Equations of Inter-Doublet Separation Explain Wave Propagation and Oscillations in Flagella

Abstract

Introduction: The mechanism of dynein coordination in cilia and flagella remains incompletely understood. In one hypothesis, the “geometric clutch” (GC) model (Lindemann, J theor Biol,1994), dynein is regulated by inter-doublet separation. A continuum mechanical model and associated partial differential equations (PDEs) of the GC model have remained lacking. Such PDEs would provide insight into the biophysics, enable mathematical analysis of the behavior, and facilitate rigorous comparison to other models. In this study equations of motion for the flagellum and its doublets are derived and analyzed to reveal mechanisms of wave propagation and instability in the GC model.Methods: A simplified mechanical model of the flagellum is considered, consisting of two pairs of doublets (Lindemann, 1994). Each doublet pair experiences external viscous forces and inter-doublet forces parallel and perpendicular to its long axis (Hines and Blum, Biophys J,1978). The equations of force and moment balance are used to derive PDEs for the shape of the flagellum and the separation between doublets. The equations of inter-doublet separation reduce to an excitable system in the form of the Extended Fisher-Kolmorogov (EFK) equation (van den Berg et al., SIAM J, 2001), which exhibits propagating solutions similar to those of reaction-diffusion equations. These equations are then coupled to the global equations of flagella motion and solved numerically.Results: The model exhibits propagation of disturbances of inter-doublet separation and dynein activity. Autonomous propulsive oscillations are seen at typical parameter values (L=12 μm; EI=500 pN/μm2; diameter a=200 nm).Transition from large-amplitude asymmetric waveforms (forward swimming) to small-amplitude symmetric waveforms (backward swimming) is achieved by varying baseline dynein activity. These results support the ability of the GC hypothesis to explain dynein coordination in flagella and provide a mathematical foundation for comparison to other models.Acknowledgements: NSF grant CMMI-1265447.