Bimodal rheotactic behavior reflects flagellar beat asymmetry in human sperm cells

Abstract

Rheotaxis, the directed response to fluid velocity gradients, has been shown to facilitate stable upstream swimming of mammalian sperm cells along solid surfaces, suggesting a robust physical mechanism for long-distance navigation during fertilization. However, the dynamics by which a human sperm orients itself relative to an ambient flow is poorly understood. Here, we combine microfluidic experiments with mathematical modeling and 3D flagellar beat reconstruction to quantify the response of individual sperm cells in time-varying flow fields. Single-cell tracking reveals two kinematically distinct swimming states that entail opposite turning behaviors under flow reversal. We constrain an effective 2D model for the turning dynamics through systematic large-scale parameter scans, and find good quantitative agreement with experiments at different shear rates and viscosities. Using a 3D reconstruction algorithm to identify the flagellar beat patterns causing left or right turning, we present comprehensive 3D data demonstrating the rolling dynamics of freely swimming sperm cells around their longitudinal axis. Contrary to current beliefs, this 3D analysis uncovers ambidextrous flagellar waveforms and shows that the cell's turning direction is not defined by the rolling direction. Instead, the different rheotactic turning behaviors are linked to a broken mirror symmetry in the midpiece section, likely arising from a buckling instability. These results challenge current theoretical models of sperm locomotion.

Keywords: fluid dynamics; microfluidics; rheotaxis; simulations; sperm swimming.

Fig. 1.

Turning behavior of human sperm under flow reversal reveals two kinematically distinct swimming states. (A) Trajectories of individual sperm cells swimming close to the channel boundary in the $\left(x,y\right)$ plane, with initial positions superimposed at time $t=0$ (viewed from inside the channel). Equal-time trajectory averages for left- and right-turning cells are shown as thick white-shaded lines. Flow was reversed at $t=0$, pointing in positive x direction for $t>0$ (white arrow). The shear velocity increases linearly in z direction. Color encodes time. (Scale bar, 200 μm.) (B) Normalized speed distributions before flow reversal at time $t=0$. Faint lines indicate left-turning cells, and the other lines indicate right-turning cells. (C) Distribution of the orientation angles $\phi \left(0\right)$, measured relative to the x axis before flow reversal at time $t=0$, signals two kinematically distinct cell populations. Colors in B and C indicate different viscosities (black, 1 cSt; red, 3 cSt; blue, 12 cSt).

Fig. 2.

Characteristics of the turning process for the cell trajectories in Fig. 1, using the same color coding for viscosities, with filled (unfilled) symbols indicating right-turning (left-turning) trajectories. (A) The persistence length $\mathrm{\Lambda }$, defined as the maximum of the x component of each mean trajectory in Fig. 1A, shows weak variation with shear rate $\dot{\gamma }$. (B) The mean turning time T, defined as the time after which the x component of a mean trajectory reaches its maximum, is approximately inversely proportional to the shear rate, $T\propto 1/\dot{\gamma }$.

Fig. S1.

The effective 2D model defined by Eqs. 1 and 2 captures the ensemble turning dynamics. (A) Thin lines show individual simulated trajectories, and thick lines show the corresponding equal time averages. For comparison, the thick white-shaded lines are experimentally measured averages from Fig. 1A. Sample size $n_L=n_R=100$. (Scale bar, 200 μm.) (B) Fit parameters used in A, obtained from systematic scans of $\mathrm{6,480}$ parameter combinations (Parameter Scans). Distribution parameters of the self-swimming speed V were directly estimated from the experiments. The same color coding as in Fig. 1 is used. Filled (unfilled) symbols are mean values for right-turning (left-turning) cells. Error bars indicate SDs when model values were sampled from distributions (Parameter Scans).

Fig. 3.

A 3D flagellar beat reconstruction reveals that a mirror symmetry breaking in the midpiece curvature separates left-turning from right-turning sperm. (A) A 2D bright-field image and tracked flagellum in the head-centered comoving frame, with arc-length s and normal line n. (Scale bar, 10 μm.) (B) A 3D beat reconstruction in the head-centered frame (Beat Reconstruction and Movie S7). (C) Typical 3D beat plane rotation for a single sperm, seen from head-on, with beat period of ∼0.08 s. The circular arrow indicates rolling direction of the flagellum. (D) The cumulative beat plane rotation θ, shown for 10 typical samples of left-turning (L) and right-turning (R) cells, implies that the rolling and turning direction are not correlated. (E) Midpiece curvature, quantified by the bend angle δ between the tangent at $s=0$ and the secant through $s_c\approx 4$ μm, correlates strongly with the turning direction (three different donors, sample size in brackets). (F) A 3D reconstruction reveals ambidextrous helicity in the first ∼70% of the flagellum.

Fig. S2.

(A) Tracked 2D flagellum in the focal plane of the microscope in the head-centered comoving frame $\left(\tilde{x},\tilde{y}\right)$, in which the $\tilde{x}$ axis is aligned with the mean swimming direction; s denotes the arc length, and n denotes the line normal to the flagellum. (Scale bar, 10 μm.) (B) Intensity cross section along the flagellum. The dashed white line, $n=0$, denotes the position of the flagellum centerline. Whenever the centerline passes through a dark (bright) region, the z coordinate is above (below) the focal plane, indicated in green (red) shaded regions. Two sample intensity profiles $S1$ and $S2$ are shown for illustration. (C) The width of the intensity maxima determines the z displacement, allowing for the 3D reconstruction of the flagellum (Movie S7).

Fig. S3.

(A) Schematic of coarse-grained double-ellipsoid approximation of 3D beat geometry. (B, Top) Direct analysis of the projected 2D beat patterns in terms of tangent angle $\alpha \left(s_{*}\right)$ results in spurious beat modes (highlighted in gray), reflected by a second peak in the power spectrum (C, Top). These spurious modes disappear when the 3D profile is analyzed in terms of the angle $\beta \left(s_{*}\right)$ between two inertial ellipsoids (B, Bottom), leaving a single dominant frequency (C, Bottom).

Fig. S4.

The 3D beat reconstruction for eight representative samples of left-turning (L) and right-turning (R) sperm, viewed from the front in the comoving frame $\left(\tilde{y},\tilde{z}\right)$. Color encodes time.

Fig. S5.

(A) Flagelloid curve for mouse sperm, reproduced from ref. . (B) Flagelloid curves for human sperm, obtained by observing the sperm head-on and tracking the motion of a single point located at $s=15$ μm.

Fig. S6.

Illustration of the planarity measure by means of the inertia ellipsoid and its two minor axes $\left|{\mathit{r}}_{+}\right|$ and $\left|{\mathit{r}}_{-}\right|$.

Fig. S7.

The time series of the planarity measure P reveals a mostly planar beat pattern. Shown are eight representative samples for left-turning (red) and right-turning (blue) sperm. Black lines denote the time-averaged planarity ${\langle P\rangle }_t\approx 0.2$.

Fig. S8.

Spatiotemporal plot of the local helicity h for eight representative samples for left- and right-turning sperm, illustrating the absence of a persistent helicity in the first ∼30 μm of the flagellum.

Fig. S9.

(A) Black lines show time-averaged local helicity, ${\langle h\left(s\right)\rangle }_t$, along the flagellum for eight representative samples. Individual data points from each frame are shown to illustrate the spread over time (red, left-turning; blue, right-turning), indicating no persistent helicity. Although a slight tendency toward positive average helicity values is observed near the posterior parts of the flagellum, this bias does not correlate with the turning direction. (B) Helicity SD $\sigma \left(h\right)$ along the flagellum.

Fig. S10.

Averaged flagellar helicity H as function of time for eight representative samples. No distinct helicity discerning right-turning (blue) from left-turning (red) sperm cells is apparent. Black solid lines indicate the time averages of H.

Fig. S11.

Average midpiece curvature, measured by the bend angle δ (see Fig. 3E), correlates with the turning behavior (eight typical samples shown): (A) Even though the individual δ-time series fluctuate considerably, (B) the distributions exhibit different mean angles for right-turning (blue) and left-turning (red) sperm cells.

Fig. 4.

Turning mechanisms implied by 3D data. Left-turning (red) and right-turning (blue) sperm cells roll their flagellum counterclockwise with a conical beat envelope, resulting in a left-turning torque. For right-turning sperm, this effect is counteracted by a larger opposing force due to the tilted head acting as a hydrofoil or “rudder.”