Nucleotidefluctuationanalysisofdim-lightvisionrhodopsingeneandmRNAsequencesinvertebratesToddHolden*,N.Gadura*,E.Cheung*,J.Rada*,P.Schneider*,G.Tremberger,Jr*,D.Sunil*,D.Lieberman*,andT.Cheung*Physicsand*BiologyDepartmentsCUNYQueensboroghCommunityCollegeBayside,NY11364USAContactemail:AbstractFractaldimensionandShannonentropywereusedtoanalyzethenucleotidefluctuationofdim-lightvisionrhodopsingene.Themaximalabsorptionwavelengthsofthestudiedgenesequencesspanarangeof480nmto526nm.Thedi-nucleotideShannonentropycorrelateswith(a)themono-nucleotideShannonentropywithR2of0.99；(b)theCG+GCcontentwithR2of0.86(N=6).TheabsorptionmaximalwavelengthcorrelationwiththeCG+GCcontentwasfoundtohaveanR2of0.82forthegenesequenceand0.91forthemRNAsequence.ThefractaldimensionsofthegenesequencesandmRNAsequenceswerefoundtobecorrelatedwithR2of0.78.SelectionpressureontheCG+GCcontentatthegeneandmRNAlevelswouldbeaconsistentobservationforphenotypedim-lightfunctionality.Keywords-component:Dim-lightvisionrhodopsingene；rhodopsinmRNA；Shannondi-nucleotideentropy；fractaldimension；correlationI.INTRODUCTIONCloselyrelatedspeciescanbeclassifiedbyacollectionofbioinformaticsmarkerswithnumbersdeterminedfrommathematicaloperationsonDNAsequence.Incaseswheresuchmarkersaresignificantlydifferentfromwhatwouldbeexpectedforrandommutations,wecangetanideaofhowmuchnaturalselectionhasinfluencedtheevolutionofaparticulargene.Itwasreportedrecentlythattherhodopsinswithvariablemaximalabsorptionwavelengthbetween480525nmhadevolvedatleast18separateoccasions1.RhodopsinisamemberoftheG-proteincoupledreceptor(GPCR)familywiththecharacteristicseven-transmembranedomainreceptorsandisverysensitivetolight.CrystalstructureandsequencecomparisonofGPCRmembersincludingrhodopsinwasreportedrecently2.Forexample,Reference2reportedthatwatermoleculesinthevicinityofhighlyconservedaminoacidsarebeingusedforstructurestabilization.Conservedregionwouldimplylessrandomnucleotidefluctuationacrossspecies.Here,weanalyzethereporteddim-lightvisionrhodopsingeneanditsmRNAsequences,usingnucleotidefrequency,mono-anddi-nucleotideentropy,andfractaldimension.Therelationshipsamongtheabovementionedbioinformaticsmarkersshowssomeevidencethatthereisselectionpressureresultinginasystematicnucleotidefluctuationtrendanditscorrelationwithdim-lightfunctionality.Thenucleotidebasepairchangesoveragenesequencecanbeviewedasafluctuationand,consequently,canbeinvestigatedwithstandardtoolsthatincludecorrelationandfractaldimensionanalysis.Forthisstudy,thenumericalsequencerepresentingthefluctuationofnucleotidesinagenesequencewasgeneratedusingtheprotonnumberofeachnucleotide.Nucleotidefluctuationhasbeenstudiedusingotherassignmentschemes3,4,5.TheuseofprotonnumberwasmotivatedpartlybytheobservationofmassfractaldimensionintheX-raydataofproteinsandribosomes6,andusingaprotonassignmentschememayrevealprotonsensitivityintheunderlyinggeneticsequencetothefoldinginducedmassfractal.Thisnumericalseriescanthenbeprocessedfurtherusingnumericalmethodssuchasamovingaverage,whichisoftenusedinstockmarkettimeseriesanalysis.Thefractaldimensionofsucharandomseriesorrandomseriesderivedfromtheoriginalatomicnumberbasedsequencecanbecomputed.Arecentcomparisonofhumanandchimpanzeegenomesrevealedthatitispossibletomeasuretheaccelerationrateoftheacceleratedregionsofthehumangenome7.Themostacceleratedregion,HAR1,wasshownbyageneexpressionexperimentinthehumanembryotobetranscriptionactiveandco-expressedwithreelin,whichisanessentialproteininvolvedinthedevelopmentofthesix-layercortexofthehumanbrain.FractalanalysiswasappliedtotheHAR1nucleotidesequenceandthehomologoussequenceinthechimpanzeegenome8.Analysisshowsthatthedifferencesinfractaldimensioncanbeusedasamarkerofevolution.The118-bpinHAR1contains18pointsubstitutionsoveranevolutionaryspanof5millionyearswhencomparingthehumantothechimpanzee.However,thesame118-bpregiononlycontainstwopointsubstitutionsoveraspanof300millionyearswhencomparingthechickentothechimpanzee.Theimplicationsofevolutionandpositiveselectionhavebeendiscussedinrecentliterature9.II.METHODSA.GeneticSequenceTheDNAsequencesweredownloadedfromGenbankusingtheaccessionnos.EU407248EU407253)listedinReference1.ThecorrespondingspeciesareAristostomiasscintillansrhodopsin(RH1)gene(1569bp),Idiacanthusantrostomusrhodopsin(RH1)gene(1547bp),Chauliodusmacounirhodopsin(RH1)gene(1608bp),Stenobrachiusleucopsarusrhodopsin(RH1)gene(1213bp),Lepidopusfitchirhodopsin(RH1-A)gene(1438bp),andLepidopusfitchirhodopsin(RH1-B)gene(1286bp).B.HiguchiFractalMethodTheprojectwaspartiallysupportedbyseveralCUNYgrantsandNIHBridgetoBaccalaureateGrant(PI:Schneider)978-1-4244-4713-8/10/$25.002010IEEETheATCGsequencewasconvertedtoanumericalsequencebyassigningtheatomicnumber,thetotalnumberofprotons,ineachnucleotide:A(70),T(66),C(58),G(78).Theassignednumberisproportionaltothenucleotidemass(ignoringisotopes).TheA-TandC-GpairsindoublestrandedDNAhavethesamevalueof136.Amongthevariousfractaldimensionmethods,theHiguchifractalmethodiswellsuitedforstudyingsignalfluctuationandhasbeenappliedtonucleotidesequencesasfollows10.ThenumericalsequenceIisusedtogenerateadifferenceseries(I(j)-I(i)fordifferentlags.Thenon-normalizedapparentlengthoftheseriescurveissimplyL(k)=|I(j)-I(i)|forall(j-i)pairsthatequalk.Thenumberoftermsinak-seriesvariesandnormalizationmustbeused.Thenormalizationvaluesareinopenliterature11.IftheI(i)isafractalfunction,thenthelog(L(k)versuslog(1/k)willbeastraightlinewiththeslopeequaltothefractaldimension.Higuchiincorporatedacalibrationdivisionstep(divisionbyk)suchthatthetheoreticalvalueofthefractaldimension,FD,iscalibratedtotherange1FD2.Whencomparingthedimensionoftwofractalforms,thepopularmethodoftakingthedifferenceofthetwoHiguchifractaldimensionvaluesisvalidtowithinaconstantregardlessofthecalibrationdivisionstep.TheHiguchifractalalgorithmusedinthisprojectwascalibratedwiththeWeierstrassfunction.ThisfunctionhastheformW(x)=a-nhcos(2anx)forallthenvalues0,1,2,3ThefractaldimensionoftheWeierstrassfunctionwasgivenby(2-h)wherehtakesonanarbitraryvaluebetweenzeroandone.TheShannonentropyofasequencecanbeusedtomonitortheleveloffunctionalconstraintsactingonthegene12.AsequencewitharelativelylownucleotidevarietywouldhavelowShannonentropy(moreconstraint)intermsofthesetof16possibledi-nucleotidepairs.Asequencesentropycanbecomputedasthesumof(pi)log(pi)overallstatesiandtheprobabilitypicanbeobtainedfromtheempiricalhistogramofthe16di-nucleotide-pairs.Themaximumentropyis4binarybitsperpairfor16possibilities(24).Themaximumentropyistwobitspermono-nucleotidewithfourpossibilities(22).III.RESULTSANDDISCUSSIONA.CGdi-nucleotidefluctationTheCG+GCdi-nucleotidepairpercentagewascalculatedandthecorrelationwiththeShannondi-nucleotideentropyisdisplayedinFigure1.TheCG+GCdi-nucleotidepairpercentagecorrelationwiththeShannondi-nucleotideentropywasnotobservedatthemRNAlevel,ascanbeseenfromthelowR2inFigure2.y=-0.6108x+2.534R2=0.855600.03.93.923.943.963.98di-nucleotideentropy(bits)CG+GCFigure1:TheCG+GCdi-nucleotidepairpercentageversustheShannondi-nucleotideentropyforthegenesequences.y=-599.1x+2482.1R2=0.524400.03.863.883.93.923.943.96di-nucleotideentropy(bits)CG+GCFigure2:TheCG+GCdi-nucleotidepairpercentageversustheShannondi-nucleotideentropyforthemRNAsequences.ThevanishingofthecorrelationofentropywithCG+GCcontentatthetranscriptionmRNAlevelwouldbeconsistentwiththerequirementofrelativelymorestablebondbetweenCandGtobelesspracticalinamRNA,atemporaryproduct.TherelativelyshortermRNAlengthsforthecorrespondingspeciesareAristostomiasscintillansrhodopsin(1041bpbp),Idiacanthusantrostomusrhodopsin(1059bpbp),Chauliodusmacounirhodopsin(1059bp),Stenobrachiusleucopsarus(1038bp),Lepidopusfitchirhodopsin(1053bp),andLepidopusfitchi(1053bp).y=0.4753x+0.1064R2=0.98671.961.971.981.9923.93.923.943.963.98di-nucleotideentropy(bits)mono-entropFigure3:TheShannonmono-entropyversusthedi-nucleotideentropyforthegenesequences.TheShannonmono-entropycorrelationwiththeShannondi-nucleotideentropywasobservedforboththeentiregenesequencesandatthemRNAlevelwithR20.95(Figures3and4).di-nucleotidevsmono-entropyy=0.5073x-0.0164R2=0.9541.951.961.971.981.993.863.883.93.923.943.96Figure4:TheShannonmono-entropyversusthedi-nucleotideentropyforthemRNAsequences.y=-0.0009x+0.5772R2=0.82100.0460480500520540wavelengthnmCG+GCFigure5:ThemaximalabsorptionwavelengthofthegeneversustheCG+GCcontentpercentage(N=5).y=-0.0009x+0.5721R2=0.905400.0460480500520540wavelengthnmCG+GCFigure6:ThemaximalabsorptionwavelengthofthemRNAsequenceversustheCG+GCcontentpercentage.TheCG+GCcontentpercentagecorrelateswellwithdim-lightfunctionalityconductedbythemRNAproducts.Theblackdragonfish(Idiacanthusantrostomus)maximalabsorptionwavelengthwasnotreportedinReference1andwasomittedintheregressionsuchthatN=5.ItappearsthatthereisasystematictrendforincludingtheappropriateCG+GCcontentthatspanthe480-525nmrange.Reference1reportedthatonly12aminoacidsitesintheopsinproteincanaccountforthevariabilityofwavelengthsensitivityforthesespecies.However,thesechangeshavealmostnoeffectontheCG+GCcontent(accountingforlessthan2%oftheobservedtrend).Thisindicatesthatsecondaryeffects,suchasgeneexpressionorproteinstability,aretheevolutionaryforcedrivingsmallgradualadaptationsfromtheCG+GCviewpoint.B.FractaldimensionfluctationThefractaldimension(FD)wascomputedusingtheHiguchimethod.Theslopewastakenusing7datapoints,consistentwithourpreviousreportofnucleotidefluctuation7.ThefractalanalysisresultisshowninFigure7.y=1.9792x+9.4164R2=0.99910246810-3-2-10Ln(1/k)LnL(kFigure7:FractaldimensionoftheAristostomiasscintillansrhodopsin(RH1)genesequenceusingthesevendatapoints.Thesequencehas1569bp.They-axiisLn(k)andthex-axisisLn(1/k).ThefractaldimensionofthegenesequenceandmRNAsequenceswerecomputed.FD-genevsFD-mRNAy=0.6544x+0.6878R2=0.78121.971.9751.981.9851.991.99521.961.971.981.992Figure8:FractaldimensionofmRNAsequencex-axisversusfractaldimensionofgenesequencey-axis.ThefractaldimensionincreaseforfiveofthestudiedmRNAsequence(exceptblackdragonfishIdiacanthusantrostomus)wouldbeconsistentwithpreviousobservationthatanexonregiongenerallywouldhavehigherfractaldimensionascomparedtoanon-codingregion13.ThemoderatecorrelationwithR-squareof0.78suggeststhatFDisafairlyrobustparameter.Onthecontrary,thereisverylittlecorrelationintheentropyparameterbetweengeneandmRNA(Figure9)geneentropyvsmRNAentropyy=0.4393x+2.169R2=0.26663.873.883.893.93.913.923.933.943.953.93.923.943.963.98Figure9:EntropyofmRNAsequencey-axisversusentropyofgenesequencex-axis.Thetranscriptionprocesslosesinformationwithlowerentropyforallthestudiedsequences；howeverthereisnosystematicpatternforentropycorrelationofgenewithmRNA.Thisfurtherreinforcedthatthefractaldimensionwhichmeasuresinformationcapacityisamorerobustparameter.Itwasreportedthatthespectraltuningsitesmaynotbethesameasthefunctionallyconservedsitesimportantfortheproperfunctioningoftheopsin14,15.Thefractaldimensionmaybeameasurerelatedtothefunctionallyconservedsites.IV.CONCLUSIONThefractaldimensionandShannonentropywereusedtoanalyzethenucleotidefluctuationofdim-lightvisionrhodopsingene.TranscriptionfromgenetomRNApreservesthefractaldimensioninsuchawaythatthereisamoderatecorrelation.TheabsorptionmaximalwavelengthcorrelationwiththeCG+GCcontentwasfoundtohaveanR-squareof0.82forthegenesequenceand0.91forthemRNAsequence.SelectionpressureontheCG+GCcontentatthegeneandmRNAlevelswouldbeaconsistentobservationfordim-lightphenotypefunctionalityevolutionintherangeof480-525nm.FuturestudiesmayincludetheRhodopsin-likegenesintheG-proteincoupledreceptorfamily.REFERENCES1ShozoYokoyama,TakashiTada,HuanZhang,andLyleBritt,“Elucidationofphenotypicadaptations:Molecularanalysesofdim-lightvisionproteinsinvertebrates”,PNAS,Vol105,1348013485,2008.2WorthCL,KleinauG,KrauseG,“ComparativeSequenceandStructuralAnalysesofG-Protein-CoupledReceptorCrystalStructuresandImplicationsforMolecularModels.PLoSONE4(9):e7011.doi:10.1371/journal.pone.0007011,20093N.N.OiwaandJ.A.Glazier,“Thefractalstructureofthemitochondrialgenomes”,PhysicaA,vol311,pp221230,20024Z.G.Yu,A.Vo,Z.M.GongandS.C.Long,“FractalsinDNAsequenceanalysis”,ChinesePhysics,vol11,pp1313-1318,2002.5H.D.Liu,Z.H.Liu,X.Sun,“StudiesofHurstIndexforDifferentRegionsofGenes”,ICBBE2007,pp238-240,20076C.Y.Lee,“MassFractalDimensionoftheRibosomeandImplicationofitsDynamicCharacteristics”,PhysicalReviewE,vol73,042901(3pages),2006.7K.S.Pollard,S.R.Salama,N.Lambert,S.Coppens,J.S.Pedersen,etal.“AnRNAgeneexpressedduringcorticaldevelopmente