The heads of live spermatozoa of the sea urchin Hemicentrotus pulcherrimus were held by suction in the tip of a micropipette mounted on a piezoelectric device and vibrated either laterally or axially with respect to the head axis. Within certain ranges of frequency and amplitude, lateral vibration of the pipette brought about a stable rhythmic beating of the flagella in the plane of vibration, with the beat frequency synchronized to the frequency of vibration [Gibbons et al. (1987), Nature 325, 351–352]. The sperm flagella, with an average natural beat frequency of 48 Hz, showed stable beating synchronized to the pipette vibration over a range of 35–90 Hz when the amplitude of vibration was about 20 microns or greater. Vibration frequencies below this range caused instability of the beat plane, often associated with irregularities in beat frequency. Frequencies above about 90 Hz caused irregular asymmetrical flagellar beating with a marked decrease in amplitude of the propagated bends and a skewing of the flagellar axis towards one side; the flagella often stopped in a cane shape. In flagella that were beating stably under imposed vibration, the wavelength was reduced at higher frequencies and increased at lower frequencies. When the beat frequency was equal to or lower than the natural beat frequency, the apparent time-averaged sliding velocity of axonemal microtubules, obtained as twice the product of frequency and bend angle, decreased with beat frequency in both the proximal and distal regions of the flagella. However, at vibration frequencies above the natural beat frequency, the sliding velocity increased with frequency only in the proximal region of the flagellum and remained essentially unchanged in more distal regions. This apparent limit to the velocity of sliding in the distal region may represent an inherent limit in the intrinsic velocity of active sliding, while the faster sliding observed in the proximal region may be a result of passive sliding or elastic distortion of the microtubules induced by the additional energy supplied by the vibrating pipette. Axial vibration with frequencies either close to or twice the natural beat frequency induced cyclic changes in the waveform, compressing and expanding the bends in the proximal region, but did not affect bends in the distal region or alter the beat frequency.
A Perturbation Analysis Approach for Studying the Effect of Swelling Kinetics on Instabilities in Hydrogel Plates
