Sensitivity Analysis of Biochemical Networks

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SensitivityAnalysisofBiochemicalNetworks: ComputerAlgebraApplicationtothe Escherichiacoli Tryptophan Operon by CaseyHenderson AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegree BachelorofArts UnderthesponsorshipofProfessorNecmettinYildirim Sarasota,Florida April,2011

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Acknowledgements IwouldliketothankProfessorNecmettinYildirimforhisinstructioninprogramming,modeling,biologyandmanymoresubjects.Heintroducedmetotheeldof MathematicalBiologyandprovideddirectionforthisproject.Iwouldliketothank ProfessorPatMcDonald,whohasprovidedguidancefrommyrstyearthroughmy lastmonth.IthankJaneandAllanClaymanfortheirinspiringsupportthroughout mylife.IthankmyMomforintroducingmetothewonderfulcommunityatNew Collegeandencouragingmealongtheway.

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SensitivityAnalysisofBiochemicalNetworks: ComputerAlgebraApplicationtothe Escherichiacoli TryptophanOperon CaseyHenderson NewCollegeofFlorida,2011 ABSTRACT Operonsarecollectionsofgeneticelementsincludingproteincodinggenesandregulatorybindingsites.Operondynamicscanbedescribedbyasystemofdi!erential equationswhichdeneasetofrelationshipsamongvariablesandparameters.The collectionofequationsmodelsregulationofgeneexpressioninacell.Mathematical techniques,suchassensitivityanalysis,canbeusedtoinvestigatepossibleregulatory mechanismsinthesesimulatedbiologicalnetworks.Sensitivityanalysisexploreshow thenetworkvariables,inthiscaseconcentrations,respondtosmallchangesinnetworkparametervalues,suchasreactionrates.TheComputerAlgebraSystemMaple generatesthesensitivityequations,whicharetransferredtoMatlab,toperformsensitivityanalysis.Thisthesisprovidesananalysisoflocalparametersensitivityinthe Mackey-Santillanmodelof E.coli tryptophanoperonusingcomputer-aidedcalculus.Themainresultisademonstrationthattryptophanconcentrationisthemost sensitivevariablewithtranscriptionalattenuationasthemostsensitiveparameter. AssistantProfessorofMathematicsNecmettinYildirim DivisionofNaturalSciences

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Contents ListofFiguresv ListofTablesvi Chapter1.Introduction1 Chapter2.MathModeling5 1.Introduction5 2.MassActionKinetics6 3.StabilityAnalysis8 4.EnzymeKinetics9 Chapter3.CellBiology13 1.EnzymeKineticsandBiochemistry13 2.TranscriptionalLevelControlofGeneExpression14 3.Operons16 Chapter4.TheTryptophanOperon18 1.TheModel19 2.TheMackey-SantillanModel20 Chapter5.SensitivityAnalysis22 1.ImplementationoftheAnalysis23 2.Results24 Chapter6.Discussion35 1.FurtherResearchandDirection36 iii

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AppendixA.EquationsandParameterDenitions38 1.SimpliedMackeySantillanModeloftheTrpOperon38 AppendixB.Code40 1.MapleSensitivityMatrixBuilder40 2.MatlabDriver42 3.MatlabModelEquations47 Bibliography57 iv

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ListofFigures 1Concentrationsfortwovaluesofparameterd6 2GraphofthereactionvelocityforthemodelgivenbyMichaelisandMenten11 3DiagramoftranscriptionalattenuationintheTRPoperon16 4Diagramofthetryptophanoperon17 5Renderingofthetryptophanoperonrepressor,trpR19 6TimeseriesandparametersensitivityforvariableOf26 7TimeseriesandparametersensitivityforvariableMf27 8TimeseriesandparametersensitivityforvariableE28 9TimeseriesandparametersensitivityforvariableE29 10Bargraphofthesensitivityscores32 11Tryptophanconcentrationsteadystatewithrespecttomaximum transcriptionalattenuationrate33 12Tryptophanconcentrationsteadystatewithrespecttotrp-trpR disassociationrate34 v

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ListofTables 1TableofNumericalResults31 2ParameterDenitions39 vi

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CHAPTER1 Introduction Thisthesisexploresabiologicalsystembyapplyingmathematicaltools,addingto theburgeoningeldofmathematicalbiology.Thegrowthofthisinterdisciplinaryeld comesfromadvancesincomputingtechnologythathavebeenpairedwithmolecular biologybreakthroughssuchassequencingDNA[ 17 ].Thesetwindevelopmentscan betracedtothe19thcenturywithCharlesBabbageandLouisPasteur,andareset tocontinueacceleratingfortheforeseeablefuture[ 18 ].Combined,theyopennew avenuesforresearchandholdthepromiseofbiologicalinsightonoldproblemsas well[ 31 ]. Vastamountsofdataarebeingproducedbyhigh-throughputmolecularbiology researchsuchasproteomicsandgenomics[ 21 ].Thevolumeofdataisimmense andtherateatwhichitgrowsisincreasing.In2008therewereanestimated92 millionindividualsequencesandthedoublingtimethenwas30months[ 2 ].This createsademandforbothresearchersandtechniquestoanalyzethisunprecedented scaleofdata.Toolsneedtobedevelopedtograpplewiththisexponentialgrowth andunderstandtheinformationproduced[ 24 ].Whilemajormilestoneshavebeen achieved,includingmappingthehumangenome,therearemanyavenuesstillleftto explore.Inparticular,theconstructionandanalysisofmathematicalmodelsleadsto openquestions.Theincreasingspeedanddecreasingcostofcomputingtechnology,as wellasthedevelopmentofnewalgorithms,catalyzesthisprogress.Thedevelopment ofinexpensivehigh-speedcomputingcoupledwithnewalgorithmdesignhasledtoa newtypeofexperimentalprotocol. Inadditiontotraditionalbiologyexperimentsconducted invivo (inanorganism)and invitro (inalaboratory),experimentscannowbeconducted insilico (in 1

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acomputerprogram)[ 23 ].Theadvancesinmolecularbiologyandcomputingboth contributetotheincredibleriseof insilico experimentation.Thisprovidesaframeworkforquantitativemodelspurportingtodescribebiologicalphenomena.Thereare currentlyprojectsunderwaytomodelarangeofbiologicalsystems,fromsub-cellular functionstosystemsofcellsinacomputersimulation.However,thereareobstacles relatedtoboththemodelingandthecomputationalaspectsofthisresearch. Thepromiseofmathmodelingtoleadtonewinsightdependsontwofactors;the modelsandthetoolsappliedtothem.First,theaccuracyofthemodelingequations oftenlimitsthequalityofthemodel'sresults.Asthesystemsincreaseincomplexity,theanalysisoftheunderlyingmodelpresentsavarietyofintellectualchallenges, incentivizingsimplifyingassumptions.Beyondthedi" cultyinfullyunderstanding biochemicalreactions,thereisatrade-o!betweenhighresolutionequationsandcomputationaltime.Second,theanalytictoolsusedtounderstandamodel'sresultscan produceartifactsandloseinformationintheprocess,especiallyinlightofamodels simplifyingassumptions[ 21 ].Thechallengeofchoosingananalyticlens'isexempliedinthecalculationofthesinglesensitivityscoreusedhereandinotherways ofmeasuringsensitivity.Anartifactoccursinsensitivityanalysiswhenascoreis normalized.Someparametersaresettoavalueofzero,andwhennormalizedthe sensitivityscoreisthenalsozero.Agoodexampleofinformationbeinglostcanbe foundinBlissetal'sappropriatelytitledFromcomplexitytowhatreallymatters' whentimeseriessolutionsaresummedupintoasinglescore[ 5 ].Thissummation occurshereaswellforcalculatingthesensitivityscore.Thisscoredoesnotconvey anyofthedynamicsofthesensitivityscore,suchasthatoftryptophanconcentration withrespecttotryptophanconsumptionbyproteinsynthesis.Theoperonswitching fromo!'toon'canbeseeninsomesensitivitytimeserieswhenthefunctionchanges sign,foraclearexampleseeparameter g inthebottomrowofFigure9. Despitethesechallenges, insilico experimentsareincreasinginpopularitybecauseoftheiradvantages[ 23 ].Theseadvantagesincludetheextremelylowcost 2

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ofexperiments,easeofreproducibilityandshorttimeframenecessarytorunexperimentsbyrunningsoftware.Theabilitytocheaplyconductmanyexperimentsquickly allowsforexploratorysciencetotakeplaceinsmallcommunities,suchasNewCollege,whereadvancedmolecularbiologyequipmentcanbeprohibitivelyexpensive. Anadditionalbenetof insilico experimentsisthelegibilityofmodelingequations comparedtotheopaquemechanismsoperatingwithinacell.Thestraightforward mathematicsofthemodelsallowforarobustunderstandingando!ersavarietyof analytictoolstoresearchers.Theprimarytoolusedhereissensitivityanalysis,the studyofinputvariationwithrespecttooutputvariation[ 13 ].Sensitivityanalysis ndsthederivativeofeachvariableconcentrationwithrespecttoeachparameter andnormalizesthesesensitivitiesforameaningfulcomparison.Alargenormalized sensitivityscoreindicatestherespectiveconcentrationwillbeinuencedmuchmore thanitwouldforperturbationsofaparameterwithasmallnormalizedsensitivity score.Theseperturbedparameterscanrepresentrealmutations,allowingthesame modeltorepresentdi!erentstrainsofthesameorganism[ 22 ]. Themodelsusedfor insilico experimentsinthisthesisarecoupledsystemsofordinarydi!erentialequations.Theseequationsdenevariableconcentrationsinterms ofconstantparameters,suchasthethosefoundinAppendixA.1.Astheresolution ofourunderstandingofcellularregulatoryprocessesincreases,sodotheparameters associatedwiththemetabolicpathway.Witheachadditionalstepinametabolic pathwaythatisdiscovered,modelsbecomemoreaccurate.Thiscomesintheform ofadditionalvariablesandparameters,oftenseveralparametersforeachvariable. Mathematicalbiologywrestleswiththeselargesystemsofequationsinvolvingmany parametersthatmodelrealmetabolism[ 11 ].Fortunatelyforthoseseekingtounderstandthesemassivesystems,mostparametershaveaminimalinuencewhileafew parametersdominatethebehaviorofthesystem[ 9 ].Thisthesispresentsatoolfor understandingsystemsintermsoftherelativeinuenceofparameters. 3

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Inthisthesisananalysisoflocalparametersensitivityisperformedforthe Mackey-Santillanmodelof E.coli tryptophanregulation[ 25 ].Themethodrelies uponcomputeralgebrasoftwareandcanbeappliedtomanysimilarsystemsofordinarydi!erentialequation.Thealgorithmisbasedincalculus,therefore,theanalysis ismoreelegantandinsightfulthandiscretelyproducedresults[ 13 ]. Theremainderofthisthesisisarrangedinvechaptersasfollows.Thesecond chapterpresentsalighttreatmentofmathematicalmodeling,providingthereader backgroundonmassactionkineticsandunderlyingassumptions.Thethirdchapter discussesmolecularbiology,theeldofstudythatdescribesthescaleandconditions insideacell.Inthesetwochapters,mathematicalmodelingisappliedtomolecular biology.Importantly,wemodelenzymekineticsandtheirroleinregulatorynetworks usingsystemsordinarydi!erentialequations.Withthisfoundation,thefourthchapterintroducesthetryptophanoperonanditsmodel,alongwiththemodel'shistory ofincreasingaccuracy.Inthenalchapter,wepresenttheanalyticaltechniquesand theirresultsforthetryptophanoperon.Thisthesisconcludeswithadiscussionand directionsforfuturework. 4

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CHAPTER2 MathModeling 1.Introduction Wewillbeinterestedinmodelingphysicalsystemswhichchangeovertime.Our fundamentaltoolwillbeordinarydi! erentialequations,thatisexpressionoftheform d dt ( X )= f ( X ( k,t ) ,k,t )(1) Where X ( k,t )isthequantitybeingsolved, k isaparameterwhichoccursinthe model, t istimeand f ( X ( t ) ,k,t )providesthedynamicsofthemodel.Wewillbe particularlyinterestedinstudyinghow X dependsontheparameter k ;di!erencesin this k valueleadstodi!erencesinthefunction X ( t )(seeFigure1).Thedi!erence betweenthevaluesof X ateach t ,fordi!erentvaluesof k ,canbecomputedinorder todeterminethemagnitudeduetothee!ectofthechangein k .Thecorresponding di !erenceapproximatesaderivative,willplayanimportantroleinthesensitivity analysis. Asanexample,consideranthranilatesynthaseconcentration[ E ]inFigure1for di !erentvaluesof d ,themaximumtryptophanuptakerate.Noticethatneartime 40,thetwocurvesintersectandthesensitivity E d ( t )switchessigns.Thispoint correspondstotheoperonswitchingfromnonproductivetoproductive.Thedi! erence betweenthesetwofunctionsisadiscretewaytomeasuresensitivity.Thecomputed sensitivitycurvecanbefoundatthebottomofFigure8.Detail,suchasthis,islost duringthecalculationofthetotalsensitivityscorefoundinTable1aswellasany methodthatproducesanumberbyaveragingovertime. 5

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! "! #!! #"! $!! !%" # #%" $ $%" & &%" '(#! ) *+ "! #!! #"! $!! !%$ !%) !%, !%# '(#! & .+ "! # !! # "! $ !! !%# !%$ !%& !%) !%" !%, !%/ 0( "! # !! # "! $ !! #! #" $! $" &! 1( Figure1. Concentrationsofanthranilatesynthasefortwovaluesfor theexternaltryptophanuptakeparameterd.Forthebluecurve,d istheestimatedvalueandtheredcurvehadexternaltryptophanuptakeincreased10%.Concentrationismeasuredin molsandtimeis measuredinminutes. 2.MassActionKinetics ThelawofMassActionstatesthatinawellmixedsolution,chemicalsexpressan a"nitynotonlybasedontheircompositionbutalsotheirconcentration[ 7 ].There aretwotermsinthismodel.Theconcentrationofthemoleculeandthereactionrate, whichisacombinationofnecessaryactivationenergyandthemolecularcollisionrate. Thismodelarisesfromdi! usion,thenaturalmovementfromhighconcentrationto lowconcentration.Thatis,thechangeinconcentrationdependsontheconcentration itself.Thisrelationshipdescribesanordinarydi!erentialequation,whichwecanthus 6

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usetomodelchemicalreactionsingeneralandenzymekinematicsspecically.We dosoasfollows. Thechangeinaconcentrationisthedi!erencebetweentheproductionandconsumptionofthechemicalasdenotedinEquation3.Bothproductionandconsumption aremodeledwiththelawofMassAction;theyareproportionaltotheproductof relevantconcentrations.Takeforexampletwomolecules,AandB,thatcombineto formaproductC.Inthelanguageofmassactionkinetics,Equation3isasecondorderequationbecausethereactionincreasesexponentiallywiththeproduct[ A ][ B ],as both[ A ]and[ B ]increase.Arstorderchemicalreactioncouldincreaselinearlyonly withanincreaseinasingleconcentration.Notethatbracketsdenoteconcentration andthatparameters k arerateconstants. A + B C (2) WecanwriteanequationsothatthechangeintheconcentrationofCisproportionaltoboththeconcentrationsofAandB,weightedbytheirreactionrates.C degradesatsomerate k 2 ,i.e.consumption. d [ C ] dt = k 1 [ A ][ B ] k 2 [ C ](3) Massactionkineticsreferstothesolutionnotbeingatasteadystate.Tostudy dynamics,weusuallybeginbyndingthesteadystate.Thesteadystatecanbe foundbysettingtheratetozero,assumingthereisnochangeintheconcentration. [ A ]+[ C ]= A 0 (4) FromEquation2andtheprinciplesofmassactionkinetics,onlyoneunitofAis neededtoproduceoneunitofC.Thus,thereisaconservationrelationshipstating [ A ]+[ C ]doesnotchange(seeEquation4).Wecansimplifythesystemandcompute 7

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steadystateswiththeconservationequation,steadystateassumptionandmodeling Equation3.Since d [ C ] dt =0,Equation3becomes k 1 [ A ][ B ]= k 2 [ C ](5) CombiningEquations4and5gives [ C ]= k 1 k 2 [ A ][ B ]= k 1 k 2 ( A 0 [ C ])[ B ]](6) k 1 k 2 [ B ][ C ]+[ C ]= k 1 k 2 A 0 [ B ](7) [ C ]= A 0 [ B ] K eq +[ B ] (8) Where[ C ]isatthesteadystateand K eq = k 2 k 1 3.StabilityAnalysis Thestabilityofsolutionscanbedeterministicallyelucidatedfromthemodeling equations.Whethervariableswilltendtooscillate,gotozeroorsomeotherlimit,or growindenitely,canallbecalculated.InEquation8, K eq istheequilibriumrate,in thiscasearatioofthetworateconstants.Consistentwithintuition,thedynamics arestablewhentheconcentrationsarebalancedrespectivetotheirreactionrates. Forexample, if K eq ismuchlessthan[B],thesolutionwillbelargelyproduct, if K eq =[B]wouldindicatethathalfofthesubstratehasbeencatalyzedinto product, andif K eq ismuchlargerthan[B],thereversereactiondominatesthesystem andthesolutionremaininglargelyunreactedsubstrate. Thesteadystateassumptionallowsustosimplifythesystembyadimension. Investigationsofsteadystatesallowustocomparereactionratesinasystem.The veryfastreactionscanbeassumedtohaveachievedsteadystateandtheirderivatives arethenzero[ 26 ].Ifavariableconcentrationistakentobeinthequasisteady 8

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stateandit'sequationsettozero,themodelisreducedbyonevariable.Thisis animportantassumptionformanymodelsandallowsustomodelsomethingas interrelatedascellularmetabolismwithreasonableaccuracy[ 25 ]. 4.EnzymeKinetics Enzymaticreactionshaveimportantcharacteristicsthatmakethemdistinctfrom otherchemicalreactions.Theircatalyticinuence,specicityandpreciseregulation makeenzymessuitableforcellularmachinery[ 11 ].Massactionkineticsneedto bemodiedtoaccuratelymodelenzymekinetics.Importantly,enzymesarenot consumedinthereactionstheycatalyze.Themassactionrelationshipimpliesthat increasesinconcentrationsleadtoincreasesinreactionrates,wheninrealityenzymes haveamaximumrate, V max .Thereactionvelocityistherateatwhichproductis created.Theenzymeconcentrationwillappearonbothsidesoftheequationinorder toconserveitstotalconcentration.Theconservationofenzymeconcentrationaswell astheconservationofmassplaysimportantrolestheanalysisofthemodelforcellular regulatoryfunctions. TheMichaelis-Mentenmodelofanenzymaticreactiondemonstratesbasicobservedbehavior[ 15 ].Thesimplestmodeltakesonesubstratethatisconvertedinto aproductbyoneenzyme.Thenewdynamicarisesfromthemiddletermwhichrepresentsthesubstrate-enzymecomplexandfromtheone-wayreactionfor[ P ].Thisis consideredone-waybecauseproductsareremovedfromthesolutionbeingmodeled (Keenernotesthatthisisnotthecase invivo [ 11 ]).Equation13istheratelimiting step.Nomattertheconcentrationofsubstrate,onlythoseboundtothexedsmall numberofenzymesin[ C ]canbeconvertedintoproduct. S + E k 1 k 2 C k 3 P + E (9) 9

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Noticetherearefourequationsdeningthefourvariables'derivatives. d [ S ] dt = k 2 [ C ] k 1 [ S ][ E ](10) d [ E ] dt =( k 2 + k 3 )[ C ] k 1 [ S ][ E ](11) d [ C ] dt = k 1 [ S ][ E ] ( k 3 + k 2 )[ C ](12) d [ P ] dt = k 3 [ C ](13) Thereareanumberofsimplifyingassumptions. Thesumofsubstratesandproductsdoesnotchangeandequalstheinitial concentrationofsubstrate. Thesumofenzymeandsubstrate-enzymecomplexconcentrationdoesnot changeandequalstheinitialconcentrationofenzymes. Themaximumconcentrationofenzyme E 0 istakentobemuchsmallerthan theconcentrationofsubstrate[ S ]. Theaboveassumptioncontributestothequasi-steadystateapproximation thattheconcentrationofsubstrate-enzymecomplexremainsconstantbecausealltheenzymesareworkingatcapacity. TheresultingexponentialcurveisdenedbytheMichaelis-Mentenequation.AlthoughtheworkofMichaelisandMentenwasfoundational,thisequationwasderived in1925byBriggsandHaldane[ 4 ].Thisimportantfunctionisusedtodeneterms suchasmaximumreactionrate,denedbythelimitingstepinEquation13,andcan befoundinFigure2.Matlabplotsoftimeseriessolutionsdemonstratethelimiting natureoftheserelationshipsintheasymptoticsteadystateconcentration.ExamplesoftheseequationscanbefoundinAppendixA.1whichmodelprocessessuchas tryptophanconsumptionby E.coli Considerationsformodelinganenzymaticreactionmustalsoincludetheinuence ofothermolecules;whethercompetitiveorcooperative.Feedbackloopsareamajor 10

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Figure2. Graphofthereactionvelocityforthemodelgivenby MichaelisandMenten.Whentheconcentrationofsubstrateequals K M ,thereactionvelocityisonehalfofmaximum[ 4 ]. regulatorymechanismincellularreactions[ 5 ].Whentheproductionofachemical stimulatesfurtherproduction,thecircuitissaidtohavepositivefeedback.This changeindynamicsleadstotheHillequation,afoundationalmodelinmolecular biology. TheQuasi-SteadyStateHypothesisstatesthatwecanreasonablyexaminethe dynamicsoftheslowerreactionsbyassumingthefasterreactionsareatequilibrium. Inthismodel,theassociationofsubstratetoenzymehappensquicklyandthetotalnumberofenzyme-substratecomplexesremainslargelyconstantafterveryearly time[ 26 ].Thesigmoidalcurveproducedbythisfunctionaccuratelymodelsmoleculara"nityinmanybiochemicalreactionsandcanbebestseeninthegraphof anthranilatesynthase(seeFigure1)[ 5 ]. Repeatingthealgorithmfromtheprevioussectiononsteadystates,wecancompute[ C ]andthemaximumreactionrate.Assuming d [ C ] dt =0,Equation12becomes ( k 3 + k 2 )[ C ]= k 1 [ S ][ E ](14) Recallthat[ E ]+[ C ]=[ E 0 ]andlet K M = k 3 + k 2 k 1 K M [ C ]=([ E 0 ] [ C ])[ S ](15) 11

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[ C ][ S ]+ K M [ C ]=[ E 0 ][ S ](16) [ C ]= [ E 0 ][ S ] [ S ]+ K M (17) SincetheratelimitingstepisgivenbyEquation13,wewrite v = d [ P ] dt = k 3 [ C ](18) v = k 3 [ C ]= k 3 [ E 0 ][ S ] [ S ]+ K M (19) Notethemaximumreactionratedependsontheinitialenzymeconcentrationsbecauseitlimitsthemaximumenzyme-substratecomplexthatcanbeformed.These dynamicsarearestatementofEquation12.Oncewehavethesteadystateapproximation,theMichaelis-Mentenequationforreactionvelocityforming[ P ]follows[ 7 ]. V max = k 3 [ E 0 ](20) v = V max [ S ] [ S ]+ K M (21) ThedynamicsofthisfunctioncanbeseeninFigure2andmodelsmanyofthe reactionsinthemolecularbiology.Bothtryptophanuptakefromtheenvironmentand tryptophanconsumptionbyproteinsynthesisin E.coli aremodeledwithMichaelisMentenequationsinfunctions F ( T,T ext )and G ( T )foundinAppendixA.1.Inthese equations, V max isgivenbytheparameters d and g respectively. 12

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CHAPTER3 CellBiology Afundamentalstructureoflife,cellsaresmall(nm)self-reproducinglipidbilayer sacs[ 16 ].Singlecelledorganismswillbethesubjectofdiscussion,howevermore complexorganisms,includinghumans,regulategeneticexpressioninsimilarways. Thesimilarityhasbeencloseenoughtoallowinsightintohumandiseases. Escherichiacoli ,atube-shapedbacterium,willbethefocusofthisstudy.Like otherprokaryotecells,geneticmaterialisnotcontainedinthenucleus;ratheritoats intheunicellularorganism'scytoplasm[ 16 ].Commonlyfoundinanimalintestines andaround2micrometerslong, E.coli iswellstudiedintermsofgenetics.In addition,muchhasbeendonebywayofthestudyof E.coli cellregulation. 1.EnzymeKineticsandBiochemistry Thecentraldogmaofmolecularbiologydescribestheowofinformationfrom geneticmaterialtocellularstructures.InlivingcellsDNAistranscribedintomRNA inaprocesscalledtranscription.Theshortlifespanofmessenger'RNAmakesit suitablefortransmittingasignalratherthanstoringinformation.Inaprocesscalled translationmRNAisencodedintoaminoacidchains[ 11 ]. DNA mRNA Protein Tryptophanisoneofthe20aminoacidsthatmayoccurinaproteinsequence.One ormoreoftheseaminoacidchainsarefolded,aprocesswhichisnotwellunderstood, intoproteins.Proteinsareinvolvedinmanycellfunctions;composingmanythings fromcellstructurestoenzymesthatcatalyzechemicalreactions.Inmostofthese cases,theshapeandtoftheproteinandothermoleculesgivestheproteinit'sunique 13

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abilities.Conformationchangesinthestructurecanleadtootherchangessuchas enablingabindingsite,asisthecaseforinhibitionofanthranilatesynthase.Proteins composeenzymesthatcanfacilitatechemicalreactions.Someproteinscan,like phosphatase,removephosphatesfromcellularstructures,rendingtheminanactive orinactivestate.Otherenzymesfosterthecovalentbondstosynthesizenecessary biochemicals. Unassistedchemicalreactionsoccurattooslowandtoorandomapacetosupportlife.Incells,enzymesaresynthesizedandusedtoregulatechemicalreactions. Enzymesareproteinsthatlowertheactivationenergyrequiredforachemicalreaction.Therearemanymechanismsthroughwhichanenzymeaccomplishesthis, includingbringingtogethertwosubstratestoformaproductwithreactionsitesthat attracttheproductsmorestronglythantheproductsattracteachother.Theyare physicallyinvolvedinthereactionandarethuslimitedinthescopeoftheire!ect. EnzymaticreactionswillbedescribedusingMassActionKineticsfromthepreceding chapter[ 11 ]. 2.TranscriptionalLevelControlofGeneExpression JacobandMonodsoughttoaddressthequestionofhowmicrobiallifeisable torespondsoquicklytoenvironmentalchanges.Theyproposedtheresponsewas duetotheexpressionofgenescontrolledbyasiteontheDNA[ 10 ].Theideaof geneticregulationbeinglocaltotheDNAstrandwasinitiallycontroversial[ 29 ]. Experimentationinprokaryotesandlatereukaryotesconrmedtheexistenceand importanceoftheirhypothesizedgeneticswitches.Fortheirworkontheoperon, theywereawardedtheNobelPrizein1965[ 20 ]. Therearemanyregulatorynetworkswithinacellusingacombinationofprotein inhibitionandsignalmolecules.Thephosphorylationofenzymes,whichinhibitsor inducestheiractivity,isperhapsthemostcommonscheme.Powerfulregulatorytools atthecell'sdisposallieatthegeneticlevel.Decidingwhichproteinsareexpressed 14

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ultimatelycanleadtocellspecializationandotherformsofgeneexpression.ThenetworkofphysicalandchemicalreactionstranscribesDNAandtranslatestheresulting mRNAintoproteinsatvaryingfrequencesbasedonsignals,suchastheconcentration oftryptophanintheoperondescribedlater.Thiscanberegulateddirectlyonthe DNAbutcontrolisfurthermodulatedbythedegradationofmRNA. Anexampleoftranscriptionallevelcontrolistranscriptionalattenuation.Transcriptionalattenuationistheabruptterminationoftranscriptiondependingonthe relativespeedoftranslationandtranscriptiontoeachother.Becausethisprocess reliesonthemRNAbeingbothtranslatedandtranscribedsimultaneously,itisonly possibleinprokaryoticcells(eukaryoticcellshaveanuclearenvelopeseparatingthese processes).Thisregulationtakesadvantageofasinglenucleotidestrand'sa" nityfor complementarystrands.ThenascentmRNAproducedbyamRNApolymeraseand stillexposedbytheribosomewillfoldoverandformnon-covalentbondsbetween complementarysectionsashairpinloops[ 30 ].Translationrequiresaminoacidsto synthesizeproteins,andtheribosomecannotmoveforwardwithoutthecoded-for aminoacid.ThisdelayallowstherstandsecondsectionsofthemRNAtoformthe hairpinstructure,whichwouldsignalacontinuationoftranscription[ 32 ].However, ifthetheribosomeisabletomoveovertheearliersectionsofmRNAthenthelatter sectionsofthestrandwillformadi!erenthairpinstructurenearertothepolymerase. ThisstructurecrowbarsthepolymeraseandunnishedmRNAstrandawayfromthe DNA,endingtranscription[ 30 ]. Anotherexampleoftranscriptionallevelcontrolisthegeneticswitch.Genetic switchesaresequencesofDNA,orgenes,whichturnonoro!theexpressionofother protein-encodinggenes.Positivelycontrolledgeneticswitchesareactivatedwhenan activatorproteinbindstothepromotorDNAregionandencouragestranscription initiation.Onceapromotorbindswiththeactivatorprotein,RNApolymeraseis abletoopenandtranscribetheoperon.RibosomestranslatetheresultingRNAinto aminoacidchainsthatarefoldedintoproteins. 15

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involveseveralsequentialstagesorevents.Thestagesused inattenuationregulationofthe trp operonof E.coli aredescribedinFigure5(LandickandYanofsky1987; Yanofsky2000).Anessentialfeatureofthisattenuation mechanismisthesynchronizationoftranslationofa 14-residueleaderpeptidecodingregion, trpL ,withtranscriptionoftheoperon'sleaderregion.Synchronization isachievedbyexploitingfeaturesoftheinitialsegmentof theleadertranscript,thesegmentoverlapping trpL .This segmentcanformanRNAhairpinstructure,designated hairpin12,calledtheanti-antiterminator.Hairpin1:2 alsoservesasatranscriptionpausesignal(seeFigs.4,5). Transcriptionalpausingisrelievedwhenaribosomebinds atthe trpL mRNAstartcodonandinitiatessynthesisof theTrpLleaderpeptide.Themovingribosomeappearsto disrupttheRNApausehairpin,releasingthepausedRNA polymerase(Fig.5,Stage1).Subsequently,eitheroftwo eventsoccurs,dependingontheavailabilityofuncharged versuschargedtRNA Trp .Whenmostofthecellular tRNA Trp isuncharged,difficultyintranslatingthetwo Trpcodonsof trpL mRNAresultsinribosomestallingat oneoftheseTrpcodons(Figs.4,5,Stage2b).Thisallows theantiterminatorstructure,hairpin23,toform(Fig.4), whichpreventsformationoftheterminatorstructure. Preventionofterminatorformationallowstranscription tocontinueintothestructuralgenesoftheoperon(Fig.5, Stage2b).WhenchargedtRNA Trp isplentiful,however, translationof trpL iscompleted,andthetranslatingribosomedissociatesatthe trpL stopcodon.Thispermitsthe leadertranscripttofoldandformtheanti-antiterminator andterminatorstructures,1:2and3:4(Fig.4),promoting transcriptiontermination(Fig.5,Stage2a).Thus,dependingontheavailabilityofchargedtRNA Trp duringtranslationof trpL ,transcriptionofthestructuralgeneregionof the trp operonwillorwillnotproceed. Manyaminoacidbiosyntheticoperons ofGram-negativebacterialspeciesare regulatedbysimilarribosome-mediated transcriptionattenuationmechanisms (Yanofsky1981).The uniquedistinguishingfeatureofeachistheinclusion ofcodonsfortherespectiveaminoacid inthecorrespondingleaderpeptide codingregion. Regulatorysubtletiesabound,however,thereforeeacheventateach stageisnotabsolute.Otherregulatory processesinfluence trp mRNAsynthesis andsurvival, trp codingregiontranslation, trp enzymefunctionandturnover,and trp enzymeactivity(Landick andYanofsky1987;Yanofskyand Crawford1987).Forexample,andmost importantly,theactivityofthefirst enzymeofthetryptophanbiosynthetic pathway,anthranilatesynthase,issubjecttofeedbackinhibitionbyLtryptophan.Thisistypicalofthefirst enzymeofmanybiosyntheticpathways. Feedbackinhibitionisaphysiologically advantageousprocessbecauseitallows aninstantaneousandreadilyreversiblereductionintheflowofcarbon andnitrogenintoapathway. trp operonorganization andregulationin B.subtilis Thegenesofthe trp operonofthis organismareorganizeddifferentlythan in E.coli (Fig.6;HennerandYanofsky FIGURE5. Thesequential,alternativeeventsregulatingtranscriptionterminationinthe leaderregionofthe trp operonof E.coli .Stage1:TheRNApolymerasemoleculethatinitiates transcriptionofthe trp operonpausesaftersynthesizingtheinitialsegmentofthetranscriptthesegmentthatformstheanti-antiterminatorpausestructure(LandickandYanofsky 1987;Yanofsky2004).Whilethepolymeraseispaused,aribosomebindsatthe trpL mRNA startcodonandinitiatessynthesisoftheleaderpeptide.Thistranslatingribosomethen disruptstheanti-antiterminatorpausestructure,releasingthepausedpolymeraseandallowing ittoresumetranscription.Stage2a:WhenthereissufficientchargedtRNA Trp inthecellto allowrapidcompletionofsynthesisoftheleaderpeptide,thetranslatingribosomeisreleased. Theanti-antiterminatorandterminatorstructuresthenform,promotingtranscription termination.Stage2b:WhenthereisadeficiencyofchargedtRNA Trp ,theribosometranslating trpL mRNAstallsatoneofitstwoTrpcodons.ThispermitstheRNAantiterminatorstructure toform,whichpreventsformationoftheterminator.Transcriptionthencontinuesintothe operon'sstructuralgenes.(ModifiedfromFig.2inYanofsky2004andreprintedwith permissionfromElsevier 2004.) Yanofsky 1144 RNA,Vol.13,No.8 Figure3. Twodi!erentwaysmRNAcanfold,where2aterminates transcriptionand2ballowsittocontinue[ 32 ]. 3.Operons Acommonandpowerfulschemeusingageneticswitchistheoperon.Oneinstanceofanoperonmaintainsahealthyleveloftryptophanandtryptophansynthesis withinan E.coli cell.Theoperonregulatorynetworkincludesasetofgeneswhich shareacommonoperatorbindingsite.Theoperatorisabindingsiteforaregulatory proteinthatblocksmRNAproduction,howeveroperonscanbeinducedaswellas repressed[ 25 ].Operonsareregulatedbyactivatorandrepressorenzymesbinding totheoperator,eitherinthepresenceorabsenceofaligand.Itisconsideredinducibleifthepresenceofaligandencouragestranscription,andrepressibleoperons arecharacterizedbyaligandwhichinhibitstranscription. Oftenoperonstatesdependonenvironmentalconditions.Forexample,anabundantproteincantriggerageneticswitchtoblockthetranscriptionofthatprotein's gene.Throughfeedback,theoperatorpreventsRNApolymerasefrombindingtothe 16

PAGE 24

biosynthesisisabiologicallyexpensive,complicatedprocess.Infact,theproductsoffourotherpathwaysare essentialcontributorsofcarbonornitrogenduringtryptophanformation(YanofskyandCrawford1987;Yanofsky etal.1999).Thus,theprincipalpathwayprecursor, chorismate,isalsotheprecursoroftheotheraromatic aminoacids,phenylalanineandtyrosine,aswellasserving astheprecursorofp-aminobenzoicacid andseveralothermetabolites.Inaddition,glutamine,phosphoribosylpyrophosphate,andL-serinecontribute nitrogenand/orcarbonduringtryptophanformation.Thus,eachorganism withtryptophan-synthesizingcapacity musthaveadoptedappropriateregulatorystrategiestoensurethatsufficientlevelsofchorismateandthese otherthreecompoundsareproduced. Inmanyorganismstryptophanserves astheprecursorofotherbiologically essentialcompounds,i.e.,niacinin mosteukaryotes,indoleaceticacidin mostplants,andindoleinmanybacteria.Thustheregulatorystrategies designedforthegenesoftryptophan biosynthesisofeachorganismhavehad tobecompatiblewithothermetabolic objectives.Aninterestingfeatureof tryptophanbiosynthesisisthatthis capabilitywasdispensedwithwhen organismsevolvedthatwerecapableof obtainingtryptophanbyfeedingon otherorganisms.Nevertheless,as mentioned,productsoftryptophan degradation/metabolismareessential intheseorganisms. Regulatorymechanismscontrolling transcriptionofthe trp operon of E.coli Thegenesrequiredfortryptophanbiosynthesisin Escherichiacoli areorganizedasasingletranscriptionalunit, the trp operon(Fig.2;Yanofskyand Crawford1987).Thisoperonhasa singlemajorpromoteratwhichtranscriptioninitiationisregulatedbya DNA-bindingprotein,theL-tryptophan-activated trp repressor(Yanofsky andCrawford1987).Thisrepressoracts bybindingatoneormoreofthree operatorsiteslocatedinthe trp operon's promoterregion(Fig.2;Lawsonetal. 2004).Thestructuresoftheinactive trp aporepressor,the tryptophan-activated trp repressor,andthe trp repressor operatorcomplexhaveallbeendetermined,andthis repressor'smechanismofactioniswellunderstood(Fig.3; Otwinowskietal.1988;JoachimiakandZhang1989; Shakkedetal.1994;Gryketal.1996).Theaporepressorand repressoraredimerscomposedof identicalhelix-turn-helix FIGURE1. Thegenes,enzymes,andreactionsofthetryptophanbiosyntheticpathway. Thesevengenes,orgeneticsegments,sevenenzymes,orenzymedomains,andsevenreactions,involvedintryptophanformationareshown(YanofskyandCrawford1987).Only oneofthereactionsisreversible.Theproductsoffourotherpathwayscontributecarbon and/ornitrogenduringtryptophanformation.Twoofthetryptophanpathwayenzymes oftenfunctionaspolypeptidecomplexes:anthranilatesynthase,consistingoftheTrpG andTrpEpolypeptides,andtryptophansynthase,consistingoftheTrpBandTrpA polypeptides. FIGURE2. Organizationofthe trp operonof E.coli .Thegenesof E.coli requiredfor tryptophanbiosynthesisfromchorismateareorganizedinasingleoperon,ortranscriptional unit(YanofskyandCrawford1987;Yanofsky2004).Twopairsofgenesarefused: trpG and trpD ,and trpC and trpF .Thestructuresofthesetwobifunctionalpolypeptidesareknown, andseparatepolypeptidesegmentsareconcernedwithcatalysisofeachreaction.Therelative orderofthesevengeneticsegments, trpEGDFCBA ,correspondsroughlytotherelativeorderof therespectiveenzymaticreactions.The trp operon'sregulatoryregion,locatedatthebeginning oftheoperon,isdesignedtosensetwosignals,L-tryptophan,andchargedvs.uncharged tRNA Trp (LandickandYanofsky1987;Yanofsky2004).Tryptophan,wheninexcess,activates the trp aporepressor,whilechargedandunchargedtRNA Trp determinewhethertranscription willorwillnotbeterminatedintheoperon'sleaderregion.Apoorlyexpressedinternal promoterprovidestranscriptsproducinglowlevelsofthelastfewenzymesofthepathway. Thispromoterisusefulwhentheprincipalpromoteristurnedoff.(p=promoter;t= terminator).(ModifiedfromFigs.1and2inYanofsky2004andreprintedwithpermission fromElsevier 2004.) Yanofsky 1142 RNA,Vol.13,No.8 Figure4. Thedi! erentgeneticelementsthatcomposeanoperon, includingtherepressorbindingsite,leadersequenceandproteincoding genes[ 32 ]. operon.UnabletotranscribetheRNA,thecell'sribosomesareunabletobindor translatethesequencesintotheaminoacidchainsthatformproteins.Thisregulatory schemefreesupresourcesforothercellfunctions.Aninducibleoperonisnaturally inhibiteduntilaproteinorothermoleculeallowsthegenetobeexpressed,causing acascadethroughthemetabolicpathway.Arepressibleoperonisexpresseduntila repressormoleculebindstotheoperonandhaltstranscription;arepressoristhereby atranscriptionfactor. Thebacteria E.coli hasagenomeconsistingofonecircularstrandofDNAencodingroughly4300proteins[ 19 ].However,noteverygeneshouldbeexpressedat alltimesandgeneticswitcheshelpfacilitatehealthycellularlife. E.coli 'sgenetic switchesaresimilartootherprokaryoticcells;theymostlydependonsurrounding resourcesandingeneralusethefewrepressionschemesdescribedhere. Transcriptioncanbeinhibitedbyarepressorproteinboundtotheoperatorand thisiscallednegativecontrol.TherepressorphysicallyblocksRNApolymerasefrom bindingtothepromotorandthuspreventsproteinsynthesis(seeFigure5foran excellentpicture).Becausethisconformationalchangeinhibitstheproductionof tryptophan,thisanegativelycontrolledrepressibleoperon[ 32 ]. 17

PAGE 25

CHAPTER4 TheTryptophanOperon Awellstudiedoperonoccurringin E.coli ,thetryptophanoperonhasbeenmodeledsuccessfully[ 23 ].Tryptophanisoneofthetwentynaturallyoccuringaminoacids andisnecessarytomakesomeproteins,whicharechainsofaminoacids.Techniques foraccuratelymeasuringtryptophanincellcultureshavebeenaroundfordecades[ 6 ]. Thesystemhastwoprimarystatesthatitwillgravitatetowardsdependingonthe initialenvironmentalconditions.Inanenvironmentdevoidoftryptophan,theoperatorbindingsiteremainsunblockedandpolymerasemaybindtoproducemRNA.The concentrationoftryptophanlevelsoutasproductionincreases,oncethereisenough tobindandactivatetherepressor.Oncebound,therepressoroccupiestheoperator bindingsiteanddownregulatesthegeneticexpression,andtherebyproduction,of tryptophan-synthesizingproteins.Thisstableconcentrationoftryptophanistherst steadystate.Thealternativeisasystemwhichbeginswithtryptophanintheenvironment.TryptophanuptakewillpreventthetrpRfromfallingintoitsaporepressor state.Ifthisenvironmentaltryptophanisfullyconsumedthenthecorepressorswill disassociatefromtheDNA,restartingtranscription. In E.coli ,thetryptophanrepressorisahomodimerwhichusesahelix-turn-helix motif[ 32 ].Thebindingoftwotryptophanmoleculestiltsthesestructuressothatthe repressorproteinmaysitproperlyintheDNA'smajorgroove.Thetwoconformations andtherepressor'stwithDNAarerenderedinFigure5.Tryptophanisconsidered theco-repressorinthismechanism.Aregulatorygeneencodestherepressorprotein andnearby,thereisabindingsitefortherepressoronthechromosome.Therepressor trpRactuallyrepressesit'sowntranscriptionaswellonlywhentryptophanispresent, 18

PAGE 26

Figure5. Renderingofthetryptophanoperonrepressor,trpR,byDr. AndrzejJoachimiak[ 32 ].Whentwomoleculesoftryptophanbindto trpR(center),thecomplexundergoesanallosterictransitionsuchthat ittsintheDNA'smajorgroove(right) forthesamephysicalreasons.Thissavesthecellforwastingaminoacidsona repressorthatisnotneeded,asimilaradvantagetorepressingthetryptophanoperon. Therearetwootherfeedbackschemesintheoperon.First,anthranilatesynthasecanbepreventedfromcatalyzingtryptophansynthesisbybindingtopresent tryptophanmolecules.Thesecondtryptophan-sensingcomponentistranscriptional attenuation.Recall,intranscriptionalattenuationthemRNAstrandproducedby thetrpgenecanformregulatorystructuresasitispolymerized[ 34 ].Theleader sequencecodesfortwotryptophanaminoacidsinarow[ 35 ].Ifaribosomeisable tondthenecessarytryptophanforthisleadersequence,thenastructurecanform thatleadstorapiddisassociationofthemRNApolymerasefromtheDNAandRNA. Iftheribosomeisstalledbyalackoftryptophan,thenitphysicallypreventsthis terminatorstructurefromformingandtranscriptionnishesnormally. 1.TheModel Ultimatelythemodelshouldbeunderstoodasaswitch.Theswitchtakesas inputtheexternaltryptophanconcentrationwhilethelevelofenzymeactivitycan 19

PAGE 27

beconsideredtheoutput.Eitherenzymesareproducingtryptophanortheyarenot, dependingonthestateofthegeneticswitch. FollowingtheworkofMichealisandMenten,theoperonwasdescribedbyJacob andMonodin1960[ 29 ].TheirconceptwasasinglesiteonDNAthatwouldregulate theexpressionofanearbygene.Mathematicalrepresentationsweredevelopedby GoodwinandGri"thby1965.TheGoodwinmodelhadthreevariables,mRNA, EnzymeandTryptophanconcentrationsandreliedheavilyonHillfunctions.This modeldidnothavetranscriptionalattenuationortimedelaysbutdidsetthestage forfurtherwork.In1982Blisspublishedanupdatedmodelincludingdelaysand manyoftheparametersusedinthecurrentmodel[ 5 ].Bliss'parameterestimations andsimplifyingassumptionssetprecedentsthathavepersistedfordecades[ 14 ].DuringthetimeBlissdevotedtothemathematicalmodel,Yanofskywasinvestigating thebiologyoftryptophanregulationanddiscoveredregulatoryschemes.Yanofsky succeededinsequencingthetryptophanoperonandprovidinggreatinsightonthe functioningoftranscriptionalattenuation[ 34 ]. WithYanofsky'sdiscoveries,Sinhapublishedarenedmodelin1988[ 28 ].This modelhadappropriateparametersforthedimerrepressoractivationbutlackedattenuationanddelayfactors[ 14 ].Themostdevelopedmodelwasrstpublishedin 2000byMackeyandSantillan,includingdelayequationsandallthreeregulatory schemes:enzymeinhibition,transcriptionalattenuationandoperonrepression[ 25 ]. 2.TheMackey-SantillanModel TheMackey-Santillanmodelissimpliedherebysettingthedelaytermstozero. Despitethelossofdetail,majordynamicsarepreserved.Otherparametershave beenexperimentallyestimatedorchosenfortheirreasonablerepresentationofthe metabolicnetwork.Theyestimateanaverageof1.6genomespercelltoaccountfor cellsthatareintheDNAreproductionorGap2phaseofthecellcycle[ 25 ]. 20

PAGE 28

Theimportantvariablesunderconsiderationthenaretheconcentrationsoffour things;operonbindingsites,mRNA,catalyzingenzymeandtryptophan[ 14 ].The modelcanbefoundinAppendixA.1.Theseareindividuallyimportant;evenafter theoperonhasbeenrepressed,themRNAmuststillbedegraded.Thecatalyzing enzymecancontinuetoproducetryptophanafterthemRNAhasbeendegraded. InEquation28,thethirdtermrelatestothechangeinconditionfromtryptophanrichmediatotryptophanstarvation.When T ext =0,theuptaketermalsoequals zero.Theremainingterms,synthesis,consumptionanddilutioncontroltryptophan dynamicsinthemodeledconditionoftryptophanstarvation.Bothoperonsand mRNAconcentrationsaretakenastheconcentrationoffreebindingsites. Importantquasi-steadystateassumptionsincludetheconcentrationoftrpRrepressor,mRNApolymeraseconcentration,ribosomalconcentrationandotherindependentlyregulatedmolecules[ 14 ].Thesevaluesareconsideredconstantbecause theirreactionsoccurveryquicklyrelativetotherestofthesystem.Theregulation oftrpRisonaseparategenebasedonit'sownconcentration.Thisenzymeachieves asteadystatesquicklyandisnotdegraded,itcanbeconsideredconstant.Similarly mRNApolymeraseandribosomesareregulatedsuchthatthecellmaintainsaconsistentconcentrationandthistoocanbeconsideredconstantforsimplicity.Allof theparameterconcentrationsareestimatedforasinglecellaveraging8 x 10 16 liters involumeinaculturethatisgrowingexponentially.Asummaryofparametersand theirestimationsappearsinAppendixA. 21

PAGE 29

CHAPTER5 SensitivityAnalysis Manye! ortshavebeenmadetowardsanalyzingsystemsintermsofparameters. Thevariationofparameterscanilluminatehowsystemsaremaintainedandcan beperturbed.Forexample,thea"nityoftryptophantotherepressorproteinis moreimportantthantherateoftranslation.Whilesensitivedependenceoninitial conditionslendsitselfmoretowardschaoticstudy,thereisafundamentaldi! erence. Instudyingchaos,questionsaboutwhatvaluesforaparameterleadtophenomena, suchasbifurcation,areimportant.Insensitivityanalysis,questionsaboutwhat parametersleadtophenomena,suchasdown-regulationofproduction,areimportant. ApopularsensitivityscoringtoolistheFourieramplitudesensitivitytest(FAST) [ 22 ].FASTisavariancebasedalgorithmappropriateforglobalanalysis.Another globalanalysisapproachusesMonteCarlocalculations[ 24 ].Someattemptsinclude perturbingaparameterbyasetpercentageandcomputingthedi!erenceinsystem outputs.Thiscanberepeatedandcomparedbutishighlysensitiveitselftothe percentchange.Thistechniqueisalsotimeconsuminganduntilrecentlyprohibitively computationallyexpensive.Hornbergetalusedmeasuressuchasintegratedareaand duration[ 9 ].Thisapproachdoesnottakeintoaccountthedi!erencesinsensitivity overtime.Thereisalsoanincreasederrorassociatedwithdiscretelycalculated di!erencesasstand-insforcontinuousprocesses. Amoreprecisemethodusescalculustoproducesymbolicsensitivityequations wherethereisanequationforthederivativeofeachvariablewithrespecttoeach parameter[ 13 ].Usingthedenitionof f ( X,t )inEquation1, d dt ( X )= f ( X ( k,t ) ,k,t ) 22

PAGE 30

wecancomputethechangein X withrespectto k .InFigure1,youcanseethe di !erencebetweentwovaluesof X atanygiventfortwovaluesofaparameter.This di !erencecanbetakenovertimeandisconsideredthesensitivityfunction X k ( t ).Because X or k maybeverylargeorsmallcomparedwith X k ,normalizingtheresults areveryimportant.Asmallchangeinabignumbershouldhaveasmallersensitivityscorethanthesamesizechangeinaverysmallnumber.Normalizingpartial derivativeswithrespecttorelevantparameterallowsformeaningfulcomparisonof resultingsensitivityscores. Startingwitheachdi!erentialequationoccurringinthemodel,wetakethepartial derivativewithrespecttoaparameterk.TheresultingODEisfoundwithachange ofbaseonthelefthandsideoftheequationandachainruleexpansionontheright (both f and X arefunctionsof k ).Thisiscalledthetotalderivative[ 5 ].From Equation1wehave: d dt X k = f ( X,k ) X X k + f ( X,k ) k (22) ThesolutiontothesystemofODEsfromEquation22isthesensitivityof X with respectto k [ 13 ].Thuswehaveaformforthederivativeofthesensitivities,where d dt X k dependson X k .TheMaplecodeinAppendixB.1followstheabovealgorithm tosymbolicallyproduce d dt X k forevery X and k inthesystem. 1.ImplementationoftheAnalysis Therearetwoprogramsusedtosolvetwopartsoftheproblem.First,inMaplewe solvesymbolicallyforthenecessarysensitivitymatricesresultingfromEquation22. ThersttermrequiresaJacobianderivativetaken, f ( X,k ) X ,because f ( X,k )isaknown function.MapleisabletotaketheJacobianofthesystemaswellascomputethe partialderivativesrequiredforthesecondterm f ( X,k ) k .Theyarecombinedwiththe unknown MxN matrix S = X k usingsymbolicmatrixmultiplicationandaddition. Maplethenproducestheresultingequationsthatdescribethesensitivity.Generally, 23

PAGE 31

for N variablesand M parameters,thereare NxM sensitivityODEsthatmustbe solvedsimultaneouslywiththemodelODEs.Inthetryptophanoperon,thereare fourvariablesand24parameters,leadingto96resultingequations.Someofthese equationsareeverywherezero,implyingavariablemaynotbesensitivetoevery parameter. Empiricaldatafromculturesgrown invitro areusedtocalculateinputvalues[ 8 ]. UsingMatlab,weplugintheempiricalparametervaluesandinitialconditionsin ordertonumericallysolvetheequationsforcomparablesensitivityscores.TheMatlab sensitivitycoe"cientresultstelluswhichparametersthesystemismostsensitiveto. Matlabcanquicklyandeasilysolvefordi!erentinitialconditions.Thecomputer revolutionmakesthispossible;thedi!erentialequationsMatlabsolveswouldbe verydi"cultandtimeconsumingtosolvesymbolicallybyhand.Mapleiscrucial totheprocessconsideringtheonehundreddi!erentialequationsneedtobesolved simultaneously.Anyderivationerrorwouldthrowthesystemo!. 2.Results Thecomputermodelprovidedbothdynamicsofthesystemaswellassensitivity scoresforeachvariableandparameter.Graphswithequalizedaxeshelpdemonstrate thedi!erencesinsensitivity.Consistentlytranscriptionalattenuationisanimportant parameterforthetryptophanoperon.Thisisnotsurprisingconsideringearlymodels oftheoperonincludedtranscriptionalattenuationwhereasotherparameterscame lateraslessinuentialrenementsofthemodel[ 25 ]. Thereareseveralpossibleresultswecanexpectfromthesoftware.Thesoftware solvesasensitivityequationproducedbythealgorithmdescribedearlierinthischapter.Forinput,thesoftwarerequiresthemodeling dX dt equationsandestimationsfor theparameters.Thesoftwareproducesamatrixofallthesolution X ( t )functions,as wellasthesetof X k ( t )sensitivitysolutionfunctions,foragivenmodelofvariable X 's andparameters.Theoutput X k functionsdescribeaparametersinuenceovertime 24

PAGE 32

foraspeciedvariableandtherearentimesmequations.Thesefunctionsmaystart largeandshrinkquickly.Somesensitivitiesarepositiveinearlytimebutnegativein latertime.Otherpossibleresultsareconstantandzerofunctionsdescribingsensitivity.Thezerofunctioncanbeexpectedforcertainvariablesthatdonotdependon thespecicparameter. Inthetryptophanoperonmodelthereare4times24,equalto96,resultingsensitivityfunctions.YoucanseeinFigures6to9thatsomefunctionsarezeroandsome changesignandsomeareonlyimportantduringcertaintimes. NotablegraphsincludeseveralfortryptophanconcentrationinFigure9.Notice nearlyeveryparameterissensitiveinearlytime.Thisisthepointwhentheculture hasjustbeenwashedofitstryptophan-richmediaandisnowbeinggrowninastate oftryptophanstarvationforthersttime.Reactionsrelatedtotryptophanareat theirhighestduringthisphase. Thegraphof forTryptophan(seethesecondfrombottomrowofFigure9)is onlypositiveduringamiddleperiodandotherwisenegative,andlargelynegativein earlytime.Atrstalargegrowthratedilutesthetryptophanandcreatesincreased demand,butperhapslaterwhentryptophanproductionisuninhibitedthereisa certaineconomy-of-scalethathelpssupplementthedemandfortryptophan,however thisdecreaseslinearlyanddoesnotlastintothesteadystate. Besidesparameterb,themaximumchanceoftranscriptionalattenuation(discussedinmoredetailbelow),parametersganddbothhaveo! -the-chartsearlytime sensitivitiesfortryptophan.Theparametergrepresentsthemaximumtryptophan consumptionratebythecell,thenaturalusageoftheaminoacidintheproductionof proteinsforregularcellfunctioning.Whetherthecellconsumesit'sleftovertryptophanstoresquicklyorslowlycriticallyinuencestheentiremetabolicpathway.The parameterdrepresentsthemaximumtryptophanuptakerate,thatistheabilityof E.coli tomoveextracellularaminoacidsintothecellandtoretaintryptophanin thecellagainstthenaturalforceofdi!usion.Thisismostlyintheformofmembrane 25

PAGE 33

! "! #! $! %! &!! &"! &#! &$! &%! "!! # '(&! # )* &!! "!! + + (((,(((( &!! "!! + + (((-(((( &!! "!! + + (( (( &!! "!! + + ( ( &!! "!! + + ((. / 0(( &!! "!! + + ((1 2 ((( &!! "!! + + (((3(((( &!! "!! + + ((4 5 ((( &!! "!! + + (((6(((( &!! "!! + + (((*(((( &!! "!! + + (((()((( &!! "!! + + (. 7 ( &!! "!! + + (. 87 ( &!! "!! + + (. 9 ( &!! "!! + + (. 89 ( &!! "!! + + (. : ( &!! "!! + + (. 8: ( &!! "!! + + (((. ; (( &!! "!! + + &!! "!! + + (( ((( & !! !! + + (((<(((( & !! !! + + ((((5((( & !! !! + + (((/(((( & !! !! + + (((4(((( Figure6. Timeseriesandparametersensitivityfortheconcentration offreeoperatorbindingsites.Thetimeseriesatthetopisin mols andtimeisinminutes.Allsensitivitiesareunitless.Bluesensitivities exceededanabsolutevalueof1atanypoint. 26

PAGE 34

! "! #! $! %! &!! &"! &#! &$! &%! "!! !'( & )*&! + ,! &!! "!! ( ( ***.**** &!! "!! ( ( ***/**** &!! "!! ( ( ** ** &!! "!! ( ( &!! "!! ( ( **0 1 2** &!! "!! ( ( **3 4 *** &!! "!! ( ( ***5**** &!! "!! ( ( **6 7 *** &!! "!! ( ( ***8**** &!! "!! ( ( ***-**** &!! "!! ( ( ****9*** &!! "!! ( ( *0 : &!! "!! ( ( *0 ;: &!! "!! ( ( *0 < &!! "!! ( ( *0 ;< &!! "!! ( ( *0 = &!! "!! ( ( *0 ;= &!! "!! ( ( ***0 > ** &!! "!! ( ( 0 &!! "!! ( ( ** *** & !! !! ( ( ***?**** & !! !! ( ( ****7*** & !! !! ( ( ***1**** & !! !! ( ( ***6**** Figure7. Timeseriesandparametersensitivityfortheconcentration offreemRNAavailabletomRNApolymerase.Thetimeseriesatthe topisin molsandtimeisinminutes.Allsensitivitiesareunitless. Bluesensitivitiesexceededanabsolutevalueof1atanypoint. 27

PAGE 35

! "! #! $! %! &!! &"! &#! &$! &%! "!! !'( & )* &!! "!! ( ( ***+**** &!! "!! ( ( ***,**** &!! "!! ( ( ** ** &!! "!! ( ( &!! "!! ( ( **. /** &!! "!! ( ( **0 1 *** &!! "!! ( ( ***2**** &!! "!! ( ( **3 4 *** &!! "!! ( ( ***5**** &!! "!! ( ( ***6**** &!! "!! ( ( ****7*** &!! "!! ( ( *8 &!! "!! ( ( *98 &!! "!! ( ( *: &!! "!! ( ( *9: &!! "!! ( ( *; &!! "!! ( ( *9; &!! "!! ( ( ***< ** &!! "!! ( ( &!! "!! ( ( ** *** & !! !! ( ( ***=**** & !! !! ( ( ****4*** & !! !! ( ( ***.**** & !! !! ( ( ***3**** Figure8. Timeseriesandparametersensitivityfortheconcentration ofanthranilatesynthase.Thetimeseriesatthetopisin molsandtime isinminutes.Allsensitivitiesareunitless.Bluesensitivitiesexceeded anabsolutevalueof1atanypoint. 28

PAGE 36

! "! #! $! %! &!! &"! &#! &$! &%! "!! &! "! '( &!! "!! ) ) (((*(((( &!! "!! ) ) (((+(((( &!! "!! ) ) (( (( &!! "!! ) ) ( ( &!! "!! ) ) ((, .(( &!! "!! ) ) ((/ 0 ((( &!! "!! ) ) (((1(((( &!! "!! ) ) ((2 3 ((( &!! "!! ) ) (((4(((( &!! "!! ) ) (((5(((( &!! "!! ) ) ((((6((( &!! "!! ) ) (, 7 ( &!! "!! ) ) (, 87 ( &!! "!! ) ) (, 9 ( &!! "!! ) ) (, 89 ( &!! "!! ) ) (, : ( &!! "!! ) ) (, 8: ( &!! "!! ) ) (((, ; (( &!! "!! ) ) &!! "!! ) ) (( ((( & !! !! ) ) (((<(((( & !! !! ) ) ((((3((( & !! !! ) ) (((-(((( & !! !! ) ) (((2(((( Figure9. Timeseriesandparametersensitivityfortheconcentration ofinternaltryptophan.Thetimeseriesatthetopisin molsandtime isinminutes.Allsensitivitiesareunitless.Bluesensitivitiesexceeded anabsolutevalueof1atanypoint. 29

PAGE 37

boundproteinsthatactasaone-wayvalve.Thehighsensitivitymaybedueto theabilityofacelltokeepinternaltryptophanonceithasbeenwashed,sincethere shouldbenoexternaltryptophantoinuencethedynamics. Foreveryvariableconcentration,bothganddchangesignsintheearlytime, althoughearlierforthefreeoperatorbindingsitesandmRNA.Thiscouldbeconsideredthepointatwhichtheswitch'isippedandthesystemgoesfromo!'to on'.Thereisareasonabledelaybeforeaconsequentincreaseintryptophanand tryptophanproducingenzymescanbeobserved. Mostsensitivitiesgetverysmallasthesystemreachesasteadystateofconstant tryptophanproductionandanassumedconstantconsumptionrate,i.e.25micromols perminute.Recallthattheparameter hasbeensettozeroandassucheachofits graphsareeverywherezero. Noticethatparameterbhasaverylargepositivegraphforfreeoperonconcentrationbuthasalargenegativegraphfortryptophanconcentration.Highlyprobable attenuationwouldfreeupmanyoperonsbutwouldpreventproductivemRNAfrom beingtranscribed,preventingtryptophanproduction. Inordertocomparethepartialdi! erentialequationsmeaningfully,wenormalize eachfunctionbyit'sownvariableconcentrationdividedbyparametervalue.This willcanceloutthenon-comparableunitsofmeasurelikemicrolitres.Comparing thesenormalizedtimeseriescanbeimportant,especiallywhenteasingoutpositive andnegativeinuencesonaparameter. NormalizedSensitivity ( t )= k X ( t ) X k ( t )(23) Score = 1 t final t final t = t 0 | NormalizedSensitivity ( t ) | (24) Equation24givestheequationforndingasinglescore.Takingtheabsolute valueofthetimeseries,summingupeachdatapointandaveragingovertimegives 30

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Table1. Summedsensitivityscoresforthe24parametersandfour concentrationsfromthecodeinAppendixB.2 RP #"k d Dn H b K g FreeOperons0.330.610.610.000.580.403.430.14 FreemRNA0.390.440.560.000.420.412.470.13 Enzyme0.480.530.530.000.520.363.020.09 Tryptophan1.481.591.590.001.550.958.990.52 TOTAL2.683.163.290.003.072.1117.910.88 efO k r k r k i k i k t 0.000.090.390.330.330.250.250.29 0.390.390.240.240.330.000.090.44 0.480.480.230.230.400.000.090.53 1.481.480.830.831.200.010.191.59 2.682.681.561.562.220.020.462.95 k t k p k cgdk TOTAL 0.290.610.610.680.002.102.130.61 15.05 0.330.440.560.590.002.092.190.56 13.70 0.400.530.530.480.002.182.100.47 14.68 1.201.591.591.500.005.514.511.59 41.76 2.223.163.293.260.0111.8710.933.22 85.19 risetoasinglesensitivityscoreforanygivenvariablewithrespecttoanygiven parameter.Thisscoreprovidesfortheclearrankingofinuence.Further,thesum ofallofasinglevariable'sorparameter'sscorescanbecomputedandthisgivemore informationaboutthemodeledsystem.ThesescoresaregiveninTable1,whichlists thesensitivityscoreswithvariableconcentrationsinrowsandparametersincolumns. SeeFigure10foravisualrepresentationofthisdata. Tryptophanhasthelargesttotalsensitivityscore.Transcriptionalattenuation hasthelargestindividualscorewiththescoreforrepressordisassociationsimilarly large.Thesetwoparametersofthe96resultingscoresarespecicallytryptophan 31

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Figure10. SummedSensitivityScoreforParameters.Eachbarrepresentsaparameter'stotalsensitivitywhileeachsectionrepresentsa specicvariable'ssensitivitytothatparameter.Thepositionalongthe xaxisrepresentsthepositiondownthelistgiveninTable2.Sensitivity isunitless. regulatingmechanismsandiftheyarethemostsensitive,tryptophanshouldbethe mostsensitivevariableinthesystem.InFigures11and12,youcanseethesteady stateoftryptophanconcentrationregulatedwithrespecttotheparameter.Although theyappeartohavesimilarslopeinthegures,parameterbcausesasimilarchange intryptophanconcentrationwithhalfthevariationinparameter k t .Astrp-trpR disassociationincreases,sodoesthetryptophansteadystateconcentration. Predictionscanbemadeaboutperturbationsusingthesensitivityscores.Changingaparameter'svaluebysomepercentshouldproducethechangesrecordedinthe sensitivityfunction'stimeseries.Ifa X k ( t )showspositivesensitivityinearlytime butnegativesensitivityinlatertime,therespectivevariable'sgraphshouldbehigher inthebeginningbutlowerthantheoriginalinlatetime.InFigure1thevalueof parameter d ,themaximumtryptophanuptakerate,hasbeenincreasedtenpercent. Itcanbeseenthatconcentrationswerechangeddi! erentlyinearlytimecomparedto latertime,correlatingwelltothesensitivityfunctionforparameter d (seethebottom rowofFigures6through9).Theconcentrations[ Of ]and[ Mf ]areinuencedrst 32

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Figure11. Tryptophanconcentrationsteadystatewithrespectto maximumtranscriptionalattenuationrate.Tryptophanismeasuredin molsandparameterbisunitless. andoccurrstinthemetabolicpathway.Then[E]and[T]areinuencedtogether asthee!ectsmoveupthroughthecentraldogmaofmolecularbiologyfromDNAto proteins.Notethatthevariousparameter k positiveandnegativeratesareequalbut oppositeintheirsensitivity. ThecomputerresultsinFigure10indicatingthissystemismostsensitivetoparameter b correlateswelltotheexperimentalresults[ 14 ].Theseexperimentsshow adi!erenceinmeasuredsteadystatescorrespondingtomutationsincreasingtranscriptionalattenuationin E.coli cultures[ 25 ].Whentheparameterisincreased,its negativesensitivityscoremirrors,overtime,thesuppressionofenzymeactivitythat ismeasuredinthelab. 33

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Figure12. Tryptophanconcentrationsteadystatewithrespectto trp-trpRdisassociationrate.Tryptophanismeasuredin molsand parameterbisunitless. Theconcentrationoftryptophanismostsensitivetoparameterchangescomparedtotheotherconcentrations,conrmingthemodeloperonprimarilyregulates tryptophanlevels. 34

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CHAPTER6 Discussion Varioussensitivitymethodsandotheranalyticaltoolsexist,howeversomemay bemorepronetoartifactsanderror.Oncesolvedandnormalized,thesolutiontothe sensitivityequationsprovidesrichinformationaboutsystemdynamics.Thesuccess witnessedhere(consistencywithlaboratoryresults,elucidationofmechanisms,possibletherapeuticinformation)maybeinpartduetonature'sconsistentlydocumented parallelswithmathematics.Itcomesasnosurprisethatacalculusbasedsensitivity analysisaccuratelyrevealsnaturaldynamics. Someanalyticaltechniqueswillproduceresultsthatindicatetranscriptionalattenuationisnegligibleasaregulatorymechanism[ 3 ].However,thisresultsfromproblemsinthetoolsusedforevaluatingtheparameters.TheJacobian-basedalgorithm producesamoreempiricallyconsistentresult.Stochasticcalculationofsensitivity usingnitedi!erenceapproximationscanproduceinconsistentresultsandaremore computationallytaxing[ 13 ]. Themethodusedherewasaconsolidationofdi! erentworks.Printingtheresults tablerequiredsomealgorithmsforcolumnsorting.OverlayinggraphsinMatlabwas achievedbyrerunningtheprogramwhilepreservingthegraphfromthelastrun. TheMatlabfunctionode15ssolvestheODEs.Thissolverisappropriateforsti! equations.Asti!systemmodelsreactionsthatoccurondi! erenttimescales,where onerequiresshortstepswhileanothercanbesolvedaccuratelywithmuchlonger steps[ 27 ]. TheinitialconditionsofvariablesareestimatedbyMackeyandSantillanwhile initialsensitivityscoresaresetto1asperLeisetal[ 25 ][ 13 ]. 35

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Themodelequationsleissplitfromtheprogramlefortworeasons.Therst isforclarityofthecodeandthesecondisforuniversalityofthemethod.Whatis writteninMapleisapplicabletoasystemofvariable X 'sandconstant k 's.Mapleis moresuitableforsymbolicoperations.Matlabismoreappropriateforthenumerical solutionofthesystem,especiallyasamatrixfocusapplication[ 1 ].Theresultsare returnedinasinglematrix.Itshouldbenotedtherearefreealternativecomputer algebrasoftwarepackages,suchasGNUOctave. 1.FurtherResearchandDirection Thereisroomforbothfutureresearchandanalysis.Themodelcouldbemade moreintricateandaccurate.Therearemanyapproachestoanalyzingmodelsaswell. Futureworkwouldrstincorporatedelaydi!erentialequationsintothe model.Thisisamoreaccuratemodelofthecellularfunctionsoftranscriptionandtranslation.Thosedelayscouldbesensitiveparametersaswell changethescoresforotherparameters. Thereareotherwaysofcalculatingthesensitivityscore,includingglobal approaches.TheseincludeGreen'smethodandtheDecouplingmethod. Thisthesisuseddeterministicmethodsbasedincalculus.Analternative approachwouldbeavariance-basedprobabilityanalysis.Notonlywould thoseresultsbeinterestinginthemselves,butthecomparisonofthetwo approachesshouldyieldinsightaswell. Amorecomplexmodelcanbedevelopedthatdoesnotapproximateculturewideconcentrations,butinsteadlocalizesconcentrationsonanx-yplane. Thismoreaccuratelysimulateschemicaldistributioninan E.coli culture. Similarresearchintoaverysmallpopulationofcellswouldbeinteresting, wherethelawoflargenumbersdoesnothold[ 14 ]. 36

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Moreworkoncurvettingwouldberequiredtodeterminethebestapproach tostatisticalissuessuchasoutliers.Aprogramcanbewrittentodetermine thebesttparametersaswellasthebestmodelingrelationships. Further,morelaboratorydatashouldbecompliedintodatabases,tocheck simulationresultsagainst.Suchadatabasecouldinstructresearcherson whatameaningfuldatasetlookslike. Thereareemergingstandardsthattakeadvantageofthiscollabrativeidea. TheSystemsBiologyMarkupLanguagee"cientlyenablesdi! erentscientists tosharetheirmodelsandexplore insilico experiments.Writingthismodel, oronewithdelaydi! erentialequationsinSBMLwouldbeacontributionto theopensourcecommunity. Collaborativeworkbetweenbiologistsandmathematiciansholdsthepromiseof anewgenerationofbreakthroughs[ 12 ].Highspeedcomputingcombinedwithhighthroughputbiotechnologycreatesanunprecedentedopportunityfordiscovery.As oldassumptionsgivewaytonewdevelopments,eacheldbenetsdespitetheneed torevisepastwork.Really,itiswhenthemodeldoesnotcapturedynamicsthatone startstolearnaboutthesystemunderstudy[ 14 ]. 37

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APPENDIXA EquationsandParameterDenitions 1.SimpliedMackeySantillanModeloftheTrpOperon dOf dt = # O # K r K r + R A ( T ) # # Of ( t )(25) dMf dt = k p # P # Of ( t ) # (1 A ( T )) ( k d D + ) # Mf ( t )(26) dE dt = 1 2 k # # # Mf ( t ) ( + ) # E ( t )(27) dT dt = K # E A ( E,T ) G ( T )+ F ( T,T ext ) # T (28) R A ( T )= R # T ( t ) T ( t )+ K t (29) A ( T )= b (1 e T ( t ) c )(30) E A ( E,T )= E ( t ) K n H i T ( t ) n H + K n H i (31) G ( T )= g T ( t ) T ( t )+ K g (32) F ( T,T ext )= d T ext e + T ext (1+ T ( t ) f ) (33) K t = k t k t (34) K i = k i k i (35) K r = k r k r (36) 38

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Table2. DescriptionandEstimationofParamters ParameterName ParameterDescription EstimatedValue&Units R TrpRConcentration .8 mols P FreemRNAPConcentration 2.6 mols # FreeRibosomeConcentration 2.9 mols Anth.Synth.DegradationConstant 0 min 1 k d D RNADegradationFactor .6 min 1 n H HillCoe" cient 1.2 b MaximumTranscriptionalAttenuation .85 K g Michaelis-MentenTerm .2 mols e IntakeParameter .9 mols f IntakeParameter 380 mols O TotalOperonConcentration 3.32x10 3 mols k r TrpR-OperonDisassociationRate 1.2 min 1 k r TrpR-OperonAssociationRate 460 mols min 1 k i Trp-Anth.Synth.DisassociationRate 720 min 1 k i Trp-Anth.Synth.AssociationRate 176 mols min 1 k t Trp-trpRDisassociationRate 2.1x10 4 min 1 k t Trp-trpRAssociationRate 348 mols min 1 k p mRNAP-DNAAssociationRate 3.9 mols min 1 k Ribsosome-mRNAAssociationRate 6.9 mols min 1 GrowthRate 1x10 2 min 1 c AttenuationFactor 4x10 2 mols g MaximumTrpConsumptionRate 25 mols min 1 d MaximumTrpUptakeRate 23 mols min 1 k MaximumTrpSynthesisRate 126.4 min 1 39

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APPENDIXB Code 1.MapleSensitivityMatrixBuilder restart;with(linalg); Var:=[Of,Mf,E,T]; NumVar:=nops(Var); Par:=[R,P,rhor,gammar,kdD,nH,b,Kg,Ee,Ff,Oo,knegr,kposr, knegi,kposi,knegt,kpost,kp,krho,mur,c,g,d,K]; NumPar:=nops(Par); f:=vector(NumVar); Kt:=knegt/kpost; Ki:=knegi/kposi; Kr:=knegr/kposr; RA:=(T*R)/(T+Kt); AT:=b(1-e^(-T/c)); EA:=E(Ki^nH/(T^nH+Ki^nH)); G:=(g*T)/(T+Kg); FText:=(d*Text)/(Ee+Text(1+(T/Ff))); f[1]:=(mur*Oo*Kr)/(Kr+RA)-mur*Of; f[2]:=kp*P*Of*(1-AT)-(kdD+mur)*Mf; f[3]:=.5*krho*rhor*Mf-(gammar+mur)*E; f[4]:=K*EA-G+FText-mur*T; 40

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R:=matrix(NumVar,NumPar,L); Jc:=jacobian(f,Var); df:=jacobian(f,Par); Eq:=evalm(Jc&R+df); 41

PAGE 49

2.MatlabDriver clearall clf % % Turn on the functions you would like 1 for on and 0 for off ', the % % features are numbered by position in the parameter operation % % default Solve ODE and graph each variable with sensitivity time % % series in it s own window ( unless 2 is on ) % % 1 Change a parameter s value ( edit below ) % % 2 Graph each variable in the same window and do not graph % % sensitivities % % 3 Plot bar graph in a new window % % 4 Print a results table text file ResultsTable.txt % % 5 Plot enzyme activity in a new window % % 6 Plot a specified parameter range against a specified variable operation=[0,0,0,0,0,0]; % % Set Parameter Values par(1)=.8; par(2)=2.6; par(3)=2.9; par(4)=0.0; par(5)=.6; par(6)=1.2; par(7)=.85; par(8)=.2; par(9)=.9; par(10)=380.0; par(11)=3.32e 3; par(12)=1.2; par(13)=460.0; par(14)=720.0; par(15)=176.0; par(16)=2.1e4; par(17)=348.0; par(18)=3.9; par(19)=6.9; par(20)=.01; par(21)=.04; par(22)=25.0; par(23)=23.5; par(24)=126.4; % % Set Labels Labs=[ Of Mf E T ]; parameterList=[ R P \ rho \ gamma k dD ... n H b K g e f O ... k { r } k { + r } k { i } k { + i } k { t } k { + t } ... 42

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' k p k {\ rho } \ mu c g d ... K ]; % % Change a Parameter Value if operation(1)==1 par(23)=par(23) 1.1; end % % Solve System for Initals with Tryptophan [T0Y0]=ode15s(@TryptophanODE,[012000],ones(1,100),[],par,400); % % Loop for large parameter variation graphs if operation(6)==1 % % Set variation range and choose varied parameter variedPar=7; count=0; for variedParRange=.85:.005:.9996 count=count+1; par(variedPar)=variedParRange; % % Set Length of Run tfinal=1000; % % Solve Again with Tryptophan Removed using Previous Results [TY]=ode15s(@TryptophanODE,[0tfinal],Y0( end ,:),[],par,0); % % Choose a variable to plot against variableChoice=4; if variableChoice==1 variableLabel= Free Operons ; elseif variableChoice==2 variableLabel= Free mRNA ; elseif variableChoice==3 variableLabel= Enzyme ; elseif variableChoice==4 variableLabel= Tryptophan ; end dataPoints(count,:)=[Y( end ,variableChoice),variedParRange]; end % % Plot the figure figure(7) plot(dataPoints(:,2),dataPoints(:,1)) xlabel(parameterList((8 variedPar 7):(8 variedPar)), Rotation ,0), ... 43

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ylabel(variableLabel, Rotation ,90) else % % Set Length of Run tfinal=200; % % Solve Again with Tryptophan Removed using Previous Results [TY]=ode15s(@TryptophanODE,[0tfinal],Y0( end ,:),[],par,0); end % % Plot Enzyme Activity if operation(5)==1 % % Set Length of Run tfinal=200; % % Solve Again with Tryptophan Removed using Previous Results [TY]=ode15s(@TryptophanODE,[0tfinal],Y0( end ,:),[],par,0); figure(6) Ki=par(14)/par(15); nH=par(6); EA=Y(:,3). (KinH)./(Y(:,4).nH+KinH); plot(T,EA, g ),holdon end % % Plot Latest Run Concentrations over Time for i=1:4 if operation(2)==1 % % plot on the same figure figure(1) subplot(2,2,i),plot(T,Y(:,i), b ),holdon ylabel(Labs(2 i 1:2 i)) else % % plot on different figures figure(i) subplot(7,1,1),plot(T,Y(:,i), k ),holdon ylabel(Labs(2 i 1:2 i), Rotation ,0) end end % % Normalize Sensitivities and Plot ( with blue lines for large scores ) for j=1:4 spacer=(j 1) 24; for i=(5+spacer):(28+spacer) NormalizedY(:,i)=Y(:,i). par(i (4+spacer))./Y(:,j); if max(abs(NormalizedY(:,i))) > 1 cc= b ; 44

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else cc= r ; end if operation(2)==0 figure(j) subplot(7,4,i spacer),plot(T,NormalizedY(:,i),cc),holdon, ... gridon ylabel(parameterList(1+8 (i (5+spacer)):8 (i (4+ ... spacer))), Rotation ,0) axis([0tfinal 55]) end end end % % Plot Bar Graph of Sensitivities by Parameter if operation(3)==1 NOf=NormalizedY(:,5:28); NMf=NormalizedY(:,29:52); NE=NormalizedY(:,53:76); NT=NormalizedY(:,77:100); figure(5) Sense=[sum(abs(NOf))',sum(abs(NMf))',sum(abs(NE))',sum(abs(NT))']; Sense=Sense./length(T); bar(Sense, stack ) legend( Of Mf E T ) xlabel( Parameters ) ylabel( Sensitivity ) end % % Chart Sensitivites of Variables by Parameters in a file ResultsTable.txt if operation(4)==1 NOf=NormalizedY(:,5:28); NMf=NormalizedY(:,29:52); NE=NormalizedY(:,53:76); NT=NormalizedY(:,77:100); entries=length(T); OFscore=sum(abs(NOf))./entries; MFscore=sum(abs(NMf))./entries; Escore=sum(abs(NE))./entries; Tscore=sum(abs(NT))./entries; TotalScore=OFscore+MFscore+Escore+Tscore; parameterList=[ R P rho gamma ... kdD nH b Kg Ee Ff ... Oo k r k + r k i k + i k t ... k + t kp krho mu c ... g d k ]; 45

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% % Print in Latex Format LatexTableArray=[OFscore,sum(OFscore);MFscore,sum(MFscore); ... Tscore,sum(Tscore);TotalScore,sum(TotalScore)]; LatexTable=latex(sym(LatexTableArray)) resultsFile=fopen( ResultsTable.txt w ); fprintf(resultsFile, % s TOTAL \ n ,parameterList); for i=1:5 score=[]; if i==1 score=OFscore; fprintf(resultsFile, Free Operons ); elseif i==2 score=MFscore; fprintf(resultsFile, Free mRNA ); elseif i==3 score=Escore; fprintf(resultsFile, Enzyme ); elseif i==4 score=Tscore; fprintf(resultsFile, Tryptophan ); elseif i==5 score=TotalScore; fprintf(resultsFile, \ n TOTAL ); end for j=1:24 fprintf(resultsFile, %+4 .2f \ t ,score(j)); end fprintf(resultsFile, %+4 .2f \ n ,sum(score)); end fclose(resultsFile); end 46

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3.MatlabModelEquations function dy=TryptophanODE(t,y,par,Text) NumVar=4; NumPar=length(par); % % Initialize Solution Matrix Y Of=y(1); Mf=y(2); E=y(3); T=y(4); R=par(1); P=par(2); rho=par(3); gamma=par(4); kdD=par(5); nH=par(6); b=par(7); Kg=par(8); Ee=par(9); Ff=par(10); Oo=par(11); knegr=par(12); kposr=par(13); knegi=par(14); kposi=par(15); knegt=par(16); kpost=par(17); kp=par(18); krho=par(19); mu=par(20); c=par(21); g=par(22); d=par(23); K=par(24); % % Set Up Parameters and Relationships for ODEs Kt=knegt/kpost; Ki=knegi/kposi; Kr=knegr/kposr; RA=R T/(T+Kt); AT=b (1 exp( T/c)); EA=E (KinH)/((TnH)+(KinH)); G=g T/(T+Kg); FText=d Text/(Ee+Text (1+T/Ff)); 47

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% % Model ODEs dOf=((mu Oo Kr)/(Kr+RA)) mu Of; dMf=kp P Of (1 AT) (kdD+mu) Mf; dE=.5 krho rho Mf (gamma+mu) E; dT=K EA G+FText mu T; dy=[dOfdMfdEdT]'; % % Set Up Sensitivity Matrix ( size NxM ) L=zeros(NumVar,NumPar); zz=1; for i=1:NumVar for j=1:NumPar L(i,j)=y(NumVar+zz); zz=zz+1; end end % % Sensivity Equations ( S = J S + dF ) Imported from Maple % dOf / dk !!!!!!!!!!!!!!!!!!!!! dy(5)= mu L(1,1) mu Oo knegr/kposr/(knegr/kposr+T/(T+ ... knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T+ ... knegt/kpost)2 R) L(4,1) mu Oo knegr/kposr/(knegr/ ... kposr+T/(T+knegt/kpost) R)2 T/(T+knegt/kpost); dy(6)= mu L(1,2) mu Oo knegr/kposr/(knegr/kposr+T/(T+ ... knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T+ ... knegt/kpost)2 R) L(4,2); dy(7)= mu L(1,3) mu Oo knegr/kposr/(knegr/kposr+T/(T+ ... knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T+ ... knegt/kpost)2 R) L(4,3); dy(8)= mu L(1,4) mu Oo knegr/kposr/(knegr/kposr+T/(T+ ... knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T+ ... knegt/kpost)2 R) L(4,4); dy(9)= mu L(1,5) mu Oo knegr/kposr/(knegr/kposr+T/(T+ ... knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T+ ... knegt/kpost)2 R) L(4,5); dy(10)= mu L(1,6) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,6); dy(11)= mu L(1,7) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... 48

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+knegt/kpost)2 R) L(4,7); dy(12)= mu L(1,8) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,8); dy(13)= mu L(1,9) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,9); dy(14)= mu L(1,10) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,10); dy(15)= mu L(1,11) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,11)+mu knegr/kposr/(knegr/ ... kposr+T/(T+knegt/kpost) R); dy(16)= mu L(1,12) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,12)+mu Oo/kposr/(knegr/ ... kposr+T/(T+knegt/kpost) R) mu Oo knegr/kposr2/ ... (knegr/kposr+T/(T+knegt/kpost) R)2; dy(17)= mu L(1,13) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,13) mu Oo knegr/kposr2/ ... (knegr/kposr+T/(T+knegt/kpost) R)+mu Oo knegr2/ ... kposr3/(knegr/kposr+T/(T+knegt/kpost) R)2; dy(18)= mu L(1,14) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,14); dy(19)= mu L(1,15) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,15); dy(20)= mu L(1,16) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,16)+mu Oo knegr/kposr/ ... (knegr/kposr+T/(T+knegt/kpost) R)2 T/(T+knegt/ ... kpost)2 R/kpost; dy(21)= mu L(1,17) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,17) mu Oo knegr/kposr/ ... (knegr/kposr+T/(T+knegt/kpost) R)2 T/(T+knegt/ ... kpost)2 R knegt/kpost2; 49

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dy(22)= mu L(1,18) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,18); dy(23)= mu L(1,19) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,19); dy(24)= mu L(1,20) mu Oo knegr/kposr/(knegr/kposr+T/(T ... +knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T/(T ... +knegt/kpost)2 R) L(4,20)+Oo knegr/kposr/(knegr/ ... kposr+T/(T+knegt/kpost) R) Of; dy(25)= mu L(1,21) mu Oo knegr/kposr/(knegr/kposr+T ... /(T+knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T ... /(T+knegt/kpost)2 R) L(4,21); dy(26)= mu L(1,22) mu Oo knegr/kposr/(knegr/kposr+T ... /(T+knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T ... /(T+knegt/kpost)2 R) L(4,22); dy(27)= mu L(1,23) mu Oo knegr/kposr/(knegr/kposr+T ... /(T+knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T ... /(T+knegt/kpost)2 R) L(4,23); dy(28)= mu L(1,24) mu Oo knegr/kposr/(knegr/kposr+T ... /(T+knegt/kpost) R)2 (0.1e1/(T+knegt/kpost) R T ... /(T+knegt/kpost)2 R) L(4,24); % dMf / dk !!!!!!!!!!!!!!!!!!!!! dy(29)=(kp P (1 b (1 exp( T/c))) L(1,1))+( kdD mu) ... L(2,1) kp P Of b (exp( T/c))/c L(4,1); dy(30)=(kp P (1 b (1 exp( T/c))) L(1,2))+( kdD mu) ... L(2,2) kp P Of b (exp( T/c))/c L(4,2)+kp Of ... (1 b (1 exp( T/c))); dy(31)=(kp P (1 b (1 exp( T/c))) L(1,3))+( kdD mu) ... L(2,3) kp P Of b (exp( T/c))/c L(4,3); dy(32)=(kp P (1 b (1 exp( T/c))) L(1,4))+( kdD mu) ... L(2,4) kp P Of b (exp( T/c))/c L(4,4); dy(33)=(kp P (1 b (1 exp( T/c))) L(1,5))+( kdD mu) ... L(2,5) kp P Of b (exp( T/c))/c L(4,5) Mf; dy(34)=(kp P (1 b (1 exp( T/c))) L(1,6))+( kdD mu) ... L(2,6) kp P Of b (exp( T/c))/c L(4,6); 50

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dy(35)=(kp P (1 b (1 exp( T/c))) L(1,7))+( kdD mu) ... L(2,7) kp P Of b (exp( T/c))/c L(4,7)+kp P ... Of ( 1+exp( T/c)); dy(36)=(kp P (1 b (1 exp( T/c))) L(1,8))+( kdD mu) ... L(2,8) kp P Of b (exp( T/c))/c L(4,8); dy(37)=(kp P (1 b (1 exp( T/c))) L(1,9))+( kdD mu) ... L(2,9) kp P Of b (exp( T/c))/c L(4,9); dy(38)=(kp P (1 b (1 exp( T/c))) L(1,10))+( kdD mu) ... L(2,10) kp P Of b (exp( T/c))/c L(4,10); dy(39)=(kp P (1 b (1 exp( T/c))) L(1,11))+( kdD mu) ... L(2,11) kp P Of b (exp( T/c))/c L(4,11); dy(40)=(kp P (1 b (1 exp( T/c))) L(1,12))+( kdD mu) ... L(2,12) kp P Of b (exp( T/c))/c L(4,12); dy(41)=(kp P (1 b (1 exp( T/c))) L(1,13))+( kdD mu) ... L(2,13) kp P Of b (exp( T/c))/c L(4,13); dy(42)=(kp P (1 b (1 exp( T/c))) L(1,14))+( kdD mu) ... L(2,14) kp P Of b (exp( T/c))/c L(4,14); dy(43)=(kp P (1 b (1 exp( T/c))) L(1,15))+( kdD mu) ... L(2,15) kp P Of b (exp( T/c))/c L(4,15); dy(44)=(kp P (1 b (1 exp( T/c))) L(1,16))+( kdD mu) ... L(2,16) kp P Of b (exp( T/c))/c L(4,16); dy(45)=(kp P (1 b (1 exp( T/c))) L(1,17))+( kdD mu) ... L(2,17) kp P Of b (exp( T/c))/c L(4,17); dy(46)=(kp P (1 b (1 exp( T/c))) L(1,18))+( kdD mu) ... L(2,18) kp P Of b (exp( T/c))/c L(4,18)+P Of ... (1 b (1 exp( T/c))); dy(47)=(kp P (1 b (1 exp( T/c))) L(1,19))+( kdD mu) ... L(2,19) kp P Of b (exp( T/c))/c L(4,19); dy(48)=(kp P (1 b (1 exp( T/c))) L(1,20))+( kdD mu) ... L(2,20) kp P Of b (exp( T/c))/c L(4,20) Mf; dy(49)=(kp P (1 b (1 exp( T/c))) L(1,21))+( kdD mu) ... L(2,21) kp P Of b (exp( T/c))/c L(4,21)+kp P ... Of b (exp( T/c)) T/(c2); dy(50)=(kp P (1 b (1 exp( T/c))) L(1,22))+( kdD mu) ... 51

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* L(2,22) kp P Of b (exp( T/c))/c L(4,22); dy(51)=(kp P (1 b (1 exp( T/c))) L(1,23))+( kdD mu) ... L(2,23) kp P Of b (exp( T/c))/c L(4,23); dy(52)=(kp P (1 b (1 exp( T/c))) L(1,24))+( kdD mu) ... L(2,24) kp P Of b (exp( T/c))/c L(4,24); % dE / dk !!!!!!!!!!!!!!!!!!!!! dy(53)=krho rho L(2,1)/0.2e1+( gamma mu) L(3,1); dy(54)=krho rho L(2,2)/0.2e1+( gamma mu) L(3,2); dy(55)=krho rho L(2,3)/0.2e1+( gamma mu) L(3,3)+krho ... Mf/0.2e1; dy(56)=krho rho L(2,4)/0.2e1+( gamma mu) L(3,4) E; dy(57)=krho rho L(2,5)/0.2e1+( gamma mu) L(3,5); dy(58)=krho rho L(2,6)/0.2e1+( gamma mu) L(3,6); dy(59)=krho rho L(2,7)/0.2e1+( gamma mu) L(3,7); dy(60)=krho rho L(2,8)/0.2e1+( gamma mu) L(3,8); dy(61)=krho rho L(2,9)/0.2e1+( gamma mu) L(3,9); dy(62)=krho rho L(2,10)/0.2e1+( gamma mu) L(3,10); dy(63)=krho rho L(2,11)/0.2e1+( gamma mu) L(3,11); dy(64)=krho rho L(2,12)/0.2e1+( gamma mu) L(3,12); dy(65)=krho rho L(2,13)/0.2e1+( gamma mu) L(3,13); dy(66)=krho rho L(2,14)/0.2e1+( gamma mu) L(3,14); dy(67)=krho rho L(2,15)/0.2e1+( gamma mu) L(3,15); dy(68)=krho rho L(2,16)/0.2e1+( gamma mu) L(3,16); dy(69)=krho rho L(2,17)/0.2e1+( gamma mu) L(3,17); dy(70)=krho rho L(2,18)/0.2e1+( gamma mu) L(3,18); dy(71)=krho rho L(2,19)/0.2e1+( gamma mu) L(3,19)+rho ... Mf/0.2e1; 52

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dy(72)=krho rho L(2,20)/0.2e1+( gamma mu) L(3,20) E; dy(73)=krho rho L(2,21)/0.2e1+( gamma mu) L(3,21); dy(74)=krho rho L(2,22)/0.2e1+( gamma mu) L(3,22); dy(75)=krho rho L(2,23)/0.2e1+( gamma mu) L(3,23); dy(76)=krho rho L(2,24)/0.2e1+( gamma mu) L(3,24); % dT / dk !!!!!!!!!!!!!!!!!!!!! dy(77)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,1)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,1); dy(78)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,2)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,2); dy(79)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,3)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,3); dy(80)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,4)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,4); dy(81)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,5)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,5); dy(82)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,6)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,6)+K E ((knegi/kposi)nH) log((knegi/ ... kposi))/(TnH+(knegi/kposi)nH) K E ((knegi/ ... kposi)nH)/((TnH+(knegi/kposi)nH)2) ((TnH) ... log(T)+((knegi/kposi)nH) log((knegi/kposi))); 53

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dy(83)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,7)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/ ... Ff) mu) L(4,7); dy(84)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,8)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/ ... Ff) mu) L(4,8)+(g T/(T+Kg)2); dy(85)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,9)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/ ... Ff) mu) L(4,9) (d Text/(Ee+Text (1+T/Ff))2); dy(86)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,10)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/ ... Ff) mu) L(4,10)+(d Text2/(Ee+Text (1+T/ ... Ff))2 T/Ff2); dy(87)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,11)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,11); dy(88)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,12)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,12); dy(89)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,13)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,13); dy(90)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,14)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,14)+K E ((knegi/kposi)nH) nH/knegi/ ... (TnH+(knegi/kposi)nH) K E (((knegi/kposi)nH)2) ... 54

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/((TnH+(knegi/kposi)nH)2) nH/knegi; dy(91)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,15)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,15) K E ((knegi/kposi)nH) nH/kposi/ ... (TnH+(knegi/kposi)nH)+K E (((knegi/kposi)nH)2) ... /((TnH+(knegi/kposi)nH)2) nH/kposi; dy(92)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,16)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,16); dy(93)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,17)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,17); dy(94)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,18)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,18); dy(95)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,19)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,19); dy(96)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,20)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,20) T; dy(97)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,21)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,21); dy(98)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,22)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... 55

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! mu) L(4,22) (T/(T+Kg)); dy(99)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,23)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,23)+(Text/(Ee+Text (1+T/Ff))); dy(100)=K ((knegi/kposi)nH)/(TnH+(knegi/kposi)nH) ... L(3,24)+( K E ((knegi/kposi)nH)/((TnH+(knegi/ ... kposi)nH)2) (TnH) nH/T (g/(T+Kg))+(g T/ ... (T+Kg)2) (d Text2/(Ee+Text (1+T/Ff))2/Ff) ... mu) L(4,24)+E ((knegi/kposi)nH)/(TnH+(knegi/ ... kposi)nH); 56

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