Pairwise hydrodynamic interactions of synchronized spermatozoa

Abstract

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. Whilst 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 coarsegraining 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.

Regularized representation of bacterial hydrodynamics

“…where U others is the velocity field generated by the other swimmers, averaged over the locations of the regularized singularities in the representation of the kth swimmer. Derived from the force and torque-free conditions appropriate for a neutrally buoyant microswimmer, analogous expressions hold for the angular velocity of the swimmers, as derived in previous works [35,38], given explicitly as…”

“…We compute, to high accuracy, the motion of a bacterial swimmer using the boundary element method [37], as implemented and verified by Walker et al [38], imposing force and torque-free conditions on the virtual swimmer. We adopt the approximate morphology of the monoflagellate P. aeruginosa [39], representing the body as a spherocylinder and the flagellum as a rigid helix rotating with constant angular velocity relative to the swimmer body, with the details of the flagellar shape as in Dasgupta et al [39], Ishimoto [40].…”

Modelling Motility: The Mathematics of Spermatozoa

“…Elastohydrodynamic modelling has since been extended to consider planar motions of flagella (Llopis et al, 2013;Goldstein et al, 2016;Taketoshi et al, 2020), with Taketoshi et al (2020) extending the boundary element method to include flagellar elasticity. This latter work again explored pairwise dynamics, numerically concluding that spermatozoa enjoy increased swimming speed when beating in synchrony, in contrast to the similar but conditional results of the prescribed-beat study of Walker et al (2019b). Further, three-dimensional studies have also been initiated (Simons et al, 2015), though there remains significant scope for the investigation of the effects of rheology, confinement, multiplicity, and the details of the driving force behind the spermatozoan beat.…”

“…The most accurate and flexible is computational fluid dynamics , which is limited in accuracy only by computational resource, machine precision constraints, and the accuracy of the underlying physics, such as the common and broadly appropriate assumption of neglecting inertia on the grounds that it is subordinate to viscous effects and induces only tiny errors. The most common of such approaches for sperm motility are the boundary element methods (Pozrikidis, 2002 ), which have been extensively exploited to study sperm dynamics (for example Phanthien et al, 1987 ; Ramia et al, 1993 ; Ishimoto and Gaffney, 2014 ; Ishimoto et al, 2016 ; Walker et al, 2019b ). However, such methodologies suffer from a relatively complex formulation, in terms of both the underlying mathematics and scientific computation.…”

“…(D) Pairwise swimming of bull spermatozoa. (A,B) reprinted from Walker et al ( 2019b ), Copyright (2019) by the American Physical Society. (D) republished with permission of the Company of Biologists Ltd from Woolley et al ( 2009 ); permission conveyed through the Copyright Clearance Centre.…”