Nonlinear instability in flagellar dynamics: a novel modulation mechanism in sperm migration?

Abstract

Throughout biology, cells and organisms use flagella and cilia to propel fluid and achieve motility. The beating of these organelles, and the corresponding ability to sense, respond to and modulate this beat is central to many processes in health and disease. While the mechanics of flagellum -fluid interaction has been the subject of extensive mathematical studies, these models have been restricted to being geometrically linear or weakly nonlinear, despite the high curvatures observed physiologically. We study the effect of geometrical nonlinearity, focusing on the spermatozoon flagellum. For a wide range of physiologically relevant parameters, the nonlinear model predicts that flagellar compression by the internal forces initiates an effective buckling behaviour, leading to a symmetry-breaking bifurcation that causes profound and complicated changes in the waveform and swimming trajectory, as well as the breakdown of the linear theory. The emergent waveform also induces curved swimming in an otherwise symmetric system, with the swimming trajectory being sensitive to head shape-no signalling or asymmetric forces are required. We conclude that nonlinear models are essential in understanding the flagellar waveform in migratory human sperm; these models will also be invaluable in understanding motile flagella and cilia in other systems.

Bimodal rheotactic behavior reflects flagellar beat asymmetry in human sperm cells

“…Future theoretical studies should focus on the interplay between rolling dynamics, wall interactions, and structural and elastic inhomogeneities in midpiece and flagellum. The basic ingredients for a bimodal rheotactic response-cell rolling and beat curvature asymmetry-are generic features of many sperm species (42,48,60,61). It will thus be important to investigate experimentally if bimodal rheotactic response occurs in other species, in particular those featuring strong head asymmetries that may critically affect the proposed hydrofoil effect.…”

The counterbend phenomenon in flagellar axonemes and cross-linked filament bundles

“…Since the discovery of the axoneme, interfilament sliding (17) has been the cornerstone of prevailing models to describe flagellar dynamics (13,14,18,(21)(22)(23)(24)(25)(26). Brokaw (18) first considered the microtubule sliding mechanism in modeling flagellar locomotion by incorporating both active sliding and passive interfilament sliding resistance.…”

“…Based on Brokaw's model, Hines and Blum (21) later derived the geometrically nonlinear elastohydrodynamic equations for the motion of a sperm flagellum. Thereafter, several studies considered the sliding filament theory (17,18) via different active control hypotheses (13,14,18,(21)(22)(23)(24)(25)(26) to investigate the generation of flagellar bending. Nevertheless, although each competing hypothesis is capable of successfully generating bending waves that "resemble" in vitro observations (13,14,18,(21)(22)(23)(24)(25)(26), without fundamentally understanding the bulk material properties arising from the flagellar structure, with a disentanglement of the trinity of contributions, viscous drag, passive structural response, and molecular motor forces, it is unclear which active control model, if any, can provide a quantitative understanding of the regulation and function of the internal mechanics.…”

“…Thereafter, several studies considered the sliding filament theory (17,18) via different active control hypotheses (13,14,18,(21)(22)(23)(24)(25)(26) to investigate the generation of flagellar bending. Nevertheless, although each competing hypothesis is capable of successfully generating bending waves that "resemble" in vitro observations (13,14,18,(21)(22)(23)(24)(25)(26), without fundamentally understanding the bulk material properties arising from the flagellar structure, with a disentanglement of the trinity of contributions, viscous drag, passive structural response, and molecular motor forces, it is unclear which active control model, if any, can provide a quantitative understanding of the regulation and function of the internal mechanics. Further uncertainties are equally associated with the use of the classical estimates of flagellar material quantities (1,2,8,10) to model the dynamical behavior of flagellar systems, because they depend on reliable estimates of the mechanical properties of the system (13,14,18,(21)(22)(23)(24)(25)(26).…”

Cooperative self-propulsion of active and passive rotors

“…We will refer to the first class of rotors driven by external torques as passive rotors, while the name of active rotors will be reserved to those that are internally driven. Several examples of active rotors are found in the living world, including sperm cells [8][9][10][11][12] , bacteria [13][14][15][16] and algae 17 near a solid surface. Various artificial swimmers, inspired by their living counterparts, have also been engineered over the past decade, and provide realizations of active rotors [18][19][20][21] .…”