A new antifungal drug $(AH)$ is being developed. It exists in two forms: a neutral form $($HA$)$

and a negatively charged form $(A^{-}).$ The pKa of the drug is $5\mathrm{.}0.$ You are provided with

the following data:

Partition coefficient of the neutral form in a model membrane system is $100$

Diffusion coefficient of the neutral form in the membrane is ${10}^{-7}c\frac{m^2}{s}$

the permeability constant $(k)$ for the charged form $(A^{-})k=5{10}^{-9}c\frac{m^2}{s}$

electrical potential difference is $-50mV$

Objective: Develop a mathematical model to describe the permeability coefficient of the

drug across the membrane at physiological $pH($pH $7\mathrm{.}4).$ Consider the influence of both

the partition coefficient and the charged state of the drug on permeability.

Assumptions:

The membrane is permeable to both neutral $(HA)$ and charged $(A^{-})$forms of the drug,

but their permeabilities differ.

The charged form $(A^{-})$has a negligible partition coefficient compared to the neutral form

due to repulsion by the charged membrane.

The total concentration of the drug remains constant throughout the experiment.

a$.$ Calculate the fraction of ionized drug $()$ at $pH7\mathrm{.}4$ using the Henderson$-$Hasselbalch

equation.

b$.$ Derive an expression for the overall permeability coefficient considering both neutral

and charged forms. Explain the physical meaning of each term in the derived

expression. Discuss the limitations of this model and potential areas for further

refinement.

c$.$ Calculate the fraction of ionized drug $()$ at $pH7\mathrm{.}4$ a different way than in part a

d$.$ Calculate the Overall Permeability Coefficient