Potential near a protein. Adapted from Problem $9\mathrm{.}8$ Physical Biology of the cell, Phillips et al$.$

Consider a protein sphere with a radius of $1\mathrm{.}8$nm$,$ and charge Q $=10$e$,$ in an aqueous solution of c $=0\mathrm{.}05$M NaCl at $250$C$.$ Consider the small ions as point charges and use the linear approximation to the Poisson$$Boltzmann equation.

$($a$)$ Derive an expression for the potential V $($r$)$ of a charge sphere in a salty solution. Assume that the electrical potential at distance r from the center of a charged sphere is:

A$$r

V$($r$)=$ eD $(1)$

r where A is constant and D the Debye length.

Determine the constant A by computing the electric field from the potential and set it equal to E$($R$)=$ Q

$4$R$2$

$($b$)$ What is the surface potential of the protein in units kBT$/$e$?$

$($c$)$ What is the concentration of Na$+$ions and of Cl$$ ions at the surface of the protein?