SUMMARY
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 nonzero curvature. To test a simple hydrodynamic theory of flagellar propulsion known as resistive force theory, we conducted highprecision 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 nonzero 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.
INTRODUCTION
Sperm cells are propelled in a liquid by regular bending waves of a whiplike 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 nonzero 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 longstanding interest (Gray and Hancock, 1955; Rikmenspoel, 1965; Brokaw, 1970; Yundt et al., 1975; Smith et al., 2009). In pioneering work, Taylor demonstrated that selfpropulsion is possible due to purely viscous forces (Taylor, 1951). Gray and Hancock introduced a local hydrodynamic theory of flagellar propulsion that neglects longrange 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 slenderbody approximations for the thin flagellum to include longrange 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; RiedelKruse et al., 2007). The nonzero 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 reinvestigated 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 counterintuitive result is due to the nonlinear 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 outofplane component of the flagellar beat, and considered the twodimensional 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 e_{1}(t) a unit vector parallel to the long axis of the sperm head. Additionally, we define a second unit vector e_{2}(t), which is obtained by rotating e_{1} in the swimming plane by an angle of π/2 in a counterclockwise fashion. With this notation, r+r_{1}e_{1} corresponds to the proximal tip of the sperm head while r–r_{1}e_{1} corresponds to the proximal end of the flagellum; here 2r_{1}≈10 μm is the length of the head along its long axis e_{1}. 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 e_{1}(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 e_{1}(t) and e_{2}(t), we can reconstruct the full flagellar beat pattern as:
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 K_{0} is defined by fitting a line K_{0}s to the zeroth mode (s); K_{0} provides a simple measure for the asymmetry of the mean shape of the flagellum. Note that the ideal case (s)=K_{0}s corresponds to a mean shape of the flagellum that is curved in a perfect arc with constant curvature K_{0}. Second, the amplitude parameter A_{0} is defined by fitting the line A_{0}s to the absolute value of the first mode (s). The ideal case =A_{0}s 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
Instantaneous velocities
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 timedependent velocity components v_{1}(t) and v_{2}(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 coarsegrained 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 v_{1}(t), but has to be determined from the averaged swimming path (t). The curvature of the averaged path is given by κ=1/r_{0}= / where r_{0} 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 longrange 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)=v_{1}(t)e_{1}(t) and v_{⊥}(s=0,t)=[v_{2}(t)+r_{2}Ω(t)]e_{2}(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 F_{head} is the hydrodynamic drag force of the head. Similarly, the total torque M acting on the sperm cell must be zero as well:
Drag force of the head
To describe sperm swimming accurately, the hydrodynamic drag force F_{head} and the torque M_{head} 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 F_{head}=ξ_{1}v_{1}e_{1}+ξ_{2}v_{2}e_{2} and M_{head}=ξ_{rot}Ωe_{1}×e_{2} (Perrin, 1934). We use 2r_{1}=10 μm for the length of the head along the long axis e_{1} and 5 μm for the length along the short axis e_{2} and obtain ξ_{1}≈40.3 pN s mm^{−1}, ξ_{2}≈46.1 pN s mm^{−1} and ξ_{rot}≈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.
Highspeed videoscopy of swimming bull sperm
Sperm cells were obtained as frozen samples (IFN Schönow, Germany) and prepared as described previously (RiedelKruse 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 phasecontrast microscopy (Axiovert 200M, Zeiss, Jena, Germany). Sperm swimming paths were recorded with a highspeed 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 custommade 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 (RiedelKruse 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, #F4375; St Louis, MO, USA). Viscosities were measured with a viscometer (Brookfield, Model DVI +; 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 highprecision tracking data for the position and orientation of the sperm head of swimming bull sperm cells, the instantaneous translational and rotational velocity components v_{1}(t), v_{2}(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 leastsquares 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 ξ_{1} and ξ_{2} 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 K_{0} 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.
RESULTS
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 timedependent velocities: translational velocities v_{1} and v_{2} 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 v_{1} and the other two components v_{2} and Ω, as only the last two change their sign when the flagellar beat pattern is reflected along the long axis e_{1} of the sperm head. Based on this symmetry argument, we infer that in the limit of small tangent angles, v_{1} should be a superposition of a nonzero 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 nonlinear 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, v_{1} indeed varies around a nonzero average value. Also, in the power spectrum S_{v1} of v_{1}, the power of the Fourier peak at frequency 2ω amounts to a considerable fraction ρ_{v}_{1}≈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 ρ_{v1}=107±54% (mean±s.d., N=6) compared with the case of normal viscosity with ρ_{v1}=22±29% (N=7). Analogously defined power ratios for the velocity components v_{2} 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 longrange hydrodynamic interactions between different parts of the moving flagellum. In our experiments, the longrange 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 longrange 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 ttest, 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 nonzero curvature (see Fig. 2A). To characterize flagellar asymmetry, we consider the mean flagellar curvature K_{0}, which is computed by fitting a line K_{0}s 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 nonzero curvature κ. Fig. 5 displays experimental data correlating mean flagellar curvature K_{0} 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 K_{0}, 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 nonlinear character of flagellar selfpropulsion at low Reynolds numbers: numerical simulations suggest that a ratio κ/K_{0} 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:
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 K_{0} 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.
DISCUSSION
In this paper, we used highspeed videoscopy and quantitative image analysis to obtain highprecision tracking data with submicrometre 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 nonunitary factor of proportionality. Thus, this theory accounts for the swimming behaviour of sperm cells near a boundary surface, both at the submicrometre scale of wiggling head movements and also on a coarsegrained length scale on which sperm cells follow circular paths. The theory accounts for the nonlinear 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; RiedelKruse et al., 2007), or even move along helical paths in threedimensional 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 (RiedelKruse et al., 2007; Brokaw, 2008). However, why these planar flagellar waves are asymmetric is not fully understood. A steadystate activity of the molecular motors in the flagellum will generate pretwist 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.
ACKNOWLEDGEMENTS
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.
APPENDIX
Flagellar propulsion in the limit of small tangent angles
For any periodic flagellar beat pattern with angular frequency ω, the instantaneous velocities v_{1}(t), v_{2}(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 ε_{0} and ε_{1} allow adjustment of the mean flagellar curvature K_{0} and the amplitude parameter A_{0}; 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 ε_{0}=K_{0}L, ε_{1}=A_{0}L. We can expand the instantaneous swimming velocities in the asymmetry factor ε_{0} and the amplitude factor ε_{1} as follows:
(A2)
Under a reflection of the beat pattern, ψ→–ψ, the velocity component v_{1}(t) remains unchanged, whereas v_{2}(t) and Ω(t) change their sign. Thus, all terms in the expansion for v_{1}(t) for which k+l is odd must vanish by symmetry; likewise, all terms in the expansions for v_{2}(t) and Ω(t) with even k+l are zero. Moreover, for any nonzero 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
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. ξ_{1}=ξ_{2}=0, ξ_{rot}=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 Y_{0,1,1}=2ωL^{2}(9+iπn)/(2πn)^{3}, Z_{0,1,1}=12ωL(3+iπn)/(2πn)^{3}, X_{1,1,1}=0. The prefactors Θ_{v}, depend on the wave number n and read Θ_{v}=(2/3)–2μ–9μ^{2}, =(3/5)–(9/2)μ^{2}–216μ^{3}, =−(2/3)+2μ+9μ^{2}, =(1/15)–6μ+(3/2)μ^{2}+432μ^{3}, where we have used the shorthand 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 K_{0} 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 K_{0}=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 K_{0}=0 was first addressed by Taylor, and Gray and Hancock (Taylor, 1952; Gray and Hancock, 1955) and reexamined 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 v_{j}(t) look different. If we had imposed the constraint Ω(t)=0 in our calculation, we would find a different prefactor Θ_{v}=(2/3)–μ/2 for the net translational speed. These differences in the expression for highlight the role of constraints in microswimming problems.
FOOTNOTES

↵* Present address: Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot, Israel

↵† Present address: Department of Bioengineering, Stanford University, Stanford, CA, USA

↵‡ howard{at}mpicbg.de

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
 A_{0}
 amplitude parameter of flagellar bending wave – defined as slope of
 K_{0}
 mean flagellar curvature – defined as slope of
 r(s,t)
 centreline of sperm flagellum as function of arc length s and time t
 λ
 wavelength of (principal) flagellar bending wave
 ψ(s,t)
 tangent angle characterizing flagellar shape
 (s)
 zeroth Fourier mode of ψ (characterizing timeaveraged shape of flagellum)
 (s)
 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
 e_{1}(t), e_{2}(t)
 unit vectors parallel and perpendicular to the long axis of the sperm head
 r(t)
 position of the center of the sperm head as function of time t
 (t)
 coarsegrained sperm swimming path that averages over subcycle motion
 v_{1}(t), v_{2}(t)
 instantaneous speed of sperm head for motion in the direction e_{1}, e_{2}, respectively
 ,
 net translational and angular speed along the path
 κ
 curvature of coarsegrained swimming path
 Ω(t)
 instantaneous angular speed of the sperm head (in the plane of swimming)
Hydrodynamic drag
 f(s,t)
 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
 © 2010.