A Solvable Model of Secondary Structure Formation

arXiv:cond-mat/0101472v1[cond-mat.dis-nn]31Jan2001ASolvableModelofSecondaryStructureFormationinRandomHetero-PolymersN.S.Skantzos,J.vanMourikandA.C.C.CoolenDepartmentofMathematics,King’sCollegeLondonTheStrand,LondonWC2R2LS,U.K.Abstract.Weproposeandsolveasimplemodeldescribingsecondarystructureformationinrandomhetero-polymers.Itdescribesmonomerswithacombinationofone-dimensionalshort-rangeinteractions(representingstericforcesandhydrogenbonds)andinfiniterangeinteractions(representingpolarityforces).Wesolveourmodelusingacombinationofmeanfieldandrandomfieldtechniques,leadingtophasediagramsexhibitingsecond-ordertransitionsbetweenfolded,partiallyfoldedandunfoldedstates,includingregionswherefoldingdependsoninitialconditions.Ourtheoreticalresults,whichareinexcellentagreementwithnumericalsimulations,leadtoanappealingphysicalpictureofthefoldingprocess:thepolarityforcesdrivethetransitiontoacollapsedstate,thestericforcesintroducemonomerspecificity,andthehydrogenbondsstabilisetheconformationbydampingthefrustration-inducedmultiplicityofstates.PACSnumbers:61.41.+e,75.10.Nr1.IntroductionProteinsarepolymericchainsofamino-acids.Thesuccessfulfunctioningofaproteininalivingorganismdependscrucially,amongotherfactors,onitsabilitytofoldintoadesiredthree-dimensionalstructure(its‘nativestate’),andtosubsequentlyattachinaveryspecificwaytoothermacro-molecules.Fromabiologicalandmedicalpointofview,itisthereforehighlydesirabletoknowwhichnativestatecorrespondstoagivenamino-acidsequence,and(conversely,fortherapeuticpurposes)toknowwhichamino-acidsequencewouldfoldintoadesirednativestate;thisrequiresaquantitativeunderstandingofthephysicalforcesunderlyingthefoldingmechanism.Adetailedidentificationofsequence-specificnativestateswillnecessarilyinvolvesophisticated(moleculardynamicsbased)computationalmethods.However,duetothelargenumberofdegreesoffreedomofproteins,thecomplicatednatureofthevarioustypesofelectro-chemicalinteractionsandtheso-called‘hard’geometricchainconstraintsofaprotein,suchcomputerprogrammesareunfortunately(asyet)extremelyslow.Thus,inordertoidentifytheroleanddegreeofimportanceofthevariousfoldingparameters,atheoretical(i.e.statisticalmechanical)analysiswouldbeverywelcome.StructureFormationinRandomHetero-Polymers2Itisgenerallyassumedthatthepresentlyobservedpopulationofrealproteinshasevolvedfromthelargerclassofrandomhetero-polymers,drivenbynaturalselection.Thissuggeststhatthestudyofrandomhetero-polymersisanaturalfirststepenroutetowardsthestatisticalmechanicalstudyofproteins.Furthermore,alreadyatanearlystageitwasrecognized[1],viaatheoreticalstudybasedontherandomenergyscheme[2],thatmanyaspectsofproteinfolding(suchastheappearanceof‘mis-folded’phases,andtransitionsbetweenfoldedandunfoldedstates)canbeunderstoodonthebasisofequilibriumstatisticalmechanicalcalculationsforrandomhetero-polymers.Evensimplemodelswithonlytwotypesofamino-acidsinteractingwiththewatersolvent,viz.hydro-phobicamino-acidsversuspolarones,cansuccessfullydescribethebasicsofproteinfolding(seee.g.theso-calledHPmodel[3]).FurtherstatisticalmechanicalapproachesincludereplicacalculationsonpolymerchainswithGaussianpairinteractions[4,5],variationalanalysesinreplicaspaces[6,7],latticemodels[8,9]andlatticegasmodels[10],tomentionbutafew.Inmostoftheseexamples,analyticalsolvabilityreliesontheabsenceofspatialstructure,whichallowsformoreorlessconventionalmean-fieldstatisticalmechanics.Inthispaperweextendtheclassofanalyticallysolvablemodelsinthisfield.Wepresentamodelforsecondarystructureformationinrandomhetero-polymersconsistingofamino-acidmonomerswhichareallowedtointeractinthreequalitativelydifferentways:(i)viaso-calledstericinteractions,whichreflectmonomer-specificgeometricconstraintsandelectricalforcesdeterminingthelocalenergylandscapefortheorientationofmonomer-connectinglinks,(ii)viahydrogen-bonding,whichactsoverlargerdistancesalongthechain,andisbelievedtoplayaroleinthestabilizationofhelix-typestructures,and(iii)viapolarity-inducedenergygradients,whichtendtopromotestatesinwhichthehydrophobicamino-acidsaremoreorlessturnedtowardsthesamesideofthepolymericchain,inordertoenableeffectiveshieldingfromwatermoleculesviafoldingofthepolymerasawhole.Interactions(i)and(ii)areofashort-rangenature,whereas(iii)islong-range.Wenotethatsecondarystructureformationhasalsobeenstudiedwithinamean-fieldapproachin[11],andthatacombinationofdifferenttypesofmonomerinteractionshasbeenproposedpreviouslyin[1].Inthelatterstudy,assumingstatisticalindependenceofenergylevels,therandomenergyschemecouldprovidequalitativeresults;however,thevalidityofthisapproachhassincethenbeenquestioned[12].Incontrast,oursolutiondoesnotemployrandomenergyconsiderations.Itisbasedonacombinationofmean-fieldandrandomtransfer-matrixtechniques,whichinone-dimensionalmodelsareknowntoreducetheevaluationofthepartitionfunctiontoarelativelysimplenumericalproblem.Duetothepresenceofadditionallong-rangeinteractions(viapolarity-inducedforces)ourmodelnolongerliesintheuniversalityclassofonedimensionalsystems,andphasetransitionsarethereforepossible(andwillindee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