Flow-induced buckling dynamics of sperm flagella

The swimming sperm of many external fertilizing marine organisms face complex fluid flows during their search for egg cells. Aided by chemotaxis, relatively weak flows are known to enhance sperm-egg fertilization rates through hydrodynamic guidance. However, strong flows have the potential to mechanically inhibit flagellar motility through elastohydrodynamic interactions—a phenomenon that remains poorly understood. Here we explore the effects of flow on the buckling dynamics of sperm flagella in an extensional flow through detailed numerical simulations, which are informed by microfluidic experiments and high-speed imaging. Compressional fluid forces lead to rich buckling dynamics of the sperm flagellum beyond a critical dimensionless sperm number, Sp, which represents the ratio of viscous force to elastic force. For nonmotile sperm, the maximum buckling curvature and the number of buckling locations, or buckling mode, increase with increasing sperm number. In contrast, motile sperm exhibit a local flagellar curvature due to the propagation of bending waves along the flagellum. In compressional flow, this preexisting curvature acts as a precursor for buckling, which enhances local curvature without creating new buckling modes and leads to asymmetric beating. However, in extensional flow, flagellar beating remains symmetric with a smaller head yawing amplitude due to tensile forces. The flagellar beating frequency also influences the maximum curvature of motile sperm by facilitating sperm reorientation relative to the compressional axis of the flow near stagnation points. These combined simulations and experiments directly illustrate the microscopic elastohydrodynamic mechanisms responsible for inhibiting flagellar motility in flow and have possible implications for our understanding of external fertilization in dynamic marine systems.

Figure 1

Top row: Time series overlay of nonmotile sperm flagella under varying strain rates from high-speed imaging experiments. The extensional flow is generated using a microfluidic cross-channel device, where the magenta plus sign indicates the location of the hyperbolic point in the extensional flow. Flagellar color represents the measured curvature. Bottom row: Sperm flagellar shapes obtained from numerical simulation under similar conditions to the above experiments. The vector field indicates the extensional flow used in the model.

Figure 2

Schematic of the model sperm geometry, where $\mathbf{r}$ represents the position vector for a point on the centerline of the flagellum. $X-Y$ coordinates denote a fixed frame and $s$ represents the arclength of flagellum measured from the cell body. Internal shear force field, which generates active flagellar beating, is shown by $f(s,t)$. Unit vectors $\mathbf{t}$ and $\mathbf{n}$ are tangent and normal vectors to the flagellar centerline, respectively. $\psi (s,t)$ is the angle between the tangent vector and the horizontal axis.

Figure 3

Time-curvature plots of nonmotile sperm flagella in an extensional flow. Panels (a) and (e) are the time-curvature plots of flagellar buckling obtained from experiments and correspond to the flagellar shapes shown in Fig. 1 and 1. The remaining plots are time-curvature plots obtained from numerical simulations for a given Sp.

Figure 4

(a) The maximum absolute curvature of the flagellum at different Sp. Red solid circle is the critical ${\mathrm{Sp}}_{\mathrm{cr}}\approx 3.8$ required for flagellar buckling. Dotted vertical lines separate the region of different buckling modes determined by identifying the buckling locations with $\left|C\right|>0.3$ in the time-curvature plots. Blue solid circle is the $C_{\max }$ of flagellum obtained from experiments and correspond to Figs. 1, 1, and 1. The small schematics of sperm represent the shape of the flagellum, when the maximum curvature occurs for a given Sp. (b) Growth rate (Re(σ)) for different eigenmodes of a sperm flagellum ($\zeta =0.2$) obtained from a linear analysis. Dotted vertical lines separate the region of different buckling modes obtained from nonlinear analysis. Constant growth rates ($\sigma =1,2$) correspond to free translation and rotation of the sperm.

Figure 5

The maximum absolute curvature decreases with initial misalignment of the flagellum from the upstream compressional axis for different Sp.

Figure 6

Time-curvature plot of numerically simulated motile sperm flagellar dynamics in an extensional flow. The internal driving shear force, $f(s,t)$, was obtained from a mathematically modeled symmetric flagellar beating pattern, ${\psi }_0(s,t)=cos(2\pi s-\mathrm{\Omega }t)$, in a quiescent fluid. Sp is constant in each column and beating frequency ($\mathrm{\Omega }$) is constant in each row.

Figure 7

The instantaneous difference between the maximum curvature of principal and reverse bend ($\mathrm{\Delta }C$) vs time (inset). The maximum value of $\mathrm{\Delta }C$ at different Sp and $\mathrm{\Omega }$. This value corresponds to the red solid circle in the inset.

Figure 8

The ratio of flagellar beating amplitude in the tensile zone of the extensional flow to the flagellar beating amplitude in the quiescent fluid vs Sp at different beating frequencies.

Figure 9

The maximum curvature of the sperm flagellum, having $f(s,t)$ obtained from a mathematical model of the waveform, at different Sp and flagellar beating frequencies ($\mathrm{\Omega }$) in the upstream (compressional) zone as well as downstream (tensile) zone of the extensional flow. We consider $0.1\le s\le 0.9$ to calculate $C_{max}$ in the tensile regime. The small schematics of sperm (except blue) depict the shape of the sperm flagellum, when the maximum curvature occurs during the compression of the flagella. The schematics colored blue represent the shape of flagellum having symmetric beating.

Figure 10

The maximum curvature of the simulated flagellum during buckling in the absence of sperm head (solid black curve). Dotted vertical lines separate Sp regimes of different buckling modes determined by identifying the buckling locations with $\left|C\right|>0.3$ in the time-curvature plots. Red solid circles are the theoretically predicted Sp in the literature [38, 79] for the transition of buckling modes of an elastic filament through a linear analysis. In the absence of a sperm cell body (head), the critical Sp of the flagellar buckling was found to be the same as the theoretical prediction for an elastic fiber, ${\mathrm{Sp}}_{\mathrm{cr}}=4.26$.

Figure 11

Time-curvature plot of nonmotile sperm flagella for different initial misalignment angles $\varphi$. Sp is constant in each row.

Figure 12

The location of the instantaneous maximum flagellar curvature in the presence of extensional flow compared to that in a quiescent fluid for the simulations shown in supplemental video 4 [77].

Figure 13

Time-curvature plots of numerically modeled motile sperm flagella, for which $f(s,t)$ has been obtained directly from high-speed imaging experiments, in an extensional flow at different Sp and beating frequencies. The flow doesn't significantly affect the curvature close to the proximal end of flagellum. Therefore, we omit data close to the head.

Figure 14

The maximum curvature of sperm flagella, having $f(s,t)$ obtained directly from experimentally measured flagellar waveforms for sea urchin sperm (A. punctulata), at different Sp and synthetic flagellar beating frequencies ($\mathrm{\Omega }$) in the upstream (compressional) zone and downstream (tensile) zone of the extensional flow. We consider $0.1\le s\le 0.9$ to calculate $C_{max}$ in the tensile regime. The small schematics of sperm depict the shape of the sperm flagellum, when the maximum curvature occurs during the compression of the flagella.