Ligand Binding and the Scatchard Plot

Question. A protein has 4 binding sites. At a given ligand concentration, 2 ligands are bound per protein molecule, on average. Find $r$, $n$, and $\Theta$.

Solution.

1. $n = 4$ (given: total binding sites per protein molecule).
2. $r = 2$ (given: average ligands bound per protein molecule).
3. $\Theta = r/n = 2/4 = 0.5$.

Answer: $\Theta = 0.5$: half the protein's binding sites are occupied, on average.

Question. An antigen (Ag) and a single-chain antibody (Ab) are studied by Scatchard analysis: bound/free ($[\text{Ab-Ag}]/[\text{Ag}]$) is plotted on the y-axis against bound ($[\text{Ab-Ag}]$) on the x-axis, giving a straight line. Which features of the line give the binding affinity, and the total antibody concentration?

Solution.

1. Write the Scatchard equation for this system: $\dfrac{[\text{Ab-Ag}]}{[\text{Ag}]} = -K_a[\text{Ab-Ag}] + K_a \cdot [\text{total binding sites}]$.
2. This is $y = mx + c$ with slope $m = -K_a$, so the slope gives the binding affinity ($K_a$).
3. Set $y = 0$ to find the x-intercept: $0 = -K_a[\text{Ab-Ag}] + K_a[\text{total binding sites}]$, so $[\text{Ab-Ag}]_{x\text{-intercept}} = [\text{total binding sites}]$.
4. A single-chain antibody has exactly one binding site per antibody molecule ($n = 1$), so total binding sites equals total antibody concentration.

Answer: Slope (affinity) and x-intercept (total antibody concentration).