(A) Sperm–wall interaction categories. Rolling sperm rotate and align with the wall upon contact to swim stably along it . Non-rolling sperm exhibiting circular motion either do not contact the wall (), detach from the wall after temporarily swimming along it , or swim slowly along the wall depending on the magnitude and direction of angular velocity. (B) Sperm detention time on the wall, that is, . (C) Sperm normalized velocity on the wall, that is, . (D) A schematic to illustrate the definition of . (E) versus . (F) Free body diagram for the , (G) , and (H) categories. (I) Rolling results in alterations between and , forming . (J) The phase curves describing the dynamics of the angle between the sperm and wall after contact. Filled dots are stable points. Note that negative and positive values correspond to the and categories, respectively. (K) Rolling can be modeled as a transition between two-phase curves obtained for . A frequent transition between results in . (L) The phase curve for a rolling sperm. The final angle between the sperm and wall is ∼10°. (M) The phase curve for a non-rolling sperm. The final angle between the sperm and wall is ∼30°. (N) The distribution of for 20 rolling sperm. is linearly related to for non-rolling sperm (inset plot).