$4\mathrm{.}13.$ Extending the previous problem, consider a specific model of protein$-$drug binding inside the cell. Let $N_D=c_D{V}_{\text{c}\text{e}\text{l}\text{l}}$ and $N_P=c_P{V}_{\text{c}\text{e}\text{l}\text{l}}$ denote the numbers of drug and protein molecules in the cell, respectively, where $c_D$ and $c_P$ are their concentrations in number per volume. For the sake of simplicity, consider the proteins to be immobile within the cell cytoskeleton. So that you can count microstates, discretize all of space into regions the size of the binding region in the protein, with volume $v,$ such that the total number of "pixels" inside a cell is $M=\frac{V_{\text{c}\text{e}\text{l}\text{l}}}{v}.$ Assume that the $N_P$ protein molecules each occupy one pixel and cannot change their location. Drug molecules can occupy any of the pixels. When a drug sits on one of the protein$-$occupied pixels, there is a favorable binding energy of $-lon$; otherwise, the energy is zero. Let $x$ denote the fraction of the $N_P$ sites with a bound drug molecule and $y$ the fraction of the remaining $(M-N_P)$ sites with a free drug molecule.

$($a$)$ Write an expression for the entropy in terms of $N_P,M,x,$ and $y.$ Use Stirling's approximation.

$($b$)$ Assuming that $N_D$ is less than the number of pixels, write an expression that relates $x$ and $y$ to each other, and to the values $N_P,N_D,$ and $M.$ Using the connection between the entropy and temperature, show that the equilibrium values of $x$ and $y$ satisfy the relation

$\frac{y(1-x)}{x(1-y)}=e^{-lo\frac{n}{k_B}T}$

Hint: Use the chain rule for the entropy derivative, e$.$g$.,$

$(del\frac{S}{d}$elE$)=(del\frac{S}{d}$elx$)(del\frac{x}{d}$elE$)$

$($c$)$ A pharmaceutical company is developing a drug to target a specific protein that exists in intracellular concentrations of ${10}^{-7}mo\frac{l}{\text{I}}.$ In order to be effective, the