Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella

Cilia and flagella are highly conserved organelles that beat rhythmically with propulsive, oscillatory waveforms. The mechanism that produces these autonomous oscillations remains a mystery. It is widely believed that dynein activity must be dynamically regulated (switched on and off, or modulated) on opposite sides of the axoneme to produce oscillations. A variety of regulation mechanisms have been proposed based on feedback from mechanical deformation to dynein force. In this paper, we show that a much simpler interaction between dynein and the passive components of the axoneme can produce coordinated, propulsive oscillations. Steady, distributed axial forces, acting in opposite directions on coupled beams in viscous fluid, lead to dynamic structural instability and oscillatory, wave-like motion. This 'flutter' instability is a dynamic analogue to the well-known static instability, buckling. Flutter also occurs in slender beams subjected to tangential axial loads, in aircraft wings exposed to steady air flow and in flexible pipes conveying fluid. By analysis of the flagellar equations of motion and simulation of structural models of flagella, we demonstrate that dynein does not need to switch direction or inactivate to produce autonomous, propulsive oscillations, but must simply pull steadily above a critical threshold force.