You have engineered a drug to combat cardiovascular disease by activating a key target

protein inside cells within atherosclerotic plaque. The drug, which is absent from the body

initially, is consistently supplied to the bloodstream at Rnmo$\frac{l}{h}.$ The drug leaves the

bloodstream through the kidneys; this process is quantitatively described with first order rate

constant $k.$ It also passively enters the cells of interest. The rate of this process is proportional to

the concentration of drug in the bloodstream. There is no mechanism for the drug to leave the

cell to re$-$enter the bloodstream. Instead, the drug can reversibly bind the target in a monovalent

manner. The number of targets per cells is denoted $T$ and can be assumed to be in significant

excess relative to the delivered drug. The drug can also be degraded via a first$-$order process

with the rate constant $k_{\text{d}\text{e}\text{g}}.$ Denote the bloodstream volume by $V_p$ and intracellular volume of all

targeted cells by $V_c.$

a$.$ Derive an expression for the dynamic concentration of drug in the bloodstream.

b$.$ What is the steady$-$state concentration of the drug in the bloodstream?

c$.$ What is the steady$-$state fraction of target proteins that are complexed?

d$.$ Derive an expression for the dynamic concentration of drug$-$receptor complex in the cells in

the atherosclerotic plaque. You can approximate that binding and unbinding are rapidly

equilibrate relative to pharmacokinetic transport.