Figures and data in Feeding rates in sessile versus motile ciliates are hydrodynamically equivalent

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Tables
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8 figures, 4 tables and 1 additional file

Figures

Figure 1

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Phylogenetic tree.

(A) Phylogenetic tree showing microorganisms known to feature cilia that generate feeding currents in either sessile (blue) or free swimming (purple) states. The class of diatoms – non-motile cells that sink when experiencing nutrient limitation – is shown for comparison. (B) Flow fields around a sessile ciliate, swimming ciliate, and sinking diatom, in lab and body frame of references. Streamlines are shown in blue in the lab frame $(X, Y, Z)$ .

Figure 2

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Stokeslet and envelope models of sessile and motile ciliates.

(A) Stokeslet model where ciliary activity is represented by a Stokeslet force $F_{c i l i a}$ is located at a distance $(L - a) / a$ outside the spherical cell surface with no-slip surface velocity. (B) Envelope model where cilia activity is distributed over the entire cell surface with slip surface velocity. (**C, D**) Fluid streamlines (white) and nutrient concentration fields (colormap) in the sessile and swimming cases. Here, $L / a = 2$ , $a = 1$ and $F_{c i l i a}$ is chosen to generate a swimming speed $U = 2 / 3$ in the motile case to ensure consistency with the envelope model. (**E, F**) Nutrient uptake in sessile and motile Stokeslet-sphere model based on calculation of clearance rate $Q$ of a fluid volume passing through an annular disk of radius $R / a = 1.1$ and Sherwood number Sh. In the latter, Pe is 100. (G) Nutrient uptake in sessile and motile envelope model based on calculation of Sherwood number Sh as a function of Pe. (H) Difference in clearance rate $Δ Q = Q_{m o t i l e} - Q_{s e s s i l e}$ and Sherwood number $Δ S h = Δ I / I_{d i f f u s i o n} = {S h}_{m o t i l e} - {S h}_{s e s s i l e}$ in the Stokeslet-sphere model for $L / a = 1.1$ and $L / a = 2$ and in the envelope model. In both metrics, the difference is less than 20%: $Δ Q$ is less than 20% the advective flux $π R^{2} U$ and $Δ I$ is less than 20% of the corresponding diffusive uptake $I_{d i f f u s i o n} = 4 π R D C_{\infty}$ . The shaded gray area denotes when the sessile strategy is advantageous.

Figure 3

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Sherwood number versus Péclet number for the sinking (green) diatom and the swimming (purple) and sessile (blue) ciliates based on the envelope model.

(A) Shifted Sherwood number (Sh - 1) versus Péclet number in the logarithmic scale for a range of Pe from 0 to 1000. Pe numbers associated with experimental observations of diatoms (square), swimming ciliates (triangle), and sessile ciliates (circle) are superimposed. Corresponding Sh numbers are calculated based on the mathematical model. Empty symbols are for oxygen diffusivity $D = 1 \times 10^{- 9} m^{2} \cdot s^{- 1}$ and the solid symbols correspond to the diffusivity $D = 4 \times 10^{- 10} m^{2} \cdot s^{- 1}$ of live bacteria (Berg, 2018). (**B–C**) Asymptotic analysis (dashed lines) of Sherwood number in the large Péclet limit (B) and small Péclet limit (C).

Figure 4

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Robustness to variations in cilia coverage and absorption fraction.

We considered a 50% cilia coverage and 50% absorption fraction located at back, middle, and front of the (A) sessile and (B) motile sphere. Concentration fields and Sherwood numbers with 100% cilia coverage and absorption area are shown in the top right corner. In all other cases, the Sh number is reported as a percentage of the full coverage/absorption case.

Appendix 1—figure 1

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Geometry of ciliates and diatoms.

(A) Representative morphologies of surveyed species of sessile ciliates (blue), swimming ciliates (purple), and sinking diatoms (green). (B) Simplified shapes of above organisms. The volumes of those shapes can be obtained as $V_{c y l i n d e r} = π D^{2} L / 4$ , $V_{c o n e} = π D^{2} L / 12$ , and $V_{s p h e r o i d} = π W^{2} L / 6$ . Later on, we further simplify those shapes to an equal volume-based sphere with radius $a$ , calculated as the formula shown in B for each geometry.

Appendix 1—figure 2

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Raw data of character flow speed and size for sessile (blue) and swimming (purple) ciliates, and sinking diatoms (green).

For motile organisms, the characteristic flow speed $U$ is the swimming speed, while for sessile organisms, the characteristic speed $U$ is the maximum flow speed reported near the organism. The characteristic length $a$ is based on the volume-equivalent spherical radius. (A) Ratio of flow speed to length scale $U / a$ in regular scale. (B) Ratio of flow speed to length scale $U / a$ in logarithmic scale. (C) Advection strength $a U$ in logarithmic scale.

Appendix 1—figure 3

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Stokeslet model.

(A) Flow field around sessile and swimming spheres. Solution in non-dimensional form for $a = 1$ and $B = F_{c i l i a} / (8 π μ a) = 1$ . (B) Fluid around a motile and sessile sphere with same point force strength at distance $L = [2 a, 5 a, 10 a]$ , respectively. (C) Normalized clearance varies as annular encounter disk $R$ for above same three point force distances. $L = [2 a, 5 a, 10 a]$

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Tables

Table 1

Survey of size $a$ and flow measurements $U$ in sessile and swimming ciliates and sinking diatoms.

Size $a$ is calculated using the volume-equivalent spherical radius. Corresponding ranges of Pe numbers are based on the diffusivity of oxygen, $D = 10^{- 9} m^{2} \cdot s^{- 1}$ , live bacteria, $D = 4 \times 10^{- 10} m^{2} \cdot s^{- 1}$ ,, and dead bacteria $D = 2 \times 10^{- 13} m^{2} \cdot s^{- 1}$ .

	Empirical measurements		Péclet number, Pe
	microorganism size	characteristic speed	oxygen diffusivity	live bacteria diffusivity	dead bacteria diffusivity
Sessile ciliates	15–60	50–2500	1–80	2–210	(5–400)×10³
Swimming ciliates	15–180	50–3200	1–160	8–390	(17–800)×10³
Sinking diatoms	10–120	40–210	0.4–23	–	–

Appendix 1—table 1

Envelope model subject to treadmill slip velocity.

Mathematical expressions of boundary conditions, fluid velocity field, pressure field, forces acting on the sphere, hydrodynamic power, and speed for each representative case: sessile ciliated sphere, freely swimming ciliated sphere, and sinking (non-ciliated) sphere. All quantities are given in dimensional form in terms of the radial distance $r$ and angular variable $μ = \cos θ$ .

	B.C. at the surface of the sphere	B.C. at infinity
Sessile ciliated sphere	${u_{θ} \|}_{r = a} = B \sqrt{1 - μ^{2}}$ , ${u_{r} \|}_{r = a} = 0$	${u \|}_{r \to \infty} = 0$
Swimming ciliated sphere	${u_{θ} \|}_{r = a} = B \sqrt{1 - μ^{2}}$ , ${u_{r} \|}_{r = a} = 0$	${u \|}_{r \to \infty} = - U e_{z}$
Sinking (non-ciliated) sphere	${u \|}_{r = a} = 0$	${u \|}_{r \to \infty} = - U e_{y}$
	Fluid velocity field
Sessile	$u_{r} (r, μ) = (\frac{a^{3}}{r^{3}} - \frac{a}{r}) B μ, u_{θ} (r, μ) = \frac{1}{2} (\frac{a^{3}}{r^{3}} + \frac{a}{r}) B \sqrt{1 - μ^{2}}$
Swimming	$u_{r} (r, μ) = (- \frac{2}{3} + \frac{2 a^{3}}{3 r^{3}}) B μ, u_{θ} (r, μ) = (\frac{2}{3} + \frac{a^{3}}{3 r^{3}}) B \sqrt{1 - μ^{2}},$
Sinking	$u_{r} (r, μ) = (- 1 + \frac{3 a}{2 r} - \frac{a^{3}}{2 r^{3}}) U μ, u_{θ} (r, μ) = (1 - \frac{3 a}{4 r} - \frac{a^{3}}{4 r^{3}}) U \sqrt{1 - μ^{2}}$
	Fluid velocity field in lab frame	Far-field signature
Sessile	same as above	Force monopole ( $u \sim 1 / r$ ) (Stokeslet)
Swimming	$u_{r} (r, μ) = \frac{2 a^{3}}{3 r^{3}} B μ, u_{θ} (r, μ) = \frac{a^{3}}{3 r^{3}} B \sqrt{1 - μ^{2}}$	Potential dipole ( $u \sim 1 / r^{3}$ )
Sinking	$\begin{array}{ll} u_{r} (r, μ) = (\frac{3 a}{2 r} - \frac{a^{3}}{2 r^{3}}) U μ, \\ u_{θ} (r, μ) = (- \frac{3 a}{4 r} - \frac{a^{3}}{4 r^{3}}) U \sqrt{1 - μ^{2}} \end{array}$ ,	Force monopole ( $u \sim 1 / r$ ) (Stokeslet)

Appendix 1—table 2

Appendix 1—table 1 extension.

	Fluid pressure field	Forces on sphere
Sessile	$p (r, μ) = p_{\infty} - η \frac{a}{r^{2}} B μ$	$F = 4 π η a B e_{z}$
Swimming	$p (r, μ) = p_{\infty} - η \frac{a}{r^{2}} B μ$	$F = 4 π η a B e_{z}$ , $D = - 6 π η a U e_{z}$
Sinking	$p (r, μ) = p_{\infty} + η \frac{3 a}{2 r^{2}} U μ$	$W = - \frac{4}{3} π a^{3} (δ ρ) g e_{y}$ , $D = 6 π η a U e_{y}$
	Hydrodynamic power	Swimming speed
Sessile	$P = 8 π a η B^{2}$	$U = 0$
Swimming	$P = 8 π a η \frac{2}{3} B^{2}$	$U = \frac{2}{3} B$
Sinking	$P = 6 π a η U^{2}$	$U = \frac{2 g a^{2} (δ ρ)}{9 η}$

Appendix 1—table 3

Expressions for Sh as a function of Pe for sessile and swimming ciliated sphere model, compared to a sinking sphere.

	Large Pe limit		Small Pe limit
	Sherwood number	Reference	Sherwood number	Reference
Sessile	$Sh = \frac{2}{\sqrt{3 π}} {P e}^{\frac{1}{2}}$	Present study	$Sh = 1 + \frac{43}{720} {P e}^{2}$	Present study
Swimming	$Sh = \frac{2}{\sqrt{3 π}} {P e}^{\frac{1}{2}}$	Magar et al., 2003; Michelin and Lauga, 2011	$S h = 1 + \frac{1}{3} P e$	Magar et al., 2003; Michelin and Lauga, 2011
Sinking	$Sh = 0.55 {P e}^{\frac{1}{3}}$	Acrivos and Goddard, 1965; Acrivos and Taylor, 1962; Guasto et al., 2012	$S h = 1 + \frac{1}{3} P e$	Acrivos and Taylor, 1962; Guasto et al., 2012

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Jingyi Liu
Yi Man
John H Costello
Eva Kanso

(2026)

Feeding rates in sessile versus motile ciliates are hydrodynamically equivalent

eLife 13:RP99003.

https://doi.org/10.7554/eLife.99003.3

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Phylogenetic tree.

Stokeslet and envelope models of sessile and motile ciliates.

Sherwood number versus Péclet number for the sinking (green) diatom and the swimming (purple) and sessile (blue) ciliates based on the envelope model.

Robustness to variations in cilia coverage and absorption fraction.

Geometry of ciliates and diatoms.

Raw data of character flow speed and size for sessile (blue) and swimming (purple) ciliates, and sinking diatoms (green).

Stokeslet model.

Survey of size a and flow measurements U in sessile and swimming ciliates and sinking diatoms.

Envelope model subject to treadmill slip velocity.

Appendix 1—table 1 extension.

Expressions for Sh as a function of Pe for sessile and swimming ciliated sphere model, compared to a sinking sphere.

MDAR checklist

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Open citations (links to open the citations from this article in various online reference manager services)

Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

Survey of size $a$ and flow measurements $U$ in sessile and swimming ciliates and sinking diatoms.