The quantitative framework describing how drugs interact with molecular targets — receptor occupancy, the Emax concentration-effect model, EC50 (potency), Hill coefficient (sigmoidal steepness), and the spectrum from full agonists through partial agonists, antagonists, and inverse agonists.
The Emax (Sigmoid) Model
The Hill (sigmoid Emax) equation is the standard model relating drug concentration
to pharmacological effect in quantitative pharmacodynamics:
E = E0 + (Emax × Cn) / (EC50n + Cn)
Where: E0 = baseline effect without drug;
Emax = maximum drug-attributable effect;
EC50 = concentration at which E = Emax/2 (potency);
n = Hill coefficient (shape/steepness of the sigmoid);
C = drug concentration.
Sigmoid Emax: effect vs log concentration — EC50 at 50% of Emax
Agonist Classification by Intrinsic Efficacy
Full agonist (α = 1)
Produces Emax at full receptor occupancy. Examples: morphine (μ-opioid), salbutamol (β2), isoprenaline (β1/β2). Concentration-effect curve reaches the physiological maximum.
Partial agonist (0 < α < 1)
Produces submaximal response even at 100% occupancy. Ceiling effect. Clinical examples: buprenorphine (μ-opioid — ceiling on respiratory depression; key advantage in OUD treatment); buspirone (5-HT1A partial agonist); aripiprazole (D2 partial agonist).
Competitive antagonist (α = 0)
Binds without activating — shifts agonist concentration-effect curve rightward (↑ EC50) without reducing Emax. Reversible by excess agonist. Examples: naloxone (μ-opioid), propranolol (β), atropine (M).
Inverse agonist (α < 0)
Stabilises the inactive receptor conformation — produces the opposite effect to the endogenous agonist. Not the same as an antagonist. β-carboline inverse agonists at GABA-A produce anxiety/seizures; used as research tools. Some antihistamines may have partial inverse agonist activity.
The partial agonist concept is clinically important for
opioid use disorder treatment
(buprenorphine ceiling on respiratory depression) and
psychiatry (aripiprazole as D2 partial agonist — provides antipsychotic activity
without full D2 blockade-related EPS). Related:
pharmacodynamic tolerance (receptor desensitisation at signalling level).
Hill Coefficient and Potency vs Efficacy
Hill coefficient n = 1: Rectangular hyperbola (classic
Michaelis-Menten shape). Single-binding-site, no cooperativity.
Most drugs behave approximately this way in the clinical concentration range.
n > 1: Positive cooperativity — steeper sigmoid. Pharmacological
examples: haemoglobin O₂ binding (n ≈ 2.7); some ion channel-linked receptors.
Clinical consequence: smaller concentration changes produce larger effect changes.
n < 1: Negative cooperativity — flatter curve. In PK-PD
modelling, n < 1 may represent receptor heterogeneity.
Potency (EC50) is NOT the same as
efficacy (Emax). A more potent drug has a lower EC50
(achieves the same effect at lower concentrations). A more efficacious drug has a
higher Emax. Example: buprenorphine is less efficacious than morphine
at the μ-receptor (lower Emax) but more potent (lower EC50
for analgesia).
What is the receptor occupancy theory of drug action?
The receptor occupancy theory (Clark, 1926) proposes that drug effect is proportional to the fraction of receptors occupied — and that maximum effect (Emax) is achieved when all receptors are occupied. The drug-receptor interaction is reversible and follows the law of mass action. Occupancy is determined by the dissociation constant (Kd): at concentrations equal to Kd, 50% of receptors are occupied. This is the foundation of the Emax model in pharmacodynamics, where effect E = Emax × C / (EC50 + C), with EC50 equivalent to Kd under simple occupancy assumptions. The theory was later refined to include the concept of efficacy (intrinsic activity) — the ability of the drug to produce a response once bound — explaining partial agonism and inverse agonism.
What is the Emax model and what is EC50?
The Emax model describes the concentration-effect relationship for drugs acting on receptors: E = Emax × C^n / (EC50^n + C^n). Emax is the maximum possible drug effect (achievable when receptor occupancy is complete); EC50 is the concentration producing 50% of Emax — a measure of drug potency (lower EC50 = more potent); n is the Hill coefficient (sigmoidal steepness — n=1 gives a rectangular hyperbola; n>1 produces a steeper, more sigmoidal relationship; n<1 is less steep). The sigmoid Emax model (Hill equation) is the standard pharmacodynamic model in drug development for relating plasma concentration to clinical or biomarker effects. EC50 is conceptually related to Kd but differs because receptor reserve and downstream signalling amplification mean EC50 can be much lower than Kd for high-efficacy agonists.
What is the difference between a full agonist, partial agonist, antagonist, and inverse agonist?
Classification by intrinsic efficacy (α): Full agonist (α=1): produces Emax equivalent to the endogenous agonist at full receptor occupancy — e.g., morphine at mu-opioid receptor. Partial agonist (0 < α < 1): produces a submaximal response even at full receptor occupancy — e.g., buprenorphine at mu-opioid receptor (ceiling effect on respiratory depression, key safety advantage; ceiling on analgesia, clinically limiting). Antagonist (α=0): binds receptor without activating it; blocks endogenous agonist — competitive (reversible, shifts EC50 right without changing Emax) or non-competitive (irreversible, reduces Emax). Inverse agonist (α < 0): binds receptor and produces the opposite effect to the endogenous agonist by stabilising the inactive receptor conformation — e.g., some benzodiazepines are inverse agonists at GABA-A receptor subtypes (anxiogenic effect).
What is receptor reserve and why does it matter clinically?
Receptor reserve (spare receptors) is the concept that maximum biological response can be achieved when only a fraction of total receptors are occupied — because each occupied receptor activates multiple downstream signalling molecules (signal amplification). Practical consequences: (1) EC50 of a full agonist is much lower than its Kd — drug achieves Emax before saturating all receptors; (2) A partial agonist with lower intrinsic efficacy may not be able to produce Emax even with 100% receptor occupancy — its ceiling depends on the size of the receptor reserve; (3) Irreversible antagonists (e.g., phenoxybenzamine at α1 receptors) can bind and eliminate receptors until the reserve is exhausted, at which point subsequent agonist doses produce less than Emax — shifting from competitive-looking behaviour to non-competitive; (4) In desensitisation/downregulation, loss of spare receptors first manifests as a shift in EC50 before Emax is reduced.