Figure 6.

A comparison of predictions of the virtual sperm flagellar waveform and observation in a highly viscous medium. (a) The sea urchin micrographs from figure 1b are reproduced, together with net plots of predicted flagellar waveforms (upper: Sp = 25, k = 6π, a = 1.3, as in figure 4a, bottom row; lower: Sp = 25, k = 4π, a = 2.09, as in figure 4a, 4th row). The naked axoneme cross-section in the plot emphasizes both that sea urchin sperm has no accessory structures and that the modelling predictions are for virtual sperm with the homogeneous non-tapering stiffness function, e₁. (b) Human sperm micrographs from figure 1b are reproduced, together with plots of simulated waveform netplots (upper: Sp = 20, k = 4π, a = 2.55, as in figure 3d, 2nd row; lower: Sp = 25, k = 6π, a = 2.21, as in figure 3d, 3rd row). The dressed human sperm flagellum cross section in the plot emphasizes both that the human sperm has accessory structures and that the modelling predictions are for virtual sperm with the tapering stiffness function, e₃. The sea urchin and human sperm micrographs are, respectively, reproduced with permission (licence no. 4543770361396) from [10] and with permission from [11]. For the observations in (a), the high viscosity was achieved by adding 2% methylcellulose to the medium with different molecular weights specified by a nominal viscosity of, respectively, 1.5 Pa.s (upper) and 4 Pa.s (lower). For (b), 1% methylcellulose was added for the upper frame and 2% methylcellulose added for the lower frame. However, the methylcellulose used had the same molecular weight, specified by a nominal viscosity of 4 Pa.s for a 2% methylcellulose aqueous solution at 20°C. The resulting methylcellulose solutions in (b) were explicitly measured using a cone-and-plate rheometer [10] with an effective viscosity given by 0.14 Pa.s (upper) and 1.6 Pa.s (lower), while the beating frequency was 11 Hz (upper) and 3 Hz (lower). Explicit measurements for the fluids in the observations of (a) are not available, though the averaged beat frequencies reported in [11] for nominal viscosities of 1.5 Pa.s (upper) and 4 Pa.s (lower) are, respectively, 3.8 Hz (a) and 2.8 Hz (b). The flagellum length reported for (a) was approximately 42 μm while for (b) it was is 50 μm. The sperm number for each case (a, upper; a, lower; b, upper; b, lower), respectively, is Sp = 23, 27, 20, 26. However, the Sp estimates for the experiments in (a) are likely to carry systematic errors as the viscosity of the medium was not explicitly measured. Micrographs in (a,b) reprinted from [3] with permission from Elsevier. (Online version in colour.)