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.     

Model-free model elimination: A new step in the model-free dynamic analysis of NMR relaxation data

Model-free analysis is a technique commonly used within the field of NMR spectroscopy to extract atomic resolution, interpretable dynamic information on multiple timescales from the R ₁, R ₂, and steady state NOE. Model-free approaches employ two disparate areas of data analysis, the discipline of mathematical optimisation, specifically the minimisation of a χ² function, and the statistical field of model selection. By searching through a large number of model-free minimisations, which were setup using synthetic relaxation data whereby the true underlying dynamics is known, certain model-free models have been identified to, at times, fail. This has been characterised as either the internal correlation times, τ _e , τ _f , or τ _s , or the global correlation time parameter, local τ _m , heading towards infinity, the result being that the final parameter values are far from the true values. In a number of cases the minimised χ² value of the failed model is significantly lower than that of all other models and, hence, will be the model which is chosen by model selection techniques. If these models are not removed prior to model selection the final model-free results could be far from the truth. By implementing a series of empirical rules involving inequalities these models can be specifically isolated and removed. Model-free analysis should therefore consist of three distinct steps: model-free minimisation, model-free model elimination, and finally model-free model selection. Failure has also been identified to affect the individual Monte Carlo simulations used within error analysis. Each simulation involves an independent randomised relaxation data set and model-free minimisation, thus simulations suffer from exactly the same types of failure as model-free models. Therefore, to prevent these outliers from causing a significant overestimation of the errors the failed Monte Carlo simulations need to be culled prior to calculating the parameter standard deviations.     

Residue Specific Ribose and Nucleobase Dynamics of the cUUCGg RNA Tetraloop Motif by MNMR 13C Relaxation

The dynamics of the nucleobase and the ribose moieties in a 14-nt RNA cUUCGg hairpin-loop uniformly labeled with ¹³C and ¹⁵N were studied by ¹³C spin relaxation experiments. R₁, R_1ρ and the ¹³C-{¹H} steady-state NOE of C₆ and C_1′ in pyrimidine and C₈ and C_1′ in purine residues were obtained at 298 K. The relaxation data were analyzed by the model-free formalism to yield dynamic information on timescales of pico-, nano- and milli-seconds. An axially symmetric diffusion tensor with an overall rotational correlation time τ_c of 2.31±0.13 ns and an axial ratio of 1.35±0.02 were determined. Both findings are in agreement with hydrodynamic calculations. For the nucleobase carbons, the validity of different reported ¹³C chemical shift anisotropy values (Stueber, D. and Grant, D. M., 2002 J. Am. Chem. Soc. 124, 10539–10551; Fiala et al., 2000 J. Biomol. NMR 16, 291–302; Sitkoff, D. and Case, D. A., 1998 Prog. NMR Spectroscopy 32, 165–190) is discussed. The resulting dynamics are in agreement with the structural features of the cUUCGg motif in that all residues are mostly rigid (0.82 < S² < 0.96) in both the nucleobase and the ribose moiety except for the nucleobase of U7, which is protruding into solution (S² = 0.76). In general, ribose mobility follows nucleobase dynamics, but is less pronounced. Nucleobase dynamics resulting from the analysis of ¹³C relaxation rates were found to be in agreement with ¹⁵N relaxation data derived dynamic information (Akke et al., 1997 RNA 3, 702–709). Electronic supplementary material Electronic supplementary material is available for this article at and accessible for authorised users.     

A suite of Mathematica notebooks for the analysis of protein main chain 15N NMR relaxation data

A suite of Mathematica notebooks has been designed to ease the analysis of protein main chain ¹⁵N NMR relaxation data collected at a single magnetic field strength. Individual notebooks were developed to perform the following tasks: nonlinear fitting of ¹⁵N-T ₁ and -T ₂ relaxation decays to a two parameter exponential decay, calculation of the principal components of the inertia tensor from protein structural coordinates, nonlinear optimization of the principal components and orientation of the axially symmetric rotational diffusion tensor, model-free analysis of ¹⁵N-T ₁, -T ₂, and {¹H}–¹⁵N NOE data, and reduced spectral density analysis of the relaxation data. The principle features of the notebooks include use of a minimal number of input files, integrated notebook data management, ease of use, cross-platform compatibility, automatic visualization of results and generation of high-quality graphics, and output of analyses in text format.L. Spyracopoulos is an AHFMR Medical Research Senior Scholar     

Accuracy of Optimized Chemical-exchange Parameters Derived by Fitting CPMG R 2 Dispersion Profiles when R 2 0a ≠ R 2 0b

The transverse relaxation rate, R₂, measured as a function of the effective field (R₂ dispersion) using a Carr-Purcell-Meiboom-Gill (CPMG) pulse train, is well suited to detect conformational exchange in proteins. The dispersion data are commonly fitted by a two-site (sites a and b) exchange model with four parameters: the relative population, p_a, the difference in chemical shifts of the two sites, δω, the correlation time for exchange, τ_ex, and the intrinsic relaxation rate (i.e., transverse relaxation rate in the absence of chemical exchange), R₂⁰. Although the intrinsic relaxation rates of the two sites, R₂^0a and R₂^0b, can differ, they are normally assumed to be the same (i.e., R₂^0a = R₂^0b = R₂⁰) when fitting dispersion data. The purpose of this investigation is to determine the magnitudes of the errors in the optimized exchange parameters that are introduced by the assumption that R₂^0a = R₂^0b. In order to accomplish this goal, we first generated synthetic constant-time CPMG R₂ dispersion data assuming two-site exchange with R₂^0a ≠ R₂^0b, and then fitted the synthetic data assuming two-site exchange with R₂⁰ = R₂^0a = R₂^0b. Although all the synthetic data generated assuming R₂^0a ≠ R₂^0b were well fitted (assuming R₂^0a = R₂^0b), the optimized values of p_a and τ _ex differed from their true values, whereas the optimized values of δω values did not. A theoretical analysis using the Carver–Richards equation explains these results, and yields simple, general equations for estimating the magnitudes of the errors in the optimized parameters, as a function of ( R₂^0a − R₂^0b).     

Negative supercoiling of DNA by gyrase is inhibited in Salmonella enterica serovar Typhimurium during adaptation to acid stress

DNA in intracellular Salmonella enterica serovar Typhimurium relaxes during growth in the acidified (pH 4–5) macrophage vacuole and DNA relaxation correlates with the upregulation of Salmonella genes involved in adaptation to the macrophage environment. Bacterial ATP levels did not increase during adaptation to acid pH unless the bacterium was deficient in MgtC, a cytoplasmic‐membrane‐located inhibitor of proton‐driven F₁F₀ ATP synthase activity. Inhibiting ATP binding by DNA gyrase and topo IV with novobiocin enhanced the effect of low pH on DNA relaxation. Bacteria expressing novobiocin‐resistant (Nov^R) derivatives of gyrase or topo IV also exhibited DNA relaxation at acid pH, although further relaxation with novobiocin was not seen in the strain with Nov^R gyrase. Thus, inhibition of the negative supercoiling activity of gyrase was the primary cause of enhanced DNA relaxation in drug‐treated bacteria. The Salmonella cytosol reaches pH 5–6 in response to an external pH of 4–5: the ATP‐dependent DNA supercoiling activity of purified gyrase was progressively inhibited by lowering the pH in this range, as was the ATP‐dependent DNA relaxation activity of topo IV. We propose that DNA relaxation in Salmonella within macrophage is due to acid‐mediated impairment of the negative supercoiling activity of gyrase.     

Nitroxide spin labels as EPR reporters of the relaxation and magnetic properties of the heme–copper site in cytochrome bo 3, E. coli

A nitroxide spin label (SL) has been used to probe the electron spin relaxation times and the magnetic states of the oxygen-binding heme–copper dinuclear site in Escherichia coli cytochrome bo _3, a quinol oxidase (QO), in different oxidation states. The spin lattice relaxation times, T ₁, of the SL are enhanced by the paramagnetic metal sites in QO and hence show a strong dependence on the oxidation state of the latter. A new, general form of equations and a computer simulation program have been developed for the calculation of relaxation enhancement by an arbitrary fast relaxing spin system of S ≥ 1/2. This has allowed us to obtain an accurate estimate of the transverse relaxation time, T ₂, of the dinuclear coupled pair Fe(III)–Cu_B(II) in the oxidized form of QO that is too short to measure directly. In the case of the F′ state, the relaxation properties of the heme–copper center have been shown to be consistent with a ferryl [Fe(IV)=O] heme and Cu_B(II) coupled by approximately 1.5–3 cm⁻¹ to a radical. The magnitude suggests that the coupling arises from a radical form of the covalently linked tyrosine–histidine ligand to Cu(II) with unpaired spin density primarily on the tyrosine component. This work demonstrates that nitroxide SLs are potentially valuable tools to probe both the relaxation and the magnetic properties of multinuclear high-spin paramagnetic active sites in proteins that are otherwise not accessible from direct EPR measurements.     

Detection of nanosecond time scale side-chain jumps in a protein dissolved in water/glycerol solvent

In solution, the correlation time of the overall protein tumbling, τ _R , plays a role of a natural dynamics cutoff—internal motions with correlation times on the order of τ _R or longer cannot be reliably identified on the basis of spin relaxation data. It has been proposed some time ago that the ‘observation window’ of solution experiments can be expanded by changing the viscosity of solvent to raise the value of τ _R . To further explore this concept, we prepared a series of samples of α-spectrin SH3 domain in solvent with increasing concentration of glycerol. In addition to the conventional ¹⁵N labeling, the protein was labeled in the Val, Leu methyl positions (¹³CHD₂ on a deuterated background). The collected relaxation data were used in asymmetric fashion: backbone ¹⁵N relaxation rates were used to determine τ _R across the series of samples, while methyl ¹³C data were used to probe local dynamics (side-chain motions). In interpreting the results, it has been initially suggested that addition of glycerol leads only to increases in τ _R , whereas local motional parameters remain unchanged. Thus the data from multiple samples can be analyzed jointly, with τ _R playing the role of experimentally controlled variable. Based on this concept, the extended model-free model was constructed with the intent to capture the effect of ns time-scale rotameric jumps in valine and leucine side chains. Using this model, we made a positive identification of nanosecond dynamics in Val-23 where ns motions were already observed earlier. In several other cases, however, only tentative identification was possible. The lack of definitive results was due to the approximate character of the model—contrary to what has been assumed, addition of glycerol led to a gradual ‘stiffening’ of the protein. This and other observations also shed light on the interaction of the protein with glycerol, which is one of the naturally occurring osmoprotectants. In particular, it has been found that the overall protein tumbling is controlled by the bulk solvent, and not by a thin solvation layer which contains a higher proportion of water.     

Off-resonance rotating-frame amide proton spin relaxation experiments measuring microsecond chemical exchange in proteins

NMR spin relaxation in the rotating frame (R_1ρ) is a unique method for atomic-resolution characterization of conformational (chemical) exchange processes occurring on the microsecond time scale. Here, we use amide ¹H off-resonance R_1ρ relaxation experiments to determine exchange parameters for processes that are significantly faster than those that can be probed using ¹⁵N or ¹³C relaxation. The new pulse sequence is validated using the E140Q mutant of the C-terminal domain of calmodulin, which exhibits significant conformational exchange contributions to the transverse relaxation rates. The ¹H off-resonance R_1ρ data sample the entire relaxation dispersion profiles for the large majority of residues in this protein, which exchanges between conformations with a time constant of approximately 20 μs. This is in contrast to the case for ¹⁵N, where additional laboratory-frame relaxation data are required to determine the exchange parameters reliably. Experiments were performed on uniformly ¹⁵N-enriched samples that were either highly enriched in ²H or fully protonated. In the latter case, dipolar cross-relaxation with aliphatic protons were effectively decoupled to first order using a selective inversion pulse. Deuterated and protonated samples gave the same results, within experimental errors. The use of deuterated samples increases the sensitivity towards exchange contributions to the ¹H transverse relaxation rates, since dipolar relaxation is greatly reduced. The exchange correlation times determined from the present ¹H off-resonance R_1ρ experiments are in excellent agreement with those determined previously using a combination of ¹⁵N laboratory-frame and off-resonance R_1ρ relaxation data, with average values of and 21 ± 3 μs, respectively.     

Dynamics in the cyclic Enterobacterial common antigen as studied by <Superscript>13</Superscript>C NMR relaxation

The motional properties of the cyclic enterobacterial common antigen (cECA), consisting of four trisaccharide repeat units, have been investigated by carbon-13 spin relaxation. R₁, R₂ and NOE relaxation parameters have been determined at three magnetic field strengths. The data were interpreted within the model-free framework to include the possibility of motional anisotropy, and overall as well as local dynamical parameters were fitted separately for each ring carbon. The motional anisotropy was addressed by assuming an axially symmetric diffusion tensor, which was fitted from the overall correlation times for each site in the sugar residues using the previously determined crystal structure. The data were found to be in agreement with an oblate shape of the molecule, and the values for D_iso and were in good agreement with translational diffusion data and an estimate based on calculation of the moment of inertia tensor, respectively. The local dynamics in cECA were found to be residue-dependent. Somewhat lower values for the order parameters, as well as longer local correlation times, were observed for the -linked ManNAcA residue compared to the two -linked residues in the trisaccharide repeat unit.     

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.     

An NMR microscopic study of grape (Vitis vinifera L.)

Summary Mature healthy grape berries and berries wound-inoculated with the fungusBotrytis cinerea were examined by¹H NMR microimaging using 2D and 3D spin echo and gradient echo procedures. These NMR images were compared with representations obtained by conventional histology, where possible using the same specimens. 3D imaging datasets from excised seeds were reconstructed by surface rendering and maximum intensity projection to allow interpretation of their internal structure. T₂-weighted spin echo images revealed the major features of the pericarp, septum and loculi of whole berries. T₁-weighted images were less discriminatory of parenchyma tissues in the fruit but revealed the endosperm in seeds as a chemically shifted feature. A non-invasive study by T₁-weighted spin echo NMR imaging of infection byB. cinerea over a 6-day period showed that the disease spread throughout the exocarp but failed to spread in the mesocarp, a result confirmed by histological examination of the same specimen. Surface rendering of 3D datasets of excised seeds revealed the two ruminations of the endosperm and the distal location of the chalaza. The position of the embryonic axis was revealed in T₂-weighted maximum intensity projections. This noninvasive study revealed the need to apply a range of imaging techniques and parameters to visualise the structural features of the different parts of the grape berry.Abbrevations BF bright field - FDA fluorescein diacetate - FI field inhomogeneity - FOV field of view - NMR nuclear magnetic resonance - RF radiofrequency - T₁ spin-lattice relaxation time - T₂ spin-spin relaxation time - TE echo time - TMS tetramethylsilane - TR repeat time     

An investigation of the dynamics of spermine bound to duplex and quadruplex DNA by <Superscript>13</Superscript>C NMR spectroscopy

A detailed analysis of the ¹³C relaxation of ¹³C-labelled spermine bound to duplex and quadruplex DNA is presented. T₁, T₂ and heteronuclear NOE data were collected at four ¹³C frequencies (75.4, 125.7, 150.9 and 201.2 MHz). The data were analyzed in terms of a frequency-dependent order parameter, S ²(ω), to estimate the generalized order parameter and the contributions to the relaxation from different motional frequencies in the picosecond–nanosecond timescale and from any exchange processes that may be occurring on the microsecond–millisecond timescale. The relaxation data was surprisingly similar for spermine bound to two different duplexes and a linear parallel quadruplex. Analysis of the relaxation data from these complexes confirmed the conclusions of previous studies that the dominant motion of spermine is independent of the macroscopic tumbling of the DNA and has an effective correlation time of ∼50 ps. In contrast, spermine bound to a folded antiparallel quadruplex had faster relaxation rates, especially R ₂. As with the other complexes, a fast internal motion of the order of 50 ps makes a substantial contribution to the relaxation. The generalized order parameter for spermine bound to duplex DNA and the linear quadruplex is small but is larger for spermine bound to the folded quadruplex. In the latter case, there is evidence for exchange between at least two populations of spermine occurring on the microsecond–millisecond timescale. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Characterization of the overall and local dynamics of a protein with intermediate rotational anisotropy: Differentiating between conformational exchange and anisotropic diffusion in the B3 domain of protein G

Because the overall tumbling provides a major contribution to protein spectral densities measured in solution, the choice of a proper model for this motion is critical for accurate analysis of protein dynamics. Here we study the overall and backbone dynamics of the B3 domain of protein G using ¹⁵N relaxation measurements and show that the picture of local motions is markedly dependent on the model of overall tumbling. The main difference is in the interpretation of the elevated R ₂ values in the -helix: the isotropic model results in conformational exchange throughout the entire helix, whereas no exchange is predicted by anisotropic models that place the longitudinal axis of diffusion tensor almost parallel to the helix axis. Due to small size (fast tumbling) of the protein, the T ₁ values have low sensitivity to NH bond orientation. The diffusion tensor derived from orientation dependence of R ₂/R ₁ is anisotropic (D _par/D _perp=1.4), with a small rhombic component. In order to distinguish the correct picture of motion, we apply model-independent methods that are sensitive to conformational exchange and do not require knowledge of protein structure or assumptions about its dynamics. A comparison of the CSA/dipolar cross-correlation rate constants with ¹⁵N relaxation rates and the estimation of R _ex terms from relaxation data at 9.4 and 14.1 T indicate no conformational exchange in the helix, in support of the anisotropic models. The experimentally derived diffusion tensor is in excellent agreement with theoretical predictions from hydrodynamic calculations; a detailed comparison with various hydrodynamic models revealed optimal parameters for hydrodynamic calculations.     

Cross-correlation suppressed T1 and NOE experiments for protein side-chain 13CH2 groups

Relaxation measurements of side-chain ¹³CH₂-groups of uniformly ¹³C labeled human ubiquitin were performed at 600 MHz and 800 MHz magnetic field strength at 30°C. Dipole-dipole cross-correlated relaxation effects in T₁ experiments were suppressed by the combination of radio-frequency pulses and pulsed field gradients during the relaxation delay leading to monoexponential relaxation decays that allow a more accurate extraction of the ¹³C T₁ relaxation times. Heteronuclear ¹H-¹³C NOEs obtained by using different proton saturation schemes indicate that the influence of cross-correlation is small. The experimental T₁ and NOE data were interpreted in a model-free way in terms of a generalized order parameter and an internal correlation time.     

NMR study of the structure of short-chain peptides in solution.

The electron-mediated spin–spin coupling constant J between the amide NH and the α-CH protons in the dipeptide fragment C^α? CO(NH? C^αH)R? C′ONH? C^α is dependent on the dihedral angle of rotation (Φ) around the N? C bond. Measurement of J in a series of zwitterionic dipeptides H₃N⁺? CHR₁? CONH? CHR₂? CO₂^? (which is conformationally similar to the dipeptide fragment) in TFA solution shows that J is independent of R₁, but dependent on the steric bulk of R₂. The data are interpreted in terms of a model that assumes that what we measure is an average value of J? a thermal average over all the possible rotamers. The groups R₁ and R₂ are, in most cases, sterically kept apart by the trans and planar amide bonds, and hence the independence of J of R₁. This model is consistent with the theoretical calculations done on the dipeptide fragment. The effect of the structural characteristics of the side chains (e.g., the effect of lengthening and branching the side chains) on the J values in dipeptides is discussed in the light of the existing results of theoretical calculations. Study of 〈J〉 values in tripeptides (C₆H₅CH₂OCONH? CHR₁? CONH? CHR₂? CO₂CH₃, essentially three linked peptide units) shows that electrostatic interaction between the two amide bonds modifies the potential energy surface and the 〈J〉 value of a dipeptide subunit in the tripeptides. Also in some cases, direct steric interaction between the two side chains in the two adjacent dipeptide subunits in the tripeptide affects the potential energy surfaces of the individual dipeptide subunits and hence the 〈J〉 values. The influence of the structural characteristics of the side chains of individual amino acids on structure formation at or beyond the dipeptide level is discussed at various points. The J(NH? ^αCH) values of CH₃CONH? CHR? CONH₂ and CH₃CONH? CHR? CO₂CH₃ with the same R are quite different for R = valine, leucine, phenylalanine, methionine, but equal for R = glycine. This, coupled with the fact that one of the carboxamide NH resonances has a chemical shift different from its counterpart in simple amides like CH₃CONH₂ and the other carboxamide NH has the same chemical shift as its counterpart in CH₃CONH₂, suggest the presence of a hydrogen bond in dipeptide CH₃CONH? CHR? CONH₂ with carboxamide NH as the donor. Theoretical evidence for two seven-membered hydrogen-bonded rings with the carboxamide NH as donor and the acetyl oxygen as acceptor is summarized. Our data cannot suggest the number of such hydrogen-bonded rings, nor can they conclude the relative proportion of these rings in a particular dipeptide. A discussion of the difficulty of interpretation is presented and the data are discussed under certain simplifying assumptions.     

Determination of <Superscript>13</Superscript>C CSA Tensors: Extension of the Model-independent Approach to an RNA Kissing Complex Undergoing Anisotropic Rotational Diffusion in Solution

Chemical shift anisotropy (CSA) tensor parameters have been determined for the protonated carbons of the purine bases in an RNA kissing complex in solution by extending the model-independent approach [Fushman, D., Cowburn, D. (1998) J. Am. Chem. Soc. 120, 7109–7110]. A strategy for determining CSA tensor parameters of heteronuclei in isolated X–H two-spin systems (X = ¹³C or ¹⁵N) in molecules undergoing anisotropic rotational diffusion is presented. The original method relies on the fact that the ratio κ₂=R₂^auto/R₂^cross of the transverse auto- and cross-correlated relaxation rates involving the X CSA and the X–H dipolar interaction is independent of parameters related to molecular motion, provided rotational diffusion is isotropic. However, if the overall motion is anisotropic κ₂ depends on the anisotropy D_||/D_⊥ of rotational diffusion. In this paper, the field dependence of both κ₂ and its longitudinal counterpart κ₁=R₁^auto/R₁^cross are determined. For anisotropic rotational diffusion, our calculations show that the average κ_av = 1/2 (κ₁+κ₂), of the ratios is largely independent of the anisotropy parameter D_||/D_⊥. The field dependence of the average ratio κ_av may thus be utilized to determine CSA tensor parameters by a generalized model-independent approach in the case of molecules with an overall motion described by an axially symmetric rotational diffusion tensor.     

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.     

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.     

Anisotropic rotational motions of poly(L-glutamic acid) in the alpha-helix conformation

The anisotropic rotational motion of the backbone and the side chains of poly(L -glutamic acid) in the α-helical structure was investigated using the ¹³C-T₁ and T₂ relaxation times of all carbon atoms with directly attached protons, obtained at a ¹³C-Larmor frequency of 67.89 MHz. The evaluation of the nmr data was carried out according to the previously derived anisotropic diffusion model, in which the macromolecule is considered a rigid rod. The rotation of the backbone is characterized by two diffusion constants, D₁ and D₃, describing the rotation perpendicular to and around the symmetry axis. The additional internal motion of the C^β-methylene group is described as a jump process with a jump rate, k₁, between two allowed rotametric states. Steric considerations indicate that the occupation of the third rotameric position is forbidden. The rotation of the C^γ-methylene group is decribed as a one-dimensional diffusion process around the C^β–C^γ bond. Investigation of the temperature dependence of the relaxation parameters led to the temperature dependence of the dynamic parameters. Activation energies were determined from these data. The dynamic parameters obtained for poly(L -glutamic acid) at 291 K are compared with the corresponding results of a previous study of poly(L -lysine). The development of an anisotropic diffusion model for the motions of the rod-shaped poly(L -lysine) α-helix and its application to the interpretation of the ¹³C-relaxation data of this molecule have already been published previously. In this model, both the overall molecular tumbling and the various internal motions have been characterized by diffusion constants or jump rates typical for each process. These dynamic parameters can be calculated from the spin–lattice relaxation times, the spin–spin relaxation times and the NOE factors of the C^α, C^β, and C^γ nuclei of the polypetide. In the present paper, we describe the application of the above-mentioned dynamic model to the interpretation of ¹³C-relaxation studies of a further homopolypeptide, poly(L -glutamic acid), in the α-helical structure. Furthermore, we studied the temperature dependence of the relaxation times of this polymer and determined the anisotropic diffusion parameters at each temperature. From their temperature dependence and from comparison of our present results with the data of our previous study of poly(L -lysine), we were able to derive new insights into the intramolecular diffusion processes and the excitation of various motions.