2.1. Propulsion of flagellated eukaryotes

First, we focus on studies providing the details of actuation in flagellated eukaryotes and spermatozoa. We assume that the waveform of their flagellar oscillations can be captured by a characteristic amplitude $h$ and wavelength $\lambda$, as sketched in figure 1(a). The database contains 23 different species of flagellated eukaryotes and 28 different species of spermatozoa for which we found measurements of $h$ and $\lambda$; note that these values were either directly reported in each paper, or were estimated by us using experimental images, see details in Rodrigues et al. (Reference Rodrigues, Lisicki and Lauga2021).

Figure 1. Aspect ratio of eukaryotic flagellar waves. (a) Sketch of flagellated eukaryotic cell with wave amplitude $h$ and wavelength $\lambda$. (b) Histogram of aspect ratios $h/\lambda$ from experiments gathered in the BOSO-Micro database (Rodrigues et al. Reference Rodrigues, Lisicki and Lauga2021). (c) Mean value of $h$ vs $\lambda$ for all flagellated eukaryotes in our database. Spermatozoa are marked with circles, while the remaining flagellated eukaryotes are marked with squares. The solid line shows the least-square fit to the data and the shaded area is the 95 % confidence interval. The dashed line shows the prediction from the optimal theory from Lauga & Eloy (Reference Lauga and Eloy2013) for $Sp=4$ and the dotted line for $Sp=6$. Cells from the same class in the tree of life are plotted using the same colour scheme, with all classes represented here listed in the inset of (b).

For all our data, we plot in figure 1(b) the histogram of the amplitude-to-wavelength aspect ratio, $h/\lambda$. This ratio does not exceed 0.6 in the data set, with most organisms showing the value below 0.25; hence the amplitude of a flagellar beat rarely exceeds a quarter of the wavelength. For further insight, we plot the amplitude $h$ against the wavelength $\lambda$ in figure 1(c). When multiple values are inferred for the same species, we plot in figure 1(c) the average value but include error bars to reflect the variability within the data reported for each species. Square markers represent flagellated eukaryotes, which are generally smaller in size as compared with spermatozoa, marked with circles.

It is clear from the data in figure 1(c) that there is a systematic increase of flagellar amplitude with wavelength. To rationalise and interpret these results, we performed a least-squares linear fit of the $h/\lambda =\text {constant}$ relationship to the data. The solid line shows the result of fitting, with the 95 % confidence interval shaded in grey. For comparison with theoretical predictions, we use the work of Lauga & Eloy (Reference Lauga and Eloy2013) who computed numerically the optimal shape of an active flagellum. They considered an internally forced, elastic planar flagellum deforming periodically and moving at a constant speed through a viscous fluid and found the shape of the flagellum that maximises the swimming speed at a fixed energetic cost, representing irreversible work of molecular motors. They concluded that the optimal shape of the beating flagellum depends on a single dimensionless combination of parameters, termed the Sperm number $Sp$ (Lauga Reference Lauga2020a), and defined as the ratio of the wavelength $\lambda$ to the elasto-hydrodynamic penetration length $\ell$,

(2.1)\begin{equation} Sp = \frac{\lambda}{\ell}, \quad\text{with}\ \ell = \left(\frac{T B}{\zeta_\perp}\right)^{1/4}, \end{equation}

where $T$ is the period of oscillatory actuation, $B$ the bending rigidity of the flagellum, and $\zeta _\perp$ the perpendicular resistance coefficient of the flagellum (i.e. the hydrodynamic force exerted on the flagellum per unit length for motion in the fluid perpendicular to its centreline). Thus the flagellum shape results from an interplay between the stiffness of the filament, viscous drag of the medium and internal forcing. Accordingly, decreasing the value of $Sp$ results in flatter, smoother kinks in the flagellar beating (Lauga & Eloy Reference Lauga and Eloy2013).

The typical values of the dimensionless $Sp$ number for eukaryotes and spermatozoa have been reported to be within the range of 1–10 (Kumar et al. Reference Kumar, Walkama, Guasto and Ardekani2019). For sea-urchin Arbacia punctulata spermatozoa, Pelle et al. (Reference Pelle, Brokaw, Lesich and Lindemann2009) reported a bending rigidity of $B\approx 0.4 \unicode{x2013} 0.9\times 10^{-21}\,{\rm Nm}^2$, while Howard (Reference Howard2001) assumes it to be the upper limit of that range. In direct measurements, Okuno & Hiramoto (Reference Okuno and Hiramoto1979) found the stiffness of echinoderm sperm flagella to be in the range of $B\approx 0.3\unicode{x2013}1.5\times 10^{-21}\,{\rm Nm}^2$, while for Strongylocentrotus purpuratus, Okuno et al. (Reference Okuno, Asai, Ogawa and Brokaw1981) estimated $B\approx 0.8\times 10^{-21}\,{\rm Nm}^2$. Mechanical measurements for the flagella of wild-type Chlamydomonas yield $B = (0.84 \pm 0.28)\times 10^{-21}\,{\rm Nm}^2$ (Xu et al. Reference Xu, Wilson, Okamoto, Shao, Dutcher and Bayly2016), with similar values for mutant strains. In water, we have $\zeta _\perp \approx 2\times 10^{-3}\,{\rm Pa}\,{\rm s}$, and the typical periods $T$ vary across species from 0.02 to 0.05 s (Brennen & Winet Reference Brennen and Winet1977; Gray Reference Gray1955; Rodrigues et al. Reference Rodrigues, Lisicki and Lauga2021). From these numbers, and typical lengths of dozens of $\mathrm {\mu }$m (Rodrigues et al. Reference Rodrigues, Lisicki and Lauga2021), we therefore estimate the values of $Sp$ to be in the range of $Sp \approx 2\unicode{x2013}7$.

To compare the experimental measurements to predictions from theory, we use the work of Eloy & Lauga (Reference Eloy and Lauga2012) where, for a given value of $Sp$, the shape of the optimal flagellum was computed; for each optimal shape, we can then extract the corresponding optimal aspect ratio, $h/\lambda$. In figure 1(c) we plot the aspect ratio as a linear curve $h/\lambda ={\rm constant}$ for $Sp=4$ ($h/\lambda =0.163$, dashed line) and $Sp=6$ ($h/\lambda =0.194$, dotted line). Both lines lie within the 95 % confidence interval of our dataset: the best fit for the data yields an aspect ratio $h/\lambda =0.188$, with the lower and upper bound of the 95 % confidence interval at $0.163$ and $0.213$, respectively ($R^2=0.562$).

2.2. Hydrodynamic drag force on ciliates

Ciliates constitute a large group of unicellular eukaryotic swimmers, characterised by the presence of hair-like cilia which cover their bodies in dense active carpets; a model organism, Paramecium, is sketched in figure 2(a). The internal structure of cilia is identical to eukaryotic flagella but they are typically shorter and used to propel cells with relatively much larger body sizes. This separation of scales has motivated mathematical models dating back to Blake (Reference Blake1971) to represent the action of cilia layers covering the cell bodies as a continuum boundary with prescribed actuation of the surrounding fluid. This surface forcing sets the fluid in motion, and leads to swimming. The balance of propulsion force and viscous drag force on the cell body determines the swimming speed $U$ (Lauga Reference Lauga2020a). Because the latter results from the overall shape of the body, one possibility for cells to increase their swimming speed is to reduce the viscous drag opposing their propulsion by choosing the appropriate drag-minimising shape, which have been characterised theoretically (Pironneau Reference Pironneau1973, Reference Pironneau1974; Montenegro-Johnson & Lauga Reference Montenegro-Johnson and Lauga2014).

Figure 2. Aspect ratio of ciliates cell body. (a) Sketch of ciliate body showing the cell width $W$ and length $B$. (b) Histograms of cell aspect ratios $W/B$ from experimental data gathered in the BOSO-Micro database (Rodrigues et al. Reference Rodrigues, Lisicki and Lauga2021). (c) Mean value of body length $B$ (in $\mathrm {\mu }$m) plotted as a function of width $W$ (in $\mathrm {\mu }$m) for all species in our database. The solid line shows the least-square fit to the data while the shaded area is the 95 % confidence interval. The dashed line shows the prediction from the theory of fixed-volume drag minimisation ($B/W=1.952$) (Pironneau Reference Pironneau1973) while the dotted line shows the drag minimum for a given surface area ($B/W=4.037$) (Montenegro-Johnson & Lauga Reference Montenegro-Johnson and Lauga2015). Cells from the same class in the tree of life are plotted using the same colour scheme, with all classes represented here listed in inset of (b).

To investigate if cells aligned with expectations from drag minimisation, we use our eukaryotic microswimmers database (Rodrigues et al. Reference Rodrigues, Lisicki and Lauga2021) to quantify the extent to which the cell bodies of ciliates minimise drag forces. In figure 2(b), we plot the histogram for the aspect ratio of the cell bodies, $B/W$, of the 91 ciliates in our database. When multiple values were available for the same species, we use their average. Colours are used to code different classes of organisms. Clearly, the reported cells are typically prolate, and most of them have a length of approximately twice their width, although more slender cell bodies are also observed.

To further explore the correlation between the two length scales characterising the cell bodies of ciliates, we present in figure 2(c) a scatter plot of the data, which suggests a linear correlation between cell body and width. We plot the linear least-square fit to the data with a solid line, corresponding to the slope $B/W=2.360$ ($R^2=0.503$), while grey shading marks the 95 % confidence interval, for aspect ratios in the range $(2.057, 2.664)$. Remarkably, these results are very close to the classical prediction of a minimal-drag shape in Stokes flow by Pironneau (Reference Pironneau1973), later computed by Bourot (Reference Bourot1974), who demonstrated numerically that the aspect ratio $B/W=1.952$ is hydrodynamically optimal for bodies with prescribed volumes (the corresponding optimal shape resembles a prolate spheroid with pointy ends). We mark this theoretical prediction with a dashed line in figure 2(c), and note that it lies just outside the confidence interval of our fit to data. In contrast, if instead one considers the shape that has minimum drag for a prescribed surface area (instead of a fixed volume), Montenegro-Johnson & Lauga (Reference Montenegro-Johnson and Lauga2014) found the optimal aspect ratio to be 4.037, marked in figure 2(c) with the dotted line; this lies well outside the data.

It is worth noting that the Stokes drag acting on an optimal fixed-volume prolate shape is only 4.5 % less than that experienced by a sphere of the same volume, suggesting that the energetic cost of diverting from the optimum is actually relatively low. It is therefore perhaps all the more remarkable that such a small energetic gain seems to be reflected in the empirical data. The small magnitude of the energetic improvement might in turn allow a larger cell shape variability and explain the diversity of observed micron-scale shape features.