Generic aspects of axonemal beating

Camalet, S.; Jülicher, Frank

doi:10.1088/1367-2630/2/1/324

Cited by 231 publications

(409 citation statements)

References 36 publications

(23 reference statements)

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Section: Introductionmentioning

confidence: 80%

“…This equation was also derived in earlier work by Machin in the context of wave propagation in the flagella of swimming microorganisms [16,17]. A similar theoretical treatment was proposed as a simple model of the sliding filament model of eukaryotic axonemal beating by coupling the elasto-hydrodynamics problem with models for the behavior of active molecular motors [18,19].The main features of this problem have been successfully exploited experimentally to measure the bending modulus of biopolymers (actin filaments and microtubules), either using thermal fluctuations [20] or using an active actuation [21,22]. Related studies include the dynamics of magnetic filaments [23][24][25], the three-dimensional actuation and *…”

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Floppy swimming: Viscous locomotion of actuated elastica

2007

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Actuating periodically an elastic filament in a viscous liquid generally breaks the constraints of Purcell's scallop theorem, resulting in the generation of a net propulsive force. This observation suggests a method to design simple swimming devices -which we call "elastic swimmers" -where the actuation mechanism is embedded in a solid body and the resulting swimmer is free to move. In this paper, we study theoretically the kinematics of elastic swimming. After discussing the basic physical picture of the phenomenon and the expected scaling relationships, we derive analytically the elastic swimming velocities in the limit of small actuation amplitude. The emphasis is on the coupling between the two unknowns of the problems -namely the shape of the elastic filament and the swimming kinematics -which have to be solved simultaneously. We then compute the performance of the resulting swimming device, and its dependance on geometry. The optimal actuation frequency and body shapes are derived and a discussion of filament shapes and internal torques is presented. Swimming using multiple elastic filaments is discussed, and simple strategies are presented which result in straight swimming trajectories. Finally, we compare the performance of elastic swimming with that of swimming microorganisms. I. INTRODUCTIONThe fluid mechanics of microorganism locomotion, pioneered more than fifty years ago by G.I. Taylor, has become one of the most successful branches of biomechanics, with success in both the basic physical understanding of flow behavior and the quantitative prediction of kinematics and energetics of locomotion [1][2][3][4][5][6][7][8].Recent technical advances have led to ever more precise fabrication at small scales (microns or less), prompting both theorists [9-13] and experimentalists [14] to design and analyze a series of simple low-Reynolds number swimmers. The experiment of Dreyfus et al. [14], in particular, reported locomotion in a sperm-like micro-swimmer, composed of a cargo (red blood cell) and a slender flexible filament made of a series of paramagnetic beads. In that case, actuation by oscillating transverse magnetic fields led to the generation of bending waves propagating along the filament and resulted in the motion of the micro-swimmer. In this system, the right-left symmetry was broken by the presence of a cargo and led to a preferential tip-to-base propagation of the bending waves, resulting in locomotion in the direction base-to-tip.An alternative way to break the symmetry in a similar system would be to build-in the asymmetry in the actuation. In particular, if an elastic filament is periodically actuated at one of its extremities in a viscous liquid, the resulting motion will lead to the propagation of bending waves and, in general, propulsive forces. This idea was originally proposed by Edward Purcell [8]. Physically, actuating an elastic filament allows one to break the constraints of the "scallop theorem" -which states that a body performing a reciprocal motion at low Reynolds number cannot p...

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Section: Introductionmentioning

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Floppy swimming: Viscous locomotion of actuated elastica

2007

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“…Previous theoretical work described the onset of flagellar oscillations as a supercritical Hopf bifurcation [27] with normal form (µ>0) [28] …”

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Active Phase and Amplitude Fluctuations of Flagellar Beating

et al. 2014

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The eukaryotic flagellum beats periodically, driven by the oscillatory dynamics of molecular motors, to propel cells and pump fluids. Small, but perceivable fluctuations in the beat of individual flagella have physiological implications for synchronization in collections of flagella as well as for hydrodynamic interactions between flagellated swimmers. Here, we characterize phase and amplitude fluctuations of flagellar bending waves using shape mode analysis and limit-cycle reconstruction.We report a quality factor of flagellar oscillations, Q = 38.0 ± 16.7 (mean±s.e.). Our analysis shows that flagellar fluctuations are dominantly of active origin. Using a minimal model of collective motor oscillations, we demonstrate how the stochastic dynamics of individual motors can give rise to active small-number fluctuations in motor-cytoskeleton systems. Here, we report direct measurements of phase and amplitude fluctuations of the flagellar beat and discuss the microscopic origin of active flagellar fluctuations using a minimal model. We further illustrate the impact of flagellar fluctuations on swimming and synchronization. Our analysis contributes to a recent interest in driven, outof-equilibrium systems and their fluctuation fingerprint [15][16][17][18] by characterizing noisy limit-cycle dynamics in an ubiquitous motility system, the flagellum.Flagellar shape analysis. We characterize flagellar beat patterns as superposition of principal shape modes. This dimensionality reduction is key to our fluctuation analysis. We analyze planar beat patterns of bull sperm swimming close to a boundary surface [19], filmed at 250 frames-per-second (corresponding to about 8 frames per beat cycle). The flagellar centerline r(s, t), tracked as function of arclength position s and time t, can be expressed with respect to a material frame of the sperm head in terms of a tangent angle ψ(s, t)Here, r h (t) denotes the sperm head center, and e 1 and e 2 are ortho-normal vectors with e 1 pointing along the long head axis, see Fig. 1A. A space-timeplot of ψ(s, t) reveals the periodicity of the flagellar beat, see Fig. 1B. This high-dimensional data set can be projected on a low dimensional 'shape space' using shape mode analysis based on principal component analysis [20]. The time-averaged tangent angle ψ 0 (s)= n i=1 ψ(s, t i )/n characterizes the mean shape of the beating flagellum (n=1024 frames in each movie). We further define a two-point correlation matrix arXiv:1401.7036v2 [q-bio.CB]

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“…On a micron scale inertia is negligible and microorganisms have to resort to a propulsion mechanism where they are constantly in motion in order to translate forward [19,26]. These propulsion mechanisms are numerous and include the flagellar breast stroke of Chlamydomonas reinhardtii [9] and the planar flagellar beating of sperm cells [5,28]. In the latter case a wave runs along the flagellum which pushes the sperm cell forward.…”

Section: Introductionmentioning

confidence: 99%

Swimming at Low Reynolds Number: From Sheets to the African Trypanosome

Babu

Schmeltzer

Stark

2012

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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Abstract. The African trypanosome is a protozoan which causes sleeping sickness in mammals. To study the dynamics of this microorganism at low Reynolds number, we implement and investigate three swimmer models, the Taylor sheet, a constanttorque swimmer, and a model for the African trypanosome. The first two swimmers are based on a semi-flexible sheet and the third is a full three-dimensional model. We simulate the viscous fluid environment of the swimmers using a technique called multi-particle collision dynamics. We verify our technique by implementing the Taylor sheet which is activated by a bending wave traveling along the sheet. Its ballistic motion turns into diffusive motion when the sheet becomes passive. For the constant-torque swimmer we apply a torque to the semi-flexible sheet which assumes a cork-screw shape and then generates the thrust force for propelling the swimmer forward. Whereas the angular velocity scales linearly with the torque, the swimming velocity displays a non-linear dependence. Finally, our trypanosome model swims with the help of a beating flagellum attached to the cell body. Since it wraps around the body, the model trypanosome displays a helical swimming trajectory. The swimming velocity displays a non-linear increase with the beating frequency of the flagellum.

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Generic aspects of axonemal beating

Cited by 231 publications

References 36 publications

Floppy swimming: Viscous locomotion of actuated elastica

Floppy swimming: Viscous locomotion of actuated elastica

Active Phase and Amplitude Fluctuations of Flagellar Beating

Swimming at Low Reynolds Number: From Sheets to the African Trypanosome

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