Theoretical relationships between axoneme distortion and internal forces and torques in ciliary beating

The axoneme is an intricate nanomachine responsible for generating the propulsive oscillations of cilia and flagella in an astonishing variety of organisms. New imaging techniques based on cryoelectron-tomography (cryo-ET) and subtomogram averaging have revealed the detailed structures of the axoneme and its components with sub-nm resolution, but the mechanical function of each component and how the assembly generates oscillations remains stubbornly unclear. Most explanations of oscillatory behavior rely on the dynamic regulation of dynein by some signal, but this may not be necessary if the system of dynein-driven slender filaments is dynamically unstable. Understanding the possibility of instability-driven oscillations requires a multifilament model of the axoneme that accounts for distortions of the axoneme as it bends. Active bending requires forces and bending moments that will tend to change the spacing and alignment of doublets. We hypothesize that components of the axoneme resist and respond to these loads in ways that are critical to beating. Specifically, we propose (i) that radial spokes provide torsional stiffness by resisting misalignment (as well as spacing) between the central pair and outer doublets, and (ii) that the kinematics of dynein arms affect the relationships between active forces and bending moments on deforming doublets. These proposed relationships enhance the ability of theoretical, multifilament models of axonemal beating to generate propulsive oscillatory waveforms via dynamic mechanical instability.