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The relation of the X-ray B-factor to protein dynamics:insights from recent dynamic solid-state NMR dataDetlef Reichert a , Tatiana Zinkevich a b , Kay Saalwächter a & Alexey Krushelnitsky aa Institut für Physik – NMR, Martin-Luther-Universität Halle-Wittenberg, Betty-Heimann-Str.7, Halle, 06120, Germanyb Zavoisky Physical-Technical Institute, Sibirsky tract, 10/7, Kazan, 420029, Russia

Version of record first published: 02 Jul 2012.

To cite this article: Detlef Reichert, Tatiana Zinkevich, Kay Saalwächter & Alexey Krushelnitsky (2012): The relation of the X-ray B-factor to protein dynamics: insights from recent dynamic solid-state NMR data, Journal of Biomolecular Structure andDynamics, 30:6, 617-627

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The relation of the X-ray B-factor to protein dynamics: insights from recent dynamicsolid-state NMR data

Detlef Reicherta*, Tatiana Zinkevicha,b, Kay Saalwächtera and Alexey Krushelnitskya

aInstitut für Physik – NMR, Martin-Luther-Universität Halle-Wittenberg, Betty-Heimann-Str. 7, Halle 06120, Germany; bZavoiskyPhysical-Technical Institute, Sibirsky tract, 10/7, Kazan 420029, Russia

Communicated by Ramaswamy H. Sarma

(Received 15 November 2011; final version received 27 March 2012)

In addressing the potential use of B-factors derived from X-ray scattering data of proteins for the understanding the(functional) dynamics of proteins, we present a comparison of B-factors of five different proteins (SH3 domain, Crh,GB1, ubiquitin and thioredoxin) with data from recent solid-state nuclear magnetic resonance experiments reflecting true(rotational) dynamics on well-defined timescales. Apart from trivial correlations involving mobile loop regions and chaintermini, we find no significant correlation of B-factors with the dynamic data on any of the investigated timescales,concluding that there is no unique and general correlation of B-factors with the internal reorientational dynamics ofproteins.

Keywords: protein dynamics; x-ray B-factor; dynamic solid-state NMR

Introduction

Most proteins carry out their function through conforma-tional transitions of their structure and thus, the dynamicproperties of the protein molecules on the molecularlevel play an important role in their biological activity.Proteins are not at all static structures; at ambient tem-peratures they undergo a variety of dynamic processes.Understanding protein function thus strongly requires tounderstand protein dynamics. Data obtained from struc-tural X-ray experiments, the so-called B-factor areemployed for this purpose. Scanning the available litera-ture and taking into account some recent solid-statenuclear magnetic resonance (NMR) data, we would liketo shed some more light on this issue

Before we turn to the experimental approaches, wewould like to consider the nature of dynamic processesin proteins and to clarify the meaning of “moleculardynamics” or “molecular flexibility”, as it is frequentlyrefered to. The most important dynamic process for thefunction seem to be transitions between different equilib-rium states which are triggered by external events likeligand binding or just change in temperature, pH, etc.We will call them type-A processes for now. However,these processes depend on the existence of dynamic

degrees of freedom on the molecular level, like local ran-dom atomic fluctuations or correlated motions of parts ofthe protein molecules within an equilibrium conforma-tion or between different equilibrium states. Thesedynamic processes, which we will call type-B processes,are equilibrium processes and are driven by the thermalenergy of a surrounding bath. To understand proteinfunction, i.e. type-A processes, it is necessary to studythe ability of the protein to perform type-B dynamics.The division of the processes in types A and B mightseem gratuitous; however, it is related to the issue ofhow different methods understand the term “flexibility ofmolecules”. X-ray crystallography, that was originallyapplied to provide structural information, was consideredsince the 70s to be able to deliver information about themolecular dynamics as well (Sternberg, Grace, & Phil-lips, 1979; Willis & Pryor, 1975). Besides the positionsof the non-hydrogen atoms, the B-factors further providea kind of dynamic information relating to the averagemean-square displacements of the nuclei. While this is apriori a by-product of the data processing, one began toview it as a tool to characterize the flexibility of the pro-tein molecules. A measure for “flexibility” here meansthe Root-Mean-Square Deviation (RMSD) within theensemble of conformations which are compatible with

*Corresponding author. Email: detlef.reichert@physik.uni-halle.de

Journal of Biomolecular Structure and DynamicsVol. 30, No. 6, 2012, 617–627

ISSN 0739-1102 print/ISSN 1538-0254 onlineCopyright � 2012 Taylor & Francishttp://dx.doi.org/10.1080/07391102.2012.689695http://www.tandfonline.com

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the experimental data. It is thus clear that the quality ofthe dynamic data depends on the quality of the data pro-cessing routines which transform the X-ray reflexionsinto the spatial structure of the scattering heavy atoms(Ringe & Petsko, 1985; Sternberg et al., 1979). Sincethese routines often include iterative refinements, theymight be a source of a significant bias. Likewise, struc-tural NMR methods (Wuthrich, 1995) employing dis-tance or/and orientation constraints end up with anumber of structures which are compatible with theexperimental data; regions of the molecule for which theatomic coordinates exhibit a larger scatter are commonlyconsidered to be more flexible. As it is obvious, theseexperimental methods do not provide a true measure ofthe dynamic degrees of freedom but a more indirect pic-ture, which is a result of data processing and refinement.They are merely a measure of the “uncertainties” inatomic coordinates resulting from the methodology ofthe structure determination procedure. We will thus termthe results of such approaches “indirect dynamic data”.

Other methods to provide dynamic information arecomputational and statistical approaches. In principle,molecular dynamics simulations are able to provide adetailed dynamic picture; however, these methods arecomputationally expensive and the computer perfor-mance nowadays available still limits this approach tothe ns range. As an alternative, coarse-grained elasticnetwork models have been used in the last decade forstudying protein dynamics. In particular the Gaussiannetwork model (GNM) (Erman, 2006; Haliloglu, Bahar,& Erman, 1997), using a single-parameter harmonicpotential has gained some success, however the choiceand optimization of the spring constants may lack a realphysical basis (Yang, Song, & Jernigan, 2009). Theseapproaches are often verified by comparisons madeagainst X-ray B-factors and NMR conformational vari-abilities, i.e. against experimental data which are not realdynamic data themselves, see above. A particularlyinsightful example is given by Yang et al. (2007): for anumber of protein structures which were both solved bysolution NMR and X-ray crystallography, informationabout the equilibrium dynamics was drawn by means ofcomputational predictions. For that, the so-called theoret-ical RMSD (considered being a measure of flexibility)was calculated by GNM from both NMR and X-raystructural data. Additionally, experimental dynamic datawere compiled from X-ray B-factors and from RMSD ofthe Cα coordinates as determined by NMR. The mainresults are: (i) the correlation between the two sets ofexperimental data (NMR vs. X-ray) is poor and (ii) theo-retical values derived from the NMR data correlate rea-sonably well with the corresponding experimental ones,while the same comparison for the X-ray case yieldsonly poor correlation. This lead the authors to concludethat the protein undergoes different types of motion

(dynamics) in the two different environments (crystaland solution). In particular it was concluded that X-raystructures contain no significant contribution from large-scale motions, while NMR models do.

The approaches discussed so far provide indirectdynamic data, i.e. they conclude about molecular flexibil-ity from optimization routines applied to structural data.In contrast, there are experimental approaches which pro-vide true dynamic data, namely correlation times ofmotion as well as amplitudes of motion. Among them,NMR relaxation and residual-dipolar coupling (RDC)experiments are the most prominent ones, for they deli-ver the kinetic parameters with a residue-based resolutionand thus permit a very detailed discussion of the dynam-ics of the molecules. There is a wealth of investigationof protein dynamics by solution-state NMR methods,featuring relaxation time experiments (Korzhnev, Billeter,Arseniev, & Orekhov, 2001; Palmer, 2001; Palmer, Kro-enke, & Loria 2001), RDCs (Tolman & Ruan, 2006),Exchange NMR (Ernst, Bodenhausen, & Wokaun, 1987;Schmidt-Rohr & Spiess, 1994) and real-time NMR (VanNuland, Forge, Balbach, & Dobson, 1998). With theexception of the latter, all these approaches yield infor-mation about type-B processes, i.e. the equilibriumdynamics. In particular for the comparison with X-raydata, it is also important to realize that they exploitNMR interactions which depend on the orientation ofthe molecules (or chemical groups) with respect to anexternal frame of reference. These NMR data thus con-tain information about reorientational dynamics only, incontrast to translational displacements from the abovediscussed approaches.

An aspect of molecular dynamics which is oftenwidely neglected is that a dynamic process, i.e. the rota-tion or the displacement of an atom or part of a molecule,is characterized not only by amplitude (reorientationangle or mean-square displacement), but also by a timeconstant, henceforth referred to as the correlation time ofmotion. This neglect of the dynamics timescale mightcome about by: (i) the fact that the time window of agiven experimental method is equally sensitive to all rele-vant modes (from picoseconds to seconds, possiblyincluding even static disorder) and can thus not distin-guish the different time scales or from (ii) the oppositefact that a given method is blind to all modes but those ina rather narrow dynamic window and thus, it can onlyaddress the presence or absence of such processes. X-raycrystallography falls in the first category, while most ofthe NMR experiments, which mainly yields spin relaxa-tion rates and are thus sensitive to the nanosecond-rangeonly (with extensions into the micro-second range byrelaxation-rate dispersion experiments) is an example forthe latter. Note that, in order to avoid the common confu-sion between spin relaxation processes on the one hand(whose time scale is set by time-dependent quantum-

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mechanical changes in the spin system, its experimentalparameter being the NMR relaxation time and its figurebeing always in the range of about 0.01ms to seconds)and the actual molecular dynamics (characterized by acorrelation time of motion and spanning about 15 ordersof magnitude in time) on the other hand, we refer to theformer as spin relaxation rates (the inverse of the NMRrelaxation time, symbol R), and to the latter as time con-stants or correlation times.

Dynamic processes in proteins happen on widelydifferent timescales, ranging from ns to ms and secondranges (Krushelnitsky & Reichert, 2005). The latter –so-called slow motions – match the dynamic range ofmany biologically important events and thus might be con-sidered particularly important for biological function.Luckily, the accessible dynamic range of dynamic NMRspans from the ps-ns (R1 relaxation) via μs (R1ρ relaxation)to ms-s (R2 dispersion in liquids and exchange NMR)ranges. The amplitude of the motion is extracted fromrelaxation-time experiments and dipolar couplingscommonly as the so-called dynamic order parameter(Lipari & Szabo, 1982a, 1982b) which is sometimesviewed as the reorientational counterpart of the B-factor(Li & Bruschweiler, 2009). However, it should be men-tioned that the classical relaxation-time experiments insolution-state NMR do not provide information aboutdynamics slower than the isotropic tumbling of the proteinmolecule, thus restricting the accessible range to ns.Except the R1ρ/R2-dispersion experiments which dependon the existence of an appreciable difference in chemicalshift of the exchanging sites, solid-state NMR provides asolution to this issue. Very convincing examples weregiven by Cole and Torchia (1991) and Yang, Tasayco, andPolenova (2009) who compared solution and solid-stateR1 values of a protein. The dispersion in the solid statewas by far larger than in solution, indicating the dominantinfluence of the overall tumbling on the relaxation data(and by thus the cutoff of slower modes) in the solutionstate experiments.

Solid-state NMR was challenged until recently by themuch poorer spectral resolution. Advances in samplepreparation, isotope labeling, and line-narrowing tech-niques eventually led to the solution of the first proteinstructure by solid-state NMR in 2002 (Castellani et al.,2002). More recent developments paved the way for fur-ther significant improvements of the site-specific spectralresolution in solid-state protein spectra by unconven-tional isotope labeling schemes such as proton depletionvia deuteration (Chevelkov, Rehbein, Diehl, & Reif,2006). Thus, only now, i.e. two decades after the firstinsight that dynamic data from solid-state NMR providebetter-defined information than liquid state (Cole & Tor-chia, 1991), it is possible to obtain true dynamic datawith a residue-based resolution over the entire range ofbiologically relevant time scales.

We like to point out that that in addition to the nature ofB-factors delivering indirect dynamic data, there is a num-ber of problems associated with B-factors as dynamicalprobes: it is also long known that the B-factor does not onlyoriginate from fluctuations of the atoms, but also reveals thestatic disorder of the crystals. It is thus not easy to disentan-gle the contributions in data sets of finite quality, however,recent refinement methods seems to be able to deal to a cer-tain degree with this problem and with the separation ofwhole body vs. internal motions. One possibility to distin-guish the overall motion from intramolecuar dynamics inX-ray data is the TLS model, which interprets the B-factorsby combining overall anisotropic rotational, translational,and screw motions (Kuriyan & Weis, 1991) However, thismodel employs a larger number of adjustable parameters(on the order of 10) and it was found that this model is to asignificant degree susceptible to overfitting (Li &Bruschweiler, 2009). Last but not least, biology requiresdynamic information of the protein being in its native envi-ronment, which a crystal clearly is not. Thus, the compari-son of dynamic data obtained from different methods likeX-ray B-factors and solution-state NMR is problematic dueto the vastly different physical states of the molecules: theprotein is in a native-like environment in solution-stateNMR experiments, while the crystal form is far from this.At best, the X-ray data might be effected by crystal-contacteffects (Li & Bruschweiler, 2009), at worst, the structures insolution and crystal are different. Thus, different dynamicdegrees of freedom might be accessible to the differentmethods (Liu, Koharudin, Gronenborn, & Bahar, 2009).This on one hand provides complementary dynamic infor-mation, on the other hand, it makes the comparison of data(as it is frequently done) difficult.

Despite the issues discussed above, comparisonsbetween B-factors and dynamic data obtained by NMRspectroscopy, are frequently discussed (Charpentieret al., 2010; Chevelkov et al., 2010; Clore & Schwiet-ers, 2006; Liu et al., 2009; Yang et al., 2007; Yang,Tasayco et al., 2009), and in References 12–21, 28from Clore and Schwieters (2006). In this contribution,we would like to discuss the question: How does truedynamic data describing molecular reorientations (asobtained by dynamic NMR) correlate with indirectdynamic data as displayed in X-ray B-factors? We col-lected site-resolved dynamic solid-state NMR data forfive different proteins available to us and correlatethem with the known X-ray B-factors. We discuss theapplicability ranges and limitations of the twoapproaches and demonstrate that apart from rather triv-ial situations, comprising mobile chain termini andloop regions, there is no direct a priori relationshipbetween molecular flexibility for which the B-factorsare taken as a measure, and the true local reorienta-tional dynamic degrees of freedom as obtained fromdynamic solid-state NMR.

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Methodological background

The principles of X-ray crystallography and dynamicNMR are described in great detail in a number of texts, forexample Drenth (1994), Ernst et al. (1987), Schmidt-Rohrand Spiess (1994). Here, we will just highlight some issueswhich are of interest for the purpose of the present paper.

X-ray scattering B-factors

One needs to recall that B-factors are derived from ran-dom displacements of specific atoms from their equilib-rium position on the protein crystal lattice. They thusrepresent and quantify in a snapshot-like fashion aninstantaneous structural disorder that can of courseresult from a dynamic process that occurs independentlyin the different protein replica on the lattice. From that itis thus not surprising that there are in fact serious objec-tions to the assumption that B-factors provide reliabledynamic information, and a number of papers in factquestion the use of experimental B-factors for this pur-pose. In particular, it was shown by Halle (2002) thatthe B-factor is essentially determined by spatial inhomo-geneities of the local packing density (the number ofnon-covalently bonded neighbour molecules), and that itprovides little information beyond that contained in themean atomic coordinates already. Hirose et al. (2010)used Normal-Mode Analyses to study both internal (thedisplacement of segments of the molecules) and externalmotions (rotational and translational motions of theprotein as a rigid body) and found that the B-factors donot relate to the internal motions to a large extent, butcorrelate most strongly with the rigid-body motions.Soheilifard, Makarov, and Rodin (2008) state, too, thatthe B-factors are dominated by rigid-body motions andare therefore poorly suited for extracting intrinsic proper-ties of the protein. They also speculate that the coinci-dence of the B-factors with results from elastic networkmodels (see ref from Soheilifard et al., 2008) might notmean that the B-factors provide true experimental datafor the protein dynamics, but that both approachessimply reflect more fundamental collective properties, asalready suggested by Halle (2002): a correlation betweentwo data sets does not automatically mean there is acausal connection between them. On the other hand, nocorrelation (as we will see below) of course does meanthe data are not related to each other at all.

Solid-state NMR

The most serious limitation of liquid-state NMR arisesfrom the fast and isotropic overall tumbling of themolecules (which in turn is necessary to achieve spectralresolution). It is hardly possible to extract information onprocesses slower than the molecular tumbling (ns-μs,depending on the size of the protein and the viscosity ofthe solvent), and internal modes on the timescale of the

tumbling are obscured. In solid-state NMR, however, thisrestriction does not apply and thus, even the relaxation rateR1 contains valuable information on a variety of possibleprocesses (Cole & Torchia, 1991). Solid-state NMR offersa wide variety of rather direct measures of proteindynamics over the whole relevant dynamic range withsite-specific resolution, combining different relaxationrates (R1, R1ρ), dipole–dipole coupling constants, line-shape analysis and recent exchange-NMR approaches.The basis of all these observables is the fact that the differ-ent NMR interactions, which mostly act as first-orderperturbations to the dominating nuclear Zeeman effect, aredependent on the orientation of a spin system with respectto the magnetic field. Any reorientational dynamic processalters the orientation of the molecule and thus leads to ameasurable alteration of the NMR interactions. The mostrelevant interactions for proteins are the chemical shiftanisotropy and the well-defined dipole–dipole couplingbetween directly bonded nuclei. In particular the dipolarcoupling between 15N and 1H of the amide group and 13Cand 1H of Cα and the side chains carbons are valuableprobes of dynamics due to their local character and thewell-known geometry. Depending on the actual correlationtime of motion and the amplitude of local orientationfluctuations, different NMR observables are affected todifferent extents. The relaxation rates R1 and R1ρ as wellas dipolar coupling probe ps to ms dynamics, whileexchange NMR covers the ms to s range (Ernst et al.,1987; Schmidt-Rohr & Spiess, 1994). Recently, suchexperimental data became available for solid proteins forthe first time (Chevelkov, Diehl, & Reif, 2008; Chevelkov,Fink, & Reif, 2009; Giraud et al., 2004; Krushelnitskyet al., 2009; Krushelnitsky, Zinkevich, Reichert, Chevel-kov, & Reif, 2010; Lewandowski et al., 2010; Schanda,Meier, & Ernst, 2010).

Initially, the starting point of our work was to com-pare the B-factors of the SH3 domain of α-spectrin withexperimental dynamic data obtained from a micro-crystalline sample of the same protein by solid-stateNMR for different dynamic ranges. We were interestedto see if the B-factors correlate with the NMR data andif this correlation is selective with respect to the dynamicrange. The residue-resolved results, shown in Figure 1,superficially reveal some correlation of 15N R1 and R1ρ

data (sensitive to ps-μs rotations of the N–H bond) withthe B-factor. This result prompted us to extend and refinethis correlation for a number of other proteins, thoughstill only limited NMR data are available to date.

Results and discussion

Figure 1 displays experimental B-factors as well asdynamic data for the SH3 domain of α-spectrin. Thoughthe preparation procedure was different for the NMR andthe X-ray samples (but the buffer conditions were of

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course identical), the obtained crystal structures matchvery well and thus, we can faithfully consider the sam-ples being equivalent (Castellani et al., 2002) The NMRdata are represented as the relaxation rates R1

(Chevelkov et al., 2008), R1ρ (Krushelnitsky et al., 2010)and 15N–1H dipolar coupling (Chevelkov et al., 2009) ofthe backbone nitrogens, which all depend on the ampli-tude of motion as well as on the correlation time ofmotion. The relaxation rates depend in a complex man-ner on the characteristic frequency of the NMR experi-ment (see below) as well as on different kineticparameters, namely the correlation time of motion andthe topology and amplitude of motion (topology meanswhether it is a diffusive or jump-like motion and for thelatter the number of sites) (Schmidt-Rohr & Spiess,1994). Both amplitude and time-scale parameters of themolecular motion are encoded in the correlation functionof motion, which determines the NMR relaxation rate. Itis thus a priori not possible to extract all relevant kineticparameters from a finite set of experimental data, thoughthe so-called model-free approach deals with this issue

on a practical level (Lipari & Szabo, 1982a, 1982b). Forthe sake of the discussion below, is sufficient to recallthat a large value of the NMR relaxation rates meansthat the corresponding residue undergoes a molecularreorientation of significant amplitude with a correlationtime of motion that is approximately equal (on the loga-rithmic time scale) to the inverse of either the Larmorfrequency (a few hundred MHz for the case of R1) orthe inverse of the nutation frequency of the so-calledspin-lock field (10–100 kHz for R1ρ). This yields charac-teristic timescales for the two experiments of nanosec-onds and microseconds, respectively. Alternativelyphrased, residues featuring a larger-than-average relaxa-tion rate exhibit substantial local flexibility/mobility withthe correlation time of motion on these timescales. Amore accurate quantitative determination of the correla-tion time of motion is basically possible; however, itdepends much on the details of the reorientational topol-ogy, for example the amplitude of motion, as well as ondetails of the NMR interactions which drive the relaxa-tion. For our purpose, increased values of R1 and R1ρ

indicate those parts of the protein backbone which fea-ture reorientational dynamic degrees of freedom in thenanosecond and microsecond range, respectively.Another dynamic probe is the heteronuclear dipolar cou-pling between backbone nitrogens and the directlybonded amide protons. The ratio of the actual value ofthe dipolar coupling to its value for a completely rigidmolecule (about 11.65 kHz) is directly related to theorder parameter, S as derived from relaxation experi-ments in the solution state (Lipari & Szabo, 1982a,1982b). It contains exclusively information about theamplitude of motion and thus, it is a complementaryinformation to the R1 data (which in contrast depend onboth the amplitude and the correlation time of motion ofthe processes happening on the ns time scale). The infor-mation which can be drawn from a value of S< 1 is (i)the molecular motion is “fast”, i.e. the correlation timeof motion of these residues is in the range of about100ns or faster and (ii) it provides a qualitative measureabout the amplitude of this fast motion: the closer S is to1, the smaller is the amplitude of motion while S = 0means a completely isotropic reorientation.

The upper four charts of Figure 1 exhibit dynamicsite-resolved X-ray B-factors as well as NMR parameters(R1, R1ρ, dipolar coupling). It appears that the residues46–50 as well as the chain termini exhibit increased B-factors in concert with increased NMR-relaxation ratesand reduced dipolar coupling, while all other residuesshow large dipolar coupling, low relaxation rates and B-factors. These results alone superficially suggest a certaincorrelation between the X-ray data and NMR relaxation/dipolar coupling. However, we notice that residues 45–51 form a loop connecting two beta sheets. Such parts ofthe backbone are commonly less well structured and

Residue Number

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Figure 1. Experimental B-factors obtained at T= 100 k(circles, pdb code: 2NUZ) and T= 293K (crossed symbols, pdbcode: 1U06) as well as dynamic data for the SH3 domain of α-spectrin. For details, see text.

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rather mobile. Thus, the increased relaxation rates andB-factors are trivial from the simple fact that in theseregions, the space available for the atoms is larger ascompared to atoms residing in secondary-structureelements which are characterized by tight bonding. Thesame argument holds for the chain termini and goesalong well with the arguments of Halle (2002) that theB-factors are basically a measure of the local packingdensity and not primarily of the backbone dynamics.

Leaving these very few residues representing thesetrivial cases aside, we may now focus on the well-defined elements of the secondary structure (α-helices and β-sheets). If we have a look at Figure 1, onemay conclude that there is not much dispersion of therelaxation rates for the different residues. However, wehave to remind ourselves that the relaxation rate is avalue to be considered on the logarithmic scale, ratherthan on the linear one. In other words, when one com-pares relaxation rates for different residues, one has tocompare relative differences rather than absolute ones.Corresponding semi-logarithmic plots of the data in Fig-ure 1 are shown in Figure 2, and indeed, these plotsreveal a large heterogeneity of the relaxation rates (ofabout one order of magnitude) over the entire backbone,not just for flexible loops and termini. And furthermore,this inhomogeneity is not at all reflected in the B-factors,as evidenced by the correlation plots (Figure 3) in whichthe relaxation parameters are again plotted logarithmi-cally (note the large range of values for the relaxationparameters, as compared to the rather narrow one for the

B-factors). These plots correlate the B-factor values witheither one of the NMR parameters for each residue. Thecorrelation coefficients R are the result of a linear regres-sion which – in the case of the relaxation rates – wereconsidered logarithmically. Excluding the trivial cases ofthe terminals and residues 44–51 (shown as open sym-bols), it is obvious that there is no correlation at allbetween the different NMR data (representing differenttime scales) and the X-ray B-factors for the well-struc-tured parts structures within the protein backbone. Forexample, correlating the dipolar coupling with the B-factors yields for all residues an R of �0.49 only whileit drops to an even smaller value of �0.16 when the triv-ial cases (open symbols in Figure 2) were excluded. Thisfinding is true for all the correlation plots. We like tonote that even the largest value of 0.59 means a bad cor-relation between the two compared values. Anotherexperimental evidence that enhanced mobility and B-fac-tors frequently occur in “non-regular” secondary struc-ture elements was recently found by Habenstein et al.(2011). Here, the reduced signal intensity of resonancesbelonging to those residues in multi-dimensional NMRexperiments was taken as evidence for dynamic degreesof freedom on the microsecond time scale, leading to areduced efficiency of the polarization transfer steps insolid-state NMR. However, the overall correlationbetween MR parameters and X-ray B-factors is as pooras in our example Figure 3.

After this initial comparison between true dynamicdata obtained from NMR and X-ray B-factor, we nowwonder if this comparison holds for the different biologi-cally relevant time scales. As said, NMR can distinguishdifferent dynamic regimes: so far (R1 and R1ρ experi-ments) delivered information about fast and intermediatemotions in the ns and μs range, respectively. As for slowmotions occurring in the millisecond range, we appliedthe so-called exchange NMR experiments, which are anexceptional case in that the correlation time of motioncan be read off directly from the experimental exchangedecay of each residue, corresponding to a site-resolvedreal-time measurement of the motional correlation func-tion (Krushelnitsky et al., 2009). The exchange data plot-ted in Figure 1 represent the normalized plateau valuesof the correlation functions (normalized exchange NMRintensity decay curves as a function of mixing time): ifthis figure is larger than 0, molecular reorientations witha correlation time of motion in the range of some milli-seconds to some seconds have occurred. The actualvalue of this figure also depends on the details of thetopology of motion and will not be discussed here anyfurther. Importantly, these results now exhibit again NOcorrelation with the B-factor data. With the given data,one thus is tempted to conclude that the dynamic pro-cesses that occur in those regions of the protein withincreased B-factors happen on the μs-ns timescale only.

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Figure 2. Logarithmic representation of R1 and R1r relaxationrates for the SH3 domain of α-spectrin, taken from Figure 1.Open symbols are those residues featuring a B-factor largerthan 15Å2.

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However, as mentioned, considering the instantaneouscharacter of X-ray diffraction experiments and the originof B-factors, the slow processes (as detectable byexchange NMR) should be reflected in the B-factors aswell, as such processes also naturally randomize atomicpositions. This finding thus is again a strong hint that theX-ray B-factor is not a genuine measure of the true localmobility of the protein backbone.

One might argue that the reason for the observednon-correlation might be due to the fact that the X-rayB-factors are for the present case governed to a signifi-cant degree accidentally by structural disorder and only

to a minor degree by the dynamic behavior (Sternberget al., 1979). The contribution of structural disorder canfor example be evidenced by X-ray experiments per-formed at different temperatures: if we assume that theamplitude of motion does increase with a slight increasein temperature but the structural disorder does not, thenthe B-factor extracted from the data taken at higher tem-peratures should increase as compared to those deter-mined from the lower-temperature data if they aremainly determined by dynamic effects. We refer to X-raydata taken at T= 100K (Chevelkov et al., 2005) andT= 293K (Chevelkov et al., 2007), and as shown inFigure 1, there is the general tendency towards larger B-factors at a higher temperature, indicating that the contri-bution of static disorder to the X-ray data is minor. Thisconfirms that the SH3 domain of α-spectrin is indeed awell-suited sample to demonstrate the correlation (oractually the lack of correlation) between internal proteindynamics and X-ray B-factors.

Apart from our findings (and previously publishedresults summarized in the introduction) that do not reveala strong correlation between dynamic data obtained fromX-ray and NMR, respectively, we were wondering aboutthe possible reasons for the differences between“dynamic” X-ray and NMR data. The possible dynamiccontribution to the X-ray B-factor as a technique whichdelivers a “snapshot picture” is about differences in theatomic coordinates, averaged over the duration of theexperiment, i.e. about translations of the scatteringatoms. The dynamic NMR data presented here, however,are governed by reorientations only. Considering themost efficient relaxation mechanism of the NMR experi-ment used here, the modulation of the heteronucleardipolar coupling between a given nitrogen atom and theattached amide proton by rotations of the N–H-bond vec-tor, it is easy to imagine a dynamic process that doeschange the orientation of this bond but does not dislocatethe nitrogen atom as the dominant scatterer to a largeextent. Actually, a simple reorientation of the bond vec-tor around the nitrogen, which is the basic molecularstep of a conformational transition, is the easiest and bestexample. Thus, a potential non-correlation among thedifferent NMR experiments is likely based on differenttimescales and amplitudes of motion while a non-correlation of NMR vs. X-ray B-factors is likely due tothe more fundamental differences, for example to theirdifferent sensitivity to reorientation vs. translation,respectively.

Based on the experimental evidence presented so farand the reasoning given above it appears difficult to fur-ther justify the general claim that X-ray B-factors pro-vide a comprehensive picture on biologically relevantdynamic events in the relevant parts of the protein mole-cule. In order to further support our line of argument, wecompiled in Figures 3 and 4 correlation plots of X-ray

1-I /Iend start

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Figure 3. Plots correlating the NMR data with the B-factors.Open symbols correspond to the open symbols in Figure 2 andindicate residues with increased B-factors. It is obvious that thereis no correlation at all between the NMR data and the B-factors.Correlation coefficients between B-factors and NMR observablesare (including all residues/including all residues except mobileones): B-factor vs. dipolar coupling: �0.49/�0.16, B-factor vs.exchange data: 0.05/�0.30, B-factor vs. R1ρ: 0.59/0.15. B-factorvs. R1: 0.56/�0.04. They were obtained from linear fits whichare shown as examples in the upper two charts.

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rebmuNeudiseRrebmuNeudiseR

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Figure 4. (a) Dependence of dynamic NMR data and B-factors on the secondary structure as well as plots correlating both data forCrh (top left, pdb code: 1MU4) (Giraud et al., 2004), GB1 (top right, pdb code: 2GI9) (Lewandowski et al., 2010), Ubiquitin (middlepanels, pdb code: 1UBQ) (Schanda et al., 2010) and thioredoxin (lower panels, pdb code: 2TRX) (Yang, Tasayco et al., 2009). Opensymbols are those with elevated B-factors; no NMR data are available for residues marked by crossed symbols. (b) Correlation plotscorrelating the B-factors and NMR parameters as shown in (a). The Correlation coefficients between B-factors and NMR observablesinclude all residues except mobile ones (open symbols in (a)). The values are 0.35 (Crh), 0.05 (GB1), �0.44 (Ubiquitin, dipolarcouplings), 0.48 (Ubiquitin, R1), �0.16 (thioredoxin, dipolar couplings and 0.54 (thioredoxin, R1).

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B-factors vs. dynamic solid-state NMR data of differentproteins which are available to date. This includes 15NR1 data for Crh in its domain-swapped dimeric form(2 × 10.4 kDa) (Giraud et al., 2004), 15N–1H dipolar cou-plings and 15N R1 data (at 19/96T) for ubiquitin(Schanda et al., 2010), 13C R1 data for GB1(Lewandowski et al., 2010) and 15N–1H dipolar cou-plings and 15N R1 data for thioredoxin (Yang, Tasayco,et al., 2009). As before, the open symbols represent datapoints belonging to residues featuring the largest B-fac-tors. As seen from the insert sketches of the secondarystructures, these residues all belong to connection regionsbetween β-sheets and obviously, these parts are particu-larly unstructured/flexible. It again supports the assump-tion that both NMR and X-ray data are sensitive to theavailable configurational space but do not form a causal

connection for the structured domains. Apart from thesefew residues, there is no visible correlation between X-ray B-factors and NMR relaxation data. We like to notethat the case of GB1 is special insofar as there is no cor-relation at all between NMR data and B-factor, evenwithout omitting the trivial cases originating from highlymobile and unstructured regions. We believe the reasonto be that GB1 is a densely packed protein (B-factorshardly reach values of 15Å2 while for the other proteins,they easily exceed values of 20Å2 and more), the B-fac-tors profile shows no abnormally increased values andthus, it is an example of a protein without distinctly flex-ible domains. The on-average larger relaxation rates (ascompared to the other examples) might be explained bythe fact that the absence of fast processes moves therelaxation rates in the well-known correlation of R1 vs.

R1, s-1 R1, s-1

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correlation time of motion (which exhibits a maximumwhen the correlation time of motion is on the order ofthe inverse Larmor frequency) closer towards the R1

maximum (Slichter, 1996).Last but not least we would like to draw the attention

to another insightful example. In Yang, Tasayco et al.(2009), the authors conducted a number of solid-stateexperiments which are sensitive to dynamic processes ondifferent time scales. Apart from the results which sup-port ours and others’ findings, like the contribution ofstatic disorder and whole-body motions to the B-factor,increased B-factors and NMR-relaxation rates in “non-regular” secondary structure elements, such as termini,loops and turns, the authors found no correlationbetween X-ray B-factor and S as well as R1. Further-more, there is no correlation between these two dynamicNMR parameters (as measures for fast motions) and theline intensity in 2D-MAS spectra which serves as a mea-sure for intermediate motions. Thus, these results are inperfect agreement with our findings.

Conclusion

Based on recently published dynamic NMR data of solidproteins, we analyzed, on a site-resolved basis, the corre-lation between the internal molecular dynamics of theseproteins and the corresponding X-ray B-factors, whichare believed to reflect dynamic information, too. Ourresults show that apart from residues in regionsconnecting adjacent rigid structural domains (α-helices orβ-sheets) and those close to the terminals, there is nocorrelation between the B-factors and the moleculardynamics of the protein backbone on the different timescales accessible to modern dynamic solid-state NMRexperiments. Therefore, the X-ray B-factor cannot beconsidered as a comprehensive probe of moleculardynamics, as it is affected by both whole-body as wellas internal motions, as it integrates over a wide dynamicrange and does not distinguish between different rangesof correlation time of motion, as it is not sensitive toreorientational dynamic degrees of freedom of the heavyatoms and on top of all, it might feature a substantialcontribution of static packing effects. We thus consider itproblematic to conclude on the dynamic signature of aprotein exclusively from X-ray crystallography, eventhough the just mentioned drawbacks could be resolvedfor specific cases. NMR, in particular solid-state NMR,provides more and clearer insight into the dynamicdegrees of freedom of a protein. In a way, X-ray andNMR can complement each other (e.g. by separatelyrevealing information about translational and reorienta-tional processes), and only when taken together, theycould provide a comprehensive picture about the trueflexibility, and can open ways for a better understandingof biological events (Brunger, 1997).

Acknowledgment

The authors thank H. Kogler (Sanofi-Aventis, Frankfurt/Germany) for raising our curiosity on this issue and B. Reif(Munich) and M.T. Stubbs (Halle) for help and discussion. Thefinancial support from the Deutsche ForschungsgemeinschaftDFG, Grant RE 1025/16 and Collaborative Research CenterSFB/TRR 102 (project A8) is gratefully acknowledged.

ReferencesBrunger, A.T. (1997). X-ray crystallography and NMR reveal

complementary views of structure and dynamics. NatureStructural Biology, 4, 862–865.

Castellani, F., van Rossum, B., Diehl, A., Schubert, M.,Rehbein, K., & Oschkinat, H. (2002). Structure of a proteindetermined by solid-state magic-angle-spinning NMR spec-troscopy. Nature, 420, 98–102.

Charpentier, T.H., Thompson, L.E., Liriano, M.A., Varney, K.M., Wilder, P.T., Pozharski, E., … Weber, D.J. (2010). Theeffects of CapZ peptide (TRTK-12) binding to S100B-Ca2+

as examined by NMR and X-ray crystallography. Journalof Molecular Biology, 396, 1227–1243.

Chevelkov, V., Diehl, A. & Reif, B. (2008). Measurement of15N-T1 relaxation rates in a perdeuterated protein by magicangle spinning solid-state nuclear magnetic resonance spec-troscopy. Journal of Chemical Physics, 128, 052316

Chevelkov, V., Faelber, K., Diehl, A., Heinemann, U.,Oschkinat, H., & Reif, B. (2005). Detection of dynamicwater molecules in a microcrystalline sample of the SH3domain of a-spectrin by MAS solid-state NMR. Journal ofBiomolecular NMR, 31, 295–310.

Chevelkov, V., Faelber, K., Schrey, A., Rehbein, K., Diehl, A.,& Reif, B. (2007). Differential line broadening in MASsolid-state NMR due to dynamic interference. Journal ofthe American Chemical Society, 129, 10195–10200.

Chevelkov, V., Fink, U., & Reif, B. (2009). Accurate determi-nation of order parameters from 1H, 15N dipolar couplingsin MAS solid-state NMR experiments. Journal of theAmerican Chemical Society, 131, 14018–14022.

Chevelkov, V., Rehbein, K., Diehl, A., & Reif, B. (2006).Ultrahigh resolution in proton solid-state NMR spectros-copy at high levels of deuteration. Angewandte Chemie-International Edition, 45, 3878–3881.

Chevelkov, V., Xue, Y., Linser, R., Skrynnikov, N. R. & Reif,B. (2010). Comparison of solid-state dipolar couplings andsolution relaxation data provides insight into protein back-bone dynamics. Journal of the American Chemical Society,132, 5015.

Clore, G.M., & Schwieters, C.D. (2006). Concordance ofresidual dipolar couplings, backbone order parametersand crystallographic B-factors for a small alpha/beta pro-tein: A unified picture of high probability, fast atomicmotions in proteins. Journal of Molecular Biology, 355,879–886.

Cole, H.B.R., & Torchia, D.A. (1991). An NMR-study of thebackbone dynamics of staphylococcal nuclease in the crys-tallien state. Chemical Physics, 158, 271–281.

Drenth, J. (1994). Principles of protein X-ray crystallography.New York, NY: Springer.

Erman, B. (2006). The Gaussian network model: Precise pre-diction of residue fluctuations and application to bindingproblems. Biophysical Journal, 91, 3589–3599.

626 D. Reichert et al.

Dow

nloa

ded

by [

Duk

e U

nive

rsity

Lib

rari

es]

at 0

3:16

12

Oct

ober

201

2

Ernst, R.R., Bodenhausen, G., & Wokaun, A. (1987). Princi-ples of magnetic resonance in one and two dimensions.Oxford: Oxford University Press.

Giraud, N., Bockmann, A., Lesage, A., Penin, F., Blackledge, M.,& Emsley, L. (2004). Site-specific backbone dynamics from acrystalline protein by solid-state NMR spectroscopy. Journalof the American Chemical Society, 126, 11422–11423.

Habenstein, B., Wasmer, C., Bousset, L., Sourigues, Y., Schutz,A., Loquet, A., … Bockmann, A. (2011). Extensive denovo solid-state NMR assignments of the 33 kDa C-terminal domain of the Ure2 prion. Journal of Biomolecu-lar NMR, 51, 235–243.

Haliloglu, T., Bahar, I., & Erman, B. (1997). Gaussian dynamicsof folded proteins. Physical Review Letters, 79, 3090–3093.

Halle, B. (2002). Flexibility and packing in proteins. Proceed-ings of the National Academy of Sciences of the UnitedStates of America, 99, 1274–1279.

Hirose, S., Yokota, K., Kuroda, Y., Wako, H., Endo, S., Kanai,S. & Noguchi, T. (2010). Prediction of protein motionsfrom amino acid sequence and its application to protein–protein interaction. BMC Structural Biology, 10: 20.Published online 13 July 2010.

Korzhnev, D.M., Billeter, M., Arseniev, A.S., & Orekhov, V.Y.(2001). NMR studies of Brownian tumbling and internalmotions in proteins. Progress in Nuclear Magnetic Reso-nance Spectroscopy, 38, 197–266.

Krushelnitsky, A., deAzevedo, E., Linser, R., Reif, B.,Saalwachter, K. & Reichert, D. (2009). Direct observationof millisecond to second motions in proteins by dipolarCODEX NMR Spectroscopy. Journal of the AmericanChemical Society, 131, 12097.

Krushelnitsky, A., & Reichert, D. (2005). Solid-state NMR andprotein dynamics. Progress in Nuclear Magnetic ResonanceSpectroscopy, 47, 1–25.

Krushelnitsky, A., Zinkevich, T., Reichert, D., Chevelkov, V.,& Reif, B. (2010). Microsecond time scale mobility in asolid protein as studied by the 15N R1ρ site-specific NMRrelaxation rates. Journal of the American Chemical Society,132, 11850–11853.

Kuriyan, J., & Weis, W.I. (1991). Rigid protein motion as amodel for crystallographic temperature factors. Proceedingsof the National Academy of Sciences of the United Statesof America, 88, 2773–2777.

Lewandowski, J. R., Sein, J., Sass, H. J., Grzesiek, S.,Blackledge, M. & Emsley, L. (2010). Measurement ofsite-specific 13C spin-lattice relaxation in a crystallineprotein. Journal of the American Chemical Society 132,8252.

Li, D.W., & Bruschweiler, R. (2009). All-atom contact modelfor understanding protein dynamics from crystallographicB-factors. Biophysical Journal, 96, 3074–3081.

Lipari, G., & Szabo, A. (1982a). Model-free approach to theinterpretation of nuclear magnetic-resonance relaxation inmacromolecules. 1. Theory and range of validity.Journal of the American Chemical Society, 104, 4546–4559.

Lipari, G., & Szabo, A. (1982b). Model-free approach to theinterpretation of nuclear magnetic-resonance relaxation inmacromolecules. 2. Analysis of experimental results.Journal of the American Chemical Society, 104, 4559–4570.

Liu, L., Koharudin, L.M.I., Gronenborn, A.M., & Bahar, I.(2009). A comparative analysis of the equilibrium dynam-ics of a designed protein inferred from NMR, X-ray, andcomputations. Proteins-Structure Function and Bioinformat-ics, 77, 927–939.

Palmer, A.G. (2001). NMR probes of molecular dynamics:Overview and comparison with other techniques. AnnualReview of Biophysics and Biomolecular Structure, 30, 129–155.

Palmer, A.G., Kroenke, C.D., & Loria, J.P. (2001). Nuclearmagnetic resonance methods for quantifying microsecond-to-millisecond motions in biological macromolecules.Nuclear Magnetic Resonance of Biological Macromole-cules, Part B, 339, 204–238.

Ringe, D., & Petsko, G.A. (1985). Mapping protein dynamicsby X-ray-diffraction. Progress in Biophysics & MolecularBiology, 45, 197–235.

Schanda, P., Meier, B.H., & Ernst, M. (2010). Quantitativeanalysis of protein backbone dynamics in microcrystallineubiquitin by solid-state NMR spectroscopy. Journal of theAmerican Chemical Society, 132, 15957–15967.

Schmidt-Rohr, K. & Spiess, H. W. (1994). MultidimensionalSolid-State NMR and Polymers. London: Academic Press.

Slichter, C.P. (1996). Principles of magnetic resonance. Berlin:Springer.

Soheilifard, R., Makarov, D. E. & Rodin, G. J. (2008). Criticalevaluation of simple network models of protein dynamicsand their comparison with crystallographic B-factors. Physi-cal Biology, 5.

Sternberg, M.J.E., Grace, D.E.P., & Phillips, D.C. (1979).Dynamic information from protein crystallography – analy-sis of temperature factors from refinement of the hen-egg-white Lysozyme structure. Journal of MolecularBiology, 130, 231–252.

Tolman, J.R., & Ruan, K. (2006). NMR residual dipolar cou-plings as probes of biomolecular dynamics. ChemicalReviews, 106, 1720–1736.

Van Nuland, N.A.J., Forge, V., Balbach, J., & Dobson, C.M.(1998). Real-time NMR studies of protein folding.Accounts of Chemical Research, 31, 773–780.

Willis, B.T.M., & Pryor, A.W. (1975). Thermal vibrations incrystallography. London: Cambridge University Press.

Wuthrich, K. (1995). NMR – this other method for protein andnucleic-acid structure determination. Acta Crystallograph-ica Section D-Biological Crystallography, 51, 249–270.

Yang, L.W., Eyal, E., Chennubhotla, C., Jee, J., Gronenborn,A.M., & Bahar, I. (2007). Insights into equilibrium dynam-ics of proteins from comparison of NMR and X-ray datawith computational predictions. Structure, 15, 741–749.

Yang, L., Song, G., & Jernigan, R.L. (2009). Protein elasticnetwork models and the ranges of cooperativity. Proceed-ings of the National Academy of Sciences of the UnitedStates of America, 106, 12347–12352.

Yang, J., Tasayco, M.L., & Polenova, T. (2009). Dynamics ofreassembled thioredoxin studied by magic angle spinningNMR: Snapshots from different time scales. Journal of theAmerican Chemical Society, 131, 13690–13702.

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