Lag, lock, sync, slip: the many "phases" of coupled flagella
LLag, lock, sync, slip: the many “phases” of coupled flagella
Kirsty Y. Wan, Kyriacos C. Leptos, Raymond E. Goldstein
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University ofCambridge, Wilberforce Road, Cambridge CB3 0WA, UK
In a multitude of life’s processes, cilia and flagella are found indispensable. Recently, the biflagellated chlorophyte alga
Chlamydomonas has become a model organism for the study of ciliary coordination and synchronization. Here, we use high-speed imaging of single pipette-held cells to quantify the rich dynamics exhibited by their flagella. Underlying this variabiltiyin behaviour, are biological dissimilarities between the two flagella – termed cis and trans , with respect to a unique eyespot.With emphasis on the wildtype, we use digital tracking with sub-beat-cycle resolution to obtain limit cycles and phases forself-sustained flagellar oscillations. Characterizing the phase-synchrony of a coupled pair, we find that during the canonicalswimming breaststroke the cis flagellum is consistently phase-lagged relative 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 flagellumtransitions to a second mode of beating with attenuated beat-envelope and increased frequency. Further, exploiting this alga’sability for flagellar regeneration, we mechanically induced removal of one or the other flagellum of the same cell to reveal astriking disparity between the beating of the cis vs trans flagellum, in isolation. This raises further questions regarding thesynchronization mechanism of Chlamydomonas flagella.
Periodicity permeates Nature and its myriad lifeforms.Oscillatory motions lie at the heart of many importantbiological and physiological processes, spanning a vastdynamic range of spatial and temporal scales. Theseoscillations seldom occur in isolation; from the pumpingof the human heart, to the pulsating electrical signalsin our nervous systems, from the locomotive gaits ofa quadruped, to cell-cycles and circadian clocks, thesedifferent oscillators couple to, entrain, or are entrainedby each other and by their surroundings. Uncoveringthe mechanisms and consequences of these entrainmentsprovides vital insight into biological function. Often, itis to the aid of quantitative mathematical tools thatwe must turn for revealing analyses of these intricatephysiological interactions.The striking, periodic flagellar beats of one particularorganism shall dictate the following discussion:
Chlamydomonas reinhardtii is a unicellular algawhose twin flagella undergo bilateral beating to elicitbreaststroke swimming. For these micron-sized cells,their motile appendages, termed flagella, are activefilaments that are actuated by internal molecular motorproteins. Each full beat cycle comprises a power stroke which generates forward propulsion, and a recoverystroke in which the flagella undergo greater curvatureexcursions, thereby overcoming reversibility of Stokesflows [1]. A single eyespot breaks cell bilateral symmetry,distinguishing the cis flagellum (closer to the eyespot),from the trans flagellum (figure 1). Subject to internalmodulation by the cell, and biochemical fluctuations inthe environs, the two flagella undergo a rich variety of tactic behaviours. For its ease of cultivation and well-studied genotype,
Chlamydomonas has become a modelorganism for biological studies of flagella/cilia-relatedmutations. For its simplistic cell-flagella configuration,
Chlamydomonas has also emerged as an idealised systemonto which more physical models of flagellar dynamicsand synchronization can be mapped [2–5]. With thisversatility in mind, the present article has two goals.First, we proffer a detailed exposition of
Chlamydomonas flagella motion as captured experimentally by high-speedimaging of live cells; second, we develop a quantitativeframework for interpreting these complex nonlinearmotions.In the light of previous work, we have found the motionof
Chlamydomonas flagella to be sufficiently regularto warrant a low-dimensional phase-reduced description[2, 3, 6, 7]. Single flagellum limit cycles are derived fromreal timeseries, and are associated with a phase (§3.b).For each cell, dynamics of the flagella pair can thus beformulated in terms of mutually coupled phase oscillators(§3.c), whose pairwise interactions can be determined tosub-beat-cycle resolution. Just as marching soldiers andOlympic hurdlers alike can have preferential footedness,we find that
Chlamydomonas is of no exception; resolvingwithin each cycle of its characteristic breaststroke gait wesee that one flagellum is consistently phase-lagged withrespect to the other. These transient episodes, previouslytermed slips [2], are to be identified with phase slipsthat occur when noisy fluctuations degrade the phase-locked synchronization of two weakly coupled oscillatorsof differing intrinsic frequencies [8]. For each cell, sampledhere over thousands of breaststroke cycles, and supportedby multi-cell statistics, we clarify the non-constancy ofsynchrony observed over a typical cycle. In particular, thetwo flagella are found to be most robustly synchronized inthe power stroke, and least synchronized at the transition a r X i v : . [ phy s i c s . b i o - ph ] D ec (a) (b) C1 C2 AB radialspokeouterdynein inner dyneinB2 cis B1 trans M4 M2D2D4
Figure 1. (a) Asymmetric cytoskeletal organization that underliesbeating differences between cis and trans flagella. Duringdevelopment, eyespot positioning delineates between the twoflagella; that closer to the eyespot is the cis flagellum, and the fartherone is the trans flagellum. (b) Inside the axoneme: a peripheralarrangement of microtubule doublets encircles a central pair, andspecialized dynein motors initiate interdoublet sliding and beatgeneration. to the succeeding recovery stroke. This trend appears tobe universal to all cells of the wildtype strain. However,the tendency for the two flagella of a given cell toexperience phase slips exhibits much greater variationacross the population (figure 6). This we take as furtherindication that
Chlamydomonas cells are highly sensitiveto fluctuating biochemical cues. Sampled across multiplecells, phase slip excursions between synchronized statescan be visualized by a dimension-reduced Poincaréreturn-map of interflagellar phase difference, showing thesynchronized state itself to be globally stable. Examiningeach flagellar phase slip in detail we show furtherthat the trans flagellum reproducibly transitions to awell-defined transient state with higher beat frequencyand attenuated waveform. This evidences an alternativemode of beating - which we conjecture to exist as one of adiscrete spectrum of modes on the eukaryotic flagellum.This second mode can also be sustained for longerdurations, and in both flagella of a particular phototaxismutant of
Chlamydomonas , as detailed elsewhere [7].Taken together, figure 3 encapsulates in a single diagramthe three possible biflagellate “gaits”, their differencesand similarities, highlighting the need for a quantitativeformulation such as that we present in this article.Intrinsic differences between the two
Chlamydomonas flagella, their construction and actuation, underlie thisrich assemblage of biflagellate phenomenology. Pastexperiments have shown such differences to exist, forexample in reactivated cell models of
Chlamydomonas [9], in which the trans flagellum has a faster beatfrequency than the cis . In contrast, we consider herethe in vivo case (§3.d); by mechanically inducingdeflagellation of either trans or cis flagellum, we renderlive wildtype cells uniflagellate. This allowed us tocompare the intrinsic beating dynamics of cis vs trans flagella. We found that whilst cis - uniflagellated cellstend to beat with the canonical breastroke-like mode (BS-mode), trans - uniflagellated cells can instead sustainthe faster mode of beating associated with the phase slip(aBS-mode) (figure 10a). Yet this cis - trans specializationis lost once the cell is allowed to regrow the lost flagellumto full-length, by which time both flagella have recoveredthe BS-mode.Flagellar tracking has enabled us to acquire true spatiallocalization of the flagellum throughout its dynamicrhythmicity, complementing recent efforts aiming in thisdirection [4, 10] The need to know precise waveformshas long been an ambition of historic works, in whichmanual light-table tracings of Chlamydomonas flagellawere used to elucidate behaviour of the wildtype [11, 12],and latterly also of flagellar mutants [13]. We hope thatthe findings and methodologies herein presented shall beof broad interest to physicists and biologists alike.
At a more fundamental level, how is beating of asingle flagellum or cilium generated, and moreover, howcan multi-ciliary arrays spontaneously synchronize? Foreach of us, or at least, for the hair-like appendageslining the epithelial cells of our respiratory tracts, thisis indeed an important question. Beating periodically,synchronously, and moreover metachronously, multitudesof these cilia drive extracellular fluid flows whichmediate mucociliary clearance. These motile cilia,and their non-motile counterparts, are regulated bycomplex biochemical networks to perform highly-specificfunctions [14, 15]. Mutations and defects in theseorganelles have been increasingly implicated in manyhuman disorders including blindness, obesity, cystickidney and liver disease, hydrocephalus, as well aslaterality defects such as situs inversus totalis [16, 17].Mice experiments in which nodal flows are artificiallydisrupted directly link mechanical flows to positioningof morphogens, which in turn trigger laterality-signallingcascades [18].Across the eukaryotic phylogeny these slender,propulsion-generating appendages possess a remarkablyconserved ultrastructure [19]. In recent decades,causality from structure to function within eukaryoticciliary/flagellar axonemes has been establishedusing sophisticated molecular genetics tools. Forthe
Chlamydomonas in particular, rapid freezing ofspecimens has made possible the capture of axonemalcomponents in near-physiological states, at molecular-level resolution [20].
Chlamydomonas flagella havea well-characterized structure of microtubuledoublets (MTD), along which are rows of the molecularmotor dynein (figure 1). These directional motorsgenerate forces parallel to filamentous MTD tracks,which slide neighbouring MTDs past each other.Anchored to the A-tubule of each MTD by theirtail domains, these dyneins detach and reattach ontothe next B-tubule, consuming ATP in the steppingprocess. Different dynein species coexist within theflagellar axoneme, with force-generation principallyprovided by outer dyneins, and modulation of flagellarwaveform by the inner dyneins. The central pair, isthought to transduce signals via the radial spokes [21].Approximately every nm this precise arrangementof dyneins, radial spokes, and linkers repeats itself[22]. Periodic elastic linkages between neighbouringMTDs called nexins provide the coupling by whichdynein-driven filament sliding produces macroscopicbending of the flagellum, which in turn propels thecell through the fluid. Treatments of axonemes whichdisrupt dynein domains have shown these nexin linkagesto function in a non-Hookean manner to provide theelastic restoring force that resists internal filament shear[23]. More recently, detailed D-tomographic analysisrevealed that nexins are in fact one and the same withthe dynein regulatory complex, collectively termed theNDRC [24]. The importance of the NDRC within thefunctioning axoneme, from those of algae to that ofhumans, has been highlighted in a recent review [25].With regard to the flagellum-cycling mechanism,consensus has been lacking. Timing-control appears tobe absent, yet much experimental evidence points toselective, periodic, dynein-activation [26]. Both powerand recovery strokes of the beat cycle involve active forcegeneration by differentially actuated dyneins. Rhythmicbeating of the flagellum may arise from dynamicalinstabilities of dynein oscillations [27, 28], and may beclosely coupled to the intrinsic geometry of the axoneme[29, 30]. With these unanswered questions in mind, thereis thus much incentive to analyze the flagellum beat invivo , as the present study seeks to demonstrate. b Molecular origins of cis - trans difference As with many species of flagellated algae, motility isessential not just for swimming, but also for cell taxis.For
Chlamydomonas , its two equal-length flagella, each − µ m long, emerge from an ellipsoidal cell body ∼ µ m in radius. Cell bilateral symmetry is brokenby the presence of an eyespot, which functions as aprimitive photosensor. Perceived directional light is thenconverted downstream via secondary messenger ions intodifferential flagellar response and hence a turn of thecell [22, 31, 32]. The two anterior basal bodies fromwhich the two flagella protrude are connected to eachother via distal striated fibres [33]. A system of fouracetylated microtubule rootlet-bundles lies beneath thecell membrane, and extends towards the cell posterior[32]. The eyespot is assembled de novo after eachcell division, breaking cell bilateral symmetry by itsassociation with the daughter four-membered rootlet(figure 1a, D4). The trans flagellum, nucleated by themother basal body (B1), has been shown in reactivatedcell models to beat at an intrinsic frequency that is ∼ − higher than that of the cis [9]. This frequencymismatch is discernible in vivo in wildtype cells that we rendered uniflagellated through mechanical deflagellation(§3.d), additionally with a discrepancy in beatingwaveform. Differential phosphorylation of an axonemaldocking complex has been suggested to underlie thedistinctive cis - trans beat [34]. In particular, differential cis-trans sensitivity to submicromolar Ca in cellmodels and in isolated axonemes [9, 36] is consistent withthe opposing flagellar modulation necessary for cells toperform phototactic turns [13].Yet, despite these intrinsic differences, the two flagellamaintain good synchrony during free swimming [4, 10,11], as well as when held stationary by a micropipette(here). Interflagellar coupling may be provided by themotion of the fluid medium [35, 37], by rocking of thecell-body [5], or further modulated internally via elasticcomponents through physical connections in the basalregion [33]. However, stochastically-induced flagellar phase slips can appear in otherwise synchronized beatingflagella in a distinctive, reproducible manner (figure 3).We find that the propensity to undergo these transientslips [13] can vary significantly even between cells of aclonal population (figure 6). In their native habitat,
Chlamydomonas cells swim in water (kinematic viscosity ν = 10 − m s − ), at speeds on the order of µ m/s,up to a maximum of µ m/s depending on strainand culture growth conditions [38]. Oscillatory flowsset up by these flagella have a frequency f = ω/ π (cid:39) Hz during the breaststroke. Stroke lengths of L = 10 µ m thus produce a tip velocity scale of U = Lω ∼ mm/s. An (oscillatory) Reynolds number Re = ωL /ν gauges the viscous and inertial force contributions to theresulting flow. Here, Re ≈ . , and cell propulsion thusrelies on viscous resistance to shearing of the fluid by theflagellar motion. To overcome the reversibility of suchflows, a breaking of spatial symmetry is essential duringthe swimming breaststroke. The rhythmic sweepingmotion of each flagellum can be partitioned into distinctpower and recovery strokes: during the power stroke,the flagella are extended to maximize interaction withthe fluid, but undergo greater bending excursions duringthe recovery stroke (figure 3a). Net swimming progressresults from the drag anisotropy of slender filamentsand the folding of flagella much closer to the cell bodyduring the recovery stroke. Interestingly, a qualitativelysimilar bilateral stroke can emerge from a theoreticaloptimisation performed on swimming gaits of biflagellatemicroorganisms [39] and on single flagella near surfaces[40]. The phase slip
Early microscopic analyses of
Chlamydomonas flagella suggested that the normal s li pb r ea s t s t r ok ea n ti ph a s e cis b r ea s t r ok e cistranstrans ciscistrans Figure 2. Three
Chlamydomonas breaststroke swimming gaits recorded at fps and shown at intervals of frames (i.e. . ms intervals).Orange dot marks cell eyespot location, red curves: cis flagellum, blue curves trans flagellum. Shown in order, in-phase synchronizedbreaststroke (both flagella in BS-mode), a phase slip in the same cell ( trans flagellum in aBS-mode), and antiphase synchronization in thephototaxis mutant ptx1 (both flagella in aBS-mode). Scale bar is µm .Figure 3. Overlaid sequences of tracked flagella showing (a) normal breaststroke (BS) ( consecutive beats), (b) a stochastic slip event ( consecutive slips), in which the trans flagellum transiently enters a different mode (aBS), and (c) in which both flagella of ptx1 sustain theaBS mode ( consecutive beats). For each flagellum, progression through the beat cycle can be tracked using an angle θ defined relativeto the cell bilateral axis (d); sample timeseries for θ are shown for each gait. The aBS-mode can clearly be seen to have an attenuatedbeat amplitude and a faster beat frequency. (e)-(f): Differences between cis and trans limit cycles are clarified in phase-space coordinates ( θ, H ( θ )) , where H ( . ) denotes the Hilbert transform (see §3.b), and the color-intensity is obtained by logarithmically scaling the probabilityof recurrence. breaststroke synchrony “may be disturbed for briefperiods” [33]. These interruptions to ‘normal’ beating,subsequently detailed in manual waveform tracings byRüffer and Nultsch, were shown to occur in the absenceof obvious stimulation, in both free-swimming cells[11] and cells affixed to micropipettes [12]. Crucially,these transient asynchronies do not significantly alterthe trajectory of swimming (unpublished observation);instead, during each such episode the cell body is seen torock back and forth slightly from frame to frame withoutaltering its prior course. These asynchronies are termed slips [2, 3] by analogy with an identical phenomenon inweakly-coupled phase oscillators. Physically, phase slipsare manifest in these coupled flagella in a strikinglyreproducible manner. Under our experimental conditions(detailed in §5), beating of the trans flagellum transitionsduring a slip to a distinct waveform, concurrently witha ∼ higher frequency [7], whilst at the same timethe cis -flagellum maintains its original mode of beatingthroughout, apparently unaffected (figure 3b). We findalso that the faster, attenuated breaststroke mode (aBS)is sustained by the trans flagellum for an integer numberof full beat cycles, after which normal synchronizedbreaststroke (BS) resumes. The antiphase
The aBS waveform assumed bythe trans -flagellum during a slip turns out to bemarkedly similar to that identified in an anti-synchronousgait displayed by a particular phototaxis mutant of
Chlamydomonas called ptx1 [41]. In recent, related work,we make these comparisons more concrete, and show thatthis gait (figure 3c) involves actuation of both flagella inaBS-mode, and in precise antiphase with each other [7].Although the precise nature of the ptx1 mutation remainsunclear, it is thought that emergence of this novel gaitin the mutant is closely associated with loss of calcium-mediated flagellar dominance. b Phase-dynamics of a single flagellum
Many biological oscillators are spatially extended andare therefore fundamentally high-dimensional dynamicalobjects, but adopting a phase-reduction approachfacilitates quantitative analyses. In such cases, stable self-sustained oscillations can be represented by dynamicson a limit cycle, for which monotonically increasingcandidate phases ϕ may be extracted.The natural or interpolated phase of an oscillator isdefined to increment linearly between successive crossingsof a Poincaré section, and as such is strongly dependenton the precise choice of the section. For discrete markerevents { t n } , and t n ≤ t ≤ t n +1 , ϕ P ( t ) = 2 π (cid:18) n + t − t n t n +1 − t n (cid:19) . (3.1)This method was used in earlier work [2, 3], toextract flagellar phases φ cis , trans , by sampling pixelintensity variations over pre-defined regions of interest, (c) π π Poincaré section π π (a) (b) Figure 4. (a) Hilbert embedding for the cis -flagellum of a sample cell,recorded over thousands of beat cycles, and coloured accordingto the equivalent transformed phase φ (via equation 3.6). Pointsof equal phase lie on isochrones, highlighted here at equi-phaseintervals. The rate of phase rotation varies systematically throughoutthe beat cycle, as indicated by the variable inter-isochrone spacing.Inset: successive zero-crossings of φ − (cid:104) φ (cid:105) map out one Poincaré-section. (b) Snapshots − show typical positions of the flagellumat phases corresponding to five representative isochrones. (c)Phase-velocity Γ( ϕ ) = dϕ H /dt is approximated by a truncatedFourier series. Shaded regions show one standard deviation offluctuations in the raw data. on individual frames of recorded video. Specifically,phase values are interpolated between successive peaksin intensity. However as beating of the flagellumcorresponds to smooth dynamics, by Poincaré sectioningthe dynamics in this way, sub-beat-cycle information islost. In the present work, we take a more continuousapproach by incorporating enhanced resolution toelucidate within-beat-cycle dynamics, and make use offlagellum tracking to define D-projections of flagellumoscillations.For this we choose an embedding via the Hilberttransform. This technique derives from the analyticsignal concept [42], and is used to unambiguously definean instantaneous phase (and amplitude) from scalarsignals with slowly-varying frequency. From a periodicscalar timeseries x ( t ) we construct its complex extension ph a s e d e v i a ti on (r a d ) Figure 5. Phase-deviation over long-timescales obtained by subtracting from unwrapped phases the linear component that scales withoscillator frequency. General trends are preserved by the phase transformation (equation 3.4), but within-cycle fluctuations are removed ζ ( t ) = x ( t ) + i ˜ x ( t ) , where ˜ x ( t ) is given by H ( x ) = ˜ x ( t ) = 1 π PV (cid:90) ∞−∞ x ( τ ) t − τ dτ . (3.2)Here, the integral to be taken in the sense of the Cauchyprincipal value. Polar angle rotation in the x − ˜ x phase-plane gives the Hilbert phase, ϕ H ( t ) = tan − (cid:18) ˜ x − ˜ x x − x (cid:19) , (3.3)which, when unwrapped, serves as a monotonically-increasing candidate phase on our limit cycle projection.The origin ( x , ˜ x ) is chosen to be strictly interior of thelimit cycle; here, x = (cid:104) x (cid:105) , ˜ x = (cid:104) ˜ x (cid:105) .Candidate phases such as the Hilbert phase do notuniformly rotate and are sensitive to cycle geometryand nonlinearity. That is, ˙ ϕ = Γ( ϕ ) is in general somenon-constant but π -periodic function. For any givenlimit cycle, there is however a unique true phase φ forwhich ω = dφ/dt is constant, the rate of rotation of φ isequal to the autonomous oscillator frequency. The desiredtransformation ϕ → φ is φ = ω (cid:90) ϕ (Γ( ϕ )) − dϕ . (3.4)Whilst φ is unique, ambiguity remains in the choice oflimit cycle that best characterizes the original phase-space. Furthermore for noisy timeseries, limit cycletrajectories do not repeat themselves exactly, so thatthe phase transformation (3.4) can only be performedin a statistical sense. Accuracy of phase estimation isimproved with longer observation time.To derive an approximation for dϕ/dt as a function of ϕ , we first sort data pairs (cid:104) ϕ, dϕ (cid:105) and then average overall ensemble realizations of ϕ (figure 4). Direct numericalapproximations for dt/dϕ are sensitive to noise, due tothe heavy-tailed nature of ratio distributions, to remedythis, we follow the approach of Revzen et al [43] andbegin by finding an N-th order truncated Fourier series approximation ˆΓ = F N [Γ] , to Γ . Next, we find a similarFourier approximation to / ˆΓ , dtdϕ ( ϕ ) ≈ F N [1 / ˆΓ] = N (cid:88) k = − N f k e ikϕ . (3.5)The zeroth coefficient f = π (cid:82) T (cid:16) dtdϕ (cid:17) dϕ = ω − where ω is the intrinsic frequency of the oscillator, so that tolowest order, ϕ and φ coincide. Substituting ( . ) into( . ) gives φ = ϕ + 2 ω N (cid:88) k =1 Im (cid:18) f k k ( e ikϕ − (cid:19) . (3.6)For flagellum oscillations, we choose scalar timeseries x = θ cis/trans (Fig. 3). For each flagellum, pairs of values x = ( x, ˜ x ) define a noisy limit cycle in phase space.Using equation. . , we can associate a phase at each x ,consistent with the notion of asymptotic phase defined inthe attracting region around a limit cycle. For differentflagella, the function Γ( ϕ ) takes a characteristic form(4c). Points of equal phase lie on isochrones, which foliatethe attracting domain (figure 4a).Perturbations that are π periodic functions of ϕ are eliminated by the transformation 3.6. Within-period oscillations are averaged out, whilst long-timescaledynamics are preserved, by virtue of the invertibilityof the transformation. This can be seen over long-timerecordings, in which the two measures of phase deviation: D φ = φ − ωt and D ϕ = ϕ H − ωt coincide (figure 5), butthe periodicity of ∆ ϕ has been smoothed out (inset). c Phase-dynamics of coupled flagella pair For microorganisms that rely on multiple flagella forswimming motility, precision of coordination is essentialto elicit high swimming efficacy. The bilateral geometricdisposition of the two
Chlamydomonas flagella facilitatesextraction of phases for an individual flagellum’soscillations, and in turn, derivation of phase synchrony b) c) (cid:0) avg=0.0524lag time ( (cid:0) /T) (cid:1) ( (cid:2) ) po w e r r ec ov e r y r ec u rr e n t no t r ec u rr e n t (cid:3) (cid:3) a) cis trans lag (cid:4) t r a n s (r a d ) (cid:5) cis (rad) Figure 6. (a) Lag synchronization in bivariate timeseries of flagella beats, shown for a typical cell. Inset: cis flagellum begins its recoverystroke fractionally ahead of trans . (b) Multi-cell similarity functions show a similar trend. (c) Noisy flagellar dynamics within a population of cells, as represented by the tendency of each cell to undergo slips. Most cells remain synchronized for more than s, whilst some exhibitfrequent asynchronies. Red to blue: from high to low probability of recurrence. relations between coupled pairs of flagella. However, atransformation function similar to equation 3.4 that isbivariate in the two phases cannot be derived fromobservations of the synchronized state alone; thereforein the following, we make use of the Hilbert phase(equation 3.3). Phase difference derivation
To monitor biflagellarsynchrony, the phase difference ϕ trans − ϕ cis is ofparticular interest. More generically, for coupled noisyphase oscillators i and j , the dynamics of each isperturbed by the motion of the other, as well as bystochastic contributions: ˙ ϕ i = f ( ϕ i ) + ε i g ( ϕ i , ϕ j ) + ξ i ( t ) , ( i = 1 , , (3.7)where g ( ϕ i , ϕ j ) is the coupling function, and ξ is a noiseterm. When ε , = 0 we recover the intrinsic motion of asingle flagellum. The two oscillators are said to be n : m phase-locked when their cyclic relative phase given by ∆ n,m = nϕ − mϕ , satisfies ∆ n,m = Const.. For noisy orchaotic systems this condition may be relaxed to ∆ n,m < Const. Here, we define the trans - cis phase difference fromthe respective angle signals θ cis ( t ) and θ trans ( t ) by ∆ = ϕ trans H ( t ) − ϕ cis H ( t ) (3.8) = tan − (cid:32) ˜ θ trans ( t ) θ cis ( t ) − θ trans ( t )˜ θ cis ( t ) θ trans ( t ) θ cis ( t ) + ˜ θ trans ( t )˜ θ cis ( t ) (cid:33) , where ˜ again denotes the Hilbert transform.We measured ∆ for a large population of cells (Fig.6). Phase-slip asynchronies are associated with rapidchanges in interflagellar phase difference, and appear asstep-like transitions that punctuate (sometimes lengthy)epochs of synchronized behaviour for which phasedifference is constant. We see that over a comparableperiod of observation time (figure 6c:i-iii), pairs offlagella can experience either perfect synchrony, few slips,or many slips . For the population as a whole, thecircular representation of figure 6 facilitates simultaneousvisualization of general trends in interflagellar phase-synchrony. Lag synchronization
Careful examination of asynchronized epoch shows that ∆ is not strictly constant,but rather fluctuates periodically about a constant value.During execution of breaststroke swimming, Poincarésectioning of the dynamics has suggested previously thatthe breaststroke gait is perfectly synchronized [2, 3].However, plotting θ cis against θ trans (figure 6a) we seea consistent lag between the two flagella, which is mostpronounced during the recovery stroke. By computingand minimizing the similarity function Λ( τ ) = (cid:115) (cid:104) ( θ trans ( t + τ ) − θ cis ( t )) (cid:105) (cid:0)(cid:10) θ trans (cid:11) (cid:10) θ cis (cid:11)(cid:1) / (cid:0) /3 - (cid:0) /6 0 (cid:0) /6 (a) (b) (c) Figure 7. Trends in cis - trans flagellar synchronization in a population of Chlamydomonas cells: each concentric annulus represents datafrom an individual cell, measured values are plotted on a circular scale → π in an anticlockwise sense. (a) Probability of stroboscopicallyobserving φ H cis ∈ [0 , π ] at discrete marker events where θ H trans reaches a minimum (i.e. start of new power stroke). (b)-(c) Difference intracked angles ( ∆ θ = θ trans − θ cis ) and Hilbert phases ( ∆ φ = φ H trans − φ H cis ), averaged over thousands of beat cycles. It is seen that ∆ θ isgreatest during the recovery stroke, and correspondingly ∆ φ becomes increasingly negative. we find this discrepancy to be indicative of lagsynchronization. Here, the periodic angle variables θ cis/trans are chosen as scalar indicators for theprogression of each flagellum through its beat cycle.In particular, the two phases are synchronized witha time lag τ min = min τ Λ( τ ) , where Λ( τ ) assumes aglobal minimum. When the oscillators are perfectlysynchronized, τ min = 0 . We calculated Λ( τ ) for multiplecells, which displayed a similar profile (figure 6b). With τ normalized by the average inter-beat period T , we seethat in every instance the minimum is displaced from (or equivalently ), with an value of . ± . ,indicative of persistent directional lag. Stability of fixed points and transients
Aphenomenological model such as equation 3.7 has aconvenient dynamical analogy. Phase difference canbe interpreted as particle in a washboard potential V (∆) = − ∆ δω + (cid:15) cos(∆) , subject to overdampeddynamics ˙∆ = − dV (∆) /d ∆ . Potential minima occurwhere ˙∆ = 0 , which requires | δω/(cid:15) | < . For noise withsufficient magnitude, the particle will have enoughenergy to overcome the potential barrier, at leasttransiently. Stochastic jumps between neighbouringpotential minima, are manifest in coupled flagella as the slip -mode (figure 3b).In the vicinity of a potential minima, the stationarydistribution of ∆ is predicted to be Gaussian(equation 3.7, with white noise). This phase distribution P (∆) , can be measured directly from experiment, andassumed to satisfy P (∆) = exp( − U (∆) /k B T ) , (3.9) from which the potential structure U (∆) can berecovered. For each well, the peak location can be usedto estimate the phase-lag of the coupled oscillators, whilepeak width is indicative of strength of noise in thesystem. We measured P (∆) for cells which did notdisplay slips for the duration of observation (figure 9a).Potential minima have a parabolic profile with a well-defined peak, on average displaced to the left of ∆ = 0 ,due to the characteristic phase-lag in the direction ofthe cis -flagellum. For certain cells, this lag is especiallypronounced during the recovery-stroke than during thepower stroke, resulting in a double-peaked minimum inthe fine-structure of the empirical potential.The stability of the synchronized state may beassessed by observing trajectories that deviate from,but eventually return to, this state. Specifically, bymeasuring ∆ during multiple flagellar phase slips weconstruct a dimension-reduced return map for the jointsystem to visualize the potential landscape that extendsbetween neighbouring minima. Slips occur with variableduration (figure 8a), in which the trans flagellum cansustain the faster aBS-mode for a variable but completenumber of beat cycles. However for an individualcell, successive slips often exhibit identical dynamics(figure 8b). Figure 9b presents the return map of ∆ associated with > slip events collected from cells,where the discretized phase-difference ∆ n is defined to be ∆ evaluated stroboscopically at the position of maximumangular extent of the trans -flagellum, during the n thbeat-cycle. We begin by approximating the return mapby a polynomial F (∆ n ) . An empirical potential function[44] can be defined by integrating the difference δ ∆ = (b) θ c i s , θ t r an s (r a d ) (a) transcis Δ / π one slipavg time un w r app e d pha s e slip Figure 8. a) Harmonics of a slip, synchrony resumes after a different(but always integer) number of beats of either flagellum. b) each slipresults in a step-like transition in phase-difference ∆ . For the samecell, successive slips overlap. Inset: forking of unwrapped phases ϕ cis H and ϕ trans H . ∆ n +1 − ∆ n ≈ F (∆ n ) − ∆ n : ˜ V (∆ n ) = − (cid:90) ∆ n δ (∆ (cid:48) ) d ∆ (cid:48) , (3.10)which we convert into a locally positive-definite function V (∆ n ) = ˜ V (∆ n ) − min ∆ ˜ V (∆) , (3.11)which satisfies V (∆) > , ∀ ∆ / ∈ argmin ˜ V (∆) . Theresulting effective potential profile (figure 9b, inset)represents the reproducible phase dynamics of a typicalflagellar phase slip, from which breaststroke synchronyre-emerges. d Coupling cis and trans flagella Pre-existing, intrinsic differences between the two
Chlamydomonas flagella are essential for control of cellreorientation, loss or reduction in cis - trans specializationmay give rise to defective phototaxis in certain mutantsof Chlamydomonas [7, 45]. Under general experimentalconditions, stochastic asynchronies which we call slipscan punctuate an otherwise synchronous breaststroke;more drastic loss 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 all theseinstances, we observe the coupled state of two flagella;in contrast, by mechanically deflagellating wildtype cells(see §5) we can now examine the intrinsic behaviour ofeach oscillator in isolation. π πΔ n Δ n + -1 -0.5 0 0.5 - l og ( P ( Δ ) Δ
510 slip (a)(b) sync V ( Δ ) F ( Δ ) - Δ ππ
012 6420 8
Figure 9. A potential analogy for wrapped phase difference,visualized a) within potential minima - ∆ exhibits local fluctuationsduring the stable synchronized gait; and b) between successivepotential minima, via a first return map of cyclic/stroboscopic relativephase. An nth-order polynomial F (∆ n ) is fit to the multicell return-map statistics, from which an empirical potential function V (∆) candefined. The ability of
Chlamydomonas to readily regeneratea lost flagellum has facilitated controlled measurementsof flagellar coupling strength as a function of flagellumlength [6]. Using the single eyespot as identifier, weremoved either the cis or trans flagellum from a pipette-captured cell and recorded the beating dynamics of theremaining flagellum. Histograms of beat-frequencies areplotted in figure 10b. On average, cis - uniflagellatedcells tend to beat at a lower frequency than trans -uniflagellated cells. A dissociation of beat-frequencyof similar magnitude has been observed previously inreactivated cell models [9]. Moreover, we find that inthe absence of the cis flagellum the trans flagellum can sustain the faster aBS-mode for thousands of beats.These differences are highlighted in Figure 10a, for asingle cell.Interestingly, the aBS-mode that we can now associatewith the intrinsic beating waveform of the trans , emergestransiently during a slip of the wildtype (figure 3b), butin both flagella during an antiphase gait of the mutant ptx1 (figure 3c, and [7]). Indeed, for ptx1 , its lack of0effectual cis - trans specialization has led to speculationthat the mutation has renders both flagella trans -like [45].Specific, structural differences known to exist betweenthe cis vs trans axonemes [34] of the wildtype, may effectthis segregation of intrinsic beating modes. For a unicellular flagellate such as
Chlamydomonas ,synchrony of its two flagella is intimately regulatedby the cell’s internal biochemistry; however, the exactmechanism by which messenger ions modulate and shapethe flagellum beat remains unclear. Our experimentaltechnique captures the motion of beating flagella in vivo ,at high resolution, and with respect to a fixed pivot,thereby permitting long-time analysis.Associating each flagellum oscillator with a continuousphase, we formulated a phase-reduced model of theperiodic dynamics. From long-time series statistics ofbivariate oscillator phases, we used phase difference totrack phase synchrony, quantifying flagellar interactionsfor a single individual, as well as across the sampledpopulation. Exquisitely sensitive to its surroundings, aflagellum can be found to undergo precise, yet dynamicchanges when executing its periodic strokes. Waveformtracking has allowed us to assess these changes in greatspatio-temporal detail.In particular, we have found the stable phase-locked breaststroke of
C. reinhardtii to exhibit a smallbut persistent cis - trans phase-lag, the magnitude anddirection of which was evaluated from statistics ofthousands of beat cycles using a similarity measure, andconfirmed for multiple cells. However, often it is notthe synchronized state itself but rather the emergenceor cessation of synchrony that is most insightful forinferring fluctuations in the physiological state of acomplex system. Phase slips are transient excursionsfrom synchrony in which, under our experimentalconditions, an alteration of beating mode is observed inthe trans flagellum only, and that appear to be initiatedby a reduced cis - trans phase lag (figure 8b). Thesereproducible events highlight the importance of cis - trans specialization of Chlamydomonas flagella. Exploring thisfurther, we mechanically removed individual flagellaof wildtype cells to obtain uniflagellated cis or trans versions, revealing significant differences between theirisolated beating behaviours (figure 10). Yet for afully-intact cell, despite these inherent differences inbeating frequency and in waveform, coupling interactionseither hydrodynamically through the surrounding fluidmedium, and/or biomechanically through elastic linkagesat the base of the flagellar protrusion, appear sufficienton the most part to enslave the beating of the trans modeto that of the cis .Ours is a very versatile technique for quantifyingflagellar synchrony not just of the wildtype system,but such a phase analysis can for instance also be used to probe defective swimming behaviours of motilitymutants. In these cases, macroscopic measurementsof population features may not be instructive tounderstanding or resolving the mutant phenotype, andwould benefit from dynamic flagellum waveform trackingand in-depth analysis at the level of an individual cell. For purposesof flagella visualization we chose two wildtype
C.reinhardtii strains, CC and CC (ChlamydomonasCenter). Stock algae maintained on TAP (Tris-Acetate Phosphate) solid agar slants, were remobilizedfor swimming motility by inoculation into TAP-liquidmedium, and cultures used for experimentation weremaintained in exponential growth phase ( − cells/ml) for optimal motility. Culture flasks are placedonto orbital shakers, and maintained at ◦ C ingrowth chambers illuminated on a 14:10 daily light/darkcycle, so as to imitate the indigenous circadian stimuli.Observation of flagellar dynamics was carried out ona Nikon TE2000-U inverted microscope, at constantbrightfield illumination. Additional experiments werealso performed with a long-pass filter ( nm) tominimize cell phototaxis . Individual cells were capturedand held on the end of tapered micropipettes (SutterInstrument Co. P- ), and repositioned with a precisionmicromanipulator (Scientifica, UK), and imaged at ratesof , - , fps (Photron Fastcam, SA ). Digital Image and Signal Processing
Recordedmovies were transferred to disk for post-processing in
MATLAB (Version . . , The Mathworks Inc. .).Flagellar waveforms were extracted from individualframes, where contiguous dark pixels that localize themoving flagellum were fit to splines. Hilbert transformswere perform in MATLAB (Signal Processing Toolbox),and further timeseries analysis performed using custom
MATLAB code.
Mechanical deflagellation of either cis or trans flagella To obtain the results described in §3.d,individual wildtype cells were first examined underwhite light to locate the unique eyespot, therebydifferentiating its cis flagellum from the trans . Oneflagellum was then carefully removed with a secondmicropipette, by exerting just enough shearing force toinduce spontaneous deflagellation by self-scission at thebasal region. That cells retain the ability for regrowthof flagella ensures basal bodies have not been damagedby our deflagellation treatment. Cells for which thebeating of the remaining flagellum became abnormalor intermittent, and also for which a clear cis - trans identification could not be made, were duly discarded.1 pd f transcis (b)(a) frequency (Hz) trans
20 40 60 80 100 12020406080100120 00.10.20.30.40.50.60.70.80.9 cis c i s
10 30 50 70 9000.5 pd f isolatedcoupled t r an s Figure 10. (a) For a single cell, cis and trans flagella were removed and in turn allowed to regrow to full length. Single-flagellum frequenciesseparate, but once regrown, lock to a common frequency ( cis : . Hz, trans : . Hz, both : . Hz). Insets: typical cis and trans waveforms, the trans waveform is reminiscent of the aBS-mode that onsets during a slip. Waveforms are overlaid on an intensity plot oflogarithmically-scaled residence times for each flagellum, over O (100) contiguous beats. (b) When beating in isolation, cis and trans flagellahave different frequencies: areas - histograms of interbeat frequencies, lines - averaged frequencies. Data
Examples of high-speed movies, flagellar timeseries, and other data referenced in this work can befound at: . We thank Marco Polin for discussions. Financial support isacknowledged from the EPSRC, ERC Advanced Investigator Grant , and a Senior Investigator Award from the Wellcome Trust(REG).
Notes Of the population of cells analysed for figure 6c, most wereobserved under white light. However a small percentage ( ) wereobserved under red light, but which for the sake of clarity, havenot been explicitly marked out in the figure. Whilst in both casesvariability in frequency of flagellar slips is observed, we find thaton average slips occur more prevalently in cells illuminated by redthan by white light (discussed further elsewhere). Whilst cell phototaxis behaviour is minimized under red-lightillumination, physiological cell motility cannot be maintained inprolonged absence of light, unless the experimental conditions wereaccordingly modified [7].
References [1] Purcell EM. 1977 Life at low reynolds-number.
AmericanJournal of Physics , , 3-11. (doi:10.1119/ 1.10903)[2] Polin M, Tuval I, Drescher K, Gollub JP, Goldstein RE.2009 Chlamydomonas swims with two "gears" in a eukaryoticversion of run-and-tumble locomotion.
Science , , 487-490.(doi:10.1126/ science.1172667)[3] Goldstein RE, Polin M, Tuval I. 2009 Noise andsynchronization in pairs of beating eukaryoticflagella. Phys. Rev. Lett. , , 168103. (doi:10.1103/PhysRevLett.103.168103)[4] Geyer VF, Jülicher F, Howard J, Friedrich BM. 2013 Cell-bodyrocking is a dominant mechanism for flagellar synchronization in a swimming alga. Proc. Natl. Acad. Sci. U.S.A. , , 18058-18063. (doi: 10.1073/pnas.1300895110)[5] Friedrich BM, Julicher F. 2012 Flagellar synchronizationindependent of hydrodynamic interactions. Phys. Rev. Lett. , , 138102. (doi:10.1103/PhysRevLett.109.138102)[6] Goldstein RE, Polin M, Tuval I. 2011 Emergenceof synchronized beating during the regrowth ofeukaryotic flagella. Phys. Rev. Lett. , , 148103.(doi:10.1103/PhysRevLett.107.148103)[7] Leptos KC, Wan KY, Polin M, Tuval I, Pesci AI, GoldsteinRE. 2013 Antiphase synchronization in a flagellar-dominancemutant of Chlamydomonas . Phys. Rev. Lett. , , 158101.(doi:10.1103/PhysRevLett.111.158101)[8] Pikovsky A, Rosenblum M, Kurths J. 2003 Synchronization:A Universal Concept in Nonlinear Sciences. , CambridgeUniversity Press.[9] Kamiya R, Witman GB. 1984 Submicromolar levels of calciumcontrol the balance of beating between the two flagellain demembranated models of
Chlamydomonas . Cell Motil.Cytoskeleton , Phys. Rev. Lett. , , 168102.(doi:10.1103/PhysRevLett.105.168102).[11] Ruffer U, Nultsch W. 1985 High-speed cinematographicanalysis of the movement of Chlamydomonas . Cell Motil.Cytoskeleton , , 251-263.[12] Ruffer U, Nultsch W. 1987 Comparison of the beatingof cis-flagella and trans-flagella of Chlamydomonas cellsheld on micropipettes.
Cell Motil. Cytoskeleton , , 87-93.(doi:10.1002/cm.970070111)[13] Ruffer U,Nultsch W. 1998 Flagellar coordination in Chlamydomonas cells held on micropipettes.
Cell Motil.Cytoskeleton , , 297-307. (doi:10.1002/(SICI)1097-0169(1998)41:4<297) [14] Smith EF, Rohatgi R. 2011 Cilia 2010: The surpriseorganelle of the decade. Sci. Signal. , , mr1.(doi:10.1126/scisignal.4155mr1)[15] Marshall WF, Nonaka S. 2006 Cilia: Tuning in tothe cellâĂŹs antenna. Curr. Biol. , , R604-R614.(doi:10.1016/j.cub.2006.07.012)[16] Ibanez-Tallon I, Heintz N, Omran H. 2003 To beat or notto beat: roles of cilia in development and disease. Hum. Mol.Genet. , , R27-R35. (doi:10.1093/hmg/ddg061)[17] Fliegauf M, Benzing T, Omran H. 2007 Mechanisms of disease- when cilia go bad: cilia defects and ciliopathies. Nat. Rev.Mol. Cell Biol. , , 880âĂŞ893. (doi:10.1038/nrm2278)[18] Nonaka S, Shiratori H, Saijoh Y, Hamada H. 2002Determination of left-right patterning of the mouseembryo by artificial nodal flow. Nature , , 96-99.(doi:10.1038/nature00849)[19] Silflow CD, Lefebvre PA. 2001 Assembly andmotility of eukaryotic cilia and flagella. lessons from Chlamydomonas reinhardtii . Plant Physiol. , , 1500-1507.(doi:10.1104/pp.010807)[20] Nicastro D, Schwartz C, Pierson J, Gaudette R, Porter ME,McIntosh JR. 2006 The molecular architecture of axonemesrevealed by cryoelectron tomography. Science , , 944-948.(doi:10.1126/science.1128618)[21] Smith EF. 2002 Regulation of flagellar dynein by theaxonemal central apparatus. Cell Motil. Cytoskeleton , , 33-42. (doi:10.1002/cm.10031)[22] Harris, EH. 2009 The Chlamydomonas sourcebook , Vol 1, 2ndedn. Academic Press.[23] Lindemann CB, Macauley LJ, Lesich KA. 2005 Thecounterbend phenomenon in dynein-disabled rat spermflagella and what it reveals about the interdoublet elasticity.
Biophys. J. , , 1165-1174. (doi:10.1529/biophysj.105.060681)[24] Heuser T, Raytchev M, Krell J, Porter ME, Nicastro D. 2009The dynein regulatory complex is the nexin link and a majorregulatory node in cilia and flagella. J. Cell Biol. , , 921-933. (doi:10.1083/jcb.200908067)[25] Wirschell M, Olbrich H, Werner C, Tritschler D, Bower R,Sale WS, Loges NT, Pennekamp P, Lindberg S. et al. 2013The nexin-dynein regulatory complex subunit drc1 is essentialfor motile cilia function in algae and humans. Nat. Genet. , ,262-8. (doi:10.1038/ng.2533)[26] Nakano I, Kobayashi T, Yoshimura M, Shingyoji C. 2003Central-pair-linked regulation of microtubule sliding bycalcium in flagellar axonemes. J. Cell Sci. , , 1627-1636.(doi:10.1242/jcs.00336)[27] Riedel-Kruse, IH, Hilfinger A, Howard J, Jülicher F. 2007 Howmolecular motors shape the flagellar beat. Hfsp Journal , ,192-208. (doi:10.2976/1.2773861)[28] Hilfinger A, Chattopadhyay AK, Jülicher F. 2009 Nonlineardynamics of cilia and flagella. Phy. Rev. E , , 051918.(doi:10.1103/PhysRevE.79.051918) [29] Brokaw CJ. 2009 Thinking about flagellar oscillation. CellMotil. Cytoskeleton , , 425-436. (doi:10.1002/cm.20313)[30] Lindemann, CB, Lesich KA. 2010 Flagellar and ciliarybeating: the proven and the possible. J. Cell Sci. , , 519-528. (doi:10.1242/jcs.051326)[31] Witman GB. 1993 Chlamydomonas phototaxis.
Trends CellBiol. , , 403-408. (doi:10.1016/0962-8924(93)90091-E)[32] Dieckmann, CL. 2003 Eyespot placement and assembly inthe green alga Chlamydomonas . Bioessays , , 410-416.(doi:10.1002/bies.10259)[33] Ringo DL. 1967 Flagellar motion and fine structure of flagellarapparatus in Chlamydomonas . J. Cell Biol. , , 543-&.(doi:10.1083/jcb.33.3.543)[34] Takada S, Kamiya R. 1997 Beat frequency difference betweenthe two flagella of Chlamydomonas depends on the attachmentsite of outer dynein arms on the outerdoublet microtubules.
Cell Motil. Cytoskeleton , , 68-75. (doi:10.1002/(SICI)1097-0169(1997)36:1<68)[35] Niedermayer T, Eckhardt B, Lenz P. 2008 Synchronization,phase locking, and metachronal wave formation in ciliarychains. Chaos , , 037128. (doi:10.1063/1.2956984)[36] Bessen M, Fay RB, Witman GB. 1980 Calcium control ofwaveform in isolated flagellar axonemes of Chlamydomonas . J. Cell Biol. , , 446-455. (doi: 10.1083/jcb.86.2.446)[37] Uchida N, Golestanian R. 2011 Generic conditions forhydrodynamic synchronization. Phys. Rev. Lett. , , 058104.(doi:10.1103/PhysRevLett.106.058104)[38] Racey TJ, Hallett R, Nickel B. 1981 A quasi-elasticlight-scattering and cinematographic investigation of motile Chlamydomonas-reinhardtii . Biophys. J. , , 557-571.[39] Tam D, Hosoi AE. 2011 Optimal feeding and swimming gaitsof biflagellated organisms. Proc. Natl. Acad. Sci. U.S.A. , ,1001-1006. (doi:10.1073/pnas.1011185108)[40] Eloy C, Lauga E. 2012 Kinematics of the mostefficient cilium. Phys. Rev. Lett. , , 038101.(doi:10.1103/PhysRevLett.109.038101)[41] Horst CJ, Witman GB. 1993 Ptx1 , a nonphototactic mutant of
Chlamydomonas , lacks control of flagellar dominance.
J. CellBiol. , , 733-741. (doi:10.1083/jcb.120.3.733)[42] Gabor D. 1946 Theory of communication. J. Inst. Electr. Eng. (London), , 429-457.[43] Revzen S, Guckenheimer JM. 2008 Estimating the phaseof synchronized oscillators. Phy. Rev. E , , 051907.(doi:10.1103/PhysRevE.78.051907)[44] Aoi S, Katayama D, Fujiki S, Tomita N, Funato T, YamashitaT, Senda K, Tsuchiya K. 2013 A stability-based mechanism forhysteresis in the walk-trot transition in quadruped locomotion. J. R. Soc. Interface , 20120908. (doi:10.1098/rsif.2012.0908)[45] Okita N, Isogai N, Hirono M, Kamiya R, Yoshimura K. 2005Phototactic activity in Chlamydomonas “non-phototactic”mutants deficient in Ca –dependent control of flagellardominance or in inner-arm dynein. J. Cell Sci. ,118