Energy Landscape of a Model Protein

arXiv:cond-mat/9904304v121Apr1999EnergyLandscapeofaModelProteinMarkA.MillerandDavidJ.WalesUniversityChemicalLaboratories,LensfieldRoad,CambridgeCB21EW,UKFebruary1,2008AbstractThepotentialenergysurfaceofanoff-latticemodelproteinischaracterizedindetailbyconstructingadisconnectivitygraphandbyexaminingtheorganisationofpathwaysonthesurface.Theresultsclearlyrevealthefrustrationexhibitedbythissystemandexplainwhyitdoesnotfoldefficientlytotheglobalpotentialenergyminimum.Incontrast,whenthefrustrationisremovedbyconstructinga‘G¯o-type’model,theresultinggraphexhibitsthecharacteristicsexpectedforafoldingfunnel.1IntroductionThepotentialenergysurface(PES)ofaninteractingsystemdeterminesitsstructural,dynamic,andthermodynamicproperties.Formally,thelinksbetweenthePESandthesepropertiesarefullydefinedbythestationarypointsonthePES,itsgradient(whichgivestheforcesontheparticles),andthepartitionfunction.However,itisonlyrelativelyrecentlythatexplicitconnectionshavebeensoughtbetweentheoverallstructureofthePES,orpotentialenergy‘landscape’,andthebehaviourofthesystemitdescribes.This1approachpromisestoprovideinsightintoanumberoffields,includingproteinfolding,globaloptimizationandglassformation.InthepresentcontributionweprovideaglobalcharacterizationofthePESforamodelheteropolymer,andshowhowthispictureexplainsthedynamicalpropertiesobservedinprevioussimulations.Intheoriginalmodel‘frustration’preventsefficientrelaxationtotheglobalpotentialenergyminimum.However,whenthefrustrationisremovedbyconstructingthecorresponding‘G¯o-like’model,thedeeptrapsdisappearandtheresult-ingsurfaceresemblesafunnel.Thetermfrustrationwasfirstusedinthecontextofspinglasses,1whereitisimpossibletosatisfyallfavourableinteractionssimultaneously.Analogouseffectsexistinproteins:2athree-dimensionalstructurethatbringstogethertwomutuallyattractiveresiduesmayinvolvegeneratingunfavourablecontactselsewhere(‘energeticfrustration’),andtheinterconversionoftwosimilarstructuresmayrequirethedisruptionofexistingfavourableinteractions(‘geometricfrustration’).Themajordifficultyinprovidingafundamentalexplanationofstructure,dynamicsandthermodynamicsintermsoftheunderlyingpotentialenergysurfaceisthatthenum-berofstationarypointsgrowsveryrapidlywiththesizeofthesystem.3Thisgrowthis,infact,thebasisofLevinthal’s‘paradox’,4whichpointsouttheapparentimpossibilityofaproteinfindingitsbiologicallyactivestateinarandomsearchamongsttheastro-nomicalnumberofavailablestructures.Someattemptstoresolvetheparadoxproposedareductioninthesearchspacefromthefullconfigurationspace.5–8Althoughitseemsunlikelythatthisreductionisthesolutiontotheparadox,thereisanimplicitrealizationinsuchapproachesthat,insomeway,thesearchisnotrandom.Intermsoftheenergylandscapetherearetworeasonsforthis.Firstly,conformationshavedifferentstatisticalweightsinthethermodynamicensemble,andsecondly,theyarenotarrangedatrandominconfigurationspace.Levinthal’sanalysisassumesthattheenergylandscapeisflat,2likeagolfcoursewithasingleholecorrespondingtothenativestate.2Byconstructingasimplemodelthatincludesanenergeticbiastowardsthenativestructure,itcanbeshownthatthesearchtimeonthefullconformationalspaceisdramaticallyreducedtophysicallymeaningfulscales.9,10OneofthefirststudiestoconsidermoreexplicitlytheorganizationoftheenergylandscapewasthatofLeopold,Montal,andOnuchic.11Theseauthorsproposedthatthelandscapeofanaturalproteinconsistsofacollectionofconvergentkineticpathwaysthatleadtoauniquenativestatewhichisthermodynamicallythemoststable.Suchaland-scapestructurewastermeda‘foldingfunnel’becauseitfocusesthemanifoldmisfoldedstatestowardsthecorrecttarget.ThisapproachhighlightsthefundamentalfallacyoftherandomsearchinLevinthal’s‘paradox’.FunneltheoryhasgainedwidespreadacceptancethroughitsdevelopmentbyWolynesandcoworkersintermsofafreeenergylandscape.2,12Thefunnelcanbedescribedintermsofthefreeenergygradienttowardsthenativestructure,andtheroughness—amea-sureofthebarrierheightsbetweenlocalfreeenergyminima,whichcanactaskinetictraps.Foldingisencouragedwhentheroughnessisnotlargecomparedwiththeenergygradient.Simulationshaveshownthatthefoldingabilitycanbemeasuredbytheratioofthefoldingtemperature,Tf,wherethenativestatebecomesthermodynamicallythemoststable,totheglasstransitiontemperature,Tg,wherethekineticsslowdowndramaticallybecauseofthefreeenergybarriers.8,13Tgisusuallydefinedasthetemperatureatwhichthefoldingtimepassesthroughacertainthreshold.FoldingiseasiestforlargeTf/Tg,sincethenativestateisthenstatisticallypopulatedattemperatureswhereitiskineticallyaccessible.Theeffectoffrustrationistoincreasetheroughnessoftheenergylandscaperelativetoitsgradienttowardsthenativestructure,therebyhinderingrelaxationtothelatter.3Wehaverecentlyshown14,15howanewvisualizationofthepotentialenergysurfaceusingdisconnectivitygraphs16revealsthefeatureswhichdeterminerelaxationofclusterstotheirglobalpotentialenergyminimum.Thisapproachhasalreadybeenusedbyotherstoexaminetheenergylandscapeofatetrapeptide16,17andtostudytheeffectsofcon-formationalconstraintsinhexapeptides18employinganall-atommodel.Inthepresentcontributionwea