Hydrodynamic Flow Patterns and Synchronization of Beating Cilia

We calculate the hydrodynamic flow field generated far from a cilium which is attached to a surface and beats periodically. In the case of two beating cilia, hydrodynamic interactions can lead to synchronization of the cilia, which are nonlinear oscillators. We present a state diagram where synchronized states occur as a function of the distance of cilia and the relative orientation of their beat. Synchronized states occur with different relative phases. In addition, asynchronous solutions exist. Our work could be relevant for the synchronized motion of cilia generating hydrodynamic flows on the surface of cells.

Figure 1

(a) Beating pattern of a cilium. The effective stroke and the recovery stroke are different (arrows) and a net flow is generated. (b) Simplified representation of the cilium by a small sphere, moving on a tilted elliptical trajectory.

Figure 2

(a) Flux lines describing the time-averaged fluid flow generated by a sphere moving on a tilted elliptical path shown in the $x\mathrm{\text{−}}y$ plane for $z/D=2$ (other parameters: $A/D=0.5$, $B/D=1$, $C/D=0.5$, $a/D=0.1$). The curves were obtained numerically using the flow field based on Eqs. (3, 4, 5). (b) Same flow field displayed in the $y\mathrm{\text{−}}z$ plane, for $x=0$. Black lines correspond to low fluid velocity, yellow lines (light gray) to high velocity. The dashed lines indicate the projections of the sphere trajectory.

Figure 3

(a) Top view of the arrangement of two elliptical trajectories which represent two beating cilia. The distance $r$ and the angle $\beta$ are indicated. (b) The state diagram as a function of the distance vector $x=x_2^0-x_1^0$, $y=y_2^0-y_1^0$, determined from the numerical solution of the full model equations. Three different regions are indicated. In the region of asynchronous beat, two frequencies occur. Two synchronous states can be distinguished with equal (${\mathrm{Sync}}^{+}$) and opposite (${\mathrm{Sync}}^{-}$) phases of both cilia in the limit of large separation $r$.

Figure 4

The phase difference $\delta$ between two spheres on elliptic trajectories in the stable stationary state as a function of their relative position in polar coordinates (angle $\beta$, distance $r$) obtained by numerical solutions to the full dynamic equations. In the limit $r\to \infty$, analytic far-field calculations show that only equal and opposite phases ($\delta =0$ and $\pm \pi$, respectively) occur. For smaller $r$, nontrivial angles $\delta$ occur. Synchronous states become unstable where the lines end (black dots). Between these points, motion is asynchronous.