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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
J Mol Biol. Author manuscript; available in PMC 2010 October 2.
Published in final edited form as:
PMCID: PMC2754388

Quantitative Determination of Site-Specific Conformational Distributions in an Unfolded Protein by Solid State Nuclear Magnetic Resonance


Solid state nuclear magnetic resonance (NMR) techniques are used to investigate the structure of the 35-residue villin headpiece subdomain (HP35) in folded, partially denatured, and fully denatured states. Experiments are carried out in frozen glycerol/water solutions, with chemical denaturation by guanidine hydrochloride (GdnHCl). Without GdnHCl, two-dimensional solid state 13C NMR spectra of samples prepared with uniform 13C labeling of selected residues show relatively sharp crosspeaks at chemical shifts that are consistent with the known three-helix bundle structure of folded HP35. At high GdnHCl concentrations, most crosspeaks broaden and shift, qualitatively indicating disruption of the folded structure and development of static conformational disorder in the frozen denatured state. Conformational distributions at one residue in each helical segment are probed quantitatively with three solid state NMR techniques that provide independent constraints on backbone ϕ and ψ torsion angles in samples with sequential pairs of carbonyl 13C labels. Without GdnHCl, the combined data are well fit by α-helical conformations. At [GdnHCl] = 4.5 M, corresponding to the approximate denaturation midpoint, the combined data are well fit by a combination of α-helical and partially extended conformations at each site, but with a site-dependent population ratio. At [GdnHCl] = 7.0 M, corresponding to the fully denatured state, the combined data are well fit by a combination of partially extended and polyproline II conformations, again with a site-dependent population ratio. Two entirely different models for conformational distributions lead to nearly the same best-fit distributions, demonstrating the robustness of these conclusions. This work represents the first quantitative investigation of site-specific conformational distributions in partially folded and unfolded states of a protein by solid state NMR.

Keywords: solid state NMR, protein folding, villin, HP35, polyproline II


Properly folded proteins are crucial to normal biological functions, while their misfolding can lead to disease. The phenomenon of protein folding, defined as the kinetic or thermodynamic progression from a structurally disordered to a structurally ordered state of a protein, is therefore centrally important to many biological and medical questions. Since protein folding involves both folded and unfolded states, the properties of both of these states can influence the thermodynamic, kinetic, and mechanistic aspects of protein folding1. The folded states of many proteins are well characterized structurally, primarily through x-ray crystallography and multidimensional nuclear magnetic resonance (NMR) of proteins in fluid solution. Structural properties of unfolded states are more difficult to characterize, because unfolded states adopt a distribution of conformations and because experimental techniques for determining conformational distributions in chemically complex macromolecules are not well established. Partial information about unfolded protein structures is available from solution NMR techniques2-5, including measurements of nuclear Overhauser effects6, chemical shifts7, scalar couplings, transverse spin relaxation rates8, paramagnetic relaxation enhancements9-11, and residual dipole-dipole couplings12-14. Among other findings, these techniques show that unfolded states may contain residual secondary structure15, sequence-specific preferential population of turn structures16, and greater long-range structural order than expected for a simple random coil9-11. Solution NMR techniques provide a mixture of site-specific and long-range conformational constraints that are averaged over multiple, rapidly exchanging conformations in an unfolded state. In certain cases, solution NMR data are well described by models for unfolded states in which local backbone conformational distributions are similar to distributions derived from folded protein structure databases17-19, perhaps with additional population of extended and polyproline II (PPII) conformations12. Additional long-range constraints on unfolded state structure are available from scattering techniques20,21, which serve as measurements of the overall dimensions of a polypeptide chain, but not the conformations at specific sites. Infrared spectroscopy with isotopic labeling can also provide site-specific structural information for unfolded states, in addition to kinetic information regarding protein folding processes22.

This paper demonstrates an alternative experimental approach to investigations of unfolded protein structure that yields quantitative constraints on site-specific conformational distributions. In this approach, solid state NMR techniques are applied to partially or fully unfolded proteins in frozen solutions, with the folding state controlled by solvent conditions such as denaturant concentration. By carrying out measurements on frozen solutions, dynamic averaging of NMR parameters is avoided and the full structural heterogeneity of an unfolded state can be explored. In addition, full (rather than residual) dipole-dipole couplings and anisotropic chemical shift tensors are available as structurally dependent nuclear spin interactions in the solid state. Multiple independent solid state NMR measurements can be carried out on a single protein sample over an extended period of time, provided that the sample remains frozen, so that the same approach would apply to freeze-trapped transient species. In principle, each data point in each measurement represents a constraint on the conformational distribution at a given site in the protein, in particular a site where isotopic labels such as 15N or 13C have been introduced. If enough data points are acquired, the full conformational distribution can in principle be determined, using appropriate models for the conformational distribution and appropriate simulations of the solid state NMR measurements. Several related studies in which solid state NMR data were used to extract conformational distributions in disordered biopolymers23-25 and synthetic polymers26-28 have been reported.

Experiments described below were carried out on the 35-residue villin headpiece subdomain (HP35), a protein that has been studied extensively as a simple model system for protein folding by a variety of experimental22,29-40 and computational41-53 methods. Earlier studies have shown that HP35 (or the 36-residue version, HP36) has the primarily helical folded state shown in Fig. 1a31,36,37,40,54, with an unfolding temperature near 70° C in the absence of chemical denaturant35, and that the time scale for equilibration of folded and unfolded states is approximately 5 μs near room temperature in aqueous solution22,34,39. In our experiments described below, partially and fully unfolded states of HP35 were prepared by addition of the denaturant guanidine hydrochloride (GdnHCl) to glass-forming glycerol/water solutions, and solid state NMR measurements were carried out at -120° C. As previously reported32 and confirmed below, two-dimensional (2D) solid state 13C NMR spectra of samples prepared with uniform 13C labeling of selected residues indicate that the three-helix folded state of HP35 is preserved in these frozen solutions in the absence of denaturant, but that addition of GdnHCl disrupts the folded state, with a denaturation midpoint at [GdnHCl]mid ≈ 4.5 M. Interestingly, this low-temperature denaturation midpoint is not significantly different from the denaturant midpoint at higher temperatures32,35, suggesting that our experiments on frozen solutions are not without relevance to properties of HP35 in fluid solutions at higher temperatures. The temperature-independence of the denaturation midpoint is supported by low-temperature circular dichroism (CD) data described below.

Figure 1
(a) Structure of folded HP3531, showing residues that are uniformly isotopically labeled for 2D spectra in Figure 2. (b) Atomic representation of the A49-V50 segment within the HP35 sequence, with 13C-labeled carbonyl sites in green. Solid state NMR measurements ...

We report results of quantitative measurements of backbone conformational distributions at one site in each helical segment (V50, A57, and K70), using three independent solid state NMR techniques55-58. These techniques are all applicable to polypeptides that are 13C-labeled at two sequential backbone carbonyl sites (A49/V50, S56/A57, and L69/K70 in the experiments reported below), and provide constraints on the ϕ and ψ torsion angles between the labeled sites, as shown in Fig. 1b. Analyses of the quantitative conformational data verify the α-helical conformations at [GdnHCl] = 0.0 M, indicate mixtures of α-helical and extended, β-strand-like conformations in the partially denatured state at [GdnHCl] = 4.5 M, and reveal conformational distributions that include both PPII-like and extended conformations in the fully denatured state at [GdnHCl] = 7.0 M. Site-specific differences in the conformational distributions in the partially and fully denatured states are discussed below.


Qualitative picture of chemical denaturation in frozen solutions from 2D solid state NMR

Chemical denaturation of HP35 in frozen solutions was first monitored at a qualitative level by 2D solid state 13C NMR spectroscopy of samples in which the protein was uniformly 15N and 13C labeled at selected residues. For these experiments, HP35 (residues 42-76 of the 76-residue chicken villin headpiece domain35, amino acid sequence L42SDEDFKAVFGMTRSAFANLPLWKQQNLKKEKGLF76, with the underlined segments forming α-helices in the folded state) was labeled at A49, G52, M53, F58, P62, and L69 (HP35-U-AGMFPL) or V50, A57 and L69 (HP35-U-VAL).

Figure 2 shows 2D spectra of HP35-U-AGMFPL and HP35-U-VAL in frozen glycerol/water solutions with [GdnHCl] = 0.0 M, 4.0 M, 5.0 M, and 7.0 M. These spectra were obtained with magic-angle spinning (MAS), proton decoupling in the two spectral dimensions, and radio-frequency-driven recoupling59 (RFDR) during the 1.71-1.79 ms mixing periods. Sample temperatures were -120° C. At [GdnHCl] = 0.0 M, crosspeaks with 2.0-2.5 ppm linewidths (full width at half maximum, FWHM) were observed that connect NMR lines of all directly-bonded 13C pairs. The relatively narrow crosspeaks, similar in width to crosspeaks in 2D solid state NMR spectra of antibody-bound peptides in frozen solution60,61 and of transmembrane peptides in phospholipid bilayers at low temperatures62 under similar measurement conditions, indicate a well-ordered, rigid protein conformation in a non-crystalline environment. 13C NMR chemical shifts extracted from the 2D spectrum at [GdnHCl] = 0.0 M are given in Table 1. These chemical shifts are consistent with the expected α-helical folded structure of HP35. In particular, secondary shifts (i.e., differences between observed 13C chemical shifts in HP35 and “random coil” shifts for unstructured peptides in fluid solutions) for CO, Cα, and Cβ sites are positive, positive, and negative, respectively, for A49, V50, A57, F58, and L69, as expected for residues in helical segments63-67. Secondary shifts for G52, M53, and P62 are non-helical, also as expected based on the folded HP35 structure. 13Cα and 13Cβ chemical shifts in Table 1 agree to within a 0.5 ppm root-mean-squared deviation with values extrapolated to low temperature from solution NMR measurements on HP35 in aqueous solution in the 10-30°C range (unpublished data).

Figure 2
2D 13C NMR spectra of HP35-U-AGMFPL (a) and HP35-U-VAL (b) in frozen glycerol/water with the indicated GdnHCl concentrations. Spectra were obtained at 100.8 MHz 13C NMR frequency. Chemical shift assignments at [GdnHCl] = 0.0 M and partial assignments ...
Table 1
13C NMR chemical shifts (ppm relative to tetramethylsilane) in the folded state of HP35 in frozen glycerol/H2O solution. Uncertainties are approximately ±0.3 ppm. Shifts in parentheses are random coil values for peptides in fluid aqueous solution, ...

With addition of GdnHCl, crosspeaks in the 2D spectra shift and broaden, but the extent of broadening is site-dependent. Signals from M53, F58, and L69 shift and broaden significantly, so that their Cα/Cβ crosspeaks are unresolved at [GdnHCl] = 7.0 M. These spectral changes reflect the disruption of the helical HP35 folded structure and the development of distributions of conformations (and hence distributions of 13C chemical shifts for individual carbon sites) in the denatured state. Signals from A49 remain relatively sharp at [GdnHCl] = 7.0 M (2.5 ppm linewidths), although the 13Cα chemical shift for A49 is reduced by 2.0 ppm and the 13Cβ chemical shift is increased by 0.6 ppm (Figure 2a). As previously reported32, signals from V50 in HP35-U-VAL have a similar behavior (Figure 2b). The CO/Cα crosspeak for G52 moves approximately 2 ppm towards a lower 13Cα chemical shift value at [GdnHCl] = 7.0 M, but is relatively narrow in the denatured state. The Cα/Cβ crosspeak of A57 is broader at [GdnHCl] = 7.0 M, shifting towards lower 13Cα chemical shift values but retaining measurable intensity at the position of the folded state. Variations in crosspeak widths in the denatured state suggest variations in conformational distributions. However, quantitative analysis of crosspeak shapes is precluded by the uncertain dependence of individual 13Cα and 13Cβ chemical shifts on sidechain torsion angles and non-local interactions.

Crosspeaks for P62 show comparatively minor changes in chemical shifts and linewidths, reflecting the fact that proline residues have limited conformational freedom even in “random coil” peptides. The 3.6 ppm and 4.2 ppm differences between Cβ and Cγ chemical shifts in the folded and unfolded states, respectively, indicate trans peptide bonds between L61 and P62 in both states68.

The evolution of crosspeak positions and shapes with increasing [GdnHCl] indicates that the denaturation midpoint lies between 4 M and 5 M. This is especially clear for the Cα/Cβ crosspeaks of A49, V50, and L69 and the CO/Cα crosspeaks of V50 and G52. Quantitative conformational measurements described below also indicate a denaturation midpoint near [GdnHCl] = 4.5 M, consistent with CD data at -40°C reported previously32. CD measurements at -55°C discussed below also indicate [GdnHCl]mid = 4.3 ±0.1 M.

Quantitative constraints on backbone conformational distributions

For quantitative conformational measurements, three samples were prepared with 13C labels at sequential backbone carbonyl sites: A49 and V50 (HP35-CO-AV), S56 and A57 (HP35-CO-SA), and L69 and K70 (HP35-CO-LK). As shown in Figure 1b, labeling of these carbonyl sites allows constraints on the ϕ and Ψ torsion angles of V50, A57, and K70 to be obtained from solid state NMR. Three techniques were used to obtain independent constraints, namely 2D MAS exchange spectroscopy57,58,69 (2DEXMAS), constant-time double-quantum-filtered dipolar dephasing56 (CTDQFD), and double-quantum chemical shift anisotropy spectroscopy55 (DQCSA). Radio-frequency (rf) pulse sequences for 2DEXMAS, CTDQFD, and DQCSA measurements are shown in Figure 3. The accuracy of these techniques in conformational studies has been demonstrated on model peptides23,55,58, and the same techniques have been applied in previous studies of peptide/antibody complexes60 and helix-forming peptides23,56 in frozen solutions, amyloid fibrils70,71, and other systems72. Measurements were carried out in frozen glycerol/water with [GdnHCl] = 0.0 M, 4.5 M, and 7.0 M, corresponding to the fully folded state, the approximate denaturation midpoint, and the fully denatured state. Experimental data for HP35-CO-AV at [GdnHCl] = 0.0 M are shown in Figure 4.

Figure 3
Rf pulse sequences used in quantitative conformational measurements. (a) 2DEXMAS sequence. Arrows mark time points that are actively synchronized to the MAS rotor phase, making either τmix and t1mix multiples of the rotor period in SI ...
Figure 4
2DEXMAS, CTDQFD, and DQCSA data for HP35-CO-AV in frozen glycerol/water with [GdnHCl] = 0.0 M. (a) 2DEXMAS spectrum at 2.50 kHz MAS frequency (left), showing crosspeaks among MAS sideband lines of A49 and V50 carbonyl sites. Crosspeak volumes are analyzed ...

The 2DEXMAS spectrum (Figure 4a), obtained with the rotor-synchronized pulse sequence in Figure 3a with a 500 ms mixing period and relatively slow MAS, contains crosspeaks that connect carbonyl MAS sideband lines. Crosspeak volumes depend on the relative orientation of the two labeled carbonyl chemical shift anisotropy (CSA) tensors (Figure 1c), which in the case of HP35-CO-AV depends on the ϕ and ψ torsion angles of V50 (Figure 1b). Thus, the crosspeak volumes in the 2DEXMAS spectrum of HP35-CO-AV place constraints on the conformational distribution of HP35 at V50. Crosspeaks connecting MAS sidebands of order 0, ±1, and ±2 had measurable intensities (signal-to-noise ≈ 10 for the strongest crosspeaks), yielding 20 data points (i.e., crosspeak volumes) from each 2DEXMAS spectrum.

CTDQFD data (Figure 4b) show the build-up and decay of double-quantum-filtered 13C NMR signal amplitudes from the carbonyl labels. Excitation and evolution of double-quantum (DQ) coherences occurs under radio-frequency-driven dipolar recoupling59 (RFDR) sequences, making CTDQFD data sensitive to the 13C-13C dipole-dipole coupling, or the 13C-13C distance, which in turn depends on ϕ. The RFDR sequences consist of trains of rotor-synchronized 13C π pulses, with one π pulse in every two MAS rotor periods. Under these experimental conditions, the RFDR recoupling mechanism depends on differences in CSA tensor orientations, making the CTDQFD data additionally dependent on ψ. CTDQFD data were obtained in a “constant-time” manner (i.e., by incrementing M - N while keeping M + N constant in Figure 3b), so that these data were insensitive to transverse 13C spin relaxation rates and could be analyzed quantitatively for 13C-13C distances up to at least 4.2 Å, covering the entire range of possible ϕ values.

DQCSA data (Figure 4c) show the dephasing of DQ coherences, excited by an RFDR sequence, under the sum of CSA interactions of the A49 and V50 labels (in the case of HP35-CO-AV). The sum of the CSA interactions for a dipole-coupled pair of carbonyl 13C labels depends on the intervening ϕ and ψ angles, making the DQCSA data sensitive to both ϕ and ψ. CSA interactions were recoupled in a constant-time manner by the three 13C π pulses in the 2τR period between DQ excitation and mixing periods in Figure 3c, making the DQCSA data insensitive to 13C transverse relaxation, isotropic chemical shifts, and inhomogeneous broadening.

Analyses assuming a single conformation

2DEXMAS, CTDQFD, and DQCSA data can be simulated accurately, allowing quantitative analyses in terms of the distribution of ϕ and ψ torsion angles. The data were first analyzed with the assumption of a single conformation at V50, A57, or K70, i.e., single populated values of ϕ and ψ. Figure 5 shows contour plots of the χ2 deviation between experimental data for HP35-CO-AV and simulated data as a function of the ϕ and ψ angles assumed in the simulations (see Materials and Methods for details of simulations and data analyses and for the definition of χ2). At [GdnHCl] = 0.0 M, χ2 plots for the individual measurements restrict the possible values of ϕ and ψ, but do not determine the best-fit values uniquely. However, when the constraints from all three measurements are combined, a single best-fit point is identified at ϕ, ψ = -70°±5°,-45°±5°, where the total χ2 has the value χ2 min = 69. This conformation for V50 in the folded state of HP35, determined from the solid state NMR data, agrees well with the values ϕ, ψ = -68°,-46° in the crystal structure of the N68H HP35 mutant at pH 6.7 (PDB file 1yrf), determined by Chiu et al.31 Similar analyses of 2DEXMAS, CTDQFD, and DQCSA data for HP35-CO-SA and HP35-CO-LK at [GdnHCl] = 0.0 M yield best-fit values ϕ, ψ = -65°±5°,-45°±5°for A57 (χ2 min = 38) and ϕ, ψ= -70°±5°,-45°±5° for K70 (χ2 min = 53). Values from the crystal structure are ϕ, ψ = -62°,48° for A57 and ϕ, ψ = -65°,-47° for K70.

Figure 5
Fits of 2DEXMAS, CTDQFD, DQCSA, and combined data for HP35-CO-AV, assuming a single backbone conformation at V50, shown as contour plots of the χ2 deviations between experimental and simulated data as functions of the ϕ and ψ torsion ...

For each sample, the total number of data points is Ndata = 48. For A57 and K70, |χ2min-(Ndata-5)| is less than or approximately equal to 2(Ndata5) as expected for a good fit with five adjustable parameters (two torsion angles and scaling factors for the three data sets). For V50, we attribute the excess in χ2 min to contributions from systematic errors. Systematic errors with magnitudes of approximately 70% of the typical random error due to noise in the solid state NMR data would explain the observed value of χ2 min. Given the high signal-to-noise ratios of the data, especially in CTDQFD and DQCSA measurements on HP35-CO-AV, systematic errors of this size are not surprising. Possible sources of systematic errors are discussed below (see Materials and Methods).

In contrast, analyses of the data at [GdnHCl] = 4.5 M with the assumption of a single conformation leads to χmin2 = 85, 102, and 112 for HP35-CO-AV, HP35-CO-SA, and HP35-CO-LK, respectively. At [GdnHCl] = 7.0 M, fitting the data to a single conformation leads to χmin2 = 128, 109, and 66, respectively. These larger values of χ2 min (with the same value of Ndata) suggest broader ranges of populated conformations.

Analyses assuming Gaussian conformational distributions

The 2DEXMAS, CTDQFD, and DQCSA data were analyzed with models in which the probability of adopting specific ϕ, ψ values at V50, A57, or K70 has the form P(ϕ,Ψ)=k=1NGhke[(ϕϕk)2+(ΨΨk)2]wk2, where NG is the number of Gaussian components in the probability distribution. For a given choice of the parameters hk, wk, ϕk, and ψk, simulated data were constructed from tables of 2DEXMAS, CTDQFD, and DQCSA simulations over a grid of ϕ and ψ values. Best-fit values of the parameters were determined by simulated annealing, using the Metropolis algorithm73 to vary these parameters (see Materials and Methods).

Table 2 summarizes the results of the simulated annealing calculations. Data at [GdnHCl] = 0.0 M were fit with a single Gaussian component (NG = 1), again resulting in best-fit ϕ and ψ values very close to the values extracted from the HP35 crystal structure, with a small Gaussian width. Attempts to fit the data at [GdnHCl] = 4.5 M and [GdnHCl] = 7.0 M with a single Gaussian component resulted in only minor reductions in χmin2 compared to the fits with a single ϕ,ψ point discussed above.

Table 2
Best fits of combined 2DEXMAS, CTDQFD, and DQCSA data with single-point and Gaussian models for ϕ,ψ distributions at V50, A57, and K70 in HP35. Uncertainties in fitting parameters are root-mean-squared deviations from best fit values, ...

At [GdnHCl] = 4.5 M, good fits were obtained with two Gaussian components (NG = 2). For all three sites, one component was centered at ϕ,ψ = -58°, ± 12°,-52° ± 3°, corresponding to α-helical conformations. The other component was centered at ϕ,ψ = -125° ± 24°,92° ± 16°, corresponding to extended, β-strand-like conformations. The best-fit population ratios of helical to extended conformations were approximately 1:1 for V50, 2:1 for A57, and 3:2 for K70. These population ratios are roughly consistent with the denaturation midpoint at [GdnHCl]mid ≈ 4.3 M indicated by the 2D 13C NMR spectra in Figure 2 and by CD data discussed below. The larger population of helical conformations for A57 is consistent with our earlier report that this site appears to lag behind other sites under GdnHCl denaturation.32

At [GdnHCl] = 7.0 M, good fits were also obtained with two Gaussian components, with one component centered at ϕ,ψ = -129° ± 13°,82° ± 8° and the other component centered at ϕ,ψ = -80° ± 5°,160° ± 5°. The first component represents extended conformations similar to those observed at [GdnHCl] = 4.5 M. The second component represents PPII conformations. Best-fit population ratios of extended to PPII conformations were approximately 3:2 for V50, 3:2 for A57, and 2:3 for K70.

These results suggest a common denaturation “pathway” for all three sites, in which the native helical conformation is first converted to a mixture of helical and extended conformations, and then to a mixture of extended and PPII conformations. Populations of PPII conformations at the denaturation midpoints are apparently low, as indicated by the fact that fits with three Gaussian components (NG = 3) did not improve the fits significantly (reduction in χ2 min less than 0.3, population in third Gaussian component less than 5%, at both [GdnHCl] = 4.5 M and [GdnHCl] = 7.0 M).

Site-specificity of conformational distributions

The best-fit two-Gaussian results suggest certain site-specific differences in the details of the conformational differences at the three sites, including differences in the relative populations of preferred conformations and differences in the precise ϕ,ψ values. However, differences in the best-fit parameters can not be interpreted definitively without an assessment of the uncertainties in these parameters. Markov Chain Monte Carlo (MCMC) simulations (see Materials and Methods) were used to determine the ranges of fitting parameters that correspond to acceptable fits to the combined 2DEXMAS, CTDQFD, and DQCSA data. Figure 6 shows the normalized sums of conformational distributions visited during MCMC simulations for V50, using a one-Gaussian model to fit data at [GdnHCl] = 0.0 M and two-Gaussian models to fit data at [GdnHCl] = 4.5 M and 7.0 M. The fact that the functions plotted in Figure 6 have clearly localized peaks demonstrates that only conformational distributions that are similar to the best-fit distributions have low χ2 values. Similar results were obtained in MCMC simulations for A57 and K70. Uncertainties given in Table 2 are root-mean-squared deviations of individual parameters from best-fit values during the MCMC simulations.

Figure 6
Results of MCMC simulations for HP35-CO-AV in frozen glycerol/water at [GdnHCl] = 0.0 M, 4.5 M, and 7.0 M (a,b,c). These simulations show the regions of the ϕ,ψ plane for V50 that are populated in two-Gaussian conformational distributions ...

To assess whether the apparent site-specific differences are statistically significant, we calculated the χ2 values for all three sites while performing the MCMC simulations for each individual site. Results are shown in Figure 7. For example, Figure 7a is a plot of Δχ2 = χ2 - χ2min values for V50, A57, and K70 at [GdnHCl] = 4.5 M during MCMC simulations that2 for sampled conformational distributions that are good fits to the V50 data. As expected, Δχ for V50 is small in Figure 7a, but the observation that Δχ2 is large for both A57 and K70 means that conformational distributions that fit the V50 data at [GdnHCl] = 4.5 M do not fit the A57 or K70 data at [GdnHCl] = 4.5 M. Therefore, the conformational distribution at V50 is significantly different from conformational distributions at A57 and K70.

Figure 7
Assessment of site-specific variations in conformational distributions. Each panel shows the values of Δχ2 [equivalent] χ2 - χ2 min for V50 (blue), A57 (green) and K70 (red) during MCMC simulations that explored the acceptable ...

At [GdnHCl] = 4.5 M, results in Figures 7a, ,7b,7b, and and7c7c show that conformational distributions at all three sites are significantly different. According to Table 2, the relative populations of helical and extended conformations are different (greatest population of extended conformations at K70), the widths of the helical component are different (greatest width at K70), and the location of the extended conformations in the ϕ,ψplane are different (especially for V50).

At [GdnHCl] = 7.0 M, results in Figures 7d and and7e7e show that the conformational distributions at V50 and A57 are not significantly different based on existing data, because all distributions with small Δχ2 for V50 also have small Δχ2 for A57 (although not all distributions with small Δχ2 for A57 also have small Δχ with small Δχ2 for V50). Results in Figure 7f show that the conformational distribution at K70 is significantly different. According to Table 2, the major difference is in the relative populations of extended and PPII conformations, with greater population of PPII conformations at K70.

Analyses assuming a multiple-point conformational distribution at V50

To test the robustness of our conclusions from one-Gaussian and two-Gaussian models, an alternative model was considered in which the conformational distributions were represented by a set of equally populated points in the ϕ,ψplane. The number of points and their locations were varied in a simulated annealing procedure (see Materials and Methods). The virtue of the multiple-point model is that it does not restrict the number or shape of the populated regions in the ϕ,ψ plane. A possible difficulty is that this model involves a large number of variable parameters, so that many distinct conformational distributions with similarly low χ2 values might in principle be found. In practice, this difficulty was not encountered, indicating that the solid state NMR data are sufficient to determine the main features of the actual conformational distributions.

Figure 8 shows the results of multiple-point fits to the combined 2DEXMAS, CTDQFD, and DQCSA data for V50. At [GdnHCl] = 0.0 M, the simulated annealing calculation converged to a single cluster of 50 points near ϕ,ψ= -70°,-50°, with χ2min = 62. At [GdnHCl] = 4.5 M, the calculation converged to three clusters, near ϕ,ψ= -70°,-50°(27 points), ϕ,ψ= -105°,80°(15 points), and ϕ,ψ = -140°,160 (two points), with χ2min = 66. At [GdnHCl] = 7.0 M, the calculation converged to four clusters, near ϕ,ψ= -120°,80°(27 points), ϕ,ψ= -75°,160°(17 points), ϕ,ψ= -75°,125°(two points), and ϕ,ψ= -170°,175°(two points), with χ2min = 63. Note that the χ2min values for the multiple-point model are nearly the same as the χ2min values in the Gaussian models. The locations of the two largest clusters at [GdnHCl] = 4.5 M and [GdnHCl] = 7.0 M are also in good agreement with the locations of the best-fit Gaussian components in two-Gaussian models. Thus, the multiple-point and Gaussian models lead to similar conclusions regarding the conformational distributions at V50. Similar agreement between multiple-point and Gaussian models was also obtained for all A57 and K70 data sets.

Figure 8
Results of multiple-point fits to the combined 2DEXMAS, CTDQFD, and DQCSA data for HP35-CO-AV in frozen glycerol/water at [GdnHCl] = 0.0 M, 4.5 M, and 7.0 M (a,b,c). Panels on the left are the best-fit configurations from simulated annealing. Final configurations ...

For A57 at [GdnHCl] = 7.0 M, MCMC simulations show that multiple-point distributions with small Δχ2 contain 12% population of α-helical conformations on the average, although the best-fit distribution has only 5% α-helical population. Similarly, MCMC simulations with a three-Gaussian model show an average α-helical population of 15%, although χ2min for the three-Gaussian model is not significantly lower than χ2min for the two-Gaussian model. This means that our data for HP35-CO-SA are compatible with partial population of α-helical conformations at A57 with [GdnHCl] = 7.0 M, as suggested by the 2D 13C NMR spectra of HP35-U-VAL in Figure 2 (see above), but do not require α-helical conformations. In contrast, MCMC simulations for HP35-CO-LK at [GdnHCl] = 7.0 M show an average α-helical population of 2% in the multiple-point model and 0% in the three-Gaussian model, ruling out significant population of α-helical conformations at K70.

Robustness of χ2 minimization by simulated annealing

With one exception, repeated simulated annealing runs (5-10 runs for each data set) converged to a unique best-fit conformational distribution for each data set. For V50 at [GdnHCl] = 7.0 M, repeated simulated annealing runs with the two-Gaussian model revealed that the combined 2DEXMAS, CTDQFD, and DQCSA data could also be fit with a sum of α-helical and extended conformations (i.e., Gaussian functions centered at ϕ,ψ = -70°,-46°and -111°,-107°, with a 3:7 population ratio, χ2min = 57). This alternative conformational distribution was discarded because it is inconsistent with the observation of a single Cα/Cβ crosspeak at a non-helical position for V50 at [GdnHCl] = 7.0 M in Fig. 2b. Population of α-helical conformations in multiple-point simulations for V50 at [GdnHCl] = 7.0 M (Figure 8f) was prevented by excluding points with both ϕ > -120° and ψ < -10°. No such restrictions were required in fits to all A57 and K70 data sets and to all other V50 data sets.

4. Discussion

Our results are summarized as follows: (1) 2D solid state NMR spectra of HP35-UAGMFPL and HP35-U-VAL (Figure 2) indicate a well folded structure in frozen solution without denaturant. 13C NMR chemical shifts are consistent with the three-helix structure determined previously by solution NMR36 and x-ray crystallography31, i.e. we find no evidence for “cold denaturation” of HP35 under our experimental conditions. The transition from folded to denatured crosspeak positions and widths occurs between 4.0 M and 5.0 M, consistent with the value [GdnHCl]mid ≈ 4.3 M inferred from CD data (see below); (2) Quantitative conformational measurements on HP35-CO-AV, HP35-CO-SA, and HP35-CO-LK with the 2DEXMAS, CTDQFD, and DQCSA techniques (Figures 3, ,44 and and5)5) yield ϕ,ψ values for V50, A57, and K70 at [GdnHCl] = 0.0 M that are in excellent agreement with the crystal structure31; (3) 2DEXMAS, CTDQFD, and DQCSA data at [GdnHCl] = 4.5 M and 7.0 M can be fit with a model in which only two regions of the ϕ,ψ plane are significantly populated, with relatively narrow Gaussian distributions in each region (Table 2). At [GdnHCl] = 4.5 M, the data are best fit by a sum of helical and partially extended conformations. At [GdnHCl] = 7.0 M, the data are fit by a combination of partially extended and PPII conformations. Thus, we identify a “pathway” for denaturation (Figure 6), in which helices are partially converted to extended segments near the denaturation midpoint, and then partially converted to PPII segments at higher denaturant concentrations; (4) Although the three sites examined in this work (one in each helix of the folded HP35 structure) follow the same pathway, our data indicate variations in the precise ratios of populated conformations and the precise ϕ,ψ values. In particular, relative populations of extended and α-helical conformations are site-dependent at [GdnHCl] = 4.5 M, relative populations of extended and PPII conformations are site-dependent at [GdnHCl] = 7.0 M, and the exact locations and widths of the populated regions in the ϕ,ψ plane are site-dependent; (5) Conclusions from the two-Gaussian fits to the 2DEXMAS, CTDQFD, and DQCSA data are corroborated by fits with a multiple-point model (Figure 8). The fact that two distinct models lead to similar conformational distributions supports the robustness of the data analysis.

Evidence for preferential population of PPII conformations in unfolded states of proteins and peptides has been obtained previously, dating back to the work of Tiffany and Krimm74, who used CD spectra as evidence that urea- and GdnHCl-denatured peptides and proteins adopt PPII conformations that are stabilized by hydrogen bonding of denaturant molecules to backbone carbonyl groups. More recent CD measurements by Whittington et al.75 on other peptides and proteins in the presence of urea support the earlier data from Tiffany and Krimm. In addition, extensive solution NMR studies of urea-denatured ubiquitin by Meier et al.12 suggest that the denatured state of a protein has site-specific conformational distributions that are similar to distributions predicted from ϕ,ψ distributions extracted from folded protein structures, but with enhanced population of both β-strand and PPII conformations. Preferential population of PPII conformations in unfolded states without chemical denaturant is supported by vibrational CD studies of poly-L-proline and poly-L-glutamate by Dukor and Keiderling76, solution NMR studies of various peptides by Kallenbach and coworkers77,78, and ultraviolet resonance Raman studies of polyalanine peptides by Asher et al.79 Computational studies of finite-length polyalanine peptides by Kentsis et al.80 indicate that formation of segments with β-strand and PPII secondary structure is more likely than formation of α-helical segments. Computational studies by Mezei et al.81 suggest that PPII conformations are preferred because PPII segments have lower solvation energies in purely aqueous solution than β-strand or α-helical segments.

The two preferred conformations indicated by our solid state NMR at [GdnHCl] = 7.0 M are similar to the enhanced β-strand and PPII conformations suggested by Meier et al. for ureadenatured ubiquitin12. As discussed above, our data additionally indicate that the PPII conformations develop primarily above the denaturation midpoint, “after” the development of extended conformations, and that the precise conformational distributions have site-specific differences. Although previous investigations of PPII conformations in unfolded and denatured proteins have been based on considerations that derive from the specific structural and chemical properties of polypeptides and their interactions with solvent and denaturant molecules, in our studies of HP35, PPII conformations emerge entirely from fitting the quantitative solid state NMR data with numerical simulations of nuclear spin dynamics. Our data analysis does not involve molecular modeling, empirical interpretations of spectroscopic signatures, structural data bases, or a priori assumptions regarding allowed or preferred conformations.

The experiments described above require low temperatures, to immobilize the protein and thereby allow solid state NMR measurements, and glycerol/water solutions, to prevent solvent crystallization that would perturb the protein structure. Sample cooling rates were approximately 10 deg/s (estimated from the time required for a 240 μl volume of glycerol/water in a transparent vial to solidify after immersion in liquid nitrogen). As the sample cools, the protein conformational distribution remains at thermal equilibrium until the sample temperature approaches the glass transition of the glycerol/water mixture. The effective temperature of our conformational measurements is then approximately -75° C. For HP35 in glycerol/water, the denaturation midpoint and thermodynamic parameters extracted from CD data were previously shown to be independent of temperature in the range from -40° C to 20° C.32 CD data for denaturation by GdnHCl in pure water and in glycerol/water are also nearly identical at 4° C.32,35 Figure 9 shows new low-temperature CD measurements that confirm the earlier results and extend these results to -55° C. Data in Figure 9a can be fit with a standard model32, yielding ΔG0 = 3.07 ± 0.08 kcal/mole and m = 0.75 ± 0.02 kcal/mole-M at 22° C and ΔG0 = 3.38 ± 0.39 kcal/mole and m = 0.79 ± 0.09 kcal/mole-M at -55° C (where ΔG0 is the free energy of unfolding at [GdnHCl] = 0.0 M and m is the slope of a linear dependence of unfolding free energy on [GdnHCl]). The denaturation midpoint, given by ΔG0/m, is 4.1 ±0.1 M at 22 °C and 4.3 ±0.2 M at -55 °C. The denaturation midpoint indicated by our solid state NMR measurements (effective temperature ≈ -75° C) is also approximately 4.5 M. The fact that [GdnHCl]mid is nearly independent of temperature from -75° C to 22° C is surprising, but suggests that our main conclusions regarding conformational distributions and denaturation pathways from solid state NMR measurements in frozen solutions may be relevant to the behavior of HP35 under more conventional conditions, although the precise populations of extended and PPII conformations may vary with solvent conditions and temperature. The increasingly positive signal at 218 nm in the CD spectrum of denatured HP35 at lower temperatures (Figure 9b) suggests an increase in PPII population74,75. The increasingly negative CD signal at 222 nm in the folded state at lower temperatures suggests a stabilization of the helical secondary structure. Additional experiments will be required to assess definitively the relevance of measurements in glycerol/water at low temperatures to conformational preferences under more conventional conditions.

Figure 9
Denaturation of HP35 monitored by CD. (a) Dependence of mean molar ellipticity at 222 nm on [GdnHCl] for HP35 in glycerol/water at 22° C (circles, solid lines) and -55° C (squares, dashed lines). Lines are fits to a standard two-state ...

Standard treatments of single-domain protein folding thermodynamics lead to a non-monotonic temperature dependence of ΔG0, with negative values both above the unfolding temperature (approximately 70° C for HP35) and below a “cold denaturation” temperature, attributable to a positive change in heat capacity ΔCp upon unfolding. Calorimetric data for HP35 indicate ΔCp ≈ 0.6 kcal/mole-K near the unfolding temperature82. If ΔG0 were decreasing towards zero with decreasing temperature, one would also expect [GdnHCl]mid to decrease. At this point, the temperature-independence of [GdnHCl]mid at low temperatures (to within ±8% from -75° C to 22° C) is mysterious, but may be related to the non-two-state nature of HP35 denaturation, temperature-dependent stabilization of the folded state by glycerol, temperature-dependent denaturing strength of GdnHCl, or a fortuitous combination of these factors.

Raleigh and coworkers have examined the unfolded state of HP36 through solution NMR and CD studies of shorter peptides38,83. They report that a 21-residue peptide that includes the N-terminal and central α-helices of HP36 has significant population of helical secondary structure and tertiary interactions similar to those in the folded state of the full-length protein. From these data, they suggest that the conformational distribution of unfolded HP36 (and presumably also HP35) may have significant population of native-like structure. Related conclusions were reached by Pande and coworkers from their analysis of molecular dynamics simulations on HP3653. Molecular dynamics simulations by Lei and Duan led to the identification a partially folded state of HP35 in which the N-terminal and central α-helices were present as an off-pathway folding intermediate, and a state in which the central and C-terminal α-helices were present as an on-pathway intermediate44. These experimental and computational studies do not involve chemical denaturation, and do not reveal preferential population of PPII and extended conformations.

Finally, we emphasize that this work represents the first quantitative determination of site-specific conformational distributions in partially and fully unfolded states of a protein by solid state NMR. The experimental techniques and approaches to data analysis described below are likely to have applications in future studies of other biopolymer systems, including studies of folding intermediates that are kinetically trapped by rapid freeze-quenching84,85.

Materials and Methods

Protein synthesis and sample preparation

HP35 samples were prepared by automated solid-phase synthesis on an Applied Biosystems 433A peptide synthesizer using fluoroenylmethoxycarbonyl (FMOC) chemistry with O-Benzotriazole-N,N,N',N'-tetramethyl-uronium-hexafluoro-phosphate/1-hydroxy-7-azabenzotriazole (HBTU/HOAt) activation and a Fmoc-Phe Wang resin (0.27 mmol/g substitution, Novabiochem), cleaved with trifluoroacetic acid/water (20:1 v/v) containing phenol, thioanisole, and 1,2-ethanedithiol as precipitated in cold t-butyl methyl ether, and purified by high-performance liquid chromatography (water/acetonitrile gradient, C18 reverse-phase column). Syntheses were carried out at the 0.1 mmol scale, using a ten-fold excess of unlabeled amino acids and a three-fold excess of isotopically labeled amino acids. Capping with acetic anhydride was performed after each coupling step. Protein purity was verified to be at least 95% by electrospray mass spectrometry. Samples for solid state NMR were prepared by dissolving approximately 5-7 mg of HP35 in 250 μl of glycerol/water (1:1 v/v) with 1 mM sodium acetate, 10 mM CuSO4, and pH 5. CuSO4 was included to reduce proton NMR spin-lattice relaxation times to approximately 1 s in the frozen solutions, as described previously.23,32,61 The folding state was controlled by addition of GdnHCl to concentrations between 0.0 M and 7.0 M. Values of [GdnHCl] for solid state NMR samples have a precision of ±0.1 M, while those in CD measurements have a precision of ±0.02 M. Approximately 240 μl of the HP35 solution was transferred into a 6 mm Varian MAS rotor and frozen by immersion in liquid nitrogen before solid state NMR measurements.

NMR experiments

NMR data were obtained at a 13C NMR frequency of 100.8 MHz (9.4 T magnetic field), using a Varian Infinity spectrometer with a Varian 6 mm MAS probe operating in two-channel (1H,13C) mode. The probe was precooled to a sample temperature of -120° C and the MAS axis was adjusted by observation of 79Br NMR signals from KBr powder at low temperature prior to loading the frozen HP35 sample into the probe.

2D 13C-13C NMR spectra of HP35-U-AGMFPL and HP35-U-VAL (Figure 2) were recorded at MAS frequencies of 6.7 kHz or 7.0 kHz, using an rf pulse sequence that included ramped-amplitude 1H-13C cross-polarization, two-pulse phase modulated (TPPM) 1H decoupling86 during t1, t2, and mixing periods, and radio-frequency-driven recoupling (RFDR) during the 1.71-1.79 ms mixing period59,87. 1H decoupling field amplitudes were 70-80 kHz. 13C π pulse lengths during RFDR were 10-30 μs, with the carrier frequency near 50 ppm. To minimize artifacts due to out-of-phase crosspeaks among MAS sidebands, the mixing period was synchronized with MAS and data were processed as described previously32,69. 13C chemical shifts are reported with respect to tetramethylsilane, based on an external adamantane reference at 38.56 ppm.

2DEXMAS measurements57,58 were carried out at MAS frequencies of 2.50 kHz and 2.75 kHz, with the latter frequency chosen to prevent overlap of MAS sideband signals of GdnHCl (natural abundance 13C) and carbonyl 13C labels. The mixing period τmix was 0.5 s, with minor variations for rotor synchronization. 2D data sets with two different modes of rotor synchronization (SI and SII in Figure 3a) were combined and processed as previously described69, yielding a single 2D spectrum in which intersite crosspeaks among MAS sidebands of labeled carbonyl sites arise from spin polarization transfer between the two carbonyl 13C labels. TPPM decoupling was applied in t1 and t2 periods with a 93 kHz 1H rf amplitude. Cross-polarization periods were approximately 1 ms, with a 56 kHz 1H rf amplitude and a ramped 13C rf amplitude from 50 kHz to 58 kHz. 13C π/2 pulse lengths were 5.0 μs. The t1 period ranged from 0.1 μs to 3175.1 μs in 25.0 μs increments. Total acquisition times for 2DEXMAS data were approximately 22 hr, with a recycle delay of 2 s.

CTDQFD measurements56 were carried out at an MAS frequency of 4.00 kHz. The RFDR sequences used for excitation and dephasing of DQ coherences were comprised of rotor-synchronized πpulses, with one πpulse centered in each 2τR period and with XY-8 phases88 The 13C cross-polarization pulse and the first 13C π/2 pulse were phase-cycled for DQ filtering, selecting 13C NMR signals from labeled carbonyl pairs, suppressing all other signals, and creating a DQ preparation period of 2LτR (see Figure 3b). As previously described56, the third and fourth 13C π/2 pulses in the CTDQFD sequence refocus evolution under 13C-13C magnetic dipole-dipole couplings, producing an effective dipolar evolution period of 2(M-N)τR. In measurements described above, L = 32, M + N = 56, and M - N was increased from 0 to 56 in increments of 8. 13C rf amplitudes were 37 kHz for π pulses and 56 kHz for π/2 pulses. 1H decoupling amplitudes were 93 kHz. Total acquisition times for CTDQFD data sets were approximately 7 hr.

DQCSA data55 were acquired at a 4.00 kHz MAS frequency, with RFDR sequences as described above and DQ preparation and mixing periods of 64τR. DQ filtering was accomplished by phase cycling of the 13C cross-polarization pulse and the first 13C π/2 pulse. CSA interactions were recoupled by three π pulses in two rotor periods, as shown in Figure 3c, with 9.0 μs pulse lengths. Total acquisition times for DQCSA data sets were approximately 17 hr.

DQ filtering efficiencies in CTDQFD and DQCSA measurements were 20%. Pulsed spin-lock detection89 improved the signal-to-noise of CTDQFD and DQCSA data by reducing the effective carbonyl 13C NMR linewidths from approximately 300 Hz to 25 Hz.

CD measurements

CD spectra were recorded with a Jasco 720 spectropolarimeter, using HP35 concentrations of approximately 60 μM and 2.0 mm path lengths. Low temperatures were achieved by placing the sample cuvette within a home-built insulated box, equipped with quartz windows, inside the sample chamber of the spectropolarimeter. The temperature inside the box was controlled with a continuous flow of cold nitrogen gas and monitored with a standard Pt resistor.

Data analyses

Tables of simulated 2DEXMAS, CTDQFD, and DQCSA data for -180° ≤ ϕ ≤ 0° and -180° ≤ ψ ≤ 180°, with 5° increments in both variables, were generated with Fortran simulation programs written specifically for this purpose (available on request from CSA tensor anisotropy and asymmetry parameters were set to δ = 151 ppm and η = 0.52, respectively. These CSA parameters gave a good fit to MAS sideband intensities in a DQ filtered one-dimensional 13C NMR spectrum of HP35-CO-AV at 2.75 kHz MAS frequency and agree well with average parameters determined for helical residues in a microcrystalline protein by Wylie et al.90 Variations in δ for backbone carbonyls are small, regardless of secondary structure90, and 2DEXMAS, CTDQFD, and DQCSA data are relatively insensitive to variations in η over the range of values found in proteins. Uncertainties in η lead to uncertainties of roughly ±5° in best-fit ϕ and ψ values. Standard peptide bond geometries and CSA tensor principal axis directions were assumed91,92. The same simulation programs have been used in previous applications of these techniques to model peptides55-58, amyloid fibrils70,71, peptide/antibody complexes60, and other systems23,72. As rf imhomogeneity significantly affects CTDQFD data, simulations of CTDQFD data were averaged over a distribution of rf amplitudes determined from 13C nutation experiments.

Crosspeaks in 2DEXMAS spectra contain both intersite contributions, arising from polarization transfers between 13C-labeled carbonyl sites during τmix and dependent on the intervening ϕ and ψ angles, and intrasite contributions, arising from 14N spin-lattice relaxation during τmix and independent of ϕ and ψ57,58. In the fitting procedures, calculated intrasite contributions were added to simulated 2DEXMAS crosspeak amplitudes with a scaling factor of 0.56 relative to the intersite contributions, based on the known value of τmix and the value T1 ≈ 0.6 s determined previously for peptides in frozen solution23. Intrasite contributions represented approximately 20% of the total crosspeak amplitudes.

Experimental data were analyzed by comparison with the simulated data tables, using scripts for MATLAB (MathWorks Inc., Natick, MA). Deviations between experimental and simulated data were represented by the quantities χq2i=1Nq[Eq,iλqSq,i]2σq2, where the index q indicates the type of measurement (2DEXMAS, CTDQFD, or DQCSA), Nq is the number of data points in the measurement, Eq,i and Sq,i are experimental and simulated data, σq is the uncertainty in the experimental data (determined from the root-mean-squared noise in the experimental NMR spectra), and λq is a scaling factor calculated to minimize χ2q as previously described57. The total deviation was given by χ2=q=13χq2, since the contribution of each measurement to χ2 was automatically properly weighted through inclusion of the σq and λq quantities in χq2. Ideally, a good fit to the experimental data is indicated by the condition χ2min(NdataNparam)±2(NdataNparam) where Nparam is the number of adjustable parameters (corresponding to roughly 90% probability that χ2min will lie in this range93). Somewhat larger values of χ2min can occur when systematic errors in experiments or simulations are not negligible compared with σq.

Three models for conformational distributions were tested against the experimental 2DEXMAS, CTDQFD, and DQCSA data. In the first and simplest model, the data were fit with a single pair of ϕ and ψ values, representing a narrow conformational distribution at V50 in HP35-CO-AV. In this case, χ2q and Sq,i were functions of ϕ and ψ. Values of Sq,i(ϕ,ψ) were taken directly from the simulated data tables. The results of the analyses are displayed as contour plots of χ2q(ϕ,ψ) and χ2(ϕ,ψ) in Figure 5.

In the second model, conformational distributions were approximated by a sum of symmetric Gaussian components, i.e., a total probability distribution with the form P(ϕ,Ψ)=k=1NGhke[(ϕϕk)2+(ΨΨk)2]wk2, where NG is the number of Gaussian components and hk, wk, ϕk, and ψk are the height, width, central ϕ value and central ψ value of the kth component. With this form, the volume of each component (representing the population of conformations near ϕkk) was πhkwk2. In this case, χq2 and Sq,i were functions of hk, wk, ϕk, and ψk, with Sq,imnP(ϕm,Ψn)S~q,i(ϕm,Ψn), where Sq,im, ψn) were derived from the simulated data tables, using bilinear interpolation to permit ϕm and ψn values in increments of 2°. Gaussian functions in P(ϕ,ψ) were truncated at 1/e4.

In the third model, conformational distributions at V50 were approximated by a large number NP of equally-populated points in the ϕ,ψ plane. In this case, χq2 and Sq,i were functions of NP and the ϕ and ψ values of each point, with Sq,i=1NPn=1NPS~q,i(ϕn,Ψn). Values of ϕn and ψn were chosen randomly during the fitting procedure, as described below, and Sq,inn) were determined by bilinear interpolation from the data tables.

Parameters that minimize χ2 were found by simulated annealing For the Gaussian model, initial values of hk, wk, ϕk, and ψk for a given NG were chosen randomly. In each attempted move of the simulated annealing algorithm, hk, wk, or both ϕk and ψk were changed by random quantities with maximum values Δh, Δw, Δϕ, and Δψ. An initial unitless temperature T was chosen to give an approximate 50% acceptance probability with large values of Δϕ, and Δψ. Moves were accepted or rejected according to the usual Metropolis criterion73, using χ2 as the energy. Simultaneously, Δh, Δw, Δϕ, and Δψ were adjusted to maintain an average acceptance probability of 50%. T was reduced by sequential multiples of 0.85 after approximately 1000 attempted moves. Simulated annealing was terminated when χ2 converged to a local minimum. Minimizations were repeated 5-10 times for each experimental data set with different initial parameter values. For the multiple-point model, NP was chosen initially to be 40 and values of ϕn and ψn were chosen randomly. In each attempted move, the position of one point was changed by Δϕ, and Δψ (also adjusted to give 50% acceptance probabilities). NP was also allowed to vary in a small range after approximately 400 attempted moves. The temperature schedule was similar to that used for the Gaussian model.

Uncertainties in the fitted parameters were defined as the variations in these parameters that increased χ2 by less than 2Ndata from its minimum value χ2min, where Ndata was the total number of data points. These variations were determined by MCMC simulations, in which the parameters were varied and the Metropolis criterion was used as described above, but with T adjusted in an iterative fashion to maintain χ2χ2min+2Ndata. Error limits reported in Table 2 are the root-mean-squared excursions of individual parameters from their best-fit values during MCMC simulations.

For Gaussian models, the value of NG can be justified by evaluation of the F statistic93:FNG,NG = {[χ2min (NG) - χ2min (NG′)] ν (NG′)}/{χ2min (NG′)[ν(NG) - ν(NG′)]}, with χ2min (NG) and ν(NG) being the best-fit χ2 value and the number of degrees of freedom for a given NG. Specifically, ν(NG) = 42, 38, or 34 for NG = 1, 2, or 3, respectively. The χ2min values from Table 2 imply F1,2 = 5.7, 7.8, and 2.8 for HP35-CO-AV, HP35-CO-SA, and HP35-CO-LK at [GdnHCl] = 7.0 M, corresponding to 0.999, 0.9999, and 0.96 confidence levels that NG = 2 is a better fit than NG = 1. F1,2 = 2.0, 7.8, and 9.5 for HP35-CO-AV, HP35-CO-SA, and HP35-COLK at [GdnHCl] = 4.5 M, corresponding to 0.89, 0.9999, and 0.99998 confidence levels that NG = 2 is a better fit than NG = 1. For all data sets at [GdnHCl] = 4.5 M and 7.0 M, F2,3 < 0.06, corresponding to confidence levels below 0.01 that NG = 3 is a better fit than NG = 2.

Analysis of systematic errors

As pointed out above (see Results), values of χ2min in Table 2 are larger than Ndata, especially for fits to experimental data for HP35-CO-AV, which had the highest signal-to-noise ratio and were therefore most sensitive to systematic errors. One conceivable source of small systematic errors is the assumption that all molecular orientations in the MAS rotor contribute equally to the experimental NMR signals (i.e., that 1H-13C cross-polarization efficiencies and spin relaxation rates are independent of molecular orientation). Under the conditions of our CTDQFD and DQCSA measurements, we have determined the carbonyl 13C transverse relaxation time T2 to be approximately 50 ms, implying that spin relaxation is too slow to contribute significantly to χ2min or to affect conformational distributions derived from our data. We have investigated whether orientation-dependent 1H-13C cross-polarization efficiencies might affect our results by performing simulations of 2DEXMAS, CTDQFD, and DQCSA experiments in which initial 13C spin polarizations were strongly orientation-dependent, but consistent with spinning sideband patterns in experimental one-dimensional 13C MAS NMR spectra. Fits to the simulated data indicate that orientation-dependent spin polarizations do not affect conformational distributions derived from the data significantly, but may increase χ2min by roughly 3-8 units.

Another conceivable source of systematic errors is the assumption in simulations that all carbonyl CSA tensors have identical anisotropy and asymmetry parameters. The work of Wylie et al. shows that anisotropy parameters are nearly constant for carbonyl CSA tensors in proteins, but that asymmetry parameters η vary between 0.45 and 0.85, with the largest values being observed in β-sheets.90 To examine effects of variations in η, we analyzed our experimental data at [GdnHCl] = 4.5 M and 7.0 M with the two-Gaussian model, using simulated data tables with either η= 0.52 (as described above) or η = 0.65, the latter value being representative of non-α-helical conformations that are not in β-sheets. Best-fit populations varied by less than 5%. Best-fit values of ϕ and ψ varied by less than 9°. χ2min values were 2-10 units higher with η= 0.65 than with η= 0.52, except in the case of HP35-CO-LK at [GdnHCl] = 4.5 M where χ2min was reduced from 55 to 47. We conclude that the choice of η may affect χ2min, but does not affect conformational distributions derived from our data significantly.

Supplementary Material








This work was supported by the Intramural Research Program of the National Institute of Diabetes and Digestive and Kidney Diseases of the National Institutes of Health.


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Supplementary Material Experimental 2DEXMAS, CTDQFD, and DQCSA data for HP35-CO-AV, HP35-CO-SA, and HP35-CO-LK at [GdnHCl] = 0.0 M, 4.5 M, and 7.0 M (Figures S1, S2, and S3); direct comparisons of experimental 2DEXMAS, CTDQFD, and DQCSA data for HP35-CO-AV, HP35-CO-SA, and HP35-CO-LK at [GdnHCl] = 7.0 with best-fit simulations (Figure S4); results of MCMC simulations for HP35-CO-SA and HP35-CO-LK, using Gaussian models and multiple-point models (Figures S5 and S6); results of simulations to investigate effects of orientation-dependent spin polarizations and variations in CSA asymmetry parameters on fits to the solid state NMR data (Tables S1 and S2).


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