The movement of spermatozoa with helical head: theoretical analysis and experimental results

The present work is concerned with the study of the swimming of flagellated microscopic organisms with a helical head and a helical pattern of flagellar beating, such as Xenopus sperms. The theoretical approach is similar to that taken by Chang and Wu (1971) in the study of helical flagellar movement. The model used in the present study allows us to determine the velocity of propulsion (U) and the frequency of rotation of the sperm head (fh) as a function of the frequency of the wave of motion (ft) traveling along the tail. The results relative to the case of helical and planar flagellar waves are compared. Our main finding is that the helical shape of the head seems to increase the efficiency of propulsion of the spermatozoon when compared with the more commonly shaped spherical head. Experimentally measured values of fh versus U may be fitted by a linear plot whose slope is much higher than that corresponding to the case of planar flagellar beating. This fact is consistent with an effectively three-dimensional (nonplanar) movement of the flagellar tail. However, the results do not fit those predicted from a circular helix, suggesting that a different shape of the flagellar beating should be considered.