$9\mathrm{.}10$ Binding to a membrane Consider a phospholipid bilayer membrane consisting of a mixture of $90\%$ uncharged lipid and $10\%$ singly charged acid lipid. Assume $0\mathrm{.}68$ nm$2$ surface area per lipid head group, and assume further that the charged lipids are uniformly distributed and immobile. The membrane is in contact with an aqueous solution of NaCl at $25$C$.$ The salt concentration is c $=100$ mM$.($a$)$ Calculate the areal charge density $($that is$,$ charge per unit surface area$)$ of the membrane. $($b$)$ Calculate the surface potential of the membrane. What is the electrostatic energy $($in kBT units$)$ of binding to the membrane of a trivalent positive ion such as spermidine PROBLEMS $381($a biologically active polyamine$)$ assuming that: $($c$)$ Binding occurs at the membrane surface? $($d$)$ Owing to steric factors, the charges of the bound spermidine stay in the water $0\mathrm{.}5$ nm distant from the membrane surface?