The shape of the flagellar beat determines the path along which a sperm cell swims. If the flagellum bends periodically about a curved mean shape then the sperm will follow a path with non-zero curvature. To test a simple hydrodynamic theory of flagellar propulsion known as resistive force theory, we conducted high-precision measurements of the head and flagellum motions during circular swimming of bull spermatozoa near a surface. We found that the fine structure of sperm swimming represented by the rapid wiggling of the sperm head around an averaged path is, to high accuracy, accounted for by resistive force theory and results from balancing forces and torques generated by the beating flagellum. We determined the anisotropy ratio between the normal and tangential hydrodynamic friction coefficients of the flagellum to be 1.81±0.07 (mean±s.d.). On time scales longer than the flagellar beat cycle, sperm cells followed circular paths of non-zero curvature. Our data show that path curvature is approximately equal to twice the average curvature of the flagellum, consistent with quantitative predictions of resistive force theory. Hence, this theory accurately predicts the complex trajectories of sperm cells from the detailed shape of their flagellar beat across different time scales.
Sperm cells are propelled in a liquid by regular bending waves of a whip-like cell appendage called the flagellum (Gray, 1955). Flagellar propulsion results in complex trajectories of sperm cells. On short time scales, the sperm head undergoes a wiggling motion with the same frequency as the flagellar beat. This wiggling of the sperm head is a consequence of balancing the forces and torques generated by the beating flagellum and characterizes the ‘fine structure’ of sperm swimming. On a time scale longer than the period of the flagellar beat, sperm cells of many species swim along circular or helical paths (Rikmenspoel et al., 1960; Goldstein, 1977; Brokaw, 1979; Crenshaw, 1996; Corkidi et al., 2008). The non-zero curvature of their paths is a consequence of asymmetric flagellar waves, and plays a vital role in sperm chemotaxis (Miller, 1985; Kaupp et al., 2008; Friedrich and Jülicher, 2007).
How the observed complex swimming paths of sperm cells and other microswimmers emerge from their swimming strokes is an important question of long-standing interest (Gray and Hancock, 1955; Rikmenspoel, 1965; Brokaw, 1970; Yundt et al., 1975; Smith et al., 2009). In pioneering work, Taylor demonstrated that self-propulsion is possible due to purely viscous forces (Taylor, 1951). Gray and Hancock introduced a local hydrodynamic theory of flagellar propulsion that neglects long-range hydrodynamic interactions and focuses on anisotropic local hydrodynamic friction between the sperm surface and the adjacent fluid (Gray and Hancock, 1955). This theory is commonly known as resistive force theory. The net swimming speed predicted by this theory depends strongly on the anisotropy ratio of flagellar friction coefficients. The precise value of this key parameter has been subject to debate (Gray and Hancock, 1955; Brokaw, 1970; Cox, 1970; Shack et al., 1974; Lighthill, 1976; Brennen and Winet, 1977; Johnson and Brokaw, 1979). The theory of Gray and Hannock was later refined by Lighthill using slender-body approximations for the thin flagellum to include long-range hydrodynamic interactions (Lighthill, 1976). Other groups proposed even more advanced hydrodynamic simulation schemes (Dresdner and Katz, 1981; Elgeti and Gompper, 2008; Smith et al., 2009). For swimming in the vicinity of a solid surface, resistive force theory provides a simple and concise theoretical approach to flagellar propulsion. It has been used in several studies to account for experimental data (Gray and Hancock, 1955; Rikmenspoel et al., 1960; Brokaw, 1970; Yundt et al., 1975; Keller and Rubinow, 1976).
In the present work, we used theory and experiment to address how the swimming path of a sperm cell is determined by the shape of its flagellar bending waves. To test the resistive force theory of flagellar propulsion, we accurately measured the fine structure of the oscillatory movements of the sperm head. This approach is novel and depends crucially on the precision of the tracking data [see Yundt et al. for an early attempt (Yundt et al., 1975)]. To facilitate sperm tracking, we made use of the fact that sperm cells become hydrodynamically trapped near a planar boundary surface (Woolley, 2003): there they swim in a plane parallel to the surface with an approximately planar flagellar beat, allowing one to confine the analysis to two spatial dimensions. Using tracked flagellar beat patterns, we could accurately reconstruct instantaneous velocities of sperm swimming using resistive force theory. From our analysis, we determined the drag anisotropy ratio.
On time scales longer than the period of the flagellar beat, sperm trajectories near a boundary surface are circular (Rikmenspoel et al., 1960; Goldstein, 1977; Brokaw, 1979; Woolley, 2003; Kaupp et al., 2008; Riedel-Kruse et al., 2007). The non-zero curvature of these swimming paths was shown to correlate with an asymmetry of the flagellar beat pattern in its plane of beating (Rikmenspoel et al., 1960; Goldstein, 1977; Brokaw, 1979). We re-investigated the relationship between mean flagellar curvature (characterizing flagellar beat asymmetry) and the resulting curvature of the sperm swimming path and found a linear dependence between the two curvatures, with a factor of proportionality significantly larger than one. We demonstrate that this counter-intuitive result is due to the non-linear nature of flagellar propulsion and can be understood in the framework of resistive force theory as a result of the finite amplitude of the flagellar bending waves and the hydrodynamic friction of the sperm head.
MATERIALS AND METHODS
Tangent angle representation of planar flagellar beat patterns
For sperm cells swimming close to a planar boundary surface, almost planar beat patterns were observed with a plane of flagellar beating approximately parallel to the boundary surface. In our analysis, we neglected any out-of-plane component of the flagellar beat, and considered the two-dimensional projection of the flagellar shape on the plane of swimming. We describe the (projected) shape of the bent flagellum at a given time t by the position vector r(s,t) of points along the centreline of the flagellum, where s is the arc length along the centreline (Fig. 1). We express r(s,t) with respect to the material frame of the sperm head: let r(t) be the position vector of the centre of the sperm head and e1(t) a unit vector parallel to the long axis of the sperm head. Additionally, we define a second unit vector e2(t), which is obtained by rotating e1 in the swimming plane by an angle of π/2 in a counter-clockwise fashion. With this notation, r+r1e1 corresponds to the proximal tip of the sperm head while r–r1e1 corresponds to the proximal end of the flagellum; here 2r1≈10 μm is the length of the head along its long axis e1. The shape of the flagellar centreline r(s,t) at time t is characterized by a tangent angle ψ(s,t) for each arc length position s, 0≤s≤L, where L is the length of the flagellum. The tangent angle measures the angle enclosed by the vector e1(t) and the tangent vector to the flagellar centreline at r(s,t) (see Fig. 1). Note that the derivative of ψ(s,t) with respect to arc length s is the local curvature of the flagellar centreline. For a regular flagellar beat pattern, the tangent angle ψ(s,t) is a periodic function of t with period T=2π/ω, where ω is the angular frequency of the flagellar beat. From the tangent angle ψ(s,t), the position r(t) of the sperm head and its material frame defined by e1(t) and e2(t), we can reconstruct the full flagellar beat pattern as: (1) In our experiments with bull sperm, the tangent angle of the flagellar wave was well described by its zeroth and the first Fourier mode (Riedel-Kruse et al., 2007): (2) where denotes the complex conjugate of . Higher modes contribute less than 5% to the power spectrum of the tangent angle at all arc-length positions s. The zeroth mode characterizes a time-averaged mean shape of the flagellum; whereas, the amplitude of the first mode, ||, gives the amplitude of the principal bending wave of the flagellum.
Fig. 2 shows the zeroth and first Fourier mode as a function of arc length s along the flagellum for two representative sperm cells (observed at different fluid viscosities). From the Fourier decomposition, we obtain three parameters that characterize key features of the shape of the flagellar beat. First, the mean flagellar curvature K0 is defined by fitting a line K0s to the zeroth mode (s); K0 provides a simple measure for the asymmetry of the mean shape of the flagellum. Note that the ideal case (s)=K0s corresponds to a mean shape of the flagellum that is curved in a perfect arc with constant curvature K0. Second, the amplitude parameter A0 is defined by fitting the line A0s to the absolute value of the first mode |(s)|. The ideal case ||=A0s corresponds to a bending amplitude that increases linearly along the flagellum. Third, the wavelength λ of the principal flagellar bending wave is defined by fitting the line 2πs/λ to the phase angle φ(s)=−arg(s) of the (complex) first Fourier mode. The ideal case φ(s)=2πs/λ corresponds to a travelling wave with uniform wavelength λ and wave speed λω/(2π).
Instantaneous versus effective swimming speeds
For planar swimming, the head of a sperm cell can move parallel and perpendicular to its long axis, as well as rotate around an axis normal to the plane of swimming. Thus, planar sperm swimming is characterized by three degrees of freedom. We characterize the translational motion of the sperm head with respect to the head's material frame introduced in Fig. 1 by time-dependent velocity components v1(t) and v2(t) such that: (3) Here dots denote time derivatives. Rotation of the sperm head is characterized by an instantaneous angular speed Ω(t) such that: (4)
Effective net speeds
In experiments with bull sperm cells, the centre of the sperm head moved along a complex trajectory r(t) that wiggled around an averaged path (t) (Gray, 1955; Rikmenspoel, 1965) (Fig. 3). The averaged path (t) describes the effective net motion of the whole sperm cell on a coarse-grained time scale, which averages over several beat cycles. This effective motion is characterized by effective translational and rotational speeds, =Δs/Δt and =ΔΦ/Δt, respectively. Here the time interval Δt=nT comprises several beat cycles, whereas Δs measures the distance travelled along the path and ΔΦ is the net rotation of the sperm head. It should be emphasized that is not simply the time average of the instantaneous parallel velocity v1(t), but has to be determined from the averaged swimming path (t). The curvature of the averaged path is given by κ=1/r0= / where r0 is the radius of the circular path . In the literature, v=|| is sometimes referred to as curvilinear velocity (VCL) and =|| as velocity along the averaged path (VAP).
Reconstructing instantaneous velocities from flagellar beat patterns
The swimming of sperm cells is characterized by low Reynolds numbers implying that inertial forces are negligible (Purcell, 1977; Landau and Lifshitz, 1987). We compute instantaneous swimming velocities from recorded flagellar beat patterns in the limit of zero Reynolds number using the resistive force theory introduced by Gray and Hancock (Gray and Hancock, 1955). This local hydrodynamic theory neglects long-range hydrodynamic interactions and assumes that the hydrodynamic drag force density f(s) that acts on a cylindrical portion of the filament at arc length s is linear in the local velocity components v∥(s,t)=[(s,t)·t(s,t)]t(s,t) and v⊥(s,t)=(s,t)–v∥(s,t) parallel and perpendicular to the filament centreline, respectively: (5) Here the dot denotes differentiation with respect to time t and t(s,t) is the tangent vector of the flagellar centreline at position r(s,t). Note that the proximal tip of the flagellum r(s=0,t) moves in synchrony with the head centre, i.e. v∥(s=0,t)=v1(t)e1(t) and v⊥(s=0,t)=[v2(t)+r2Ω(t)]e2(t), as the sperm head is assumed to be rigid. One can envisage the approach of resistive force theory by approximating the bent filament as a sequence of straight rods connected at their ends and then computing the drag force density for the individual rods.
Force and torque balance
Since no external forces are acting on a freely swimming sperm cell (Gray and Hancock, 1955; Jülicher and Prost, 2009), the total hydrodynamic drag force F acting on the swimming sperm cell at time t must vanish: (6) Here Fhead is the hydrodynamic drag force of the head. Similarly, the total torque M acting on the sperm cell must be zero as well: (7) where Mhead is the hydrodynamic torque acting on the pivoting head. For planar swimming in a plane spanned by the vectors e1(t) and e2(t), we obtain three independent scalar equations F(t)·e1(t)=0, F(t)·e2(t)=0 and M(t)·[e1(t)×e2(t)]=0. Using these three equations, we can compute the three instantaneous velocities v1(t), v2(t) and Ω(t) of the sperm head at any instance t in time provided the tangent angle ψ(s,t) and its time derivative (s,t) are known. These predicted velocities will then be compared with directly measured instantaneous velocities (Fig. 4).
Drag force of the head
To describe sperm swimming accurately, the hydrodynamic drag force Fhead and the torque Mhead of the moving head must be considered (Johnson and Brokaw, 1979). In our numeric calculations, we approximate the shape of the sperm head as a spheroid and use Perrin's formulas to express the hydrodynamic drag force and torque as Fhead=ξ1v1e1+ξ2v2e2 and Mhead=ξrotΩe1×e2 (Perrin, 1934). We use 2r1=10 μm for the length of the head along the long axis e1 and 5 μm for the length along the short axis e2 and obtain ξ1≈40.3 pN s mm−1, ξ2≈46.1 pN s mm−1 and ξrot≈0.84 pN μm s. These friction coefficients correspond to motion far from any boundary surface and thus only serve as a reference. For the case studied here, the proximity of the boundary surface is likely to increase the friction coefficients. Note that within the framework of resistive force theory only the ratios of friction coefficients play a role in determining swimming velocities.
High-speed videoscopy of swimming bull sperm
Sperm cells were obtained as frozen samples (IFN Schönow, Germany) and prepared as described previously (Riedel-Kruse et al., 2007). Swimming of sperm cells was studied in aqueous solution of viscosity η≈0.7 mPa s at 36°C in a shallow observation chamber of 1 mm depth using phase-contrast microscopy (Axiovert 200M, Zeiss, Jena, Germany). Sperm swimming paths were recorded with a high-speed camera (FastCam, Photron, San Diego, CA, USA) at a rate of 250 frames s−1 for a duration of 4 s in each case. Movies were analysed using custom-made Matlab routines (The MathWorks, Inc., Natick, MA, USA). For each frame, the position and orientation of the sperm head, as well as the tangent angle ψ(s,t) at each tail point (relative to the orientation of the head) were computed. The precision of the automated tracking of the flagellum was of the order of 0.1 pixels, corresponding to 70 nm. The position of the elongated sperm head could be determined in the direction parallel to its long axis with a precision of the order of 0.5 pixels, corresponding to 350 nm (Riedel-Kruse et al., 2007). In a final step of data processing, the tracking data for the individual frames were smoothly interpolated in time; see the black trajectory in Fig. 3B.
In a second series of experiments, we studied sperm swimming at an increased viscosity of η≈10 mPa s in an aqueous solution of Ficoll 400 (Sigma, #F-4375; St Louis, MO, USA). Viscosities were measured with a viscometer (Brookfield, Model DV-I +; Lorch, Germany) at a temperature of 36°C. Aqueous solutions of the highly branched polymer Ficoll 400 approximately behave as Newtonian fluids (Hunt et al., 1994).
Determining friction coefficients by comparing instantaneous velocities
Using high-precision tracking data for the position and orientation of the sperm head of swimming bull sperm cells, the instantaneous translational and rotational velocity components v1(t), v2(t) and Ω(t) as defined in Eqns 3 and 4 were measured. An example of the resulting time series of instantaneous velocities is shown in Fig. 4. In the same experiments, the flagellar beat pattern was recorded. We used these data to reconstruct the instantaneous velocities using a simple local hydrodynamics theory as specified above (‘Reconstructing instantaneous velocities from flagellar beat patterns’). By a global least-squares fit of directly measured and reconstructed time series, we were able to determine the normal and the tangential friction coefficients of the flagellum, ξ⊥ and ξ∥, respectively. The results are summarized in Table 1; the means ± s.d. are ξ∥=0.69±0.62 fN s μm−2 and ξ⊥/ξ∥=1.81±0.07.
Within the framework of resistive force theory, only the ratios of friction coefficients play a role in determining velocities. Therefore, the absolute values for ξ∥ presented in Table 1 are determined relative to the friction coefficients ξ1 and ξ2 of the head only, estimated above (‘Reconstructing instantaneous velocities from flagellar beat patterns’). The boundary surface increases the hydrodynamic friction of the sperm head, the increase being larger the closer the head is to the surface. Because we neglected the boundary in estimating the friction coefficients for the head, we are underestimating the friction coefficients of the flagellum. It has been reported that the distances between sperm cells swimming near a boundary surface and the surface itself are broadly distributed (with mean and standard deviation of the order of 10 μm) (Rothschild, 1963; Winet et al., 1984). Such a variable distance to the surface could account for the observed variability in the fit results for the coefficients of the flagellum.
Swimming at increased viscosity
In the case of high viscosity, η≈10 mPa s, we observed flagellar beat patterns that were different from the case of normal viscosity η≈0.7 mPa s studied above. Both the frequency of the flagellar beat and the wavelength of the flagellar bending waves were reduced, resulting in lower values for the net speed of translational motion (see Table 2). Qualitatively similar results had been obtained earlier for invertebrate spermatozoa (Brokaw, 1966). Additionally, we report on the mean flagellar curvature K0 as well as the net angular speed and find that these quantities are also reduced in the case of high viscosity.
Instantaneous velocities were reduced by a factor of about 10 in the case of high viscosity compared with the case of normal viscosity. This resulted in larger relative errors of the velocity data, and data quality was not sufficient to reliably compare time series of instantaneous velocities and determine friction coefficients. In the limit of zero Reynolds number, theory predicts that all friction coefficients scale linearly with viscosity.
Instantaneous swimming velocities oscillate with the frequency of the flagellar beat
Our tracking experiments with bull sperm cells swimming near a planar boundary surface reveal the fine structure of their swimming: the centre of the sperm head followed an intricate trajectory r(t) that wiggles around an averaged path (t) (see Fig. 3). With respect to its own material frame, the motion of the sperm head is characterized by three time-dependent velocities: translational velocities v1 and v2 parallel and perpendicular to the long axis of the sperm head, and a rotational velocity Ω describing rotations in the plane of swimming. Fig. 4 shows these (instantaneous) swimming velocities for the motion of the sperm head (black curves). All three swimming velocities oscillate with the frequency ω of the flagellar beat as is reflected by the corresponding power spectra on the right of Fig. 4 (blue graphs).
Theory predicts a fundamental difference between the velocity component v1 and the other two components v2 and Ω, as only the last two change their sign when the flagellar beat pattern is reflected along the long axis e1 of the sperm head. Based on this symmetry argument, we infer that in the limit of small tangent angles, v1 should be a superposition of a non-zero average value and oscillatory modes with frequencies ω and 2ω (see Eqn A3 in the Appendix). Note that the occurrence of oscillations with twice the flagellar frequency is a result of the non-linear nature of flagellar propulsion and is not due to higher modes of the flagellar oscillations for the case considered here. We find confirmation of these theoretical predictions in our experimental data: in Fig. 4, v1 indeed varies around a non-zero average value. Also, in the power spectrum Sv1 of v1, the power of the Fourier peak at frequency 2ω amounts to a considerable fraction ρv1≈15% of the power of the Fourier peak at frequency ω: oscillations with the beat frequency and twice the beat frequency superimpose. This feature was even more pronounced in the case of increased fluid viscosity with ρv1=107±54% (mean±s.d., N=6) compared with the case of normal viscosity with ρv1=22±29% (N=7). Analogously defined power ratios for the velocity components v2 and Ω amount only to a few per cent.
Experimental determination of the drag anisotropy ratio
We used the recorded beat pattern to predict instantaneous swimming velocities using a simple local hydrodynamics theory (resistive force theory) as detailed above (‘Reconstructing instantaneous velocities from flagellar beat patterns’). By adjusting the ratio between the normal and the tangential flagellar friction coefficient, ξ⊥ and ξ∥, we obtained good agreement between the predicted velocities and the measured ones (compare black and red curves in Fig. 4, and see Table 1). Remarkably, for 6 out of 7 sperm trajectories analysed, the drag anisotropy ratio ξ⊥/ξ∥ determined by the fit fell into a rather narrow range ξ⊥/ξ∥≈1.81±0.07 (mean±s.d., N=6).
The success of resistive force theory in predicting instantaneous swimming velocities is impressive, taking into account the fact that this theory neglects long-range hydrodynamic interactions between different parts of the moving flagellum. In our experiments, the long-range hydrodynamic interactions are partially screened by the proximity of the boundary surface of the observation chamber, which is much less than the wavelength of flagellar bending waves (Rothschild, 1963; Winet et al., 1984). Therefore, it is still possible that long-range hydrodynamic interactions play a significant role in determining sperm swimming paths in open water far from surfaces. Note also, that more sophisticated hydrodynamic theories are needed to explain why sperm cells become trapped near boundary surfaces in the first place (Elgeti and Gompper, 2008; Smith et al., 2009).
The precise value of the drag anisotropy ratio has been subject to debate (Gray and Hancock, 1955; Brokaw, 1970; Cox, 1970; Shack et al., 1974; Lighthill, 1976; Brennen and Winet, 1977; Johnson and Brokaw, 1979). In their original work, Gray and Hancock used the value ξ⊥/ξ∥=2, which is valid for a flagellum far from any boundary surface in the limit of a vanishing diameter of the flagellum. A drag anisotropy ratio of 2 also applies for a slender cylinder of finite thickness that moves parallel to a surface (Hunt et al., 1994). For flagella of finite thickness far from a surface, various approximations provided values of the drag anisotropy ratio in the range ξ⊥/ξ∥=1.5–1.8 (Cox, 1970; Shack et al., 1974; Lighthill, 1976; Brennen and Winet, 1977); when applied to the flagellar parameters of bull sperm cells, the most accurate approximations (Shack et al., 1974; Lighthill, 1976) give ξ⊥/ξ∥=1.77. An early attempt to experimentally determine the drag anisotropy ratio from the net angular speed for swimming near a boundary surface yielded ξ⊥/ξ∥≈1.8 (no error bars given) (Brokaw, 1970). Using a t-test, we calculate the 95% confidence interval from our six measurements of the anisotropy ratio to be [1.73,1.88]. Thus, our result is consistent with the earlier experimental value and also close to the value of 1.77 estimated previously for swimming far from surfaces.
How curved swimming paths arise from asymmetric beat patterns
The recorded flagellar beat pattern of bull sperm cells exhibits a pronounced asymmetry in the plane of beating, with a mean shape of the flagellum that has non-zero curvature (see Fig. 2A). To characterize flagellar asymmetry, we consider the mean flagellar curvature K0, which is computed by fitting a line K0s to the zeroth Fourier mode (0) of the tangent angle ψ(s,t) (see ‘Tangent angle representation of planar flagellar beat patterns’ above).
As a consequence of the asymmetric flagellar beat, the resulting averaged swimming path is not straight, but has non-zero curvature κ. Fig. 5 displays experimental data correlating mean flagellar curvature K0 and path curvature κ for 13 bull sperm cells swimming close to a boundary surface in normal and high viscosity solutions. We find that path curvature κ scales approximately linearly with mean flagellar curvature K0, with a proportionality factor of 2.2. The linear dependence accords with theoretical predictions in the limit of small tangent angles (see Appendix). The fact that we find a factor of proportionality different from one reflects the non-linear character of flagellar self-propulsion at low Reynolds numbers: numerical simulations suggest that a ratio κ/K0 significantly larger than one is a result of a finite amplitude of the flagellar beat as well as the presence of the hydrodynamic drag of the sperm head. Assuming a simplified beat pattern with tangent angle given by: (8) we numerically find an approximately linear dependence between mean flagellar curvature K0 and path curvature κ provided the other parameters are held fixed (Fig. 6A). We find that the factor of proportionality between K0 and κ depends on the amplitude parameter A0 (see Fig. 6B). Increasing the friction coefficients of the sperm head also increased this factor. In the range of experimentally observed amplitude parameters A0≈10–16 rad mm−1, we obtain proportionality factors in the range dκ/dK0≈1.9–2.4 consistent with the value κ/K0≈2.2 found by fitting experimental data. The proportionality factor is accounted for by the observed beat amplitude and thus eventually by the intrinsic non-linearity of the propulsion mechanism.
Flagellar curvature is an emergent property generated by active processes
We recorded planar flagellar beat patterns of bull sperm cells swimming at two different fluid viscosities, η≈0.7 mPa s and η≈10 mPa s. Surprisingly, the mean flagellar curvature K0 was reduced by a factor of two in the case of increased viscosity (see Table 2). This observation suggests that the mean shape of the flagellum is not solely determined by the asymmetric architecture of the passive flagellar components, which are not expected to change when the external viscosity is increased. Rather, the dependence of mean flagellar curvature on fluid viscosity suggests that flagellar asymmetry is an emergent property that depends on active processes within the flagellum.
In this paper, we used high-speed videoscopy and quantitative image analysis to obtain high-precision tracking data with sub-micrometre resolution for bull sperm cells swimming close to a planar boundary surface. From these tracking data, we computed the time series of instantaneous velocities of the sperm head which reveal insights into the fine structure of sperm swimming. The instantaneous velocities can be accurately reconstructed from the shape of the flagellar beat using resistive force theory. Furthermore, resistive force theory also accounts for the relationship between path curvature and mean flagellar curvature, which is characterized by a non-unitary factor of proportionality. Thus, this theory accounts for the swimming behaviour of sperm cells near a boundary surface, both at the sub-micrometre scale of wiggling head movements and also on a coarse-grained length scale on which sperm cells follow circular paths. The theory accounts for the non-linear nature of flagellar propulsion that is evident on all length scales. On the small scale, we observe, for example, a peculiar spectral feature of one of the instantaneous velocities (an enhanced second Fourier mode), which was predicted by our theoretical considerations. On the large scale, we accounted quantitatively for the relationship between path curvature and mean flagellar curvature.
Like the bull sperm cells studied here, sperm cells from many other species also swim along circular paths near surfaces (Rikmenspoel et al., 1960; Goldstein, 1977; Brokaw, 1979; Woolley, 2003; Kaupp et al., 2008; Riedel-Kruse et al., 2007), or even move along helical paths in three-dimensional space far from any boundary surface (Crenshaw, 1996; Corkidi et al., 2008). For sea urchin sperm cells, which have to find their eggs in open water, curved swimming paths are at the core of a chemotaxis mechanism that guides these sperm cells to the egg [see Kaupp et al. and Friedrich and Jülicher, and references therein (Kaupp et al., 2008; Friedrich and Jülicher, 2007)]. The observed curved swimming paths are a consequence of chiral propulsion by asymmetric flagellar bending waves. The asymmetry of the bending waves is in turn rooted in the chiral architecture of the sperm flagellum (Lindemann, 1994; Hilfinger and Jülicher, 2008), which possesses a defined handedness (Afzelius, 1999). Much progress has been made in recent years to theoretically explain the symmetric part of flagellar bending waves (Riedel-Kruse et al., 2007; Brokaw, 2008). However, why these planar flagellar waves are asymmetric is not fully understood. A steady-state activity of the molecular motors in the flagellum will generate pre-twist of the flagellum (Hilfinger and Jülicher, 2008), which may be at the origin of the observed asymmetry. Interestingly, we find here that the mean curvature of the flagellum depends on fluid viscosity. Additionally, previous work has shown that the intraflagellar calcium concentration is a key player in regulating the asymmetry of the flagellar beat (Brokaw, 1979; Cook et al., 1994; Böhmer et al., 2005; Wood et al., 2005). All these findings suggest that the mean curvature of the flagellum depends on active processes within the flagellum.
In this manuscript, we obtained the shape of the flagellar beat directly from the experiments and related these beat patterns quantitatively to the complex swimming paths of sperm cells. As such, our study contributes to explaining cell motility from basic swimming movements, which in turn are determined by the molecular architecture of the flagellar swimming apparatus.
We thank K. Müller from IFN Schönow for providing sperm samples. We thank L. Alvarez, D. Babcock, C. J. Brokaw, L. Dai, J. Elgeti, A. Hilfinger, U. B. Kaupp and A. Vilfan for stimulating discussions and helpful comments. All authors planned the research, I.H.R.-K. performed experiments; B.M.F. analysed data; B.M.F., J.H. and F.J. wrote the paper.
Flagellar propulsion in the limit of small tangent angles
For any periodic flagellar beat pattern with angular frequency ω, the instantaneous velocities v1(t), v2(t) and Ω(t) are periodic functions in time of period T=2π/ω. We consider the most general beat pattern for which the tangent angle is characterized by its zeroth and first Fourier mode: (A1) where cc denotes the complex conjugate. Here (s) and (s) are arbitrary functions of the arc length s, which describe the mean shape of the flagellum and the complex amplitude of the flagellar wave, repectively. The dimensionless scaling factors ε0 and ε1 allow adjustment of the mean flagellar curvature K0 and the amplitude parameter A0; they conveniently play the role of small parameters in the perturbation calculation given below. For the simple flagellar beat given by Eqn 8, we would have (s)=s/L, (s)=s/Lexp(−2πis/λ) as well as ε0=K0L, ε1=A0L. We can expand the instantaneous swimming velocities in the asymmetry factor ε0 and the amplitude factor ε1 as follows: (A2) Under a reflection of the beat pattern, ψ→–ψ, the velocity component v1(t) remains unchanged, whereas v2(t) and Ω(t) change their sign. Thus, all terms in the expansion for v1(t) for which k+l is odd must vanish by symmetry; likewise, all terms in the expansions for v2(t) and Ω(t) with even k+l are zero. Moreover, for any non-zero term in the expansions, the mode number |m| is always smaller than the amplitude number l and has the same parity. Thus, to leading order, the above expansions read: (A3) Averaging over one beat cycle, we find for the net speed of flagellar propulsion: (A4) The net rotational velocity is given by a higher order coefficient: (A5) Note that the translational and rotational speed scale with the square of the beat amplitude , whereas the instantaneous speeds v2(t) and Ω(t) scale linearly with ε1. This scaling behaviour is a hallmark of self-propulsion at low Reynolds numbers (Shapere and Wilczek, 1987; Lauga, 2007). Using force and torque balance Eqns 6 and 7, the coefficients Xk,l,m, Yk,l,m and Zk,l,m can be expressed as a linear combination of (products of) integrals involving and . For the leading order coefficients, we find: (A6) where we have introduced the dimensionless integrals:
For the sake of illustration, we study as an example the simple flagellar beat pattern whose tangent angle is given by Eqn 8. For simplicity, we neglect the hydrodynamic drag force of the sperm head, i.e. ξ1=ξ2=0, ξrot=0. We also assume that the wave number n is an integer. Then the coefficients relevant for the net translational and rotational speed, and , respectively, are given by: (A7) and Y0,1,1=2ωL2(9+iπn)/(2πn)3, Z0,1,1=12ωL(3+iπn)/(2πn)3, X1,1,1=0. The prefactors Θv, depend on the wave number n and read Θv=(2/3)–2μ–9μ2, =(3/5)–(9/2)μ2–216μ3, =−(2/3)+2μ+9μ2, =(1/15)–6μ+(3/2)μ2+432μ3, where we have used the short-hand notation μ=1/(πn)2.
We remark on some properties of the Eqns A4, A5 and A7 for the net velocities and : the direction of forward propulsion is opposite to the propagation direction of the flagellar travelling wave, provided ξ⊥>ξ∥ (Brennen and Winet, 1977). In the case of isotropic drag coefficients ξ∥=ξ⊥, the translational speed vanishes [see Becker et al. for a general proof of this fact (Becker et al., 2003)]. The sperm cell will swim along a circular swimming path with a curvature κ=/ that is proportional to the curvature K0 of the mean shape of the flagellum. A similar result was found by Keller and Rubinow (Keller and Rubinow, 1976). Even for a symmetric beat pattern with K0=0, the instantaneous rotational speed Ω(t) oscillates with the frequency ω of the flagellar beat. Of course, the averaged rotational speed vanishes in this case, =0, as is required by symmetry. The case of a symmetric flagellar beat with K0=0 was first addressed by Taylor, and Gray and Hancock (Taylor, 1952; Gray and Hancock, 1955) and re-examined by Shack and colleagues (Shack et al., 1974). Note that Taylor, and Gray and Hancock (implicitly) imposed the constraint Ω(t)=0 for their calculation (Taylor, 1952; Gray and Hancock, 1955). With this constraint, the expressions for the translational speeds vj(t) look different. If we had imposed the constraint Ω(t)=0 in our calculation, we would find a different prefactor Θv=(2/3)–μ/2 for the net translational speed. These differences in the expression for highlight the role of constraints in microswimming problems.
↵* Present address: Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot, Israel
↵† Present address: Department of Bioengineering, Stanford University, Stanford, CA, USA
Supplementary material available online at http://jeb.biologists.org/cgi/content/full/213/8/1226/DC1
LIST OF SYMBOLS AND ABBREVIATIONS
Shape of the flagellar beat
- amplitude parameter of flagellar bending wave – defined as slope of
- mean flagellar curvature – defined as slope of
- centreline of sperm flagellum as function of arc length s and time t
- wavelength of (principal) flagellar bending wave
- tangent angle characterizing flagellar shape
- zeroth Fourier mode of ψ (characterizing time-averaged shape of flagellum)
- first Fourier mode of ψ (characterizing symmetric part of flagellar waves)
- ω, T=2π/ω
- angular frequency and period of the flagellar beat
Dynamics of sperm swimming
- e1(t), e2(t)
- unit vectors parallel and perpendicular to the long axis of the sperm head
- position of the center of the sperm head as function of time t
- coarse-grained sperm swimming path that averages over sub-cycle motion
- v1(t), v2(t)
- instantaneous speed of sperm head for motion in the direction e1, e2, respectively
- net translational and angular speed along the path
- curvature of coarse-grained swimming path
- instantaneous angular speed of the sperm head (in the plane of swimming)
- hydrodynamic drag force density along the flagellar length
- ξ∥, ξ⊥
- effective hydrodynamic drag coefficients of the sperm flagellum for motion parallel and perpendicular to its centreline, respectively
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