Final answer:
To find the fraction of protein that is unfolded at equilibrium with a stability of 7.1 kcal/mole, we use the provided equation to calculate Keq and then express the unfolded fraction as [U]/[T]. Calculation shows that approximately 0.001% of the protein is unfolded at equilibrium.
Step-by-step explanation:
To calculate the fraction of a protein that is unfolded at equilibrium at 37°C with a stability of 7.1 kcal/mole, we need to use the given equilibrium constant (Keq) equation and relate it to the free energy change (ΔG°). The given equation for Keq in terms of ΔG° is Keq = 10-ΔG°/1.42. We can rearrange this equation to solve for the unfolded fraction of the protein (U/F).
First, calculate Keq using the stability of the protein:
Keq = 10-7.1 kcal/mole / 1.42
Keq = 10-5
Since Keq is the ratio of unfolded to folded protein ([U]/[F]), we can write the following:
Keq = [U]/[F]
By definition, at equilibrium, the total amount of protein ([T]) must be the sum of the folded and unfolded forms:
[T] = [F] + [U]
We can also express [U] as Keq multiplied by [F]:
[U] = Keq * [F]
Substituting [U] back into the total protein equation:
[T] = [F] + Keq * [F]
Since we are interested in the fraction that is unfolded ([U]/[T]), we can rearrange the previous equation to solve for [F]/[T]:
[F]/[T] = 1 / (1 + Keq)
Substitute Keq:
[F]/[T] = 1 / (1 + 10-5)
Now, solve for the unfolded fraction ([U]/[T]):
[U]/[T] = 1 - [F]/[T]
[U]/[T] = 1 - 1 / (1 + 10-5)
And upon calculation:
[U]/[T] ≈ 1 - 0.99999 = 0.00001
Therefore, the fraction of the protein that is unfolded at equilibrium at 37°C is approximately 0.00001 or 0.001%.