Solve the coupled pheromone–Bar1 reaction–diffusion PDE system

Determine the solution to the coupled partial differential equations governing the spatiotemporal dynamics of α-factor pheromone concentration α(x,y,z,t) and Bar1 protease concentration B(x,y,z,t), where α evolves with diffusion, autodegradation, a source term, and a reaction term proportional to B (modeling Bar1-mediated pheromone degradation), and B evolves with diffusion, autodegradation, and a source term. Establish this solution under physically relevant initial and boundary conditions to enable end-to-end channel characterization without neglecting Bar1 effects.

Background

The paper models pheromone propagation using a reaction–diffusion equation that includes diffusion (Dα), autodegradation (kα), a source term (Sα), and a reaction term −k_re B α representing degradation by the Bar1 protease. Bar1 itself diffuses with its own reaction–diffusion dynamics characterized by diffusion (DB), autodegradation (kB), and a source term (SB). These equations are coupled through the reaction term involving both α and B.

Because the coupled PDE system is analytically challenging, the authors simplify by setting B=0, thereby removing the reaction term and enabling a tractable channel impulse response via Green’s function methods. They explicitly state that solving the full coupled system with Bar1 is an open problem, which is essential for a more accurate end-to-end model of yeast molecular communication that captures Bar1-mediated pheromone degradation.