An elastohydrodynamical simulation study of filament and spermatozoan swimming driven by internal couples

Eukaryotic cell swimming is frequently actuated via the flagellum, which is a slender flexible appendage driven by waves of internal couples generated by dynein molecular motors. Here we adapt a regularized elastohydrodynamic model of flagella by Simons et al. (2015, A fully three-dimensional model of the interaction of driven elastic filaments in a Stokes flow with applications to sperm motility, J. Biomech., 48(9, SI), 1639--1651) to consider active filament models of sperm swimming given the patterns of dynein couples indicated by the analysis of flagellate digital microscopy, with a further consideration of helically beating flagella. We additionally consider whether boundary accumulation and rheotaxis, as predicted by modelling studies with prescribed flagellum waveforms, are inherited once flagellum elasticity is accommodated with the flagellum deforming in response to the mechanical forces it experiences. However, the simulations presented here are limited to filament morphologies and thus either headless sperm or those with a filamentous head. We find that simple patterns of dynein contraction generate flagellar waveforms that are qualitatively similar to observation and also that flagellum buckling instabilities predicted by resistive force theory elastohydrodynamical models need not occur. Furthermore, modelling simulations with prescribed helical waveforms qualitatively match elastohydrodynamic modelling predictions in the context of boundary accumulation but flagellum elasticity is predicted to ameliorate the impact of background flows for flagellates with a helically beating flagellum near a surface. In contrast, elastohydrodynamic modelling predictions for boundary accumulating flagellates with planar waveforms indicate that boundary behaviours are subtle for these cells and fully reproducing observed behaviours, while accommodating dynamic flagellar deformation, raises further modelling challenges in the study of swimming cells.