4.12: Laboratory Determination of the Thermodynamic Parameters for Protein Denaturation

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Learning Goals 

(Learning goals written by Claude, Sonnet 4.6, Anthropic)

Quantitative Analysis of Protein Denaturation Curves

• Derive the expressions for fractional denaturation (fD) and fractional native state (fN) from experimentally measured observables (absorbance, fluorescence, viscosity, CD signal), explain the physical assumption underlying the two-state model (N ↔ D), and calculate Keq and ΔG° at each point along a sigmoidal denaturation curve using the relationship ΔG° = −RT ln Keq.
• Explain why ΔG° and Keq for the N ↔ D transition are easily measured in the transition zone but not at low denaturant concentrations or low temperatures, and describe how extrapolation of the linear ΔG° vs. [urea] plot to zero denaturant concentration yields ΔG°w — the free energy of unfolding in water — along with the assumptions and limitations of this long extrapolation.

Thermodynamic Parameters from Thermal Denaturation

• Apply the van't Hoff equation (d ln Keq / d(1/T) = −ΔH°/R) to extract ΔH°_vHoff from the slope of a semi-log plot of ln Keq vs. 1/T for thermal denaturation data, and calculate ΔS° at a given temperature from ΔG° = ΔH° − TΔS° using the independently determined values of ΔG° and ΔH°.
• Compare the van't Hoff enthalpy (ΔH°_vHoff) obtained from equilibrium thermal denaturation curves with the calorimetric enthalpy (ΔH_cal) obtained from the area under a differential scanning calorimetry Cp vs. T curve — explaining what agreement or discrepancy between these values implies about the validity of the two-state model for a given protein — and identify the melting temperature Tm as the point where Keq = 1, ΔG° = 0, and the Cp vs. T curve reaches its maximum.

Introduction

Multiple methods can be used to investigate protein denaturation. These include UV, fluorescence, CD, and viscosity measurements. 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 concentration (such as urea or guanidine hydrochloride). From these curves, we want to calculate the standard free energy of unfolding (ΔG^O) for the protein (for the reaction N ↔ D). It is relatively easy 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 denaturation curve is shown below in Figure \(\PageIndex{1}\).

Figure \(\PageIndex{1}\): Graphical analysis of protein denaturation

A more realistic denaturation curve might show a small linear change in the dependent variable (fluorescence intensity, for example) with temperature or denaturant concentration, well before the major unfolding transition and above the concentration at which it is unfolded. In these cases, the mathematical analyses presented at the end are required.

Denaturation with urea or guanidine hydrochloride

For each curve, the value of y (either A₂₈₀, fluorescence intensity, viscosity, etc.) can be considered the sum contributed by the native state and the denatured states, which are both present in different fractional concentrations from 0 to 1. Hence, the following equation should be reasonably intuitive.

\begin{equation}
y=\left(f_N y_N\right)+\left(f_D y_D\right)
\end{equation}

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. y_D is the contribution to the dependent variable y from the denatured state. Conservation gives the following equation.

\begin{equation}
1=f_N+f_D \text { or } f_N=1-f_D
\end{equation}

Substituting (2) into (1) gives

\begin{equation}
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)
\end{equation}

Rearranging this equation gives

\begin{equation}
\mathrm{f}_{\mathrm{D}}=\frac{\mathrm{y}-\mathrm{y}_{\mathrm{N}}}{\mathrm{y}_{\mathrm{D}}-\mathrm{y}_{\mathrm{N}}}
\end{equation}

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

By substituting (2) and (3) into the expression for the equilibrium constant for the reaction N ↔ D, we get:

\begin{equation}
K_{e q}=\frac{[D]_{e q}}{[N]_{e q}}=\frac{f_D}{f_N}=\frac{f_D}{1-f_D}
\end{equation}

From this, we can calculate ΔG⁰.

\begin{equation}
\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]
\end{equation}

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 with temperature and solvent conditions. At low temperatures 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). The denatured state is favored at high temperatures and urea/guanidine-HCl concentrations and ΔG^O < 0.

At some temperature or urea/guanidine concentration value, both the native and denatured states would be equally favored. At this value, K_eq = 1 and ΔG^O = 0.

The temperature at the maximal C_p value is called the protein's melting temperature (Tm). It is analogous to the T_m in the heat capacity vs temperature graphs for protein denaturation, as we saw in Chapter 4.9. This is illustrated in Figure \(\PageIndex{2}\).

Figure \(\PageIndex{2}\): T_m for thermal denaturation of a protein

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 protein concentration 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 ΔG^o_w for N ↔ D in water without 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}\).

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

It is important to know the K_eq and ΔG⁰ for the N ↔ D transition in the absence of urea and under "physiological conditions." A comparison of the calculated ΔG⁰_w values in the absence of urea for a series of similar proteins (e.g., differing by a single amino acid, prepared by site-specific mutagenesis of the wild-type gene) would indicate the extent to which the mutants are stabilized or destabilized relative to the wild-type protein. ΔG⁰_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 reasonable values can be obtained with high-quality data and a high correlation coefficient for the linear regression analysis of the best-fit line.

Denaturation with heat

Calculation of ΔH^o and ΔS^o for N <=>D at room temperature

K_eq values can be calculated from thermal denaturation curves, as described above, using urea as the denaturant by monitoring the change in an observable (e.g., a spectral signal) as a function of temperature. Knowing K_eq and ΔH⁰, DS⁰ can be calculated from the equation below since a semi-log plot of lnK_eq vs 1/T is a straight line with a slope of - ΔH⁰R and a y-intercept of + ΔS⁰/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 thermodynamic constants (ΔG⁰, ΔH⁰, and ΔS⁰ ) for the N ↔ D transition can be calculated from thermal denaturation curves.

Equation (8) below shows that the derivative of equation (7) with respect to 1/T (i.e. the slope of equation 7 plotted as lnK_eq vs 1/T) is indeed -ΔH⁰/R. Equation (8) is the van't Hoff equation, and the calculated enthalpy change is termed the van't Hoff enthalpy, ΔH⁰v_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 helpful to compare the van 't Hoff enthalpy, ΔH⁰v_Hoff, with the enthalpy 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 or kJ. The ΔH⁰ for the unfolding is inversely proportional to the width of the curve.)

In contrast to the long extrapolation of the ΔG⁰ vs [urea] to [urea] = 0 to get ΔG⁰ (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 ΔS⁰/R, the y-intercept, has little meaning since the 1/T value at the y-intercept is 0, which occurs when T approaches infinity. ΔS⁰ can be calculated at any reasonable temperature from the calculated value of ΔG⁰ at that temperature and the calculated ΔH⁰_vHoff.

Summary

(Summary written by Claude, Sonnet 4.6, Anthropic)

This short but methodologically essential chapter provides the quantitative framework for extracting thermodynamic parameters from experimental protein denaturation data — converting raw spectroscopic or calorimetric measurements into ΔG°, ΔH°, and ΔS° for the native-to-denatured transition, which are the fundamental quantities needed to compare protein stabilities, interpret mutational effects, and test models of folding.

Chemical denaturation with urea or guanidine hydrochloride provides a practical route to measuring ΔG° for the N ↔ D equilibrium. The key assumption is the two-state model: at any given denaturant concentration, a protein molecule is either fully native or fully denatured, with no significant population of partially folded intermediates. Under this assumption, the experimentally measured observable y — whether absorbance at 280 nm, fluorescence intensity, CD signal, or viscosity — can be expressed as the weighted sum of contributions from the native state (yN) and the denatured state (yD), each scaled by their respective fractional populations (fN and fD, where fN + fD = 1). Rearranging these relationships yields fD as a function of y, yN, and yD, all of which are measurable from the denaturation curve. The equilibrium constant Keq = fD/fN is then calculated at each denaturant concentration in the transition zone, and ΔG° = −RT ln Keq is determined at each point. Outside the transition zone — at low denaturant concentrations where the protein is almost entirely native, or at high concentrations where it is almost entirely denatured — fD and fN approach 0 or 1 respectively, making Keq extremely small or large and difficult to determine accurately. Within the transition zone, however, both states are substantially populated and the ratio is readily measurable.

For a cooperative two-state transition, a plot of ΔG° vs. [urea] is empirically linear, described by ΔG° = ΔG°w − m[urea], where ΔG°w is the free energy of unfolding in water (in the absence of denaturant) and m is a constant reflecting the cooperativity of unfolding and the increase in solvent-accessible surface area upon denaturation. Extrapolating this line to [urea] = 0 gives ΔG°w — the quantity of primary interest because it reflects the intrinsic stability of the protein under physiological conditions in the absence of the denaturing agent. Although this extrapolation is long (typically from ~4–8 M urea to 0 M), it yields physically meaningful and reproducible values when data quality is high and the linear fit is robust. Comparison of ΔG°w values across a series of mutants differing by a single amino acid (produced by site-directed mutagenesis) directly quantifies the contribution of specific residues to protein stability — the approach used in the mutagenesis studies of hydrophobic and hydrogen-bonding contributions discussed in Chapter 4.9.

Thermal denaturation provides complementary thermodynamic information. Keq values can be derived from thermal denaturation curves, as from chemical denaturation curves, by monitoring an observable as a function of temperature. The van't Hoff equation — ln Keq = −ΔH°/RT + ΔS°/R, or equivalently d ln Keq/d(1/T) = −ΔH°/R — shows that a semi-log plot of ln Keq vs. 1/T yields a straight line whose slope is −ΔH°/R and whose y-intercept is +ΔS°/R. The slope directly gives the van't Hoff enthalpy ΔH°vHoff, and ΔS° at any biologically relevant temperature can then be calculated from ΔG° = ΔH° − TΔS°. Importantly, while the y-intercept of the van't Hoff plot nominally gives ΔS°/R, the physical meaning of this intercept is limited because it corresponds to 1/T = 0 (i.e., T → ∞), making the extrapolation physically unreasonable; ΔS° is therefore better calculated indirectly.

The melting temperature Tm — the temperature at which the protein is half-denatured (fD = fN = 0.5), so that Keq = 1 and ΔG° = 0 — is identified as the peak of the Cp vs. T curve in differential scanning calorimetry (DSC). The area under the DSC Cp curve gives the calorimetric enthalpy ΔHcal. Comparison of ΔHvHoff (from the van't Hoff plot) with ΔHcal (from DSC) is a powerful test of the two-state model: if they are equal, the transition is genuinely two-state with no significantly populated intermediates; if ΔHcal < ΔHvHoff, there may be multiple independent units unfolding; and if ΔHcal > ΔHvHoff, intermediates or aggregation may complicate the transition. Together, chemical and thermal denaturation methods provide a comprehensive experimental toolkit for determining all thermodynamic parameters characterizing a protein's native-state stability.