18.2: Two-State Thermodynamics

Here we describe the basic thermodynamics of two-state systems, which are commonly used for processes such as protein folding, binding, and DNA hybridization. Working with the example of protein folding analyzed through the temperature-dependent folded protein content.

\[K=\dfrac{k_f}{k_u} = \dfrac{[F]}{[U]} = \dfrac{\phi_F}{1-\phi_F} \nonumber \]

where φ_F is the fraction of protein that is folded, and the fraction that is unfolded is (1 ‒ φ_F).

\[ \begin{aligned} &\phi_F =\dfrac{K}{K+1} \\ &K= e^{-\Delta G^0/RT} \\ &\phi_F = \dfrac{1}{1+e^{-\Delta G^0/RT}} = \dfrac{1}{1+e^{\Delta H^0/RT} e^{-\Delta S^0/R}} \end{aligned} \]

Define the melting temperature T_m as the temperature at which φ_F= 0.5. Then at T_m, \(\Delta G^0=0 \mathrm{or} T_m = \Delta H^0/\Delta S^0 \). Characteristic melting curves for T_m= 300 K are below:

We can analyze the slope of curve at T_m using a van’t Hoff analysis:

\[ \begin{aligned} \dfrac{d\phi_F}{dT}&=\dfrac{d\phi_F}{dK} \cdot \dfrac{dK}{dT}=\dfrac{d\phi_F}{dK}\cdot K \dfrac{d\ln{K}}{dT}\\ \dfrac{d\ln{K}}{dT} &= \dfrac{\Delta H^0}{RT^2} \\ \dfrac{d\phi_F}{dK} &=K^{-2}(1+K)^{-2} \\ \left( \dfrac{d\phi_F}{dT} \right)_{T=T_m} &= \dfrac{\Delta H^0}{4RT^2_m} \quad \mathrm{since} \; K=1 \; \mathrm{at} \; T_m \end{aligned} \]

This analysis assumes that there is no temperature dependence to ΔH, although we know well that it does from our earlier discussion of hydrophobicity. A more realistic two-state model will allow for a change in heat capacity between the U and F states that describes the temperature dependence of the enthalpy and entropy.

\[ \Delta G^0(T) = \Delta H^0(T_m)-T\Delta S^0(T_m)+ \Delta C_p \left[ T-T_m-T \ln{\dfrac{T}{T_m}} \right] \nonumber \]