Two harmonics shape the flagellar beat

We monitored the flagellar beat of human sperm in a shallow recording chamber (150 µm depth) filled with an aqueous buffer solution (viscosity ~0.7 mPa s at 37 °C; Fig. 1a). While swimming near a surface at low viscosities (<20 mPa·s), human sperm undergo a rolling motion8,31. Rolling occurs despite the fact that the flagellar beat is almost planar8. We prevent cell rolling by tethering sperm with their head to the recording chamber. Under these conditions, the beat plane remains parallel to the surface, 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 ω o of about 20 Hz, but also higher-frequency components, mainly the second harmonic (Fig. 1e).

Fig. 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 Ω(t). Each color corresponds to a different snapshot taken at the indicated time. White and gray lines below the blue snapshot show the tracked flagellum at consecutive frames acquired at 2 ms intervals. Scale bar represents 5 µm. b Rotation velocity of the cell around the tethering point. The gray area indicates the time interval that is further analyzed in c, 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 d. d Curvature of the flagellum at segments located at 15 μm (blue) and 25 μm (red) from the head. The curvature displays a sawtooth-like profile. e Power spectrum of the curvature at 15 μm and 25 μm. The fundamental frequency is ω o = 20 Hz Full size image

Principal component analysis32 provides further support for the presence of a second harmonic (Fig. 2). We decomposed the curvature profile C(s,t) into normal modes Γ n (s) (Fig. 2a):

$$C\left( {s,t} \right) = \mathop {\sum}\limits_{\mathrm{n}} {{\mathrm{\chi }}_{\mathrm{n}}(t){\rm{\Gamma}} _{\mathrm{n}}(s)} ,$$ (1)

Fig. 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 λ, 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 χ 1 and χ 2 for a representative recording. Of note, χ 2 lags behind χ 1 by a phase of π/2. The symbols (blue circles, red squares, blue triangles, red stars) indicate the modes shown in panel c, and correspond to the beat phase depicted in 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(χ 1 ,χ 2 ), averaged for 1.5 s from 26 different cells. During a full beat cycle, the phase α = arctan(χ 2 /χ 1 ) varies between 0 and 2π. The symbols indicate the phase of the amplitudes in b. e Beat frequency at a fixed phase for each cell as ω(α) = ⟨∂ t α|α⟩. The gray 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 ω(α) is nearly constant (right panel) Full size image

where χ 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 section 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.

The mode amplitudes χ 1,2 (t) encode the time dependence of the flagellar shape. The probability P(χ 1, χ 2 ) of observing a particular combination of mode amplitudes χ 1 and χ 2 adopts a typical limit-cycle shape (Fig. 2d)32,33,34. For a single-frequency cycle, the probability density would adopt an isotropic ring-like distribution, characterized by (χ 1 ,χ 2 ) = a(cos(ω o t), sin(ω o t)). 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 α|α⟩, reveals that the frequency smoothly oscillates between about 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 τ/2 (i.e. C(s,t) = −C(s,t + τ/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”2,35, the waveform of a flagellum oriented on average parallel to the x-axis can be described by a superposition of first and second harmonics:

$$y\left( {x,t} \right) = y_1{\mathrm{sin}}\left( {kx - {{\omega}}_{\rm o}t} \right) + y_2{\mathrm{sin}}\left( {kx - 2{ {\omega}}_{\rm o}t + {\rm{\phi}} } \right),$$ (2)

where k is the wave vector, ϕ is the phase shift between the two modes, and ω o is the fundamental frequency. Note that Eq. (2) can be rewritten as y(x,t) = Y(t)sin(kx−ω o t + Φ(t)), where the amplitude Y and phase Φ are functions of time. At any instant in time t o , the shape y(x,t o ) is still a sine wave, and is thus mirror symmetric in space. Therefore, no average flagellar curvature is produced. By contrast, because the amplitude Y(t) is time-modulated, at any given point x o the temporal dependence of y(x o ,t) is asymmetric in time. Such an asymmetric beat pattern allows steering, because the hydrodynamic drag forces during the two halves of the beat cycle do not cancel. For amplitude ratios y 2 /y 1 ≲0.3, y(x o ,t) resembles an asymmetric sawtooth-like profile as in Fig. 2b.

The hydrodynamic drag on the flagellum is anisotropic, i.e. the drag coefficients in the perpendicular (ξ ⊥ ) and tangential directions (ξ ∥ ) are not equal. Each point along the flagellum is subjected to the drag force f(x,t) = −ξ ⊥ v ⊥ −ξ ∥ v ∥ , where v(x,t) = (0, ∂ t y) is the velocity of the filament at time t and position x 2,7,36,37. The net perpendicular force, averaged over one beat cycle is (see Supplementary Note)

$$f_{\mathrm{y}}\left( x \right) = {\mathrm{\omega }}_{\mathrm{o}}k^2\left( {{{{\rm{\xi}} }}_ \bot - {{{\rm{\xi}} }}_{||}} \right)y_1^2y_2{\mathrm{cos}}\left( {kx - {\rm{\phi}} } \right)$$ (3)

In the presence of a second harmonic (y 2 ≠ 0), the force f y , integrated over the whole flagellum, does not vanish. The rotation velocity Ω around the tethering point is obtained by torque balance: the torque generated around the tethering point \(T_{\mathrm{a}} \approx {\int}_0^L \,{xf_{\mathrm{y}}} \left( x \right)\cdot {\mathrm{d}}x\) equals the viscous torque. For comparison with experiments, it is more useful to describe the waveform in terms of local curvature C(s,t), with amplitudes C 1 and C 2 instead of y(x,t) with amplitudes y 1 and y 2 in Eq. (2). In addition, a small-curvature approximation is more accurate for larger beat amplitudes. The general result for Ω (Supplementary Note) depends on wavelength λ and flagellum length L; for λ → L, a simple relation is obtained:

$${{\Omega}} = - {{\omega}} _{\mathrm{o}}\frac{{3L^3}}{{4\left( {2{\rm{\pi}} } \right)^4}}\frac{{{\rm{\xi}} _ \bot - {\rm{\xi}} _{||}}}{{{\rm{\xi}} _ \bot }}C_1^2C_2\sin {\rm{\phi}} .$$ (4)

Eq. (4) illustrates that rotation results from the superposition of the first and the second harmonics coupled to the anisotropic drag in a similar way as Eq. (3). Note that Ω depends both on the amplitude C 2 of the second harmonic and on the phase shift ϕ between the two modes. We refer to C 2 sin(ϕ) as the “second-harmonic intensity”.

Second-harmonic intensity and rotation velocity correlate

The experimentally observed rotation velocity slowly varies with time (Fig. 1b), providing the means to test the predictions from Eq. (4). We determined the phase (ϕ) and amplitude (C 2 ) 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)/ω o and second harmonic intensity C 2 (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 Eq. (4), the phase ϕ(t) is corrected by a constant phase shift ϕ o to yield ϕ eff (t) = ϕ(t) + ϕ o . The constant shift is chosen such as to maximize the correlation coefficient R. We find that the second-harmonic intensity and the rotation velocity are highly correlated (Fig. 3a–c) (R = 0.91 ± 0.13, mean ± SD).

Fig. 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/ω o , C 2 sin(ϕ 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 Δt = 4 ms (left) and Δt = 6 ms (right) intervals. Scale bar represents 10 µm. 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 SD Full size image

Alternatively, an average intrinsic curvature (C o ) of the flagellum might contribute to the rotation. An intrinsic curvature, which can generate an asymmetric beat, has been observed for some cilia and flagella6,10,11. The small-curvature calculation (Methods and Supplementary Note) predicts that

$${{\Omega}} = {{\omega}} _{\mathrm{o}}\frac{{L^3}}{{\left( {2{\rm{\pi}} } \right)^6}}\frac{{{\rm{\xi}} _ \bot - {\rm{\xi}} _{||}}}{{{\rm{\xi}} _ \bot }}C_1^2C_0 \cdot {\rm{\pi}} \left( {{\rm{\pi}} ^2 - 3} \right).$$ (5)

Thus, for equal magnitudes of C o and C 2 , both mechanisms contribute alike to the rotation frequency (compare Eqs. (4) and (5)). However, the average intrinsic curvature of the flagellum is usually much smaller than the amplitude of the second harmonic (|C o |/|C 2 | = 0.13; Supplementary Note and Supplementary Fig. 5). Therefore, we conclude that the second-harmonic contribution to rotation velocity dominates. Accordingly, we find that the correlation of C o with the rotation frequency is weak (R = 0.13 ± 0.65; n = 35, mean ± SD). 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.

Changes in [Ca2+] control second-harmonic intensity

The flagellar beat of sperm and the steering response is controlled by changes in [Ca2+] i 38. The female sex hormone progesterone evokes robust Ca2+ entry into human sperm by activating the CatSper Ca2+ channel39,40. The ensuing change in the flagellar beat pattern has been proposed to underlie hyperactivated motility and chemotaxis13,41,42,43. We used progesterone stimulation to examine whether Ca2+ modulates the second-harmonic contribution. Sperm were imaged before and after photo-release of progesterone from a caged derivative (Fig. 3d)27. In Fig. 3e, 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.

Sperm can navigate by adjusting the phase between harmonics

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 an actively beating filament of bending rigidity κ; local hydrodynamic interactions resulting from the dynamic coupling between different portions of the flagellum via the induced fluid flow are taken into account by anisotropic drag6,35,37. The filament is driven by active bending torques T(s, t), assuming a superposition of two traveling waves,

$$T\left( {s,t} \right) = T_1\sin \left( {ks - {{\omega}}_{\rm o}t} \right) + T_2\sin \left( {ks - 2{{\omega}} _{\rm o}t + {\rm{\psi}} } \right).$$ (6)

Due to hydrodynamic and boundary effects, the phase shift ψ of the torque can be different from the phase shift ϕ of the flagellar curvature in Eq. (4). All parameters in Eq. (6) and the bending rigidity κ were derived by fitting to experimental data, including flagellar waveform, rotation velocity, and normal modes (Supplementary Note). The simulation, which reproduces the beat pattern reasonably well (Fig. 4a, Supplementary Movie 1), provides several insights. First, constant torque amplitudes T 1 and T 2 along the flagellar arclength suffice to account for the experimentally observed 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, simulations confirm two predictions from the small-curvature approximation Eq. (4): The rotation velocity Ω scales linearly both with T 2 (Fig. 4c, d) and 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.

Fig. 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 Δt = 2 ms. Simulation parameters: κ = 1.9 nN μm2, T 1 ~ 0.65 nN μm, T 2 = 0.15T 1 , ψ = 2.26, ω o = 30 Hz, L = 41 μm, ξ ⊥ /ξ ∥ = 1.81, ξ ∥ = 0.69 fNs μm−2 and λ/L = 0.65 (Supplementary Movie 1). Scale bar represents 10 µm. b Representative simulation of flagellar beat with a second-harmonic amplitude T 2 = 0.3T 1 . The mid-piece is aligned for visualization. The time interval between snapshots is Δt = 4 ms. The red thick line shows the non-symmetric trajectory of the flagellum tip. Scale bar represents 10 µm. c Rotation velocity Ω vs. normalized wavelength λ. Note that Ω has been normalized to the second-harmonic torque amplitude. The inset shows that Ω scales linearly with T* = T 2 /T 1 . d Average curvature < C > vs. phase ψ of the second-harmonic torque. Note that the curvature has been normalized by T* = T 2 /T 1 . 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). Scale bar represents 20 µm. f Average dissipated power P d (blue) vs. generated power P g (red) in simulations, and average dissipated power measured in experiments (gray lines). The simulated dissipated power shows good agreement with the experimental results. Of note, power is relocated along the flagellum Full size image

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 waves33,44,45,46,47,48,49. For propulsion, not all beating gaits offer the same efficiency of energy consumption50,51,52. In fact, the flagellar beat pattern can be predicted from optimal swimming efficiency51. However, quantitative estimates of how power is used for bending and how power is dissipated along the flagellum of microswimmers are lacking53,54,55.