Cell signalling formulas

For a simple 1:1 interaction. $K_d$ has units of concentration ($k_{off}$ in ${\text{s}}^{-1}$, $k_{on}$ in ${\text{M}}^{-1}{\text{s}}^{-1}$). A smaller $K_d$ means tighter (higher-affinity) binding. The association constant is $K_a=1/K_d$.

At $[L]=K_d$, $\theta =0.5$. Rearranged, $[L]=K_d\frac{\theta }{1-\theta }$, so moving from 10% to 90% occupancy needs an 81-fold rise in $[L]$ (from $K_d/9$ to $9K_d$). Valid only for one independent binding site per receptor.

$B$ = bound ligand ($[RL]$), $F$ = free ligand, $B_{max}$ = total receptor concentration. For a single class of independent sites, plotting $B/F$ against $B$ gives a straight line with slope $-1/K_d$ and x-intercept $B_{max}$. A curved Scatchard plot indicates cooperativity or more than one class of binding site.

Here $K_d$ has units of concentration${}^n$ and equals $(K_{0.5}{)}^n$, where $K_{0.5}$ is the ligand concentration that gives half saturation. $n$ is the Hill coefficient: $n>1$ means positive cooperativity, $n<1$ means negative cooperativity (or several classes of site), and $n=1$ reduces to the single-site equation. $n$ is an empirical measure: it cannot exceed the number of binding sites and is usually smaller.

A straight line whose slope is the Hill coefficient $n$.