Nonextensive Statistical Mechanics Application to Vibrational Dynamics of Protein Folding

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arXiv:cond-mat/0703408v3 [cond-mat.stat-mech] 24 Sep 2007NONEXTENSIVESTATISTICALMECHANICSAPPLICATIONTOVIBRATIONALDYNAMICSOFPROTEINFOLDING

EthemAkt¨urk∗andHandanArkın†DepartmentofPhysicsEngineering,HacettepeUniversity,06800Ankara,Turkey

ThevibrationaldynamicsofproteinfoldingisanalyzedintheframeworkofTsallisthermostatistics.Thegeneralizedpartitionfunctions,internalenergies,freeenergiesandtemperaturefactor(orDebye-Wallerfactor)arecalculated.Ithasalsobeenobservedthatthetemperaturefactorisdependentonthenon-extensiveparameterqwhichbehaveslikeascaleparameterintheharmonicoscillatormodel.Asq→1,wealsoshowthattheseapproximationsagreewiththeresultofGaussiannetworkmodel.Keywords:Tsallisthermostatistics,HarmonicOscillator,ProteinFolding,Tem-peratureFactor

PACSnumbers:05.20.-y,87.10.te,05.70.-a

I.INTRODUCTIONThestatisticalmechanicsparadigmbasedonBoltzmann-Gibbsentropywhichisgreatsuccesslooksafterbeincapabletodealwithmanyinterestingphysicalsystems[1].In1988,TsallisadvancedanonextensivegeneralizationofBoltzamann-Gibbsentropicmea-sure.TherearemanyapplicationsforTsallisformalismsuchasthespecificheatoftheharmonicoscillator[2],one-dimensionalIsingmodel[3],theBoltzmannH-theorem[4],theEhrenfesttheorem[5],quantumstatistics[6],paramagneticsystems[7],Cerclemaps[8],Henonmap[9],Haldaneexclusionstatistics[10],q-expectationvalue[11],quantumme-chanicaltreatmentofconstraints[12].Inthispaper,wewillfocusonthermodynamicsand2vibrationpropertiesofproteinsintheframeworkofageneralizedthermostatistics.Beforeproceedingwithgeneralizedgaussiannetworkmodel,wedescribedfundamentaldefinitionsofTsallisthermostatisticsinSectionII.WethenstudygeneralizedgaussiannetworkmodelandresultinsectionIII.ResultswillbediscussedinsectionIV.

II.FUNDAMENTALDEFINITIONSOFTSALLISTHERMOSTATISTICSThegeneralizationwhichhasbeensuccessfullynonextensivestatisticalmechanicsisbasedonthefollowingexpression[13]

Sq=k1−Tr(ρq)

Zq

[1−(1−q)βH]1/(1−q),(2)

whereZq=󰀆N󰀅n=1dpndxn[1−(1−q)βH]1/(1−q),(3)isthepartitionfunctionwithq∈Randβ≡1/T(withk=1).Eq.(2)isobtainedmaximizingtheTsallisentropyusinglagrangemultiplymethods[13,14].Theinternalenergyisdescribedby

Uq=󰀆N󰀅n=1dpndxnρqH(4)where󰀆N󰀅

n=1dpndxnρq=1.(5)

InFreeenergynotation,Eq.(4)appearsasUq=Fq−T∂Fq/∂T,(6)whereFqisfreeenergyisdescribedasFq=(Zq−1q−1)/[(1−q)β].(7)TheexpectationofanobservableAisAq=󰀈A󰀉q=󰀆N󰀅n=1dpndxnρq.(8)3III.GENERALIZEDGAUSSIANNETWORKMODELANDRESULTSWefocusourdiscussionontheclassicalTsallisstatisticsbyconsideringproteinfoldingandharmonicapproximation.ConsiderthefollowingHamiltonianforGaussianNetworkModel(GNM)basedonharmonicapproximation[15]

H=N󰀃i=1p2i2∆RTiL∆Ri,(9)wherethefirsttermisthekineticenergyofthesystemandγisthestrengthofthesprings.Inthismodel,springsareassumedashomogeneous.Riand∆RiisalsodefinedastheequilibriumpositionandthedisplacementwithrespecttoRiofthei-thCαatoms.ThemodeliseventuallydescribedbythecontactmatrixLwithentries:Lij=1ifthedistance|Ri−Rj|betweentwoCα’s,inthenativeconformation,isbelowthecutoffR0,whileis0otherwise.FromEq.(3),partitionfunctionfortheGaussianNetworkmodel[15]whichhasabovehamiltonianisgivenby

Zq=󰀆N󰀅i=1dpid∆Ri[1−(1−q)β×󰀏N󰀃i=1p2i2∆RTiL∆Ri󰀁󰀄1/(1−q).(10)weintroducethevariablesL=VTλV,x=V∆Randthus∆RTL∆R=xTλxifλiseigenvaluesoftheLandViseigenvectorsofL.Wecanrewritepartitionfunctionas:

Zq=󰀆N󰀅i=1dpidxi[1−(1−q)β×󰀏N󰀃i=1p2i2λix2i󰀁󰀄1/(1−q).(11)Tocalculatethisintegral,wedefinitionnewvariables:yi=[(1−q)γλiβ/2]1/2xiandyN+i=[(1−q)β/(2m)]1/2pi[16],wherei=1,2,3,...,N.SubstitutingthesevariablesintoZqgives

Zq=󰀋N󰀅i󰀎2γ)1/2󰀐󰀍

×󰀆2N󰀅n=1dyn󰀏1−2N󰀃k=1y2k󰀁1/(1−q).(12)4Byusinghypersphericalcoordinateswithu=(󰀈2Nn=1y2n)1/2andcalculatingtheintegralovertheangularvariables[16],weobtain

Zq=󰀋N󰀅n=1󰀎2γ)1/2󰀐󰀍Ω2N

(1−q)λnβ(m1−q

+1

󰀉

1−q+1+N󰀉

=󰀎󰀊2−q(1−q)λnβ(m

(a−1)!isthePochhammersymbol[18].Freeenergyofthesystemisgivenby

Fq=−1

βlnq󰀎󰀊2−q(1−q)λnβ(m

βlnq󰀎󰀊2−q(1−q)β(m

βZ1−qqN(1−q)β(m1−q󰀌N󰀐−1

󰀍

1−q

.(18)

Hence,weobtainedthermodynamicpropertiesofthegeneralizedGaussiannetworkmodel.Ontheotherhand,themostpropertiesofGaussiannetworkmodeliscalledtemperaturefactor.ThecrystallographictemperaturefactorsorB-factors,whichisdefinedasintrinsicfluctuationsoftheatomsincrystal,isdirectlyrelatedtothisfluctuations.TheX-ray