Interproton distance bounds from 2D NOE intensities: Effect of experimental noise and peak integration errors

Summary The effect of experimental and integration errors on the calculations in interproton distances from NOE intensities is examined. It is shown that NOE intensity errors can have a large impact on the distances determined. When multiple spin (spin diffusion) effects are significant, the calculated distances are often underestimated, even when using a complete relaxation matrix analysis. In this case, the bias of distances to smaller values is due to the random errors in the NOE intensities. We show here that accurate upper and lower bounds of the distances can be obtained if the intensity errors are properly accounted for in the complete relaxation matrix calculations, specifically the MARDIGRAS algorithm. The basic MARDIGRAS algorithm has been previously described [Borgias, B.A. and James, T.L. (1990) J. Magn. Reson., 87, 475–487]. It has been shown to provide reasonably good interproton distance bounds, but experimental errors can compromise the quality of the resulting restraints, especially for weak cross peaks. In a new approach introduced here, termed RANDMARDI (random error MARDIGRAS), errors due to random noise and integration errors are mimicked by the addition of random numbers from within a specified range to each input intensity. Interproton distances are then calculated for the modified intensity set using MARDIGRAS. The distribution of distances that define the upper and lower distance bounds is obtained by using N randomly modified intensity sets. RANDMARDI has been used in the solution structure determination of the interstrand cross-link (XL) formed between 4-hydroxymethyl-4,5,8-trimethylpsoralen (HMT) and the DNA oligomer d(5-GCGTACGC-3)₂ [Spielmann, H.P. et al. (1995) Biochemistry, 34, 12937–12953]. RANDMARDI generates accurate distance bounds from the experimental NOESY cross-peak intensities for the fixed (known) interproton distances in XL. This provides an independent internal check for the ability of RANDMARDI to accurately fit the experimental data. The XL structure determined using RANDMARDI-generated restrains is in good agreement with other biophysical data that indicate that there is no bend introduced into the DNA by the cross-link. In contrast, isolated spin-pair approximation calculations give distance restraints that, when applied in a restrained molecular dynamics protocol, produce a bent structure.Abbreviations NOE nuclear Overhauser effect - SD standard deviation - HMT 4-hydroxymethyl-4,5,8-trimethylpsoralen - XL psoralen-DNA interstrand cross-link     

Structural homology of spinach acyl carrier protein and Escherichia coli acyl carrier protein based on NMR data

Acyl carrier proteins (ACPs) from spinach and from Escherichia coli have been used to demonstrate the utility of proton NMR for comparison of homologous structures. The structure of E. coli ACP had been previously determined and modeled as a rapid equilibrium among multiple conformational forms (Kim and Prestegard, Biochemistry 28:8792–8797, 1989). Spinach ACP showed two slowly exchanging forms and could be manipulated into one form for structural study. Here we compare this single form to postulated multiple forms of E. coli ACP using the limited amount of NOE data available for the spinach protein. A number of long-range NOE contacts were present between homologous residues in both spinach and E. coli ACP, suggesting tertiary structural homology. To allow a more definitive structural comparison, a method was developed to use spinach ACP NOE constraints to search for regions of structural divergence from two postulated forms of E. coli ACP. The homologous regions of the two protein sequences were aligned, additional distance constraints were extracted from the E. coli structure, and these were mapped onto the spinach sequence. These distance constraints were combined with experimental NOE constraints and a distance geometry simulated annealing protocol was used to test for compatibility of the constraints. All of the experimental spinach NOE constraints could be successfully combined with the E. coli data, confirming the general hypothesis of structural homology. A better fit was obtained with one form, suggesting a preferential stabilization of that form in the spinach case. Proteins 27:131–143 © 1997 Wiley-Liss, Inc.     

SESAME: A least-squares approach to the evaluation of protein structures computed from NMR data

Summary A method is proposed for defining a probability distribution on an ensemble of protein conformations from a 2D NOE spectrum, while at the same time back-calculating the experimental spectrum from the ensemble. This enables one to assess the relative quality and significance of the conformations, and to test the consistency of the ensemble as a whole with the experimental spectrum. The method eliminates the need to integrate the cross-peak intensities and is surprisingly insensitive to random noise in the spectrum. In this communication, these advantages are demonstrated by applying the method to simulated data, for which the correct result is already known.Abbreviations NOE nuclear Overhauser effect - BPTI basic pancreatic trypsin inhibitor - rmsd root-mean-square deviation     

Calculation of symmetric multimer structures from NMR data using a priori knowledge of the monomer structure, co-monomer restraints, and interface mapping: The case of leucine zippers

Summary NMR studies of symmetric multimers are problematic due to the difficulty in distinguishing between intra-, inter-, and co-monomer (mixed) NOE signals. Previously, one of us described a general calculation strategy called dynamic assignment by which this difficulty can be overcome [Nilges, M. (1993) Proteins, 17, 297–309]. Here we describe extensions to the method for handling many co-monomer NOEs and for taking advantege of prior knowledge of monomer structures. The new protocol was developed for the particularly difficult case of leucine zipper (LZ) homodimers, for which the previous protocol proved inefficient. In addition to the problem of dimer symmetry, LZs have a particularly high proportion of co-monomer NOE signals and a high degree of repetition in sequence and structure, leading to significant spectral overlap. Furthermore, the leucine zipper is a rather extended (as opposed to globular) protein domain; accurately determining such a structure based only on the very short distances obtainable by NMR is clearly a challenge to the NMR structure determination method. We have previously shown that, for LZ homodimers, many of the backbone-backbone NOESY cross peaks can be unambiguously assigned as intra-monomer, enabling approximate monomer structures to be calculated. Using model and experimental data sets, we verified that the new protocol converges to the correct dimer structure. The results show that short-range NMR distance data can be sufficient to define accurately the extended LZ. The protocol has been used to derive a novel solution structure of the c-Jun LZ domain. Based on these calculations, we propose the protocol as a prototype for the general case of symmetric multimers where the monomer structure is known.Abbreviations 3D three-dimensional - GCN4-c crystal structure of the GCN4 LZ homodimer - GCN4-s solution structures of GCN4 - GSYM global symmetry - Jun-m model structure of the Jun LZ homodimer - Jun-s solution structure of Jun - LZ leucine zipper - MFP mean force potential - MDSA molecular dynamical simulated annealing - NCS noncrystallographic symmetry - NOE nuclear Overhauser enhancement - rmsd root-mean-square deviation - vdW van der Waals     

“Ensemble” iterative relaxation matrix approach: A new NMR refinement protocol applied to the solution structure of crambin

The structure in solution of crambin, a small protein of 46 residues, has been determined from 2D NMR data using an iterative relaxation matrix approach (IRMA) together with distance geometry, distance bound driven dynamics, molecular dynamics, and energy minimization. A new protocol based on an “ensemble” approach is proposed and compared to the more standard initial rate analysis approach and a “single structure” relaxation matrix approach. The effects of fast local motions are included and R-factor calculations are performed on NOE build-ups to describe the quality of agreement between theory and experiment. A new method for stereospecific assignment of prochiral groups, based on a comparison of theoretical and experimental NOE intensities, has been applied. The solution structure of crambin could be determined with a precision (rmsd from the average structure) of 0.7 Å on backbone atoms and 1.1 Å on all heavy atoms and is largely similar to the crystal structure with a small difference observed in the position of the side chain of Tyr-29 which is determined in solution by both J-coupling and NOE data. Regions of higher structural variability (suggesting higher mobility) are found hi the solution structure, in particular for the loop between the two helices (Gly-20 to Pro-22). © 1993 Wiley-Liss, Inc.     

An estimate of spin diffusion in a spin subset: Application to iterative distance calculation from 3D 15N NOESY-HMQC

Summary A method for quantification of distances between amide hydrogens using only the 3D NOESY-HMQC experiment recorded on a ¹⁵N-labelled protein is presented. This method is based on an approximate expression of the NOE intensities between amide hydrogens obtained from continuum modelling of the non-amide spins; this expression is used in a distance calculation algorithm. The algorithm has been named CROWD, standing for Continuum approximation of Relaxati On path Ways between Dilute spins. This approximation as well as the CROWD algorithm are tested on a simulated case; the CROWD algorithm is then applied to experimental data, measured on a fragment of bacteriorhodopsin.     

Validation of the use of intermolecular NOE constraints for obtaining docked structures of protein-ligand complexes

Summary The use of intermolecular NOEs for docking a small ligand molecule into its target protein has been investigated with the aim of determining the effectiveness and methodology of this type of NOE docking calculation. A high-resolution X-ray structure of a protein-ligand complex has been used to simulate loose distance constraints of varying degrees of quality, typical of those estimated from experimental NOE intensities. These simulated data were used to examine the effect of the number, distribution and representation of the experimental constraints on the precision and accuracy of the calculated structures. A standard simulated annealing protocol was used, as well as a more novel method based on rigid-body dynamics. The results showed some analogies with those from similar studies on complete protein NMR structure determinations, but it was found that more constraints per torsion angle are required to define docked structures of similar quality. The effectiveness of different NOE-constraint averaging methods was explored and the benefits of using R^–6 averaging rather than centre averaging with small sets of NOE constraints were shown. The starting protein structure used in docking calculations was obtained from previous X-ray or NMR structure studies on a related complex. The effects on the calculated conformations of introducing structural differences into the binding site of the initial protein structure were also considered.To whom correspondence should be addressed.     

Graph-theoretical assignment of secondary structure in multidimensional protein NMR spectra: Application to the lac repressor headpiece

Summary A novel procedure is presented for the automatic identification of secondary structures in proteins from their corresponding NOE data. The method uses a branch of mathematics known as graph theory to identify prescribed NOE connectivity patterns characteristic of the regular secondary structures. Resonance assignment is achieved by connecting these patterns of secondary structure together, thereby matching the connected spin systems to specific segments of the protein sequence. The method known as SERENDIPITY refers to a set of routines developed in a modular fashion, where each program has one or several well-defined tasks. NOE templates for several secondary structure motifs have been developed and the method has been successfully applied to data obtained from NOESY-type spectra. The present report describes the application of the SERENDIPITY protocol to a 3D NOESY-HMQC spectrum of the ¹⁵N-labelled lac repressor headpiece protein. The application demonstrates that, under favourable conditions, fully automated identification of secondary structures and semi-automated assignment are feasible.Abbreviations 2D, 3D two-, three-dimensional - NOESY nuclear Overhauser enhancement spectroscopy - HMQC heteronuclear multiple quantum coherence - SSE secondary structure element - SERENDIPITY SEcondary structuRE ideNtification in multiDImensional ProteIn specTra analYsis Supplementary Material available from the authors: Two tables containing the total number of mappings resulting from the graph search procedure for simulated and experimental NOE data.     

Construction of hypothetical three-dimensional structure of P2Y1 receptor based on Fourier transform analysis

G protein-coupled receptors constitute a large family of homologous transmembrane proteins that represents one of the most important classes of confirmed drug targets. For novel drug discovery, the 3D structure of target protein is indispensable. To construct hypothetical 3D structures of G protein-coupled receptors, several prediction methods have been proposed. But none of the them has confirmed a correct ligand binding site. In this study we constructed the 3D structure of bovine rhodopsin using the prediction method proposed by Donnelly et al., with some modification. We found that our 3D model showed a good agreement with the reported retinal binding site. Using the similar method, we constructed the 3D structure of the P2Y₁ receptor; one of the G protein-coupled receptors, and showed a binding site of an endogenous ligand, ADP, on the basis of the 3D model and in vitro experimental data. These results should be valuable for design of a specific antagonist for P2Y₁ receptor.     

Direct nuclear Overhauser effect refinement of crambin from two-dimensional nmr data using a slow-cooling annealing protocol

The solution structure of crambin has been refined using a direct nuclear Overhauser effect (NOE) simulation approach (DINOSAUR) following a slow-cooling simulated annealing protocol starting from eight previously obtained nmr and the x-ray structures of crambin. Theoretical NOE intensities calculated with inclusion of local motions were directly compared to the experimental nmr data and forces were derived using a simple first-order approximation for the calculation of the NOE gradient. A dynamic assignment procedure was applied for the peaks involving unassigned diastereotopic proton pairs or equivalent aromatic protons. With this approach, R factors could be minimized in a reasonable simulation time to low values (around 0.26) while deviations from ideal bond lengths and angles are still acceptable. The improvement in R factors is accompanied by an improvement of the precision of the structures, the rms deviations (rmsd; from the average) calculated on the ensemble of nine structures decreasing from 0.65 to 0.55 Å for backbone atoms and from 1.0 to 0.85 Å for all heavy atoms. The solution structure is significantly different from the x-ray structure with rmsd for all atoms of 1.35 Å compared to 0.85 Å between solution structures. The largest differences are found for residues Thr-21 and Pro-22 in the loop region between the two α-helices and for the side chain of Tyr-29. © 1994 John Wiley & Sons, Inc.     

The determination of the conformational properties of nucleic acids in solution from NMR data

A program, NUCFIT, has been written for simulating the effects of conformational averaging on nuclear Overhauser enhancement (NOE) intensities for the spin systems found in nucleic acids. Arbitrary structures can be generated, and the NOE time courses can be calculated for truncated one-dimensional NOEs, two-dimensional NOE and rotating frame NOE spectroscopy (NOESY and ROESY) experiments. Both isotropic and anisotropic molecular rotation can be treated, using Woessner's formalism (J. Chem. Phys. (1962) 37, 647-654). The effects of slow conformational averaging are simulated by taking population-weighted means of the conformations present. Rapid motions are allowed for by using order parameters which can be supplied by the user, or calculated for specific motional models using the formalism of Tropp (J. Chem. Phys. (1980) 72, 6035-6043). NOE time courses have been simulated for a wide variety of conformations and used to determine the quality of structure determinations using NMR data for nucleic acids. The program also allows grid-searching with least-squares fitting of structures to experimental data, including the effects of spin-diffusion, conformational averaging and rapid internal motions. The effects of variation of intra and internucleotide conformational parameters on NOE intensities has been systematically explored. It is found that (i) the conformation of nucleotides is well determined by realistic NOE data sets, (ii) some of the helical parameters, particularly the base pair roll, are poorly determined even for extensive, noise-free data sets, (iii) conformational averaging of the sugars by pseudorotation has at most second-order influence on the determination of other parameters and (iv) averaging about the glycosidic torsion bond also has, in most cases, an insignificant effect on the determination of the conformation of nucleotides.     

Simulation of NOESY spectra of DNA segments: A new scaling procedure for iterative comparison of calculated and experimental NOE intensities

Summary A new algorithm for simulation of two-dimensional NOESY spectra of DNA segments has been developed. For any given structure, NOE intensities are calculated using the relaxation matrix approach and a new realistic procedure is suggested for 1:1 comparison of calculated and experimental intensities. The procedure involves a novel method for scaling of calculated NOE intensities to represent volumes of digitised cross peaks in NOESY spectra. A data base of fine structures of all the relevant cross peaks with Lorentzian line shapes and in-phase components, is generated in a digitised manner by two-dimensional Fourier transformation of simulated time domain data, assuming a total intensity of 1.0 for each of the cross peaks. With this procedure, it is shown that the integrated volumes of these digitised cross peaks above any given threshold scale exactly as the total intensity of the respective peaks. This procedure eliminates the repetitive generation of digitised cross peaks by two-dimensional Fourier transformation during the iterative process of structure alteration and NOE intensity calculation and thus enhances the speed of DNA structure optimization. Illustrative fits of experimental and calculated spectra obtained using the new procedure are shown.[/p]     

SANE (Structure Assisted NOE Evaluation): An automated model-based approach for NOE assignment

A reliable automated approach for assignment of NOESY spectra would allow more rapid determination of protein structures by NMR. In this paper we describe a semi-automated procedure for complete NOESY assignment (SANE, Structure Assisted NOE Evaluation), coupled to an iterative procedure for NMR structure determination where the user is directly involved. Our method is similar to ARIA [Nilges et al. (1997) J. Mol. Biol., 269, 408–422], but is compatible with the molecular dynamics suites AMBER and DYANA. The method is ideal for systems where an initial model or crystal structure is available, but has also been used successfully for ab initio structure determination. Use of this semi-automated iterative approach assists in the identification of errors in the NOE assignments to short-cut the path to an NMR solution structure.     

Discrimination Between A-type and B-type Conformations of Double Helical Nucleic Acid Fragments in Solution by Means of Two-dimensional Nuclear Overhauser Experiments

Abstract

The solution structure of two double helical nucleic acid fragments, viz. r(CGCGCG) and d(CGCGCG), was probed by means of two-dimensional nuclear Overhauser effect spectroscopy. The two compounds were selected as models for the A-type and B-type double helical conformations, respectively, and it is shown that for each of the two model compounds the intensities of the NOE cross peaks between base- and H2′ (deoxy)ribose proteins are qualitatively in correspondence with the relative NOE intensities expected on basis of the supposed duplex conformations. Thus our results indicate that NOE-data can be used to differentiate between A- and B-type double helical conformations in solution.

Coupling constant data show that, except for G(6), all ribose rings in r(CGCGCG) adopt pure N (C3′-endo) conformations thereby manifesting that this molecule takes up a regular A-type double helical conformation in solution. In contrast, the deoxyribose rings in d(CGCGCG) retain conformational freedom in the duplex state, albeit that the N/S- equilibrium is biased towards the S (C2′-endo) sugar conformation. This finding indicates that in solution the B-DNA backbone is highly dynamic.     

Modelling of Structural and Vibrational Properties of Poly(p-Phenylene) and Polypyrrole Using Molecular Orbital Methods

Abstract

This paper concentrates on two very important conducting polymers poly(p-phenylene) and polypyrrole. Detailed atomistic molecular models have been developed with the help of ab initio and semi-empirical quantum mechanical calculations using the Cerius² and WinMOPAC (version 6.0) programs.

Their optimised geometry had been calculated and compared with experimental X-ray diffraction data. The simulated and experimental vibrational spectra of biphenyl as well as isolated pyrrole monomers and oligomers from n = 1 and 2, where n is the number of structural repeat units used, have been computed using the ab initio 3–21G basis set. The results obtained are compared with experimental data for the case of biphenyl and for oligomers with n = 2 to 5 for both neutral benzenoid and quinonoid oligopyrroles, from semi-empirical predictions obtained by AM1 and PM3. The trends in the computed harmonic force fields, vibrational frequencies and intensities are monitored as a function of the chain length. The data are analyzed in conjunction with the trends in computed equilibrium geometries.     

Structure prediction of protein complexes by an NMR-based protein docking algorithm

Protein docking algorithms can be used to study the driving forces and reaction mechanisms of docking processes. They are also able to speed up the lengthy process of experimental structure elucidation of protein complexes by proposing potential structures. In this paper, we are discussing a variant of the protein-protein docking problem, where the input consists of the tertiary structures of proteins A and B plus an unassigned one-dimensional ¹H-NMR spectrum of the complex AB. We present a new scoring function for evaluating and ranking potential complex structures produced by a docking algorithm. The scoring function computes a `theoretical' ¹H-NMR spectrum for each tentative complex structure and subtracts the calculated spectrum from the experimental one. The absolute areas of the difference spectra are then used to rank the potential complex structures. In contrast to formerly published approaches (e.g. [Morelli et al. (2000) Biochemistry, 39, 2530–2537]) we do not use distance constraints (intermolecular NOE constraints). We have tested the approach with four protein complexes whose three-dimensional structures are stored in the PDB data bank [Bernstein et al. (1977)] and whose ¹H-NMR shift assignments are available from the BMRB database. The best result was obtained for an example, where all standard scoring functions failed completely. Here, our new scoring function achieved an almost perfect separation between good approximations of the true complex structure and false positives.     

A computationally efficient method for evaluating the gradient of 2D NOESY intensities

Summary A computationally efficient method for calculating the derivative of NOE intensities with respect to any parameter is presented. This method is based on an integral expression representing the gradient. We will derive this expression from first principles using standard perturbation expansion techniques, and show it to be equivalent to an analytical expression [Yip, P. and Case, D.A. (1989) J. Magn. Reson., 83, 643] Implementation of this method in a refinement scheme (NOE-MD) is also briefly mentioned.     

Untenability of the Heteronomous DNA Model for Poly(dA) · Poly(dT) in Solution. This DNA Adopts a Right-Handed B-DNA Duplex in Which the Two Strands are Conformationally Equivalent. A 500 MHz NMR Study Using One Dimensional NOE

Abstract

ID NOE ¹H NMR spectroscopy at 500 MHz was employed to examine the structure of poly(dA)·poly(dT) in solution. NOE experiments were conducted as a function of presaturation pulse length (50, 30, 20 and 10 msec) and.power (19 and 20 db) to distinguish the primary NOEs from spin diffusion. The 10 msec NOE experiments took 49 hrs and over 55,000 scans for each case and the difference spectra were almost free from diffusion.

The spin diffused NOE difference spectra as well as difference NOE spectra in 90% H₂O + 10% D₂O in which TNH3 was presaturated enabled to make a complete assignment of the base and sugar protons. It is shown that poly(dA) ·poly(dT) melts in a fashion in which single stranded bubbles are formed with increasing temperature.

Extremely strong primary NOEs were observed at H2′/H2″ when AH8 and TH6 were presaturated. The observed NOEs at AH2′ and that AH2″ were very similar as were the NOEs at TH2′ and TH2″. The observed NOEs at AH2′ and AH2″when AH8 was presaturated were very similar to those observed at TH2′ and TH2″ when TH6 was presaturated. In addition, presaturation of H1′ of A and T residues resulted in similar NOEs at AH2′/H2″ and TH2′/H2″ region and these NOEs at H2′ and H2″ were distinctly asymmetric as expected in a C2′-endo sugar pucker. There was not a trace of NOE at AH8 and TH6 when AH3′ and TH3′ were presaturated indicating that C3′-endo, × = 30–40° conformation is not valid for this DNA. From these NOE data, chemical shift shielding calculations and stereochemistry based computer modellings, we conclude that poly(dA)·poly(dT) in solution adopts a right- handed B-DNA duplex in which both dA and dT strands are conformationally equivalent with C2′-endo sugar pucker and a glycosyl torsion, ×, of ?73°, the remaining backbone torsion angles being φ′ = 221°, ω′ = 212°, ω = 310°, φ = 149°, ψ = 42°, ψ′ = 139°. The experimental data are in total disagreement with the heteronomous DNA model of Arnott et. al. proposed for the fibrous state. (Arnott, S., Chandrasekaran, R., Hall, I.H., and Puigjaner, L.C., Nucl. Acid Res. 11, 4141, 1983).     

The structure of ColE1 rop in solution

Summary The structure of the ColE1 repressor of primer (rop) protein in solution was determined from the proton nuclear magnetic resonance data by a combined use of distance geometry and restrained molecular dynamics calculations. A set of structures was determined with low internal energy and virtually no violations of the experimental distance restraints. Rop forms homodimers: Two helical hairpins are arranged as an antiparallel four helix bundle with a left-handed rope-like twist of the helix axes and with left-handed bundle topology. The very compact packing of the side chains in the helix interfaces of the rop coiled-coil structure may well account for its high stability. Overall, the solution structure is highly similar to the recently determined X-ray structure (Banner, D.W., Kokkinidis, M. and Tsernoglou, D. (1987)J. Mol. Biol.,196, 657–675), although there are minor differences in regions where packing forces appear to influence the crystal structure.Abbreviations rop repressor of primer - NMR nuclear magnetic resonance - NOE nuclear Overhauser enhancement - NOESY NOE spectroscopy - RAN Set Structures generated from random choice of the dihedrai angles - HEL Set Structures generated from random choice of the dihedral angles restricted to ranges allowed for helices - MD molecular dynamics - EM energy minimization - RMSD root-mean-square deviation of atomic positions     

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.