Abstract：
Cilia are hairlike microactuators whose cyclic motion is specialized to propel extracellular fluids at low Reynolds numbers. Clusters of these organelles can form synchronized beating patterns, called metachronal waves, which presumably arise from hydrodynamic interactions. We model hydrodynamically interacting cilia by microspheres elastically bound to circular orbits, whose inclinations with respect to a no-slip wall model the ciliary power and recovery stroke, resulting in an anisotropy of the viscous flow. We derive a coupled phase-oscillator description by reducing the microsphere dynamics to the slow timescale of synchronization and determine analytical metachronal wave solutions and their stability in a periodic chain setting. In this framework, a simple intuition for the hydrodynamic coupling between phase oscillators is established by relating the geometry of flow near the surface of a cell or tissue to the directionality of the hydrodynamic coupling functions. This intuition naturally explains the properties of the linear stability of metachronal waves. The flow near the surface stabilizes metachronal waves with long wavelengths propagating in the direction of the power stroke and, moreover, metachronal waves with short wavelengths propagating perpendicularly to the power stroke. Performing simulations of phase-oscillator chains with periodic boundary conditions, we indeed find that both wave types emerge with a variety of linearly stable wave numbers. In open chains of phase oscillators, the dynamics of metachronal waves is fundamentally different. Here the elasticity of the model cilia controls the wave direction and selects a particular wave number: At large elasticity, waves traveling in the direction of the power stroke are stable, whereas at smaller elasticity waves in the opposite direction are stable. For intermediate elasticity both wave directions coexist. In this regime, waves propagating towards both ends of the chain form, but only one wave direction prevails, depending on the elasticity and initial conditions.
摘要：
纤毛是毛状微致动器，其循环运动专门用于在低雷诺数下推动细胞外液。这些细胞器的集群可以形成同步的跳动模式，称为元时波，这可能是由流体动力学相互作用引起的。我们通过与圆形轨道弹性结合的微球对流体动力学相互作用的纤毛进行建模，其相对于无滑移壁的倾斜度模型为纤毛功率和恢复行程，导致粘性流的各向异性。我们通过将微球动力学简化为同步的慢时间尺度来得出耦合的相位振荡器描述，并确定解析的历时波解及其在周期性链设置中的稳定性。在这个框架中,通过将细胞或组织表面附近的流动几何形状与流体动力耦合函数的方向性相关联，可以建立相位振荡器之间的流体动力耦合的简单直觉。这种直觉自然地解释了元时波的线性稳定性的性质。表面附近的流动稳定了沿动力冲程方向传播的长波长的变波，此外,短波波垂直于动力冲程传播。执行具有周期性边界条件的相位振荡器链的模拟，我们确实发现两种波类型都出现了各种线性稳定的波数。在相位振荡器的开放链中，变征波的动力学是根本不同的。在这里，模型纤毛的弹性控制波方向并选择特定的波数：在大弹性时，在动力冲程方向上传播的波是稳定的，而在较小的弹性波在相反的方向是稳定的。对于中等弹性,两个波方向共存。在这个制度中,波向链的两端传播，但是只有一个波浪方向占上风，取决于弹性和初始条件。
