Evidence for a self-organized compliant mechanism for the spontaneous steady beating of cilia

Cilia or eukaryotic flagella are slender 200-nm-diameter organelles that move the immersing fluid relative to a cell and sense the environment. Their core structure is nine doublet microtubules (DMTs) arranged around a central-pair. When motile, thousands of tiny motors slide the DMTs relative to each other to facilitate traveling waves of bending along the cilium's length. These motors provide the energy to change the shape of the cilium and overcome the viscous forces of moving in the surrounding fluid. In planar beating, motors walk toward where the cilium is attached to the cell body. Traveling waves are initiated by motors bending the elastic cilium back and forth, a self-organized mechanical oscillator. We found remarkably that the energy in a wave is nearly constant over a wide range of (ATP) and medium viscosities and inter-doublet springs operate only in the central and not in the basal region. Since the energy in a wave does not depend on its rate of formation, the control mechanism is likely purely mechanical. Further the torque per length generated by the motors acting on the doublets is proportional to and nearly in phase with the microtubule sliding velocity with magnitude dependent on the medium. We determined the frequency-dependent elastic moduli and strain energies of beating cilia. Incorporation of these in an energy-based model explains the beating frequency, wavelength, limiting of the wave amplitude and the overall energy of the traveling wave. Our model describes the intricacies of the basal-wave initiation as well as the traveling wave.