Question

Proteins are polymers of amino acids that can exist in ordered structures stabilized by a variety of molecular interactions. However, when certain conditions are changed, the compact structure of a polypeptide chain may collapse into a random coil. This structural change may be regarded as a phase transition occurring at a characteristic transition temperature, the melting temperature, Tm, which increases with the strength and number of intermolecular interactions in the chain. A thermodynamic treatment allows predictions to be made of the temperature Tm for the unfolding of a helical polypeptide held together by hydrogen bonds into a random coil. If a polypeptide has N amino acids, N-4 hydrogen bonds are formed to form an a-helix, the most common type of helix in naturally occurring proteins. Because the first and last residues in the chain are free to move, N-2 residues form the compact helix and have restricted motion. Based on these ideas, the molar Gibbs energy of unfolding of a polypeptide with N 25 may be written as Ayold G=(N - 44 H -(N-2)TA.S where AnbH and Ansare, respectively, the molar enthalpy and entropy of dissociation of hydrogen bonds in the polypeptide. (a) (4 points) Justify the form of the equation for the Gibbs energy of unfolding. That is, why are the enthalpy and entropy terms written as (N - 4)AnpH and (N-2)Anos, respectively? (N-4)4, H (b) (3 points) Show that Tm may be written as I =* (N-2)4,5 (c) (4 points) Plot Tm/AnH/AS) for 5 S N s 100. It is easier to do this in excel. At which value do you predict it will converge to? Can you provide a mathematical argument?

Answer 1

Answer to part a: Since, enthalpy change is related to the number of bonds (the difference between the the energy required to break the bonds and energy gained by the formation of new bonds). Here (N-4) hydrogen bonds are present. The enthalpy term is written as .

(N – 4)AhbH

The entropy is a state of function. Therefore, entropy would depend on number of molecules take part in the reactant. Therefore, the entropy change is written as .

(N - 2) Ahos

Answer to part b:

At melting temperature T_m the system is in equilibrium. Thus .

AunfaldG = OandT = Tim

(N – 4)AhH - (N - 2)TmAh S=0

(N – 4) AhbH (N - 2) AhbS

Answer to part c:

y (Tm) vs.x ((N-4)/(N-2)) graph o 20 40 80 100 120 60 (N-4)/(N-2)

When N = 4, the graph will converge because then T_m = 0.