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 $10-7$ cm$2/$s

the permeability constant $($k$)$ for the charged form $($A$)$ k $=5$X$10-9$ cm$2/$s

electrical potential difference is $-50$ mV

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 $(\backslash$alpha $)$ at pH $7\mathrm{.}4$ 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 $(\backslash$alpha $)$ at pH $7\mathrm{.}4$

d$.$ Calculate the Overall Permeability Coefficient