# 4.12: Appendix - Laboratory Determination of the Thermodynamic Parameters for Protein Denaturation

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## Introduction

Multiple methods can be used to investigate the denaturation of a protein. These include UV, fluorescence, CD, and viscosity measurement. In all these 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 (ΔG^{O}) for the protein (for the reaction N ↔ D). It is relatively to calculate if the denaturation curves show a sigmoidal, cooperative transition from the native to the denatured state, indicating a two-state transition. The dependent variable can also be normalized to show fractional denaturation (fD). An idealized example of an experimental denaturation curve is shown in Figure \(\PageIndex{1}\) below.

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 at the end, is required.

## Denaturation with urea or guanidine hydrochloride

For each curve, the value of \(y\) (either A_{280}, gluorescence 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.

\[y=\left(f_{N} y_{N}\right)+\left(f_{D} y_{D}\right) \label{1}\]

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 gives the following equation.

\[1=f_{N}+f_{D} \text { or } f_{N}=1-f_{D} \label{2}\]

Substituting \ref{2} into \ref{1} gives

\[y=\left(1-f_{D}\right) y_{N}+\left(f_{D} y_{D}\right)=y_{N}-f_{D} y_{N}+\left(f_{D} y_{D}\right)\]

Rearranging this equation gives

\[\mathrm{f}_{\mathrm{D}}=\frac{\mathrm{y}-\mathrm{y}_{\mathrm{N}}}{\mathrm{y}_{\mathrm{D}}-\mathrm{y}_{\mathrm{N}}} \label{3}\]

Notice the right hand side of the equations contains variables that are easily measured.

By substituting \ref{2} and \ref{3} into the expression for the equilibrium constant for the reaction \(\ce{N <=> D}\) we get:

\[K_{e q}=\frac{[D]_{e q}}{[N]_{e q}}=\frac{f_{D}}{f_{N}}=\frac{f_{D}}{1-f_{D}}\]

From this, we can calculate ΔG^{0}.

\[\Delta \mathrm{G}^{0}=-\mathrm{R} \operatorname{Tln} \mathrm{K}_{\mathrm{eq}}=-\mathrm{R} \operatorname{Tln}\left[\frac{\mathrm{f}_{\mathrm{D}}}{1-\mathrm{f}_{\mathrm{D}}}\right]\]

Remember that ΔG^{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 ↔ D transition, ΔG^{O} > 0 (i.e. denaturation is NOT favored). At high temperature and urea/guanidine HCl concentration, the denatured state is favored, and ΔG^{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 ΔG^{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 protein denaturation we saw in Chapter 4.9. This is Illustrated in Figure \(\PageIndex{2}\) below.

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 ΔG^{o} for the reaction N ↔ D. However, in the range where the protein denatures (either with urea or increasing temperature), it is possible to measure f_{D}/f_{N}.and hence ΔG^{o} at each urea or temperature.

Now we can calculate the Δ**G**^{o}_{w}^{ }for N ↔ D in water in the absence of urea. For a simple two state N ↔ D, a plot of ΔG^{o} vs [urea] is linear, and given by the following equation, which should be evident from (\PageIndex{1}\).

\[\Delta \mathrm{G}^{0}=\Delta \mathrm{G}_{\mathrm{w}}^{0}-\mathrm{m}[\text { urea }] \]

It is important to know the K_{eq} and ΔG^{0} for the N ↔ D transition in the absence of urea and under "physiological conditions". A comparison of the calculated values of ΔG^{0}_{w} 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 to what extent the mutants were stabilized or destabilized compared to the wild-type protein. ΔG^{0}_{w} for the N ↔ D transition of 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

Calculation of Δ**H**^{o} and Δ**S**^{o} for N <=>D at room temperature

K_{eq} values can be calculated from thermal denaturation curves in the same way as described above using urea as a denaturant by monitoring change in an observable (spectra signal for example) vs temperature. Knowing K_{eq}, ΔH^{0}, DS^{0} can be calculated from equation x below since a semi-log plot of lnK_{eq}_{ }vs 1/T is a straight line with a slope of - ΔH^{0}R and a y intercept of + ΔS^{0}/R, where R is the ideal gas constant.

\begin{equation}

\begin{gathered}

\Delta \mathrm{G}^{0}=\Delta \mathrm{H}^{0}-\mathrm{T} \Delta \mathrm{S}^{0}=-\mathrm{RTln} \mathrm{K}_{\mathrm{eq}} \\

\ln \mathrm{K}_{\mathrm{eq}}=-\frac{\Delta \mathrm{H}^{0}-\mathrm{T} \Delta \mathrm{S}^{0}}{\mathrm{RT}} \\

\ln \mathrm{K}_{\mathrm{eq}}=-\frac{\Delta \mathrm{H}^{0}}{\mathrm{RT}}+\frac{\Delta \mathrm{S}^{0}}{\mathrm{R}}

\end{gathered}

\end{equation}

From these equations, it should be evident that all the major thermodynamics constants (ΔG^{0}, ΔH^{0} and ΔS^{0} ) for the N ↔ D transition can be calculated from thermal denaturation curves.

Equation (9) below shows that the derivative of equation (8) with respect to 1/T (i.e. the slope of equation 8 plotted as lnK_{eq} vs 1/T) is indeed -ΔH^{0}/R. Equation (9) is the van 't Hoff equation, and the calculated value of the enthalpy change is termed the van 't Hoff enthalpy, ΔH^{0}v_{Hoff}.

\begin{equation}

\frac{d \ln \mathrm{K}_{\mathrm{eq}}}{d(1 / \mathrm{T})}=-\frac{\Delta \mathrm{H}^{0}}{\mathrm{R}}=-\frac{\Delta \mathrm{H}_{\mathrm{vHoff}}^{0}}{\mathrm{R}}

\end{equation}

It is useful to compare the van 't Hoff enthalpy, ΔH^{0}v_{Hoff}, with the enthaly change determined directly using differential scanning calorimetry by analyzing a plot of Cp vs T. (Note that the area under the Cp vs T curve as the protein transitions to the unfolded state has units of kcal. The ΔH^{0} for the unfolding is inversely proportional to the width of the curve.) .

In contrast to the long extrapolation of the ΔG^{0} vs [urea] to [urea] = 0 to get ΔG^{0} (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 ΔS^{0}/R, the y intercept, has little meaning since the 1/T value at the y intercept is 0, which occurs when T approaches infinity. ΔS^{0} can be calculated at any reasonable temperature from the the calculated value of ΔG^{0} at that temperature and the calculated ΔH^{0}_{vHoff}.