Analysis of unstable modes distinguishes mathematical models of flagellar motion

Bayly, Philip V.; Wilson, Kate

doi:10.1098/rsif.2015.0124

Cited by 42 publications

(73 citation statements)

References 34 publications

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“…Here, we complement the seminal work by Machin [3] and Goldstein [5] on the dynamics of passive filaments, and demonstrate how the nanometric cross-linking proteins that are present in passive cross-linked filament bundles instigate novel dynamical counterbend phenomena. This is in contrast with previous models on flagellar wave coordination [4,8,36,37,40,41,[43][44][45][46][47][48][49], which incorporate the cross-linking interaction in conjunction with molecular motor dynamics. We consider the dynamical situation in which only the structural passive elements are present.…”

Section: Introductionmentioning

confidence: 59%

The counterbend dynamics of cross-linked filament bundles and flagella

Coy

Gadêlha

2017

J. R. Soc. Interface.

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Cross-linked filament bundles, such as in cilia and flagella, are ubiquitous in biology. They are considered in textbooks as simple filaments with larger stiffness. Recent observations of flagellar counterbend, however, show that induction of curvature in one section of a passive flagellum instigates a compensatory counter-curvature elsewhere, exposing the intricate role of the diminutive cross-linking proteins at large-scales. We show that this effect, a material property of the cross-linking mechanics, modifies the bundle dynamics non-trivially, and induces a bimodal L 2 − L 3 length-dependent material response that departs from the Euler-Bernoulli theory. Hence, the use of simpler theories to analyse experiments can result in paradoxical interpretations. Remarkably, the counterbend dynamics instigates counter-waves in opposition to driven oscillations in distant parts of the bundle, with potential impact on the regulation of flagellar bending waves. These results have a range of physical and biological applications, including the empirical disentanglement of material quantities via counterbend dynamics.

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Section: Introductionmentioning

confidence: 59%

The counterbend dynamics of cross-linked filament bundles and flagella

Coy

Gadêlha

2017

J. R. Soc. Interface.

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“…The origin of the bending wave instability can be understood as a consequence of the antagonistic action of dyneins competing along the flagellum (15,39). The instability is then further stabilized by the nonlinear coupling between dynein activity and flagellum shape, without the need to invoke a nonlinear axonemal response to account for the saturation of the unstable modes, in contrast to previous studies (23,24). Moreover, the governing equations (Eqs.…”

Section: Discussionmentioning

confidence: 97%

Nonlinear amplitude dynamics in flagellar beating

2017

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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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“…The model (21) was solved numerically using an implicit finite-difference scheme described in detail by Tornberg and…”

Section: B Numerical Studies Of a Single Filamentmentioning

confidence: 99%

Elastohydrodynamic synchronization of adjacent beating flagella

et al. 2016

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It is now well established that nearby beating pairs of eukaryotic flagella or cilia typically synchronize in phase. A substantial body of evidence supports the hypothesis that hydrodynamic coupling between the active filaments, combined with waveform compliance, provides a robust mechanism for synchrony. This elastohydrodynamic mechanism has been incorporated into 'bead-spring' models in which the beating flagella are represented by microspheres tethered by radial springs as they are driven about orbits by internal forces. While these low-dimensional models reproduce the phenomenon of synchrony, their parameters are not readily relatable to those of the filaments they represent. More realistic models which reflect the underlying elasticity of the axonemes and the active force generation, take the form of fourth-order nonlinear PDEs. While computational studies have shown the occurrence of synchrony, the effects of hydrodynamic coupling between nearby filaments governed by such continuum models have been theoretically examined only in the regime of interflagellar distances large compared to flagellar length. Yet, in many biological situations ≪ 1. Here, we first present an asymptotic analysis of the hydrodynamic coupling between two extended filaments in the regime ≪ 1, and find that the form of the coupling is independent of the microscopic details of the internal forces that govern the motion of the individual filaments. The analysis is analogous to that yielding the localized induction approximation for vortex filament motion, extended to the case of mutual induction. In order to understand how the elastohydrodynamic coupling mechanism leads to synchrony of extended objects, we introduce a heuristic model of flagellar beating. The model takes the form of a single fourth-order nonlinear PDE whose form is derived from symmetry considerations, the physics of elasticity, and the overdamped nature of the dynamics. Analytical and numerical studies of this model illustrate how synchrony between a pair of filaments is achieved through the asymptotic coupling.

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Analysis of unstable modes distinguishes mathematical models of flagellar motion

Cited by 42 publications

References 34 publications

The counterbend dynamics of cross-linked filament bundles and flagella

The counterbend dynamics of cross-linked filament bundles and flagella

Nonlinear amplitude dynamics in flagellar beating

Elastohydrodynamic synchronization of adjacent beating flagella

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