Takumi Ogawa, Shuji Koyama, Toshihiro Omori, and Kenji KikuchiDepartment of Finemechanics, Tohoku University,6-6-01 Aoba, Aramaki, Aoba-ku, Sendai 980-8579, JapanHélène de MalepradeInstitut Jean Le Rond d’Alembert, Sorbonne Université, CNRS UMR 7190, 75005 Paris, FranceRaymond E. GoldsteinDepartment of Applied Mathematics and Theoretical Physics,University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United KingdomTakuji IshikawaDepartment of Biomedical Engineering, Tohoku University,6-6-01 Aoba, Aramaki, Aoba-ku, Sendai 980-Sponges, the basalmost members of the animal kingdom, exhibit a range of complex architecturesin which microfluidic channels connect multitudes of spherical chambers lined with choanocytes,flagellated filter-feeding cells. Choanocyte chambers can possess scores or even hundreds of suchcells, which drive complex flows entering through porous walls and exiting into the sponge channels.One of the mysteries of the choanocyte chamber is its spherical shape, as it seems inappropriate forinducing directional transport since many choanocyte flagella beat in opposition to such a flow. Herewe combine direct imaging of choanocyte chambers in living sponges with computational studies ofmany-flagella models to understand the connection between chamber architecture and directionalflow.
We studied the effect of changing the flagellar wave number while keeping the flagellar length fixed. Interestingly, as shown in Fig. 8, the outlet flow rate and mechanical pumping efficiency exhibit peaks at intermediate values of k, while the particular value of the peak k differs between the two. The mechanical pumping efficiency reaches a maximum when at the relatively low wave num7 !(a)(b)(c)(d)(e)"!"#"%"[]"[]"[]"[]!=10!=45!=90 FIG. 9. Correlation between pumping function and outlet opening angle. (a) Computational domain for three values of a. (b-e) Chamber properties as a function of opening angle, keeping area fraction in narrow ranges: red: 0.074 0.093, blue: 0.31 0.34, and green: 0.39 0.49. (b) Outlet flow rate. (c) Work rate of flagella. (d) Maximum pressure in the chamber. (e) Mechanical pumping efficiency.
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That higher wave numbers lead to reduced efficiency, despite the reduced work rate of the flagella, arises from an effect similar to that found when the chamber radius is reduced; Since L is fixed, shrinks at higher wave numbers, increasing the space at the chamber center from which flagella are absent, and the central pressure reduces with higher wave number. The wave number associated with the efficiency is maximized is compared with the experimental results in Section V. E. Efficiency is maximized at intermediate a
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While the choanocyte chamber diameter, apopyle area, and diameter were examined in previous studies [22, 51, 52], it has remained unclear how the apopyles aperture ratio affects the pumping function of the chamber. We examined the pumping function a [10, 90] while keeping the area fraction within each of three narrow ranges, adjusting the number of flagella with a accordingly. Figure 9(b-e) shows the variation in pumping functions with a. With all three flagella densities, similar 4 5 2 6 (a)(b) 3 FIG. 10. Wave number of flagella. (a) Mechanical pumping efficiency as a function of the wave number with color scheme as in Fig. 9 for in the ranges 0.074 0.093 (red), 0.31 0.34 (blue), and 0.39 0.49 (green). (b) Flagellar motion of a choanocyte of E. muelleri. White arrows indicate flagella. Colored curves represent extracted flagellar waveforms.
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The outlet flow rates reaches a maximum at a 40 60. On the other hand, the maximum pressure decreases monotonically with a, an effect that arises from the reduction in the the number of flagella directed against the bulk flow as a increases. Thus, the spherical shape of the choanocyte chamber has the effect of increasing pressure. Since the pumping efficiency is the product of the maximum pressure and the flow rate, its peak in Fig. 9(e) shifts to the lower a regime of 20 50 compared to the case of the flow rate (cf. Fig. 9(b)). From these results, we conclude that flagella around apopyles, which seem to disturb unidirectional flow, contribute to creating the high pressure rise and that the choanocytes with intermediate but small a can achieve high mechanical pumping efficiency.
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