Pairwise hydrodynamic interactions of synchronized spermatozoa

The journey of mammalian spermatozoa in nature is well known to be reliant on their individual motility. Often swimming in crowded microenvironments, the progress of any single swimmer is likely dependent on their interactions with other nearby swimmers. While the complex dynamics of lone spermatozoa have been well-studied, the detailed effects of hydrodynamic interactions between neighbors remain unclear, with inherent nonlinearity in the pairwise dynamics and potential dependence on the details of swimmer morphology. In this study, we will attempt to elucidate the pairwise swimming behaviors of virtual spermatozoa, forming a computational representation of an unbound swimming pair and evaluating the details of their interactions via a high-accuracy boundary element method. We have explored extensive regions of parameter space to determine the pairwise interactions of synchronized spermatozoa, with synchronized swimmers often being noted in experimental observations, and have found that two-dimensional reduced autonomous dynamical systems capture the anisotropic nature of the swimming speed and stability arising from near-field hydrodynamic interactions. Focusing on two initial configurations of spermatozoa, namely those with swimmers located side-by-side or above and below one another, we have found that side-by-side cells attract each other, and the trajectories in the phase plane are well captured by a recently proposed coarse-graining method of microswimmer dynamics via superposed regularized Stokeslets. In contrast, the above-below pair exhibit a remarkable stable pairwise swimming behavior, corresponding to a stable configuration of the plane autonomous system with swimmers lying approximately parallel to one another. At further reduced swimmer separations, we additionally observe a marked increase in swimming velocity over individual swimmers in the bulk, potentially suggesting a competitive advantage to cooperative swimming. These latter observations are not captured by the coarse-grained regularized Stokeslet modeling or simple singularity representations, emphasizing the complexity of near-field cell-cell hydrodynamic interactions and their inherent anisotropy.

Figure 1

Computational meshes representing spermatozoan geometry. (a) Views of a triangular mesh representing the head of a spermatozoon, with vertices created in ellipsoidal bands and equally spaced around each ellipsoidal cross section, following the parametrization given in the Appendix. (b) An example of adaptive meshing based on swimmer proximity, with vertices shown as points and elements omitted. The distal portions of the flagella of both swimmers have been partially remeshed to increase local accuracy while retaining computational efficiency. The mesh is shown at reduced resolution for visual clarity, though the numerous vertices at each cross section of the flagellum are effectively amalgamated into one ellipsoidal marker point due to the resolution of the image. Finally, note that these flagella correspond to a flagellum length of $56\mu \mathrm{m}$, a standard scale for human spermatozoa [43].

Figure 2

Schematic of virtual spermatozoa configurations and accompanying parametrizations, with free-space headings being shown as dashed lines and their separation $h$ represented by a dotted line. (a) Swimmers share a common beat plane in the side-by-side configuration, with $\theta$ defined as the signed planar angle between their free-space headings. (b) In the above-below configuration, the swimmers' relative heading is defined as the signed angle between their beat planes, as shown, where we are making the approximation that the relative motion of the swimmers is expressible solely by the quantities $h$ and $\theta$, justified a posteriori in Secs. 3c and 3b. In both configurations, $\theta =0$ corresponds to the virtual spermatozoa sharing both beat plane orientation and free-space heading, and we adopt the sign convention such that $\theta >0$ denotes spermatozoa facing away from one another.

Figure 3

The transient dynamics of side-by-side swimming, with trajectory start and end points shown as red circles and blue diamonds, respectively. (a) Each thin curve shows the positions of the head-tail junctions ${\widehat{\mathit{x}}}_0^i$ of a pair of spermatozoa in the laboratory frame ${\widehat{x}}_1{\widehat{x}}_2{\widehat{x}}_3$ for side-by-side swimmers with initial conditions $(h,\theta )=(35\mu \mathrm{m},0.1\pi )$ and $(h,\theta )=(40\mu \mathrm{m},0)$ for the upper and lower panels, respectively, and simulated over 20 periods of the flagellar beat. Averaged swimmer paths are shown as heavy black curves, highlighting separation in the upper panel and convergence of the swimmers in the lower panel. Shown in gray are the swimmers in their original configurations, with the cell bodies omitted here for clarity. (b) Pairwise dynamics in the $h\text{−}\theta$ space, with the projections of sample long-time simulations, typically simulated for 20 periods of the flagellar beat, shown as solid red curves. Solutions of the approximate plane autonomous system are shown as dotted curves with the same initial conditions as the long-time simulations, demonstrating very good agreement between the full and reduced systems, serving as a posteriori validation of the reduction to $h\text{−}\theta$ space. Further, the dynamics as predicted by the phase plane exhibit no stable pairwise swimming, merely the behaviors of eventual collision and separation off into the bulk, though sperm initially heading away from one another are predicted to sometimes still be on a path that will result in eventual collision.

Figure 4

The approximate dynamics of above-below swimming, highlighting a pairwise swimming mode that corresponds to stable long-time motion in close proximity, represented here by the existence of an attracting stable spiral around $h\approx 19.1\mu \mathrm{m}$, $\theta \approx 0.0076\pi$. Projections of long-time simulations, computed for 20 periods of the flagellar beat and serving as a posteriori validation of the phase plane, are shown as red curves, with start and end points displayed as red circles and blue diamonds, respectively. In particular, a long-time simulation initiated from approximately the stationary point of the autonomous system shows that swimmers indeed undergo negligible relative motion, verifying the existence of a stable swimming configuration. Solutions of the approximate plane autonomous system are shown as dotted curves with the same initial conditions as the long-time simulations, demonstrating very good agreement between the full and reduced systems. Trajectories can be seen to spiral inward, approaching the stationary point on timescales on the order of hundreds of beat cycles.

Figure 5

Change in swimmer separation and relative heading over a single beating period as a function of initial separation $h$, denoted $\mathrm{\Delta }h$ and $\mathrm{\Delta }\theta$, respectively. Initially the swimmers are situated directly above or beside one another, and with $\theta =0$. Above-below swimmers, shown as solid curves, can be seen to undergo reduced changes in relative heading and separation when compared to side-by-side swimmers, shown as dot-dashed curves, highlighting a disparity in the timescales of relative motion between swimmer configurations.

Figure 6

The linear velocity of a swimmer over a single beating period as a function of separation, $h$, in both above-below and side-by-side configurations. With swimmers initially directly above/beside one another, and shown here with $\theta =0$, observed for both configurations is a reduction in swimming speed with separation between swimmer head-tail junctions, though the magnitude of such reduction is notably greater in the case of side-by-side swimming than for above-below swimmers, the change in the latter being difficult to observe on the scale of the above plot. At further-reduced separations, below approximately $10\mu \mathrm{m}$, remarkably, virtual spermatozoa in an above-below configuration can be seen to increase in speed, surpassing even their free-space swimming speed of approximately $77\mu \mathrm{m}{\mathrm{s}}^{-1}$.

Figure 7

Instantaneous flow field in the plane of beating for an individual virtual spermatozoon in the laboratory frame ${\widehat{x}}_1{\widehat{x}}_2{\widehat{x}}_3$, with the swimmer shown in red and situated at the origin of the laboratory frame. The flow field as computed by the boundary element method is given in (a), with the corresponding regularized singularity representation flow field shown in (b). (c) An illustration of the low-dimensional regularized singularity representation, comprised of the lowest two PCA modes and the mean over a beating period, with arrow size and direction corresponding to the strength and direction of the regularized Stokeslets. The circle size corresponds to the regularization parameter taken for each regularized Stokeslet, and inset is the normalized weighting of each of the PCA modes over a single beating period. Each regularized Stokeslet lies on the line ${\widehat{x}}_2=0$ by the symmetry of the flagellar beat, with those corresponding to the mean flow here being shown offset for visual clarity. Qualitative details of the instantaneous flow field can be seen to be well-approximated by the low-order regularized singularity representation. The magnitude of each flow field is given by the intensity of the shading, with projected streamlines shown in white.

Figure 8

Comparison of swimmer behaviors as predicted by the boundary element method and PCA-derived singularity representations. (a) The PCA-derived singularity representations (SR), shown here as blue curves, agree qualitatively with the boundary element method (BEM) predictions for linear swimming speed in the case of side-by-side swimming, though no such agreement is present in close proximity for swimmers in an above-below configuration. (b), (c) Evolution in $h\text{−}\theta$ space for (b) side-by-side and (c) above-below swimmers as predicted by singularity representations, shown as black arrows, with trajectories of the plane autonomous system shown as dotted black curves. Superimposed as solid red curves are projections of boundary element simulations onto the reduced space, with start and end points shown as red circles and blue diamonds, respectively. We note good agreement in the case of side-by-side swimming, though there is little correspondence between the predictions of the boundary element method and the singularity representations for swimmers in an above-below configuration. Levels of agreement are further evidenced by comparison with Figs. 3 and 4.