How Negative Feedback from Filamentous Actin Affects Cell Shapes and Motility

J. M. Hughes, J. Algorta, L. Edelstein-Keshet (Authors)
J. Algorta, J. M. Hughes (Contributors)
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Exploring cell shapes, motility, and transitions using numerical PDE bifurcation analysis and Morpheus CPM simulations

Introduction

The crawling motility of many eukaryotic cells is driven by filamentous actin (F-actin), and regulated by a complex signaling network of small GTPases. The feedbacks between active GTPase, which recruits actin-polymerization machinery, and F-actin, which can recruit GAPs that in turn inactivate the GTPase, give rise to dynamic patterns known as ‘actin waves’.

This model suite explores how cell shapes and motility are regulated by the negative feedback rate ($s$) and the cell perimeter ($L$) of a mass-conserved reaction-diffusion partial differential equation (PDE) model. Guided by numerical PDE bifurcation analysis, we couple the reaction-diffusion equations on the 1D cell perimeter to a 2D Cellular Potts model (CPM) to simulate various cell states, including directed motion, edge-ruffling, wobbly oscillations, back-and-forth turning, and random amoeboid-like motility.

**Figure 1.** Bifurcation diagram for cell perimeter $L = 3\lambda$ showing the existence and stability of polar and traveling wave solutions as a function of the negative feedback parameter $s$. Shaded regions mark the parameter regimes for directed polarization (yellow), ruffling (purple), turning (brown), and their co-existence (orange and red) that govern the shapes and motion of simulated cells.
Figure 1. Bifurcation diagram for cell perimeter $L = 3\lambda$ showing the existence and stability of polar and traveling wave solutions as a function of the negative feedback parameter $s$. Shaded regions mark the parameter regimes for directed polarization (yellow), ruffling (purple), turning (brown), and their co-existence (orange and red) that govern the shapes and motion of simulated cells.

Reproducibility of Article Figures

The Morpheus XML files for each of these models are provided directly on this page for simulation in the Morpheus GUI. For the complete computational pipeline, including Python analysis scripts, Jupyter notebooks, and the raw simulated cell trajectory data, please visit the GitLab repository.

Mathematical Model

We model the cell perimeter as a 1D membrane domain where active GTPase ($u$), inactive GTPase ($v$), and filamentous actin ($F$) evolve under reaction-diffusion dynamics:

$$ \frac{\partial u}{\partial t} = (b + \gamma u^2) v - (1 + u^2 + s F) u + D_u \frac{\partial^2 u}{\partial x^2} $$

$$ \frac{\partial v}{\partial t} = -(b + \gamma u^2) v + (1 + u^2 + s F) u + D_v \frac{\partial^2 v}{\partial x^2} $$

$$ \frac{\partial F}{\partial t} = \epsilon (p_0 + p_1 u - F) + D_F \frac{\partial^2 F}{\partial x^2} $$

Where:

  • $u$ represents the active form of a GTPase (e.g. RhoA or Rac) bound to the membrane.
  • $v$ represents the inactive cytosolic form.
  • $F$ represents the F-actin downstream effector.
  • $s$ is the F-actin-dependent GTPase inactivation rate (negative feedback rate).
  • $b$ and $\gamma$ represent basal and auto-activation rates respectively.
  • $\epsilon$, $p_0$, and $p_1$ govern F-actin assembly and disassembly rates.
  • $D_u$, $D_v$, and $D_F$ are the respective diffusion coefficients on the cell perimeter.

The local value of F-actin ($F$) along the cell boundary is coupled to the CPM, favoring local membrane protrusions where $F$ is high, driving cell deformation and motility.

Motility Phenotypes and Simulations

Note: All figures and simulation configurations on this page are reproduced or adapted from the original publication (Hughes et al., 2026).

1. Polar Cells (Directed Migration)

When parameter values lie on the stable unipolar branch, the cell maintains a single, stable peak of active GTPase. This concentrates F-actin at the cell front and drives persistent, directed migration.

  • Simulation files: polar_s_0.376.xml, polar_s_0.475.xml, polar_s_0.519.xml
  • Key behavior: As $s$ increases, the GTPase zone at the front narrows and active GTPase becomes more highly concentrated, increasing F-actin levels and propelling the cell faster (Figure 2).
**Figure 2.** Shapes and trajectories of polar cells on the unipolar branch ($L = 3\lambda$). As the negative feedback rate $s$ increases ($s = 0.376, 0.475, 0.519$), the active GTPase zone narrows, concentrating active protein at the front. This drives higher F-actin polymerization and faster cell speed, causing shapes to transition from canoe-like to D-like keratocyte forms.
Figure 2. Shapes and trajectories of polar cells on the unipolar branch ($L = 3\lambda$). As the negative feedback rate $s$ increases ($s = 0.376, 0.475, 0.519$), the active GTPase zone narrows, concentrating active protein at the front. This drives higher F-actin polymerization and faster cell speed, causing shapes to transition from canoe-like to D-like keratocyte forms.

2. Edge-Ruffling Cells (3 Traveling Waves)

Along the traveling wave branch, multiple peaks can propagate along the perimeter.

  • Simulation files: tw_s_0.413.xml, tw_s_0.475.xml, tw_s_0.607.xml
  • Key behavior: The cell perimeter supports three traveling waves, creating three competing protrusions. Because the forces pull in opposite directions, the cell undergoes no directed migration, displaying a ruffling phenotype.
**Figure 3.** Edge-ruffling (3TW) cell motility along the traveling wave branch ($L = 3\lambda$). For parameter values $s = 0.413, 0.475, 0.607$, the cell boundary supports three competing traveling waves. The resulting protrusions pull in opposite directions, preventing directed migration and yielding a stationary ruffling phenotype.
Figure 3. Edge-ruffling (3TW) cell motility along the traveling wave branch ($L = 3\lambda$). For parameter values $s = 0.413, 0.475, 0.607$, the cell boundary supports three competing traveling waves. The resulting protrusions pull in opposite directions, preventing directed migration and yielding a stationary ruffling phenotype.

3. Destabilized Polar Cells (Wobbly Motion)

Just after the onset of polar pattern instability, the spatial pattern of GTPase oscillates periodically.

  • Simulation file: Wobby_s_0.600.xml (s = 0.54)
  • Key behavior: The active GTPase zone remains localized in one direction, maintaining directed motion, but oscillates in shape and concentration. This causes the cell shape to shift periodically between oval, D-like, and kidney shapes, and its migration speed fluctuates (Figure 4).
**Figure 4.** Destabilized polar cells exhibiting wobbly motion at $s = 0.54$, after the onset of polar pattern instability. While the active GTPase zone remains localized in one direction to maintain directed migration, the shape oscillates periodically between D-like, oval, and kidney-like forms, accompanied by speed oscillations.
Figure 4. Destabilized polar cells exhibiting wobbly motion at $s = 0.54$, after the onset of polar pattern instability. While the active GTPase zone remains localized in one direction to maintain directed migration, the shape oscillates periodically between D-like, oval, and kidney-like forms, accompanied by speed oscillations.

4. Coexisting Motility Modes & Exotic Trajectories

For smaller cells ($L = 2\lambda$), multiple modes can coexist and interact dynamically.

  • Simulation files: CW_s_0.600.xml, CW_s_0.8_noise.xml
  • Key behavior: At $s = 0.6$, the cell supports counter-propagating waves, moving back and forth (Figure 5). At $s = 0.8$, adding noise can disrupt one of the counter-propagating peaks, leaving a single traveling wave that leads to small circular turning paths or reversal of motion (Figure 6).
**Figure 5.** Coexisting motility modes in smaller cells ($s = 0.6$, $L = 2\lambda$). The cell is initialized in a polar state, but the distribution destabilizes into a pair of counter-propagating traveling waves. These waves drive back-and-forth cell movement as they propagate and periodically split/merge along the rim.
Figure 5. Coexisting motility modes in smaller cells ($s = 0.6$, $L = 2\lambda$). The cell is initialized in a polar state, but the distribution destabilizes into a pair of counter-propagating traveling waves. These waves drive back-and-forth cell movement as they propagate and periodically split/merge along the rim.
**Figure 6.** Exotic cell trajectories under temporary noise ($s = 0.8$, $L = 2\lambda$). Following application of spatial noise (red stripe), the counter-propagating waves transition to a single-peak traveling wave (top, leading to circular turning trajectories) or experience an axis shift/reversal of back-and-forth motion (bottom).
Figure 6. Exotic cell trajectories under temporary noise ($s = 0.8$, $L = 2\lambda$). Following application of spatial noise (red stripe), the counter-propagating waves transition to a single-peak traveling wave (top, leading to circular turning trajectories) or experience an axis shift/reversal of back-and-forth motion (bottom).

5. Transitions under Time-Dependent Feedback

By sweeping the negative feedback parameter $s(t)$ dynamically in time, cells can be driven across parity-breaking bifurcations.

  • Simulation files: POLAR2TW_main.xml (directed to turning), TW2POLAR.xml (turning to directed)
  • Key behavior: A linear increase in $s$ causes a polar cell to transition into a single traveling wave and start turning (Figure 7). A linear decrease in $s$ brings the cell back from turning to directed migration (Figure 8).
**Figure 7.** Transition from directed cell motion to cell turning. As the feedback rate $s(t)$ grows linearly from $0.475$ to $0.6$ (sweeping across the parity-breaking bifurcation), the polar distribution shifts into a single traveling wave, causing the cell to veer into a circular path.
Figure 7. Transition from directed cell motion to cell turning. As the feedback rate $s(t)$ grows linearly from $0.475$ to $0.6$ (sweeping across the parity-breaking bifurcation), the polar distribution shifts into a single traveling wave, causing the cell to veer into a circular path.
**Figure 8.** Transition from cell turning to directed cell motion. In the reverse of the sweep, decreasing the feedback rate $s(t)$ from $0.6$ back to $0.475$ slows the traveling wave to zero velocity, allowing a polar distribution to re-emerge and driving the cell along a straight, polarized path.
Figure 8. Transition from cell turning to directed cell motion. In the reverse of the sweep, decreasing the feedback rate $s(t)$ from $0.6$ back to $0.475$ slows the traveling wave to zero velocity, allowing a polar distribution to re-emerge and driving the cell along a straight, polarized path.

6. Random Motion

  • Simulation file: Random_motion.xml
  • Key behavior: By allowing $s(t)$ to undergo a random walk (Wiener process with reflective boundaries in the range $[0.475, 0.6]$), the cell transitions randomly between polar, counter-propagating peaks, and traveling waves, generating irregular trajectories.
**Figure 9.** Cell motility under a randomly changing feedback parameter $s(t)$ executing a random walk in the range $[0.475, 0.6]$. The cell exhibits random switching between directed polar migration, counter-propagating peaks, and traveling waves.
Figure 9. Cell motility under a randomly changing feedback parameter $s(t)$ executing a random walk in the range $[0.475, 0.6]$. The cell exhibits random switching between directed polar migration, counter-propagating peaks, and traveling waves.

7. Continual Space-Time Noise

  • Simulation file: Dicty.xml
  • Key behavior: In larger cells ($L = 4\lambda$) with $s = 0.8$, continual space-time noise disrupts regular traveling waves. This leads to competing pseudopods where one randomly dominates the other, producing amoeboid-like motility trajectories that resemble Dictyostelium discoideum.
**Figure 10.** Cell shapes and pseudopod competition under continual space-time distributed noise ($s = 0.8$, $L = 4\lambda$). The noise disrupts regular counter-propagating peaks, causing alternating pseudopod dominance and random turning trajectories that resemble the amoeboid motility phenotype of *Dictyostelium*.
Figure 10. Cell shapes and pseudopod competition under continual space-time distributed noise ($s = 0.8$, $L = 4\lambda$). The noise disrupts regular counter-propagating peaks, causing alternating pseudopod dominance and random turning trajectories that resemble the amoeboid motility phenotype of Dictyostelium.

Reference

This model is the original used in the publication, up to technical updates:

J. M. Hughes, J. Algorta, L. Edelstein-Keshet: How negative feedback from filamentous actin affects cell shapes and motility. arXiv 2602.08779, 2026.

The model originates from a preprint and thus has not been formally peer-reviewed.

Model

Get this model via:

  • Morpheus-Link or
  •  Download: POLAR2TW_main.xml

    • XML Preview 

    <MorpheusModel version="4">
    <Description>
        <Title>M2074 Transition from Directed Cell Motion to Cell Turning</Title>
        <Details>Model ID:	https://identifiers.org/morpheus/M2074
  File type:	Main model
  Software:	Morpheus (open source). Download from: https://morpheus.gitlab.io
Full title:	How Negative Feedback from Filamentous Actin Affects Cell Shapes and Motility
Authors:	J. M. Hughes, J. Algorta, L. Edelstein-Keshet
Submitter:	J. Algorta
Curator:	D. Jahn
Date:	07.06.2026
Reference:	This model is the original used in the publication, up to technical updates:
	J. M. Hughes, J. Algorta, L. Edelstein-Keshet: How negative feedback from filamentous actin affects cell shapes and motility. arXiv 2602.08779, 2026.
	https://doi.org/10.48550/arXiv.2602.08779
Comment:	This model simulates the transition from directed cell motion to cell turning due to a linearly increasing negative feedback rate s(t), simulated in Morpheus for Figure 12 of the publication. PDE model for F-actin negative feedback to GTPases, and GTPase promotion of F-actin. The reaction-diffusion PDEs are implemented on the cell edge. u = active GTPase (Rac), v = inactive GTPase, F = F-actin.</Details>
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    Model Graph
    Model Graph

    Model Files Included

    Main Model Configurations

    • POLAR2TW_main.xml: Transition from directed motion to turning (increasing $s(t)$, Figure 12)
    • TW2POLAR.xml: Transition from turning to directed motion (decreasing $s(t)$, Figure 13)
    • CW_s_0.600.xml: Coexisting back-and-forth motility ($s = 0.600, L = 2\lambda$, Figure 10)
    • CW_s_0.8_noise.xml: Exotic trajectories under noise ($s = 0.800, L = 2\lambda$, Figure 11)
    • Wobby_s_0.600.xml: Wobbly shape and speed oscillations ($s = 0.540$, Figure 9)
    • Random_motion.xml: Random motion under fluctuating parameter $s(t)$ (Figure 14)
    • Dicty.xml: Amoeboid-like motility with continual noise ($s = 0.800, L = 4\lambda$, Figure 16)

    Polar Cell Motility Branch (Figure 7)

    • polar_s_0.376.xml: Polar cell state at $s = 0.376$
    • polar_s_0.475.xml: Polar cell state at $s = 0.475$
    • polar_s_0.519.xml: Polar cell state at $s = 0.519$

    Traveling Wave Ruffling Cell Branch (Figure 8)

    • tw_s_0.413.xml: Edge-ruffling cell state at $s = 0.413$
    • tw_s_0.475.xml: Edge-ruffling cell state at $s = 0.475$
    • tw_s_0.607.xml: Edge-ruffling cell state at $s = 0.607$

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    Files associated with this model:

    1D Membrane Actin Waves Bifurcation Analysis Cell Culture Cell Deformation Cell Motility Cell Perimeter Cell Polarity Cell Trajectory Cell Turning Cellular Potts Model CPM Destabilized Polar Cell Dicty Dictyostelium Dictyostelium Discoideum Directed Motion Edge-ruffling F-actin Feedback Mechanism GTPase Inhibitor PDE Polarization Published Models Random Motility Reaction-Diffusion Reaction-Diffusion Dynamics Reaction-diffusion System Ruffling Traveling Waves Turning Wave-Pinning Wobbly Motion Wobbly Oscillation
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    Last updated on Wed, 22 Jul 2026

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