Finite element models of flagella with sliding radial spokes and interdoublet links exhibit propagating waves under steady dynein loading.

It remains unclear how flagella generate propulsive, oscillatory waveforms. While it is well known that dynein motors, in combination with passive cytoskeletal elements, drive the bending of the axoneme by applying shearing forces and bending moments to microtubule doublets, the origin of rhythmicity is still mysterious. Most conceptual models of flagellar oscillation involve dynein regulation or switching, so that dynein activity first on one side of the axoneme, then the other, drives bending. In contrast, a "viscoelastic flutter" mechanism has recently been proposed, based on a dynamic structural instability. Simple mathematical models of coupled elastic beams in viscous fluid, subjected to steady, axially distributed, dynein forces of sufficient magnitude, can exhibit oscillatory motion without any switching or dynamic regulation. Here we introduce more realistic finite element (FE) models of 6-doublet and 9-doublet flagella, with radial spokes and interdoublet links that slide along the central pair or corresponding doublet. These models demonstrate the viscoelastic flutter mechanism. Above a critical force threshold, these models exhibit an abrupt onset of propulsive, wavelike oscillations typical of flutter instability. Changes in the magnitude and spatial distribution of steady dynein force, or to viscous resistance, lead to behavior qualitatively consistent with experimental observations. This study demonstrates the ability of FE models to simulate nonlinear interactions between axonemal components during flagellar beating, and supports the plausibility of viscoelastic flutter as a mechanism of flagellar oscillation.