Cilia are slender micro-organelles (200 nm diameter, 10 µm long) that generate propagating waves to propel cells or move fluid. The cytoskeletal structure of the cilium (the axoneme) consists of 9 outer microtubule doublets and 2 central microtubule singlets. Outer doublets are connected by active motor proteins (dyneins) and passive components (nexin links). The structure of the cilium is well known by electron microscopy, but the mechanism by which cilia oscillate remains uncertain. Traditional theories propose that the intermittent, “on-off” activation of dyneins causes the axoneme to bend back and forth. However, it is possible that instability caused by a constant follower load causes or contributes to the oscillation. By building a 3D finite-element model of the axoneme, we can investigate the constant load theory of cilia motion and evaluate the effects of system parameters on stability. In fact, using this model, we can estimate coupling parameters that we are not able to measure experimentally in these nanoscale structures.

While the actual structure of a cilium is more intricate, we can model the biological system with beam structures, including a four-outer-doublet system (4-beam model) as a theoretical baseline and a six-outer-doublet with a central-singlet system (6+1 beam model) as a more realistic model.  The commercial finite element software Comsol Multiphysics (version 5.4, Comsol Inc., Burlington, MA) is used to build the models. Both of the models are further simplified by limiting loading to two active pairs connected by passive links. For the active pairs, “follower” loads modeling dynein forces are applied along a direction that is the average of the tangent vectors of the beams within the active pairs. Reaction moments are also applied to maintain force-moment equilibrium in each component. The passive inter-doublet links are modeled as nonlinear viscoelastic springs, using Comsol’s “extrusion couplings” to map relative positions of different beams and calculate average tangential vectors for the active pairs. This model allows us to investigate the effects of physical parameters such as dynein force, dynein moments, nexin link stiffness and damping, on instabilities like flutter and divergence that can create large, dynamic bends.

Three predominant behaviors were observed through parametric sweeps: (i) stable equilibria; (ii) flutter; and (iii) divergence. In the stable regime, dynein forces are not enough to drive the system to motion, leaving the system in a static equilibrium with no bending. In the flutter region, the system oscillates periodically; this behavior corresponds to complex conjugate pairs of eigenvalues in the least stable mode. In the divergence region, bending deformations grow without oscillation; the least stable mode has a real eigenvalue.

Using these simplified 3D cilium models, we observed how varying parameters can cause transitions between these three behaviors. We compared the results with a theoretical model that incorporated as small-deformation approximation. As these theoretical models approach the actual structure of the cilium, we can estimate a reasonable range of values for parameters that are difficult to measure experimentally, greatly enhancing our understanding of this miniscule structure. The full 3D cilium model opens several questions for future work, most notably: Does one of these instabilities (flutter or divergence) or a combination of both underlie the true mechanism behind the wavelike oscillation of cilia? 


Effects of Coupling Parameters on Flutter and Divergence Instabilities in 3-D Models of Cilia

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Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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