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PROTEINS:Structure,Function,andGenetics29:240–251(1997)

PredictionofProteinConformationalFreedomFromDistanceConstraints

B.L.deGroot,1D.M.F.vanAalten,2R.M.Scheek,1A.Amadei,1G.Vriend,3andH.J.C.Berendsen1*1GroningenBiomolecularSciencesandBiotechnologyInstitute(GBB),DepartmentofBiophysicalChemistry,UniversityofGroningen,Groningen,TheNetherlands

2KeckStructuralBiology,ColdSpringHarborLaboratory,ColdSpringHarbor,NewYork3EMBL,Heidelberg,Germany

ABSTRACTAmethodispresentedthatgeneratesrandomproteinstructuresthatfulﬁlasetofupperandlowerinteratomicdistancelimits.Theselimitsdependondistancesmea-suredinexperimentalstructuresandthestrengthoftheinteratomicinteraction.Struc-turaldifferencesbetweengeneratedstruc-turesaresimilartothoseobtainedfromexperi-mentandfromMDsimulation.Althoughdetailedaspectsofdynamicalmechanismsarenotcoveredandtheextentofvariationsareonlyestimatedinarelativesense,applicationstoanIgG-bindingdomain,anSH3bindingdomain,HPr,calmodulin,andlysozymearepresentedwhichillustratetheuseofthemethodasafastandsimplewaytopredictstructuralvariabilityinproteins.Themethodmaybeusedtosupportthedesignofmutants,whenstructuralﬂuctuationsforalargenum-berofmutantsaretobescreened.Theresultssuggestthatmotionalfreedominproteinsisruledlargelybyasetofsimplegeometricconstraints.Proteins29:240–251,1997.௠1997Wiley-Liss,Inc.Keywords:moleculardynamics;essentialdy-namics;proteindynamics;NMR

INTRODUCTION

StructuralstudieslikeX-raycrystallographyandNMRspectroscopyoftenprovideinsightintothefunctionofaprotein.However,detailedquestionsonmanydynamicaspectsofenzymaticmechanismssuchasregulationorsubstrateentry,remainunan-sweredwhenonlystaticstructuresareavailable.Dynamicprocessesarecrucialstepsinthefunction-ingofenzymes.Therefore,detailedinformationonthedynamicsofaproteinisnecessaryforacompleteunderstandingofitsfunction.

Simulationtechniquescanhelptoobtaindynamicinformationthatcannotbeprovidedbyexperimen-taltechniquesinastraightforwardmanner.Anum-berofcomputationaltechniqueshasbeendevelopedtogaininformationonproteindynamicsandstruc-turalﬂuctuations.MolecularDynamics(MD)andMonteCarlo(MC)techniquesarethemostpopularones.Theaccuracyofthesetechniquesdependson௠1997WILEY-LISS,INC.

theprotocolsused(force-ﬁeld,molecularrepresenta-tionetc.)andonthesimulationlength.Usingthemostrealisticforceﬁelds,atmostafewnanosecondsforasmallproteininanaqueousenvironmentcanbesimulatedwithinacceptablecomputertime.1,2Thistimescaleisafewordersofmagnitudesmallerthanthatonwhichmostbiologicalprocessestakeplace,leavingtheMDtechniquewithasigniﬁcantsam-plingproblem.3,4TheefficiencyofMCcalculationsiscomparabletothatofMDduetothepresenceofinternalbarriers.5EssentialDynamics

EssentialDynamics(ED),6–9equivalenttoPrinci-palComponent10,11analysesofMDtrajectorieshaveshownthatmost(morethan90%)ofthesimulatedatomicﬂuctuationsusuallycanbedescribedbyafewlarge-scaleconcertedmotions.EDanalysesofMDtrajectoriesdeterminetheeigenvectorsofthecovari-ancematrixofatomicﬂuctuations.Diagonalisationofthismatrixyieldsasetofeigenvectorsandeigen-valuesandtheeigenvectorswithlargesteigenvalues(usuallyatypicalnumberoftensuffices)describealllarge-scaleconcertedﬂuctuations.Iftheeigenvec-torsareseenasvectorsthatspanacomplexspacethenthefew‘‘essential’’eigenvectorswithlargesteigenvaluesspanasubspace,theessentialsubspace,andalllargeconcertedmotionstakeplaceinthissubspace.Itisassumedthatalsothetrueconﬁgura-tionalspaceofmostproteinscontainsalow-dimen-sionalsubspaceinwhichmostpositionalﬂuctua-tionstakeplace.Theessentialsubspaceobtainedfromsimulationisanapproximationofthatsub-space.EDanalysesofMDtrajectorieshavebeenhelpfulinanumberofcasestostudyfunctionalmotionsandpredictmutants.7,8,12Asthetrajectoryofeachsimulationcanbeconsideredasadiffusionalpaththroughapartoftheavailablespacespannedbytheﬁrstfeweigenvectors,13,14thedeﬁnitionof

*Correspondenceto:H.J.C.Berendsen,GroningenBiomolecu-larSciencesandBiotechnologyInstitute,DepartmentofBio-physicalChemistry,UniversityofGroningen,Ny¨enborgh4,9747AG,Groningen,TheNetherlands.E-mail:berends@chem.rug.nl

Received2August1996;Accepted6May1997

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individualeigenvectorsspanningthissubspacefromasimulationhasnotconvergedinthesimulatedtime,3,4butthedeﬁnitionofthesubspaceitselfapproximatelyhas.15,16Thismeansthatthehigheigenvalue-eigenvectorsconstructedfromindepen-dent(piecesof)simulation(s)arerotatedwithre-specttoeachotherbutonlyinasubspacewithlimiteddimension.Thefactthatthedynamicbehav-iorofsimulatedproteinscanbecapturedbyonlyafewdirectionsinconﬁgurationalspacecanbeusedtoimprovesamplingefficiencyinMDsimulationsbydrivingasecondMDrunalongeigenvectorsex-tractedfromaninitialMDrun.13,14,17

CurrentlytheeigenvectorsthatapproximatetheessentialsubspacecanonlybedeterminedfromcovarianceanalysesoflongMDruns,requiringconsiderablecomputationaleffort.Inthepresentstudy,however,anattemptismadetoobtainthesemostprominentcollectivestructurevariationsinaverysimpliﬁedway.

AnalogyWithStructureDeterminationFromNMRData

StructuresolutionbyNMRismainlybasedontheconversionofforce-ﬁeldderivedandexperimentallydetermineddistances(fromNOEdata)intoasetofthree-dimensionalcoordinates.Theavailabledataisofteninsufficienttoreachauniquesolution,aprob-lemthatisusuallycircumventedbyprovidinganensembleofstructures.Largelocalconformationaldifferencesbetweengeneratedstructurescanrepre-sentstructuralﬂexibilitybutareoftentheresultofalackofexperimentaldata.18,19

Herewecarrythisideaabitfurther.Ifalldis-tancesareknown,andtheirupperandlowerboundsaresettophysicallyrealisticvalues,thentheresult-ingstructuresareclosetorealisticconﬁgurationsthatshould,inprinciple,bereachable(duringanMDsimulation).AnEDanalysisofsuchasetofstructureswill,iftheensembleofgeneratedstruc-turesislargeenough,yielddirectionsdescribingﬂuctuationsthatarepossiblewithintheselecteddistancelimits.Ifthedistancelimitsarechoseninasensiblemanner,thentheobservedﬂuctuationscorrespondtorealisticconﬁgurationalfreedomandtheEDresultscouldbeusedtoimprovethesam-plingduringMDsimulation.13,14,17

Atechniquehasbeendevelopedtogenerateran-domstructures,limitedbydistancecriteria.Themethodhasbeenappliedtoanumberofproteins(theB1IgG-bindingdomainofstreptococcalproteinG,thechicken␣-spectrinSH3domain,HPrfromEsch-erichiacoli,bacteriophageT4lysozymeandrattestiscalmodulin).Theseapplicationsindicatethattheapplieddistancerestrictionsarecompatiblewithacceptableproteinstructuresandthatthediffer-encesbetweenthesestructurescanbeusedtoex-tractinformationonthestructuralvariabilityoftheproteinsstudied.

METHODS

DistanceBounds

Themethodinitscurrentimplementationisbasedonacovarianceanalysisofrandomlygeneratedstructuresthatfulﬁlasetofdistanceconstraints.Theﬁrststepistomeasureallpairwiseinteratomicdistancesinthe(known)experimentalstructureoftheproteintobestudied.ThedistancelimitsarenowsetatthisdistanceplusorminusDnanometers,whereDissmallfortightlyinteractingatompairsandlargerforweakerinteractions.ThedifferenttypesofinteractionsthatwereconsideredarelistedinTableIanddistancelimitsDaregiveninTableII.Forallcovalent1–4pairs,theupperandlowerboundsarecorrectedsuchthattheirdistanceisalwaysbetweenthedistancescalculatedinthe‘cis’resp.‘trans’conformation.Thereisaspecialgroupforatompairsthatarepartofthesamesecondarystructureelementtomakesuresecondarystructure(helix,strand)ispreservedinthegeneratedstruc-tures.Thisway,atotalof4697distancerestrictions(3.3%ofthetotalnumberofdistances)couldbedeﬁnedfortheB1IgG-bindingdomain.Thisnumberwas4197(2.5%)forSH3,7333(2.4%)forHPr,14388(1.5%)forcalmodulinand17818(0.6%)forlysozyme,respectively(seeTableIIforthedistributionofdistancesoverthedifferentclasses).

Tospeedupthesearchforstructuresthatfulﬁlalldistancecriteria,upperandlowerboundsarede-ﬁnedforallatompairsthatarenotexplicitlymen-tionedinTableI.TherangeoffreedomDgiventothesepairs(0.5nm)ismuchlargerthanforallotherpairs(TableII)(thelowerdistancelimitsforthesepairsarecorrectedsuchthattheyarenotlowerthanthesumofthevanderWaalsradiioftheatomsinvolved).Ifthisupperlimitisrelaxed,thespeedofconvergenceisstronglyreducedbuttheresultingstructuresarevirtuallyunchanged.

ForallstudiedproteinsexceptHPr,distanceswerecalculatedfromtheexperimental(X-ray)struc-tures(pdbentries1pgb,201shg,213cln22and2lzm,23respectively).ForHPr,asnapshotfromanequili-bratedMDsimulation24(initiatedfromtheNMRstructurewithpdbentry1hdn25)wasusedtoextractthedistances.AllstructureswereenergyminimisedusingtheGROMOS26forceﬁeldbeforedistanceswerecalculated.Allnonpolarhydrogenatomswereincludedwithinunitedcarbonatoms(exceptforaromatichydrogensinthecaseoflysozyme).PolarhydrogenswereplacedusingstandardGROMOShydrogenplacement.Thisresultedin535atomsfortheIgG-bindingdomain(56residues),583forSH3(57residues),785forHPr(85residues),1364forcalmodulin(143residues;residues1–4and148wereexcluded,sincetheywerenotobservableinthecrystallographicdata)and1703forlysozyme(164residues).

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B.L.DEGROOTETAL.

TABLEI.DifferentClassesofInteractingPairs

1–21–3Rings

Side-chaindoublebondsOmega

Pairsthatarecovalentlybonded.

Ifatom1and2andatom2and3arecovalentlybonded.Allatompairsthatarepartofringsystems.

ASN,GLNandARGhaveoneormore(partiallydelocalized)doublebond(s)inthesidechain.Tor-sionanglesaroundthesebondsarerestricted,making1–4pairs(atom1–2,2–3and3–4arecovalentlybonded)aroundthesebondsmorerestrictedthanothers.

DistancesbetweenC␣atomsfromneighboringresiduesdependonthe␻dihedralangle,whichis

morerigidthanthe␾and␺torsionanglesduetoconjugationofthecarbonylbondalongthepeptidebond,whichcausesthepeptideunittoberigidandplanar(other1–4pairsdeﬁnedbythistorsionanglealsofallinthiscategory).

DistancesbetweenbackboneNatomsdependon␺dihedralangles,whereasdistancesbetween

backbonecarbonylCatomsdependon␾dihedralangles(other1–4pairsdeﬁnedby␾and␺alsofallinthiscategory).␾/␺restrictedpairsaresubdividedinthreegroups:

—Tight␾/␺:pairsofneighboringresiduesofwhichoneisaprolineandpairsthatarepartofthesamesecondarystructureelement(helixorstrand).BackbonedihedralsarerelativelymorerigidinprolineresiduesbecausetheNandC␣arepartofaringsystem.Residuesinhelixandstrandconformationhavewell-deﬁnedpositionsintheRamachandranplot,fromwhichlittledeviationisusuallyobserved.

—Loose␾/␺:pairsofneighboringresiduesofwhichoneisaglycineandpairsofresiduesinloopregions.Glycineresidueshaverelativelymuchrotationalfreedomaroundtheir␾and␺tor-sionanglesbecausethereisnosidechainthatinducesspeciﬁcpreferenceforcertain␾and␺combinationsoverothers.Loopregionsareknowntohavearelativelypoorlydeﬁnedstruc-ture,indicativeofconformationalﬂexibility.—Other␾/␺:allother␾/␺restrictedpairs.

Other1–4dihedralanglerestrictedpairs,involvingside-chainatoms.

Pairsofbackboneatomsthatarepartofthesamesecondarystructureelement(helixorstrand)

andarenotmorethan4residuesapart.

Oppositelychargedgroups(allatomsfromsuchagrouparerestricted)incloseproximity(Ͻ4Å).Donor–acceptordistanceshouldnotexceed3.5Å,thehydrogen–acceptorshouldnotexceed2.5Å

andthedonor–hydrogen–acceptorangleshouldbeminimally90°.

Pairsofatomsbetweenwhichtheinteratomicdistanceissmallerthanthesumofthevander

Waalsradiioftheinvolvedatomsplus0.5Åthatdonotfallinoneoftheabovecategories.Identicaltotighthydrophobic,butnowpairsareincludedofwhichtheinteratomicdistanceis

smallerthanthesumofthevanderWaalsradiioftheinvolvedatomsplus1.0Å.

Phi/psi

1–4

SecondarystructureSaltbridgeHydrogenbondTighthydrophobicLoosehydrophobic

ThevaluesgiveninTableIIwereobtainedfromananalysisofthedistanceﬂuctuationsinMDsimula-tionsoftheB1IgG-bindingdomainofstreptococcalproteinG.ThelimitswerechosensuchthatthemajorityoftheMD-generateddistancesiscontainedwithinthelimits.

GenerationofStructures

Havingdeﬁneddistanceboundsforallpairsofatoms,thenextstepistoﬁndstructures,otherthanthereferencestructure,thatfulﬁlallconstraints.Wehavedevelopedanew,iterativeprocedurethatgener-atesstructuresfulﬁllingtherequirementthatalldistancesfallbetweentheirlowerandupperbound.Startingfromrandomcoordinates,correctionsareappliediterativelytothepositionsofthoseatomsthatareinvolvedininteratomicdistancesthatvio-latetheupperorlowerdistancebound.Correctionsareappliedsuchthatforeachviolatingpair,thedistanceisputrandomlybetweentheupperandlowerbound(bothatomsinvolvedaredisplacedbyanequalamount).Thesumofviolationsdecreaseswiththenumberofiterations.Theprocedureisstoppedwhenthesumofviolationsiszero.Conver-genceisusuallyreachedafter100–300iterationsofNsteps(Nisthenumberofviolations).Occasionally,thealgorithmdoesnotconvergetoastructuresatisfyingalldistanceconstraints.Whenthenum-berofiterationsexceedsacriterion(typically500),thealgorithmisstoppedandrestartedwithadiffer-entsetofrandomstartingcoordinates.Sincenoinformationonchiralityisincludedinthedistancebounds,bothmirrorimagesaregenerated.ThegeneratedD-aminoacidenantiomersareconvertedintotheLformbysimplytakingthemirrorimage.Themethod,calledCONCOORD(fromCONstraintstoCOORDinates)resemblesamethodproposedbyCrippen27butdiffersfromitinthewaythedistancecorrectionsareapplied.

Sinceinitialcoordinatesarechosenrandomly(fromacubewithedgesof2nm)anddistancecorrectionsareappliedbychoosingdistancesrandomlybetweentheirupperandlowerbounds,biasintheresultsisminimal.Thereisnocorrelationbetweenanytwostructuresthataregenerated,andtherefore,theaccessiblespacedeﬁnedbythedistanceboundsismoreefficientlysampledthanbyproceduresinwhichsuchcorrelationispresent(likeMD).

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TABLEII.ParametersUsedintheCONCOORDMethod*No.ofatoms

Type

1234567891011121314Total15

1–21–3Ring

doublebond1–4Omega1–4

Tightphi/psi1–4Loosephi/psi1–4Otherphi/psi1–4Other1–4Sec.str.SaltbridgesHydrogenbondsTighthydrophobicLoosehydrophobicAllotherpairs

pgb535

D(nm)0.0020.0050.010.010.010.020.040.030.040.050.0750.050.050.10.5

5417806816220272120322541556

8472785054697

59285588362241901925627659611603536654194SH3583

HPrcal7851364No.ofpairs79211373440336422180443552776

1544487147333

137619627396568762265726246622

2102741113214388

lys1703172325106291726528932887674574713986963157117818

No.

*Valuesindicatethedegreeoffreedomininteratomicdistancesrelativetotheexperimentalstructures.Thenumberofdistancesforallproteinsstudiedineachcategoryarelisted.Abbreviations:pgb,theB1IgG-binding;cal,calmodulin;lys,lysozyme.

Forallproteinsstudied,500structuresweregen-eratedwiththeCONCOORDmethod.FortheIgGbindingdomain(56residues),ϳ1hourofCPUtimeonaPentium100processorwasrequired(forcom-parison:anumberofweekswouldberequiredforanMDsimulationof1ns).Thespeedcouldbeimprovedbyintroductionofacutoffradiusforinteratomicdistancesorothermethodsthatreducethenumberofpairsthatneedtobecorrectedeveryiterationstep.However,themethodinitspresentimplemen-tationisfastenoughforallpracticalpurposes.Startingfromcoordinatesotherthanrandomlycho-senonesmayalsoenhanceconvergencespeed,butsincethecorrectionalgorithmisparticularlyeffi-cientintheinitialstageandbecausewewanttominimizetheamountofbiasintheresults,wepreferredrandomstartingcoordinates.

Allinformationonstructuralvariabilityisstoredintheupperandlowerdistancebounds.Therefore,itshouldinprinciplebepossibletoextractthisinforma-tiondirectlyfromthedistancebounds,withoutﬁrstgeneratingstructures.Wehavenotbeenabletoderiveananalyticalsolution,butanapproximationispossible.Givenaninteractionfunction,awaytogaininsightinthemostprominentmodesofmotionisbydiagonalizationofthe(mass-weighted)Hessianmatrix,asinNormalModes(NM)analyses.28–30Thematrixelementscorrespondtosecondderivativesofthepotentialenergywithrespecttothecoordinates.Thesimplestwaytoimplementdistancerestrictionsinsuchaninteractionfunctionistomodelallpairinteractionsbyharmonicpotentials,withthemini-mumdeﬁnedatthedistancemeasuredintheexperi-mentalstructureandtheforceconstantinversely

proportionaltothedifferencebetweenupperandlowerdistancebound(allmassesareputto1.0).EigenvectorsoftheHessianmatrixthathavethesmallesteigenvalues(apartfromthosethatcorre-spondtooverallrotationandtranslation)aredirec-tionsinconﬁgurationalspacethatrepresenttheslowestvibrationsofamolecularsystem.Inade-tailedforceﬁeld,thesedirectionshavebeenshowntobesimilartotheeigenvectorswithlargesteigenval-uesfromPrincipalComponentanalysesofMDtrajec-tories,15,31althoughnormalmodeshavetherestric-tionofharmonicity.

Startingfromthesamedistancebounds,diagonal-izationoftheHessianmatrixwillyieldresultsthataresomewhatdifferentfromthoseobtainedfromdiagonalisationofthecovariancematrixofposi-tionalﬂuctuationsfromgeneratedstructuresforanumberofreasons.First,duringgenerationofstruc-tures,somedistanceboundswillneverbereachedbecausetheyareexcludedbythepresenceofotherdistancelimits.Therefore,boundsmoothingonthetriangulationlevel32hadtobeperformedbeforecalculationoftheHessianmatrix.Second,distribu-tionsofdistancesareassumedtobeGaussianintheharmonicapproximation,whereasnosuchassump-tionismadeduringthegenerationofstructuresinCONCOORD,wherethedistancedistributionmayevenbeasymmetric.AnalysisTechniques

EssentialDynamics6analyseswereusedforcom-parisonofstructuralfreedominproteins.ThemethodconsistsofdiagonalizationofthecovariancematrixCofatomicﬂuctuations,afterremovalofoverall

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B.L.DEGROOTETAL.

TABLEIII.AverageGeometricalPropertiesforStructuresGeneratedbyMDandCONCOORD,Compared

WiththeValuesObtainedFromExperimentalStructures

RMSD

pgbPDBpgbMD

pgbCONCOORDSH3PDBSH3MD

SH3CONCOORDHPrPDBHPrMD

HPrCONCOORDcalPDBcalMD

calCONCOORDlysPDBlysMD

lysCONCOORD

0.001.431.040.001.290.810.001.390.900.002.651.930.001.811.57

NRC8.09.67.314.014.813.312.014.112.120.021.317.916.020.419.7

HBO39.044.242.338.040.044.574.067.973.2110.099.4108.9122.0122.3124.8

ACC339138403673366540513858484050314892935598519788867588638585

GYR1.0211.0231.0231.0121.0261.0011.1461.1471.1262.0952.1132.0911.5901.5621.581

DIH1.02.681.863.02.032.945.05.094.525.010.425.465.07.217.91

QUALϪ0.083Ϫ0.662Ϫ0.337Ϫ0.668Ϫ1.231Ϫ0.639Ϫ0.553Ϫ0.741Ϫ0.540Ϫ0.160Ϫ0.728Ϫ0.509Ϫ0.228Ϫ0.953Ϫ0.891

ENEϪ2241Ϫ2005Ϫ2140Ϫ2975Ϫ2816Ϫ2811Ϫ4237Ϫ4252Ϫ4223Ϫ7428Ϫ7505Ϫ7484Ϫ10869Ϫ10740Ϫ10493

Abbreviations:pgb,theB1IgG-bindingdomain;cal,calmodulin;lys,lysozyme;RMSD,rootmeansquaredeviation,expressedinÅ;NRC,numberofresiduesinrandomcoilconformation,accordingtoDSSP34;HBO,numberofmainchainhydrogenbonds(DSSP);ACC,totalsolventaccessiblesurfaceinÅ2(DSSP);GYR,radiusofgyrationinnm;DIH,numberofresiduesinunfavorableregionsinRamachandranplot35,44;QUAL,WHATIFindexindicatingthenormalityofpacking45;ENE,potentialenergyafterenergyminimizationintheGROMOSforceﬁeld.

translationandrotation:

Cijϭ7(xiϪ7xi8)(xjϪ7xj8)8

(1)

wherexarecartesianatomiccoordinates.Resultingeigenvectorsaredirectionsinconﬁgurationalspaceofwhichthecorrespondingeigenvaluesgivethemeansquareﬂuctuationofthedisplacementineachdirection.EDanalysescanbeappliedtoany(sub)setofcoordinatesofthestudiedmolecularsystem.OnlyC␣atomswereincludedinEDanalysespresentedherebecauseithasbeenshown6,8,15thatthisap-proachbestdetectsthelarge-scaleconcertedmo-tionsinproteins.

ThesoftwareforthegenerationofstructureswillbeavailableontheWWW(http://rugmd0.chem.rug.nl)andisimplementedintheWHATIF33package.EDandallotherstructuralanalyseswereperformedusinganinterfaceinthemolecularmodelingpackageWHATIF.33Secondarystructureanalysesandacces-siblesurfacecalculationswereperformedwithDSSP.34DihedralanglecriteriaweretakenfromPROCHECK.35

RESULTS

AllCONCOORDstructuresweresubjectedtoanumberofstructuralanalysestoassesshowphysi-callyrealisticthegeneratedstructuresare(TableIII).Thesameanalyseswereperformedonstruc-turessampledbyMD(forsimulationdetails:theIgG-bindingdomain16(1ns),SH39(1ns),HPr24(300ps),calmodulin36(500ps),andlysozyme(submittedforpublication)(1ns).AllMDsimulationswereperformedinexplicitsolventatroomtemperature.ComparisonwithcrystalstructuresandMDshows

thatinCONCOORD,withthepresentsetofparam-eters,structuresgenerallyaremoresimilartotheirrespectiveexperimentalstructurethaninMD.ThereisgoodcorrespondencebetweenthevaluesobtainedfromMDandCONCOORDforallpropertiestakenintoaccount.MeansquareatomicﬂuctuationsofC␣atomsareplottedinFigure1forbothCONCOORDandMD.ThereisreasonablequalitativecorrelationbetweencurvesobtainedfromCONCOORDandMD(correlationcoefficientsbetween0.501and0.871).Forallmoleculesstudiedtheensemblesofconfor-mationsgeneratedbyMDandCONCOORDweresubjectedtoessentialdynamicsanalyses.Inallcasesonlyafeweigenvectorswerefoundwithsigniﬁ-canteigenvalues.TheseeigenvaluesareshowninFigure2(eigenvalueshavebeensortedbydecreas-ingvalue).Eigenvaluecurvesfrombothtechniquesareequallysteepforallproteins,indicatingthatalsofromtheCONCOORDresultsonlyafewcollectiveﬂuctuationsemergewithappreciablefreedom.

InnerproductsbetweeneigenvectorsfromMDandCONCOORDwerecalculatedtoevaluatewhethereigenvectorsobtainedfrombothtechniquesrepre-sentsimilarﬂuctuations.SquaredinnerproductsareshownforeverypairofeigenvectorsfromMDandCONCOORDfortheB1IgG-bindingdomaininFigure3a.Allhighinnerproductsarefoundclosetothediagonal,meaningthatforbothtechniques,directionsinconﬁgurationalspaceareorderedsimi-larlywithrespecttotheamountofﬂuctuation,thatis,directionsthatshowlargeﬂuctuationsinMDalsoshowrelativelylargeﬂuctuationsinCONCOORD,andviceversa.Figure3bshowsthesquaredinnerproductsbetweeneigenvectorsobtainedfromtwohalvesofanMDsimulationof1ns.Theoverlap

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Fig.1.MeansquarepositionalﬂuctuationofC␣atoms.ThecorrelationcoefficientbetweenthecurvesobtainedfromMDandCONCOORDisshownnexttotheﬁgures.

betweenthetwoeigenvectorsetsfromMDissimilartothatbetweenMDandCONCOORD.InFigure3cthesamecomparisonismadefortwosetsofstruc-turesobtainedbyCONCOORD.Twoindependentsetsof250structureswereusedintheEDanalyses.Figure3showsthattheoverlapbetweenMDandCONCOORDisespeciallyhighintheessentialsub-space(deﬁnedarbitrarilyasthesubspacespannedbythe10eigenvectorswithlargesteigenvalues).TheoverlapoftheessentialsubspacesfromMDandCONCOORDhasbeenevaluatedinamorequantita-tivewaybecausetheessentialsubspaceisofparticu-larinterest(about80%oftheobservedstructuralﬂuctuationusuallyoccursinthissubspace).Figure4showsthemeancumulativesquaredinnerproductsbetweeneigenvectors(fromMDandCONCOORD)spanningthissubspaceandtheﬁrst50eigenvectorsfromindependentMD/CONCOORDruns,fortheIgGbindingdomain.Overlapisconcentratedintheinitialpart.Forexample,80%ofoverlapwiththeﬁrst10CONCOORDeigenvectorsisreachedwithintheﬁrst20MDeigenvectors,indicatingthatallessentialdirectionsfoundbyCONCOORDarealsoaccessibleinMD.TheoverlapbetweeneigenvectorsfromtwoindependentMDrunsisverysimilartotheoverlapbetweenCONCOORDandMD,whereastheoverlapbetweentwoindependentCONCOORDrunsisveryclosetothemaximumpossibleoverlap,indicatinganalmostcompleteconvergence.

Themeansquaredinnerproductsbetweenthe10eigenvectorswithlargesteigenvaluesfromMDandCONCOORDaregiveninTableIV,forallproteinsstudied.Theoverlapbetweentheessentialsub-spacesobtainedbyMDandCONCOORDiscompa-rabletotheoverlapobtainedfromthetwohalvesofeachMDtrajectory.Atypicaloverlapofϳ0.5isobtainedforallproteins(avalueof1.0wouldbeobtainedifthetwosetsareidentical).OverlapbetweeneigenvectorsobtainedfromtwopartsoftheclustersproducedbyCONCOORDissigniﬁcantlylargerforallproteins.

Overlapofthe10CONCOORDeigenvectorswithlargesteigenvalueswiththe10lowestfrequency-eigenvectorsobtainedfromdiagonalizationoftheHessianmatrixwascalculatedtobe0.678fortheB1IgG-bindingdomain(C␣componentswereextractedfromtheeigenvectorsoftheHessianmatrixandtheobtainedvectorswererenormalisedbeforetheanaly-

246

B.L.DEGROOTETAL.

Fig.2.EigenvaluesobtainedfromMDtrajectoriesandensemblesofstructuresgeneratedbyCONCOORD.Onlythe50largesteigenvaluesareshownoutof168(pgb,B1IgG-bindingdomain),171(SH3),255(HPr),429(cal),and492(lys),respectively.

sis).Thisvalueissomewhatsmallerthantheover-lapbetweeneigenvectorsobtainedfromtwoclustersofCONCOORDstructures(0.866),indicatingthatsmalldeviationsfromtheconvergedCONCOORDresultsemergeinthisapproximation.TheoverlapoftheHessianeigenvectorswithMDeigenvectorswascalculatedtobe0.486.ThisisslightlylowerthantheoverlapoftheeigenvectorsobtainedfromCONCOORDstructureswithMDeigenvectors(0.532).

ThedifferenceinthewaytheconformationalspaceissampledinCONCOORDandMDisillustratedinFigure5.InMD(Fig.5a),asinglepathisfollowedthatresemblesarandomwalk,13,14,17whereasinCONCOORD(Fig.5b),arandomsamplingtakesplace,witheachpositionindependentfromtheprevi-ousone.ToinvestigateinmoredetailtowhichextentthemodesofmotionpredictedbyCONCOORDareaccessibleinMD,anextendedMDsimulationwithconstraintsonthetwoCONCOORDeigenvectorswithlargesteigenvalueswasperformed.Thewayinwhichtheseconstraintsareappliedmakesitpos-sibletoefficientlyassesstheportionoftheconforma-tionalspacethatisaccessibletoMD.13,14,17AscanbeseenfromFigure5c,theregionsampledbythistechniqueissimilartotheregionsampledbyCONCOORD.

StructurescollectedalongthemostimportantdirectionsdeﬁnedbyCONCOORDareshowninFigure6forcalmodulinandlysozyme.TheCONCOORDeigenvectorwithlargesteigenvalueforcalmodulincorrespondstoacombinationofabendandatwistoftheinterdomainhelix,resultinginarotationofonedomainwithrespecttotheother(Fig.6a).Fromexperiments(hydrogenexchangemeasurements,37NMRrelaxationdata38andNMRNOEdata39fromwhichdisorderinthesetofNMRstructures40emerged),thehelixisknowntobreakinthemiddle,whichwasalsoobservedinMDandNormalModesanalyses.36

Forlysozyme,theCONCOORDeigenvectorwithsecondlargesteigenvaluecorrespondstoaﬂuctua-tionthatissimilartostructuraldifferencesthathavebeenobservedbycrystallographyofanumberofmutants41(Fig.6b).Themaindomainﬂuctuationconsistsofarotationofthetwodomainswithrespecttoeachother,initiatedbyacombinedtwistingand

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Fig.3.SquaredinnerproductmatricesfortheB1IgG-bindingdomain.A:EigenvectorsfromMD(1ns,y-axis)arecomparedtothosefromCONCOORD(500independentstructures,x-axis).B:EigenvectorsfromthetwohalvesoftheMDrun(500pseach)arecomparedtoeachother.C:ThesameisdoneforCONCOORD(250independentstructureswereusedineachanalysis).

bendingoftheinterdomainhelix.Thedifferencebetweenthemostopen41andthemostclosed42X-raystructurealongthisrotationaxisisasmuchas49°.TheangulardifferencebetweenthemostopenandmostclosedCONCOORDstructurewas33°;forMDthisvaluewas28°.BothCONCOORDandMDdonotreachthemostopenexperimentalconﬁguration.

DISCUSSION

TheresultsshowthattherearemanysimilaritiesbetweenMDandCONCOORD.However,thereisalsoanumberofapparentdiscrepancies.InFigure1,anumberofpeaksareonlyobservedinthecurvesobtainedfromCONCOORDandnotfromMD,orviceversa.Thebroadpeaknearresidue48(locatedintheturnconnecting␤strands3and4)fortheB1IgG-bindingdomaininCONCOORDthatisnotpresentinthecurvefromMDrepresentsﬂuctua-tionsthataredominatingtheCONCOORDeigenvec-torwithlargesteigenvalue.ThisdirectionisnotpresentwithintheﬁrsttwoeigenvectorsfromMD,butisrepresented75%bytheﬁrstsixMDeigenvec-tors,indicatingthatthismotionisalsoaccessibleinMD.Likewise,thepeaknearresidue39forcalmodu-lin(asurfaceloopconnectinghelices2and3)inMDismostlytheresultfromthemotionalongtheﬁrstMDeigenvector.ThismodeofmotionshowslittleoverlapwiththeﬁrstﬁveeigenvectorsofCONCOORDbutiscontainedfor75%intheﬁrst15CONCOORDeigenvectors,indicativeofsigniﬁcantﬂuctuationinthecloudofCONCOORDstructures.

ThesimilarityoftheMDandCONCOORDresultsisremarkable,sincebothtechniquesdifferonsev-eralfundamentalpoints.First,theinteractionfunc-tionbetweenparticlesismuchmorecomplexinMDthaninCONCOORD,intermsofthenumberofparametersthatdeterminetheamountandkindofﬂuctuationsthatareaccessible.Inthecurrentimple-

248

B.L.DEGROOTETAL.

Fig.4.Cumulativemeansquareinnerproductsbetweenthe10eigenvectorswithlargesteigenvaluesobtainedfromMD/CONCOORDandalleigenvectorsobtainedfromdifferentMD/CONCOORDruns.Afterdivisionby10,allcurvesconvergeto1.0,sinceeveryeigenvectorfromonesetiscontainedinthecompletesetofvectorsfromanotherset.Thesolidlinecorrespondstothemaximumobtainableoverlap.pgbdenotestheB1IgG-bindingdomain.

TABLEIV.MeanSquaredInnerProductsBetweenSubsetsContainingthe10EigenvectorsWith

LargestEigenvalues

MeancumulativesquareinnerproductMD–MD–CONCOORD–ProteinCONCOORDMDCONCOORD

pgb0.532

0.5600.866SH30.4460.4940.809HPr0.4160.3870.904cal0.4400.5320.802lys

0.454

0.487

0.910

TheﬁrstcolumncontainsacomparisonbetweenMDandCONCOORD,thesecondcolumncomparestwohalvesofeachMDtrajectory,whichisdoneinthethirdcolumnforCONCOORD.pgbdenotestheB1IgG-bindingdomain.

mentation,atotalofonly15parametersissufficient.Second,inCONCOORDonlyshort-rangeinterac-tions(roughlysmallerthan6Å)withintheproteinmakeaseriouscontribution,whereasinMDlong-rangeinteractionsandinteractionswithsolventarealsoincluded.Additionally,allinteractionsareimple-mentedintheformofdistanceconstraintsinCONCOORD.InMD,usuallyonlybondlengthsaredescribedthisway.AnotherimportantdifferencebetweenMDandCONCOORDisthewayinwhichstructuresaregenerated.InMD,theequationsofmotionsareintegratednumericallytoyieldauniquepathinconﬁgurationalspace,whereeachstructureisadeterministicresultofthepreviousone.InCONCOORD,structuresaregeneratedbyarandomsearchmethodthatsearchesforsolutionsinapre-deﬁnedcoordinatespace.Incompletesamplingisoneofthedominatingreasonsforerrorsinthedeﬁnition

ofanessentialsubspacefromMDsimulation.3,4,16ThefactthattheoverlapbetweenCONCOORDandMDissimilartotheoverlapbetweendifferentpartsofMDsimulationssuggeststhattheseerrorsareofthesameorderofmagnitudeastheerrorsmadeinCONCOORDduetoatoosimplemodel.

ThedifferencesbetweenMDandCONCOORDimplythatnotallthedatathatcanbeobtainedbyMDcanalsobeobtainedbyCONCOORD.Dynamic(time-dependent)information,forexample,cannotbederivedfromCONCOORDdata.Also,theampli-tudeofpredictedﬂuctuationscanonlybederivedinarelativesense,thatis,themethodonlypredictscertainmodestobemoreaccessiblethanothers.Forexample,thehingebendingmodeinlysozymewasnotsampledinthesamerangeasinexperiment.However,thisalsoholdsforanMDsimulationof1ns.Thelocalcauseofalargeoverallstructurevariationcannotbededucedreliablyfromananaly-sisofCONCOORDresults.Themainmotionincalmodulin,forexample,isknowntobetheresultofthebreakingoftheinterdomainhelix.Sucharigor-ouseventisnotallowedwithinthedistanceboundsastheyaredeﬁnednow.However,itisinterestingtonotethateveninthecaseofsuchlargeconforma-tionalchanges,theﬁrststageofsuchchangesisalreadysampledand,inthecaseofcalmodulin,emergesastheﬂuctuationwithlargestamplitude.ThecomparisonofeigenvectorsobtainedfromdiagonalizationoftheHessianmatrixwiththosefromCONCOORDandMDindicatesthatevenwith-outthegenerationofstructures,aroughapproxima-tioncanbeobtainedofthesubspaceinwhichallsigniﬁcantbackbonemotionstakeplace.Diagonaliza-tionoftheHessianmatrixisfasterthanthegenera-tionofalargeenoughsetofstructuresbyCONCO-ORDforacovarianceanalysis.Inmostcasesthegenerationofstructuresistobepreferred,however,sincetheproducedstructurescanalsobeusedforotheranalyses,andtheCONCOORDeigenvectorsshowbetteroverlapwithMD.

TheparametersusedforCONCOORD(TableII)weregeneratedfortheB1IgG-bindingdomain,buttheywereapplicablewithoutmodiﬁcationsfortheotherproteinsandgavemeaningfulresults.ThevaluesinTableIIIindicatethatasetofphysicallyrealisticstructureshasbeengeneratedbyCONCOORDforallproteinsstudied.

StructuralVariationinClustersofNMRStructures

AsigniﬁcantlevelofcorrelationbetweenessentialdirectionsdeﬁnedfromMDandfromclustersofNMRstructureshasbeenfoundforanumberofproteins(unpublishedobservations).FortheB1IgG-bindingdomainofstreptococcalproteinGforinstance,thesummedsquareinnerproductsofthe10eigenvectorswithlargesteigenvaluesfromMDandNMRwasfoundtobe0.35,comparabletothe

PROTEINCONFORMATIONSFROMDISTANCECONSTRAINTS

249

Fig.6.Stereorepresentationofextremestructures(thinlineandthindashedline)alongCONCOORDeigenvectors,togetherwithaveragestructures(boldline).A:Calmodulin,eigenvector1.B:Lysozyme,eigenvector2.

valuesinTableIV.Inarecentstudy,asimilarobservationwasreported43forBPTI.TheamountofdynamicinformationthatcanbederivedfromNMR/NOEdatahasbeensubjectofdiscussion.Ithasbeenargued18,19thattheamountofinformationusuallyusedforstructuregenerationfromNMRdataisgenerallytoolimitedtoyieldinformationontheconformationalﬂexibilityofmacromolecules.Inlinewiththeresultspresentedinthispaper,however,methodsthatprovideasetofproteinstructuresinwhichallstructuralconstraintsarefulﬁlledcanbeexpectedtogiveinsightintotheconformationalﬂexibilityofthesemolecularsystems.Theinforma-tionderivedfromaclusterofNMRstructuresisonlypartiallytheresultoftheexperimentaldatausedintheanalysis.InNMRstructurereﬁnement,notonlytheexperimentallyderived(distance)restrictionsareusedfortheanalyses,alsoknowledgeof,for

Fig.5.a(top):ProjectionoftheMDtrajectoryoftheIgGbindingdomainandb(middle):thecollectionofCONCOORDstructuresontotheplanesdeﬁnedbythetwoeigenvectorswithlargesteigenvaluesfrombothtechniques.c(bottom):ProjectionofCONCOORD(smallcircles)andextendedMD(continuousline)structuresontotheplanedeﬁnedbythetwoCONCOORDeigen-vectorswithlargesteigenvalues.

250

B.L.DEGROOTETAL.

instance,bondlengthsandanglesisusuallyin-cludedtogeneratestructures.Thecollectionoftheserestraintsrestrictsthegeneratedconﬁgurationstosuchanextentthatmeaningfulinformationabout(thefew)importantcollectivedegreesoffreedommaybederivedfromsuchanalyses.

CONCLUSIONS

WehaveshownthatthemajorﬂuctuationsinproteinstructuresthatarepredictedbyCONCOORDareconcentratedinafewdirectionsinconﬁgura-tionalspace.Apparently,theboundsoninteratomicdistances,whichareononehanddeﬁnedbytheconnectivitiesinthestructure(covalentbonds)andontheotherhandbythewaytheproteinisfolded(hydrogenbonds,saltbridges,hydrophobiccon-tacts),restricttheconformationalfreedomofthesesystemssuchthatonlyafewcollectivedegreesoffreedomﬂuctuatesigniﬁcantly.Apartfromthedisad-vantagesthatnotimedependentinformationisobtainedandthattheextentandstructuralcauseoftheﬂuctuationscannotbedetermined,analmostconvergeddescriptionofthemostimportantcollec-tivedegreesoffreedomisobtainedwhenonlyalimitednumberofstructureshasbeengenerated.Ithasbeenshownthatitisnotnecessarytousesophisticatedatomicinteractionfunctionstoobtainbasicknowledgeaboutthestructuralﬂuctuationsofproteinsinsolution.Thesumofallinteractionsinproteinsmakesﬂuctuationstobeconcentratedinafewcollectivedegreesoffreedomwhichcanbeobtainedbyastraightforwardmethod.Theminimalcomputationaleffortinvolvedallowsforthescreen-ingofﬂuctuationsinmanyconﬁgurations,whichcould,forexample,facilitatethedesignofmutants,orenhancethecapabilitiesofhomologyprediction.

ACKNOWLEDGMENTS

WethankDavidvanderSpoelforkindlyprovidingthecalmodulinMDtrajectory,NicovanNulandfortheHPrMDtrajectory,andMichaelNilges(EMBL,Heidelberg)forcriticallyreadingthemanuscript.

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