*3.1. Steady-State Measurements of Global Thermodynamic Parameters with 1D High-Pressure NMR*

The good dispersion usually observed in the <sup>1</sup>H-NMR spectrum of a folded protein is essentially due to the extreme sensibility of the proton NMR resonances (or chemical shifts) to through-space effects of neighboring groups. These effects vanished when the protein unfolds, and the <sup>1</sup>H-NMR spectrum becomes poorly resolved. For instance, the well-resolved regions characteristic of resonances of the methyl groups, or of the amide groups, in the <sup>1</sup>H spectrum of a folded protein will collapse upon unfolding (Figure 1).

**Figure 1.** Evolution upon pressure of the 1D <sup>1</sup>H-NMR spectrum of Titin I27 Ig-like domain. Stacked plot of 1D spectra recorded as a function of pressure at 600 MHz and 298 K on a 1 mM sample of Titin I27 in Tris buffer pH 7.0, 1 mM DTT. A 1.7 M sub-denaturing concentration of GuHCl has been added to the sample in order to decrease the protein stability and to observe complete unfolding in the 1–2500 bar pressure range allowed by the experimental set-up (zirconium oxide ceramic tubes, Daedalus InnovationTM). The solid-line frames delimit the regions corresponding to HN amide (red frame) and CH<sup>3</sup> methyl group resonances (blue frame). The insert corresponds to a zoom on the indole resonances region (black dashed-line frame) showing the decrease with pressure of the HN indole resonance of Trp-34 in the folded state (F) and the concomitant increase of the same resonance in the unfolded states (U).

≈ Thus, the evolution of the 1D NMR spectrum allows us to monitor the high-pressure denaturation of a protein, as depicted in Figure 1 for the I27 Immunoglobin-like domain of the sarcomeric protein Titin [44]. As an effect of the energy barrier between the folded and unfolded protein (≈2 kcal/mol in the experimental conditions reported in Figure 1), these species are in slow exchange with regard to the NMR timescale: we observe the disappearance of resonances belonging to the native state, with the concomitant appearance of new peaks that correspond to the spectrum of the unfolded states.

As it can be observed for the indole resonance of Trp-34 (Figure 2), the decrease in pressure of the peak corresponding to the folded (F) state, as well as the increase of the peak corresponding to the unfolded states (U), can be generally well-fitted by a sigmoidal curve [45,46] characteristic of a two-state equilibrium in the form of:

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F U

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−∆

**Figure 2.** Monitoring the unfolding reaction of titin I27 domain with 1D HP-NMR spectroscopy (**A**): 1D HN indole region of the proton NMR spectra of Titin I27 recorded at increasing pressure. F stands for the resonance in the folded state, U for the resonance in the unfolded state. (**B**): denaturation curves obtained from the fit of the evolution with pressure of the native (open circle) and the denatured (filled circle) indole resonance of tryptophan-34 with a two-state equilibrium equation. Similar values of ∆*V* 0 are found for both fits.

∆ ∆ ∆ with a characteristic F *<sup>K</sup><sup>u</sup>* ⇋ *Kf* U equilibrium constant of:

$$K\_{\rm eq} = k\_{\rm f(p)} \mu\_{\rm u(p)} = \text{[U]} \text{[F]} \tag{1}$$

∆β − δ δ ∆ where *k*f(p) and *k*u(p) stands for the folding and the unfolding rate constants at a given pressure p. *K*eq can be also expressed from the Boltzmann equation as:

$$K\_{\rm eq} = \exp(-\Delta G\_{\rm eq}/\mathcal{R}T) \tag{2}$$

where the free energy change can be expressed as a Taylor expansion, truncated at the second order term:

$$
\Delta G\_{\rm eq} = G\_{\rm U} - G\_{\rm F} = \Delta G^0 + \Delta V^0 (\mathbf{p} - \mathbf{p}\_0) - 1/2 \,\Delta \beta V^0 (\mathbf{p} - \mathbf{p}\_0)^2 \tag{3}
$$

Here ∆*G*eq and ∆*G* <sup>0</sup> are the Gibbs-free energy changes from F to U at pressure p and p<sup>0</sup> (p<sup>0</sup> = 1 bar), respectively; ∆*V* 0 is the partial molar volume change; ∆β is the change in compressibility coefficient (∆β = −(1/*V* 0 ) \* δ*V*/δp), R is the gas constant, and *T* is the absolute temperature. It has been shown that for proteins the difference in compressibility between native and denatured states is negligible [47]. Thus, the expression of ∆*G*eq simplifies to:

$$
\Delta G\_{\rm eq} = \Delta G^0 + \Delta V^0 (\mathbf{p} - \mathbf{p}\_0) \tag{4}
$$

Using NMR spectroscopy, the observable will be *I*, either the intensity (peak height) or the integral of a peak corresponding to either the folded species or of the unfolded species. In the present case (Figure 2), we chose to follow either the decrease of the peak intensity corresponding to the HN indole resonance of Trp-34 in the folded species or the increase of the corresponding resonance in the unfolded species. Alternatively, one can follow the increase of the peak at 0.86 ppm corresponding to the resonances of methyl groups in the unfolded species. Thus, the equilibrium constant can be written as:

$$K\_{eq} = \frac{[\mathcal{U}]}{[F]} = \frac{I\_F - I}{I - I\_{\mathcal{U}}} \tag{5}$$

If we choose to follow the increase with pressure of a resonance corresponding to the unfolded species in the 1D NMR spectrum, *I<sup>F</sup>* stands for the intensity of the corresponding NMR line in the folded spectrum at 1 bar (*I<sup>F</sup>* = *Imin*), whereas *I<sup>U</sup>* corresponds to the intensity of the same line at high pressure, when the protein is fully unfolded (*I<sup>U</sup>* = *Imax*). Combining this equation with Equations (2) and (4) gives the characteristic equation for a two-state equilibrium:

$$I = \frac{I\_F + I\_{II}e^{-\left[\Delta G^0 + (p - p0)\Delta V^0\right]/RT}}{1 + e^{-\left[\Delta G^0 + (p - p0)\Delta V^0\right]/RT}}\tag{6}$$

Fitting either the sigmoidal decrease with pressure of the indole resonance in the folded state or the sigmoidal increase of the indole resonance in the unfolded state (Figure 2) yield "global" values for ∆*V* 0 of unfolding (−84 ± 5 mL/mol and −82.4 ± 5 mL/mol, respectively) and for ∆*G* 0 (2.14 ± 0.12 kcal/mol and 2.01 ± 0.13 kcal/mol, respectively), under the conditions of the study (pH = 7, 25 ◦C, 1.7M GuHCl). Note that the values extracted from the two different fits fall within experimental uncertainties, confirming the two-state equilibrium for the folding/unfolding reaction. Slightly different values (∆*V* <sup>0</sup> = −78.8 ± 4 mL/mol, ∆*G* <sup>0</sup> = 2.26 ± 0.26 kcal/mol) can be measured at equilibrium when referring to the resonance corresponding to methyl groups in the unfolded states (0.96 ppm). This indicates that we are measuring "apparent" values for these thermodynamic parameters that depend of course on the global stability of the protein but that are also influenced by the local stability sensed by a given resonance in a given environment. As we will see further, this is of paramount importance for the description of protein folding pathways.
