How negative feedback from filamentous actin affects cell shapes and motility

TL;DR

The paper investigates how negative feedback from F-actin onto a GTPase-driven signaling module shapes cell shapes and motility by coupling a mass-conserved uvF reaction-diffusion system to a Cellular Potts model. Through linear and nonlinear bifurcation analyses of the PDEs and Morpheus-based edge simulations, it reveals coexisting polar and traveling-wave states, transitions between directed motion, turning, and ruffling, and rich dynamics such as counter-propagating waves in smaller cells. Key findings show that the F-actin–mediated inactivation strength $s$ and cell perimeter (domain length) $L$ control when polar states persist, when traveling waves emerge via parity-breaking bifurcations, and how time-dependent $s(t)$ can switch modes. The work provides a mechanistic, parameter-driven framework linking intracellular actin signaling to emergent cell motility, using accessible open-source tools and offering testable predictions for real-cell behaviors across keratocytes and Dictyostelium-like systems.

Abstract

The crawling motility of many eukaryotic cells is driven by filamentous actin (F-actin), and regulated by a network of signaling proteins and lipids (including small GTPases). The tangle of positive and negative feedback loops gives rise to various experimentally observed dynamic patterns (``actin waves''). Here we consider a recent prototypical model for actin waves in which F-actin exerts negative feedback onto a GTPase. Guided by recent numerical PDE bifurcation analysis in Hughes (2025) and Hughes et al (2026), we explore cell shapes and motility associated with polar, oscillatory, and traveling waves solutions of a mass-conserved partial differential equation (PDE) model. We use Morpheus (cellular Potts) simulations to investigate the implications of such regimes of behavior on the shapes and motion of cells, and on transitions between modes of behavior. The model demonstrates various cell states, including resting (spatially uniform GTPase), polar cells (static ``zones'' of GTPase), and traveling waves along the cell edge. In some parameter regimes, such states can coexist, so that cells can transition from one behavior to another in response to noisy stimuli.

Figures (19)

• Figure 1: Schematic diagram of models: (a) Typical cycling between active (red) and inactive (green) Rho GTPase with positive feedback from the active form to its own activation. The model by Mori2008 for polarization ("wave-pinning", WP) was based on this circuit. (b) A downstream effector (such as F-actin) is promoted by the GTPase, and then exerts negative feedback (enhances the inactivation of Rho). Arrow colors represent GAP (red), GEF (green), downstream target (purple), positive (solid black) and negative (gray) feedback. The dot-dashed gray arrow in (b) was proposed as a simplification of the saturating kinetics used in holmes2012. (c) The distribution of active GTPase on a thin sheet-like cell has been simulated elsewhere in 2D (e.g., in maree2012cells). Here, our geometry is a periodic 1D domain along the cell edge (white dashed curve, and contour shown on the right).
• Figure 2: Two-parameter bifurcation diagram of uniform steady states: Parameters $b,s$ were used to produce the two-parameter plane shown above. In hughes2024travelling, two-parameter continuation was applied to \ref{['eq:model']} to trace the onsets of bifurcations of the homogeneous steady states (HSSs). The bifurcations shown are finite wavenumber Hopfs (WB), long wavelengths (LW), homogeneous Hopfs (HB), and saddle-nodes (SN). The black-striped subset of the yellow region corresponds to the bistability region, where two HSSs out of three are stable. In the solely yellow-shaded region, there is at most one stable HSS. Note that the curves only indicate onsets of bifurcations, not regions of existence of a given pattern. The dashed lines denote $b$-value slices corresponding to values used in the parameter sweeps of the model \ref{['eq:model']} given in \ref{['fig:sbKymoScan', 'fig:sbKymoScan 2L']}.
• Figure 3: Coexistence of polar and ruffling states in time-dependent simulations. Same idea as in \ref{['fig:TwoParBifHughesLiu']} but showing the full PDE solutions as the parameters $s,b$ are scanned with $L=3\lambda\approx9.28$, where $\lambda$ is the wavelength that induces the finite wavenumber Hopf instability of the upper HSS. Solutions are shown as space-time kymographs of the active GTPase, $u(x,t)$ as a heat map, for several values of the basal GTPase activation rate, $b$ (0, 0.067, 0.15, 0.3 on vertical axis) and various values of the negative feedback parameter $s$ (horizontal axis, $0.3\le s \le 0.9$). The system is initiated with $u(x,0)=0.75-0.5 \cos(2\pi x/L)$, $v(x,0)=1.25-0.1 \cos(2\pi x/L)$, and $F(x,t)=3.5 - 2 \cos(2\pi x/L)$, and simulated to $t=400$. For $160<t<240$, white noise is added to the $du/dt$ equation and subtracted from the $dv/dt$ equation (to avoid changing the total mass $M$). The noise results in several transitions between coexisting states such as polar, TWs, uniform steady states, and more complex time-periodic dynamics. If $s$ is too small or too large, only the uniform state exists. Profiles of $u, v, F$ at $t=400$ corresponding to these kymographs are shown in \ref{['fig:sbProfiles']}. Other parameter values as in \ref{['tab:par valuesSims']}.
• Figure 4: Final solution profiles from \ref{['fig:sbKymoScan']}. Line plots showing the $u,v,F$ profiles at $t=400$ for the simulations in \ref{['fig:sbKymoScan']}. We see uniform states for small and large values of $s$, traveling waves of 3, 2, or 1 wavelengths for intermediate values of $s$, and polar patterns for $s=0.4$ and small $b$. The inactive GTPase (green) is nearly uniform due to its relatively high rate of diffusion. The arrows indicate the direction of propagation. When $b=0$ and $s=0.7$, counter-propagating waves are observed and the arrows demonstrate the direction each wave travels.
• Figure 5: Coexistence of polar and ruffling states: Single parameter bifurcation diagram with respect to the negative feedback parameter, $s$, for $b=0.067$ and $L=3\lambda$, where $\lambda\approx3.09$ is the wavelength that leads to the instability of the upper HSS. The solutions are projected onto the maximum value of the active GTPase, $u_{\max}$. Solid lines denote linearly stable solutions (i.e., observable in the long run) and dashed lines denote unstable solutions. Uniform HSSs (solid black curves) represent an unpolarized "resting cells" with spatially uniform GTPase activity around its perimeter. Polarized states (PP$_{3\lambda}$) are stable along the green solid curve. Waves with 3 peaks (3TW$_\lambda$, "ruffling") are stable along the solid blue curve. As $s$ increases, transitions in behavior are predicted: from uniform to polar, from polar to ruffling, and from ruffling to low uniform stable state. However, not all possible solution types are illustrated. For example, traveling waves with 1 and 2 peaks and bipolar and tripolar patterns also exist in this regime (see hughes2024travelling and \ref{['fig:codim2 bif 3L full']} for more details). In the orange shaded region, both polar and ruffling states coexist, so transitions between them can take place. Typical solutions are shown in the $b=0.067$ row of \ref{['fig:sbKymoScan']}. The shapes and motility behavior of cells corresponding to points along these curves are shown in \ref{['fig:Polar_Solutions', 'fig:WobblyCell', 'fig:3TW_Solutions']}.