*3.2. Measurements of Global Kinetic Parameters of the Folding*/*Unfolding Reaction with 1D High-Pressure NMR*

∆*V* <sup>0</sup> and ∆*G* <sup>0</sup> are thermodynamic parameters at atmospheric pressure, characteristics of the system at equilibrium. But a kinetic analysis of the folding/unfolding reaction is needed to obtain information on the transition state (usually described not as a unique conformer but as an ensemble of conformers, hence the term of "Transition State Ensemble" (TSE) used for proteins) of the reaction. Even though characterizing transition states in protein folding constitutes an essential step in the puzzle [48], the relations between the protein sequences, their 3D structures, and the structure (at least the hydration state) of their TSE are not yet well understood. Thus, the HP-NMR comparative study of the folding of Titin I27 module and DEN4-ED3 domain from the viral envelope of the dengue virus, two proteins with unrelated sequences but sharing a common Ig-like fold, shows similar folding intermediates but very different TSE, the transition state of Titin I27 being considerably less hydrated than the one of DEN4-ED3 [49]. Such analysis relies on the measurement of kinetic parameters after perturbation (*P*-jump) of the thermodynamic equilibrium between the folded and unfolded conformers of the protein at a given pressure, yielding the rates of folding and unfolding at atmospheric pressure. Moreover, these studies give access to the values of the activation volume between the folded or unfolded state and the TSE, related to the hydration state of the TSE.

Due to the very large volumes of activation involved in the folding/unfolding reaction, high pressure can considerably slow down the rate of folding and also possibly unfolding [13,50]. Thus, although the completion of a folding/unfolding reaction is usually a few seconds at atmospheric

pressure, it can take up to a few hours at high pressure (about 12 h for ∆ + PHS SNase at pressure above 1 kbar [12]). This is more than enough for the use of real-time 1D NMR to follow the folding/unfolding reaction after a *P*-jump, until steady state is achieved. Real-time 1D NMR spectroscopy consists in recording with time a series of 1D NMR spectrum at a constant repeating rate. After a *P*-jump, it is then possible to observe the exponential decrease with time of a resonance corresponding to the folded species, or the exponential growth with time of a resonance corresponding to the unfolded states, until the steady state is reached. In the case of Titin I27, we observed the exponential growth with time of the resonance at 0.96 ppm corresponding to the methyl groups of the unfolded states (Figure 3A). Alternatively, the exponential decrease with time after a positive *P*-jump of the well-resolved resonances of the shielded methyl protons can be used as a probe for such measurements, giving residue-specific information on the folding kinetics related to the local hydration of the TSE (see further) [32].

τ τ τ ∆ ‡ ∆௨ ‡ ∆ ‡ ∆ ∆ ‡ ∆ **Figure 3.** Measuring global kinetic parameters for the folding/unfolding reaction with real-time 1D HP-NMR spectroscopy. (**A**) Two 1D NMR proton spectra (methyl groups resonances) recorded on Titin I27 (same conditions as in Figure 1) just after a 700 to 900 bar P-Jump (black trace) and 2 h after the P-jump (red trace) 1). These two spectra represent the extreme points of a series of sixty spectra of 2 min each recorded over a period of 2 h. The arrow indicates the increase of the resonance at 0.96 ppm that corresponds to methyl groups in the unfolded species. (**B**) Measurement of the relaxation time, τ, at 900 bar, through the fit of the exponential growth of this methyl resonance. (**C**) Exponential growths of the resonance corresponding to the methyl groups in the unfolded states after successive 200 bar P-Jumps between 300 and 1900 bar, the pressure range where Titin I27 unfolds. Relaxation times τ(p) can be measured from these experiments for the different pressures. (**D**) "Chevron plot" of ln(τ) measured at different pressures: the fit with Equation (8) allows to extract the folding or unfolding kinetic rate constants kf0 and ku0, respectively, and the activation volume of folding ∆*V* ‡ *f* 0 or of unfolding ∆*V* ‡ *u*0 at atmospheric pressure. (**E**) Volumetric diagram obtained for Titin I27 domain, displaying average values of ∆*V* ‡ *f* 0 (plain bar) and ∆*V* 0 (open bar). The value of the ratio ∆*V* ‡ *f* 0 /∆*V* 0 close to 1 deduced from this diagram indicates a dehydrated TSE.

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*τ*

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*τ*(p) = 1/( u(p) + f(p))

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Concerning the experimental aspects for the realization of the *P*-jump, these experiments are more demanding than those used for the steady-state analysis. Indeed, the time needed for the sample pressurization should be negligible with respect to the time needed by the folding/unfolding reaction to reach the plateau. For instance, the 200 bar *P*-jumps used for acquiring the data presented in Figure 3 needed about 10 s when performed with the Deadalus InnovationTM Xtreme electric HP-pump. In this particular case, the time needed to reach the steady state after equilibrium was about 40 min (Figure 3C), so that the pressurization time can be safely neglected. For proteins with shorter folding relaxation times, Kremer et al. [51] have circumvented this limitation by pre-pressurizing a reservoir, upstream of the high-pressure cell, containing a large volume of mineral oil, much greater than the volume corresponding to the pressurization line and the high-pressure cell itself. Thus, opening an electric valve placed in between the reservoir and the cell allows an almost immediate (in the millisecond range) equilibration of the pressure between the reservoir and the sample cell. Since the volume of pressurization liquid is far greater in the reservoir than in the rest of the set-up, the final pressure reached in the sample cell when opening the valve is virtually the initial pressure in the reservoir.

The relaxation time characteristic of the kinetics after a given *P*-jump (τ (p) = 1/(*k*u(p) + *k*f(p)), where *k*u(p) and *k*f(p) are the unfolding and the folding rates at the pressure p reached at the end of the *P*-Jump, can be extracted from the fit of the exponential growth of this resonance (Figure 3B). Then, it becomes possible to extract the values at atmospheric pressure of *k*u0 and *k*f0, as well as those of the activation volume of unfolding ∆*V* ‡ *u*0 (or folding, ∆*V* ‡ *f* 0 ), by measuring this relaxation time after different p-jumps, between different pressures in the range where the protein unfolds (Figure 3C):

At a given pressure: τ(p) = 1/(*k*u(p) + *k*f(p)) with *k <sup>f</sup>* (*p*) = *k <sup>f</sup>* <sup>0</sup> *e* −*p*∆*V* ‡ *f* 0 /*RT* and *<sup>k</sup>u*(*p*) <sup>=</sup> *<sup>k</sup>u*0*<sup>e</sup>* −*p*∆*V* ‡ *u*0 /*RT* τ(p) can be rewritten as:

$$\pi\_{(p)} = \left[ k\_{l0} e^{(\frac{-p\Delta V\_{p0}^{\ddagger}}{RT})} + k\_{f0} e^{(\frac{-p\Delta V\_{f0}^{\ddagger}}{RT})} \right]^{-1} \tag{7}$$

The value of ∆*V* 0 , the volume difference between the folded and unfolded states measured at equilibrium ∆*V* 0 (= ∆*V*<sup>f</sup> − ∆*V*u), and *K*eq (= *k*f0/*k*u0) can be measured from the steady state experiments described above. One can then decrease the number of parameters for the fit:

$$\pi\_{(p)} = \left[ k\_{\rm u0} e^{(\frac{-p\Delta V\_{\rm u0}^{\ddagger}}{RT})} + k\_{\rm u0} K\_{\rm cq} e^{(\frac{-p(\Delta V^{0} + \Delta V\_{\rm u0}^{\ddagger})}{RT})} \right]^{-1} \tag{8}$$

Only two variables need to be fitted: *k*u0 (or *k*f0), the unfolding (or folding) rate at atmospheric pressure, and ∆*V* ‡ *u*0 (or ∆*V* ‡ *f* 0 ), the activation volume for unfolding (or folding) at atmospheric pressure.

The fit is usually performed on a plot of ln(τ) as a function of pressure, displaying the characteristic "chevron plot" pattern (Figure 3D). In the case of Titin I27, the activation volume ∆*V* ‡ *f* 0 is close to the equilibrium ∆V <sup>0</sup> value (Figure 3E), suggesting a dehydrated TSE where most of the native voids are present.
