Effects Of Inter-Doublet Coupling On Flagellar Beating

Cilia and flagella are slender organelles that exhibit wavelike oscillations to propel cells or move fluid. The cytoskeletal structure (the axoneme) of the cilium is well known, comprising arrays of the motor protein dynein between 9 pairs of microtubule doublets connected by nexin links and surrounding a central pair of microtubules. However, the mechanism by which dynein creates the oscillatory, propulsive motion remains uncertain. Are dynein forces dynamically regulated, or do oscillatory motions occur spontaneously due to dynamic instabilities under steady dynein activity? How do changes in physical parameters, such as stiffness and nonlinearity of radial spokes (RS) and nexin-dynein regulatory complex (N-DRC), affect the emergence and character of these oscillations? In the current work, we have created a custom finite-element code that allows us to simulate active, unstable, and nonlinear behavior on the nanoscale, unconstrained by the limits of commercial FEM packages. The complex eigenvalues of the system allow us to determine critical values that cause the system to oscillate. Time-domain simulations allow us to compare model behavior and waveforms with those of cilia and allow us to observe the effects of system nonlinearities. We first modeled a single filament under distributed axial follower load to find critical values for dynamic instability and study the effects of mechanical parameters in individual doublets. We then moved to a two-filament model to study the effect of filament-coupling parameters on oscillatory behavior of multiple filaments coupled by passive elements (RS and N-DRC) and active elements (dyneins). Finally, we created a four-doublet, three-dimensional model to show the complex behavior of multiple filaments interacting through internal forces and moments as well as passive elastic and viscous coupling. Using biologically-plausible parameter values, all systems exhibited dynamic instability, leading to oscillations like those observed in real cilia and flagella.