= Antibody equilibrium occupancy
{title2=$\theta$}
<Antibody equilibrium occupancy> counts occupied <antibody binding sites>, rather than retained assay label. For an original 1:1 model, let $A$ denote an unoccupied site, $E$ free <antigen> bearing one accessible <epitope>, and $AE$ their complex. At <chemical equilibrium>, write
$$
A+E\rightleftharpoons AE,\qquad c=[E],\qquad K_d=\frac{[A]c}{[AE]},\qquad A_T=[A]+[AE],\qquad \theta=\frac{[AE]}{A_T}=\frac{c}{K_d+c}.
$$
Square brackets mean <concentrations>. The <dissociation constant> $K_d$ here has <concentration> units. This is the same independent-site calculation as <receptor binding occupancy>, now applied to an <antibody binding site>. The last equality follows by substituting $[AE]=[A]c/K_d$ into the occupied fraction and cancelling $[A]$.
Assume identical independent sites, reversible 1:1 binding, dilute <chemical solution> behavior, fixed <temperature> and <buffer solution>, and enough time for <chemical equilibrium>. Exclude two-arm bridging, <cross-reactivity> and inaccessible sites from this model. Crucially, $c$ is free <antigen> <concentration>; substituting the amount supplied per volume requires negligible depletion by binding. Independent <IgG> arms could obey this model site by site, but it does not describe <avidity> from bridging a surface.
Stipulate $K_d=20\,\mathrm{nM}$, with $1\,\mathrm{nM}=10^{-9}\,\mathrm{mol\,L^{-1}}$. Free <antigen> <concentrations> $2$, $20$ and $200\,\mathrm{nM}$ give $\theta=1/11$, $1/2$ and $10/11$. A second stipulated <dissociation constant> of $200\,\mathrm{nM}$ instead gives $\theta=1/11$ at $c=20\,\mathrm{nM}$. Both curves approach one at large $c$: a nearly saturated readout can conceal an <antibody affinity> difference. These are invented values, with no universal threshold for a useful <antibody>.
\Image[/antibody-equilibrium-occupancy-and-assay-readout.png]
{title=<Antibody equilibrium occupancy> and the additional steps in an assay readout}
{description=Original analytic curves for stipulated <dissociation constants> of $20$ and $200\,\mathrm{nM}$; marked examples belong to the first curve. The adjacent schematic separates <epitope> accessibility, binding, washing and detection. It contains no measured sensorgrams, fitted assay parameters or experimental signals. A <logarithmic scale> changes the display, not the independent-site assumptions.}
A <Western blot> readout additionally depends on <Western blot transfer>, accessible <epitopes>, loss during washing and <Western blot antibody detection>. <Western blot response saturation> can flatten the final signal even when binding itself remains responsive. An <ELISA> also reports retained label through an <enzyme> readout, rather than automatically reporting $\theta$. Thus half of an assay's maximum signal need not occur at $c=K_d$. Varying free <antigen> through <concentrations> near the candidate <dissociation constant>, while checking depletion and the readout's response, is more informative than comparing a single saturated point.
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