Functionals linear in curvature and statistics of helical proteins

A.Feoli∗

DipartimentodiIngegneria,Universit`adelSannio,CorsoGaribaldin.107,PalazzoBoscoLucarelli,

82100Benevento,ItalyandINFNSezionediNapoli,GruppocollegatodiSalerno

80126Napoli,Italy

V.V.Nesterenko†

BogoliubovLaboratoryofTheoreticalPhysics

JointInstituteforNuclearResearch

141980Dubna,Russia

G.Scarpetta‡

DipartimentodiFisica”E.R.Caianiello”–Universit`adiSalerno

84081Baronissi(SA),Italy

andINFNSezionediNapoli,GruppocollegatodiSalerno

80126Napoli,Italy

Abstract

TheeffectivefreeenergyofglobularproteinchainisconsideredtobeafunctionaldefinedonsmoothcurvesinthreedimensionalEu-clideanspace.Fromtherequirementofgeometricalinvariance,to-getherwithbasicfactsonconformationofhelicalproteinsanddynam-icalcharacteristicsoftheproteinchains,weareabletodetermine,inauniqueway,theexactformofthefreeenergyfunctional.Namely,thefreeenergydensityshouldbealinearfunctionofthecurvature

ofcurvesonwhichthefreeenergyfunctionalisdefined.WebrieflydiscussthepossibilityofusingthemodelproposedinMonteCarlosimulationsofexhaustivesearchingthenativestablestateofthepro-teinchain.Therelationofthemodelproposedtotherigidrelativisticparticlesandstringsisalsoconsidered.

PACSnumbers:87.15.-v,87.15.Cc,05.10.Ln,02.40.-k

Keywords:proteinfolding,semi-flexiblepolymers,geometryofcurves,helices,rigidstring,particleswithrigidity.

2

1Introduction

whereβisinversetemperatureandwehavesplittheproteinradiusvec-torxintotheclassicalpartxclandthequantumpartxqu.InordertousethisformulatheenergyfunctionalEforanindividualproteinchainshouldbespecified.Theproblemunderconsiderationisclosetothestatisticsofsemi-flexiblepolymers.Inthelastcaseanumberofmodels(i.e.,particularfunctionalsE)havebeenproposedwhichsubstantiallyrelyonthedifferentialgeometry(Kratky-Porodmodel[5],Diracmodel[6,7],tubemodel[8]andsoon).Differentialgeometryprovidesadirectrelationbetweenphysicsandmathematicsandtherebyopensastraightrouteforapplyingthemathemat-icalanalysistothisphysicalproblem[9,10].Ifk(s)andκ(s)arerespectivelythecurvatureandthetorsionofapolymerchainthentheenergyEcanberepresentedasapolynomialinthesegeometricalinvariants,forexample,in

1

Afascinatingandopenquestionchallengingphysics,biochemistryandevengeometryisthepresenceofhighlyregularmotifssuchasα−helicesandβ−sheetsinthefoldedstateofbiopolymersandproteins.Awiderangeofapproacheshavebeenproposedtorationalizetheexistenceofsuchsecondarystructures(see,forexample,reviews[1,2,3]andreferencestherein).

Inprinciple,inordertofindthestablenativestateofaprotein,oneshouldcompute,foreverypossibleconformationofthechain,thesumofthefreeenergiesoftheatomicinteractionswithintheproteinaswellaswiththesolventandthenfindtheconformationwiththelowestfreeenergy.However,itisnotfeasible,becausethenumberofconformationsofaproteinchaingrowsexponentiallywiththechainlength.

Inthepresentpaperwearegoingtoproposeapuregeometricalapproachtodescribetheeffectivefreeenergyofproteins,proceedingfromthemostgeneralinvariancerequirementsandbasicexperimentalfactsconcerningtheproteinconformation.ItshouldbenotedherethatweshallconsidernottheHamiltonianH(orenergy)ofanindividualproteinmolecule,buttheeffectivefreeenergyofanassemblyofnoninteractingproteinchains.WedefinethisenergyassuchafunctionalFdependingontheradiusvectorx(s)ofaproteinchain,thestationaryvaluesofwhich(theextrema)areprovidedonlybytheobservedconfigurationsofproteins.Obviously,theeffectivefreeenergydefinedinsuchawayisanalogoustotheeffectiveactioninquantumfieldtheory[4].

LetE[x(s)]betheenergyofanindividualproteinchainthentheeffectivefreeenergyFweareinterestedinisapparentlydeterminedbythefunctionalintegral󰀈

e−βF[xcl]=Dxque−βE[xcl+xqu],(1.1)

theform[9]

E=

󰀈

L

ds

0

󰀎

A

4

k4−αk2κ+

β

D-dimensionalEuclideanspaceuptoitsrotationsanddisplacementsasawhole.Thecurvaturekα(s)isexpressedintermsofthederivatives,withrespecttos,oftheradiusvectorofacurvex(s)tillthe(α+1)-thorderincluded.

Thefirstprincipalcurvature,orsimplythecurvature,k1(s)=k(s)ofacurvecharacterizesthelocalbendingofthecurveatthepoints.Hence,thedependenceoffreeenergydensityFonk(s)specifiestheresistanceofaproteinchaintobebent.Thesecondcurvatureortorsionκ(s)isdeterminedbytherelativerotation,aroundthetangentdx(s)/dsatthepoints,oftwoneighborinfinitelyshortelementsoftheproteinchain.Itiswellknown[1]that,inthecaseofproteinmolecules,sucharotationisquiteeasy,asitrequireslittleeffort.Inotherwords,thisrotationresultsinsmallenergydifferences,allowingmanyoverallconformationsofaproteinchaintoarise.ThusthedependenceofthefreeenergydensityFontorsionκ(s)canbeneglectedatleastasafirstapproximation.SomecommentsconcerningthisrestrictionwillbegiveninSection4.FinallyonecanconsiderthefreeenergydensityFtobeafunctiononlyofthecurvaturek(s),i.e.F=F(k(s)).Inwhatfollowsweshalltrytospecifythisdependenceexplicitlykeepinginmindthedescriptionofglobularproteinconformation.

Apeculiarityofconformationofglobularproteinsisthattheycanbeorderedassemblieseitherofhelicesorofsheetsaswellasamixtureofhe-licesandsheets[1].Inthephenomenologicalmacroscopicapproach,whichisdevelopedhere,thepresenceofsheetsinthespatialstructureofglobu-larproteinsimpliesthenecessitytointroduce,inadditiontospacecurvesx(s),newdynamicalvariablesy(s,s′)describingsurfacesinambientspace.1Obviouslysuchanextensionoftheproblemsettingwouldcomplicateconsid-erablyourconsideration.Thereforeweconfineourselvestohelicalproteinsandtrytoanswerthequestion:Isitpossibletospecify󰀋thefunctionF(k(s))insuchawaythattheextremalsofthefunctionalF=Fdswouldbeonlyhelices?Theanswertothisquestionturnsouttobepositiveandunique,namely,thedensityofthefreeenergyF(k(s))shouldbealinearfunctionofthecurvaturek(s):

F(k)=α−|β|k(s),(1.4)wheretheconstantsαandβaretheparametersspecifyingthephenomeno-logicalmodelproposed,α>0,β<0.Arigorousproofofthisassertionis

themainresultofthepresentpaper.

Thelayoutofthepaperisasfollows.InSection2theEuler-Lagrange

variationalequationsforthefunctionalFarereformulatedintermsoftheprincipalcurvatures.InSection3theexplicitintegrabilityoftheseequa-tionswillbeshown.Inotherwords,foranarbitraryfunctionF(k(s)),thecorrespondingEuler-Lagrangeequationsareintegrablebyquadratures.Thephysicalmeaningoftherelevantconstantsofintegrationsisalsodiscussed.InSection4therequirementthatthesolutionsoftheEuler-Lagrangeequa-tionsdescribeonlyhelicesimmediatelyconfinestheadmissiblefreeenergydensityF(k(s))tobealinearfunction(1.4).Inconclusion(Section5),theobtainedresultsandtheirphysicalimplicationsarebrieflydiscussed.

2

Euler-Lagrangeequationsintermsofprin-cipalcurvatures

Keepinginmindthedescriptionoftheproteinconformation,oneshouldconsidertheprobleminthreedimensionalEuclideanspace.However,fromthemathematicalstandpoint,itisconvenienttoformulatetheequationswearelookingfor,firstinD-dimensionalEuclideanspaceandonlyafterthattogotothethreedimensionalambientspace.

Weshallfollowourpapers[13,14],wheretheanalogousproblemhasbeenconsideredinMinkowskispace-timeandtheresultsobtainedhavebeenappliedtothemodelsofrelativisticparticlesdescribedbytheLagrangiansdependentonhigherderivativesoftheparticlecoordinates.Unfortunately,thetransitiontotheEuclideanspaceinthefinalformulasofRef.[13]isnotobvious.2ThereforewederivetheequationsfortheprincipalcurvaturesandtherelevantintegralsofmotioninEuclideanspaceanew.Inthissectionweshalluseanaturalparametrizationofthecurvex(s)intheD-dimensionalEuclideanspace:xi(s),i=1,2,...,D.Inthisparametrization

dxi

ds

=(x′x′)=1,

(2.1)

orinanotherwayds2=dxidxi=(dxdx).Asusualthesumovertherepeatedindexesisassumedinthecorrespondingrangeand,forshortening,thedifferentiationwithrespecttothenaturalparameterswillbedenotedbyaprime.

Thecurvexi(s),i=1,2,...,DhasD−1principal(orexternal)curva-tureswhichdeterminethiscurveuptoitsmotionasawholeinembedding

space.Thefirstcurvature,orsimplycurvature,isgivenby

k21(s)

=k2

(s)=

d2xi

ds2

=(x′′x′′).

(2.2)

ForanarbitraryfunctionalFdefinedoncurvesxi(s)inD-dimensional

spacetheEuler-LagrangeequationsareasetofexactlyDequations

δF

ds

=ωabeb,ωab+ωba=0.(2.6)

TheunitvectoroftheFrenetbasiseα(s)isexpressedintermsofthederiva-tivesofthepositionvectorx(s)withrespecttotheparameterstilltheorderαincluded,α=1,2,....,D.Thefirstvectore1isdirectedalongthetangentofthecurveatthepoints

e1(s)=

dx

curvatureisdefinedinEq.(2.2).Besides,weshallusetheexplicitexpres-sion,throughthederivativesofx(s),onlyforthesecondcurvatureortorsionk2(s)=κ(s).

Thevariationofthespaceformoftheproteinmolecule

δx(s)=εa(s)ea(s),

a=1,...,D

(2.9)

resultsinthefollowingvariationofthefreeenergyfunctionalFdefinedinEq.(2.4)

δF=δF1+δF2,(2.10)where

δF1=

󰀈

dsF′(k1)δk1(s),󰀈

F(k1)δds.

(2.11)(2.12)

δF2=

InwhatfollowstheprimeonthefreeenergydensityF(k1)willdenotethe

differentiationwithrespecttoitsargumentk1.

Thevariationδdsiscalculatedinastraightforwardway

δds=δ

√

ds

=dxi

Weareinterestedinstatisticalapplicationsoftheultimateequations,thereforethecontributionsdependingonthepositionoftheproteinedgesaredropped.

3

6

ThecalculationofthevariationδF1,definedinEq.(2.11),ismorecom-plicated.Firstwefindthevariationofthecurvatureδk1(s).Definition(2.2)andtheFrenetequations(2.6)give

k1(s)δk1(s)=(x′′δx′′)=(e′1δx′′)=k1(e2δx′′).

Hence

δk1=(e2δx′′).

dx(x′dδx)

ds

(2.17)

ApplyingEq.(2.13)stepbysteponederivesδx=δ

′

dx

ds

δx−

δx−x′(x′

d

ds

δx−e1(e1

d

δx).ds(2.19)

Inviewof(2.17),thelastterminEq.(2.19)doesnotcontributetoδk1.Sub-stitutingEq.(2.19)into(2.17)andusingtheFrenetequations(2.6)togetherwithexpansion(2.9)weobtain

ds

ds2

ds

′22′

δk1(s)=ε1k1+ε2+ε2(k1−k2)−2ε′3k2−ε3k2−ε4k2k3.

=

d

=

d2

δx)−e1

d

(2.20)

Asonecouldexpectformula(2.20)forδk1differsfromitsanalogueinMinkowskispace(seeEq.(2.18)inRef.[13]onlybysignsofsometerms.However,withoutthedirectcalculationofδk1inEuclideanspacetheruleofsignchangesinthisexpressionisnotobvious.

Thus,thevariationofthefirstcurvature,δk1(s),dependsonthevaria-tionsoftheworldlinecoordinatesonlyalongthedirectionse1,e2,e3,ande4(onthefourarbitraryfunctionsεa(s),a=1,2,3,4)andonthefirstthreecurvaturesk1,k2,andk3.

SubstitutingEq.(2.20)into(2.11),integratingthelatterequationbyparts,andtakingintoaccount(2.16)weobtain

󰀈󰀆󰀎

󰀒2󰀔′d22

δF=dsk1−k2F(k1)+

󰀑󰀉

′

(F′(k1)k2)−k2F′(k1)ε3(s)−F′(k1)k2k3ε4(s)=0.(2.21)dsThetermsinδF1andδF2,dependingonthevariationε1(s)alongthetan-genttothecurve,aremutuallycancelled,andthevariationδFdepends,asonecouldexpect,onlyonthenormalvariationofthecurvex(s),ormore

7

precisely,onthevariationofx(s)alongthethreenormalse2,e3,ande4.ApparentlythelastpropertyofthevariationδFisduetoaspecialtypeoffunctionals(2.4)underconsideration,whichdependonlyonthecurvatureofacurve.FromEq.(2.21)threeequationsforprincipalcurvaturesfollow

d2

′

(F′(k1)k2)=k2F′(k1),

ds

F′(k1)k2k3=0.

(2.23)(2.24)

Atfirstglance,Eqs.(2.22)–(2.24)areobviouslyinsufficientinordertodetermineallD−1principalcurvaturesofthecurvex(s)embeddedinD-dimensionalEuclideanspacewitharbitraryD.However,itisnotthecase.WestarttheanalysisoftheobtainedequationsfromEq.(2.24).Themostgeneralconditionontheprincipalcurvatures,followingfromthisequation,istherequirementofvanishingthethirdcurvature

k3(s)=0.

(2.25)

Thepointisthattheprincipalcurvatures,duetotheirconstruction,obeytheconditions[17,18]:ifacurvaturekj(s)doesnotvanishatthepoints,thenatthispointthecurvatureskα(s)withα=1,2,....,j−1arealsodifferentfromzero.Butthevanishingofkj(s)atapoints,kj(s)=0,impliesthatkα(s)=0forα=j+1,j+2,...,D−1.Inviewofthis,Eq.(2.25)entailsthefollowingconditions

k4(s)=k5(s)=...=kD−1(s)=0.

(2.26)

Thus,intheproblemunderconsiderationtherearetwonontrivialequa-tions(2.22)and(2.23)forthecurvaturesk1(s)andk2(s).Equation(2.23)canbeintegratedwitharbitraryfreeenergydensityF(k1).Actually,fork2=0onecanrewritethisequationintheform

′k2

orHence

F′(k1)

=0

2d[lnF′(k1)]+d(lnk2)=0.

(F′(k1))k2=C,

2

(2.27)

whereCisanintegrationconstant.

8

Relation(2.27)enablesonetoeliminatethetorsionk2(s)fromEq.(2.22).Asaresultweareleftwithonenonlineardifferentialequationofthesecondorderforthecurvaturek1(s)

󰀌2

d′

F(k1)−k1F(k1)=0.(2.28)

(F′(k1))4Havingresolvedthisequationfork1(s),onecandeterminetherestofcurvaturesbymakinguseofEqs.(2.27),(2.25),and(2.26).IntegrationoftheFrenetequations(2.6)withprincipalcurvaturesfoundenablesonetorecoverthecurvex(s)itself.

Notwithstandingitsnonlinearcharacter,Eq.(2.28)canbeintegratedinquadraturesforarbitraryfunctionF(k1).Toshowthis,thefirstintegralforthisequationwillbeconstructedproceedingfromthesymmetrypropertiesofthevariationalproblemunderstudy.

3

ExactintegrabilityoftheEuler-Lagrangeequationsforprincipalcurvatures

Thefunctional(2.4)possessesaquitelargesetofsymmetries,theanalysisofwhichwillallowustoconstructthefirstintegralfornonlinearequation(2.28).Inthissectionweshalluseanarbitraryparametrizationofthecurvex(τ).Nowthefunctional(2.4)assumestheform

󰀈√F=F(k1)

˙2)3(x

.(3.2)

Functional(3.1)isinvariantunderthefollowingtransformations:i)translationsofthecurvecoordinatesbyaconstantvector

x→x+a,

a=const.

(3.3)

ii)SO(D)-rotationsoftheambientspacecoordinates

xi→xi+ωijxj,

ωij=−ωji,9

i,j=1,2,...,D;

(3.4)

iii)reparametrization

τ→f(τ)

(3.5)

withanarbitraryfunctionf(τ)subjectedtotheconditionf˙(τ)=0.AccordingtothefirstNoethertheorem,theinvarianceofthefunctional(3.1)underthetranslations(3.3)entailstheconservation,underthemotionalongthecurvex(s),ofthe‘momentum’vector

P=

i

d

˙2F(k1)x˙2F(k1)x

dτ

Pi=0,

i=1,2,...,D.(3.7)

Itisclearthattherelations(3.6),duetotheirvectorcharacter,cannot

beuseddirectlyasintegralsforEqs.(2.22)–(2.24)determiningtheprincipalcurvaturekα.Simply,thevectorPicannotbeexpressedonlyintermsoftheinvariantskα(s).However,onecanhopethattheinvariant

2

P2=P20≡M,

(3.8)

whereM2isanewnonnegativeconstant,canbeexpressedthroughthe

principalcurvatureskα(s)only.Itturnsoutthatthisisreallythecase.Toshowthis,wepassinEqs.(3.6)fromthedifferentiationwithrespecttoarbitraryparameterτtothedifferentiationwithrespecttothenaturalparameters(thelengthofacurve).Forthispurposetheformulas

d

˙2x

d

ds2

=

˙2x¨−(x˙x¨)x˙x

¨∂x

=1ds2

,

k1

∂k1

˙2)4(x

areuseful.Asaresult,Eq.(3.6)forthevectorP,constantalongthecurve

x(s),acquirestheform

󰀊′

F(k1)F′(k1)dx′

+.(3.10)P=[2F(k1)k1−F(k1)]

dsds2ds3

󰀍󰀂󰀅󰀏2222˙x¨)−2x˙x¨x˙−x˙(x˙x¨)x¨3(x

(3.9)

10

Thedefinitionsoftheparameters(2.1)andthecurvaturek1(2.2)entailtherelations

󰀊󰀌󰀊󰀌󰀊2󰀌dxdxdxdk12

=0,=−k,=k11

ds2ds3ds3

ds

,

dx

ds3

2

󰀌

,(3.12)

wheredetG(a,b,c)istheGrammdeterminantforvectorsa,b,andc[19].

ThankstoEq.(2.27),weobtainfrom(3.12)

󰀌2󰀊3󰀊

dxdk1224

.(3.13)+k1k2=k1+

ds(F′(k1))4SquaringEq.(3.10)andusingEqs.(3.11)and(3.13),wehave

M=(F(k1)k1−F(k1))+

2

′

2

C2

ds󰀎

M2−

=±

󰀇

C2

(F′′(k1))2

󰀇

Thus,ifthefreeenergydensityF(k1)obeysthecondition(3.15),thenthecurvaturek1(s)andthetorsionk2(s)ofthestationarycurvex(s)arethefunctionsoftheparametersdefinedbyEqs.(3.18)and(2.27).Thecasewhenthecondition(3.15)isnotsatisfied,i.e.,whenthefreeenergydensityF(k1)isalinearfunctionofthecurvaturek1(s),willbeconsideredinthenextsection.

Closingthissectionwebrieflyconsidertheconsequencesoftheinvarianceofthefreeenergyunderthetransformations(3.4)and(3.5).AccordingtothefirstNoethertheorem,therotationinvarianceofthefunctional(2.4)entailstheconservationoftheangularmomentumtensor

󰀃2Mij=

(qiσpjσ−qjσpiσ),

iσ=1

wherethecanonicalvariablesqσandpσ,σ=1,2aredefinedasfollows[20]

q1=x,q2=x˙,p∂(√dp1=P=−

∂x

˙−

2

x

˙2F)M2

,(3.21)

where

W=

1

inthetheoryofrelativisticstrings[23],onecanshowthattheprojectionof

˙(τ)isidenticallyequaltozeroEqs.(2.3)onthetangentvectorx

δF

2k1+k2

,2

d=

2π|k2|

αβ

4

,3

k2=

C

ItisinterestingtonotethatforanarbitrarydimensionDoftheambientspace,the

curveswithconstantprincipalcurvatureshaveadrasticallydifferentbehaviorinthelargedependingonwhetherthedimensionDisevenorodd.IfDiseven,thenthecurvesunderconsiderationaresituatedinarestrictedpartofthespace.ForoddD,suchcurvesgotoinfiniteinonedirection.

13

Sincek1(s)=|x′′(s)|>0,theconstantsαandβshouldhaveoppositesigns.Itisnaturaltoputα>0andβ<0.

ForthefreeenergydensityF(k1),linearincurvature,theintegral(3.14)givesjusttherelationbetweentheintegrationconstantsM2andS2(orbetweenM2andC2)

α22

M=

2d

󰀌󰀄

,C=αβ

2πR

consideredtobeageneralizationoftherelativisticmodelofapointparticlewithpossibleapplicationtostringtheory[27],boson-fermiontransmutations[28],anyonmodels[29],randomwalks[30]andsoon.ItisworthnotingthattheLagrangefunctionslinearinacurvatureoftheworldlineprovetobedistinguishedinthisfieldtoo.TherichestsetofappealingpropertiesisprovidedbythePlyushchaymodel[31,32],whereα=0.Inthiscasetheactionisscaleinvariant.Acompletesetofconstraintsinthephasespaceofthismodelpossessesmanysymmetries,specifically,localW3symmetry.Thismodeldescribesthemasslessparticles,thehelicityofwhichacquires,uponquantization,integerandhalf-odd-integervalues.Itisinterestingthatclassicalsolutionsinthismodelarethespacelikehelicalcurves[31].

5Conclusion

Proceedingfromrathergeneralprinciplesandmakinguseofthebasicfactsconcerningtheconformationofglobularproteinswehaveobtained,inauniqueway,ageometricalmodelforphenomenologicaldescriptionofthefreeenergyofhelicalproteins.Itisworthnotingthatourfunctional(4.6)shouldbeconsideredasaneffectivefreeenergyofthehelicalproteinwhichalreadytakesintoaccountthenatomicinteractionswithintheproteinandwiththesolvent.Hence,thereisnoneedtoquantizeit,asoneproceedsintherandomwalkstudies[30].

Certainlyoursimplemodeldoesnotpretendtodescribealltheaspectsoftheproteinfolding.However,onecanhopethatitcouldbeemployed,forexample,inMonteCarlosimulationtosearchforastablenativestateoftheprotein.Inthiscasethemodelcanbeusedforthedescriptionofthefreeenergyofindividualparts(blocks)ofaproteinchainthathavethehelicalform.Withoutanydoubt,itshouldresultinsimplificationandaccelerationoftheexhaustingsearchingofthenativestablestateoftheproteinchainbyacomputer[1].

Inthegeneralproblemoftheproteinfolding,aselfcontainingtaskistorevealthemechanismoftheproteinchaintransitionintothestablenativestate.Atypicaltimeofthisprocessissuchthatitiscompletelyinsufficienttoshowthemutualinfluenceofallthepartsoftheproteinchainduringthefoldingtothestablestate.Presumablythefunctionalsproposedinthispapercanbeusefulheretoo.Forthispurposethecoordinatesoftheproteinchainxshouldbeconsideredastimedependent,i.e.,x=x(t,s).Duringitsmotion,theproteinchainsweepsoutatwodimensionalsurfaceinambientspacedescribedparametricallybyitscoordinatesx(t,s).Inthiscasetheanalogueofthelinecurvaturek(s)isthelocalgeometricalinvariantofasurface,its

15

externalormeancurvatureH(t,s)[11].Astraightforwardgeneralizationofthefunctional(2.4)tothedynamicalproblemathandistheactionSlinearinexternalcurvatureofthesurface

󰀈󰀈󰀈󰀈S=adσ+bHdσ,(5.1)wheredσisadifferentialelementofthesurfacesweptoutbyaproteinchain

inthecourseofitsmotion,aandbareconstants.

Intheelementaryparticletheorythemodelslike(5.1)areknownastherigidrelativisticstring[23,33].Oneofthemotivationstoconsidersuchstringswastheattempttodevelopstringdescriptionforquantumchromo-dynamics.Apeculiarityoftheclassicaldynamicsofsuchstringmodelsisitsinstability[34].Thereasonofthisisthedependenceoftheaction(5.1)notonlyonthevelocitydx(t,s)/dtbutalsoontheaccelerationd2x(t,s)/dt2oftheproteinchain.Asaresult,theenergyofsuchsystemsprovestobeunboundedfrombelow[35].Itisverylikelythatthisinstabilitycouldbecru-cialindescribing,intheframeworkofthestringmodel(5.1),thetransitionoftheproteinchaintothestablenativestate.

Acknowledgements

Thisstudyhasbeenconductedduringthestayofoneoftheauthors(V.V.N.)atSalernoUniversity.HewouldliketothankG.Scarpetta,G.Lambiase,andA.Feoliforthehospitalityextendedtohim.Oneoftheauthors(A.F.)wishestothankA.Ramponeforusefuldiscussionsandreferencesabouttheproteinfoldingproblem.ThefinancialsupportofINFNisacknowledged.V.V.N.waspartiallysupportedbytheRussianFoundationforBasicResearch(GrantNo.03-01-00025)andbytheInternationalScienceandTechnologyCenter(ProjectNo.840).

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