Simulated and Experimental Bending Dynamics in DNA with and without A-Tracts

AlexeyK.MazurandDimitriE.Kamashev

LaboratoiredeBiochimieTh´eorique,CNRSUPR9080

InstitutdeBiologiePhysico-Chimique

13,ruePierreetMarieCurie,Paris,75005,France.FAX:+33[0]1.58.41.50.26.Email:alexey@ibpc.fr

arXiv:physics/0108038v2 [physics.bio-ph] 20 Sep 2001ThemacroscopiccurvatureofdoublehelicalDNAin-ducedbyregularlyrepeatedadeninetractsiswell-knownbutstillpuzzling.Itsphysicaloriginremainscontroversialeventhoughitisperhapsthebest-documentedsequencemodula-tionofDNAstructure.Wereportheretheresultsofcompara-tivetheoreticalandexperimentalstudiesofbendingdynamicsin35-merDNAfragments.Thislengthappearslargeenoughforthecurvaturetobedistinguishedbygelelectrophoresis.TwoDNAfragments,withidenticalbasepaircomposition,butdifferentsequencesarecompared.Inthefirstone,asingleA-tractmotifwasfourtimesrepeatedinphasewiththeheli-calscrewwhereasthesecondsequencewas”random”.BothcalculationsandexperimentsindicatethattheA-tractDNAisdistinguishedbythelargestaticcurvatureandcharacter-isticbendingdynamics,suggestingthatthecomputedeffectcorrespondstotheexperimentalphenomenon.TheresultspoorlyagreewithearlierviewsthatattributedadecisiveroleinDNAbendingtosequencespecificbasepairstackingorbindingofsolventcounterions,butlendadditionalsupporttothehypothesisofacompressedfrustratedstateoftheback-boneastheprincipalphysicalcauseofthestaticcurvature.Wediscussthepossiblewaysofexperimentalverificationofthishypothesis.

INTRODUCTION

ItisgenerallyacceptedthatthebasepairsequencecanaffecttheoverallformoftheDNAdoublehelix.In-trinsicDNAbendingisthesimplestsucheffect.NaturalstaticcurvaturewasdiscoverednearlytwentyyearsagoinDNAcontainingregularrepeatsofAnTm,withn+m>3,calledA-tracts1,2,3.Sincethenthisintrigu-ingphenomenonhasbeenactivelystudied,withsev-eralprofoundreviewsoftheresultspublishedindifferentyears4,5,6,7,8,9.ItisknownthatthecurvatureisdirectedtowardstheminorgroovesofA-tractsand/orthemajorgroovesofthejunctionzonesbetweenthem,andthatitsmagnitudeisaround18◦perA-tract.However,theexactsitesandthecharacteroflocalbendsremainamatterofdebateaswellastheirmechanismandphysicalorigin.AlreadythepioneeringconformationalcalculationsoftheseventiesshowedthattheDNAdoublehelixexhibitssignificantbendabilitywhichisanisotropicandsequencedependent10,11.BasedupontheseviewsthewedgemodelofferedtheveryfirstexplanationofbendinginducedbyA-tractsbypostulatingthatstackinginApAstepsisin-trinsicallynon-parallel12.Modifiedversionsofthisthe-1

oryaccountedforasubstantialpartofavailableexper-imentaldata,withgoodscoresofcurvaturepredictionfromsetsoffittedwedgeangles13,14,15.Atthesametime,clearexperimentalcounter-examplesexistwherebendingcouldnotresultfromsimpleaccumulationofwedges16,17.Thejunctionmodel18,2betterthanothertheoriesex-plainedexperimentaldataongelretardationofcurvedDNA.ItoriginatedfromanideathatabendshouldoccurwhentwodifferentDNAformsarestacked19.Ifpoly-dAdoublehelixhadaspecialB’formassuggestedbysomedata20thehelicalaxisshouldbekinkedwhenanA-tractisinterruptedbyarandomsequence.Inturn,theX-raydataarebestinterpretedwithanalternativetheorythatpostulatesthatbendingisintrinsicinmostDNAsequencesexceptA-tractswhicharestraight21,22,23.An-otherinterestingmodelattractedattentionintherecentyears,namely,bendingbyelectrostaticforcesthatre-sultfromneutralizationofphosphatesbysolventcationstrappedintheminorgroovesofA-tracts24.Thisprob-lemisofgeneralimportancebecausetheaccumulatedlargevolumeofapparentlyparadoxicalobservationssug-geststhatsomeessentialfeaturesarestillunknownthatmaybeessentialforthefinestructureandthebiologicalfunctionoftheDNAmolecule.

OneofushasrecentlyproposedanewhypothesisofthephysicaloriginofintrinsicbendsindoublehelicalDNA25.Accordingtoit,thesugar-phosphatebackboneinphysiologicalconditionsisslightlycompressed,thatistheequilibriumspecificlengthofthecorrespondingfreepolymerinthesamesolventislargerthanthatinthecanonicalB-form.Therefore,thebackbone“pushes”stackedbasepairs,forcingthemtoincreasethehelicaltwistandrisewhilethestackinginteractionsopposethis.Asaresult,thebackboneincreasesitslengthbydeviatingfromitsregularspiraltraceandwandersalongtheheli-calsurfacecausingquasi-sinusoidalmodulationsofDNAgrooves.Concomitantbasestackingperturbationsresultinmacroscopicstaticcurvaturewhencertainpropertiesofbasepairsalternatealongthesequenceinphasewiththehelicalscrew.

DrewandTravers(1984,1985)apparentlywerethefirsttonoticethatnarrowingofbothDNAgroovesattheinneredgeofabendisanecessaryandsufficientconditionofbending,andthatanunusuallocalgroovewidthshouldbeaccompaniedbystructuralperturbationsbeyondthisregion.They,andlaterBurkhoffandTul-lius(1987),consideredthepreferenceofnarrowandwideminorgrooveprofilesbycertainsequencesasthepos-

sibleoriginalcauseofthiseffect.Sprousetal(1999)proposedasimilarideawithinthecontextofthejunc-tionmodel.Inacertainsense,thecompressedback-bonetheorycontinuedthesamelineofthinking.Unlikeothermodels,itnaturallyexplainswell-knownenviron-mentaleffectupontheA-tractcurvature,notably,itsreductionwithtemperature26,27,28andadditionofdehy-dratingagents29,30,31(seediscussioninMazur(2000)).Thistheorycertainlyneedsfurtherexaminationandcrit-icalcomparisonwithothermodelsinbothcalculationsandexperiments.

Conformationalmodelingearlierhelpedtoshedlightuponmanyaspectsoftheaboveproblems.ConstructionofspatialDNAtracesfromlocalwedgeparameterscom-binedwithMonteCarlosimulationsofloopclosurewasappliedtocheckdifferenthypothesesandtoestimatelo-calbendinganglesfromexperimentaldata18,32.Energycalculationsrevealedthatbendingmaybeeasieratsomedinucleotidestepsandincertainspecificdirections11,33,withexperimentalsequenceeffectsreproducedinsomeremarkableexamples34.DNAwasshowntohavelocalenergyminimainbentconformationscorrespondingtothejunctionmodel35,36.AllatomMonteCarlocalcula-tionsshowedthatnarrowingoftheA-tractminorgroovewithafewNMR-derivedrestraintsmaybesufficienttoprovokethecurvature37.

Thesimplestset-upformodelingDNAbendingistotakeastraightsymmetricaldoublehelixandletitbendspontaneouslywithnoextraforcesapplied,thatisduetogenericatom-atominteractions.This“naive”approachhasrecentlybecomepossibleowingtotheprogressinmethodologyofmoleculardynamics(MD)calculationsofnucleicacids38,whichwasdemonstratedbysuccess-fulsimulationsofseveralcurvedandstraightDNAfrag-mentsinrealisticenvironmentincludingexplicitwaterandcounterions39,40.Thecharacterofbendingqualita-tivelyagreedwiththetheoriesoutlinedabovesothatnoneofthemcouldbepreferred.Thoroughdiscrimi-nativetestingwouldrequiremoreextensivesamplingofbendingevents,whichshouldbecomepossibleinfuture.DetailedstructuresofshortA-tractfragmentshavealsobeenstudiedbyMD41,42.

ThemajorobstacleinfreeMDsimulationsofintrinsiccurvatureisthelimitedcapacityofsamplingofbendingevents.Thephysicaltimeoftransitionbetweenstraightandbentconformationsmaybetoolongforastatisticallysignificantnumberofsucheventstobeaccumulatedinsimulations.Moreover,experimentaleffectsmaynotap-pearduringinfinitelylongMDbecausemodelsareneverperfect.Tocircumventthesedifficulties,weemployedadifferentstrategy.Wefirstlookedfor,andfoundashortA-tractmotifthatcouldreproduciblyinducesta-blebendsinDNAduringafewnanosecondsofMDwithasimplifiedmodelofB-DNA.Weusedthismotiftocon-structlongerdoubleheliceswithintrinsiccurvatureinsilicoandwecouldincreasethelengthofDNAfragmentsincalculationsto35bp,whichmakespossibleadirectcomparisonwithexperimentsinvitro.

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Thetwo35-merDNAfragmentswestudyherehaveidenticalbasepaircompositionanddifferonlybytheirsequences.ThefirstfragmentisthedesignedA-tractrepeatwhiletheothersequenceis“random”.AllMDtrajectoriesstartfromcanonicalstraightA-andB-DNAconformations.FortheA-tractDNAfragmenttheycon-vergedtoasinglestaticallybentstatewithplanarcur-vaturetowardsthenarrowedminorgroovesat3’endsofA-tracts.Themagnitudeofbendingisclosetotheexperimentalestimates.Therandomfragmentwasnotstraightaswell,butitscurvaturewasmuchlesssignif-icantandlessplanar.Ingelmigrationassaysthetwomoleculesproducewell-resolveddistinctbands,withtheA-tractsequencedemonstratingareducedmobilitychar-acteristicofcurvedDNA.TheseresultssuggestthattheintrinsicDNAcurvaturereproducedincalculationscor-respondstotheexperimentalphenomenon.Thebendingdynamicsqualitativelyagreeswiththecompressedback-bonetheory,butitcannotbeaccountedforbyothermodels.

MATERIALSANDMETHODS

Calculations

Moleculardynamicssimulationshavebeenperformedbytheinternalcoordinatemethod(ICMD)43,44includ-ingspecialtechniqueforflexiblesugarrings45,withAMBER9446,47forcefieldandTIP3Pwater48.Allcal-culationswerecarriedoutwithoutcut-offsandbound-aryconditions.Thetimestepwas10fsec.Theso-calledminimalmodelofB-DNAwasused49,50.Itin-cludesonlyapartialhydrationshellandtreatscounterionandlongrangesolvationeffectsimplicitly.Advantagesaswellaslimitationsofthisapproachhavebeenreviewedelsewhere38.Themodelhasnootherbiastowardsbentornon-bentconformationsexceptthebasepairsequence.ThestartingfiberA-andB-DNAmodelswerecon-structedfromthepublishedatomcoordinates51.Thehy-drationprotocolsweresameasbefore,25withanidenti-calnumberofexplicitwatermoleculesinA-andB-DNAstarts.ProgramsCurves,52XmMol,53andMathematicabyWolframResearchInc.wereemployedforgraphicsanddataanalysis.

Thetwo35pbDNAfragmentsarereferredtobelowasAtandnAt,fortheA-tractrepeatandthenon-A-tractDNA,respectively.ForbothfragmentstwolongMDtrajectorieswerecomputedstartingfromeitherA-orB-canonicalDNAforms.Thesefourtrajectoriesarere-ferredtoasAt-A,At-B,nAt-A,andnAt-B,respectively,wherethelastcharacterindicatesthestartingstate.Alltrajectorieswerecontinuedto20nsexceptAt-Bwhichwasstoppedatabout12nsbecauseithadclearlycon-vergedlongbefore.

Experimental Sequence5’−...CAAAAATGTCAAAAAATAGGCAAAAAATG...3’RESULTSANDDISCUSSIONConstructionofDNAFragments

Bend centre135bp A−tract repeat5’−AAAATAGGCTATTTTAGGCTATTTTAGGCTATTTT−3’3’−TTTTATCCGATAAAATCCGATAAAATCCGATAAAA−5’234Reference random fragment5’−TTAGATAGTATGACTATCTATGATCATGTATGATA−3’3’−AATCTATCATACTGATAGATACTAGTACATACTAT−5’FIG.1.Constructionof35bpdoublestrandedDNAfrag-ments.ThetopsequencewiththeboxedheptamermotifAAAATAGistakenfromthetrypanosomekinetoplastDNA2.TheA-tractsarenumberedandtheircentersareseparatedbyapproximately10bp.Thereferencerandomfragmenthasthesamebasepaircontentasthe35-merrepeat,butitssequencehasbeenmanuallyre-shuffledtoexcludeanyA-tractmotives.Oligonucleotidesandconstructionof5’-labeledDNA

probes

Figure1explainshowthetwoDNAfragmentsusedinourstudyhavebeenconstructed.TheA-tractmo-tifAAAATAGoriginallyattractedourattentioninMDsimulationsofthenaturalDNAshowninFig.154,whichisthefirstcurvedDNAlocusstudiedinvitro2.The35bpA-tractfragmentwasconstructedbyrepeatingthismotiffourtimesandithadtobeinvertedtomakethetwoDNAterminisymmetrical.Suchinversionshouldnotaffectbending,55butisessentialforsimulationsbe-causethe3’-and5’-endA-tractsmayrepresentqualita-tivelydifferentboundaries.InrepeatedsimulationswiththisandsimilarA-tractfragments,thestaticcurvatureemergedspontaneouslyanditbecamemoreevidentasthechainlengthincreased25.Toobtainareferencenon-A-tractDNA,wehavere-shuffledmanuallybasepairsoftheA-tractrepeat.Wepreferredthisrandomizedse-quencetocommonlyusedGC-richstraightfragmentsinordertokeepthebasepaircontentidenticalandreducethenoisethatcouldcausesmallvariationsingelmobilityandhidethesubtledifferencesweweregoingtodetect.

SpontaneousDevelopmentofCurvaturein

Simulations

Thesequencesofthe35nt-longsyntheticoligonu-cleotidesusedhereareshowninFig.1.Thedou-blestrandedDNAswereobtainedbyannealingofthetwocomplementaryoligonucleotides,oneofthemla-beledwithT4polynucleotidekinaseand[32-P]-ATP.Theannealingwascarriedoutbyincubatingtheoligonu-cleotides(300nM)for3minat80◦Cin20mMTris-HCl(pH8.0),400mMNaCl,0.2mMEDTAandthenallowingthemtocoolslowly.

Gelmobilityassays

MobilityoftheDNAfragmentswasanalyzedin16%gels(acrylamidetobis-acrylamide,29:1)bufferedwith90mMTris-borate,1mMEDTA,pH8.6.Gelswerepre-rununderconstantpoweruntilstabilizationofcurrent.End-labeledDNAinabuffercontaining20mMTris-HCL,50mMNaCL,7%glycerol,pH8.0andbromophenol-bluewasloadedontothegel.Theelectrophoresiswasper-formedunderconstantpowerandconstanttemperatureof8◦C.Thedriedgelswereexposedtostoragephos-phorscreensandvisualizedona400SPhosphorImager(MolecularDynamics).

AllfourtrajectoriesexhibitedstabledynamicswithDNAstructuresclosetotheBform.TableIshowspa-rametersofthefinal1ns-averageconformations.TheyallhaveremarkablysimilarhelicoidalscorrespondingtoatypicalB-DNA.Forexample,theaveragehelicaltwistestimatedfromthebest-fitB-DNAexperimentalvalues56gives34.0±0.2◦and33.8±0.2◦fortheA-tractfragmentandtherandomizedsequence,respectively.Atthesametime,thermsdeviationsfromthecanonicalstructuresvarymoresignificantly.

AsshowninFig.2,duringthefirstfewnanoseconds,thermsdfromthecanonicalB-DNAquicklyleveledataround4˚Ainallfourtrajectories.FortheA-DNAstartthiscorrespondstoarapidtransitiontoB-formwithre-ductionofrmsdfromtheinitial10.7˚A.ThesubsequentdynamicsisremarkablydifferentfortheAtandnAttra-jectories.InAt-AandAt-B,aftersomedelay,thermsdvaluedrasticallyincreasedandstabilizedatahigherlevelofaround6˚A.Thetracesofthebendangleandtheaxisshorteningindicatethatthiswasatransitiontoasignifi-cantlylargercurvature.Incontrast,fornAttrajectories,Fig.2exhibitsonlyfluctuationsatroughlythesamelevelasinAt-AandAt-Bbeforethetransition.

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TABLEI.SomestructuralparametersofstandardandcomputedDNAconformations.ThehelicoidalsarethesequenceaveragedvaluescomputedwithprogramCurves52.Alldistancesareinangstr¨omsandanglesindegrees.A-DNAB-DNAAt-AAt-BnAt-AnAt-B

-5.4-0.7+0.1-0.4-0.1-0.1

+19.1-6.0-4.0-5.2-4.2-4.7

2.63.43.53.53.53.5

32.736.034.234.534.334.4

0.010.711.611.610.611.2

10.70.05.96.83.84.1

At−AX20A100510150T TT TA TC GG AT TT TA TC GG AT TT TA TC GG AT AA AA0nAt−AX10500520nsT TT TA TC GG AT TT TA TC GG AT TT TA TC GG AT AA AA20ns1015A TA GT AT GT AC TA GT AT CT AT CA GT AT GA TA GA TT20ns420-2YY0510101520ns15A TA GT AT GT AC TA GT AT CT AT CA GT AT GA TA GA TTFIG.3.ThetimeevolutionoftheoverallshapeofthehelicalaxisinAt-AandnAt-A.Theaxisofthecurveddoublehelixiscomputedasthebestfitcommonaxisofcoaxialcylindricalsurfacespassingthroughsugaratoms,whichgivessolutionsclosetothoseproducedbytheCurvesalgorithm52.ThetwosurfaceplotslabeledXandYareconstructedbyusingprojectionsofthecurvedaxisupontheXOZandYOZplanes,respectively,oftheglobalCartesianframeshowninFig.4a.Anytimesectionofthesesurfacesgivesthecorrespondingprojectionaveragedoveratimewindowof400ps.Thehorizontaldeviationisgiveninangstr¨omsand,forclarity,itsrelativescaleistwotimesincreasedwithrespecttothetrueDNAlength.Shownontherightarethecorrespondingviewsofthefinal1ns-averageconformations.TheATbasepairsareshownbythickerlines.

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TheoriginofthisdifferenceisanalyzedinFig.3.ItdisplaysdynamicsoftheoverallDNAshapebyusingtwoorthogonalprojectionsofthehelicalaxis.AplanarbendwouldgiveaplaneintheYprojectionandacurvedsur-faceintheXprojection.AsharpincreaseofcurvatureinAt-Aafterthe13thnanosecondisevident.AnalogouseventoccurredinAt-Bafterabout3ns.InagreementwithFig.2,thetwonAtsurfacesshowfluctuationswithamplitudessimilartothoseduringthefirst13nsofAt-A.Thispatternprobablycorrespondstoagenerictypeofdynamicscharacteristicofarbitrary35-merDNAfrag-ments.

ComparisonofthethreecolumnsofplotsinFig.2in-dicatesthatfluctuationsusuallyoccurredsimultaneouslyinallthreeparameters,whichmeansthatthebendingdynamicsmakesamajorcontributiontothermsdfromB-DNA.ItsvaluesshowninTableIareactuallymuchlargerthanwouldbeforstraightconformationswiththesamehelicalparameters.Forinstance,thermsdbetweentheAt-AandAt-BstructuresinTableIwas2.3˚Aonlybecause,asweshowbelow,theywerebentinthesamedirection.

ConvergenceofTrajectories

(a)JLZϕO’KXYO(b)180120600-60-120-180180120600-60-120-180180120600-60-120-180180120600-60-120-180At−A05101520ADNAmoleculewithdetectablestaticcurvaturecaneitherhaveaminimumofpotentialenergyinabentstateoritsenergyvalleyshouldhaveaspecialshapesuchthatabentformhaslargerconformationalentropy57.InbothcasessuchstaterepresentsafreeenergyminimumwhereMDtrajectoriesshouldbetrapped.Thequestionis,how-ever,howlongarealMDtrajectoryshouldstayinabentconformationtoberepresentative.SomeexperimentssuggestthatbendingdynamicsinDNAfragmentsofonly100basepairsmayinvolverelaxationtimeslongerthanamicrosecond58,59,therefore,nopracticalprocedureexiststoproverigorouslythatcomputedconformationsarerep-resentative.Nevertheless,ifseveraltrajectoriesconvergetothesamestatefromverydifferentstartingpoints,onecanarguethatthisstateisanattractorintheconfor-mationalspace,whichisanecessaryconditionofthestaticcurvature.Thereciprocalconvergenceoftrajecto-riesstartingfromcanonicalA-andB-DNA,therefore,isaveryimportantaspectofthesesimulations.Themini-malB-DNAmodelisnotexpectedtogivestableA-DNAstructuresandwedidnottrytoequilibratetheinitialA-DNAstates.ThestartformtheA-formisimportantbecauseitprovidesanindependentdynamicassaywithaverydifferententrytotheB-DNAfamily,whichal-lowsonetoverifyconvergenceoftrajectoriestospecificconformations.Weanalyzeseparatelytwolevelsofstruc-turalconvergence.

At−B024681012nAt−A05101520nAt−B05101520Time (ns)FIG.4.(a)GeometricconstructionsusedforevaluatingtheDNAbending.ThetwocoordinateframesshownaretheglobalCartesiancoordinates(OXYZ),andthelocalframeconstructedinthemiddlepointofthecurvedDNAaxisac-cordingtotheCambridgeconvention(O’JKL)88.ThecurveisrotatedwithtwoitsendsfixedattheZ-axistoputthemid-dlepointinplaneXOZ.ThebendingdirectionismeasuredbyangleϕbetweenthisplaneandvectorJofthelocalframe.Bydefinition,thisvectorpointstothemajorDNAgroovealongtheshortaxisofthereferencebasepair88.Consequently,thezeroϕvaluecorrespondstotheoverallbendtowardsthemi-norgrooveinthemiddleoftheDNAfragment.(b)Thetimeevolutionofthebendingdirectionasmeasuredbytheϕangleinplate(a)(indegrees).Thetraceshavebeensmoothedbyaveragingwithawindowof75psinAt-Band150psother-wise.

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Overallshapes

ThermsdcomparisonbetweenAt-AandAt-BisshowninFig.5.ItclearlydemonstratesthatAt-AandAt-Btrajectoriesmanagedtocomeveryclosetoeachothereventhoughtheirstartingpointsweresignificantlysepa-ratedinconformationalspace.Theinitialrmsdof10.7˚Abetweenthecanonical35-merA-andB-DNAformseven-tuallywentdowntoaslowas1.3˚A.Thefinalfallofthermsdoccurredwhenthecurvaturedrasticallyincreased(compareFigs.2and5).Moreover,duringthelastnanosecondsthebendingdirectionwasvirtuallyiden-ticalinAt-AandAt-Bandessentiallyfixedataround90◦(seeFig.4b),whichexplainstheoriginoftheblackrectangleintheupperrightcornerofFig.5.Thisdi-rectioncorrespondstobendingtowardstheminorgrooveatapproximatelythreebasepairstepsfromthemiddleGCpair(seeFig.4a),thatisatthe3’endofthethirdA-tractinFig.1.

ThenAttrajectoriesexhibitedqualitativelydifferentfeatures.ThermsdcomparisonofanytwolongintervalsofnAt-AandnAt-Bgivesfluctuationsbetween3and6˚Awithoutanycleartimetrend.Figure2showsthatthermsdfromB-DNAalsofluctuatedbetween3and6˚Aandthatitcorrelatedwithbendingparameters.AsseeninFig.3themoleculereallywasnotstraight.Ac-cordingtoFig.4bthebendingdirectionsinnAt-AandnAt-Bwerewelldefinedbutslightlydifferent.Theynei-therdivergednorconverged,remainingataround100◦fromeachother.Themoleculeshowsnosignsofslowstraightening,whichwouldgiveadecreaseoffluctua-tionsinFig.2andanincreaseinscatteringofdirectionsinFig.4b.Allthissuggeststhatbentshapesarefa-voredoverstraightones,andthattherearemanystablebends,withtransitionsbetweenthembeingtooraretobesampledbyoursimulations.

Grooveprofilesandlocalstructures

ularoscillationsoftheminorgroovewidthsareobservedonlyinA-tractrepeats60,andthisstructuralperiodicityiscertainlyrelatedtothatofthesequence.However,suchbehaviorisexactlywhatoneshouldexpectifthewavingofthebackboneresultsfromitsintrinsiccompression.Inthiscase,thegroovemodulationsshouldoccurregardlessofthebasepairsequenceandtheircharacteristicwavelengthsshouldbedeterminedbythebackbonestiffnessaswellasoverallhelicalpitchanddiameter.Thisex-plainswhythewavesintheleft-handandtheright-handplatesinFig.6haveroughlysimilarscales,eventhoughonlytheA-tractsequenceisperiodical.Inexperiment,however,suchmodulationscanbeobservedonlyiftheirphasesarefixedintime,whichisthecaseofA-tractre-peats.Forrandomsequences,liketheoneweuseasareference,thefinestructureshouldbesmoothedoutonaveragingoverthewholeensemble.

Figure7comparesBI/BIIbackbonedynamicsinthetwoAttrajectories.Therearemanysimilaritiesindy-namicsaswellasinthefinalconfigurations.Theconver-genceisbetternearbothendsandwithinA-tracts.ThedissimilardistributionsoftheconformersinthemiddlecorrespondstothedifferenceinminorgrooveprofilesinFig.6.InA-tracts,theBIIconformersareveryrareinT-strandsandtendtoalternatewithBIinA-strands.Figure8compareslocalhelicalparametersinthelastav-eragestructures.OnlytheBuckleandPropellertracesexhibitlargescalemodulationsphasedwiththehelicalscrew.Allparametersstronglyfluctuateandthesefluc-tuationsareapparentlychaoticwithratherdissimilarphasesinthetwostructures.

Couplingbetweenthelevels

DynamicsoftheminorgrooveprofilesisshowninFig.6.Thereareevidentqualitativeresemblanceaswellassomesubtledifferencesbetweenthesefoursurfaces.InAt-B,thecharacteristicregulargrooveshapehasestab-lishedearly,withsignificantwideningsinthethreezonesbetweentheA-tracts.Theveryleftwideningissome-whatdifferentprobablybecauseitoccursbetweenanti-parallelA-tracts.InAt-A,theprofilestronglychangedatthebeginning,butalsoestablishedbytheendofthe10thnanosecond.AlthoughthefinalAt-AandAt-Bpro-filesarenotidentical,theyareclearlysimilar,withgoodcorrespondenceoflocalwideningsandnarrowings.

ThetwonAtsurfacesshowlittlesimilaritybetweeneachother,butqualitativelytheirshapesarenotverydifferentfromthosefortheA-tractfragment,withmodu-lationsofsimilarwavelengthsandamplitudes.Thislookssomewhatcounter-intuitivebecause,inexperiments,reg-

Figures5and4demonstratethatAt-AandAt-Bar-rivedatthesamestaticallybentstate.ThisdynamicscontraststhoseofthetwonAttrajectoriesanditstronglysuggestthatthecurvedDNAshapeoftheA-tractfrag-mentisanattractoroftrajectorieswithameta-basinofattractioncomprisingbothcanonicalAandBDNAforms.Figures6-8showthatthebendingconvergenceisaccompaniedbysomecleartrendsinlocalconformationaldynamics.Theselocalfeaturesareprobablycoupledtobending,however,acloselookrevealsthatthiscouplingisveryloose.Theconvergenceoftheminorgroovepro-filesinFig.6isatbestqualitative.Figure7indicatesthatactivebackbonedynamicscontinuedafterthecur-vaturehasestablishedandthatonecanpickupratherdifferentdistributionsofconformersfromtheensembleofbentstructures.ThenoisytracesinFig.8obtainedbyaveragingovertwosimilarlybentensemblessuggestthatthehelicalparametersarefarfrombeingconstant.Thenaturalconclusionfollowsthatconvergenceofthebendingdynamicsdoesnotrequireuniquespecificlocalconformations,i.e.thatthebentstateismicrohetero-geneous.

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129At−A6305101520At−BFIG.5.A2DdensityplotofthermsdifferencebetweenAt-AandAt-B.Conformationsspacedby2.5psintervalswerefirstaveragedoverof50and25psintervalsinAt-AandAt-B,respectively,andtheresultingstructurescomparedbetweenthetrajectories.Darkershadingimpliessmallerrmsdvalues.Thelowerleftcornercorrespondstotheinitialstructures,thatisthecanonicalA-andB-forms,withrmsdabout10.7˚A.Theshadedrectangleintheupperrightcornerdemonstratesconvergence

Aarenotshadedwhereasinthedarkestzonesitfallsdownofthetwotrajectoriestothesamebentstate.Thevaluesabove4˚

˚to1.3A.Theblackverticalbandatapproximately10nsindicatesthatAt-Ashortlyvisitedthefinalstate3nsbeforethe

definitetransition.

At−A048121620ns10A86nAt−A048121620ns10A86AATAGGCTATTTTAGGCTATTTTAGGCTATT4AGATAGTATGACTATCTATGATCATGTATGA4At−B024681012ns10A86nAt−B121620ns10A86AGATAGTATGACTATCTATGATCATGTATGA4AATAGGCTATTTTAGGCTATTTTAGGCTATT4FIG.6.Thetimeevolutionoftheprofileoftheminorgrooveinthefourtrajectories.Thesurfaceplotsareformedbytime-averagedsuccessiveminorgrooveprofiles,withthatonthefrontfacecorrespondingtothefinalDNAconformation.ThegroovewidthisevaluatedbyusingspacetracesofC5’atoms.Itsvalueisgiveninangstr¨omsandthecorrespondingcanonicalB-DNAlevelof7.7˚Aismarkedbythethinstraightlinesonthefacesofthebox.ThesequencesareshownforthecorrespondingtopstrandsinFig.1withthe5’-endsontheleft.TheA-tractsareunderlined.Notethatthegroovewidthcanbemeasuredonlystartingfromthethirdbasepairfrombothtermini.

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(a)At−AAAAATAGGCTATTTTAGGCTATTTTAGGCTATTTT00At−BAAAATAGGCTATTTTAGGCTATTTTAGGCTATTTT4147.7872144.0154Time (ns)Time (ns)-210.74586128-228.714101620TTTTATCCGATAAAATCCGATAAAATCCGATAAAA12TTTTATCCGATAAAATCCGATAAAATCCGATAAAA(b)AAAATAGGCTATTTTAGGCTATTTTAGGCTATTTTAt-AAt-BTTTTATCCGATAAAATCCGATAAAATCCGATAAAAFIG.7.(a)DynamicsofBIandBIIbackboneconformersinAt-AandAt-B.TheBIandBIIconformationsaredistinguishedbythevaluesoftwoconsecutivebackbonetorsions,εandζ.Inatransition,theychangeconcertedlyfrom(t,g−)to(g−,t).Thedifferenceζ−εis,therefore,positiveinBIstateandnegativeinBII,anditisusedasamonitoringindicator,withthecorrespondinggrayscalelevelsshownontheright.Eachbasepairstepischaracterizedbyacolumnconsistingoftwosub-columns,withtheleftsub-columnsreferringtothesequencewrittenatthetopin5’-3’directionfromlefttoright.Therightsub-columnsrefertothecomplementarysequenceshownatthebottom.(b)ComparisonofthefinaldistributionsofBIandBIIbackboneconformersinAt-AandAt-Bshowninthesamewayasinplate(a).

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TheMagnitudeandTheCharacterofBendingin

theA-tractRepeat

TheexperimentalmagnitudeofbendingcausedbyA-tractswasearlierestimatedbyseveralgroupswithdiffer-entapproaches61,21,62,32.Thereportedbendangleswerebetween11◦and28◦perA-tract,and18◦ispresentlyconsideredasthemostreasonableestimate9.Thecur-vaturesomewhatvarieswiththebasepairsequenceanddependsuponenvironmentalconditionssuchasthetem-perature,theconcentrationofcounterionsetc.Althoughincalculationsallthesedetailscannotyetbeproperlytakenintoaccountaquantitativecomparisonwithex-perimentisinstructive.

Whenthecurvaturehasestablished,thatisafter13nsofdynamicsinAt-Aandafter3nsinAt-B,thebendangleoscillatedaround60◦(seeFig.2).Intheconsec-utive1ns-averagedconformationsitsvaluewasbetween42◦and74◦,withtheaverageof54◦for16suchstruc-tures.Thisvaluecorrespondsto54/4=13.5◦perA-tract,thatisclosetothelowerexperimentalestimate.Alargervalueof54/3=18◦results,however,ifoneassumes,assuggestedbysomeexperimentalobservations23,63thattheA-tractsarestraight,andthatthebendingactuallyoccursinthethreezonesbetweenthem.Yetanotherestimateisobtainedfromtheincreaseofbendingwithrespecttotheshorter25-merfragmentstudiedearlier25.ItappearsthatoneadditionalA-tractandjunctionzoneincreasetheoverallbendby20-22◦.Weseethatthemagnitudeofbendinginsimulationsisratherclosetoexperimentalestimates,andthattheagreementisbetterifthecurvatureisreallylocalizedinthejunctionzonesbetweenA-tracts.

Figure9presentsacloserlookathowthelocalcur-vatureisdistributedinthelast1-nsaveragestructureofAt-B.Thetotalbendingangleisabout50◦.Threezonescontributemorethanothertotheoverallbend.ThetwojunctionsbetweenA-tracts2,3and4arebentinanidenticaldirectionwhichisclosetothatofthewholestructure.Togethertheycontributearound40◦totheto-talbend,whichisthelargestlocalpositivecontribution.Incontrast,thestronglycurvedfourthA-tractmakesanegativecontributionbecauseitsdirectiondivergesbymorethan90◦.ThethirdA-tractisvirtuallystraight.Finally,A-tracts1and2andthejunctionzonebetweenthemexhibitasmoothcurvaturewithastable“good”directionandcontributetheremaining20◦ofthetotalbend.

AAAATAGGCTATTTTAGGCTATTTTAGGCTATTTT20100-10-20-30105RollTiltSlideTwistRisePropellerBuckle0-5-101.510.50-0.5-1504030204.2543.753.53.25350-5-10-15-20-25-3020100-10-20-30TTTTATCCGATAAAATCCGATAAAATCCGATAAAAFIG.8.Sequencevariationsofhelicoidalparametersinthelast1ns-averagestructuresofAt-AandAt-B.Thesequenceofthefirststrandisshownonthetopin5’–3’direction.Thecomplementarysequenceofthesecondstrandiswrittenonthebottomintheoppositedirection.AllparameterswereevaluatedwiththeCurvesprogram52andaregivenindegreesandangstr¨oms.At-A–solidline,At-B–dashedline.

10

(a)(b)4Local angle202341AAAATAGGCTATTTTAGGCTATTTTAGGCTATTTT1030TTTTATCCGATAAAATCCGATAAAATCCGATAAAA(c)6810ns220151050AAAATAGGCTATTTTAGGCTATTTTAGGCTATTTTAAAATAGCTATTTT1FIG.9.(a)Thelast1-nsaveragestructureofAt-BshownintheXOZprojectionaccordingFig.4a.TheATbasepairs

arehighlighted.(b)Thequantifieddistributionofcurvatureinthestructureshowninplate(a).Thelocalbendingangleisevaluatedbymovingaslidingwindowalongthehelicalaxis.Thewindowsizewas3basepairsteps,withthemeasuredvaluesassignedtoitscenter.ThesequencesofstrandsaregivenasinFig.1withtheA-tractsunderlinedandnumbered.(c)DynamicsoflocalbendinginAt-B.Thesurfaceplotisformedbytime-averagedsuccessiveprofileslikethatinplate(b),withthefrontfaceoftheboxcorrespondingtotheendofthetrajectory.

11

Theforegoinganalysiscertainlyisnotfreefrompit-falls.Forinstance,theapparentsmoothcurvaturecanresultfromtimeaveragingofseveralalternativelocalbends.Nevertheless,Fig.9indicatesthattherearezonesinthisDNAfragmentthatarebentmorethanotherandthattwosuchzonesaredistinguishablebetweenA-tracts.Figure9cdisplaysthelocalbendingdynamicsinAt-B.Itisseenthatthemainfeaturesnoticedinplates(a)and(b)werequitevisibleduringthewholetrajectory.Moreover,thezonebetweenthefirsttwoA-tractsalsosometimescarriedanincreasedcurvature.However,itwouldbein-correcttoconcludethatA-tractsarestraight.Theyjustexhibitgenerallysmalleranddistributedcurvaturethanthejunctionzones.Thiscurvatureisusuallydirectedto-wardstheminorgroove,therefore,itdoesnotcanceloutinaveragedstructures.

Theforegoingpatternagreesqualitativelywiththere-centNMRandX-raydata65aswellasthecharacterofbendingearlierobservedincalculations39,40.Manyear-lierreportedX-raystructuresofA-tractssuggestedthattheyproduceanintrinsicallystraightDNAcomparedtoothersequences63.OurcalculationsdonotcontradicttheseobservationsbecausethecrystalA-tractstructuresshouldbeadditionallystraightenedduetospecialcrys-tallizationconditions30,31,66,andbecauseasingleshortA-tractmayinfactbesomewhatlesscurvedthanthatinsertedinalongDNAfragment.

VerificationofCurvaturebyGelElectrophoresis

Gel startThesequenceinducedstaticDNAcurvaturewasfirstnoticedowingtoreducedmigrationrateofcurvedDNAfragmentsingelelectrophoresis1.LatergelmigrationstudiesprovidedawealthofinformationoncurvatureinA-tractrepeats7.Thedifferenceingelmobilitybe-tweenstraightandcurvedDNArapidlygrowswithchainlength,therefore,thecurvaturewasusuallystudiedinratherlongDNAfragments.Dataforsequencesshorterthan50bparerare27,and,toourbestknowledge,ithasneverbeenshownthatcurvedandstraight35-merscouldbedistinguished.Nevertheless,subtlesequenceeffectsindoublestrandedoligomersofaround10bpweredetectedwithhighergelconcentration67,andonecouldhopethatthiswouldworkforsomewhatlongersequencesaswell.Alternatively,theeffectofbendingcouldbeenhancedbyinsertingconstructedfragmentsintoalongstretchofstraightDNA2,butthiswouldcomplicatefurtheranaly-sisbecausetheDNAmoleculesusedinexperimentsandcalculationscouldnolongerbeidentical.

Figure10showscomparisonoftheacrylamidegelmo-bilityofthesetwofragments.Asexpected,theA-tractrepeatexhibitsareducedrateofmigration.Thediffer-enceisquitesignificantsothatthetwomoleculesarewellresolvedbothinseparatelanesandwhenmixedinthesamesample.Owingtotheidenticalbasepaircontent,theminorfactorssuchasthenumberoftightlybound

AtnAtnAtAtAt+nAtFIG.10.Gelmobilityassay.Thetwo32P-labeled35bpDNAconstructs(AtandnAt)wereelectrophoresedin16%plyacrylamidegelbufferedwithTris-borate,pH8.6.Thegelwasdriedandautoradiographed.ThelaneslabeledAt,nAt,andAt+nAtcorrespondtotheA-tractrepeat,therandomsequence,andtheirmixture,respectively.BandsassignedtoeachDNAfragmentaremarkedbyarrows.

12

counterionsandwatermoleculesisreducedheretothepossibleminimumand,mostprobably,theobserveddif-ferenceisentirelyduetothecurvatureintheA-tractfragment.

DISCUSSION

ComparisonwithEarlierStudies

tionsandwebelievethisapproachpresentsconsiderableinterestforfuturestudies.SequenceeffectsinDNAfrag-mentsof35-50basepairscanbeprobedinbothMDsimulationsandgelelectrophoresis.Suchexperimentsarerapidandinexpensive,whichprogressivelybecomesthecaseforMDsimulationsaswell.

ComparisonwithTheoriesofDNABending

Toourknowledge,theonlyearliersuccessfulunbiasedsimulationsaimedatreproducingA-tractinducedcur-vatureinDNAhavebeenreportedbyDavidBeveridgegroup39,40.Thesesimulationswerecarriedoutinfullwaterenvironmentwithexplicitcounterions.Thechar-acterofthephasedA-tractbendingappearedoscillatorywithaperiodofatleast3to4ns39.Becausedurationoftrajectorieswasonly5ns,itwasdifficulttoconfirmthestaticcharacterofbendinganddistinguishbetweenessentialandoccasionalobservations.Therefore,con-clusionsconcerningapplicabilityofdifferentmodelswerenotrestrictiveandleftroomformanytheories.Oursim-ulationshavethesamegoalandasimilarsetup,butweuseasimplermodelsystem.TheprimarilylongterminterestinB-DNAmodelswithimplicitorsemi-implicitrepresentationofenvironmentisconnectedwithapprox-imatesimulationsofverylongDNAmolecules38.Asshownheretheminimalmodelcanalsocapture,atleastqualitatively,importantsequenceeffectsliketheA-tractinducedcurvature.

Severalfeaturesinourcalculationscorrespondwelltothoseobservedearlier,notably,spontaneousdevelop-mentofquasi-sinusoidalminorgrooveprofilesinbothA-tractandnon-A-tractsequencesandstrongbendsinjunctionzonesbetweenA-tracts.Incontrasttoearliersimulations,however,thecurvaturehereemergedafterseveralnanosecondsofdynamicsandthedifferencebe-tweentheA-tractandnon-A-tractstructuresdidnotreducewithtime.OneshouldnotealsothattheA-tractstructurescomputedwiththeminimalmodelareveryclosetoexperimentaldataasregardsthehelicalpitchandtheabsolutegroovesizes25,50.InstandardAMBERandCHARMMsimulations,B-DNAalwaysap-pearssomewhatunderwoundandthenarrowestA-tractminorgroovesremain1-2˚Awiderthaninexperimental

39,40,42

structures.Theoriginofthissubtlebiasremainsunclear,andattemptstoreduceithavebeenmadeintheveryrecentmodificationsoftheAMBERforcefield47,68.Intheminimalmodel,thisbiaswascompensatedbyfit-tingreducedphosphatecharges,whichapparentlyim-provedtheA-tractstructuresandstabilizedthecurva-ture.

Asignificantenforcementofthepresentresultscom-paredtoourpreviousreports25,54consistsinthedirectcomparisonofbendinginsilicoandinvitro,whichbe-camepossibleowingtoincreasedlengthoftheDNAfrag-ment.SuchapossibilityisratheruniqueforMDsimula-

TheoriginofintrinsiccurvatureinDNAremainsun-clear.Theoriesthatexplainitalwaysassumesomespe-cificbalanceofinteractionsintheDNAstructure,andthatiswhythesetheoriesareperhapsmoreimportantthantheparticularroleofA-tracts.Thelistofavailableinteractionsiswell-known,butthequestioniswhichofthemisthedrivingforce.Belowwebrieflyanalyzeourresultsincontextsofsometheories.

BasePairStackingModels

Accordingtoanymechanismthatstartsfrombasepairstacking,likethewedgeorthejunctionmodels12,18,2,acurvedDNAmustbebuiltoutofasymmetricblocks,withtheirstructuresdeterminedbybasepairsequence.Thebending,therefore,mustbeaccompaniedbyrep-etitionoflocalstructuresinidenticalsequencefrag-ments.Thisfundamentaltheoreticalpredictionfailsforthestaticbendsobservedhere,whichconfirmsearlierconclusions25,54.Thestructuresofsequencerepeatsinthebentstatearemicroscopicallyheterogeneousandcon-vergencetospecificlocalconformationsisnotnecessaryforbending.Asshownabove,theA-tracttrajectoriesar-riveatasinglebentstate,buttheminorgrooveprofilesinFig.6areonlysimilar,notidenticalaswellaslocalhelicalparametersandbackboneconformationsinFigs.7and8.

CounterionElectrostaticModels

AnalternativemodelthatrecentlyattractedmuchattentionconsiderssolventcationstrappedinA-tractminorgroovesastheinitialcauseofbending24.Theroleofcounterionsinthisphenomenonisrathercontroversial69,65,70,andafewgeneralcommentsarenec-essarybeforeconsideringourresults.BecausestraightDNAstructurescorrespondtosymmetricminimaofelec-trostaticenergybendscanresultfromsymmetrybreak-inginthechargedistribution,namely,ifpositiveexternalchargesaccumulateatoneDNAsideitshouldbendto-wardsthem71,72,73,74,24.However,thesamesituationiswellinterpretedbyothermodelsofbending.Namely,inacurveddoublehelix,thephosphategroupsattheinneredgemustapproach,whichcreatesregionsoflowpotentialthatshouldbepopulatedbycounterionsifthey13

areavailable61.Inthefirstcasethecounterion-DNAin-teractionsaresequencespecificandtheycausebending.Inthesecondcasetheyarestructurespecificandtheystabilizepre-existingcurvature.

Twophysicallydifferentmodelsofsequence-specificcounterioninvolvementcanbedistinguished.Inthefirstonethecounterionsactlocally.WhenacounterionisplacedinoneoftheDNAgroovesbetweentwophos-phategroupstheirelectrostaticinteractionbecomesat-tractive,whichnarrowsthegroove74.Asinsomeear-liermodels75,60,theglobalcurvatureresultsfromagen-eralmechanicallinkbetweengroovedeformationsandbending.Incontrast,thesecondmodelispurelyelectro-static.HeretheminorgroovesofA-tractsactasflexibleionophores24,76andtrapcounterions.SinceinphasedsequencestheyoccuratthesameDNAsidethedoublehelixbendstowardsthemtorelaxthelongrangephos-phaterepulsionattheoppositeside.

ThesecondmodelemploysthegeneralideainitiallyproposedforproteinDNAinteractions71andconfirmedexperimentallyforfreeDNA72.However,itqualitativelydisagreeswithacornerstoneexperimentalobservationconcerningtheA-tractinducedbending,namely,thatanA-tractcanbecharacterizedbyadefinitebendan-gleregardlessofitslengthandthedistancefromotherA-tracts.WhenthelengthofanA-tractexceedsoneheli-calturnbothsidesofthedoublehelixappearneutralized.Asaresult,thecurvatureshoulddecreaseintheseries(A12N9)n−(A14N7)n−(A16N5)nbecausethelengthofthenon-neutralizedN-tractsisreduced,andfurthermore,insequence(A16N5)nthebendangleperA-tractshouldbedrasticallyreducedwithrespecttothatin(A6N5)n,forexample,becausethedistancebetweentherepulsiveN-tractsisincreased.Thesepredictionsapparentlydis-agreewiththeexperimentaltrends77althoughadditionalexperimentsareperhapsnecessarytocheckthem.

ThefirstmodelcannotexplaintheoriginoftheA-tractcurvaturebecauseonlymultivalentcounterionscancausesignificantbends74whereasbendingiscommonlyobservedinbufferscontainingEDTAandotherchelat-ingagents.Also,theoptimalcounterionpositionforthistypeofbendisattheentranceofthegrooveandnotinside,therefore,itcannotbebothstrongandsequencespecific.ThelastargumentagreeswiththerecentMDstudiesofcorrelationsbetweentheminorgroovewidthandpositioningofcounterions.Notably,thereisnosuchcorrelationwhenonlycounterionsinteractingwithbasesareconsidered70.Incontrast,acorrelationexistsforcounterionpositionsatthegrooveentrance78.Thelastobservationcorrespondstothestructurespecificbind-ingbetterthantothesequencespecificone.Structure-specificinteractionscanexplainallexperimentalresultsconcerningthepreferentialbindingofcounterionsinA-tracts79,24,80,whichmakessuchdataintrinsicallyneutralasregardsdifferentmodelsofbending.

Inourcalculationsallcounterioneffectsareconsiderednon-specific,andtheresultsobtainedindicatethatmod-ulationsofDNAgroovesandstaticbendingarephys-14

icallypossiblewithoutbreakingthechargesymmetryaroundDNA.Althoughsimulationsalonecannotprovetherealmechanismallexperimentalandcomputationalobservationstakentogethersuggestthatsolventcounte-rionsarehardlyresponsiblefortheintrinsiccurvatureinDNA,whichbynomeansquestionstheirimportantroleinDNAstructureandfunction.

CompressedBackboneTheory

Themainideaofthecompressedbackbonetheory25wasoutlinedinIntroduction.AllseeminglyparadoxicalMDobservationsfromwhichitoriginallyemergedareconfirmedhere.Notably,thistheorypredictsthat,withanybasepairsequence,thebackbonestiffnessshouldcausesmoothmodulationsofDNAgrooves.Thehelicalsymmetrybecomesbrokenwiththebasepairstackingperturbed,whichcreatesregionsofintrinsiccurvature.Ina“random”DNA,thelocalcurvaturechangesitsdi-rectionwithtimebecausegroovewideningsandnarrow-ingsmigrateslowlyalongthedoublehelix.Asaresult,thegenericDNAappearsstraightonaveragealthoughitiscurvedlocally.Insequenceswherecertainbasepairpropertiesstronglyalternate,thephasesofbackboneos-cillationsappearfixed.Inthiscasethelocalcurvaturecansumuptogivemacroscopicstaticbends,asinA-tractrepeats.ThistheoryconsidersamacroscopicallycurvedDNAasan“idioform”characterizedbytopolog-icalattributes,ratherthanastructurewithfixedatompositions.Theseattributesarethebenddirectionandthephaseofgroovemodulations.Themicroheterogene-ityofthebentstateshouldbeexpectedbecausethesamewavingbackboneprofileiscompatiblewithmanyalter-nativelocalconformations.

Acompetitionbetweenthestackinginteractionsandthebackbonecompressionpostulatedbythistheoryischaracteristicofphysicalsystemscalledfrustrated81.Considerthecommontextbookexampleofthreeanti-ferromagneticspinsinatriangleconfiguration.Theop-timalorientationofeachpairisanti-parallel,butallthreepairscannotbeanti-parallelinatriangle.Thereisalwaysatleastoneparallelpairandthegroundstateappearsde-generate.NowconsideracircularduplexDNAwithaho-mopolymersequence.Thecompressedbackbonecausesgroovemodulations,buttherearenopreferableregionsfornarrowingsandwideningsandthegroundstateap-pearsstronglydegenerate.Thesimilaritybetweenthesetwoexamplesisevident.Oneusualphysicalconsequenceoffrustrationisveryimportantforbiology,namely,thepossibilityofaglassystatewheremicroscopictransitionsaredramaticallysloweddown.TransitionsbetweenwavybackboneconfigurationsinalongDNAcanbeveryslowbecausemanygroovenarrowingsandwideningsmustbemovedconcertedly.Thismayexplainobservationsofsupra-microsecondrelaxationtimesinbendingdynam-icsofrelativelyshortDNAfragments58,59.

Finally,thecompressedbackbonetheoryoffersanewviewofsomeenvironmentaleffectsuponthecurvature.Commonphysicalfactorslikethetemperature,counte-rions,andvariousdehydratingagentsarelong-knowntochangeslightlythehelicalpitchofDNA82,83,84,whichcanreasonablybeattributedtothedependenceofthestateoftheDNAbackboneuponthesolventscreeningofphos-phates.Thesesamefactorsmodulatesignificantlythese-quencespecificityofnucleasesprobablybychangingtheshapeofDNAgrooves85,andproducecomplexeffectsupontheintrinsiccurvature26,27,29,30,31,28.Itseemswisetopostponeanydetailedinterpretationofthesefactsforfuturestudies,butonecanjustnotethatwithintrinsicfrustrationoutlinedaboveaverysmallchangeinthepar-tialspecificbackbonelengthcaninducesignificantglobalchangesintheDNAstructure.

PossibilitiesofExperimentalVerificationof

BackboneCompression

Thethecompressedbackbonetheorydoesnotgivesimplerulesforaprioricalculationofcurvatureinanysequence.Nevertheless,itofferssomepredictionsthatcanbecheckedinexperiments.Itsuggests,forexample,thattheA-tractcurvaturecanberelaxedbyintroducingsingle-strandbreaks.Itseemsinterestingalsotoexaminethepossiblerelationshipbetweenthebackbonecompres-sionandsupercoiling.Thereisaconsensusthatintrin-sicbendsaffecttheshapeofthesuperhelicalDNA86,87.Unlikeothermodels,however,thecompressedbackbonetheorypredictsthattheintrinsiccurvatureshouldvaryundersuperhelicalstressinaratherspecialway.Namely,withapositivedensity,thebackboneisstretchedandthecurvatureofaninternalA-tractrepeatshouldbere-duced.Conversely,thecurvatureshouldincreasewhenthesuperhelicaldensityisnegative.DiekmannandWangearlierobservedthattheA-tractstructurechangesun-dersuperhelicalstress26,andtheirapproachmayserveforamorespecificexperimentalverificationoftheabovepredictions.

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