Healthcare & Clinical Formulas
Scatchard Plot Linear Regression: Dissociation Constant Kd & Bmax
Extract equilibrium receptor-ligand dissociation constant K_d = -1 / SLOPE(B/F, B) (nM) and total receptor site density B_max = -INTERCEPT(B/F, B) / SLOPE(B/F, B) from radioligand or SPR saturation binding assays.
How George Scatchard (1949) turned hyperbolic receptor binding into a straight line!
Direct radioligand or surface-plasmon-resonance saturation curves Bound = B_max * [Free] / (K_d + [Free]) approach B_max asymptotically, making visual estimation of total receptor density treacherous! George Scatchard (Annals of the New York Academy of Sciences, 1949) rearranged mass-action equilibrium into the linear equation Bound / Free = -(1 / K_d) * Bound + (B_max / K_d): plotting B / F on the y-axis against Bound B on the x-axis yields slope = -1 / K_d and x-intercept = B_max!
Using Excel’s SLOPE(y_range, x_range) and INTERCEPT(y_range, x_range) on a 4-point saturation titration (Free F = [1.25, 5.0, 15.0, 45.0] nM, Bound B = [24, 60, 90, 108] fmol/mg, B/F = [19.2, 12.0, 6.0, 2.4]): SLOPE(C2:C5, B2:B5) = -0.2000 extracts affinity K_d = -1 / -0.2000 = 5.00 nM, while -INTERCEPT / SLOPE = 24.0 / 0.2000 recovers B_max = 120.0 fmol/mg!
| Free Ligand F (nM) | Specific Bound B (fmol/mg) | Bound / Free Ratio (B / F) | Scatchard Parameter Extracted |
|---|---|---|---|
| 1.25 nM -> 5.00 nM | 24.0 -> 60.0 (= 0.5 Bmax) | 19.20 -> 12.00 | SLOPE = -0.2000 -> Kd = 5.00 nM |
| 15.00 nM -> 45.00 nM | 90.0 -> 108.0 | 6.00 -> 2.40 | X-Intercept = Bmax = 120.0 fmol/mg |
| Non-Negative Slope (>= 0) | Positive Cooperativity / Artifact | "Invalid Scatchard Slope" | Single-Site Mass Action Requires Slope < 0 |
Equilibrium Dissociation Constant K_d = -1 / SLOPE(B/F, B) (nM)
Verifies a negative Scatchard slope across C2:C5 (B/F) vs B2:B5 (Bound B) and computes K_d = -1 / SLOPE.
=IF(SLOPE(C2:C5, B2:B5) >= 0, "Invalid Scatchard Slope", ROUND(-1 / SLOPE(C2:C5, B2:B5), 2))Maximum Specific Receptor Density B_max = -INTERCEPT / SLOPE (fmol/mg)
Finds the horizontal x-axis intercept where B / F extrapolates to 0 at infinite free ligand concentration.
=IF(SLOPE(C2:C5, B2:B5) >= 0, "Invalid Scatchard Slope", ROUND(-INTERCEPT(C2:C5, B2:B5) / SLOPE(C2:C5, B2:B5), 1))Try the example
Use the original values or change them to test your understanding.