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Flagellar motility is critical to natural and many forms of assistedreproduction. Rhythmic beating and wave propagation by the flagellum propelssperm through fluid and enables modulation between penetrative progressivemotion, activated side-to-side yaw and hyperactivated motility associated withdetachment from epithelial binding. These motility changes occur in response tothe properties of the surrounding fluid environment, biochemical activationstate, and physiological ligands, however a parsimonious mechanisticexplanation of flagellar beat generation that can explain motility modulationis lacking. In this paper we present the Axonemal Regulation of Curvature,Hysteretic model (ARCH), a curvature control-type theory based on switching ofactive moment by local curvature, embedded within a geometrically nonlinearelastic model of the flagellum exhibiting planar flagellar beats, together withnonlocal viscous fluid dynamics. The biophysical system is parameterisedcompletely by four dimensionless parameter groupings. The effect of parametervariation is explored through computational simulation, revealing beat patternsthat are qualitatively representative of penetrative (straight progressive),activated (highly yawing) and hyperactivated (non-progressive) modes. Anaylsisof the flagellar limit cycles and associated swimming velocity reveals a cuspcatastrophe between progressive and non-progressive modes, and hysteresis inthe response to changes in critical curvature parameter. Quantitativecomparison to experimental data on human sperm exhibiting typical penetrative,activated and hyperactivated beats shows a good fit to the time-averageabsolute curvature profile along the flagellum, providing evidence that themodel is capable of providing a framework for quantitative interpretation ofimaging data.

Axonemal regulation by curvature explains sperm flagellar waveform modulation