Separation of anisotropy and exchange broadening using (15)N CSA-(15)N-(1)H dipole-dipole relaxation cross-correlation experiments

Based on the measurement of cross-correlation rates between (15)N CSA and (15)N-(1)H dipole-dipole relaxation we propose a procedure for separating exchange contributions to transverse relaxation rates (R(2) = 1/T(2)) from effects caused by anisotropic rotational diffusion of the protein molecule. This approach determines the influence of anisotropy and chemical exchange processes independently and therefore circumvents difficulties associated with the currently standard use of T(1)/T(2) ratios to determine the rotational diffusion tensor. We find from computer simulations that, in the presence of even small amounts of internal flexibility, fitting T(1)/T(2) ratios tends to underestimate the anisotropy of overall tumbling. An additional problem exists when the N-H bond vector directions are not distributed homogeneously over the surface of a unit sphere, such as in helix bundles or beta-sheets. Such a case was found in segment 4 of the gelation factor (ABP 120), an F-actin cross-linking protein, in which the diffusion tensor cannot be calculated from T(1)/T(2) ratios. The (15)N CSA tensor of the residues for this beta-sheet protein was found to vary even within secondary structure elements. The use of a common value for the whole protein molecule therefore might be an oversimplification. Using our approach it is immediately apparent that no exchange broadening exists for segment 4 although strongly reduced T(2) relaxation times for several residues could be mistaken as indications for exchange processes.     

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.     

Nuclear relaxation and motional anisotropy of liquid s-triazine

The dynamic structure of liquid s-triazine has been studied by analysing the deuterium and nitrogen-14 quadrupolar relaxation data of d ₃-s-triazine.

The molecular motions are markedly anisotropic with: (a) fast, large angle jump, inertial type in-plane motions of almost zero activation enthalpy and large negative activation entropy; (b) comparatively slow, small angle jump, rotational diffusion type, out of plane motions of higher activation enthalpy and small activation entropy. Comparison of the data on the dynamic behaviour of pyridine [3] and benzene [4] with the present ones on s-triazine leads to a general picture of molecular motions of planar hexagonal rotors in the liquid state (at atmospheric pressure). The behaviour of pyridine, which has a dipole moment departs somewhat from the more similar (and more anisotropic) behaviour of benzene and s-triazine. These results also support our previous finding of motional anisotropy in liquid pyridine [3].

A pictorial representation of the motional anisotropy in benzene, pyridine and s-triazine is giving using motional ellipsoids whose axes lengths are proportional to the diffusion constants.     

Improvement of duty-cycle heating compensation in NMR spin relaxation experiments

To reliably measure NMR relaxation properties of macromolecules is a prerequisite for precise experiments that identify subtle variations in relaxation rates, as required for the determination of rotational diffusion anisotropy, CSA tensor determination, advanced motional modeling or entropy difference estimations. An underlying problem with current NMR relaxation measurement protocols is maintaining constant sample temperature throughout the execution of the relaxation series especially when rapid data acquisition is required. Here, it is proposed to use a combination of a heating compensation and a proton saturation sequence at the beginning of the NMR relaxation pulse scheme. This simple extension allows reproducible, robust and rapid acquisition of NMR spin relaxation data sets. The method is verified with (15)N spin relaxation measurements for human ubiquitin.     

A simple method to measure 13CH2 heteronuclear dipolar cross-correlation spectral densities

Here, we report a method to simultaneously determine CH2 cross-correlation spectral densities and T1 relaxation times in the laboratory and rotating frames. To accomplish this, we have employed an indirect approach that is based on measurement of differences in relaxation rates acquired with and without cross-correlation terms. The new method, which can be employed using multidimensional NMR and standard relaxation pulse sequences, is validated experimentally by investigation of a selectively 13C-enriched hexadecapeptide and the uniformly 13C-enriched immunoglobulin-binding domain of streptococcal protein G (GB1). Use of this approach makes determination of CH2 cross-correlation spectral densities in uniformly 13C-enriched proteins now routine and provides novel information concerning their internal motions.     

Determination of the rotational diffusion tensor of macromolecules in solution from nmr relaxation data with a combination of exact and approximate methods--application to the determination of interdomain orientation in multidomain proteins

In this paper we present a method for determining the rotational diffusion tensor from NMR relaxation data using a combination of approximate and exact methods. The approximate method, which is computationally less intensive, computes values of the principal components of the diffusion tensor and estimates the Euler angles, which relate the principal axis frame of the diffusion tensor to the molecular frame. The approximate values of the principal components are then used as starting points for an exact calculation by a downhill simplex search for the principal components of the tensor over a grid of the space of Euler angles relating the diffusion tensor frame to the molecular frame. The search space of Euler angles is restricted using the tensor orientations calculated using the approximate method. The utility of this approach is demonstrated using both simulated and experimental relaxation data. A quality factor that determines the extent of the agreement between the measured and predicted relaxation data is provided. This approach is then used to estimate the relative orientation of SH3 and SH2 domains in the SH(32) dual-domain construct of Abelson kinase complexed with a consolidated ligand.     

A Simple Approach to Analyzing Protein Side-Chain Dynamics fromC NMR Relaxation Data

A simple approach to deriving motional dynamics information of protein and peptide side chains by using¹³C NMR relaxation data is presented. By using linear approximation of internal rotational correlation functions, simple equations for relating side-chain conformation, bond rotational amplitudes, and rotational correlation coefficients with different NMR relaxation parameters have been obtained. Auto- and cross-correlation spectral densities are considered, and it is shown that proton-coupled¹³C NMR relaxation measurements allow detailed motional information to be obtained.     

Characterization of hydrogen bond lengths in Watson-Crick base pairs by cross-correlated relaxation

Hydrogen bond lengths in Watson-Crick base pairs can be characterized by cross-correlated relaxation between 1H chemical shift anisotropy and dipole-dipole coupling of 1H and its hydrogen bond acceptor 15N. As a reference, the cross-correlated relaxation between 1H chemical shift anisotropy and dipole-dipole coupling of 1H and its hydrogen bond donor 15N is used. With the two measured cross-correlated relaxation rates, an apparent hydrogen bond length can be determined, which is composed by the hydrogen bond length multiplied by a term representing the amplitude of inter-base motions. Data are presented for the 15N3-1H3...15N1 hydrogen bonds in A=T base pairs of the Antennapedia homeodomain-DNA complex with a correlation time of global rotational diffusion of 20 ns.     

Sodium ions in ordered environments in biological systems: analysis of 23Na NMR spectra.

Experimentsthat selectively excite I = 32 nuclei exhibiting residual quadrupolar splittings are used to acquire 23Na NMR spectra from a range of biologically relevant samples containing sodium in ordered environments. Three complementary approaches to the analysis of such spectra are described: (i) measurement of relaxation rates, (ii) extraction of homogeneous linewidths from two-dimensional Jeener-Broekaert spectra, and (iii) simultaneous fitting of detailed theoretical functions to a series of one-dimensional Jeener-Broekaert spectra. Analysis of relaxation rates provides evidence for compartmentation in bovine nasal cartilage. Each approach is used to demonstrate the presence of anisotropy in transverse relaxation in porcine tendon. For certain samples containing collagen, a good theoretical fit to the spectra was obtained using a model that allows for anisotropic relaxation by including the effects of slow lateral and radial diffusion.     

Efficient and accurate determination of the overall rotational diffusion tensor of a molecule from (15)N relaxation data using computer program ROTDIF

We present a computer program ROTDIF for efficient determination of a complete rotational diffusion tensor of a molecule from NMR relaxation data. The derivation of the rotational diffusion tensor in the case of a fully anisotropic model is based on a six-dimensional search, which could be very time consuming, particularly if a grid search in the Euler angle space is involved. Here, we use an efficient Levenberg-Marquardt algorithm combined with Monte Carlo generation of initial guesses. The result is a dramatic, up to 50-fold improvement in the computational efficiency over the previous approaches. This method is demonstrated on a computer-generated and real protein systems. We also address the issue of sensitivity of the diffusion tensor determination from (15)N relaxation measurements to experimental errors in the relaxation rates and discuss possible artifacts from applying higher-symmetry tensor model and how to recognize them.     

HYDRONMR: prediction of NMR relaxation of globular proteins from atomic-level structures and hydrodynamic calculations

The heteronuclear NMR relaxation of globular proteins depends on the anisotropic rotational diffusion tensor. Using our previous developments for prediction of hydrodynamic properties of arbitrarily shaped particles, by means of bead models, we have constructed a computational procedure to calculate the rotational diffusion tensor and other properties of proteins from their detailed, atomic-level structure. From the atomic coordinates file used to build the bead model, the orientation of the pertinent dipoles can be extracted and combined with the hydrodynamic information to predict, for each residue in the protein, the relaxation times. All of these developments have been implemented in a computer program, HYDRONMR, which will be of public domain.     

A General Method for Constrained Analysis of Fluorescence Anisotropy Decay: Application of the Steady-State Anisotropy

Time-resolved fluorescence anisotropy is an invaluable method for investigating the internal and rotational dynamics of biomolecules. The range of rotational motions detectable by anisotropy decay is limited by the fluorescence lifetime; typically, a depolarizing motion may be resolved if the associated correlation time is between 0.1 and 10 times the intensity decay lifetime. To extend that range and to improve the recovery of anisotropy decay parameters, a general analytical method has been developed. This procedure utilizes a modification of Lagrange multiplier methods to constrain the values of the iterated kinetic parameters during nonlinear least-squares analysis of anisotropy decay data. The form of the constraint equation is derived from the classic relationship between the decay parameters and the steady-state anisotropy, which can be simply and accurately measured. Application of the constraint to analyses of synthetic data sets increased the accuracy of recovery by decreasing the uncertainty in the iterated parameters. The constraint also enabled the accurate recovery of correlation times that were a factor of 30 greater than the fluorescence lifetime, although it did not improve recovery of correlation times that were much shorter than the lifetime. Using this technique, it should now be possible to characterize the dynamics of larger macromolecules and assemblies than those that can currently be studied by fluorescence anisotropy decay.     

Sodium Ions in Ordered Environments in Biological Systems: Analysis of Na NMR Spectra

Experimentsthat selectively excite I = nuclei exhibiting residual quadrupolar splittings are used to acquire ²³Na NMR spectra from a range of biologically relevant samples containing sodium in ordered environments. Three complementary approaches to the analysis of such spectra are described: (i) measurement of relaxation rates, (ii) extraction of homogeneous linewidths from two-dimensional Jeener–Broekaert spectra, and (iii) simultaneous fitting of detailed theoretical functions to a series of one-dimensional Jeener–Broekaert spectra. Analysis of relaxation rates provides evidence for compartmentation in bovine nasal cartilage. Each approach is used to demonstrate the presence of anisotropy in transverse relaxation in porcine tendon. For certain samples containing collagen, a good theoretical fit to the spectra was obtained using a model that allows for anisotropic relaxation by including the effects of slow lateral and radial diffusion.     

Pulsed-Field-Gradient NMR Study of Anisotropic Molecular Translational Diffusion in nOCB Liquid Crystals

Pulsed-field-gradient nuclear magnetic resonance (NMR) combined with magic echo decoupling is applied to study anisotropic diffusion in samples with strong static dipolar spin interactions. The approach, due to its moderate demands on the NMR hardware, can be implemented on standard commercial equipment for routine diffusion studies of liquid crystals. Using a microimaging probe, measurement of diffusion in arbitrary spatial direction is possible. Hence, the principal components of the diffusion tensor are directly obtained. Anisotropic diffusion is investigated in the thermotropic mesophases of a homologous series of nOCB liquid crystals and an analogous compound with hydroxyl groups. The geometric average diffusion coefficient changes continuously at the isotropic–nematic phase transition. Experimental data are described in terms of the molecular translation models in the nematic phase and for the second-order nematic–smectic A phase transition. The diffusion anisotropy is higher for the sample with terminal hydroxyl groups suggesting significant molecular association via hydrogen bonding.     

Nematic ordering of suspension of charged anisotropic colloids detected by multinuclear quadrupolar spectra and 1H PGSE-NMR measurements

The structure of aqueous dispersion of charged anisotropic nano-composites (synthetic Laponite clays) have been studied by NMR and numerical simulations based on a multi-scale statistical analysis have been used to interpret the mobility of the confined water molecule diffusing within dense Laponite aqueous dispersions (29-52% w/w) prepared by uniaxial compression. Firstly, the lineshape detected by NMR quadrupolar spectroscopy of the counterions ((23)Na or (7)Li) exhibits a large residual splitting Delta nu which is the fingerprint of the macroscopic nematic ordering of the anisotropic particles. Secondly, these results are also confirmed by the anisotropy of the self-diffusion tensor of the water molecule measured by (1)H Pulsed Gradient Spin Echo NMR. This self-diffusion anisotropy increases with the suspension density. Thirdly, the multi-scale statistical analysis of the water mobility bridges the gap between the time-scale (ps) accessible by Molecular Dynamics simulations and the time-scale (micros) accessible by Brownian Dynamics, leading to macroscopic behaviour comparable with PGSE-NMR data measurements.     

Acceleration of multi-dimensional propagator measurements with compressed sensing

NMR can probe the microstructures of anisotropic materials such as liquid crystals, stretched polymers and biological tissues through measurement of the diffusion propagator, where internal structures are indicated by restricted diffusion. Multi-dimensional measurements can probe the microscopic anisotropy, but full sampling can then quickly become prohibitively time consuming. However, for incompletely sampled data, compressed sensing is an effective reconstruction technique to enable accelerated acquisition. We demonstrate that with a compressed sensing scheme, one can greatly reduce the sampling and the experimental time with minimal effect on the reconstruction of the diffusion propagator with an example of anisotropic diffusion. We compare full sampling down to 64× sub-sampling for the 2D propagator measurement and reduce the acquisition time for the 3D experiment by a factor of 32 from ～80 days to ～2.5 days.     

A Simple Derivation of the Luminescence Anisotropy Decay from Randomly Distributed Cylinders Rotating About a Single Axis

A simple derivation is given of the expression describing the anisotropy decay of luminescence for a solution of molecules that can only undergo rotational diffusion about a single cylindrical axis. The usual derivations of the anisotropy decay for this cylindrical model have simply taken limiting cases of the equations resulting from the general treatment of the anisotropy decay of a completely anisotropic rotator or the rotation of an ellipsoid. The arguments presented here can be understood without the mathematical sophistication required to follow the general derivations for the rotational diffusion of a completely anisotropic rotator or ellipsoids. The underlying physical mechanisms leading to a multiple exponential decay of the fluorescence anisotropy signal from a single axis rotating cylinder are clearly shown by following this derivation. The resulting expression for the anisotropy decay is not new. However, the derivation is easily understood, and this article is meant as an introduction to the more advanced treatments of anisotropy decay by rotational diffusion. After presenting the derivation of the rotating cylinder, the corresponding steps of these general treatments and this simple model are indicated. The model is of special interest for describing the anisotropy decay resulting from rotations of proteins within membranes.     

1H and13C nuclear magnetic relaxation and local dynamics of lysozyme and synthetic polypeptides

The combined analysis of¹H and¹³C NMR relaxation data in solid lysozyme and some typical homopolypeptides was carried out by using “model-free” approach. Three types of relaxation transitions (γ’, γ and β) were revealed in the temperature range investigated. The microdynamical parameters of these motions were determined. From the comparison of these parameters with those of selected synthetic polymers it follows that the molecular motions in proteins and synthetic polymers are of the same nature. All these motions show pronounced anisotropic character. In the investigated temperature range no molecular motions corresponding to α-relaxation (liquid-like) transition were revealed. The hydration effects on parameters of the motions in proteins were considered. The most pronounced effect takes place for β-transition. The effect of Brownian rotation of protein molecule in solution on measured correlation function of local motions was also discussed.     

Angular anisotropy of group averaged absorption coefficient and its effect on the behavior of diffusion approach in radiative transfer

Multi-group method, a generally accepted procedure for handling the radiative transfer equation, is accompanied by the group averaged absorption coefficient, and actually the coefficient is angularly anisotropic. In the paper, we present a brief discussion how the anisotropy of the coefficient makes the material absorb photons at different rate in each direction, also study its effect on the diffusion approach by comparing the results calculated using multi-group diffusion and multi-group discrete ordinate SN for the isotropic and anisotropic group averaged absorption coefficients respectively, and find that the anisotropy deteriorates the behavior of diffusion approach.