📡 Cell Signaling Cascade — 2D Dose-Response

A ligand binds its receptor following a Hill-equation dose-response curve, then amplifies through Ras → Raf → MEK → ERK via Michaelis-Menten-style saturating kinetics, with ERK negative feedback onto the receptor.

Ligand concentration [L]: 50% Receptor affinity (1/Kd): 50% Amplification factor: 50% Feedback strength: 40%

Receptor—%

Ras-GTP—%

Raf*—%

MEK**—%

ERK**—%

Signal amp—×

Response—

Feedback—%

f(L) = Lⁿ / (Kdⁿ + Lⁿ), n = 2 │ d[X*]/dt = k_cat·f(upstream)·amp − k_off·[X*]

How this model works

Receptor binding follows a Hill equation: the fraction of occupied/active receptor rises with ligand concentration L and saturates once L passes the dissociation constant Kd (set by the affinity slider — higher affinity = lower Kd = a leftward-shifted, steeper curve). Ras, Raf, MEK and ERK each activate with their own Michaelis-Menten-style saturating production term driven by the tier below them, balanced against a constant deactivation (phosphatase) rate — the same production-vs-decay balance real kinase cascades run on. The amplification factor scales gain at every tier, mimicking scaffold proteins that boost local kinase concentration. Active ERK drives negative feedback that suppresses receptor signalling, producing the pulse-then-adapt response typical of growth-factor pathways.