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

This study employs bifurcation analysis and numerical simulations of a mass-conserved reaction-diffusion model to reveal how codimension-2 instabilities and domain size variations drive the coexistence and transitions between steady wave-pinning and travelling wave states, thereby offering mechanistic insights into diverse eukaryotic cell motility patterns.

Original authors: Jack M. Hughes, Saar Modai, Leah Edelstein-Keshet, Arik Yochelis

Published 2026-04-13

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CC BY 4.0

Original authors: Jack M. Hughes, Saar Modai, Leah Edelstein-Keshet, Arik Yochelis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a cell not as a static blob, but as a bustling city with a flexible, moving border. This border is made of a protein called F-actin, which acts like the city's construction crew, constantly building and rebuilding roads and walls to help the cell move, turn, or change shape.

But who tells the construction crew where to build? The answer lies in a group of molecular "managers" called Rho-GTPases. These managers have two modes: "Active" (the boss is awake and giving orders) and "Inactive" (the boss is asleep).

This paper is a mathematical story about how these managers and the construction crew interact to create different "traffic patterns" inside the cell. The authors built a simplified computer model to figure out why cells sometimes move in a straight line, sometimes spin in circles, and sometimes ruffle like a dog shaking off water.

Here is the breakdown of their discovery using everyday analogies:

1. The Rules of the Game: A Closed System

The most important rule in this model is Mass Conservation. Imagine the city has a fixed number of construction workers and managers. They can't be created or destroyed; they can only move around or change their job titles (from active to inactive).

The Mechanism: When a manager becomes "Active," it recruits the construction crew (F-actin). However, the construction crew then sends a signal to put the manager back to sleep (inactivation).
The Result: This creates a tug-of-war. If the crew gets too big, they shut down the manager, causing the crew to shrink. If the manager is too strong, the crew grows. This feedback loop is the engine of all the patterns.

2. The Two Main "Traffic Patterns"

The researchers found that depending on the size of the cell (the "domain") and the strength of the feedback, the system settles into two very different, stable states. Think of these as two different driving modes for the cell:

A. The "Wave-Pinning" State (The Polarized Cell)

What it looks like: Imagine a long, straight road. Suddenly, a massive traffic jam forms at one end, while the rest of the road is empty. This jam stays put.
The Cell Behavior: This represents a cell that has picked a direction. It has a clear "front" (where the jam/actin is) and a "back." This is how a cell crawls forward in a straight line toward a target (like a white blood cell chasing bacteria).
The Metaphor: It's like a surfer who has caught a wave and is riding it steadily toward the shore. The wave is "pinned" to the surfer.

B. The "Travelling Wave" State (The Moving Cell)

What it looks like: Imagine a wave of traffic that moves smoothly down the road, never stopping. The cars (actin) move, but the pattern of the traffic jam keeps traveling.
The Cell Behavior: This represents a cell that is turning or spinning. Instead of a static front, the "front" of the cell is constantly moving around the perimeter.
The Metaphor: It's like a conga line or a wave in a stadium. The people (actin) are moving, but the pattern of the wave travels around the circle.

3. The Big Surprise: Bistability (The "Switch")

The most exciting part of the paper is the discovery of Bistability.
Usually, you might think a system is either in one state or the other. But the authors found that on "medium-sized" domains (like the size of a real cell), the system can be unstable. It can exist in either the "Pinned" state or the "Travelling" state, and it can flip between them.

The Analogy: Imagine a light switch that is stuck in the middle. It's not quite "On" and not quite "Off." A tiny nudge (a small change in the environment) can flip it to "On" (straight crawling), while a different nudge flips it to "Off" (spinning/ruffling).
Why it matters: This explains how a single cell can be incredibly versatile. It can decide to move straight, then suddenly decide to turn, or start ruffling its edges to explore its surroundings, all based on tiny fluctuations in its internal chemistry.

4. The Role of Domain Size (The "Room" Matters)

The paper emphasizes that the size of the room (the cell's perimeter) changes the rules.

Small Room: The patterns might not fit well, or they might just be static.
Huge Room: The patterns might get messy or unstable.
Just Right (Medium Room): This is where the magic happens. The cell is just the right size to support both the "Pinned" state and the "Travelling" state simultaneously. This allows the cell to be robust and adaptable.

5. Real-World Implications

Why should we care?

Cell Migration: This explains how immune cells find their way to infections. They can switch from "crawling straight" to "turning" to navigate complex environments.
Cancer: Cancer cells often lose their ability to control these patterns. They might get stuck in a "ruffling" state (moving chaotically) or fail to polarize, which helps them spread (metastasize).
Development: When an embryo is forming, cells need to know exactly when to stop moving and when to start dividing. This "switch" mechanism is likely crucial for those decisions.

Summary

In simple terms, this paper shows that cells are like smart traffic systems. By using a simple set of rules (a fixed number of workers who can switch jobs), they can spontaneously create complex patterns. They can lock into a straight line to run a marathon, or switch to a spinning wave to dance. The key to this flexibility is that the system is "bistable"—it has two stable modes it can jump between, allowing the cell to be a master of adaptation.

1. Problem Statement and Motivation

The paper addresses the complex dynamic patterns observed in eukaryotic cells, specifically the behavior of filamentous actin (F-actin) and its regulators, Rho-GTPases. These molecules drive cell motility, polarization, and division. Experimental observations reveal diverse patterns, including:

Steady polarization: A static "front" and "rear" (directed migration).
Travelling waves (TWs): Rhythmic waves of F-actin that can cause cell turning or circular crawling.
Ruffling: Multiple protrusions circulating around the cell perimeter.

Existing mathematical models for these "actin waves" are often highly complex, relying heavily on numerical simulations that obscure generic mechanisms. The authors aim to identify a prototypical, analytically tractable model that captures the essential mass-conserved interactions between GTPases and F-actin. The core research questions are:

Can a mechanism for the bistability between steady wave-pinning (polarization) and travelling wave states be identified in non-gradient reaction-diffusion (RD) models with mass conservation?
How do these patterns relate to distinct cellular motility modes (directed migration, turning, ruffling)?

2. Methodology

The authors employ a multi-faceted approach combining mathematical modeling, bifurcation theory, and numerical simulations.

Model Formulation:
- They start with the mass-conserved Mori et al. model for cell polarization and extend it with the Holmes et al. "GTPase-actin" model, which introduces F-actin as a slow negative feedback variable.
- Key Modification: To facilitate analysis, they replace the original Hill-function nonlinearities with polynomial nonlinearities. The resulting 3-variable system (Active GTPase $u$ , Inactive GTPase $v$ , F-actin $F$ ) conserves the total mass of the GTPase ( $M = u+v$ ) and includes mass conservation for F-actin (with negligible diffusion).
- The system is analyzed on a 1D periodic domain (representing a cell edge) and a 1D domain with Neumann boundaries (representing a cell diameter).
Analytical Techniques:
- Linear Stability Analysis: Determining the stability of Homogeneous Steady States (HSS) by analyzing the dispersion relation $\sigma(q)$ . They identify specific instability onsets: Long Wavelength (LW), Finite Wavenumber Hopf (WB), and Homogeneous Hopf (HB).
- Weakly Nonlinear Analysis: Using amplitude equations (multiple timescales method) near the codimension-2 LW/WB instability point to determine the criticality (supercritical vs. subcritical) of emerging travelling and standing waves.
- Numerical Continuation: Using the software AUTO to track solution branches (steady states, pulses, waves) as bifurcation parameters (specifically the feedback strength $s$ and basal activation $b$ ) vary. They compute Sobolev norms and propagation speeds to classify stability.
- Direct Numerical Integration: Using Julia (Method of Lines) to simulate time-dependent dynamics, testing the robustness of solutions against large perturbations and initial condition superpositions.

3. Key Contributions and Results

A. Identification of a Codimension-2 Instability

The analysis centers on a codimension-2 point where a Long Wavelength (LW) instability (steady state) and a Finite Wavenumber Hopf (WB) instability (oscillatory) emerge simultaneously. This point acts as an organizing center for the system's dynamics.

At this point, the system exhibits a rich structure of solutions, including steady wave-pinning states, propagating fronts, excitable pulses, travelling waves (TWs), and standing waves (SWs).

B. Bistability Between Wave-Pinning and Travelling Waves

The most significant finding is the demonstration of bistability between:

Wave-Pinning (WP) States: Steady, spatially localized patterns (mesa states) representing cell polarization.
Travelling Waves (TWs): Coherent propagating solutions.

This bistability is highly dependent on domain size and total mass ( $M$ ):

Large Domains ( $L \gg \lambda$ ): The stability regions of WP and Excitable Pulses (EPs) are generally separated. WP solutions are often unstable or coexist with EPs in a complex T-point structure.
Moderate Domains ( $L \sim O(m\lambda)$ ): Crucially, on domain sizes comparable to biological cells (e.g., $L = 2\lambda$ or $3\lambda$ ), the authors find a robust bistable region where stable WP states and stable TWs coexist. The system can settle into either a polarized state or a travelling wave state depending on initial conditions.

C. Mechanisms of Pattern Selection

Mass Conservation Role: Mass conservation restricts the flow in phase space, leading to non-hyperbolic steady states and specific bifurcation structures (e.g., Maxwell construction for WP states).
Domain Length Unfolding: By varying the domain length, the authors show how the system transitions between different motility modes.
- Single TW: Corresponds to cell turning.
- Multiple TWs (e.g., 2 or 3 wavelengths): Corresponds to "ruffling" behavior.
- WP: Corresponds to directed migration.

D. Robustness and Transitions

Perturbation Analysis: Time-dependent simulations show that WP states are robust to small perturbations but can transition to TWs under large perturbations or specific initial superpositions.
Invasion Dynamics: When a WP state and a TW state are placed adjacently, the outcome depends on the feedback strength $s$ . In some regimes, TWs invade and dominate; in others, the WP state persists.

4. Significance and Implications

Theoretical Advancement: The paper provides a rigorous mathematical framework for understanding how mass conservation in non-gradient RD systems leads to distinct pattern formation mechanisms. It highlights that mass conservation is not just a constraint but a driver of complex bifurcation structures (e.g., codimension-2 and potential codimension-3 points).
Biological Relevance: The model successfully maps mathematical solutions to biological phenotypes:
- Wave-Pinning $\rightarrow$ Directed Migration (stable front/rear).
- Single Travelling Wave $\rightarrow$ Turning/Circular Motion.
- Multi-wavelength Waves $\rightarrow$ Ruffling (multiple protrusions).
Mechanism for Cell Decision Making: The discovery of bistability between WP and TW states on moderate domain sizes suggests a mechanism for how cells might switch between different motility modes (e.g., from crawling to turning) without changing internal parameters, but rather by responding to perturbations or geometric constraints.
Future Directions: The work opens avenues for investigating higher-order instabilities (codimension-3, T-points, Shil'nikov bifurcations) and the interplay between reaction-diffusion dynamics and mechanical cell deformation.

Conclusion

This study bridges the gap between abstract mathematical bifurcation theory and biological cell motility. By simplifying the actin-GTPase system into a polynomial mass-conserved model, the authors demonstrate that bistability between steady polarization and travelling waves is a generic feature of such systems on biologically relevant scales. This provides a fundamental explanation for the coexistence and transitions of diverse cell motility modes observed in nature.