Asymmetry in cilia configuration induces hydrodynamic phase locking

To gain insight into the nature of biological synchronization at the microscopic scale, we here investigate the hydrodynamic synchronization between conically rotating objects termed nodal cilia. A mechanical model of three rotating cilia is proposed with consideration of variation in their shapes and geometrical arrangement. We conduct numerical estimations of both near-field and far-field hydrodynamic interactions, and we apply a conventional averaging method for weakly coupled oscillators. In the nonidentical case, the three cilia showed stable locked-phase differences around $\pm \pi /2$. However, such phase locking also occurred with three identical cilia when allocated in a triangle except for the equilateral triangle. The effects of inhomogeneity in cilia shapes and geometrical arrangement on such asymmetric interaction is discussed to understand the role of biological variation in synchronization via hydrodynamic interactions.

Figure 1

Schematic illustration of two kinds of motile cilia. (a) Beating motion, consisting of an effective stroke and a recovery stroke. (b) Rotating motion in a node; L, R, A, and P indicate the left, right, anterior, and posterior sides of the node cavity in mice, respectively.

Figure 2

Coordinate system for the rotational movement of cilia. L, R, A, and P denote the left, right, anterior, and posterior sides in a node cavity, respectively.

Figure 3

Numerical estimation of viscous drag coefficients in the two-cilium system. (a) Setup. (b) Drag coefficients $J_i$ for an isolated cilium. (c)–(f) Drag coefficients $K_{ij}^{\prime }$ as an interaction function from cilium $j$ to $i$. The diagonal lines are included as a visual guide in (d) and (e) (see Sec. 4). The parameter values of each cilium are ${\alpha }_i=\pi /6,{\beta }_i=\pi /4,{\eta }_i=\pi /2,r_i=0.1l_0,l_1=1.5l_0,l_2=l_0=1,T_1=0.140$, and $T_2=0.0494$, respectively. The driving torque $T_i$ is set so that the natural frequency ${\omega }_i$ is 52.4 [rad/s]. The root positions of cilia 1 and 2 are set to the origin and $(x_2,y_2)=(2l_0,0)$, respectively. The drag coefficients are normalized by $\mu \omega l_0^3$, where $\mu$ is the viscosity.

Figure 4

Numerical estimation of viscous drag coefficients in the three-cilium system. (a) Setup. (b) Drag coefficients $J_i$. (c)–(k) Drag coefficients $K_{ij}^{\prime }$. Here, the sections of $K_{ij}^{\prime }$ at ${\theta }_1=2\pi ,{\theta }_2=2\pi$, and ${\theta }_3=0,2\pi$ are displayed. The parameter values of cilium 3 are ${\alpha }_3=\pi /6,{\beta }_3=\pi /4,{\eta }_3=\pi /2,r_3=0.1l_0,l_3=l_0=1,T_3=0.0494$, and $(x_3,y_3)=(4l_0,0)$, in addition to those of Fig. 3. The drag coefficients are normalized by $\mu \omega l_0^3$, where $\mu$ is the viscosity.

Figure 5

Time evolution of the phases of rotating cilia. (a) Evolution of the original phases ${\theta }_1$ and ${\theta }_2$ and their difference ${\varphi }_{12}={\theta }_2-{\theta }_1$ in the two-cilium system. (b) Evolution of the original phases ${\theta }_1,{\theta }_2$, and ${\theta }_3$ in the three-cilium system. (c) Evolution of the phase differences ${\varphi }_{12}={\theta }_2-{\theta }_1$ and ${\varphi }_{23}={\theta }_3-{\theta }_2$. The cilium parameter values are the same as those defined in Figs. 3 and 4 in the two- and three-cilium systems, respectively.

Figure 6

Phase portrait of the average phase difference ${\mathrm{\Phi }}_{12}$ in the $({\mathrm{\Phi }}_{12},{\dot{\mathrm{\Phi }}}_{12})$ plane. Stable and unstable fixed points are indicated by filled and open circles, respectively. The parameter values of the cilia are the same as those given in Fig. 3.

Figure 7

Dependence of phase-locked states on arrangement and shapes in the two-cilium model. (a), (b) Spatial dependence in the identical and nonidentical (different cilia lengths) settings. Cilium 1 was placed at the origin and cilium 2 as placed at an arbitrary position. The white area indicates the asynchronous states. (c)–(e) Phase-locked properties with a change in length. Cilium 2 was placed at the points with radius $3l_0$ and angle $n\pi /4(n=0,...,7)$. The cilium parameter values are the same as those described in Fig. 3 except for $l_1=l_0$ and $T_1=0.0494$ in (a).

Figure 8

Phase portrait of the average phase difference in the three-cilium system. (a), (b) Phase portraits of the average phase differences ${\mathrm{\Phi }}_{12}$ and ${\mathrm{\Phi }}_{23}$ in the $({\mathrm{\Phi }}_{12},{\mathrm{\Phi }}_{23},{\dot{\mathrm{\Phi }}}_{12})$ and $({\mathrm{\Phi }}_{12},{\mathrm{\Phi }}_{23},{\dot{\mathrm{\Phi }}}_{23})$ spaces, respectively. (c) Vector field in the phase space $({\mathrm{\Phi }}_{12},{\mathrm{\Phi }}_{23})$, summarizing the results of phase space analyses in (a) and (b). The filled and open circles represent stable and unstable foci, respectively. Open triangles denote saddlepoints. The cilium parameter values are the same as those given in Fig. 4.

Figure 9

Effects of geometrical arrangement on phase locking. (a) Spatial arrangement of the three cilia. (b)–(d) Results for the identical settings. (e)–(g) Results for the nonidentical (different cilia lengths) settings. (b), (e) The largest real part of the eigenvalues. (c), (f) Range of the natural frequency difference. Open and filled circles indicate asynchronous and phase-locked states, respectively. (d), (g) Average locked-phase difference. The parameter values of the cilia are the same as those in Fig. 4 except for $l_1=l_0$ and $T_1=0.0494$ in the identical situation, and $l_1=l_0,T_1=0.0494,l_3=1.5l_0$, and $T_3=0.140$ in the nonidentical situation. In both cases, the position of cilium 3 was $(x_3,y_3)=(2l_{0}cos{\theta }_{\mathrm{rot}}+2l_0,2l_{0}sin{\theta }_{\mathrm{rot}})$.

Figure 10

Schematic illustration of the arrangement of three identical cilia for symmetric and asymmetric interactions. (a) Equilateral-triangular arrangement. (b) Isosceles-triangular arrangement.