A Bayesian statistical method for the detection and quantification of rotational diffusion anisotropy from NMR relaxation data

It has recently become more widely appreciated that the presence of rotational diffusional anisotropy in proteins and other macromolecules can have a significant affect on the interpretation of NMR relaxation data in terms of molecular motion. In this paper, we show how commonly used NMR relaxation data (R(1), R(2), and NOE) obtained at two spectrometer frequencies can be analyzed using a Bayesian statistical approach to reliably detect and quantify the degree of rotational diffusion anisotropy. Our approach differs from previous methods in that it does not make assumptions concerning the internal motions experienced by the residues which are used to quantify the diffusion anisotropy, but rather averages the results over all internal motions consistent with the data. We demonstrate our method using synthetic data corresponding to isotropic, axially symmetric anisotropic, and fully asymmetric anisotropic rotational diffusion, as well as experimental NMR data. We compare the Bayesian statistical approach with a widely used method for extracting tumbling parameters using both synthetic and experimental data. While it can be difficult to separate the effects of chemical exchange from rotational anisotropy using this "standard" method, these effects are readily separated using Bayesian statistics. In addition, we find that the Bayesian statistical approach requires considerably less CPU time than an equivalent standard analysis.     

Evaluation of decay times in coupled spaces: reliability analysis of Bayeisan decay time estimation

This paper discusses quantitative tools to evaluate the reliability of "decay time estimates" and inter-relationships between multiple decay times for estimates made within a Bayesian framework. Previous works [Xiang and Goggans, J. Acoust. Soc. Am. 110, 1415-1424 (2001); 113, 2685-2697 (2003)] have applied Bayesian framework to cope with the demanding tasks in estimating multiple decay times from Schroeder decay functions measured in acoustically coupled spaces. A parametric model of Schroeder decay function [Xiang, J. Acoust. Soc. Am. 98, 2112-2121 (1995)] has been used for the Bayesian model-based analysis. The relevance of this work is that architectural acousticians need to know how well determined are the estimated decay times calculated within Bayesian framework using Schroeder decay function data. This paper will first address the estimation of global variance of the residual errors between the Schroeder function data and its model. Moreover, this paper discusses how the "landscape" shape of the posterior probability density function over the decay parameter space influences the individual decay time estimates, their associated variances, and their inter-relationships. This paper uses experimental results from measured room impulse responses in real halls to describe a model-based sampling method for an efficient estimation of decay times, and their individual variances. These parameters along with decay times are relevant decay parameters for evaluation and understanding of acoustically coupled spaces.     

Processing of heteronuclear NMR relaxation data with the new software DASHA

The new program DASHA is an efficient implementation of common data processing steps for the protein internal dynamic analysis. The “model-free” parameters and their uncertainties (Lipari G., Szabo A.: J. Am. Chem. Soc.104, 4546–4559 (1982) can be calculated from an arbitrary combination of experimental data sets (i.e. heteronuclear¹H?¹⁵N or¹H?¹³C relaxation times and NOE values at different spectrometer frequencies). Anisotropy of the molecular rotational diffusion could be also taken into account without introduction of the new adjustable parameters into the spectral density functionJ(ω), provided the structure of the molecule is known. Parameters of chemical (conformational) exchange can be estimated from the CPMG spin-lock frequency dependences (Bloomet al.: J. Chem. Phys.42, 1615–1624 (1965); Orekhovet al.: Eur. J. Biochem.219, 887–896 (1994). The program can be used both in the interactive and batch modes. It has sophisticated PostScript plotting facilities.     

Systematic bias in the model-free analysis of heteronuclear relaxation

Several theoretical and practical aspects of the use of heteronuclear relaxation to characterize the internal motion of biomacromolecules are examined. The treatment is cast in the terms of the popular model-free theory of G. Lipari and A. Szabo (J. Am. Chem. Soc.104, 4546 (1982)). In particular, the measurement of longitudinal relaxation in fully coupled systems is found to systematically bias obtained generalized order parameters. It is concluded that it is generally appropriate to utilize an experimental approach which ensures that heteronuclei are decoupled frorn bonded protons during sampling of relaxation. This restricts the types of experiments that can be used to sample relaxation by indirectly detected two-dimensional spectroscopy. The influence of a limited sampling of relaxation, typically required by practical limitations when two-dimensional spectroscopy is employed, is also examined. Finally, the relative importance of T₁ and NOE in determining model-free parameters is explored in conjunction with the impact of commonly used assumptions on the analysis.     

Evaluation of decay times in coupled spaces: an efficient search algorithm within the Bayesian framework

This paper discusses an efficient method for evaluating multiple decay times within the Bayesian framework. Previous works [N. Xiang and P. M. Goggans, J. Acoust. Soc. Am. 110, 1415-1424 (2001); 113, 2685-2697 (2003); N. Xiang, P. M. Goggans, T. Jasa, and M. Kleiner, 117, 3707-3715 (2005)] have applied the Bayesian inference to cope with demanding tasks in estimating multiple decay times from Schroeder decay functions measured or calculated in acoustically coupled spaces. Since then a number of recent works call for efficient estimation methods within the Bayesian framework. An efficient analysis is of practical significance for better understanding and modeling the sound energy decay process in acoustically coupled spaces or even in single spaces for reverberation time estimation. This paper will first formulate the Bayesian posterior probability distribution function (PPDF) in a matrix form to reduce the dimensionality as applied to the decay time evaluation. Based on existence of only global extremes of PPDFs as observed from extensive experimental data, this paper describes a dedicated search algorithm for an efficient estimation of decay times.     

Evaluation of decay times in coupled spaces: Bayesian decay model selection

This paper applies Bayesian probability theory to determination of the decay times in coupled spaces. A previous paper [N. Xiang and P. M. Goggans, J. Acoust. Soc. Am. 110, 1415-1424 (2001)] discussed determination of the decay times in coupled spaces from Schroeder's decay functions using Bayesian parameter estimation. To this end, the previous paper described the extension of an existing decay model [N. Xiang, I. Acoust. Soc. Am. 98, 2112-2121 (1995)] to incorporate one or more decay modes for use with Bayesian inference. Bayesian decay time estimation will obtain reasonable results only when it employs an appropriate decay model with the correct number of decay modes. However, in architectural acoustics practice, the number of decay modes may not be known when evaluating Schroeder's decay functions. The present paper continues the endeavor of the previous paper to apply Bayesian probability inference for comparison and selection of an appropriate decay model based upon measured data. Following a summary of Bayesian model comparison and selection, it discusses selection of a decay model in terms of experimentally measured Schroeder's decay functions. The present paper, along with the Bayesian decay time estimation described previously, suggests that Bayesian probability inference presents a suitable approach to the evaluation of decay times in coupled spaces.     

A new approach to visualizing spectral density functions and deriving motional correlation time distributions: applications to an alpha-helix-forming peptide and to a well-folded protein

A new approach to visualizing spectral densities and analyzing NMR relaxation data has been developed. By plotting the spectral density function, J(omega), as F(omega)=2 omega J(omega) on the log-log scale, the distribution of motional correlation times can be easily visualized. F(omega) is calculated from experimental data using a multi-Lorentzian expansion that is insensitive to the number of Lorentzians used and allows contributions from overall tumbling and internal motions to be separated without explicitly determining values for correlation times and their weighting coefficients. To demonstrate the approach, (15)N and (13)C NMR relaxation data have been analyzed for backbone NH and C(alpha)H groups in an alpha-helix-forming peptide 17mer and in a well-folded 138-residue protein, and the functions F(omega) have been calculated and deconvoluted for contributions from overall tumbling and internal motions. Overall tumbling correlation time distribution maxima yield essentially the same overall correlation times obtained using the Lipari-Szabo model and other standard NMR relaxation data analyses. Internal motional correlational times for NH and C(alpha)H bond motions fall in the range from 100 ps to about 1 ns. Slower overall molecular tumbling leads to better separation of internal motional correlation time distributions from those of overall tumbling. The usefulness of the approach rests in its ability to visualize spectral densities and to define and separate frequency distributions for molecular motions.     

Indirect detection of nitrogen-14 in solids via protons by nuclear magnetic resonance spectroscopy

This Communication describes the indirect detection of 14N nuclei (spin I=1) in solids by nuclear magnetic resonance (NMR) spectroscopy. The two-dimensional correlation method used here is closely related to the heteronuclear multiple quantum correlation (HMQC) experiment introduced in 1979 to study molecules in liquids, which has recently been used to study solids spinning at the magic angle. The difference is that the coherence transfer from neighboring 1H nuclei to 14N is achieved via a combination of J couplings and residual dipolar splittings (RDS). Projections of the two-dimensional correlation spectra onto the 14N dimension yield powder patterns which reflect the 14N quadrupolar interaction. In contrast to the indirect detection of 14N via 13C nuclei that was recently demonstrated [Gan, J. Am. Chem. Soc. 128 (2006) 6040; Cavadini et. al., J. Am. Chem. Soc., 128 (2006) 7706], this approach may benefit from enhanced sensitivity, and does not require isotopic enrichment in 13C, although the 1H line-widths may have to be reduced upon selective deuteration.     

Quantitative analysis of 17O exchange and T1 relaxation data: application to zirconium tungstate

The theoretical basis behind a recent quantitative analysis of 17O exchange in ZrW2O8 [M.R. Hampson, J.S.O. Evans, P. Hodgkinson, J. Am. Chem. Soc. 127 (2005) 15175-15181] is set out. Despite the complexities of combining the multi-exponential relaxation of half-integer quadrupolar nuclei with chemical exchange, it is shown how magnetisation transfer experiments can be analysed to obtain estimates of absolute exchange rates. The multi-exponential relaxation is best modelled using a magnetic mechanism, i.e. the rapid T1 relaxation observed, particularly at high temperatures, can be directly related to the relatively high degree of 17O labelling employed. The combination of the 1D EXSY results with T1 values as a function of temperature provides exchange rates and activation barriers over a wide temperature range (40-226 degrees C).     

Uncertainty analysis in matched-field geoacoustic inversions

Quantifying uncertainty for parameter estimates obtained from matched-field geoacoustic inversions using a Bayesian approach requires estimation of the uncertainties in the data due to ambient noise as well as modeling errors. In this study, the variance parameter of the Gaussian error model, hereafter called error variance, is assumed to describe the data uncertainty. In practice, this parameter is not known a priori, and choosing a particular value is often difficult. Hence, to account for the uncertainty in error variance, several methods are introduced for implementing both the full and empirical Bayesian approaches. A full Bayesian approach that permits uncertainty of the error variance to propagate through the parameter estimation processes is a natural way of incorporating the uncertainty of error variance. Due to the large number of unknown parameters in the full Bayesian uncertainty analysis, an alternative, the empirical Bayesian approach, is developed, in which the posterior distributions of model parameters are conditioned on a point estimate of the error variance. Comparisons between the full and empirical Bayesian inferences of model parameters are presented using both synthetic and experimental data.     

A multifrequency scintillation method for ocean flow measurement.

The transverse flow of inhomogeneous fluid produces fluctuation of the acoustic signal passing through it. The coherence of frequency-spaced signal fluctuation is related to the advection of the inhomogeneous medium through the sound path, thus providing a basis for the current velocity measurement. This method can be considered to be the "frequency-domain" version of the conventional scintillation approach to the current velocity registration based on the measurement of the signal correlation transmitted from the source to the two separated in space receivers (space-domain scintillation) [S. Clifford and D. Farmer, J. Acoust. Soc. Am. 74, 1826-1832 (1983)]. The sensitivity of the method depends on the features of the ocean fine structure, which is determined mainly by the internal waves and turbulence. To estimate the sensitivity of the multifrequency method of transverse current probing, the coherence function of two signals propagating through a frozen and moving internal wave field and through the turbulence is considered. The application of the multifrequency signal allows estimation of the fine-structure parameters as well as the current velocity.     

Selective NMR Experiments on Macromolecules: Implementation and Analysis of QUIET-NOESY

The QUIET-NOESY experiment (Zwahlenet al., J. Am. Chem Soc.116, 362–368, 1994) is applied to measure the mobility of the flexible extensions in the large aggregate (800 kDa) of a small heat-shock protein. The proper choices of the experimental protocol and parameters are discussed in order to employ a simplified data analysis procedure. Further experimental verification of the proposed strategy is also presented using the cyclic peptide gramicidin S as a model compound. Under suitable conditions, the determinations based on the analysis of QUIET-NOESY data are affected to a negligible extent by the approximations that are introduced by the proposed approach.     

Differential matrix formalism for depolarizing anisotropic media

Azzam's differential matrix formalism [J. Opt. Soc. Am. 68, 1756 (1978)], originally developed for longitudinally inhomogeneous anisotropic nondepolarizing media, is extended to include depolarizing media. The generalization is physically interpreted in terms of means and uncertainties of the elementary optical properties of the medium, as well as of three anisotropy absorption parameters introduced to describe the depolarization. The formalism results in a particularly simple mathematical procedure for the retrieval of the elementary properties of a generally depolarizing anisotropic medium, assumed to be globally homogeneous, from its experimental Mueller matrix. The approach is illustrated on literature data and the conditions of its validity are identified and discussed.     

A pseudo-inverse algorithm for simultaneous measurements using multiple acoustical sources

Simultaneous multiple acoustical sources measurement (SMASM) has been proposed for more effective and reliable identification of acoustical systems under critical conditions [N. Xiang and M. R. Schroeder, J. Acoust. Soc. Am. 113, 2754-2761 (2003); N. Xiang, J. N. Daigle, and M. Kleiner, J. Acoust. Soc. Am. 117, 1889-1894 (2005)]. This paper presents a pseudo-inverse algorithm for the SMASM correlation technique as an alternative way of extracting impulse responses of acoustical channels. Simulations and room acoustics experiments are carried out and the results prove the feasibility of the proposed algorithm.     

Analytical solution to the Lipari-Szabo model based on the reduced spectral density approximation offers a novel protocol for extracting motional parameters.

An analytical solution to the Lipari-Szabo model is derived for isotropic overall tumbling. The parameters of the original Lipari-Szabo model, the order parameter S2 and the effective internal correlation time tau(e), are calculated from two values of the spectral density function. If additionally the spectral density value J(0) is known, the exchange contribution R(ex) term can also be determined. The overall tumbling time tau(c) must be determined in advance, for example, from T1/T2 ratios. The required spectral density values are obtained by reduced spectral density mapping from T1, T2, and NOE measurements. Our computer simulations show that the reduced spectral density mapping is a very good approximation in almost all cases in which the Lipari-Szabo model is applicable. The robustness of the analytical formula to experimental errors is also investigated by extensive computer simulations and is found to be similar to that of the fitting procedures. The derived formulas were applied to the experimental 15N relaxation data of ubiquitin. Our results agree well with the published parameter values of S2 and tau(e), which were obtained from standard fitting procedures. The analytical approach to extract parameters of molecular motions may be more robust than standard analyses and provides a safeguard against spurious fitting results, especially for determining the exchange contribution R(ex).     

Simple and accurate determination of global tau(R) in proteins using (13)C or (15)N relaxation data

In the study of protein dynamics by (13)C or (15)N relaxation measurements different models from the Lipari-Szabo formalism are used in order to determine the motion parameters. The global rotational correlation time tau(R) of the molecule must be estimated prior to the analysis. In this Communication, the authors propose a new approach in determining an accurate value for tau(R) in order to realize the best fit of R(2) for the whole sequence of the protein, regardless of the different type of motions atoms may experience. The method first determines the highly structured regions of the sequence. For each corresponding site, the Lipari-Szabo parameters are calculated for R(1) and NOE, using an arbitrary value for tau(R). The chi(2) for R(2), summed over the selected sites, shows a clear minimum, as a function of tau(R). This minimum is used to better estimate a proper value for tau(R).     

A Metropolis Monte Carlo Implementation of Bayesian Time-Domain Parameter Estimation: Application to Coupling Constant Estimation from Antiphase Multiplets

The Bayesian perspective on statistics asserts that it makes sense to speak of a probability of an unknown parameter having a particular value. Given a model for an observed, noise-corrupted signal, we may use Bayesian methods to estimate not only the most probable value for each parameter but also their distributions. We present an implementation of the Bayesian parameter estimation formalism developed by G. L. Bretthorst (1990,J. Magn. Reson.88, 533) using the Metropolis Monte Carlo sampling algorithm to perform the parameter and error estimation. This allows us to make very few assumptions about the shape of the posterior distribution, and allows the easy introduction of prior knowledge about constraints among the model parameters. We present evidence that the error estimates obtained in this manner are realistic, and that the Monte Carlo approach can be used to accurately estimate coupling constants from antiphase doublets in synthetic and experimental data.     

Fine-tuning molecular acoustic models: sensitivity of the predicted attenuation to the Lennard-Jones parameters

In a previous paper [Y. Dain and R. M. Lueptow, J. Acoust. Soc. Am. 109, 1955 (2001)], a model of acoustic attenuation due to vibration-translation and vibration-vibration relaxation in multiple polyatomic gas mixtures was developed. In this paper, the model is improved by treating binary molecular collisions via fully pairwise vibrational transition probabilities. The sensitivity of the model to small variations in the Lennard-Jones parameters--collision diameter (sigma) and potential depth (epsilon)--is investigated for nitrogen-water-methane mixtures. For a N2(98.97%)-H2O(338 ppm)-CH4(1%) test mixture, the transition probabilities and acoustic absorption curves are much more sensitive to sigma than they are to epsilon. Additionally, when the 1% methane is replaced by nitrogen, the resulting mixture [N2(99.97%)-H2O(338 ppm)] becomes considerably more sensitive to changes of sigma(water). The current model minimizes the underprediction of the acoustic absorption peak magnitudes reported by S. G. Ejakov et al. [J. Acoust. Soc. Am. 113, 1871 (2003)].