Doing more with less: The flagellar end piece enhances the propulsive effectiveness of human spermatozoa

Spermatozoa self-propel by propagating bending waves along a predominantly active elastic flagellum. The organized structure of the $9+2$ axoneme is lost in the most-distal few microns of the flagellum, and therefore this region is unlikely to have the ability to generate active bending; as such it has been largely neglected in biophysical studies. Through elastohydrodynamic modeling of humanlike sperm we show that an inactive distal region confers significant advantages, in both propulsive thrust and swimming efficiency, when compared with a fully active flagellum of the same total length. The beneficial effect of the inactive end piece on these statistics can be as small as a few percent but can be above 430%. The optimal inactive length, between 2% and 18% of the total length, depends on both wave number and viscous-elastic ratio, and therefore is likely to vary in different species. Potential implications in evolutionary biology and clinical assessment are discussed.

Figure 1

Schematic of an idealized sperm cell. (a) Cross sections (redrawn from [10]) showing the internal flagellar structure, where SMT denotes singlet microtubules, DMT doublet microtubules, and CP the central microtubule pair. (b) Sketch of the model sperm highlighting the position of the centerline $\mathit{X}(s,t)$ [parametrized by the arc length $s\in [0,L]$ and tangent angle $\theta (s,t)$], head surface $\partial H$ (blue ellipsoid), head angle $\varphi \left(t\right)$, and flagellum-neck junction ${\mathit{X}}_0\left(t\right)$. The active component of the flagellum ($s\in [0,\ell )$) is shown in dark yellow, with the inactive distal end ($s\in [\ell ,L]$) shown by the black dashed line. (c) A tapering fibrous sheath surrounds the core of the flagellum, modeled through the varying elastic stiffness function (2).

Figure 2

Values of the actuation parameter $\mathcal{M}$, optimizing (a) swimming speed VAL and (b) the Lighthill efficiency $\eta$ for $\mathcal{S}\in [9,18]$ and $k\in [3\pi ,5\pi ]$.

Figure 3

Effect of the inactive end piece length on swimming speed and efficiency of propulsion, for simulations actuated with $\mathcal{M}={\mathcal{M}}_{\text{VAL}}$. (a) swimming speed VAL, (b) Lighthill efficiency $\eta$, and relative increase in (c) VAL and (d) efficiency when comparing the optimally active and fully active flagella, for viscous-elastic parameter choices $\mathcal{S}\in [9,18]$ and wave numbers $k\in [3\pi ,5\pi ]$.

Figure 4

(a) Normalized VAL and (b) normalized Lighthill efficiency $\eta$ for $\mathcal{S}\in [9,18]$ versus $\ell \in [0.5,1]$, shown for five choices of dimensionless wave number $k$. Values in each plot are normalized with respect to either (a) the maximum VAL or (b) the maximum $\eta$ for each $(\mathcal{S},k)$ pair. The optimal choices of $\ell$ are highlighted as red dots. (c) Active length that optimizes VAL (${\ell }_{\text{VAL}}$) and (d) active length that optimizes the Lighthill efficiency ${\ell }_{\eta }$, for $\mathcal{S}\in [9,18]$ and $k\in [3\pi ,5\pi ]$. Simulations are actuated with $\mathcal{M}={\mathcal{M}}_{\text{VAL}}$.

Figure 5

Effect of the inactive end piece length on swimming speed and efficiency of propulsion, for simulations actuated with $\mathcal{M}={\mathcal{M}}_{\eta }$. (a) swimming speed VAL, (b) Lighthill efficiency $\eta$, and the relative increase in (c) VAL and (d) efficiency when comparing the optimally active and fully active flagella, for viscous-elastic parameter choices $\mathcal{S}\in [9,18]$ and wave numbers $k\in [3\pi ,5\pi ]$.

Figure 6

(a) Normalized VAL and (b) normalized Lighthill efficiency $\eta$ for $\mathcal{S}\in [9,18]$ versus $\ell \in [0.5,1]$, shown for five choices of dimensionless wave number $k$. Values in each plot are normalized with respect to either (a) the maximum VAL or (b) the maximum $\eta$ for each $(\mathcal{S},k)$ pair. The optimal choices of $\ell$ are highlighted as red dots. (c) Active length that optimizes VAL (${\ell }_{\text{VAL}}$) and (d) active length that optimizes the Lighthill efficiency ${\ell }_{\eta }$, for $\mathcal{S}\in [9,18]$ and $k\in [3\pi ,5\pi ]$. Simulations are actuated with $\mathcal{M}={\mathcal{M}}_{\eta }$.

Figure 7

Swimming track following the head for both fully active and optimally inactive sperm (top and bottom of each pair, respectively) for a selection of parameter pairs ($\mathcal{S},k$) actuated with $\mathcal{M}={\mathcal{M}}_{\text{VAL}}$ and simulated for ten cycles of the active moment. In each pair the sperm is plotted at the final time point with the optimally inactive length $\ell ={\ell }_{\text{VAL}}$, shown in orange.

Figure 8

Flagellar waveform for a fully active and optimally inactive sperm (left and right of each pair, respectively), simulated for a selection of parameter pairs ($\mathcal{S},k$) and actuated with $\mathcal{M}={\mathcal{M}}_{\text{VAL}}$. The color of the flagellum indicates progression through a single beat, from blue to yellow. For each pair the sperm are plotted with the optimally inactive length ${\ell }_{\text{VAL}}$ shown in orange.

Figure 9

Swimming track following the head for both fully active and optimally inactive sperm (top and bottom of each pair, respectively) for a selection of parameter pairs ($\mathcal{S},k$) actuated with $\mathcal{M}={\mathcal{M}}_{\eta }$ and simulated for ten cycles of the active moment. For each pair the sperm are plotted at the final time point with the optimally inactive length ($\ell ={\ell }_{\eta }$) shown in orange.

Figure 10

Flagellar waveform for a fully active and optimally inactive sperm (left and right of each pair, respectively) simulated for a selection of parameter pairs ($\mathcal{S},k$) and actuated with $\mathcal{M}={\mathcal{M}}_{\eta }$. The color of the flagellum indicates progression through a single beat, from blue to yellow. For each pair the sperm are plotted with the optimally inactive length $\ell ={\ell }_{\eta }$, shown in orange.

Figure 11

Comparison of the flow fields around a simulated sperm at $t=8\pi$ with $\mathcal{S}=13.5$ and $k=4\pi$ for (a) a cell actuated with $\mathcal{M}={\mathcal{M}}_{\text{VAL}}$ and (b) $\mathcal{M}={\mathcal{M}}_{\eta }$. In each case we plot (i) a cell with a fully active flagellum, (ii) an optimally inactive cell, and (iii) the difference between flow fields, with positive values corresponding to the optimally inactive case having a faster fluid velocity. (i) and (ii) Instantaneous fluid streamlines are shown in white with arrows indicating direction and the fluid magnitude is indicated by the colorbar for each panel. In (ii) and (iii) the inactive part of the flagellum is shown in black. For additional time points see Appendix pp2.

Figure 12

Comparison between simulated and experimental data to verify the choice of elastic stiffness parameters. (a) The mean absolute value of curvature for both a cell simulated with $(\mathcal{S},k)=(18,5\pi )$ (blue) and an experimentally tracked sperm (magenta) is plotted against arc length. A waveform comparison is shown for the (b) simulated and (c) experimental cell (tracked with fast [15], scale bar denotes $5\mu \mathrm{m}$). In the case of the simulated waveform, the inactive end piece is shown in orange.

Figure 13

Comparison of the flow fields around a simulated sperm actuated with $\mathcal{M}={\mathcal{M}}_{\text{VAL}}$ at $t=8\pi$ with $\mathcal{S}=13.5$ and $k=4\pi$, for (a) a cell with a fully active flagellum and (b) an optimally inactive cell, each shown at four time points in the final simulated beat. (c) Difference between flow fields, with positive values corresponding to the optimally inactive case having a faster fluid velocity. (a) and (b) Instantaneous fluid streamlines are shown in white with arrows indicating direction and the fluid magnitude is indicated by the colorbar for each panel. In (b) the inactive part of the flagellum is shown in black.

Figure 14

Comparison of the flow fields around a simulated sperm actuated with $\mathcal{M}={\mathcal{M}}_{\eta }$ at $t=8\pi$ with $\mathcal{S}=13.5$ and $k=4\pi$, for (a) a cell with a fully active flagellum and (b) an optimally inactive cell, each shown at four time points in the final simulated beat. (c) Difference between flow fields, with positive values corresponding to the optimally inactive case having a faster fluid velocity. (a) and (b) Instantaneous fluid streamlines are shown in white with arrows indicating direction and the fluid magnitude is indicated by the colorbar for each panel. In (b) the inactive part of the flagellum is shown in black.