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.     

Set theory formulation of the model-free problem and the diffusion seeded model-free paradigm

Model-free analysis of NMR relaxation data, which describes the motion of individual atoms, is a problem intricately linked to the Brownian rotational diffusion of the macromolecule. The diffusion tensor parameters strongly influence the optimisation of the various model-free models and the subsequent model selection between them. Finding the optimal model of the dynamics of the system among the numerous diffusion and model-free models is hence quite complex. Using set theory, the entirety of this global problem has been encapsulated by the universal set Ll, and its resolution mathematically formulated as the universal solution Ll. Ever since the original Lipari and Szabo papers the model-free dynamics of a molecule has most often been solved by initially estimating the diffusion tensor. The model-free models which depend on the diffusion parameter values are then optimised and the best model is chosen to represent the dynamics of the residue. Finally, the global model of all diffusion and model-free parameters is optimised. These steps are repeated until convergence. For simplicity this approach to Ll will be labelled the diffusion seeded model-free paradigm. Although this technique suffers from a number of problems many have been solved. All aspects of the diffusion seeded paradigm and its consequences, together with a few alternatives to the paradigm, will be reviewed through the use of set notation.     

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.     

Optimisation of NMR dynamic models II. A new methodology for the dual optimisation of the model-free parameters and the Brownian rotational diffusion tensor

Finding the dynamics of an entire macromolecule is a complex problem as the model-free parameter values are intricately linked to the Brownian rotational diffusion of the molecule, mathematically through the autocorrelation function of the motion and statistically through model selection. The solution to this problem was formulated using set theory as an element of the universal set —the union of all model-free spaces (d’Auvergne EJ and Gooley PR (2007) Mol BioSyst 3(7), 483–494). The current procedure commonly used to find the universal solution is to initially estimate the diffusion tensor parameters, to optimise the model-free parameters of numerous models, and then to choose the best model via model selection. The global model is then optimised and the procedure repeated until convergence. In this paper a new methodology is presented which takes a different approach to this diffusion seeded model-free paradigm. Rather than starting with the diffusion tensor this iterative protocol begins by optimising the model-free parameters in the absence of any global model parameters, selecting between all the model-free models, and finally optimising the diffusion tensor. The new model-free optimisation protocol will be validated using synthetic data from Schurr JM et al. (1994) J Magn Reson B 105(3), 211–224 and the relaxation data of the bacteriorhodopsin (1–36)BR fragment from Orekhov VY (1999) J Biomol NMR 14(4), 345–356. To demonstrate the importance of this new procedure the NMR relaxation data of the Olfactory Marker Protein (OMP) of Gitti R et al. (2005) Biochem 44(28), 9673–9679 is reanalysed. The result is that the dynamics for certain secondary structural elements is very different from those originally reported. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Anisotropic rotational diffusion in model-free analysis for a ternary DHFR complex

Model-free analysis has been extensively used to extract information on motions in proteins over a wide range of timescales from NMR relaxation data. We present a detailed analysis of the effects of rotational anisotropy on the model-free analysis of a ternary complex for dihydrofolate reductase (DHFR). Our findings show that the small degree of anisotropy exhibited by DHFR (D_||/D=1.18) introduces erroneous motional models, mostly exchange terms, to over 50% of the NH spins analyzed when isotropic tumbling is assumed. Moreover, there is a systematic change in S², as large as 0.08 for some residues. The significant effects of anisotropic rotational diffusion on model-free motional parameters are in marked contrast to previous studies and are accentuated by lowering of the effective correlation time using isotropic tumbling methods. This is caused by the preponderance of NH vectors aligned perpendicular to the principal diffusion tensor axis and is readily detected because of the high quality of the relaxation data. A novel procedure, COPED (COmparison of Predicted and Experimental Diffusion tensors) is presented for distinguishing genuine motions from the effects of anisotropy by comparing experimental relaxation data and data predicted from hydrodynamic analyses. The procedure shows excellent agreement with the slow motions detected from the axially symmetric model-free analysis and represents an independent procedure for determining rotational diffusion and slow motions that can confirm or refute established procedures that rely on relaxation data. Our findings show that neglect of even small degrees of rotational diffusion anisotropy can introduce significant errors in model-free analysis when the data is of high quality. These errors can hinder our understanding of the role of internal motions in protein function.     

Solution structure and backbone dynamics of the defunct domain of calcium vector protein.

CaVP (calcium vector protein) is a Ca(2+) sensor of the EF-hand protein family which is highly abundant in the muscle of Amphioxus. Its three-dimensional structure is not known, but according to the sequence analysis, the protein is composed of two domains, each containing a pair of EF-hand motifs. We determined recently the solution structure of the C-terminal domain (Trp81-Ser161) and characterized the large conformational and dynamic changes induced by Ca(2+) binding. In contrast, the N-terminal domain (Ala1-Asp86) has lost the capacity to bind the metal ion due to critical mutations and insertions in the two calcium loops. In this paper, we report the solution structure of the N-terminal domain and its backbone dynamics based on NMR spectroscopy, nuclear relaxation, and molecular modeling. The well-resolved three-dimensional structure is typical of a pair of EF-hand motifs, joined together by a short antiparallel beta-sheet. The tertiary arrangement of the two EF-hands results in a closed-type conformation, with near-antiparallel alpha-helices, similar to other EF-hand pairs in the absence of calcium ions. To characterize the internal dynamics of the protein, we measured the (15)N nuclear relaxation rates and the heteronuclear NOE effect in (15)N-labeled N-CaVP at a magnetic field of 11.74 T and 298 K. The domain is mainly monomeric in solution and undergoes an isotropic Brownian rotational diffusion with a correlation time of 7.1 ns, in good agreement with the fluorescence anisotropy decay measurements. Data analysis using a model-free procedure showed that the amide backbone groups in the alpha-helices and beta-strands undergo highly restricted movements on a picosecond to nanosecond time scale. The amide groups in Ca(2+) binding loops and in the linker fragment also display rapid fluctuations with slightly increased amplitudes.     

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     

Assessing potential bias in the determination of rotational correlation times of proteins by NMR relaxation

The various factors that influence the reliable and efficient determination of the correlation time describing molecular reorientation of proteins by NMR relaxation methods are examined. Nuclear Overhauser effects, spin-lattice, and spin-spin relaxation parameters of 15N NMR relaxation in ubiquitin have been determined at 17.6, 14.1, 11.7 and 9.4 Tesla. This unusually broad set of relaxation parameters has allowed the examination of the influence of chemical shift anisotropy, the functional form of the model-free spectral density, and the reliability of determined spin- spin relaxation parameters on the characterization of global tumbling of the protein. Treating the 15N chemical shift anisotropy (CSA) as an adjustable parameter, a consensus value of –170 ± 15ppm for the breadth of the chemical shift tensor and a global isotropic correlation time of 4.1ns are found when using the model-free spectral density to fit T1 and NOE data from all fields. The inclusion of T2 relaxation parameters in the determination of the global correlation time results in its increase to 4.6ns. This apparent inconsistency may explain a large portion of the discrepancy often found between NMR- and fluorescence-derived m values for proteins. The near identity of observed T2 and T1 values suggests that contributions from slow motions are not the origin of the apparent inconsistency with obtained T1 and NOE data. Various considerations suggest that the origin of this apparent discrepancy may reside in a contribution to the spectral density at zero frequency that is not represented by the simple model-free formalism in addition to the usual experimental difficulties associated with the measurement of these relaxation parameters. Finally, an axially symmetric diffusion tensor for ubiquitin is obtained using exclusively T1 and NOE data. A recommendation is reached on the types and combinations of relaxation data that can be used to reliably determine m values. It is also noted that the reliable determination of m values from 15N T1 and NOE relaxation parameters will become increasingly difficult as m increases.     

Study of the rotational diffusion properties of peptides in solution

In this paper we examine the theory and method for obtaining rotational diffusion coefficients for peptides in dilute solution from ¹³C-nmr spin-lattice relaxation data. We show that even for the case of nearly equal observed relaxation times of chemically and magnetically nonequivalent alpha-carbons marked rotational anisotropy will be the usual case. We describe two interactive, minicomputer programs which are of general use in this type of work. The implications of this study on spectral density-based conformational determinations of peptides is discussed.     

Backbone dynamics of barstar: a (15)N NMR relaxation study

Backbone dynamics of uniformly (15)N-labeled barstar have been studied at 32 degrees C, pH 6.7, by using (15)N relaxation data obtained from proton-detected 2D (1)H-(15)N NMR spectroscopy. (15)N spin-lattice relaxation rate constants (R(1)), spin-spin relaxation rate constants (R(2)), and steady-state heteronuclear (1)H-(15)N NOEs have been determined for 69 of the 86 (excluding two prolines and the N-terminal residue) backbone amide (15)N at a magnetic field strength of 14.1 Tesla. The primary relaxation data have been analyzed by using the model-free formalism of molecular dynamics, using both isotropic and axially symmetric diffusion of the molecule, to determine the overall rotational correlation time (tau(m)), the generalized order parameter (S(2)), the effective correlation time for internal motions (tau(e)), and NH exchange broadening contributions (R(ex)) for each residue. As per the axially symmetric diffusion, the ratio of diffusion rates about the unique and perpendicular axes (D( parallel)/D( perpendicular)) is 0.82 +/- 0.03. The two results have only marginal differences. The relaxation data have also been used to map reduced spectral densities for the NH vectors of these residues at three frequencies: 0, omega(H), and omega(N), where omega(H),(N) are proton and nitrogen Larmor frequencies. The value of tau(m) obtained from model-free analysis of the relaxation data is 5.2 ns. The reduced spectral density analysis, however, yields a value of 5.7 ns. The tau(m) determined here is different from that calculated previously from time-resolved fluorescence data (4.1 ns). The order parameter ranges from 0.68 to 0.98, with an average value of 0.85 +/- 0.02. A comparison of the order parameters with the X-ray B-factors for the backbone nitrogens of wild-type barstar does not show any considerable correlation. Model-free analysis of the relaxation data for seven residues required the inclusion of an exchange broadening term, the magnitude of which ranges from 2 to 9.1 s(-1), indicating the presence of conformational averaging motions only for a small subset of residues.     

Rotational dynamics of transfer ribonucleic acid: effect of ionic strength and concentration

We have investigated the influence of ionic strength and nucleic acid concentration on the rotational Brownian motion of Escherichia coli tRNA1Val by studying the decay of the fluorescence polarization anisotropy (FPA) of intercalated ethidium on a nanosecond time scale. The rotational relaxation time tau R remains essentially constant as the ionic strength is varied from 2 to 100 mM at a tRNA concentration of 54 mg/mL. tau R also remains practically unchanged as the tRNA concentration is varied from 0.3 to 54 mg/mL at an ionic strength of 130 mM. Present hydrodynamic theories generally predict a more pronounced concentration dependence for rotational diffusion than we observe. This disagreement may result from a nonrandom distribution of the tRNA molecules in solution due to electrostatic interactions. By combining independent data from time-resolved nuclear Overhauser effect (NOE) cross-relaxation experiments and FPA experiments on the same tRNA, we are able to estimate the interproton spacing for the guanine N1-H and the uracil N3-H of the GU-50 base pair in E. coli tRNA1Val. This distance is 0.272 nm.     

Fluorescence correlation spectroscopy and Brownian rotational diffusion

A theroy relating rotational Brownian motion to the time autocorrelation function of the intensity of radiation from a fluorescent system composed of spherical rotors is presented. The calculation shows three relaxation times, two associated with the rotational diffusion, and the third associated with the natural decay of the fluorescence. The correlation function contains terms that relax independently of the fluorescence decay time, thus arbitrarily extending the time range over which rotational diffusion can be studied by fluorescence.     

Simulation of the conformation and dynamics of a double-helical model for DNA

We propose a partially flexible, double-helical model for describing the conformational and dynamic properties of DNA. In this model, each nucleotide is represented by one element (bead), and the known geometrical features of the double helix are incorporated in the equilibrium conformation. Each bead is connected to a few neighbor beads in both strands by means of stiff springs that maintain the connectivity but still allow for some extent of flexibility and internal motion. We have used Brownian dynamics simulation to sample the conformational space and monitor the overall and internal dynamics of short DNA pieces, with up to 20 basepairs. From Brownian trajectories, we calculate the dimensions of the helix and estimate its persistence length. We obtain translational diffusion coefficient and various rotational relaxation times, including both overall rotation and internal motion. Although we have not carried out a detailed parameterization of the model, the calculated properties agree rather well with experimental data available for those oligomers.     

Study of molecular dynamics of bovine carbonic anhydrase B by 13C-NMR in two strongly polarizing magnetic fields. I. Intramolecular mobility of polypeptide chains of the native enzyme

The study of the dynamics of enzyme segmental movement is of considerable importance in the understanding of the physics of the catalytic function of these macromolecules, which cannot be adequately described without introduction of intramolecular mobility of their polypeptide chains. At present high resolution [13C]NMR is mostly used as an effective and selective method for the observation of spectral and relaxation parameters that are sensitive to structure, conformation and local motion. The molecular dynamics of bovine carbonic anhydrase B (carbonate hydrolase EC. 4.2.1.1) in the native form was studied. Measurements of the relaxation parameters (T1, T2 and NOE) of the alpha-carbons of the polypeptide chain in two high magnetic fields (4.7 and 11.7 T) were carried out. The model-free approach of Lipari and Szabo to the interpretation of these experimental data show a satisfactory agreement between theory and experiment for these carbon nuclei if an internal degree of motion such as libration or restricted diffusion in a cone with angular amplitude in the 10 degrees less than theta less than or equal to 20 degrees range and an effective correlation time tau e approximately equal to 6 to 7 x 10(-11) S in addition to the tau R = 3 x 10(-8) S reorientation correlation time of the whole molecular is introduced.     

NMR structure and backbone dynamics of the extended second transmembrane domain of the human neuronal glycine receptor alpha1 subunit

The structure and backbone dynamics of an extended second transmembrane segment (TM2e) of the human neuronal glycine receptor alpha(1) subunit in sodium dodecyl sulfate micelles were studied by (1)H and (15)N solution-state NMR. The 28-amino acid segment contained the consensus TM2 domain plus part of the linker between the second and third transmembrane domains. The presence of a well-structured helical region of at least 13 amino acids long and an unstructured region near the linker was evident from the proton chemical shifts and the pattern of midrange nuclear Overhauser effects (NOE). (15)N relaxation rate constants, R(1) and R(2), and (15)N-[(1)H] NOE indicated restricted internal motions in the helical region with NOE values between 0.6 and 0.8. The squared order parameter (S(2)), the effective correlation time for fast internal motions (tau(e)), and the global rotational correlation time (tau(m)) were calculated for all TM2e backbone N-H bonds using the model-free approach. The S(2) values ranged about 0.75-0.86, and the tau(e) values were below 100 ps for most of the residues in the helical region. The tau(m) value, calculated from the dynamics of the helical region, was 5.1 ns. The S(2) values decreased to 0.1, and the tau(e) values sharply increased up to 1.2 ns at the linker near the C-terminus, indicating that the motion of this region is unrestricted. The results suggest a relatively high degree of motional freedom of TM2e in micelles and different propensities of the N- and C-terminal moieties of the transmembrane domain to assume stable helical structures.     

Trehalose effect on low temperature protein dynamics: fluctuation and relaxation phenomena

We performed spectral diffusion experiments in trehalose-enriched glycerol/buffer-glass on horseradish peroxidase where the heme was replaced by metal-free mesoporphyrin IX, and compared them with the respective behavior in a pure glycerol/buffer-glass (Schlichter et al., J. Chem. Phys. 2000, 112:3045-3050). Trehalose has a significant influence: spectral diffusion broadening speeds up compared to the trehalose-free glass. This speeding up is attributed to a shortening of the correlation time of the frequency fluctuations most probably by preventing water molecules from leaving the protein interior. Superimposed to the frequency fluctuation dynamics is a relaxation dynamics that manifests itself as an aging process in the spectral diffusion broadening. Although the trehalose environment speeds up the fluctuations, it does not have any influence on the relaxation. Both relaxation and fluctuations are governed by power laws in time. The respective exponents do not seem to change with the protein environment. From the spectral dynamics, the mean square displacement in conformation space can be determined. It is governed by anomalous diffusion. The associated frequency correlation time is incredibly long, demonstrating that proteins at low temperatures are truly nonergodic systems.     

Secondary structure and dynamics of an intrinsically unstructured linker domain

The transient secondary structure and dynamics of an intrinsically unstructured linker domain from the 70 kDa subunit of human replication protein A was investigated using solution state NMR. Stable secondary structure, inferred from large secondary chemical shifts, was observed for a segment of the intrinsically unstructured linker domain when it is attached to an N-terminal protein interaction domain. Results from NMR relaxation experiments showed the rotational diffusion for this segment of the intrinsically unstructured linker domain to be correlated with the N-terminal protein interaction domain. When the N-terminal domain is removed, the stable secondary structure is lost and faster rotational diffusion is observed. The large secondary chemical shifts were used to calculate phi and psi dihedral angles and these dihedral angles were used to build a backbone structural model. Restrained molecular dynamics were performed on this new structure using the chemical shift based dihedral angles and a single NOE distance as restraints. In the resulting family of structures a large, solvent exposed loop was observed for the segment of the intrinsically unstructured linker domain that had large secondary chemical shifts.     

A variable target intensity-restrained global optimization (VARTIGO) procedure for determining three-dimensional structures of polypeptides from NOESY data: Application to gramicidin-S

Summary A global optimization method for intensity-restrained structure refinement, based on variable target function (VTF) analysis, is illustrated using experimental data on a model peptide, gramicidin-S (GS) dissolved in DMSO. The method (referred to as VARTIGO for variable target intensity-restrained global optimization) involves minimization of a target function in which the range of NOE contacts is gradually increased in successive cycles of optimization in dihedral angle space. Several different starting conformations (including all-trans) have been tested to establish the validity of the method. Not all optimizations were successful, but these were readily identifiable from their large NOE R-factors. We also show that it is possible to simultaneously optimize the rotational correlation time along with the dihedral angles. The structural features of GS thus obtained from the successful optimizations are in excellent agreement with the available experimental data. A comparison is made with structures generated from an intensity-restrained single target function (STF) analysis. The results on GS suggest that VARTIGO refinement is capable of yielding better quality structures. Our work also underscores the need for a simultaneous analysis of different NOE R-factors in judging the quality of optimized structures. The NOESY data on GS in DMSO appear to provide evidence for the presence of two orientations for the ornithine side chain, in fast exchange. The NOESY spectra for this case were analyzed using a relaxation rate matrix which is a weighted average of the relaxation rate matrices for the individual conformations.     

Backbone dynamics of apocytochrome b5 in its native, partially folded state

The backbone dynamics in the native state of apocytochrome b5 were studied using 15N nuclear magnetic spin relaxation measurements. The field (11.7 and 14.1 T) and temperature (10-25 degrees C) dependence of the relaxation parameters (R1, R2, and R1rho) and the 1H-15N NOE established that the protein undergoes multiple time scale internal motions related to the secondary structure. The relaxation data were analyzed with the reduced spectral density mapping approach and within the extended model-free framework. The apoprotein was confirmed to contain a disordered heme-binding loop of approximately 30 residues with dynamics on the sub-nanosecond time scale (0.6 < S2 < 0.7, 100 ps < taue < 500 ps). This loop is attached to a structured hydrophobic core, rigid on the picosecond time scale (S2 > 0.75, taue < 50 ps). The inability to fit the data for several residues with the model-free protocol revealed the presence of correlated motion. An exchange contribution was detected in the transverse relaxation rate (R2) of all residues. The differential temperature response of R2 along the backbone supported slower exchange rates for residues in the loop (tauex > 300 micros) than for the folded polypeptide chain (tauex < 150 micros). The distribution of the reduced spectral densities at the 1H and 15N frequencies followed the dynamic trend and predicted the slowing of the internal motions at 10 degrees C. Comparison of the dynamics with those of the holoprotein [Dangi, B., Sarma, S., Yan, C., Banville, D. L., and Guiles, R. D. (1998) Biochemistry 37, 8289-8302] demonstrated that binding of the heme alters the time scale of motions both in the heme-binding loop and in the structured hydrophobic core.