Collective nonlinear dynamics of cilia and flagella: from n=2 to n>>2 interacting cilia

纤毛地毯中的自组织异时波代表了活性物质中时空秩序自发出现的模型系统。每一个跳动的纤毛代表一个非线性的生物振荡器。这些振荡器中的几个的动力学通过周围的流体耦合，这促进了集体动力学。纤毛毯存在于生物微泳者的表面，如团藻或草履虫，以及高等动物的气道，输卵管和脑室。此外，纤毛地毯代表了一个最小的系统集体同步的研究中观察到的生物microswimmer与一个单一的纤毛（如精子细胞），其特征在于额外的平移和旋转自由度的集合。在这个项目中，我们提出了一个新的理论描述纤毛地毯集体非线性动力学，特别强调纤毛方向的排列和异时波的出现之间的关系。因此，我们将了解空间顺序（纤毛排列）和时间顺序（集体同步）在一个重要的模型系统之间的相互作用。为了实现这一目标，我们提出了一个新的理论框架，题为：“拉格朗日力学的主动系统（LAMAS）”，描述流体-结构-相互作用的主动，弹性结构，如纤毛。这种形式主义推广经典拉格朗日力学的保守和耗散系统的积极的系统。在第一个项目阶段已经针对n=2纤毛的情况开发了这种形式的关键要素，现在将扩展到n>>2纤毛的情况。到目前为止所获得的结果包括纤毛跳动的主动波动的理论，一个单独的纤毛的负载响应的表征，以及n=2纤毛的流体动力学同步的理论。通过发展纤毛地毯的模型系统，特别强调纤毛方向的旋转自由度的建议的形式主义，我们期待新的物理集体非线性动力学的生物microswimmer。我们采用创新的方法，如快速多极边界元法的斯托克斯方程，它管理这个系统的流体动力学的数值解。纤毛之间的流体动力学相互作用处理使用广义法克森定律。纤毛跳动的波形顺应性占通过将主变形模式与动态振幅。我们的新的理论方法结合了目前在SPP中使用的不同方法，包括最小的理论描述与最小的自由度，细粒度，但昂贵的模拟，允许对集体动力学的定量预测。该项目还将有助于从生物微泳者中汲取灵感，合理设计人工活性纤毛阵列。

Self-organized metachronal waves in cilia carpets represent a model system for the spontaneous emergence of spatio-temporal order in active matter. Each beating cilium represents a nonlinear, biological oscillator. The dynamics of several of these oscillators is coupled by the surrounding fluid, which facilitates collective dynamics. Cilia carpets are found on the surface of biological microswimmers, such as Volvox or Paramecium, as well as in the airways, oviducts and brain ventricles of higher animals. Furthermore, cilia carpets represent a minimal system for the study of collective synchronization as observed in collections of biological microswimmers with a single cilium (such as sperm cells), which are characterized by additional translation and rotation degrees of freedom. In this project, we propose a new theoretical description of collective nonlinear dynamics in cilia carpets, with special emphasis on the relationship between alignment of cilia orientation and the emergence of metachronal waves. Thereby, we will understand the interplay between spatial order (cilia alignment) and temporal order (collective synchronization) in an important model system. To achieve this aim, we propose a novel theoretical framework, titled: "Lagrangian mechanics of active systems (LAMAS)", to describe fluid-structure-interactions for active, elastic structures, such as cilia. This formalism generalizes classical Lagrangian mechanics for conservative and dissipative systems to active systems. Key elements of this formalism have been already developed in the first project phase for the case of n=2 cilia, and will now be extended to the case of n>>2 cilia. The results obtained so far include a theory of active fluctuations of the cilia beat, a characterization of the load-response of an individual cilium, as well as a theory of hydrodynamic synchronization for n=2 cilia. By developing the proposed formalism for the model system of cilia carpets, with special emphasis on rotational degrees of freedom of cilia orientation, we expect novel physics of collective nonlinear dynamics of biological microswimmers. We employ innovative methods, such as fast multipole boundary element methods for numerical solution of the Stokes equation, which governs the hydrodynamics of this system. Hydrodynamic interactions between cilia are treated using a generalized Faxen's law. The waveform compliance of the cilia beat is accounted for by incorporating principal deformation modes with dynamic amplitude. Our novel theoretical approach combines different methodologies currently in use in the SPP, including minimal theoretical descriptions with minimal number of degrees of freedom, and fine-grained, yet expensive simulations that allow quantitative predictions on collective dynamics. This project will also contribute to the rational design of artificial arrays of active cilia, drawing inspiration from biological microswimmers.