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

TL;DR

The paper develops a mass-conserved three-variable RD model for active/inactive GTPase ($u,v$) and F-actin ($F$) on a 1D cell-edge domain to study transitions between wave-pinning and traveling waves. Through linear and nonlinear bifurcation analyses, it reveals how codimension-2 LW/WB instabilities organize a rich repertoire of patterns, including WP, TW, EPs, and LWO, with domain size and total mass $M$ controlling which states coexist or compete. Amplitude-equation reductions around the LW/WB point quantify the onset and relative amplitudes of TWs and SWs, while numerical continuation and time simulations map out the nonlinear regime across large and moderate domains. The results provide mechanistic insights into how mass conservation shapes actin-wave patterns and cell motility modes, highlighting robust WP–TW coexistence and domain-length–dependent transitions, with broader implications for mass-conserving RD systems and potential higher-codimension instabilities.

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.

Figures (25)

• Figure 1: Schematic representation of cells and corresponding F-actin dynamics at the cell edge. Top row: Four cell behaviours showing typical cell shapes and dynamics, together with F-actin distribution along the cell edge; light (dark) colour indicates high (low) F-actin concentration. Bottom row: Corresponding numerical solutions on a one-dimensional (1D) domain with periodic boundary conditions. (a) Stable high uniform F-actin distribution depicting an unpolarized ("resting’’) cell. (b) Polar distribution associated with directed cell migration. (c) Travelling wave (TW) with a single wavelength in the periodic domain. This TW would lead the cell to crawl/turn in a circular arc. (d) Travelling wave with three wavelengths, resulting in a "ruffling" behaviour, with three protrusions circulating around its edge.
• Figure 1: The linear onsets of bifurcations along homogeneous steady states (HSSs) are shown as curves in the $(s,b)$ parameter plane, where $s$ is the strength of F-actin negative feedback and $b$ is the GTPase basal activation rate. The bifurcation onsets are computed using \ref{['eq:disper eigen']}, where LW - long wavelength (green), WB - finite wavenumber Hopf (blue), and HB - homogeneous Hopf (red). (a) Low vs (b) high total GTPase concentration ($M$) differ since in (a) LW occurs before HB while in (b) it is reversed (note that in (b) we use a logarithmic dependence of $b$). Three HSSs coexist in the entire yellow region. The region is bounded by saddle-nodes SN that end at a cusp bifurcation at (a) $(s,b)\approx(0.64,0.092)$ and (b) $(s,b)\approx(2.5,0.11)$. 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. The horizontal dotted line in (a) at $b=b_c\approx0.067$, corresponds to a one-parameter slice shown in \ref{['fig:2par bif 1par disper']}, where a codimension-2 LW/WB instability occurs. See \ref{['fig:2 par appendix i', 'fig:2 par appendix ii', 'fig:2 par appendix iii', 'fig:2 par appendix v']} for more one-parameter slices of \ref{['fig:2par bif HSS']}a. Other parameter values as in \ref{['tab:par values']}.
• Figure 1: (a) Bifurcating branches of the primary travelling and standing waves (with wavelength $\lambda\approx3.09$) obtained at the codimension-2 onset at $s=s_{WB}=s_c^t\approx 0.409$ and $b=b_c\approx 0.067$ (see the slice in \ref{['fig:2par bif HSS M2']}), hereafter $\text{TW}_{\lambda}$ and $\text{SW}_{\lambda}$, respectively. The solutions are projected onto the maximum F-actin concentration, $F_{\max}$. We compare results from the amplitude equations \ref{['eq:F-field']} and \ref{['eq:rel_KnoblochAmps']} to numerical continuation, over spatial wavelength $\lambda$ and, in the case of SW, also over the temporal period $T$ (also obtained at the onset): Linearly stable TWs (from continuation: solid blue; from amplitude equations: dashed-dotted blue) and linearly unstable SWs (from continuation: dashed orange; from amplitude equations: dotted orange). The inset shows the difference between the total amplitudes of the TWs and SWs according to \ref{['eq:rel_KnoblochAmps']} ($\Delta\tilde{\rho}={\widetilde{\rho}}_{TW}-{\widetilde{\rho}}_{SW}$), indicating that TWs are larger than SWs. (b) Same as (a) but with TWs being projected onto the phase wave speed $c$ (i.e., the initial propagation speed, left vertical axis) and SWs onto the time-period $T$ (right vertical axis), see \ref{['eq:c_T_amp_eq']} for the corresponding expressions from the amplitude equations. Coefficients for the amplitude equations are obtained numerically using Mathematica: $\alpha = 2.13528 - 1.66874i$, $\gamma = 0.06019 + 0.01434i$, $\eta = 0.06500 + 0.16646i$, $\widehat{a} \approx -0.00482-0.15211 i$, and $\widehat{b} \approx -0.06019 -0.01434i$. Panels (c,d) provide the same comparison but at $b=0.25$, where the finite wavenumber Hopf onset occurring at $s=s_{WB}\approx 0.522$ with a critical wavelength $\lambda \approx3.13$, and respective coefficients: $\alpha = 1.26708 - 0.46261i$, $\gamma = 0.14570 - 0.12758i$, $\eta = 0.38413 - 0.11440i$, $\widehat{a} \approx -0.23843 -0.01318i$, and $\widehat{b} \approx -0.14570 + 0.12758i$. Other parameters as in \ref{['tab:par values']} with $M=2$.
• Figure 1: Bifurcation diagrams computed via numerical continuation as a function of $s$, showing excitable pulses (EPs), travelling fronts (TFs), wave-pinning (WP) solutions, homogeneous steady states $Q^\ast_{0,1,2}$, and the long wavelength oscillatory (LWO) solutions that emerge from the homogeneous Hopf bifurcation (HB) for $M=4.5$ and $L=1000\gg \lambda\approx{1.78}$ (where $\lambda$ is the wavelength leading to the instability of $Q_2^\ast$ in this regime). The solution branches are projected with the Sobolev norm \ref{['eq:Sobolev norm']} in (a) and the propagation speed $c$ in (b); solid (dashed) lines denote linearly stable (unstable) solutions. The inset in (a) zooms into the stable region of EPs while the insets in (b) zoom into the T-point regime. The critical values $(s_{WB},s_P,s_{HB},s_T,s_{LW}^t)=({1.023},1.167,1.781,1.804,2.044)$ represent the finite wavenumber Hopf (WB) instability of $Q_2^\ast$, the parity-breaking bifurcation (black triangle) of stationary pulses to EPs, the homogeneous Hopf (HB) bifurcation onset, the T-point (red square), and the top ($Q_2^\ast$) long wavelength (LW) bifurcation, respectively. After the T-point, $s=s_T$, EPs bi-asymptotic to $Q_2^\ast$ become bi-asymptotic to $Q^\ast_1$, and then LWO solutions after an additional homoclinic bifurcation (for details, see Yochelis2022) with a speed of $c\approx 410$ at $s=s_{HB}$ but is cropped in (b). The symbols (circle, diamond, etc.) represent selected locations of solution profiles given in \ref{['fig:EP M45 solutions']}. Other parameter values as in \ref{['tab:par values']} with $b=0.00067$.
• Figure 1: Direct numerical integration of \ref{['eq:model']} with periodic boundary conditions, initialized with a convex combination \ref{['eq:convex']} of wave-pinning and travelling waves (left panels). Spacetime plots ("kymographs’’, right panels) showing $F$ as a heat map for (a) $\delta=0.25$, (b) $\delta=0.5$, and (c) $\delta=0.75$. In (b), we demonstrate a distinct kind of modulated solution that persists also at a much longer simulation time, $t=50000$; see also a supplementary movie_Fig14b. Parameter values as in \ref{['tab:par values']} with $M=2, b=b_c\approx0.067$, $s\approx0.50$, and $L=3\lambda\approx3\cdot3.09$.