To quantify and compare nutrient uptake across microorganisms, we approximated the cell body by a sphere of radius a, as typically done in modeling sessile and swimming ciliates (Blake, 1971b; Magar et al., 2003; Michelin and Lauga, 2011; Andersen and Kiørboe, 2020) and sinking diatoms (Riley, 1952; Karp-Boss et al., 1996; Kanso et al., 2021; Figure 2).
The fluid velocity u around the sphere is governed by the incompressible Stokes equations, −∇p+η∇2u=0 and ∇⋅u=0, where p is the pressure field and η is viscosity. We solved these equations in spherical coordinates (r,θ,ϕ), considering axisymmetry in ϕ and proper boundary conditions. In the motile case, we solved for the fluid velocity field u in body frame by superimposing a uniform flow of speed U equal to the swimming speed past the sphere; we calculated the value of U from force balance considerations (Dölger et al., 2017; Andersen and Kiørboe, 2020) (see SI for details).
We solved the Stokes equations for two models of cilia activity: cilia represented as a Stokeslet force Fcilia placed at a distance L and pointing towards the center of the sphere and no-slip velocity at the spherical surface (Blake, 1971a; Wróbel et al., 2016; Andersen and Kiørboe, 2020; Kim and Karrila, 1991; Figure 2A), and densely packed cilia defining an envelope model with a slip velocity u|r=a=Usinθ at the spherical surface where all Cilia exert tangential forces pointing from one end of the sphere to the opposite end (Blake, 1971b; Michelin and Lauga, 2010; Michelin and Lauga, 2011; Figure 2B). Detailed expressions of the flow fields and governing equations in both models are included in the SI (Appendix 1—tables 1 and 2). In dimensionless form, we set the cell’s length scale a=1 and tangential velocity scale U=1 in the envelope model, and we set the ciliary force Fcilia in the Stokeslet model to produce the same swimming speed (U=2/3) as in the envelope model when the sphere is motile.
To evaluate the steady-state concentration of dissolved nutrients around the cell surface, we numerically solved the dimensionless advection-diffusion equation Peu⋅∇C=ΔC in the context of the Stokeslet and envelope models. Here, the advective and diffusive rates of change of the nutrient concentration field C, normalized by its far-field value C∞, are given by Peu⋅∇C and ΔC, respectively, with ∇C the concentration gradient. At the surface of the sphere, the concentration is set to zero to reflect that nutrient absorption at the surface of the microorganism greatly exceeds transport rates of molecular diffusion (Berg and Purcell, 1977; Bialek, 2012; Short et al., 2006).
In Figure 2C and D, flow streamlines (white) and concentration fields (colormap at Pe = 100) are shown in the Stokeslet and envelope models. In the sessile sphere, ciliary flows drive fresh nutrient concentration from the far field towards the ciliated surface. These fresh nutrients thin the concentration boundary layer at the leading surface of the sphere, where typically the cytostome or feeding apparatus is found in sessile ciliates, with a trailing plume or ‘tail’ of nutrient depletion. Similar concentration fields are obtained in the swimming case, albeit with a narrower trailing plume.
To assess the effects of these cilia-generated flows on the transport of nutrients to the cell surface, we used two common metrics of feeding. First, we quantified fluid flux or clearance rate Q through an encounter zone near the organism’s oral surface (Christensen-Dalsgaard and Fenchel, 2003; Pepper et al., 2013; Shekhar et al., 2023). Namely, following Andersen and Kiørboe, 2020, we defined the clearance rate Q=−2π∫aRu⋅ez|z=0rdr, normalized by the advective flux πR2U, over an annular encounter zone of radius R extending radially away from the cell surface (Figure 2A). Second, we quantified the concentration flux of dissolved nutrients at the cell surface (Kanso et al., 2021; Michelin and Lauga, 2010; Michelin and Lauga, 2011). To this end, we integrated the inward concentration flux I=∫SD∇C⋅n^dS, normalized by the diffusive nutrient uptake Idiffusion=4πaDC∞ to get the Sherwood number Sh=I/Idiffusion. We applied both metrics to each of the Stokeslet and envelope models.
In Figure 2E, we report the clearance rate Q in the context of the Stokeslet model as a function of the ciliary force location L/a for a small annular encounter zone of radius R=1.1a extending away from the cell surface. Swimming is always more beneficial. However, the increase in clearance rate due to swimming is less than 10%. This is in contrast to the several-fold advantage obtained in Andersen and Kiørboe, 2020 for L=4a and R=10a. (results of Andersen and Kiørboe, 2020 are reproduced in Fig. S3). We employed the same metric Q in the envelope model and found that motility is also more advantageous, albeit at less than 5% benefit (Figure 2H).
A few comments on the choice of the size of the encounter zone are in order. Nutrient encounter and feeding in ciliates occur near the leading edge of the ciliary band (Gilmour, 1978; Thomazo et al., 2021; Jiang and Buskey, 2025a; Jiang and Buskey, 2025b). Cilia are typically of the order of 10 microns in length, and the cell body of a ciliate is typically in the range of 10–1000 microns. We chose R=1.1a indicating encounter within an annular protrusion extending 10% beyond the body radius because it falls within the biological range and because a larger encounter zone would induce additional drag on the body that needs to be accounted for in the model. In contrast, Andersen and Kiørboe, 2020 chose an encounter zone extending up to 900% the body radius, without accounting for the drag that such a large collection area would add to a swimming body. This also exceeds biological considerations in most ciliates and flagellates, even in Choanoflagellates (Nielsen et al., 2017) and Chlamydomonas (Nielsen et al., 2017), where the flagellum length could be up to six times the cell radius.
In Figure 2F and G, we report the Sh number based on the Stokelet and envelope models, respectively. In the Stokeslet model (Figure 2F), sessile spheres do better when the cilia force is close to the cell surface (L−a)/a⪅1.25. In the envelope model (Figure 2G), motile spheres do slightly better for all Pe⪅100. The difference ΔSh between the sessile and motile spheres favors, by less than 20%, the sessile strategy in the Stokeslet model and the swimming strategy in the envelope model (Figure 2H).
Comparing Sh between the Stokeslet and envelope models (Figure 2C and D), we found that, at Pe = 100, Sh = 2.7 (sessile) and 2.6 (motile) in the Stokeslet model compared to Sh = 6.7 (sessile) and 6.9 (motile) in the envelope model. This is over a twofold enhancement in nutrient uptake at the same swimming speed U=2/3 simply by distributing the ciliary force over the entire surface of the cell! Indeed, this improvement occurs because the ciliary motion in the envelope model significantly thins the concentration boundary layer along the entire cell surface as opposed to only near where the cilia force is concentrated in the Stokeslet model.
In our survey of sessile and motile ciliates (Figure 1), cilia are clearly distributed over the cell surface. Thus, we next explored in the context of the envelope model the behavior of the Sh number across a range of Pe values that reflect empirical values experienced by the surveyed ciliates (Table 1).