Living cells are surrounded by a plasma membrane composed of amphipathic phospholipids. These molecules self-assemble into a bilayer in aqueous solvent so as to point inwards their hydrophobic tails, while their highly polar head groups are exposed to water. Many peptides-very small polymers of amino acids-are thought to be useful as therapeutics for their interactions with these cell membranes. For example, they may disrupt bacterial membranes and exhibit antibacterial activity. Consider identical peptides that are membrane-associated. Assume that each can either sit at the extracellular interface in the hydrophilic region or insert into the hydrophobic bilayer, as in Fig. 4.8.

a) You can view this problem as one in which two systems can freely exchange peptide molecules, one corresponding to the hydrophilic extracellular region and one corresponding to the hydrophobic bilayer interior. According to the entropy maximization principle, what condition must be satisfied for equilibrium of peptide exchange?

b) Consider a highly simplified model in which the change in energy for going from the extracellular interface to the bilayer region is given by ?. Find an expression for the fraction of the peptides that are in the bilayer as a function of temperature.

c) An approximate energy change for moving a single glutamine amino acid from the bilayer interface to a hydrocarbon environment is 0.2 kcal/mol. Assume that a homopeptide made of glutamines then has a transfer energy of n × (0.2 kcal/mol). What is the maximum length of such a peptide for which at least 10% of the peptides will be present inside the bilayer? Assume that the temperature is T = 300 K.