Limits...
Nonlinear amplitude dynamics in flagellar beating

View Article: PubMed Central - PubMed

ABSTRACT

The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating.

No MeSH data available.


Related in: MedlinePlus

Bending initiation and transient dynamics. (a) Tangent angle kymograph for an eigenmode initial condition. (b) Tangent angle kymograph for a sinusoidal initial condition for the tangent angle with n+(0)=n−(0)=0.1. (c) Bound motor time evolution for the plus (solid line) and minus (dashed line) dynein populations at  for the case of a sinusoidal initial condition. Inset: flagella profiles at different times in ms. (d) Snapshots of the flagellar shape for the sinusoidal initial condition up to t=t′′ (white dashed line in figure 5b) at equal time intervals (20 ms). At t=t′, wave interference changes the direction of wave propagation. The full movie can be seen in electronic supplementary material, movie S3. Sp=5, μ=100, μa=2000, η=0.14, ζ=0.4, L=50 μm and τ0= 50 ms. Arrows indicate the direction of wave propagation.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC5383814&req=5

RSOS160698F5: Bending initiation and transient dynamics. (a) Tangent angle kymograph for an eigenmode initial condition. (b) Tangent angle kymograph for a sinusoidal initial condition for the tangent angle with n+(0)=n−(0)=0.1. (c) Bound motor time evolution for the plus (solid line) and minus (dashed line) dynein populations at for the case of a sinusoidal initial condition. Inset: flagella profiles at different times in ms. (d) Snapshots of the flagellar shape for the sinusoidal initial condition up to t=t′′ (white dashed line in figure 5b) at equal time intervals (20 ms). At t=t′, wave interference changes the direction of wave propagation. The full movie can be seen in electronic supplementary material, movie S3. Sp=5, μ=100, μa=2000, η=0.14, ζ=0.4, L=50 μm and τ0= 50 ms. Arrows indicate the direction of wave propagation.

Mentions: Finally, we study bending initiation and transient dynamics for two different initial conditions, in order to understand the selection of the unstable modes. In figure 5a,b the spatio-temporal transient dynamics are shown for the case of an initial eigenmode solution corresponding to the maximum eigenvalue (figure 5a) and an initial sine perturbation in ϕ, with equal constant bound motor densities (figure 5b). In case (b), travelling waves initially propagate in both directions and interfere at t=t′ (figure 5b,d and electronic supplementary material, movie S3). However, in the steady state, both the eigenmode and sine cases reach the same steady-state solution, despite the sinusoidal initial condition being a superposition of eigenmodes. This result provides strong evidence that the fastest-growing mode is the one that takes over and saturates in the steady state. In figure 5c, the transient dynamics for case (b) are shown for plus- and minus-bound dynein populations close to the tail (). Both populations decay exponentially with characteristic time to n0 and begin oscillating in anti-phase around this value, in a tug-of-war competition.Figure 5


Nonlinear amplitude dynamics in flagellar beating
Bending initiation and transient dynamics. (a) Tangent angle kymograph for an eigenmode initial condition. (b) Tangent angle kymograph for a sinusoidal initial condition for the tangent angle with n+(0)=n−(0)=0.1. (c) Bound motor time evolution for the plus (solid line) and minus (dashed line) dynein populations at  for the case of a sinusoidal initial condition. Inset: flagella profiles at different times in ms. (d) Snapshots of the flagellar shape for the sinusoidal initial condition up to t=t′′ (white dashed line in figure 5b) at equal time intervals (20 ms). At t=t′, wave interference changes the direction of wave propagation. The full movie can be seen in electronic supplementary material, movie S3. Sp=5, μ=100, μa=2000, η=0.14, ζ=0.4, L=50 μm and τ0= 50 ms. Arrows indicate the direction of wave propagation.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5383814&req=5

RSOS160698F5: Bending initiation and transient dynamics. (a) Tangent angle kymograph for an eigenmode initial condition. (b) Tangent angle kymograph for a sinusoidal initial condition for the tangent angle with n+(0)=n−(0)=0.1. (c) Bound motor time evolution for the plus (solid line) and minus (dashed line) dynein populations at for the case of a sinusoidal initial condition. Inset: flagella profiles at different times in ms. (d) Snapshots of the flagellar shape for the sinusoidal initial condition up to t=t′′ (white dashed line in figure 5b) at equal time intervals (20 ms). At t=t′, wave interference changes the direction of wave propagation. The full movie can be seen in electronic supplementary material, movie S3. Sp=5, μ=100, μa=2000, η=0.14, ζ=0.4, L=50 μm and τ0= 50 ms. Arrows indicate the direction of wave propagation.
Mentions: Finally, we study bending initiation and transient dynamics for two different initial conditions, in order to understand the selection of the unstable modes. In figure 5a,b the spatio-temporal transient dynamics are shown for the case of an initial eigenmode solution corresponding to the maximum eigenvalue (figure 5a) and an initial sine perturbation in ϕ, with equal constant bound motor densities (figure 5b). In case (b), travelling waves initially propagate in both directions and interfere at t=t′ (figure 5b,d and electronic supplementary material, movie S3). However, in the steady state, both the eigenmode and sine cases reach the same steady-state solution, despite the sinusoidal initial condition being a superposition of eigenmodes. This result provides strong evidence that the fastest-growing mode is the one that takes over and saturates in the steady state. In figure 5c, the transient dynamics for case (b) are shown for plus- and minus-bound dynein populations close to the tail (). Both populations decay exponentially with characteristic time to n0 and begin oscillating in anti-phase around this value, in a tug-of-war competition.Figure 5

View Article: PubMed Central - PubMed

ABSTRACT

The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating.

No MeSH data available.


Related in: MedlinePlus