The MAPK cascade amplifies signals through sequential phosphorylation. The Goldbeter-Koshland function (1984) describes a covalent modification cycle where kinase (E1) and phosphatase (E2) both operate near saturation (Km ≪ total substrate).
At zero-order (saturated enzymes), the response becomes ultrasensitive: the effective Hill coefficient n_H can exceed 2 even for simple Michaelis-Menten kinetics. With dual phosphorylation (MAPKK then MAPK, each requiring two steps), the cascade generates bistability — two stable steady states coexist for the same input.
Left: cascade diagram with real-time phosphorylation levels. Right: bifurcation diagram showing bistable region (S-shaped steady-state curve). The system hysteresis means cell fate decisions can be irreversible.