The purpose of this exercise is to examine the effects of immobilization on observed binding

and dissociation kinetics. Consider a polystyrene bead of diameter $5m$ on which either a

protein $(150$kDa$)$ or its small$-$molecule ligand $(150Da$ can be immobilized. In either case, the

soluble protein or ligand can be labeled so as to track the number of complexes on the bead as

a function of time. In solution, the association rate constant for the protein$-$ligand complex

formation is $k_{on}={10}^6{M}^{-1}{s}^{-1},$ and the dissociation rate constant is $k_{\text{o}\text{f}\text{f}}={10}^{-3}{s}^{-1}.$ You may

assume that the density of the ligand and protein is $1\mathrm{.}1\frac{g}{c}{m}^{-3}.$

a$.$ Plot the observed association rate constant $k_f$ versus the number of molecules immobilized

per bead for both the case of immobilized protein and the case of immobilized ligand.

b$.$ Plot the observed dissociation rate constant $k_r$ versus the number of molecules immobilized

per bead for both the case of immobilized protein and the case of immobilized ligand.

c$.$ Under what immobilization conditions $($i$.$e$.$ protein or ligand surface density$)$ will the observed

rate constants $(k_f,k_r)$ reflect the intrinsic rate constants $(k_{on},k_{\text{o}\text{f}\text{f}})?$

$($Hint: $k_f=\frac{4D{r}_0{k}_{on}}{4D{r}_0+k_{on}},$ but if the partner of binding has multiple binding sites $($it can be either

protein or ligand in this problem$),$ the equation should be modified to $k_f=\frac{4D{r}_{0}N{k}_{on}}{4D{r}_0+N{k}_{on}}.$ You can

use the same approach for $k_r$ by using $k_r=\frac{4D{r}_{0}N{k}_{\text{o}\text{f}\text{f}}}{4D{r}_0+N{k}_{on}}.$ Here, $N$ is the number of immobilized

molecules $($either ligand or protein$)$ per bead particle, $r_0$ becomes the radius of the particle $+$

radius of $($ligand$+$protein$),$ but the particle is much larger, so $r_0$~~ radius of the particle. Diffusion

of the particle is negligible compared to the free protein or ligand, therefore, you can assume $D$

is that of free protein or free ligand. You can use Stokes$-$Einstein equation to estimate D$.$ You

will need the molecular weight and density information to estimate the radius of protein or

ligand$)$