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

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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.