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ENGINEERING that now-existing motile ciliates and microalgae have survived by ate with different metachronal waves are shown in Fig. 1C, for acquiring optimal propulsion efficiency. 2a/L = 20 and N = 636, where a is the body radius, L is the cil- ium length, and N is the number of cilia (the results of in-phase Flow around the Model Ciliate and prolate body cases are shown in SI Appendix). As shown Consider a ciliate swimming via individual ciliary motion in an in the figure, undulating metachronal waves generate complex infinite fluid, which is otherwise at rest. Because Re scaled by fluid flows around the ciliate, and the ciliate exhibits oscillatory ciliary motion or swimming are much smaller than unity (Re ∼ motion (cf. Fig. 1 C and D). When the anterior-sided cilia beat 10−2), we neglect the inertial effect. Details of the governing in effective strokes (t/T = 0.3 in the symplectic and t/T = 0.9 equations and numerical method can be found in our published in the antiplectic waves), the swimming velocity becomes neg- paper (14) and SI Appendix. The time-dependent profile of each ative, while it is positive when the posterior-sided cilia have ciliary motion vcilia is derived by the mathematical formula of Ful- effective strokes (t/T = 0.9 and t/T = 0.1, respectively). Dur- ford and Blake (28) (cf. Fig. 1A and SI Appendix). Neighboring ing the effective stroke, the mechanical power induced by the cilia beat collectively—i.e., a metachronal wave—and two types ciliate also becomes large, as shown in Fig. 1E. Although the of metachronal waves are employed (5, 10) (Fig. 1B): symplectic ciliate shows back-and-forth motions, the time-integrated swim- (the beat direction and that of wave transmission coincide) and ming velocities are positive for both metachronal waves; thus, it antiplectic (the two directions are opposite). Cilia are located on propels forward. the vertices or element centers of the surface triangle meshes, Here, we introduce a theory derived by Stone and Samuel (29) and they are distributed almost homogeneously (cf. Fig. 1B). in order to compare the swimming velocity. Let (v, q) be the The surface triangles are made by repeatedly dividing the tri- velocity and the traction force on the organism surface under angles of icosahedron; one triangle was dividing into four small free-swimming condition. Also, let (v0, q0) be the solution for triangles. The temporal velocity fields around a spherical cili- translation of the same-shaped object with an external force F0.

Fig. 1. Swimming ciliate with two different metachronal waves. (A) Each cilium repeats effective and recovery strokes within the period (solid and dotted lines, respectively). (B) A symplectic wave (Symp); the beat direction and that of wave transmission coincide, whereas the two directions are opposed in an antiplectic wave (Anti). Red indicates the effective stroke. (C) Flow field around the ciliate (Lab frame); red represents the flow magnitude. (D and E) Time change of swimming velocity and power with two different metachronal waves. For the comparison with Stone and Samuel (29), analytical predictions are also plotted as square symbols in D. Time t is normalized by the period T.

30202 | www.pnas.org/cgi/doi/10.1073/pnas.2011146117 Omori et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 ih-adsd fEq. of side right-hand et epniua otesimn ieto r zero; are direction compo- swimming velocity the F and to force perpendicular swimmer, the nents of axisymmetry to Due decomposed be velocity can v organism translational the a of into velocity surface The face. where i.2. Fig. between relationship a provides and theorem reciprocal The mr tal. et Omori F (C coefficient. = steda force, drag the is z 0 = U (v U + 0 S , cln ihtebd eghudrtecniino osatnme fcilia of number constant of condition the under length body the with Scaling y q v = stesraeo h raim nldn h iir sur- ciliary the including organism, the of surface the is cilia 0 sfollows: as ), U ieaeae oe.(E power. Time-averaged (D) velocity. swimming Time-averaged ) Using . z 0 = U n Eq. and , stesimn speed, swimming the is U F Z v x 0 0 1 · = = q U a evnse,adw have we and vanished, be can 0 − U · = vdS 2 Z − n h oc-recniin the condition, force-free the and a erwitnas rewritten be can U q Z = 0 n iir motion ciliary a and · Z q v cilia 0 · q v dS cilia · L v 0 steclaylength, ciliary the is /F dS dS x 0 , . . v cilia µ —i.e., stevsoiy and viscosity, the is (v, F y [3] [2] [1] 0 q) = wmigefiinywt ifrn eahoa waves. metachronal different with efficiency Swimming ) ftebd,aduiomtneta stress tangential uniform and body, the of al. et here. Ishikawa briefly in only A them appendix explain in we so found (30), be forces. ciliary can the details express model to The surface shear spherical body tangential the of above mean slightly model a stress prescribes theoretical which a squirmer, introduce stress-given we states 2C. mechanism, to law Fig. proportional the Stokes’ is in because drag shown counterintuitive, viscous as that be two, may minus result about This of slope increasing with by decreases, However, force, law. drag The drag as average. scaled time be the can indicates thus, bar the and velocity, coefficient drag radius 2B, body the Fig. in shown As o aiu oyszs keeping sizes, efficiency body and various power, velocity, for swimming the investigate next Cilia. We of Number Constant with Efficiency Swimming with coincide always predictions. results analytical numerical the our that see We symbols). Eq. of predictions Analytical hr san-lpbudr at boundary no-slip a is There T N steperiod. the is h oylnt hne rm2 a/L from changes length body The (A) . PNAS a where , | F eebr1 2020 1, December ∝ F aU steda force, drag the is 3 hc orsod oteStokes’ the to corresponds which , r lopotdi i.1D Fig. in plotted also are L and a r a wmigvlct rapidly velocity swimming , = ahrthan rather , F N a | ¯ where , U ¯ o.117 vol. osat(f i.2A). Fig. (cf. constant −1 f θ U spootoa to proportional is 0t 0 (B 30. to 10 = sapidt the to applied is a steswimming the is stebd radius, body the is | a a o 48 no. steradius the is 2 oclarify To . (square | Drag ) 30203

ENGINEERING fluid at radius a0 = a + L. The squirmer swims freely in the θ = 0 polynomial. U is the unknown swimming speed, and the first direction, so the force and torque exerted on it by the fluid are term in Eq. 5 represents the uniform stream at infinity, relative (2) (2) zero, where θ is the polar angle relative to the squirmer. The to the squirmer, and Cn and Dn are zero for all n, so the con- r = a v = v = 0 v v (2) boundary condition at are r θ , where r and θ are tribution to the velocity at infinity tends to zero. Moreover, A r θ r = a 1 the velocity in the , directions, respectively. At 0, con- is also zero, because this is the Stokeslet term, proportional to v v −p + σ tinuity of r , θ, and the normal stress rr are assumed, the net force on the sphere, which is zero. The constant f can be where p is the pressure, and σ is the stress tensor. The stress θ also expanded in a series of the Vn , where f is the same constant ∆σ r = a f θ jump rθ at 0 is given by θ. Then, the problem is solved on both sides and could be canceled. with respect to unknown U under given fθ condition. Con- sider the flow in two regions; inside the ciliary layer, a < r < ∞ a0, and outside the ciliary layer, a0 < r < ∞, and represent it X by stream functions ψ(i)(r, θ), i = 1 and 2, respectively. The fθ = fθ Fn Vn (θ), [6] solution of the axisymmetric Stokes equation in each layer is n=1 given by 1 3 where F2m = 0, and F2m+1 = (4m + 3)Γ(m + 2 )Γ(m + 2 )/ ∞ " n−2 n 4Γ(m + 1)Γ(m + 2). The swimming speed U is given by the first (1) X 1 (1) a0  (1) a0  ψ = sin θVn (θ) An + Bn [4] mode of the equation; hence, we need to use only the n = 1 2 r r n=1 terms; for example, the relevant contribution to fθ is F1 = 3π/8.  n+1  n+3# A(1) B (1) C (1) D (1) B (2) (1) r (1) r The constants 1 , 1 , 1 , 1 , and 1 are determined +Cn + Dn , by the boundary conditions, and the result can be written as: a0 a0

and πf L 4ε2 + 9ε + 6 U = θ , [7] µ 24(1 + ε) ∞ " n−2 (2) 1 2 2 X 1 (2) a0  ψ = − Ur sin θ + sin θV (θ) A [5] 2 2 n n r n=1 where ε = L/a. Taking the limit of ε → 0, the swimming speed U n+1 n+3# converges to a constant value, if the stress f is constant. How-  a n  r   r  θ +B (2) 0 + C (2) + D (2) , ever, when the number of cilia is constant, total ciliary force n r n a n a 0 0 is fixed, the density fθ decreases as the radius increases, and the swimming speed should decay U ∝ a−2. Theoretical solu- 2 0 tion also predicts that swimming speed decreases as the power of where Vn (θ) = sin θP (cos θ), Pn is the Legendre poly- n(n+2) n minus two when the total ciliary force is fixed: This agrees with (i) (i) (i) (i) nomial, and An , Bn , Cn , and Dn are constants of the our numerical results.

Fig. 3. Effects of number of cilia. (A and B) Flow around the ciliate in effective stoke. The color represents the flow magnitude. (C) Averaged swimming velocity; the body length is fixed to 2a/L = 20. C, Inset indicates the power as a function of N.(D and E) Swimming efficiency with various N and body radius a. Filled symbols indicate the maximum efficiency. Anti, antiplectic; symp, symplectic.

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ENGINEERING ellum (32), while the sperm model can swim forward by beating a flagellum like a whip (33) (geometrical data are shown in SI Appendix). In Fig. 5A, η of the three types of swimmers as a function of their body size (head size for the sperm model) are shown. Note that the optimal η of the ciliate model is also plot- ted in the figure. In the small body-length regime (2a/L ≤ 10−1), η becomes large for both the bacteria and sperm models. Typi- cal sizes of Escherichia coli and human sperm are also plotted in the figure, which indicate that E. coli and human sperm exist in the high-η regime. As body size increases, η of bacteria and sperm rapidly decrease, approximately as (a/L)−3 in the large-a regime. In the ciliate model, however, the optimal η is propor- tional to (a/L)−1, even in the large-body regime. This result suggests that η is dramatically enhanced by multiple cilia on the body. Such a strategy should be helpful for ciliates to evolve into larger forms. We here compare mechanical power and physiological meta- bolic rate. In Katsu-Kimura et al. (8), they investigated stan- dard metabolic rate of an unicellular ciliate, Paramecium, by measuring oxygen-consumption rate, which was estimated as 3.28 × 10−10 J·s−1·cell−1. We assume the body length to be 2a/L = 10, the cilium length L = 10 µm, the frequency 30 Hz, and the viscosity 10−3 Pa·s, the optimum mechanical power would be 1.26 × 10−11 J·s−1. Thus, the power for locomotion should be much smaller than the physiological metabolic rate; 3.8% of oxygen consumption may convert to the mechanical power. Lastly, we show the optimal number density of cilia, which maximizes η, and compare the results with actual microorgan- isms. Fig. 5B shows N for achieving the optimal η. Because the optimal interciliary distance is almost constant in all cases, the optimal value of N is proportional to (a/L)2, regardless of the metachronal wave mode. The parameters of actual cili- Fig. 5. Comparison with various types of swimmers. (A) Efficiency of sperm- ates (Blepharisma, Loxodes, Paramecium, and Tetrahymena) and type swimmers, bacteria-type swimmers, and ciliates. Typical length ratio of Volvocaceae (Chlamydomonas, Gonium, Eudorina, Pleodorina, human sperm and E. coli are indicated. Filled symbols represent the results of ciliates with optimal efficiency, while open symbols are the results of and Volvox) are also plotted in Fig. 5B (morphological data are sperm-type or bacteria-type swimmers. ( ) Number of cilia as a function of taken from refs. 16–27). N of actual microorganisms increases B 2 body length; optimal number of cilia (filled symbols) increases with (a/L)2, roughly with (a/L) , so we can confirm quantitative agreement which coincides with species of microalgae and ciliates found in nature between our prediction and actual microorganisms. Drescher (open circles). et al. (34) stated that V. carteri consists of thousands of biflagellated somatic cells, and cilia are arranged in rows. In other species—Tetrahymena, for instance—somatic cilia are studies [the cilia are embedded on a moving rigid sphere in this also arranged into longitudinal rows (25). To see the effect study, whereas they are on an infinite flat plane in Blake (31)]. of placement of cilia, we calculate the swimming efficiency These results suggest that the number of cilia N increases with with cilia arranged in the spherical coordinate, and results are a2 in the optimal swimming. Accordingly, the optimal power 2 shown in SI Appendix. We confirm that the maximum effi- increases with a (cf. Fig. 4A), since the power is scaled by N . On ciency is achieved when cilia are distributed homogeneously. the other hand, optimal swimming speed is almost invariant with The resultant maximum efficiency is similar to Fig. 5A, and the body size (cf. Fig. 4B). Such a tendency coincides with the the optimal number of cilia is proportional to (a/L)2, even constant fθ condition in Ishikawa et al. (30). The optimal swim- when cilia are arranged in the spherical coordinate. We con- ming efficiency then can be scaled as a(U opt)2/P opt ' a−1 (cf. −3 clude, therefore, that motile microorganisms in nature have Fig. 5A), though η ∝ a in the fixed N conditions. These results achieved the optimal η by acquiring the optimal N on their body illustrate that the hydrodynamic interactions among motile cilia surfaces. govern the ciliate’s dynamics, and multiple cilia can enhance the efficiency η. Data Availability. All study data are included in the article and SI Appendix. We now compare η for various types of microswimmers: bac- teria, sperm, and ciliate-type swimmers. The bacterial model has ACKNOWLEDGMENTS. This work was supported by Japan Society for the a helical rigid flagellum, and it can propel by rotating the flag- Promotion of Science Grants 17H00853 and 18K18354.

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