New modification of model-free approach to analysis of nuclear magnetic relaxation data in proteins

A formal approach to the analysis of 13C magnetic relaxation data in proteins has been developed. It is based on the concepts of one of the authors on the internal motions in solid polymers (Fedotov, V.D., Pulse NMR in bulk polymers, Doctoral dissertation, Kazan, USSR, 1981). According to this approach the intermolecular motions in proteins are considered as anisotropic ones and described in terms of a spectrum of correlation times. To characterize the motions a set of formal microdynamic parameters has been introduced. They are: the anisotropy parameter (a measure of spatial restriction of motion), the most probable correlation time, the parameter of the correlation time distribution width. The analysis of protonated carbon relaxation in globular proteins (bovine pancreatic trypsin inhibitor and ribonuclease S) and polymers has been carried out by the model-free approach. Microdynamic parameters of CH3-, CH2-, aromatic CH-groups have been considered within the framework of the diffusional rotation-oscillation models. To explain the backbone CH-group relaxation the model of the defect diffusion has been applied. The distinctive feature of the results obtained is the broad correlation time distribution for all groups of any type. The causes of nonexponential correlation function of local motion have been discussed. To elucidate the nature of the correlation time the carbon magnetization decays in the wide range of microdynamic parameter values imitating various experimental conditions have been calculated.     

Dynamic Structure of Proteins in Solid State. 1H and 13C NMR Relaxation Study

Abstract

Temperature dependencies of ¹H non-selective NMR T₁ and T₂ relaxation times measured at two resonance frequencies and natural abundance ^l3C NMR relaxation times T_l and T_lr measured at room temperature have been studied in a set of dry and wet solid proteins—;Bacterial RNase, lysozyme and Bovine serum albumin (BSA). The proton and carbon data were interpreted in terms of a model supposing three kinds of internal motions in a protein. These are rotation of the methyl protons around the axis of symmetry of the methyl group, and fast and slow oscillations of all atoms. The correlation times of these motions in solid state are found around 10^?11, 10^?9 and 10^?6 s, respectively. All kinds of motion are characterized by the inhomogeneous distribution of the correlation times. The protein dehydration affects only the slow internal motion. The amplitude of the slow motion obtained from the carbon data is substantially less than that obtained from the proton data. This difference can be explained by taking into account different relative inter- and intra- chemical group contributions to the proton and carbon second moments. The comparison of the solid state and solution proton relaxation data showed that the internal protein dynamics in these states is different: the slow motion seems to be few orders of magnitude faster in solution.     

Model-free analysis for large proteins at high magnetic field strengths

Protein backbone dynamics is often characterized using model-free analysis of three sets of ¹⁵N relaxation data: longitudinal relaxation rate (R ₁), transverse relaxation rate (R ₂), and ¹⁵N–{H} NOE values. Since the experimental data is limited, a simplified model-free spectral density function is often used that contains one Lorentzian describing overall rotational correlation but not one describing internal motion. The simplified spectral density function may be also used in estimating the overall rotational correlation time, by making the R ₂/R ₁ largely insensitive to internal motions, as well as used as one of the choices in the model selection protocol. However, such approximation may not be valid for analysis of relaxation data of large proteins recorded at high magnetic field strengths since the contribution to longitudinal relaxation from the Lorentzian describing the overall rotational diffusion of the molecule is comparably small relative to that describing internal motion. Here, we quantitatively estimate the errors introduced by the use of the simplified spectral density in model-free analysis for large proteins at high magnetic field strength.     

Analysis of Sub-τc and Supra-τc Motions in Protein Gβ1 Using Molecular Dynamics Simulations

The functions of proteins depend on the dynamical behavior of their native states on a wide range of timescales. To investigate these dynamics in the case of the small protein Gβ1, we analyzed molecular dynamics simulations with the model-free approach of nuclear magnetic relaxation. We found amplitudes of fast timescale motions (sub-τ_c, where τ_c is the rotational correlation time) consistent with S² obtained from spin relaxation measurements as well as amplitudes of slow timescale motions (supra-τ_c) in quantitative agreement with S² order parameters derived from residual dipolar coupling measurements. The slow timescale motions are associated with the large variations of the ³J couplings that follow transitions between different conformational substates. These results provide further characterization of the large structural fluctuations in the native states of proteins that occur on timescales longer than the rotational correlation time.     

Hydration dependence of backbone and side chain polylysine dynamics: a 13C solid-state NMR and IR spectroscopy study

The molecular dynamics of solid poly-L-lysine has been studied by the following natural abundance (13)C-NMR relaxation methods: measurements of the relaxation times T(1) at two resonance frequencies, off-resonance T(1rho) at two spin-lock frequencies, and proton-decoupled T(1rho). Experiments were performed at different temperatures and hydration levels (up to 17% H(2)O by weight). The natural abundance (13)C-CPMAS spectrum of polylysine provides spectral resolution of all types of backbone and side chain carbons and thus, dynamic parameters could be determined separately for each of them. At the same time, the conformational properties of polylysine were investigated by Fourier transform infrared spectroscopy. The data obtained from the different NMR experiments were simultaneously analyzed using the correlation function formalism and model-free approach. The results indicate that in dry polylysine both backbone and side chains take part in two low amplitude motions with correlation times of the order of 10(-4) s and 10(-9) s. Upon hydration, the dynamic parameters of the backbone remain almost constant except for the amplitude of the slower process that increases moderately. The side chain dynamics reveals a much stronger hydration response: the amplitudes of both slow and fast motions increase significantly and the correlation time of the slow motion shortens by about five orders of magnitude, and at hydration levels of more than 10% H(2)O fast and slow side chain motions are experimentally indistinguishable. These changes in the molecular dynamics cannot be ascribed to any hydration-dependent conformational transitions of polylysine because IR spectra reveal almost no hydration dependence in either backbone or side chain absorption domains. The physical nature of the fast and slow motions, their correlation time distributions, and hydration dependence of microdynamic parameters are discussed.     

Determination of protein rotational correlation time from NMR relaxation data at various solvent viscosities

An accurate determination of the overall rotation of a protein plays a crucial role in the investigation of its internal motions by NMR. In the present work, an innovative approach to the determination of the protein rotational correlation time _R from the heteronuclear relaxation data is proposed. The approach is based on a joint fit of relaxation data acquired at several viscosities of a protein solution. The method has been tested on computer simulated relaxation data as compared to the traditional _R determination method from T₁/T₂ ratio. The approach has been applied to ribonuclease barnase from Bacillus amyloliquefaciens dissolved in an aqueous solution and deuterated glycerol as a viscous component. The resulting rotational correlation time of 5.56 ± 0.01 ns and other rotational diffusion tensor parameters are in good agreement with those determined from T₁/T₂ ratio.     

Backbone dynamics of free barnase and its complex with barstar determined by 15N NMR relaxation study

Backbone dynamics of uniformly ¹⁵N-labeled free barnase and its complex with unlabelled barstar have been studied at 40°C, pH 6.6, using ¹⁵N relaxation data obtained from proton-detected 2D {¹H}-¹⁵N NMR spectroscopy. ¹⁵N spin-lattice relaxation rate constants (R₁), spin-spin relaxation rate constants (R₂), and steady-state heteronuclear {¹H}-¹⁵N NOEs have been measured at a magnetic field strength of 14.1 Tesla for 91 residues of free barnase and for 90 residues out of a total of 106 in the complex (excluding three prolines and the N-terminal residue) backbone amide ¹⁵N sites of barnase. The primary relaxation data for both the cases have been analyzed in the framework of the model-free formalism using both isotropic and axially symmetric models of the rotational diffusion tensor. As per the latter, the overall rotational correlation times (_m) are 5.0 and 9.5 ns for the free and complexed barnase, respectively. The average order parameter is found to be 0.80 for free barnase and 0.86 for the complex. However, the changes are not uniform along the backbone and for about 5 residues near the binding interface there is actually a significant decrease in the order parameters on complex formation. These residues are not involved in the actual binding. For the residues where the order parameter increases, the magnitudes vary significantly. It is observed that the complex has much less internal mobility, compared to free barnase. From the changes in the order parameters, the entropic contribution of NH bond vector motion to the free energy of complex formation has been calculated. It is apparent that these motions cause significant unfavorable contributions and therefore must be compensated by many other favorable contributions to effect tight complex formation. The observed variations in the motion and their different locations with regard to the binding interface may have important implications for remote effects and regulation of the enzyme action.     

A NOESY-HSQC simulation program, SPIRIT

A program SPIRIT (Simulation Program considering Incomplete Recovery of z magnetization and INEPT Transfer efficiency) has been developed to simulate three-dimensional NOESY-HSQC spectra. This program takes into account (1) different transfer efficiency during INEPT and reverse INEPT durations due to differential relaxation rates and¹ J coupling constants; (2) the different effect of the sensitivity-enhancement scheme on CH, CH₂ and CH₃ systems; and (3) incomplete recovery of longitudinal magnetization between scans. The simulation program incorporates anisotropic tumbling mode for symmetric tops, and allows for differential external relaxation rates for protons. Some well-defined internal motions, such as the fast rotation of methyl groups, are taken into account. The simulation program also allows for input of multiple conformations and their relative populations to calculate the average relaxation matrix to account for slow internal motions. With the SPIRIT program, the sensitivity-enhanced NOESY-HSQC experiment can be used directly in the evaluation of the accuracy of structures, which can potentially be improved by direct refinement against the primary data. Abbreviations: NOESY, nuclear Overhauser enhancement spectroscopy; HSQC, heteronuclear single quantum correlation; INEPT, insensitive nuclei enhanced by polarization transfer.     

A comparison of 15N NMR relaxation measurements with a molecular dynamics simulation: Backbone dynamics of the glucocorticoid receptor DNA-binding domain

The rapid motions of the backbone of the DNA-binding domain of the glucocorticoid receptor (GR DBD) have been investigated using proton-detected heteronuclear NMR experiments on ¹⁵N-labeled protein at pH 6.0 and with a 200 psec molecular dynamics simulation of hydrated GR DBD. The experimental data were interpreted in terms of a generalized order parameter (S²) and an effective correlation time (τ_e) for the internal motion of each amide bond. A back calculation, using the same model, yielded the {¹H}-¹⁵N nuclear Overhauser effects (NOEs) and the ¹⁵N spin-lattice relaxation times (T₁) from the simulated data. The rapid motions of the backbone turned out to be rather limited and uniform throughout the protein, with a somewhat reduced mobility in the two major α-helical regions and a slightly enhanced flexibility for some residues in the first zinc coordinating region. The agreement between the experimental and simulated S²-values was as good as quantitative for most of the residues, except for some residues that were subject to a more large-scale, and in the simulation thus poorly sampled, motion. Examples of such motions that were found in the simulation include jumps of the amide bond of Ile-487 between the charged oxygens of the side chain of Asp-485 and less distinct large scale motions for some of the residues in the extended regions, that were shown to give rise to noisy and/or fast decaying internal reorientational correlation functions. For these residues large differences in the simulated and experimental τ_e-values were found, indicating that motions on different time scales were dominating in the experimental and simulated values. The lower (<0.7) experimental NOEs for these residues could not be reproduced in the simulation and were shown to be a consequence of the lower τ_e-values estimated in the simulation. By combining information from the simulation and the experiment a more complete picture of the motions for these residues can be obtained as is illustrated with an estimation of the jump angle and jump frequency for the amide bond of Ile-487. © 1993 Wiley-Liss, Inc.     

NMR relaxation processes of 31P in macromolecules

³¹P-Nmr relaxation parameters (spin-lattice relaxation time, linewidth, and nuclear Overhauser effect) were obtained at three different frequencies for poly(U) and a well-defined (145 ± 3 base-pair) fragment of DNA in solution. Data sets for the two samples were analyzed by theories which included relaxation by the mechanisms of ³¹P chemical shift anisotropy as well as by ¹H-³¹P dipole–dipole interaction. Neither data set could be satisfactorily described by a single correlation time. A model of a rigid rotor most nearly fits the data for the DNA molecule. Parameters obtained from the least-square fit indicate (1) that the DNA undergoes anisotropic reorientation with a correlation time τ₀ = 6.5 × 10^?7 sec for the end-to-end motion, (2) the ratio of diffusion constants D_∥/D_⊥ is 91, and (3) that the linewidth is due to chemical shift dispersion to the extent of 0.5 ppm. Some deviations of the calculated from the observed values suggested that significant torsional and bending motions may also take place for this DNA. Another model which contains isotropic motion but with a broad distribution of correlation times was required to fit the data for poly(U). A log ? χ² distribution function of correlation times [Scheafer, J. (1973) Macromolecules 6 , 881–888] described well the motion of poly(U) with the average correlation time τ = 3.3 × 10^?9 sec and a distribution parameter p = 14.     

Fast evaluation of protein dynamics from deficient <Superscript>15</Superscript>N relaxation data

Simple and convenient method of protein dynamics evaluation from the insufficient experimental ¹⁵N relaxation data is presented basing on the ratios, products, and differences of longitudinal and transverse ¹⁵N relaxation rates obtained at a single magnetic field. Firstly, the proposed approach allows evaluating overall tumbling correlation time (nanosecond time scale). Next, local parameters of the model-free approach characterizing local mobility of backbone amide N–H vectors on two different time scales, S² and R _ex , can be elucidated. The generalized order parameter, S², describes motions on the time scale faster than the overall tumbling correlation time (pico- to nanoseconds), while the chemical exchange term, R _ex , identifies processes slower than the overall tumbling correlation time (micro- to milliseconds). Advantages and disadvantages of different methods of data handling are thoroughly discussed.     

Intramolecular and overall motion of proline: the influence of viscosity on carbon-13 spin-lattice relaxation times.

Nuclear magnetic resonance of ¹³C is used to probe the overall and internal motions of proline. Spin-lattice relaxation times (T₁) are reported for proline monomer dissolved in water/glycerol mixtures. Rates of overall molecular motion and internal motion depend on solvent composition but to different degrees. The effective correlation times (τ_eff) of the various proton-bearing carbon atoms in proline vary linearly as a function of solvent composition (%v/v) rather than of solution viscosity. The effective correlation time for molecular motion (τ_eff) is separated into contributions from overall molecular motion (τ_mol) and internal motion (τ_int). The γ-carbon of proline shows the smallest dependence of τ_int on solvent composition. The data indicate a high degree of intramolecular motion for the γ-carbon of proline. Inclusion of anisotropic molecular reorientation in the data analysis was found not to affect the above conclusions. The observed values of τ_eff indicate that the rotational diffusion model of molecular reorientations should apply to proline. The values of τ_eff calculated for proline using the Stokes-Einstein relation are larger than those observed; the discrepancy is discussed in terms of solvent-solute interactions.     

Solvent-induced anomeric diastereoselectivity switching using a single glycosyl donor

Highly diastereoselective glycosylation reactions have been developed; however, not all glycosylation reactions are diastereoselective and these reactions have probably not been reported. For some fucosylation reactions, unusually low or abnormally opposite selectivities have been demonstrated. In the present study, the fucosylation reaction of long-chain hydrocarbon alcohols, ethyl 9-hydroxynonanoate and decanol using a series of the 2-O-benzyl-protected fucopyranosyl donors were investigated. The resulting products demonstrated the solvent-induced diastereoselectivity switching using diethyl ether (Et₂O) or dichloromethane (CH₂Cl₂). Practical α-selectivities were observed using ether solvents. In contrast, practical β-selectivities were observed using CH₂Cl₂. The anomeric diastereoselectivity switching was similarly observed in the alcohol galactosylation reaction. The larger spin-lattice relaxation time constant (T₁) actually indicated that molecular motion of ethyl 9-hydroxynonanoate was more vigorous in Et₂O than in CH₂Cl₂, suggesting its dissociation in Et₂O and association in CH₂Cl₂. The bulkiness of the associated alcohols is most likely responsible for the observed diastereoselectivity.     

Backbone dynamics of the natively unfolded pro-peptide of subtilisin by heteronuclear NMR relaxation studies

The dynamics of the natively unfolded form of the pro-peptide of subtilisin (PPS) have been characterized at two different pHs (6.0 and 3.0) by ¹⁵N relaxation experiments. ¹⁵N relaxation data is obtained at multiple field strengths and a detailed comparison of spectral density mapping, the model free approach and the recently proposed Cole–Cole model free (CC-MF) analysis is presented. The CC-MF analysis provides a better fit to the observed magnetic field dependence of ¹⁵N relaxation data of unfolded PPS than conventional model free approaches and shows that fluctuations in R₂ may be accounted for by a distribution of correlation times on the nanosecond timescale. A new parameter derives from the analysis and represents the width of the distribution function and the heterogeneity of the dynamics on the nanosecond timescale at a particular site. Particularly interesting is the observation that is sensitive to pH changes and that PPS samples a wider distribution of nanosecond time scale motions at less acidic pHs than at more acidic pHs. These results suggest that PPS experiences a higher degree of correlated motion at pH 6.0 and that electrostatic interactions may be important for inducing correlated motions on the nanosecond timescale in unfolded PPS.     

High resolution nuclear magnetic resonance studies of nigeran

Polymer motion in solution can be studied by ¹³CNMR relaxation methods, which provide information about the correlation time for C-H vectors. ¹³C-Relaxation and Nuclear Overhauser Enhancement (NOE) data may frequently be combined to determine the dipole-dipole relaxation contribution. An alternative method is proposed based on a comparison of the proton spin-lattice relaxation rates of the centre proton resonances of an unlabelled molecule with the relaxation rates of the ¹³C satellites (from ¹³C labelled molecules).Selectively labelled nigeran which is an alternating $\text{1 → 3 and 1 → 4 α-}\text{d}\text{-glucan}$ has been investigated. The discussion in terms of the occurrence of different motions for each of the two units of the polymer requires an unambiguous assignment of the two anomeric carbons. For this reason a detailed assignment of the ¹H and ¹³C Nuclear Magnetic Resonance (NMR) spectra of nigeran in dimethylsulphoxide-d₆ is described, based on T₁ and NOE measurements in addition to selective homonuclear and heteronuclear spin decoupling experiments. These values are correlated with a conformation estimated by HSEA hard-spheres calculation. The measurements of the relaxation parameters for labelled and unlabelled compounds which provide an alternative determination of the ¹³C-¹H dipole-dipole relaxation contribution in a macromolecule agree well with ¹³C-{¹H} NOE experiments.     

Internal mobility of the ologonucleotide duplexes d(TCGCG)2 and d(CGCGCG)2 in aqueous solution from molecular dynamics simulations

Summary In this paper we present longitudinal relaxation times, order parameters and effective correlation times for the base and sugar carbons in both strands of the oligonucleotide duplexes d(TCGCG)₂ and d(CGCGCG)₂, as calculated from 400 ps molecular dynamics trajectories in aqueous solution. The model-free approach (Lipari and Szabo, 1982) was used to determine the amplitudes and time scales of the internal motion. Comparisons were made with NMR relaxation measurements (Borer et al., 1994). The order parameters could acceptably be reproduced, and the effective correlation times were found to be lower than the experimental estimates. Reasonable T₁ relaxation times were obtained in comparison with experiment for the nonterminal nucleosides. The T₁ relaxation times were found to depend mainly on the order parameters and overall rotational correlation time.Abbreviations MD molecular dynamics - CSA chemical shift anisotropy To whom correspondence should be addressed.     

Relating side-chain mobility in proteins to rotameric transitions: Insights from molecular dynamics simulations and NMR

The dynamic aspect of proteins is fundamental to understanding protein stability and function. One of the goals of NMR studies of side-chain dynamics in proteins is to relate spin relaxation rates to discrete conformational states and the timescales of interconversion between those states. Reported here is a physical analysis of side-chain dynamics that occur on a timescale commensurate with monitoring by ²H spin relaxation within methyl groups. Motivated by observations made from tens-of-nanoseconds long MD simulations on the small protein eglin c in explicit solvent, we propose a simple molecular mechanics-based model for the motions of side-chain methyl groups. By using a Boltzmann distribution within rotamers, and by considering the transitions between different rotamer states, the model semi-quantitatively correlates the population of rotamer states with ‘model-free’ order parameters typically fitted from NMR relaxation experiments. Two easy-to-use, analytical expressions are given for converting S²_axis’ values (order parameter for C–CH₃ bond) into side-chain rotamer populations. These predict that S²_axis’ values below 0.8 result from population of more than one rotameric state. The relations are shown to predict rotameric sampling with reasonable accuracy on the ps–ns timescale for eglin c and are validated for longer timescales on ubiquitin, for which side-chain residual dipolar coupling (RDC) data have been collected.     

Model-free Analysis of Protein Dynamics: Assessment of Accuracy and Model Selection Protocols Based on Molecular Dynamics Simulation

The popular model-free approach to analyze NMR relaxation measurements has been examined using artificial amide (15)N relaxation data sets generated from a 10 nanosecond molecular dynamics trajectory of a dihydrofolate reductase ternary complex in explicit water. With access to a detailed picture of the underlying internal motions, the efficacy of model-free analysis and impact of model selection protocols on the interpretation of NMR data can be studied. In the limit of uncorrelated global tumbling and internal motions, fitting the relaxation data to the model-free models can recover a significant amount of quantitative information on the internal dynamics. Despite a slight overestimation, the generalized order parameter is quite accurately determined. However, the model-free analysis appears to be insensitive to the presence of nanosecond time scale motions with relatively small magnitude. For such cases, the effective correlation time can be significantly underestimated. As a result, proteins appear to be more rigid than they really are. The model selection protocols have a major impact on the information one can reliably obtain. The commonly employed protocol based on step-up hypothesis testing has severe drawbacks of oversimplification and underfitting. The consequences are that the order parameter is more severely overestimated and the correlation time more severely underestimated. Instead, model selection based on Bayesian Information Criteria (BIC), recently introduced to the model-free analysis by d'Auvergne and Gooley (2003), provides a better balance between bias and variance. More appropriate models can be selected, leading to improved estimate of both the order parameter and correlation time. In addition, the computational cost is significantly reduced and subjective parameters such as the significance level are unnecessary.     

Phospholipid Reorientation at the Lipid/Water Interface Measured by High Resolution P Field Cycling NMR Spectroscopy

The magnetic field dependence of the ³¹P spin-lattice relaxation rate, R₁, of phospholipids can be used to differentiate motions for these molecules in a variety of unilamellar vesicles. In particular, internal motion with a 5- to 10-ns correlation time has been attributed to diffusion-in-a-cone of the phosphodiester region, analogous to motion of a cylinder in a liquid hydrocarbon. We use the temperature dependence of ³¹P R₁ at low field (0.03-0.08 T), which reflects this correlation time, to explore the energy barriers associated with this motion. Most phospholipids exhibit a similar energy barrier of 13.2 ± 1.9 kJ/mol at temperatures above that associated with their gel-to-liquid-crystalline transition (T_m); at temperatures below T_m, this barrier increases dramatically to 68.5 ± 7.3 kJ/mol. This temperature dependence is broadly interpreted as arising from diffusive motion of the lipid axis in a spatially rough potential energy landscape. The inclusion of cholesterol in these vesicles has only moderate effects for phospholipids at temperatures above their T_m, but significantly reduces the energy barrier (to 17 ± 4 kJ/mol) at temperatures below the T_m of the pure lipid. Very-low-field R₁ data indicate that cholesterol inclusion alters the averaged disposition of the phosphorus-to-glycerol-proton vector (both its average length and its average angle with respect to the membrane normal) that determines the ³¹P relaxation.     

Motional timescale predictions by molecular dynamics simulations: Case study using proline and hydroxyproline sidechain dynamics

We propose a new approach for force field optimizations which aims at reproducing dynamics characteristics using biomolecular MD simulations, in addition to improved prediction of motionally averaged structural properties available from experiment. As the source of experimental data for dynamics fittings, we use ¹³C NMR spin‐lattice relaxation times T₁ of backbone and sidechain carbons, which allow to determine correlation times of both overall molecular and intramolecular motions. For structural fittings, we use motionally averaged experimental values of NMR J couplings. The proline residue and its derivative 4‐hydroxyproline with relatively simple cyclic structure and sidechain dynamics were chosen for the assessment of the new approach in this work. Initially, grid search and simplexed MD simulations identified large number of parameter sets which fit equally well experimental J couplings. Using the Arrhenius‐type relationship between the force constant and the correlation time, the available MD data for a series of parameter sets were analyzed to predict the value of the force constant that best reproduces experimental timescale of the sidechain dynamics. Verification of the new force‐field (termed as AMBER99SB‐ILDNP) against NMR J couplings and correlation times showed consistent and significant improvements compared to the original force field in reproducing both structural and dynamics properties. The results suggest that matching experimental timescales of motions together with motionally averaged characteristics is the valid approach for force field parameter optimization. Such a comprehensive approach is not restricted to cyclic residues and can be extended to other amino acid residues, as well as to the backbone. Proteins 2014; 82:195–215. © 2013 Wiley Periodicals, Inc.