Traveling waves in coupled reaction-diffusion models with degenerate sources

Front and pulse solutions for a system of reaction-diffusion equations with degenerate source terms

Motivated by several biological models such as the SIS model from epidemiology and the Tuckwell-Miura model describing cortical spreading depression, we investigate the types of wave solutions that can exist for reaction-diffusion systems of two equations in which the reaction terms are degenerate in the sense that they are linearly dependent. In particular, we show that there are surprising differences between the types of waves that occur in a single reaction-diffusion equation and the types of waves that occur in a degenerate system of two equations. Importantly, and in contrast to previously published results, we demonstrate that nonstationary pulse solutions can exist for a degenerate system of two equations but cannot exist for a single reaction-diffusion equation. We show that this has important consequences for the minimal model that can generate the types of waves observed in cortical spreading depression. On the other hand, stationary fronts can exist for both single reaction-diffusion equations and degenerate systems. However, for degenerate systems, such solutions cannot be accessed when perturbing a uniform rest state with a localized perturbation unless the diffusion coefficients of the two species are equal. We also give an explicit condition on the source term in a degenerate reaction-diffusion system that guarantees the existence of nonstationary and stationary pulse and front solutions. We use this approach to provide several examples of reaction terms that have analytical pulse and front solutions. We also show that the case in which one species cannot diffuse is singular in the sense that the degenerate reaction-diffusion system can admit infinite families of stationary piecewise constant solutions. We further show how such solutions can be accessed by perturbing a constant rest state with a localized continuous disturbance.

ASYMPTOTIC AND BIFURCATION ANALYSIS OF WAVE-PINNING IN A REACTION-DIFFUSION MODEL FOR CELL POLARIZATION

We describe and analyze a bistable reaction-diffusion (RD) model for two interconverting chemical species that exhibits a phenomenon of wave-pinning: a wave of activation of one of the species is initiated at one end of the domain, moves into the domain, decelerates, and eventually stops inside the domain, forming a stationary front. The second ("inactive") species is depleted in this process. This behavior arises in a model for chemical polarization of a cell by Rho GTPases in response to stimulation. The initially spatially homogeneous concentration profile (representative of a resting cell) develops into an asymmetric stationary front profile (typical of a polarized cell). Wave-pinning here is based on three properties: (1) mass conservation in a finite domain, (2) nonlinear reaction kinetics allowing for multiple stable steady states, and (3) a sufficiently large difference in diffusion of the two species. Using matched asymptotic analysis, we explain the mathematical basis of wave-pinning, and predict the speed and pinned position of the wave. An analysis of the bifurcation of the pinned front solution reveals how the wave-pinning regime depends on parameters such as rates of diffusion and total mass of the species. We describe two ways in which the pinned solution can be lost depending on the details of the reaction kinetics: a saddle-node or a pitchfork bifurcation.

Wave-pinning and cell polarity from a bistable reaction-diffusion system

Motile eukaryotic cells polarize in response to external signals. Numerous mechanisms have been suggested to account for this symmetry breaking and for the ensuing robust polarization. Implicated in this process are various proteins that are recruited to the plasma membrane and segregate at an emergent front or back of the polarizing cell. Among these are PI3K, PTEN, and members of the Rho family GTPases such as Cdc42, Rac, and Rho. Many such proteins, including the Rho GTPases, cycle between active membrane-bound forms and inactive cytosolic forms. In previous work, we have shown that this property, together with appropriate crosstalk, endows a biochemical circuit (Cdc42, Rac, and Rho) with the property of inherent polarizability. Here we show that this property is present in an even simpler system comprised of a single active/inactive protein pair with positive feedback to its own activation. The simplicity of this minimal system also allows us to explain the mechanism using insights from mathematical analysis. The basic idea resides in a well-known property of reaction-diffusion systems with bistable kinetics, namely, propagation of fronts. However, it crucially depends on exchange between active and inactive forms of the chemicals with unequal rates of diffusion, and overall conservation to pin the waves into a stable polar distribution. We refer to these dynamics as wave-pinning and we show that this phenomenon is distinct from Turing-instability-generated pattern formation that occurs in reaction-diffusion systems that appear to be very similar. We explain the mathematical basis of the phenomenon, relate it to spatial segregation of Rho GTPases, and show how it can account for spatial amplification and maintenance of polarity, as well as sensitivity to new stimuli typical in polarization of eukaryotic cells.