What Contributions to Protein Side-chain Dynamics are Probed by NMR Experiments? A Molecular...

doi:10.1016/j.jmb.2005.03.001 J. Mol. Biol. (2005) 349, 185–203

What Contributions to Protein Side-chain Dynamics areProbed by NMR Experiments? A Molecular DynamicsSimulation Analysis

Robert B. Best1, Jane Clarke1 and Martin Karplus2,3*

1MRC Centre for ProteinEngineering, Department ofChemistry, Lensfield RoadCambridge, CB2 1EW, UK

2Laboratoire de ChimieBiophysique, ISIS, UniversiteLouis Pasteur, 4 rue BlaisePascal, 67000 StrasbourgFrance

3Department of Chemistry andBiological Chemistry, HarvardUniversity, 12 Oxford StreetCambridge, MA 02138, USA

0022-2836/$ - see front matter q 2005 E

Present address: R. B. Best, LaborPhysics, NIDDK, National InstituteMD 20892-0520, USAAbbreviations used: MD, molecu

fibronectin type III; rmsd, root-meaE-mail address of the correspond

[email protected]

Molecular dynamics simulations of the structurally homologous proteinsTNfn3 and FNfn10 have been used to investigate the contributions to side-chain dynamics measured by NMR relaxation experiments. The resultsreproduce the variation in core side-chain dynamics observed by NMR andhighlight the relevance of anharmonic motion and transitions betweenlocal minima for explaining NMR side-chain order parameters. A methodis described for calculating converged order parameters by use of replicaexchange molecular dynamics in conjunction with an implicit solventmodel. These simulations allow the influence of various factors, such as theflexibility of side-chains and their free volume, on the mobility to be testedby perturbing the system. Deletion mutations are found to have the largesteffect on the more densely packed FNfn10. Some counterintuitive effectsare seen, such as an increase in order parameters close to deletion mutationsites, but these can be rationalized in terms of direct interactions with themodified side-chains. A statistical analysis of published order parameterssupports the conclusions drawn from the simulations.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: order parameter; replica exchange; fibronectin type III; side-chain dynamics; normal mode analysis
*Corresponding author
Introduction

Despite the fact that the hydrophobic cores ofprotein crystal structures are densely packed,1–3 ithas long been known that even buried residues aremobile.4 For example, early studies using nuclearmagnetic resonance (NMR) experiments andmolecular dynamics (MD) simulations revealedthe 1808 flipping of buried tyrosine residues, inwhich the two conformers are crystallographicallyidentical due to the tyrosine symmetry.5–7 NMRrelaxation experiments for monitoring thedynamics of methyl groups provide another probefor side-chain motion at many different sitesthroughout a protein;8–10 similarly, analysis ofvicinal coupling constants involving methyl groupsdemonstrated the existence of alternative confor-mers in the protein core.11,12 The recent development

lsevier Ltd. All rights reserve

atory of Chemicals of Health, Bethesda,

lar dynamics; fnIII,n-square deviation.ing author:

of methods for probing methyl dynamics using 2Hrelaxation13–15 has greatly facilitated such studies,including applications to understanding the effectsof mutations16–18 and ligand-binding19,20 ondynamics, the dynamics of unfolded proteins,21 thedynamics of homologous proteins22 and even esti-mates of the “residual entropy” in the native state.23

Given the widespread use of methyl side-chaindynamics experiments in investigating biologicalproblems, it is important to understandwhat factorsgovern the dynamics probed by the experiments.The dynamics of side-chains are more complex

than those of the backbone. The variation in theamplitude of backbone dynamics, at least on thepico- to nanosecond time-scale, is often not large,and backbone order parameters can be predictedreasonably accurately on the basis of harmonicanalysis24,25 or simple heuristic contact models26

(long loops are an important exception). Similarmethods have also been used to predict crystal-lographic Debye–Waller factors from structure.27

On the other hand, the prediction of side-chainorder parameters from single protein structures(beyond what is expected for the types of resi-dues concerned) has proved more difficult,28,29

suggesting greater complexity. A reasonable

d.

186 Interpretation of Side-chain Dynamics Experiments

correlation of prediction with experiment (corre-lation coefficient z0.65)29 has been obtained usingheavy-atom contacts and the number of side-chainrotatable bonds as input. However, this correlationis calculated by considering all amino acid typestogether, and much lower correlations are obtainedfor the variations of the order parameter withineach type of residue.28 One aspect of the complexityof the dynamics is anharmonic motion, includingtransitions between local minima as inferred fromexperimental analyses;12,30 such transitions wererecently shown to be important in explaining thedistribution of side-chain order parameters.31 Thesefeatures of side-chain motion invite a moredetailed analysis that combines moleculardynamics simulation results with experimentaldata for appropriate systems.

In a previous experimental study, we comparedthe core dynamics of two structurally homologousdomains.22 The third fibronectin type III domainfrom human tenascin (TNfn3) and the tenthfibronectin type III domain of human fibronectin(FNfn10) are both members of the fibronectin typeIII (fnIII) superfamily (Figure 1): this group ofstructurally related proteins has seven b-strandsarranged in two sheets of three and four strandseach. We showed by means of spin-relaxationexperiments on 2H-labelled methyl groups that,despite their similar fold, there are substantialdifferences in the mobility of core side-chains inthese domains:22 the amplitude of the orientationalfluctuations of side-chain methyl groups, asmeasured by the order parameter for the methylaxis S2axis is unusually high for many buried (core)residues in TNfn3, even relative to the averagevalues expected for the types of residues found inthe core (notable examples are V70, I48, I59 andL72), while the corresponding sites in FNfn10 aremore rigid. Thus, these two proteins provide amodel system for examining the variation in thecore dynamics of structurally similar proteins.

Figure 1. Ribbon diagrams of (a) TNfn3 and (b) FNfn10produced with MOLSCRIPT.87 The location of eachstrand and the FG loop are indicated. These two membersof the fibronectin type III (fnIII) superfamily have only23% sequence identity, but a backbone rmsd of 1.3 A overresidues in secondary structure.

Differences between the order parameters in thetwo domains were analyzed in terms of the residuecomposition of the hydrophobic core and packingdensity; however, the importance of these factorshas yet to be tested and quantified.

Here, the side-chain dynamics of the two proteinsare investigated using molecular dynamics (MD)simulations. Although such simulations are gener-ally limited by requirements of computer time tomotions in the pico- to nanosecond range, this issufficient for comparison with spin-relaxationexperiments,32,33 which characterize motion on acorresponding time-scale. Side-chain 2H relaxationexperiments are particularly well suited for provid-ing the required information because the slowertime-scale (micro- to millisecond) chemicalexchange contribution to relaxation is small due tothe rapid quadrupolar R2 relaxation rate and thesmall 2H chemical shift range.34 There are only afew molecular dynamics studies analyzing side-chain dynamics33,35 (although there are manyconcerned with backbone dynamics36), and onlyone to our knowledge comparing simulations with2H spin relaxation experiments;37 the latterfocussed on the estimation of entropy associatedwith the measured order parameters.

All-atom MD simulations with explicit solventare shown to reproduce both the magnitudes andtime-scales of the observed side-chain motions inthe two proteins with reasonable accuracy. Notably,the unusually large mobility for certain coreresidues found in the experiments is also seen inthe simulations. By separating the harmonic andanharmonic contributions to the motions, we showthat the latter is especially important in describingside-chain motions and in explaining side-chainorder parameters. Since the simulations reproducethe order parameters, we are able to test hypothesesabout which factors are important in determiningthe observed dynamics by exploiting the fact thatthe potential function employed in the simulationsis under our control. We have combined enhancedsampling techniques (which ensure better con-vergence) with perturbations to the system, suchas mutations and constraints on side-chain flexi-bility; this allows us to test explicitly the influence ofthese factors on the dynamics. Finally, we analyze adatabase of published side-chain dynamics data toinvestigate the effect of local packing on orderparameter correlations.

Results

All-atom MD simulations of TNfn3 and FNfn10were run in explicit solvent with periodic boundaryconditions for 3 ns, following an initial 40 psequilibration (see Methods). Figure 2 shows theroot-mean-square deviation (rmsd) of eachprotein from its minimized crystal structure (theequilibration period is omitted). Only heavyatoms in the secondary structure have been usedfor this calculation, as inclusion of loops gives less

Figure 2. Top: rmsd of backbone heavy atoms inb-strands from the minimized crystal structure over thecourse of simulations of (a) TNfn3 and (b) FNfn10.Middle: per-residue backbone rmsd of simulation fromminimized crystal structure for (c) TNfn3 and (d) FNfn10.Bottom: per-residue fluctuations from the average struc-ture for (e) TNfn3 and (f) FNfn10. Backbone fluctuationsare given by the lightly shaded area, and side-chainfluctuations by the heavily shaded area. The rmsfluctuations computed from the crystal structure B-factorsas (3B/8p2)1/2 are shown as broken lines in (e) and (f),offset by 2 A for clarity.

Interpretation of Side-chain Dynamics Experiments 187

meaningful results for FNfn10: the long FG loopdeviates by up to 8 A from the crystal structure, andthere are also large deviations in the CC 0 loop, bothof which may well correspond to flexibility of theseregions in solution. Also, the magnitude of theseloop displacements is in accord with crystallo-graphic results showing differences between thetwo FNfn10 crystal structures 1FNF38 and 1FNA39

of up to 6 A in the FG loop. The overall rmsd of eachprotein from the crystal structure in Figure 2 hasalmost stabilized over the equilibration period.There is a small further increase in the first 50 psof the FNfn10 simulation and a slower increase overthe next 500 ps. It was found that the exclusion ofthis 500 ps period had almost no effect on the results(e.g. calculated order parameters), so the entire 3 ns“production” phase was used for analysis. FNfn10has higher backbone rmsd (1.5 A) than TNfn3(1.0 A) even when only residues in b-strands areconsidered. This is mainly due to the increaseddeviation of residues near the strand termini, whichis related to the loop flexibility; the FG loop is four

residues longer and the CC 0 loop is more glycine-rich in FNfn10 than TNfn3.

Methyl and side-chain rotameric transitions

One of the main sources of anharmonic motion infolded proteins arises from transitions betweendifferent side-chain (and occasionally, backbone)rotamers. This is important because it can make acontribution to the order parameters measured byNMR spin relaxation experiments (as has recentlybeen demonstrated experimentally),30 and becauselocal librations within a minimum and transitionsbetween rotamers have a qualitatively differenteffect on the methyl order parameters.22 This is alsorelevant to protein thermodynamics (via its contri-bution to the configurational entropy of the nativestate (“residual entropy”).40,41 In addition, if thereare transitions, their slow rate relative to motionswithin a rotamer will determine whether thecalculated correlation functions relevant to NMR(see Methods) converge in standard simulations.Transition statistics were computed for rotations

of each methyl group about its axis. To distinguishlarge fluctuations and partial barrier crossings fromtrue transitions, the dihedral angle was averagedover a block of 10 ps, and a transition was said tohave occurred if its next value differed bymore than808 from the running average, where the “nextvalue” is not a running average, but a specific time-point. The residence times (average time betweentransitions) span a wide range, from one methylgroup that makes no transitions (T46 g2 in TNfn3)to one making hundreds (M79 3 in TNfn3) (a listingfor each methyl group is available as supple-mentary data). There is no simple correlation ofresidence time with burial: for example, in TNfn3L62 is deeply buried, but has methyl rotationstatistics similar to L26, which is partially exposed(mean residence times 118 ps and 124 ps, respec-tively). Likewise, in FNfn10, the methyl groups ofthe buried V72 and solvent-exposed V45 havesimilar rotation rates (residence times 95 ps and126 ps). Chatfield et al. found that steric (van derWaals) interactions with neighbouring residueswere most important in determining methylrotation rates,33 and that a Boltzmann weightedaverage of the adiabatic energy barriers calculatedfrom different structures from an MD simulationwas able to predict methyl correlation times quitewell,42 in accord with earlier results.43 Indeed, asimple calculation of barriers from rigid rotationabout the dihedral angle shows that the buriedmethyl groups that are rotating rapidly do not havesignificant barriers to rotation arising from van derWaals interactions. A few examples of time-seriesare shown in Figure 3(a). The gmethyl group of I38makes less frequent dihedral transitions than the dmethyl group. This trend is seen for all isoleucinemethyl groups. T66 and M79 are shown asexamples of the extremes of methyl rotation ratesobserved in different environments.Rotameric transitions are observed for many

Figure 3. (a) Dihedral time-series for rotational motionabout the methyl axis (defined as the preceding carbon–carbon bond) for four representative methyl groupsundergoing motion at different rates from TNfn3.(b) Representative side-chain rotameric transitions (for aselection of side-chains that make transitions in TNfn3).L26 is an exposed Leu that populates the two mostcommon combinations of rotamers observed in crystalstructures; L34 is a buried Leu that makes transitions onlyin c2, while L72 is a buried leucine whose c2 transitionsare correlated with those of L34 c2.

Table 1. Number of side-chain dihedrals undergoing rotame

TNfn3

Dihedral Buried Expos

Val (c1) 1/2 (2) 2/2 (3,Leu (c1) 1/5 (1) 1/3 (2Leu (c2) 4/5 (4,23,15,2) 3/3 (13,Ile (c1) 0/5 2/3 (1,Ile (c2) 4/5 (51,48,59,3) 3/3 (51,7Thr (c1) 0/1 2/11 (2Met (c1) 1/1 (3Met (c2) 1/1 (9Met (c3) 1/1 (5

Buried residues were defined as those whose side-chains had less thafraction of dihedrals in each category that make transitions, and thethose cases.


side-chains. Table 1 briefly summarizes the side-chain transition statistics by residue type andburial. The effect of burial is more marked forthese side-chain dihedrals than for the methylgroups, presumably because of the larger freevolume required to move a side-chain in the core.This is particularly notable for the c1 dihedrals ofthe b-branched Val and Ile residues, which are mostsensitive to a crowded environment. For Val, one offive buried residues (over both proteins) makes c1

transitions, versus five out of ten exposed residues,while for Ile none of the nine buried residues makesc1 transitions, and two of the five surface residuesdo. The statistics for c2 dihedrals of Ile, by contrast,are similar for buried and exposed residues; this islikely related to the smaller free volume required tomove only the d carbon atom when rotating aboutc2 as compared with the more extensive side-chainmotion inherent in rotating about c1. A similareffect can be seen for c1 and c2 in Leu residues,although in this case there are instances of buriedresidues populating multiple c1 rotamers. Thesingle surface Met of TNfn3 makes transitions inall three dihedrals, consistent with the highmobility generally seen for Met residues in NMRrelaxation studies.31 Few transitions are seen forThr. The frequency of transitions in cases wherethey occur is clearly larger for the c2 torsion anglesfor the reasons mentioned above. There does notappear to be a significant correlation betweentransition frequency and burial in the presentresults.

It is known that the two side-chain dihedrals ofLeu are generally correlated (from crystal structuresand energetic calculations).44 In the case of onesurface-exposed Leu (L26 of TNfn3), the transitionsof c1 and c2 were observed to be highly correlated,hopping between the two predominant rotamersseen in the crystallographic database in a simul-taneous “crankshaft”-like motion (Figure 3(b)). Thecorrelation for the other Leu was not as strong,especially for buried residues where such a“crankshaft” transition is precluded by excludedvolume constraints.

ric transitions, classified by residue type and burial

FNFn10

ed Buried Exposed

1) 0/3 3/8 (1,1,1)) 1/3 (1) 0/19,4) 2/3 (3,22) 1/1 (16)1) 0/4 0/2,16) 4/4 (104,2,91,38) 1/2 (38),1) 0/1 1/11 (1)))9)

n 10% solvent-accessible surface area. The bold fractions give thenumbers in parentheses are the number of transitions in each of

Table 2. Contributions to rms fluctuations for variousatom selections

rms fluctuations (A)

Atom selectionNormalmodes

MDsim.

%Anharm. DL

TNfn3All atoms 0.485 1.159 82.5 0.21Backbone atoms 0.418 0.943 80.4 0.17Backbone atomsin b-strands

0.359 0.641 68.6 0.12

Side-chain atoms 0.545 1.348 83.7 0.25Core methylcarbon atoms

0.501 1.140 80.7 0.21

Non-core methylcarbon atoms

0.659 1.536 81.6 0.28

FNfn10All atoms 0.535 1.496 87.2 0.27Backbone atoms 0.471 1.332 87.4 0.24Backbone atomsin b-strands

0.362 0.754 76.9 0.14

Side-chain atoms 0.601 1.664 87.0 0.30Core methylcarbon atoms

0.524 0.978 71.3 0.18

Non-core methylcarbon atoms

0.501 1.441 82.8 0.26

The percentage harmonicity is defined as the ratio of the mean-square fluctuations obtained from a normal mode analysis tothose from the MD simulations; the percentage anharmonicity isthe remaining fraction. DL is the generalized Lindemann meltingcriterion.50


There are not many significant correlationsbetween dihedral transitions in different side-chains. Figure 3(b) shows one example, in whichthere is an apparent correlation between transitionsof L34 c2 with L72 c2 over the short time-windowshown. This is reasonable, since these two residuesare in contact in the core. However, such corre-lations generally do not persist over the wholesimulation, and the correlation between dihedralangles of different side-chains is quite low (!0.3 inmost cases).

Residue V70 is an exceptionally mobile residue inthe TNfn3 core, which was shown to have very loworder parameters in the side-chain dynamicsstudy.22 Although it visits the three c1 rotamers inthe simulation, each is sampled only once, so anaccurate estimate of their relative populationscannot be made. This is a problem in evaluatingthe order parameters from the simulations whichoccurs for all side-chains that make only a smallnumber of transitions in the simulation.

Few of the aromatic side-chains in either proteinmake any rotameric transitions (ring-flips), and theonly transitions observed are for partially exposedresidues (e.g. F23 in TNfn3 or F48 in FNfn10).However, this may be related to the limited lengthof the simulation, as such transitions have beenobserved experimentally and in simulations ofother systems.5,7,45

Harmonic and anharmonic contributions tofluctuations

An approximate description of the dynamics of aprotein within a local energy minimum is obtainedfrom normal mode analysis. In this approach, thepotential energy surface around a minimum isapproximated as a harmonic well, by evaluatingthe second derivatives of the force-field energy atthe minimum. This is computationally convenient,but ignores anharmonic effects within a well andtransitions between different minima, which areknown to be important features of protein dynamicsand play a role in the “glass transition”.46 Since theharmonic approximation has been shown toprovide a reasonable description of backbonedynamics,25,47 it is of interest to probe the validityof the harmonic approximation for side-chains. Togain insight into the relative importance of harmo-nic and anharmonic dynamics, normal modes andfrequencies were calculated from the minimizedcrystal structures of TNfn3 and FNfn10, respect-ively. The same analysis was done on minimizedstructures selected from the MD trajectory, but theresults were similar (data not shown), demonstrat-ing that the modes are generally not sensitive to thedetails of the local minima.48 From these results, it ispossible to calculate the mean-square fluctuationsfor harmonic motion at 298 K (see Methods), thesame temperature as the simulations, and thefraction of the total atomic fluctuations found inthe simulation arising from harmonic motion(within a single minimum). Table 2 details the rms

displacements from the normal modes and MDsimulation for various atom selections, and thefraction of the mean-square fluctuations due toanharmonic motion.The fraction of the harmonic contribution to the

fluctuations is defined as the ratio of the mean-square harmonic fluctuations to the total from themolecular dynamics and the anharmonic contri-bution is equal to the difference. For each type ofatom selection, the harmonic contributions aresimilar for the two proteins, though generallysomewhat higher for FNfn10. However, FNfn10has significantly greater total rms fluctuations in theMD simulation (1.496 compared with 1.159 forTNfn3), resulting in a larger “anharmonic” contri-bution to fluctuations. This trend is seen both for thebackbone and side-chain atoms. An importantdifference, though, is that the anharmonic contri-bution to side-chain atom motion is greater thanthat for the backbone in TNfn3 (as seen before forBPTI);49 in FNfn10, the backbone has a slightlyhigher anharmonic contribution. This is mainly dueto the long FG loop in the latter: exclusion of thisloop (residues 77–87) from the calculation reducesthe anharmonic contribution to the backbonemotion to 85%. In general, the anharmonic contri-bution to loops is greater; restriction to backboneatoms in b-strands reduces the anharmonic contri-butions to 69% and 77% for TNfn3 and FNfn10,respectively.The core atoms tend to have a slightly lower

anharmonic contribution than the rest of theprotein, as would be expected from the reduction


in conformational transitions in the more crowdedcore environment. This effect is clearly seen inFNfn10: the amplitude of fluctuations for buriedmethyl carbon atoms (defined as those in side-chains with less than 10% solvent-accessible surfacearea) is 65% of the average fluctuation for all atoms.However, the equivalent ratio for TNfn3 is 98%,indicating that the core methyl carbon fluctuationsare comparable to the average for all atoms in thatprotein. The TNfn3 core methyl fluctuations alsohave a larger anharmonic component, similar tothat for the whole protein (80.7%), while theanharmonic contribution to FNfn10 core methylfluctuations is much less than that for the wholeprotein (71.3%). This is qualitatively the sameconclusion as obtained from the NMR experiments,where analysis of the order parameters showed thatthe buried (“core”) residues of TNfn3 were moremobile than the equivalent positions in FNfn10.

The Lindemann melting criterion is an empiricaldisorder parameter widely used to characterizeliquid–solid transitions; a generalized version hasbeen applied to protein motions.50 The generalizedLindemann criterion, DL, is defined as the ratio ofmean-square atomic fluctuations to the most prob-able nearest-neighbour distance, a 0. The criticalvalue of the criterion for the melting transition isbetween 0.1 and 0.15. For TNfn3 and FNfn10, a 0 wastaken from the maximum in the radial distributionfunction of distances between heavy atoms,approximately 5.5 A, and the calculated parametersare listed in Table 2. In both proteins, the coremethyl carbon atoms are less liquid-like than thoseon the surface, conforming to the model of proteinsas a “surface-molten solid”, seen in other cases.50

Figure 4. Definition of S2axis for a methyl group from afragment of a side-chain: the order parameter, S2

measured directly by 2H relaxation measures the ampli-tude of the C–D bond shown. Since this is dominated bythe rapid rotation about the methyl axis (red broken line)defined by the preceding C–C bond, an “axial” orderparameter is often calculated for the methyl axis, S2axis,assuming that the contribution from rapid methyl groupspinning can be factored out (this contribution is given byð1=2ð3 cos2 qK1ÞÞ2 when the angle between the C–D bondand the methyl axis is q).

However, the buried methyl groups in TNfn3 are, onaverage, more “liquid-like” than those in FNfn10.

Model-free analysis

The comparison of NMR relaxation measure-ments with MD simulations is facilitated by usingthe “model-free” formalism.51 In this approach, themotion of each internuclear vector is characterizedby an order parameter, S2, describing the amplitudeof the motion, and a correlation time te giving itsrate. These quantities are related to the correlationfunction of the internuclear vector, as described inMethods. For the backbone, the order parametersconsidered here refer to the orientation of thebackbone NH bond vectors, and for the side-chainsto the orientation of the methyl axis bond vector(defined in the simulations as the C–C bondpreceding the methyl group). An illustration ofthe definition of the methyl axis and its relation tothe spin pair probed by the NMR experiment isgiven in Figure 4.

It is well-known33 that the slow rate of transitionsabout side-chain dihedral angles and the relativelylimited duration of current MD simulations canresult in a lack of convergence in the calculatedNMR correlation functions (equation (4)). If tran-sitions occur sufficiently infrequently, convergedorder parameters cannot be calculated for the nucleiinvolved. In Figure 5(a) are shown correlationfunctions for three representative methyl groupaxes from TNfn3: V13 g1 (A strand), A18 b (Bstrand) and L34 d2 (C strand). The correlationfunction for A18 b is well converged, approachingan almost perfect plateau value; L34 d2 hasconverged fairly well (it has stopped decreasing,although there are significant oscillations due to theinaccuracy of the correlation function at long times),while V13 g1 has certainly not converged to aplateau; the last case is clearly due to poorlysampled c1 dihedral transitions (compare (a) and(c) of Figure 5), causing “kinks” in the correlationfunction. Note that the time at which the kinksoccur in the correlation function in Figure 5(a)correspond to the time-intervals between jumps inthe dihedral time-series in Figure 5(c).52

A simple operational criterion was used toidentify correlation functions that have converged.The order parameter from equation (7) was calcu-lated cumulatively ðS2axisðtÞÞ over the course of thesimulation. If the correlation function has con-verged, the value of the order parameter shouldapproach some limiting value after sufficient timehas elapsed. This avoids the statistical inaccuracy ofusing the correlation function as a criterion, since itis most inaccurate at the critical long time values(compare, for example, the correlation function forL34 d2 in Figure 5(a) with the cumulative orderparameter in Figure 5(b)). The difference betweenthe maximum and minimum values of S2axisðtÞ forthe second half of the simulation (tO1.5 ns) wasused as the convergence criterion; i.e. if thedifference was greater than 0.05, the correlation

Figure 6. Experimental backbone order parameters22

(continuous line) compared with different theoreticalcalculations (symbols): (a) TNfn3 order parameterscalculated from crystal structure normal mode analysis;(b) TNfn3 order parameters from quasiharmonic analysis;(c) TNfn3 order parameters from the full MD simulation,filled symbols indicate order parameters deemed to haveconverged; (d), (e) and (f) analogous results for FNfn10.The positions of the b-strands are indicated above (a) and(c). Independently determined sets of experimentalbackbone order parameters53 are shown as broken linesin (c) and (f).

Figure 5. Convergence of correlation functions formethyl axes. Correlation functions for three examplemethyl groups, A18 b, V13 g1 and L34 d2 are shown in (a),and the corresponding cumulative methyl axis orderparameters S2axisðtÞ in (b). Note that only the first 1250 ps ofthe correlation function is shown in (a), since the latterportion is statistically inaccurate. While the correlationfunctions for the A18 b and L34 d2 methyl groups arerelatively well converged, several kinks are evident in thecorrelation function of V13 g1, principally due torotations about the c1 dihedral (part (c)).


function was deemed to have failed to converge.By this standard, 46% and 58% of the correlationfunctions for methyl axes have converged in TNfn3and FNfn10, respectively. This is a relatively strictcut-off; increasing the cut-off to a difference in orderparameter of 0.1 increased the number of “con-converged” correlation functions for TNfn3 by only8%, and increasing it to 0.2 resulted in a 19%increase. If a nucleus passes this test, it does notmean convergence has been achieved over theentire native state ensemble, since there might, forexample, be some conformers that are not sampledat all during the simulation.

Backbone

Backbone generalized order parameters from the

normal mode analysis24 are given in Figure 6(a) and(d). Good agreement is obtained with experiment22

for residues in b-strands, but the method generallyoverestimates order parameters in loops. Thissuggests that additional anharmonic motion playsa role in lowering the experimental order para-meters. Surprisingly good agreement is obtainedwith the experimental order parameters in the longFG loop of FNfn10. However, in this case, theexperimental data were fitted with a model thatincludes conformational exchange on a millisecondtime-scale,22,53 and the order parameters were usedto represent only the motion on a pico- tonanosecond time-scale. The fact that these orderparameters agree well with the harmonic approxi-mation supports the experimental separation of themotion into a fast (approximately harmonic) com-ponent, and a slower conformational exchange.Using the MD simulations for calculating the

order parameters (Figure 6(c) and (f)) again givesgood agreement in the strands (where a cluster


analysis usually groups the N–H vectors into asingle cluster), but the simulation order parametersare too low in the loops. The order parametersarising from converged correlation functions (asdetermined by the criterion given in the previoussection) are shown as filled squares. Most of thedeviation from the experimental values involvesvectors whose correlation functions have not yetconverged, and must therefore be considered tohave large associated errors. However, there aresome loop amides with apparently convergedcorrelation functions whose order parametersfrom simulation are significantly too low (e.g.residue 43 in TNfn3). This result suggests thatthe simulations introduce too much flexibility in theloop regions. For both proteins there is an inde-pendently determined set of backbone order para-meters;53 they are shown in Figure 6(c) and (f) bybroken lines. In the CC 0 and FG loops of FNfn10, theorder parameters from the simulations are closer to,although still lower than, those from the secondexperimental data set. It should also be noted thatthe specific model used to fit the order parametersto experimental measurements could affect theresults. By generating synthetic NMR relaxationdata from long molecular dynamics simulations,a recent study has shown that the conventional“step-up” hypothesis testing for choosing themodels to fit to NMR data can lead to over-estimation of the order parameter when nanose-cond time-scale motions are involved; a situationthat is most likely to occur in loops54 (the step-uptesting consists of fitting increasingly sophisticatedmodels until a point is reached at which theimprovement in the fit is not statisticallysignificant).

An intermediate approximation, known asquasiharmonic analysis, is to assume that theenergy surface is harmonic, but to derive the secondderivatives from the covariance matrix of fluctu-ations obtained from the MD simulation rather thandirectly from the second derivative of the potentialenergy.55 When order parameters are calculatedwith this assumption, results very similar to the MDsimulation are obtained (Figure 6(b) and (e)). Thus,the anharmonic effects from the MD simulation canbe approximated quite well (as far as orderparameters are concerned) by an effective harmonicforce-field.

In the analysis of the anharmonic contribution tothe motion, it is important to distinguish betweenanharmonic effects coming from transitionsbetween minima and those arising from deviationsfrom harmonic behaviour within a single mini-mum. Although both the backbone and side-chainscontain significant contributions from anharmonicmotion (Table 2), it is the transitions betweenminima that are the important difference, relativeto the backbone, in explaining the side-chain orderparameters. A cluster analysis of backbone N–Hbond vectors (data not shown) indicates that inmost cases the data are described well by a singlecluster, but when the same analysis is applied to the

side-chainmethyl axes, several clusters are requiredfor many of the side-chains. Calculation of orderparameters for any one cluster of bond vectors(for either backbone or side-chain) results in high(0.8–0.9) order parameters. Since the clusters aregenerally found to be associated with rotamericminima, this indicates that transitions betweenrotamers are required to explain the low orderparameters seen for many side-chains.

Side-chains

For the side-chain order parameters, an approachanalogous to that adopted for the backbone wasused to resolve the order parameters for the methylgroup axis into the relative contributions fromfluctuations within a minimum and transitionsbetween minima. The results are summarized inFigure 7. It is immediately clear that, in contrast tothe backbone, motion within a single minimumcontributes only a small amount to the value of theorder parameter. If only this type of motion werepossible, all side-chain order parameters would besimilar to those of the backbone (even for long side-chains like leucine). Harmonic motion (or motionwithin any cluster) generally yields order para-meters in the range 0.8–0.9, with a few closer to 0.7.Only by including transitions are the smallerexperimentally observed order parametersreproduced.

There are several methyl groups that are wellfitted without transitions (e.g. I29 g2, d1, L62 d1, d2and T66 g2 in TNfn3), and for which transitionsbetween minima play little role in determining theorder parameters. However, inclusion of transitionsas shown by comparison of the full MD results withthe harmonic values substantially lowers the orderparameters and improves agreement with experi-ment for most methyl groups, although in manycases the order parameters have not converged.A few exceptions stand out, such as V13 (close tothe protein surface) in TNfn3, which agrees betterwith experiment when transitions in the simulationare left out. Most of the differences betweenexperiment and MD calculation are for methylgroups whose correlation functions are not con-verged, suggesting that the sampling, and not theforce-field, is inadequate.

As is the case for the backbone, the calculation ofside-chain order parameters within the quasi-harmonic approximation yields results similar tothose from the molecular dynamics (Figure 7(b)and (e)).

The agreement of the order parameters from thefull MD simulation with experimental data hasbeen quantified by means of linear correlationcoefficients and rmsd. The linear correlation coeffi-cients are 0.62 and 0.51 for TNfn3 and FNfn10,respectively. This difference is due mainly to thedifference in the amino acid composition of the twoproteins and not to the results being more accuratefor TNfn3. Specifically, the additional leucine andmethionine residues in TNfn3 tend to have low

Figure 7. Experimental side-chain methyl order parameters22 (continuous line) compared with different theoreticalcalculations (symbols): (a) TNfn3 order parameters calculated from the crystal structure normal mode analysis using themethod of Henry and Szabo;24 (b) order parameters from a quasiharmonic analysis; (c) order parameters for TNfn3calculated from the full MD simulation, filled circles indicate order parameters deemed to have converged; (d), (e) and (f)analogous results for FNfn10.


order parameters. These residues improve thecorrelation coefficient by spreading the data over alarger range of order parameters, even if the errorsin each value are approximately the same; i.e.simply assigning the mean order parameter for eachtype of residue also gives a better correlation withthe experimental data for TNfn3 than FNfn10. We,therefore, regard the rmsd as a better indicator: foreach protein the rmsd from experiment is 0.23 when

all data are included, while for converged residuesonly it is reduced to 0.19 for TNfn3 and 0.22 forFNfn10.Correlation times for the internal motions, te,

have been calculated from the intramolecularcorrelation function for the C–D bond (see equation(4)) and the mean values for each type of methylgroup are listed in Table 3 (a detailed listing isavailable as supplementary data). The values are in

Table 3. Comparison of mean correlation times (inpicoseconds) for intramolecular motion for each type ofmethyl group from experiment (texp) and simulation(tsim); the average is taken over methyl groups in bothTNfn3 and FNfn10

Methyl type texp tsim

Ala b 56.3 (27.7) 94.6 (40.2)Thr g2 69.4 (17.3) 93.2 (40.0)Val g1, g2 58.8 (23.7) 55.3 (27.9)Ile g2 55.5 (19.8) 85.3 (42.9)Ile d1 29.6 (7.4) 14.6 (5.9)Leu d1, d2 49.7 (17.7) 44.6 (14.3)Met 3 14.4 (0.0) 8.0 (0.0)

A detailed site-by-site listing of the correlation times is availableas supplementary data.

Figure 8. Improvement of convergence using replicaexchange. In the upper panels, the order parameters fortwo independent 8 ns simulations of each protein at 300 Kin the EEF1 force field are plotted against each other for(a) TNfn3 and (b) FNfn10. In the lower panels are plottedthe analogous comparisons for two independent 8 nsreplica exchange simulations (only the 300 K slice is used)for (c) TNfn3 and (d) FNfn10.


the range (10–150 ps) found experimentally, and arerelatively well correlated with the experimentalresults (correlation coefficient 0.51 for data fromboth proteins). Although methyl correlation timesfor Ala b and Thr g2 are, on average, overestimatedby the simulation, those for the other types ofmethyl groups are in accord with the experimentalaverages; e.g. the correct trend for Ile g2 and d1methyl group is obtained. The barrier for Alamethyl rotation is too high in CHARMM 22,by about 1 kcal molK1,56 which explains the dis-crepancy with experiment in this case. A calculationof the potential of mean force for methyl rotation forblocked dipeptides in explicit water (results notshown) confirms that the torsional barrier in Ala isapproximately 4.7 kcal molK1, while those for othertypes of methyl group are close to 4.0 kcal molK1.

Figure 9. Temperature dependence of selected methylaxis order parameters from TNfn3. The examples shownare Ala 4 b (black squares), Thr 87 g2 (red diamonds), Ile73 d1 (blue circles) and Leu 92 d1 (green triangles).

Perturbations to core dynamics

The above simulation results reproduce experi-mental data relatively well, and mirror the patternof order parameters seen in experiment. However,they do not by themselves explain what localfactors, beyond the identity of a particular side-chain, are important in determining order para-meters and in turn the differences in dynamicsobserved in the two proteins. On the basis ofcorrelations with structural properties, analysis ofthe experimental results identified core compo-sition (i.e. more flexible side-chains will cause therest of the core to be more mobile) and packingdensity (i.e. more free volume permits greater side-chain mobility) as possible causes, although theseare not entirely independent effects. Since the MDsimulations reproduce the phenomenon, we usethem to test such hypotheses by altering thepotential function, which is under our control.Specifically, we have perturbed the system by(i) fixing very flexible side-chains (e.g. leucine) ina single rotamer to mimic a more rigid side-chainand by (ii) creating deletion “mutants” to createmore free volume.

Since repeating the full simulation with explicitsolvent for each case is computationally prohibitive,a force-field with an implicit solvent (EEF1)57 was

used instead. However, this still leaves the problemof convergence, as no meaningful comparison canbe made between unconverged results. To circum-vent this, replica exchange simulations (seeMethods)58 were used to enhance the sampling,together with harmonic backbone constraints toprevent partial unfolding at higher temperatures.Use of these conditions results in a markedimprovement of convergence. Figure 8 comparesorder parameters from two independent 8 nssimulations in EEF1 at 300 K (Figure 8(a) and (b))


and from the 300 K slice (see Methods) of twoindependent replica exchange simulations (Figure8(c) and (d)). Since very similar order parametersare obtained from independent simulations withreplica exchange, useful comparisons to simu-lations performed with various perturbations canbe made. This would not have been possible withconventional MD at 300 K.

Figure 9 shows the dependence of order para-meter on temperature for several methyl groups.The order parameters vary approximately linearlywith temperature. This is in accord with experi-ments on the temperature-dependence of calmodu-lin order parameters.59

The order parameters from simulations with andwithout perturbations are compared as scatter plotsin Figure 10. The statistical error in the calculatedorder parameter must be accounted for whenmaking such comparisons. This has been estimatedfrom the above-mentioned pairs of independent,unperturbed simulations for each protein: i.e. thestandard deviation of the distribution of differencesin order parameter from these two simulationshas been used to estimate statistical uncertainty.The grey shaded region in Figure 10 represents twosuch standard deviations above and below the line

Figure 10. Change of order parameters calculated fromreplica exchange simulations when the proteins areperturbed; left column shows results for TNfn3 andright column for FNfn10: (a) and (b) order parameterswith all Leu residues restrained to a single rotamerplotted against unrestrained simulations (Leu excluded);(c), (d) order parameters on mutation of Ile 20 to Alaplotted against those for wild-type (residue 20 excluded).The statistical uncertainty of the results is estimated fromcomparison of two independent unperturbed simulationsof each protein: the solid line indicates where theperturbation has no effect and the grey shaded arearepresents two standard deviations (calculated from thedistribution of differences between the two independentunperturbed simulations) on either side of this.

corresponding to equal order parameters. Anychanges lying outside this region are likely to bestatistically significant.Restraining leucine residues might be expected to

cause a general increase in order parameters ofother residues, due to “rigidification” of the core.However, this effect appears to be small. Apart fromthe methyl groups lying outside of two standarddeviations from zero change (shaded region inFigure 10(a) and (b)), there is no obvious shifttowards higher order parameters. The averagechange in order parameter, excluding methylgroups with large changes, is an increase of 0.004and 0.002 for TNfn3 and FNfn10, respectively; theratio of the non-leucine methyl groups whose orderparameters increase to those that decrease is 21:15for TNfn3 and 20:14 for FNfn10 (compared to 29:28and 26:30 for two independent, unrestrainedsimulations of these proteins, respectively).There are several residues for which the change

in order parameters is outside of two standarddeviations. Inspection of the protein structureshows that these are all residues, which directlycontact Leu residues. Within this set, there areresidues with both increases and decreases in theirorder parameters. The fact that some methyl groupshave lower order parameters when the leucineresidues are restrained can be rationalized: forexample, T21 g2 in TNfn3 contacts L19 in theTNfn3 crystal structure. In Figure 11(a), the residue–residue interaction energy between L19 and T21 inEEF1 is plotted as a function of the T21 c1 dihedralangle (this is a static calculation in which only the c1

angle is varied). The black curve shows theinteraction energy when L19 is in the crystal-lographic rotamer (tgC) and the red curve when itis in the other most favourable leucine rotamer(gKt). The crystallographic L19 conformer thereforeslightly favours the T21 gC rotamer over gK relativeto the other rotamer; hence, fixing the leucine in thisconformation causes a larger population of theminor gC rotamer (Figure 11(b) and (c)). Since thegC and gK populations are closer to being equalwhen L19 is constrained, the order parameter of T21is lower (e.g. for a bond vector undergoing jumpsbetween two tetrahedrally displaced rotamerswhose population ratio is r, the order parameter isgiven by S2Z1K(8/3){r/(1Cr)2}, which is reducedby increasing r towards 1, i.e. making the popu-lations closer to being equal).10 The interpretationfor side-chains with a marked increase in orderparameter is analogous (in this case the Leuconstraint causes an increase in population of thealready dominant rotamer).In Figure 10(c) and (d) is shown the effect of

mutating Ile20 to Ala. In TNfn3 (Figure 10(c)), themutation has almost no effect on the orderparameter distribution, with only a very slightshift of the order parameters (excluding I59 g2)to lower values (mean change K0.009). The oneoutlier is Ile59 g2, which is in contact with Ile20.The cause of this effect is that the mutation makesthe dominant rotamer even more favourable as

Figure 11. Rationalization of changes in order para-meters on perturbation. The residue–residue interactionenergy between residues T21 and L19 is shown in (a) forL19 in the crystallographic tgC (black line) and othermajor rotamer gKt (red line), while the distributions ofT21 c1 for unrestrained and leucine-restrained simu-lations are given in (b) and (c), respectively. In (d) isshown the residue–residue interaction energy in the EEF1force field between residues 20 and 72 as a function of theV72 c1 torsion angle for all atoms (black line) and onlyatoms that remain in the truncation mutant (red line). Thedistribution of V72 c1 in the wild-type and mutant replicaexchange simulations (300 K slice) are shown in (e) and(f), respectively.


discussed above. In FNfn10, there is also only asmall average shift of order parameters to smallervalues (mean change K0.004). However, a numberof methyl groups undergo a significant shift tolower order parameter values, namely I34 g2, I59d1, I88 g2 and V72 g1,g2. These are all side-chainsthat are in contact with residue 20, where thecreation of the cavity permits greater side-chainmobility, hence lower order parameters. Ile59 is anexception, for the same reason as in TNfn3. Overall,therefore, this mutation appears to have a muchlarger impact on the FNfn10 core. It is plausiblethat, since there is already more free volumeavailable in the core of TNfn3, the dynamics arenot affected as much by a small increase from themutation.

The effect of the cavity on the FNfn10 mutants isillustrated for V72 in Figure 11(d), where theinteraction energies of V72 with I20 (black line)

and A20 (red line) are indicated. Mutation to Alalowers the energy of the V72 gC rotamer, resultingin it being more populated. Since the rotamerpopulations are more equal, the order parameteris lower, as described above.

Recent experimental studies of the effects ofdeletion mutations on side-chain order parametersfor the Fyn SH3 domain16 and eglin c17 have shownchanges that are qualitatively similar to thosepresented here; i.e. the largest changes in orderparameter occur for residues in contact with themutated residue. In addition, in both studies, butparticularly for eglin c, mutations were found toincrease the order parameters for some residuessignificantly.

Correlation analysis of published orderparameters

A database of published order parameters for 18proteins for which structures are also available wasused to determine spatial correlations betweenorder parameters.31 Some correlation of orderparameters that are close in space is expectedfrom considerations of the density of proteinpacking alone. It is also expected on the basis ofthe above effects on order parameters.

To examine spatial correlations, an order para-meter pair correlation function, g(r) was con-structed. Since the order parameter distributionsfor each type of amino acid have different meanand standard deviation, order parameters werefirst normalized to have zero mean and unitstandard deviation by computing a Z-score asZðS2i ÞZ ðS2i KmÞ=s; where S2i is the order parameterof methyl group i and m and s are the mean andstandard deviation of order parameters for this typeof methyl group.

The correlation function g(r) was then defined as:

gðrÞZ hZðS2i ÞZðS2j ÞirijZr (1)

The subscript indicates that the average is takenover all pairs of methyl groups i,j, such that theseparation of methyl carbon atoms in the mini-mized crystal structure, rij, is equal to r (in practice,between r and rCDr). The resulting correlationfunction is plotted in Figure 12, showing a decay asa function of methyl separation. A very similarresult, obtained for order parameters calculatedfrom 5 ns MD simulations of the same proteins,31 isalso shown.

Although the experimental data require the useof order parameters, a more straightforward corre-lation function can be calculated from the MDsimulation: the correlation of atomic fluctuationsis defined as gðrÞZ hdri,drjirijZr; where dri is thenormalized displacement of atom i.60,61 The atomicfluctuation correlation function for methyl carbonatoms, calculated from the MD simulations, is alsoshown in Figure 12, and closely matches the orderparameter correlation functions. Thus, the experi-mental order parameters can be used to infer

Figure 12. Spatial correlation of order parameters for 18proteins for which side-chain order parameters have beenmeasured.31 The black squares indicate the correlationfunction gðrÞZ hZðS2i ÞZðS

2j ÞirijZr of order parameter

Z-scores (where ZðS2i Þ is the Z-score for methyl-group i)calculated from the experimental data (only averagescomputed from more than 100 data points are plotted);the corresponding result for order parameter Z-scorescalculated from 5 ns MD simulations of the sameproteins31 is shown as red circles. For comparison, theaverage correlation of methyl carbon atomic displace-ments from the same MD simulations, gðrÞZ hdridrjirijZr isshown as a green line, where dri is the normalizeddisplacement of atom i. In all cases the distance rij is takenas the distance between the carbon atoms of methylgroups i and j in the minimized crystal structure.


information on correlated motions, which is inagreement with the results of MD simulations.A recent study of side-chain order parameters in theprotein GB1 has also revealed correlations betweenchanges of order parameter upon mutation,although this was not calculated as a function ofmethyl separation.18 Due to the noise in the orderparameter correlation function, a large data set suchas the one used here is necessary for this approach.The combined data in Figure 12 can be approxi-mately fitted to an exponential decay with acorrelation length of 8.5 A. However, we note thatthe atomic fluctuation correlation function is knownto be non-exponential,60 since it first decays to anegative value before rising towards to a limit nearzero at large separation.

Discussion

The previous experimental study of the coredynamics of TNfn3 and FNfn1022 stronglysuggested that many residues in the core of TNfn3were more mobile on a pico- to nanosecond time-scale on the basis of methyl axis order parametersderived from NMR 2H relaxation experiments. Thisdifference was manifested in relatively low orderparameters in the TNfn3 core, compared withFNfn10. Part of the explanation for this is simplythe residue composition of the core (i.e. the TNfn3

core is comprised of more leucine and fewer valineand tyrosine residues), but a weak correlation ofdynamics with packing was also found. MDsimulations have been used to construct a detailedpicture of both backbone and side-chain dynamicsand to explain the differences between the twoproteins. The results confirm earlier findings thatbackbone motion is described well as fluctuationwithin a single minimum, with the exception ofloop motion. Consequently, as we show, a normalmode analysis is able to describe backbone orderparameters quite accurately. The fact that backbonemotion can be described well as arising from asingle minimum also explains the finding thatbackbone residual dipolar coupling data can bemodelled well using single-copy structure refine-ment and does not require averaging over morethan two copies.62,63 Use of the full MD simulationsto calculate order parameters results in relativelygood agreement with experimental data for thoseresidues whose correlation functions have con-verged; those that have not converged are mainlyin loops and appear to involve multiple minima.Overall, on the pico- to nanosecond time-scale,neither the simulations nor the experiment indicatean obvious difference between the backbonedynamics of the two proteins.The dynamics of the side-chains are more

complex. In contrast to the backbone, the mean-square fluctuations of the core side-chain methylgroups of TNfn3 are larger and contain a largercontribution from anharmonic motion than those ofFNfn10. The dominant anharmonic contributionarises from jumps between minima. The distri-bution of mean-square fluctuations mirrors thepattern of experimental methyl axis order para-meters, with the most mobile side-chains beinglocated in the centre of the TNfn3 core and towardthe periphery of the FNfn10 core. The core methylgroups of TNfn3 experience large rms fluctuations(1.140 A), similar to the average for the wholeprotein, while those of FNfn10 are smaller (0.978 A)and well below the average for the whole protein.A wide range of methyl rotation rates (as

measured by residence times or correlation times)is observed and there is quite good agreementgenerally between the simulations and experiment.A study employing an earlier CHARMM all-atomforce-field found that the methyl torsional barrierswere too high (compared with ab initio calculations)and in many cases gave correlation times that weretoo low.33 The CHARMM 22 force-field is animprovement in this respect, although some recentstudies have found the torsional rotation barrier isstill too high by about 1 kcal molK1.56,64 There is nosimple relationship between structure (for example,degree of burial) and rotation rate. This resultsupports the hypothesis that the rate of hoppingbetween methyl rotamers is very sensitive to thelocal environment.64

Numerous side-chain rotameric transitionsoccurred in the simulations, even for buriedresidues. However, as expected, a higher fraction


of surface side-chains was observed to maketransitions, due to the less sterically crowdedenvironment. Buried leucine side-chains mademainly c2 transitions, while the surface L26 ofTNfn3, for example, made correlated c1 and c2

transitions in a crankshaft-like manner. Notably, theburied L72 in the TNfn3 core also makes a singlecrankshaft-like motion during the simulation(not shown), consistent with a more fluid TNfn3core; no such motion is seen for buried leucineresidues in FNfn10.

Fewer transitions were seen for buriedb-branched side-chains such as Val and Ile thanfor Leu. It is interesting to compare the observedtransitions with the crystallographic evidence.Valine 70 of TNfn3 is present in two alternate c1

rotamers in the crystal structure of 1TEN (46% gC

and 54% gK), and is seen to populate both these andthe remaining t rotamer in the simulations. In thetwo crystal structures of FNfn10, 1FNF and 1FNA,many of the surface residues are in differentconformations. Although this is not surprising,there are also several such buried residues; i.e.L18, I34 and I88. Of these, L18 and I34 are observedto make c2 transitions in the simulations. Thedifference in I88 conformation may be permittedby the truncated A strand in the 1FNA construct,which contacts the G strand in which I88 is located.Of the aromatic residues, only a few exposed side-chains undergo any flips during the course of thesimulation. However, ring flipping often occurs ona much longer time-scale and would need to betested for experimentally.5,7 This would be particu-larly useful for filling in dynamic information forresidues lacking methyl groups.

Side-chain order parameters calculated fromharmonic analysis agreed only with a small fractionof the NMR dynamics data. The fact that mostresidues required the full MD simulation to obtaina reasonable agreement with experimental dataemphasizes the importance of transitions betweenminima (e.g. rotamers) in explaining side-chainorder parameters.

A connection between order parameters andconfigurational entropy has been proposed:23,65,66

since the order parameter is related to motionalamplitudes, it is clearly related to the entropy of itsassociated bond vector. This implies that orderparameters can be used to estimate some fraction ofthe native state configurational entropy, which isknown to be significant.40,55 However, the questionremains of what that fraction is and whether thatfraction varies from protein to protein. Wrabl et al.suggested from an analysis of MD simulations thatthe entropy related to amide order parameters iscorrelated with the total configurational entropycalculated from a quasiharmonic analysis.67 In fact,the entropy per mode (from the quantum mech-anical expression48) estimated from the quasi-harmonic analysis is almost identical for the twoproteins (1.103 cal molK1 KK1 modeK1 for TNfn3and 1.101 cal molK1 KK1 modeK1 for FNfn10),despite the substantial differences in their side-

chain order parameters. Nevertheless, order par-ameters may still be useful for giving qualitativeinsights into protein configurational entropychanges upon small perturbations such as ligand-binding,68 although of course such measurementsgive no information on changes in total entropy.

Possible effects on order parameters were testedby making selected side-chains more rigid, orintroducing deletion mutants that create cavitiesin the core. Although it is not possible to run longenough simulations to obtain converged correlationfunctions to compare the effects of perturbations,convergence was achieved using a system withimplicit solvent in conjunction with replicaexchange (parallel tempering) to enhance sampling.This allows the calculation of order parametersfrom the appropriate temperature slice of thesimulation, although all time-dependent infor-mation is lost. A criticism that might be applied tothis approach is that it implicitly assumes that theorder parameter is an equilibrium average overthe full native state ensemble, whereas in practicethe averaging in the experiment is over a (shorter)nanosecond time-scale. Thus, one would anticipatethat the simulation would predict lower orderparameters than experiment. This is certainly apossible effect, but a recent experimental studysuggests that it is a minor contribution for manyresidues:30 comparison of order parameters deter-mined from residual dipolar couplings, which areaveraged on a millisecond time-scale, with thosedetermined from spin relaxation, indicates goodagreement for most residues. There is only a smallsubset of residues for which the order parametersderived from dipolar couplings are lower thanthose from spin relaxation, where rotamericaveraging on a slower time-scale appears to beimportant. Furthermore, our primary interest is inthe changes in order parameter on perturbation,rather than the absolute values.

The results of the simulations with perturbationsindicate that the effect of rigidifying side-chains ormaking deletion mutants is localized to residuesthat are adjacent to the site of the change. Effectsoutside of this region are generally small and notlikely to be significant. Notably, making side-chainsmore rigid led to both increases and decreases oforder parameter. This can be rationalized in termsof the relative effect of the perturbation on theenergy of different rotamers. The complexity ofthe results of such small changes explains thedifficulty in prediction of side-chain order para-meters using simple contact-based models.

The response of TNfn3 and FNfn10 to hydro-phobic core deletion mutants is known to be quitedifferent: the loss of stability in TNfn3 is close towhat is predicted from a structural analysis (e.g. bycounting the number of contacts lost on mutation),while similar mutations in FNfn10 cause a muchsmaller loss than expected. A possible interpre-tation for this is that the core of FNfn10 isoverpacked, resulting in an increase in coremobility on mutation, which offsets the enthalpic


loss due to the mutation. This is supported by thelarger response of FNfn10 to the I20Amutation seenin the replica exchange simulations.

Conclusion

We have shown by molecular dynamics simu-lations that anharmonic effects, principally hoppingbetween rotameric minima, are the most importantcontribution to the effect of side-chain mobility onmethyl axis order parameters. This contrasts withbackbone dynamics, which is well described asmotion within a single minimum, as demonstratedby normal mode analysis. The results of thesimulations of the homologous proteins TNfn3and FNfn10 indicate that the core side-chains ofTNfn3 are more mobile and have a greateranharmonic character compared with those inFNfn10, supporting the experimental conclusionthat the TNfn3 core is more fluid-like. We haveinvestigated the local effects that differentiate themobility of the same type of side-chain in differentenvironments by performing perturbed moleculardynamics simulations of the proteins. As expected,a deletion mutant in the more densely packedFNfn10 results primarily in a decrease in the orderparameters of residues close to the mutation.Intriguingly, however, there are adjacent residueswhose order parameters increase; this effect isshown to be related to the alteration of the preferredrotamers by the mutation, in the context of the side-chain packing. Similarly, when Leu side-chains areartificially constrained, both increases anddecreases in order parameter are observed foradjacent residues. This suggests that attempts to“predict” order parameters from structure will haveto consider which rotamers are accessible and theirrelative weights. We anticipate that the approachwe have introduced for examining perturbationswill prove generally useful in understanding theeffect of mutation on side-chain order parametersand other motional properties studied experi-mentally.16,17 Finally, we have shown that there isa spatial correlation between order parameters ofmethyl groups in different residues, which corre-sponds closely to the spatial correlation of atomicfluctuations found in MD simulations.

Methods

All-atom molecular dynamics simulations

All simulations were done with the programCHARMM69 using the all-atom CHARMM 22 parameterset for proteins.70 An orthorhombic cell of TIP3P71 watermolecules of dimensions 68A!55A!48 A (obtained byadding 24 A, or twice the non-bonded cut-off, to thedimensions of the protein) was created by translation of asmaller equilibrated box of 125 water molecules, andtrimming to the required size. This system was set-upwith orthorhombic periodic boundary conditions,minimized for 40 steps (including optimization of cell

dimensions), and equilibrated in two stages. The first wasa constant volume heating from 10 K to 298 K over 40 psand the second a constant pressure volume adjustmentover a further 40 ps at 298 K and 1 atm (101,325 Pa), usinga modification of the extended system method.72 Thismethod adds a damping term to the dynamics of thepressure degree of freedom in order to prevent “ringing”of the box volume. The resulting water density was1.003 g cmK3, close to the value expected for TIP3P waterat 298 K.73

The crystal structures 1TEN74 and 1FNF38 were used asinitial protein coordinates for the simulations of TNfn3and FNfn10, respectively. The 1TEN structure representsthe “short” 1–90 form of TNfn3; since a C-terminalextension of two residues has been shown to influence thestability and dynamics of the protein,75,76 and since the“extended” form was used in the NMR experiments,the C-terminal Gly-Leu was added in an extendedconformation. The coordinates for FNfn10 were takenfrom the four-domain structure 1FNF, using residues1416–1509. This 94 residue domain is the equivalent of the“short” form of TNfn3 when the sequences are aligned,and corresponds to the NMR construct. Hydrogen atomswere added to each protein using the HBUILD facility ofCHARMM,77 and ionizable side-chains were protonatedassuming a neutral pH; neither protein contains any Hisresidue.Each protein was placed within the equilibrated water

box, with its principal axes aligned to the cartesian axes,and all water molecules whose oxygen atoms were within2.76 A of a protein heavy-atom were deleted. The waterwas allowed to adjust about the protein for a further 40 psof constant pressure dynamics78,72 at 1 atm pressure and atemperature of 298 K, during which all protein heavy-atoms were harmonically constrained to their initialvalues. Finally, the constraints on the protein atomswere released gradually during a further 40 ps ofequilibration. The simulations were then run for 3 ns ata constant pressure of 1 atm and a constant temperatureof 300 K. Bonds to hydrogen atoms and the TIP3P watergeometry were constrained using SHAKE,79 and theequations of motion were integrated with the Nose-Hoover algorithm using a time-step of 2 fs.80,81 Thesimulations each took approximately one month usingeight processors on a Sun E10000.

Replica exchange simulations

The EEF1 implicit solvent57 was used for all replicaexchange simulations with the solvent temperature set to300 K in all replicas. A harmonic constraint on thebackbone heavy-atoms in elements of secondary struc-ture, with a force constant of 1.0 kcal AK2, was used toprevent any slight unfolding at higher temperatures. TheMD replica exchange algorithm of Sugita and Okamoto58

was implemented in a modified version of CHARMM(R.B.B., unpublished results). Trial simulations at severaltemperatures were used to estimate the dependence ofpotential energy on temperature, from which a distri-bution of temperatures was chosen to achieve a goodprobability of exchange82 (the average fraction of success-ful attempts was approximately 40%). The same tem-peratures were used for both proteins: 300.0, 309.7, 319.7,330.1, 340.8, 351.8, 363.2, 375.0, 387.1, 399.6, 412.6, 425.9 K.The replicas were slowly heated from an initial tempera-ture of 50 K to the desired final temperature at an averagerate of 5 K psK1, and equilibrated for a further 200 ps.Following this, replica exchange MD was run for 8 ns perreplica, with attempts to swap replicas every 2 ps. At the


end of the simulations, all the 300 K snapshots werecombined into a single trajectory for analysis.In addition, two simulations were run with different

perturbations for each protein: (i) all leucine residuesharmonically constrained to a single rotamer; and (ii) theI20A mutant.

Harmonic and quasiharmonic analysis

The all-atom CHARMM 22 representation of theprotein70 was employed for all normal mode (harmonicanalysis) calculations, with a distance-dependent dielec-tric to approximately screen electrostatic interactions. Forstructures taken from the MD simulations, the solventwater was initially stripped away. All structures wereprepared by minimizing by steepest descent for 2000steps and then by adopted basis Newton Raphson untilthe change in gradient was less than 1!10K6, or, failingthat, 10,000 steps.The normal modes were calculated by diagonalization

of the full inverse mass-weighted Hessian to solveequation (2), using the vibration analysis module ofCHARMM.83 The normal modes are the eigenvectors ofthis matrix and their corresponding frequencies are thesquare-roots of the eigenvalues li.

jMK1=2ðV2EÞMK1=2 KlIjZ 0 (2)

In this expression, the diagonal matrix M has elementsMiiZmi, wheremi is the mass of atom i, V2E is the Hessianmatrix and I is the identity matrix.The correlations of the cartesian components of the

bond vector displacements, hDkDli, (required for orderparameter calculation below) are obtained from thenormal modes yi from:

hDkDliZ kBTXi

ðyipk KyiqkÞðyipl KyiqlÞ

M1=2p M1=2

q u2i

(3)

These equations refer to the displacements of the bondvector between atoms p and q from the mean; yipk refers tothe kth cartesian component (x,y,z) of atom p in the ithmode, Mp is the mass of atom p and the sum is overall non-translational and non-rotational modes i withfrequencies ui.A quasiharmonic analysis was also done on each MD

trajectory, in which an effective harmonic force-field isderived from the fluctuation matrix (i.e. hDk Dli) from theMD simulations.83

Calculation of NMR model-free parameters

The quadrupolar relaxation of the 2H nucleus and thedipolar relaxation of the nitrogen-15 nucleus are eachdescribed by the correlation function:

CðtÞZ hP2ðmð0Þ$mðtÞÞi (4)

where m is a unit vector along the relevant C–D or N–Hbond and P2 is the second-order Legendre polynomial.51

If the internal motions and overall tumbling of the proteinare assumed to be independent, then C(t) can be factoredinto a contribution from tumbling, Co(t) and one fromintramolecular motion Ci(t): the intramolecular com-ponent is also calculated from equation (4) but with mdefined as a unit vector along the C–D or N–H bondwithin a coordinate frame fixed to the protein.In the “so-called” model-free formalism, Ci(t) is

approximated as a single exponential decay from unityto a limiting value S2, with time constant te. That is, S

2 is

defined by:

S2 Z limt/N

CiðtÞZ limt/N

hP2ðmð0Þ$mðtÞÞi (5)

where m is a unit vector along the relevant bond in theintramolecular coordinate frame.51 Defining the 3!3tensor F as described in equation (6), it can be shownthat the order parameter can be calculated from theensemble average of F by equation (7):24

Fij Zrirj

r2(6)

S2 Z3

2trhFi2 K

1

2ðtrhFiÞ2 (7)

where the ri (iZ1,2,3) are the x,y and z components of theinstantaneous internuclear vector r in the coordinatesystem fixed to the protein. The order parameter can beobtained from a molecular dynamics simulation bycomputing hFi over the whole trajectory, provided thatCi(t) has converged within the length of the simulation.Note that overall tumbling must first be removed fromthe trajectory by reorienting it to a reference coordinateset.84,85 The motion of methyl groups is dominated byspinning about the methyl axis; if this is independent ofthe axial motion, the overall order parameter can bewritten:

S2 Z S2rot S2axis (8)

The rotational contribution S2rot is given by ð1=2ð3 cos2

qK1ÞÞ2; where q is the angle between the C–D bond andthe methyl axis (Figure 4). From the MD simulations, thetotal order parameter S2 can be calculated using a unitvector along the C–D bond, and the methyl axis orderparameter S2axis from a unit vector along the precedingC–C bond.The relaxation time for Ci(t) is obtained (by definition)

from integration of the correlation function:

te Z1

1KS2

ðN0CiðtÞKS2 dt (9)

In practice, this is done by integrating over all theavailable data (correlation functions were calculated to athird of the total simulation length) rather than to infinity,or if Ci(t) becomes negative at any point, the first time itreaches zero. The variance in the calculated te wasestimated from equation (10), in which T is the length ofthe simulation:33,86

sðteÞZ teffiffiffiffiffiffiffiffiffiffiffiffi2te=T

p(10)

The formalism developed by Henry & Szabo24 wasused to calculate order parameters from the instan-taneous displacements, D, derived from a normal modeanalysis. A first-order Taylor expansion is used to showthat hFi can be approximated as:

hFijiZ zizj C1

RzihDjiCzjhDiiK5zizj

Xk

zkhDki

!

C1

R2hDiDjiKzizj

Xk

hD2k iC16zizj

Xk;l

zkzlhDkDli

K5 ziXk

zkhDjDkiCzjXk

zkhDiDki

" #!

(11)

where the ziZRi/R are the direction cosines of the mean


internuclear vector R and the Di are the individualcartesian components of the displacements. Since thepotential is (assumed to be) harmonic, the averagedisplacements Di are all zero. The auto-ðhD2

k iÞ and cross(hDk Dli) correlation terms are obtained from the normalmodes yi using equation (3).In the case of quasiharmonic modes, the above

approximation results in unphysical values (because ofthe large effective changes of bond lengths). The orderparameters were therefore estimated numerically: anensemble of structures was generated by superposingprojections along all non-translational and non-rotationalmodes with random amplitudes drawn from the Gaus-sian distribution expected at 300 K (i.e. the displacementsalong mode yi was given by aiyi where ai is randomlychosen from a Gaussian distribution with zero mean andstandard deviation (2kBT/li)

1/2). The order parameterswere calculated over the ensemble thus generated.

Acknowledgements

We thank the Cambridge Commonwealth Trust(to R.B.B.) and Wellcome Trust (to J.C.) for support,and the National Institutes of Health for partialsupport of the research done at Harvard University.

Supplementary Data

Supplementary data associated with this articlecan be found, in the online version, at doi:10.1016/j.jmb.2005.03.001

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Edited by A. G. Palmer III

(Received 6 December 2004; received in revised form 28 February 2005; accepted 1 March 2005)

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