The Sigmoidal Relationship
Nearly every drug follows a sigmoidal (S-shaped) dose-response curve when plotted on a log-concentration axis. At low concentrations, receptors are mostly unoccupied and the effect is minimal. As concentration rises through the EC50 region, receptor occupancy — and therefore effect — increases rapidly. At high concentrations, receptors saturate and the response plateaus at Emax. This universal pattern reflects the law of mass action governing drug-receptor binding.
The Hill Equation
Archibald Hill originally derived his equation in 1910 to describe oxygen binding to hemoglobin. It has since become the standard model for dose-response relationships across pharmacology: E = Emax × C^n / (EC50^n + C^n). The Hill coefficient n captures cooperativity — the degree to which binding of one molecule influences binding of the next. For most single-receptor drugs, n is close to 1.
Potency vs. Efficacy
Two drugs can have identical dose-response curve shapes but differ in their horizontal position (potency, EC50) or vertical height (efficacy, Emax). A partial agonist has lower Emax than a full agonist regardless of dose. This distinction is clinically critical: buprenorphine is a partial opioid agonist with a ceiling effect that limits overdose risk, while fentanyl is a full agonist with no such ceiling.
From Bench to Bedside
Dose-response curves guide every stage of drug development. In preclinical screens, EC50 values rank compound potency. In clinical trials, they determine starting doses and dose-escalation schedules. In clinical practice, understanding the dose-response relationship helps physicians balance therapeutic effect against toxicity — the fundamental challenge of pharmacotherapy.