Q. how to calculate \(K_d\)

Goal. Explain what \(K_{d}\) means and show step by step how to calculate it in the common situations.  Keep each mathematical step explicit.

Definition. \(K_{d}\) is the equilibrium dissociation constant for a simple bimolecular binding equilibrium between a protein P and a ligand L forming a complex PL.  Write the equilibrium verbally as P plus L at equilibrium with PL.  The equilibrium expression is

\[ K_{d} \;=\; \frac{[\mathrm{P}]\,[\mathrm{L}]}{[\mathrm{PL}]} \]

Interpretation. \(K_{d}\) has units of concentration.  A smaller \(K_{d}\) means tighter binding.  If you know the kinetic rate constants for association and dissociation then

\[ K_{d} \;=\; \frac{k_{\mathrm{off}}}{k_{\mathrm{on}}} \]

Method 1. Calculate \(K_{d}\) from measured equilibrium concentrations.  Step by step procedure.

Step 1. Measure the equilibrium concentrations of free protein, free ligand, and complex.  Denote these as \( [\mathrm{P}] \), \( [\mathrm{L}] \), and \( [\mathrm{PL}] \) respectively.

Step 2. Plug the measured values into the equilibrium expression and evaluate the fraction.

Worked numerical example for Method 1.  Suppose the measured concentrations at equilibrium are \( [\mathrm{P}] = 0.8\ \mu\mathrm{M} \), \( [\mathrm{L}] = 10\ \mu\mathrm{M} \), and \( [\mathrm{PL}] = 0.2\ \mu\mathrm{M} \).  Compute \(K_{d}\) as follows.

\[ K_{d} \;=\; \frac{0.8\ \mu\mathrm{M} \times 10\ \mu\mathrm{M}}{0.2\ \mu\mathrm{M}} \;=\; 40\ \mu\mathrm{M} \]

Note. One of the concentration units cancels, leaving the result in units of concentration.  In this example \(K_{d}=40\ \mu\mathrm{M}\).

Method 2. Calculate \(K_{d}\) from kinetic measurements.  Step by step.

Step 1. Measure the association rate constant \(k_{\mathrm{on}}\) and the dissociation rate constant \(k_{\mathrm{off}}\).  Step 2. Compute

\[ K_{d} \;=\; \frac{k_{\mathrm{off}}}{k_{\mathrm{on}}} \]

Method 3. Determine \(K_{d}\) from a binding isotherm or saturation curve.  This is commonly used in equilibrium binding assays where you vary ligand concentration and measure the fraction of protein bound.

For a single site binding model the fraction bound Y is given by

\[ Y \;=\; \frac{[\mathrm{PL}]}{[\mathrm{P}]_{0}} \;=\; \frac{[\mathrm{L}]}{[\mathrm{L}] + K_{d}} \]

To solve for \(K_{d}\) algebraically from a measured Y at a known free ligand concentration do the following steps.  Rearrange the equation for \(K_{d}\).

\[ Y \;=\; \frac{[\mathrm{L}]}{[\mathrm{L}] + K_{d}} \]

Rearrange to isolate \(K_{d}\).  Multiply both sides by the denominator and then solve for \(K_{d}\).

\[ Y\bigl([\mathrm{L}] + K_{d}\bigr) \;=\; [\mathrm{L}] \]

\[ Y[\mathrm{L}] + YK_{d} \;=\; [\mathrm{L}] \]

\[ YK_{d} \;=\; [\mathrm{L}]\bigl(1 – Y\bigr) \]

\[ K_{d} \;=\; [\mathrm{L}]\left(\frac{1 – Y}{Y}\right) \]

Practical shortcut for binding curves.  When you measure a full binding curve and fit it to the single site isotherm, the ligand concentration that gives half maximal binding, that is Y equal to 0.5, equals \(K_{d}\).  So the concentration at which the response is half maximal is a direct experimental estimate of \(K_{d}\).

Summary checklist for calculating \(K_{d}\).  Choose the method that matches your data.  If you have equilibrium concentrations use the ratio \(K_{d} = \frac{[\mathrm{P}][\mathrm{L}]}{[\mathrm{PL}]}\).  If you have kinetic rates use \(K_{d} = \frac{k_{\mathrm{off}}}{k_{\mathrm{on}}}\).  If you have a binding curve fit the single site isotherm or read off the half maximal ligand concentration.