ABSTRACT Biological systems are robust to perturbations at both the genetic and environmental levels, although these same perturbations can elicit variation in behaviour. The interplay

between functional robustness and behavioural variability is exemplified at the organellar level by the beating of cilia and flagella. Cilia are motile despite wide genetic diversity between

and within species, differences in intracellular concentrations of ATP and calcium, and considerable environment fluctuations in temperature and viscosity. At the same time, these

perturbations result in a variety of spatio-temporal patterns that span a rich behavioural space. To investigate this behavioural space we analysed the dynamics of isolated cilia from the

mechanochemical model in the low-viscosity regime. This allowed us to associate waveform shape variability with changes in only the curvature response coefficients of the dynein motors.

SIMILAR CONTENT BEING VIEWED BY OTHERS TWIST–TORSION COUPLING IN BEATING AXONEMES Article Open access 24 February 2025 _C. ELEGANS_ EPISODIC SWIMMING IS DRIVEN BY MULTIFRACTAL KINETICS

Article Open access 08 September 2020 VERSATILE PROPERTIES OF DYNEIN MOLECULES UNDERLYING REGULATION IN FLAGELLAR OSCILLATION Article Open access 29 June 2023 MAIN Biological systems can

function despite genetic and environmental perturbations in their molecular components. These perturbations, in turn, affect behaviour at higher levels, from organelles through cells to

whole organisms. A key finding is that behavioural spaces, the spaces that embed the whole repertoire of behaviours, tend to be low-dimensional. Examples are the shapes of moving

nematodes1,2, the swimming trajectories of ciliates and bacteria3,4, the postures of walking flies5 and mice6 and the activities of _Hydra_7, which can all be reduced to low-dimensional

behavioural spaces. An important conclusion of these analyses is that only a few features define each behaviour, although there can be considerable variation of these features among

individuals within that behaviour. A challenge is to map these behavioural spaces onto underlying molecular and biophysical mechanisms8. Cilia are motile organelles that undergo complex

shape changes. These changes drive motion relative to the surrounding fluid and allow cilia to respond to external cues. Cilia, therefore, display a simple form of behaviour that makes them

a potentially good model system for relating behaviour to underlying molecular mechanisms. A key feature of cilia is that they can beat in the presence of a wide range of genetic mutations

of key proteins9,10,11 and can operate over a wide range of temperatures12, viscosities12,13, ATP concentrations14 and buffer conditions such as pH and calcium concentration15,16,17. Thus,

cilia exemplify conserved function18, namely beating, in the face of large genetic and environmental perturbations. In this work we use variations in beat shapes elicited by these

perturbations to construct the behavioural space of cilia. The beating motion of cilia is powered by thousands of dynein motors belonging to several different classes19. These motors drive

the sliding of adjacent doublet microtubules20 within the axoneme, the structural core of cilia21. Sliding is converted to bending by proteins that constrain the shearing of doublets (for

example, nexin links or radial spokes)20,22,23,24 and sliding at the base25. This biomechanical and structural knowledge forms the basis of mechanochemical models that can successfully

recapitulate the oscillatory motion of cilia26,27,28,29,30,31,32 and synthetic filament bundles33,34. The combination of mechanistic models, cited in this paragraph, with phenotypic

variability in response to perturbations, cited in the previous paragraph, makes the cilium an ideal model system to study the connection between behavioural spaces and molecular mechanisms.

In this work, we have asked: what is the geometry of the behavioural space of beating cilia subject to a wide range of perturbations, and how do the individual components of this space

relate to the underlying mechanochemistry. RESULTS QUANTIFICATION OF THE CILIARY BEAT We measured the waveforms of isolated and reactivated _Chlamydomonas reinhardtii_ axonemes (Fig. 1a)

with high temporal and spatial precision (see ref. 32 and Methods). The waveforms, discretized in 25 points along the arc length of the axoneme, were tracked over time for up to 200 beat

cycles using filament tracking software35 (one beat cycle is shown in Fig. 1b). The periodic beat of the axoneme was parameterized by the tangent angle _ψ_(_s_, _t_) (with _s_ the arc length

and _t_ the time) relative to the co-swimming frame36 (Fig. 1b). The power spectrum of the tangent angle (Fig. 1d) showed that typically <10% of the power was in the modes _n_ ≥ 2, which

we therefore neglected. After subtracting the static mode _n_ = 0, discussed in refs. 32,37, we parameterized the beat dynamics as a travelling wave: $$\psi (s,t)=a(s)\cos (2\uppi

ft+\varphi (s))\,,$$ (1) where _a_(_s_) is the amplitude profile, _φ_(_s_) is the phase profile and _f_ is the beat frequency. The arc length was normalized by the total length _L_ of the

axoneme, so that _s_ ∈ [0, 1]. Although _a_(_s_) and _φ_(_s_) are often approximated by scalars, such as the mean amplitude \({a}_{0}\equiv \int\nolimits_{0}^{1}a(s){\mathrm{d}}s\) and the

wavelength \(\lambda \equiv -\frac{\uppi L}{6}\int\nolimits_{0}^{1}(s-1/2)\varphi (s){\mathrm{d}}s\), here we are interested in their arc-length dependence, which defines the shape of the

beat. Our goal is to characterize variations in the set of parameters \({{{\mathcal{P}}}}=\{a(s),\varphi (s),f,L\}\), which we define as the behavioural space of beating. A subset of these

parameters, {_a_(_s_)/_a_0, _φ_(_s_)}, defines the shape. We explored the variability of \({{{\mathcal{P}}}}\) by observing groups of axonemes under several environmental perturbations

(temperature, viscosity, different ATP concentrations, Ca2+ and taxol) and genetic perturbations (_oda1_, _ida3_, _ida5_, _mbo2_ and _tpg1_) (Fig. 2). Our dataset comprises 498 axonemes

undergoing a total of ~40,000 beat cycles. Under all conditions we found beating axonemes, with a percentage of reactivating axonemes greater than 70%. The range of conditions resulted in a

wide range of beat shapes and lengths (Fig. 2a), beat frequencies (Fig. 2b), amplitude profiles (Fig. 2c) and phase profiles (Fig. 2d). Therefore, although the beating of cilia is robust

against perturbations, the features of its waveform display substantial variations. We now set out to quantify these variations. VARIATIONS IN FREQUENCY, MEAN AMPLITUDE AND WAVELENGTH The

beat frequency increases with ATP concentration and temperature and decreases with viscosity (Fig. 3a). This variation encompasses a frequency range of about one order of magnitude, and the

between-conditions variance is about seven times larger than the within-condition variance. By contrast, the mean amplitude varies little between these conditions, although it shows large

variations within conditions: the between-conditions variance in amplitude is about one-fifth the within-conditions variance. Thus, ATP concentration, temperature and viscosity lead to large

changes in frequency with little effect on mean amplitude, in agreement with studies cited in the Introduction. The mutations _oda1_ (which lacks outer-arm dyneins38), _ida3_ and _ida5_

(which lack inner dynein arms f and a, c, d, e, respectively39,40), _mbo2_ (which lacks microtubule inner proteins and has a symmetric beat41) and _tpg1_ (which has reduced

polyglutamamylation required for axonemal integrity42,43), as well as addition of the ion Ca2+ and the microtubule poison taxol, did not appreciably broaden the range of beat frequencies

(Fig. 3b, 15 Hz to 160 Hz). We note, however, that the _oda1_ mutant has a twofold lower beat frequency over all temperatures, as reported in earlier studies (Introduction). The _mbo2_

mutation and taxol increased the range of mean amplitudes (Fig. 3b). Interestingly, in _mbo2_ the amplitude is high but the beat frequency is low, whereas in taxol (which has not been

studied before) the amplitude is low but the beat frequency is high. This reciprocal variation of amplitude and frequency may reflect the energetics of the beat (Discussion). The variations

in the beat frequency and mean amplitude within individual axonemes are small compared to the axoneme-to-axoneme variation. Over the times of analysis, which were typically ~50 cycles, the

variances of the frequency and the mean amplitude were considerably smaller than the axoneme-to-axoneme variance (Supplementary Fig. 1 and Supplementary Table 1). It is nonetheless possible

that variability within a given condition is due to long-term dynamic effects, for example due to the sampling of axonemes in different states of rundown. Fluctuations in beat frequency over

time are so small that the _Q_-factor of the oscillations can reach values as high as 150 (Fig. 3b, inset), confirming earlier observations44. This _Q_-factor, one of the highest in any

biological system, is exceeded by the tuning of sonar responses in the inner ear of the moustached bat (_Q_ up to 1,000 (ref. 45)). Thus, the variation in beat frequency and amplitude is not

due to short-term variation within individual axonemes, but rather due to differences between axonemes and their response to perturbations. The lengths of axonemes vary twofold over the

entire dataset, from 7.2 μm to 15.4 μm (Fig. 3c). This variation is primarily due to the shorter _mbo2_ axonemes and the longer _oda1_ axonemes. Remarkably, the wavelengths under all these

conditions are almost equal to the lengths, with _λ_/_L_ = 0.97 ± 0.08 (mean ± standard deviation for all 498 cilia) (see analysis relating to Fig. 5). ASYMMETRY AND PARABOLICITY ARE

DOMINANT WAVEFORM TRAITS To characterize the variation in the amplitude and phase profiles of the tangent angle, we decomposed both into shifted Legendre polynomials of increasing order

(Methods and Fig. 4a–c). For the amplitude, the lowest order (_a_0) equals the mean amplitude, the first order (_a_1) corresponds to the linear deviation from constant amplitude and is a

measure of the proximal–distal asymmetry, and the second order (_a_2) measures the quadratic deviation. For the phase, we set _φ_0 to zero (as it is arbitrary), the first order is _φ_1 = 

−π_L_/_λ_ by definition of _λ_ (the linear component of the phase profile), and _φ_2 corresponds to the quadratic deviation from the line. Because we are interested in shape, we normalized

the amplitude coefficients: \({\bar{a}}_{1}={a}_{1}/{a}_{0}\) and \({\bar{a}}_{2}={a}_{2}/{a}_{0}\). Figure 4d displays the space of variations of parabolicity and asymmetry of the

amplitude. Most points cluster around \({\bar{a}}_{1}\approx 0.0\) and \({\bar{a}}_{2}\approx 0.3\), which corresponds to a symmetric and convex amplitude profile. The exceptions include

taxol, _ida5_ and _tpg1_, which have \({\bar{a}}_{1} < 0\) corresponding to decreasing amplitudes towards the distal tip, and Ca2+, which has \({\bar{a}}_{1} > 0\) corresponding to

increasing amplitude towards the distal tip. The _oda1_ waveforms have a small parabolicity \({\bar{a}}_{2}\) corresponding to their linear amplitude profiles. The _ida5_ waveforms have

smaller parabolicity and negative asymmetry. Thus, despite the robustness of the beat, both genetic and environmental perturbations led to variations in the waveforms. Parabolicity and

asymmetry, the polynomial terms that together contribute 80% of the variance (Fig. 4e), are also the main features of the data: they strongly correlate with the first two principal

components \({a}_{1}^{{{{\rm{pc}}}}}\) and \({a}_{2}^{{{{\rm{pc}}}}}\) (Fig. 4f, left boxes). Furthermore, the asymmetry of the amplitude profile, \({\bar{a}}_{1}\), is anti-correlated with

the parabolicity of the phase profile, _φ_2, whereas the parabolicity of the amplitude profile, \({\bar{a}}_{2}\), is correlated with the asymmetry of the phase profile, _φ_1 (Fig. 4f, right

boxes). Thus, the dominant shape features are the parabolicity and asymmetry of the amplitude and phase profiles. MOTOR RESPONSE CONTROLS WAVEFORM VARIABILITY The low-dimensionality of

shapes suggests that the perturbations, irrespective of their molecular mechanisms, affect only a few collective properties of the axoneme. To explore this possibility, we asked whether a

mechanical model of the ciliary beat, based on that in ref. 32, could account for the diversity of waveforms. The model, described in the Methods, contains four non-dimensional coefficients:

_β_′ and _β_″, which characterize the motors’ dependence on the instantaneous curvature (instantaneous response) and the rate of change of curvature (dynamic response), respectively; _k_,

which is the shear stiffness of the axoneme; and \(\overline{\mathrm{Ma}}\), which is a constant that comes from non-dimensionalizing Machin’s equation (see equation (4) in ref. 26).

\(\overline{\mathrm{Ma}}=2\uppi f{\xi }_{\mathrm{n}}{L}^{4}/\kappa\), where _κ_ is the flexural rigidity of the axoneme and _ξ_n is the normal friction coefficient. For _Chlamydomonas_,

\(\overline{\mathrm{Ma}} \approx 50\), using the parameters in the Methods. We fitted the mechanical model to the data (see example in Fig. 1e,f). The curve-fitting reduced the

dimensionality of the waveform description from 48, _a_(_s__i_) and _φ_(_s__i_) with _i_ = 1, …, 24 points along the length, to just 3, the model parameters (Fig. 5a). The agreement of the

model with the data was excellent, with 92% of the axonemes having _R_2 > 0.9 (Fig. 5b). Figure 5c shows that one of the parameters, the sliding stiffness _k_, is strongly correlated with

another parameter, the dynamic motor response _β_″. The observed relationship between these parameters can be recovered analytically by noting that the ratio of friction to bending forces,

which is proportional to the Machin number \({\mathrm{Ma}}\equiv \overline{\mathrm{Ma}}/{(2\uppi )}^{4}\), is ≪1. Taking the limit Ma → 0 gives the curved dashed line in Fig. 5c (Methods).

The good match between this ‘dry friction’ limit and the full theory supports the hypothesis that _Chlamydomonas_ operate in a low-friction regime (Discussion). Therefore, the waveform is

controlled by just the two motor response coefficients, which correlate with the asymmetry and parabolicity of the amplitude (Fig. 5c, inset). In agreement with the above argument, Fig. 5d

recapitulates the variability in waveform features observed in Fig. 4d. For example, the taxol and Ca2+ data lie far apart from each other in the (_β_′, _β_″) space, just as in the

\(({\bar{a}}_{1},{\bar{a}}_{2})\) space. Furthermore, because amplitude and phase are correlated, the motor response coefficients also correlate with the phase features (Fig. 5d, inset).

Parameters _β_′ and _β_″ not only control the asymmetry and parabolicity of the amplitude, respectively, but _β_″ also controls the wavelength. Specifically, in the limit Ma → 0, we can show

that the curvature wavelength is _λ_ = −4π_L_/_β_″, and therefore _λ_/_L_ = 1, which is typically observed, requires _β_″ = −4π, in agreement with the average values in Fig. 5c (dotted

line) and Fig. 5d. Thus, the space spanned by the motor response coefficients accurately recapitulates the space of amplitude and phase features, which suggests that all genetic and

environmental perturbations are buffered into changes of the curvature regulation of the motors. DISCUSSION THE DIMENSIONALITY OF THE CILIARY BEAT IS LOW By using a wide variety of

environmental and genetic perturbations, we have constructed the behavioural space of ciliary swimming waveforms. Asymmetry (\({\bar{a}}_{1}\) and _φ_1) and parabolicity (\({\bar{a}}_{2}\)

and _φ_2) of the amplitude and phase profiles suffice to describe about 80% of the variation between and within conditions. The dimensionality of swimming behaviour for isolated axonemes is

therefore very low: beat frequency, mean amplitude, shape asymmetry, shape parabolicity and axoneme length. Low-dimensional phenotypic spaces have been observed in other behaviours

(Introduction), but the range of conditions probed in our study makes the low-dimensionality even more remarkable. It remains open how the dimensionality of isolated swimming axonemes

relates to that of intact cells. THE SHAPE SPACE IS SPANNED BY THE CURVATURE SENSITIVITY OF THE DYNEINS The geometry of the shape space (the amplitude and phase profiles) accords with a

simple biomechanical model of ciliary beat, which is an extension of a previous model32. In this model, dynein motors respond to curvature with an instantaneous coefficient _β_′ and a

dynamic coefficient _β_″. The model also includes a passive stiffness to the sliding of doublets, _k_. Note that the typical value _k_ = 20 (non-dimensionalized from Fig. 5c), which

corresponds to 120 pN, is in good agreement with the measured value of 80 pN from ref. 46. The connection between the shape and biomechanical spaces enabled a remarkable finding: the

curvature coefficients _β_′ and _β_″ strongly correlate with the shape parameters, \({\bar{a}}_{1}\), \({\bar{a}}_{2}\), _φ_1 and _φ_2: $${\bar{a}}_{1} \sim \beta ^{\prime} \sim -{\varphi

}_{2}\quad {\mathrm{and}}\quad {\bar{a}}_{2} \sim \beta ^{\prime\prime} \sim +{\varphi }_{1}\propto \lambda /L\,,$$ (2) where ‘~’ denotes correlation. Thus, the different shapes correspond

to different curvature sensitivities. Wild-type cilia have symmetric and convex amplitude profiles, \({\bar{a}}_{1}\approx 0.0\) and \({\bar{a}}_{2}\approx 0.3\), corresponding to _β_′ = 0

and _β_″ ≈ −10. Using equation (2), we can ascribe deviations from this typical behaviour to changes in motor properties. For example, taxol, _tpg1_ and _ida5_ have \({\bar{a}}_{1} < 0\),

corresponding to a beat whose amplitude decays distally. Despite the different nature of these perturbations, we find that all three perturbations decrease the instantaneous response

coefficient, _β_′, and increase the dynamic response coefficient, _β_″. This means that the sensitivity to instantaneous curvature is increased, whereas the sensitivity to dynamic curvature

is decreased. In other words, the phase of the curvature response is altered. Conversely, Ca2+ leads to \({\bar{a}}_{1} > 0\), and so _β_′ > 0. The _oda1_ mutation has a flat profile,

with \({\bar{a}}_{2}\approx 0\) and \({\bar{a}}_{1}\approx 0\), and so _β_′ ≈ 0 and _β_″ ≈ −12. This counters the general belief that outer-arm dyneins affect only beat frequency47. The

_ida3_ mutation has a high parabolicity, that is, a larger dip in the middle, which reflects a less negative value of the dynamic curvature coefficient. The _mbo2_ mutation has a waveform

very similar to that of wild type, which is remarkable given the absence of a static mode in the waveform of this mutant37,41 but is consistent with the observation that the _mbo2_ mutations

do not affect dyneins47. Thus, changes in the instantaneous and dynamic curvature sensitivity of the dyneins can account for the effects of environmental and genetic perturbations, although

a causal connection between the perturbations and dynein activity has not been proven. Studying the relationship between shape and model parameters for alternative motor mechanisms (for

example, ‘geometric clutch’28 or compression instability29,48) may give further insight in this direction. ELASTICITY DOMINATES _CHLAMYDOMONAS_ SWIMMING Our results contain three arguments

that viscous forces are smaller than elastic forces for _Chlamydomonas_ axonemes. First, the Machin number, which is the ratio of viscous to elastic forces and is related to the Weissenberg

number49, is much smaller than unity (Ma ≪ 1). For a plane wave with wavenumber _q_ = 2π/_λ_, we have $${\mathrm{Ma}}=\frac{{\xi }_{\mathrm{n}}\omega }{\kappa {q}^{4}}=\frac{1}{{(2\uppi

)}^{4}}\overline{\mathrm{Ma}}\,,$$ (3) which ranges from 0.02 to 0.14 under all conditions. Note, however, that there is considerable uncertainty in _κ_, especially in the presence of

taxol50 (even if the flexural rigidity of the axoneme were reduced 10-fold by taxol, our models shows that there would be only a small effect on the shape). Second, the data collapse of the

fitted _k_ and _β_′′ in Fig. 5c almost coincides with that predicted by the low-viscosity relation (dashed line). This extends our earlier finding that a low Ma accords with wild-type and

_mbo2_ axonemes32. Third, despite the wide variability in frequency and mean amplitude, the total range is ~10-fold for both, there is an absence of high-amplitude and high-frequency beats.

The dissipation of elastic energy is proportional to \({a}_{0}^{2}f\) whereas the dissipation of viscous energy is proportional to \({a}_{0}^{2}{f}\,^{2}\) (ref. 26). Thus, if elastic or

viscous energy were limited by the energy available from the hydrolysis of ATP by the motors, then _a_0 ~ _f_ −1/2 or _a_0 ~ _f_ −1, respectively. The former provides a better bound on the

data than the latter (Fig. 3b dashed and dotted curves, respectively), as expected when elastic dissipation dominates. A puzzling finding is that the Machin number measured experimentally is

relatively insensitive to changing viscosity because increasing viscosity decreases the beat frequency, compensating for much of the expected change in Ma. This highlights our lack of

understanding of how the beat frequency is selected. The low-viscosity regime is also supported by the literature. In ref. 33, the ATPase rate of axonemes was shown to increase in proportion

to beat frequency, as predicted for elastic dissipation, and in ref. 51 measurements of the flow field around _Chlamydomonas_ cilia showed that friction forces are smaller than the

estimated bending forces. Furthermore, a low Ma may also explain the different relationship between curvature and sliding force reported in ref. 52 relative to that of our model32, as the

larger Ma of sperm may allow for a different motor control mechanism31. We conclude that _Chlamydomonas_ swims in the regime of low Ma, that is, low friction. Because the Reynolds number Re,

which is the ratio of inertial to viscous forces, is also small, we have Re ≪ Ma ≪ 1. PERSPECTIVE The functional robustness of the ciliary beat, evidenced by its persistence in the face of

genetic and environmental perturbations, indicates that the ciliary beat is a highly canalized process53. Such processes can tolerate genetic variations, pre-adaptations, which can be

selected for if large environmental changes occur54,55,56. This may underlie the diversity of axonemal forms. For example, the number of doublet microtubules in motile axonemes ranges from

as few as three57 to as many as hundreds58. Furthermore, radial spokes and the central pair are missing in some motile cilia, and additional asymmetrically localized molecules lead to planar

or asymmetric beat patterns59. Axonemes can be long or short, can be solitary or arrayed and can drive cell motility or fluid flow. This diversity of axonemal structure and function likely

originates, at least in part, in the strong tendency of motor-driven bending of cytoskeletal filaments to produce oscillations when the bending feeds back on the activity of the motors, as

we postulate for dynein. The propensity for oscillation likely underlies the successful reconstitution of cilia-like motility using heterologous motors and filaments33,34. METHODS

EXPERIMENTS PREPARATION AND REACTIVATION OF AXONEMES Axonemes from _Chlamydomonas reinhardtii_ cells (received from http://chlamy.org) were purified and reactivated. The procedures described

here are detailed in ref. 60. Chemicals were purchased from Sigma Aldrich unless stated otherwise. In brief, cells were grown in tris–acetate–phosphate medium with additional phosphate

(TAP+P medium) under conditions of continuous illumination (from two 75 W fluorescent bulbs) and of air bubbling at 24 °C over the course of 2 days, to a final density of 106 cells per

millilitre. Cilia were isolated using dibucaine, then purified on a 25% sucrose cushion and demembranated in HMDEK buffer (30 mM HEPES-KOH, 5 mM MgSO4, 1 mM dithiothreitol, 1 mM EGTA and 50 

mM potassium acetate, at pH 7.4) augmented with 1% (v/v) IGEPAL detergent and 0.2 mM Pefabloc SC protease inhibitor. The membrane-free axonemes were resuspended in HMDEK plus 1% (w/v)

polyethylene glycol (molecular weight 20 kDa), 30% sucrose and 0.2 mM Pefabloc, and were stored at −80 °C. Prior to reactivation, axonemes were thawed at room temperature, then kept on ice.

Thawed axonemes were used for up to 2 h. Reactivation was performed in flow chambers with a depth of 100 μm built from easy-clean glass and double-sided sticky tape. Thawed axonemes were

diluted in HMDEKP (HMDEK augmented with 1% (w/v) PEG 20.000) reactivation buffer. Unless stated otherwise, a standard reactivation buffer containing 1 mM ATP and an ATP-regeneration system

(5 units per millilitre creatine kinase and 6 mM creatine phosphate) was used to maintain a constant ATP concentration. The axoneme dilution was infused into a glass chamber, which was

blocked using casein solution (from bovine milk, 2 mg mL−1) for 10 min and then sealed with vacuum grease. Prior to imaging, the sample was equilibrated on the microscope for 5 min and data

were collected for a maximum time of 20 min. SPECIAL REACTIVATION CONDITIONS For the temperature series, the temperature was controlled using an objective heater from Bioptech. Unless stated

otherwise, the sample temperature was kept constant at 24 °C. The temperature series was acquired by increasing the temperature in 2 °C steps and letting the system equilibrate for 10 min.

After equilibration, the target temperature was checked using an inbuilt reference thermistor. For the ATP series, the standard buffer (without ATP) was augmented with different amounts of

ATP (50, 66, 100, 240, 370, 500, 750 and 1,000 μM). For the viscosity series, the standard buffer was augmented with Ficoll 400 (1%, 5% 10% (w/v)), and then axonemes were added to this

solution. For the calcium, we used a Ca2+ buffered reactivation solution with a concentration of free Ca2+ of 100 μM (calculated with the program Maxchelator

https://somapp.ucdmc.ucdavis.edu/pharmacology/bers/maxchelator). This concentration was chosen as it converts the asymmetric beat into a symmetric one61. For the taxol, the standard buffer

was augmented with 10 μM taxol, a concentration that stabilizes polymerized microtubules62, and then axonemes were added to this solution. IMAGING OF AXONEMES The reactivated axonemes were

imaged by phase-contrast microscopy, set up on an inverted Zeiss Axiovert S100 TV or Zeiss Observer Z1 microscope using a Zeiss 63× Plan-Apochromat NA 1.4 or a 40× Plan-Neofluar NA 1.3 Phase

3 oil lens in combination with a 1.6× tube lens and a Zeiss oil condenser (NA 1.4). Movies were acquired using an EoSens 3CL CMOS high-speed camera. The effective pixel size was 139 nm or

219 nm per pixel. Movies of up to 3,000 frames were recorded at a frame rate of 1,000 fps. DATA ANALYSIS HIGH-PRECISION TRACKING OF ISOLATED AXONEMES To track the shape of the axoneme in

each movie frame with nanometre precision, the MATLAB-based software tool FIESTA Version 1.03 was used35. Prior to tracking, movies were background-subtracted to remove static

inhomogeneities arising from uneven illumination and from dirt particles. The background image contained the mean intensity in each pixel calculated over the entire movie. This procedure

increased the signal-to-noise ratio by a factor of three60. Phase-contrast images were inverted. For tracking, a segment size of 733 nm (approximately 5 × 5 pixels) was used, corresponding

to program settings of a full width at half maximum of 750 nm and of a ‘reduced box size for tracking especially curved filaments’ of 30%. Along the arc length of each filament, 25 equally

spaced segments were fitted using two-dimensional Gaussian functions. FOURIER ANALYSIS OF TANGENT ANGLE Using the _x_, _y_ positions of the filament shape we calculated the tangent angle

_ψ_(_s_, _t_) at every arc-length position in time. To study only the dynamic modes of the waveform we subtracted the static mode (the 0th Fourier mode)37 and Fourier-decomposed the tangent

angle. The spectrum of dynamic modes of the tangent angle shows a peak at the fundamental frequency (Fig. 1d, first Fourier mode). Because the fundamental mode accounts for >90% of the

total power, we neglected the higher harmonics (_n_ = 2, 3, 4, …) in all further analysis (see ref. 32 for more detail) and considered only the first Fourier mode of _ψ_(_s_, _t_), which is

denoted as _ψ_(_s_) and will be used hereafter for all dynamic quantities. We calculated the frequency _f_ = 〈_f_(_s_)〉 and the arc-length-dependent amplitude _a_(_s_) = ∣_ψ_(_s_)∣ and phase

\(\varphi (s)=\arg \psi (s)\) profiles of the fundamental Fourier mode. POLYNOMIAL DECOMPOSITION We use shifted Legendre polynomials, _ℓ__n_(_s_), with normalization

\(\int\nolimits_{0}^{1}{\ell }_{n}(s){\ell }_{m}(s){\mathrm{d}}s=\frac{1}{2n+1}{\delta }_{mn}\), to characterize amplitude and phase profiles. Any function _g_(_s_) with support [0, 1] can

be written as _g_(_s_) = ∑_n_≥0 _g__n__ℓ__n_(_s_), with \({g}_{n}=(2n+1)\int\nolimits_{0}^{1}g(s){\ell }_{n}(s){\mathrm{d}}s\). With this definition,

\({g}_{0}=\int\nolimits_{0}^{1}g(s){\mathrm{d}}s\) is the mean over the arc length. THEORY MECHANICAL MODEL OF THE AXONEME The dynamics of the axoneme is characterized by a balance of

hydrodynamic, elastic and internal sliding forces26,63. The sliding forces can be active, exerted by motors, or passive, associated with structures such as nexin links and radial spokes in

addition to the motors. To linear order and in frequency space, the force balance equation, which we call the Machin equation, is given by $$i\overline{\mathrm{Ma}}\psi = -{\partial

}_{s}^{4}\psi +{\partial }_{s}^{2}{f}_{\mathrm{sl}}\,,$$ (4) where _f_sl is the dimensionless sliding force density and \(\overline{\mathrm{Ma}}=2\uppi f{\xi }_{\mathrm{n}}{L}^{4}/\kappa\)

is a dimensionless constant related to the Machin number by \(\overline{\mathrm{Ma}} = (2 \pi)^4{\rm{Ma}}\). A global balance of sliding forces also holds,

\(\int\nolimits_{0}^{1}{f}_{\mathrm{sl}}(s){\mathrm{d}}s={F}_{\mathrm{b}}\), where the basal force is given by \(F_{\mathrm{b}}=\chi_{\mathrm{b}}{\varDelta_{\mathrm{b}}}\) with

\(\varDelta_{\mathrm{b}}\) the basal sliding and \({\chi }_{\mathrm{b}}={\chi }_{\mathrm{b}}^{\prime} +i{\chi }_{\mathrm{b}}^{\prime\prime}\) the complex basal response coefficient with

\({\chi }_{\mathrm{b}}^{\prime},{\chi }_{\mathrm{b}}^{\prime\prime} > 0\). We take the sliding force density _f_sl(_s_) generated by motors to depend on the local curvature ∂_s__ψ_(_s_)

and sliding \(\varDelta(s)\) = \(\varDelta_{\mathrm{b}}\) + _ψ_(_s_) − _ψ_(0) of doublets32. In particular, $${f}_{\mathrm{sl}}=k{{\varDelta }}+\beta {\partial }_{s}\psi \,,$$ (5) where _k_ 

> 0 is a passive sliding stiffness and _β_ = _β_′ + _i__β_′′ is a complex curvature response coefficient. For _β_′′ < 0 there is forward (base-to-tip) wave propagation64. Note that,

unlike in ref. 64, we allow for instantaneous curvature response via _β_′, which is key for characterizing amplitude asymmetry (in ref. 32, _β_′ = 0 was sufficient because only the symmetric

wild-type and _mbo2_ axonemes were studied). The model does not speak to the mechanism of curvature sensing, which must be very sensitive given that axonemal curvatures are low. Such

curvature sensitivity might be realized by strains that distort the cross-section of the axoneme, for example due to the ‘geometric clutch’ model28 or to the coupling between twist and

bending64. As boundary conditions we use torque and force balances: ∂_s__ψ_(0) = _F_b, ∂_s__ψ_(1) = 0, \({\partial }_{s}^{2}\psi (0)={f}_{\mathrm{sl}}(0)\), and \({\partial }_{s}^{2}\psi

(1)={f}_{\mathrm{sl}}(1)\). Equations (4) and (5) correspond to the solution of a non-linear problem near the point of instability63. Note that we have not considered the role of

three-dimensional components to the beat, for example via twist64. MODEL FITTING In previous works, we have used the Machin equation and its variants to reproduce the observed beating

patterns of cilia31,64. For a given value of \(\overline{\mathrm{Ma}}\), estimated from experiments, and values for the sliding response coefficients _k_, _β_′ and _β_′′, there exists a

discrete set of solutions to the system of equations posed by equations (4) and (5), the global sliding balance and the boundary conditions36,63. We keep the solution with the lowest

wavenumber, because for simple dynamical non-linear motor models it has been shown to be the first one to be excited (see Section 5.2.2 in ref. 65), so that for a given set of motor response

parameters we can determine a unique theoretical waveform, \(\{k,\beta ^{\prime} ,\beta ^{\prime\prime} \}\to {\psi }_{\mathrm{the}}(s)\). The procedure to fit the model to the data has

been extensively described in ref. 36. In brief, we define a score between the theoretically predicted waveform and the experimentally observed one, _R_2(_ψ_the, _ψ_exp), so that _R_2 = 1

corresponds to a perfect matching of theory and experiments36,64. We then maximize this score with respect to the three parameters {_k_, _β_′, _β_′′} using a principal axis algorithm with

three different initial points: {3π2, 0, −4π}, which corresponds to the plane wave approximation, and two perturbations around it, {1.7, −2.1, −5.8} and {3π2, 0.15, −9.6}. In addition, to

calculate \(\overline{\mathrm{Ma}}\), we used _κ_ = 580 pN μm2 and _ξ_n = 0.0049 pN s μm−2 (ref. 64). There is a considerable range of values of these parameters in the literature. For

example, in ref. 66 the normal friction coefficient was estimated to be _ξ_n = 0.0015 pN s μm−2 based on measurements in intact cells, and in ref. 46 a bending modulus of _κ_ = 800 pN μm2

was obtained using optical tweezers. Because we are in the low-Machin-number regime, our analysis is quite insensitive to changes in these two parameters, which was confirmed by increasing

and decreasing the Machin number 10-fold. In the Ficoll datasets we adjusted _ξ_n following our own measurements by factors of 1.1, 1.6 and 3.2 for dilutions of 1%, 5% and 10%, respectively.

LOW FRICTION LIMIT Although equations (4) and (5) are linear, the boundary conditions result in complicated relationships between the exponential scales in the solution and the motor

parameters36. We found that such complications are greatly reduced in the limit \(\overline{\mathrm{Ma}}\to 0\). It is particularly convenient to work in terms of curvature ∂_s__ψ_, which

can be shown to take the form $${\partial }_{s}\psi ={\mathrm{e}}^{cs}({\mathrm{e}}^{cd(s-1)}-{\mathrm{e}}^{-cd(s-1)})\,,$$ (6) with _c_ = _β_/2 and \(d=\sqrt{1+4k/{\beta }^{2}}\) where the

sliding stiffness, _k_, is now allowed to be complex, with the imaginary component corresponding to sliding friction. The general condition for the existence of a solution in this limit is

simply $$\tanh (cd)=-\frac{{\chi }_{\mathrm{b}}cd}{{\chi }_{\mathrm{b}}c+k }\,,$$ (7) As the observed values for the basal stiffness arising from the fits are small, we explore the limit

_χ_b → 0. In this limit equation (7) becomes tanh(_cd_) = 0, which implies _c__d_ = _i__n_π, with _n_ = 1, 2, … From this we obtain $${\partial }_{s}\psi =\exp \left(\beta ^{\prime}

s/2\right)\sin \left(n\uppi (s-1)\right)\exp \left(i\beta ^{\prime\prime} s/2\right)\,,$$ (8) with the sliding stiffness and curvature response coefficients related by \(\beta

\sqrt{1+4k/{\beta }^{2}}=i2\uppi n\). Note that, in the case that _k_ is real, we have _β_′ = 0. The dashed line in Fig. 5 corresponds to the previous equation when _n_ = 1, which provides

good agreement with the fits for the full model (finite viscosity, non-zero basal compliance and non-zero asymmetry). Note that in equation (8) the amplitude asymmetry is directly related to

_β_′, with _β_′ < 0 corresponding to decreasing amplitude. Furthermore, the wavelength can be directly read off from this expression, and is given by _λ_ = −4π_L_/_β_′′. This is also in

agreement with observations from fitting the full model. DATA AVAILABILITY The data shown in the main figures are appended to this article in csv format. In addition, all tracked waveforms

are available on Dryad (https://doi.org/10.5061/dryad.0gb5mkm2j). CHANGE HISTORY * _ 23 FEBRUARY 2022 A Correction to this paper has been published:

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reinhardtii_. _Biophys. J._ 100, 2716–2725 (2011). Article  ADS  Google Scholar  Download references ACKNOWLEDGEMENTS V.F.G., P.S. and J.H. acknowledge support from the Max Planck Society

during the early phases of this work. J.H. would also like to thank Yale University for start-up funds used to support this project. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * B

CUBE–Center for Molecular Bioengineering, Technische Universität Dresden, Dresden, Germany Veikko F. Geyer * Department of Molecular Biophysics and Biochemistry, Yale University, New Haven,

CT, USA Jonathon Howard * Instituto Gulbenkian de Ciência, Oeiras, Portugal Pablo Sartori Authors * Veikko F. Geyer View author publications You can also search for this author inPubMed 

Google Scholar * Jonathon Howard View author publications You can also search for this author inPubMed Google Scholar * Pablo Sartori View author publications You can also search for this

author inPubMed Google Scholar CONTRIBUTIONS V.F.G. performed the experiments, P.S. and V.F.G. carried out data analysis, P.S. was responsible for modelling, P.S., V.F.G and J.H.

conceptualized the work, and wrote the paper together. CORRESPONDING AUTHORS Correspondence to Veikko F. Geyer, Jonathon Howard or Pablo Sartori. ETHICS DECLARATIONS COMPETING INTERESTS The

authors have no competing interests. PEER REVIEW PEER REVIEW INFORMATION _Nature Physics_ thanks Philip Bayly and Kirsty Wan for their contribution to the peer review of this work.

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SUPPLEMENTARY INFORMATION Supplementary Fig. 1, Table 1 and description of data file. SUPPLEMENTARY DATA Post-processed waveform data. The first row contains descriptors of each column,

whereas the subsequent 498 rows correspond to the axonemes analysed in this work. More details on the structure of this file are found in the Supplementary Information. RIGHTS AND

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Howard, J. & Sartori, P. Ciliary beating patterns map onto a low-dimensional behavioural space. _Nat. Phys._ 18, 332–337 (2022). https://doi.org/10.1038/s41567-021-01446-2 Download

citation * Received: 28 May 2021 * Accepted: 01 November 2021 * Published: 10 January 2022 * Issue Date: March 2022 * DOI: https://doi.org/10.1038/s41567-021-01446-2 SHARE THIS ARTICLE

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