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Some mathematical models for flagellar activation mechanisms

François Alouges, Irene Anello, Antonio DeSimone, Aline Lefebvre-Lepot, Jessie Levillain

TL;DR

The paper develops a hierarchy of motor-assembly models for flagellar bending, progressing from a one-row to a two-row and then to an N-row formulation. It proves well-posedness (existence and uniqueness) for the two-row PDE system and demonstrates a supercritical Hopf bifurcation driven by ATP concentration, using center-manifold reduction and Fourier analysis to obtain the oscillation frequency and amplitude. The authors supplement theory with an upwind numerical scheme, illustrating amplitude, frequency sensitivity, and synchronization patterns across N-row configurations, including N=8 that exhibit phase-related groupings. The work provides mathematical grounding for dynein-driven axonemal beating, clarifies the role of symmetry in equilibrium, and offers a framework for exploring curvature-regulation and coupling to elastic flagellum mechanics. Future directions include incorporating the central microtubule pair and fully coupled elastic-flagellum dynamics to capture three-dimensional beating patterns.

Abstract

This paper focuses on studying a model for molecular motors responsible for the bending of the axoneme in the flagella of microorganisms. The model is a coupled system of partial differential equations inspired by Jülicher et al. or Camalet, incorporating two rows of molecular motors between microtubules filaments. Existence and uniqueness of a solution is proved, together with the presence of a supercritical Hopf bifurcation. Additionally, numerical simulations are provided to illustrate the theoretical results. A brief study on the generalization to N-rows is also included.

Some mathematical models for flagellar activation mechanisms

TL;DR

The paper develops a hierarchy of motor-assembly models for flagellar bending, progressing from a one-row to a two-row and then to an N-row formulation. It proves well-posedness (existence and uniqueness) for the two-row PDE system and demonstrates a supercritical Hopf bifurcation driven by ATP concentration, using center-manifold reduction and Fourier analysis to obtain the oscillation frequency and amplitude. The authors supplement theory with an upwind numerical scheme, illustrating amplitude, frequency sensitivity, and synchronization patterns across N-row configurations, including N=8 that exhibit phase-related groupings. The work provides mathematical grounding for dynein-driven axonemal beating, clarifies the role of symmetry in equilibrium, and offers a framework for exploring curvature-regulation and coupling to elastic flagellum mechanics. Future directions include incorporating the central microtubule pair and fully coupled elastic-flagellum dynamics to capture three-dimensional beating patterns.

Abstract

This paper focuses on studying a model for molecular motors responsible for the bending of the axoneme in the flagella of microorganisms. The model is a coupled system of partial differential equations inspired by Jülicher et al. or Camalet, incorporating two rows of molecular motors between microtubules filaments. Existence and uniqueness of a solution is proved, together with the presence of a supercritical Hopf bifurcation. Additionally, numerical simulations are provided to illustrate the theoretical results. A brief study on the generalization to N-rows is also included.
Paper Structure (18 sections, 4 theorems, 87 equations, 9 figures, 1 table)

This paper contains 18 sections, 4 theorems, 87 equations, 9 figures, 1 table.

Table of Contents

  1. Motivation and Modeling
  2. The one-row model
  3. The two-row model
  4. The $N$-row model
  5. Theoretical results
  6. Proof of Theorem \ref{['th: Ex and Uniqueness']} (existence and uniqueness)
  7. Proof of Theorem \ref{['th: Hopf bifurcation']} (Hopf bifurcation)
  8. $N$-row model: theoretical considerations
  9. Numerics
  10. Choice for parameters
  11. $2$-row: numerical simulations
  12. Amplitude of the oscillations
  13. Parameters sensitivity
  14. $N$-row model: Numerical results
  15. Conclusion and outlook
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let us fix $\ell>0$. Assume $\omega_A(\xi;\Omega)$ and $\omega_B(\xi;\Omega)$ as in eq: uniformity om1+om2. Moreover, assume $\omega_B$ and $\Delta W$ to be at least $C_{\#}^1([0,\ell])$. If the initial data $P(\xi, 0)$ and $Q(\xi, 0)$ are $C_{\#}^1([0,\ell])$ and $x(0)=0$, then the system of equati

Figures (9)

  • Figure 1: A microscopic slice of the axoneme (on the right), zoomed in from the whole filament (on the left). The periodic structure is highlighted by an alternation of green and blue sections, modeling preferred binding sites for molecular motors.
  • Figure 2: (a) Cross section of the axoneme. (b) Cross section of the axoneme where we group the filaments 1-4 (blue) and 5-9 (red) together. (c) Unfolding of the axoneme in (b) with two opposite microtubule pairs.
  • Figure 3: Simplified axoneme with $9$ microtubule doublets and no central pair Yagi1995.
  • Figure 4: Motors and tubule shift in the $N$-row model (here the horizontal axis $x$ shows absolute tubule displacement instead of motor position $\xi$ in a periodicity cell).
  • Figure 5: Two-row model ((a),(b)) and one-row model ((c),(d)): (a) Probability density $P_1(\xi,t)$ of the top layer of being in state $1$ over time in a periodicity cell, in steady-state regime. The probability density $P_2(\xi,t)$ of the bottom layer of being in state $1$ is similar to $P_1(\xi, t)$. (b) Relative tubule shift $x$ over time. (c) Probability density $P(\xi,t)$ of the moving layer in the one-row model. (d) Relative tubule shift $x$ over time in the one-row model.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Theorem 2.1: Existence and uniqueness
  • Theorem 2.2: Hopf bifurcation
  • Proposition 2.1: First order coefficients
  • proof
  • Proposition 2.2: Higher order coefficients
  • proof