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

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.