Measurement of structural distribution functions in disordered systems: A general approach for sensitivity estimation

Statistical learning of NMR tensors from 2D isotropic/anisotropic correlation nuclear magnetic resonance spectra

Many linear inversion problems involving Fredholm integrals of the first kind are frequently encountered in the field of magnetic resonance. One important application is the direct inversion of a solid-state nuclear magnetic resonance (NMR) spectrum containing multiple overlapping anisotropic subspectra to obtain a distribution of the tensor parameters. Because of the ill-conditioned nature of this inverse problem, we investigate the use of the truncated singular value decomposition and the smooth least absolute shrinkage and selection operator based regularization methods, which (a) stabilize the solution and (b) promote sparsity and smoothness in the solution. We also propose an unambiguous representation for the anisotropy parameters using a piecewise polar coordinate system to minimize rank deficiency in the inversion kernel. To obtain the optimum tensor parameter distribution, we implement the k-fold cross-validation, a statistical learning method, to determine the hyperparameters of the regularized inverse problem. In this article, we provide the details of the linear-inversion method along with numerous illustrative applications on purely anisotropic NMR spectra, both synthetic and experimental two-dimensional spectra correlating the isotropic and anisotropic frequencies.

What can solid state NMR contribute to our understanding of protein folding?

Complete understanding of the folding process that connects a structurally disordered state of a protein to an ordered, biochemically functional state requires detailed characterization of intermediate structural states with high resolution and site specificity. While the intrinsically inhomogeneous and dynamic nature of unfolded and partially folded states limits the efficacy of traditional X-ray diffraction and solution NMR in structural studies, solid state NMR methods applied to frozen solutions can circumvent the complications due to molecular motions and conformational exchange encountered in unfolded and partially folded states. Moreover, solid state NMR methods can provide both qualitative and quantitative structural information at the site-specific level, even in the presence of structural inhomogeneity. This article reviews relevant solid state NMR methods and their initial applications to protein folding studies. Using either chemical denaturation to prepare unfolded states at equilibrium or a rapid freezing apparatus to trap non-equilibrium, transient structural states on a sub-millisecond time scale, recent results demonstrate that solid state NMR can contribute essential information about folding processes that is not available from more familiar biophysical methods.

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 (13)C NMR spectra of samples prepared with uniform (13)C labeling of selected residues show relatively sharp cross-peaks at chemical shifts that are consistent with the known three-helix bundle structure of folded HP35. At high GdnHCl concentrations, most cross-peaks 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 varphi and psi torsion angles in samples with sequential pairs of carbonyl (13)C labels. Without GdnHCl, the combined data are well fit by alpha-helical conformations. At [GdnHCl]=4.5 M, corresponding to the approximate denaturation midpoint, the combined data are well fit by a combination of alpha-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.

Characterization of molecular disorder in vapor-deposited thin films of aluminum tris(quinoline-8-olate) by one-dimensional Al27 NMR under magic angle spinning

Aluminum tris (quinoline-8-olate) (Alq3) is used as an electron-transport layer in organic light-emitting diodes. The material can be obtained in a wide range of different solid phases, both crystalline and amorphous, by deposition from the vapor phase or from solution under controlled conditions. While the structure of the crystalline polymorphs of Alq3 has been investigated thoroughly by x-ray diffraction as well as solid-state NMR, very little information is currently available on the amount of structural disorder in the amorphous forms of Alq3. In the present contribution, we report the use of 27Al NMR spectroscopy in the solid state under magic angle spinning to extract such information from amorphous vapor deposits of Alq3. The NMR spectra obtained from these samples exhibit different degrees of broadening, reflecting distributions of the electric-field gradient tensor at the site of the aluminum ion. These distributions can be obtained from the NMR spectra by solving the corresponding inverse problem. From these results, the magnitude of structural disorder in terms of molecular geometry has been estimated by density-functional theory calculations. It was found that the electric-field gradient anisotropy delta follows a bimodal distribution. Its majority component is centered around delta values comparable to the meridianal alpha crystal polymorph and has a width of about 10%, corresponding to distortions of the molecular geometry of a few degrees in the orientation of the ligands. Alq3 samples obtained at higher deposition rates exhibit higher degrees of disorder. The minor component, present at about 7%, has a much smaller anisotropy, suggesting that it may be due to the facial isomer of Alq3.

A DOQSY approach for the elucidation of torsion angle distributions in biopolymers: application to silk

Silk from the wild silkworm Samia cynthia ricini with a molecular mass of about 300kDa consists of alternating repeats of nano-crystalline poly-(Ala) and non-crystalline glycine-rich domains. The backbone torsion angles between pairs of these two amino acids is determined by DOQSY solid-state NMR spectroscopy: the alanine-rich domains are predominantly in a beta-sheet conformation, whereas the glycine-rich domains are found to be partially in an extended beta-sheet conformation and partially in an approximately 3(1)-helical conformation. In the cast film from liquid silk significantly different secondary structures were found: the alanine-rich domains are alpha-helical conformation, whereas the results for glycine-labelled sample are explained by a random-coil state. A detailed error analysis of the technique is presented.

Quantification of global orientational order in organic solids by magic-angle spinning deuterium NMR with rotor synchronization

A new method for the characterization of orientational order in organic solids based on magic-angle spinning NMR spectroscopy is introduced. The method is related to the rotor-synchronized magic-angle spinning experiment proposed by Harbison and Spiess [Chem. Phys. Lett. 124, 128 (1986)], but exploits the anisotropy of the deuterium quadrupolar coupling instead of the carbon-13 chemical shielding anisotropy. Magic-angle spinning provides a sensitivity advantage over pseudostatic techniques; using the deuterium quadrupolar coupling makes the method applicable to systems that do not exhibit large carbon chemical shift anisotropies, such as aliphatic polymers. Due to the magnitude of the deuterium quadrupolar coupling, a large number of spinning sidebands can be reliably observed, allowing for a precise determination of the orientational distribution function. Experimental data are analyzed in terms of Wigner matrix basis functions as well as the conjugate orthogonal functions framework. Unidirectionally cold-drawn poly(ethylene) is used as an example to demonstrate the method.

Inverse methods in two-dimensional NMR spectral analysis

Solid-state NMR is a valuable technique for the study of disordered materials. Analysis of such spectra usually involves solution of so-called ill-posed inverse problems. Here we present a strategy for the analysis of two-parameter two-dimensional NMR problems and test it on 2D DECODER and DOQSY experiments. Using Monte Carlo tests, constraints are determined for the resolution and accuracy of the analysis for both experiments. The methods are finally applied to spectra of spider dragline silk, a heterogeneous solid fibrous protein.

Biomolecular solid state NMR: advances in structural methodology and applications to peptide and protein fibrils

Solid state nuclear magnetic resonance (NMR) methods can provide atomic-level structural constraints on peptides and proteins in forms that are not amenable to characterization by other high-resolution structural techniques, owing to insolubility, high molecular weight, noncrystallinity, or other characteristics. Important examples include peptide and protein fibrils and membrane-bound peptides and proteins. Recent advances in solid state NMR methodology aimed at structural problems in biological systems are reviewed. The power of these methods is illustrated by experimental results on amyloid fibrils and other protein fibrils.