= Equilibrium dialysis
{title2=$M=(V_1+V_2)L+V_1B$}
\b[<Equilibrium dialysis>] measures partitioning or binding using two compartments separated by a <semipermeable membrane>. In a common ligand-binding design, a small <ligand> can cross while the binding macromolecule is retained on one side. Under matched solution conditions, the free ligand equilibrates; retained bound ligand can make the total ligand concentrations different.
A historically motivating assay measured inducer binding to the <lac repressor> during its isolation by <Walter Gilbert> and <Benno Müller-Hill>. The primary paper describes equilibrium dialysis against radioactive inducer as a way to detect the regulatory protein.<Isolation of the lac repressor by Walter Gilbert and Benno Müller-Hill>[printed p. 1891, assay description and Design of the Experiment]{ref} Detecting binding was an experimentally tractable step even before every part of the repression mechanism was known.
For an original one-site model, let free ligand concentration be $L$, retained site concentration $R_T$, and dissociation constant $K_d$. Bound concentration is
$$
B=R_T\frac{L}{K_d+L}.
$$
If compartment volumes are $V_1,V_2$, and protein is confined to compartment 1, total ligand amount $M$ obeys
$$
M=(V_1+V_2)L+V_1B.
$$
Free concentration is not generally the initial concentration: binding can substantially deplete the free pool.
Take equal 1.00 mL compartments, $R_T=4.00\,\mathrm{\mu mol\,L^{-1}}$, $K_d=1.00\,\mathrm{\mu mol\,L^{-1}}$ and total ligand 8.00 nmol. Expressing $L$ numerically in micromoles per litre gives $2L+4L/(1+L)=8$, hence
$$
L=\frac{1+\sqrt{17}}2\approx2.562,\qquad B\approx2.877.
$$
The total concentrations are about 5.438 and 2.562 micromoles per litre, although free ligand equilibrates. Substituting the initial average concentration 4.00 for $L$ would overestimate bound concentration in this model.
The interpretation requires adequate equilibration, membrane retention of the macromolecule, known accessible volumes and suitable blank measurements. Nonspecific membrane adsorption or unequal chemical conditions can imitate or modify an apparent binding difference. Equal free concentrations are an assumption appropriate to the stated ideal setup; equilibrium more generally concerns compatible <chemical potentials>, not arbitrary equality of total concentrations.
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