Mathematical modeling of Erk activity waves in regenerating zebrafish scales

Abstract

Erk signaling regulates cellular decisions in many biological contexts. Recently, we have reported a series of Erk activity traveling waves that coordinate regeneration of osteoblast tissue in zebrafish scales. These waves originate from a central source region, propagate as expanding rings, and impart cell growth, thus controlling tissue morphogenesis. Here, we present a minimal reaction-diffusion model for Erk activity waves. The model considers three components: Erk, a diffusible Erk-activator, and an Erk-inhibitor. Erk stimulates both its activator and inhibitor, forming a positive and negative feedback loop, respectively. Our model shows that this system can be excitable and propagate Erk activity waves. Waves originate from a pulsatile source which is modeled by adding a localized basal production of the activator that switches the source region from an excitable to an oscillatory state. As Erk activity periodically rises in the source, it can trigger an excitable wave which travels across the entire tissue. Analysis of the model finds that positive feedback controls the properties of the traveling wavefront and that negative feedback controls the duration of Erk activity peak and the period of Erk activity waves. The geometrical properties of the waves facilitate constraints on the effective diffusivity of the activator, indicating that waves are an efficient mechanism to transfer growth factor signaling rapidly across a large tissue.Significance statementSignaling waves represent a possible mechanism of spatiotemporal organization of multicellular tissues. We have recently shown that waves of activity of the kinase Erk control osteoblast regeneration in adult zebrafish scales. Here, we present a detailed characterization of a mathematical model of these signaling waves. We show that a source region poised in an oscillatory state can broadcast traveling waves of Erk activity in the surrounding excitable tissue. The dynamics of the source control the number and frequency of waves. Geometrical arguments support the notion that excitable Erk waves are an effective mechanism to transport growth factor signaling across a large regenerating tissue.