Human sperm steer with second harmonics of the flagellar beat

Sperm are propelled by bending waves traveling along their flagellum. For steering in gradients of sensory cues, sperm adjust the flagellar waveform. Symmetric and asymmetric waveforms result in straight and curved swimming paths, respectively. Two mechanisms causing spatially asymmetric waveforms have been proposed: an average flagellar curvature and buckling. We image flagella of human sperm tethered with the head to a surface. The waveform is characterized by a fundamental beat frequency and its second harmonic. The superposition of harmonics breaks the beat symmetry temporally rather than spatially. As a result, sperm rotate around the tethering point. The rotation velocity is determined by the second-harmonic amplitude and phase. Stimulation with the female sex hormone progesterone enhances the second-harmonic contribution and, thereby, modulates sperm rotation. Higher beat frequency components exist in other flagellated cells; therefore, this steering mechanism might be widespread and could inspire the design of synthetic microswimmers.

Many microorganisms and cells are propelled by motile flagella or cilia, i.e. hair-like protrusions that extend from the cell surface [1][2][3] .The beat patterns of flagella vary among cells.
The green algae Chlamydomonas reinhardtii navigates by a breaststroke-like movement of a pair of flagella that alternate between in-phase and anti-phase states of the beat 4 .The bacterium Escherichia coli is propelled by the rotary movement of a helical bundle of flagella; when the rotary direction reverts, the flagellar bundle disentangles and the cell randomly adopts a new swimming direction 5 .Most animal sperm swim by means of bending waves travelling from the head to the tip of the flagellum.Near a surface, sperm swim on a curvilinear path 6,7 , which is thought to result from the spatial asymmetry of bending waves.Two different mechanisms have been proposed that could generate a flagellar asymmetry: a dynamic buckling instability resulting from flagellar compression by internal forces 8,9 or an average intrinsic curvature 6,10,11 .
The flagellum not only propels sperm, but also serves as antenna that integrates sensory cues as diverse as chemoattractant molecules, fluid flow, or temperature that modify the flagellar beat pattern.Gradients of such chemical or physical cues guide sperm to the egg 12,13 .The sensory stimulation gives rise to changes in intracellular Ca 2+ concentration ([Ca 2+ ]i) that modulates the flagellar beat and, thereby, swimming direction [13][14][15][16][17] .However, it is not known whether an intrinsic flagellar curvature, or a buckling instability, or some other mechanism is used for steering and how sensory cues modulate any of these mechanisms.
Here we study the flagellar beat pattern of tethered human sperm by high-speed, high-precision video microscopy and by theoretical and computational analysis.We show that the beat pattern is characterized by a superposition of two bending waveswith a fundamental frequency and its second harmonictravelling down the flagellum.The second harmonic breaks the symmetry of the overall waveform by a temporal rather than by a spatial mechanism.The sexual hormone progesterone, which evokes a Ca 2+ influx, enhances the second-harmonic contribution and changes the rotation velocity of the cell.Our analysis suggests a novel temporal steering mechanism of sperm and uniflagellated eukaryotic microswimmers in general.

Results
Two harmonics shape the flagellar beat.In a narrow recording chamber, we monitored the flagellar beat of human sperm (Fig. 1a).While swimming near a surface, sperm undergo a rolling motion.We prevent rolling by tethering sperm with their head to the recording chamber.
Near a surface, the beat pattern is almost planar and parallel to the surface 8 , which facilitates tracking and imaging of the flagellar motion.Sperm revolve around the tethered head with a rotation velocity Ω that varied smoothly over time between 0 and 0.5 Hz (Fig. 1a,b).The flagellar shape, extracted by image processing, was characterized by the local curvature C(s,t) at time t and arclength s along the flagellum.The curvature profile shows the well-known bending wave propagating along the flagellum from the mid-piece to the tip (Fig. 1c).However, at a fixed arclength position, the curvature deviates in time from a perfect sinusoidal wave; instead, the curvature displays an asymmetric sawtooth-like profile in time, suggesting that multiple beat frequencies contribute to the overall waveform (Fig. 1d).Fourier analysis of the beat pattern reveals a fundamental beat frequency ωo of approximately 20 Hz, but also higher-frequency components, mainly the second harmonic (Fig. 1e).
Principal-component analysis 18 provides further support for the presence of a second harmonic (Fig. 2).We decomposed the curvature profile C(s,t) into normal modes Γn (s) (Fig. 2a): where n(t) is the amplitude of the n-th mode (Fig. 2b,c).The curvature is sufficiently well described using the first two modes (Fig. 2a), that account for about 90% of the signal (35 cells; Methods and Supplementary Fig. 2).The beat wavelength varied among sperm.However, after rescaling the arclength by the beat wavelength, the first two modes from different cells can be superimposed (Fig. 2a).Thus, the superposition of two eigenmodes recapitulates fairly well the beat pattern of human sperm.
However, the measured probability density displays two regions with higher probabilities at the "north" and "south" poles of the beat cycle, indicating the presence of a second harmonic (Fig. 2d).The average phase velocity for a given phase, ⟨∂t|⟩, reveals that the frequency smoothly oscillates between approximately 20 Ηz and 40 Hz during each beat cycle (Fig. 2e).
Thus, the beat pattern is indeed characterized by a fundamental frequency and its second harmonic.A second-harmonic mode breaks the mirror symmetry temporally.A planar beat with a single frequency becomes its mirror image after half a beat period /2 (i.e.C(s,t) = -C(s,t + /2)).Sperm using such a mirror-symmetric flagellar beat would swim on a straight path, and tethered sperm would not rotate.The second harmonic breaks this mirror symmetry; consequently, curved swimming paths and tethered-sperm rotation becomes possible.We examined theoretically how this broken symmetry generates the torque that drives rotation.
With the "small-amplitude approximation" 2,21 , the waveform of a flagellum oriented on average parallel to the x-axis can be described by a superposition of first and second harmonics: where k is the wave vector,  is the phase shift between the two modes, and ωo is the fundamental frequency.Note that equation ( 2 The hydrodynamic drag on the flagellum is anisotropic, i.e. the drag coefficients in the perpendicular (⊥) and tangential directions (∥) are not equal.Each point along the flagellum is subjected to the drag force f(x,t) = -⊥v⊥-∥v∥, where v(x,t) = (0, ∂ty) is the velocity of the filament at time t and position x 2,7,22,23 .The net perpendicular force, averaged over one beat cycle is (see Supplementary Notes) In the presence of a second harmonic (y2 ≠ 0), the force fy, integrated over the whole flagellum, does not vanish.The rotation velocity around the tethering point is obtained by torque balance: the torque generated around the tethering point equals the viscous torque.For comparison with experiments, it is more useful to describe the waveform y(x,t) in terms of local curvature C(s,t), with amplitudes C1 and C2 instead of y1 and y2 in equation ( 2).
The general result for  (Supplementary Notes) depends on wavelength λ and flagellum length L; for  → L, a simple relation is obtained: Equation ( 4) illustrates that rotation results from the superposition of the first and the second harmonics coupled to the anisotropic drag (equation ( 2)).Note that  depends both on the amplitude C2 of the second harmonic and on the phase shift ϕ between the two modes.We refer to C2 sin(ϕ) as the "second-harmonic intensity".
Second-harmonic intensity and rotation velocity are correlated.The experimental rotation velocity slowly varies with time (Fig. 1b), providing the means to test the predictions from equation ( 4).We determined the phase (ϕ) and amplitude (C2) of the second harmonic from the spectrogram of the flagellar curvature (Methods and Supplementary Fig. 3) and compared the second-harmonic intensity with the rotation velocity Ω (Fig. 3a).For each cell (n = 35), the correlation coefficient R between the normalized rotation velocity Ω(t)/ωo and second harmonic intensity C2(t)sin(ϕ(t)) was calculated by time averaging over the course of the experiment.To account for the approximations introduced for the derivation of equation ( 4), the phase ϕ(t) is corrected by a constant phase shift ϕo to yield ϕeff(t) = ϕ(t) + ϕo.The constant shift is chosen such as to maximize the correlation coefficient R. We find that the secondharmonic intensity and the rotation velocity are highly correlated (Fig. 3a-c) (R = 0.91 ± 0.13).
Alternatively, an average intrinsic curvature (Co) of the flagellum might contribute to the rotation.An intrinsic curvature, which can generate an asymmetric beat, has been observed for some cilia and flagella 6,10,11 .The small-curvature calculation (Methods and Supplementary Notes) predicts that, for equal magnitudes of Co and C2, both mechanisms contribute equally to the rotation frequency.However, the average intrinsic curvature of the flagellum is usually much smaller than the amplitude of the second harmonic (|Co|/|C2|= 0.13; Supplementary Notes and Supplementary Fig. 5).Therefore, we conclude that the second-harmonic contribution dominates.Accordingly, we find that the correlation of Co with the rotation frequency is weak (R = 0.13 ± 0.65; n = 35).However, sometimes the average curvature, second harmonic intensity, and rotation velocity display a similar time course (Fig. 3a).In summary, these results support the hypothesis that human sperm steer with the second harmonic.channel 24,25 .The ensuing change in the flagellar beat pattern has been proposed to underlie hyperactivated motility and chemotaxis [26][27][28][29] .We used progesterone stimulation to examine whether Ca 2+ modulates the second-harmonic contribution.Sperm were imaged before and after photo-release of progesterone from a caged derivative (Fig. 3d) 14 .In Fig. 3e and f, we compare the beat pattern during 0.5 s before and after the release of progesterone.Although progesterone slowed down the beat frequency of human sperm (Fig. 3e), the rotation around the tethering point was enhanced (Fig. 3f).A direct comparison of the second-harmonic contribution before and after the release (Fig. 3c,g) demonstrates that progesterone modulates the second-harmonic intensity and, thereby, the rotation velocity; moreover, both measures are highly correlated (Fig. 3b,c,f,g).The strong second-harmonic component might thus represent the mechanism of hyperactivated beating of human sperm upon progesterone stimulation.
An active elastic-filament model predicts that beating with two harmonics produces an intrinsic flagellar curvature.Beyond a purely geometric description of the shape, we study by simulation the elasticity, forces, and the power generated or dissipated during a flagellar beat.A sperm cell is modeled as a tethered, actively beating filament of bending rigidity κ; hydrodynamic interactions are taken into account via anisotropic drag.The filament is driven by active bending torques T(s, t), assuming a superposition of two traveling waves, T(s,t) = T1sin(ks -ot) + T2sin(ks -2ot + ψ). ( Due to hydrodynamic boundary effects, the phase shift ψ of the torque can be different from the phase shift ϕ of the flagellar curvature in equation ( 4).All parameters in equation ( 5) and the bending rigidity κ were derived by fitting to experimental data, including flagellar waveform, rotation velocity, and normal modes (Supplementary Notes).The simulation, which reproduces the beat pattern reasonably well (Fig. 4a, Supplementary Movie 1), provides several insights.First, constant torque amplitudes T1 and T2 along the flagellar arclength suffice to account for the experimental beat shapes, including the very high curvature of the end-piece.
Thus, no structural inhomogeneity or differential motor activity along the flagellum is needed to account for this peculiarity of the flagellar beat shape.Second, although the bending forces are mirror-symmetric with respect to the filament displacement, a small average curvature is generated by the superposition of two harmonics that breaks the mirror symmetry of the beat waveform in both time (second harmonic) and in space (average curvature) (Fig. 4b,d).Third, the simulations confirm two predictions from equation ( 4): The rotation velocity scales linearly with T2 (Fig. 4c,d), and scales with the sine of the phase ψ (Fig. 4d, Supplementary Fig. 7).Fourth, for wavelength λ < L, the rotation velocity is largely independent of the wavelength; however, for longer wavelengths, the rotation velocity decreases (Fig. 4c).Finally, simulations of freely swimming sperm in 2D show that the curvature of the swimming path is controlled by the phase ψ (Fig. 4e), i.e. sperm could navigate by adjusting the phase ψ between the two harmonics.Energy consumption and dissipation.Several aspects regarding the energetics of motile cilia and flagella have been studied, including traveling waves, power-and-recovery stroke, and metachronal waves 19,[30][31][32][33][34][35][36] .For propulsion, not all beating gaits offer the same efficiency of energy consumption [37][38][39] .In fact, the flagellar beat pattern can be predicted from optimal swimming efficiency 38 .However, quantitative estimates of how power is used for bending and how power is dissipated along the flagellum of microswimmers are lacking [40][41][42] .
Our simulations provide insight into the energetics of beating.Comparison of experimental results with simulations is only possible for power Pd dissipated due to drag forces.The dissipated power Pd (Supplementary Notes) increases along the flagellum within about 10 µm from the head, stays roughly constant for 25 µm along the entire principal piece, and then steeply rises towards the end piece, where the flagellum moves faster (Fig. 4f).The experimental recordings, which are restricted to a flagellar section between 7 and 35 µm from the head center, agree reasonably well with the simulations (Fig. 4f).From the simulations, we can also estimate the power Pg generated by the instantaneous local torques described by equation ( 5) (Supplementary Notes).The generated power Pg increases steadily, becomes maximal at about 30 µm down the flagellum, and decreases again towards the tip region.The distribution of generated and dissipated power differ along the flagellum: During the steady increase of Pg to its maximum, the dissipated power Pd stays almost constant; then Pg quickly drops, whereas Pd steeply increases thereafter (Fig. 4f).We conclude that the effects of local torques add up in order to generate large beating amplitudes and velocities in the tip region; by contrast amplitudes and velocities are smaller in the mid-piece region due to the tethering constraint and the head drag.Thus, although bending forces in eukaryotic flagella are locally generated along the length of the axoneme, power dissipation due to fluid drag is not equally distributed, yet relocated towards the tip region of the flagellum.

Discussion
Several mechanisms have been proposed that can produce asymmetric waveforms of the ciliary or flagellar beat.Symmetry breaking can emerge from structural features such as the central apparatus of the axoneme, or elastic filaments, or dynein motors that vary along the circumference or the long axis of the axoneme [43][44][45] .Here we identify a novel mechanism of symmetry breaking that is dynamic rather than static: two travelling waves of fundamental and second-harmonic frequency determine the beat asymmetry by their phase relation and the relative amplitude of each wave.In principle, this mechanism does not require a spatial or structural asymmetry that gives rise to an intrinsic curvature of the flagellum.Furthermore, simulations using homogeneous constant torque amplitudes along the flagellar arclength suffice to account for the experimental beat shapes (equation ( 5), Fig. 4a, Supplementary Movie 1).Hydrodynamic simulations suggest that intrinsic curvature of the midpiece affects swimming path curvature 10 .However, an intrinsic curvature and a dynamic component produced by the second-harmonic mechanism are not mutually exclusive.In fact, Fourier analysis of beat waveforms from different sperm species and Chlamydomonas 11 reveals a zero component or intrinsic curvature component 6 and at least two other components: a principal and a second harmonic 6,8 .However, in human sperm, the average curvature is small (Fig. 3a,c, Supplementary Fig. 5) and modulation of intrinsic curvature was not favored as a steering mechanism during rheotaxis 8 .
Alternatively, buckling instabilities have been proposed to produce asymmetric beating 8,9 .
These instabilities are enhanced at higher shear forces and flagellar compression 9 .However, progesterone stimulation slows down the beat frequency considerably, whereas the secondharmonic intensity and the rotation frequency are enhanced (Fig. 3e-g).These experiments, therefore, argue against dynamic buckling instabilities underlying mirror-symmetry breaking in human sperm.
Simulation of the flagellar beat shows that the second-harmonic intensity can control the swimming path of freely moving sperm and the rotation velocity of tethered sperm (Fig. 4e, Supplementary Movie 2).Furthermore, we find that progesterone-evoked Ca 2+ influx enhances the relative contribution of the second harmonic to the overall beat.Thus, the dynamics of principal and second-harmonic travelling waves could steer sperm across gradients of sensory cues that modulate the Ca 2+ concentration.
The mechanisms underlying the second-harmonic and its modulation by Ca 2+ are not known.
However, dynein arms behave as endogenous oscillators that slide microtubules with a frequency set by the ATP concentration 46 .Thus, principal and second harmonics could be inherent properties of different dynein motors.Consistent with this idea, it has been shown that axonemal models from sea urchin sperm and flagella from Chlamydomonas mutants lacking the outer dynein arms beat at about half the frequency, indicating that inner and outer dynein arms could be tuned to produce different beat frequencies 47 .Furthermore, isolated Chlamydomonas flagella that were reactivated with varying ATP concentrations display beat amplitudes with two peak resonances at 30 and 60 Hz 48 , and higher harmonics have been suggested to control steering during phototaxis of Chlamydomonas 49 .Future studies need to address the molecular mechanisms by which a second-harmonic mode is created and tuned for sperm steering.

Sperm preparation
Samples of human semen were from healthy donors with their consent.Sperm were purified by a "swim-up" procedure in human tubal fluid containing (in mM): 97.

Sperm motility
Single sperm cells were imaged in custom-made observation chambers of 150 µm depth.To gently tether the head of sperm cells to the glass surface, the HSA concentration in the buffer was reduced to 1 µg/ml, resulting in a large fraction of cells tethered to the surface with the head, but the flagellum was freely beating.The flagellar beat was recorded under an inverted microscope (IX71; Olympus) equipped with a dark-field condenser, a 20x objective (UPLANFL; NA 0.5), and additional 1.6x magnification lenses (32x final magnification).The temperature of the microscope was adjusted to 37 °C using an incubator (Life Imaging Services).Illumination was achieved using a red LED (M660L3-C1; Thorlabs), and a custommade power supply.Images were collected at 500 frames per second using a high-speed CMOS camera (Dimax HD; PCO).For release of progesterone from its caged derivative (1 µM) 14 a brief flash (100 ms) of UV light was used (365-nm LED; M365L2-C; Thorlabs).UV light reached the sample through the backport of the microscope and a 380 nm long-pass dichroic filter (380 DCLP; Chroma).Tracking of the flagellum was achieved with custom-made programs written in MATLAB (Mathworks).The program identified the best threshold for binarization of the image by iteratively reducing the threshold until the expected cell area and coarse flagellar length in the image was achieved.This was followed by a skeleton operation to identify the flagellum.The position of the head was determined by fitting an ellipse around the tethering point.

Rotation velocity
The rotation velocity Ω(t) is obtained as the time derivative of the angle  between the x-axis and the vector connecting the head tethering point with the first tracked flagellar point.The angle  was filtered with a Gaussian of width 1 s to remove oscillations due to the fast beat.

Flagellar curvature
An arclength s sampled every Δs = 0.9 µm from head to tip, is assigned to the tracked flagellum.
Because the number of tracked flagellar points can differ from frame to frame (compare panels a and b in Supplementary Fig. 1), we analyse only the part of the flagellum that has been tracked for all frames.The curvature C(s,t) is computed as the inverse of the radius of the circle that connects three contiguous points of the tracked flagellum, and is positive (negative) for counter-clockwise (clockwise) bends.

Principal-component analysis
The curvature C(s,t) is a matrix of about 30 (in arclength) by 10,000 (in time) entries (Supplementary Fig. 1).We reduce the dimensionality of the dataset by principal-component analysis 18  describes fairly well the experimental data (compare Supplementary Fig. 1a and 1b), and was used for further analysis.
To identify whether the same modes underlie all observed beat patterns (see Fig. 2), we rotate and mirror the first two eigenmodes, 1 Γ and 2 Γ , of each experiment to maximize similarity with a reference pair of modes, 1   and 2 .
  Because arclength is measured in wavelength units Δ Δ / , ss  modes are interpolated to correct for the different tracking point density in the scaled representation.To avoid potential artefacts due to rotation and mirroring of modes, we use untransformed modes for other analyses.

Curvature spectrogram
We perform a discrete Fourier transform of the curvature C(t, s = so) every 30 frames in the time window (to -W/2,to + W/2) at fixed arclength so and time window width W = 250 frames.
W is a compromise between time and frequency resolution.The peak at the fundamental frequency ωo is clearly identified (Supplementary Fig. 3).The second-harmonic amplitude C2 and phase  are measured at 2ωo.

Correlation coefficient R
The correlation coefficient R between rotation velocity and second-harmonic intensity is defined as where   t  is obtained directly by Fast-Fourier Transform of C(t, s = so) (see Fig. 1c).The phase offset o is chosen such as to maximize R. Note that this is just one constant offset for each experiment with about 1,000 data points.We test whether the value of o is independent of so by comparing R25, estimated from the curvature at 25 µm, with R15, estimated from the curvature at 15 µm, using the phase estimated for R25.The results are virtually identical (Supplementary Fig. 4).The correlation coefficient is centred around the values This choice agrees with the expectation that in the absence of a second harmonic the rotation velocity is zero.
In particular, for ϵ = 1 and 2/k → L The torque Ta generated by the second harmonic is balanced by the torque generated by the perpendicular viscous drag Tv.In line with the small-amplitude approximation, we estimate the viscous torque as the torque acting on a straight rod that is tethered at one end and rotates with angular velocity Ω,   Torque balance (Tv + Ta = 0) finally yields Small-curvature approximation Second, we consider a description of the flagellar shape in an expansion of small local curvature.This has the advantage that larger perpendicular deviation amplitudes can be included.The flagellar curvature C(s,t) is written as: where C1, and C2 are the curvature amplitudes of the two harmonics and  is the secondharmonic phase.
In the limit of L  and for ò = 1, this simplifies to     Comparison of second harmonic and average curvature Supplementary Fig. 5 shows the probability distribution of the ratio |Co|/|C2|.The secondharmonic contribution is always larger than that of the average curvature; the mean ratio is 0.13 for unstimulated and 0.16 for stimulated sperm.Because the ratio is much smaller than unity, the second harmonic dominates sperm rotation.

Trajectory curvature
The second harmonic generates a rotation of sperm around its tethering point.The rotation velocity  depends on the flagellar curvature amplitudes C1 and C2, the phase ϕ, and the difference between drag coefficients, ⊥-∥.We estimate how these parameters affect the trajectory of freely swimming sperm.
The time needed for sperm to complete a rotation around its center of rotation is T = 2π / where  is approximately given by equation (S8) for tethered sperm.Assuming that freely swimming sperm have the same center of rotation, they move during the same time along a trajectory of length vT, with velocity v = fx / ⊥.Because  << ω, we can ignore the fast wiggling motion due to the beat.

Figure 1 |
Figure 1 | The flagellar beat pattern of human sperm displays a second-harmonic component.(a) Four snapshots of a tethered human sperm that rotates clockwise around the tethering point with rotation velocity (t).Each color corresponds to a different snapshot taken at the indicated time.White and grey lines below the blue snapshot show the tracked flagellum at consecutive frames acquired at 2 ms intervals.(b) Rotation velocity of the cell around the tethering point.The grey area indicates the time interval that is further analyzed in panels c and d.(c) Kymograph of the flagellar curvature during approximately 10 beat cycles (0.5 s).The curvature corresponding to the two horizontal lines is plotted in panel d.(d) Curvature of the flagellum at segments located at 15 m (blue) and 25 m (red) from the head.The curvature displays a sawtooth-like profile.(e) Power spectrum of the curvature at 15 m and 25 m.The fundamental frequency is o = 20 Hz.

Figure 2 |
Figure 2 | Principal-component analysis of the flagellar beat.(a) Superposition of the two main normal modes of the flagellar beat for n = 35 human sperm cells.Each trace corresponds to a different cell.Although each cell has a different set of eigenmodes, when rescaled by the wavelength, the individual modes collapse onto a common curve (solid blue and red lines; see Methods).The wavelength appears longer than the flagellum, because it is traced only partially.(b) Time evolution of the mode amplitudes 1 and 2 for a representative recording.Of note, 2 lags behind 1 by a phase of /2.The symbols (•,■,▲,★) indicate the modes shown in panel c, and correspond to the beat phase depicted in panel d.(c) Illustration of the composition of the beat by principal modes.The peak of the wave travels from left to right.(d) Histogram of the joint probability P(1,2), averaged for 1.5 s from 26 different cells.During a full beat cycle, the phase = arctan(2/1) varies between 0 and 2π.The symbols indicate the phase of the amplitudes in panel b.(e) Beat frequency at a fixed phase for each cell as () = ⟨∂t|⟩.The grey stripes highlight the phases of minimal and maximal frequency (20 Hz and 40 Hz, respectively).The black line indicates the median.The standard deviation of ( is nearly constant (right panel).

Figure 3 |
Figure 3 | Second-harmonic intensity correlates with rotation velocity and is enhanced by progesterone.(a) Normalized rotation velocity (blue line), second-harmonic intensity (red line), and average curvature (green line) for a representative sperm cell.(b) Histogram of the correlation R(Ω/o, C2 sin(eff)).Red bars refer to unstimulated human sperm, blue bars to sperm stimulated with progesterone.We never observed anti-correlation (R<0).(c) Normalized rotation velocity (blue line) and second-harmonic intensity (red line), and average curvature (green line) 2.0 s before and after the release of progesterone with a flash of UV light (at t = 0).(d) Stroboscopic views of a sperm cell before (left) and after (right) stimulation with progesterone.Flagellar snapshots were recorded at t = 4 ms (left) and t = 6 ms (right) intervals.(e-g) Cell-by-cell comparison of the beat frequency (e), the rotation frequency (f), and the second-harmonic intensity (g), before and after progesterone release.Average values during 0.5 s before and after the stimulus.Points inside the colored areas correspond to an increase after the stimulation.Error bars are s.d.

Figure 4 |
Figure 4 | Simulations reproduce the beat and steering dynamics.(a) Stroboscopic view of experimental (red) and simulated (blue) beat pattern using an active semi-flexible filament and anisotropic drag force.Time interval between snapshots (fading lines) is t = 2 ms.Simulation parameters: κ = 1.9 nN μm 2 , T1 ~ 0.65 nN μm, T2 = 0.15T1, ψ = 2.26, ωo = 30 Hz, L = 41 μm, ξ⊥/ξ∥ = 1.81, ξ∥ = 0.69 fNs/μm 2 and λ/L = 0.65 (Supplementary Movie 1).(b) Representative simulation of flagellar beat with a second-harmonic amplitude T2 = 0.3T1.The mid-piece is aligned for visualization.The time interval between snapshots is t = 4 ms.The red thick line shows the non-symmetric trajectory of the flagellum tip.(c) Rotation velocity  versus normalized wavelength λ.Note that has been normalized to the second-harmonic torque amplitude.The inset shows that  scales linearly with T* = T2/T1.(d) Average curvature <C> versus phase ψ of the second-harmonic torque.Note that the curvature has been normalized by T* = T2/T1.(e) Simulated sperm trajectory resulting from a slowly changing phase ψ over time (phase indicated by the color of the trajectory).By modulating the phase, sperm swim on curvilinear paths (Supplementary Movie 2).(f) Average dissipated power Pd (blue) versus generated power Pg (red) in simulations, and average dissipated power measured in experiments (grey lines).The simulated dissipated power shows good agreement with the experimental results.Of note, power is relocated along the flagellum.
first term in the numerator is usually much larger than the second term.From torque balance and assuming a viscous torque as in equation (S7), we approximate 3 a 3/ TL     by the expression given in equation (4) in the Main Text.Similarly, the average torque generated by an average flagellar curvature Co is obtained ) and (S15) together with torque balance demonstrate that the rotation frequency is linear in both the second-harmonic amplitude C2 and mean curvature Co.Furthermore, for similar values of |Co| and |C2| the two terms contribute about equally.
to filter out white noise.The normal modes Γn of the curvature are the eigenvectors is the local angle between the flagellar tangent and the xaxis.Here, we assume that sperm are clamped at their head such that o(t) = 0.By If a net force is generated, it has to arise from the second term, which is proportional to the friction anisotropy .The active torque around the tethering point is then obtained (to leading order in ϵ) to be Given the flagellar curvature C(s,t) at time t, its spatial coordinates r(s,t) are || .      