Rikmenspoel R
Biophys J. 1978 Aug;23(2):177-206. doi: 10.1016/S0006-3495(78)85442-3.
The equation of motion for sperm flagella, in which the elastic bending moment and the active contractile moment are balanced by the moment from the viscous resistance of the surrounding fluid, is solved for a wave solution that superimposes partial solutions. Substitution of the expression for the wave solution into the equation leads to an expression for the active contractile moment. This active moment can be decomposed into two parts. The first part describes an active moment that travels over the flagellum with the mechanical flagellar wave, the second part represents a moment in phase over the entire length of the flagellum, which decreases linearly towards the distal tip. The linear synchronous moment, to which an amount of traveling moment has been added as a perturbation, leads to wave solutions that closely resemble flagellar waves. Properties such as wavelength and wave amplitudes and also the shape of the waves in sea urchin sperm flagella at different frequencies are accurately described by the theory. The change in wave shape in sea urchin sperm flagella at raised viscosity is predicted well by the theory. The different wave properties caused in bull sperm flagella by different boundary conditions at the proximal junction are explained. When only a traveling active moment is present in a flagellum, the wave solutions describe waves of a small wave length in a long flagellum. Some properties of the wave motion of sperm flagella are derived from the theory and verified experimentally.
精子鞭毛的运动方程是通过叠加部分解得到波动解来求解的，在该方程中，弹性弯曲力矩和主动收缩力矩由周围流体粘性阻力产生的力矩平衡。将波动解的表达式代入方程可得到主动收缩力矩的表达式。这个主动力矩可分解为两部分。第一部分描述了一个随机械鞭毛波在鞭毛上传播的主动力矩，第二部分表示在鞭毛整个长度上同相的力矩，该力矩朝着远端末梢呈线性减小。线性同步力矩加上作为微扰的行进力矩量，会产生与鞭毛波非常相似的波动解。该理论准确地描述了海胆精子鞭毛在不同频率下的波长、波幅等特性以及波形。该理论很好地预测了海胆精子鞭毛在粘度升高时波形的变化。解释了在近端连接处不同边界条件在公牛精子鞭毛中引起的不同波动特性。当鞭毛中仅存在行进的主动力矩时，波动解描述了长鞭毛中小波长的波。精子鞭毛波动运动的一些特性是从该理论推导出来并经过实验验证的。
