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.