Quantitative Receptor Pharmacology
Gaddum Competitive vs Non-Competitive Occupancy Formula
Compare J.H. Gaddum’s (1937) Competitive Orthosteric Occupancy [A] / ([A] + K_A*(1 + [B]/K_B)) against Non-Competitive Allosteric Insurmountable Occupancy ([A] / ([A] + K_A)) / (1 + [B]/K_B) and Ceiling 1 / (1 + [B]/K_B).
See why saturating agonist overcomes competitive blockers yet hits an insurmountable non-competitive ceiling!
Sir John Henry Gaddum (1937, Proceedings of the Royal Society) formulated the foundational mass-action equation for two ligands competing for a receptor pool! When Agonist [A] (dissociation constant K_A) faces an Orthosteric Competitive Antagonist [B] (dissociation constant K_B), fractional active receptor occupancy is rho_comp = [A] / ([A] + K_A * (1 + [B] / K_B)): flooding the tissue with high [A] >> K_A * (1 + [B] / K_B) still drives rho_comp -> 1.0000 ("Surmountable" parallel shift)! By contrast, when [B] binds an independent Non-Competitive / Allosteric Inactivating Site regardless of whether [A] is bound, active occupancy collapses to rho_noncomp = ([A] / ([A] + K_A)) / (1 + [B] / K_B), imposing an Insurmountable Ceiling rho_max = 1 / (1 + [B] / K_B) < 1 no matter how high [A] climbs!
Verify on Agonist_Conc [A] = 100.0 nM (B2), Agonist_KA = 25.0 nM (B3 -> Unblocked Control Occupancy = 100 / 125 = 0.8000 in E2), Antagonist_Conc [B] = 30.0 nM (B4), and Antagonist_KB = 10.0 nM (B5 -> [B]/K_B = 3.0): Competitive Occupancy is 100 / (100 + 25 * 4) = 0.5000 (50.00% in E3); Non-Competitive Occupancy drops to 0.8000 / 4.0 = 0.2000 (20.00% in E4); and Non-Competitive Insurmountable Ceiling is 1 / (1 + 3.0) = 0.2500 (25.00% in E5)! Now flood Agonist [A] = 99,900 nM in cell B2: Competitive Occupancy rebounds to 0.9990 (99.90%), whereas Non-Competitive Occupancy remains trapped at 0.2499 (<= 25.00% ceiling)!
| Gaddum Receptor Metric ([A]=100 nM, KA=25 nM, [B]=30 nM, KB=10 nM) | Formula | Verified Output |
|---|---|---|
| Control Unblocked Occupancy | [A] / ([A] + K_A) | 0.8000 (80.00%) |
| Gaddum Competitive Occupancy | [A] / ([A] + K_A * (1 + [B] / K_B)) | 0.5000 (0.9990 @ 99.9 uM) |
| Non-Competitive Occupancy | ([A] / ([A] + K_A)) / (1 + [B] / K_B) | 0.2000 (0.2499 @ 99.9 uM) |
| Insurmountable E_max Ceiling | 1 / (1 + [B] / K_B) | 0.2500 (25.00% Hard Cap) |
Gaddum Competitive vs Non-Competitive Occupancy Formula
Validate [A] >= 0, K_A > 0, [B] >= 0, and K_B > 0!
=LET(ag_a, B2, ka_d, B3, an_b, B4, kb_d, B5, IF(OR(ag_a < 0, ka_d <= 0, an_b < 0, kb_d <= 0), "Invalid Ligand Input", ROUND(ag_a / (ag_a + ka_d * (1 + an_b / kb_d)), 4)))Insurmountable Non-Competitive Occupancy & Ceiling Formula
Comparing E3 and E4 at escalating agonist doses is the classic clinical bioassay test distinguishing reversible orthosteric blockers from irreversible/allosteric receptor inactivators!
=IF(ISTEXT(E3), E3, ROUND((B2 / (B2 + B3)) / (1 + B4 / B5), 4))Try the example
Use the original values or change them to test your understanding.