A generic theory of metachronal waves

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Any machine which performs work continuously has a well defined periodic duty cycle. In particular, this includes molecular scale machines such as molecular motors, and cellular scale machines such as beating cilia.The continuous progression through the duty cycle of a machine can be quantified by a phase variable.We propose to complement the generic theories for active fluids, which have been successfully used to describe the mechanics of the cell cytoskeleton, growing tissues and suspension of swimming bacteria by such a phase variable. This will enable us to study phase synchronization and traveling phase waves in active fluids.This project is motivated by several biologically relevant processes in which phase synchronization and phase waves are observed.In particular the theory we propose to develop will inform the study of metachronal waves in beating cilia.Beating cilia play an important role in many biological processes. Examples are the of pumping mucus in the human lungs and brains.Moreover, they are involved in the motility of microorganisms like {\it Volvox} or {\it Paramecium}.During the development of vertebrate organisms, they are involved in breaking the left-right symmetry of the organism. Deficient cilia are responsible for a large variety of human diseases.One of the main task of cilia is to collectively generate fluid flows.In healthy organisms they do so efficiently by synchronizing their beating patterns into metachronal surface waves. How this synchronization occurs is so far poorly understood. The generic theory for active fluids which we propose would incorporate the active mechanics of the flows, using the methods which we presented in our earlier work on active chiral fluids. We will expand this theory by a phase variable that describes the phase of rotation of the beating cilia.This theory would allow to study the prerequisites for the formation of metachronal waves and the forces and torques metachronal waves exert on the surrounding fluid, shedding light on the principles that underly fluid transport by ciliary carpets.