Non-Sinusoidal Bending Waves of Sperm Flagella*

ABSTRACT Photomicrographs giving greater resolution than previously available have been obtained for the spermatozoa of a sea urchin, a tunicate, and an annelid. The bending waves are not sine waves. They contain circular arcs, in which the amount of bending is constant, separated by shorter regions which are nearly straight. The flagellum must be effectively very stiff to maintain this waveform as it moves through the medium, but this stiffness must be greatly reduced at the junctions between straight and circular regions, where the flagellum is actively bending and unbending. This type of wave pattern may be generated by a simple ‘on-or-off’ activation of bending elements, but the amount of bending which occurs can be influenced by environmental conditions, such as the viscosity of the medium. A mathematical analysis of the wave movement can be carried out more easily than with sinusoidal waves, and expressions have been obtained for the forward swimming speed, the distribution of bending moment along the flagellum, and the energy expenditure for movement, in terms of the parameters of the waves. The expression for the forward swimming speed accurately predicts the swimming speed of headless spermatozoa, confirming the validity of the hydrodynamic assumptions used in this analysis.