals θcis(t) and θtrans(t) by

∆ =ϕtransH (t)− ϕcis

H (t) (3.8)

= tan−1

(θtrans(t)θcis(t)− θtrans(t)θcis(t)

θtrans(t)θcis(t) + θtrans(t)θcis(t)

),

where ˜ again denotes the Hilbert transform.

We measured ∆ for a large population of cells(Fig. 6). Phase-slip asynchronies are associatedwith rapid changes in interflagellar phase difference,and appear as step-like transitions that punctuate(sometimes lengthy) epochs of synchronizedbehaviour for which phase difference is constant.We see that over a comparable period of observationtime (figure 6c:i-iii), pairs of flagella can experienceeither perfect synchrony, few slips, or many slips1. For the population as a whole, the circularrepresentation of figure 6 facilitates simultaneousvisualization of general trends in interflagellarphase-synchrony.

Lag synchronization Careful examination of asynchronized epoch shows that ∆ is not strictlyconstant, but rather fluctuates periodically abouta constant value. During execution of breaststrokeswimming, Poincaré sectioning of the dynamics hassuggested previously that the breaststroke gait isperfectly synchronized [2, 3]. However, plotting θcisagainst θtrans (figure 6a) we see a consistent lagbetween the two flagella, which is most pronouncedduring the recovery stroke. By computing andminimizing the similarity function

Λ(τ) =

√〈(θtrans(t+ τ)− θcis(t))2〉(⟨

θ2trans

⟩ ⟨θ2cis⟩)1/2

we find this discrepancy to be indicative of lagsynchronization. Here, the periodic angle variablesθcis/trans are chosen as scalar indicators for theprogression of each flagellum through its beat cycle.In particular, the two phases are synchronized with

8

0.05 0.01 0.002 0.0004 -0.2 0.2 0.6 -π/3 -π/6 0 π/6

(a) (b) (c)

0.25

Figure 7. Trends in cis-trans flagellar synchronization in a population of Chlamydomonas cells: each concentric annulusrepresents data from an individual cell, measured values are plotted on a circular scale 0→ 2π in an anticlockwise sense.(a) Probability of stroboscopically observing φHcis ∈ [0, 2π] at discrete marker events where θHtrans reaches a minimum (i.e. start ofnew power stroke). (b)-(c) Difference in tracked angles (∆θ = θtrans − θcis) and Hilbert phases (∆φ = φHtrans − φHcis), averaged overthousands of beat cycles. It is seen that ∆θ is greatest during the recovery stroke, and correspondingly ∆φ becomes increasinglynegative.

a time lag τmin = minτ Λ(τ), where Λ(τ) assumes aglobal minimum. When the oscillators are perfectlysynchronized, τmin = 0. We calculated Λ(τ) formultiple cells, which displayed a similar profile(figure 6b). With τ normalized by the averageinter-beat period T , we see that in every instancethe minimum is displaced from 0 (or equivalently1), with an value of 0.0524± 0.01, indicative ofpersistent directional lag.

Stability of fixed points and transients Aphenomenological model such as equation 3.7 hasa convenient dynamical analogy. Phase differencecan be interpreted as particle in a washboardpotential V (∆) =−∆δω + ε cos(∆), subject tooverdamped dynamics ∆ =−dV (∆)/d∆. Potentialminima occur where ∆ = 0, which requires |δω/ε|<1. For noise with sufficient magnitude, theparticle will have enough energy to overcome thepotential barrier, at least transiently. Stochasticjumps between neighbouring potential minima,are manifest in coupled flagella as the slip-mode(figure 3b).In the vicinity of a potential minima, the

stationary distribution of ∆ is predicted to beGaussian (equation 3.7, with white noise). Thisphase distribution P (∆), can be measured directlyfrom experiment, and assumed to satisfy

P (∆) = exp(−U(∆)/kBT ) , (3.9)

from which the potential structure U(∆) can berecovered. For each well, the peak location canbe used to estimate the phase-lag of the coupledoscillators, while peak width is indicative of strengthof noise in the system. We measured P (∆) for 18cells which did not display slips for the durationof observation (figure 9a). Potential minima havea parabolic profile with a well-defined peak, on

(b)

0

1

1

1

1

2

0

2

0

2

0

2

θci

s, θ

trans

(rad

)(a) transcis

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Figure 8. a) Harmonics of a slip, synchrony resumes aftera different (but always integer) number of beats of eitherflagellum. b) each slip results in a step-like transition in phase-difference ∆. For the same cell, successive slips overlap.Inset: forking of unwrapped phases ϕcis

H and ϕtransH .

average displaced to the left of ∆ = 0, due to thecharacteristic phase-lag in the direction of the cis-flagellum. For certain cells, this lag is especiallypronounced during the recovery-stroke than duringthe power stroke, resulting in a double-peakedminimum in the fine-structure of the empiricalpotential.The stability of the synchronized state may be

assessed by observing trajectories that deviate from,but eventually return to, this state. Specifically, by

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Figure 9. A potential analogy for wrapped phase difference,visualized a) within potential minima - ∆ exhibits localfluctuations during the stable synchronized gait; and b)between successive potential minima, via a first return mapof cyclic/stroboscopic relative phase. An nth-order polynomialF (∆n) is fit to the multicell return-map statistics, from whichan empirical potential function V (∆) can defined.

measuring ∆ during multiple flagellar phase slipswe construct a dimension-reduced return map forthe joint system to visualize the potential landscapethat extends between neighbouring minima. Slipsoccur with variable duration (figure 8a), inwhich the trans flagellum can sustain the fasteraBS-mode for a variable but complete numberof beat cycles. However for an individual cell,successive slips often exhibit identical dynamics(figure 8b). Figure 9b presents the return map of∆ associated with > 500 slip events collected from70 cells, where the discretized phase-difference ∆n

is defined to be ∆ evaluated stroboscopically at theposition of maximum angular extent of the trans-flagellum, during the nth beat-cycle. We begin byapproximating the return map by a polynomialF (∆n). An empirical potential function [44] can bedefined by integrating the difference δ∆ = ∆n+1 −∆n ≈ F (∆n)−∆n:

V (∆n) =−∫∆n

0δ(∆′) d∆′ , (3.10)

which we convert into a locally positive-definitefunction

V (∆n) = V (∆n)−min∆

V (∆) , (3.11)

which satisfies V (∆)> 0, ∀∆ /∈ argmin V (∆). Theresulting effective potential profile (figure 9b, inset)represents the reproducible phase dynamics of atypical flagellar phase slip, from which breaststrokesynchrony re-emerges.

d Coupling cis and trans flagella

Pre-existing, intrinsic differences between the twoChlamydomonas flagella are essential for controlof cell reorientation, loss or reduction in cis-transspecialization may give rise to defective phototaxisin certain mutants of Chlamydomonas [7, 45].Under general experimental conditions, stochasticasynchronies which we call slips can punctuate anotherwise synchronous breaststroke; more drasticloss of interflagellar synchrony can lead to drifts,which over time, can result in a diffusive randomwalk in the trajectory of an individual cell [2]. In allthese instances, we observe the coupled state of twoflagella; in contrast, by mechanically deflagellatingwildtype cells (see §5) we can now examine theintrinsic behaviour of each oscillator in isolation.The ability of Chlamydomonas to readily

regenerate a lost flagellum has facilitated controlledmeasurements of flagellar coupling strength asa function of flagellum length [6]. Using thesingle eyespot as identifier, we removed either thecis or trans flagellum from a pipette-capturedcell and recorded the beating dynamics of theremaining flagellum. Histograms of beat-frequenciesare plotted in figure 10b. On average, cis-uniflagellated cells tend to beat at a lower frequencythan trans- uniflagellated cells. A dissociationof beat-frequency of similar magnitude has beenobserved previously in reactivated cell models [9].Moreover, we find that in the absence of the cisflagellum the trans flagellum can sustain the fasteraBS-mode for thousands of beats. These differencesare highlighted in Figure 10a, for a single cell.Interestingly, the aBS-mode that we can now

associate with the intrinsic beating waveform ofthe trans, emerges transiently during a slip of thewildtype (figure 3b), but in both flagella duringan antiphase gait of the mutant ptx1 (figure 3c,and [7]). Indeed, for ptx1, its lack of effectual cis-trans specialization has led to speculation thatthe mutation has renders both flagella trans-like[45]. Specific, structural differences known to existbetween the cis vs trans axonemes [34] of thewildtype, may effect this segregation of intrinsicbeating modes.

4 Discussion

For a unicellular flagellate such as Chlamydomonas,synchrony of its two flagella is intimately regulated

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Figure 10. (a) For a single cell, cis and trans flagella were removed and in turn allowed to regrow to full length. Single-flagellumfrequencies separate, but once regrown, lock to a common frequency (cis: 57.12 Hz, trans: 80.52 Hz, both: 63.38 Hz). Insets:typical cis and trans waveforms, the trans waveform is reminiscent of the aBS-mode that onsets during a slip. Waveforms areoverlaid on an intensity plot of logarithmically-scaled residence times for each flagellum, over O(100) contiguous beats. (b)When beating in isolation, cis and trans flagella have different frequencies: areas - histograms of interbeat frequencies, lines -averaged frequencies.

by the cell’s internal biochemistry; however, theexact mechanism by which messenger ions modulateand shape the flagellum beat remains unclear.Our experimental technique captures the motion ofbeating flagella in vivo, at high resolution, and withrespect to a fixed pivot, thereby permitting long-time analysis.Associating each flagellum oscillator with a

continuous phase, we formulated a phase-reducedmodel of the periodic dynamics. From long-timeseries statistics of bivariate oscillator phases, weused phase difference to track phase synchrony,quantifying flagellar interactions for a singleindividual, as well as across the sampled population.Exquisitely sensitive to its surroundings, a flagellumcan be found to undergo precise, yet dynamicchanges when executing its periodic strokes.Waveform tracking has allowed us to assess thesechanges in great spatio-temporal detail.In particular, we have found the stable phase-

locked breaststroke of C. reinhardtii to exhibita small but persistent cis-trans phase-lag, themagnitude and direction of which was evaluatedfrom statistics of thousands of beat cycles usinga similarity measure, and confirmed for multiplecells. However, often it is not the synchronizedstate itself but rather the emergence or cessationof synchrony that is most insightful for inferringfluctuations in the physiological state of a complexsystem. Phase slips are transient excursions fromsynchrony in which, under our experimentalconditions, an alteration of beating mode isobserved in the trans flagellum only, and thatappear to be initiated by a reduced cis-transphase lag (figure 8b). These reproducible eventshighlight the importance of cis-trans specializationof Chlamydomonas flagella. Exploring this further,we mechanically removed individual flagella ofwildtype cells to obtain uniflagellated cis or transversions, revealing significant differences betweentheir isolated beating behaviours (figure 10). Yet fora fully-intact cell, despite these inherent differencesin beating frequency and in waveform, coupling

interactions either hydrodynamically through thesurrounding fluid medium, and/or biomechanicallythrough elastic linkages at the base of the flagellarprotrusion, appear sufficient on the most part toenslave the beating of the trans mode to that ofthe cis.Ours is a very versatile technique for quantifying

flagellar synchrony not just of the wildtype system,but such a phase analysis can for instance alsobe used to probe defective swimming behavioursof motility mutants. In these cases, macroscopicmeasurements of population features may notbe instructive to understanding or resolving themutant phenotype, and would benefit from dynamicflagellum waveform tracking and in-depth analysisat the level of an individual cell.

5 Methods and techniques

Single algal cells on micropipettes Forpurposes of flagella visualization we chose twowildtype C. reinhardtii strains, CC124 and CC125(Chlamydomonas Center). Stock algae maintainedon 2% TAP (Tris-Acetate Phosphate) solid agarslants, were remobilized for swimming motilityby inoculation into TAP-liquid medium, andcultures used for experimentation were maintainedin exponential growth phase (105 − 106 cells/ml)for optimal motility. Culture flasks are placedonto orbital shakers, and maintained at 24 ◦Cin growth chambers illuminated on a 14:10 dailylight/dark cycle, so as to imitate the indigenouscircadian stimuli. Observation of flagellar dynamicswas carried out on a Nikon TE2000-U invertedmicroscope, at constant brightfield illumination.Additional experiments were also performed witha long-pass filter (622 nm) to minimize cellphototaxis2. Individual cells were captured andheld on the end of tapered micropipettes (SutterInstrument Co. P-97), and repositioned with aprecision micromanipulator (Scientifica, UK), and

11

imaged at rates of 1, 000-3, 000 fps (PhotronFastcam, SA3).

Digital Image and Signal ProcessingRecorded movies were transferred to disk forpost-processing in MATLAB (Version 8.1.0, TheMathworks Inc. 2013.). Flagellar waveforms wereextracted from individual frames, where contiguousdark pixels that localize the moving flagellum werefit to splines. Hilbert transforms were perform inMATLAB (Signal Processing Toolbox), and furthertimeseries analysis performed using custom MATLABcode.

Mechanical deflagellation of either cis ortrans flagella To obtain the results described in§3.d, individual wildtype cells were first examinedunder white light to locate the unique eyespot,thereby differentiating its cis flagellum from thetrans. One flagellum was then carefully removedwith a second micropipette, by exerting just enoughshearing force to induce spontaneous deflagellationby self-scission at the basal region. That cellsretain the ability for regrowth of flagella ensuresbasal bodies have not been damaged by ourdeflagellation treatment. Cells for which the beatingof the remaining flagellum became abnormal orintermittent, and also for which a clear cis-trans identification could not be made, were dulydiscarded.

Data Examples of high-speed movies, flagellartime series, and other data referenced in this workcan be found at: http://www.damtp.cam.ac.uk/user/gold/datarequests.html.

We thank Marco Polin for discussions. Financial support isacknowledged from the EPSRC, ERC Advanced InvestigatorGrant 247333, and a Senior Investigator Award from theWellcome Trust (REG).

Notes

1Of the population of cells analysed for figure 6c, mostwere observed under white light. However a small percentage(10%) were observed under red light, but which for the sakeof clarity, have not been explicitly marked out in the figure.Whilst in both cases variability in frequency of flagellarslips is observed, we find that on average slips occur moreprevalently in cells illuminated by red than by white light(discussed further elsewhere).

2Whilst cell phototaxis behaviour is minimized underred-light illumination, physiological cell motility cannotbe maintained in prolonged absence of light, unless theexperimental conditions were accordingly modified [7].

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