Eukaryotic organisms rely on the coordinated beating of motile cilia for a multitude of fundamental reasons. In smaller organisms, such as Paramecium and the single cell alga Chlamydomonas reinhardtii, it is a matter of propulsion, to swim towards a higher concentration of nutrients or away from damaging environments. Larger organisms use instead the coordinated motion of cilia to push fluid along an epithelium: examples common to mammals are the circulation of cerebrospinal fluid in the brain, the transport of ovules in the fallopian tubes, and breaking the left/right symmetry in the embryo. Another notable example, and one that is central to this thesis, is mucociliary clearance in human airways: A carpet of motile cilia helps keeping the cell surface free from pathogens and foreign particles by constantly evacuating from lungs, bronchi, and trachea a barrier of mucus.

The question of how motile cilia interact with one another to beat in a coordinated fashion is an open and pressing one, with immediate implications for the medical community. In order for the fluid propulsion to be effective, the motion of cilia needs to be phase-locked across significant distances, in the form of travelling waves (``metachronal waves''). It is still not known how this long-range coordination emerges from local rules, as there is no central node regulating the coordination among cilia.

In the first part of this thesis I will focus on studying the coordination in carpets of cilia with a top-down approach, by proposing, implementing, and applying a new method of analysing microscope videos of ciliated epithelia.

Chapter 1 provides the reader with an introduction on motile cilia and flagella, treating their structure and motion and reporting the different open questions currently tackled by the scientific community, with particular interest in the coordination mechanisms of cilia and the mucociliary clearance apparatus.

Chapter 2 introduces Differential Dynamic Microscopy (DDM), a powerful and versatile image analysis tool that bridges the gap between spectroscopy and microscopy by allowing to perform scattering experiments on a microscope. The most interesting aspects of DDM for this work are that it can be applied to microscope videos where it is not possible to resolve individual objects in the field of view, and it requires no user input. These two characteristics make DDM a perfect candidate for analysing several hundred microscope videos of weakly scattering filaments such as cilia.

In Chapter 3 I will present how it is possible to employ DDM to extract a wealth of often-overlooked information from videos of ciliated epithelia: DDM can successfully probe the ciliary beat frequency (CBF) in a sample, measure the direction of beating of the cilia, and detect metachronal waves and read their direction and wavelength. In vitro ciliated epithelia however often do not show perfect coordination or alignment among cilia. For the analysis of these samples, where the metachronal coordination might not be evident, we developed a new approach, called multiscale DDM (multiDDM), to measure a coordination length scale, a characteristic length of the system over which the coordination between cilia is lost.

The new technique of multiDDM is employed in Chapter 4 to study how the coordination among cilia changes as a response to changes in the rheology of the mucous layer. In particular, we show that cilia beating under a thick, gel-like mucus layer show a larger coordination length scale, as if the mucus acted as an elastic raft effectively coupling cilia over long distances. This is corroborated by the coordination length scale being larger in samples from patients affected by Cystic Fibrosis than in healthy samples, and much shorter when the mucus layer is washed and cilia therefore beat in a near-Newtonian fluid. We then show how it is possible to employ multiDDM to measure the effectiveness of drugs in recovering, in CF samples, a coordination length scale typical of a healthy phenotype.

In the second part I will focus instead on the single cilium scale, showing how we can attempt to link the beating pattern of cilia to numerical simulations studying synchronisation in a model system.

In particular in Chapter 5 I will describe our approach to quantitatively describe the beating pattern of single cilia obtained from human airway cells of either healthy individuals or patients affected by Primary Ciliary Dyskinesia. Our description of the beating pattern, and the selection of a few meaningful, summary parameters, are then shown to be accurate enough to discriminate between different mutations within Primary Ciliary Dyskinesia.

In Chapter 6 instead I report the results obtained by coarse-graining the ciliary beat pattern into a model system consisting of two ``rotors''. The rotors are simulated colloidal particles driven along closed trajectories while leaving their phase free. In my study, the trajectories followed by the rotors are analytical fits of experimental trajectories of the centre of drag of real cilia. The rotors, that are coupled only via hydrodynamics interactions, are seen to phase-lock, and the shape of the trajectory they are driven along is seen to influence the steady state of the system.