Question

4. Now let's say that the spherical and disc-shaped proteins can interconvert, i.e., each protein has two possible states. The spherical state has a lower energy relative to the disc-shaped state. For definiteness, the spherical state has energy 0, while the disc-shaped state has energy & > 0.

A) The partition function for a collection of N two-state proteins can be written as: N Z(T,V,N)=Z, (T,V)
J=0 Here, Z, is the partition function for a collection of Jidentical spherical proteins and N-J identical disc- shaped proteins, assuming the proteins cannot interconvert. Explain why this expression for Z makes sense.
B) Find Z,(T,V).
C) At some value of J (which you do not need to find), Z, is maximal. What is the physical significance of this value of J?

Answer 1

The partition function Z accounts for all possible states of two-state proteins, reflecting a sum over Boltzmann statistics. Zj can be calculated using individual energies and Boltzmann factors. The maximal Zj reflects the most probable distribution of protein states at equilibrium.

Step-by-step explanation:

Explanation of the Partition Function Z

The partition function Z(T,V,N) for a collection of N two-state proteins is given as a sum over all possible microstates, where each microstate is characterized by J spherical and N-J disc-shaped proteins. This expression makes sense because the partition function takes into account all possible combinations of the two states across the N proteins. Considering the given energy states (0 for spherical and >0 for disc-shaped), this partition function captures the Boltzmann statistics for such a two-state system.

Finding Zj(T,V)

To find Zj(T,V), you would need the energies of the individual states and the Boltzmann factor. However, since the question does not provide specific details about the energies other than the relative difference, and asks to ignore any typos or irrelevant parts, Zj would be the sum over all configurations of J spherical proteins at energy 0 and N-J disc-shaped proteins at energy >0, each weighted by the Boltzmann factor e-βε where β=1/(kT), k is the Boltzmann constant, T is the temperature, and ε is the energy difference between the two states.

Physical Significance of Maximal Zj

The value of J at which Zj is maximal has significant physical meaning. It represents the most probable distribution of proteins between the two states at a given temperature, because at equilibrium, a system tends to occupy the macrostate that maximizes the partition function, which correlates to the maximum entropy and the most statistically likely distribution of states.