week ending 14 JUNE 2013
PHYSICAL REVIEW LETTERS
PRL 110, 248107 (2013)
Hydrodynamic Phase Locking in Mouse Node Cilia Atsuko Takamatsu,1 Kyosuke Shinohara,2 Takuji Ishikawa,3 and Hiroshi Hamada2 1
Department of Electrical Engineering and Bioscience, Waseda University, Shinjuku-ku, Tokyo 169-8555, Japan Developmental Genetics Group, Graduate School of Frontier Biosciences, Osaka University, Suita Osaka 565-0871, Japan 3 Department of Bioengineering and Robotics, Tohoku University, Sendai, Miyagi 980-8579, Japan (Received 24 January 2012; revised manuscript received 27 April 2013; published 14 June 2013)
2
Rotational movement of mouse node cilia generates leftward fluid flow in the node cavity, playing an important role in left-right determination in the embryo. Although rotation of numerous cilia was believed necessary to trigger the determination, recent reports indicate the action of two cilia to be sufficient. We examine cooperative cilia movement via hydrodynamic interaction. Results show cilia to be cooperative, having phases locked in a certain relation; a system with a pair of nonidentical cilia can achieve phaselocked states more easily than one with a pair of identical cilia. DOI: 10.1103/PhysRevLett.110.248107
PACS numbers: 87.16.Qp, 05.45.Xt, 47.63.b
Hydrodynamic interaction among flagella or cilia is considered to play an important role in their cooperative movement [1]. Examples are found in swimming protozoan and metachronal waves of cilia in epithelium, in which flagella or cilia are arranged densely on the cell surface. In contrast, in monociliated cells, single cells with a single cilium, details of the effects of hydrodynamic interaction among cilia have not been studied because cilia intervals are much larger. There are a few reports in which experiments particularly addressing the interaction among sparse elements have been described [2–5]. In this Letter, we investigate rotational movement of mouse node monocilia and present experimental evidence of phase locking in a broad sense. Then, using phase reduction theory [6] combined with computational fluid dynamics, we explain that a phase-locked state can exist. A node cavity located on the ventral side of a mouse embryo at an early developmental stage (7.75 days after fertilization) is filled with extra embryonic fluid. It comprises a few hundred node cells, on each of which a motile cilium rotates clockwise (Fig. 1). The rotational movement plays an important role in left-right (LR) symmetry breaking in mice. The movement generates a leftward laminar flow of up to 1–10 m=s. Unidirectional flow can be achieved by tilting the rotational axis of respective cilia, which results in an effective stroke generating leftward flow and a recovery stroke generating slower rightward flow because of a no-slip boundary condition on the cell surface [Fig. 1(b)]. The mechanism underlying the conversion of the flow to a left-determinant signal has not been elucidated, although a few hypotheses exist [7,8]. Simultaneous movement of numerous cilia is explained in any hypothesis as necessary to generate a sufficiently strong signal. Interestingly, Shinohara et al. discovered that local leftward flow generated with only two rotating cilia is sufficient for LR determination using a mutant-mice impaired cilia structure [9]. We explored what simply two cilia could achieve by particularly examining cilia phase locking. 0031-9007=13=110(24)=248107(4)
Rotational movements of an isolated pair of cilia aligned along the RL direction obtained from a ciliary mutant Dpcd [9,10] were recorded using high-speed imaging for 2 s (100 frames=s; HAS-500M, Ditect Co. Ltd.) [9]. Tip portions of the cilia were tracked manually to obtain a two-dimensional projection on the x-y plane [Fig. 1(c); inset of Fig. 2(a)]. The phase of each rotating cilium was estimated as that around the rotational axis of the cilium after an ellipse correction was applied [11,12] because the projected trajectory is elliptic as a result of tilting of cilia. The phase lag between the two cilia, 2 1 , was calculated [Fig. 2(a)]. For a measure of the degree of phase locking, we defined the phase-locking probability [13] (PLP) as the fraction of time spent at the most probable phase lag. As presented in Fig. 2(b), this is measured as the area of the mode bar (colored bars) in the normalized histogram of interciliary phase lags [14]. Figure 2(a) presents an example of a phase-locked state in a broad sense: The phase lag is almost fixed at around =2, but it fluctuates with a period that is approximately equivalent to the period of the cilia rotation [13]. The PLP was estimated as 0.36, which is significantly higher than that for a phase-drift case [13]. The mode phase lag in high PLP can be characterized as a locked phase lag, as shown in Fig. 2(b), in which the lag was estimated as around =2. Figure 2(c) summarizes the relation between PLP and the mode of the phase-lag distribution, indicating that when PLP > 0:2, the phase lags are locked at around =2 (or more precisely, distributed around [ , 0]), whereas when PLP 0:2, the mode phase lags are distributed across the complete interval [ , ]. The cases of PLP > 0:2 clearly suggest phase-locked states. Figure 2(d) also supports the conclusion presented above because the difference between the rotational phase speeds of the two cilia is small when PLP > 0:2, being estimated as ! < 2:5 ðrad=sÞ against a mean speed ! ¼ 58:3 ðrad=sÞ (!=! < 4%) [13]. To date, no biological interaction has been found among node cells such as gap-junction and Notch-Delta
248107-1
Ó 2013 American Physical Society
PHYSICAL REVIEW LETTERS
PRL 110, 248107 (2013) (a) R CC
(b)
cilium NC
L cilia (c)
NF
RS
cilium 2 A R
L P
R
β
x
θ1
α
NC ES
L
z cilium 1
θ2 y P
FIG. 1. (a) and (b) Node cilia. (c) Setup of cilia for computational fluid dynamics. L, R, A, and P, respectively, denote left, right, anterior, and posterior directions in a node. NC, node cell; CC, crown cell; NF, node flow; ES, effective stroke; RS, recovery stroke.
connections. Therefore, we assume only a hydrodynamic interaction. Several mathematical models for cooperative movement via hydrodynamic interaction have been proposed [15]. However, many of those models target beating cilia (moving mainly back and forth). Few models are applicable to rotating cilia [16–18]. In those systems, cilia were modeled as rigid beads moving on certain trajectories, which is a good approximation when two cilia are far apart. When two cilia are placed in the near field, however, details of the cilia shape strongly influence their hydrodynamic interactions, which might influence their state of synchronization. Consequently, we model cilium ið¼ 1; 2Þ as a rigid cylinder with length li and diameter ri . The cylinder can rotate clockwise around the rotational axis tilting to a direction i [19] with an angle of i along a conical face with an open angle of i [Fig. 1(c)]. The two cilia align along the x axis (RL direction) with an interval of d. The phase of cilium i, i , is measured from
the position at which the cilium points to the posterior (P) direction. Then, the phase vector is defined as ¼ ð1 ; 2 ÞT . No background flow of the surrounding fluid is assumed. The rotating cylinder is subjected to a viscous drag force for which the tangential component balances the driving force F ¼ ½F1 ð1 Þ; F2 ð2 ÞT if the inertia of both fluid and cilia is negligible. Then, the equation of rotational move_ where ment in the Stokes flow regime is written as F ¼ K, ! K11 K12 K¼ K21 K22 is a resistance matrix and Kij ¼ Kij ði ; j Þ is the coefficient of the drag force from cilium j to cilium i. Each component of K is calculated numerically using the boundary element method [12,13,20]. By exploiting the linearity of the problem, K11 and K21 are calculable independently from K12 and K22 . For instance, K21 can be derived from the drag force exerted on cilium 2 when cilium 1 rotates with unit rotational velocity while cilium 2 is fixed in space. The geometry of cilium i is set, in principle, as ri ¼ 0:1l0 , i ¼ =6, i ¼ =4, and d ¼ 2l0 (where l0 is a certain unit length related to cilium length, which was set as 1 in the numerical calculations.), based on experimental observation [7,8]. Each cilium surface is discretized by 594 triangle elements. The Stokes flow around the two nearby cilia is computed under given velocity conditions. The force distribution on the two cilia is obtained by using the boundary element analysis, which is used to obtain K. To derive phase equations, jKj 0 was assumed; this assumption holds in the numerical calculation above. Then, _ ¼ K1 F is obtained. The drag coefficient Kii is separated into a term for an isolated single cilium, Ji ði Þ (the numerical results of which are depicted in Fig. 3 of Ref. [12]) and a term depending on the presence of the other: Kii ¼ Ji ði Þ þ Kii0 ði ; j Þ. The other coefficient is rewritten as Kij ¼ Kij0 ði ; j Þ. The coefficient represents the interaction strength, which can depend on the distance between the cilia [13]. Consequently, is maximized at a possible minimum distance [13], where the tip portions of cilia never mutually touch. Then, it is found that 0max . Herein, it is assumed that ¼ 1 when d ¼ 2l0 , as defined above. Hence, the equations for the phase speed can be rewritten as _ i ¼
FIG. 2 (color online). Examples of phase locking represented in time courses (a) and distributions of phase lag (b). The inset in (a) represents the projected trajectories of the two cilia. (c) Relation between PLP and mode phase lag. (d) Relation between PLP and observed phase speed. Open circles and closed triangles, respectively, show data for cilium 1 and 2. The standard deviations of phase speed for the data are 3:3–7:5 ðrad=sÞ. Data in (c) and (d) were obtained from 29 trials under almost identical settings [13].
week ending 14 JUNE 2013
0 Fi Jj ðj Þ þ ðKjj Fi Kij0 Fj Þ : jKj
(1)
The driving force is defined as Fi ði Þ¼F0ðiÞ f1FrðiÞ cosði FsðiÞ Þg, F0ðiÞ > 0, 0 FrðiÞ < 1, 0 l2 [Fig. 5(a)] and l1 < l2 [Fig. 5(b)]. The phase speed of the cilia with interaction in the latter half of the leftward stroke ( < 2 < 3=2) increases 13% against that without interaction when l1 > l2 (and an increase is also observed when 1 > 2 or 1 > 2 ). Although the increase in the speed is not always observed as seen in the case of l1 < l2 (and also for the case of 1 < 2 or 1 < 2 ), it is noteworthy that pairs of cilia can generate faster flow than isolated single cilia do. In conclusion, we demonstrated the existence of phase locking in a two-cilium system of a mutant mouse node experimentally and proposed a model that is compatible with experimental observations. The phase-locked state
[1] D. Bray, Cell Movements: From Molecules to Motility (Garland Science, New York, 2000), 2nd ed. [2] M. Polin, I. Tuval, K. Drescher, J. P. Gollub, and R. E. Goldstein, Science 325, 487 (2009). [3] J. Kotar, M. Leoni, B. Bassetti, M. C. Lagomarsino, and P. Cicuta, Proc. Natl. Acad. Sci. U.S.A. 107, 7669 (2010). [4] R. E. Goldstein, M. Polin, and I. Tuval, Phys. Rev. Lett. 107, 148103 (2011). [5] D. R. Brumley, M. Polin, T. J. Pedley, and R. E. Goldstein, Phys. Rev. Lett. 109, 268102 (2012). [6] Y. Kuramoto, Prog. Theor. Phys. Suppl. 79, 223 (1984). [7] W. F. Marshall and S. Nonaka, Curr. Biol. 16, R604 (2006). [8] H. Shiratori and H. Hamada, Development (Cambridge, U.K.) 133, 2095 (2006). [9] K. Shinohara, A. Kawasumi, A. Takamatsu, S. Yoshiba, Y. Botilde, N. Motoyama, W. Reith, B. Durand, H. Shiratori, and H. Hamada, Nat. Commun. 3, 622 (2012). [10] Y. Kobayashi, M. Watanabe, Y. Okada, H. Sawa, H. Takai, M. Nakanishi, Y. Kawase, H. Suzuki, K. Nagashima, K. Ikeda, and N. Motoyama, Mol. Cell. Biol. 22, 2769 (2002). [11] Y. Okada, S. Takeda, Y. Tanaka, J.-C. I. Belmonte, and N. Hirokawa, Cell 121, 633 (2005). [12] A. Takamatsu, T. Ishikawa, K. Shinohara, and H. Hamada, arXiv:1212.3081. [13] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.110.248107 for phase-lock probability, fluctuation of phase locked state, cilia parameters in Fig. 2, calculation of drag coefficients, approximation for , setting of the minimum distance between cilia, effect of difference in the tilting direction, and effect of orbit flexibility. [14] The minimum value of PLP is 1=24 0:042 because the bin width was set as 2=24. [15] R. Golestanian, J. M. Yeomans, and N. Uchida, Soft Matter 7, 3074 (2011). [16] A. Vilfan and F. Ju¨licher, Phys. Rev. Lett. 96, 058102 (2006). [17] P. Lenz and A. Ryskin, Phys. Biol. 3, 285 (2006). [18] N. Uchida and R. Golestanian, Phys. Rev. Lett. 106, 058104 (2011). [19] The tilting direction i is measured from x axis direction. Some examples are i ¼0ðLÞ, =2ðAÞ, ðRÞ, =2ðPÞ. [20] T. Ishikawa, M. P. Simmonds, and T. J. Pedley, J. Fluid Mech. 568, 119 (2006). [21] Results of numerical calculations satisfy this. We did not use this approximation for the calculation for Figs. 3(b), 3(c), and 4. [22] Furthermore, i implicitly includes parameters determining intrinsic phase-speed with Ji ði Þ and Fi ði Þ. [23] Values F0ð1;2Þ are tuned so that !1 !2 .
248107-4
PHYSICAL REVIEW LETTERS
PRL 110, 248107 (2013)
Hydrodynamic Phase Locking in Mouse Node Cilia Atsuko Takamatsu,1 Kyosuke Shinohara,2 Takuji Ishikawa,3 and Hiroshi Hamada2 1
Department of Electrical Engineering and Bioscience, Waseda University, Shinjuku-ku, Tokyo 169-8555, Japan Developmental Genetics Group, Graduate School of Frontier Biosciences, Osaka University, Suita Osaka 565-0871, Japan 3 Department of Bioengineering and Robotics, Tohoku University, Sendai, Miyagi 980-8579, Japan (Received 24 January 2012; revised manuscript received 27 April 2013; published 14 June 2013)
2
Rotational movement of mouse node cilia generates leftward fluid flow in the node cavity, playing an important role in left-right determination in the embryo. Although rotation of numerous cilia was believed necessary to trigger the determination, recent reports indicate the action of two cilia to be sufficient. We examine cooperative cilia movement via hydrodynamic interaction. Results show cilia to be cooperative, having phases locked in a certain relation; a system with a pair of nonidentical cilia can achieve phaselocked states more easily than one with a pair of identical cilia. DOI: 10.1103/PhysRevLett.110.248107
PACS numbers: 87.16.Qp, 05.45.Xt, 47.63.b
Hydrodynamic interaction among flagella or cilia is considered to play an important role in their cooperative movement [1]. Examples are found in swimming protozoan and metachronal waves of cilia in epithelium, in which flagella or cilia are arranged densely on the cell surface. In contrast, in monociliated cells, single cells with a single cilium, details of the effects of hydrodynamic interaction among cilia have not been studied because cilia intervals are much larger. There are a few reports in which experiments particularly addressing the interaction among sparse elements have been described [2–5]. In this Letter, we investigate rotational movement of mouse node monocilia and present experimental evidence of phase locking in a broad sense. Then, using phase reduction theory [6] combined with computational fluid dynamics, we explain that a phase-locked state can exist. A node cavity located on the ventral side of a mouse embryo at an early developmental stage (7.75 days after fertilization) is filled with extra embryonic fluid. It comprises a few hundred node cells, on each of which a motile cilium rotates clockwise (Fig. 1). The rotational movement plays an important role in left-right (LR) symmetry breaking in mice. The movement generates a leftward laminar flow of up to 1–10 m=s. Unidirectional flow can be achieved by tilting the rotational axis of respective cilia, which results in an effective stroke generating leftward flow and a recovery stroke generating slower rightward flow because of a no-slip boundary condition on the cell surface [Fig. 1(b)]. The mechanism underlying the conversion of the flow to a left-determinant signal has not been elucidated, although a few hypotheses exist [7,8]. Simultaneous movement of numerous cilia is explained in any hypothesis as necessary to generate a sufficiently strong signal. Interestingly, Shinohara et al. discovered that local leftward flow generated with only two rotating cilia is sufficient for LR determination using a mutant-mice impaired cilia structure [9]. We explored what simply two cilia could achieve by particularly examining cilia phase locking. 0031-9007=13=110(24)=248107(4)
Rotational movements of an isolated pair of cilia aligned along the RL direction obtained from a ciliary mutant Dpcd [9,10] were recorded using high-speed imaging for 2 s (100 frames=s; HAS-500M, Ditect Co. Ltd.) [9]. Tip portions of the cilia were tracked manually to obtain a two-dimensional projection on the x-y plane [Fig. 1(c); inset of Fig. 2(a)]. The phase of each rotating cilium was estimated as that around the rotational axis of the cilium after an ellipse correction was applied [11,12] because the projected trajectory is elliptic as a result of tilting of cilia. The phase lag between the two cilia, 2 1 , was calculated [Fig. 2(a)]. For a measure of the degree of phase locking, we defined the phase-locking probability [13] (PLP) as the fraction of time spent at the most probable phase lag. As presented in Fig. 2(b), this is measured as the area of the mode bar (colored bars) in the normalized histogram of interciliary phase lags [14]. Figure 2(a) presents an example of a phase-locked state in a broad sense: The phase lag is almost fixed at around =2, but it fluctuates with a period that is approximately equivalent to the period of the cilia rotation [13]. The PLP was estimated as 0.36, which is significantly higher than that for a phase-drift case [13]. The mode phase lag in high PLP can be characterized as a locked phase lag, as shown in Fig. 2(b), in which the lag was estimated as around =2. Figure 2(c) summarizes the relation between PLP and the mode of the phase-lag distribution, indicating that when PLP > 0:2, the phase lags are locked at around =2 (or more precisely, distributed around [ , 0]), whereas when PLP 0:2, the mode phase lags are distributed across the complete interval [ , ]. The cases of PLP > 0:2 clearly suggest phase-locked states. Figure 2(d) also supports the conclusion presented above because the difference between the rotational phase speeds of the two cilia is small when PLP > 0:2, being estimated as ! < 2:5 ðrad=sÞ against a mean speed ! ¼ 58:3 ðrad=sÞ (!=! < 4%) [13]. To date, no biological interaction has been found among node cells such as gap-junction and Notch-Delta
248107-1
Ó 2013 American Physical Society
PHYSICAL REVIEW LETTERS
PRL 110, 248107 (2013) (a) R CC
(b)
cilium NC
L cilia (c)
NF
RS
cilium 2 A R
L P
R
β
x
θ1
α
NC ES
L
z cilium 1
θ2 y P
FIG. 1. (a) and (b) Node cilia. (c) Setup of cilia for computational fluid dynamics. L, R, A, and P, respectively, denote left, right, anterior, and posterior directions in a node. NC, node cell; CC, crown cell; NF, node flow; ES, effective stroke; RS, recovery stroke.
connections. Therefore, we assume only a hydrodynamic interaction. Several mathematical models for cooperative movement via hydrodynamic interaction have been proposed [15]. However, many of those models target beating cilia (moving mainly back and forth). Few models are applicable to rotating cilia [16–18]. In those systems, cilia were modeled as rigid beads moving on certain trajectories, which is a good approximation when two cilia are far apart. When two cilia are placed in the near field, however, details of the cilia shape strongly influence their hydrodynamic interactions, which might influence their state of synchronization. Consequently, we model cilium ið¼ 1; 2Þ as a rigid cylinder with length li and diameter ri . The cylinder can rotate clockwise around the rotational axis tilting to a direction i [19] with an angle of i along a conical face with an open angle of i [Fig. 1(c)]. The two cilia align along the x axis (RL direction) with an interval of d. The phase of cilium i, i , is measured from
the position at which the cilium points to the posterior (P) direction. Then, the phase vector is defined as ¼ ð1 ; 2 ÞT . No background flow of the surrounding fluid is assumed. The rotating cylinder is subjected to a viscous drag force for which the tangential component balances the driving force F ¼ ½F1 ð1 Þ; F2 ð2 ÞT if the inertia of both fluid and cilia is negligible. Then, the equation of rotational move_ where ment in the Stokes flow regime is written as F ¼ K, ! K11 K12 K¼ K21 K22 is a resistance matrix and Kij ¼ Kij ði ; j Þ is the coefficient of the drag force from cilium j to cilium i. Each component of K is calculated numerically using the boundary element method [12,13,20]. By exploiting the linearity of the problem, K11 and K21 are calculable independently from K12 and K22 . For instance, K21 can be derived from the drag force exerted on cilium 2 when cilium 1 rotates with unit rotational velocity while cilium 2 is fixed in space. The geometry of cilium i is set, in principle, as ri ¼ 0:1l0 , i ¼ =6, i ¼ =4, and d ¼ 2l0 (where l0 is a certain unit length related to cilium length, which was set as 1 in the numerical calculations.), based on experimental observation [7,8]. Each cilium surface is discretized by 594 triangle elements. The Stokes flow around the two nearby cilia is computed under given velocity conditions. The force distribution on the two cilia is obtained by using the boundary element analysis, which is used to obtain K. To derive phase equations, jKj 0 was assumed; this assumption holds in the numerical calculation above. Then, _ ¼ K1 F is obtained. The drag coefficient Kii is separated into a term for an isolated single cilium, Ji ði Þ (the numerical results of which are depicted in Fig. 3 of Ref. [12]) and a term depending on the presence of the other: Kii ¼ Ji ði Þ þ Kii0 ði ; j Þ. The other coefficient is rewritten as Kij ¼ Kij0 ði ; j Þ. The coefficient represents the interaction strength, which can depend on the distance between the cilia [13]. Consequently, is maximized at a possible minimum distance [13], where the tip portions of cilia never mutually touch. Then, it is found that 0max . Herein, it is assumed that ¼ 1 when d ¼ 2l0 , as defined above. Hence, the equations for the phase speed can be rewritten as _ i ¼
FIG. 2 (color online). Examples of phase locking represented in time courses (a) and distributions of phase lag (b). The inset in (a) represents the projected trajectories of the two cilia. (c) Relation between PLP and mode phase lag. (d) Relation between PLP and observed phase speed. Open circles and closed triangles, respectively, show data for cilium 1 and 2. The standard deviations of phase speed for the data are 3:3–7:5 ðrad=sÞ. Data in (c) and (d) were obtained from 29 trials under almost identical settings [13].
week ending 14 JUNE 2013
0 Fi Jj ðj Þ þ ðKjj Fi Kij0 Fj Þ : jKj
(1)
The driving force is defined as Fi ði Þ¼F0ðiÞ f1FrðiÞ cosði FsðiÞ Þg, F0ðiÞ > 0, 0 FrðiÞ < 1, 0 l2 [Fig. 5(a)] and l1 < l2 [Fig. 5(b)]. The phase speed of the cilia with interaction in the latter half of the leftward stroke ( < 2 < 3=2) increases 13% against that without interaction when l1 > l2 (and an increase is also observed when 1 > 2 or 1 > 2 ). Although the increase in the speed is not always observed as seen in the case of l1 < l2 (and also for the case of 1 < 2 or 1 < 2 ), it is noteworthy that pairs of cilia can generate faster flow than isolated single cilia do. In conclusion, we demonstrated the existence of phase locking in a two-cilium system of a mutant mouse node experimentally and proposed a model that is compatible with experimental observations. The phase-locked state
[1] D. Bray, Cell Movements: From Molecules to Motility (Garland Science, New York, 2000), 2nd ed. [2] M. Polin, I. Tuval, K. Drescher, J. P. Gollub, and R. E. Goldstein, Science 325, 487 (2009). [3] J. Kotar, M. Leoni, B. Bassetti, M. C. Lagomarsino, and P. Cicuta, Proc. Natl. Acad. Sci. U.S.A. 107, 7669 (2010). [4] R. E. Goldstein, M. Polin, and I. Tuval, Phys. Rev. Lett. 107, 148103 (2011). [5] D. R. Brumley, M. Polin, T. J. Pedley, and R. E. Goldstein, Phys. Rev. Lett. 109, 268102 (2012). [6] Y. Kuramoto, Prog. Theor. Phys. Suppl. 79, 223 (1984). [7] W. F. Marshall and S. Nonaka, Curr. Biol. 16, R604 (2006). [8] H. Shiratori and H. Hamada, Development (Cambridge, U.K.) 133, 2095 (2006). [9] K. Shinohara, A. Kawasumi, A. Takamatsu, S. Yoshiba, Y. Botilde, N. Motoyama, W. Reith, B. Durand, H. Shiratori, and H. Hamada, Nat. Commun. 3, 622 (2012). [10] Y. Kobayashi, M. Watanabe, Y. Okada, H. Sawa, H. Takai, M. Nakanishi, Y. Kawase, H. Suzuki, K. Nagashima, K. Ikeda, and N. Motoyama, Mol. Cell. Biol. 22, 2769 (2002). [11] Y. Okada, S. Takeda, Y. Tanaka, J.-C. I. Belmonte, and N. Hirokawa, Cell 121, 633 (2005). [12] A. Takamatsu, T. Ishikawa, K. Shinohara, and H. Hamada, arXiv:1212.3081. [13] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.110.248107 for phase-lock probability, fluctuation of phase locked state, cilia parameters in Fig. 2, calculation of drag coefficients, approximation for , setting of the minimum distance between cilia, effect of difference in the tilting direction, and effect of orbit flexibility. [14] The minimum value of PLP is 1=24 0:042 because the bin width was set as 2=24. [15] R. Golestanian, J. M. Yeomans, and N. Uchida, Soft Matter 7, 3074 (2011). [16] A. Vilfan and F. Ju¨licher, Phys. Rev. Lett. 96, 058102 (2006). [17] P. Lenz and A. Ryskin, Phys. Biol. 3, 285 (2006). [18] N. Uchida and R. Golestanian, Phys. Rev. Lett. 106, 058104 (2011). [19] The tilting direction i is measured from x axis direction. Some examples are i ¼0ðLÞ, =2ðAÞ, ðRÞ, =2ðPÞ. [20] T. Ishikawa, M. P. Simmonds, and T. J. Pedley, J. Fluid Mech. 568, 119 (2006). [21] Results of numerical calculations satisfy this. We did not use this approximation for the calculation for Figs. 3(b), 3(c), and 4. [22] Furthermore, i implicitly includes parameters determining intrinsic phase-speed with Ji ði Þ and Fi ði Þ. [23] Values F0ð1;2Þ are tuned so that !1 !2 .
248107-4
