Ligand Binding and the Scatchard Plot

Ligand binding equilibrium#

A ligand is any molecule or ion that binds reversibly to a protein. Ions, nucleic acids, and other proteins can all act as ligands.

$$[P] + [L] \underset{k_d}{\overset{k_a}{\rightleftharpoons}} [PL]$$

• $[P]$: free protein concentration, $\text{M}$
• $[L]$: free ligand concentration, $\text{M}$
• $[PL]$: protein-ligand complex concentration, $\text{M}$
• $k_a$: rate constant for complex formation (association), $\text{M}^{-1}\text{s}^{-1}$
• $k_d$: rate constant for complex breakdown (dissociation), $\text{s}^{-1}$

Association and dissociation constants#

$$K_a = \frac{[PL]}{[P][L]} \qquad\qquad K_d = \frac{[P][L]}{[PL]} = \frac{1}{K_a}$$

• $K_a$: association constant, $\text{M}^{-1}$
• $K_d$: dissociation constant, $\text{M}$
• $[P]$, $[L]$, $[PL]$: free protein, free ligand, and complex concentrations, $\text{M}$ each

A high $K_d$ means low affinity between $[P]$ and $[L]$: the complex falls apart easily. A high $K_a$ means high affinity.

Scatchard plot#

Scatchard plot is a graphical method for reading two numbers off a ligand-binding saturation experiment: the binding affinity ($K_a$ or $K_d$) and the total number of binding sites ($n$), both read from a straight line without curve-fitting.

[Image to come: protein with a single binding site, vs. a multi-site binder (e.g. an antibody) with two binding sites]

Fractional saturation#

• $\Theta$: fractional saturation, the fraction of a protein's total binding sites occupied by ligand at equilibrium, dimensionless (0 to 1). Some texts write this as $Y$ instead of $\Theta$.
• $r$: average number of ligand molecules bound per protein molecule, dimensionless (can exceed 1 for a multi-site protein).

$$\Theta = \frac{r}{n}$$

• $\Theta$: fractional saturation, dimensionless
• $r$: average ligand molecules bound per protein molecule, dimensionless
• $n$: total binding sites per protein molecule, dimensionless

Worked example: fractional saturation from binding-site count#

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.

The Scatchard equation#

$$\frac{\Theta}{[L]} = \frac{n}{K_d} - \frac{\Theta}{K_d}$$

• $\Theta$: fractional saturation, dimensionless
• $[L]$: free (unbound) ligand concentration, $\text{M}$
• $n$: total binding sites per protein molecule, dimensionless
• $K_d$: dissociation constant, $\text{M}$

The experimentally measured version replaces $\Theta$ with $r$ (bound ligand per protein) and uses $K_a$:

$$\frac{r}{[L]} = nK_a - rK_a = K_a(n - r)$$

• $r$: average ligand molecules bound per protein molecule, dimensionless
• $[L]$: free ligand concentration, $\text{M}$
• $n$: total binding sites per protein molecule, dimensionless
• $K_a$: association constant, $\text{M}^{-1}$

Plotting $r/[L]$ against $r$ gives a straight line, $y = mx + c$, with $y = r/[L]$, $x = r$, slope $m = -K_a$, and intercept $c = nK_a$.

[Image to come: Scatchard plot: y-axis $r/[L]$, x-axis $r$, straight line with y-intercept $nK_a$, x-intercept $n$, slope $-K_a$]

Reading the plot#

Feature	Value	Gives
Slope	$-K_a$	association constant $K_a$ (binding affinity)
Y-intercept	$nK_a$	combined with the slope, solves for $n$
X-intercept	$n$	total binding sites per protein molecule, directly

A steeper (more negative) slope means a higher $K_a$: tighter binding. The x-intercept reads off $n$ directly, without needing the y-intercept at all.

Worked example: antigen-antibody Scatchard analysis#

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).

Key points#

• $K_a = 1/K_d$: a high $K_d$ means low binding affinity; a high $K_a$ means high binding affinity.
• The Scatchard plot linearizes saturation binding data: plot $r/[L]$ (or $\Theta/[L]$) against $r$ (or $\Theta$) to read off affinity and binding-site count without curve-fitting.
• Slope $= -K_a$, y-intercept $= nK_a$, x-intercept $= n$.
• $\Theta = r/n$ is fractional saturation, always between 0 and 1; $r$ itself is not bounded by 1 for a multi-site protein.
• For a protein with a single binding site per molecule ($n = 1$), total binding-site concentration equals total protein concentration.

Common confusion#

$\Theta$ (fractional saturation) and $r$ (average ligand bound per protein) are often used interchangeably in speech but are not the same number: $r$ is an absolute count of bound ligand per protein molecule, while $\Theta = r/n$ normalizes that count to a 0–1 fraction of the available binding sites. They coincide only when $n = 1$.