A Multifaceted Approach to the Interpretation of NMR Order Parameters: A Case Studyof a Dynamic r-Helix†

Eric Johnson, Scott A. Showalter, and Rafael Bru1schweiler*Department of Chemistry and Biochemistry and National High Magnetic Field Laboratory,Florida State UniVersity, Tallahassee, Florida 32306

ReceiVed: NoVember 24, 2007; In Final Form: February 4, 2008

The model-free approach (Lipari, G.; Szabo, A.J. Am. Chem. Soc.1982, 104, 4546-4559; 4559-4570) isthe standard method for the analysis of NMR relaxation data from proteins. The method assigns to eachinteraction vector a generalized order parameterS2, which is a measure of the spatial restriction that thevector experiences in a molecular reference frame. A variety of techniques for the interpretation ofS2 valuesand their underlying assumptions are tested here for a dynamicR-helix represented by an ensemble of peptidestructures that satisfies a set of predefined constraints. A self-consistent picture is developed that characterizesthe influence of major determinants, including backbone dihedral angle fluctuations and their correlations,separability of internal and overall motion, the preservation of hydrogen bonds, restricted end-to-end distancefluctuations, locally anisotropic dynamics, and local contacts between interaction vector atoms and theirenvironment. Many of the features parallel experimental NMR and computational observations of helices inproteins. The understanding gained from this model system is expected to contribute toward the ever moredetailed interpretation of experimental order parameter profiles.

Introduction

NMR spin relaxation experiments of proteins provide uniquelydetailed and quantitative information about conformationaldynamics on the pico- to nanosecond time scale.1-3 Over thepast two decades, sophisticated spin relaxation experiments of15N, 13C, or 2H labeled proteins have become available for themeasurement of longitudinal T1, transverse T2, the heteronuclearNOE, and other relaxation pathways. Interpretation of theseparameters has become intimately connected to the model-freeformalism by Lipari and Szabo.4,5 The power behind the model-free approach, which is also referred to as Lipari-Szaboapproach, is that it separates (using minimal assumptions) spatialdynamics from time scale effects by means of a generalizedorder parameterS2, an effective internal correlation timeτint,and an overall tumbling correlation timeτc. S2 is a measure ofthe spatial restriction of the reorientational motion of thedominant spin-interaction vector in a molecular reference frame,such as the15N-1H bond vector in the case of backbone15Nrelaxation data.6 More recently, it has become possible todetermineS2 values from residual dipolar couplings measuredin multiple alignment media reporting on pico- to millisecondtime scale motions.7,8

The Lipari-Szabo approach is widely used and is generallythe first step in the analysis of experimental NMR relaxationdata. In a recent survey of the literature, it was found that nearly90% of the published NMR backbone relaxation dynamicsstudies use a model-free analysis of the data.9 The appeal ofthe model-free approach lies in its conceptual simplicity thatlargely avoids the risk of overinterpretation of the experimentaldata. In fact, the model-free formalism was introduced (andhence its name) to overcome the arbitrary character of certain

analytical motional models for the interpretation of relaxationdata. These features explain the unabated popularity of theapproach 25 years after its inception.

In a more general sense, a model-free analysis is the beginningof a dynamic description of a protein rather than the end. As asingle real-valued quantity, theS2 order parameter does notprovide a complete description of the underlying reorientationalmotional distribution. This is not a deficiency of the model-free approach per se but rather reflects the actual dynamicsinformation that relaxation data provide for a given nucleuswithout further interpretation involving additional assumptions.The interpretation of multiple order parameters (or relaxationparameters) for a whole protein or protein fragment, such as anR-helix, within a self-consistent structural/biophysical frame-work requires a more concrete description. For instance, amolecular fragment in a protein, such as an aromatic ring or apeptide plane, reorients anisotropically relative to its surround-ings, which requires the simultaneous interpretation of orderparameters from multiple interaction vectors together with theirrelative orientations. Also, the effect of correlated motionsbetween interaction vectors belonging to different residues, forexample across a loop region, needs motional models that gobeyond a standard model-free description.

To meet these demands, several methods have been developedthat enhance the information provided by individual orderparameters or by complete order parameter profiles. Suchapproaches include molecular dynamics (MD) computersimulations,10-18 normal-mode analysis,19,20 quasi-harmonicanalysis,21-25 elastic network models,26-28 local contact mod-els,27,29,30MD-derived analytical models,31,32and thermodynamicmodels.33-37 Some of these approaches can also be applieddirectly to the interpretation of spectral densities extracted fromthe relaxation data38-42 or to the raw relaxation data them-selves.18

† Part of the “Attila Szabo Festschrift”.* To whom correspondence should be addressed. Tel.: 850-644-1768.

Fax: 850-644-8281. E-mail: bruschweiler@magnet.fsu.edu.

6203J. Phys. Chem. B2008,112,6203-6210

10.1021/jp711160t CCC: $40.75 © 2008 American Chemical SocietyPublished on Web 04/01/2008

The generalized order parameterS2 of an interaction vectoris defined by4

where the polar angles (θ,j) define the direction of the vectorin a molecular reference frame andY2m(θ,æ) are the normalizedspherical harmonics of rank 2. The angular brackets in eq 1indicate thatS2 is defined either as a time average or over anensemble of snapshots. Thus, experimental order parametersdirectly reflect the underlying conformational ensemble. It isimportant to note that the model-free method assumes thatoverall and internal motions are statistically independent. Thisassumption is difficult to test both experimentally and compu-tationally. Alternative approaches to the model-free method havebeen developed that specifically treat correlations betweenoverall and internal motions.43-45 Extended MD simulations arealso increasingly common, which will facilitate future tests ofthe assumptions underlying the model-free approach.

MD simulations offer several advantages, including theexplicit consideration of time scale effects.18 The relativecomplexity of the MD model, however, makes it difficult torelate order parameter profiles to specific constraints imposedon secondary structural elements. To establish relationshipsbetweenS2 values and other structural and dynamic descriptors,we analyze here finite conformational ensembles of a modelsystem. We focus on the backbone dynamics of anR-helicalmodel peptide consisting of 12 amino acids. A database analysisof experimental order parameters shows that the backbone15N-1H S2 profiles of helices are nonuniform exhibiting characteristictrends especially toward the ends of the helix.46 Here, varioushelical ensembles are generated by systematically modifyingφ/ψ dihedral angles both with respect to fluctuation amplitudesand their mutual correlations. From these relationships, one canestablish guidelines for solving the inverse problem, which isthe deduction of dynamic information about these internaldegrees of freedom from experimental order parameters. Theprimary advantage of this approach is that it relies upon minimalassumptions, providing an unobstructed view of the underlyingdynamics.

Order parameters are calculated for this system using fivedifferent methods. In addition to the standard definition of eq1, these methods include: (1) the isotropic reorientationaleigenmode dynamics method (iRED), (2) a first-order expansionin local variances and covariances, (3) the three-dimensional(3D) Gaussian axial fluctuation (GAF) method, and (4) the localcontact model. Each of these methods illuminates a differentaspect of order parameters and how they reflect their dynamicenvironment.

This work focuses on the effects of anisotropic motions,correlated motions, and the presence of short-range versus long-range motions on order parameters.47 Short-range motionmodulates the local dihedral angles in the vicinity of aninteraction vector, such as theφi/ψi-1 dihedral angle dynamicsin the case of a Ni-Hi vector. Long-range motion, on the otherhand, reorients an interaction vector via dihedral angles thatare further apart, as in the case of a hinge bending motion of awhole protein fragment or fraying at the end of a polypeptidechain. It is shown how the combined use of multiple methodsof analysis can provide an integrated description of these variousfactors.

Computational Methods

A 12 residue peptide was constructed using the CHARMM22topology file.48 The local backbone geometry of the Leucineresidue was selected for all residue subunits. A referenceR-helixwas generated by setting the average backbone dihedral anglesto φref ) -57°, ψref ) -47°. Ensembles were generated byrandomly varying allφ andψ angles according to a Gaussianprobability distribution centered around the reference angles witha standard deviationσ in all angles. In addition, the distributionswere skewed in such a way that they adopt a predefined linearcorrelation coefficientr from 0 to-1 between angleφ of residuei and angleψ of residue (i - 1). This procedure requires thatone specifyσ, r, and a tilt angle that transforms the (φ,ψ)distribution to a set of normal coordinates. The details of thisprocedure are outlined by eqs A1-A4 in the Appendix of Bremiet al.49 A representative ensemble of 50 structures is providedin Figure 1 for the case in whichσ ) 10° andr ) -0.5. Thereference helix is represented below the ensemble for compari-son.

Order parameters are calculated using eq 1 after all conform-ers were superimposed (by translation and rotation) with respectto the reference helix. As a second method, the isotropicreorientational eigenmode dynamics (iRED) method is ap-plied.50,51 In this approach, where the conformers do not needto be superimposed, the isotropically averaged covariance matrixM is computed with elements

whereei and ej are the normalized interaction vectorsi and jthat belong to the same conformer, and the angular bracketsindicate averaging over all conformers of the ensemble. Ninetydifferent backbone vectors are included in the analysis, corre-sponding to the 11 N-H, 11 N-CR, 11 N-C′, 11 CR-C′, 11C′-O, 12 CR-HR, 12 CR-Câ bond vectors as well as the 11cross-products of the N-H and N-CR vectors. The 11 cross-product vectors are included in addition to the in-plane N-H,N-CR, N-C′, CR-C′, and C′-O vectors to comprehensivelyprobe reorientational motion of the peptide planes. Next, aprincipal component analysis is performed by solving theeigenvalue problemM |m⟩ ) λm|m⟩ wherem ) 1, ...,N, andNis the number of vectors.λm are the eigenvalues sorted accordingto size (first eigenvalue is largest), and|m⟩ are the associatedeigenvectors. TheSj

2 value for vectorj was then calculated as50

Figure 1. Helical ensemble (upper panel) consisting of 50 structuresfor the case in whichσ ) 10° and r ) -0.5, which is representativefor the average reference helix (lower panel).

Mij ) 12

⟨3(eiej)2 - 1⟩ (2)

S2 )4π

5∑

m)-2

2

⟨Y2m(θ,æ)⟩⟨Y*2m (θ,æ)⟩ (1)

6204 J. Phys. Chem. B, Vol. 112, No. 19, 2008 Johnson et al.

where the sum extends over the eigenvectors|m⟩ that belongto the N - 5 smallest eigenvaluesλm, which correspond tointernal motions. There is no need to align the conformersbecause the covariance matrixM contains only inner productsand therefore is rotationally invariant. In fact, one of the mainadvantages of the iRED method is that it makes no assumptionsregarding the separability between internal and overall tumblingmotions. A separability indexg follows naturally from theanalysis and was calculated as

A largeg value reflects good separability between internal andoverall motions, which is typically found for globular proteins.In contrast, smallg values are observed in disordered systemswhere internal and overall tumbling motions are not independent.Other issues related to separability have been discussed byMeirovitch et al.43

The 3D GAF model provides an analytical description ofanisotropic peptide plane motion.31,32The model represents theinternal dynamics of the peptide plane in terms of independentGaussian axial fluctuations about three orthogonal axesR, â,and γ. Reorientational motion can then be visualized asellipsoids with principal axes parallel toR, â, andγ and lengthsthat are proportional to the fluctuation amplitudesσR, σâ, andσγ. In this respect, the 3D GAF model acts as the NMR analogueto the anisotropic temperature factor in X-ray crystallography,although the latter is also sensitive to translational motion. The3D GAF model has recently been applied to the interpretationof residual dipolar couplings52 as well as13C′-13CR cross-relaxation rate constants.53

The local contact model was originally developed to predictthe order parameter directly from a 3D protein structure.29 Inthe case of the N-H bond vector for residuei, the calculationis based on the close contacts experienced by the amide H andthe carbonyl O of the preceding residue with the surroundingheavy atomsk

where di-1,kO and di,k

H denote the distances (in angstroms)between heavy atomk and the carbonyl O of amino acid (i -1) and the amide H of amino acidi, respectively, shortened by1.2 Å.

Results

Order Parameter Profiles. Ensembles were generated forthe 12 residue peptide first with a standard deviation in theφ/ψdihedral angles ofσ ) 10°. Dihedral angle fluctuations of thismagnitude are consistent with a recent MD simulation ofubiquitin at 300 K.18 In that simulation, of which the first 10nanoseconds are analyzed in the present study, the standarddeviation in the dihedral angles of the central helix (residues23-34) varies between 7.7 and 19.7° for φ and 7.3 and 15.4°for ψ. SiRED

2 values for the 8 different bond vector types areplotted as a function of residue number in Figure 2 for fourdifferent ensembles, each consisting of 10 000 conformers andσ ) 10°, but with different linear correlation coefficientsrbetween the dihedral anglesφi andψi-1. It should be noted that

these ensembles include neither local librational motions54 norzero-point vibrational motions,19,20 both of which are presentin the experiment and which will reduce theS2 values. Therefore,the computedS2 values presented here tend to be higher thantheir experimental counterparts.

In the uncorrelated case (r ) 0), all of the backbone bondvectors exhibit similarS2 profiles with gradually reducedS2

values toward both termini; a maximumS2 value of around 0.9occurs near the middle of the sequence, while a minimum valueof around 0.7 is observed at the ends (Figure 2a). This frayingeffect is accompanied by a relatively low separability index (g) 5.5) between overall and internal rotation, which suggeststhat the capability to distinguish between internal motion andoverall orientation is limited. In contrast, when anticorrelationsin φi/ψi-1 dihedral angles are introduced the fraying decreasesand the separability increases (Figure 2b-d). Anticorrelatedvariations inφi/ψi-1 are commonly referred to as crankshaftmotions,55-60 which are structure-conserving in the sense thatthey minimize distortions relative to the reference helix. In thelimiting case whereφi/ψi-1 are perfectly anticorrelated (r )-1.0; Figure 2d), there is little or no fraying and the separabilityindex is notably high (g ) 31.0). In this case, theS2 values fallinto two clusters that reorient differently as a result ofφ/ψfluctuations. One cluster exhibitsS2 values near 1.0 and includesall four bond vectors that are attached to a CR atom (N-CR,CR-C′, CR-HR, and CR-Câ). Despite variations inφ/ψ (σ )10°), the anticorrelated nature of the fluctuations results in verylittle reorientational motion of these bond vectors. The secondcluster exhibits order parameters well below 1.0 and includesall of the bond vectors that are not attached to a CR atom (N-H, N-C′, C′-O, and (N-H) × (N-CR)). Among this secondclass of bond vectors, the order parameters are around 0.94 forthe N-H, N-C′, and C′-O vectors and 0.93 for the (N-H) ×(N-CR) vectors. These observations are consistent with crank-shaft motions that keep the CR positions fixed with respect toboth translation and reorientation and allow for limited fluctua-tions of the peptide plane.

The limiting cases represented in Figure 2a,d provide usefulreference points, but they are not observed in MD simulations,which typically yield r values of around-0.5 for R-helices.Figure 2b,c illustrate theS2 profiles for partial correlations ofr) -0.5 and-0.75. Both ensembles exhibit relatively highseparability between overall and internal rotation (g ) 8.8 andg ) 13.4, respectively). Some fraying is observed even atr )-0.75, although it is significantly attenuated relative to theuncorrelated case.

Experimental observations indicate that proximity to helixtermini does in fact correlate with lower average order param-eters, but the magnitude of the effect is small. Goodman et al.compiled a database of 1855 order parameters from 20independent NMR relaxation studies.46 They found that thechange inS2 per residue is around 0.003 and 0.01 for the sixresidues near the N- and C- termini, respectively, ofR-helices.One likely cause is the fact that the N-H groups of the firstfour residues and the C′-O groups of the last four residues arenot hydrogen bonded, leading to increased flexibility.

As mentioned above, the iRED method allows calculationof S2 values without requiring superposition of the conformers.For ther ) -0.5 ensemble, order parameters for the N-H bondvectors were determined using both the iRED method as wellas using the standard relationship from eq 1 that requires explicitsuperposition. Structures were superimposed on the referencehelix by a least-squares method using the N, CR, C′, and Oatoms. Figure 3 demonstrates that theS2 values obtained by eq

Sj2 ) 1 - ∑

m)6

N

λm||m⟩j|2 (3)

g ) ∑m)1

N

λm/ ∑m)6

N

λm (4)

Si2 ) tanh[0.8∑

k

exp(-di-1,kO ) + 0.8exp(-di,k

H )] - 0.1 (5)

NMR Order Parameter inR-Helices J. Phys. Chem. B, Vol. 112, No. 19, 20086205

1 are slightly lower than the iRED order parameters. The sametrend, although on a smaller scale, is observed in the MDsimulation of ubiquitin. In both cases, the effect is mostpronounced in the flexible termini. The MD simulation yieldsa higher separability index than ther ) -0.5 ensemble (g )9.6 and 8.8, respectively), which contributes to the fact thatthe discrepancy is smaller for the MD ensemble. Ther )-0.5 ensemble reflects the transition toward the situation whereoverall and internal motions are no longer separable. Theprincipal component analysis that underlies the iRED approachattributes the maximal amount of motion to overall reorientation(5 largest modes) so that the remaining internal modes yield ahigherS2 value.

Correlations Introduced by Structural Constraints. In theprevious ensembles, the introduction ofφ/ψ correlations gener-ates ensembles with little fraying and relatively high separabilitybetween overall and internal rotation. Alternatively, a muchlarger ensemble with 106 conformers was generated withσ )

10° andr ) 0 that was trimmed according to structural criteria,such as backbone-backbone hydrogen bonds or the end-to-end distance. These criteria were intended to mimic theconstraints imposed on the helix by the surrounding parts ofthe protein. The ensemble was first trimmed by selecting onlythose conformers in which all of the O and H atoms thatparticipate in native (i, i + 4) hydrogen bonds are within adistance of 1.6 and 2.6 Å. This criterion selects 7639 structuresfrom the original ensemble. The relatively small number ofstructures that were selected reveals that, in the absence of anydihedral angle correlations,φ/ψ fluctuations of this magnitude(σ ) 10°) generally do not permit the preservation of allhydrogen bonds. Moreover, the sculpted ensemble that fulfillsthe hydrogen-bonding criterion exhibits a small but statisticallysignificant correlation betweenφi andψi-1 (r ) -0.28). As aresult of the trimming procedure, the average standard deviationsin φ andψ also decrease slightly to 8.7 and 9.4°, respectively.

In a next step, the ensemble was further trimmed by selectingonly those structures whose end-to-end distance is( 1 Å fromthat of the reference helix. This second step selects 5848 of the7639 structures that met the initial hydrogen-bonding criterion.The standard deviations inφ andψ are essentially unchangedas a result of this second trimming step (σ ) 8.6 and 9.2°),whereas the amount of anticorrelation increases to a value ofr) -0.33. TheSiRED

2 profile for this double trimmed ensembleis shown in Figure 4. As a consequence of the trimmingprocedures, the fraying effect is relatively small and comparableto what is observed for the case in which theφi/ψi-1 correlationwas explicitly set to-0.75 (Figure 2c) with almost identicalseparability indices (g ) 13.2 for the double trimmed ensembleversus 13.4 for ther ) -0.75 ensemble). The trimmingprocedure, therefore, produces a reasonable order parameterprofile, although experimental profiles for helices in intact

Figure 2. The iRED order parameterS2 plotted as a function of the peptide plane number forφi/ψi-1 correlationsr ) 0 (a), r ) -0.5 (b), r )-0.75 (c), andr ) -1 (d). The following bond vectors are represented by the symbols in parentheses: N-H (+), N-CR (circle), N-C′ (x), CR-C′(down triangle), C′-O (up triangle), CR-HR (square), CR-Câ (diamond), and (N-H) × (N-CR) (*). The peptide planes are numbered by thenumber of the amino acid residue that contributes the N atom.

Figure 3. A correlation plot between theS2 value calculated using eq1 that requires superposition and the iREDS2 value for the N-H vectorin the r ) -0.5 case.

6206 J. Phys. Chem. B, Vol. 112, No. 19, 2008 Johnson et al.

proteins can show small, additional features arising from tertiaryinteractions that are not captured by the current trimming criteria.The trimming procedure is also consistent with the moleculardynamics (MD) simulation of ubiquitin in that it accuratelyreflects the effects of reorientational motion on the differentbond vector types. Average order parameters for each bondvector type were calculated from both the MD simulation ofubiquitin and the double trimmed (DT) model. The averagevalues from the MD simulation exhibit the following trend:

which is largely consistent with that observed from the doubletrimmed ensemble

The main difference between the two ensembles is the averageorder parameter for the cross-product vectors (N-H) × (N-CR). A factor that contributes to the low⟨S(N-H)×(N-CR)

2 ⟩MD

value is the fact that HN atom experiences an out-of-planemotion that is absent in the double trimmed ensemble. Analternative cross-product vector (C′-O) × (N-CR) is lesssusceptible to this effect. For example, the average orderparameter over the central region of the ubiquitinR-helix(peptide planes 27-31) for the (N-H) × (N-CR) vector is 0.90,whereas the corresponding value for the (C′-O) × (N-CR)vectors is 0.93.

The trimming procedure demonstrates that structures canexhibit relatively large amplitude fluctuations, yet still retainspecific geometric features (H-bonds, end-to-end distance)provided that the dihedral angles are suitably correlated. Toexplore such correlations, Figure 5 plots the correlation coef-ficients betweenφ7/φ7+k (a) andφ7/ψ7+k (b) as a function ofthe relative amino-acid positionk for the double trimmed 5848conformer ensemble. Figure 5 also shows the correspondingcorrelations that are observed in the molecular dynamics (MD)simulation of ubiquitin. Dihedral angleφ29 was selected for thiscomparison as it is located in the middle of ubiquitin’s centralhelix. Neither the double trimmed ensemble, nor the MD-derivedensemble exhibits significantφ/φ correlations. Both ensembles,however, exhibit a crankshaft motion betweenφi andψi-1. Inaddition to this crankshaft motion, there is a relatively largeφi/ψi anticorrelation. More distant correlations are weak andare damped out relatively quickly. Theφi/ψi correlation has been

noted previously,56,57although it is not as widely recognized asthe correlation betweenφi andψi-1. The MD results indicatethat theφi/ψi anticorrelation is most prevalent inR-helices andfar less pronounced inâ-sheets. This trend is not unique toubiquitin, but has also been observed in MD simulations of otherproteins that we have investigated, including MDM2 and RBD1(unpublished). These results indicate that, while theâ-sheetstructure is maintained largely byφi/ψi-1 crankshaft motions,the R-helical structure exhibits an additionalφi/ψi correlation.

The absence of significant long-range correlations suggeststhat it might be possible to predict theS2 values using a smallset of local variances and covariances.47 In Figure 6a, thevariancesµφiφi andµψiψi and covariancesµφiψi-1, µφiψi, andµψiψi-1

are plotted as a function of residue number. The variances andcovariances that are observed in the N- and C-terminal residuesare very close to the original values in the untrimmed ensemble.The central residues, however, are perturbed from their originalvalues. The average variances and covariances across the fivecentral residues (5-9) are⟨µφiφi⟩ ) 65 (deg2), ⟨µψiψi⟩ ) 81 (deg2),⟨µφiψi-1⟩ ) -35 (deg2), ⟨µφiψi⟩ ) -11 (deg2), and ⟨µψiψi-1⟩ )-14 (deg2). As a comparison, the MD simulation of ubiquitingives values of⟨µφiφi⟩ ) 77 (deg2), ⟨µψiψi⟩ ) 70 (deg2), ⟨µφiψi-1⟩) -43 (deg2), ⟨µφiψi⟩ ) -28 (deg2), and⟨µψiψi-1⟩ ) 0.3 (deg2).The double trimmed and MD-derived ensembles again providea consistent overall trend, particularly in the largest variancesand covariancesµφiφi, µψiψi, and µφiψi-1. Using this set ofvariances and covariances, the order parameter is expressed as

where the 3 coefficientscψi-1ψi-1, cφiφi, andcψi-1φi are determinedby a linear least-squares optimization using singular valuedecomposition with the iREDSi

2 values and theµ values given

Figure 4. The iREDS2 value plotted as a function of peptide planenumber for the double trimmed ensemble of 5848 structures that meetthe hydrogen bonding and end-to-end distance constraints. The symbolsare the same as those used in Figure 1.

Figure 5. The correlation coefficients betweenφ7/φ7+k (a) andφ7/ψ7+k (b) plotted as a function ofk for the double trimmed ensemble(circles). The correlation coefficients betweenφ29/φ29+k (a) andφ29/ψ29+k (b) plotted as a function ofk from an MD simulation of ubiquitin(squares).

Si2 ) 1 - cψi-1ψi-1

µψi-1ψi-1- cφiφi

µφiφi- cψi-1φi

µψi-1φi(6)

⟨SN-CR2 ⟩MD > ⟨SCR-C′

2 ⟩MD > ⟨SN-C′2 ⟩MD > ⟨SCR-Câ

2 ⟩MD >

⟨SCR-HR2 ⟩MD > ⟨SC′-O

2 ⟩MD > ⟨SN-H2 ⟩MD > ⟨S(N-H)×(N - CR)

2 ⟩MD

⟨SN-CR2 ⟩DT > ⟨SCR-C′

2 ⟩DT > ⟨S (N-H)× (N-CR)2 ⟩DT > ⟨SCR-Câ

2 ⟩DT >

⟨SN-C′2 ⟩DT > ⟨SCR-HR

2 ⟩DT > ⟨SN-H2 ⟩DT > ⟨SC′-O

2 ⟩DT

NMR Order Parameter inR-Helices J. Phys. Chem. B, Vol. 112, No. 19, 20086207

in Figure 6a. A correlation plot of the iREDSi2 values and the

values obtained from the first-order expansion in eq 6 is shownin Figure 6b for the N-H bond vectors usingcψi-1ψi-1 ) -4.6×10-4,cφiφi ) 2.0 × 10-3, cψi-1φi ) 7.1 × 10-4. Although thedifferences between residues are relatively subtle in Figure 6a,the variances and covariances represented therein provide anaccurate estimate of the order parameter. The rmsd betweenthe two sets is 0.007, while the linear correlation coefficient is0.97. The inclusion of additional terms improves the fit onlyminimally. For example, the addition of thecψiψiµψiψi andcφiψiµφiψi terms decreases the rmsd to 0.005 and increases thecorrelation coefficient to 0.99. Equation 6 captures the orderparameters for the MD-derived ensemble of ubiquitin well (rmsdof 0.02 and linear correlation coefficient of 0.93), after the 3coefficients of eq 6 have been suitably adjusted.

3D GAF Order Parameters.Peptide planes of the ensemblesstudied here generally do not reorient “symmetrically” as ismanifested in the order parameter differences within theindividual peptide planes (Figures 2 and 4). Additional insightinto the factors governing these differences is provided by a3D GAF analysis. In Figure 7a,σR, σâ, and σγ (collectivelyreferred to asσ3DGAF) are plotted as a function of peptide planenumber for the double trimmed ensemble. The trend is similarto that observed among the variances and covariances in Figure6a in that the fluctuation amplitudes increase near the termini.Note, however, that a correspondence between Figures 6a and7a is not necessary. As an example, consider Figure 7b in whichthe σ3DGAF profile is provided for the ensemble in which theφi/ψi-1 correlation coefficient is explicitly set to-0.75. Recallthat this ensemble yields a similarS2 profile as the doubletrimmed ensemble. A comparison of Figures 7a and 7b indicatesthat theσ3DGAF profiles for the two ensembles share similarfeatures. Despite these similarities, the two ensembles exhibitvery different fluctuations in their internal (dihedral angle)coordinates. The trimmed ensemble exhibits variations across

the sequence that are missing in ther ) -0.75 ensemble. Inthe r ) -0.75 ensemble, the variances and covariances inφ/ψare uniform (µφiφi ) µψiψi ) 100 deg2, µφiψi-1 ) -75deg2). Thisobservation implies that theS2 variations in ther ) -0.75ensemble are due to longer range motions that propagate overmultiple residues. Theσ3DGAF values are sensitive not only tothese long-range motions, but also to short-range motions thatmodulate the local internal coordinates; thus 3D GAF motionsinclude both of these contributions. They represent a cumulativeeffect in which a large number of motional modes are projectedonto the individual peptide planes. As result, ensembles withrather differentφ/ψ fluctuations can yield similarσ3DGAF (andS2) profiles.

Order parameters are calculated in the 3D GAF analysis fromthe fluctuation amplitudes and the mean bond vector orientationin the 3D GAF reference frame. A correlation plot of the iREDand 3D GAFS2 values is provided in Figure 8 for the N-Hbond vectors. The double trimmed andr ) -0.75 ensembles

Figure 6. (a) The variancesµφiφi (squares) andµψiψi (circles) andcovariancesµφiψi-1 (diamonds),µφiψi (+), andµψiψi-1 (up triangle) plottedas a function of residue number for the double trimmed ensemble. (b)A correlation plot between theS2 value estimated from eq 6 and theiRED S2 value for the N-H vector.

Figure 7. The 3D GAF fluctuation amplitudesσR (circles),σâ (squares),andσγ (diamonds) plotted as a function of peptide plane number forthe double trimmed (a) andr ) -0.75 (b) ensembles.

Figure 8. A correlation plot between the 3D GAF and iREDS2 valuesfor the N-H vector in the double trimmed (circles) andr ) -0.75(squares) ensembles.

6208 J. Phys. Chem. B, Vol. 112, No. 19, 2008 Johnson et al.

are represented as circles and squares, respectively. The cor-relation between the iRED and 3D GAF methods is high inboth cases. The double trimmed ensemble gives an rmsd of0.014 and a linear correlation coefficient of 0.94, while ther )-0.75 ensemble gives an rmsd of 0.015 and a correlationcoefficient of 0.82. The largest discrepancies between the twomethods are observed among the two to three peptide planestoward the N- or the C-terminus.

Local Contact Model Calculations.A local contact modelcan account for many features of backbone order parameters inglobular proteins.29 The contact model is generally applied eitherto a high-resolution X-ray structure or to a small ensemble ofNMR structures. In the present context, the local contact modelis applied to all members of the helical ensemble and theresultingS2 values are averaged over the ensemble. A com-parison between the N-H order parameters by the local contactmodel and the ones by the iRED method is given in Figure 9.The results for the double trimmed andr ) -0.75 ensemblesare represented by circles and squares, respectively. The doubletrimmed ensemble gives a correlation coefficient of 0.94 andthe r ) -0.75 ensemble a value of 0.88. The contact modelexhibits a plateau atS2 ) 0.9, which reflects the experimentalS2 values that are typically observed in a rigid protein segment.The helical peptide system exhibits a slightly higher baselineorder parameter, indicating that it is not entirely representativeof globular proteins for which the local contact model wasparametrized. As noted above, it lacks zero-point vibrationalcontributions that affect the order parameter baseline.19,20

Conclusions

An S2 order parameter represents quantitative informationabout the reorientational fluctuation of an interaction vector atan individual site. Because of its general mathematical nature,a given S2 value can be consistent with a large number ofdifferent possible motional models. This ambiguity can beovercome by placingS2 into context: the present workdemonstrates how an extensive set of order parameters belongingto a helical segment reflects upon structural and dynamicproperties and constraints, such as end-to-end distance con-straints, hydrogen bonds, local contacts, locally anisotropicdynamics, and local backboneæ,ψ dihedral angle motionalamplitudes and their mutual correlations. This informationprovides the basis to address the inverse problem, which is theextraction of this kind of structural and dynamic informationfrom experimentally accessible order parameters. The minimal-istic assumptions made for the generation of the helicalensembles and the fact that many of the structural dynamic

features observed mirror those found in all-atom MD simulationsin explicit solvent suggest that these patterns are of a generalnature and therefore rather broadly applicable to helices inproteins.

Order parameters have been calculated here for ensemblesof a helical model peptide by five complementary methods: (1)the standard eq 1, (2) iRED, (3) a first-order expansion in localvariances and covariances, (4) the 3D GAF method, and (5)the local contact model. Despite their differences, all fivemethods provide consistent results for the scenarios consideredhere. This observation is noteworthy given the fact that the orderparameter variations are relatively small in several of theensembles. Each method offers complementary information. (1)The standard definition provides a useful reference. (2) TheiRED method allows one to assess separability between internaland overall rotation, which is a basic assumption of the model-free approach. (3) The first-order expansion specifically accountsfor the contributions from local dihedral angle fluctuations. (4)The 3D GAF model provides a full description of anisotropicpeptide plane motion, and (5) the local contact model demon-strates an intrinsic link between protein structure and dynamics.By applying all five of these methods to a simple model systemwhose dynamics is governed by minimal assumptions, oneobtains a comprehensive view of the motional processes thatgive rise to order parameter variations. For this purpose, it isinstructive to note that method 3 is unique in that it is insensitiveto contributions from long-range motion. The fact that method3 is successful when applied to the double trimmed ensembleindicates that long-range motions do not strongly contribute tothe order parameters in that case. The structural constraints thatare present in the double trimmed ensemble introduce localvariations in theφ/ψ fluctuations. These effects, which are localin origin, are reflected in the order parameter variations acrossthe sequence. This scenario corresponds to what is commonlyobserved in the core of globular proteins. The situation is quitedifferent, however, for the ensembles in which theφi/ψi-1

correlation is explicitly defined. In these cases, method 3 failsto describe changes ofS2 along the helix due to the uniformityof the localφ/ψ fluctuations. Variations in the order parameter,therefore, by necessity reflect contributions from long-rangemotion. This effect is included in the other four methods, whichproject the contribution from long-range motion onto the localenvironment. For example, the contact model indicates that thenumber of local contacts decreases near the termini for ther )-0.75 ensemble. Likewise, the 3D GAF model demonstratesthat theσ3DGAF values increase near the termini. Although theorigin of the effect is nonlocal, it is seen locally viaS2. Thesefactors give rise to the fraying effect that is often observed inthe order parameter profiles of globular proteins toward thetermini of the polypeptide chain. The ability to discriminatebetween these various scenarios is one of the benefits of theapproach presented here. This case study demonstrates that aself-consistent view of reorientational motion emerges whenmultiple complementary methods are applied to the samesystem. Generalization of the properties for anR-helix describedhere to other structural motifs will further enhance the powerof NMR order parameter analysis of proteins.

Acknowledgment. E.J. is the recipient of a postdoctoralfellowship from the American Heart Association, GreaterSoutheast Affiliate. S.A.S. is the recipient of an NIH postdoc-toral fellowship. This work was supported by the NSF (Grant0621482).

Figure 9. A correlation plot between the contact model and iREDS2

values for the N-H vector in the double trimmed (circles) andr )-0.75 (squares) ensembles.

NMR Order Parameter inR-Helices J. Phys. Chem. B, Vol. 112, No. 19, 20086209

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