To quantify the relationship between rolling and progressive motion, we first characterized asymmetry in the flagellar beating pattern. In agreement with Friedrich et al., 2010, measuring bending in the midpiece of the flagellum in time (Figure 2A) and the corresponding normalized fast Fourier transform $\left(P^{*}\left(t\right)\right)$ revealed the presence of a zeroth harmonic within the frequency domain of flagellation (Figure 2B). Note that the constant offset in the normalized power spectrum is white noise in our measurement system. Furthermore, the experimental noise coming from our measurement system (e.g., image processing) is also included in the peak width around the first harmonic frequency. That is, the peak width centered at $\omega$ includes the intrinsic noise originated from flagellar sources, as well as the noise associated with our measurement system.

Figure 2 with 6 supplements see all

Download asset Open asset

Writing the flagellar beating pattern in Fourier series ansatz (Equation 1) and further modeling the flagellar beating pattern (SI, Section II), we found that $\sigma =\frac{a_1-a_0}{a_1}$ approximately presents the asymmetry level in the beating pattern. That is, $\sigma =1$ presents symmetric beating whereas lower values correspond to higher asymmetry in the beating pattern:

In Equation 1, the $x$ axis was set parallel to the flagellum at its straight-line form, $k=\frac{2\pi }{\lambda }$ (with $\lambda \approx L$) is the wave number, $\omega$ is the main frequency, and $a_n$ is the amplitude of the nth harmonic. Note that the amplitude and frequencies of beating include delta-correlated Gaussian noise that can be described as $a_n={\tilde{a}}_n\left(1+{\eta }_n\left(t\right)\right)$ and ${\omega }_n={\tilde{\omega }}_n\left(1+\xi \left(t\right)\right)$, respectively, with $⟨{\eta }_n\left(t\right)⟩=⟨\xi \left(t\right)⟩=0$, $⟨{\eta }_n\left(t\right){\eta }_{n^{\text{'}}}\left(t\text{'}\right)⟩=D_n{\delta }_{n{n}^{\text{'}}}\delta (t-t\text{'})$, and $⟨\xi \left(t\right)\xi \left(t\text{'}\right)⟩=D_{\omega }\delta (t-t\text{'})$. The tilde sign above a quantity represents its time average value. Following Saggiorato et al., 2017; Friedrich et al., 2010; Elgeti et al., 2015, we applied small amplitude (i.e., $\forall n\to a_n\ll L$) and length preservation constraints and used resistive force theory (SI, Sections II–IV), to find that the zeroth harmonic in the beating pattern yields a toque as follows:

Note that ${\xi }_T$ and ${\xi }_N$ are drag coefficients in the tangential and normal directions, respectively, and $L$ is the sperm length. Although the propulsive force produced by the flagellum correlates with the characteristics of the first and higher harmonics, the amplitude of the zeroth harmonic is involved solely in the torque produced (Equation 2). Applying the zero net torque and force constraints, we found that propulsive $\left({\tilde{V}}_P\right)$ and angular $\left(\tilde{\mathrm{\Omega }}\right)$ velocities were related through $a_0$:

To verify Equation 3, we measured the angular velocity of the sperm and plotted a normalized path curvature $\left(L\mathrm{\Omega }{V}_P^{-1}\right)$ with respect to the normalized amplitude of the zeroth harmonic $\left(a_0{L}^{-1}\right)$ in TALP, TALP + 4% PVP (viscous), and TALP + 1% PAM (viscoelastic) solutions (Figure 2C and D). The linear correlation between $L\mathrm{\Omega }{V}_P^{-1}$ and $a_0{L}^{-1}$ is consistent with our mathematical arguments (Equations 1–3), confirming that $a_0$ modulates the curvature of the non-rolling sperm path. Furthermore, the linear relationship between $L\mathrm{\Omega }{V}_P^{-1}$ and $a_0{L}^{-1}$ predicted by the resistive force theory is preserved for sperm motion in a viscoelastic solution whereas the slope differs from that observed for sperm motion in standard and viscous solutions (Zhang and Goldman, 2014; Teran et al., 2010; Li and Ardekani, 2015; Riley and Lauga, 2017; Spagnolie et al., 2013).

The resistive force theory, as a mean field approach, cannot explain the observed inconsistency in the circular path. The inconsistency in the circular path, however, can be characterized by quantifying the random fluctuations at the center of the circular path (Ma et al., 2014). We noticed that fluctuations at the center, and thus circular motion, are diffusive in character as the mean square displacement (MSD) of the center is proportional to the elapsed time: $\mathrm{M}\mathrm{S}\mathrm{D}~t$ (Figure 2E). This diffusive motion can be quantified by the diffusion coefficient of the center determined by the intercept of MSD, or alternatively, the center’s speed distribution (Figure 2F).

We measured the head light intensity (HLI) of the sperm over time to characterize rollings during progressive motility (Figure 2G). HLI is a pulse-type quantity, with the pulse duration shorter than the time that elapses between two consecutive pulses, $T_{SR}$. Therefore, we defined the edge-sensitive function $\mathrm{\Pi }\left(t\right)$ with $\mathrm{\Pi }\left(0\right)=1$, such that $\mathrm{\Pi }\left(t\right)$ is multiplied by −1 at each positive edge of HLI (Figure 2G). Note that $\mathrm{\Pi }\left(t\right)$ captures the rolling component as a rapid switch in the direction of asymmetry (Figure 1I); rolling can therefore be incorporated into Equation 1 with $\mathrm{\Pi }\left(t\right)$:

Solving the equations of motion using Equation 4, we found that, depending on ${\displaystyle \widetilde{\mathrm{\Omega }{T}_{SR}}}$, sperm swim along varying pathways with average progressive velocity of:

(Figure 2—figure supplement 4A and B). At the frequent rolling limit $\left(\widetilde{\mathrm{\Omega }{T}_{SR}}\to 0\right)$, $\stackrel{-}{V}$ approaches ${\tilde{V}}_P$, which means that frequent rolling asymptotically yields progressive motion with the average path velocity equal to the propulsive velocity even though $\tilde{\mathrm{\Omega }}\ne 0$. Our experimental measurements are consistent with this finding because ${\displaystyle \widetilde{\mathrm{\Omega }{T}_{SR}}}$ is less than $2\pi$ for rolling sperm, whereas for circular motion with infrequent rolling it is greater than $2\pi$ (Figure 2—figure supplement 4C). Note that for non-rolling sperm, ${\displaystyle \widetilde{\mathrm{\Omega }{T}_{SR}}\to \mathrm{\infty }}$.

Even though ${\displaystyle \widetilde{\mathrm{\Omega }{T}_{SR}}}$ determines the average path, how do deviations of $T_{SR}$ from the mean during motion influence the trajectory? Our measurements of $T_{SR}$ for one rolling sperm for ~40 s indicate that the mean and standard deviation of this random variable are ${\displaystyle {\tilde{T}}_{SR}\approx 0.11\mathrm{s},{\sigma }_{SR}\approx 0.02\mathrm{s}}$, respectively (Figure 2H). Combining arbitrary values for $\tilde{\mathrm{\Omega }}$ with $T_{SR}$ distribution, we found that the direction of the average path obeys a similar distribution with mean of 0 and standard deviation of ${\sqrt{n_{SR}}\sigma }_{SR}\tilde{\mathrm{\Omega }}$ (Figure 2I), in which $n_{SR}$ is the number of rolling occurrences.

Considering rolling as a switch in the direction of asymmetry, we investigated how rolling influenced sperm rheotaxis. The sperm-rheotactic behavior and angular velocity of upstream orientation is modeled using an Adler-type Equation 10:

in which $\gamma$ is the shear rate, $A$ is a constant, and $\theta$ is the relative orientation of the sperm with respect to the external fluid stream. For sperm with intrinsic angular velocity $\left(\mathrm{\Omega }\right)$, the net angular velocity under fluid flow is ${\Omega }_{RH}+\mathrm{\Omega }$, which orients the sperm with respect to the flow during upstream motion $\left({\theta }_{UP}\right)$:

Including the rolling component in Equation 7 using $\Pi \left(t\right)$, the average orientation of the sperm with respect to flow during upstream motion is:

Notably, ${\displaystyle \widetilde{\mathrm{\Pi }\left(t\right)}\ll 1}$. Equation 8 predicts that rollings result in significant decay in ${\displaystyle {\tilde{\theta }}_{UP}}$. Consequently, Equations 9 and 10 respectively present sperm net velocity in the upstream direction (${\tilde{V}}_{UP}$) in the presence or absence of frequent rollings:

In Equations 9 and 10, $V_P$ and $V_N$ are sperm propulsive and perpendicular velocities (corresponding to the angular velocity), respectively, whereas $V_F$ is the external flow velocity (SI, Section V). To experimentally confirm the prediction obtained from Equations 9 and 10, we back-tracked and analyzed sperm motion under flow prior to their entry into the quiescent zone. We observed that ${\displaystyle {\tilde{\theta }}_{UP}}$ is greater for non-rolling sperm $\left({\tilde{\theta }}_{UP}=40°\pm 10°\right)$ than for those exhibiting the rolling motion $\left({\tilde{\theta }}_{UP}=10°\pm 5°\right)$. Furthermore, the average upstream velocity of rolling sperm was much higher ${\displaystyle \left({\tilde{V}}_{UP}=60\pm 10\mu \mathrm{m}/\mathrm{s}\right)}$ than that of non-rolling sperm ${\displaystyle \left({\tilde{V}}_{UP}=30\pm 10\mu \mathrm{m}/\mathrm{s}\right)}$ (SI, Section V). These results indicate that, even though the rolling component is not required for sperm rheotaxis, it facilitates rheotactic behavior by minimizing the angle between sperm orientation and the external fluid flow, thus maximizing the upstream component of their motion.

Non-rolling sperm swim along diffusive circular paths and explore the surface (Ma et al., 2014). To characterize the diffusivity in circular motion (Figure 2E and F), we overlayed consecutive images of one sperm with 0.08 s intervals over different times (Figure 3A). The thickness of the combined circular paths $\left({\delta }_{1-4}\right)$ increases over time in all directions, ${\delta }_{1-4}\propto \sqrt{t}$ (Figure 3B) as predicted by the MSD of the circular path’s center (Figure 2E and F). For any sperm that diffused more rapidly, cell tracking followed by circle fitting was used to characterize the motion of circular path’s center (Figure 3C).

Figure 3 with 4 supplements see all

Download asset Open asset

To identify the flagellar source of the center’s diffusion, we solved equations of motion where the amplitude and phase of all harmonics included white Gaussian noises, as described in Equation 1 (the results are shown in Figure 3D). Our simulations suggest that noise in the amplitudes and the phases of first and higher harmonics yielded a noisy $V_P$ without resulting in fluctuations of the center (SI, Section VI). By contrast, the noise in the amplitude of the zeroth harmonic yielded a similar noise in $\mathrm{\Omega }$ but not in $V_P$. Therefore, $\mathrm{\Omega }$ and $\kappa$ can be expressed as $\mathrm{\Omega }\left(t\right)=\tilde{\mathrm{\Omega }}(1+{\eta }_0(t\left)\right)$ and $\kappa \left(t\right)=\tilde{\kappa }(1+{\eta }_0(t\left)\right)$, respectively, such that lower signal-to-noise ratios $\left(\mathrm{S}\mathrm{N}\mathrm{R}=\frac{{\tilde{\kappa }}^2}{⟨{\kappa }^2⟩-{\tilde{\kappa }}^2}\right)$ for identical mean curvatures yielded faster diffusion, whereas higher SNRs yielded more consistent pathways (Figure 3D). We measured the normalized mean step size $⟨{\delta }^2{\kappa }^2⟩$ and the corresponding SNR and found that our experimental data were consistent with the simulation-based results, except for a constant as shown in Figure 3E and F. That is, as suggested by simulations, the normalized mean step size was found to be inversely correlated with the SNRs: $⟨{\delta }^2{\kappa }^2⟩\propto {\mathrm{S}\mathrm{N}\mathrm{R}}^{-1}$.

As previously published (Ma et al., 2014; Goldstein et al., 2009), these fluctuations in the circular path’s center correspond to non-thermal noise in the sperm flagellar beating, rather than thermally driven fluctuations. Approximating sperm as a rod with length of ∼80 µm and diameter of ∼5 µm, the rotational diffusion caused by thermal fluctuations is in the other of 10⁻⁶s⁻¹, which yields a $SNR~{10}^7$ for $\tilde{\kappa }~(5\times {10}^{-3}){\mu m}^{-1}$. This order of $SNR$ estimated for thermal fluctuations are far greater than our measured values and the experimental resolution; therefore, the contribution of thermal noise to the sperm diffusive circular motion is negligible.

Diffusive circular motion might be abruptly interrupted by infrequent rollings (Figure 1I, Figure 3G and H). In this case, the center of the circular path not only diffuses in time but also relocates ballistically upon rolling in a random direction with $V_r$ (Figure 3H,I), such that $\frac{D}{V_r}$∼10⁻⁶ −1 µm (SI, Section VII). This two-phase motion is an intermittent search, with greater efficiency in surface exploration than with normal diffusion (Bénichou et al., 2011). This increase in the efficiency of exploration may be interpreted as a decrease in the probability of revisiting previously swept spots. More precisely, the average area swept by the sperm through diffusion is proportional to the square root of diffusion time $⟨A\left(t\right)⟩\propto \sqrt{t}$. Assuming that $T_i$ is the time frame between the (i−1)th and ith relocations, the areas swept with and without relocation are proportional to $\sum _i\sqrt{T_i}$ and $\sqrt{\sum _i{T}_i}$, respectively. Because $\sqrt{\sum _i{T}_i}\mathrm{}\le \sum _i\sqrt{T_i}$, infrequent rolling increases the area swept by the sperm (see SI, Section VII).

In addition to active swimming and external flow, the surrounding walls contribute to sperm motion (Elgeti et al., 2010). For a sperm that is positioned away from the walls (distance from the wall > the sperm length), the wall’s effect on the swimmer is known to be a drift velocity that can be either attractive or repulsive depending on the swimmer’s angle with respect to the wall (Elgeti et al., 2010). The model proposed for studying this drift velocity is based on positing the swimmer as a force dipole that includes the propulsive force provided by the flagellum as well as the corresponding drag force (Elgeti et al., 2010). However, this model can be used when the sperm exhibits fully symmetric flagellation. We therefore developed a new swimmer model that includes the torque caused by asymmetric beating. The proposed model is depicted in Figure 4—figure supplement 1A, in which $f$ is the propulsive force, $f^{\text{'}\text{'}}$ is the perpendicular force corresponding to the torque, and $f^{\text{'}}$ is the drag force required for the torque-free condition. We then carried out finite element method simulations in a cylindrical domain, like our microfluidic quiescent reservoir, and solved the Stokes and mass conservation equations for our model to find the velocity field imposed by the sperm active swimming.

The velocity field imposed by flagellar beating for ${\displaystyle \widetilde{\mathrm{\Delta }\theta }=0-{15}^{\circ }}$ is shown in Figure 4—figure supplement 1B. Integrating net flow in the y direction imposed on the sperm body that is caused by no-slip walls, we calculated the drift velocity toward the wall (Figure 4—figure supplement 1C). The simulation results indicated that the motion of the microswimmer is influenced to a lesser degree by nearby boundaries as the circular component emerges in the motility (SI, Section VIII). This decay in the drift velocity caused by the circular component of motion was also predicted by the analytical solution derived from a Stokeslet (Ardekani and Stocker, 2010; Li and Ardekani, 2014) description (SI, Section IX). The attraction of the sperm toward the wall in the presence of circular behavior can be described by the following equation:

where $U_w$ and $U_w^p$ are the drift velocities with and without the circular motion. Calculating the average far-field drift velocity imposed on the sperm during one round of circulation, we found that the average magnitude of the drift velocity in one round (<1 µm/s) is much smaller than random fluctuations ((>540 µm/s) of the circular path’s center (Figure 4—figure supplement 1D and SI, Section X); therefore, the sperm’s circular motion is more strictly controlled by the level of asymmetry in its flagellum and diffusivity than by distant walls.

Frequent rollings, however, negate the contribution of circular components of the motion, producing a drift velocity as if the flagellation is symmetric and the sperm behaves like a dipole swimmer,

We reiterate that ${\displaystyle \widetilde{\mathrm{\Pi }\left(\mathrm{t}\right)}\ll 1}$. Equation 12 states that frequent rollings make the sperm more readily disposed to physical boundaries at the far field.

Sperm near-field interactions with its nearby walls fit in one of the four categories shown in Figure 4A. A rolling sperm reorients upon wall contact and swims along the wall ($S_1$, Video 7), whereas with a non-rolling sperm, depending on $\mathrm{\Omega }$ and the location of the circular path’s center at the contact point, three other behaviors were observed. At a high magnitude of $\mathrm{\Omega }$, the sperm do not contact the wall, maintaining their circular motion ($S_2$, Video 8). At lower magnitudes of $\mathrm{\Omega }$, the sperm contact the wall and, depending on the location of the path’s center relative to the contact point, either swim near the wall temporarily and detach ($S_3$, Video 9) or swim more slowly along the wall (compared with $S_1$) with a tilted orientation with respect to the wall ($S_4$, Video 10). We compared these four types of sperm–wall interactions quantitatively using the time of sperm detention on the wall ($T_D$, Figure 4B), sperm velocity on the wall divided by its velocity before wall contact ($V^{*}$, Figure 4C), and the normalized length of detention ${\rho }^{*}$ (Figure 4D,E). Based on the measurements shown in Figure 4B–E, we found that the $S_1$ and $S_3$ categories yield similar $V^{*}$ values (close to 1), whereas for $S_4$, $V^{*}$ is smaller than 1 and for ${\displaystyle \mathrm{\Omega }>1.5{\mathrm{s}}^{-1}}$ the sperm do not migrate along the wall. Furthermore, $S_1$ and $S_4$ yield similar ${\rho }^{*}$ values, close to 1, whereas for $S_3$, ${\rho }^{*}$ is much smaller than 1, indicating temporary detention on the wall. A simple yet insightful approach to understanding these four categories involves surface contact force analysis (Marion, 2013), where the sperm contact with the wall can be modeled by a positive force that is perpendicular to the surface $N$.

Figure 4 with 6 supplements see all

Download asset Open asset

Video 7

Download asset

Video 8

Download asset

Video 9

Download asset

Video 10

Download asset

Suppose that a sperm swims along a wall at an angle $\beta$ (Figure 4F). Under a zero net force constraint, the normal surface force becomes ${\displaystyle N=F_T\sin \left(\beta \right)-F_N\cos \left(\beta \right)}$, where $F_T$ and $F_N$ are the propulsive and perpendicular forces produced by the sperm, respectively. The threshold angle (${\beta }_{th}$) that corresponds to the $N=0$ situation is equal to ${\tan }^{-1}\gamma$, where $\gamma =\frac{F_N}{F_T}$. For $\beta <{\beta }_{th}$, $N$ becomes negative and no contact occurs ($S_2$). Because an increase in $\gamma$ leads to higher ${\beta }_{th}$, sperm with greater $\gamma$ values are less likely to contact and follow the wall. For $\beta$ values greater than ${\beta }_{th}$, where sperm–wall contact occurs, greater $F_N$ yields a smaller $N$ and leads to easier detachment from the wall ($S_3$). When the direction of the sperm perpendicular force at the contact point is against the wall ${\displaystyle \left(N=F_T\sin \left(\beta \right)+F_N\cos \left({\beta }_s\right)\right)}$, a greater surface force is exerted, yielding a stronger attachment to the wall (Figure 4G). In this configuration, however, the parallel-to-the-wall components of the perpendicular and propulsive forces thwart each other and cause slower sperm motion along the wall. Therefore, asymmetric flagellation perturbs sperm motion along the wall by decreasing either its detention time ($S_3$) or velocity ($S_4$) on the wall. However, the rolling component functions as a switch between $S_3$ and $S_4$, and not only guarantees longer detention but also increases sperm velocity along the wall (Figure 4H,I).

Based on previous theoretical and computational studies on hydrodynamic interactions of a sperm (or a bacteria) with a wall (Drescher et al., 2011; Schaar et al., 2015; Elgeti and Gompper, 2013; Rode et al., 2019; Elgeti and Gompper, 2016), we developed a simple hydrodynamic model of sperm–wall interaction at the lubrication limit. Solving Stokes and mass conservation equations (SI, Section XI), we plotted the phase curve ${\displaystyle \left(\dot{\beta }\mathrm{v}\mathrm{s}\beta \right)}$ of sperm dynamics after contacting the wall (Figure 4J). Note that $\dot{\beta }$ is the effect of the wall superimposed with intrinsic $\mathrm{\Omega }$. For $\mathrm{\Omega }=0$, stability ($\dot{\beta }=0$) occurs at ${\beta }_S=0$, implying that the final orientation of a symmetrically beating sperm with respect to the wall is 0. For $\mathrm{\Omega }>0$, corresponding to $S_4$, ${\beta }_S>0$ is the final angle between the sperm and the wall. For $\mathrm{\Omega }<0$, corresponding to $S_3$, ${\beta }_S<0$ and $\frac{d\beta }{dt}>0$ at $\beta =0$, suggesting that swimming parallel to the wall is unstable, and the sperm detaches from the wall with ${\beta }_S$. Figure 4J shows that the instability that occurs at ${\beta }_U=\frac{\pi }{2}$ (for $\mathrm{\Omega }=0$) changes with $\mathrm{\Omega }$ as well.

To include the contribution of rolling in sperm dynamics after wall contact, we modeled rollings as transitions between two curves in the phase space with positive and negative values of $\mathrm{\Omega }$ (Figure 4K). We then can rewrite $\dot{\beta }$ and include $\mathrm{\Pi }\left(t\right)$ such that:

Note that $g\left(\beta \right)$ is the curve in the phase space that corresponds to $\mathrm{\Omega }=0$. Insofar as stability occurs at $\dot{\beta }=0$,

in which ${{\beta }_S}^{*}$ and ${\beta }_S$ are the stable points with and without taking rolling into account. Because ${\displaystyle \widetilde{\mathrm{\Pi }\left(\mathrm{t}\right)}\ll 1}$, the average of ${\beta }_S^{*}\left(t\right)$ is:

Equation 15 suggests that, at the frequent rolling limit where ${\displaystyle \widetilde{\mathrm{\Pi }\left(\mathrm{t}\right)}}$ approaches 0, ${\displaystyle \widetilde{{\beta }_S^{\ast }}}$ approaches 0 as well. Consequently, frequent rollings mitigate the destructive role of $\mathrm{\Omega }$ in sperm motion along the wall, thereby yielding faster and longer-lasting swimming along the wall (SI, Section XII). Because at the frequent rolling limit ${\displaystyle \widetilde{{\beta }_S^{\ast }}}$ is close to 0, we posit that ${\displaystyle g\left(\widetilde{{\beta }_S^{\ast }}\right)\approx 0}$, and thus $\dot{\beta }$ near the stable point can be written as:

Based on Equation 16, $\beta \left(t\right)$ is a triangular function near the stable point in the form of:

We measured $\dot{\beta }$ experimentally to verify the results obtained from the lubrication theory (Figure 4J). The net rotation ($\dot{\beta }$) with respect to the angle between the sperm and wall ($\beta$) that corresponds to $S_1$ and $S_4$ is consistent with the results obtained from our model (Figure 4J,L,M). Furthermore, the frequent rollings observed in the $S_1$ category decreased ${\displaystyle \widetilde{{\beta }_S}}$ significantly to ∼10° (Figure 4N) in comparison with ${\displaystyle \widetilde{{\beta }_S}}$ for $S_4$, which was greater and linearly correlated with $\frac{\mathrm{\Omega }}{V}$ (Figure 4N, inset plot). This measurement supports our prediction that rolling enables the sperm to swim stably along the wall by decreasing ${\displaystyle \widetilde{{\beta }_s}}$. Measuring $\beta \left(t\right)$ during stable swimming along the wall for 25 sperm that belong to the $S_1$ category, we found that $\beta \left(t\right)$ is indeed a triangular function near the stable point, as predicted by Equation 17 (SI, Section XII).

A unified picture of sperm motion within the quiescent reservoir can be obtained by developing a state diagram and identifying the transitions between the $S_{1-4}$ states, possibly through the diffusivity of the circular motion and rolling. Suppose that, within the quiescent reservoir (radius of $R$), the non-rolling sperm swims in circle (with a radius of ${\displaystyle R^{\prime }}$), such that the distance between the centers of the two circles is $d\left(t\right)$, which evolves through diffusion (Figure 5A). Defining ${\displaystyle s\left(t\right)=\frac{1}{2}\left(R^2+R^{\prime 2}-d{\left(t\right)}^2\right)}$, the circular path does not intersect the reservoir for $s\left(t\right)>RR\text{'}$ (category $S_2$), whereas for ${\displaystyle 0<s\left(t\right)\le R{R}^{\prime }}$ the angle between the two circles at the intersection point (sperm incidence angle) is greater than ${\beta }_u$ (category $S_3$); for $s<0$, the angle at the intersection point is less than ${\beta }_u$ (category $S_4$). Assuming that at $t=0$ and the sperm is in the $S_2$ state, the circular path starts to diffuse over time, which leads to transitions between the $S_2$ and $S_3$ states. The average time for the first occurrence of a transition is ${\displaystyle ⟨T⟩=\frac{{\left(R-R^{\prime }\right)}^2}{2D}}$ (Figure 5B). One might expect the diffusion process to continue until $s\left(t\right)$ becomes negative and the transition to $S_4$ occurs (Figure 5B). However, the angle at which the sperm detaches from the wall after the first contact $\left({\beta }_s\right)$ depends on $\mathrm{\Omega }$ rather than the location and angle at which contact occurs (Figure 5C). Therefore, after first contact, the sperm returns to the wall at an incidence angle that is identical to the angle at which it detaches from the wall ${\beta }_s)$. Therefore, the transition from $S_3$ to $S_4$ occurs if ${\beta }_s>{\beta }_u$. Given that ${\displaystyle {\beta }_s\approx \frac{\mathrm{\Omega }}{\left|g^{\prime }\left(0\right)\right|}}$, ${\beta }_u\approx \frac{\pi }{2}+\frac{\mathrm{\Omega }}{g^{\text{'}}\left(\frac{\pi }{2}\right)}$, and ${\displaystyle \left|g^{\prime }(0)\right|\ll g^{\prime }\left(\frac{\pi }{2}\right)}$, the condition of such a transition reduces to:

Figure 5

Download asset Open asset

The prime indicates derivative with respect to $\beta$. The corresponding curvature that satisfies Equation 18 is κ ∼ 0.1 µm⁻¹, which is much greater than the experimentally observed values. The state diagram of non-rolling sperm–wall interactions and the possible transitions are summarized in Figure 5D.

For rolling sperm, the state diagram shown in Figure 5D alters significantly. First, infrequent rollings result in abrupt changes in $d\left(t\right)$ and thus $s\left(t\right)$, causing reversible transitions between $S_2$, $S_3$, and $S_4$ (Figure 5F). Second, at the frequent rollings limit, one may write $s\left(t\right)\approx \mathrm{\Pi }\left(t\right)\left|s\right|$, so that ${\displaystyle \widetilde{s\left(t\right)}}$ approaches 0 (Figure 5E). Consequently, in the presence of frequent rollings, all states transition to $S_1$, where the sperm swims stably along the wall (Figure 5F).

Commercially available cryopreserved bovine semen samples taken from two mature black and white Holstein bulls (5.5 and 6 years of age) were kindly donated by Genex Cooperative (Ithaca, NY) in milk and egg yolk–based extender in plastic straws. The ejaculate concentration was 2.9 and 3 billion cells/mL, respectively, and had a pre-freeze motility of 65%. The semen was thawed at 37°C in a water bath and diluted in a 1:4 ratio with TALP. After dilution, the viscosity of the samples was ~5 mPas. The initial sperm concentration in the thawed semen samples was ~200 million/mL, which was diluted to ~40 million/mL with TALP. The motility of the semen sample after dilution decreased to 20–30%. We used 10 separate semen samples in both the milk and egg yolk–based extender. At least three replicates were performed for each experiment to validate the accuracy of data and obtain a valid value for error bars.

TALP was prepared as follows: NaCl (110 mM), KCl (2.68 mM), NaH₂PO₄ (0.36 mM), NaHCO₃ (25 mM), MgCl₂ (0.49 mM), CaCl₂ (2.4 mM), HEPES buffer (25 mM), glucose (5.56 mM), pyruvic acid (1.0 mM), penicillin G (0.006% or 3 mg/500 mL), and bovine serum albumin (20 mg/mL). To tether the sperm head to the glass surface, we reduced the concentration of bovine serum albumin to 5 mg/mL. To increase the viscosity and viscoelasticity of TALP, we added 1–4% of PVP (weight percent) and 0.25–1% PAM (weight percent).

Dynamic rheological measurements were performed with a rheometer (MCR 501, Anton Paar, Stuttgart, Germany) with a 50 mm parallel plate at a gap of 0.5 mm. The amplitude sweep was conducted from 0.01% to 100% strain with a 1 Hz angular frequency to identify the linear viscoelastic region. The frequency sweep was performed from 1 to 100 s⁻¹ (Tung et al., 2017) with a constant 1% strain (within the linear viscoelastic region). The viscosity was measured using the steady shear mode with the shear rate from 0.01 to 100 s⁻¹. Samples were characterized at 37°C.

The microfluidic device was made of polydimethylsiloxane using a standard soft lithography protocol. The diameter of the circular quiescent zone was 500 µm and the height of the chamber was 25 µm. Diluted semen was injected into the microfluidic device using gravity and the flow generated in the channel was controlled by changing the height of the semen container. Because sperm rheotaxis occurs under a very low shear rate (0.6 s⁻¹), using gravity instead of conventional syringe pumps is more efficient for obtaining and controlling low flow rates.

To isolate motile bovine sperm inside the quiescent reservoir, we used a microfluidic corral system that isolated motile swimmers based on their ability to move upstream. As we injected the sample at an injection rate of 1.2 µLh⁻¹, sperm with motilities higher than 53.2 µms⁻¹ could swim upstream and enter the quiescent zone, which was filled with TALP, allowing us to study sperm movement with minimal fluid mechanical noise. Sperm movement was observed with a Nikon Eclipse TE300 inverted phase-contrast microscope (20× and 40× magnifications) and recorded with an Andor Zyla 5.5 sCMOS camera (25 and 50 frames/s).

Sperm trajectories and other motility-related characteristics were analyzed using ImageJ and MATLAB. To identify the harmonics of midpiece bending, we first removed noise and background using Gaussian filter and image subtraction. We then binarized the images taken from the sperm at 25 frames/s and measured the deviation of the midpiece (i.e., the segment located at 10 ± 1 µm from the head) from the centerline using the optical flow Flareback method. Taking the fast Fourier transform of the bending, we identified the zeroth, first, and second harmonics of the bending signal. A simple method for measuring the amplitude of the zeroth harmonic involves measuring the maximum bending toward the left $\left(y_L\right)$ and right $\left(y_R\right)$ sides of the swimmer. Therefore, the magnitude of the zeroth harmonic can be calculated using the following equation:

The main advantage of our method is its simplicity, as tracking the whole flagellum was not required for measurement of zeroth harmonic.

To increase the viscosity and viscoelasticity of the standard TALP medium, we added 1–4% of PVP and 0.25–1% of PAM. The rheological properties of the prepared solutions are shown in Figure 1—figure supplement 1. Note that $G^{\text{'}}$ of PVP-based solutions with percentages lower than 4% was very low and not detected by the rheometer.

To model beating patterns that resemble those of sperm flagella, we studied the pattern in one cycle of flagellar beating using a traveling sine wave with a temporal phase of $\varphi \left(t\right)=\omega t$ (with $\omega =40$ Hz) in the range of $\pi -{\varphi }_0\le \varphi \left(t\right)\le {\varphi }_0$, such that ${\varphi }_0\in [\pi ,2\pi ]$. Notably, ${\varphi }_0=2\pi$ corresponds to a fully symmetric beating pattern and thus, $\sigma =1$, whereas lower values of ${\varphi }_0$ correspond to lower $\sigma$ values and higher asymmetry in the beating pattern.

Positing that the beating pattern will identically repeat in time, we constructed the even extension of the partial sine wave to form the flagellar beating function over time. Figure 2—figure supplement 1 shows the modeled beating patterns of untethered sperm, which resemble the flagellar beating observed inside the quiescent zone, especially that of the mid- and principal piece, which we are interested in. To model the beating pattern, we considered the length preservation constraint as well because the sine wave form does not preserve the length in one beat.

In this section, we used resistive force theory to derive equations describing the forces produced by each segment of the flagellum and follow the presentation used by Friedrich et al. and Saggiorato et al. The velocity of each segment in the y direction ($V$) can be decomposed into its tangential and normal components $V_T$ and $V_N$, respectively, using $\alpha$, which is the tangential angle (Equations S14–S17):

According to the resistive force theory, the forces produced by each element in the tangential and normal direction are linearly related to the velocity in those directions:

where ${\xi }_T$ and ${\xi }_N$ are drag coefficients in the tangential and normal direction, respectively. Because the amplitudes of all harmonics are small in comparison to sperm length, we may posit the following assumptions:

Using the approximations in Equations S20 and S21, we can write the tangential and normal velocities and forces using Equations S22–S25:

The force produced by each segment in the x and y directions can be described as follows:

Plugging Equations S24 and S25 into Equations S26 and S27:

yields Equations S30 and S31, which describe the forces produced by the flagellum in the x and y directions.