Citation
Fast Dynamics in Protein Folding: Time-Resolved Fluorescence and Absorbance Studies of Polypeptide Reconfigurations

Material Information

Title:
Fast Dynamics in Protein Folding: Time-Resolved Fluorescence and Absorbance Studies of Polypeptide Reconfigurations
Creator:
PABIT, SERSITA SUZETTE A. ( Author, Primary )
Copyright Date:
2008

Subjects

Subjects / Keywords:
Amino acids ( jstor )
Fluorescence ( jstor )
Internal friction ( jstor )
Kinetics ( jstor )
Lasers ( jstor )
Molecules ( jstor )
Protein folding ( jstor )
Quenching ( jstor )
Solvents ( jstor )
Viscosity ( jstor )

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Sersita Suzette A. Pabit. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
4/30/2009
Resource Identifier:
436098103 ( OCLC )

Downloads

This item is only available as the following downloads:


Full Text

PAGE 1

FASTDYNAMICSINPROTEINFOLDING: TIME-RESOLVEDFLUORESCENCEANDABSORBANCESTUDIES OFPOLYPEPTIDERECONFIGURATIONS By SERSITASUZETTEA.PABIT ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2004

PAGE 2

Copyright2004 by SersitaSuzetteA.Pabit

PAGE 3

DedicatedtoEd,toDaddy, andtothememoryofNinongBudds

PAGE 4

ACKNOWLEDGMENTS IwouldliketoexpressmygratitudeandappreciationtoDr.St eveHagen,my adviser,forbeingapatientandinspiringmentor;andforteac hingbyhisexample. Iconsideritagreathonortohaveworkedwithaveryinsightful andverytalented experimentalist.Ioweamajorpartofmyscienti¯cgrowthtohis guidance.I thankhimforintroducingmetothewonderfulproblemofprot einfolding;and forshowingmehowexcitinglifecouldbeintheinterfaceofph ysics,biology,and chemistry. IthankDr.ArtEdison,Dr.TomMareci,Dr.SergeiObukhov,andD r.David Tanner,forservinginmysupervisorycommittee.Iappreciatet heirtime,e®orts, andinvaluablecomments.Ialsowanttothank:Dr.AdrianRoitbe rgandDr. RobertCohenforusefuldiscussions;Dr.AndrewRinzler,Dr.ArtHeba rdand Dr.FredShari¯,fortheirassistanceinsiliconmicroprocessingdu ringtheinitial stagesofmyinvestigation;Dr.CherianZachariahforsynthesis oftheparvalbumin mutant;andDr.HeinrichRoderoftheFoxChaseCancerCenterfo rcollaborative e®ortsintheinvestigationofthecompactmetastablestateoffe rrocytochrome c .I amgratefultomylabmates,Dr.LinlinQiu,OmjoyGanesh,andPr iiyankShukla, foralltheirhelpandtheircompany.Iwouldliketoacknowle dgetheThreadgill ScholarshipProgramFundforsupportwhileIwritemydissertati on.Lastbutnot theleast,Ithankmyhusband,EdPabit,forhisloveandencoura gement. iv

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .............................iv LISTOFTABLES .................................viii LISTOFFIGURES ................................ix ABSTRACT ....................................xi CHAPTER1GENERALINTRODUCTION .......................1 1.1Overview:TheProtein-FoldingProblem ..............1 1.2PhysicsandProteinFolding .....................4 1.3ScopeofthisDissertation .......................5 2RECENTADVANCESINPROTEINFOLDING .............7 2.1HistoricalPerspective .........................7 2.2FastTechniquesinProteinFolding .................8 2.2.1RapidMixing .........................8 2.2.2PhotochemicalTriggering ...................9 2.2.3Temperature-JumpSpectroscopy ...............11 2.2.4Pressure-JumpSpectroscopy .................12 2.3TheoreticalAdvancesinProteinFolding ..............12 2.3.1KramersRateTheory .....................12 2.3.2EnergyLandscapeTheory ..................16 2.3.3ProteinFoldingRatesandContactOrder ..........18 2.4WhatControlstheSpeedofProteinFolding? ............20 3FASTLAMINAR-FLOWFLUIDMIXER .................23 3.1Introduction ..............................23 3.1.1Turbulentvs.LaminarFlow .................24 3.1.2Turbulent-°owMixersvs.Laminar-°owMixers .......25 3.2BuildingaRapidLaminar-FlowMixer ...............27 3.2.1MixerDesignandOperation .................27 3.2.2Di®usionModelofMixerBehavior ..............31 3.3CharacterizationofMixerPerformance ...............34 3.3.1QuantitativeFluorescenceMeasurement ...........35 3.3.2MeasurementofReactionRates ...............37 v

PAGE 6

3.3.3DeadTimeofMixing .....................43 3.4Conclusions ..............................44 4SMALLPROTEINTRYPTOPHANCAGE ................46 4.1Introduction ..............................46 4.2Two-StateProteinFolding ......................47 4.3Trp-cageStructure ..........................48 4.4EquilibriumStudiesonTrp-cage ...................49 4.4.1Far-UVCircularDichroismMeasurements ..........49 4.4.2FluorescenceMeasurements ..................51 4.4.3AnalysisofTrp-cageTwo-StateFolding ...........55 4.5Trp-cageFoldingKinetics ......................57 4.6DiscussiononTrp-cageFolding ...................58 4.7Conclusions ..............................63 5INTERNALFRICTIONCONTROLSFOLDINGRATES ........64 5.1Introduction ..............................64 5.2Background ..............................65 5.2.1KramersRateTheoryandFoldingDynamics ........65 5.2.2ProteinInternalFriction ...................66 5.3FoldingfromaCompactStateofFerrocytochrome c .......69 5.3.1The M -COandthe M States ................70 5.3.2TransientAbsorptionSpectroscopy ..............74 5.3.3 M ! N :StabilityandFolding ................77 5.4DiscussiononFoldingfromaCompactState ............80 5.5Conclusions ..............................85 5.6FutureDirection ............................86 6GENERALCONCLUSIONS ........................88 APPENDIX ...................................90 AAMINOACIDS ................................90 BNUMERICALMETHODS ..........................91 B.1KineticModellingofLaminar-FlowCoaxialMixing ........91 B.1.1SolutionoftheRadialDi®erentialEquation .........91 B.1.2CalculationofReactionRates ................93 B.2KineticModellingusingSingularValueDecomposition ......94 CFUTUREDIRECTIONOFINTERNALFRICTIONSTUDIES .....98 C.1Introduction ..............................98 C.2Triplet-TripletEnergyTransfer ...................98 C.3ApplicationtoInternalFrictionStudies ...............99 vi

PAGE 7

C.4Modi¯edTransientAbsorptionSpectrometer ............100 REFERENCES ...................................103 BIOGRAPHICALSKETCH ............................113 vii

PAGE 8

LISTOFTABLES Table page 4{1Foldingratesoffastfoldingproteins ...................60 A{1Listof20aminoacidsandtheirproperties ...............90 viii

PAGE 9

LISTOFFIGURES Figure page 1{1Hierarchicalnatureofproteinstructure .................3 2{1Photochemicaltriggeringinitiatesfolding ................10 2{2Foldinginvolvesbarriercrossing .....................15 2{3Foldingenergylandscape .........................17 3{1Laminar-°owcoaxialmixer ........................28 3{2Operationofthelaminarcoaxialmixer .................30 3{3Numericalmodelofthemixingscheme .................32 3{4Laminarmixingincylindricalvs.rectangulargeometrie s .......33 3{5Quantitative°uorescencemeasurementinthecoaxialmixer ......36 3{6KineticsoftheNATA-NBSquenchingreactioninthecoaxialm ixer .38 3{7Quenchingratedependsonradiusofinnerstream ...........40 3{8Kineticsindicatingmixingdeadtime ..................43 4{1Trp-cagestructure .............................48 4{2CDMeasurementsofTrp-cage ......................50 4{3FluorescenceemissionspectrainTrp-cage. ...............52 4{4TemperaturedependenceofTrp-cageandNATA°uorescence. ....53 4{5Temperaturedependenceunderdi®erentsolventcondition s. ......54 4{6EquilibriumunfoldingofTrp-cage. ....................56 4{7KineticsofTrp-cagefolding. .......................58 5{1Foldingfromacollapsedstate. ......................69 5{2Horseheartcytochrome c . ........................71 5{3Absorptionspectraofferrocytochrome c : M -CO ! M ! N states. .72 5{4Transientabsorptionspectrometer. ...................74 ix

PAGE 10

5{5Time-resolvedopticalabsorptiondi®erencespectra ..........76 5{6AmplitudesSV( t )ofspectralchangesobtainedbySVD ........77 5{7Foldingrateisconstantatlowdenaturant ...............78 5{8Viscositydependenceof M ! N foldingtime ..............80 5{9Temperatureandviscositye®ectson k ¡ 1 f ................81 B{1Transientabsorptionspectraofferrocytochrome c ...........96 C{1Transientspectrometerwithpulseddyelaser ..............101 x

PAGE 11

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFul¯llmentofthe RequirementsfortheDegreeofDoctorofPhilosophy FASTDYNAMICSINPROTEINFOLDING: TIME-RESOLVEDFLUORESCENCEANDABSORBANCESTUDIES OFPOLYPEPTIDERECONFIGURATIONS By SersitaSuzetteA.Pabit May2004 Chair:StephenJ.HagenMajorDepartment:Physics Weareatanerawherewearebeginningtounderstandthephysica laspects ofproteinfolding.Theenergylandscapetheoryofproteinfo ldingexplainswhy protein-foldingreactionsaresorapidcomparedtorandomsea rch.Thehigh frictionlimitofKramerstheoryfordi®usion-drivenreactio nsbestdescribesprotein foldingkinetics.However,keybiophysicalquestionsremain,p articularlyinfolding dynamicsatfasttimescales.Whydosomeproteinmoleculesfoldso fast?What physicalparameterscontrolthefoldingratesnearthespeedl imit?Wecontributed tounderstandingproteinfoldingatfasttimescalesbydevelop ingandimproving techniquestoprobesubmillisecondkineticsandbystudyingth efastest-folding proteinsystems:tryptophancageandthecompactlate-stageint ermediateof ferrocytochrome c . Westudiedfastdynamicsinproteinfoldingbytime-resolved°uo rescence andabsorbancemeasurements.Wefabricatedandcharacterized asubmillisecond laminar-°owmixingdevicethatallowsUV-excitationandobserv ationofkinetic °uorescencechangesinpeptideswithpicomolarsampleconsumpt ion.Together xi

PAGE 12

withequilibriumcirculardichroismand°uorescencemeasureme nts,weused temperature-jumpdatatocharacterizethetwo-statefoldin gofthedesigned miniproteintryptophancage.Wehaveappliedlaser°ashphotol ysistothehemeCObondandusednanosecond-resolvedtransientabsorptionspectr oscopytolook atthefast M ! N foldingtransitionsinferrocytochrome c .Wearecurrently developingexperimentsbasedontriplet-tripletenergytra nsfertodotime-resolved opticalabsorptionmeasurementsonrecon¯gurationsoftrypto phan-containing proteinsandpeptides. Theminiproteintryptophancagefoldsin4microsecondsandse tstheconditionsforfastfolding:atwo-statereaction,aweakfoldingac tivationenergybarrier, anearlyoptimizedfreeenergylandscape,andpre-organized structuresintheunfoldedstate.Inferrocytochrome c ,thefoldingtimefromacompactcon¯guration is12microsecondsinwater.Analysisofthesolventviscosity-dep endenceofthe foldingtimeusingamodelbasedonKramersratetheoryallowed ustoidentifytwo limitingtimescalesinproteinfolding:thetimescaleforsolv ent-coupledreorganizationsandthetimescalecontrolledbytheinternalfrict ionwithintheprotein molecule. xii

PAGE 13

CHAPTER1 GENERALINTRODUCTION 1.1Overview:TheProtein-FoldingProblem Thequestionofhowproteinmoleculesassembleintotheirdistin ctbiologically activeformsiscalledtheprotein-foldingproblem.Inthep ost-genomicerawhere completegenomeshavebeensequenced,proteinfoldingremai nsasoneofthemost excitingproblemsinscience,involvingcollaborationamon gthefundamentaldisciplinesofphysics,chemistry,biochemistry,andcomputerscie nce.Understanding howproteinsfoldhassigni¯cantimplicationsfortheadvance mentofstructuraland molecularbiology,biochemistry,drug-design,andmedicine . Theprotein-foldingproblemaddresseshowproteinsfoldinto astable,distinct structurefromaseeminglyrandomheteropolymerchain.Befor efurtherdiscussion, fourpointsaboutproteinsshouldbemadeclearfromthebegin ning: Proteinsarebiomolecularmachines. Theyarethemajorfunctional componentsofalllivingcells.Enzymesareproteinsthatcat alyzebiological reactions.Regulatoryproteinscontrolgeneexpression;andst ructuralproteinsform theframeworktowhichotherproteinsareattached.Therear etransferproteins, membraneproteins,transportproteins,andimmunoproteins[ 1 ].Withoutproteins, thereisnobiology|lifewillnotexistasweknowit. Proteinsmustadoptanessentiallyunique\native"structu rein ordertofunction. Eachspeci¯cproteinhasadistinctstructurecalleda\fold" orthe\native"state.Fromadisorderedchainofaminoacids,pr oteinsmustreach thisstructureinabiologicallyrelevanttimeframetoperfo rmaspeci¯cfunctionin themolecularprocessesoflife.Proteins,therefore,needtof oldinthecorrectway inordertobebiologicallyactive.Proteinmisfoldingorinc orrectproteinfoldingis 1

PAGE 14

2 recognizedasafactorinmajordiseaseslikeAlzheimer'sandBo vineSpongiform Encephalopathy(BSE),commonlyknownasMadCowDisease[ 2 ]. Proteinsarepolymers. Theyaresynthesizedinthecell,from20molecular buildingblockscalledaminoacids.Aminoacids(listedinAppen dixA)arejoined togetherbythepeptidebond,formingapolypeptidechain.T hepolypeptidechain hasacovalentbackboneconsistingofonenitrogenatom,follo wedbytwocarbon atoms,andsoon: ¡ N ¡ C ® ¡ C ¡ .Bondedtothe C ® isaaminoacidsidechain. Eachsidechainisdi®erentforthe20di®erentaminoacids.Thenu mber,position andsequenceofaminoacidsinthechainareencodedintheDNA.Th ehuman genomealoneencodesnearly30,000proteinsequences[ 3 ]. Theparticularsequenceofaminoacidsinthechainiscalledt heprimary structure.Whenaproteinmoleculeisunfolded,onlytheprim arystructure remainsintact.Inafoldedprotein,crossconnectionsarefo rmedbyhydrogen bonds,disul¯debonds,andionicsaltbridgesthatallowsecondary ,tertiary, andquaternarystructures.Figure 1{1 showsthehierarchicalnatureofprotein structure.Ingeneral,hydrophobicamino-acidresiduesform thehydrophobicinner core,whilehydrophilicresiduesaresolventexposed. Proteinstructuresareencodedinthesequence. Thethree-dimensional spatialstructureofaproteinisencodedintheone-dimensiona lsequence[ 5 ]. Therefore,mostsmallwater-soluble,globularproteins( · 300aminoacidresidues) canfoldspontaneously invitro tothenativestategivenfavorableconditions(the righttemperature,pH,solvent,etc.).Insidethecell,somepro teinsneedthehelp ofmolecularchaperonesinordertofoldtothenativestate,w hileotherproteins remainintrinsicallyunstructureduntilitcomesintocontac twiththerightpartner molecules. Therearetwomainaspectsoftheprotein-foldingproblem.On eisprediction ofthethree-dimensionalstructurefromtheone-dimensionalse quence.Thisrelates

PAGE 15

3A. PrimaryStructureB. Secondary Structure C. Tertiary Structure D. Quaternary Structure Multiple-domain proteins Hemoglobin Lambda crorepressor K A T N C (CH)NH 242 C O N C C CH 3 O N C CH(OH)CH 3 C H H H H H H O Polypeptide Chain Peptide Bond a -helix b -sheet turns Single-domain proteins CI2 Cyt c Figure1{1:Proteinstructureiscomplexandisbest-described byitshierarchy. (A)Primarystructureisthelinearsequenceofaminoacids:thep olypeptidechain joinedtogetherbythepeptidebond.Whenaproteinisunfold ed,theprimary structureremainsintact.(B)Secondarystructuresaresmallh ydrogen-bonded structuralunitsformedbyaminoacidsclosetogetherinsequen ce: ® -helix, ¯ -sheet andturns.(C)Tertiarystructurede¯nestheoverallnativefol dofsingle-chain proteins,includingthespatialrelationshipofaminoacidsth atareincontactin thefoldedstatebutarefarapartinsequence.Shownherearepo lypeptidebackbonestructuresoffoldedchymotrypsininhibitor2(CI2)andc ytochromec(Cyt c ).(D)Quaternarystructuresarestructuresofmulti-domainp roteins|several foldedchainsformingonelargefunctionalnetwork.Hemoglo binhas4subunits whilelambdacrorepressorhas2.Theproteinstructuresshownhe rearedrawn usingProteinExplorer(MDLChime),fromthestructuralcoord inatesreportedin theProteinDataBank[ 4 ].

PAGE 16

4 tosequencehomologyandproteinfolds,proteinfunction,pro teinengineering, anddrugdesign.Theotheristounderstandthemechanismofthef oldingprocess itself[ 6 ].Thisapproachfocusesonhowaverylargemolecule¯ndsitsna tivestate inabiologicallyrelevanttimescalewithoutpassingthroughc ountlessstructural alternatives.Ourstudyisconcernedwiththelatterproblem. 1.2PhysicsandProteinFolding Evensmallproteinmolecules,withlengthsintheorderof100a minoacid residues,aremadeupof » 1000atoms.Sinceproteinsaresynthesizedaslinear heteropolymerchains,apurelyrandomsearchthroughallpossib leconformations wouldnotallowfoldingtothenativestructureinaphysiologi caltimescale.In fact,Levinthal[ 7 ]showed,inwhathascometobeknownastheLevinthalparadox, thataproteinwith150residueswouldhaveatleast10 90 possiblecon¯gurations. Samplingallpossibleconformationsatarateof10 12 s ¡ 1 wouldrequire10 70 years, » 60ordersofmagnitudemorethantheentirelifetimeoftheun iverse.Nevertheless,proteinsfoldinminutes,inseconds,orinasmallfractionof asecond. Physicistsworkingontheprotein-foldingproblemareintere stedinfolding timescales.Onegoalistounderstandwhatmakesthisfoldingre actionsorapid comparedtorandomsearch.Theobservationthatdi®erentprote insfoldona varietyoftimescales(secondstomillisecondstomicroseconds) inspiresalotof questions:Whyaresomefoldingreactionsfasterthanothers?Wha tcontrols foldingrates?Isthereafoldingspeedlimit? Weareinterestedinfoldingdynamicsatfasttimescales.Wealsow ishto understandthephysicallimitsofthemostrapidproteinfoldin greactions.Tobe abletodosoentailslookingatthefastesteventsinproteinfol ding:identifyingthe mostelementarystepsinproteinfolding,andstudyingtheforc esandprocesses thatleadtothenativestate.Thus,ourmainthemeisthestudyof fastdynamics inproteinfolding:toshowthatweareabletodevelopandimpr oveonexisting

PAGE 17

5 techniquesofprobingprotein-foldingdynamicsatnanoseco nd-to-microsecondtime scales;andtorelateexperimentaldatatoexistingphysicalmod elsoffolding. 1.3ScopeofthisDissertation Wehaveconductedexperimentalstudiestoimproveunderstand ingoffast eventsinproteinfolding.Wehaveusedrapid-mixingandlaser -basedoptical triggeringtoinitiateandobservefoldingatfasttimescales.W ehavecharacterized twofastfoldingsystems:thetryptophancageminiprotein,oneo fthefastest foldingprotein-likemoleculesmeasuredtodate,andthecom pactlate-stage intermediateofcytochrome c . Thisdissertationpresentsthreeindependentprojects:(1)the developmentof alaminar-°owcapillarymixerforuseintime-resolved°uorescen cekineticstudies ofproteinfolding,(2)thecharacterizationoftwo-statefo ldingoftryptophan cageusingequilibriumcirculardichroismand°uorescencestudi estogetherwith kineticdatafromtemperature-jumpmeasurementsand(3)the studyofinternal frictioninacompactmetastableformofthehemeproteincyto chrome c usinglaser °ashphotolysisandtime-resolvedopticalabsorptionspectroscop y.Wehavealso describedtheextensionofinternalfrictionstudiesofotherp olypeptideandprotein systemsusingthetripletstateofthenaturalaminoacidtryptop hanasprobe. Thedissertationisthereforeorganizedtopresenttheindepen dentprojectsina coherentmanner.Followingthisgeneralintroduction,Cha pter2presentsasurvey ofrecentadvancesinthe¯eldofproteinfolding.Particular emphasisisgivento experimentaltechniquesthatprobefast(submillisecond)eve ntsinproteinfolding; andtorecentdevelopmentsintheoriesandmodelsthatgivep hysicalunderstanding oftheratesoftheprotein-foldingreactions. Chapters3,4,and5arestand-alonechaptersonthe3independ entprojects undertaken.Chapter3isonthefabricationofalaminar-°owm ixer.Chapter4is acasestudyonthetryptophancageminiprotein.Chapter5discu ssesthelimiting

PAGE 18

6 e®ectofproteininternalfrictiononfoldingrates.Chapter6 providesasummary andgeneralconclusions.Severalappendiceshavealsobeenpro videdtosupplement informationonthesubjectsbeingdiscussed.AppendixAliststhep roteinbuilding blocks:the20aminoacidsandtheirproperties.AppendixBdiscu ssesingreater detailthenumericalmethodsemployedinChapters3and5.La stly,AppendixC describestheextensionoftime-resolvedopticalabsorptionex perimentsusingthe tripletstateoftryptophan.

PAGE 19

CHAPTER2 RECENTADVANCESINPROTEINFOLDING 2.1HistoricalPerspective ChristianB.An¯nsen[ 5 ]wasawardedtheNobelPrizeinChemistryin1972, forshowingthatchemicallydenaturedproteinsrefold invitro tothenativestate whenbroughtintothecorrectfoldingconditions.An¯nsenandco workersputforth thegoverningprincipleinunderstandinghowproteinsfold. Theyshowedthatthe amino-acidsequenceencodesthebiologicallyactivenative stateandthisadvanced thethermodynamichypothesis:Thereversibilityoffoldingto thenativestate meansthatthefoldednativestateisthermodynamicallystabl e.Ofalltheavailable structures,thenativestatehastheminimumfreeenergy. Despitemorethan30yearsofworkafterAn¯nsen's,theprotein-fo lding problemcontinuestobeoneofthemissinglinksinthe°owofinfo rmationbetween thegenesequenceandthethree-dimensionalstructureofthepr otein[ 8 ].Key biophysicalquestionsremain.Whyisthefoldingtimesofastcom paredtorandom search?Whataretheforcesthatgovernthefoldingspeed?What isthespeedlimit ofproteinfolding?Whatisthecompletemolecularpictureo ftheprotein-folding reaction? However,thepastdecadehaspresentedmajoradvancesinthe¯eld ofprotein folding.Anumberofexperimentaltechniqueswithverygood timeresolution havebeendevelopedthatallowobservationofproteinfoldin gonthenanosecond andmicrosecondtimescales.Newtheoreticaladvancesalsoillum inatetheroad tounderstandingproteinfolding.Thekeyistocombineresult sfromdi®erent experimentaltechniquesandtoclarifytheearliestfolding events[ 8 , 9 ]. 7

PAGE 20

8 Thefollowingsectionsreviewthefastestexperimentaltechni quesinprobing theprotein-foldingreaction,andthemostrecenttheoretic aladvancesinprotein folding. 2.2FastTechniquesinProteinFolding Thedevelopmentoffastexperimentalprobeswithmicrosecond tonanosecond timeresolutionhasopenedthetimewindowforobservingandun derstanding thefastesteventsinproteinfolding.Thegoalofallfasttechn iquesistoinitiate, inaveryshorttimeframe,astrongthermodynamicperturbatio ntounfoldor refoldaprotein.Thesubsequentfoldingandunfoldingkineti cscanthenbe observedspectroscopically.Therearefourwaystoinitiatefa stfoldingorunfolding: rapidmixing,photochemicaltriggering,temperature-jum p,andpressure-jump spectroscopies.Thereisnosinglemethodofchoiceinmonitorin gfasteventsin proteinfolding.Thevarioustechniquescomplementeachot herandarenecessaryin buildingaglobalunderstandingoftheproblem.2.2.1RapidMixing Inamixingexperiment,proteinfoldingorunfoldingisinit iatedbyrapidly changingthesolventproperties.Tofoldaprotein,asuddenshif tfromgoodsolvent tobadsolventmustbeachieved.(Throughoutthetext,weadopt polymerphysics terminologyof\goodsolvent"and\badsolvent".Agoodsolvent ,typicallyan addedchemicaldenaturant,unfoldsorextendsthepolypept idechain,whileabad solventfoldsorcollapsesthechain.)Forexample,anunfolde dproteindissolvedin ahighconcentrationofchemicaldenaturantwillstarttofol doncethedenaturant concentrationislowered.Likewise,suddenincreaseordecrea seofhydrogenion concentrationscanleadtoarapidchangeinpHtoonethatfav orsthefolded state[ 10 ].Rapidmixingprovidesthemostrobustwaytoperturbaprotei n system.Mixingdirectlychangessolutionconditions,andalmost anyprotein canbeunfoldedinasolventwithhighconcentrationofchemic aldenaturant[ 6 ].

PAGE 21

9 Commercialstopped-°owmixersareavailablewithobservation timewindow beginningfromthemillisecondtimeregime,hence,foldinge ventsoccurringbefore 1msarenotobserved.Therefore,themajordrawbackinmixing experiments remainstimeresolutionandsampleconsumption. Continuous-°owmixershavebeendevelopedwithmicrosecondti meresolution andwithmixingdeadtimesintheorderof50 ¹ s.However,continuous-°owmixing technologybasedonturbulentmixingstillconsumesvastquanti tiesofsample.A silicon-basedlaminar-°owmixerusinghydrodynamicfocusingto facilitatefast mixingbydi®usionhaslowersampleconsumption[ 11 ].Unfortunately,silicon isopaquetofar-UV-based°uorescencespectroscopycommonlyusedto monitor proteinfolding.ThedevelopmentofaUV-transparent,laminar -°owmicrosecond mixerisdiscussedinChapter3.2.2.2PhotochemicalTriggering Laser-basedtriggersarenaturallythefastestwaytoinitiatep roteinfolding. Inaphotochemicaltriggeringexperiment,lightpulsesareu sedtotriggerchemical reactions.Photonsfromnanosecond-pulsedlaserscanbreakspec i¯cbondsor facilitatefastelectrontransferthatleadsproteinmolecul estothenativestate[ 10 ]. The¯rstfast-foldingexperimentsusedananosecondlaser°ashtopho todissociatecarbonmonoxide(CO)fromthehemeofunfoldedcytochrom e c [ 12 ].This approachtakesadvantageofthefactthatCObindspreferent iallytothehemein theunfoldedstateandpreventstheproteinfromfolding.Whe ntheCOisphotodissociatedfromthehemebyalaser°ash,themoleculeisfreeto formnative ornonnativecontacts,allowingobservationandmeasuremento fratesofinitial contactformationsoftransientspeciesastheproteinsamples itsconformational space[ 12 , 13 ].Figure 2{1 illustratesphotochemicaltriggeringincytochrome c and inadesignedpolyalaninepeptide[ 10 ].

PAGE 22

10 laser flash photolysis folding native contact nonnative contact CO laser flash photolysis A.B. Figure2{1:Inaphotochemicaltriggeringexperiment,phot onsfromalaser°ash areusedtoinitiatechemicalreactions.(A)TheCOboundtotheh emeofunfolded cytochrome c isphotodissociatedbyalaser°ash.Thisallowsobservationoftra nsientbindingbetweennativeandnonnativecontacts[ 14 ].(B)Anonnativedisul¯de bondforcesthepolyalaninepeptide(farleft)intoalessord eredstructure.Laser °ashphotolysisofthedisul¯debondinitiatesfoldingtowardthe nativehelical structure[ 10 ]. Themainadvantageofusinganopticaltriggerliesinopening ahugetime windowforobservationofproteinrecon¯gurationandfolding :fromnanoseconds tomicrosecondstoseconds,aprocessthatspansupto10ordersofm agnitudein time[ 14 ].However,mostfast-foldingstudiesusingaphotochemicaltrig gerrequire signi¯cantamountsofchemicaldenaturants[ 10 ].InChapter5,weshowthat optically-triggeredfoldingfromacompactstateofcytochr ome c canbestudied atverylowdenaturantconcentrations.Thisisofinterestbec auseproteininternal frictione®ectsdominatethefoldingratewhenfoldingfroma collapsedstate. Recently,photochemically-triggeredfastdynamicsexperi mentshavebeen extendedfromhemeproteinstomodelpeptidesystems.Asdemonstr atedin Figure 2{1 B,Volkandcoworkers[ 10 ]usedalaser°ashtocleaveanonnative disul¯delinkthatforceshelicalpeptidestoformanonnative conformation.

PAGE 23

11 Kiefhaberandcoworkers[ 15 , 16 ]useddi®usion-controlledtriplet-tripletenergy transferfromnonnativedonorandacceptormoleculestolook attheinitial rateofcontactformationinpeptides.Lapidus etal. [ 17 ]madeuseofnaturally occurringaminoacidsincontactformationexperiments:the excitedtripletstateof tryptophanisquenchedbyacysteineresidue.Quenchingofthe tryptophantriplet statebycysteine,however,isnotadi®usion-limitedreaction[ 18 ].Investigationsof tryptophantripletquenchingbynaphthalenearediscussedinAp pendixC. 2.2.3Temperature-JumpSpectroscopy Proteinsandpeptidesarestableonlywithinalimitedtemper aturerange. Therefore,anotherwaytoopticallytriggerfastfoldingoru nfoldingistogenerate asuddentemperaturejumpof5to20 ± C.Mosttemperature-jumpspectrometers useintensenanosecondlaserpulsesatawavelengtheasilyabsorbed bywater, pulsesfromaRamanshifted1064-nmNd:YAGlaser.Thesolutionish eated inafewnanosecondsbyvibrationalexcitationoftheO-Hstret chingovertone ofwater[ 6 ].Resistiveheatingwithanelectricdischargehasalsobeenused to facilitatetemperaturechange.Temperature-jumpexperim entsusingresistive heatinghaveatimeresolutionofintheorderof10 ¹ s. Thelaser-inducedtemperaturejumpismoregenerallyapplic ablethana photochemicaltrigger,becausemostproteinscanbeperturbe dbyarapidtemperaturechange.Theobservationtimewindow,however,isli mitedtothethe timerangewhenthesolutionremainsatanelevatedtemperatu re:nanoseconds tomilliseconds,or3to4ordersofmagnitudeintime.Temperat ure-jumpspectroscopyhasbeenusefulinthestudyofthehelix-coiltransitio n,collapsedynamics ofproteinsandproteinfragments,andthefoldingofpeptide sandsmallproteins[ 19 , 20 , 21 , 22 ].ThefoldingtimeoftheTrp-cageminiprotein,thesubjecto f Chapter4,wasmeasuredusingatemperature-jumpspectrometer .

PAGE 24

12 2.2.4Pressure-JumpSpectroscopy Arapidchangeinpressurealsoproducesenoughthermodynamicp erturbation totriggerproteinfoldingandunfolding.Schmidandcowork ers[ 23 ]useastack ofpiezoelectriccrystalstoapply10to20MPaofpressuretoa50 ¹ Lcoldshock proteinsolutionin50to100 ¹ s.Foldingdynamicscanbeobservedfrom70 ¹ sto 70s,anaccessibletimerangeof6ordersofmagnitude.Forcompl eteunfoldingof aprotein,however,pressuresgreaterthan1000MPaarenecessar y.Pressure-jump experimentsrequirethesampletobeclosetoathermallyorden aturant-induced foldingtransition. 2.3TheoreticalAdvancesinProteinFolding Foldingtothenativestateinvolvescomplexmolecularrecog nitionwitha largenumberofweaknon-covalentinteractionsinvolvingt housandsofatoms.The complexityoftheprocessmakesitdi±culttobuildaglobalthe oreticalframework thatspansbothkineticsandenergetics.Thetwomostimportant theoretical advancesarediscussedbelow:theapplicationofKramersreact ion-ratetheory toprotein-foldingkinetics;andthedescriptionofthefoldi ngprocessintermsof amultidimensionalenergylandscape.Recentphenomenologic alobservationof thecorrelationbetweenaquantitycalled\contactorder"a ndfoldingrateisalso presented.Fittingly,Chapter2endswiththeyet-unanswered questionofwhatare thephysicallimitstospeedofproteinfolding,ananalysisofp reviouswork,and whatliesahead.2.3.1ApplicationofKramersReactionRateTheorytoProtein Folding Experimentsshowthatmostsmall, » 100aminoacidsinlength,singledomainproteinsfoldinasimpletwo-stateprocesswithoutdete ctablestructural intermediates.Inthissimplestcase,thereareonlytwostatespr esentinthe system:anexperimentallyindistinguishableensembleofunfold edstructurescalled theunfoldedstate, U ,andthenativefoldedstate, N .Foldingproceedsthrougha

PAGE 25

13 cooperativetransition U k f ;k u ­ N: (2.1) Timedependenceisexponential[ 3 , 24 ];and k f and k u arefoldingandunfolding rateconstants,respectively. Theobservationofcooperativityandexponentialtime-depe ndenceleadsto theinterpretationthatasinglefreeenergybarrierispresen talongthereaction coordinate;andtotheapplicationofreaction-ratetheory totwo-stateprotein folding[ 25 ].Thisissupportedbytheoreticalsimulationsoflatticemod els[ 26 ]. Likewise,exponentialormulti-exponentialtimedependenc eisobservedforcomplex multistateproteinfoldingreactions,extendingtheinterpr etationtoseveralenergy barriersalongamorecomplexreactioncoordinate[ 25 ]. Transitionstatetheory. Conventionaltransition-statetheory(TST) remainsthesimplestandmostcommonanalysisappliedtotwo-stat efolding.In classicalTST,therate, k ,ofthefoldingreactionisgivenby k TST =( k B T=h )exp( ¡ ¢ G u !z = ( RT ))(2.2) where k B istheBoltzmannconstant, h isthePlanckconstant, T istemperature, R isthemolargasconstantinkJ/mol,and¢ G u !z istheGibbsfreeenergyof activationforformingthetransitionoractivatedstate[ 3 ]. AlthoughtheTSTisstillcommonlyusedforinitialanalysisoffo ldingreaction rates,itdoesnotaccuratelycharacterizetheprotein-fold ingreaction.TSTwas formulatedtodescribeelementarychemicalreactionsofsmal lmolecules[ 27 ].Its pre-exponentialfactor, k B T=h ,correspondstovibrationalmotionsofthechemical bond,ontheorderof6 £ 10 12 s ¡ 1 atroomtemperature,grosslyoverestimating foldingrates.Itdoesnottakeintoaccountdampinge®ectsdue tothesolventand thefactthatprotein-foldingreactionsinvolvesimultaneo usmakingandbreaking ofmanyweaknon-covalentbondsinsolution[ 3 , 28 ].ThekeyassumptioninTST

PAGE 26

14 isthatthereisaone-timebarrier-crossingeventfromreacta nttoproductalong thereactioncoordinate.Laterrecrossingsareconsideredunr elatedevents[ 29 ], althoughEyring[ 27 ]introducedan adhoc fudgefactor · · 1tomake k = · ¢ k TST . Thisreducesthereactionrateandcorrectsforreactivetra jectoriesthatrecrossthe transitionstate. Kramersratetheory. Abetter-suitedapproachforproteinfoldingreactionsisthereaction-rateformalismdevelopedbyKramersin 1940[ 30 ].Kramers theoryincludestheroleofdi®usionandBrownianmotionasdri venbythermal forcesinbarrier-crossingevents.Foralongtime,Kramers'wor kwasonlyappreciatedbyafewtheorists.Recentlyhowever,ithasbeenext endedbyHÄanggi andcoworkerstounderstandkineticsofcondensedphases[ 31 ].HÄanggi etal. [ 31 ] presentsamodernderivationofKramersresultusingNewton'seq uationofmotion intheformofaLangevinequation. M Ä x = ¡ U 0 ( x ) ¡ °M _ x + » ( t )(2.3) M isthemassofaclassicalparticlemovinginaone-dimensionalasy mmetric doublewellpotential U ( x )(Fig. 2{2 )and x isthereactioncoordinate.Inasimple modelofKramerstheory,alltheremainingdegreesoffreedo maredescribedbya random°uctuatingforce, » ( t ),whichobeysthe°uctuation-dissipationtheorem; andalineardampingforce, ¡ °M _ x where ° providestheconstantdampingrate. Forreactionsinsolution,thee®ectofsolventmoleculesisinc orporatedinthe frictionterm ° .Forstrongreactionfriction,thesystemisover-damped;the M Ä x termisverysmallandcanbeeliminatedfromthecalculations. Asopposedto TST,multiplebarrier-crossingeventsareconsideredinratec alculations. Thebarrier-crossingrate, k + = j=n a ,isgivenbythe°ux, j ,overthepopulationatA, n a .The°uxandthepopulationarecalculatedfromaprobability density,whichobeysthestationaryFokker-Planckequation, aroundthebarrier.In

PAGE 27

15 Ux () A B C x wawbwc E b + k + k Figure2{2:Potentialenergy U ( x )withtwometastablestates A and C .Transitions fromstate A to C atrate k + entailscrossingtheenergybarrier, E + b . thehigh-friction( ° À ! b )limit,whichisapplicabletoreactionsinsolution,the famousKramers'resultis k overdamped + = ! a ! b 2 ¼° exp( ¡ ¯E b ) : (2.4) E + b istheactivationenergy, E + b = U ( x b ) ¡ U ( x a ), ! a istheundampedangular frequencyoftheminimumatA,and ! b isthefrequencyofthetransitionstate atB[ 25 ]. ¯ caneitherbe( k B T ) ¡ 1 iftheenergyisinkJ,or( RT ) ¡ 1 forenergyin kJ/mol,whichismorecommonlyusedinbiochemistry.For T =20 ± C=293.15K, RT =2 : 436kJ/mol. DirectapplicationofEquation 2.4 toproteinfoldingposeschallengesin actuallyde¯ningthecurvatureofthepotential,asdescribed by ! a and ! b .Nevertheless,Kramerstheorystillprovidesabetteralternative toTST.Itincludes di®usivee®ectsduetofriction,whichoccuraslargemolecules undergoconformationalchangesinsolution.Inafoldingsimulationstudy[ 32 ]thatincludesthe frictioncoe±cient ° ,Kramerstheoryofbarriercrossingadequatelydescribed foldingrates.

PAGE 28

16 Ifthefriction ° inEquation 2.4 arisesmainlyfromsolventdynamicviscosity, ´ S ,thee®ectsoffrictiononfoldingdynamicsareeasilyprobede xperimentallyby addingexternalviscosity-generatingagents.Experimentsdo showthatfoldingrates followaninversedependenceonsolventviscosity[ 33 ].Thus,foldingratescanbe treatedas k u ! f / 1 ´ S exp( ¡ ¢ G a = ( RT )) : (2.5) ¢ G a istheheightoftheactivationenergybarrier.However,weca nnotexpect k u ! f togrowwithoutlimitas ´ S declines.TheapplicationofKramerstheorytoprotein foldingreactionsandthelimitinge®ectofproteininternal frictioninfastfolding systemsarebothaddressedinChapter5.2.3.2EnergyLandscapeTheory IntheKramerspicture,aone-dimensionaldouble-wellpoten tialseemssu±cientindescribingtherateoftheprotein-foldingreaction .Themultidimensionality oftheprocessisaccountedforbyfriction, ° ,andrandom°uctuations, » ( t )[ 25 ]. ThismakestheKramersformalismapowerfultoolinstudyingpr otein-folding kinetics.However,afullunderstandingofthefoldingprocessst illrequiresaglobal viewoftheenergylandscape. Theenergylandscapetheoryisastatisticalmechanicalmodelt odescribe theprotein'spotentialenergysurface.Theenergylandscape itselfcanbede¯ned asamappingofthepolymerchainconformationtoitsfreeene rgy[ 34 ].For arandomheteropolymer,theenergylandscapeisveryrough.Ho wever,fora protein,whichisakineticallyandthermodynamicallyfold ableheteropolymer, theenergylandscape,thoughalsorough,hasafunnel-shapedre gionthatleads tothenativestate[ 35 ].Thenativestate, N ,correspondstothesmallensemble ofconformationalstructuresattheminimumoftheroughfold ingfunnel,while theunfoldedstate, U ,hasahugeensembleofaccessibleconformationswithlarge

PAGE 29

17 conformationalentropy.Athree-dimensionalprojectionof amultidimensional landscapeisshowninFigure 2{3 . Free Energy Configuration spaceU N Unfoldedprotein Figure2{3:Thefoldingenergylandscape.Theensembleofunfo ldedproteinsat thetopofthefunnelmayfoldtothenativestate, N ,byavarietyofdi®erent routes. Priortotheenergylandscapetheory,proteinfoldingwasdesc ribedinterms ofspeci¯cfoldingpathwaysandintermediatestates.Theenergy landscapetheory showsthatfoldingisaprogressivereorganizationofanensembl eofpartiallyfolded structuresusingnumerousroutestothenativestate.Somepath ways,however, maybemorepopulatedthanothers.Thefunnelshapedescribesco nformations thatleadtothenativestate,andisnecessarytoovercometheLe vinthalparadox[ 7 , 34 ].Theone-dimensionalreactioncoordinatecanbeviewedasa projection ofthemultidimensionalpotentialenergysurfaceintoaparti cularreactioncoordinate(nativecontacts,radiusofgyration,solvent-accessible surfacearea,etc.)[ 25 ].

PAGE 30

18 Theapplicationoftheenergylandscapeformalismisoneofthe majorbreakthroughsinthetheoryofproteinfolding.Energylandscapet heorygivesaclearer pictureofprotein-foldingenergetics.Foldingisaprogressi velossofcon¯gurational entropy,leadingtotheminimumfreeenergy.Socci,Onuchic andWolynes[ 36 ] showedthatkineticsoffoldingcanbedescribedasadi®usivefun nel-likeprocess representedbyasimpledi®usiveratetheory,Kramerstheoryext endedtotakeinto accounttheaccounttheruggednessofthefreeenergylandscap e.Inthesimplest scenario,foldingtime ¿ F canbewrittenas[ 37 ] ¿ F = ¿ C exp( ¯ ¢ G z )(2.6) where ¿ C isthetime-scaleforcon¯gurationalreorganizationofaprot eininthe reactioncoordinate,takingintoaccountdampinge®ectsand theruggedenergy landscape. G z isafreeenergybarrierforthefoldingtransition.Quantify ing ¿ and G z ,however,isnottrivialbecausemanyfactorsdeterminingfo ldingratesarestill notknown.2.3.3ProteinFoldingRatesandContactOrder Inastudyofgeneralequilibriumpropertiesofsingle-domain proteinsthatfold withasimpletwo-statetransition,Plaxco etal .[ 38 ]haveshownthattopological complexityhasamuchstrongercorrelationtoproteinfoldin gratesthandochain length,stability,unfoldingfreeenergy,andtransitionstat eplacement.They quantifytopologicalcomplexityintermsofaparametercal ledrelativecontact order( CO ), CO = 1 L ¢ N N X ¢ S ij : (2.7) N isthenumberofcontacts, L isthetotalnumberofamino-acidresiduesinthe protein,and¢ S ij isthenumberofaminoacidsseparatingtheinteractingresidu es, i and j .Tworesiduesareconsideredincontactwhentheyhavenon-hyd rogen atomsthatarewithin6.0 ºAapart.Adjacentresiduesthatareincontactare

PAGE 31

19 assumedtohave¢ S i;i +1 =1.Relativecontactordersarecalculatedfromstructural coordinatespublishedontheProteinDataBank[ 4 ].Ahigh-contact-orderstructure wouldhavemorenonlocalinteractions(highlyfolded).The rearecontactsbetween morepairsofresiduesthatarefartherapartinthechain.Pro teinswithlower contact-orderstructuresexhibitfasterfolding.Theprotei nsstudiedhaveno disul¯debondsandno cis proline 1 residues. Thecorrelationcoe±cientbetweenrelativecontactorder, CO ,andthenatural logarithmoftherefoldingrateinwater,ln( k ),is ¡ 0 : 81.Thefoldingratesofthe proteinsconsideredhavebeendeterminedusingdi®erentmetho dsinthepresence ofdenaturantandthenextrapolatedtowatersolvent.Correl ationcoe±cients betweenln( k )andotherequilibriumpropertiesareverypoor.Forthefre eenergy ofunfolding,whichdeterminesnativestatestability,corre lationcoe±cientis0.13. Thecorrelationcoe±cientbetweenchainlengthandln( k )is ¡ 0 : 20[ 38 ]. Although,thecontactorderparameterseemstobesuccessfulinde scribingfoldingrates,fastfoldingpeptides,likethetryptophanc age(discussedin Chapter4),havelarge CO andareobviousoutliersintherelativecontactorder correlationplot.Recently,otherparametershavebeenfou ndthatcorrelateslightly betterthan CO [ 39 , 40 ].Absolutecontactorder,arenormalizationof CO ,isproposedtoincludemultistatefoldingproteinsandshortpeptide s[ 40 ].Allthese, however,areempiricalcorrelationsbasedonexperimentald ata.Althoughthereisa 1 Inthepolypeptidechain,thepeptidebondsenergeticallyf avorthe trans conformation.Anexceptiontothisruleisthepeptidebondimmed iatelyprecedinga prolineresiduewhich,inthenativestate,canadopteitherth e cis or trans conformationalmostisoenergetically.Inunfoldingandrefoldinge xperiments,proteins with cis prolinescouldhaveslow(10to20s)refoldingphasesdueto cis-trans isomerization.Intheunfoldedstates,mixturesof cis and trans prolinesarepresent. Inthefoldedstate,however,apeptidebondimmediatelyprec edingaprolinehas onlyoneform,either cis or trans [ 3 ].

PAGE 32

20 clearlinkbetweencomplexityandtopologytofoldingspeedsh owingthatentropic searchisimportant,theunderlyingphysicsisnotknownexact ly[ 25 ].Nevertheless, thecorrelationsshowthatfoldingrateandtopologicalcomp lexityaremorerelated thanotherequilibriumpropertiesofproteins.Qualitative ly,long-rangeordernarrowstheconformationsearchspaceandreducestheentropicco ntributionofthe folding-energybarrier.Therefore,thereisneedforquant itativetheoriesrelating foldingkineticsandtopology,likethetopomersearchmodel fortwo-stateprotein folding[ 41 ].Inaddition,itisimportanttonotethatvariationsdueto site-directed mutagenesis,thoughmaintainingthestructureoftheprotein, changethefolding ratesdramatically[ 24 ].Thismeansthatfactorsotherthantopologycontribute signi¯cantlytofoldingrates. 2.4WhatControlstheSpeedofProteinFolding? Fromtherelativecontactordercorrelation,ithasbeenshow nthatfolding ratessomehowdependontheoveralltopologyoftheproteinmo lecule.But,what actuallylimitsthespeedofproteinfolding? Proteinsfoldinawiderangeoftimescales:microsecondstomin utes.The rateorspeedoffoldingfromtheunfoldedtothenativestateis determinedbythe slowestevent,therate-limitingstep.Veryslowfoldingreacti onsarelimitedby prolineisomerizationandoligomerization.Thelowerbound tothefoldingspeedof proteinsisdeterminedbybiology.Aeucaryoticcelladdsab out2aminoacidsto apolypeptidechainpersecond[ 42 ].Bacterialcellsmanufacturesproteinsfaster. E.coli cansynthesizepolypeptidechainsatarateof10to20aminoaci dsper second.Proteinmoleculesinthecellmustfoldimmediatelyaf tersynthesisorrisk beingcleavedbyproteases.Anaverage-sizedproteinmoleculeo f » 300amino acidscanbesynthesizedin » 20seconds.Thissetsalowerlimittofoldingspeed ofapproximately1/min[ 6 ].However,knownfoldingratesofsimplesingle-domain two-statefoldingproteinsaremuchfasterthan1minute[ 24 ].Therefore,theupper

PAGE 33

21 boundtothefoldingspeedisnolongerdeterminedbybiologic alneedbutislimited byintramolecularphysicalinteractions. Thephysicalinteractionsinfoldingfromadenaturedrandom -coilstatetothe nativecon¯gurationinvolveacomplexseriesofevents(initi alpolypeptidecollapse, loopandcontactformation,developmentofsecondarystructu re)thechronology ofwhichisstillnotverywellunderstood.Inrecentyears,howe ver,therehasbeen anincreaseintime-resolvedexperimentsthatprovideaframe workforprobingthe timescalesofformationofthebasicstructuralelements.Alphah elicesformin about400to800ns[ 43 , 44 ],beta-hairpinsfoldontheorderof1to10 ¹ s[ 45 , 46 ] andcontactformationindisorderedloopsoccursin10to200n s[ 16 , 47 ]. Theinitialestimatetotheupperlimitoffoldingspeedwasmad ebyHagen et al. [ 13 ]byscalingtheirmeasuredrateofinitialcontactformationi nanunfolded proteintothesimplestatisticalmechanicaltheoryofSzabo,S chultenandSchulten ontheend-to-endcontactratesforidealpolymerchains(SS Stheory)[ 48 ].They claimthattheinitialcollapseandfoldingtimeofaproteinc annotbefasterthan thetimeforashortloopformation( » 1 ¹ sfora » 10-residueloop).Other estimates,alsobasedoncontactformationexperiments,yieldfa sterrates( » 100ns fora10-residueloop)[ 47 ].Thefastestfoldingproteinsmeasured,the20-residue tryptophancage,the35-residuevillinheadpeacesubdomain, andthe23-residue BBA5,foldin4.1 ¹ s[ 49 ],4.3 ¹ s[ 50 ],and7.5 ¹ s[ 51 ]atroomtemperatures, respectively.The73-residue ® 3 D3-helixbundlefoldsin3.2 ¹ s[ 52 ]at50 ± C. Despitetheseadvances,thefactorscontrollingthefoldingra tesinthisupper limitarestillnotwellunderstood.Muchcanbelearnedfromfo ldingtrajectories ofall-atommolecular-dynamicssimulationsthatarebeginn ingtoapproach microsecondtimescales.Someproponentssaythatthefastestfold ingproteinslose theirfreeenergybarrierandfollowthe\downhill"folding scenarioofBryngelson et al. [ 53 , 54 ].InChapter5,weshowthatiftheinitialstageofcollapseisby passed

PAGE 34

22 andfoldingcommencesfromacompactlate-stageintermediat e,thenprotein internalfrictioncontrolsthefoldingspeed.

PAGE 35

CHAPTER3 LAMINAR-FLOWFLUIDMIXERFOR FASTFLUORESCENCEKINETICSTUDIES 3.1Introduction Thedevelopmentofnewexperimentaltechniquesfortrigger ingandprobing fastconformationaldynamicsandfoldinginproteinshasgre atlyadvancedthe understandingofprotein-foldingmechanisms[ 6 , 55 ].Theexperimentaltechniques withgreatestimpactincludelaser-basedopticaltriggersfor folding,temperaturejumpspectrometers,andultra-rapid°uid-mixingdevices[ 56 ].Althoughlaser-based triggeringwillalwaysprovidethefastesttimeresolutionink ineticstudies,ultrarapidmixingo®ersanimportantadvantage:arapidsolventcha ngecangenerate alargethermodynamicperturbationonvirtuallyanybiomol ecularsystem.Hence, continuedimprovementofrapidmixingtechnologiesremain sanimportantgoal inthestudyofproteininteractionsandfolding.Inthischap ter,wediscussthe fabricationandcharacterizationofalaminar-°owcoaxialj et°uidmixerforfast °uorescencekineticstudies[ 57 ]. Atthemolecularlevel,themixingoftwo°uidsreliesondi®usio ntogeneratea uniformspatialdistributioninsoluteconcentrations.Timesca lesfordi®usionvary asthesquareofthedistanceoverwhichsolutesmustdi®use,inoned imension t = (¢ x ) 2 2 D (3.1) where¢ x isthecharacteristiclengthscaletravelledbythesolutemole culesat time t and D isthecoe±cientofdi®usion.Therefore,rapidmixingispossible onlyifthelengthsscalesoftheinitialheterogeneityareve ryshort[ 58 ].Typical 23

PAGE 36

24 smallmoleculesoluteshavedi®usioncoe±cientsof D ¼ 10 ¡ 5 cm 2 /s,mixingat submillisecondsrequiresthelengthscaleofthe°ow,¢ x ,notexceed ¼ 1 ¹ m. Di®usioncanbeenhancedbyturbulent°owbecausetheactionoftu rbulent eddiescanpremixthesolutiontomicroscopiclengthscalesand signi¯cantly acceleratemixing[ 59 ].Hence,oneapproachtogeneratefastmixingistocreate turbulencewithinthemixer.Thatis,onecandrivethe°owathi ghReynolds number.Tobeabletodiscusstheadvantagesanddisadvantageso fmixingwith highorlowReynoldsnumber,abriefreviewof°uiddynamicsis calledfor. 3.1.1Turbulentvs.LaminarFlow Thebasicequationthatgovernsthedynamicsofincompressible °uidsisthe Navier-Stokesequation, ½ µ @~u @t +( ~u ¢r ) ~u ¶ = ¡r P + ´ r 2 ~u (3.2) where ½ isthe°uiddensity, ~u isthe°uidvelocity, P isthepressureand ´ isthe °uiddynamicviscosity.Equation 3.2 issimplyanexpressionofNewton'slawfor °uidswiththeinertialtermsontheleft-handsideandtheforc esperunitvolume, duetoapressuregradientandviscosity,ontheright-handside.Ad ditionalforces actingonthe°uid,ifconsidered,addtotheright-handside[ 58 ]. Adimensionlessparameter,theReynoldsnumber,determinesm anyproperties ofthe°ow.TheReynoldsnumber, < ,measures,inorderofmagnitudes,theratio oftheinertialforces, ½ ¯¯¯¯ @~u @t +( ~u ¢r ) ~u ¯¯¯¯ » ½ µ u ¢ t + u 2 ¢ l ¶ » ½u 2 ¢ l ; (3.3) totheviscousforceactingonthe°uid, ´ jr 2 ~u j» ´u ¢ l 2 : (3.4)

PAGE 37

25 Therefore,theReynoldsnumber[ 60 ]isgivenby < = inertialforces viscousforces = ½u ¢ l ´ (3.5) where¢ l isthecharacteristicspatialdimension,suchasthediameterof the°ow channel.LowReynoldsnumber(
PAGE 38

26 mixerswithmixingdeadtimesof » 100 ¹ sorbetter,haveatypicalsampleconsumptionof » 0 : 5to2mL/s.Infact,proteinconsumptioninturbulentmixing studieshasbeenashighas8mg/s[ 66 ].Thescalingof Q with < suggeststhatall butthesmallestturbulentmixerswillprobablyconsumelargea mountsofsample. Todriveanaqueoussolutiontothethresholdofturbulence(i. e.,to <» 2 £ 10 3 ) whileconsumingonly1mL/minofsample,wouldrequireforacyl indricalchannel, adiameteroforder d ¼ 10 ¹ manda°owspeed u ¼ 210m/s.Themanytechnical andpracticaldi±cultiespredictedbythisroughestimate,to getherwithaninterest inperformingrapidmixingexperimentsonpoorly-expressed( orotherwisescarce) proteins,motivatesthesearchforalternativestoturbulent mixing.Laminar-°ow mixingisanidealoptionbecausethemajorrequirementforla minar°owisvery lowReynoldsnumberandconsequently,verylow°owvelocities. Rapidmixingcanbeaccomplishedunderlaminar°owconditions, ifrapid di®usioncanbepromotedbydrawingthemixingsolutionsintoth inparallellayers (themicro°uidicapproach).Knight etal. [ 11 ]constructedsuchamixerusing siliconmicroprocessing:photolithographyandreactiveionetching.Theyused arectangularhydrodynamic-focusinggeometry,inwhichath insheetof°owing sampleispinchedbetweentwostreamsofdilutingbu®er;solutes di®useintothe sheetinalargelyone-dimensionalmanner.Pollackandcowork ersusedthismixer designinsmall-anglex-rayscattering(SAXS)experimentsofpro teinandRNA foldingdynamics[ 67 , 68 , 69 ],withtimeresolutionintheorderof » 350 ¹ s. Althoughtheseadvancesdemonstratesubmillisecondmixing,the ysuggest thattheonlyapproachtolaminar-°owmixingexperimentsare elaboratedetection techniques,likeSAXS,andsiliconorsilicon-relatedmicroproc essing.Onerelated processisreplicamoldingofreliefpatternsinsiliconusingt heelastomericpolymer poly-dimethylsiloxane(PDMS)[ 70 ].Whilemicro-machiningandandreplica moldingpatternsinsiliconisnotcomplicated,thereareissue sinapplyingthese

PAGE 39

27 techniquesinproteinfoldingexperiments|likethemechan icalweaknessofsilicon andcomplicationsinthe°owandsealing.Themostcompellingof theseissues istheinabilityofsilicon-basedandPDMS-basedmixerstoallo wobservationof mixingunderultraviolet(UV)illumination.BecausePDMSandsi liconmixers arenotUV-transparent,thesemixersarenoteasilyadaptedtokin eticstudies thatprobetheexcitation(at260to290nm)andemission( » 350nm)ofthe naturalaminoacidtryptophan,anon-prosthetic°uorophorea ndastandard biophysicalprobefor°uorescencespectroscopystudiesofprotei nfoldingdynamics. Therefore,wehavesoughttodevelopasample-e±cientdeviceth atmixes°uids withmicroseconddeadtimesunderlaminar°owconditions,whil eatthesametime allowingbroadUV-visibleopticalaccesstothesample. 3.2BuildingaRapidLaminar-FlowMixer 3.2.1MixerDesignandOperation Design. Wehaveconstructedalaminar-°ow,sample-e±cient,fastmixer fromfusedsilicacapillariesusingthecoaxialjetgeometry(F ig. 3{1 ).Theuseof fusedsilicacapillaries(PolymicroTechnologies,Phoenix,AZ )allowsexcitation anddetectionintheUVregion.Themixerisassembledunderthem icroscope byinsertingthe¯rstsilicacapillary,20 ¹ minnerdiameter(ID)and66 ¹ mouter diameter(OD),intoalargersilicacapillary.Theoutercapi llaryhasasquare cross-section,withaninsidewidthof100 ¹ m.Thecylindricalinnercapillary carriesathinoutercoatingofpolyimide(90 ¹ mOD)thatactsasaspacer,keeping theinnerandoutercapillariesconcentric.A°uorescentsampl e(e.g., N -acetyltryptophan-amide)°owsoutoftheinnercapillaryandisrele asedintothecenterof a°owofdilutingbu®erthatpassesthroughalargeroutercapilla ry.Theresulting °uorescentstreamishydrodynamicallyfocusedtoanarrowcrossse ctionasitis acceleratedbythefasteroutercapillary°ow.Astheinnerstrea misthusdrawn

PAGE 40

28 2 a OD: 66mID: 20m m m Inner capillary Outer capillary OD: 300mID: 100m m m Spacer90m m Inner capillary stream flowing into the outer capillary solution Figure3{1:Laminar-°owcoaxialmixer.Astreamof°uorescentsam pleexitingtheinnercapillaryishydrodynamicallyfocusedbythefa ster°owofdiluting bu®er(non-°uorescent)passingthroughtheoutercapillary.Thi sdrawsthesample streamintoacolumnofradius a ¼ 1 ¹ m,allowingrapiddi®usionofthesurroundingbu®erinthesampleregion.Theuseoffusedsilicacapillarypr ovidesUVtransparency.Thediagramontheleftshowsmixerdimensions.Theima geontheright isthebackground-subtractedCCDimageofthe°uorescentsample . outintoathincolumn,radialdi®usionofsolutesbetweenthesam pleandthe dilutingbu®ercanoccurrapidly,providingfastmixing. Thisdesignrepresentsamicro°uidicanalogofthelargercoaxi aljetmixer describedbyChristianandcoworkersfor°owcytometryapplica tions[ 71 , 72 ]. Thatdevicemixestwoconcentricstreamsofliquidthroughou tthevolumeofthe observationchannel,ratherthanjustalongthecentralaxis( asinourmixer). Therefore,whenoperatingunderlaminarconditions,thelar germixerrequired di®usionacrosstherelativelywide(hundredsofmicrons)chann el,generating mixingtimesof » 10swithsampleconsumptionof60 ¹ L/min.Underturbulent °owconditions,thoseauthorsobtainedadeadtimeof » 55ms[ 71 ].Ourdesign reducesthephysicaldimensionsandcon¯nesthemixingregiont othecentralaxis ofthe°ow,whichgeneratesatleasta100-foldimprovementind eadtime(to » 400

PAGE 41

29 ¹ s;seebelow),andatleasta160-foldreductioninsampleconsumpt ion,while maintaininglaminar°ow. Operation. Themixeroperatesincontinuous°ow.Tryptophan°uorescence isexcitedbyfocusingthe266-nmfourthharmonicofaquasi-co ntinuousNd:YAG laser(2mW,NanoUV,JDSUniphase,SanJose,CA)throughaspatial¯ltera nd acylindricallensontothemixer(Fig. 3{2 A).The350-nm°uorescenceemission ofthemixingstreamiscollected90 ± fromtheexcitationsourcebya10 £ longworking-distancemicroscopeobjective(0.3N.A., 1 -correctedfromOlympus America,Melville,NY).TheemissionpassesthroughaWG-320Schott glass¯lter 2 andsilicatubelensbeforebeingimagedontoaUV-sensitive1300 £ 1030pixel CCDarray(Micromax,RoperScienti¯c,Princeton,NJ).Thepix elsizeoftheCCD is6.7¹ m £ 6.7¹ m.The10 £ -objectivemagni¯estheimagetoaspatialresolution of0.67¹ m £ 0.67¹ m.Thespatialpositionofapointinthe°owstreamisalinear functionofthetimeelapsedsincemixing.Thisimpliesanomin altimeresolution of3.4 ¹ sata°owspeedof20cm/s,althoughouractualresolutionisreduc edby binningofthepixels. Thecoaxialmixer,assembledunderamicroscope,isheldinamac hined sampleholderwithLuerconnections(Fig. 3{2 B).Aprogrammablesyringepump suppliesa¯xed°owofbu®erthroughtheoutercapillary.Themuch smaller°ow ratesthroughtheinnercapillaryaremaintainedbygaspressu recontrolledby aprecisionregulator(OmegaEngineering,Meriden,CT).Fig ure 3{2 Cshowsa typicalbackground-subtracted°uorescenceimageoftheopera tingmixer.The °uorescentstreamdoesnotdi®usefarfromthecentralaxisofthe°o wbefore passingoutofthemixer.The°owremainscon¯nedwithinthemixer atall 2 TheWG-320¯lterpreventswavelengthslessthan320nmtoreach theCCD camera.Thelaserexcitationwavelengthis266nm.

PAGE 42

30 UV Laser Mixer Cylindrical Microscope objective Schott WG320 filter CCD A B Outer Capillary Syringe Pump Inner Capillary Regulator N gas 2 Observation Region C (Top view)Lens Sample Reservoir Spatial Filter Inner Capillary Mixing region Flow Figure3{2:(A)Schematicof°uorescenceexcitationandcollec tionoptics:the °owingsampleisilluminatedbyaUVlaserandimagedontoaCCD.(B )Sample deliverysystem,consistingofaN 2 pressureregulatorfordrivinginner-capillary °owandaprogrammablesyringepumpforcontrolling°owofdilut ingbu®erin theoutercapillary.TubingconnectionsaremadewithLuer¯t tingsandmicrobore tubing(500 ¹ mID).(C)Background-subtractedCCDimageofa°uorescentdye °owingoutoftheinnercapillaryintothecenterofthe°owofdi lutingbu®erpassingthroughthelargeoutercapillary.Thebu®erisnon-°uoresc ent,hence,invisible intheimage.times.Theuseoffusedsilicacapillarymakesthemixingandreac tionregion UV-transparentandpromotesobservationoftheentiremixingpr ocess. Becauseofthenonuniformvelocitypro¯leofaviscous°uid°owing througha rectangularchannel,the°uidvelocityalongthecentralaxi softheoutercapillary exceedsthecross-sectionalaverage°owvelocitysetbythesyring epump[ 58 , 60 ].

PAGE 43

31 Forasquarecapillary,theratio 3 ofthemaximumspeedatthecenterofthe capillarytotheaveragespeedis » 1.6.Weusethiscentralaxisvelocity,ratherthan theaveragevelocity,asthescalefactortoconvertspatial°uo rescencedata(from theCCD)intotime-dependent°uorescencedecaydata.Thetypi caloperating parametersare » 5mL/h°owrateintheoutercapillary(foranaxial°owvelocity of » 22cm/s)anddrivingpressures 4 of » 5to25psiN 2 fortheinnercapillary.The mixerconsumes1.4 ¹ L/softhedilutingbu®erand » 1nL/sof°uorescentsample. CCDexposuretimesof10to45saresu±cientformeasuringquenchi ngrates.The °owpattern,whichischaracterizedby <¼ 14,isstableandtimeindependent. Abackgroundimageissubtractedfromall°uorescenceimages.Mi xingdata presentedhereareobtainedat22 ± C. 3.2.2Di®usionModelofMixerBehavior Thesimplecylindricalcoaxialjetgeometryallowsnumerica lmodellingofthe mixerbehaviorwitharadialdi®usionequation: @C ( r;t ) @t = D r µ @C @r + r @ 2 C @r 2 ¶ (3.7) C ( r;t )istheconcentrationoftheoutercapillarysolute(e.g.,th edilutingbu®er) atradius r (distancefromthemixeraxis)andtime t . D isthesolutedi®usion 3 TheratioiscalculatedfromthesolutiontotheStokesEquati on(Eq. 3.6 )fora squarepipeofside2 a [ 60 ]: u ( x;y )= ¡ 16 a 2 ¼ 3 ´ dP dz 1 X n =1 ; 3 ; 5 ( ¡ 1) ( n ¡ 1) = 2 µ 1 ¡ cosh( n¼x= 2 a ) cosh( n¼= 2) ¶ cos( n¼y= 2 a ) n 3 where x and y arecoordinatesinthecross-sectionofthechannelandthe z directionisthedirectionof°ow. 4 1psi(poundspersquareinch)=51.715torr=6894.757Pa

PAGE 44

32 constant.Weconsideramixingexperimentwithinitialcondit ions, C ( r; 0)= 0if r · a C 0 if r>a , (3.8) where C 0 istheinitialconcentrationofdi®usablesolutesintheouterc apillary bu®erand a istheradiusoftheinnercapillarystream. ThenumericalsolutiontoEquation 3.7 iscalculatedusingMATLAB(Mathworks,Natick,MA).DetailsofthecalculationareshowninAppend ixB,Section B.1 .Figure 3{3 showstheconcentrationpro¯les C ( r;t )fromthenumerical solutionofEquation 3.7 .Theconcentrationat r =0risesfrom C =0to C ¼ (0 : 5) C 0 inatime t ¼ a 2 = 3 D andto C ¼ (0 : 9) C 0 inatime t ¼ 2 a 2 =D .This meansthatfor a » 0 : 5 ¹ mand D » 7 £ 10 ¡ 6 cm 2 /s,mixingmustbe90%complete in ¼ 700 ¹ s.Thetime-dependenceofconcentrationisshowninFigure 3{4 B. 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 r/aC(r,t)/C 0Dt/a 2 = 0.001 Figure3{3:Numericalsolutionstothecylindricaldi®usionequ ation(Eq. 3.7 )for di®usionofasolutefromadilutingbu®erintoasamplestreamofrad ius a .The curvesrepresentconcentrationpro¯les, C ( r;t ),ofthedilutingbu®eratdimensionlesstimes Dt=a 2 =0.001,0.01,0.06,0.1,0.2,0.3,0.6,0.9,1.5,and3.Detai lsofthe numericalcalculationsaregiveninAppendixB. Wealsocomparethetheoreticalperformanceofthelaminarco axialjetmixer withalaminarmixerofrectangulargeometry|the°atsheetsili conmixerof

PAGE 45

33 Knight etal. [ 11 ].Mixinginarectangulargeometrycanbemodelledbyaonedimensionaldi®usionequation @C ( x;t ) @t = D @ 2 C @x 2 (3.9) fora°atsheetwithsolutesmixingintotheinnerstreamfrom2side s,asdiagrammedinFigure 3{4 A.Theinitialconditionsare C ( x; 0)= 0if ¡ 2 a · x · 2 a C 0 if j x j > 2 a , (3.10) Numericalsolutionsfortherectangulargeometryaregenerat edthesamewayas thesolutionsforthecylindricalsystem. Dt/a 20 0.2 0.4 0.6 0.8 1C(r =,t)/C 0 00.010.1 100 10 1B CylindricalGeometry RectangularGeometry A 2 a Figure3{4:(A)Cartooncomparisonoflaminarmixingincylind ricalversusrectangulargeometries:cross-sectionofthemixingregionwith°owp erpendicularto thepage,arrowsindicatedi®usiondirectionofdilutingsolut esintotheinner°uorescentstream(darkregions).(B)Riseofsoluteconcentrations atthecenterofthe innerstreamasafunctionofdimensionlesstimes.Di®usionofsolut esinamixer withcoaxialgeometry(|)isfasterthaninarectangularmixe r( ¡¢¡ ). Forthesamelengthscale a ,di®usioninacoaxialgeometryisfasterthan therectangulargeometry.ThisisillustratedinFigure 3{4 A,inacylindrical

PAGE 46

34 geometry,solutesdi®usetothecentralregionfromalldirecti ons.Ontheother hand,di®usiontothecentralregioninarectangulargeometry onlyoccurson 2sides.Therisetimeintheconcentrationofdilutingsolutesat thecenterof themixingstreamsinbothcylindricalandrectangulargeome triesareshown inFigure 3{4 B.Cylindricaldi®usionismoreabrupt,riseofconcentrationo f outercapillarysolutes,to90%of C 0 attheinnercapillarycentralaxis,is » 20 £ fastercomparedtothesameriseinconcentrationatthecentral region( x =0)oftherectangulargeometry.Experimentsrequiringawe llde¯ned¯nal soluteconcentration C 0 wouldtendtobene¯tmorefromthisgeometry.The hydrodynamicsofthecoaxialdesignarealsosimplerinprincip le,becausethe °uorescentstreamremainsinthecenterofthechannelandavoid sthestationary boundarylayeratthewalls. 3.3CharacterizationofMixerPerformance Intheprevioussection,wehaveshownthatoncecaninprincipl eattainsubmilliseconddi®usioninalaminarcoaxialjetmixerdesign.Itis nowimportant toexperimentallydemonstratefastmixing.Wedothisby°owing a°uorescent sampleintheinnercapillaryandexaminingtheactionofquen chersolutes°owing intheoutercapillary.The°uorescentsampleusedis N -acetyl-tryptophan-amide (NATA),astablederivativeofthenaturallyoccurringaminoac idtryptophan. Quenchersusedareiodideions(I ¡ )andN-bromosuccinimide(NBS).Mixing ofNATAwithsodiumiodide(NaI)resultsinquenchingofthetrypt ophan°uorescencebytheiodideionsinadi®usion-limitedreaction,wi thabimolecular rateof » 4 £ 10 9 (M ¢ s) ¡ 1 [ 73 ].Thisallowsustoobservemixingasithappens.ThequenchingofNATA°uorescencebyNBS 5 proceedsataslowerrate 5 NBSreactswithtryptophaninanoxidationreactionthatconv ertstheindole chromophoreoftryptophantooxindole[ 74 , 75 ].

PAGE 47

35 ¼ (7 : 3to7 : 9) £ 10 5 (M ¢ s) ¡ 1 [ 64 , 76 ].Thisallowsustomonitorthetimeresolution ofthemixingdevice.3.3.1QuantitativeFluorescenceMeasurement InFigure 3{5 ,weshow°uorescencequenchingreactionsinthemixer.Panels AandBarebackground-subtractedCCDimagesof°uorescentNATAst reams °owingoutoftheinnercapillaryintotheouter-capillarybu ®ersolution.Weuse50 ¹ MNATAin100mMphosphatebu®er(pH7).Theoutercapillarysoluti onisnot °uorescentandtherefore,isinvisibleinthebackground-subtr actedCCDimages. Figure 3{5 Ashowstheimageofthecontrol,°uorescentNATAcomingoutofthe innercapillary°owsintoanon-quenchingphosphatebu®ersolut ion.Theimage inFigure 3{5 Busesthesame°owconditionsandNATAconcentrationasinA, butwithaquenchingsolution,34mMNaI,°owingintheoutercapi llary.Since mixingofNATAwiththeNaIresultsinquenchingofthetryptopha n°uorescence bytheiodideionsinadi®usion-limitedreaction,thismanife stsasaloweringof °uorescenceintensityofthe°uorescentstreaminFigure 3{5 Bascomparedto Figure 3{5 A. ForFigure 3{5 ,AandB,theimagesofNATA-phosphateandNATA-iodide mixingarecollectedunderconstantoutercapillary°owratea ndconstantinner capillarydrivingpressure.Immediatelyaftereachpicturei staken,abackground imageiscollectedbysuppressing°owintheinnercapillarywhil eallowingthe outercapillary°owtocontinue.Theimagesweresubtractedin theCCDdata acquisitionsoftware(WinView,RoperScienti¯c,Princeton,NJ) .ThebackgroundsubtractedrawimagedataaretransferredtoMATLABforfurther analysis. ThequenchingofNATAbyI ¡ isquanti¯edbycomparingthe°uorescenceintensitypro¯leinthequenching-reactionimage, I reaction (Fig. 3{5 B:NATA-iodide), andthe°uorescenceintensitypro¯leinthecontrolimage, I control (Fig. 3{5 A:

PAGE 48

36 0 200 400 600 800 1000 0 0.5 1 PixelI rel 0 50 100 150 200 0 1 2 3 4 5 [NaI] (mM) 1ms A B CD Region of Interest 100m m Mix intoPhosphate Mix intoNaI 1/(equilibrium) IrelFigure3{5:Quantitative°uorescencemeasurementinthecoaxi almixer.(A)Image ofcontrolexperimentinwhichNATA°owingfromtheinnercapil larymixeswith (inert)phosphate,pH7,intheoutercapillary.Drivingpressu refortheinnercapillary°owis6psiandoutercapillary°owrateis5mL/h.Backgro undhasbeen subtractedfromtheimage.(B)Imageofquenchingexperiment inwhichNATA mixeswith34mMNaI°owingintheoutercapillary,underthesame °owconditionsasabove.(C)Intensityratio I rel calculatedfromaboveimages,asafunction ofdistancedownstreamfrompointofmixing(inpixels).The¯gur eindicates I rel of50%,asexpectedfromequilibriumdata.(D)Stern-Volmer plotoftheNATANaIquenchingreactionfromequilibrium°uorescencemeasureme nts(SeeEq. 3.12 ) indicatingthat I rel shouldbe50%at34mMNaI.

PAGE 49

37 NATA-phosphate).Theintensitypro¯lesareobtainedbybinningt hepixelsdownstreamofthemixingregiontocovertheentirewidthofthe°uor escentstream.The regionofinterest(Fig. 3{5 A)isalongtheverticalspatialdirectionwhichisthe directionof°ow(timeaxis).Therelative°uorescenceintensity , I rel ,iscalculated foreachbinnedtimealongtheregionofinterest, I rel = I reaction I control : (3.11) Figure 3{5 Cshowstheplotoftherelativeintensityratio I rel topixelposition (time)alongthedirectionof°ow.Theplotindicatesthat I rel ¼ 0 : 5,giving quantitativeagreementwiththeequilibrium°uorescencedat ainFigure 3{5 D. Figure 3{5 Dalsoshowsthatquenchingoftryptophan°uorescencebyiodidei ons followstheStern-Volmerequation[ 73 ], I control I reaction = 1 I rel =1+ K Q [ Q ](3.12) where I reaction isthemeasured°uorescenceintensity, I control inthe°uorescence intensitywithoutquencher, K Q istheStern-Volmerconstantand[ Q ]isthe quencherconcentration.3.3.2MeasurementofReactionRates Thetime-resolutionofmixersisbestmeasuredwithareactiont hatoccurs moreslowlythanthedi®usion-limitedNATA-iodidequenchingrea ction.The quenchingofNATA°uorescencebyN-bromosuccinimide(NBS)procee dsata bimolecularrate k bi ¼ (7 : 3to7 : 9) £ 10 5 (M ¢ s) ¡ 1 atroomtemperature[ 64 , 76 ]. Figure 3{6 showsamixingexperimentinwhich50 ¹ MNATA°owsintheinner capillaryand1.5mMNBS°owsintheoutercapillary,underdi®er entvaluesof drivingpressuresfortheinnercapillary.Theoutercapillar y°owrateremains¯xed at5mL/h.Thequenchingreactionattheseconcentrationsshou ldprogresswith

PAGE 50

38 A P = 13.5 psi inner Region of Interest 0 0.2 0.4 0.6 0.8 pixel PixelP= 9.1 psi inner DIrel020040060080010000200400600P= 22.2 psi inner P= 13.5 psi inner B C0 0.2 0.4 0.6 0.8 0 0.2 0.3 0.4 0.50.1IrelIrelFigure3{6:(A)CCDimageofNATA(50 ¹ M)°owingfromtheinnercapillary(at 13.5psidrivingpressure)andmixingwith1.5mMNBSintheouterc apillary(at5 mL/h).TheNATA°uorescencedecayswithdistancedownstreamfrom thepointof mixing.The¯gurealsoshowsrelative°uorescence I rel asafunctionofpositionfor di®erentinnercapillarypressures:22.2psi(B),13.5psi(C),and 9.1psi(D).The decayrate s isobtainedfromexponential¯tsto I rel (solidlines).

PAGE 51

39 pseudo-¯rst-orderkineticsatarate k 0 ¼ [NBS] ¢ k bi =1100to1200s ¡ 1 withNBSin excess. TheintensityplotsinFigure 3{6 ,panelsBtoD,showexponentialdecayof I rel asafunctionofdistancedownstreamfromthepointofmixing.T hedecayrate s ,obtainedfromthesingleexponential¯t,isinunitsofpixels ¡ 1 andneedstobe convertedtoactualunitsoftime.TheresolutionoftheCCDca mera, ±x ,perpixel is0.67 ¹ mduetothe10 £ -magni¯cation.Sincetheratioofmaximum°owspeed atthecenterofthecapillarytotheaveragespeedis » 1 : 6forasquarecapillary andweusecapillariesofinnerdimensionof100 ¹ Mwithacontrolled°owrateof5 mL/h,theaxialvelocity u =21cm/s.Thentheapparentquenchingrate, k ,with unitsofs ¡ 1 ,isgivenby k = s ¢ u=±x . Thequenchingreactionisfasterwhenthepressuredrivingthei nnercapillary °ow, P inner ,isreduced,abehaviorthatisanticipatedasaconsequenceo flaminar mixing.Lowerpressuresintheinnercapillarydriveasmallerv olumeof°uorescent materialintothemixer,resultinginanarrower°uorescentstre am,fastermolecular di®usion,andmorerapidquenchingofthe°uorescence.Highpressur es,by contrast,resultinabroaderstreamofNATA,whichslowsthedi®usion ofNBSand delaysthequenchingreaction.Thustheobservedquenchingr ateathighpressure iscontrolledbythespeedofmoleculardi®usion,whereasinthe limitofverylow pressuresweexpecttheintrinsicreactionrate k 0 tocontroltheapparentquenching rate.Figure 3{7 Ademonstratesthatthisisindeedthecase.Asthedriving pressuredecreasestoward5psi, k approaches k 0 ,andtheobserved°uorescence decayprovidesanaccuratemeasurementofthesub-millisecond reactionrate. Figure 3{7 Bshowsthatthetotal°uorescenceintensityinthestreamgrowsli nearly withdrivingpressure P inner whenaninert,non-quenchingbu®erisusedinthe outercapillary.Thisindicatesthatthecross-sectionalarea ofthe°uorescent streamgrowsinproportionto P inner .

PAGE 52

400 200 400 600 800 1000Int. (counts)0 500 1000 1500Quench rate (1/s)Est. Radius (m) m 0510152025 Pressure (psi)B 0 1 1.5 2 2.5 AFigure3{7:(A)Variationoftheapparentrateof°uorescencequ enchingof 50 ¹ MNATAby1.5mMNBSwithinner-capillarydrivingpressure.Lower driving pressuresreducethevolumeof°ow,shrinkingtheradiusoftheNAT Astreamand allowingfasterdi®usionofNBS.Theapparentquenchingratethe napproachesthe expectedvalue(dottedline)forthisbimolecularreaction , k =1190s ¡ 1 .Athigher pressures,the°uorescentstreamwidensandthequenchingratebec omesdi®usion controlled.Thesolidcurveisthebehaviorpredictedbythen umericalmodeldiscussedindetailinAppendixB.(B)Theoverall°uorescenceintensi tyoftheinner stream(inanon-quenchingbu®er)versusinner-capillarydriv ingpressure.Intensity growslinearlywithpressure,because°owvolumeisproportiona ltothedriving pressuregradient.The°owcutso®whenthedrivingpressureequal sthepressure P 0 =3.5psiintheoutercapillaryatthepointofmixing. Theproblemofaxi-symmetric°owinacircularpipe,¯rststudiedb yHagen in1839[ 77 ]andPoiseuillein1840[ 78 ],hasbeensolveddirectlyfromtheStokes equation(Eq. 3.6 )andiscalledtheHagen-Poiseuille°ow. Q = ¡ ¼R 4 8 ´ @P @z (3.13)

PAGE 53

41 where Q isthevolume°owrateinthecircularpipeofradius R drivenbya pressuregradient @P=@z .Onethenexpectsthe°owcomingoutoftheinner capillarytobe Q inner = ¼R 4 8 ´ P inner ¡ P 0 L (3.14) where P 0 isthepressureattheterminusoftheinnercapillaryoflength L .From theinterceptinFigure 3{7 B, P 0 ¼ 3.5psi.Theinnercapillaryusedhas R = 10 ¹ mand L =105mm.Therefore,atourlowestpressures, P inner ¼ 5psi, Q inner =0 : 3nL/s. Conditionsofhydrodynamicfocusingimpliesthatradiusoft heinnerstream ispinchedfromalldirectionsbythefastersyringe-pumpcont rolledoutercapillary °ow, Q outer =1.4 ¹ L/s.Wecanestimatetheinnerstreamradius a , a = µ Q inner ¼u ¶ 1 = 2 (3.15) where u istheaxial°owspeedoftheoutercapillarysolution.Theestima ted radiifordi®erentdrivingpressuresareshownintheupper x -axisofFigure 3{7 A. Reactionratesupto3 D=a 2 ¼ 3000s ¡ 1 shouldbemeasurableunderthese conditionsif D ¼ 7 £ 10 ¡ 6 cm 2 /s,whereashigherrateswouldbeaccessiblewith lowerpressuresorwithfasterdi®usingspecies(suchasH + inapH-jumpinduced foldingreaction). Takingadvantageofthesimplephysics,weextendedthenumeric almodel 6 ofcylindricaldi®usionforthegenerationofthesamplestreamo fNATA,withthe simultaneousinwarddi®usionandreactionofNBS,topredictthe apparentrateof °uorescencequenching, k app .TheconcentrationofNATAdecreasesexponentially 6 Theextendednumericalmodel,whichincludesquenchingofNA TA°uorescence byNBS,isalsodiscussedinAppendixB,Subsection B.1.2 .

PAGE 54

42 uponcontactwithNBS, [NATA]( r;t )=[NATA] 0 exp µ ¡ k bi Z t 0 C ( r;t 0 ) dt 0 ¶ (3.16) where[NATA] 0 isinitialNATAconcentration. C ( r;t )=[NBS]( r;t )fromSubsection 3.2.2 andAppendixB(Sec. B.1 ).The°uorescenceintensityofNATAdependson theradius a oftheNATAstream.TheinitialconditionsforNATA°uorescenceat t =0isgivenby F 0 = Z a 0 [NATA] 0 dr: (3.17) Therelative°uorescenceintensityatanytime t is F rel ( t )= R R À a 0 [NATA]( r;t ) dr R a 0 [NATA] 0 dr : (3.18) Weexpecttherelative°uorescence F rel todecaywithtime, F rel ( t )=exp( ¡ k app ¢ t ) : (3.19) Single-exponential¯tsto F rel ( t )fordi®erentvaluesofestimatedradii a ata giveninitialNBSconcentrationgeneratetheapparentrate, k app ,fordi®erent a .As expectedforlargevaluesof a , k app
PAGE 55

43 moreslowlythanthesolutes,wewouldexpecttheactualperform ancetomatchthe modelledcurvemoreclosely.3.3.3DeadTimeofMixing Characterizationofmixingbehaviorisnotcompletewithou tmeasurementof thedeadtimeofthemixer.Thedeadtime, t dead ,istheelapsedtimebetweenthe initiationofmixingandthe¯rstobservablepointwheremixing iscomplete[ 76 ]. Iftherelative°uorescencedataofFigure 3{6 BtoDareextrapolatedbackwards tothepointwhere I rel =1,andthatpointisde¯nedas t =0,thenthe¯rst observablepointinthe°uorescencedecaycurveoccursat t dead .Figure 3{8 shows thisextrapolationfortwomeasurementsofthequenchingof5 0 ¹ MNATAby0.75 mMand1.5mMNBS,inthefast(i.e.,low P inner )limitofmixeroperation.The0 0.2 0.4 0.6 0.8 1 Time (ms)0 12 3IrelFigure3{8:Kineticsofthe°uorescencequenchingreactionbe tweenNATA,at50 ¹ M,andNBSat1.5mM(¯lledtriangle)and0.75mM(¯lledcircle).B ackward extrapolationofthe¯tsto I rel =1providesanestimateofthemixingdeadtime, t dead =425 ¹ s. °uorescencedecaysindicatequenchingratesof575 § 52s ¡ 1 and1086 § 107s ¡ 1 , respectively,insatisfactoryagreementwiththeexpectedval ues540to590s ¡ 1 and 1100to1200s ¡ 1 .Furthermore,the¯gureindicates t dead ¼ 425 ¹ s.Becauseeach measurementconsumedlessthan » 1 ¹ Lof50 ¹ MNATA,the¯gureshowsthat

PAGE 56

44 quantitativeUV-°uorescencestudiesofmicrosecondreactionsca nbeperformed whileusingonlypicomolequantitiesofmaterial. Weexpectthatgeometricalparametersde¯nethedeadtimeoft hismixing device.The°owdynamicsanddi®usionaltransportnearthetermi nusofthe innercapillaryarenonuniformandquitecomplex,becauseth e°uorescentstream broadensasitexitstheinnercapillaryandthennarrowsasi tisacceleratedbythe faster°owofsurroundingbu®er.Thisregionthusrepresentsane®e ctivelydead volumeforthemixer.However,the°owleavesthisregionandat tainitssteadystate°owpro¯leoveranentrylength x ¼ 0 : 05 d < =0 : 05 ½ud 2 =´ ¼ 75 ¹ m[ 60 ], where d =100 ¹ misthewidthoftheoutercapillaryand u =14cm/sisthe (typical)mean°owspeed.Thissuggeststhatthemixingdeadtime willbe x=v ¼ 0 : 05 ½d 2 =´ ¼ 500 ¹ s,independentof°owspeed.Thisestimateisconsistent withthedataofFigure 3{8 .Moreover,thefactor d 2 indicatesthatfurtherreducing thedimensionoftheoutercapillarynotonlyisaroutetoimpr ovingsample e±ciency,butalsocouldsubstantiallyreducethedeadtime.Bec ausesilica capillarytubingiscommerciallyavailableoverawiderang eofsizes,signi¯cant improvementinmixerperformancemaybeattainablewithout useoflithographic techniques. 3.4Conclusions Thefabricationandcontinuingimprovementofultrafast°uid mixingdevices isoneofthedrivingforcesbehindprogressintheunderstandi ngofconformational dynamicsandfoldingofproteins.Laminar(low < )mixingrepresentsanimportant newdirectionformixingtechnologies,becauseitcanleadtosu bmillisecond deadtimestogetherwithmicroliterornanolitersampleconsu mption.Herewe haveshownthatalaminarcoaxialjetisaparticularlysimplem ixinggeometry thatprovidesreadyaccesstomicrosecondchemicalkineticra testhroughUV °uorescencespectroscopy.Themethodprovidessu±cientsignal-to -noiseratioto

PAGE 57

45 measurefast°uorescencedecayswith » 400 ¹ sdeadtime,whileconsumingonly 0.2to6nL/sofsampleintheinnercapillaryand1.4 ¹ L/sintheoutercapillary.

PAGE 58

CHAPTER4 SMALLPROTEINTRYPTOPHANCAGE: EQUILIBRIUMMEASUREMENTSANDTWO-STATEFOLDING 4.1Introduction Proteinsfoldonavarietyoftimescales:secondstomillisecond stomicroseconds.The¯nitespeedofBrowniandi®usionsuggeststhatfoldingspee dcannot exceedtherateof » 10 6 s ¡ 1 [ 13 , 14 , 15 , 17 ].Theexactlimitsandtheconditionsin whichmoleculesmayapproachthemremainopenquestions.Toga inunderstanding ofthephysicallimitationstofoldingspeedthatproteinsmust eventuallyencounter, acasestudyofafastfoldingproteinisnecessary. Thischapterfocusesonadesignedminiprotein,commonlyknow nasthe \tryptophancage"or\Trp-cage"[ 79 ].Weshowequilibriumcirculardichroismand °uorescencestudiesofthe20-aminoacidlongTrp-cageminipro tein.Equilibrium measurementsindicatethatfoldinginTrp-cageishighlycoo perative,¯tting verywelltoanidealtwo-statesystem.Foldingratesweremeasur edusinga temperature-jumpspectrometer[ 49 ]. Todate,Trp-cageisoneofthesmallestandfastestfoldingprote in-like moleculesknown,withafoldingtimeof4microsecondsatroom temperature.The foldingspeedofTrp-cageapproachestherateofdi®usionalloo pformationwithin theunfoldedchain.Trp-cageisaspecialproteinbecauseitap pearstohavea perfectlyoptimizedfreeenergylandscape.Atthesametime,i tisanoutlierinthe contactorderplotforpredictionoffoldingratesoftwo-sta teproteins.Thesmall sizeandexceedinglyfastfoldingofTrp-cagealsomakeitanimp ortantbenchmark proteinformoleculardynamicssimulation. 46

PAGE 59

47 4.2Two-StateProteinFolding Manysmall,single-domainproteinsfoldwithsimpletwo-statek inetics.The characteristicsoftwo-statefoldinghavebeenwell-documen ted[ 24 , 80 ].Fora two-statefoldingreaction,onlytheunfoldedstate, U ,andthenativestate, N ,are populatedalongthefoldingpathway.Theunfoldingandrefo ldingaremonophasic barrier-crossingprocessesthatarecharacterizedbythefoldi ngrate, k f ,andthe unfoldingrate, k u . U k f ;k u ­ N: (4.1) Foratwo-statesystemattemperature T ,thefractionofunfoldedmolecules,  unf , inthepopulationisgivenby  unf = P U P U + P N = e ¡ G U =RT e ¡ G U =RT + e ¡ G N =RT = 1 1+ e ( G U ¡ G N ) =RT (4.2) where P U and P N arethepopulationsoftheunfoldedandthenativestates, and G U and G N aretheGibbsfreeenergyfortheunfoldedandnativestates, respectively.Thefreeenergychangeuponunfoldingis¢ G = G U ¡ G N = ¢ H ¡ T ¢ S .Thus,Equation 4.2 canbewrittenas  unf ( T )= 1 1+exp ¡ ¢ H RT ¡ ¢ S R ¢ : (4.3) Theunfoldingenthalpyandentropy,andthetemperaturedep endenceofthe unfoldedfraction,  unf ,canbedeterminedfromequilibriummeasurements.We canalsodeterminethemeltingtemperature, T m .At T m ,thepopulationofthe foldedandunfoldedstatesareequal,andthefreeenergyofun folding,¢ G ,is zero.Averyimportantcriterionforatwo-statesystemistheag reementbetween thermodynamicparametersmeasuredwithdi®erentequilibriu mprobes.InTrpcage,wehaveusedfar-UVcirculardichroism(CD)spectroscopyan dtryptophan °uorescencetomonitorthethermalfoldingandunfolding.The agreementbetween CDand°uorescencedatashowthatTrp-cageisanidealtwo-statef oldingsystem.

PAGE 60

48 4.3Trp-cageStructure Trp-cageisa20-aminoacidlongminiproteindesignedbyAnder son'sgroup throughtruncationandmutationofa39-residueGilamonstersa livapeptide[ 79 ]. TheconstructwecallTrp-cageistheirTC5bmutantwithsequen ceNLYIQ WLKDGGPSSGRPPPS.Trp-cage(Fig. 4{1 )hasalltheattributesofalarger, morecomplexfoldableprotein:multiplesecondarystructure elements,tertiary interactions,side-chaintoside-chainpacking,abackboneam ideprotectiongreater thanexpectedfromsecondarystructurealone,spontaneousfol ding,cooperativity andresistancetothermalunfolding. AB C DE FN C Figure4{1:Trp-cagestructure.(A)Thefamilyof38structures determinedfrom NMR,shownintheskeletalmodel.(B)to(F)showsonlyonestructur e,allinthe sameorientationusingdi®erentmodels.(B)Trp-cageinaball-a nd-stickmodel showingheavyatoms.(C)Trp-cageinaribbon-modeloftheback bone.Notethe ® -helix(residues2-8)andthesmaller3 10 -helix(residues11-14).(D)Trp-cage structurewithonlythepolymerbackboneshown.(E)Trp-cagep olymerbackbone withthesalt-bridge-formingresidues,asparticacid(position 9)andarginine(position16)instickmodel.(F)Trp-cagepolymerbackbonewitht heresiduesthat formthehydrophobiccore,instickmodel:tyrosine(position3 )andtryptophan (position6)packingagainstaglycine(position11)andprolin eresidues(positions 12,18and19).

PAGE 61

49 TheNuclearMagneticResonance(NMR)structureisreportedbyNei digh etal. ,theProteinDataBankcodeis1L2Y.Thehydrophobiccoreform s atryptophan-in-a-cagemotif.Itiscomposedofaromaticside chainstyrosine (position3)andtryptophan(position6)packingagainstaglyc ine(position11) andprolineresidues(positions12,18and19).Thesecondarystr uctureelements includean ® -helix(residues2-8),a3 10 -helix(residues11-14)andapoly-proline IIhelixattheC-terminus.Neidigh'sstabilitydataalsosuggesta nasparticacid (position9)andarginine(position16)saltbridge[ 79 ]. 4.4EquilibriumStudiesonTrp-cage Weusefar-UVcirculardichroism(CD)spectroscopyandtryptopha n°uorescenceatneutralandacidicpHtomonitorthethermalfoldi ngandunfolding oftheTrp-cageminiprotein.TheTrp-cagesamplesaresynthesi zedusingFMOC chemistrybyAlfredChungandassociatesoftheUFInterdisciplina ryCenterfor BiotechnologyResearch(ICBR).Trp-cagesamplesforbothequ ilibriumandkineticstudiesarepreparedbydirectlydissolvingweighedamou ntsoflyophilized proteininbu®er.Theconcentration, C 1 ,ischeckedusingaUV-Visabsorption spectrophotometer(ShimadzuCorporation,Columbia,MD).4.4.1Far-UVCircularDichroismMeasurements Far-UVCirculardichroism(CD)spectraquantifysecondarystruc turecontent ofproteinsandnucleicacids.Proteinsareasymmetricmolecu les.Hence,they exhibitapreferenceinabsorptionofeitherrightorleftcir cularlypolarizedlight. CDisameasureofthedi®erenceinabsorption,¢ A = A LCP ¡ A RCP where A LCP istheabsorbanceforleftcircularlypolarizedlight, A RCP istheabsorbanceforright 1 C = A ¸ =" ¸ l where " ¸ istheextinctioncoe±cient, l isopticalpathlengthand A ¸ isabsorbance(measuredopticaldensity)attheindicatedwave length.Fora moleculewithonetryptophanandonetyrosine, " 278 =6760(M ¢ cm) ¡ 1 at278nm.

PAGE 62

50 circularpolarizationand A istheabsorbanceofunpolarizedlight.Typical ® -helical proteinshaveCDspectrawithminimaat208and222nm.TheCDspe ctrafor ® helix, ¯ -sheetandrandom-coilpolypeptideconformationsareveryd i®erent.Thus, CDmeasurementsareusedtofollowproteinsecondarystructurec onformational changesduetoexternalperturbations:temperature-orpH-ch angeorpresenceof denaturantmolecules[ 81 ]. 200 210 220 230 240 250 -15000 -10000 -5000 0 CD (deg-cm/dmol)2Wavelength (nm) pH 3 pH 7 0 20 40 60 80 -12000 -10000 -8000 -6000 -4000 -2000 Temperature (C) °CD(deg-cm/dmol)2 2 2 2A B pH 3 pH 7 Figure4{2:CDMeasurementsofTrp-cage.(A)CDspectraofTrp-c ageinpH7 (phosphatebu®er)andpH3(citricacidbu®er)at25 ± C.ThecurveatneutralpH isattributedtoaveryhighlevelofhelicity.(B)ChangeinC Dspectralvaluesat 222nmforbothneutralandacidicTrp-cagesamplesdemonstrat ehowhelicity islostduetothermalunfolding.ThesigmoidalshapeofthepH7curvesuggests cooperativeunfolding.

PAGE 63

51 Figure 4{2 Ashowsstabilityandstrong ® -helicalcontentofTrp-cageatneutral pH(pH7,Trp-cageinphosphatebu®er)androomtemperature(25 ± C).Thermal unfoldingismonitoredbyplottingCDspectralvaluesat222n mversustemperature(Fig. 4{2 B).LossofCDsignalat222nmcorrespondstolossofhelicity.The sigmoidalshapeoftheneutralpHcurvesuggestscooperativethe rmalunfolding (discussedbelow).AtpH3(Trp-cageincitricacidbu®er),theca rboxylicacid group(COO ¡ )oftheasparticacidresidue(position9)gainsaH + ion.Consequently,theasparticacidandargininesaltbridgeisbroken. Itisinterestingto notethathelicalstructureisalsolostuponbreakingofthesalt bridge(Figure 4{2 ). TheCDmeasurementsaretakenusingacommerciallyavailableC Dspectrometer(AvivInstruments,Lakewood,NJ).CDspectralvaluesis reportedin millidegreesbecause A LCP ¡ A RCP isproportionaltoellipticity 2 (inmdeg). Normalizationinconcentration,pathlengthoftheCDcuvett eandnumberof polypeptideresiduesallowexpressionofCDspectralvaluesinc ommonlyusedunits ofresiduemolarellipticity(deg ¢ cm 2 /dmol). 4.4.2FluorescenceMeasurements Far-UVCDspectroscopymeasuressecondarystructurecontent.Flu orescence spectroscopyisanequilibriumtechniquethatiscomplimenta rytoCDbecauseit 2 Ellipticity µ isameasureofopticalactivity[ 81 ].Planepolarizedlightabsorbed byanasymmetricmoleculeischangedtoellipticallypolariz edlightduetodi®erenceinabsorptionof LCP and RCP light. µ =tan ¡ 1 ( a=b ) where a istheminoraxisand b isthemajoraxisoftheellipsetracedbytheellipticallypolarizedlight.Circulardichroismisgivenby ¢ A = A LCP ¡ A RCP = 4 ¼µ (2 : 303)(180) .

PAGE 64

52 canbeusedasanindicatorofthedegreeofcompactnessofthemo lecule.ForTrpcage,wetakeadvantageofthenaturaltryptophan°uorescence anditsdependence onthelocalenvironment.Equilibrium°uorescencemeasuremen tsaretakenwith aJASCO°uorescencespectrometer(Easton,MD).Wehavechosen266n mas excitationwavelengthtomatchthefourthharmonicoftheND: YAGlaserusedin the°uorescencekineticsexperiment.RepresentativeTrp-cag e°uorescenceemission spectraatpH7areshowninFigure 4{3 . 300 320 340 360 380 400 420 0 10 20 30 40 10, 30, 40C ° 60C ° 80C °Emission Wavelength (nm)Fluorescence Intensity (cts) Figure4{3:Fluorescenceemissionspectraoftheaminoacidtryp tophaninTrpcage,266-nmexcitation,pH7(phosphatebu®er).Fluorescence intensitydecreases morestronglyathighertemperatureswhereTrp-cageistherm allyunfoldedandthe tryptophanisnolongerburiedinthehydrophobiccore. Theintensityoftryptophan°uorescenceinTrp-cagedecreasesm orestrongly athighertemperatures( > 40 ± C).At60{80 ± C,moreTrp-cagemoleculesinthe ensembleareexpectedtobethermallyunfolded,withtryptop hannolongerburied inthehydrophobiccore.Figure 4{4 showsacomparisonofthetemperaturedependenceoftryptophanintegrated°uorescence(normalize dinconcentration)in Trp-cagewithfreetryptophaninsolution, N -acetyl-tryptophan-amide(NATA), underthesameneutralbu®erconditions(pH7,phosphatebu®er). Thereported °uorescencevaluesinFigures 4{4 and 4{5 areinnormalizedintegrated°uorescence,°uorescenceintensityintegratedoverthewavelength, F = R I ( ¸ ) d¸ ,and

PAGE 65

53 0 20 40 60 80 5.5 6 6.5 7 7.5 8 log(/concentration) FTemperature (°C) NATA Trp Cage Figure4{4:Temperaturedependenceoftheintegrated°uoresc enceoftryptophan inTrp-cageandinfreesolution,NATA.Theintegrated°uorescenc e, F ,isnormalizedinconcentration.BothTrp-cageandNATAareprepare dusingphosphate bu®er,pH7.TheintensityofTrp-cage°uorescencedecreaseswith temperaturebut notasstronglyandnotmonotonicallyasfreetryptophan.normalizedwithrespecttoconcentration.NATA°uorescencedec reasesmonotonicallywithtemperature(slope »¡ 22 : 2counts/ ± C).Trp-cage°uorescence,onthe otherhand,hasaveryweaknegativeslopefrom » 5 ¡ 20 ± C,averyweakpositive slopeat » 20 ¡ 40 ± C,andasteadydecreasein°uorescencefrom40 ± C,although thisisaweakerdecreasecomparedtofreetryptophan. Inacompactfoldedstate,theaminoacidtryptophanisburied inthehydrophobiccore.Figure 4{5 characterizestryptophanburialbylookingattryptophan°uorescenceinTrp-cageatacidicpHandinthepresenceo fahighconcentrationof°uorescencequenchingiodideions.FromtheCDda tashownin Figure 4{2 ,Trp-cagepossessessigni¯cantlyless ® -helicalcontentatpH3than pH7.Thisisane®ectofthebreakingoftheasparticacid-argin inesaltbridge atlowpH.Consequently,atpH3moreTrp-cagemoleculesareless compactand haveasolvent-exposedtryptophan.Hence,Trp-cage°uorescence atpH3followsa NATA-liketemperaturedependence.

PAGE 66

54 0 10 20 30 40 50 60 70 80 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 Temperature (°C)log( Integrated Fluorescence / Concentration ) NATA Trp Cage, pH 7 Trp Cage, pH 3Trp Cage, NaI Figure4{5:Temperaturedependenceoftheintegrated°uoresc enceoftryptophan inTrp-cage( » 5to7 ¹ M)atpH3,pH7andatpH7inthepresenceofexcess iodideions(47mMNaI).Reference°uorescenceoffreetryptoph aninsolution (NATA)isalsoshown. Anotherwaytocharacterizetryptophanburialisbylookinga tquenchingby iodide.Iodideionsquenchtryptophan°uorescenceinadi®usion -limitedreaction withabimolecularrateconstantof » 4 £ 10 9 (M ¢ s) ¡ 1 .Trp-cage°uorescence atpH7inexcessiodideionsolvent(Fig. 4{5 )canbedescribedbytwoslopes ( »¡ 4 : 6counts/ ± Cat < 40 ± Cand »¡ 5 : 2counts/ ± Cat > 40 ± C).Tobesusceptible toiodidequenching,thetryptophaninTrp-cagemustalready beoutofthehydrophobiccore,evenattemperaturesbelow40 ± C.Thisobservation,togetherwith thesigmoidalchangein ® -helicalcontentwithtemperatureforpH7(Fig. 4{2 B),

PAGE 67

55 suggestsanall-or-nothingorcooperativefoldingtransition .ForTrp-cagetoexperiencecooperativefolding,thereshouldbeonlytwopopulat ionsinsolution,the indistinguishableensembleofunfoldedstructures, U ,andthenativefoldedstate, N ,withnodetectableintermediates. 4.4.3AnalysisofTrp-cageTwo-StateFolding Forproteinsundergoingatwo-statefoldingtransition,thet emperature dependenceoftheunfoldedfraction,  unf ,canbedeterminedfromequilibrium data.We¯tthepH7CDspectraat222nm(Fig. 4{2 B)and°uorescencedata (Fig. 4{4 )toEquation 4.3 todeterminetheunfoldingthermodynamicparameters: enthalpy¢ H ,entropy¢ S ,andmeltingtemperature T m . Twoscalingparameters, A s and A o areusedtoconvertbackground-subtracted CDdata,CD 222 ,totheunfoldedfraction,  unf = A s ¢ (CD 222 ¡ A o ) : (4.4) Calculationof  unf fromthe°uorescencedataentailsmodellingthetemperature dependenceofthe°uorescenceofthefoldedandunfoldedstates, F fold and F unf , respectively. F fold = A f ¢ exp( ¡ a f T ) F unf = A u ¢ exp( ¡ a u T ) : (4.5) Normalized°uorescence 3 F N isgivenby F N = F fold ¢ (1 ¡  unf )+ F unf ¢  unf : (4.6) Thescalingparameters A s , A o , A f , a f , A u ,and a u ,togetherwith¢ H and¢ S are derivedfromsimultaneousleastsquares¯ttoEquations 4.3 , 4.4 , 4.5 ,and 4.6 using 3 F N istheTrp-cage°uorescenceintensityintegratedoverthewave lengthand normalizedwithrespecttointegrated°uorescenceofNATAat5 ± C.

PAGE 68

56 MATLAB.TheequilibriumthermalunfoldingcurvesfromCDand °uorescence dataoverlapeachotherasshowninFigure 4{6 .Thesolid-lineistheleastsquares ¯tresultwhichgives¢ H ¼ 48.6kJ/moland¢ S ¼ 155J/mol-Kforunfolding. Theunfoldingmidpointoccursat T m ¼ 41 ± Cwhere¢ G =0andthepopulation betweenfoldedandunfoldedstatesareequal.TheoverlapofC Dand°uorescence datasigni¯esthatequilibriumthermalunfoldingisindepend entoftheprobe. Thisobservationtogetherwiththequalityofthe¯ttoEquatio n 4.3 satis¯esthe requirementsfortwo-statefolding[ 80 ].Neidigh etal. reportsthesamecooperative transitionwiththermalunfoldingmidpointof42 ± CfromCDandNMRchemicalshift-dispersionanalysis[ 79 ]. CD at 222 nm Trp fluorescencecunf 1 0.5 0 02040 60 80 T (C) o Figure4{6:EquilibriumunfoldingofTrp-cage.Fractionun foldedinphosphate bu®er,pH7,asderivedfrom222nmcirculardichroismdataandb yTrp°uorescencedata(266nmexcitation).Fittotwo-statetransition (solidcurve)gives ¢ H ¼ 48.6kJ/moland¢ S ¼ 155J/mol-Kforunfolding. ThesigmoidalcurveinFigure 4{6 isasignatureofanall-or-nothingfolding transitionforanidealtwo-statesystem.Proteinfoldingandun foldingisvery di®erentfromcoil-globuleandglobule-coiltransitionsinh omopolymers,where thereisgradualdecreaseorincreaseofthevolumeofthemolec ule.Forproteinsin atwo-statesystem,therearetwoequallystablestates,twopopula tionsinsolution, theensembleofunfoldedmolecules U andthefoldedmolecules N ,withvirtually nosemi-nativeormisfoldedstructures.Thetwo-statetransition isanalogousto

PAGE 69

57 ¯rst-orderphasetransitions(e.g.,sublimationofasolid,crysta lmelting)albeit withalargetransitionwidth[ 1 ].Therefore, U and N areseparatedbyanenergy barrierandthefavoredpopulationisdeterminedbyexterna lconditionslike temperature,pressure,pHordenaturantconcentration. 4.5Trp-cageFoldingKinetics Trp-cagefoldingkineticsweremeasuredusinglaser-inducedt emperature jump( T -jump)spectroscopy.Thelaser T -jumpmethodusesa » 7nsinfrared laserpulsetoheattheproteinsamplein20to30ns.The1.89 ¹ m-infraredpulse isgeneratedfroma1.064 ¹ mNd:YAGlaserpulse(ContinuumSurelite,Santa Clara,CA)thatisRamanshiftedafterpassingthrougha1-mH 2 cell(Light AgeInc.,Somerset,NJ).Thesampleremainsattheelevatedtempe raturefor » 10 2 milliseconds.Tryptophan°uorescenceisthenmonitoredattime delay t (30 nsto » 100 ¹ s)afterthetemperatureperturbation.The°uorescenceisexci ted usinga5-ns266-nmpulse,fromthefourthharmonicofanotherNd :YAGpulsed laser(ContinuumMinilite,SantaClara,CA),thesameexcitati onwavelength usedinthetheequilibrium°uorescencemeasurements.Amicroscop eobjective at90 ± projects°uorescenceemissionintoaphotomultiplier.Thekine ticpro¯leis assembledfrom ¼ 100di®erentpump/probetimedelayscollectedinrandomorde r. Thelaserinduced5to20 ± C T -jumpsstimulateTrp-cageunfolding.The measuredrelaxationtime ¿ afterthe T -jump, ¿ » 1to10 ¹ s,varieswith¯nal sampletemperature(butnottheinitialtemperature),and¯ts verywelltoasingle exponentialfunction[ 46 , 49 ].Otherthanthermalrecovery,weobservenoother relaxationsfollowingthemicrosecondprocess.Thissupportst heequilibriumresult thatTrp-cage,likemostsmallproteins,foldsintwo-statekine tics.Thefoldingand unfoldingrates, k f and k u ,arederivedfromthemeasuredrelaxationrate ¿ ¡ 1 , ¿ ¡ 1 ( T )= k f ( T )+ k u ( T )(4.7)

PAGE 70

581020304050 10 4 10 5 10 6 T (C)oRate (s)-1 k f k u 1/t Figure4{7:KineticsofTrp-cagefolding.Theinvertedtria nglesaretheobserved relaxationrates(1 =¿ )measuredafterthelaser-inducedtemperaturejump(inphosphatebu®er,pH7).Folding( k f )andunfoldingrates( k u )calculatedfrom ¿ and  unf .Thesolidanddottedlinesarethe¯tstoEquation 4.9 . andtheunfoldedfraction,  unf ,fromequilibriummeasurements  unf ( T )= k u ( T ) k f ( T )+ k u ( T ) : (4.8) At T =23.5 ± C,theobservedrelaxationtime ¿ ¼ 3 : 1 ¹ sgivesafoldingrate k f =240 ; 000s ¡ 1 =(4 : 1 ¹ s) ¡ 1 andunfoldingrateof k u =81 ; 500s ¡ 1 =(12 ¹ s) ¡ 1 . Thetemperaturedependenceof ¿ , k f and k u areshowninFigure 4{7 .Thefolding andunfoldingratesare¯ttoanArrhenius-typesingle-barrierc rossingevent, k f = k 0 ;f exp( ¡ H a;f =RT ) k u = k 0 ;u exp( ¡ H a;u =RT ) : (4.9) Fromthe¯t,theactivationenergybarrierforfoldingis27 § 1kJ/moland76 § 5 kJ/molforunfolding. 4.6DiscussiononTrp-cageFolding Trp-cageisanIdealTwo-stateSystem. Equilibriumandkinetic measurementsshowthatTrp-cageexhibitsanidealtwo-statefo ldingreaction.

PAGE 71

59 Ifanintermediatestateliesbetweenthefoldedandtheunfol dedcon¯gurations, weexpectittogenerateadditional°uorescencerelaxationaf terthemicrosecond process.Thisisnotobservedinthekineticdata.Theobservedki neticrelaxation amplitudematchesexpectationsfromequilibriummeasureme nts.Althoughwe cannotruleoutanintermediatewithasmall-amplituderelax ationontheslow ( » ms)timescaleofthermalrecovery,thehighlycooperativetra nsitionobserved inequilibriumtogetherwiththequalityofthe¯ttoatwo-stat emodel(Eqs. 4.3 to 4.8 )maketheexistenceoffoldingintermediatesveryunlikely. Manysmallproteins( < 100residues)dofoldcooperativelywithtwo-state kinetics[ 24 ].Athoroughunderstandingofthethermodynamicsandkineti csof thesesmallproteinsmayyielda`minimalistic'answeronhownat uresolvesthe protein-foldingproblem[ 26 ].Moreover,anunderstandingofthefoldingoflarge, complicatedproteinswouldbeimpossiblewithoutthoroughun derstandingofthe foldingofthesimplestones.Trp-cage,beingaverysmallprotei nof20residues, providesthisframework.Itssmallsizealsomakesitveryaccessi bletoall-atom moleculardynamicssimulations.Simmerling etal. [ 82 ]havealreadyreported thecorrectstructurepredictionandfoldingsimulationprio rtothereleaseofthe Neidigh'sNMRcoordinates[ 79 ]. Trp-cagefoldsin4.1 ¹ satroomtemperature. Knownratesfor proteinsthatfoldwithtwo-statekineticsrangefromseconds tomillisecondsto microseconds.Trp-cagebelongstothefamilyofultrafastfold erswith( k f ) ¡ 1 inthe ¹ srange.Infact,Trp-cagefoldingrateexceedsobservedrate sofother ultrafastfoldingproteinsatroomtemperature.Theclosestar ethe35-residue villinheadpeacesubdomain,thesmallestnaturallyoccurring peptidethatfolds spontaneously[ 50 ],whichfoldsin4.3 ¹ sat27 ± C,andthe denovo designed 73-residuethree-helixbundleprotein,whichfoldsin3.2 ¹ satamuchhigher temperatureof50 ± C[ 52 ].TheTrp-cage,thevillinheadpeacesubdomain,andthe

PAGE 72

60 ® 3 Dthree-helixbundleallhavepredominantlyhelicalstructu res.Anotherultrafast folderistheBBA5designedminiprotein,a23-residue ¯ -hairpin/turn/ ® -helix motif,whichfoldsin7.5 ¹ sat27 ± C[ 51 ].Asummaryofratesofknownfast-folding proteinsisshowninTable 4{1 . Table4{1:Foldingratesoffastfoldingproteins. ProteinNameNo.ofResidues k ¡ 1 f ( ¹ s) T ( ± C)Ref. Trp-cage204.123.5[ 49 ] Villinheadpiecesubdomain354.327[ 50 ] BBA5237.525[ 51 ] ® 3 D3-helixbundle733.250[ 52 ] WWdomainFBP28W30A372425[ 83 ] Engrailedhomodomain542625[ 84 ] ¸ -repressor 6 ¡ 85 A37G804457[ 85 ] Ofthefast-foldingproteinsthatarelistedinTable 4{1 ,onlytheWWdomain FBP28W30Amutanthasapredominantly ¯ -sheetsecondarystructure.This agreeswiththeanalysisofKubelka etal. [ 54 ]that ® -helicalproteinsfoldslightly fasterthan ¯ or ®¯ proteins.ThisissupportedbytheZagrovicandPande suggestion[ 86 ]that ® -helicalsegmentsfoldfastbecause ® -helicesaregeometrically closetotheaverageidealrandom-°ightchainsoftheunfolded state. Theseultrafastfolderspavethewayforveri¯cationoffolding kineticssimulationresults.ThePandegrouphaveusedmoleculardynamicssi mulationsvia distributedcomputingtoanalyzefoldingkineticsofTrp-ca ge[ 87 ],villinheadpiece subdomain[ 88 ]andBBA5[ 51 ].Snow etal. correctlyestimatesTrp-cagefolding ratetobeintherange1.5to6 : 9 ¹ s[ 87 ]. Trp-cagemaysettheconditionsforultrafastfolding. Forfolding toberapid,thefoldingenergybarriermustbesmall(Eq. 4.9 )andtheproteinis expectedtofollowasmoothenergysurfacetothefoldedstate.T rp-cageencounters averyweakenergeticbarrierforfolding.Theactivationen ergyforfoldingis

PAGE 73

61 H a =27 § 1kJ/molbuttheenergyassociatedwiththeviscosityofsolventis » 17kJ/mol.Therefore,Trp-cagehasanetbarrierof » 10kJ/mol ¼ 4 k B T . ThefoldingspeedofTrp-cageapproachestherateofdi®usional loopformation,whichlikelyposesaphysicallimittofoldingspeed.Prev iousstudiesonthe rateofdi®usionoffreepolypeptidechainsinsolution[ 13 , 14 , 15 , 17 ]showthat twoendpointsofa20-residuechainshoulddi®useintocontacton atimescaleof » 0 : 2 ¹ s,or » 20-foldfasterthanTrp-cagefolding.ThisimpliesthatTrpcagehas anearlyoptimizedenergylandscape. InTrp-cage,residualinteractionsintheunfoldedstatedoes notexhibit random-coillikeNMRchemicalshiftdispersions[ 79 , 89 ].Thisislikelydueto formationofhydrophobicclusterswhichmayreducetheentro piccostoffolding. Simplemodelsofproteinfoldingclaimthatahighdensityofl ocalinterresidue interactionsprovidesthenecessaryentropyloss,andsubsequent loweringofthe freeenergybarrier,asthechainfolds[ 38 , 90 ].Correspondingly,thepresenceofthe alwaysrigidpoly-prolineIIhelixatthe C -terminusdecreasesthecon¯gurational entropyoftheunfoldedstate. MoleculardynamicssimulationofTrp-cagefreeenergylandsc apeinexplicit waterbyZhou[ 91 ]showthatlandscapeissmoothandfunnel-like.However,Zhou's simulationdatagivesameltingtemperatureof167 ± C,signi¯cantlyhigherthanthe experimentalresult, T m ¼ 41 ± C. Trp-cagefoldingrateisnotpredictedbyContactOrdercorr elation. Relativecontactorder, CO ,isatopologicalparameterthatcorrelatesverywell withthelogarithmsofin-waterfoldingrates,ln( k f ),ofmosttwo-statefolding proteins(Sec. 2.3.3 ).Theparameter CO issmallforproteinsthataremainly stabilizedbylocalinteractions.Large CO proteinshaveinteractionsbetween residuesfarapartinsequence.Thestrongnegativecorrelatio nbetween CO and ln( k f )implythatproteinswithsmall CO foldfaster.Contrarytothatempirical

PAGE 74

62 observation,Trp-cagefoldsveryfastalthoughithasalarger elativecontact order CO =0 : 19,moretypicalofslowlyfoldingproteinswith k f ¼ 10s ¡ 1 =(100ms) ¡ 1 [ 38 ].Ignoringmanyindolecontactsinthehydrophobiccorered uces therelativecontactorderto CO =0 : 17.Thisstillpredictsaslowfoldingrateof k f ¼ 100s ¡ 1 =(10ms) ¡ 1 ,a2500-folddi®erencefromtheexperimentalvalues.A fewotherproteinsalsoexceedthepredictionsofthe CO correlation,albeitbya smallerfactorof » 50 £ . Weexpecttheempiricalcorrelationtoseverelyunderestimat ethefoldingof Trp-cagebecauseitrequiresstatisticalindependenceofmostl ong-rangetertiary contactsinthefoldingtransitionstate.Thisisimprobablef ora20-residueprotein withapersistencelengthof4-5residues.Eventually,di®usionbe comesthelimiting factorinapproachingthecorrecttopology[ 92 ].Di®usiontimeislongerforlonger chainsbutnotsoforshortpolypeptides.Infact,originalestim atesoffoldingtimes frompolymercollapsetheoryinvolveslengthdependence[ 22 , 93 ].Thefailureof contactordercorrelationtopredictfoldingratesofmulti statefoldingproteins andshortpolypeptidesleadstotheresurgenceofresearchonle ngthdependent topologicalparameterstoexplainfoldingrates.Forexampl e,Ivankov etal. have triedtocorrectforlengthdependenceusingthetopological parametercalled size-modi¯edcontactorder( SMCO ): SMCO = CO £ L P where L isthetotal chainlengthandthepower P is0 · P · 1,at P =0, SMCO = CO [ 40 ]. Theirresultshowshighestcorrelationofln( k f )(includingmultistateproteinsand otherpeptides)with L 0 : 70 ,incloseagreementwithanempiricalscalingof L 0 : 61 of KogaandTakada[ 94 ]usingsimpli¯edo®-latticefoldingsimulations.Ultimately, contactordermodelsormodelsthatrelatefoldingratestost ructuremustaccount forchain-lengthand°exibility[ 40 , 94 , 95 , 96 , 97 ].Thetopomer-searchmodel ofMakarovandPlaxco[ 92 ]andthee®ectivecontactordermodelofDilland

PAGE 75

63 coworkers[ 98 ]providedetailed,mechanisticdescriptionsoftherelation between topology,foldingmechanisms,andfoldingrates. 4.7Conclusions WehavecharacterizedthefoldingofTrp-cage,asmall20-resi duelongdesigned miniprotein.EquilibriumCDand°uorescencemeasurementsdem onstrateahigh degreeofcooperativityinthethermalunfoldingcurve.The equilibriumandkinetic experiments,takenfrom T -jumpmeasurements,demonstrateexcellentagreementof Trp-cagefoldingtoatwo-state`all-or-nothing'model.Fol dingisveryfast,4.1 ¹ sat 23.5 ± Candtheactivationenergybarrierforfoldingisalow27kJ/ mol.Trp-cageis oneofthefastestfoldingprotein-likemoleculesatroomtemp erature. Trp-cagesetsconditionsforfastproteinfolding:atwo-state reaction,aweak foldingactivationenergybarrier,anearlyoptimizedfree energylandscape,some pre-organizedstructuresintheunfoldedstate(amongotherf actorsyettobe determined).Trp-cagealsoservesasagoodbenchmarkmolecul eforall-atom simulations.Furthermore,foldingdataonTrp-cagecontribu testothegrowthof researchonthecorrelationoftopologyofproteinstructuret ofoldingrates.

PAGE 76

CHAPTER5 INTERNALFRICTIONCONTROLSTHESPEEDOF PROTEINFOLDINGFROMACOMPACTCONFIGURATION 5.1Introduction Aproteinfoldingreactioninvolvesmotionsofamacromolec uleinsolution undergoingweaknon-covalentinteractionsatsmalllengthsc ales.Constant molecularcollisionsandfrictionfromthesolventenvironme ntcompletelydominate thedynamicsofapolypeptidechainasitmovesacrossitsfree energysurface andtowardsthenativeconformation.Thus,proteinsalwaysf oldinastrongly dampedenvironment.Kramersreaction-ratetheorytakesin toaccountdamping e®ectsduetothemediumandisusedtodescribedi®usion-drivenba rrier-crossing reactions[ 30 , 31 ].Therefore,thehigh-frictionKramersdescriptionisanat ural frameworkforanalyzingproteinfoldingkinetics[ 32 , 99 , 100 ]. Theexternalmediumhassuchastronge®ectonproteinfoldingki neticsthat rateofpolypeptidechaindi®usionthroughtheviscoussolventse tsanupperlimit tothespeedofproteinfolding.However,proteinscanbeconsid eredasamedium inthemselves[ 28 ].Inthischapter,wegiveproofofinternaldampinge®ects (weakercomparedtoexternale®ectsduetosolvent)owingtoth einternaldegrees offreedomwithintheprotein.Ifthesolventviscositybecomes signi¯cantlysmaller oriffoldingproceedsfromasu±cientlycompactcon¯guration, otherinteractions associatedwithreorganizationofthecompactmoleculeshould setadi®erentlimit onfoldingspeed.Weuselaserspectroscopytomeasurethislimitby studying thein°uenceofsolventviscosityontherapid( » ¹ s)foldingofaproteinfroma highlycompact,late-stageintermediatecon¯guration.Wesho wthatregardlessof theviscogenicagentaddedtothesolvent,thefoldingrateext rapolatestowarda 64

PAGE 77

65 common¯nitevalue » 10 5 s ¡ 1 inthelimitofdecreasingsolventviscosity.\Internal friction"withinthecompactdenaturedchainsetsthetimesca leforthefoldingas solventfrictiondeclines.Furthermore,thetimescaleforthe sesolvent-independent dynamicsvariesstronglywithtemperature,suggestingthatla rgeintra-chain interactionenergiescontrolthedynamicsofcon¯gurationa ldi®usionwithinthe compact,near-nativestatesoftheprotein. 5.2Background 5.2.1KramersRateTheoryandFoldingDynamics Foldingofsmall,singledomainproteinsiswell-describedbya two-statemodel withsingleexponentialkinetics,wherethenative, N ,andunfolded, U ,statesare separatedbyanenergybarrier.Sinceafoldingreactioninvo lvesmacromolecules insolutionundergoingmanyconformationalchangeswithwea kintra-andintermolecularinteractions,theoreticalmodelssuggestthatdi®usi vechainmotions limitthespeedofproteinfolding[ 32 , 37 , 101 ].Kramerstheorymodelsreactions asdi®usionalbarrier-crossingevents.Thehigh-frictionlimit ofKramerstheoryis well-suitedfortreatmentofproteinfoldingreactions(Sec . 2.3.1 ).Asappliedto proteinfolding,Kramerstheoryassertsthatthefoldingrate , k f ,dependsonboth theactivationfreeenergy,¢ G a ,andthereactionfriction, ° , k f = A ° exp( ¡ ¢ G a =k B T )(5.1) where A isaconstantpre-exponentialfactorde¯ningthecurvatureof thepotential energybarrier[ 29 ]. Ifdi®usionofthepolypeptidechainacrosstheenergybarrieri sstrongly coupledtointeractionswithsolventmolecules,thenthereac tionfriction ° is dominatedbysolventdynamicviscosity, ´ S .Ifthatisso,Kramerstheoryimplies thatthefoldingrateshouldaccelerateininverseproportion to ´ S ( k f / ´ ¡ 1 S ). Severalexperimentsthathavetestedthisprediction[ 33 , 100 , 102 , 103 , 104 , 105 ,

PAGE 78

66 106 , 107 , 108 ]andfoundtheexpectedinverse-viscositybehaviorinfolding rates,if thenative-statestabilityisheldconstant 1 ,while ´ S varies[ 33 , 100 , 102 , 103 ]. Theobservationthat k f / ´ ¡ 1 S doesnotmeanthatthefoldingratewill increasewithoutlimitasthesolventviscositydecreases.Nordoes thisimplythat thereactionfriction ° ! 0as ´ S ! 0.Foldingratesprobeonlytheratelimiting process,andifdi®usionalpassageovertheprimarybarriercontin uestoaccelerate as ´ S decreases,late-stageinternalmotionsofthechainandside-ch aindynamics thatcouplelessstronglytosolventforcesmustbegintocontrol k f atsu±ciently smallvaluesof ´ S . 5.2.2ProteinInternalFriction Theoryandsimulationsuggestthatinteractionswithinthemol eculeitself maybegintodominatethedynamicsatlow ´ S [ 28 , 99 , 109 , 110 , 111 ].Thereare severalmechanismsbywhichapolymerchaincouldexperienced ragforcesand kinetice®ectsthatdonotsimplyscaleinproportiontothesolve ntviscosity[ 109 , 110 ].These\internalfriction"e®ectscanincludepotentialene rgybarriersto backbonerotations,long-range(sequence-distant)inter-re sidueinteractions,and theaccessibilityoffreevolumeinanon-continuumsolvent.At low ´ S ,internal frictioncansetanupperlimittofoldingspeed,unrelatedtot hedi®usionalspeed limitsimposedbyloopformation,hydrophobiccollapse,ando therbulkmotions ofthechainthatiscoupledtothesolvent[ 13 , 15 , 17 , 22 , 112 ].Someauthorshave suggestedthat,forproteinsthatfoldatsu±cientlyhighrates,t hesee®ectscould evenin°uencethedynamicsatordinaryaqueoussolventviscosit ies[ 113 ]. 1 Theimpositionofthestabilityconditionisnecessarybecausevi scogens(solvent viscosityincreasingcosolventssuchasglucose,glycerol,ethyl eneglycol,etc.)alter thenetstabilityofthenativestateandthefreeenergyoffold ing[ 33 ].

PAGE 79

67 Proteininternalfrictione®ectsmaycausefoldingratestode partfrom k f / ´ ¡ 1 S andapproachalimitingvalueatverylow ´ S .Experimentally,wecannot probefoldingat ´ S ¿ ´ water exceptbyextrapolationfromhigh ´ S .Extrapolating experimentaldatato ´ S ! 0doesnotimplythatthesystemleavesthehighfrictionlimitofKramerstheory.Constantcollisionswithsol ventmoleculesand amongmoleculesinthepolypeptidechaindampenthepolypep tidemotionso stronglythatfrictionalforceswouldcontinuetooverwhelm inertiale®ectsinfolding evenif ´ S fellbyordersofmagnitude. Additiveinternalfrictionmodel. Ansariandcoworkers[ 114 ]used Kramerstheoryfordescribingnanosecondconformationaldyn amicsinafolded protein,myoglobin.Theyproposedanempirical,additivefr ictionmodelto quantifythestrengthofaninternalfrictioncontributiont ochaindynamics: k = A ¾ + ´ S exp( ¡ ¢ G=k B T )(5.2) where k istherateofsmall,globalconformationalchangeswithinth eproteinand ¾ hasunitsofviscosity.Theparameter ¾ canbeinterpretedasthecontributionof proteininternalfrictiontothetotalfriction.Theiranal ysisledtoalarge\internal viscosity"fromthemyoglobinmolecule: ¾ =4 : 1 § 1 : 3mPa ¢ s,about4 £ largerthan theviscosityofwater. MorerecentauthorswhohaveappliedEquation 5.2 toproteinfoldinghave foundnoclearevidencefor ¾ ,eveninproteinsthatfoldthroughcompacttransition states.InthestudyofthefoldingofProteinL,PlaxcoandBaker [ 33 ]found ¾ ¼¡ 0 : 1 § 0 : 2mPa ¢ s.InthefoldingofcoldshockproteinCspB,Jacob etal. [ 102 ] observedthatthefoldingtime k ¡ 1 f appearstoextrapolatetowardszeroas ´ S ! 0, implyingasimilarlysmall ¾ forCspB. Timescaleforinternal-friction-controlledreorganizat ion. The observationthat ¾ ¼ 0forproteinLandCspBseemstosuggestthat(atleast

PAGE 80

68 overtheviscosityrangestudied)thespeedofconformationalsea rchdepends moreondi®usivemotionsofsolvent-exposedregionsofthechain thanoninternal dynamicsinportionsofthemoleculedecoupledfromthesolve nt.However,kinetic experimentsprobeonlytherate-limitingstageoftheprocess. Onewouldnot readilydetectevenalargeinternalfriction ¾ associatedwithafaststageoffolding ifaslowerprocess,withdynamicsthatarestronglycoupledtoth esolvent,limits theoverallfoldingrate. Moreover,di®erentmodelsfortheinternalfrictionleadtod i®erentexplanationsofdynamics.Ansariandcoworkers[ 28 , 114 ]usedthemostcommon interpretationofinternalfriction,whichtreatsreactio nfriction ° (ofEquation 5.1 ) asasumofcontributionsfromsolventandfrominternalchainm otions[ 109 ].Thus, totalreactionfriction ° wasexpressedas ° / ° int + ° S where ° int and ° S represent frictioncontributionfromthemoleculeitselfandfromtheso lventenvironment(as inEquation 5.2 ).However,MankeandWilliams[ 110 ]havesuggestedthatrateof con¯gurationalchangesshoulddependontheproductofsolvent -dependentand solvent-independentparameters.Amultiplicativerelation shipgives ° / ° int ¢ ° S . Inthelattercase,afoldingexperimentthatsuggests k ¡ 1 f ! 0atlow ´ S doesnot necessarilyprovetheabsenceofinternalfrictione®ects. Therefore,lookingfortheinternalviscosityparameter ¾ maynotbethe bestwaytoquantifyinternalfrictione®ects.Instead,wemustin vestigateif internalfrictioneverlimitstheobservedrateofproteinfo lding.Weneedto measurethetimescaleforinternal-friction-controlledreo rganizationsafterslower solvent-coupledevents,likechaincollapse,havealreadyocc urred.Noexperiment hasidenti¯edthistimescaleforacompactpolypeptide.Howeve r,thestudyof viscosity-dependenceofChymotrypsinInhibitorCI2foldingsu ggeststhatinternal frictione®ectsmayslowthedynamicsofacompactchain[ 108 ].Asamatterof fact, k ¡ 1 f ! 0as ´ S ! 0doesnotprovetheabsenceofinternalfrictione®ects

PAGE 81

69 asimpliedinthefoldingstudyofProteinLandCspB.Theirdata havesu±cient timeresolutiononlytoshowthatthelimitingvaluefor k ¡ 1 f doesnotexceeda fewmilliseconds.Accordingly,nanosecond-resolvedexperimen tshaveamuch betterprobabilityofelucidatinginternal-friction-con trolledtimescales.Inthenext section,weshowthatinternalfrictionprocessesdoexistandlim itthefoldingrate ofasu±cientlycompactproteinfast-foldingproteinmolecule 5.3FoldingfromaCompactStateofFerrocytochrome c Wehavemeasuredtheinternalfrictione®ectonarapidprotein foldingreactionthatbeginsafterthecollapseandtherate-limitinglar ge-scalereorganization ofthepolypeptidechain.Figure 5{1 showsacartoonthatsummarizesthefoldingexperiment.Weuseahemeprotein,cytochome c ,foldingofwhichiseasily triggeredphotochemicallybecausecarbonmonoxide(CO)bin dspreferentially tothereducedhemeironintheunfoldedstate.Thecompactlat e-stagefolding collapse by dilution laser photolysis foldingk f COM -CO U -CO M NFigure5{1:Studyoffoldingfromacollapsedstate: U -CO ! M -CO ! M ! N . UnfoldedCO-cytochromecathighGdnHCl( U -CO)collapsestothemetastable M -COstateupondilutionintobu®er.Laser°ashphotolysisdissociate stheCO moleculefromthehemetogivethe M state,whichfoldstothenativestate N at rate k f . intermediateofferrocytochrome c ,the M -COstate,ispreparedbyloweringthe

PAGE 82

70 chemicaldenaturant 2 concentrationinthepresenceofCO.Laser°ashphotolysis oftheCO-ironbondinthismetastableintermediatetriggers thefoldingreaction tothenativestate( M ! N ).Foldingratesaremeasuredbynanosecond-resolved transientabsorptionspectroscopy.Thesolvent-viscositydepend enceofthisrapid ( k f » 10 5 s ¡ 1 )reactionindicatesthatdi®usionalmotionsintheinterioro fthe molecule,whichisveryweaklycoupledtothesolventenviron ment,largelycontrol therateofconformationaltransitionbetweenverycompactc on¯gurations, M and N . 5.3.1The M -COandthe M States Cytochrome c . Horseheartcytochrome c isasmall,singledomainhelical proteinwith104amino-acidresidues[ 115 ].Ithasahydrophobiccorewithaheme ligandcovalentlyattachedtocysteineresiduesatpositions1 4and17.Inthefolded state,theironatomatthecenterofthehemegroupiscovalent lyboundtotwo otheraminoacidresidues,histidineatposition18andmethioni neatposition 80.Inreducedferrocytochrome c ,theironatomhasanoxidationstateof+2.In oxidizedferricytochrome c ,theironatomhasanoxidationstateof+3.Figure 5{2 isthebackbonestructureofcytochrome c withthehemegroupinspace-¯lled modelandseveralaminoacidsshownasstickmodels:tryptophana tposition59, histidineresiduesatpositions18,26and33,andmethioninere siduesatpositions 65and80(ProteinDataBankcode:1AKK).Duetothecloseproxim ityof tryptophanandthehemeinthefoldedstate,tryptophan°uoresc enceisquenched byFÄorsterenergytransfertotheheme. 2 Thechemicaldenaturantusedtounfoldcytochrome c isGuanidineHydrochloride(SigmaChemicalCo.,St.Louis,MO),referredtointhete xtasGdnHCl.

PAGE 83

71 Met Met Trp His His Figure5{2:Backbonestructureofhorseheartcytochrome c .Thehemegroupat thecenterofthehydrophobiccoreisshownasaspace-¯lledmode l.Severalamino acidsareshownasstickmodels:tryptophan(position59),histid ines(positions18, 26and33)andmethionines(position65and80). Unfoldingferrocytochrome c athighdenaturantconcentrationbreaksthe covalentlinkbetweenthehemeironanditsnativeligand(th esulfurofmethionine80),andallowsexogenousligandssuchascarbonmonoxide(CO )tobindtothe hemeiron[ 12 , 116 ].IntheabsenceofCO,ferrocytochrome c undergoesahighly cooperativeunfoldingtransitionatveryhighdenaturantco ncentrationsindicative ofanunusuallystableprotein.Theunfoldingfreeenergyisla rge, » 71kJ/mol, givingadenaturanttransitionmidpointat5.1MGdnHCl.Additi onofCOresults inadramatic » 38kJ/moldecreaseinstability.Thedenaturantconcentratio n transitionmidpointisalsoloweredto3.95MGdnHCl[ 117 ].COadditionalso causesmajordeviationsfromatwo-stateunfoldingtransition [ 12 , 118 ]. M -COisacompact,metastableintermediatestate. CObinds preferentiallytothehemeinunfoldedferrocytochrome c .Wecallthisthe U -CO state.Whenthedenaturantconcentrationisdilutedoutbybu ®erinthepresence ofCO,ahighly-populated,compact,metastable,intermedia testateforms:the M -COstate.Detailedequilibriumstudies[ 117 ]haveshownthatthe M -COstate hasnear-nativefar-UVCDspectrumandweaktryptophan-59°uor escencedueto

PAGE 84

72 FÄorstertransfertotheheme,indicativeofacompactprotein .Absorptionspectra con¯rmthatCOremainsboundtothehemeinthe M -COstate.Furthermore, recentNMRmeasurements[ 118 ]showthatthe M -COstatehasextensivechemical shiftperturbationssimilartonativeferrocytochrome c . M -CO ! M . Thus,CD,°uorescence,absorption,andNMRmeasurements showthatthe M -COisacompact,highly-structuredstatewithnative-likehe lix content,butnonnativetertiarystructureandhemecoordina tion.Itismetastable atroomtemperatureunderlowdenaturantconcentrations.Th eCOthermally dissociatesandescapesfromtheprotein,withatimeconstant ¿ ¼ 32min at24 ± C(Fig. 5{3 )and ¿ ¼ 2hat15 ± C.Wecanalsoapplyanexternallight 390 400 410 420 430 440 0 0.2 0.4 0.6 0.8 1 Wavelength (nm)ODM -CO N Figure5{3:Absorptionspectraofferrocytochrome c : M -CO ! M ! N states. Themetastable M -COstateispreparedunderCOatmospherewith0.2MGdnHCl at24 ± Candplacedinasealedquartzcuvetteunderathicklayerofmi neraloil. TheCOthermallydissociates( ¿ ¼ 32min)and M foldsquicklytothenativestate N ,whichisstronglyfavoredatlowdenaturantequilibriumcon ditions.TheabsorptionspectrashownaretakenwithaShimadzuspectrophotomer:1 ,10,40,100and 170minutesaftersamplepreparation.pulsetobreaktheheme-CObond,allowingtheCOtoescape.This converts M -COintoastructurallyequivalentpenta-coordinatestate, M ,whichthen foldsrapidlytothenative, N ,state.Thisisincontrasttopreviousstudies thatfoundnoevidenceforstructuralintermediatesinthefo ldingprocessof

PAGE 85

73 ferrocytochrome c [ 117 , 119 ].RecentrapidmixingresultsofMaki etal. [ 120 ]show cleardeviationsfromtwo-statefolding/unfoldingkinetic sbothatlowandhigh denaturantconcentrationsconsistentwitha4-statemechanism featuringbothearly andlatefoldingintermediates.Theirresultscon¯rmthatanat ive-likestate, M , occursevenintheabsenceofCO. Samplepreparationanddelivery. Topreparethe M -COstateforkinetic experiments,lyophilizedhorsecytochrome c (SigmaChemicalCo.,St.Louis,MO) isdissolvedintoasolutionof6MGdnHCl,0.1MTrispH7.0anddeox ygenatedby a°owofO 2 -freeCOat1atm,withtraceamounts( · 2mM)ofsodiumdithionite addedtoreducethehemeirontoFe 2+ .DilutingthismixtureintoaCO-saturated bu®ercollapsestheproteinintothe M -COstate.Thedilutionbu®eralsocontains cosolutes(glycerol,ethyleneglycol,orglucose)atconcent rationssu±cientto raisethesolventdynamicviscosity, ´ S ,5to6fold,andaglucose/oxidase/catalase mixturetoscavengeanyremainingO 2 [ 121 ].Solutionsforfoldingstudiestherefore contained50 ¹ Mcytochrome c ,0.5MGdnHCl,aviscogeniccosolute,theenzyme system( · 0 : 32 ¹ Mglucoseoxidase, · 3 : 5 ¹ Mcatalase,and0.3%glucose),and tracesodiumdithionite.Thekinematicviscosity, ´ S =½ ,ofeachmixtureislater measuredwithacalibratedCannon-Fenskeviscometerfullyimm ersedinawater bath,foranaccuracyof § 0 : 1%. Immediatelyafterpreparation,theproteinsolutionistran sferredtoagas tightHamiltonsyringeandkeptat · 10 ± Cpriortodeliverytoa0.5-mmquartz °owcell(NSGPrecisionCells,Farmingdale,NY)usingasyringepumpt hrough microboretubing(°owrate:8mL/h).Laser°ashphotolysisoftheC O-ironbond inthe M -COstatetriggersthefoldingtothenativestate.Thefolding processis observedbycollectingtransientabsorptionspectra.The°owcel lissecuredina thermallyregulatedaluminumholderandcoveredbyaninsula tedPVCboxwith N 2 -°owandwindowsforentranceandexitofthepumplaserandprob ebeams.

PAGE 86

74 Thetemperatureiscontrolledby°uid°owintothealuminumhol derfroma circulatingbath.Thetemperaturesarestableto § 0 : 2 ± Candaremonitoredby¯ne thermocouples(OmegaEngineering,Stamford,CT)attached toboththeholder andthe°owcell.5.3.2TransientAbsorptionSpectroscopy Transientspectrometer. Thetransientspectrometer[ 14 ]usesthesecond harmonicofapulsedNd:YAGlaser(Spectra-Physics,MountainView ,CA),5-7ns pulses, · 8mJ/pulse,532-nm,tophotodissociatetheCOfromthehemeiron .Ata Photolyzed region hit by the laser pulse Xe lamp flashes at timeafter laser pulse t Sample cell Xe flashlamp BSM M BS M Nd:YAG 532 nm ICCD Spectrometer Syringe pump Sample AB Figure5{4:Transientabsorptionspectrometer.(A)Opticallay -outofthespectrometer,bird'seyeview.A532-nmlaser°ashphotodissociatest heCOfromthe heme.Atatimedelay t afterthelaser°ash,transmittedlightfromaXe°ashlamp iscollectedfromphotolyzedandunphotolyzedregionsofth esampleanddelivered tothespectrometerandICCDtogeneratevisibleabsorptiondi®e rencespectra.M standsformirror,BSforbeamsplitter.(B)TheXelightpulseissp litandfocused ontwospots,photolyzedandunphotolyzedregionsofthesample . timedelay t afterthelaser°ash( » 10ns · t · 1s),amicrosecondXe°ashlamp (EG&GOptoelectronics,Gaithersburg,MD)istriggeredprodu cingabroadband

PAGE 87

75 probelight.Theprobelightissplitbya50-50beamsplitteran dfocusedinto twopointsinthesample°owcell.Onepointcoincideswiththel aser°ashand theotherfallsinanunphotolyzedregionofthesample°owcell .Togenerate visibledi®erenceabsorptionspectra,imagingopticscollectt hetransmittedlight fromboththephotolyzedandunphotolyzedregionsanddirec titthroughan imagingspectrometer(ActonResearchCorporation,Acton,MA)an dontoa gatedintensi¯edCCDcamera(RoperScienti¯c,Trenton,NJ).The nanosecond gatingoftheICCDprovidesatimeresolutionof10nsorbetter inspiteofthe » 1 ¹ sdurationoftheXe°ash.A1.5-secondintervalbetweenlaser°ashes allow fullreplacementoffresh M -COdeliveredbythesyringepumptothe°owcell. Thespectrometerandtheassociatedelectronicsareoperatedt hroughaLabView (NationalInstruments,Austin,TX)softwareinterfacewithaprogr amwrittenby ErikSjolander[ 14 ].Theopticallay-outofthetransientspectrometerisshownin Figure 5{4 . Kineticmodellingofspectra. Anaverageof10time-resolvedoptical absorptiondi®erencespectra,¢OD( ¸;t )= A ( ¸;t ) ¡ A ( ¸; 0 ¡ ),iscollectedper timedelay t bytheWinSpecsoftware(PrincetonInstruments,RoperScienti ¯c, Trenton,NJ). A ( ¸;t )istheopticalabsorbanceatwavelength ¸ (wavelengthrange: 390to450nm)andtimedelay t fromthephotolyzedregion,and A ( ¸; 0 ¡ )is theopticalabsorbanceatthesamewavelengthfromtheunphoto lyzedregion. MATLABisusedtoprocessandsortthedi®erencespectrawithrespect to » 50 to70logarithmicallyspacedtimedelaysbetween10nsto > 1stakeninrandom orderfollowingphotolysis.Representativespectraareshownin Figure 5{5 . Weusesingularvaluedecomposition(SVD),apowefulmatrixtech nique,to ¯lteroutexperimentalnoiseandidentifytheindependentlye volvingtransient speciesfromthedi®erencespectra.Wehaveadetaileddiscussiono fSVDandthe kineticmodellingofdi®erencespectrainAppendixB(Sec. B.2 ).UsingSVD,we

PAGE 88

76 Wavelength (nm) log(Time delay [s]) 10 D OD Figure5{5:Time-resolvedopticalabsorptiondi®erencespectr afollowingphotodissociationofCOwitha532-nmNd:YAGlaser°ash.Sampleshownis 50 ¹ M ferrocytochrome c in1.2MGdnHCl, T =20 ± C. reformulateeachdataset,¢OD,asaproductofthreematrices, ¢OD( ¸;t )= U ( ¸ ) ¢ S ¢ V ( t ) T : (5.3) Thecolumnsof U ( ¸ )representstheminimalsetoforthonormalbasisspectra, thecolumnsof V ( t )arethecorrespondingamplitudesasafunctionoftime,and thediagonalelementsof S areameasureofthecontributionofthecorresponding basisspectratotheobservedspectra.Atlowdenaturantconcent rations( · 2 : 4M GdnHCl),SVDanalysisshowstwodominantSVDcomponents,eachdescr ibing thesametworelaxations.Thisallowsidenti¯cationof3states:t heunphotolyzed M -COstate,thephotoproduct M stateandthenativestate N whichisstrongly favoredatequilibrium.Relaxationrateswereobtainedbysi multaneously¯tting the¯rsttwoSVDcomponents, S i V i ( t )( i =1 ; 2)toasumof2exponentials: S i V i ( t )= C i 1 exp( ¡ k 1 ¢ t )+ C i 2 exp( ¡ k 2 ¢ t ) : (5.4)

PAGE 89

77 The¯rstrelaxationrate, k 1 = k f ,representsspectralchangescorrespondingtothe foldingtransition M ! N ,sinceequilibriumstronglyfavorsthefolded N stateover thecompactdenaturedstates M and M -COatlowdenaturantconcentrations.The secondrelaxationrate, k 2 ,isduetosample°ow,thereplacementof N by M -CO, asasyringepumpsuppliesfreshsampletotheoptical°owcell.Fig ure 5{6 shows representativeamplitudesSV 1 ( t )andSV 2 ( t )ofspectralchangesobtainedbySVD oftransientspectrafollowingphotolysis,andbiexponential¯t . 10-8 -0.2 -0.1 0 Time (s)SV2SVx2SV110-210-410-6100 12s m Figure5{6:AmplitudesSV( t )ofspectralchangesobtainedbySVDoftransient spectrafollowingphotolysis.Solidlinesarethesimultaneous biexponential¯tto SV 1 ( t )andSV 2 ( t )(Eq. 5.4 ).Sampleshownis50 ¹ Mferrocytochrome c in1.2M GdnHCl, T =20 ± C.The¯rstrelaxationrate( k f ¼ 12 ¹ s)correspondstofolding, M ! N . 5.3.3 M ! N :StabilityandFolding TherelaxationrateobservedafterphotolysisofCOcorrespond stotherateof folding( k f )withminimalcontributionsfromunfolding. M k f ! N: (5.5) Afterthelaser°ashphotolysisoftheCOfromtheheme,twothingsh appen. M eitherfoldstothenativestate N ,withfoldingrate k f .COrebindingtothe M statetoreform M -COisalsoobserved.Thelatterprocessiscompletein » 50ns. Unfoldingof N tometastablestate M isnotobserved,itishighlyunfavorableat lowdenaturantconcentrations( · 2 : 4MGdnHCl).

PAGE 90

78 Foldingof M ! N requiresbothcon¯gurationaldi®usionofthecompact, denaturedprotein,andachemicalreactionbetweentheheme anditsnativeligand, thesulfuratomofmethionineatposition80.Thatchemicalrea ctionoccurswith ageminaterate k g ¼ 4 £ 10 10 s ¡ 1 at22 ± C(and k g ¼ 10 11 s ¡ 1 at40 ± C[ 122 ]),far exceedingtheobservedfoldingrates.Therefore,chemicalbo ndformationdoesnot limitthemeasuredfoldingrate, k f .Instead, k f characterizesthechainmotions associatedwithalate-stagefoldingtransitionthatbringsmet hionine-80into contactwiththepenta-coordinateheme,leadingtofolding to N . Stabilitye®ectin M ! N folding. Previousstudiesofviscositye®ects onfoldinghavetakenintoaccountthefactthataddingviscog eniccosolutesto thefoldingbu®erusuallyaltersthestabilityofthenativestat e,shifting¢ G a andgeneratinganextraperturbationto k f [ 33 , 100 , 102 , 103 , 104 , 105 , 106 , 107 , 108 , 123 ].Plaxco etal. [ 33 ]havecompensatedforthese¢ G a shiftsinProteinL byaddingdenaturantsimultaneouslywiththeviscogens.Jacob etal. [ 102 ]have chosenexperimentalconditionswherethecosolutedoesnota®e ct¢ G a ofCspB. 0 12 3Gdn-HCl Conc. (M)0 5 10 15 20 1000 1500 2000kf -1(s) m Figure5{7:Foldingtime, k ¡ 1 f ,isconstantupto » 2.5MGdnHCl, T =20 ± C. Therefore,theactivationenergyoffolding,¢ G a ,isuna®ectedbyGdnHCland othercosolutesbelow2.5MGdnHCl

PAGE 91

79 Figure 5{7 showsthatforferrocytochrome c ,the M ! N foldingtime k ¡ 1 f remains » 12 ¹ supto » 2.5MGdnHCl.Theveryweake®ectofdenaturanton k f ( @ ln k f =@ [GdnHCl] ¼ (0 : 02 § 0 : 05)M ¡ 1 )indicatesthatcosolutes,including thestrongchemicaldenaturantGdnHCl,donotshifttheactivat ionfreeenergyof folding,¢ G a .Evenasthefreeenergyoffolding¢ G = G M ¡ G N decreasesfrom » 17kJ/molto0[ 118 ],thebarrier-height¢ G a remainsuna®ected.Thereforethe datadonotrequirecompensationforviscogene®ectson¢ G a . Anotherimportantimplicationofthisveryweake®ectofdenat urantconcentrationsonfoldingratesistheuniformityofsolvent-ex posedsurfaceareaas theproteinpassesfromthe M statetothefoldingtransitionstate.Athigher denaturantconcentrations( > 2 : 5MGdnHCl),themoleculeopensupandmore complicateddynamicsoccur.AnearlylookatFigure 5{8 (inset)showsthatthe slightincreaseinthesolventviscosity(1.09to1.17mPa ¢ s)duetoincreaseindenaturantconcentrationfrom0.5to2.4MGdnHCl,mayevenacco untforthesmall variationof k f . ViscosityDependenceofFoldingTime. Threedi®erentviscogens (glycerol,glucose,ethyleneglycol)haveasimilarslowinge®e ctonthefoldingtime, causingthesamelinearincreasein k ¡ 1 f withsolventviscosity(Fig. 5{8 ).Although consistentwiththeexpectationofalineardependence, k ¡ 1 f » ´ S ,thedataclearly indicatea¯nitelimitingvalue k ¡ 1 f (0) ¼ 8 : 10 § 0 : 63 ¹ sat20 : 0 § 0 : 2 ± Cas ´ S ! 0. Thelimitingfoldingrate k f (0)onlymoderatelyexceeds(by50%)therateinwater ( ´ water =1 : 002mPa ¢ sat20 ± C).Althoughwatermoleculespresentintheinteriorof theglobulemayin°uence k ¡ 1 f (0),thefoldingdynamicscoupleonlyweaklytothe bulkviscosityoftheenvironment. TemperatureandViscosityE®ectsontheFoldingTime. Temperaturehasaprofounde®ectontheviscosity-dependenceoftherat easdemonstrated inFigure 5{9 .Theindicatedfoldingratesanddynamicviscositiesaremeasu red

PAGE 92

80 0 1 2 3 4 5 0 10 20 30 40 Solvent Viscosity,(mPas) h · 1 1.1 1.2 10 15 h GdnHClGlycerolEthylene GlycolGlucose S Skf -1(s) m Figure5{8:Viscositydependenceof M ! N foldingtimeat20 ± C.Datafordifferentviscogensallfollowthesamelineardependence.Theco solventsare:glycerol(¯lledcircles),ethyleneglycol(triangles)andglucose( squares)allin0.5M GdnHCl,and0.5{2.4MGdnHCl(opencircles)withnoextraviscoge n.Thedottedlineisalinear¯ttoalldatapoints.Insetmagni¯esthelowvi scosityregion, showingdataatvariableGdnHClconcentration.experimentallyattheindicatedtemperatures:17,20,25,an d30 ± C.Notehowa changeintemperaturehasamoresubstantiale®ectontheinterc eptat k ¡ 1 f (0)than ontheslope.Thenatureofthedotted-line¯tsinFigure 5{9 isexplainedindetail intheDiscussionsection. 5.4DiscussiononFoldingfromaCompactState Wehaveobservedfoldingfromthecompactnear-nativestate M ofhorse heartproteinferrocytochrome c .Figures 5{8 and 5{9 showthatthefoldingtimes accelerateinproportiontosolventviscosity, k ¡ 1 f / ´ S ,witha¯nitelimitingvalue k ¡ 1 f (0)as ´ S ! 0.WeanalyzethedatausingaKramerstheorydescriptionofa foldingreactionandproposeamodeltoquantifytheinternal frictioncontribution tofastfoldingdynamics.

PAGE 93

81 0 2 4 6 0 5 10 15 20 25 30 35 40 45 T = 17 °C 30 °C 25 °C 20 °C 135Solvent Viscosity,(mPas) hs· kf -1(s) mFigure5{9:Temperatureandviscositye®ectsonthefoldingtim e, k ¡ 1 f .Indicated viscositiesrefertothedynamicviscosityofthesamplesolvent,d irectlymeasuredat theindicatedtemperatures.Temperaturechangeshaveamore substantiale®ecton theinterceptat k ¡ 1 f (0)thanontheslope.Dottedlinescorrespondtoasimultaneous4-parameter¯tusingEquation 5.8 ,formulatedintheDiscussionsection. Kramerstheoryandinternalfrictione®ects. AKramerstheory descriptionofaproteinfoldingreactionassertsthatthefold ingratevariesinversely withthereactionfriction ° ( k ¡ 1 f / ° ).Becausetheproteinmoleculesmovein viscoussolvent,weexpectthefoldingratealsotovaryinversel ywiththesolvent friction(solventdynamicviscosity, ´ S ).Ourresultsshowthat k ¡ 1 f isproportional to ´ S regardlessoftheviscogeniccosolventusedwitha¯nitenon-zero intercept (Fig. 5{8 and 5{9 ).The¯niteinterceptisevidencefora¯nitefoldingspeedatlo w viscosityasshowninFigures 5{8 and 5{9 .Itsupportstheinitialassumptionthat internalfrictionprocessesdoexistandthattheylimitthespee dofproteinfolding, atleastforsu±cientlycompact(andfastfolding)molecules.

PAGE 94

82 InSection5.2.2,weintroducedanempiricalmodel(Eq. 5.2 )ofadditive internalfrictionproposedbyAnsariandcoworkers[ 114 ].Theyparameterized internalfrictione®ectsusinganinternalviscosity ¾ ,whichwaslargeformyoglobin ( ¾ =4 : 1 § 1 : 3mPa ¢ s)butsmall( ¾ ¼ 0)forProteinL[ 33 ]andCspB[ 102 ].For the M ! N foldingofferrocytochome c ,however,thedataofFigure 5{8 clearly indicateasubstantialvalue ¾ ¼ 2 : 1 § 0 : 3mPa ¢ sat20 ± C. Thisdoesnotimplythatfrictionvariesdramaticallyfromo neproteinto another,becausefoldingkineticsprobeonlytherate-limit ingstageoftheprocess. They-interceptofFigure 5{8 (thefoldingtimeinthelimitofvanishingsolvent friction)presentsalessambiguousgaugeoffrictionthandoe s ¾ .Forexample, althoughthedataofrefs.[ 33 ]and[ 102 ]implythat ¾ =0inthefoldingofprotein LandCspB,theycertainlydonotsuggestthat k ¡ 1 f ! 0as ´ S ! 0.Theirdata havesu±cienttimeresolutiononlytoshowthatthelimitingvalu efor k ¡ 1 f doesnot exceedafewmilliseconds.Theycertainlydonotruleoutavalu e k f » 10 5 s ¡ 1 as seeninFigures 5{8 and 5{9 . Newmodelforinterpretingfoldingtime. Havingestablishedthe importanceofthelimitingtimescaleinthefoldingdata,wep roposeanewmodel tointerpretthefoldingtimeassumoftwoseparaterelaxation times: k ¡ 1 f = ¿ S + ¿ int (5.6) where ¿ S standsforthetime-scaleforsolvent-coupledreorganization sand ¿ int isforinternal-friction-controlled(solvent-independen t)timescales.Weaddthe relaxationtimesratherthantheratesbecausethefoldingra te k f remains¯niteas ´ S ! 0,whiletherate1 =¿ S woulddivergeinthislimit.Inourmodel, ¿ S takesinto accountthelineardependenceof k ¡ 1 f on ´ S anddescribesdynamicswhichhasa Kramers-likeresponsetosolventviscosity: ¿ S / ´ S exp(¢ G S =k B T ) : (5.7)

PAGE 95

83 Thesecondrelaxationtime, ¿ int ,accountsforthefundamentaltimescalefor polypeptidemotionsthatdonotcoupletothesolventviscosity ´ S .Itcharacterizes internalfrictione®ectsandsetsthe¯nitelimitingvalueof k f . Whatdoesanadditiverelaxationtimemodeltellusaboutpro teinfolding? Itseparatesthetimescaleduetosolvent-dependentreorganiz ationofthesolventexposedregionsofthemoleculefromthetimescaleduetointer nalfrictione®ects: therecon¯gurationswithinthecompactinteriorwithminima linteractionsfromthe surroundingsolvent.Dependingonthemagnitudeof ´ S ,eithersetofmotionsmay controltheoveralltimeforfolding. Mathematically,Equations 5.6 and 5.7 areequivalenttotheproteinfriction modelofAnsari etal. (Eq. 5.2 )|theybothdescribe k ¡ 1 f asalinearfunctionof ´ S witha¯niteintercept.However,theydi®ersigni¯cantlyininter pretation.In Equation 5.2 ,theemphasisisonviscosity ¾ .Ourmodel,basedonEquations 5.6 and 5.7 ,identi¯estheinternalfriction-controlledtimescale.Fro mtheexample ofmyoglobin,ProteinL,CspBandourowndatafromcytochrome c , ¾ varies inconsistentlyfrom ¡ 0 : 1 § 0 : 2mPa ¢ sto4 : 1 § 1 : 3mPa ¢ s.However,noneofthese resultsareinconsistentwithvaluesof ¿ int » ns{ ¹ s. AcloseexaminationofFigure 5{9 (Temperatureandviscositye®ectson k ¡ 1 f ) separatesthetwotimescales: ¿ S isthetimescalecontrollingtheslopesofthe viscosity-dependentratesmeasuredatdi®erentsampletemperat ureswhile ¿ int correspondstothetimescalede¯nedbytheintercept, k ¡ 1 f ( ´ S ! 0).Notetheslight decreaseintheslopeasthetemperaturerisesfrom17 ± to30 ± .Atthesametime, theinterceptalsodecreases3-foldinthistemperaturerange .Weanalyzethese relationshipsquantitativelybyassuminganexplicitlyArrhen ius T -dependencefor

PAGE 96

84 ¿ S and ¿ int and¯ttingthedataofFigure 5{9 to: k ¡ 1 f = ¿ S + ¿ int ¿ S ( T;´ S )= A´ S exp(¢ H=k B T ) ¿ int ( T )= B exp(¢ E=k B T ) : (5.8) ThedottedlinesinFigure 5{9 showresultsofthesimultaneous4-parameter-¯t toalldatapoints.The¯tparametersgive¢ H =19 § 7kJ/moland¢ E = 67 § 16kJ/mol,with A ¼ 1 : 88ns/(mPa ¢ s)and B ¼ 8 : 31 £ 10 ¡ 18 s.Usingthese valuesandEquation 5.8 tocalculatetherelevanttimescalesinwater( ´ S =1 : 002) at20 ± C,we¯ndthat ¿ int ¼ 6 : 2 ¹ salreadyexceeds ¿ S ¼ 5 ¹ s.Therefore,evenin water-solvent,recon¯gurationsinternaltotheproteinmole culealreadydominate thefoldingtime. Fromthe¯ttoEquation 5.8 ,¢ E ¼ 67kJ/molwhile¢ H ¼ 19kJ/mol. Since¢ E isassociatedwith ¿ int ,wecanassertthatasubstantialactivation enthalpycontrolstheinternalreorganizationinthecompa ctstate,evenifcompact con¯gurationsofthemoleculemayalreadylienearthebottom oftheenergy surface.Bycontrast, ¿ S variesonlyweaklyintemperature,indicatingthatthat thesolvent-dependentbarriertofoldingisprimarilyentro pic.Thisisverydi®erent fromreorganizationsinafullyunfoldedcon¯gurationwhere di®usionalmotionsof thepolypeptidescaledirectlywithsolventviscosityandshowli ttleevidencethat intra-chaininteractionenergiesa®ectthedynamics[ 14 ]. Roughnessoftheenergylandscape. Infoldingfromacompactstate, solvent-independentinternalrecon¯gurationscontrolthef oldingtime.Thecharacteristictimeforinternalinteractions, ¿ int ,dependssensitivelyontheenergetic costsofconformationaldi®usionamongthecompactcon¯guratio ns.Thisresemblesanenergy-landscapedescriptionoffoldingasdi®usionona roughenergy surface[ 53 , 36 ],wheretheconformationalsearchtimescalesinverselywitht he e®ectivedi®usioncoe±cient D ¤ .Intheroughlandscapemodel, D ¤ dependsina

PAGE 97

85 non-Arrheniusmannerontheenergyscale " oftheroughness[ 124 ], D ¤ = D exp[ ¡ ( "=k B T ) 2 ] : (5.9) If D ¤ controlsthereorganizationandfoldingtimes, ¿ int ,wecan¯tthedataof Figure 5{9 usingthenon-Arrheniustemperaturedependencefor ¿ int , ¿ int = C exp[( "=k B T ) 2 ] : (5.10) Fromthe¯ttothedata,weobtainthe"roughness" " ,whichgives " =9 : 8 § 1 : 0kJ/moland C ¼ 6 : 2ps.The¯tusingEquation 5.10 isvirtuallyindistinguishablefromthedotted-linesshowninFigure 5{9 . 5.5Conclusions Therateofdi®usionalmotionofanunfoldedpolypeptidechain throughits solventplacesonephysicallimitonthespeedofproteinfoldin g.Reducingthe solventviscositycanthereforeacceleratefolding,butonlyu ntilotherphenomena (lessstronglycoupledtothesolvent)begintocontrolthefold ingrate.Wehave shownthat,fora104residueprotein,di®usionalreorganizatio nofacompact, late-stagefoldingintermediateimposesasolventviscosity-in dependent,although stronglytemperature-dependent,upperlimitofabout10 5 s ¡ 1 tothefoldingspeed. Thisobservedreorganizationrateanditstemperaturedepen denceprovidea windowintofundamentaltimeandenergyscalesassociatedwith theprotein's conformationalsearchthroughthelowervalleysofitsfreee nergysurface. Kramerstheory,asappliedtoproteinfolding,providesasat isfactorydescriptionforthesolventviscositydependenceofmanyslowerfolding eventscoupledto thesolvent.However,thefastestfoldingdynamicsinacompactsy stemappear todecouplefromthesolventatlowviscosities.Thisgivesriseto afoldingspeed limitofafewmicrosecondsduetointernalreorganizationsw ithinthecompact

PAGE 98

86 molecule.Thisisfarslowerthantherelaxationtimeofabout 100ns,foranideal non-interactingpolypeptideofsimilarlength[ 22 ]. Thefastrateofthesedynamicstendstoexplainwhypreviousstu diesof proteinfoldingatmillisecondtimeresolutionfailedtodete ctclearevidencefor anyviscosity-independent,or\internalfriction"controll ed,dynamics.Since mostbulkdi®usionalmotions,includingcontactformationandc haincollapse, occurontimescalesofatleasttensorhundredsofmicroseconds orlonger,the presentdatasuggestthatinternalfrictione®ectswilllimitth efoldingofonlythe fastestfoldingproteins.Shorterpeptidesappeartomeetlitt leinternalfriction informingelementalstructures[ 107 , 111 ],butsimulations[ 113 ]suggestthat longerchainsfoldingunderparticularlyfavorablecondit ionsmaywellencounter internalfrictionale®ectsintheirdynamics.Recentexperim entalstudiesof ¸ 6 ¡ 85 foldingappeartocon¯rmthatevenan80-residueproteincanfo ldontimescales approachingthefastestmolecularreorganizationsofthecha in[ 85 ].Forsuch proteins,refoldinginsolventsofvaryingviscositycouldcause ashiftintherelative importanceofdi®erentfoldingpathways,ifsomepathwayscoup lemorestrongly tothesolventthandoothers.Ourresultsalsosupportthepossibili tythatinternal frictioncoulda®ectthedynamicsofunfoldedproteinsarti¯c iallycon¯nedto su±cientlysmallvolumes,suchasbysolventcrowding[ 108 ]orwithinmolecular chaperones. 5.6FutureDirection Wewouldliketoextendinternalfrictionstudiestootherpro teinsand polypeptidesystems.Thisentailsobservationoffastdynamicsi nthenanosecondstomicrosecondsregimeinsystemswithouttheheme-COtype ofinteraction seeninthefoldingofcytochrome c .Wewouldliketotakeadvantageofthetriplet stateofthenaturallyoccurringaminoacidtryptophan[ 17 ]todotime-resolved opticalabsorptionexperimentsonothernaturalorsynthetic tryptophan-containing

PAGE 99

87 proteinsorpeptides.Atpresent,wehavecon¯guredthetransien tabsorption spectrometerwithananosecond-pulseddyelasersystememittinga t280nm toexcitethetryptophantripletstate(SeeAppendixCformore information). Contactoftheexcitedtryptophanwithatripletacceptormo leculequenchesthe tryptophantripletstateandfacilitatesabsorbancechange. Wewouldliketouse thissystemtostudyhowproteininternalfrictiona®ectsinitia ltertiarycontact formation.Thetargetsystemsincludesyntheticpolypeptides withtripletdonor andacceptormoleculeinvaryingpositionswithinthepolype ptidechainandthe tryptophan-mutantoftheproteinparvalbumin.

PAGE 100

CHAPTER6 GENERALCONCLUSIONS Weareatanerawherewearebeginningtounderstandthephysica laspects ofproteinfoldingandproteindynamics.Theenergylandscape theoryofprotein foldingseemstoanswerthequestionofwhatmakesproteinfoldi ngreactionsso rapidcomparedtorandomsearch.Thehigh-frictionlimitofK ramerstheoryfor di®usiondrivenreactionsbestdescribesproteinfoldingkinet ics.However,key biophysicalquestionsregardingthephysicallimitsoffoldin gatfasttimescales remain.Inthisstudy,wehavecontributedtothatunderstandi ngbydeveloping andimprovingtechniquestolookatfastdynamicsandbystudyi ngthefastestfoldingproteinsystems:tryptophancageandthecompactlatestageintermediate offerrocytochrome c . Wepresentedseveralwaystostudyfastdynamicsinproteinfoldi ng.We havedevelopedandcharacterizedasubmillisecondlaminar-°o wmixingdevice thatallowsUV-excitationandobservationofkinetic°uorescenc echangesin proteinswithpicomolarsampleconsumption.Togetherwitheq uilibriumCDand °uorescencemeasurements,wehaveusedtemperature-jumpdatato characterize thetwo-statefoldingofthedesignedminiproteintryptophan cage.Wehave appliedlaser°ashphotolysistotheheme-CObondandusedtransien tabsorption spectroscopytolookatthefast M ! N foldingtransitionsinferrocytochrome c . Wearecurrentlydevelopingatriplet-tripletenergytransf erexperimenttodo nanosecond-resolvedopticalabsorptionexperimentsontrypt ophancontaining proteinsandpeptides. Ourlaminar-°owcoaxialmixercontributedtothecontinuing improvement ofultrafastmixingdevices.Theuselow-Reynolds-number°owisa nimportant 88

PAGE 101

89 directionformixingtechnologies,becauseitprovidessubmil liseconddeadtimes withmicrolitertonanolitersampleconsumption.Wemeasuredf ast°uorescence decayswith » 400 ¹ sdeadtime,whileconsumingonly0.2to6nL/sofsamplein theinnercapillaryand1.4 ¹ L/sintheoutercapillary. Theminiproteintryptophancagefoldsin4 ¹ s.Theequilibriumandkinetic measurementsshowexcellentagreementwithatwo-stateall-or -nothingfolding reaction.Trp-cagesetstheconditionsforfastfolding:atwo -statereaction,aweak foldingactivationenergybarrier,anearlyoptimizedfree energylandscape,and pre-organizedstructuresintheunfoldedstate.Duetoitssmal lsizeandfastfolding rate,Trp-cageservesagoodbenchmarkmoleculeforall-atom simulations.Asan outlierinthecontact-ordercorrelationplot,Trp-cageal socontributestothegrowth ofresearchonthecorrelationoftopologytofoldingrates. Inferrocytochrome c ,thefoldingtimefromacompactcon¯guration M ! N is12 ¹ sinwater.Analysisofthesolventviscosity-dependenceofthefo ldingtime usingamodelbasedonKramersratetheoryallowedustoidentif ytwolimiting timescalesinproteinfolding:thetimescaleforsolvent-coup ledreorganizations andthetimescalecontrolledbytheinternalfrictionwithin theproteinmolecule.

PAGE 102

APPENDIXA AMINOACIDS Welistthe20aminoacidsinorderofhydrobicity,fromthemost totheleast hydrophobic.WefollowthescaleofEngelman,SteitzandGold man[ 125 ].Wealso listthe3-letterand1-letterabbreviationsthatisusedtoid entifytheaminoacids, andsomegeneralamino-acidproperties. TableA{1:Listof20aminoacids,inorderofhydrophobicity,an dtheirproperties AminoacidsAbbreviationProperties PhenylalaninePhe,FVeryhydrophobic,aromaticMethionineMet,MVeryhydrophobic,with{S{Isoleucinelle,IVeryhydrophobicLeucineLeu,LVeryhydrophobicValineVal,VVeryhydrophobicCysteineCys,CVeryhydrophobic,with{SHTryptophanTrp,WVeryhydrophobic,aromatic,°uorescentAlanineAla,AHydrophobicThreonineThr,THydrophobic,with{OHGlycineGly,GHydrophobicSerineSer,SHydrophobic,with{OHProlinePro,PHydrophobic,cyclicTyrosineTyr,YHydrophobic,aromatic,with{OHHistidineHis,HBasic,hydrophobicGlutamineGln,QAcidicAsparagineAsn,NAcidicGlutamicacidGlu,EAcidicLysineLys,KBasicAsparticacidAsp,DAcidicArginineArg,RBasic 90

PAGE 103

APPENDIXB NUMERICALMETHODS B.1KineticModellingofLaminar-FlowCoaxialMixing B.1.1SolutionoftheRadialDi®erentialEquation InChapter2,wedescribedthelaminar-°ow°uidmixerforfast°uor escence kineticstudies.A°uorescentsample°owsoutoftheinnercapillar yandisreleased inthecenterofdilutingbu®erpassingthroughthelargerouter capillary.Mixing isdrivenbydi®usionoftheoutercapillarysolutessurrounding thethinstreamof °uorescent.Thesimplecylindricalcoaxialjetgeometryallow snumericalmodelling ofthemixerbehaviorwitharadialdi®usionequation: @C ( r;t ) @t = D r µ @C @r + r @ 2 C @r 2 ¶ (B.1) C ( r;t )istheconcentrationoftheoutercapillarysolute(e.g.,th edilutingbu®er) atradius r (distancefromthemixeraxis)andtime t . D isthesolutedi®usion constant.Weconsideramixingexperimentwithinitialcondit ions, C ( r; 0)= 0if r · a C 0 if r>a , (B.2) where C 0 istheinitialconcentrationoftheoutercapillarybu®erand a istheradius oftheinnercapillarystream. ThenumericalsolutiontoEquation B.1 iscalculatedusingMATLAB.The generalsolutionfortheconcentrationofsolutesisdetermin edbyconstructinga N r £ N t concentrationmatrix C where N r isthenumberofradialpositiongrids and N t isthenumberoftimegrids.Eachelement C i;j isadiscreteconcentrationat radialposition r i andtime t j .Thegridspacinginpositionisaconstantgivenby 91

PAGE 104

92 annularelement ±r = a=N where N isanintegerand r N r ¸ 8 a .Theminimumand maximumtimepointsaregivenby t 1 =0 : 003 a 2 =D and t N t =6 a 2 =D ,respectively. Theradiusoftheinnerstream a andthedi®usionconstant D areuser-de¯ned parameters.Thetimeelement ±t = t 1 . The¯nite-di®erencerepresentationofEquation B.1 isgivenby dC i dt = D r i µ C i +1 ¡ C i ±r + r i C i +1 ¡ 2 C i + C i ¡ 1 ( ±r ) 2 ¶ : (B.3) Symbolically, dC d¿ = K ¢ C (B.4) where K isan N r £ N r tridiagonalmatrixofrateconstants, d¿ isthedimensionless timeelementgivenby d¿ = dt ( D=±r 2 ).Thedimensionlessratematrix K hasthe followingelements, for i =2to N r ¡ 1 0BBBB@ K i;i ¡ 1 = r i =r i K i;i = ¡ (2 r i + ±r ) =r i K i;i +1 =( r i + ±r ) =r i 1CCCCA (B.5) for i =1to2and N r ¡ 1to N r 0BBBBBBB@ K 1 ; 1 = ¡ ( r 1 + ±r ) =r 1 K 1 ; 2 =( r 1 + ±r ) =r 1 K N r ;N r ¡ 1 = r N r =r N r K N r ;N r = ¡ r N r =r N r 1CCCCCCCA where r i isthepositionat i , ±r isthegridspacingand N r isthenumberofposition grids.Theeigenvalues, ¸ ,andeigenvectors, V ,oftheratematrix K isdetermined. Since K isdimensionless,weneedtomultiply ¸ bythefundamentaltimescale D=±r 2 ,togeneratethetrueeigenvalues.FromEquation B.4 ,weexpectthegeneral solutionof C tobe C » C init exp µ D=±r 2 Z ¸dt ¶ (B.6)

PAGE 105

93 where C init speci¯estheinitialcondition.Weconstructthe N r £ N t matrix T given by T l;m =exp( D=±r 2 ¸ l ¢ t m )(B.7) where t m isthetimepointat m .Weimposetheinitialvalueconditionby¯ndingthematrixofamplitudes A ( C init = V ¢ A ).Thegeneralsolutionforthe concentrationofsolutes, C ,isgivenby C i;j = N r X n =1 V T i;n T n;j A n : (B.8) Theconcentrationofsolutesasafunctionofspaceandtimeissh owninFigure 3{3 . B.1.2CalculationofReactionRates Inordertocharacterizetheperformanceofthecoaxialmixe r,the°uorescence ofNATA°owingoutoftheinnercapillaryisquenchedbyNBS.Weca nextendthe numericalmodelforcylindricaldi®usionshownaboveforgener ationofthesample streamofNATA,withthesimultaneousinwarddi®usionandreaction ofNBS,to predicttheapparentrateof°uorescencequenching, k app .Theconcentrationof NATAasfunctionofpositionandtimeisgivenby [NATA]( r;t )=[NATA] 0 exp µ ¡ k bi Z t 0 C ( r;t 0 ) dt ¶ (B.9) where C ( r;t )=[NBS]( r;t ).TheNATAconcentrationmatrixisrepresentedbythe N r £ N t matrix W withelements W i;j = W 0 exp à ¡ k bi ±t i X l =1 C l;j ! : (B.10) ThisisthematrixanalogofEquation B.9 . W 0 imposesinitialNATAconcentration (0at t =0for r>a ).Therelative°uorescenceintensity F isarowmatrixof length N t givenbyelements F i = r T n W n;i r T m W m; 1 : (B.11)

PAGE 106

94 Single-exponential¯tsto F fordi®erentvaluesofestimatedradii a generatethe apparentrate, k app ,fordi®erentvaluesof a .Thesepredictedratesareshownasthe solidlineinFig. 3{7 A;andapproachtheactualrate k = k bi ¢ [NBS],atlow a . B.2KineticModellingusingSingularValueDecomposition InChapter5,weuseatransientabsorptionspectrometertomonit orfolding fromacompactstateofferrocytochrome c afterlaser°ashphotolysis.Wecollect time-resolvedopticalabsorptiondi®erencespectra,¢OD,anduse singularvalue decomposition(SVD)toanalyzethedata.SVDisamatrixtechniq uethatallows noise-¯lteringandassessmentoftheminimumnumberofindependen tlyevolving spectralpro¯lesneededtodescribephysicallysigni¯cantspectra lchanges[ 126 , 127 ]. Foratypicaltime-resolvedopticalabsorptionexperiment,¢O Disa N ¸ £ N t matrixofopticalabsorptiondi®erencespectra( N ¸ =97and N t =48or67). N ¸ isthenumberofwavelengthpoints.Thewavelengthrangeusedi ntheexperiment isfrom » 390to » 450nmwithspectralresolutionof0.124nm. N t isthenumber oflogarithmically-spacedtimedelaysfrom » 10nsto > 1s.Attime t ,each di®erencespectrumisgivenby A ( ¸;t ) ¡ A ( ¸; 0 ¡ ),thedi®erenceinopticaldensity betweenthephotoproductandtheunphotolyzedsamplemeasure dasfunctionof wavelength.Therefore,eachcolumnin¢ODisadi®erencespectr umattimedelay t .¢ODcolumnsvaryonlyinwavelengthandeachrowcorrespondst oasingle wavelength. Forsimplicity,wedenote¢ODas D .WeusetheSVDalgorithminMATLAB tofactor D intoaproductof3matrices. D = U ¢ S ¢ V T : (B.12) U and V arethematrixofeigenvectorsfor DD T and D T D ,respectively.The singularvaluesof D arethesquarerootsofthesharedeigenvalues E of DD T and

PAGE 107

95 D T D ,whicharethediagonalelementsof S .Themathematicalrelationshipsare: DD T = U ¢ E ¢ U T ; D T D = V ¢ E ¢ V T ; and S = E 1 = 2 : (B.13) ForanintuitiveunderstandingofSVD,notethat N ¸ £ N ¸ matrix DD T containsoverlapofkineticvectorsforallpairsofwavelen gths.Likewise N t £ N t matrix D T D containstime-pairwiseoverlapofspectralvectors.Hence,the columnsof U containstheminimalsetoforthonormalbasisfortherowspaceo f D .Likewise,thecolumnsof V containstheorthonormalbasisforthecolumnspace of D .Thus,SVDidenti¯esthemathematicallyindependentbasesofth edata, spectrallyin U ( ¸ )andtemporallyin V ( t )[ 127 ]. Weselectusablecomponentsof U ( ¸ )and V ( t )fromthemagnitudesofthe singularvaluesof D (thediagonalelementsof S ).Intheabsenceofmeasurement noise,thenumberofnon-zerosingularvaluesisthenumber r oflinearlyindependentcomponentspectratodescribethedataset D .Inrealexperimentaldata, allsingularvaluesarenon-zero.Hence,identi¯cationofrele vantsingularvalues dependsoninstrumentsignal-to-noiseratio. IntheSVDof¢OD,thetime-resolvedabsorptionspectraafterphot olysis ofthe M -COstateofferrocytochrome c ,weidentifyonlytwodominantsingular valuesatlowdenaturantconcentrations( · 2 : 4MGdnHCl).Thus,weonlyhave tworelevantSVDcomponents( r =2):columnvectors U 1 and V 1 associatedwith singularvalue S 11 ,andcolumnvectors U 2 and V 2 associatedwithsingularvalue S 22 .Weareabletoreducethe¢ODmatrixfrom N ¸ £ N t to N ¸ £ r .Hence,wecan expressthedi®erencespectraasatruncatedmatrix, ¢OD r ( ¸;t )= U 1 ¢ V T 1 +( S 22 =S 11 ) U 2 ¢ V T 2 : (B.14)

PAGE 108

96 Thecomponentswith U n>r , V n>r ,and S nn ( n>r )arediscardedasinstrument noise. Wecanthenconstructthenoise-¯lteredabsorptionspectraatany delaytime t , A r ( ¸;t )=¢OD r ( ¸;t )+ A ( ¸; 0 ¡ ) : (B.15) Wecanalsoidentifytheabsorptionspectraofthephotolyzedtr ansientspecies, 390 400 410 420 430 440 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Wavelength (nm)ODN M M -CO FigureB{1:Transientabsorptionspectra, A photo ( ¸;t ),offerrocytochrome c .The M stateformsimmediatelyafterphotolysisoftheCOfromthehem eandfoldsinto the N statewith k ¡ 1 f ¼ 12 ¹ s.Thesyringepumpthendeliversfresh M -COtothe sample°owcell.Timecourserangefrom » 10nsafterthelaser°ash,wherethe dominantspeciesisthe M -state,to > 1safterthelaser°ash,wherethedominant speciesinthe M -COstate.Sampleshownis50 ¹ Mferrocytochrome c in1.2M GdnHCl, T =20 ± C.Theassumedphotolysisyield f is18.6%(Eq. B.16 ). A photo ,byassumingaphotolysisfractionalyield, f .(COrebindingtothe M -state occurs » 50nsafterphotolysisandaccountsforthelowyield( f< 20%)ofthe photoproduct M .) A r ( ¸;t )= f ¢ A photo ( ¸;t )+(1 ¡ f ) ¢ A ( ¸; 0 ¡ ) : (B.16)

PAGE 109

97 Figure B{1 showstheconstructedtransientabsorptionspectra, A photo .Weidentify thephotoproduct M state,thefolded N state,andtheunphotolyzed M -COstate asitisdeliveredbythesyringepumpforthenextlaser°ash. Relaxationratesforthefoldingtransition M ! N andtheforreplacementof N by M -CO(duetosample°ow)wereobtainedbysimultaneously¯ttingthe ¯rst twotimecomponents,weightedbythesingularvalues,toasumof2 exponentials: S 11 V 1 ( t )= C 11 exp( ¡ k 1 ¢ t )+ C 12 exp( ¡ k 2 ¢ t ) S 22 V 2 ( t )= C 21 exp( ¡ k 1 ¢ t )+ C 22 exp( ¡ k 2 ¢ t ) : (B.17) The¯rstrelaxationrate, k 1 = k f ,representsspectralchangescorrespondingtothe foldingtransition M ! N ,sinceequilibriumstronglyfavorsthefolded N stateover thecompactdenaturedstates M and M -COatlowdenaturantconcentrations.The secondrelaxationrate, k 2 ,isduetosample°ow,thereplacementof N by M -CO, asasyringepumpsuppliesfreshsampletotheoptical°owcell.Fig ure 5{6 in Chapter5showsrepresentativeamplitudesSV 1 ( t )andSV 2 ( t )ofspectralchanges obtainedbySVDoftransientspectrafollowingphotolysis,andbi exponential¯t.

PAGE 110

APPENDIXC FUTUREDIRECTIONOFINTERNALFRICTIONSTUDIES: POLYPEPTIDERECONFIGURATIONSVIAENERGYTRANSFER C.1Introduction Forextensionofinternalfrictionstudiestootherproteinsa ndpolypeptide systems,weneedtoobservefastdynamicsinthenanosecondstomicr oseconds regimeinsystemswithouttheheme-COscheme.Todothis,wewould liketouse triplet-tripletenergytransfer(TTET)tostudyhowproteini nternalfrictiona®ects initialtertiarycontactformation.Wewouldliketotakead vantageofthetriplet stateofthenaturallyoccurringaminoacidtryptophan[ 17 ]todotime-resolved opticalabsorptionexperimentsonothernaturalorsynthetic tryptophan-containing proteinsorpeptides. C.2Triplet-TripletEnergyTransfer Therearethreestatesthatplayadominantroleinthephotoph ysicsofmost organicmoleculesinsolution:thegroundstate(thelowestsing letstate),thelowest excitedstate(the¯rstexcitedtripletstate),andthe¯rstexcite dsingletstate[ 128 ]. Theexcitedtripletstateislowerinenergythantheexcitedsi ngletstate,but electronictransitionsfromthegroundstatetothetripletsta teareforbidden. However,organicmoleculesinthegroundstatethatareexcite dopticallytothe ¯rstexcitedsingletstatecandecaytothethetripletstate(thel owestexcited state).Alternatively,theycanundergoelectronictransitio nsbacktotheground stateeitherby°uorescenceorbyradiationlessprocesses. Thetripletstatelifetimecanbeorderofmagnitudeslongert hattheexcited singletstate.Therefore,onceinthetripletstate,molecules canstayexcited untilthetripletstateenergyistransferredtotripletaccep tormolecules,oruntil 98

PAGE 111

99 physicallyquenchedbyparamagneticmolecules(e.g.,O 2 ,NO,innertransitional metalions)[ 129 ].Thus,wecanusetriplet-tripletenergytransfer(TTET)to measureratesofcontactformationbetweentripletdonorand tripletacceptor moleculesinsertedwithinthepolypeptidechain.Contactof thedonormolecule, whichhasbeenopticallyexcitedtoitstripletstate,withth eacceptormolecule leadstoenergytransferthatexcitestheacceptormoleculet oitsowntriplet stateandfacilitatesabsorbancechange.Energytransferhapp ensviatheDexter mechanism,anelectron-exchangemechanism,whichrequiresv anderWaalscontact betweendonorandacceptormolecules[ 15 ].Theabsorbancechangeswhichoccur ondonor-acceptorcontactcanbetime-resolvedusingamodi¯ed versionofthe transientabsorptionspectrometerusedinChapter5. Oneoftherequirementsforagoodtriplet-tripletenergy-t ransferpairisa higherdonortripletstateenergycomparedtoacceptortripl etstateenergy[ 129 ]. Forend-to-endcontactformationexperimentsinsyntheticp olypeptides,Keifhaber andcoworkers[ 15 , 16 ]usedthioxanthone(withenergyofthetripletstate, E T ¼ 265kJ/mol)andxanthone( E T ¼ 310kJ/mol)asdonormolecules;and napthalene( E T ¼ 253kJ/mol)astheacceptormolecule.Lapidus etal. used thetripletstateoftryptophanwithaminoacidcysteineasthe tripletquencher. Theuseofnaturally-occurringaminoacidsisadvantageousb ecausee®ectsof largemolecularprobes(likethioxanthoneandxanthone)th atarenotnativeto thepolypeptidechainsareavoided.However,quenchingofth etripletstateof tryptophanbycysteineisnotadi®usion-limitedprocess[ 18 ]. C.3ApplicationtoInternalFrictionStudies Twoindependentgroups,Keifhaberandcoworkers[ 15 ]andEatonandcoworkers[ 18 ],haveobserved k ¡ 1 / ´ S inend-to-endcontactformationexperimentsin disorderedpolypeptides.However,noonehasactuallyexplore dtheroleofprotein internalfrictioninsuchsystems.

PAGE 112

100 Wearecurrentlytestingtriplet-tripletenergytransferwit htryptophanas donormoleculeandanonnativeacceptormolecule,napthale neoranapthalenederivative.Fromthevaluesoftheirtripletenergies(tryp tophan, E T ¼ 301kJ/mol andnapthalene, E T ¼ 253kJ/mol),tryptophanandnapthaleneseemtobeasuitabledonor-acceptorpair.Weexpectthetransferprocesstobe di®usion-controlled andeasilyfollowedbythetripletstateabsorbancechanges.Mor eover,tryptophan andthenon-naturalaminoacidnapthyl-alanine(anapthale nederivative)can beinsertedanywherewithinapolypeptidechain,allowingsma ll-loopformation experimentsinlongchains. Wewouldliketoinvestigateifwecanobserveinternalfrictio ne®ectsinthe loopformationofdisorderedpolypeptidechainswithtrypto phanandnapthylalanineinsertedindi®erentlocationswithinthechainsequen ce.Wearealso interestedinexploringinternalfrictione®ectsinintramol ecularcontactformation inproteins.Ourtargetmoleculeisthetryptophan-mutantof parvalbumin,F102W. ParvalbuminF102W(ProteinDataBankcode:1B8R)isa108-re sidueproteinwith alonetryptophan(position102)andalonecysteine(position1 8). C.4Modi¯edTransientAbsorptionSpectrometer Atpresent,wehavecon¯guredthetransientabsorptionspectrome terwitha nanosecond-pulseddyelasersystem(JaguarCDyeLaser,Continuum ,SantaClara, CA)emittingat281nmtoexcitethetryptophantripletstate.F igure C{1 shows thenewopticallay-out.WeusethesecondharmonicofapulsedNd: YAGlaser (532nm,100mJ/pulse, » 7-nanosecondpulses)topumpadyelasercontaining rhodaminedye(0.09gRhodamine6Gperlitermethanol).Agra zingincidence gratingwithatuningmirrorsetsthedyelaseroutputwaveleng thto562nm. Optimalalignmentoftheassociateddyelaseropticsproduces2 0%conversion e±ciencyoftheNd:YAGlaserenergy.

PAGE 113

101 Xe flashlamp BSM M BS M Nd:YAG 532 nm ICCD Spectrometer Sample DyeLaser562 nm Freq.Doubler 281 nm Schott GG400 filter 562 nm FigureC{1:Opticallay-outofthetransientabsorptionspectr ometercon¯gured withapulseddyelaser.A281-nmdye-laser°ashexcitesthetriple tstateoftryptophan.Attimedelay t afterthelaser°ash,transmittedlight(400to650nm) fromaXe°ashlampiscollectedfromexcitedandunexcitedregio nsofthesampleanddeliveredtothespectrometer/ICCDsystemtomonitorop ticalabsorption changesuponcontactoftheexcitedtripletstatewithaquenc herorwithanacceptormolecule. The562-nmoutputofthedyelaserpassesthroughasecond-harmon icgeneratingcrystalyielding281nmforuseintryptophantripl etexcitation.Since both562-nmand281-nmlaserlightcomeoutofthecrystalfrequ encydoubler, weuseasilicaprismtoselectthe281-nmpulsedlaserlight,whichi sfocusedinto thesamplecell.Attimedelay t afterthe281-nmlaser°ash,aXe°ashlamp¯res toprobeabsorbancechanges.AsinChapter5,theprobelightissp lit50-50and focusedinto2pointsinthesamplecell:theregionexcitedbyt he281-nmlaser °ashandinanunexcitedregionofthesamplecell. Togeneratedi®erenceabsorptionspectraduetotriplet-tripl etenergytransfer, imagingopticscollectthetransmittedlightfromUV-exciteda ndunexcitedregions anddirectsitthrougha3-mmGG-400Schottglass¯lterandthen throughthe imagingspectrometerandthegatedintensi¯edCCDcamera.TheG G-400Schott glass¯lterpreventswavelengthslessthan400nmfromreaching thecamera,

PAGE 114

102 ¯lteringoutmosttryptophan°uorescenceemission.(Tryptophan°u orescence emissionpeaksat » 350nmanddecaysinafewnanosecondswhilethetriplet stateabsorptionpeaksat » 450nmandhasareportedlifetimeof » 40 ¹ s[ 17 ].) Thespectrometerandassociatedelectronicsareoperatedbyat hesameLabView softwareinterfaceusedinChapter5.Likewise,di®erenceabsorp tionspectraper timedelay t arecollectedbytheWinSpecsoftware. Initialtransientabsorptionexperimentsusingfreetryptoph an(100 ¹ MNATA inpH7trisbu®er,deoxygenatedwithelectronscavengerN 2 O)insolutionand usingfreetryptophanwithfreenapthalene(100 ¹ MNATAand100 ¹ M1-napthylaceticacidinpH7trisbu®er,alsodeoxygenatedwithelectron scavengerN 2 O) yieldpromisingresults.Weobservetripletstateexcitationoft ryptophanwith apparentenergytransfertonapthalene.However,lowsignal-t o-noiseremainsabig issueandneedstobeaddressedbeforemoreexperimentsaremade.

PAGE 115

REFERENCES [1]A.V.FinkelsteinandO.B.Ptitsyn. ProteinPhysics .AcademicPress,London, 2002. 1.1 , 4.4.3 [2]C.M.Dobson.Proteinfoldingandmisfolding. Nature ,426:884{890,2003. 1.1 [3]R.H.Pain,editor. MechanismsofProteinFolding .FrontiersinMolecular Biology.OxfordUniversityPress,NewYork,secondedition,2000. 1.1 , 2.3.1 , 2.3.1 , 1 [4]H.M.Berman,J.Westbrook,Z.Feng,G.Gilliland,T.N.Bhat, H.Weissig, I.N.Shindyalov,andP.E.Bourne.Theproteindatabank. NucleicAcids Research ,28:235{242,2000. 1{1 , 2.3.3 [5]C.B.An¯nsen.Principlesthatgovernfoldingofproteincha ins. Science , 181:223{230,1973. 1.1 , 2.1 [6]W.A.Eaton,V.Mu~noz,S.J.Hagen,G.S.Jas,L.J.Lapidus,E.R. Henry,and J.Hofrichter.Fastkineticsandmechanismsinproteinfolding . Annu.Rev. Biophys.Biomol.Struct. ,29:327{359,2000. 1.1 , 2.2.1 , 2.2.3 , 2.4 , 3.1 [7]C.Levinthal.Aretherepathwaysforproteinfolding? J.Chim.Phys. ,65:44, 1968. 1.2 , 2.3.2 [8]S.E.Radford.Proteinfolding:Progressmadeandpromises ahead. Trends Biochem.Sci. ,25:611{618,2000. 2.1 [9]S.YehandD.L.Rousseau.Hierarchicalfoldingofcytochrom e c . Nature Struct.Biol. ,7:443{445,2000. 2.1 [10]M.Volk.Fastinitiationofpeptideandproteinfoldingp rocesses. Eur.J.Org. Chem. ,2001:2605{2621,2001. 2.2.1 , 2.2.2 , 2{1 , 2.2.2 [11]J.B.Knight,A.Vishwanath,J.P.Brody,andR.H.Austin.Hydrod ynamic focusingonasiliconchip:mixingnanolitersinmicroseconds. Phys.Rev. Lett. ,80:3863{3866,1998. 2.2.1 , 3.1.2 , 3.2.2 [12]C.M.Jones,E.R.Henry,Y.Yu,C.Chan,S.D.Luck,A.Bhuyan,H. Roder, J.Hofrichter,andW.A.Eaton.Fasteventsinproteinfoldingin itiatedby nanosecondlaserphotolysis. Proc.NatlAcad.Sci.USA ,90:11860{11864, 1993. 2.2.2 , 5.3.1 103

PAGE 116

104 [13]S.J.Hagen,J.Hofrichter,A.Szabo,andW.A.Eaton.Di®usionlimitcontact formationinunfoldedcytochrome c :Estimatingthemaximumrateofprotein folding. Proc.NatlAcad.Sci. ,93:11615{11617,1996. 2.2.2 , 2.4 , 4.1 , 4.6 , 5.2.2 [14]S.J.Hagen,C.W.Carswell,andE.M.Sjolander.Rateofint rachaincontact formationinanunfoldedprotein:temperatureanddenatura nte®ects. J. Mol.Biol. ,305:1161{1171,2001. 2{1 , 2.2.2 , 4.1 , 4.6 , 5.3.2 , 5.3.2 , 5.4 [15]O.Bieri,J.Wirz,B.Hellrung,M.Schutkowski,M.Drewell o,and T.Kiefhaber.Thespeedlimitforproteinfoldingmeasuredbyt riplettripletenergytransfer. Proc.NatlAcad.Sci.USA ,96:9597{9601,1999. 2.2.2 , 4.1 , 4.6 , 5.2.2 , C.2 , C.3 [16]F.Krieger,B.Fierz,O.Bieri,M.Drewello,andT.Keifh aber.Dynamicsof unfoldedpolypeptidechainsasmodelfortheearlieststepsin proteinfolding. J.Mol.Biol. ,332:265{274,2003. 2.2.2 , 2.4 , C.2 [17]L.J.Lapidus,W.A.Eaton,andJ.Hofrichter.Measuringther ateofintramolecularcontactformationinpolypeptides. Proc.NatlAcad.Sci.USA , 97:7220{7225,2000. 2.2.2 , 4.1 , 4.6 , 5.2.2 , 5.6 , C.1 , C.4 [18]L.J.Lapidus,P.J.Steinbach,W.A.Eaton,A.Szabo,andJ.Ho frichter.E®ect ofchainsti®nessinloopformationinpolypeptides.Appendix:Te stinga 1-dimensionaldi®usionmodelforpeptidedynamics. J.Phys.Chem.B , 106:11628{11640,2002. 2.2.2 , C.2 , C.3 [19]R.Gilmanshin,S.Williams,R.H.Callender,W.H.Woodru®,R. B.Dyer, andM.C.R.Shastry.Fasteventsinproteinfolding:Relaxatio ndynamicsof secondaryandtertiarystructureinnativeapomyoglobin. Proc.NatlAcad. Sci.USA ,94:3709{3713,1997. 2.2.3 [20]M.Gruebele,J.Sabelko,R.Ballew,andJ.Ervin.Laserte mperaturejump inducedproteinrefolding. Acc.Chem.Res. ,31:699{707,1998. 2.2.3 [21]S.J.HagenandW.A.Eaton.Two-stateexpansionandcollapseo fapolypeptide. J.Mol.Biol. ,301:1019{1027,2000. 2.2.3 [22]L.Qiu,C.Zachariah,andS.J.Hagen.Fastchaincontracti onduringfolding: \foldability"andcollapsedynamics. Phys.Rev.Lett. ,90:1681031{4,2003. 2.2.3 , 4.6 , 5.2.2 , 5.5 [23]M.Jacob,G.Holtermann,D.Perl,J.Reinstein,T.Schindl er,M.A.Geeves, andF.X.Schmid.Microsecondfoldingofthecoldshockproteinm easuredby pressure-jumptechnique. Biochemistry ,38:2882{2891,1999. 2.2.4 [24]S.E.Jackson.Howdosmallsingle-domainproteinsfold? Fold.Des. , 3:R81{R91,1998. 2.3.1 , 2.3.3 , 2.4 , 4.2 , 4.6

PAGE 117

105 [25]O.BilselandC.R.Matthews.Barriersinproteinfoldingr eactions. Adv.Prot. Chem. ,53:153{207,2000. 2.3.1 , 2.3.1 , 2.3.2 , 2.3.2 , 2.3.3 [26]E.I.Shakhnovich.Theoreticalstudiesofprotein-fold ingthermodynamicsand kinetics. Curr.Opin.Struct.Biol. ,7:29{40,1997. 2.3.1 , 4.6 [27]H.Eyring.Theactivatedcomplexinchemicalreactions. J.Chem.Phys. , 3:107{115,1935. 2.3.1 [28]A.Ansari.Langevinmodesanalysisofmyoglobin. J.Chem.Phys. ,110:1774{ 1780,1999. 2.3.1 , 5.1 , 5.2.2 , 5.2.2 [29]H.FrauenfelderandP.G.Wolynes.Ratetheoriesandpuzzl esofhemeprotein kinetics. Science ,229:337{345,1985. 2.3.1 , 5.2.1 [30]H.A.Kramers.Brownianmotionina¯eldofforceandthedi®usio nmodelof chemicalreactions. Physica ,7:284{304,1940. 2.3.1 , 5.1 [31]P.HÄanggi,P.Talkner,andM.Borkovec.Reactionrateth eory:Fiftyyears afterKramers. Rev.Mod.Phys. ,62:215{341,1990. 2.3.1 , 5.1 [32]D.K.KlimovandD.Thirumalai.Viscositydependenceofthe foldingratesof proteins. Phys.Rev.Lett. ,79:317{320,1997. 2.3.1 , 5.1 , 5.2.1 [33]K.W.PlaxcoandD.Baker.Limitedinternalfrictionint herate-limiting stepofatwo-stateproteinfoldingreaction. Proc.NatlAcad.Sci.USA , 95:13591{13596,1998. 2.3.1 , 5.2.1 , 1 , 5.2.2 , 5.3.3 , 5.4 [34]S.S.PlotkinandN.Onuchic.Understandingproteinfoldin gwithenergy landscapetheory.PartI:Basicconcepts. QuarterlyReviewsofBiophysics , 35:111{167,2002. 2.3.2 , 2.3.2 [35]J.N.Onuchic,Z.Luthey-Schulten,andP.G.Wolynes.Theo ryofprotein folding:Theenergylandscapeperspective. Annu.Rev.Phys.Chem. ,48:545{ 600,1997. 2.3.2 [36]N.D.Socci,J.N.Onuchic,andP.G.Wolynes.Di®usivedynamic softhe reactioncoordinateforproteinfoldingfunnels. J.Chem.Phys. ,104:5860{ 5868,1996. 2.3.2 , 5.4 [37]S.TakadaandP.G.Wolynes.Microscopictheoryofcritica lfoldingnuclei andrecon¯gurationactivationbarriersinfoldingproteins. J.Chem.Phys. , 107:9585{9598,1997. 2.3.2 , 5.2.1 [38]K.W.Plaxco,K.T.Simmons,andD.Baker.Contactorder,t ransitionstate placementandtherefoldingratesofsingledomainproteins. J.Mol.Biol. , 277:985{994,1998. 2.3.3 , 2.3.3 , 4.6 , 4.6

PAGE 118

106 [39]H.ZhouandY.Zhou.Foldingratepredictionusingtotalcon tactdistance. Biophys.J. ,82:458{463,2002. 2.3.3 [40]D.N.Ivankov,S.O.Garbuzynskiy,E.Alm,K.W.Plaxco,D.Ba ker,andA.V. Finkelstein.Contactorderrevisited:In°uenceofproteinsize onfoldingrate. Prot.Sci. ,12:2057{2062,2003. 2.3.3 , 4.6 [41]D.E.MakarovandK.W.Plaxco.Thetopomersearchmodel:A simple, quantitativetheoryoftwo-stateproteinfoldingkinetics. Prot.Sci. ,12:17{26, 2003. 2.3.3 [42]B.Alberts,D.Bray,A.Johnson,J.Lewis,M.Ra®,K.Roberts,and P.Walter. EssentialCellBiology:Anintroductiontothemolecularbio logyofthe cell .GarlandPublishing,Inc.,NewYork,1998. 2.4 [43]P.A.Thomson,V.Mu~noz,G.S.Jas,E.R.Henry,W.A.Eaton,and J.Hofrichter.Thehelix-coilkineticsofaheteropeptide. J.Phys.Chem. B ,104:378{339,2000. 2.4 [44]J.H.Wernere,R.B.Dyer,R.M.Fesinmayer,andN.H.Andersen.D ynamicsof theprimaryprocessesofproteinfolding:Helixnucleation. J.Phys.Chem.B , 106:487{494,2002. 2.4 [45]Y.Xu,R.Oyola,andF.Gai.Infraredstudyofthestabilityan dfolding kineticsofa15-residue ¯ -hairpin. J.Am.Chem.Soc. ,125:15388{15394,2003. 2.4 [46]L.Qiu. Laserinducedtemperaturejumpinvestigationsoffastprot einfolding dynamics .PhDthesis,UniversityofFlorida,Gainesville,2003. 2.4 , 4.5 [47]I.J.Chang,J.C.Lee,J.R.Winkler,andH.B.Gray.Thepro teinfolding speedlimit:Intrachaindi®usiontimessetbytheelectron-tran sferratesin denaturedRu(NH 3 ) 5 (His-33)-Zn-cyctochrome c . Proc.NatlAcad.Sci.USA , 100:3838{3840,2003. 2.4 [48]A.Szabo,K.Schulten,andZ.Schulten.Firstpassagetimeap proachto di®usioncontrolledreactions. J.Chem.Phys. ,72:4350{4357,1980. 2.4 [49]L.L.Qiu,S.A.Pabit,A.E.Roitberg,andS.J.Hagen.Smalle randfaster:The 20-residuetrpcageproteinfoldsin4 ¹ s. J.Am.Chem.Soc. ,124:12952{12953, 2002. 2.4 , 4.1 , 4.5 , 4{1 [50]J.Kubelka,W.A.Eaton,andJ.Hofrichter.Experimentalt estsofvillin subdomainfoldingsimulations. J.Mol.Biol. ,329:625{630,2003. 2.4 , 4.6 , 4{1 [51]C.D.Snow,N.Nguyen,V.S.Pande,andM.Gruebele.Absoluteco mparison ofsimulatedandexperimentalprotein-foldingdynamics. Nature ,420:102{106, 2002. 2.4 , 4.6 , 4{1 , 4.6

PAGE 119

107 [52]Y.Zhu,D.O.V.Alonso,K.Maki,C-Y.Huang,S.J.Lahr,V.Daggett , H.Roder,W.F.,andF.Gai.Ultrafastfoldingof ® 3 D:A denovo designed three-helixbundleprotein. Proc.NatlAcad.Sci.USA ,100:15486{15491, 2003. 2.4 , 4.6 , 4{1 [53]J.D.Bryngelson,J.N.Onuchic,N.D.Socci,andP.G.Wolyne s.Funnels, pathways,andtheenergylandscapeofproteinfolding:Asynthe sis. Proteins , 21:167{195,1995. 2.4 , 5.4 [54]J.Kubelka,J.Hofrichter,andW.A.Eaton.Theproteinfol ding`speedlimit'. Curr.Opin.Struct.Biol. ,14:76{88,2004. 2.4 , 4.6 [55]R.H.Callender,R.B.Dyer,R.Gilmanshin,andW.H.Woodru®. Fastevents inproteinfolding:Thetimeevolutionofprimaryprocesses. Annu.Rev.Phys. Chem. ,49:173{202,1998. 3.1 [56]H.RoderandM.C.R.Shastry.Methodsforexploringearlye ventsinprotein folding. Curr.Opin.Struct.Biol. ,9:620{626,1999. 3.1 [57]S.A.PabitandS.J.Hagen.Laminar-°ow°uidmixerforfast°uor escence kineticsstudies. Biophys.J. ,83:2872{2878,2003. 3.1 [58]J.P.Brody,P.Yager,R.E.Goldstein,andR.H.Austin.Biote chnologyatlow Reynoldsnumbers. Biophys.J. ,71:3430{3441,1996. 3.1 , 3.1.1 , 3.1.1 , 3.2.1 [59]P.Regenfuss,R.M.Clegg,M.J.Fulwyler,F.J.Barrantes,a ndT.M.Jovin. Mixingliquidsinmicroseconds. Rev.Sci.Instr. ,56:283{290,1985. 3.1 [60]V.N.Constantinescu. Laminarviscous°ow .Springer-Verlag,NewYork,1995. 3.1.1 , 3.2.1 , 3 , 3.3.3 [61]G.W.MoscowitzandR.L.Bowman.Multicapillarymixerof solutions. Science ,153:428{429,1966. 3.1.2 [62]S.Takahashi,Y.C.Ching,J.Wang,andD.L.Rousseau.Microse cond generationofoxygen-boundcytochrome c oxidasebyrapidsolutionmixing. J.Biol.Chem. ,270:8405{8407,1995. 3.1.2 [63]K.Chan,Y.Hu,S.Takahashi,D.Rousseau,andW.Eaton.Submil lisecond proteinfoldingkineticsbyultrarapid-mixing. Proc.NatlAcad.Sci.USA , 94:1779{1784,1997. 3.1.2 [64]M.C.R.Shastry,S.D.Luck,andH.Roder.Acontinuous-°owca pillarymixing methodtomonitorreactionsonthemicrosecondtimescale. Biophys.J. , 74:2714{2721,1998. 3.1.2 , 3.3 , 3.3.2 [65]D.BÄokenkamp,A.Desai,X.Yang,Y.C.Tai,E.M.Marzlu®,andS .L.Mayo. Microfabricatedsiliconmixersforsubmillisecondquench-°ow analysis. Anal. Chem. ,70:232{236,1998. 3.1.2

PAGE 120

108 [66]S.Akiyama,S.Takahashi,T.Kimura,K.Ishimori,I.Morishi ma, Y.Nishikawa,andT.Fujisawa.Conformationallandscapeofcytoc hrome c foldingstudiedbymicrosecond-resolvedsmall-anglex-rayscat tering. Proc. NatlAcad.Sci. ,99:1329{1334,2002. 3.1.2 [67]L.Pollack,M.W.Tate,A.C.Finnefrock,C.Kalidas,S.Tro tter,N.C.Darnton, L.Lurio,R.H.Austin,C.A.Batt,S.M.Gruner,andS.G.J.Mochrie .Timeresolvedcollapseofafoldingproteinobservedwithsmallanglexrayscattering. Phys.Rev.Lett. ,86:4962{4965,2000. 3.1.2 [68]L.Pollack,M.W.Tate,N.C.Darnton,J.B.Knight,S.M.Gr uner,W.A.Eaton, andR.H.Austin.Compactnessofthedenaturedstateofafast-foldi ng proteinmeasuredbysubmillisecondsmall-anglex-rayscatterin g. Proc.Natl Acad.Sci.USA ,96:10115{10117,1999. 3.1.2 [69]R.Russell,I.S.Millett,M.W.Tate,L.W.Kwok,B.Nakatani ,S.M.Gruner, S.G.J.Mochrie,V.Pande,S.Doniach,D.Herschlag,andL.Polla ck.Rapid compactionduringrnafolding. Proc.NatlAcad.Sci.USA ,99:4266{4271, 2002. 3.1.2 [70]J.C.McDonald,D.C.Dufty,J.R.Anderson,D.T.Chiu,H.Wu, O.J.A. Schueller,andG.M.Whitesides.Fabricationofmicro°uidicsyst emsin poly(dimethylsiloxane). Electrophoresis ,21:27{40,2000. 3.1.2 [71]L.D.Scampavia,G.Blankenstein,J.Ruzicka,andG.D.Ch ristian.A coaxialjetmixerforrapidkineticanalysisof°owinjectiona nd°owinjection cytometry. Anal.Chem. ,67:2743{2749,1995. 3.2.1 [72]V.P.Andreev,S.B.Koleshko,D.A.Holman,L.D.Scampavia,an dG.D. Christian.Hydrodynamicsandmasstransferofthecoaxialjetmi xerin°ow injectionanalysis. Anal.Chem. ,71:2199{2204,1999. 3.2.1 [73]S.S.Lehrer.Soluteperturbationofprotein°uorescence :Thequenchingof tryptophyl°uorescenceofmodelcompoundsandlysozymebyiodi deion. Biochemistry ,17:3254{3263,1971. 3.3 , 3.3.1 [74]P.C.Hsieh,B.C.Shenoy,F.C.Haase,J.E.Jentoft,andN.F.B. Phillips. Involvementoftryptophan(s)attheactivesiteofpolyphospha te/ATP glucokinasefrom Mycobacteriumtuberculosis . Biochemistry ,32:6243{6249, 1993. 5 [75]Y.Yu,R.Li,C.Xu,K.Ruan,Y.Shen,andGovindjee.N-bromosuc cinimide modi¯cationoftryptophan241attheC-terminusofthemangan esestabilizingproteinofplantphotosystemIIin°uencesitsstructurea ndfunction. PhysiologicaPlantarum ,111:108{115,2001. 5 [76]B.F.Peterman.Measurementofthedead-timeofa°uorescen cestopped-°ow instrument. Anal.Biochem. ,93:442{444,1979. 3.3 , 3.3.2 , 3.3.3

PAGE 121

109 [77]G.Hagen.Ueberdiebewegungdeswassersinengencylindrishc henrohren. PoggendorfsAnn.Phys.Chem. ,46:423{442,1839. 3.3.2 [78]J.L.M.Poiseuille.Reserchesexp¶erimentalessurlemouv ementdesliquides danslestubesdetrµes-petitsdiamµetres. CRAcad.Sci.Paris ,11:961{ 967,1041{1049,1840. 3.3.2 [79]J.W.Neidigh,R.M.Fesinmey,andN.H.Anderson.Designinga20residue protein. NatureStruct.Biol. ,9:425{430,2002. 4.1 , 4.3 , 4.3 , 4.4.3 , 4.6 , 4.6 [80]S.E.Jackson.Foldingofchymotrpsininhibitor2:1.evid enceforatwo-sate transition. Biochemistry ,30:10428{10435,1991. 4.2 , 4.4.3 [81]C.R.CantorandP.R.Schimmel. Biophysicalchemistry:Techniquesforthe studyofbiologicalstructureandfunction ,volume2.W.H.Freemanand Company,SanFrancisco,1980. 4.4.1 , 2 [82]C.Simmerling,B.Strockbin,andA.E.Roitberg.All-atom structure predictionandfoldingsimulationsofastableprotein. J.Am.Chem. Soc ,124:11258{11259,2002. 4.6 [83]N.Ferguson,C.M.Johnson,M.Macias,H.Oshkinat,andA.R.Fersh t. UltrafastfoldingofWWdomainswithoutstructuredaromaticcl ustersinthe denaturedstate. Proc.NatlAcad.Sci.USA ,98:13002{13007,1995. 4{1 [84]U.Mayor,C.M.Johnson,V.Daggett,andA.R.Fersht.Proteinf oldingand unfoldinginmicrosecondstonanosecondsbyexperimentandsim ulation. Proc.NatlAcad.Sci.USA ,97:13518{13522,2000. 4{1 [85]W.Y.YangandM.Gruebele.Foldingatthespeedlimit. Nature ,423:193{ 197,2003. 4{1 , 5.5 [86]B.ZagrovicandV.Pande.Structurecorrespondencebetwe enthe ® -helixand therandom-°ightchainresolveshowunfoldedproteinscanhav enative-like properties. NatureStruct.Biol. ,10:955{961,2003. 4.6 [87]C.D.Snow,B.Zagrovic,andV.S.Pande.Thetrpcage:Fold ingkineticsand unfoldedstate. NatureStruct.Biol. ,9:425{430,2002. 4.6 [88]B.Zagrovic,C.Snow,M.Shirts,andV.Pande.Simulationo ffoldingofa smallalpha-helicalproteininatomisticdetailusingworldwi dedistributed computing. J.Mol.Bio. ,323:927{937,2002. 4.6 [89]S.H.GellmanandD.N.Woolfson.Mini-proteinsTrptheligh tfanstastic. NatureStruct.Biol. ,9:408{410,2002. 4.6 [90]V.Mu~nozandW.A.Eaton.Asimplemodelforcalculatingthe kineticsof proteinfoldingfromthree-dimensionalstructures. Proc.NatlAcad.Sci.USA , 96:11311{11316,1999. 4.6

PAGE 122

110 [91]R.Zhou.Trp-cage:Foldingfreeenergylandscapeinexpl icitwater. Proc. NatlAcad.Sci.USA ,100:13280{13285,2003. 4.6 [92]D.E.MakarovandK.W.Plaxco.Thetopomersearchmodel:A simple, quantitativetheoryoftwo-stateproteinfoldingkinetics. Prot.Sci. ,12:17{26, 2003. 4.6 [93]E.Pitard.In°uenceofhydrodynamicsonthedynamicsofa homopolymer. Eur.Phys.J.B ,7:665{673,1999. 4.6 [94]N.KogaandS.Takada.Rolesofnativetopologyandchainlengthscalingin proteinfolding:AsimulationstudywiththeG¹omodel. Science ,281:253{256, 1998. 4.6 [95]H.KayaandH.S.Chan.Solvatione®ectsanddrivingforcesf orprotein thermodynamicsandkineticcooperativity:Howadequateisn ative-centric topologicalmodeling? J.Mol.Biol. ,326:911{931,2003. 4.6 [96]BengtNÄolting,W.SchÄalike,P.Hampel,F.Grundig,S.Ga ntert,N.Sips, W.Bandlow,andP.X.Qi.Structuraldeterminantsoftherateo fprotein folding. J.Theoret.Biol. ,223:299{307,2003. 4.6 [97]M.S.Li,D.K.Klimov,andD.Thirumalai.Thermaldenatu rationandfolding ratesofsingledomainproteins:sizematters. Polymer ,45:573{579,2004. 4.6 [98]T.R.WeiklandK.A.Dill.Foldingratesandlow-entropylossroutesof two-stateproteins. J.Mol.Biol. ,329:585{598,2003. 4.6 [99]J.J.Portman,S.Takada,andP.G.Wolynes.Microscopicth eoryofprotein foldingrates.ii.localreactioncoordinatesandchaindyna mics. J.Chem. Phys. ,114:5082{5096,2001. 5.1 , 5.2.2 [100]M.JacobandF.X.Schmid.Proteinfoldingasadi®usionalp rocess. Biochemistry ,38:13773{13779,1999. 5.1 , 5.2.1 , 5.3.3 [101]M.KarplusandD.Weaver.Proteinfoldingdynamics:The di®usion-collision modelandexperimentaldata. Prot.Sci. ,3:650{668,1994. 5.2.1 [102]M.Jacob,M.Geeves,G.Holtermann,andF.X.Schmid.Di®usio nalbarrier crossinginatwo-stateproteinfoldingreaction. NatureStruct.Biol. ,6:923{ 926,1999. 5.2.1 , 5.2.2 , 5.3.3 , 5.4 [103]M.Jacob,T.Schindler,J.Balbach,andF.X.Schmid.Di®u sioncontrolinan elementaryproteinfoldingreaction. Proc.NatlAcad.SciUSA ,94:5622{5627, 1997. 5.2.1 , 5.3.3 [104]T.Y.TsongandR.L.Baldwin.E®ectofsolventviscosityanddi ®erent guanidinesaltsonkineticsofribonucleaseAchainfolding. Biopolymers , 17:1669{1678,1978. 5.2.1 , 5.3.3

PAGE 123

111 [105]R.P.BhattacharyyaandT.R.Sosnick.Viscositydependenc eofthefolding kineticsofadimericandmonomericcoiledcoil. Biochemistry ,38:2601{2609, 1999. 5.2.1 , 5.3.3 [106]A.G.Ladurner,,andA.R.Fersht.Upperlimitofthetimescal efordi®usion andchaincollapseinchymotrypsininhibitor2. NatureStruct.Biol. ,6:28{31, 1999. 5.2.1 , 5.3.3 [107]G.S.Jas,W.A.Eaton,andJ.Hofrichter.E®ectofviscosityon thekineticsof ® -helixand ¯ -hairpinformation. J.Phys.Chem.B ,105:261{272,2001. 5.2.1 , 5.3.3 , 5.5 [108]M.SilowandM.Oliveberg.Highconcentrationsofviscog ensdecreasethe proteinfoldingrateconstantbyprematurelycollapsingthec oil. J.Mol.Biol. , 326:263{271,2003. 5.2.1 , 5.2.2 , 5.3.3 , 5.5 [109]P.G.deGennes. Scalingconceptsinpolymerphysics .CornellUniversity Press,IthacaandLondon,1979. 5.2.2 , 5.2.2 [110]C.W.MankeandM.C.Williams.Internalviscosityofpolym ersandtherole ofsolventresistance. Macromolecules ,18:2045{2051,1985. 5.2.2 , 5.2.2 [111]B.ZagrovicandV.Pande.Solventviscositydependenceof thefolding rateofasmallprotein:Distributedcomputingstudy. J.Comput.Chem. , 24:1432{1436,2003. 5.2.2 , 5.5 [112]I.J.Chang,J.C.Lee,J.R.Winkler,andH.B.Gray.Thepr otein-foldingspeed limit:Intrachaindi®usiontimessetbyelectron-transferrate sindenatured ru(nh 3 ) 5 (his-33)-zn-cytochrome c . Proc.NatlAcad.Sci.USA ,100:3838{3840, 2003. 5.2.2 [113]H.KayaandH.S.Chan.Towardsaconsistentmodelingofprot einthermodynamicandkineticcooperativity:Howapplicableisthetra nsitionstate picturetofoldingandunfolding? J.Mol.Biol. ,315:899{909,2002. 5.2.2 , 5.5 [114]A.Ansari,C.M.Jones,E.R.Henry,J.Hofrichter,andW.A.Eato n.The roleofsolventviscosityinthedynamicsofproteinconformati onalchanges. Science ,256:1796{1798,1992. 5.2.2 , 5.2.2 , 5.4 [115]S.Akiyama,S.Takahashi,K.Ishimori,andI.Morishima.St epwiseformation of ® -helicesduringcytochrome c folding. NatureStruct.Biol. ,7:514{520, 2000. 5.3.1 [116]P.GeorgeandA.Schejter.Thereactivityofferrocytoc hromecwithironbindingligands. J.Biol.Chem. ,239:1504{1508,1964. 5.3.1 [117]A.K.BhuyanandR.Kumar.Kineticbarrierstothefoldin gofhorse cytochromecinthereducedstate. Biochemistry ,41:12821{12834,2002. 5.3.1 , 5.3.1 , 5.3.1

PAGE 124

112 [118]R.F.Latypov,K.Maki,H.Cheng,andH.Roder.Foldingmec hanismof reducedcytochrome c :Equilibriumpropertiesinthepresenceandabsenceof carbonmonoxide.Tobesubmitted,2004. 5.3.1 , 5.3.1 , 5.3.3 [119]T.Pascher,J.P.Chesick,J.R.Winkler,andH.B.Gray.Pro teinfolding triggeredbyelectrontransfer. Science ,271:1558{1560,1996. 5.3.1 [120]K.Maki,H.ChengR.F.Latypov,S.Luck,S.A.Pabit,S.J.Ha gen,and H.Roder.Foldingmechanismofreducedcytochrome c :Kineticbehaviorin thepresenceandabsenceofcarbonmonoxide.Tobesubmitted,20 04. 5.3.1 [121]S.W.Englander,D.B.Calhoun,andJ.J.Englander.Bio chemistrywithout oxygen. Anal.Biochem. ,164:300{306,1987. 5.3.1 [122]S.J.Hagen,J.Hofrichter,andW.A.Eaton.Rateofintrach aindi®usionof unfoldedcytochrome c . J.Phys.Chem.B ,101:2352{2365,1997. 5.3.3 [123]C.D.Waldburger,T.Jonsson,andR.T.Sauer.Barriersto proteinfolding:Formationofburiedpolarinteractionsisaslowstepinac quisitionof structure. Proc.NatlAcad.Sci.USA ,93:2629{2634,1996. 5.3.3 [124]R.Zwanzig.Di®usioninaroughpotential. Proc.NatlAcad.Sci.USA , 85:2029{2030,1988. 5.4 [125]C.K.Matthews,K.E.vanHolde,andK.G.Allen. Biochemistry .Addison WesleyLongman,Inc.,SanFrancisco,CA,3rdedition,2000. A [126]R.A.GoldbeckandD.S.Kliger.Nanosecondtime-resolveda bsorptionand polarizationdichroismspectroscopies. MethodsinEnzymology ,226:147{177, 1993. B.2 [127]E.R.HenryandJ.Hofrichter.Singularvaluedecompositi on-applicationto analysisofexperimental-data. MethodsinEnzymology ,210:129{192,1992. B.2 , B.2 [128]S.L.Murov,I.Carmichael,andG.L.Hug. HandbookofPhotochemistry . MarcelDekker,Inc.,NewYork,secondedition,1993. C.2 [129]G.PorterandF.Wilkinson.Energytransferfromthetrip letstate. Proc. Roy.Soc.A ,264:1{18,1961. C.2

PAGE 125

BIOGRAPHICALSKETCH SersitaSuzetteAtienzaPabitwasbornonJune27,1974,toSer gioAtienza andTeresitaQuesada,inthePhilippines.Suzettewasraisedint hecoastalprovince ofBatangas.SheattendedtheprestigiousPhilippineScience HighSchoolin QuezonCity,Philippines;andspentayearasanexchangestuden tatMatsue NorthHighSchoolinShimane,Japan.In1997,shegraduatedwith aB.S.in PhysicsfromtheUniversityofthePhilippinesatDiliman.Shet aughtfor2years asInstructorofPhysicsattheUniversityofthePhilippinesbef oreattendingthe UniversityofFloridain1999.Alwaysattractedtointerdiscipl inaryphenomena,she joinedthelaboratoryofProf.StephenJ.Hagenin2000towork ontheproteinfoldingproblem.ShemarriedEdgardoPabitin2001.Sherece ivedherM.S.in PhysicsfromtheUniversityofFloridain2002,andcompletedhe rPh.D.in2004. 113