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arxiv: 2410.12213 · v4 · submitted 2024-10-16 · 🌊 nlin.PS · q-bio.CB

Bistability of travelling waves and wave-pinning states in a mass-conserved reaction-diffusion system: From bifurcations to implications for actin waves

Jack M. Hughes , Saar Modai , Leah Edelstein-Keshet , Arik Yochelis This is my paper

Pith reviewed 2026-05-23 19:22 UTC · model grok-4.3

classification 🌊 nlin.PS q-bio.CB
keywords bistabilitywave-pinningtravelling wavesmass conservationreaction-diffusionactin wavesRho-GTPaseHopf instability
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The pith

In mass-conserved models of Rho-GTPase and F-actin, moderate domain lengths create bistability between wave-pinning states and travelling waves.

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

A reaction-diffusion model conserves total Rho-GTPase mass while active forms promote F-actin and F-actin feeds back to inactivate Rho-GTPase. On a one-dimensional periodic domain, spatial-dynamics bifurcation analysis identifies a codimension-2 long-wavelength and finite-wavenumber Hopf instability. This organizes steady wave-pinning solutions that obey the Maxwell construction, fronts, excitable pulses, and both travelling and standing waves. The analysis shows that bistability between the pinned states and travelling waves appears specifically on moderate domain sizes, with the bistable region unfolding as domain length is varied. Such coexistence offers possible mechanisms for cells to switch among directed migration, turning, and ruffling behaviors.

Core claim

The codimension-2 Hopf instability organizes a family of solutions distinguished by mass-conservation regimes and classified by domain size: steady wave-pinning states (mesas obeying the Maxwell construction), propagating fronts and excitable pulses, and travelling and standing waves; in particular, bistability between wave-pinning and travelling waves unfolds through domain length on moderate domains.

What carries the argument

Codimension-2 long-wavelength and finite-wavenumber Hopf instability on the mass-conserved 1D periodic domain.

Load-bearing premise

The specific negative-feedback loop in which F-actin inactivates Rho-GTPase while Rho-GTPase promotes F-actin, together with strict mass conservation, holds on the 1D periodic domain.

What would settle it

Numerical continuation or simulation on moderate domains showing that the region of bistability between wave-pinning and travelling waves does not unfold with domain length.

Figures

Figures reproduced from arXiv: 2410.12213 by Arik Yochelis, Jack M. Hughes, Leah Edelstein-Keshet, Saar Modai.

Figure 1
Figure 1. Figure 1: Schematic representation of cells and corresponding F-actin dynamics at the cell [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic diagrams of several Rho-GTPase feedback-loop models. (a) The Mori [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The linear onsets of bifurcations along homogeneous steady states (HSSs) are shown [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: One-parameter bifurcation diagram for the F-actin component as a function of the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Bifurcating branches of the primary travelling and standing waves (with wave [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Bifurcation diagrams computed via numerical continuation as a function of [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Solution profiles at selected locations in [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) Bifurcation diagrams computed via numerical continuation as a function of [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Bifurcation diagrams computed via numerical continuation as a function of [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) Bifurcation diagram computed via numerical continuation as a function of [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Solution profiles at selected locations along the (a) EP, (b) SP, and (c) WP branches [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Bifurcation diagrams computed via numerical continuation as a function of [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Solution profiles at selected locations in [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: (a) Bifurcation diagram computed via numerical continuation as a function of [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: (a) Bifurcation diagrams computed via numerical continuation as a function of [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Direct numerical integration of (2.3) with periodic boundary conditions, initialized with a convex combination (6.1) of wave-pinning and travelling waves (left panels). Spacetime plots (“kymographs”, right panels) showing F as a heat map for (a) δ = 0.25, (b) δ = 0.5, and (c) δ = 0.75. In (b), we demonstrate a distinct kind of modulated solution that persists also at a much longer simulation time, t = 500… view at source ↗
Figure 17
Figure 17. Figure 17: Direct numerical integration of (2.3) with periodic boundary conditions, initialized with wave pinning (WP) solutions and 3-period travelling waves (TWs) of wavelength λ, on adjoining parts of the same domain (left panels). Spacetime plots (right panels) showing F as a heat map for (a) s ≈ 0.45, (b) s ≈ 0.52, and (c) s ≈ 0.53. In (b) the asymptotic TWs are with a wavelength of 6λ/5. Parameter values as in… view at source ↗
Figure 18
Figure 18. Figure 18: Bifurcation diagram computed via numerical continuation as a function of [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: (a-h) Spacetime plots showing F as a heat map obtained via direct numerical integration of (2.3) with periodic boundary conditions, initialized with wave pinning (WP) solutions obtained from selected locations given by symbols in [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: (a) As in Figure 3a but with a dotted line slice at b ≈ 0.14, where an￾other codimension-2 long wavelength/finite wavenumber Hopf bifurcation emerges. (b) One￾parameter bifurcation diagram for the F-actin component of the homogeneous steady states as a function of s, along the dotted line slice in (a). Solid (dashed) lines denote linear stability (instability). The values (sW B, sLW , st HB, sb HB, sb c )… view at source ↗
Figure 21
Figure 21. Figure 21: (a) As in Figure 3a but with a dotted line slice at b ≈ 0.11, where the locus of finite wavenumber Hopf bifurcations terminates. (b) One-parameter bifurcation diagram for the F-actin component of the homogeneous steady states as a function of s, along the dotted line slice in (a). Solid (dashed) lines denote linear stability (instability). The values (sW B, st LW , st HB, sSP , sb LW ) ≈ (0.433, 0.451, 0.… view at source ↗
Figure 22
Figure 22. Figure 22: (a) As in Figure 3a but with a dotted line slice at b ≈ 0.08, where a codimension-2 Bogdanov-Takens bifurcation emerges. (b) One-parameter bifurcation diagram for the F-actin component of the homogeneous steady states as a function of s, along the dotted line slice in (a), where Q∗ 0,1,2 are homogeneous steady states; solid (dashed) lines denote linear sta￾bility (instability). The values (sW B, st LW , s… view at source ↗
Figure 23
Figure 23. Figure 23: (a) As in Figure 3a but with a dotted line slice at b ≈ 0.02, where a codimension-2 long wavelength/saddle-node instability emerges. (b) One-parameter bifurcation diagram for the F-actin component of the homogeneous steady states as a function of s, along the dotted line slice in (a), where Q∗ 0,1,2 are homogeneous steady states; solid (dashed) lines denote linear stability (instability). The values (s t … view at source ↗
Figure 24
Figure 24. Figure 24: Bifurcation diagrams of (B.1) with respect to total mass M, showing the ho￾mogeneous steady states Q∗ 0,1,2 , travelling waves (TWs) with critical wavelength λ (TWλ), wave-pinning (WP) solutions, and excitable pulses (EPs) for L = 1000 ≫ λ ≈ 4.02. The solution branches are projected with the Sobolev norm (5.1) and the propagation speed c; solid (dashed) lines denote linearly stable (unstable) solutions. T… view at source ↗
Figure 25
Figure 25. Figure 25: Solution profiles at selected locations in [PITH_FULL_IMAGE:figures/full_fig_p043_25.png] view at source ↗
read the original abstract

Eukaryotic cells demonstrate a wide variety of dynamic patterns of filamentous actin (F-actin) and its regulators. Some of these patterns play important roles in cell functions, such as distinct motility modes, which motivate this study. We devise a mass-conserved reaction-diffusion model for active and inactive Rho-GTPase and F-actin in the cell cortex. The mass-conserved Rho-GTPase system promotes F-actin, which feeds back to inactivate the former. We study the model on a 1D periodic domain (edge of thin sheet-like cell) using bifurcation theory in the framework of spatial dynamics, complemented with numerical simulations. Among several discussed bifurcations, the analysis centers on the study of the codimension-2 long wavelength and finite wavenumber Hopf instability, in which we describe a rich structure of steady wave-pinning states (a.k.a. mesas, obeying the Maxwell construction), propagating coherent solutions (fronts and excitable pulses), and travelling and standing waves, all distinguished by mass conservation regimes and classified by domain sizes. Specifically, we highlight the unexpected conditions for bistability between steady wave-pinning and travelling wave states on moderate domain sizes, i.e., unfolding through domain length. These results uncover and exemplify possible mechanisms of coexistence, robustness, and transitions between distinct cellular motility modes, including directed migration, turning, and ruffling. More broadly, the results indicate that non-gradient reaction-diffusion models comprising mass conservation have distinct pattern formation mechanisms that motivate further investigations, such as the unfolding of codimension-3 instabilities and T-points.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript presents a mass-conserved reaction-diffusion model coupling active/inactive Rho-GTPase with F-actin, where the Rho system promotes F-actin and F-actin inactivates Rho-GTPase. On a 1D periodic domain, spatial-dynamics bifurcation analysis (centered on the codim-2 long-wavelength/finite-wavenumber Hopf point) combined with numerical simulations classifies steady wave-pinning (mesa) states obeying the Maxwell construction, fronts, excitable pulses, travelling/standing waves, and identifies bistability between wave-pinning and travelling waves that unfolds with domain length at moderate sizes, all distinguished by mass-conservation regimes.

Significance. If the central bifurcation results hold, the work demonstrates how mass conservation in non-gradient RD systems produces distinct mechanisms for pattern coexistence and transitions, with direct implications for cellular motility modes (directed migration, turning, ruffling). The explicit classification of solutions by domain size via spatial dynamics, together with the identification of the codim-2 unfolding, constitutes a clear advance over generic Turing or wave-pinning analyses.

minor comments (3)
  1. [Abstract] Abstract: the description of the feedback loop and the codim-2 point would be strengthened by a single displayed model equation or parameter list so that the Maxwell-construction claim can be immediately verified against the conservation law.
  2. [Numerical methods] The numerical continuation results that confirm the bistability region on moderate domains should include a brief statement of the discretization scheme, tolerance, and how the periodic boundary conditions are enforced, to allow reproduction of the domain-length unfolding.
  3. [Figures] Figure captions for the bifurcation diagrams should explicitly label which branches correspond to the wave-pinning versus travelling-wave states and indicate the mass-conservation regime for each panel.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work. The recommendation for minor revision is noted; we will prepare a revised manuscript accordingly. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation applies standard spatial-dynamics bifurcation analysis (codim-2 long-wavelength/finite-wavenumber Hopf point) and direct numerical simulation to the stated mass-conserved RD system on a 1D periodic domain. All reported states (wave-pinning mesas obeying Maxwell construction, fronts, pulses, travelling/standing waves) are obtained from the model's PDEs and boundary conditions without any fitted parameter being relabeled as a prediction, without self-definitional closure, and without load-bearing reliance on prior self-citations whose validity is presupposed. The bistability result is an output of the unfolding with domain length, not an input. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields limited parameter information; the model is built on mass conservation and a specific inhibitory feedback from actin, treated here as domain assumptions rather than fitted quantities.

axioms (2)
  • domain assumption The Rho-GTPase system is strictly mass-conserved and promotes F-actin which in turn inactivates Rho-GTPase.
    Stated in the model description within the abstract.
  • domain assumption Analysis is performed on a 1D periodic domain representing the edge of a thin sheet-like cell.
    Explicitly stated as the setting for bifurcation study.

pith-pipeline@v0.9.0 · 5839 in / 1250 out tokens · 23883 ms · 2026-05-23T19:22:39.780700+00:00 · methodology

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