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Nonlinear amplitude dynamics in flagellar beating

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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.

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Nonlinear analysis. (a) Nonlinear flagella steady-state profiles for μa= 4300 (left) and μa=5600 (right), considering the respective eigenmodes as initial conditions. The beating cycles are divided into 10 frames as in figure 2a,b (top panels). (b) ϕ (solid lines) and δn (dashed lines) evaluated at  for the profiles in (a), respectively. (c) Maximum absolute tangent angle evaluated at  and dimensionless frequency ω as a function of the relative distance to the bifurcation . Sp=10, μ=50, η=0.14, ζ=0.4, , L=50 μm and τ0=50 ms.
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RSOS160698F3: Nonlinear analysis. (a) Nonlinear flagella steady-state profiles for μa= 4300 (left) and μa=5600 (right), considering the respective eigenmodes as initial conditions. The beating cycles are divided into 10 frames as in figure 2a,b (top panels). (b) ϕ (solid lines) and δn (dashed lines) evaluated at for the profiles in (a), respectively. (c) Maximum absolute tangent angle evaluated at and dimensionless frequency ω as a function of the relative distance to the bifurcation . Sp=10, μ=50, η=0.14, ζ=0.4, , L=50 μm and τ0=50 ms.

Mentions: In this section, we study the nonlinear dynamics of the flagellum in the limit of small curvature by numerically solving equations (2.10) and (2.11) using a second-order accurate implicit-explicit numerical scheme (see electronic supplementary material). The unstable modes presented in §4 follow an initial exponential growth and eventually saturate at the steady state owing to the nonlinearities in the system. In figure 3, two different saturated amplitude solutions are shown. Figure 3a (left) corresponds to a case where the system is found close to the Hopf bifurcation, whereas figure 3a (right) corresponds to a regime far from the bifurcation. We note that the marginal solution obtained in the linear stability analysis (figure 2b, top panel) gives a very good estimate of the nonlinear profile close to the bifurcation point, although it does not provide the magnitude of ϕ or δn. Frequencies are approximately 10 Hz and maximum amplitudes are found to be small, around ≃4% of the total flagellum length. However, the oscillation amplitude for high motor activity is more than double in respect to the case of low activity (figure 3a). The colour code in figure 3a indicates the value of the semi-difference of plus- and minus-bound motors δn. Plus-bound motors are predominant in regions of positive curvature (ϕs>0) along the flagellum and vice versa. Despite the low duty ratio of dynein motors [35], approximately 2% bound dyneins along the flagellum are sufficient to produce micrometre-sized amplitude oscillations. The full flagella dynamics corresponding to figure 3a are provided in the supplementary material, movies S1 and S2.Figure 3


Nonlinear amplitude dynamics in flagellar beating
Nonlinear analysis. (a) Nonlinear flagella steady-state profiles for μa= 4300 (left) and μa=5600 (right), considering the respective eigenmodes as initial conditions. The beating cycles are divided into 10 frames as in figure 2a,b (top panels). (b) ϕ (solid lines) and δn (dashed lines) evaluated at  for the profiles in (a), respectively. (c) Maximum absolute tangent angle evaluated at  and dimensionless frequency ω as a function of the relative distance to the bifurcation . Sp=10, μ=50, η=0.14, ζ=0.4, , L=50 μm and τ0=50 ms.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC5383814&req=5

RSOS160698F3: Nonlinear analysis. (a) Nonlinear flagella steady-state profiles for μa= 4300 (left) and μa=5600 (right), considering the respective eigenmodes as initial conditions. The beating cycles are divided into 10 frames as in figure 2a,b (top panels). (b) ϕ (solid lines) and δn (dashed lines) evaluated at for the profiles in (a), respectively. (c) Maximum absolute tangent angle evaluated at and dimensionless frequency ω as a function of the relative distance to the bifurcation . Sp=10, μ=50, η=0.14, ζ=0.4, , L=50 μm and τ0=50 ms.
Mentions: In this section, we study the nonlinear dynamics of the flagellum in the limit of small curvature by numerically solving equations (2.10) and (2.11) using a second-order accurate implicit-explicit numerical scheme (see electronic supplementary material). The unstable modes presented in §4 follow an initial exponential growth and eventually saturate at the steady state owing to the nonlinearities in the system. In figure 3, two different saturated amplitude solutions are shown. Figure 3a (left) corresponds to a case where the system is found close to the Hopf bifurcation, whereas figure 3a (right) corresponds to a regime far from the bifurcation. We note that the marginal solution obtained in the linear stability analysis (figure 2b, top panel) gives a very good estimate of the nonlinear profile close to the bifurcation point, although it does not provide the magnitude of ϕ or δn. Frequencies are approximately 10 Hz and maximum amplitudes are found to be small, around ≃4% of the total flagellum length. However, the oscillation amplitude for high motor activity is more than double in respect to the case of low activity (figure 3a). The colour code in figure 3a indicates the value of the semi-difference of plus- and minus-bound motors δn. Plus-bound motors are predominant in regions of positive curvature (ϕs>0) along the flagellum and vice versa. Despite the low duty ratio of dynein motors [35], approximately 2% bound dyneins along the flagellum are sufficient to produce micrometre-sized amplitude oscillations. The full flagella dynamics corresponding to figure 3a are provided in the supplementary material, movies S1 and S2.Figure 3

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