Introduction

Nutrient transport is fundamental to life; without it, cells cannot sustain the vital processes required for cell function and growth. Early life forms, like bacteria and archaea, depended on diffusion to transport dissolved nutrients to and across the cell membrane1,2; a physical process that strictly limited cell size3,4,5. The evolution of phagotrophy—the ability of cells to ingest organic particles as food (Fig. 1)—defined one of the most significant events in the history of life6,7,8. Phagotrophy was accompanied by development of specialized mechanisms to both the cell exterior—for food particle collection - and to the cell interior - for material transport within the cell7,9. These external and internal features fundamentally distinguished single-celled eukaryotes from earlier archaea and bacteria10, allowing them to attain much larger size and structural complexity. In turn, these changes led to the emergence of diverse unicellular predation strategies7, the evolution of multicellularity11, and the development of the modern food webs that determine Earth’s habitability5,12,13. However, despite its key role in the evolution of life, the link between intracellular and extracellular transport in single-celled eukaryotes remains largely unexplored.

Fig. 1: Ciliates utilise phagotrophy to ingest food particles.
Fig. 1: Ciliates utilise phagotrophy to ingest food particles.
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A An illustration of swimming (purple) and sessile (blue) ciliates in a drop of water, showing that ciliates can be over 1000 times bigger than bacterial cells. Magnified illustration of (B). Swimming Paramecium and (C). Attached Stentor, where cilia beat in concert to direct food particles and prey to the cell feeding apparatus. Cilia densely cover the Paramecium cell surface, while the Stentor generates feeding currents using ciliary bands that are localised around the feeding apparatus.

In this study, we explore this critical link in ciliates. Ciliates are important aquatic protistan grazers, characterised by the presence of hair-like organelles called cilia. Cilia protrude from the cell surface and act as microactuators that move fluid across the cell surface to enable the cell to swim or generate feeding currents (Fig. 1)5,14. The ciliary activity directs fluid towards the cell's “mouth”15 or cytostome, where food particles are ingested and subsequently processed into usable molecules that get transported within the cell16. Intracellular transport in eukaryotes is powered by molecular motors along complex microtubular and actin highways and is a highly organised process that is conserved across cell types and evolutionary lineages9. Interestingly, the same materials – microtubules and dynein motors – compose the highly conserved axoneme structure that powers the cilia responsible for driving extracellular transport, implying an intimate co-evolution of intracellular and extracellular transport. However, while the internal feeding and ciliary structures are conserved, the arrangements of cilia in cells that swim are markedly different from those in cells that attach and generate feeding currents. The latter exhibit ciliary bands that are localized around the feeding apparatus (Fig. 1C). This raises the following question: do optimal cilia arrangements vary according to the life strategy of the cell, swimming versus attached?

Results

Energetic feasibility of flow-driven nutrient transport in microbal cells

Bacterial cells, which are commonly preyed upon by ciliates, are often 1000-fold smaller in volume (Fig. 1A). Bacteria rely on diffusion for both intracellular transport2,17,18 and the transport of dissolved nutrients to the cell surface1,13 (Fig. 2A). Diffusive transport is optimal when molecule-specific receptors are distributed randomly but uniformly over the entire cell surface1,2 (Fig. 2A, B). No swimming or fluid movement can enhance transport to small bacterial cells beyond the diffusive limit, a restriction dictated by cell size1. This physical constraint may initially seem surprising given that both bacteria and ciliates occupy a fluid environment where viscosity is dominant; the Reynolds number Re, ratio of inertial to viscous forces, is small. For a cell swimming at a speed U = 103 μm⋅ s−1 in water, Re does not exceed 0.1 even for a cell as large as a = 103 μm in length, such as the Stentor, and for a bacterial cell of length a = 1 μm, Re ≤ 10−4. However, for cells inhabiting this viscous regime, the mechanisms of nutrient transport can vary from relying solely on diffusion to actively generating fluid movement.

Fig. 2: Nutrient transport by diffusion versus cilia-driven feeding currents.
Fig. 2: Nutrient transport by diffusion versus cilia-driven feeding currents.
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A–C In small bacterial cells, both intracellular transport and transport to the cell surface are limited by diffusion. A When a large number of molecule-specific receptors (medium gray) are randomly but uniformly distributed over the cell surface (light gray), the nutrient uptake is nearly maximal; equivalent to that of a fully-absorbing cell1. B Clustering receptors over a fraction of the cell surface reduces diffusive flux. C Concentration (colormap) and diffusive flux (white arrows) when only half the cell surface is absorbent. D–G In ciliates, intracellular transport relies on cytoskeletal molecular motors; extracellular transport is driven by the coordinated activity of cilia (dark gray) covering (D) Ac = 90% and E Ac = 20% of the outer cell surface; nutrient uptake over a fraction Af = 10% of the cell surface (medium gray). Flow streamlines (blue lines) and speed (colormap) for (F) swimming (shown in lab and cell frame) and (G) fixed cells. H Cell design space (Ac, Af) describes the fractions of the cell surface dedicated, respectively, to the mouth and cilia coverage.

Enhancing nutrient transport to the cell surface through fluid movement—whether by swimming or attaching and creating feeding currents—depends on the Péclet number Pe = aU/D. Transport improves when the fluid motion over a length a occurs on a time scale a/U that is significantly shorter than the diffusion time scale a2/D, where D is nutrient diffusivity. According to Pe, this benefit increases linearly with cell size, but it actually depends more strongly on cell size than suggested by Pe alone. This becomes apparent when considering the cost of energy dissipated in the viscous fluid, an energy that the cell itself must supply1. The rate of dissipation energy \({{\mathcal{P}}}\) per cell volume \({{\mathcal{V}}}\) required to improve the cell’s nutrient uptake by fluid movement scales with ηD2/a4, while that required to propel the same cell at speed U scales with ηU2/a2, where η is the fluid viscosity (Methods, Supplementary Table. S.1). The ratio defines a relative cost \({{\mathcal{C}}} \sim {{\mbox{Pe}}}^{-2}\) of improving nutrient uptake through fluid movement: the cost decreases as an inverse square law with increasing Pe and thus cell size a. Compared to a bacterial cell of size a = 1 μm swimming at speed U = 10a, for a ciliate of size a = 100 μm also swimming at speed U = 10a, the cost of improving nutrient transport by moving fluid reduces by a factor of 104. At around cell size a ≈ 10 μm, the cost \({{\mathcal{C}}}\) decreases below 1, marking the size beyond which increasing nutrient intake shifts from being “hopelessly futile” to potentially beneficial (Methods, Supplementary Fig. S.1)1.

Essentially, as cell size increases from a few microns to several tens of microns, the energetic cost of moving fluid for enhancing the cell nutrient uptake, via ciliary-driven feeding current, decreases dramatically, making it an energetically feasible tactic for both swimming and attached cells. From an energetic perspective, this justifies the investment that ciliates made in maintaining active cilia that drive fluid across the cell surface. Meanwhile, relying on diffusion only for internal transport within the cell becomes prohibitively time-consuming as cell size increases, requiring an investment in active cytoskeletal transport. As intracellular distances exceed 10 μm, active cytoskeletal transport becomes more time-efficient than diffusion (Fig. 2, Methods, Supplementary Fig. S.1). That is, the fundamental principles of physics, when applied to the energetics and duration of transport, coupled with the fact that the cell uses the same materials for driving transport within and outside the cell, strongly imply a co-evolution of active intracellular and extracellular transport systems in ciliates. However, while phagotrophy and orchestrated intracellular material transport favor designation of a specific portion of the cell surface as a ‘mouth’ for efficient food intake, the optimal allocation of cell surface regions dedicated for food intake versus fluid movement remains unclear. We thus asked: what optimal arrangements of cilia and feeding regions maximize the cell nutrient uptake?

Optimal cell design and trade-offs between ciliary and feeding structures

To address this question, we modeled the cell as a sphere of radius a, immersed in an incompressible viscous fluid, with a constant energy budget \({{\mathcal{P}}}\) dedicated to powering cilia. We prohibited overlap between cilia and mouth: cilia covered a fraction Ac of the cell surface non-overlapping and adjacent to a surface fraction Af allocated for food intake, where, by design, Ac + Af ≤ 100% (Fig. 2H). Ciliates often feature cilia in the oral structure, but these cilia specialize in transporting food particles through the cytosome, exhibiting beating patterns that are different from cilia that drive external flows15; we thus ignored their effects on external flows. To emulate cilia activity, we adapted the spherical envelope model19,20,21,22,23 and imposed a tangential force density on the ciliated fraction Ac of the cell surface that drives flows. We solved the Stokes equations, together with a force balance on the sphere, in non-dimensional form, where we selected a and \({{\mathcal{U}}}=\sqrt{{{\mathcal{P}}}/8\pi \eta a}\) as the characteristic length and velocity scales and normalized both to unity, thereby keeping \({{\mathcal{P}}}\) and \({{\mathcal{V}}}\) constant throughout (Methods, Supplementary Tables S.1 and S.2). This spherical model bridges the two archetypes of motile and sessile cells in a concise way and enables efficient exploration of the entire cell design space spanned by (Ac, Af). In Fig. 2D, E are depictions of a motile sphere with Af = 10% and Ac = 90% and a sessile sphere with Af  = 10% and Ac = 20% representing mathematical abstractions of the motile and sessile ciliates in Fig. 1B and C; the corresponding cilia-driven flows are shown in Fig. 2F (motile) and Fig. 2G (sessile).

We next introduced two measures of feeding performance: the feeding flux Q defined as the integrated fluid velocity over a cross-sectional area leading to the cell’s designated mouth (Fig. 3A, inset)24,25,26,27,28, and the Sherwood number Sh, defined as the ratio of total nutrient flux directed to the mouth region divided by the corresponding diffusive flux3,5,21,22,23,29,30,31. Notably, diffusion here captures the Brownian motion of food particulates, not dissolved molecules. Calculating Sh required not only solving for the fluid velocity field but also coupling the fluid solution to an advection-diffusion equation governing the concentration C of a nutrient, which in non-dimensional form depends on a single parameter, the Pe number (Methods, Supplementary Fig. S.2). Fixing Af = 10% in both the sessile and motile cells, we found that optimal feeding occurred when cilia are arranged in a narrow band around the feeding apparatus (Fig. 3A, B). At Af = 30%, optimal feeding in sessile cells followed a similar arrangement, with cilia covering only a narrow fraction Ac of the cell surface, while in motile cells, feeding became optimal when cilia covered the remaining cell surface, Ac = 100% − Af (Fig. 3C, D). These results aligned with calculations based on a closed-loop optimization approach for maximizing Sh (Methods, Supplementary Fig. S.2).

Fig. 3: Optimal cilia coverage in sessile and motile cells at given feeding fractions.
Fig. 3: Optimal cilia coverage in sessile and motile cells at given feeding fractions.
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Feeding flux Q and Sherwood number Sh versus cilia coverage Ac for (A, B). feeding fraction Af = 10% and (C, D). feeding fraction Af = 30%. Feeding flux Q is obtained by integrating fluid velocity over a disk aligned with the feeding region (A inset). Maximal Q and Sh (marked by open circles and squares, respectively) occur at the same cilia fraction Ac, representing a ciliary band around the feeding fraction for both motile and sessile ciliate, at Af = 10%, whereas at Af = 30%, optimal cilia fraction bifurcates from ciliary bands for sessile cells to ciliary covering the remaining cell surface in motile cells. Q-optimal flow streamlines (blue lines) and speed (colormap) are shown in lab frame to the right of panels (A, C); Sh-optimal streamlines (white lines) and concentration field (colormap) are shown in body frame to the right of panels (B, D). Scale bars in all panels represent a length of 100 μm.

The value of Ac that maximizes Q (or Sh) at a given choice of feeding fraction Af is a local optimum in that it is tailored to that specific Af. Next, we computed Q and Sh for the entire range of Af and Ac from 5% to 95% (Fig. 4A–D) and identified for each Af, the cilia coverage Ac that locally maximized nutrient uptake (Fig. 4, solid yellow lines). We discovered that, regardless of the choice of performance metric, Q or Sh, the local optima of Ac depended on cell motility: for Af ⪅ 20%, the local minima overlapped for sessile and motile cells, with ciliary bands around the feeding fraction optimized feeding efficiency. At around Af ≈ 20%, the local optima in motile cells bifurcated into a new branch where cilia covered the remaining cell surface not dedicated to feeding (Fig. 4A, C), whereas in sessile cells, no such bifurcation occurred. The branch of local optima continued smoothly (Fig. 4B, D). The combination (Ac, Af) that maximized Q (or Sh) over the entire design space is a global optimum (Fig. 4, solid yellow markers). In sessile cells, the global optimum corresponds to ciliary bands, while in motile cells, it features cilia spread over the entire non-oral cell surface.

Fig. 4: Optimal arrangements of cilia and feeding structures in sessile and motile cells.
Fig. 4: Optimal arrangements of cilia and feeding structures in sessile and motile cells.
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A, B Feeding flux Q and (C, D). Sherwood number Sh as colormaps over the morphospace (Ac, Af) of motile (top) and sessile (bottom) cells; darker colors indicate larger values, white contour lines indicate designs that produce same Q or Sh values, solid yellow lines indicate local optima at each Af; filled circles and squares indicate global optima of sessile and motile designs over the entire design space. Insets in (A, B): contour lines of Q = 104 in units of bodyvolume per hour for motile (purple) and sessile (blue); Insets in (C, D): contour lines of Sh = 2 for motile (purple) and sessile (blue) shows that Sh is more than doubled nearly everywhere in the morphospace. E Cell swimming speed U increases with increasing cilia coverage. Inset: swimming speed in units of bodylength per second, highlighting contour lines of 10 and 30 body lengths per second, with maximal value of 50 body lengths per second in the rightmost corner. F Force required to keep the cell attached increases with cilia coverage. All calculations are done at constant hydrodynamic power \({{\mathcal{P}}}\), cell volume \({{\mathcal{V}}}\), and Pe = 100 chosen based on typical particle diffusivity and flow speed measurements in ciliates.

Thus far, we focused, by design, on adjacent, non-overlapping cilia and feeding regions. When we relaxed the adjacency constraint—allowing a non-absorbing, inactive gap between the cilia and feeding areas—nutrient uptake decreased monotonically in both sessile and swimming cells (Methods, Supplementary Fig. S.3). This demonstrates that feeding efficiency is indeed optimal when cilia are positioned directly adjacent to the feeding apparatus.

A closer examination of the design space (Ac, Af) shows that at the global optima, the surface fraction Af allocated to the feeding apparatus is <50% in both sessile and motile cells (Fig. 4, solid yellow markers, and Supplementary Fig. S.4 for Pe = 500). That is, unlike in diffusive transport, where maximal intake is achieved when receptors are randomly distributed over the entire cell surface (Fig. 2B)1, in flow-driven transport, the designation of a specific portion of the cell surface as a ‘mouth’ not only facilitates the intracellular processing of ingested food particles, but also optimizes external transport to the cell surface. The Q - and Sh-landscapes in both sessile and swimming cells are broad peaked. This feature implies that nutrient uptake is robust to small variations in cell design and insensitive to perturbations in the specific feeding and ciliary arrangements. To underscore the significance of this feature, we calculated the contour lines corresponding to Q = 104 body volumes/h and highlighted the regions of the cell morphospace (Ac, Af), where Q exceeding 104 body volumes/h. We choose this Q value because empirical observations in ciliates range from Q = 103 to 105 body volumes/h (Methods, Supplementary Table S.5)32. To map our dimensionless model results to units of body volumes per hour, we used a maximal speed of 50 body lengths/s, consistent with empirical measurements in ciliates (Supplementary Table S.5, Fig. 4E, inset)33. Our model predicts clearance rates higher than Q = 104 body volumes/h over broad regions of (Ac, Af) space (Fig. 4A, B inset). Likewise, for Sh number, over twofold increase in nutrient uptake is attainable nearly everywhere in the cell morphospace (Fig. 4C, D inset).

Because ciliary activity in motile cells serves to also power the cell motion, we explored how the arrangements of feeding and cilia coverage affect the cell swimming speed U. Not surprisingly, U is maximum at maximal cilia coverage (Fig. 4E). In sessile cells, ciliary activity exerts a force on the surrounded fluid that should be balanced by an attachment force, sustained by a specialized stalk22,23,34; we thus computed the attachment force over the entire cell morphospace and found that this force is minimized at minimal cilia coverage (Fig. 4F). That is, in motile cells, optimal designs that maximize Q and Sh by distributing cilia across the entire surface fraction unoccupied by the feeding apparatus also improve the propulsive force and swimming speed U of the cell (Fig. 4E), while in sessile cells, optimal designs that maximal Q and Sh by concentrating ciliary activity in a band around the cell’s feeding area also reduce the force needed to keep the cell attached (Fig. 4F).

To rigorously account for these multiple objectives, we defined a multi-objective optimization problem using the weighted-sum method, where the objective function \({{\mathcal{J}}}\) depended on cell motility: in motile cells, \({{\mathcal{J}}}=\alpha {{\rm{Sh}}}+(1-\alpha )U\) aimed to maximize both feeding rate Sh and swimming speed U, whereas in sessile cells, \({{\mathcal{J}}}=\alpha {{\rm{Sh}}}-(1-\alpha )F\) aimed to maximize feeding while minimizing the attachment force F. The parameter α describes the weight given to nutrient uptake; at α = 1, \({{\mathcal{J}}}\) account for only feeding, recovering the optimization problem in Fig. 4C, D. At α = 0, feeding does not enter in the optimization. As we varied α from 0 to 1, the optimal solutions describe a Pareto front35. In motile cells, the Pareto front starts at maximal cilia coverage at α = 0, which maximizes swimming speed, and ends at maximal nutrient uptake at α = 1, thus favoring maximal cilia coverage over the entire cell surface not allocated to feeding (Fig. 5A). In sessile cells, the Pareto front jumps abruptly from minimal cilia coverage at α = 0, when feeding is discounted, to the branch of optimal solutions characterized by a localized ciliary band surrounding the feeding area, ending with maximal nutrient uptake at α = 1 (Fig. 5B). That is, the Pareto front in sessile cells follows closely the Sh-optimal branch, indicating no conflict between the two objectives of maximizing uptake and minimizing attachment force.

Fig. 5: Optimal designs explain the diversity of cilia arrangements in sessile and swimming ciliates.
Fig. 5: Optimal designs explain the diversity of cilia arrangements in sessile and swimming ciliates.
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A, B Solid yellow lines are local optima at each Af from Fig. 4. Pareto fronts that maximize \({{\mathcal{J}}}=(\alpha {{\rm{Sh}}}+(1-\alpha ){{\rm{U}}})\) for swimming cells (purple solid line) and \({{\mathcal{J}}}=(\alpha {{\rm{Sh}}}-(1-\alpha )F)\) for sessile cells (blue solid line). Here, we normalized Sh, U and F to each lie in the range from 0 to 1, and for each value of α ∈ [0, 1], we computed the corresponding optimal cell design (Ac, Af) to obtain the Pareto front as α increases from 0 to 1. We surveyed 31 motile (purple and red) and sessile (blue and green) ciliates, listed by their genus (Supplementary Tables S.3 and S.4). Empirical measurements are mapped onto the design space (Ac, Af): sessile ciliates cluster in one region of the morphospace, characterized by a ciliary crown surrounding the feeding apparatus, while motile ciliates exhibit two types of cell morphologies, one similar to the sessile ciliates and one that maximizes cilia coverage over the entire cell surface. C All surveyed ciliates lie in a region of the morphospace where the model predicts a feeding rate Q of over 104 body volumes per hour, consistent with experimental measurements (Supplementary Table S.5) (contour lines of Q and optimal designs taken from Fig. 4 (A, B). D Likewise, all surveyed ciliates correspond to over twofold increase in nutrient uptake compared to diffusion alone at Pe = 100; (contour lines of Sh and optimal designs taken from Fig. 4C, D. Superimposing optimal solutions, including the Pareto fronts, show remarkable consistency between optimal model predictions and surveyed ciliates.

Evaluating optimal predictions and empirical observations in ciliates

How do these model predictions help elucidate the diversity in cilia arrangements observed in sessile and swimming ciliates? To answer this question, we surveyed 31 genera of ciliates within the phylum Ciliophora, encompassing the classes Oligohymenophorea, Heterotrichea, Litostomatea, Spirotrichea, and Phyllopharyngea listed in Fig. 5 and Supplementary Table S.3. Ciliates that attach and generate feeding currents, such as Stentor36,37,38,39, Vorticella28,40,41, Opercularia42,43,44 and Pseudovorticella5,45,46, feature ciliary bands localized around the feeding apparatus. According to our model, this design maximizes feeding efficiency while minimizing the required attachment force. Of the swimming ciliates, Didinium47,48 and Strombidium49,50,51 feature localized ciliary bands while the entire non-oral cell surface of Paramecium52,53 and Tetrahymena54,55,56,57 is covered by cilia. We calculated the equivalent spherical volume and radius a and estimated the associated Af and Ac for all 31 surveyed genera (Supplementary Tables S.3 and S.4)22. Empirical measurements of feeding currents and swimming speeds yielded a range of Pe values <250 considering the diffusivity of dead bacterial cells (Supplementary Table S.5); thus validating our choice of Pe = 100. Mapping these empirical observations onto our morphospace (Ac, Af), we found that sessile ciliates (blue and green circles) grouped into a single cluster, characterized by localized ciliary bands surrounding the feeding apparatus, whereas motile ciliates (red and purple circles) formed two distinct clusters, one characterized by localized ciliary bands and the other characterized by cilia covering the entire cell surface (Fig. 5A, B). The motile and sessile ciliates that feature localized ciliary bands followed closely the branches of local optima of Q and Sh predicted by our model, which in sessile cells also coincide with the Pareto front and lead to the global optimum of nutrient uptake. The motile ciliates, such as the Paramecium and Tetrahymena, closely followed the predictions of our Pareto front for motile cells. These results indicate that cilia allocation is a balance between swimming and feeding.

Theoretically, for the same energetic cost, flow physics permits larger increase in nutrient uptake for the swimming cells that follow the Pareto front but at the expense of reducing their swimming speed. Larger increase in both nutrient uptake and swimming speed are possible for the swimming cells that exhibit localized ciliary bands around their feeding apparatus, yet at the cost of having to build more cilia. Such improvements are possible at no additional energetic cost for powering the cilia. This is evident because our modeling results are all obtained at the same hydrodynamic power \({{\mathcal{P}}}\) and cell volume \({{\mathcal{V}}}\). Why don’t all swimming cells then cluster at the Pareto front that optimizes feeding and swimming?

To explore this question, we superimposed all surveyed ciliates, both motile and sessile, onto our model predictions of nutrient flux Q and Sherwood number Sh (Fig. 5C, D). We found that all surveyed ciliates, whether motile or sessile, with ciliary bands or covered by cilia, occupied regions of the morphospace where nutrient uptake was >104 body volumes per hour, consistent with empirical observations (Supplementary Table S.5); in these regions, nutrient uptake was more than doubled compared to diffusion only. That is, swimming and sessile ciliates achieve similar levels of nutrient uptake through different body plans, and beyond a threshold of doubling nutrient uptake, further increases in Sh number do not seem to be a dominant selective force in cell design. Because of the coevolution of extracellular and intracellular transport mechanisms in eukaryotic cells, nutrient acquisition does not depend solely upon external transport. Intracellular processing of transported nutrients may have played a critical role in cell adaptations. Such considerations support the hypothesis that, unlike cells that rely solely on diffusion, the limiting factor for nutrient uptake in unicellular eukaryotes may not be the external transport of nutrients to the cell surface, but rather the cell’s capacity to process these transported nutrients58. This is far from being the only plausible explanation, evolutionary processes other than biophysical optimality may have limited further increase in nutrient transport. This is particularly relevant for organisms like protists, which typically have small effective population sizes, making them more influenced by evolutionary processes driven by genetic drift59,60,61. Another factor could be the energetic costs associated with the formation and maintenance of the ciliary structures. Since the energetic cost of structural investments scale with cell size, evolutionary adaptations must balance the benefits of increased nutrient uptake against the costs of building and maintaining additional ciliary structures62. Or simply, a twofold increase in nutrient acquisition over diffusion might be sufficient for survival, and further increases might not confer a practical selective advantage.

Discussion

We established a quantitative map linking nutrient uptake in ciliates to the cell’s design or surface allocation to feeding structures and cilia activity. The evolutionary transition from small prokaryotic cells, which relied on diffusion for nutrient transport1,13, to larger eukaryotic cells, capable of consuming particulates and smaller cells via phagotrophy7, entailed increases in cell volume of up to eight orders of magnitude63. Sustaining these larger cells required expanding the mechanisms for nutrient transport to the cell surface beyond simple diffusion3,7. We showed that the energetic cost of moving fluid for enhancing nutrient uptake, for example by ciliary activity, decreases dramatically in larger cells, as an inverse square law in Péclet number, making it an energetically viable strategy for attached and swimming unicellular eukaryotes. The designation of a small surface area as a ‘mouth’ enhances nutrient transport to the cell surface, thanks to the cell’s ability to actively direct fluid to the designated mouth. Importantly, the diversity of cilia arrangements relative to the oral area in ciliates is not accidental but, instead, is shaped by flow physics. Sessile ciliates maximize prey encounter while also minimizing the force necessary to keep the ciliate attached to the substrate by concentrating their cilia in a band around their oral area. Swimming ciliates display more diverse designs: some have localized ciliary bands around their oral regions, while others are entirely covered with cilia. At first glance, this diversity is puzzling because of the view that optimal feeding corresponds to optimal swimming21,22,64 and localized ciliary bands are less effective for swimming. But the flow physics of nutrient transport at the microscopic scale is more nuanced.

Our study uncovered an important, previously unrecognized, connection between the cell surface area allocated to feeding and optimal cilia coverage, and showed a bifurcation in optimal designs of swimming ciliates from localized ciliary bands to cells covered in cilia. These bifurcated designs achieve similar benefits in nutrient uptake. The implications are important for swimming - ciliates with localized ciliary bands may not reach the same swimming speeds as those covered in cilia, but both can achieve similar levels of nutrient uptake. Indeed, surveyed sessile and swimming ciliates lay within the same range of nutrient uptake, despite the theoretical potential for further increases of nutrient uptake in swimming ciliates, at no additional energetic cost. These findings indicate that, unlike smaller prokaryotes1, the limiting factor for unicellular eukaryotes is not the energy required for generating feeding currents. The limiting factor could be the cell’s feeding structures and their capacity to process transported nutrients58 or the investment in building and maintaining more cilia62. Alternatively, evolutionary processes other than biophysical optimality might have been relevant59,60,61, or simply that a twofold increase in nutrient uptake is “good enough” for survival.

Adaptations to the surrounding fluids for nutritional transport by individual cells capable of phagotrophy, as we note here for ciliates, has shaped the lifestyles of these protists as free-living unicellular predators consuming other organisms7 and enabled their proliferation as one of the most influential planktonic protistan lineages. These adaptations also formed the building blocks that led to more complex collections of cells comprising colonies3,4,25,36. Cooperation means sharing of resources among individuals with different abilities36, and is facilitated when having an abundance of these resources65,66. Our findings, which strongly suggest that ciliates have the energetic capacity to transport more nutrient-laden fluids than they are able to individually process, offer a fresh perspective and opportunities for future research on the origins of cooperation and teamwork between unicellular protists3,4,5,25,36,67.

Methods

Morphological data

We collected morphological data from a total of 31 ciliates (Supplementary Tables S.3 and S.4). For each ciliate, we estimated from multiple images the spherical equivalent radius and the cell surface fractions allocated to feeding Af and cilia coverage Ac are shown in Supplementary Table S.3. In addition to directly measuring Ac from images, we also approximated its value by considering that the bandwidth of the cilia ring is 10–20 μm, which is twice the length of a single cilium (5–10 μm). A summary of the mean and standard deviation of these measurements is shown in Supplementary Table S.4. Meanwhile, wherever available, we collected measurements of maximal flow speeds in sessile ciliates and average swimming speeds in motile ciliates, and computed the associated Pe values using the average value of the equivalent sphere radius, along with diffusivity coefficients for Oxygen and dead bacteria (Supplementary Table S.5).

Intracellular transport rates

The average time a microscopic particle undergoing diffusion in 3D space spends to move over a short distance Δr inside the cell is given by 〈Δτ〉 = Δr2/6D, where the diffusion coefficient D is estimated experimentally in refs. 68,69 (Fig. S.1C, gray region). In eukaryotic cells, nutrients are typically transported within membrane-covered vacuoles using molecular motors walking on filamentous highways9,70 following a biased 1D random walk. For a biased random walk, the average time that a particle takes to move a distance Δr is estimated as 〈Δτ〉 = Δr/vb, where the average walking velocity vb is experimentally measured and summarized in71 (Fig. S.1D, pink lines). Comparing average time of diffusive and advective transport shows that, for traversing larger distance Δr in larger cells, the transport rate by molecular motion exceeds the transport rate by Brownian motion. Note that with more complex motion, such as when molecular motors jump between filaments or change moving directions, the average time scales as Δτ ~ Δrα, where 1 < α < 2, which still increases the rate of nutrients transport in large cells compared to pure diffusion70.

Mathematical modeling of ciliates and cilia-driven flows

We modeled the cell as a sphere of radius a and we introduce the Cartesian coordinate (x, y, z) and the spherical coordinates (r, θ, ϕ), where r is measured from the cell center and θ is measured from the z-axis (Supplementary Fig. S.2A). The fluid velocity field u(r, θ, ϕ, t) and the fluid pressure field p(r, θ, ϕ, t) in the domain bounded internally by the sphere are governed by the incompressible Stokes equation72,

$$-\nabla p+\eta {\nabla }^{2}{{\bf{u}}}=0,\qquad \nabla \cdot {{\bf{u}}}=0.$$
(1)

We considered slip boundary conditions at the cell surface, consisting of a tangential surface velocity field that is axisymmetric, that is, independent of ϕ. We also considered proper decay of the velocity field at ∞19.

Analytical solution in terms of Legendre polynomials

We solved for the components (ur, uθ) of the fluid velocity field following the classic envelope model19. Namely, we decomposed the surface velocity in terms of the Legendre polynomials by introducing the nonlinear transformation of variable \(\mu=\cos \theta\), such that

$${\left.{{\bf{u}}}\right\vert }_{r=a}={\sum }_{n=1}^{\infty }{B}_{n}{V}_{n}(\mu ){{{\bf{e}}}}_{\theta },\qquad {{\bf{u}}}{| }_{r\to \infty }=-\!U{{{\bf{e}}}}_{z}.$$
(2)

Here, U = 0 for a sessile sphere and U ≠ 0 is the swimming speed of a motile sphere. The surface velocity emulating cilia activity is given in terms of constant coefficients Bn associated with the polynomial basis \({V}_{n}(\mu )=\frac{2}{n(n+1)}\sqrt{1-{\mu }^{2}}P{{\prime} }_{n}(\mu )\), n = 1, …, ∞, given in terms of the Legendre polynomials Pn(μ) and their first derivatives \(P{{\prime} }_{n}(\mu )=d{P}_{n}(\mu )/d\mu\). The analytical solutions of fluid velocity and pressure fields, along with the energy dissipation rate, for the sessile and motile model ciliates are listed in Supplementary Table S.2; see in refs. 19,20,21,22,23 for details.

Describing surface velocity induced by ciliary bands

We described ciliary activity over a ciliary band covering a fraction of the spherical surface, specified by r = a = 1 and μ ∈ [μ2, μ1], where μ2 and μ1 are the limit values. Namely, we wrote \({{\bf{u}}}{| }_{r=a}=\left(g(\mu )/G\right){{{\bf{e}}}}_{\theta }\), where G is a constant and g(μ) is given by

$$g(\mu )=\left\{\begin{array}{ll}\sin \left(\pi \frac{{\mu }_{1}-\mu }{{\mu }_{1}-{\mu }_{2}}\right),&\in [{\mu }_{2},{\mu }_{1}].\\ 0,& \in [-1,{\mu }_{2})\cup ({\mu }_{1},1].\end{array}\right.$$
(3)

We projected g(μ)/G on the velocity basis Vn(μ),

$${B}_{n}=\frac{n(n+1)(2n+1)}{8}\int_{-1}^{1}\frac{g(\mu )}{G}{V}_{n}(\mu )d\mu .$$
(4)

We considered a total of N velocity modes, n = 1, …, N, where the N is selected to ensure that the relative error between the prescribed and reconstructed surface velocities remained below 10−4. We determined the constant G by imposing the constraint of constant energy dissipation rate

$${{\mathcal{P}}}={{\rm{constant}}}=8\pi \eta a \, \left\{\begin{array}{ll}{\sum }_{n=1}^{\infty }\frac{2{B}_{n}^{2}}{n(n+1)},&\,{\mbox{sessile}}\, {\mbox{cell}}\,,\\ \left(\frac{2}{3}{B}_{1}^{2}+{\sum }_{n=2}^{\infty }\frac{2{B}_{n}^{2}}{n(n+1)}\right),&\,{\mbox{motile}}\, {\mbox{cell}}\,.\end{array}\right.$$
(5)

Non-dimensional form

We wrote the solution in non-dimensional form, choosing a and \({{\mathcal{U}}}=\sqrt{{{\mathcal{P}}}/(8\pi \eta a)}\) as the characteristic length and velocity scales, which we normalized to unity.

Advection-diffusion equation

To analyze the nutrient concentration around the sphere, we considered the steady-state advection-diffusion equation for a concentration field C of nutrients with diffusivity coefficient D,

$${{\bf{u}}}\cdot \nabla C=D\Delta C,\quad {\left.C\right\vert }_{r\to \infty }={C}_{\infty },\quad \left\{\begin{array}{ll}C=0,\hfill&\,{\mbox{surface}}\, {\mbox{fraction}}\, {\mbox{designated}}\, {\mbox{as}}\,{\hbox{`}} {\mbox{mouth}}\,{\hbox{'}},\\ {\nabla }_{r}C=0,&\,{\mbox{remaining}}\, {\mbox{cell}}\, {\mbox{surface}}\hfill\,.\end{array}\right.$$
(6)

We normalized the concentration field by the far-field background value C∞, and considered the dimensionless concentration c = (C∞ − C)/C∞ governed by

$${{\rm{Pe}}}\ {{\bf{u}}}\cdot \nabla c=\Delta c,\quad {\left.c\right\vert }_{r\to \infty }=0,\quad \left\{\begin{array}{ll}c=0,\hfill&\,{\mbox{surface}}\, {\mbox{fraction}}\, {\mbox{designated}}\, {\mbox{as}}\,{\hbox{`}} {\mbox{mouth}}\,{\hbox{'}},\\ {\nabla }_{r}c=0,&\,{\mbox{remaining}}\, {\mbox{cell}}\, {\mbox{surface}}\hfill\,.\end{array}\right.$$
(7)

Here, the Péclet number \(\,{\mbox{Pe}}\,=a{{\mathcal{U}}}/D\) is defined based on the characteristic length scale a and the characteristic speed \({{\mathcal{U}}}\).

Surface fraction designated as ‘mouth’

We defined the absorbing fraction of the surface as

$${{{\rm{A}}}}_{{{\rm{f}}}}=\frac{1}{4\pi {a}^{2}}\int_{0}^{2\pi }\int_{0}^{{\theta }_{1}}{a}^{2}\sin \theta d\theta d\phi=\frac{1}{2}\int_{\cos {\theta }_{1}}^{1}d(\cos \theta )=\frac{1-{\mu }_{1}}{2},$$
(8)

where the ‘mouth’ area always starts from the pole of the spherical cell at which μ = 1 and ends at μ1 (Supplementary Fig. S.2A). The adjacent ciliary band coverage near the ‘mouth’ is in the range μ ∈ [μ1, μ2].

Numerical approach

We solved for the nutrient concentration field c numerically by discretizing the governing equation Eq. (7) using the a finite difference method. To reach the far-field boundary condition, we used a non-uniform radial mesh such that the grid is denser near the sphere surface and more sparse in the far field. We chose to discretize θ on a uniform mesh; see20,21,22,23 for more details.

Feeding flux

To estimate the feeding rate, we calculated the flux Q through a disk placed at a distance 0.1a away from the cell ’mouth.’ The disk is of radius \({R}_{d}=a\sqrt{(1-{\mu }_{1}^{2})}\) (with rdrdϕ being an area element)

$$Q=\int_{0}^{2\pi }\int_{0}^{{R}_{d}}{\left.{u}_{z}\right\vert }_{z=1.1a}\, rdrd\phi .$$
(9)

Sherwood number

By using Fick’s law, the nutrient flux at the sphere surface is equal to the area integral of the concentration flux over the spherical surface,

$${{\rm{I}}}=-\int{\int}_{\hbox{`} {\mbox{mouth}}{\hbox{'}} }{\hat{{{\bf{e}}}}}_{r}\cdot (-D{\nabla }_{r}C)\,{\mbox{d}}\,S=\int_{0}^{{\theta }_{2}}D\frac{\partial C}{\partial r}2\pi {a}^{2}\sin \theta d\theta .$$
(10)

The sign convention is to ensure positive I when the cell takes up nutrients. In the case of pure diffusion, the inward flux Idiffusion due to molecular diffusion can be obtained by numerically solving the diffusion equation with appropriate boundary conditions. Accounting for both advective and diffusive transport, the Sherwood number Sh = I/Idiffusion is defined as dimensionless nutrient flux through the sphere surface.

Closed-loop optimization of cilia range that maximizes feeding flux

To search for optimal surface motions that maximize feeding in the ciliated sphere model with partial oral fraction, we considered an optimization method based on variational analysis and steepest descent21,23. The problem consists of a PDE-constrained optimization problem, where the goal is to find an optimal range of cilia coverage from μ1 to μ2 that maximizes the nutrients flux, subject to the concentration field c satisfying the advection-diffusion equation and surface velocity satisfying the constant energy constraint. The value of μ1 is dictated by the end of the ‘mouth’ area to satisfy the adjacency and no overlap condition between ‘mouth’ and cilia. The goal is thus to find the optimal cilia endpoint μ2,

$${\max}_{{\mu }_{2},c}\,{{\rm{I}}}({\mu }_{2},c),\qquad {{\rm{subject\ to}}}\quad {{\mathcal{L}}}[c]=0\quad {{\rm{and}}}\quad {\sum }_{n=1}^{N}{\beta }_{n}^{2}=1.$$
(11)

Here, the linear operator \({{\mathcal{L}}}={{\rm{Pe}}}{{\bf{u}}}\cdot \nabla -{\nabla }^{2}\) is that of the advection-diffusion equation along with the corresponding boundary conditions. The coefficients \({\beta }_{n}={B}_{n}\sqrt{2/(n(n+1))}\) are weighted coefficients, satisfying the constraint on the energy dissipation rate, \({\sum }_{n}\, {\beta }_{n}^{2}=1\).

The variation in nutrient flux δI due to variation of the endpoint δμ2 is given by

$$\frac{\delta {{\rm{I}}}}{\delta {\mu }_{2}}=-2\pi {{\rm{Pe}}}\int_{1}^{\infty }\int_{-1}^{1}c\left(\frac{\delta {{\bf{u}}}}{\delta {\mu }_{2}}\cdot \nabla h\right)\ {r}^{2}drd\mu,$$
(12)

where h(r, μ) is the adjoint function associated with the advection-diffusion equation; h(r, μ) satisfies

$${{\rm{Pe}}}{{\bf{u}}}\cdot \nabla h+{\nabla }^{2}h=0,\quad h(r=1)=1,\quad h(r\to \infty )=0,$$
(13)

and the consistency equation

$${\int}_{S}\nabla \delta c\cdot {{\bf{n}}}dS=-\!{{\rm{Pe}}}{\int}_{V}c\delta {{\bf{u}}}\cdot \nabla h\ dV.$$
(14)

From functional analysis, we have

$$\frac{\delta {{\bf{u}}}}{\delta {\mu }_{2}}={\sum }_{n=1}^{N}\left[\left(\frac{\delta {u}_{r}}{\delta {B}_{n}}\frac{\delta {B}_{n}}{\delta {\mu }_{2}}\right){{{\bf{e}}}}_{r}+\left(\frac{\delta {u}_{\theta }}{\delta {B}_{n}}\frac{\delta {B}_{n}}{\delta {\mu }_{2}}\right){{{\bf{e}}}}_{\theta }\right].$$
(15)

Using the Leibniz integral rule and recalling (3) and (4), the variation in velocity coefficients Bn generated by variation in end point μ2 is given by

$$\begin{array}{rcl}\frac{\delta {B}_{n}}{\delta {\mu }_{2}}&=&\frac{n(n+1)(2n+1)}{8}\left(-g{| }_{\mu={\mu }_{2}}{V}_{n}(\mu )+\int_{{\mu }_{2}}^{{\mu }_{1}}\frac{\partial g}{\partial {\mu }_{2}}{V}_{n}(\mu )d\mu \right)\\ &=&\frac{n(n+1)(2n+1)}{8}\int_{{\mu }_{2}}^{{\mu }_{1}}\frac{\pi ({\mu }_{1}-\mu )}{{({\mu }_{1}-{\mu }_{2})}^{2}}\cos \left(\frac{\pi ({\mu }_{1}-\mu )}{{\mu }_{1}-{\mu }_{2}}\right){V}_{n}(\mu )d\mu . \hfill\end{array}$$
(16)

By substituting (16) into (15), we obtain the derivative of the velocity field with respect to μ2. At each iteration for updating μ2, we solved for the adjoint function h(r, μ) numerically by solving the associated adjoint equation along with boundary conditions (13). By substituting h together with (15) into (12), we obtained the expression for the gradient δI/δμ2, which we used to update the value of I in order to identify μ2 that maximizes I. The step size s for updating μ2 was chosen dynamically at each iteration step j,

$${\mu }_{2}^{(j+1)}={\mu }_{2}^{(j)}+s{\left(\frac{\delta {{\rm{I}}}}{\delta {\mu }_{2}}\right)}^{(j)},$$
(17)

such that \(s=({\mu }_{2}^{(j)}-{\mu }_{2}^{(j-1)})/({\frac{\delta {{\rm{I}}}}{\delta {\mu }_{2}}}^{(j)}-{\frac{\delta {{\rm{I}}}}{\delta {\mu }_{2}}}^{(j-1)})\). This choice is based on the Barzilai-Borwein method to accelerate the computation time.

Reporting summary

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