Pith. sign in

REVIEW 3 major objections 5 minor 38 references

Beating of eukaryotic flagella via Hopf bifurcation of a system of stalled molecular motors

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A two-row system of antagonistic molecular motors is shown to undergo a Hopf bifurcation as ATP concentration rises, converting a stalled axoneme into an oscillating, beating flagellum.

desk verdict A novel two-row motor model with a clean linear analysis, but the simulations as described validate a different model. read the letter →

arxiv 2412.06067 v1 pith:TH3WBUCF submitted 2024-12-08 cond-mat.soft nlin.AOphysics.bio-ph

Irene Anello , François Alouges , Antonio De Simone This is my paper · ORCID
classification cond-mat.softnlin.AOphysics.bio-ph MSC 37G1592C1092C37
keywords flagellarbeatingHopfbifurcationmolecularmotorsdyneintug-of-warslidingfeedbackaxonemechemo-mechanicalmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that planar flagellar beating can arise from a Hopf bifurcation in the dynamics of two rows of stalled molecular motors pulling against each other. The authors build a multiscale model, the µ-chemoEH model, in which dynein motors attached to each of two filaments bind and unbind stochastically, and couple that microscopic chemistry to the bending of an elastic filament pair in a viscous fluid. Below a critical ATP concentration $\Omega_c$ the symmetric stalled state is stable; above it, the first bending mode becomes unstable and the two motor rows enter an oscillatory tug-of-war that produces alternating filament sliding and a traveling bending wave. Linear stability analysis locates the onset, and fully nonlinear simulations at bull-sperm parameters reproduce the predicted critical ATP and the 20.6 Hz beating frequency. If right, the model supplies a single mechanism connecting molecular motor kinetics to the emergence of spontaneous flagellar oscillations.

What carries the argument

The load-bearing object is the µ-chemoEH model, a two-state mechanochemical motor model in which each motor is bound or unbound, with sinusoidal potential difference $\Delta W(\xi)=U\cos(2\pi\xi/\ell)$, a constant rate sum $\omega_1+\omega_2=\alpha$, and a sinusoidal unbinding rate $\omega_2(\xi)$. Because only the first Fourier mode of the binding probabilities couples to sliding, the infinite system collapses to a four-ODE system plus the force balance for the sliding displacement $u$; the key derived quantity is the linear response coefficient $\chi(\Omega, \sigma)=2(\lambda\sigma+K-\rho N k_{cb}\Omega \sigma/(\alpha+\sigma))$, which connects active force to sliding at frequency $\sigma$. The flagellar bending equation with clamped-free boundary conditions and this $\chi$ gives a characteristic equation whose first mode crosses the imaginary axis at a critical ATP concentration, the Hopf point. The same structure yields a cubic approximation and supports the comparison with the chemoEH model.

What would settle it

Measure dynein transition rates as a function of applied load and sliding speed in a controlled assay: if the attachment-plus-detachment rate varies with load over the operating range, the four-ODE reduction and its predicted critical ATP for bull-sperm parameters fail quantitatively. Alternatively, observe the onset of beating: a gradual rise of oscillation amplitude from zero with no discrete threshold ATP would contradict a Hopf bifurcation.

Watch

Extended reading notes

Core claim

The central claim is that no external oscillator or curvature control is needed: the interaction between two symmetric rows of antagonistic molecular motors and the elastic-viscous flagellum is itself sufficient to lose stability. At the stalled equilibrium both motor rows exert equal and opposite forces, so the net active force vanishes. Raising ATP concentration $\Omega$ destabilizes that equilibrium; the first eigenvalue of the linearized flagellum-plus-motors system crosses the imaginary axis at $\Omega_c \approx 3.7\times 10^{-3}$ for bull-sperm parameters with critical frequency $\theta_c = 20.6$ Hz, a Hopf bifurcation. Nonlinear simulations near the threshold produce self-sustained oscillations whose frequency and waveform match the linear prediction, and the reconstructed motor probabilities show the two rows alternating dominance in a tug-of-war. The paper argues this is why sufficient ATP alone can convert a stalled axoneme into a beating flagellum.

Load-bearing premise

The whole reduction rests on assuming that the motor attachment and detachment rates keep a constant sum in space and time, with a sinusoidal detachment rate; if real dynein rates vary with load in a way that breaks this, as the paper notes the chemoEH model does, the predicted ATP threshold no longer describes the actual motor population.

Editorial extensions

If this is right

  • Sufficient ATP is, on this model, a control parameter: crossing $\Omega_c$ turns a quiescent axoneme into a self-oscillating one without any pacemaker or external signal.
  • The bifurcation frequency is set by the motor kinetics and flagellar mechanics, so parameter changes such as length, bending stiffness, and viscosity shift the observed beat frequency in a predicted way through the sperm number Sp.
  • At long-flagellum parameters the unstable mode is a retrograde traveling wave, and the wave speed increases with Sp, matching the known transition from standing-wave to traveling-wave beats.
  • The cubic model inherits the same linear onset and can serve as a reduced description, but its nonlinearities are weaker: the µ-chemoEH model allows self-coiling-like shapes that the cubic model suppresses.
  • The tug-of-war picture predicts that the first Fourier coefficients of the two motor-row probabilities oscillate with $p^+=q^-$ and $p^-=q^+$, a signature that could be looked for in motor-fluorescence experiments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: if the Hopf mechanism is generic, it suggests a unifying principle for cilia and flagella of different lengths, with the same two-row motor kinetics producing standing, traveling, or even direction-reversed waves depending on where the system sits relative to the bifurcation.
  • Beyond the paper: the model's distinction between the µ-chemoEH and chemoEH nonlinearities predicts that wave-propagation direction can reverse with distance from threshold; amplitude-resolved experiments on Chlamydomonas flagella could test this without waiting for near-threshold behavior.
  • Beyond the paper: the predicted microscopic spatial phase shift, $P^-(\xi,t)=-P^+(3\ell/4-\xi,t)$, is a concrete signature that could be sought in high-resolution measurements of dynein binding along the axoneme during a beat.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper formulates a nonlinear sliding-feedback model of flagellar beating by coupling two planar elastic filaments to a two-row population of molecular motors. The motor kinetics are reduced by Fourier expansion to a four-ODE system plus an algebraic force balance, and a cubic approximation and the chemoEH model are presented for comparison. Linear stability analysis of the straight flagellum gives a mode that crosses the imaginary axis at Ωc ≈ 3.7×10^-3 with θc = 20.6 Hz for bull sperm parameters, and fully nonlinear simulations are reported to show spontaneous oscillations near this threshold. The paper also compares the three models for short flagella (Chlamydomonas parameters) and analyzes the microscopic tug-of-war in a motor unit.

Significance. If the validation issues are fixed, this is a useful contribution: it gives an explicit microscopic-to-macroscopic bridge, a clean Hopf bifurcation mechanism, and a direct comparison among three sliding-feedback models. The Fourier reduction and the closed-form linear response coefficient in Eq. (26) are valuable, and the reconstructed probability distributions inside a tug-of-war unit offer a microscopic picture that many flagellar models lack. The quantitative claims, however, currently rest on a force-halving procedure that is not reflected in the linear analysis, and the beat frequency is fitted by adjusting λ; the novelty therefore lies primarily in the nonlinear dynamics and the model comparison rather than in an independently predicted beat frequency.

major comments (3)
  1. [§5.1, Eq. (16), Eq. (26)] The manuscript states that the µ-chemoEH model is simulated 'by halving the motor force f(t) defined in (16)', but the linear stability prediction χ(Ω,σ) = 2(λσ + K − ρN kcbΩσ/(α+σ)) is derived for the unhalved force. If the halving enters the dynamics, the linear response coefficient of the simulated system is different from Eq. (26), so the quoted threshold Ωc ≈ 3.7×10^-3 and frequency θc = 20.6 Hz do not apply to that system. The authors should either clarify that the halving is applied only when displaying forces for comparison with [16], or re-derive the Hopf threshold and frequency for the halved model and recompute the corresponding panels of Figures 3 and 4. The same ambiguity appears in §6.1, where f(t)/2 is said to be the total motor force for the µ-chemoEH and cubic models.
  2. [§5.1, Table 1] The claimed critical frequency is not an independent prediction: λ is set to 7 pN·s/µm² precisely so that the linear analysis reproduces the 20 Hz value taken from [17]. This calibration should be stated as such, and the paper would be materially strengthened by reporting how Ωc and θc vary with λ and by testing against an independent beat frequency rather than the one used for fitting.
  3. [§5.1, Figure 4] The statement that the nonlinear simulations 'reproduce correctly' the linear predictions is supported only by a single run at ε = 0.01; no limit-cycle frequency, no measured amplitude, and no sweep across Ω below and above Ωc are reported. A bifurcation diagram showing the simulated oscillation amplitude and frequency versus Ω, together with the linear prediction, is needed to substantiate the quantitative agreement claimed in the text.
minor comments (5)
  1. [§5.1] The parameter β in the initial conditions p±(0,s) = β/ℓ (1 ± 0.01) and q±(0,s) = β/ℓ (1 ∓ 0.01) is not defined; the authors should specify its value and relation to η.
  2. [§5.2, §6.1] The model comparisons are made at different distances from the bifurcation (ε = 2 vs ε = 4 for the cubic model; ε = 0.15 vs ε = 1.2 vs ε = 2 across models), so the observed differences may reflect the different ε values rather than model structure alone; this should be stated explicitly in the captions or text.
  3. [§3.1, Eq. (13)] The text calls η the average fraction of bound motors, but p0 has units of inverse length, and the equilibrium value is p0 = η/ℓ; the notation should be clarified to avoid a dimensional inconsistency.
  4. [§4.1] In the Ma → 0 limit the linear response coefficient χ(Ω,iθ) tends to 2K, not K; the sentence 'the linear response coefficient reduces to the non-dimensional elastic term K' is inconsistent with Eq. (26) unless a factor of two is absorbed elsewhere, which should be stated.
  5. [Figure 3(c)] The axis notation 'ωc ≈ 129.5' uses rad/s while the text reports θc = 20.6 Hz; the caption should define the units and the relationship ω = 2πθ to prevent confusion.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline critical frequency is calibrated by tuning the internal friction to 20 Hz, and the nonlinear 'validation' uses a halved motor force whose linear response is [16]'s, not the analyzed Eq. (26); the central Hopf-bifurcation derivation itself remains self-contained.

  1. fitted input called prediction [Section 5.1, near Table 1 and Figure 3(c)]
    "To reproduce a beating frequency close to 20 · 2π rad/s – which is the one shown in [17] – we adjust the internal friction λ to 7 pN · s/µm2 (which is comparable to the one from [16])."

    The internal friction λ is not independently measured or constrained; it is explicitly adjusted so that the linear analysis yields approximately the 20 Hz beating frequency taken from [17]. The subsequent result, 'The first mode crosses the imaginary axis with a critical frequency θc = 20.6Hz', is therefore not an independent prediction: the frequency is forced by the calibration of λ. The ATP threshold Ωc is still a genuine output, but the headline frequency, used later to compute the Machin number, reduces by construction to the input target frequency.

  2. other [Section 5.1, before Figure 4; repeated in Section 6.1]
    "To facilitate the comparison with the literature we simulate the µ-chemoEH model by halving the motor force f (t) defined in (16); in this way we obtain that the linear analysis matches the one in [16]. ... The non-linear simulations reproduce correctly the linear predictions of the critical activation Ωc and of the frequency of oscillation θc."

    The linear stability analysis of Section 4 uses the two-row response χ(Ω, σ) = 2(λσ + K − ρN kcbΩσ/(α+σ)) in Eq. (26), which the paper itself notes is 'twice the one defined in [16]'. The nonlinear simulations, however, halve the motor force f(t), so the numerically integrated model has the single-row linear response of [16], not the χ of Eq. (26). Claiming that these simulations reproduce the linear predictions of Ωc and θc is therefore not a test of the analyzed two-row model: the simulated model was pre-adjusted to match [16]'s linear analysis. The validation, as stated, reduces by construction to the imported linear result rather than providing an independent check of the two-row Hopf prediction.

full rationale

The paper's core derivation — the two-row motor model, the Fourier reduction to the four-ODE system (16), the linear stability analysis leading to Eq. (26), and the isolated-axoneme Hopf analysis in Appendix A.2 — is self-contained and does not rely on circular self-citation. The constitutive closure (13) is taken from independent references [16,31,32], and the paper is transparent about the constant-sum transition-rate assumption; this is a modeling choice, not a circularity. The self-citation to [28] is descriptive rather than load-bearing: it says the two-row model was described there, but the present paper derives the equations it uses. The statement that Eq. (26) is twice the [16] coefficient is explicit and does not disguise an imported result. However, two quantitative steps undermine the paper's central validation. First, the critical frequency θc=20.6 Hz is not an independent prediction: the internal friction λ is adjusted precisely to reproduce the 20 Hz frequency from [17], so the reported frequency is an input restated as an output. Second, the nonlinear simulations halve the motor force, making their linear response equal to that of [16] rather than to the analyzed Eq. (26); the claimed agreement between simulations and the linear predictions is therefore not a self-consistent test of the two-row model. These two issues affect the headline numerical validation but not the existence of the Hopf-bifurcation mechanism itself, so partial circularity is the appropriate verdict.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a small number of standard modeling assumptions plus two ad-hoc closures: the sinusoidal transition-rate choice in Eq. (13) that makes the Fourier reduction exact at first order, and the quasi-static approximation used to derive the cubic model. The comparison models require additional hand-chosen parameters (K, v0) for Chlamydomonas. No new physical entities are postulated.

free parameters (4)
  • internal viscosity λ (bull sperm) = 7 pN·s/µm²
    Adjusted in Section 5.1 so that the linear analysis gives a beating frequency close to 20·2π rad/s from [17]; the model's predicted frequency is therefore partly fitted.
  • internal viscosity λ (Chlamydomonas) = 4 pN·s/µm²
    Chosen for short flagella in Table 2 without a stated fitting target; affects the bifurcation threshold and frequency.
  • internal elasticity K (Chlamydomonas) = 25×10³ pN/µm²
    Defined in Section 6.1 so that a²K = 90 pN, a value chosen to be close to the range used in [23]; this choice affects the Hopf threshold and the force-velocity loop.
  • zero-load velocity v0 (chemoEH comparison) = 60 µm/s
    Selected in Section 6.1 so that ζ = a/(τ0 v0) = 0.4, to align with short-flagella parameters in [23]; this parameter sets the chemoEH comparison but not the µ-chemoEH central model.
assumptions (6)
  • domain assumption The axoneme can be represented as two inextensible filaments at fixed distance a, with sliding displacement u related to curvature by u' = aφ' (Eq. 4).
    This geometric reduction is standard in the sliding-feedback literature (e.g., [16]) but ignores basal sliding, filament stretching, and 3D structure.
  • domain assumption Resistive force theory with constant coefficients ξn and ξt describes the hydrodynamic drag (Section 2.2).
    Neglects hydrodynamic interactions between the filaments and the far-field flow; standard for slender flagella at low Reynolds number.
  • ad hoc to paper Motor transition rates satisfy ω1+ω2 = α constant and ω2 = α(η + Ω/(2π²)(cos+sin)) (Eq. 13).
    This specific functional form is chosen to make the Fourier expansion truncate at first order and to symmetrize the two motor rows; it is not derived from a measured rate law and is the key closure enabling the four-ODE reduction.
  • ad hoc to paper The quasi-static approximation in the cubic model, 0 = -α fa + 2ρN kcb Ω εv(1-(εv/γ)²), holds when the rate of change of v is small compared to α (Section 3.2).
    This is an uncontrolled approximation used to replace the motor ODE by an algebraic force-velocity relation; its validity for large-amplitude beating is not established.
  • standard math Higher Fourier modes n>1 of the motor probabilities decay to zero and can be neglected (Appendix A.1).
    The linear time-varying system for modes n>1 has eigenvalues -α ± i(2π/ℓ)u_dot, so solutions decay as e^{-αt}; valid for bounded sliding velocity.
  • domain assumption Clamped-free boundary conditions with no basal sliding, φ(0)=0 and u(0)=0 (Eq. 6, Section 2.2).
    Standard for sperm flagella, but basal sliding can occur in some flagella and would alter the instability.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Beating of eukaryotic flagella via Hopf bifurcation of a system of stalled molecular motors." pith.science (2026). https://pith.science/paper/TH3WBUCF

@misc{pith2026241206067,
  author       = {Pith},
  title        = {Pith review of: Beating of eukaryotic flagella via Hopf bifurcation of a system of stalled molecular motors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TH3WBUCF}},
  note         = {Machine review of arXiv:2412.06067}
}
read the original abstract

The modeling of the beating of cilia and flagella in fluids is a particularly active field of study, given the biological relevance of these organelles. Various mathematical models have been proposed to represent the nonlinear dynamics of flagella, whose motion is powered by the work of molecular motors attached to filaments composing the axoneme. Here, we formulate and solve a nonlinear model of activation based on the sliding feedback mechanism, capturing the chemical and configurational changes of molecular motors driving axonemal motion. This multiscale model bridges microscopic motor dynamics with macroscopic flagellar motion, providing insight into the emergence of oscillatory beating. We validate the framework through linear stability analysis and fully nonlinear numerical simulations, showing the onset of spontaneous oscillations. To make the analysis more comprehensive, we compare our approach with two established sliding feedback models.

Figures

Figures reproduced from arXiv: 2412.06067 by the authors.

Figure 1
Figure 1. From the axoneme to the two rows of molecular motors. (A) Cross section of the axoneme [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Unfolding of the flagellum two-filament structure in 1(B) and (C). The red filaments are [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Linear study for the two row model. (a), (b) Critical ATP concentration 10 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Fully non-linear simulations for the µ-chemoEH model with ε = 0.01. (a) Deformed configuration during a beating cycle. The legend bar on the right indicates the time normalized over the period of oscillation t/T. (b) Oscillations in time of the first order for the prob…
Figure 5
Figure 5. Figure 5: Kymographs of the tangent angle φ varying the sperm number Sp ∈ {1, 5, 10}. The legend bar on the right indicates the amplitude of φ. The tangent angle is plotted over the dimensionless arc-length and three period of oscillations. The wave speed increases by increasing…
Figure 6
Figure 6. Figure 6: Fully non-linear simulations for the deformed configuration of bull sperm over a beating [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 8
Figure 8. Figure 8: For the µ-chemoEH, even at ε = 1, simulations show that the filaments start to twist on themselves. The deformed configurations plotted in Figures 8(a) and [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 7
Figure 7. Figure 7: Comparison between three models at ε = 0.01: µ-chemoEH (first column), chemoEH model (second column), cubic-model (third column). For each model, we show the deformed configuration (first row), the limit cycles between active force and velocity at different arc-length …
Figure 8
Figure 8. Figure 8: Comparison between three models in the non linear regime: same as Figure 7 with [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: Tug-of-war between two rows of motors as the bifurcation parameter [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: Active force along the deformed configuration at [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 29 canonical work pages

  1. [16]

    Generic aspects of axonemal beating

    S´ ebastien Camalet and Frank J¨ ulicher. “Generic aspects of axonemal beating”. In: New J. Phys. 2.1 (2000). doi: 10.1088/1367-2630/2/1/324

    work page doi:10.1088/1367-2630/2/1/324 2000

  2. [17]

    How molec- ular motors shape the flagellar beat

    Ingmar H Riedel-Kruse, Andreas Hilfinger, Jonathon Howard, and Frank J¨ ulicher. “How molec- ular motors shape the flagellar beat”. In: HFSP j. 1.3 (2007). doi: 10.2976/1.2773861

    work page doi:10.2976/1.2773861 2007

  3. [1]

    Life at low Reynolds number

    E. M. Purcell. “Life at low Reynolds number”. In: Am. J. Phys. 45.1 (1977). doi: 10.1119/ 1.10903

    work page 1977

  4. [2]

    The hydrodynamics of swimming microorganisms

    Eric Lauga and Thomas R Powers. “The hydrodynamics of swimming microorganisms”. In: Rep. Prog. Phys. 72.9 (2009). doi: 10.1088/0034-4885/72/9/096601

    doi:10.1088/0034-4885/72/9/096601 2009

  5. [3]

    Mammalian sperm motility: observation and theory

    Eamonn A Gaffney, Hermes Gadˆ elha, David J Smith, John R Blake, and Jackson C Kirkman- Brown. “Mammalian sperm motility: observation and theory”. In: Annu. Rev. Fluid Mec. 43.1 (2011). doi: 10.1146/annurev-fluid-121108-145442

    doi:10.1146/annurev-fluid-121108-145442 2011

  6. [4]

    The radial spokes and central apparatus: mechano- chemical transducers that regulate flagellar motility

    Elizabeth F Smith and Pinfen Yang. “The radial spokes and central apparatus: mechano- chemical transducers that regulate flagellar motility”. In: Cell Motil. Cytoskeleton 57.1 (2004). doi: 10.1002/cm.10155

    work page doi:10.1002/cm.10155 2004

  7. [5]

    The dynein regulatory complex is the nexin link and a major regulatory node in cilia and flagella

    Thomas Heuser, Milen Raytchev, Jeremy Krell, Mary E Porter, and Daniela Nicastro. “The dynein regulatory complex is the nexin link and a major regulatory node in cilia and flagella”. In: JCB 187.6 (2009). doi: 10.1083/jcb.200908067

    work page doi:10.1083/jcb.200908067 2009

  8. [6]

    The bank of swimming organisms at the micron scale (BOSO-Micro)

    Marcos F. Velho Rodrigues, Maciej Lisicki, and Eric Lauga. “The bank of swimming organisms at the micron scale (BOSO-Micro)”. In: PLoS One 16.6 (2021). doi: doi . org / 10 . 1371 / journal.pone.0252291

    work page 2021

Show all 38 references
  1. [7]

    Eukaryotic Flagella: Variations in Form, Function, and Composition during Evolution

    Jonathan Moran, Paul G. McKean, and Michael L. Ginger. “Eukaryotic Flagella: Variations in Form, Function, and Composition during Evolution”. In: BioSci. 64.12 (2014). doi: 10.1093/ biosci/biu175

    2014

  2. [8]

    Asymmetric distribution and spatial switching of dynein activity generates ciliary motility

    Jianfeng Lin and Daniela Nicastro. “Asymmetric distribution and spatial switching of dynein activity generates ciliary motility”. In: Sci. 360.6387 (2018). doi: 10.1126/science.aar1968. 26

    2018 doi

  3. [9]

    Wave propagation along flagella

    KE Machin. “Wave propagation along flagella”. In: J. Exp. Biol. 35.4 (1958). doi: 10.1242/ jeb.35.4.796

    1958

  4. [10]

    Predicting the locations of force-generating dyneins in beating cilia and flagella

    Jonathon Howard, Alexander Chasteen, Xiaoyi Ouyang, Veikko F Geyer, and Pablo Sartori. “Predicting the locations of force-generating dyneins in beating cilia and flagella”. In: Front. cell dev. biol. 10 (2022). doi: 10.3389/fcell.2022.995847

    2022

  5. [11]

    Computer simulation of flagellar movement. VI. Simple curvature-controlled models are incompletely specified

    CJ Brokaw. “Computer simulation of flagellar movement. VI. Simple curvature-controlled models are incompletely specified”. In: Biophys. J. 48.4 (1985). doi: 10.1016/S0006-3495(85) 83819-4

    1985 doi

  6. [12]

    Bend propagation by a sliding filament model for flagella

    Charles J Brokaw. “Bend propagation by a sliding filament model for flagella”. In: J. Exp. Biol. 55.2 (1971). doi: 10.1242/jeb.55.2.289

    1971 doi

  7. [13]

    Axonemal regulation by curvature explains sperm flagellar waveform modulation

    Meurig T Gallagher, Jackson C Kirkman-Brown, and David J Smith. “Axonemal regulation by curvature explains sperm flagellar waveform modulation”. In: PNAS nexus 2.3 (2023). doi: 10.1093/pnasnexus/pgad072

    2023 doi

  8. [14]

    Ciliary beating patterns map onto a low-dimensional behavioural space

    Veikko F Geyer, Jonathon Howard, and Pablo Sartori. “Ciliary beating patterns map onto a low-dimensional behavioural space”. In: Nat. Phys. 18.3 (2022). doi: 10.1038/s41567-021- 01446-2

    2022 doi

  9. [15]

    Internal friction controls active ciliary oscillations near the instability threshold

    Debasmita Mondal, Ronojoy Adhikari, and Prerna Sharma. “Internal friction controls active ciliary oscillations near the instability threshold”. In: Sci. Adv. 6.33 (2020). doi: 10.1126/ sciadv.abb0503

    2020

  10. [18]

    Nonlinear amplitude dynamics in flagellar beating

    David Oriola, Hermes Gadˆ elha, and Jaume Casademunt. “Nonlinear amplitude dynamics in flagellar beating”. In: R. Soc. Open Sci. 4.3 (2017). doi: 10.1098/rsos.160698

    2017 doi

  11. [19]

    A ”Geometric Clutch

    C. B. Lindemann. “A ”Geometric Clutch” Hypothesis to Explain Oscillations of the Axoneme of Cilia and Flagella”. In: J. Theor. Biol. 168.2 (1994). doi: 10.1006/jtbi.1994.1097

    1994

  12. [20]

    A model of flagellar and ciliary functioning which uses the forces transverse to the axoneme as the regulator of dynein activation

    Charles B Lindemann. “A model of flagellar and ciliary functioning which uses the forces transverse to the axoneme as the regulator of dynein activation”. In: Cell Motil. Cytoskeleton 29.2 (1994). doi: 10.1002/cm.970290206

    1994 doi

  13. [21]

    Equations of interdoublet separation during flagella motion reveal mechanisms of wave propagation and instability

    Philip V Bayly and Kate S Wilson. “Equations of interdoublet separation during flagella motion reveal mechanisms of wave propagation and instability”. In: Biophys. J. 107.7 (2014). doi: 10.1016/j.bpj.2014.07.064

    2014 doi

  14. [22]

    Dy- namic curvature regulation accounts for the symmetric and asymmetric beats of Chlamy- domonas flagella

    Pablo Sartori, Veikko F Geyer, Andre Scholich, Frank J¨ ulicher, and Jonathon Howard. “Dy- namic curvature regulation accounts for the symmetric and asymmetric beats of Chlamy- domonas flagella”. In: Elife 5 (2016). doi: 10.7554/eLife.13258

    2016 doi

  15. [23]

    The reaction-diffusion basis of animated patterns in eukaryotic flagella

    James F Cass and Hermes Bloomfield-Gadˆ elha. “The reaction-diffusion basis of animated patterns in eukaryotic flagella”. In: Nat. Commun. 14.1 (2023). doi: 10.1038/s41467-023- 40338-2

    2023 doi

  16. [24]

    Analysis of unstable modes distinguishes mathematical models of flagellar motion

    PV Bayly and KS Wilson. “Analysis of unstable modes distinguishes mathematical models of flagellar motion”. In: J. R. Soc. Interface 12.106 (2015). doi: 10.1098/rsif.2015.0124

    2015

  17. [25]

    Bifurcations and nonlinear dynamics of the follower force model for active filaments

    Bethany Clarke, Yongyun Hwang, and Eric E. Keaveny. “Bifurcations and nonlinear dynamics of the follower force model for active filaments”. In: Phys. Rev. Fluids 9 (7 2024). doi: 10. 1103/PhysRevFluids.9.073101. 27

    2024

  18. [26]

    Instability-driven oscillations of elastic microfila- ments

    Feng Ling, Hanliang Guo, and Eva Kanso. “Instability-driven oscillations of elastic microfila- ments”. In: Journal of The Royal Society Interface 15.149 (2018). doi: 10.1098/rsif.2018. 0594

    2018 doi

  19. [27]

    Cooperative molecular motors

    Frank J¨ ulicher and Jacques Prost. “Cooperative molecular motors”. In: PRL 75.13 (1995). doi: 10.1103/PhysRevLett.75.2618

    1995 doi

  20. [28]

    Some mathematical models for flagellar activation mechanisms

    Fran¸ cois Alouges, Irene Anello, Antonio DeSimone, Aline Lefebvre-Lepot, and Jessie Levillain. Some mathematical models for flagellar activation mechanisms . 2024. arXiv: 2409.03506

    2024 arXiv

  21. [29]

    Modeling molecular motors

    Frank J¨ ulicher, Armand Ajdari, and Jacques Prost. “Modeling molecular motors”. In: RMP 69.4 (1997). doi: 10.1103/RevModPhys.69.1269

    1997 doi

  22. [30]

    The propulsion of sea-urchin spermatozoa

    James Gray and Gregory J Hancock. “The propulsion of sea-urchin spermatozoa”. In: J. Exp. Biol. 32.4 (1955)

    1955

  23. [31]

    Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models

    Thomas Gu´ erin, J Prost, and J -F Joanny. “Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models”. In:Eur. Phys. J. E 34 (2011). doi: 10.1140/epje/i2011- 11060-5

    2011 doi

  24. [32]

    Bidirectional motion of motor assemblies and the weak- noise escape problem

    T Gu´ erin, J Prost, and J-F Joanny. “Bidirectional motion of motor assemblies and the weak- noise escape problem”. In: Phys. Rev. E Stat. Nonlin. Soft Matter Physics 84.4 (2011). doi: 10.1103/PhysRevE.84.041901

    2011 doi

  25. [33]

    Nonlinear dynamics of cilia and flagella

    Andreas Hilfinger, Amit K Chattopadhyay, and Frank J¨ ulicher. “Nonlinear dynamics of cilia and flagella”. In: Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79.5 (2009). doi: 10.1103/ PhysRevE.79.051918

    2009

  26. [34]

    Theory of mitotic spindle oscillations

    Stephan W Grill, Karsten Kruse, and Frank J¨ ulicher. “Theory of mitotic spindle oscillations”. In: PRL 94.10 (2005). doi: 10.1103/PhysRevLett.94.108104

    2005 doi

  27. [35]

    Force and motion generation of molecular motors: A generic description

    Frank J¨ ulicher. “Force and motion generation of molecular motors: A generic description”. In: Transport and Structure. 1999. doi: 10.1007/BFb0104221

    1999 doi

  28. [36]

    Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (2nd ed.) CRC press, 2015

    Steven H Strogatz. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (2nd ed.) CRC press, 2015. doi: 10.1201/9780429492563

    2015 doi

  29. [37]

    Molecular mechanism for oscillation in flagella and muscle

    Charles J Brokaw. “Molecular mechanism for oscillation in flagella and muscle.” In: PNAS 72.8 (1975). doi: 10.1073/pnas.72.8.310

    1975 doi

  30. [38]

    The biomechanical role of extra-axonemal structures in shaping the flagellar beat of Euglena gracilis

    Giancarlo Cicconofri, Giovanni Noselli, and Antonio DeSimone. “The biomechanical role of extra-axonemal structures in shaping the flagellar beat of Euglena gracilis”. In:Elife 10 (2021). doi: 10.7554/eLife.58610. 28

    2021 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.