# Determination of ΔG of Unfolding

Multiple methods can be used to investigate the denaturation of a protein. These include UV, fluorescence, CD, and viscosity measurement.

### How to Calculate ΔG from Protein Denaturation Curves

In all these experimental methods, the dependent variable (\(y\)) is measured as a function of the independent variable, which is often temperature (for thermal denaturation curves) or denaturant (such as urea, guanidine hydrochloride) concentration. From these curves we would like to calculate the standard free energy of unfolding (\(\Delta G^o\)) for the protein (for the reaction \(N \rightleftharpoons D\)). The denaturation curves usually show a sigmoidal, cooperative transition from the native to the denatured state. The dependent variable can also be normalized to show fractional denaturation (\(f_D\)). An idealized example of an experimental denaturation curve is shown in the figure below:

Figure: DENATURATION CURVE FOR PROTEINS

A more realistic denaturation curve might show a linear change in the values of the dependent variable (fluorescence intensity for example) for values of temperature or denaturant concentration well below that at which the protein starts to unfold, or above that at which it is unfolded. In these cases, the mathematical analysis, presented below, is a bit more complicated.

For each curve, the value of \(y\) (either \(A_{280}\), Fluorescence intensity, viscosity, etc) can be thought of as the sum contributed by the native state and from the denatured state, which are present in different fractional concentrations from 0 - 1. Hence the following equation should be reasonably intuitive.

\[ y = f_N y_N + f_D y_D \tag{1}\]

where \(f_N\) is the fraction native and \(y_N\) is the contribution to the dependent variable \(y\) from the native state, and \(f_D\) is the fraction denatured and \(y_D\) is the contribution to the dependent variable y from the denatured state. Conservation give equation 2.

\[1 = f_N + f_D\] or \[f_N = 1 - f_D. \tag{2}\]

Substituting 2 into 1 gives:

\[ y = ( 1 - f_D) y_N + f_D y_D.\]

\[ y = y_N - f_Dy_N + f_D y_D.\]

\[y - y_N = f_D (y_D - y_N).\]

Rearranging this equation gives

\[f_D = \dfrac{y - y_N}{y_D - y_N} \tag{3}\]

Notice the right hand side of the equations contains variables that are easily measured. By substituting equation 2 and equation 3 into the expression for the equilibrium constant for the reaction \(N \rightleftharpoons D\) we get:

\[K_{eq} = \dfrac{[D]_{eq}}{[N]_{eq}} = \dfrac{f_D}{f_N}\]

or

\[K_{eq} = \dfrac{f_D}{1 - f_D} \tag{4}\]

and

\[ \Delta G_ = -RT\; \ln\, K_{eq} = -RT\, \ln \left( \dfrac{f_D}{1 - f_D} \right ) \tag{5a}\]

Remember that \(\Delta G^o\) (and hence \(K_{eq}\)) depends only on the intrinsic stability of the native vs. denatured state for a given set of conditions. They vary as a function of temperature and solvent conditions. At low temperature and low urea/guanidine HCl concentration, the native state is favored, and for the \(N \rightleftharpoons D\) transition, \(\Delta G^o > 0\) (i.e., denaturation is NOT favored). At high temperature and urea/guanidine HCl concentration, the denatured state is favored, and \(\Delta G^o < 0\). At some value of temperature or urea/guanidine concentration, both the native and denatured state would be equally favored. At this point, \(K_{eq} = 1\) and \(\Delta G^o=0\). If temperature is the denaturing agent, the temperature at this point is called the melting point (\(T_m\)) of the protein, which is analogous to the \(T_m\) (in the heat capacity vs. temperature graphs) for the gel to liquid crystalline phase transition of phospholipid vesicles.

**Figure**: gel to liquid crystalline phase transition of phospholipid vesicles

Figure: gel to liquid crystalline phase transition of phospholipid vesicles

Ordinarily, at a temperature much below the \(T_m\) for the protein or at a low urea concentration, so little of the protein would be in the \(D\) state that it would be extremely difficult to determine the concentration of the protein in the \(D\) state. Hence it would be difficult to determine the \(K_{eq}\) or \(\Delta G^o\) for the reaction \(N \rightleftharpoons D\). However, in the range where the protein denatures (either with urea or increasing temperature), it is possible to measure \( \frac{f_D}{f_N}\) and hence \(\Delta G^o\) at each urea or temperature.

### Denaturation with chemical perturbants (such as urea)

Calculation of \(\Delta G^o\) for \(N \rightleftharpoons D\) in the absence of urea. A plot of \(\Delta G^o\) vs. [urea] is linear, and given by the following equation, which should be evident from the beginning figure in this section:

\[\Delta^o_D = \Delta^o_D \, \text{(w/o urea)} - m[urea] \tag{5b} \]

Although it is nice to know the \(K_{eq}\) and \(\Delta G^o\) for the \(N \rightleftharpoons D\) transition in the presence of various urea concentrations, it would be even more useful to determine those parameters in the absence of urea, that is, under "physiological conditions." A comparison of the calculated values of \(\Delta G^o\) in the absence of urea for a series of similar proteins (such as those varying by a single amino acid prepared by site-specific mutagenesis of the normal or wild-type gene, would indicate how the mutants were stabilized or destabilized compared to the wild-type protein. In experimental cases in which the denaturant is a substance such as urea or guanidine HCl, the \(\Delta^o_D\) for the protein in the absence of denaturant (i.e., in water) can be determined by extrapolating the straight line to [urea] = 0. Admittedly, this is a long extrapolation, but with high quality data and a high correlation coefficient for the linear regression analysis of the best-fit line, reasonable values can be obtained.

### Denaturation with Heat

\(K_D\) values, calculated from thermal denaturation curves as described above, can be used to calculate the \(\Delta S^o\) and \(\Delta H^o\) for the \(N \rightleftharpoons D\) transition, assuming again that the denaturation is cooperative (no intermediates) and that \(\Delta H^o\) is independent of temperature over the ranges studied. The relationship between \(K_{eq}\) (denoted \(K_D\) below since at the moment my program won't allow me to change it), calculated as describe above from the fraction denatured and fraction native, and \(1/T\) in equation 8 is particularly useful since a semi-log plot of \(\ln\, K_{eq}\) vs. \(1/T\) is a straight line with a slope of \(- \Delta H^o/R\) and a \(y\)-intercept of \(\Delta S^o/R\).

\[ \Delta G^o = \Delta H^o - T \Delta S^o = -RT\, \ln \, K_D \tag{6}\]

\[ \ln\, K_D = -\dfrac{\Delta H^o-T\Delta S^o}{RT} \tag{7}\]

\[ \ln\, K_D = - \dfrac{\Delta H^o}{RT} + \dfrac{\Delta S^o}{R} \tag{8}\]

\[ \dfrac{d \, \ln \, K_D}{d (1/T)} = -\dfrac{\Delta H^o}{R} = -\dfrac{\Delta H^0_{vHoff}}{R} \tag{9}\] or

\[ \dfrac{d \, \ln \, K_D}{dT}=\dfrac{\Delta H^0_{vHoff}}{RT^2} \tag{10}\]

Hence from Equations 6 and 8, it should be evident that all the major thermodynamics parameters (\(\Delta G^o\), \(\Delta H^o\) and \(\Delta S^o\)) for the \(N \rightleftharpoons D\) transition can be calculated from an ideal thermal denaturation curve. Equation 9 shows that the derivative of equation (8) with respect to \(1/T\) (i.e., the slope of equation 8 plotted as \(\ ln\, K_D\) vs. \(1/T\)) is indeed \( -\Delta H^o/R\).

Equation 8 is the **van't Hoff equation**, and the calculated value of the enthalpy change is termed the van't Hoff enthalpy, \(\Delta H^o_{vHoff}\). Equation 10 calculates the derivative of \(\ln\, K_{eq}\) with respect to \(T\) instead of \(1/T\). This equation will be useful later when we compare the enthalpies calculated using the van't Hoff equation with those determined directly using differential scanning calorimetry.

In contrast to the long extrapolation by evaluating the limit of [urea] = 0 for the plot of \(\Delta G^o\) vs. [urea] discussed above to estimate \(\Delta G^o\) (the y-intercept) in the absence of urea, which has some physical meaning, extrapolation of the straight line from the van't Hoff plot from equation 8 to get \(\Delta S^o/R\), the y-intercept, has little meaning since the \(1/T\) value at the y-intercept is 0, which occurs when \(T\) approaches infinity. \(\Delta S^o\) can be calculated at any reasonable temperature from the the calculated value of \( \Delta G^o\) at that temperature and the calculated \(\Delta H^o_{vHoff}\).