Coordination of elastic cilia

Cilia facilitate fundamental biological functions by generating fluid transport in eukaryotic life forms. They are hair-like cell appendages whose collective cyclic beating exhibits self-organized coordination phenomena, such as synchronization and metachronal waves. An explanation for the emergence of coordinated beating considers microscopic flows in the extracellular medium in combination with an elastic compliance in the beating of the cilia. This thesis uses mathematical modeling to reduce the dynamics of the cilia to phases that rotate around a compliant orbit, coupled by hydrodynamic interactions. This phase-oscillator model is employed with the aim of contributing to the understanding of cilia coordination in three ways: (i) Observations show that metachronal waves near cellular surfaces exhibit specific directions and wavelengths, which differ across biological systems. The mechanism underlying the regulation of metachronal wave characteristics is currently debated. In this thesis, the regulation of wave characteristics by surface effects is established for an elastic compliance model. Linear stability analysis is performed analytically in a periodic chain setting, revealing a specific signature of stable metachronal waves. An intuitive picture of the mechanism that regulates the stability of these waves is given by relating the geometry of flow near a surface to the directedness of the hydrodynamic coupling between the phase oscillators. In contrast to the periodic case, where multiple waves are stable, simulations of a chain with free ends exhibit the selection of a specific wave mode, with the magnitude of elastic compliance serving as a control parameter. (ii) Recent direct measurements of flow around ciliated organisms demonstrate that considerable unsteady effects occur in the oscillatory flow created by the cilia beat. How unsteady flow affects coordination in elastic compliance models is unknown. This thesis reveals the phenomenological consequences of the interplay between elastic compliance and unsteady flow in the free space. It is shown that a delay in the hydrodynamic coupling between the phase oscillators results from the vorticity diffusion timescale associated with the unsteady flow. This delay fundamentally changes the coordination phenomena of phase-oscillator pairs. Likewise, the linear stability and dispersion of metachronal waves in periodic chains are fundamentally affected. In addition, numerical simulations of chains with free ends demonstrate that besides the magnitude of elastic compliance, a further control parameter for the regulation of coordination modes results from the unsteady flow. (iii) In addition to elastic compliance, a variable rate of beating can substantially contribute to the coordination of the cilia. The model classification of this thesis shows that the mechanisms that regulate metachronal waves systematically differ between the elastic compliance and variable beat models. A comparison of the model predictions with biological data suggests their significance in different biological systems. Elastic compliance can explain the observations of metachronal waves in sparse arrangements of cilia. Variable beating offers a mechanism that explains the observed waves in dense arrangements.