On the reliability of NMR relaxation data analyses: a Markov Chain Monte Carlo approach

Bayesian NMR petrophysical characterization

Identification of reservoir rock types is necessary for the exploration and recovery of oil and gas. It involves determining the petrophysical properties of rocks such as porosity and permeability which play a significant role in developing reservoir models, estimating the volumes of oil and gas reserves, and planning production methods. Nuclear magnetic resonance (NMR) technology is a fast and accurate tool for petrophysical rock characterization. The distributions of relaxation times (T2 distributions) offer valuable insights into the distribution of pore sizes in rocks, and these distributions are closely linked to important petrophysical parameters like porosity, permeability, and bound fluid volume (BFV). This work introduces a Bayesian estimation method for analyzing NMR data. The Bayesian approach uses prior knowledge of T2 distributions in the form of the prior mean and covariance. The Bayesian approach combines prior knowledge with observed data to obtain improved estimation. We use the Bayesian estimation method where prior information regarding the rock sample type, for example shale, is available. The estimators were evaluated on decay data simulated from synthesized distributions that replicate the features of experimental T2 distributions of three types of reservoir rocks. We compared the performance of the Bayesian method with two existing methods using porosity, bound fluid volume (BFV) geometric mean (T2LM) and root mean square error (RMSE) of the estimated T2 distribution as evaluation criteria. Additional experiments were carried out using experimental T2 distributions to validate the results. The performance of the Bayesian methods was also tested using mismatched priors. The experimental results illustrate that the Bayesian estimator outperforms other estimators in estimating the T2 distribution. The Bayesian method also outperforms the ILT method in estimating derived petrophysical properties except in cases where the noise level is below 0.1 and the T2 distributions are associated with short relaxation times.

Deconvolution and Analysis of the 1H NMR Spectra of Crude Reaction Mixtures

Nuclear magnetic resonance (NMR) spectroscopy is an important analytical technique in synthetic organic chemistry, but its integration into high-throughput experimentation workflows has been limited by the necessity of manually analyzing the NMR spectra of new chemical entities. Current efforts to automate the analysis of NMR spectra rely on comparisons to databases of reported spectra for known compounds and, therefore, are incompatible with the exploration of new chemical space. By reframing the NMR spectrum of a reaction mixture as a joint probability distribution, we have used Hamiltonian Monte Carlo Markov Chain and density functional theory to fit the predicted NMR spectra to those of crude reaction mixtures. This approach enables the deconvolution and analysis of the spectra of mixtures of compounds without relying on reported spectra. The utility of our approach to analyze crude reaction mixtures is demonstrated with the experimental spectra of reactions that generate a mixture of isomers, such as Wittig olefination and C-H functionalization reactions. The correct identification of compounds in a reaction mixture and their relative concentrations is achieved with a mean absolute error as low as 1%.

Molecular Dynamics-Assisted Optimization of Protein NMR Relaxation Analysis

NMR relaxation analysis of the mobile residues in globular proteins is sensitive to the form of the experimentally fitted internal autocorrelation function, which is used to represent that motion. Different order parameter representations can precisely fit the same set of ¹⁵N R₁, R₂, and heteronuclear NOE measurements while yielding significantly divergent predictions of the underlying autocorrelation functions, indicating the insufficiency of these experimental relaxation data for assessing which order parameter representation provides the most physically realistic predictions. Molecular dynamics simulations offer an unparalleled capability for discriminating among different order parameter representations to assess which representation can most accurately model a wide range of physically realistic autocorrelation functions. Six currently utilized AMBER and CHARMM force fields were applied to calculate autocorrelation functions for the backbone H–N bond vectors of ubiquitin as an operational test set. An optimized time constant-constrained triexponential (TCCT) representation was shown to markedly outperform the widely used (S_f²,τ_s,S²) extended Lipari–Szabo representation and the more closely related (S_f²,S_H², S_N²) Larmor frequency-selective representation. Optimization of the TCCT representation at both 600 and 900 MHz ¹H converged to the same parameterization. The higher magnetic field yielded systematically larger deviations in the back-prediction of the autocorrelation functions for the mobile amides, indicating little added benefit from multiple field measurements in analyzing amides that lack slower (∼ms) exchange line-broadening effects. Experimental ¹⁵N relaxation data efficiently distinguished among the different force fields with regard to their prediction of ubiquitin backbone conformational dynamics in the ps–ns time frame. While the earlier AMBER 99SB and CHARMM27 force fields underestimate the scale of backbone dynamics, which occur in this time frame, AMBER 14SB provided the most consistent predictions for the well-averaged highly mobile C-terminal residues of ubiquitin.

Bootstrap Aggregation for Model Selection in the Model-free Formalism

The ability to make robust inferences about the dynamics of biological macromolecules using NMR spectroscopy depends heavily on the application of appropriate theoretical models for nuclear spin relaxation. Data analysis for NMR laboratory-frame relaxation experiments typically involves selecting one of several model-free spectral density functions using a bias-corrected fitness test. Here, advances in statistical model selection theory, termed bootstrap aggregation or bagging, are applied to ¹⁵N spin relaxation data, developing a multimodel inference solution to the model-free selection problem. The approach is illustrated using data sets recorded at four static magnetic fields for the bZip domain of the S. cerevisiae transcription factor GCN4.

13C NMR Relaxation Analysis of Protein GB3 for the Assessment of Side Chain Dynamics Predictions by Current AMBER and CHARMM Force Fields

Molecular simulations with seven current AMBER- and CHARMM-based force fields yield markedly differing internal bond vector autocorrelation function predictions for many of the 223 methine and methylene H-C bonds of the 56-residue protein GB3. To enable quantification of accuracy, ¹³C R₁, R₂, and heteronuclear NOE relaxation rates have been determined for the methine and stereochemically assigned methylene C^α and C^β positions. With only three experimental relaxation values for each bond vector, central to this analysis is the accuracy with which MD-derived autocorrelation curves can be represented by a 3-parameter equation which, in turn, maps onto the NMR relaxation values. In contrast to the more widely used extended Lipari-Szabo order parameter representation, 95% of these MD-derived internal autocorrelation curves for GB3 can be fitted to within 1.0% rmsd over the time frame from 30 ps to 4 ns by a biexponential Larmor frequency-selective representation (LF-S²). Applying the LF-S² representation to the experimental relaxation rates and uncertainties serves to determine the boundary range for the autocorrelation function of each bond vector consistent with the experimental data. Not surprisingly, all seven force fields predict the autocorrelation functions for the more motionally restricted ¹H^α-¹³C^α and ¹H^β-¹³C^β bond vectors with reasonable accuracy. However, for the ¹H^β-¹³C^β bond vectors exhibiting aggregate order parameter S² values less than 0.85, only 1% of the MD-derived predictions lie with 1 σ of the experimentally determined autocorrelation functions and only 7% within 2 σ. On the other hand, substantial residue type-specific improvements in predictive performance were observed among the recent AMBER force fields. This analysis indicates considerable potential for the use of ¹³C relaxation measurements in guiding the optimization of the side chain dynamics characteristics of protein molecular simulations.

Analysis of NMR Spin-Relaxation Data Using an Inverse Gaussian Distribution Function

Spin relaxation in solution-state NMR spectroscopy is a powerful approach to explore the conformational dynamics of biological macromolecules. Probability distribution functions for overall or internal correlation times have been used previously to model spectral density functions central to spin-relaxation theory. Applications to biological macromolecules rely on transverse relaxation rate constants, and when studying nanosecond timescale motions, sampling at ultralow frequencies is often necessary. Consequently, appropriate distribution functions necessitate spectral density functions that are accurate and convergent as frequencies approach zero. In this work, the inverse Gaussian probability distribution function is derived from general properties of spectral density functions at low and high frequencies for macromolecules in solution, using the principle of maximal entropy. This normalized distribution function is first used to calculate the correlation function, followed by the spectral density function. The resulting model-free spectral density functions are finite at a frequency of zero and can be used to describe distributions of either overall or internal correlation times using the model-free ansatz. To validate the approach, ¹⁵N spin-relaxation data for the bZip transcription factor domain of the Saccharomyces cerevisiae protein GCN4, in the absence of cognate DNA, were analyzed using the inverse Gaussian probability distribution for intramolecular correlation times. The results extend previous models for the conformational dynamics of the intrinsically disordered, DNA-binding region of the bZip transcription factor domain.

Dynamic domain arrangement of CheA-CheY complex regulates bacterial thermotaxis, as revealed by NMR

Bacteria utilize thermotaxis signal transduction proteins, including CheA, and CheY, to switch the direction of the cell movement. However, the thermally responsive machinery enabling warm-seeking behavior has not been identified. Here we examined the effects of temperature on the structure and dynamics of the full-length CheA and CheY complex, by NMR. Our studies revealed that the CheA-CheY complex exists in equilibrium between multiple states, including one state that is preferable for the autophosphorylation of CheA, and another state that is preferable for the phosphotransfer from CheA to CheY. With increasing temperature, the equilibrium shifts toward the latter state. The temperature-dependent population shift of the dynamic domain arrangement of the CheA-CheY complex induced changes in the concentrations of phosphorylated CheY that are comparable to those induced by chemical attractants or repellents. Therefore, the dynamic domain arrangement of the CheA-CheY complex functions as the primary thermally responsive machinery in warm-seeking behavior.