Receptor Pharmacology & Bioassay
Arunlakshana-Schild Dose Ratio & pA2 Affinity Formula
Calculate Competitive Antagonist Dose Ratio DR = EC50_antag / EC50_ctrl, Schild Ordinate LOG10(DR - 1), Dissociation Constant K_B = [B] / (DR - 1), and negative log affinity pA_2 = -LOG10(K_B).
Quantify competitive receptor antagonism from parallel EC50 rightward shifts!
When a reversible competitive antagonist at molar concentration [B] competes with an agonist for the same orthosteric receptor binding site without depressing maximal response E_max, Arunlakshana and Heinz Otto Schild (1959, British Journal of Pharmacology) proved that the Agonist Equieffective Dose Ratio DR = EC50_shifted / EC50_control obeys the linear Schild Equation: DR - 1 = [B] / K_B, or in log-log Schild-plot coordinates LOG10(DR - 1) = LOG10([B]) - LOG10(K_B)! Defining pA_2 = -LOG10(K_B) gives the negative base-10 logarithm of the molar antagonist concentration required to force a 2-fold rightward shift (DR = 2, so DR - 1 = 1) in the agonist concentration-response curve!
Verify on Control_Agonist_EC50 = 10.0 nM (B2), Shifted_Agonist_EC50 = 1,010.0 nM (B3), and Antagonist_Molar_Conc [B] = 1e-7 M (100 nM, B4): Agonist Dose Ratio is DR = 1,010 / 10 = 101.00 (E2); Schild Plot Ordinate LOG10(101 - 1) = LOG10(100) = 2.0000 (E3); Antagonist Dissociation Constant in nanomolar is K_B = 1e9 * (1e-7 / 100) = 1.00 nM (E4); and Schild Affinity Index is pA_2 = -LOG10(1e-9) = 9.0000 (E5—confirming that at [B] = 1e-9 M = 1 nM, DR = 1 + 1/1 = 2.0x exact)! Test Shifted_EC50 = 20.0 nM at [B] = 1e-9 M: DR = 2.00, LOG10(DR - 1) = 0.0000, and pA_2 remains an invariant 9.0000!
| Schild Receptor Bioassay Metric (EC50: 10 nM -> 1,010 nM @ [B]=100 nM) | Formula | Verified Output |
|---|---|---|
| Agonist Dose Ratio (DR) | EC50_shifted / EC50_control | 101.0000x |
| Schild Ordinate LOG10(DR - 1) | LOG10(DR - 1) | 2.0000 |
| Antagonist K_B (nanomolar) | 10^9 * [B] / (DR - 1) | 1.00 nM (1.0e-9 M) |
| Schild Affinity Constant pA_2 | -LOG10([B] / (DR - 1)) | 9.0000 |
Schild Dissociation Constant K_B & pA_2 Affinity Formula
Enforce EC50_shifted > EC50_control (DR > 1): if EC50_shifted <= EC50_control, no antagonism occurred (allosteric potentiation or experimental noise) and LOG10(DR - 1) is undefined!
=LET(ec_0, B2, ec_b, B3, cn_b, B4, IF(OR(ec_0 <= 0, ec_b <= ec_0, cn_b <= 0), "Invalid Schild Shift (DR <= 1)", LET(dr_v, ec_b / ec_0, kb_m, cn_b / (dr_v - 1), ROUND(-LOG10(kb_m), 4))))Nanomolar K_B Conversion Formula
Convert molar K_B to nanomolar via 10^9 * K_B so assay pharmacologists can read nanomolar potency alongside dimensionless pA_2!
=IF(ISTEXT(E5), E5, ROUND(1000000000 * (B4 / (B3 / B2 - 1)), 4))Try the example
Use the original values or change them to test your understanding.