Figure 2. A simulation of sperm movement with helical waveform.

(A) A snapshot of a model spermatozoon with helical waveform (Supplemental Movie 2). The red and blue spots indicate the edges of the widest part of the flattened head. (B) Time course of the simulated swimming spermatozoon with the same helical waveform as illustrated in A, viewed from the head side (Supplemental Movie 2). Each snapshot was taken at intervals of 0.4 beat cycle, and was vertically shifted so that the centers of the head are horizontally aligned. The red and blue spots correspond to the same points shown in A, demonstrating that the head rotates clockwise as time progresses. (C) Simulation of a single swimming spermatozoon connected to an egg. The translation and rotation after 10 seconds are shown, though the flagellum is emphatically illustrated for visualization. The spermatozoon exerts force and torque, and in turn, the egg is moved and rotated clockwise when viewed from the head. The egg rotation can be recognized by the marked points. The estimated rotation velocity by a single spermatozoon is 0.34 rpm, which might be insufficient for recognition in real observation. (D) Schematic pictures of the force and torque exerting on the egg (yellow) in sample cases with simple sperm configuration (black), viewed from above. The red arrows indicate the total force exerting on the egg, and the rotation direction by the whole torque is shown by blue arrows. When two spermatozoa attached from opposite directions, both the net force and torque vanish (second from the left), while the egg may rotate right or left by the binding of a third spermatozoon.