Keller J B, Rubinow S I
Biophys J. 1976 Feb;16(2 Pt 1):151-70. doi: 10.1016/s0006-3495(76)85672-x.
The swimming motion of a microorganism with a single flagellum is investigated for both helical and planar flagellar motion. First the force and torque exerted on the organism by the surrounding fluid are calculated in terms of the specified flagellar motion and the unknown linear and angular velocity of the whole organism. Then these unknown velocities are determined by the condition that the net force and torque on the organism are zero. Using these velocities, the trajectory of the organism is found. In the case of helical flagellar motion, the path of the entire organism is found to be a helix of small radius. The axis of the flagellum is not parallel to the axis of the helical path, but makes a small angle with it and precesses around it. If the flagellar motion is planar and sinusoidal, then the trajectory of the organism is found to be a straight line with small oscillations about it. Each point of the flagellum also oscillates longitudinally with double the frequency of the transverse oscillation, producing a figure eight motion. However if the flagellar motion is planar and asymmetric, then the trajectory is found to be a circle with small superposed oscillations. These conclusions account for the observed helical and circular trajectories of sperm, and for the figure eight motion of the tip of the flagellum in the planar case.
针对具有单个鞭毛的微生物的螺旋状和平面状鞭毛运动，研究了其游动方式。首先，根据特定的鞭毛运动以及整个生物体未知的线速度和角速度，计算周围流体施加在生物体上的力和扭矩。然后，通过生物体上的合力和扭矩为零这一条件来确定这些未知速度。利用这些速度，得出生物体的轨迹。在螺旋状鞭毛运动的情况下，整个生物体的路径是一个小半径的螺旋线。鞭毛的轴不与螺旋路径的轴平行，而是与其成一个小角度并绕其做进动。如果鞭毛运动是平面且正弦式的，那么生物体的轨迹是一条带有围绕其的小振荡的直线。鞭毛的每个点也以横向振荡频率的两倍进行纵向振荡，产生一种“8”字形运动。然而，如果鞭毛运动是平面且不对称的，那么轨迹是一个带有小叠加振荡的圆。这些结论解释了所观察到的精子的螺旋状和圆形轨迹，以及在平面情况下鞭毛尖端的“8”字形运动。
