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Dynamics of Single-Substrate Continuous Cultures

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Title:
Dynamics of Single-Substrate Continuous Cultures
Creator:
GUPTA, SHAKTI ( Author, Primary )
Copyright Date:
2008

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Subjects / Keywords:
Cell growth ( jstor )
Enzyme substrates ( jstor )
Enzymes ( jstor )
Kinetics ( jstor )
Nucleotides ( jstor )
Protein synthesis ( jstor )
Ribosomes ( jstor )
RNA ( jstor )
Simulations ( jstor )
Substrate specificity ( jstor )

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University of Florida
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University of Florida
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Copyright Shakti Gupta. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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8/31/2009
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439083747 ( OCLC )

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Itakethisopportunitytostatemydeepsenseofgratitudetomymentor,Dr.AtulNarang,forintroducingmetotheexcitingeldofmodelingofbiologicalsystems.Iamdeeplyindebtedtohisguidanceandconstantencouragement.Iwouldliketothankmyco-advisor,Dr.SergeiS.Pilyugin,forhisguidanceandencouragement.Iwouldalsoliketothankthemembersofmycommittee,Dr.RanganathanNarayanan,Dr.LewisE.Johns,Dr.SpyrosSvoronosandDr.BenKoopman,fortheiradviceandavailability.IamverygratefultoJasonNoelandBrentCoxfortheirvaluablehelpandlaboratoryassistance.IwouldliketothankDr.KarthikSubramanian,EricMay,VedSharmafortheirconstantsupportandfriendshipinandoutsidethelab.Thisacknowledgementcannotbecompletedwithoutthesenames:Saurabh,Rishabh,Krishna,Neeli,Gunjan,Ashish,Priyank,Himanshu,Dushyant,Samir,ArvindandallmyotherfriendsformakingmefeelcomfortableandmakingmystayatGainesvilleablissfulexperience.IwouldalsoliketothankShirley,NancyandDebbiefortheirtirelessassis-tancethroughoutmygraduatestudies.Lastbutnottheleast,Iwanttoexpressthedeepsenseofmyemotionformyparents,wife,brother,sisterandmyotherfamilymembers,whoalwaysstoodbesideme,throughthickandthin. iv

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page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. vii LISTOFFIGURES ................................ viii ABSTRACT .................................... x CHAPTER 1INTRODUCTION .............................. 1 1.1GeneralObjectives .......................... 1 1.2SpecicAims ............................. 2 1.3LiteratureReview ........................... 3 1.3.1RegulationofKeyProcesses ................. 3 1.3.2ExperimentalData ...................... 7 1.3.2.1Steadystates ...................... 7 1.3.2.2Transients ........................ 11 2THEROLEOFTRANSPORTENZYME ................. 21 2.1TheModel ............................... 22 2.2Simulations .............................. 25 2.2.1SteadyStates ......................... 25 2.2.2Transients ........................... 26 2.2.2.1Dilutionrateshifts ................... 26 2.2.2.2Feedswitches ...................... 28 2.3Discussion ............................... 30 3THEROLEOFRIBOSOMES ........................ 33 3.1TheModel ............................... 36 3.2Simulations .............................. 40 3.2.1SteadyStates ......................... 41 3.2.1.1Persistencesteadystate ................ 41 3.2.1.2Washoutsteadystate ................. 45 3.2.2Transients ........................... 46 3.2.2.1Continuous-to-batchshifts ............... 47 3.2.2.2Substrateswitch .................... 50 3.2.2.3Dilutionrateshift-down ................ 56 v

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.................. 59 3.3Discussion ............................... 64 4THEROLEOFADENINENUCLEOTIDE ................ 68 4.1TheModel ............................... 69 4.2Simulations .............................. 75 4.2.1SteadyStates ......................... 75 4.2.2Transients ........................... 77 4.2.2.1Substratepulse ..................... 78 4.2.2.2Continuous-to-batchshifts ............... 82 4.2.2.3Substrateswitch .................... 84 4.2.2.4Dilutionrateshift-up .................. 84 4.2.2.5Starvation ........................ 87 4.2.2.6Resumptionofthegrowthinthestarvationstate .. 89 4.2.2.7Slowsupplyofthesubstrateduringstarvation .... 92 4.3Discussion ............................... 93 5CONCLUSIONS ............................... 95 REFERENCES ................................... 97 BIOGRAPHICALSKETCH ............................ 103 vi

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Table page 2{1Parametervaluesusedinthetransportenzymemodelsimulations. .. 25 3{1Parametervaluesusedintheribosomemodelsimulations. ....... 40 4{1ParametervaluesusedintheATPmodelsimulations. ......... 75 vii

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Figure page 1{1Theproposedmodelofmicrobialgrowth. ................ 4 1{2Controlofproteinsynthesis,glycogenstorageandacetateexcretion. . 6 1{3Variationofsteadystateconcentrationswithrespecttodilutionrateincarbon-limitedcultures. ....................... 8 1{4Responseofnitrogen-limitedculturetodilutionrateshift-up ..... 12 1{5Responsetoaswitchintheidentityofthegrowth-limitingsubstratefromglucosetonitrilotriaceticacid .................. 13 1{6Initialresponseofglucose-limitedsteadystatecontinuousculturesgrowingatvariousdilutionratestosupersaturatingglucosecon-centrations ............................... 16 1{7AutocatalytickineticsofRNAsynthesis ................ 17 1{8Transientresponsetoasubstratepulseinminutestimescale. ..... 19 1{9Transientresponsetoasubstratepulseinsecondstimescale. ..... 20 2{1Kineticschemeofthetransportenzymemodel. ............. 23 2{2Variationofsteadystatesoftransportenzymemodelwithdilutionrate. ................................... 27 2{3Trajectoriesfordilutionrateshift-up .................. 28 2{4TrajectoriesforaswitchintheidentityofthesubstrateatK3=104 30 2{5TrajectoriesforaswitchintheidentityofthesubstrateatK3=5104 3{1Transientresponseofaglucose-limitedculturetoadilutionrateshift-up .................................... 34 3{2Kineticschemeoftheribosomemodel .................. 37 3{3Variationofthesteadystateswithdilutionrate ............ 42 3{4Initialresponsefollowingacontinuous-to-batchshift .......... 48 viii

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.............................. 51 3{6Phaseportraitsoftheslowmotionduringafeedswitches. ...... 52 3{7Transientresponseofribosomemodeltoadilutionrateshift-down .. 57 3{8Responseofaglucose-limitedculturetoadilutionrateshift-down .. 59 3{9Transientresponseofribosomemodeltodilutionrateshift-ups. ... 61 3{10Phaseportraitsoftheslowmotionduringdilutionrateshifts. .... 62 3{11Responseofaglycerol-limitedcultureofK.aerogenestoadilutionrateshift-up. .............................. 64 4{1KineticschemeoftheATPmodel .................... 70 4{2Variationofthesteadystateswithdilutionrates ............ 76 4{3Transientresponsetoasubstratepulseinsecondstimescale. ..... 79 4{4Transientresponsetoasubstratepulseinminutestimescale. ..... 81 4{5Initialresponsefollowingacontinuous-to-batchshift. ......... 83 4{6Transientresponseforaswitchintheidentityofthesubstrate. .... 85 4{7Transientresponsetoadilutionrateshift-up. ............. 86 4{8Simulationresultsofthebatchcultureduringexhaustionofthesub-strate. .................................. 88 4{9Simulationresultsofthebatchculturewithresupplyofthesubstrateduringstarvation. ............................ 90 4{10Transientresponseoftheglucoselimitedbatchculturewithsupplyofglucoseafter45minutesofstarvation. ............... 91 4{11Simulationresultsofthebatchculturewiththeslowsupplyofthesubstrateduringstarvation. ...................... 92 4{12Transientresponseofaglucoselimitedbatchculturewiththeslowsupplyofglucoseduringstarvationtomaintainenergycharge. ... 93 ix

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Whenachemostatisperturbedfromitssteadystate,itdisplayscomplexdynamics.Forinstance,iftheidentityofthegrowth-limitingsubstrateisswitchedabruptly,thesubstrateconcentrationandcelldensityundergoapronouncedexcursionfromthesteadystatethatcanlastseveraldays.Thesedynamicsoccurbecausecertainphysiologicalvariablesrespondslowly.Intheliterature,severalphysiologicalvariableshavebeenpostulatedaspotentialsourcesoftheslowresponse.Theseincludetransportenzymesandribosomes. Westudiedtheroleoftransportenzymesbyconsideringexperimentsinwhichlowlevelsofthetransportenzymelimitsthegrowth.Itwasshownthatthelonglagscouldoccurbecausetransportenzymesynthesisisautocatalytic.Amodelwasdevelopedtoaccountfortransportenzymesynthesis,whichcapturedthesetransientsquantitatively.Wethenextendedthemodeltoaccountforexperimentsinwhichthelagspersistedevenifthetransportenzymelevelwashigh.Inthiscase,thegrowthratesappeartobelimitedbytheproteinsyntheticmachinery(ribosomes).Weshowedthatanextendedmodeltakingdueaccountofribosomessynthesiscouldcapturethesetransientsqualitatively. x

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xi

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1.1 GeneralObjectives Thechemostats(CSTR)arecommonlyusedinindustrytogrowthecellattheexpenseofnutrientconsumption.Theimportanceofthechemostatliesinvariousapplications.Theenvironmentalindustriesusethechemostattoremovethetoxiccompoundsfromthewaste.Inthiscase,thecellconsumesthetoxicmaterialsandproducesbenigncells.Inpharmaceuticalindustries,thecellsaregrownattheexpenseoflowvaluefoodandexcretehighvalueproducts(e.g.,penicillin).Whenachemostatisexposedtoabruptdisturbances,i.e.,changeintheowrate,feedconcentrationsetc.,thecellstaketimetoadapttotheenvironmentanddisplaycomplextransientresponseinvolvingovershootinthesubstrateconcentration,undershootincelldensityandexcretionofpartiallyoxidizedmetabolitesinthereactor.Thesedynamicsoftencauseregulatoryviolationsinwastewatertreatmentplantsandlossordeteriorationofproductinindustrialfermenters.Abetterunderstandingoftheseprocessesisessentialfordevelopingrationaloperatingprotocolsandmodel-basedcontrolstrategies. Thecomplexdynamicsofachemostatcanbeattributedtotheadaptivenatureofcells.Whenenvironmentalconditionsarechanged,thecellaltersthein-tracellularmachineryinordertoadjusttothenewenvironment.Theslowresponseofthecelloccursbecauseoftheslowsynthesisofcertainintracellularcomponents.Thegeneralobjectiveofthisworkistoanalyzethevariousexperimentaldata,identifytheslowphysiologicalvariablesresponsibleforcomplextransientsandunderstandthekineticsoftheseslowphysiologicalvariableevolutionsintermsoffundamentalbiochemistryandmolecularbiology.

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1.2 SpecicAims Thespecicaimofthisworkistodevelopanintegratedmodelofbacterialcellonthebasisofvastexperimentaldataobservedintheliteratureusingonlyfewkeyvariablesandtoverifythemodelviaexperimentation.Thehypothesisbehindthisapproachisthatdespitethecomplexityofthemetabolicpathways,onlyafewslowphysiologicalvariablesplayanimportantroleinthedynamicsofthechemostat.Anextensivereviewoftheliteratureindicatesthatwecandescribemosttransientsobservedinexperimentsintwocategories: Wehaveattributedtheaboveresponsestotwoslowvariables,(1)theribosomeswhichareresponsibleforproteinsynthesis(2)andtheperipheralenzymesthatmediatetransportofthegrowth-limitingsubstratefromtheenvironmenttoinsidethecell.Whentheenvironmentischangedabruptly,theslowvariablesgenerallyrespondinabiphasicmanner.Initially,theirsynthesisratesacceleraterapidlyintimescaleofseconds.Thisrapidresponseistriggeredbytwofastvariables,freeaminoacidsor/andnucleotidephosphates.Andsubsequentlyslowresponsesonatimescaleofhoursbecauseoftheautocatalyticsynthesisoftheslowvariables. Afterdevelopingthemodelcontainingtransportenzymeandribosomes,thedistributionoftheincomingsubstrateindierentmetabolicpathwaysraisedan-otherquestion.Whenthecellpicksupthesubstrate,itcandistributetheincomingsubstrateinthedierentpathways(Figure 1{1 ),i.e.,growth,respiration,storageorexcretion.Thenhowdoesthecellmakesthedecisionaboutthedistributionoftheincomingsubstrate?Inordertoovercomethisobstacle,wehaveaddedanarticialintelligencefeaturetothemodelcell.Thus,themodelwasfurtherextendedto

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accountforthemetabolicuxdistributionduringgrowthandstarvationbyaddingadeninenucleotidesasenergymolecules. Thegoalofthemodelingisnotmerelytotthedata,buttogainfurtherinsightsintothedynamicsofthecellbyanalyzingthemodelwiththehelpoftheoreticalandnumericaltoolsdrawnfromdynamicalsystemsandbifurcationtheory. InSection 1.3 ,wesummarizetheliteratureandformulateaconceptualmodel(Figure 1{1 )thatprovidesaqualitativeexplanationofalltheseobservations. 1.3 LiteratureReview Theexhaustiveliteraturereviewshowsthatthekeyprocessesthatdeterminethedynamicresponseofmicrobialcellsaresubstrateuptake,proteinsynthesis,storage,andexcretion.Toaccountfortheseprocesses,weproposetheschematicmodelasshowninFigure 1{1 .Thegoalofthissectionistoshowthattheliter-atureprovidesstrongexperimentaljusticationforthisscheme.InSection 1.3.1 ,wedenethekeyprocessesandsummarizetheliteratureontheirregulations.InSection 1.3.2 ,wereviewtheexperimentalliteratureonthesteadystatesanddynamicsofthevariables. 1.3.1 RegulationofKeyProcesses Thegoalofthisbriefoverviewistoshowthatthemodelschemecapturestheessentialfeaturesofthemechanismsthatcontrolthegrowth. Substrateuptake [ 1 ] .Thespecicsubstrateuptakerateisdeterminedbytheconcentrationofthegrowth-limitingsubstrateandtheactivityoftheperipheralenzymesthatcatalyzethesubstrateuptakeandperipheralcatabolism.Thekeymechanismscontrollingperipheralenzymesynthesisareinductionandcataboliterepression.Themolecularmechanismforinductionofperipheralenzymesisuni-versal.TheJacob-Monodmodel,orsomevariantofit,appliestomostsystems.

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Figure1{1: Theproposedmodelofmicrobialgrowth.Here,Sdenotesthegrowthlimitingcarbonandenergysource,Edenotesthetransportenzyme,XdenotestheinternalizedsubstratewhichinducesthesynthesisofE,PdenotestheprecursorsproducedbycatabolismofX,Psdenotesstoredcompounds,Pxdenotesexcretedmetabolites,Emdenotesbiosyntheticenzyme(s),Mdenotesaminoacidmonomers,Rdenotesribosomes,Cdenotesproteins,andNTP/NDPdenotenucleotidetrianddiphosphates. However,thereisconsiderablevariationinthemechanismsofcataboliterepres-sion[ 1 ].ThebestknownistheclassicalcAMPsysteminwhichtheuptakeofglucosebythephosphotransferasesystemlowersthepoolofcAMP,whichinhibitsthetranscriptionofthemRNAforenzymesthattransportandmetabolizeothercarbonsources. Proteinsynthesis [ 2 ] .Thespecicproteinsynthesisrateisdeterminedbytheconcentrationoffreeaminoacidsandribosomes(Figure 1{2 a).RibosomesareassembledfromribosomalRNA(rRNA)andribosomalproteins(r-proteins).Anegativefeedbackmechanismensuresthattherateofr-proteinsynthesisisinsynchronywiththeprevailingrateofrRNAsynthesis[ 3 ].Theexcessr-proteinsbindtothemRNAsthatcodeforthem.ThisinactivatesthemRNAandstops

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furthersynthesisofr-proteins.Therateofribosomalsynthesisis,therefore,completelydeterminedbytherateofrRNAsynthesis. ItiswidelybelievedthattherRNAsynthesisrateiscontrolledbytheconcen-trationsoffreenucleotidephosphates(NTPs),aminoacidsandRNApolymerase(RNAP),butthereisconsiderabledisagreementonthemolecularmechanismoftheiraction[ 4 ].JensenandPedersenproposedamodelbasedonthefollowinghypotheses[ 5 ]: RecentexperimentsprovidestrongevidencesupportingallthreehypothesesoftheJensenandPetersenmodel[ 4 ]. Storage [ 6 , 7 ] .Microbialcellsstorecarbonascarbohydrates(glycogen)orlipids(polyhydroxyalkanoates).Althoughdistinctpathwaysareinvolvedinthesynthesisofglycogenandpolyhydroxyalkanoates,theirregulatorycircuitssuggestacommonprinciple.Inbothcases,storageisactivatedonlyifbothprecursorsandenergyareathighconcentrations.Glycogen,forinstance,isproducedbythe

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(b) (c) Controlofproteinsynthesis,glycogenstorageandacetateexcretion:(a)Proteinsynthesisisactivatedbyaminoacids(b)Acetateexcretionisactivatedbyprecursors(pyruvate)butinhibitedbyenergy(NADH,ATP)(c)Acetateexcre-tionisactivatedbyprecursors(pyruvate)butinhibitedbyenergy(NADH,ATP)Glycogenstorageisactivatedbyprecursors(fructose-1,6-diphosphate)andenergy(ATP). consecutiveactionofADP-glucosepyrophosphorylaseandglycogensynthetase(Figure 1{2 b).Inprokaryote,ADP-glucosepyrophosphorylaseistheregulatedenzyme.Itisactivatedbyfructose-1,6-diphosphateandATP,andinhibitedbyAMP[ 6 ]. Excretion [ 8 ] .Cellsexcreteawidevarietyofcompounds,dependingonthegrowth-limitingsubstrate[ 9 ].Incarbon-limitedgrowth,acetateisthepredominantexcretedcompound[ 8 ].Itisproducedbytheconsecutiveactionoftwoenzymes,

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phosphotransacetylaseandacetatekinase(Figure 1{2 c).Theregulatedenzymeofthispathwayisphosphotransacetylase.Itsregulationisconsistentwiththefactthatacetateexcretionresultsinthenetgenerationofenergy(onemoleofATPforeverymoleofacetateexcreted).Itisactivatedbypyruvate,whichsignalsanabundantsupplyofprecursors,butinhibitedbyNADHandATP,whichsignalanenergysurplus[ 10 ]. 1.3.2 ExperimentalData Inthissection,wehavecollatedthedataonthesteadystatesanddynamicsofcontinuouscultures. 1.3.2.1 Steadystates Insingle-substrategrowth,theexperimentalistcanvaryonlytwocontrolparameters|theowrateandthefeedconcentration.Ifthefeedconcentrationisincreasedataxeddilutionrate,thecelldensityincreasessuchthatthesteadystatevaluesofallothervariablesarerestoredtotheirpreviousvalues[ 11 ].Asshownbelow,thesteadystatevariationsareconsiderablymorecomplexifthedilutionrateischangedataxedfeedconcentration. Celldensityandsubstrateconcentration .Thecellswashoutatbothsmallandlargedilutionrates.ThewashoutatlargedilutionratesisuniversallyacceptedandhasbeenextensivelystudiedbeginningwiththeclassicalworkofHerbertetal.[ 18 ].Thewashoutatlowdilutionrates,oftencalledtheminimumgrowthrate,iswidelyaccepted,buttherearefewsystematicstudies[ 19 ].SchulzeandLipeshowedthatglucose-limitedculturesofE.coliwerewashedoutatdilutionratesbelow0.021/hr[ 20 ].Theyarguedthatthiswashoutoccurredbecauseatsuchlowdilutionrates,thespecicuptakerateofthegrowth-limitingcarbonsource(glucose)becamelessthanthemaintenancecoecient(0.055g/(gdw-hr)).Tempestetal.determinedthesteadystatesofglycerol-limitedculturesofK.aerogenesatdilutionratesdownto0.0041/hr[ 12 ].Undertheseconditions,the

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(b) (c) (d) Variationofsteadystateconcentrationswithrespecttodilutionrateincarbon-limitedculturesofK.aerogenes:(a)Celldensity,substrate(glycerol)concentrationandyieldofbiomass[ 12 , 13 ](b)Peripheralandbiosyntheticenzymelevels[ 14 { 16 ](c)Adeninenucleotides[ 17 ]andviability[ 12 ](d)RNA,proteinandcarbohydratecontents[ 12 , 13 ]. viabilityoftheculturesdroppedto40%(Figure 1{3 c),andthespecicgrowthrateoftheviableorganismsasymptoticallyapproachedaminimumvalueof0.0091/hr. Betweenthetwowashoutdilutionrates,thecelldensitypassesthroughamaximum(Figure 1{3 a).Sincethesubstrateconcentrationequalsthefeedconcentrationatdilutionratesaboveandbelowthetwowashoutdilutionrates,itshouldpassthroughaminimumbetweenthesedilutionrates.Unfortunately,nodataareavailableexceptatrelativelylargedilutionratesneartheupperwashoutdilutionrate.Atsuchlargedilutionrates,thesubstrateconcentrationisanincreasingfunctionofthedilutionrate(Figure 1{3 a). Peripheralandbiosyntheticenzymelevels [ 21 { 23 ] .Basedonacomprehensivereviewoftheliterature,weconcluded(seeFigure 1{3 b)that(1)theactivityof

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22 ]. Theexistenceofamaximuminthesteadystateproleofinducibleperipheralenzymesisanoutcomeofacompetitionbetweentheopposingeectsofenzymeinductionandcataboliterepression[ 15 ].Atlowdilutionrates,enzymeinductiondominates,sothattheenzymelevelisanincreasingfunctionofthedilutionrate.Conversely,athighdilutionrates,enzymeinductionsaturatessothatcataboliterepressiondominates,andtheenzymelevelisadecreasingfunctionofthedilutionrate.Thisargumentalsoexplainsthedecreasingtrendofconstitutiveperipheralenzymes.Insuchcases,cataboliterepressionistheonlymechanisminuencingtheenzymeactivity,sinceinductionofconstitutiveenzymesoccursataconstantrate. Adeninenucleotides .Theconcentrationsofadeninenucleotidesareconstantatdilutionratesabove0.11/hr[ 17 ].Atlowerdilutionrates,theconcentrationofATPdecreases,theconcentrationofAMPincreases,andtheconcentrationofADPpassesthroughamaximum(Figure 1{3 c). Toexplainthesetrends,itsucestounderstandtheconcentrationproleofATP|theconcentrationsofADPandAMParecompletelydeterminedbytheconcentrationofATP.ThisisbecauseATP,ADPandAMPareinaquasi-equilibriumdeterminedbyadenylatekinase;i.e.,[ATP][AMP]=[ADP]2=Kandtheenergycharge,denedastheratio[ATP]+0:5[ADP]=([ATP]+[ADP]+[ADP]),is0:9atallbutthesmallestdilutionrates[ 24 ]. Freeaminoacidpool .Althoughthedataforfreeaminoacidpoolsaresparse,theyshowthattheirtotalconcentrationincreaseswithrespecttothedilutionrate.Inammonia-limitedculturesofA.aerogenes,thefreeaminoacidpool

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concentrationswere5.5,8.0and14.7mMatdilutionratesof0.1,0.3and0.71/hr,respectively[ 25 ].Inglucose-limitedculturesofS.cerevisiae,theconcentrationswere380and529mMatdilutionratesof0.1and0.251/hr,respectively[ 26 ].Glutamateisbyfarthemostabundantaminoacid(60{70%ofthepool),whichisconsistentwiththefactthatglutamatesupplies85%oftheaminogroupsrequiredforproteinsynthesis. RNAandprotein .RNAandproteinarethemostpredominantcellularconstituents(Figure 1{3 d).Theyconstituteroughly80-85%ofthedryweightofthecellatalldilutionratesbetween0.004and0.851/hr[ 12 , 13 ].Asthedilutionrateincreases,theRNAcontentincreasesthree-foldfrom6%at0.0041/hrto18%at0.851/hr.Theproteincontentundergoesacorrespondingdecreasefromroughly80%to70%ofthedryweight.Thus,asthedilutionrateincreases,theRNAcontentgrowsattheexpenseofproteins. Athighdilutionrates,theribosome-to-proteinratioisproportionaltothedilutionrate,whichimpliesthatthespecicrateofproteinsynthesisperribosomeortheribosomaleciencyisconstant[ 27 ].Thissuggeststhatathighdilutionrates,thespeedwithwhichribosomesaddaminoacidstoapeptidechainisconstant.Itfollowsthatathighdilutionrates,furtherimprovementsinthespecicproteinsynthesisrate,andhencethespecicgrowthrate,canbeobtainedonlybyincreasingtheconcentrationoftheribosomes.AsimilarconclusionisreachedbasedontransientexperimentsdescribedinSection 1.3.2.2 . Carbohydratesandexcretedmetabolites .Thecarbohydratecontentvariesdramaticallywithrespecttotheidentityofthemicrobialspeciesandthecarbonsource.However,foragivenmicrobialspeciesandcarbonsource,thecarbohydratecontentisalmostindependentofthedilutionrate(Figure 1{3 d).Inglycerol-limitedculturesofK.aerogenes,thecarbohydratecontentis1.6{3.4%atalldilutionrates

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between0.004and0.851/hr[ 12 , 13 ].Inglucose-limitedculturesofT.utilis,thecarbohydratecontentis20%atalldilutionratesbetween0.04and0.51/hr[ 28 ]. Theextracellularacetateconcentrationalsovariessignicantlydependingontheidentityofthemicrobialspeciesandthecarbonsource.Inglycerol-limitedculturesofK.aerogenes,thereisnomeasurableexcretionatalldilutionratesbetween0.004and0.851/hr[ 12 ].Inglucose-andpyruvate-limitedculturesofE.coli,thereispronouncedexcretionofacetate(seedashedlineinFigure 1{6 d),butonlyatsucientlyhighdilutionrates[ 29 { 32 ]. 1.3.2.2 Transients Chemostatshavebeensubjectedtofourtypesofperturbations:Dilutionrateshifts,continuous-to-batchshifts,substratepulsesandsubstrateswitches.Eachoftheseperturbationshasyieldeduniqueinsightsintothevariablesthatcontrolthegrowthdynamics. Dilutionrateshift .Indilutionrateshift-upexperiments,thechemostatisallowedtoreachsteadystateatsomeinitialdilutionrate,say,D0,andthedilutionrateisabruptlyincreasedtoD>D0.Theseexperimentsshowthatchemostatsdisplayexcitabledynamics;i.e.,theyreturntothesteadystaterapidlywhensubjectedtosmallperturbations,butundergoalargeandprolongedexcursionwhensubjectedtolargeperturbations[ 33 ]. Matelesetal.demonstratedexcitabledynamicsinachemostatbysubjectingnitrogen-limitedculturesofE.coliBtodilutionrateshift-ups[ 34 ].Theyobservedthatthespecicgrowthrateincreasesinstantlyinresponsetothedilutionrateshift-up,regardlessoftheinitialdilutionrate.Iftheshift-up,DD0,issmall,theenhancedspecicgrowthrateimmediatelyequalsthenewdilutionrate.Ifthedilutionrateshift-upislarge,theenhancedspecicgrowthratecannotmatchthenewdilutionrate(Figure 1{4 ).Therenowbeginsaperiodduringwhichthespecicgrowthrateincreasesslowly.Duringthisperiod,thecelldensity

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Figure1{4: Responseofnitrogen-limitedcultureofE.coliBtoadilutionrateshiftfromD=0:411/hrtoD=0:831/hrataxedfeedconcentrationof100mg/L(from[ 34 ]). decreasesandthesubstrateconcentrationincreases.Thislargeexcursionofthesubstrateconcentration,whichcanlastafewhours,isnallycontainedwhenthespecicgrowthratebecomesequaltothenewdilutionrate.Atthisinstant(^t)thesubstrateconcentrationreachesamaximum(^s)andthecelldensityreachesaminimum.Thisisfollowedbyaperiodduringwhichthespecicgrowthrateexceedsthenewdilutionrate.Duringthisperiod,thecelldensityincreasesandthesubstrateconcentrationdecreasesuntiltheyreachvaluesconsistentwiththenewsteadystate.Thetimetakentoreturntothesteadystate(t1)canbeaslongas12.5hours. Similartransientshavebeenobtainedincarbon-limited[ 12 , 35 { 37 ]andphosphate-limitedcultures[ 38 , 39 ]. Substrateswitch .Thesubstrateswitchexperimentsrevealthatifthepe-ripheralenzymeforthegrowth-limitingsubstrateareinducibleandandiftheconcentrationleveloftheenzymeissmall,thegrowthislimitedbythesubstrateuptakerate.Herein,thecultureisallowedtoreachsteadystateatagivendilutionrateandfeedconcentrationononesubstrate.Oncethesteadystateisachieved,theidentityofthegrowth-limitingsubstrateisswitchedabruptlytoanyothersubstrate.

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(b) (c) Responsetoaswitchintheidentityofthegrowth-limitingsub-stratefromglucosetonitrilotriaceticacid(NTA)inacultureofC.heintzii(datafrom[ 40 ],simulationsfrom[ 41 ]).Thedilutionratewasheldxedat0.0061/hr.Thedashedlinein(a,b)showstheprolethatwouldbeobtainediftherewasnosubstrateconsumption. BallyandEgliperformedsubstrateswitchexperimentswithC.heintziiculturesgrowinginachemostatunderglucoseornitrilotriaceticacid(NTA)limitation[ 40 ].WhenthecarbonsourcewasswitchedfromNTAtoglucose,therapidgrowthwasobservedafter2{3hours.Incontrast,whenthecarbonsourcewasswitchedfromglucosetoNTA,therewasalmostnosubstrateconsumptionfornearly30hours(Figure 1{5 a,b).Thedramaticdierenceinthetimerequiredtoconsumesignicantamountsofsubstrateisduetothedierenceintheinitialperipheralenzymelevels.Theperipheralenzymesforglucose,beingconstitutive,arealwayspresentathighlevels.Consequently,whenthesubstrateisswitchedfromNTAtoglucose,consumptionofglucosebeginsimmediately.Ontheotherhand,theperipheralenzymesforNTA,beinginducible,havevanishinglysmall

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concentrationsduringgrowthonglucose.WhenthecultureisshiftedfromglucosetoNTA,ittakesseveraltensofhoursbeforetheperipheralenzymesforNTAbuilduptosucientlyhighlevels(Figure 1{5 c). Similarresultswereobtainedwhenbacterialcultureswereshiftedfromglu-cosetoxyloseandviceversa[ 37 , 42 ].Switchingfromglucosetoxyloseledtoaprolongeddecreaseinthecelldensitylastingover1dayduringwhichxyloseaccu-mulatedinthechemostat.Switchingfromxylosetoglucosecausednomeasurableaccumulationofglucose. Insubstrateswitches,thegrowthrateislimitedbysubstrateuptakeonlybecausethecellshavenotbeengrowingonthesubstrate,sothattheinitiallevelsoftheperipheralenzymesarevanishinglysmall.However,ifthecellshavebeengrowingonthesubstrateatsomedilutionrate,theinitialenzymelevelisrelativelylargeatallbutthesmallestdilutionrates(seecurvefor-galactosidaseinFigure 1{3 b).Itfollowsthatthetransientresponseofcellsgrowingatverylowdilutionrateswillbecontrolledbyperipheralenzymesynthesis.Moreover,thetransientwillbecharacterizedbylongrecoverytimes(ontheorderofdays)andtheabsenceofexcretionandstorage[ 12 ].Asystematicstudyhasneverbeendone,butitseemslikelythatthetransientresponseofcellsgrowingatmediumtolargedilutionrateswillbecontrolledbyrRNAsynthesis.Thistransientwillbecharacterizedbyshortrecoverytimes(ontheorderofhours)anditwillbeaccompaniedbysignicantexcretionor/andstorage. Continuous-to-batchshift .Indilutionrateshift-ups,thesubstrateconcen-trationinstantlyincreasestosupersaturatinglevels.Thespecicgrowthratealsoincreasesinstantlybutdoesnotreachthemaximumspecicgrowthratedespitethepresenceofsupersaturatingsubstrateconcentrations.Thisraisestwoquestions:(1)Whatisresponsiblefortheinstantaneousincreaseinthespecicgrowthrate?

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(2)Whatpreventsthespecicgrowthratefrominstantlyachievingthemaximumvalue?Theanswerstobothquestionsareprovidedbycontinuous-to-batchshifts. Intheseexperiments,asampleofcellsiswithdrawnfromasteadystatecontinuousculturegrowingatacertaindilutionrate.Thesampleisimmediatelyexposedtosupersaturatingsubstrateconcentrationsinabatchreactor,andtheinitialratesofvariousprocesses(substrateuptake,growth,respirationandexcretion)aremeasured.Theseexperimentsshowthatregardlessofthedilutionrateatwhichcellsaregrowingbeforetheirwithdrawalfromthechemostat: 43 { 45 ](Figure 1{6 a). 14 , 46 ](Figure 1{6 b). Itfollowsthatthereisanadequatesupplyofprecursorsandenergyimmediatelyaftertheshift-up. Incontrast,thespecicbiosynthesisrate(ofRNAandprotein)dependscruciallyonthedilutionrateatwhichcellsaregrowingbeforetheirwithdrawalfromthechemostat[ 14 ].Whenexposedtosupersaturatingsubstrateconcentrations(Figure 1{6 c) Thisanswersthesecondquestionraisedabove.ItistheslowsynthesisofRNAandproteinsthatpreventsthespecicgrowthratefrominstantlyachievingitsmaximumvalue.Alsoconsistentwiththisconclusionistheobservationthat 29 , 45 ](Figure 1{6 d). 14 ].

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(b) (c) (d) Initialresponseofglucose-limitedsteadystatecontinuousculturesgrowingatvariousdilutionratestosupersaturatingglucoseconcentrations.Thedashedlineshowstherateofaprocessduringsteadystategrowthinachemo-stat.Thefulllineshowstheinitialrateofthesameprocesswhenthecultureisabruptlyexposedtosupersaturatingglucoseconcentrations.(a)Specicsubstrateuptakerate[ 45 ](b)Specicrespirationrate[ 46 ](c)Specicbiosynthesisrate[ 14 ](d)Specicexcretionrate[ 29 , 45 ]).Thedatain(c)wasobtainedwithglycogenlessmutantsofE.coliB.Allotherdatawasobtainedwithwild-typeK.aerogenes. Thisindicatesthatthesubstrateuptakeandrespirationratesimmediatelyincreasetotheirmaximallevels,whereasthespecicbiosynthesisrateincreasesonlypartially.Consequently,thereisrapidaccumulationofprecursors,resultinginstorageandexcretion.TheinstantaneousimprovementinthespecicgrowthrateofculturesgrowingatlowdilutionratesisduetorapidsynthesisofRNA,protein,andglycogen.Theinstantaneousimprovementinthespecicgrowthrateofculturesgrowingathighdilutionratesisduetorapidsynthesisofglycogen,sincetheRNAandproteinsynthesisratesofsuchculturesincreaseonlyafteralagof30{60minutes.Basedontheseexperimentaldataandthemechanismsdescribedin

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Figure1{7: AutocatalytickineticsofRNAsynthesisinglucose-limitedculturesofAzotobactervinelandiiinresponsecontinuous-to-batchtransitions[ 47 ].Thethreecurvesshowncorrespondtoinitialdilutionratesof0.1,0.15and0.201/hr.Themaximumspecicgrowthrateonglucoseis0.2751/hr. Section 1.3.1 ,wecansynthesizethefollowingpictureofthechemostattransientsinresponsetosubstrate-excessconditions. Inculturesgrowingatlowdilutionrates,biosyntheticenzymes,RNApoly-merase(RNAP)andribosomesaresubsaturatedwithrespecttoprecursors,nu-cleosidetriphosphates(NTPs)andaminoacids,respectively.Whentheexogenoussubstrateconcentrationincreasesinresponsetoashift-up,thespecicsubstraterateimmediatelyincreasescausingrapidaccelerationinthesynthesisratesofalltheintracellularcomponents.Duringthisfasttransient,theconcentrationsofthefastvariables(precursors,NTPs,andaminoacids)increaseinstantaneously,whiletheconcentrationsoftheslowvariables(biosyntheticenzymes,RNAP,ribosomes,RNAandproteins)remainessentiallyconstant.TherapidincreaseofNTPsor/andaminoacidstriggersrRNAsynthesisbythemechanismsdescribedinSection 1.3.1 .ThefasttransientisfollowedbyaslowdynamicduringwhichtheconcentrationofRNAincreasessigmoidally(Figure 1{7 )[ 14 , 47 , 48 ].ThesigmoidalkineticsreecttheautocatalytickineticsofRNAsynthesis.Indeed,anincreaseintheconcentra-tionofRNAimpliesanincreaseintheconcentrationofribosomes.ThispromotesthesynthesisofproteinsingeneralandRNAPinparticular,whichresultsinthesynthesisofevenmoreRNA.

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Inculturesgrowingathighdilutionrates,theRNAandproteinsynthesisratesshownoimprovementonthefasttimescale.Itfollowsthateitherthebiosyn-theticenzymesaresaturatedwithprecursorsortheribosomesaresaturatedwithaminoacids.Thereisevidencesupportingbothconclusions.Harveyhasshownthatthespecicgrowthrateisproportionaltotheactivityofglutamatedehydro-genasethroughoutthetransientresponseinacontinuous-to-batchshift[ 14 ].Ontheotherhand,ithasbeenshownthatwhenaminoacidsareaddedtoaculturegrowingonexcessglucose,thereisnochangeintheproteinsynthesisrateuntiladditionalribosomesaresynthesized[ 2 , 14 , 49 ].Althoughbothhypothesesaretenable,theyimplydierentconsequences.Ifthebiosyntheticenzyme(s)aresatu-ratedwithprecursors,thereshouldbenochangeintheconcentrationofaminoacidmonomers.However,thereshouldbearapidaccumulationofprecursors(leadingtoexcretionor/andstorage)andNTPs.AccumulationofNTPswouldstimulateRNAsynthesiswhichinturnwouldpromotesynthesisofthebiosyntheticenzyme(s).Iftheribosomesaresaturatedwithaminoacidmonomers,thereshouldbeaccumu-lationofaminoacidmonomerswhichinturnwouldstimulatesynthesisofrRNA.Highconcentrationsofaminoacidsshouldpersistuntilribosomalconcentrationsbuilduptosucientlylargelevels. Substratepulse .Inthesubstratepulseexperiments,thereactorismaintainedatsteadystateataxeddilutionrate.Asmallperturbationisgiventotheglucoseconcentrationbyinjectingapulseofextraglucoseinthereactor.Theglucoseconcentrationinstantlyincreasestosupersaturatinglevel,andthecelldisplaysthedynamicssimilartocontinuoustobatchshiftsforrstfewminutes.Asglucoseisbeingconsumedandwashingouttogether,theglucoseconcentrationdecreasescontinuouslyandbecamesubsaturating.Thus,thereactorcomesbacktotheinitialsteadystateinthenextfewhours.Thistypeofperturbationhasbeenwidelyusedtostudytheroleofadeninenucleotides(ATP)asenergymoleculesinthecell.

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(b) Transientresponsetoasubstratepulseinminutestimescale.Att<0,theculturewasatsteadystatecorrespondingtodilutionrated=0:21/hr.Glucosewasinjectedattimet=0tothechemostat. ThecellusesATPasenergymoleculeintheprotein,polysaccharidessyn-thesisetc,andasrawmaterialintheRNA,DNA,Proteinsynthesis.TheATPconcentrationisverysmall6-10mole=gdwformostbacterialcellandyeastcellatsteadystate.AtkinsonhascalculatedtheATPturnoverratesforSalmonellatyphimuriumandEscherichiacoliandconcludedthatATPexhibitsfastdynamics.TheATPpoolwasconsumedandfurtherreplenishedinlessthan38381secondsatdierentgrowthrates[ 50 ]. HarrisonandMitrahaveusedasubstratepulseexperimentstoexaminethestatusofATPconcentrationinsidethecellfortherst10mins[ 17 ].Astheinitialtransientsaresameascontinuoustobatchshift,thesubstrateuptakerateandrespirationratejumpsinstantaneously.Hence,theATPconcentrationshouldalsoincrease,howeverthefallinATPconcentrationwasobserved(Figure 1{8 ).Atinitialconsideration,thefallinATPconcentrationwithanincreasedavailabilityofsubstrateandincreaserespirationrateseemsanomalous.ButitwasexplainedbyanincreasedutilizationofATPforthesynthesisofpolysaccharidesandothercellcomponents. Tocapturethefastdynamicsinsecondstimescale,thepulseexperimentshavebeendoneusingrapidsamplecollectiontechnique[ 51 , 52 ].ThedropinATP

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(b) (c) (d) Transientresponsetoasubstratepulseinsecondstimescale.Att<0,theculturewasatsteadystatecorrespondingtodilutionrated=0:11/hr.Glucosewasinjectedattimet=0tothechemostatculture.Thegureshowtheevolutionofthe(a)Adeninetriphosphate,(b)Adeninediphosphate,(c)Adeninemonophos-phateconcentrationand(d)Energycharge.EnergychargewascalculatedbyusingATP,ADPandAMPconcentration. concentrationwithsubsequentrecoverywasobservedinrst40seconds(Figure 1{9 ).ThedropinATPconcentrationimmediatelyafterincreasedinjectionoftheglucosepulsewasattributedtosuddenincreaseinATPconsumptioncausedbyincreasedintheglucosetransportprocess.Atsteadystate,thetransportenzymesaresubsaturatedwithrespecttoglucose.Whentheglucoseconcentrationincreasesinresponsetoapulse,thetransportenzymebecomessaturatedresultinginimme-diateincreaseinthespecicsubstrateuptakerate.Glucosegetsphosphorylatedinglucose-6-phosphatebyconsumingphosphatefromATPintheprocessoftransport.ThisATPconsumptionresultsinthedecreaseinATPconcentration,therebyincreasingtheADPconcentrationanddecreasingtheenergycharge.

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Overshootinthesubstrateconcentrationoccurswhenthestarvedcellsaresuddenlyexposedtolargesubstrateconcentrationsbyincreasingtheowrateorthesubstrateconcentrationinthefeed.Twofactorscouldcontributetothisphenomenon.Eitherthestarvedcellslacktheenzymewhichtransportsthesubstratefromthemediumintothecell,andaconsiderablelengthoftimeisrequiredtosynthesizetheenzymetoalevelthatishighenoughtomatchtheincreasedsupplyofsubstrate.Orthestarvedcellslacktheribosomalmachineryrequiredtoconvertthecatabolicproductsderivedfromthesubstrateintobiomass.Theresultantaccumulationofthecatabolicproductsrepressessubstrateuptakebyinhibitingthetransportenzyme.Furthermore,ittakesalongtimetobuildtheribosomalmachinerytoalevelconsistentwiththehigherspecicgrowthratethatisrequiredtomatchtheincreasedsupplyofsubstrate[ 2 ]. Weareaddressingthisquestionbysystematicallyinvestigatingtheroleofpotentialcandidates.Inthischapter,westudiedtheroleofperipheralenzymes,i.e.,theenzymesthatcatalyzethetransportandperipheralcatabolismofsub-strates[ 41 ].Thisstudywasmotivatedbythedynamicsofsubstrateswitches.Figure 1{5 showsanexampleofsuchanexperiment.Whenthegrowth-limitingcarbonsourceofaC.heintziicultureisswitchedfromglucosetonitrilotri-aceticacid(NTA),thereisalmostnosubstrateuptake,andhence,nogrowthfor30hours(seeFigures 1{5 a,b).WearguedthatthisisbecausethesynthesisoftheperipheralenzymesforNTAisinducible,andhence,autocatalytic.SincethecellshavenotseenNTAuntilthesubstrateisswitched,theinitialleveloftheperipheralenzymesforNTAisvanishinglysmall(Figure 1{5 c).Becauseinducibleenzyme 21

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synthesisisautocatalytic,ittakesseveralhourstobuildsucientlyhighlevelsoftheseenzymes. Severalstructuredmodelshavebeenproposedinordertocapturesingle-substratedynamics.Theseincludehighlystructuredmodelsaccountingformaintenance,storage,andsynthesisofribosomesandvariousenzymes[ 53 , 54 ].Thegoalofthisworkistostudytheroleoftransportenzymeinsingle-substrategrowth.Themodelhasbeendevelopedtoexplainthedilutionrateshift-upandsubstrateswitchexperiments,wherethesluggishresponseofachemostat,wheneveritoccurs,isexclusivelyduetolowtransportenzymelevelsinthecells.Weassume,thatthereisnomaintenance,and\excess"ribosomesarealwaysavailableforthesecases.Thisassumptionisnottodenytherolesofmaintenanceandribosomalsynthesis,buttodeterminetheextenttowhichenzymesynthesisalonecanaccountfortheobservedlags.ExperimentsshowthatinE.coli,theassumptionof\excess"ribosomesisvalidfordilutionratesupto0:41/hr. 2.1 TheModel ThekineticschemeisshowninFigure 2{1 .Here,Sdenotesthesubstrate,Edenotestheinducibleenzymeor\lumped"systemofinducibleenzymescat-alyzingtheuptakeandperipheralcatabolismofS,XdenotestheinducerforE,Pdenotesthe\lumped"poolofbiosyntheticprecursors,andCdenotesallconstituentsofcellmassexceptE,X,andP.TheentirecellconsistingofE,X,PandCisdenotedbyC.Theconcentrationsoftheseentitiesaredenotedbythecorrespondinglower-caseletterss,c,e,x,p,andc.Here,sg/Landcgdw/Larebasedonthevolumeofthechemostat,whereasx;e;pandcg/gdwarebasedonthedryweightofthebiomass.Steadystateandquasi-steadystateconcentrationsaredenotedbyoverlayingtheseletterswithand,respectively(forinstance,~xandx).Theyield,denotedY,isthemassofPproducedperunitmassofX. Thefollowingassumptionsaremadeaboutthekineticsofthevariousprocesses

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Figure2{1: Kineticschemeofthetransportenzymemodel.Here,Sdenotesthesubstrate,Xdenotestheinducer,Edenotestheinducibleenzyme(s)catalyzingtheuptakeandperipheralcatabolismofS,PdenotesthebiosyntheticprecursorsderivedfromcatabolismofX,Cdenotestheremainingcomponentsofbiomass,andCdenotestheentirecellconsistingofE,X,PandC.Thepositivefeedbacklooprepresentstheinductionofenzymesynthesis. Thespecicsubstrateuptakerate,denotedrs,satisesthekineticlawrsVses Ks+s 55 ].Thus,K1istheequilibriumconstantforbindingofarepressortooneinducermolecule,K2istheequilibriumconstantforbindingofarepressortotwoinducer

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molecules,andK3isproportionaltotheequilibriumconstantforbindingofarepressortoanoperator. Inwhatfollows,weshallappealtothefollowingtwofacts.First,sincerepressor-operatorbindingisnotperfectlytight,ie.,K3isnite,theenzymeissynthesizedevenintheabsenceoftheinducer;thisphenomenonisreferredtoasconstitutiveenzymesynthesis.Second,ifK2K21,asisthecaseforthelacoperon,bindingoftherstinducermoleculetoarepressorfacilitatesthebindingofthesecondinducermolecule,resultingincooperativeorsigmoidalkinetics. Thespecicrateofenzymedegradation,denotedrd,followsrst-orderkineticsrdkde Amassbalanceonthestatevariablesyields dt=D(sfs)rsc dt=rsrxD+1 dtx dt=Yrxre+rdrpD+1 dtp dt=rerdD+1 dte dtc wherethelasttermin( 2.2 { 2.5 )representsthedilutionofX,P,E,andC,respectively,byeuentowandgrowth,sfdenotestheconcentrationofSinthefeed,andDdenotesthedilutionrate.Itisshownin[ 56 ]thatsince

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Themassfractionofallintracellularentitiesequalsunityx+p+e+c=1g=gdw dt=dp dt=0)rxrs;rcYrx 2.1 { 2.5 )areapproximatedbythefollowingequations dt=D(sfs)Vses Ks+sc dt=Ve1+K1x+K2x2 Ks+s+kde dt=YVses Ks+sDc Ks+s Ks+s forallbutanegligiblysmallinitialtimeinterval. 2.2 Simulations ThesimulationsweredonewithparametervaluesinTable 2{1 . Table2{1: Parametervaluesusedinthetransportenzymemodelsimulations. gdwhrKs=102g LY=0:4gdw gkx=9001 gdwhrK1=105gdw gK2=1011gdw g2K3=105kd=1021 SteadyStates Equations( 2.6 { 2.10 )admittwotypesofsteadystates,namely,thepersistence(~c6=0)andthewashout(~c=0)steadystates;wedenotethemby1and0,respectively.

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Thepersistencesteadystate,1,isunique,nomatterwhattheparametervalues.Atthepersistencesteadystate,thespecicgrowthrateequalsthedilutionrate;thatis Ks+~s=D(2.11) Itimmediatelyfollowsthat ~x(D)=D Ykx ~p(D)=D kg ~e(D)=1 ~s(D)=KsD YVs~eD ~c(D;sf)=Y(sf~s) (2.16) Atfeedconcentrations(sf=1:5g/L),thereisauniquewashoutdilutionrateatDw=0:81/hr,andthebifurcationdiagramisformallysimilartothebifurcationdiagramfortheMonodmodel(Figure 2{2 ).Belowthe(unique)washoutdilutionrate,thepersistencesteadystateisgloballystable;abovethewashoutdilutionrate,thewashoutsteadystateisgloballystable. 2.2.2 Transients 2.2.2.1 Dilutionrateshifts Intheseexperiments,att<0,thecultureisinasteadystateatsomedilutionrate,sayD0,andfeedconcentration,sf.Att=0,thedilutionrateisshifteduptoanewvalue,D>D0,whilethefeedconcentration,sf,isheldxed.Iftheshift-up,DD0,issmall,thehigherspecicgrowthratethusachievedequalsthehigherdilutionrate,D.Inthiscase,thechangeinthesubstrateconcentrationandcelldensityisimperceptiblysmall.Iftheshift-up,DD0,islarge,thehigherspecicgrowthrateachievedonthefasttimescalefallsshortofthedilutionrate,D.Therapidincreaseofthespecicgrowthrateisfollowedbyaperiodofslowincrease

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Figure2{2: VariationofsteadystateswithDatsf=1:5g/L.Thewashoutbi-furcationpointisdenotedbythesymbol.Stableandunstablesteadystatesaredenotedbyfullanddashedlines,respectively. duringwhichsincreasesandcdecreases.Afterseveralhoursordays,thespecicgrowthrateoftheculturecatchesupwiththeincreaseddilutionrate.Atthisinstant,sreachesamaximumandcattainsaminimum.Thereafter,thespecicgrowthrateishigherthanD;duringthisperiod,sdecreasesandcincreasesuntiltheculturereachesanewsteadystateconsistentwiththenewdilutionrateD. Weshowthatallthreefeaturesaremirroredbythemodel.IfD0issmallcomparedtoD,theinitialenzymelevelissosmallthatthespecicgrowthrate

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cannotmatchthenewdilutionrateevenafterthesubstrateconcentrationhasreachedsupersaturatinglevels(sKs).Tomitigatethisgrowthratedecit,thecellsbegintheslowprocessofenzymesynthesisundersupersaturatingsubstrateconcentrations.Duringthisperiod,whichlastsseveralhours(seeFigure 2{3 ),sincreasesandcdecreases.ThiscontinuesuntilthespecicgrowthratebecomesequaltoD.TheorbitthenenterstheregioninwhichthespecicgrowthrateexceedsD,andbeginsitsnalapproachtothesteadystate.Duringthisperiod,econtinuestogrow,butsdecreasesuntilthesteadystateisachieved. Figure2{3: Trajectoriesfordilutionrateshiftstothenaldilutionrate,D=0:61/hr,frominitialdilutionrates,D0=0:031/hr(fulllines)andD0=0:31/hr(dashedlines);thefeedconcentration,sf,isheldxedat1:5g/L. 2.2.2.2 Feedswitches Intheseexperiments,attimet<0,thecultureisallowedtoreachasteadystate.Attimet=0,theidentityofthegrowth-limitingsubstrateischangedwhile

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holdingthedilutionrateandthefeedconcentrationxedattheiroriginalvalues.Undersuchperturbations,itisobservedthat[ 37 , 40 , 43 ] Ifthenewsubstratesupportsarelativelylargeconstitutiveenzymesynthesisrate,theculturerecoversrapidlyandthereisnoobservablesubstrateovershoot;ifthenewsubstratesupportsarelativelysmallconstitutiveenzymesynthesisrate,thereisaverypronouncedlagbeforetheexcursionofthesubstrateconcentrationcanbecontained. Toconsiderthedynamicsofthemodel,thefullsystemconsistingofequations( 2.6 { 2.8 )mustbeconsideredalongwiththeinitialconditionss(0)=0;e(0)=e0=Ve SimulationsshowthatthetimerequiredforthechemostattoachievethenewsteadystatedependsverystronglyonK3.WhenK3=105,ittakes1000hoursforthesubstrateconcentrationtoreachvaluescomparabletoKs.Suchlonglagshave,andwill,neverbeobserved;therefore,weassumedthatK3104.WhenK3=104,therecoverytimeisontheorderof10hours(Figure 2{4 );whenK3=5104,therecoverytimeincreasesbyanorderofmagnitude(Figure 2{5 ).Bothguresalsoshowthattherecoverytimedependsonthedilutionrateatwhichtheidentityofthesubstrateisswitched.Intermsofourmodel,thisphenomenonhasasimpleexplanation.Atlargedilutionrates,themassowrateofthenewsubstrateis

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higherandtheinitialenzymelevelislower;thatis,ahighersubstrateburdenisimposeduponcellsthatareevenlesscapableofconsumingthesubstrate. Figure2{4: TrajectoriesforaswitchintheidentityofthesubstrateatD=0:011/hr(fulllines)andD=0:11/hr(dashedlines);K3wasassumedtobe104. 2.3 Discussion Thegoalofthisworkistoadvanceourunderstandingofdynamicsofachemostatinresponsetosuddenchangesintheowrate,oridentityofasinglegrowth-limitingsubstrate.Tothisend,wehaveanalyzedasimplemodelaccount-ingforinductionanddilutionofthetransportenzyme.Theparametervaluesofthemodelarebasedonthelacoperon.Despitethesimplicityofthemodel,theresultsarepromising.WeshowthatAthighfeedconcentrations(sf1g/L),thesimulationsagreequalitativelywithknownexperimentaldataforperturbations,namely,abruptincreasesinowrate,andchangeintheidentityofthesubstrate.Wendthatinallcases,thesubstrateconcentrationincreasesforseveralhours

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Figure2{5: TrajectoriesforaswitchintheidentityofthesubstrateatD=0:011/hr(fulllines)andD=0:11/hr(dashedlines);K3wasassumedtobe5104. beforeitcanbecontained.Accordingtoourmodel,thisoccursbecausetheinitialtransportenzymelevelsaretoosmalltocopewiththeincreasedsubstratesupply.Themagnitudeofthesubstrateexcursionismoststrikingwhenonechangestheidentityofthesubstratefeedingintothechemostat.Inthiscase,theinitialenzymelevelisatitssmallestsincethereisnoprioradaptationtothenewsubstrate;theresultinglagsareontheorderofseveraldays. Moredataisneededtotestquantitativepredictionsofthemodel.WeknowthatPowell'smodel,whichisformallysimilartothemodelpresentedhere,yieldsresultsthatagreequantitativelywiththedata,providedtheparametervaluesforsynthesisofthe\Q-substance"aresuitablychosen[ 36 , 57 ].However,torigorouslytestthequantitativepredictionsofthemodel,itisnecessarytoobtaintransientdataforasubstratesuchaslactosewhoseenzymeinductionkinetics

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areindependentlymeasurable.Worktothiseectiscurrentlyinprogressinourlaboratory.

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Inpreviouschapter,westudiedtheroleoftransportenzymestoexplainthedilutionrateshift-upandsubstrateswitchexperimentsinwhichthelowleveloftransportenzymelimitsthegrowth.Itwasshownthatthelonglagcouldoccurbecauseofautocatalyticsynthesisoftransportenzyme,andthemodelaccountingfortransportenzymesynthesiscancapturetheexperimentaldataquantitatively. Itiswellknownthatsimilardynamicsareobservedeveniftheglucose-limitedchemostatissubjectedtodilutionrateshift-ups.Whenaglucose-limitedcultureofE.coliissubjectedtoadilutionrateshift-up,thecelldensitydecreasesandthesubstrateconcentrationincreases(Figure 3{1 ).ButthesteadystateperipheralenzymelevelsarehighatallD>0:11/hr(Figure 1{3 b).Infact,theenzymelevelsareevenhigherthanthelevelsobservedatthemaximum(washout)dilutionrate.Theexistenceofsuchhighperipheralenzymelevelssuggeststhatthedynamicsofdilutionrateshift-upsstartingfromsucientlylargeinitialdilutionratescannotbeduetoslowsubstrateuptake.Thenwhatisresponsibleforthesluggishresponseinsuchdilutionrateshift-ups? Thesourceoftheslowdynamicsindilutionrateshift-upsisrevealedbyexaminingtheinitialresponsetocontinuous-to-batchshifts.Intheseexperiments,thecells,maintainedatsteadystateinachemostat,areabruptlyexposedtoexcesssubstrateconcentrations.Theinitialresponseisobtainedbymeasuringtheratesofvariousprocesseswithin10{15minutesoftheshifttosubstrate-excessconditions.Figure 1{6 showstheinitialresponseofthesubstrateuptake,biosynthesis,respiration,andexcretionincontinuous-to-batchshiftsofglucose-limitedcells.Itisevidentthatthespecicsubstrateuptakerateincreasestothe 33

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(b) (c) Transientresponseofaglucose-limitedcultureofE.coliK12toadilutionrateshift-up(from[ 48 ]).Att<0,thecultureisatthesteadystatecorre-spondingtothedilutionrate,D0=0:21/hr,andfeedconcentration,sf=5g/L.Att=0,thedilutionrateisshifteduptoD=0:61/hr,whilethefeedcon-centrationisheldconstant.Theguresshowtheevolutionofthe(a)Cellden-sityandsubstrateconcentration(b)Ribosomelevel(c)Specicsubstrateuptakeandgrowthratescalculatedfromthecurvesin(a)byappealingtotheformulas,rs=[D(sfs)ds=dt]=candrg=(dc=dtDc)=c,whichfollowfromthemassbalancesforthesubstrateandcells. maximallevelsobtainednearthewashoutdilutionrate(Figure 1{6 a).However,thespecicrateofRNAandproteinsynthesisincreasesonlypartiallyiftheculturehasbeengrowingatlowdilutionrates,andshowsnoperceptibleincreaseiftheculturehasbeengrowingathighdilutionrates(Figure 1{6 b).Itfollowsthatwhencellsgrowingatsteadystateinachemostatareexposedtosubstrate-excessconditions,thesubstrateentersthecellatnear-maximalrates,butthecatabolicproductsderivedfromitare,atbest,onlypartiallychanneledintobiosynthesis.Theexcesssubstrateiseliminatedbyinstantlyincreasingtheratesofrespiration

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(Figure 1{6 c),excretion(Figure 1{6 d),andstorage[ 12 , 14 ].Thus,weconcludethattheresponseissluggishindilutionrateshift-upsbecausethespecicRNAandproteinsynthesisratesdonotachievemaximallevelsinstantly. Theinitialresponseofcontinuous-to-batchshiftsrevealstheidentityofthevariablesthatpreventthebiosynthesisratefromincreasinginstantly,butshedsnolightonthereasonfortheslowresponseofthevariables.Wegainsomeinsightintothemechanismoftheslowresponsebyexaminingtheevolutionofribosomelevelsincontinuous-to-batchshifts(Figures 1{7 ).Thesetransientssuggestthatthesynthesisofribosomesisautocatalytic.Thesynthesisratesaresmallinitially,acceleratesubsequently,andsubsidenallyafterpassingthroughaninectionpoint.ItisconceivablethattheseautocatalytickineticsoccurbecauseanincreaseintheactivityofRNAresultsinelevatedsynthesisofprotein,whichactivatesthesynthesisofribosomalRNAandribosomes[ 4 , 5 , 58 ].Inwhatfollows,weassumethatsynthesisofRNAisautocatalyticwithoutmakinganyattempttomodeltheunderlyingmolecularmechanism. 1{6 , 3{1 andadditionaldatadiscussedbelow.Webeginbyformulatingtheextendedmodel(Section 3.1 ).Wethensimulateandanalyze 59 ]andKoch[ 60 ],whoassumedthattheproteinsynthesisratewasproportionaltotheconcentrationofRNA,andtheRNAsynthesisrate,inturn,wasproportionaltotheconcentrationofproteins.ItwasmadeexplicitlybyPowell[ 57 ]andYunetal[ 48 ],whosimplyassumedthatthesynthesisrateofRNAisproportionaltotheconcentrationofRNA.However,BremerandKochdidnotaccountforthedynam-icsofthesubstrateconcentration.PowellandYunetalaccountedforsubstratedynamicsbyassumingaconstantyield.Thiscouplessubstrateuptakeandgrowth,thusprecludingthecelldensityovershootobservedindilutionrateshift-downs(see[ 48 ,Figure6]).

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themodeltoshowthatityieldsresultsinqualitativeagreementwiththedata(Section 3.2 ).Finally,wediscusstheextenttowhichthemodelcapturesthekeyresultsoftheexperimentalliterature(Section 3.3 ). 3.1 TheModel Figure 3{2 showsthekineticschemeofthemodel.Here,Sdenotesthegrowth-limitingcarbonandenergysource,EdenotestheperipheralenzymesthatcatalyzethetransportandperipheralcatabolismofthecarbonsourceandXdenotestheinternalizedformofthesubstratethatinducesthesynthesisofE;PdenotesthepoolofprecursorsproducedbycatabolismofX;RdenotesribosomesorribosomalRNA(rRNA); Throughoutthiswork,theinstantaneousconcentrationsofthevariablesaredenotedbythecorrespondinglower-caseletterss;e,x,p,r,c,andc,whilesteadystateandquasisteadystateconcentrationsaredenotedbyoverlayingtheletterswithand,respectively(forinstance,~xandx).Theconcentrationsoftheenvironmentalvariables,sandc,arebasedonthevolumeofchemostat(g/Landgdw/L,respectively),andtheconcentrationsofthephysiologicalvariables,e,x,p,r,andcarebasedonthedryweightofbiomass(g/gdw). Wemakethefollowingassumptionsregardingthekineticsoftheprocesses. 3 ].

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Figure3{2: Kineticschemeofthemodel.Here,Sdenotesthesubstrate,Ede-notestheinducibleenzyme(s)catalyzingtheuptakeandperipheralcatabolismofS,XdenotestheinducerforE,PdenotesthebiosyntheticprecursorsderivedfromcatabolismofX,RdenotesrRNA,Cdenotesproteins,andCdenotestheentirecellconsistingofE,X,P,R,andC.ThepositivefeedbackloopsrepresentinductionofenzymesynthesisandautocatalyticsynthesisofRNA.Thenegativefeedbacklooprepresentstheinhibitionofsubstrateuptakebyprecursors. 1. Thespecicsubstrateuptakerate,denotedrs,satisesthekineticlawrsVses Ks+s1 1+p=Ki 1{3 b),rsshouldbethree-foldhigherforculturesgrowingatD=0:21/hrcomparedtoculturesgrowingatD=0:61/hr.ButFigure 1{6 ashowsthatrsisthesameforculturesgrowingatdilutionratesbetween0.2and0.61/hr.Thissuggeststhatinculturesgrowingatlowdilutionrates,feedbackinhibitionactstopreventthespecicsubstrateratefromexceedingitsvalueathighdilutionrates. 2. ThespecicrateofbreakdownofXintoenergyandprecursorsP,denotedrx,isgivenbyrxkxx Thespecicrateofrespirationisrco2kco2p

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4. Thespecicrateofenzymesynthesis,denotedr+e,is Ke+r1+K2x2 UnderlyingthesekineticsisthefactthattheenzymesynthesisrateispreciselytherateatwhichthemessengerRNA(mRNA)fortheenzymeistranslatedbytheribosomes.Weassumethatthespecictranslationrateoftheenzymeiskemr=(Ke+r),wheremandrdenotetheconcentrationofmRNAandribosomes,respectively.Now,theevolutionofmRNAisgivenbytheequationdm=dt=r+mrmrgm,wherer+mVm(1+K2x2)=(K3+K2x2)isthespecictranscriptionrate, 61 ],itrapidlyreachesthequasisteadystateconcentration,mr+m=km.Substitutingthisconcentrationintheexpressionforthespecictranslationrateyields( 3.1 )withVekeVm=km. 5. Thespecicrateof(ribosomal)RNAsynthesisisgivenbyr+rk+rrp 6. Thespecicrateofprotein(C)synthesisisgivenbyr+cVcrp Kc+p 7. Thespecicratesofperipheralenzyme,rRNA,andproteindegradation,denotedre,rr,andrc,respectively,aregivenbyrekee;rrkrr;rckcc 55 ].Thus,K2istheequilibriumconstantforbindingofarepressortotwoinducermolecules,andK3isproportionaltotheequilibriumconstantforbind-ingoftherepressortotheoperator.Notethatsincerepressor-operatorbindingisnotperfect,K3isnite,sothatmRNAistranscribedevenintheabsenceoftheinducer(r+mjx=0=(Vm=K3)>0).Thisphenomenoniscalledconstitutiveenzymesynthesis.

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8. Thesynthesisofperipheralenzymes,rRNAandproteinsdepletesthepoolofprecursors.Likewise,theirdegradationreplenishesthepoolofprecursors. Amassbalanceonthestatevariablesyields dt=D(sfs)rsc dtc dt=r+ereD+1 dte dt=r+rrrD+1 dtr dt=rsrxD+1 dtx dt=rxrco2(r+crc)(r+ere)(r+rrr)D+1 dtp wheresfdenotestheconcentrationofSinthefeed,andDdenotesthedilutionrate.Theseequationsdene(1=c)(dc=dt)implicitlyintermsoftheotherderiva-tives.Wecansolvefor(1=c)(dc=dt)explicitlybyobservingthatthemassfractionofallintracellularentitiesequalsunity,i.e. Hence,additionof( 3.3 { 3.7 )yieldstheequation dt=rsrco2(3.9) whichcanberewritteninthemorefamiliarformdc dt=(rgD)c

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Equations( 3.9 )and( 3.10 )implythatthelastterminequations( 3.3 { 3.7 )repre-sentsthedilutionofthecorrespondingphysiologicalconcentrationduetogrowth. Thus,wearriveattheequations dt=D(sfs)rsc dt=(rgD)c dt=r+ererge dt=r+rrrrgr dt=rsrxrgx dt=rxrco2(r+crc)(r+ere)(r+rrr)rgp wherergisgivenby( 3.10 ).Itisworthnotingthat( 3.8 )impliesthatequa-tions( 3.13 { 3.17 )arelinearlydependent.Hence,wecanreplaceanyoneoftheseequationswith( 3.8 ). 3.2 Simulations ThesimulationsweredonewiththeparametervaluesinTable 3{1 . Table3{1: Parametervaluesusedintheribosomemodelsimulations. ghrKs=102g LKi=5103kx=103g ghrVe=5104g gdwhrKe=0:1g gdwK2=1010gdw g2K3=5103ke=0:075g ghrk+r=100gdw ghrkr=0:1g ghrVc=3g ghrKc=0:002g gdwkc=0:05g ghrkco2=150g ghr

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3.2.1 SteadyStates Toanalyzethesteadystates,wereplace( 3.17 )with( 3.8 ).Thus,weconsidertheequations dt=D(sfs)Vses Ks+s1 1+p=Kic=0 (3.18) dt=(rgD)c=0 (3.19) Kc+prg+kcc=0 (3.20) dt=Ver Ke+r1+K2x2 (3.21) dt=k+rprgkrr=0 (3.22) dt=Vses Ks+s1 1+p=Ki(rg+kx)x=0 (3.23) (3.24) wherergisgivenby( 3.10 ).Theseequationsadmittwotypesofsteadystates|thepersistencesteadystate(~c>0),andthewashoutsteadystate(~c=0).Foragivenmicrobialspeciesandgrowth-limitingsubstrate,thesesteadystatescandependontwoparameters|thedilutionrateandthefeedconcentration. 3.2.1.1 Persistencesteadystate Thevariationofthepersistencesteadystatewithrespecttothefeedconcen-trationissimple.Ifthefeedconcentrationisincreasedataxeddilutionrate,thecelldensityincreases,butthereisnoperceptiblechangeinthesubstrateconcen-tration[ 11 , 62 ].Thereisnodataonthevariationofthephysiologicalsteadystateswithrespecttothefeedconcentration.However,sincethephysiologicalstateiscompletelydeterminedbythesubstrateconcentration,thesesteadystatesshouldalsobeindependentofthefeedconcentration.Weshowbelowthatthispropertyisinherentinthemodel. Thevariationofthepersistencesteadystatewithrespecttothedilutionrateismorecomplex.Thesubstrateconcentrationandribosomelevelsincrease

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(b) (c) (d) (e) (f) VariationofthesteadystateswithDatsf=5g/L.Thepersistence,andwashoutarerepresentedbyblack,andredlines,respectively.Stableandunsta-blesteadystatesaredenotedbyfullanddashedlines,respectively.Thefullcircleshowsthebifurcationpointatwhichthepersistenceandwashoutsteadystateexchangestability.

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monotonically(Figure 1{3 a,d).Ontheotherhand,thecelldensityandperipheralenzymelevelspassthroughamaximum(Figure 1{3 a,b).Figure 3{3 showsthatthemodelsimulationsareingoodagreementwiththisdata.Inwhatfollows,weshedmorelightonthesesimulationsbyderivingexplicitexpressionsforthesteadystatevaluesofallthevariables. Tothisend,observethatsince~c;~r>0atthepersistencesteadystate,( 3.19 )and( 3.22 )implythat~rg=Dand ~p(D)=D+kr Thus,~pincreaseslinearlywithD(Figure 3{3 f).Interestingly,asDtendstozero,~papproachesapositivelimit.Thus,weobtain\maintenanceeects,"eventhoughnomaintenancewasexplicitlypostulated.Indeed,substituting( 3.25 )in( 3.10 )yields ~rs=D+~rco2=D+kco2~p=D+kco2D+kr where^Yk+r=(k+r+kco2)isthemaximumyieldofbiomass,andmkco2(kr=k+r)isthemaintenancecoecient(whichreectsthe\futilecycling"ofrRNAatvanishinglysmalldilutionrates). Itfollowsimmediatelyfrom( 3.23 )and( 3.26 )that ~x(D)=~rs kx+DD=^Y+m kx(3.27) sothat~xincreaseslinearlywiththedilutionrate(Figure 3{3 e).Tondtheconcentrationsoftheremainingphysiologicalvariables,weobservethat( 3.20 )and( 3.21 )implythat ~c(~r)=~rc Kc+~p;~e(~r)=~r+e Ke+~r1+K2~x2

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Substitutingtheserelationsin( 3.24 )yields ~r+Vc Kc+~p+Ve Ke+~r1+K2~x2 whichisaquadraticin~r.SincetheLHSisanincreasingfunctionof~r,andtheRHSisindependentof~r,( 3.29 )hasatmostonepositiverootatanygivenD.Asimpleapproximationto~r(D)isobtainedbyrecognizingthat~e;~x;~p1.Inthiscase,( 3.29 )yieldstherelation ~r Athighdilutionrates,Dkc;krandproteinsynthesisissaturated(pKc),sothat~r Vc+D 3{3 d).Thesteadystateenzymelevelisobtainedbysubstituting~r(D)in( 3.28 ).ItpassesthroughamaximumsincetheenzymesynthesisratesaturatesatlargeD,i.e.,~r+eVe(Figure 3{3 c). Thesteadystatesubstrateconcentrationcanbederivedbyappealingtothedenitionofrs.Thus ~s Ks+~s=~rs Wegainsomeinsightintothevariationof~swithrespecttoD(Figure 3{3 a)byrecallingthat~rsand~pincreaselinearlywithD.Athighdilutionrates,~eVe=D,sothat~s=(Ks+~s),andhence,~s,increaseswithD. ThevariationofthesteadystatecelldensitywithD(Figure 3{3 b)followsimmediatelyfrom( 3.18 ).Thus ~c=D(sf~s) ~rs(3.32)

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Atlowdilutionrates,~ssfand~rsm;hence,~c(sf=m)D.Athighdilutionrates,~rsD=^Y,sothat~c^Y(sf~s). Notethatthesteadystateconcentrationsofthesubstrateandthephysiolog-icalvariablesareindependentofthefeedconcentration,sf.ThisisbecausethesteadystateconcentrationsofSandthevephysiologicalvariables(C,E,R,X,andP)arecompletelydeterminedbythesixequations( 3.19 { 3.24 ),whichareindependentofthefeedconcentration. 3.2.1.2 Washoutsteadystate Asnotedabove,athighdilutionrates,thesubstrateconcentrationcorre-spondingtothepersistentsteadysteadyisanincreasingfunctionofD.WhenDbecomessucientlylarge,~s=sfand~c=0,i.e.,thecellswashoutofthechemostat.Werefertothissteadystateasthewashoutsteadystate,andthecorre-spondingdilutionrate,denotedDw,asthewashoutdilutionrate.Thephysiologicalvariablescorrespondingtothissteadystatearedeterminedbyequations( 3.20 { 3.24 )withrg=rsrco2and~s=sf.Itfollowsthatthewashoutsteadystateisindependentofthedilutionrate. Sincethewashoutsteadystateisachievedwhenthesubstrateconcentrationcorrespondingtothepersistentsteadystateequalssf,thewashoutdilutionratesatisesthefollowingequationobtainedbyletting~s=sfin( 3.31 ) where~p,~rs,and~earegivenby( 3.25 ),( 3.26 )and( 3.28 ),respectively.Now,intypicalexperiments,sf1g=L,sothatsf=(Ks+sf)1.Moreover,atlargedilutionrates(DDw),~p,~rs,and~ecanbeapproximatedbytherelations~pD k+r;~rsD

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sincethelossofenzymeandrRNAbydegradationisnegligiblecomparedtotheirlossbydilutionduetogrowth(ke;krD);substrateconsumptionformaintenance(asopposedtogrowth)isnegligiblysmall(mD=^Y);andenzymeinductionisnear-maximal(r+eVe).Wesubstitutetheserelationsin( 3.33 )toconcludethatthewashoutdilutionratesatisestheapproximaterelation1 3.2.2 Transients Tosimulateandanalyzethetransients,wereplace( 3.13 )with( 3.8 ).Thus,weconsidertheequations.ds dt=D(sfs)rscdc dt=(rgD)cde dt=r+erergedr dt=r+rrrrgrdx dt=rsrxrgxdp dt=rxrco2(r+crc)(r+ere)(r+rrr)rgpc=1erxp dt=dp dt=0 thequasi-stateimpliesthatforallbutanegligiblysmalltimeinterval,theseequationscanbeapproximatedbythereducedequations

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dt=D(sfs)Vses Ks+s1 1+p=Kic dt=(rgD)c dt=Ver Ke+r1+K2x2 dt=k+rrpkrrrgr 0Vses Ks+s1 1+p=Kikxx 0kxxkco2prg wherergisgivenby Kc+pkc(1r)+k+rrpkrr:(3.40) Thatis,thespecicgrowthrateiseectivelyequaltothespecicrateofnetrRNAandproteinbiosynthesis. 3.2.2.1 Continuous-to-batchshifts Incontinuous-to-batchshifts,theinitialratesofvariousprocessesaremea-suredwithin10{15minutesofexposingtheculturetosubstrate-excessconditions(Figure 1{6 ).Tosimulatetheseexperimentswiththemodel,weobservethatinthisshorttimeperiod,theperipheralenzymeandribosomelevelsremainessen-tiallyunchanged,buttheinducerandprecursorconcentrationsrapidlymovetothenewquasisteadystatecorrespondingtosubstrate-excessconditions.Thisnewquasisteadystateisobtainedbylettings=(Ks+s)1in( 3.38 { 3.39 ),whiletheperipheralenzymeandribosomelevelsareheldattheirinitialsteadystatevalues,e0=~e(D)andr0=~r(D).Thus xVse0=(1+p=Ki)

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(b) (c) (d) (e) (f) (g) (h) Initialresponsefollowingacontinuous-to-batchshift.Thedashedlineshowstheinitialsteadystatevalue.Thefulllineshowsthevalueafterthecellshavebeenexposedtosubstrate-excessconditions:(a)Specicsubstrateuptakerate(b)Specicrespirationrate(c)Specicgrowthrate(d)Yieldofbiomass(e)Netspecicproteinsynthesisrate(rcr+crc)(f)NetspecicrRNAsynthesisrate(rrr+rrr)(g)Precursorconcentration(h)Inducerconcentration.

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andpsatisestheequation 1+p=Kikco2p+Vcr0p Kc+pkc(1r0)+k+rr0pkrr0(3.42) IfweassumethatthedegradationratesofrRNAandproteinarenegligiblysmallcomparedtotheirsynthesisrates,andproteinsynthesisissaturated(pKc),( 3.42 )canbesolvedtoyieldtheapproximatesolution pKi 1+Vcr0 whichisaccuratetowithin10%oftheexact(numerical)solutionforalle0andr0.Theratesofvariousprocessesimmediatelyafteracontinuous-to-batchshiftcannowbeobtainedbysubstitutingtheaboveconcentrationsintotheappropriatekineticexpressions(forexample,rs=Vse0=(1+p=Ki)andrco2=kco2p). Figure 3{4 showstheratesofvariousprocessesandthequasisteadystateconcentrations(x,p)immediatelyafterthecontinuous-to-batchshift.ThesimulatedratesareconsistentwiththeexperimentaldatashowninFigure 1{6 .Atallbutthesmallestdilutionrates,thespecicsubstrateuptakeandrespirationratesimmediatelyjumptohighlevels(Figure 3{4 a,b),butthespecicgrowthrateincreasesonlypartially(Figure 3{4 c).Thus,substrateuptakeandgrowthbecomeuncoupledimmediatelyaftertheshift,resultinginareductionoftheinstantaneousyield,Yrg=rs(Figure 3{4 d).Theincreaseinthespecicsubstrateuptakeandgrowthratesislargestatintermediatedilutionrates,anddecreasesatlowandhighdilutionrates.Athighdilutionrates,substrateuptakeisalreadysaturatedbeforethecellsareexposedtosubstrate-excessconditions(s0Ks),sothatfurtherprovisionofthesubstrateprovokesnoadditionalresponse.Atlowdilutionrates,theinitialperipheralenzymearetoosmalltosupportasubstantialincrementin

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thespecicsubstrateuptakerate,andtheinitialrRNAlevelsaretoosmalltosupportasubstantialincrementintheproteinandrRNAsynthesisrates(Figure 3{4 e,f). 3.2.2.2 Substrateswitch Thetransientresponsetosubstrateswitchescouldbecapturedbyourear-liermodel(seeFigure 1{5 ).Simulationsoftheextendedmodelpreservethesetransients.Thespecicsubstrateuptakeandgrowthratesremainatvanishinglysmalllevelsfortherst20hours(Figure 3{5 c,f).Consequently,thesubstrateconcentrationandcelldensitygothroughamassiveovershootandundershoot,re-spectively(Figure 3{5 a,b).Theperipheralenzymelevelshowsabiphasicresponse:Afteraninitialincrease,itbecomesmoreorlessconstantatt25hoursbeforeincreasingonceagain(Figure 3{5 d).ThesesimulationsareingoodagreementwiththedatashowninFigure 1{5 .Inadditiontothesedynamics,theextendedmodelalsodescribestheevolutionofrRNA(Figure 3{5 e). Wecandecomposethesedynamicsintofourphases.Itisworthexaminingthesephasesinmoredetail.Asweshowlater,thedynamicsofdilutionrateshift-upsandshift-downsreproducethedynamicsofoneormoreofthesephases. Phase1 .Duringtherstphase,thesubstrateattainssupersaturatingconcen-trationswithin10{15minutes.Indeed,sincetheinitialenzymelevelisnegligiblysmall[e0=(Ve=K3)(D+ke)r0=(Ke+r0)106g/gdw],soistheinitialsub-strateuptakerate.Theinitialmotionofthesubstrateconcentrationis,therefore,approximatedbytheequation dtDsfDs;s(0)=0)s(t)sf[1exp(Dt)]:(3.44)

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(b) (c) (d) (e) (f) TransientresponseforaswitchintheidentityofthesubstrateatD=0:21/hr.Beforetheswitch,thecultureisinasteadystatecorrespondingtotheabsenceofthesubstrate.TheverticaldashedlinesmarktheendofPhases1,2and3.

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(b) Phaseportraitsoftheslowmotionduringafeedswitches.(a)Theslowmotiontowardbalancedgrowth.Thelinewithshortdashesshowsthenull-clinefore;thelinewithlongdashesshowsthenull-clineforr;theintersectionofthetwonull-clinesrepresentstheconcentrationofeandratbalancedgrowth;thefulllineshowstheapproachofeandrtowardthestateofbalancedgrowth.(b)Theslowmotionawayfrombalancedgrowthtotheultimatesteadystate.Thelinewithshortdashesshowsthenull-clineforc;thelinewithlongdashesshowsthenull-clineforr;theintersectionofthetwonull-clinesrepresentsthesteadystateconcentrationsofcandratD=0:21/hr;andthefulllineshowsthemotionofcandrfrombalancedgrowthtowardthenalsteadystate. ThisequationdescribesthetheoreticalwashoutcurveshownasadashedcurveinFigure 1{6 a.Itimpliesthatthesubstrateconcentrationincreasestosupersaturat-inglevels(sKs)withinln(1Ks=sf) 3.41 )and 3.43 ). Phase2 .Duringthesecondphase,whichlastsabout30hours,thesubstrateconcentrationissupersaturating,i.e.,s=(Ks+s)1.Undertheseconditions,

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thesubstrateconcentrationandcelldensitychange,butthesechangescannotbediscernedbythecells,sincetheyseetheenvironmentthroughtheratios=(Ks+s),andthisratioisapproximatelyconstant.Giventhisquasiconstantenvironment,thephysiologicalvariablesevolvetowardaquasisteadystate.Weshallrefertothistransientassubstrate-sucientbatchdynamicsbecausethequasisteadystatereachedbythephysiologicalvariablesisidenticaltothephysiologicalstateattainedbyasubstrate-sucientbatchcultureduringtheexponentialgrowthphase:Allthephysiologicalvariablesareconstant,andthecellsgrowexponentiallyattheirmaximumspecicgrowthrate(Figure 3{5 f).Inthemicrobiologicalliterature,thisquasisteadystateisfrequentlyreferredtoasthestateofbalancedgrowth. xVse=(1+p=Ki) pKi 1+Vcr Themotiontowardbalancedgrowthis,therefore,two-dimensional:Theslowvariables,eandr,evolvegraduallyaccordingtoequations( 3.36 { 3.37 ),whilethefastvariables,xandp,constantlyadjusttotheslowmotioninaccordancewith( 3.45 { 3.46 ).Figure 3{6 ashowsthephaseportraitforthistwo-dimensionalmotion.Notethat

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correspondingtobalancedgrowthareimaginarywithnegativerealparts(1;2=1:10:5).Thequasisteadystateis,therefore,astablefocus,i.e.,thephasepathspiralsintothequasisteadystate. Thesedynamicsreectthefactthatthespecicenzymesynthesisrateincreasesrst,followedbythespecicribosomesynthesisrate,thespecicproteinsynthesisrate,andnally,thespecicgrowthrate.Thus,theperipheralenzymeandribosomelevelsgothroughmaximabecausetheirsynthesisratesincreasetoomuchbeforetheirdilutionratescatchup,andtheperipheralenzymelevelreachesitsmaximumbeforetheribosomelevelbecausethesynthesisrateoftheenzymeincreasesbeforethesynthesisrateoftheribosomes. Phase3 .Duringthethirdphase,thesubstrateconcentrationswitchesfromsupersaturatingtosubsaturatinglevels.Thistransition,whichmarkstheendofbalancedgrowth,occursonatimescaleof1min.Toseethis,observethatatthebeginningofthisphase,thesubstrateconcentrationisontheorderofKs,andthesubstrateuptakerate,rsc,isontheorderof1g/(L-hr).ItfollowsthatwithinKs=(rsc)0:01hrsoftheendofbalancedgrowth,thesubstratereachesthequasisteadystateconcentrationdenedbytheequation Atthesametime,theinducerandprecursorlevels,whichconstantlyadjusttotheexogenoussubstrateconcentration,achievethecorrespondingquasisteadystatevalues xDsf=c kx(3.48)

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pKc Dsf=cVcr obtainedbysubstituting( 3.47 )inequations( 3.38 { 3.39 ).Theachievementofthesequasisteadystatesissorapidthatthecelldensity,theperipheralenzymelevelandtheribosomelevelremainatthevaluesachievedattheendofbalancedgrowth. Phase4 .Duringthelastphase,s,xandpremaininthequasisteadystatesdenedby( 3.47 { 3.49 ),whilee,r,andcevolveslowlyaccordingtotheequationsde dtVe1+K2x2 dtk+rrpkrrrgr dt(rgD)c wherergisgivenby( 3.40 )andandx;paregivenby( 3.48 { 3.49 ). Thestudyofthesedynamicsisfacilitatedbyobservingthatpisindependentoftheenzymelevel[see( 3.48 { 3.49 )].Thus,wecanstudytheslowmotionduringthisphasebyconningourattentiontothetwo-dimensionalsystem( 3.51 { 3.52 ).Attheheartofthissimplicationisthatthefactthatduringthisphase,thecellsconsumenearlyallthesubstrateenteringthechemostat.Thisensuresthatthespecicsubstrateuptakerateiseectivelyindependentoftheperipheralenzymelevel(rsDsf=c).Now,theprecursorlevelssensetheenzymelevelthroughthespecicsubstrateuptakerate[see( 3.39 )].Sincethespecicsubstrateuptakerateisindependentoftheenzymelevel,soisp.Thus,theribosomelevelandcelldensityevolvewithoutseeingthechangesintheperipheralenzymelevel. Figure 3{6 bshowsthemotionofcandronthephaseplane.Notethat 1. Thephasepathintersectsbothnull-clinesbeforereachingbalancedgrowth.Hence,bothcandrpassthroughextremaastheyapproachthesteadystate.Herealso,thesteadystateisastablefocussincethetheeigenvaluesofthelinearizationaboutthesteadystateare1;2=0:20:1.

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2. Whenthephasepathcrossesthenull-clineforc,thesignof_cchangesfrompositivetonegative.Whenitcrossesthenull-clineforr,thesignof_rchangesfromnegativetopositive.Thisimpliesthatcpassesthroughamaximumwhilerpassesthroughaminimum. 3. Thephasepathintersectsthenull-clineforcbeforeitintersectsthenull-clineforr.Itfollowsthatcreachesitsmaximumbeforerattainsitsminimum. Thesedynamicsreectthatfactthatthespecicribosomesynthesisratedecreasesrst,followedbythespecicproteinsynthesisandgrowthrates.Thus,atthebeginningofPhase4,_r<0becausethesynthesisrateofribosomeshasincreasedmuchmorethantheirdilutionrate,and_c>0becauseoftheinertiainthespecicgrowthrate.Asrdecreases,sodoesthespecicgrowthrateuntilitbecomesequaltoD=0:21/hr.Thisisthepointatwhichthecelldensitydisplaysamaximum.Astheribosomelevels,andhence,thespecicgrowthrate,decreasefurther,thedilutionrateoftheribosomesprogressivelydecreasesuntiltheirsynthesisanddilutionratesbecomeequal.Thisisthepointatwhichtheribosomesdisplayaminimum. Weshallrefertothesetransientsascell-sucientfed-batchdynamicsbecausethefundamentalpropertyunderlyingthesedynamics|thecompleteconsumptionofalltheinuentsubstrate(Dsfrsc)|ischaracteristicofhigh-densityfed-batchcultures. 3.2.2.3 Dilutionrateshift-down Figure 3{7 showsthedynamicsofadilutionrateshift-down.Weassumethattheinitialdilutionrateisclosetothewashoutdilutionrate,butnottooclosetoit,sothattheinitialsubstrateconcentrationsatisestheinequalityKss0sf.Therstinequality,s0Ks,impliesthattheinitialsubstrateconcentrationisatsupersaturatinglevels.Consequently,thereisaninitialphaseduringwhichthecellscontinuetoconsumesubstrateandgrowattheirpre-shiftrates,rs;0andD0.Theevolutionofthesubstrateconcentrationandcelldensityduringthisinitialgrowth

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(b) (c) (d) (e) (f) Transientresponsetoadilutionrateshift-down.Att<0,thecultureisatthesteadystatecorrespondingtothedilutionrate,D0=0:81/hr,andfeedconcentration,sf=5g/L.Att=0,thedilutionrateisshifteddowntoD=0:21/hr,whilethefeedconcentrationisheldconstant.

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phaseisapproximatedbytheequationsds dtD(sfs)rs;0cdc dt(D0D)c: Thesecondinequality,s0sf,impliesthattheinitialcelldensity,c0^Y(sfs0)^Ysf,islargecomparedtotheinitialsubstrateconcentration.Theinitialgrowthphaseshould,therefore,lastforarelativelyshortperiodoftime.Thisisindeedthecase.Toseethis,observethattheinitialexponentialgrowthphasepersistsuntilthesubstrateconcentrationdropstolevelscomparabletothesaturationconstant(sKs).Itfollowsfrom( 3.53 )thatthetimetakentoreachsubsaturatingsubstrateconcentrationsisgivenbytheexpression 1 whichis0:2hrsifs0=sf=0:1andD0D=0:51/hr. OncethesubstrateconcentrationreacheslevelsontheorderofKs,thereactorreplicatesthedynamicsofphase3andphase4above.Thecelldensity,theperipheralenzymelevelandtheribosomeconcentrationevolveslowlyfromthevaluesachievedattheendofbalancedgrowthtotheirultimatesteadystatevalues,whilethesubstrate,inducer,andprecursorlevelsremainatthequasisteadystateconcentrationsdenedby( 3.47 { 3.49 ). ThesimulationsareingoodagreementwiththedatashowninFigure 3{8 ,theonlydierencebeingthatthepredictedtimeintervaloftheinitialexponentialgrowthphaseissmallerthantheobservedtimeintervalofthisphase.Thisis

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(b) (c) Responseofaglucose-limitedcultureofE.coliK12toadilutionrateshift-down(from[ 48 ]).Att<0,thecultureisatthesteadystatecorrespondingtothedilutionrate,D0=0:61/hr,andfeedconcentration,sf=5g/L.Att=0,thedilutionrateisshifteddowntoD=0:21/hr,whilethefeedconcentrationisheldconstant.Theguresshowtheevolutionofthe(a)Celldensityandsubstrateconcentration(b)RNAlevel(c)Specicsubstrateuptakegrowthratescalculatedfromthecurvesin(a). becauseinthisexperimentalsystem,theinitialsubstrateconcentrationandthesaturationconstantarerelativelylarge(s01g/L,Ks=0:1g/L).Thelargeinitialsubstrateconcentrationincreasesthedurationoftheinitialexponentialgrowthphase[see( 3.55 )].Thehighsaturationconstantincreasesthetimerequiredforthesubstratetoswitchfromsupersaturatingtosubsaturatinglevels. 3.2.2.4 Dilutionrateshift-up Theresponsetodilutionrateshift-upsalsoreplicatesthedynamicsobservedinfeedswitches.Figure 3{9 showsthesimulationsofthreedilutionrateshift-upsfromD0=0:11/hrtoD=0:41/hr,D=0:61/hr,andD=0:81/hr,respectively.Inallthreecases,thesubstrateconcentration,thespecicsubstrateuptake,and

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thespecicgrowthraterapidlyincreasewithin10{15minutesFigure( 3{9 a,c,f).However,thesubsequentbehaviorofthethreetransientsisquitedierent. Iftheshift-upissmall(D=0:41/hr),thesubstrateconcentrationneverachievessupersaturatingconcentrations.Thisisbecausetheshift-upissosmallthattherapidincrementofthespecicsubstrateuptakerateissucienttomatchtheincreaseinthesubstrateinputrate,Dsf.Thus,within10{15minutes,thesubstratereachesthequasisteadystateconcentrationgivenby( 3.47 ).Thesubsequentevolutionisgivenbyequations( 3.35 { 3.37 ),whiles,xandp,areinthequasisteadystatesgivenby( 3.47 { 3.49 ),respectively.ThephaseportraitforthismotionisshowninFigure 3{10 a. Iftheshift-upislarge(D=0:81/hr),thesubstrateconcentrationrapidlyreachessupersaturatinglevels,andremainsattheselevelsthroughoutthesub-sequenttransient.Inthiscase,theshift-upissolargethatthesubstrateuptakerate,rsc0,cannotmatchthesubstrateinput,Dsf,evenwhenthesubstratecon-centrationhasreachedsupersaturatinglevels.Thus,thesubstrateconcentrationcontinuestoincrease,andremainsatsupersaturatinglevels.Therapidattainmentofsupersaturatingsubstrateconcentrationsalsostimulatesthespecicgrowthrate.Asexpectedfromtheearlieranalysisofcontinuous-to-batchshifts(Figure 3{4 ),thespecicgrowthimmediatelyincreasesto0.31/hr(Figure 3{10 f).Butthisenhancedspecicgrowthratefallswellbelowthenewdilutionrate(D=0:81/hr).Thus,inadditiontoaccumulationofthesubstrate,thereisapronounceddeclineinthecelldensitywhichcontinuesuntiltheribosomelevelshavebeenbuiltuptosucientlyhighlevels.ThesedynamicsaresimilartothePhase1andPhase2dynamicsofthefeedswitch.ThephaseportraitforthisevolutionisshowninFigure 3{10 b. Iftheshift-upisintermediate(D=0:61/hr),thesubstrateconcentrationsaturates,andthephysiologicalvariablesbegintheirapproachtobalancedgrowth,i.e.,thecellsgothroughPhase1andenterPhase2.Butbeforebalancedgrowthis

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(b) (c) (d) (e) (f) Transientresponsetodilutionrateshift-ups.Att<0,thecultureisatthesteadystatecorrespondingtothedilutionrate,D0=0:11/hr,andfeedconcen-tration,sf=5g/L.Att=0,thedilutionrateisshifteduptoD=0:41/hr(--),D=0:61/hr({{),andD=0:81/hr(||).

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(b) (c) Phaseportraitsoftheslowmotionduringdilutionrateshifts.(a)Theslowmotioncorrespondingtothedilutionrateshift-upfromD0=0:11/hrtoD=0:41/hr.(b)Theslowmotioncorrespondingtothelargedilutionrateshift-upfromD0=0:11/hrtoD=0:81/hr.(c)Classicationofdilutionrateshift-updynamics. reached,thesubstrateconcentrationreturnstosubsaturatinglevels,andthecellsswitchtoPhase3andPhase4dynamics.Similartransientshavebeenobservedinexperiments.Figure 3{11 showstheresponseofaglycerol-limitedcultureofK.aerogenestoadilutionrateshift-upfromD0=0:0041/hrtoD=0:241/hr.Theribosomelevel,thespecicgrowthrate,andthespecicsubstrateuptakerateincreaseinamannerconsistentwiththeapproachtobalancedgrowth(Figures 3{11 b,c).However,balancedgrowthisnotfullyattainedbecausethesubstrateconcentrationdropstosubsaturatinglevelswellbeforethecellsachievebalancedgrowth.Indeed,thesubstrateconcentrationdropstosubsaturatinglevelsatt10hrs(Figure 3{11 a).Atthispoint,thespecicgrowthrateis0:61/hr,whichislessthanthemaximumspecicgrowthrateof0.81/hr.

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Intuitionsuggeststhatthemagnitudeofthedilutionrateshift-uprequiredtoprovokethesedynamicsdependsontheinitialdilutionrate.For,iftheinitialdilutionrateisverysmall,soistheinitialenzymelevel.Inthiscase,relativelysmallshift-upsshouldyieldsupersaturatingsubstrateconcentrations,resultinginPhase1andPhase2dynamics.Ontheotherhand,ifthedilutionrateislarge,theperipherallevelsaresohighthatthesubstrateconcentrationshouldremainsubsaturatingthroughoutthetransient,leadingtoPhase3andPhase4dynamics.Thisisindeedthecase.Infact,giventheresponsetocontinuous-to-batchshifts(Figure 1{6 ),onecanpredicttheclassofdynamicscorrespondingtoanypre-assignedinitialandnaldilutionrates.Toseethis,observethatthesubstrateconcentrationattainsthequasisteadystategivenby( 3.47 )ifandonlyifrs(D0)c0>Dsf,wherers(D0)denotesthespecicsubstrateuptakerateachievedimmediatelyafterthecellshavebeenexposedtosupersaturatingconcentrations(seeFigure 3{4 a).Butitfollowsfrom( 3.19 )thatc0D0sf=~rs(D0)atallbutthehighestdilutionrates.WeconcludethatsattainsquasisteadystateifandonlyifrsD0sf ~rs(D0) Inwords,thesubstrateconcentrationattainsquasisteadystateifandonlyiftheratiobywhichthedilutionrateincreases,D=D0,islessthanratiobywhichthespecicuptakerateincreasesinresponsetoacontinuous-to-batchtransition,rs(D0)=~rs(D0).Thus,wecanclassifythedynamicsbyplottingthetransitiondilutionrateDtD0rs(D0) ~rs(D0) asafunctionoftheinitialdilutionrate,D0(Figure 3{10 c).GivenanyD0,Phase1and2dynamicsareobtainedifDliesabovethecurve;Phase3and4dynamicsareobtainedifDliesbelowthecurve.Inparticular,onecanseethatthesimulationsshowninFigure 3{9 areconsistentwiththisclassication.

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(b) (c) Responseofaglycerol-limitedcultureofK.aerogenestoadilutionrateshift-up(from[ 12 ]).Att<0,thecultureisatthesteadystatecorrespondingtothedilutionrate,D0=0:0041/hr,andfeedconcentration,sf=15g/L.Att=0,thedilutionrateisshifteduptoD=0:241/hr,whilethefeedconcentrationisheldconstant.Theguresshowtheevolutionofthe(a)Celldensityandsub-strateconcentration(b)RNAlevel(c)Specicsubstrateuptakeandgrowthratescalculatedfromthecurvesin(a). 3.3 Discussion Chemostatdynamicshavebeenthesubjectofnumerousexperimentalstudies.Fromthisformidablebodyofliterature,Daigger&Grady[ 63 ]andDubocetal[ 64 ]havedistilledthekeygeneralizations.Webeginbyreiteratingthesegeneralizationsandinterpretingthemintermsofourmodel. Basedonanextensivereviewoftheliteraturepriorto1980,Daigger&Gradyclassiedthetransientsintotwocategories:growthresponsesandstorageresponses.Sinceourmodeldoesnotaccountforstorage,weconneattentiontothegrowthresponses.Theystatethat

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1. Thegrowthresponsespossessthefollowingproperties (a) Themodelexhibitsgrowthratehysteresis.Indeed,Figure 3{7 fshowsthatimmediatelyafterthedilutionrateshift-down,theculturesettlesintoaspecicgrowthratethatishigherthanitsnalvalue.Conversely,thespecicgrowthrateimmediatelyafteradilutionrateshift-upislowerthanitsultimatevalue(Figure 3{9 f). (b) Simulationsofthecontinuous-to-batchshiftsshowthatthespecicsubstrateuptakeandgrowthratesdoincreaseimmediately(Figure 3{4 a,c).Moreover,theincrementsintheseratesincreaseasthedilutionrateoftheculturedecreases.Butthisistrueonlyuptoapoint.Atsucientlylowdilutionrates,theincrementsdecreaseonceagainbecauseatverylowdilutionrates,theperipheralenzymeandribosomelevelsbecomesmall. 2. Inmanyinstances,bothphenomenaareobserved.Forexample,thespecicgrowthrateimmediatelyafterexposuretosubstrate-excessconditionsisgreaterthantheratebefore(AGP),butlessthanthemaximumratewhichtheculturecouldattain.However,withthepassageoftime,thespecicgrowthrategraduallyincreases,andeventuallyreachesthemaximumrate.Throughoutthisperiod,thespecicgrowthrateislessthanthemaximumspecicgrowthrate(GRH). ThispropertyisevidentinFigures 3{5 fand 3{9 f. 3. Theconceptofphysiologicaladaptationmaybeusedtounderstandthesegrowthresponses.Whenamicrobialcultureisgrownatasubmaximalspecicgrowthrate,thecultureadaptstothatgrowthenvironment.Inparticular,theRNAcontent,theenzymelevels,andconcentrationofin-tracellularmetabolitesarelowerthanthevaluesattainedatthemaximumspecicgrowthrate.Whenthecellsareexposedtoasubstrate-excessenvi-ronment,thecultureisunabletogrowatitsmaximumrate.Thisisbecausethephysiologicaladaptationisnotcomplete,sothattheculturepossessesonlyalimitedabilitytoimmediatelyincreaseitsspecicgrowthrate(AGP).Astimepasses,thelevelsofRNA,enzymes,andintracellularmetabolites

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graduallyincrease.Thus,thespecicgrowthrateofthecultureincreasesprogressivelyuntiliteventuallyreachesitsmaximumvalue(GRH). Thistrainofeventsispreciselywhattranspiresinthemodel(seeFigures 3{5 and 3{9 ). Morerecently,Dubocetalarrivedatthefollowingconclusionsbasedonextensivedilutionrateshift-upexperimentswithS.cerevisiae[ 64 ]. 1. Thereisanimmediateincreaseinthecatabolicactivity.Moreover,theincreaseinthecatabolicactivityisalwayslargerthantheincreaseinthegrowthrate,leadingtoatransientuncouplingofcatabolismandgrowth. ThismodeldisplaysthischaracteristicasisevidentfromFigures 3{4 a,c,d.Tocapturethedierentresponsetimesofthecatabolicandanabolicrates,Dubocetalassumedthattheseratesfollowedthephenomenologicalequa-tionscatdrcat anduniquetimeconstants,catandgwerechosenforeachexperimentinordertotthedata.Ourmodelmaybeviewedasanattempttogiveaphysiologicalbasisfortheforegoingphenomenologicalequations. 2. Insomeexperiments,suchasthedilutionrateshift-upofaerobiccultureslimitedbyaceticacid,thetransientbehaviorofthecultureissimilartotheadaptationofthemetabolismduringabatchexperimentwheresubstrateisinlargeexcessanddoesnotinuencethespecicgrowthrate.Inyetotherexperiments,suchasthedilutionrateshift-upofglucose-limitedaerobiccultures,thesubstratedidnotaccumulate. Accordingtoourmodel,thesetwotransientsreectPhase1/Phase2andPhase3/Phase4dynamics,respectively.Sincethemaximumspecicgrowthrateonaceticacidismuchlowerthanthemaximumspecicgrowthrateonglucose,theverysamedilutionrateshift-up,i.e.,theverysameD0andD,couldleadtosubstrate-sucientbatchdynamicsintherstcase,andcell-sucientfed-batchdynamicsinthesecondcase. Themodelyieldsfurtherinsightsintotheseobservations. 1. Thesubstrateconcentrationexistsineitheroneoftwostates.Itiseitheratsupersaturatinglevelsoratthesubsaturatinglevelsattainedwhenthesubstratereachesquasisteadystate.Transitionsbetweenthesetwostatesaresofast(ontheorderofminutes)thattheymaybeignoredonthetimescaleofinterest(ontheorderofhours). 2. Thedynamicscorrespondingtothesetwostatesarecanonicalinthefollowingsense.Thetransientsobservedinresponsetoawidevarietyofperturbationsareacombinationofthesetwoclassesofdynamics.Thus,indilutionrateshift-ups,thesubstrateissupersaturatingthroughoutthetransientifthenal

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dilutionrateisclosetothemaximumspecicgrowthrate.Itissubsaturatingthroughoutthetransientifthenaldilutionrateisclosetotheinitialdilutionrate.Itswitchesfromsupersaturatingtosubsaturatinglevelswhentheshift-upsaremoderate.Thelatterisalsothecaseinfeedswitcheswherethesubstrateconcentrationissupersaturatingatrst,butswitchestosubsaturatingconcentrationslater.Itfollowsthatacompleteunderstandingofthesetwoclassesofdynamicsissucientforunderstandingallotherdynamics. 3. Thesetwoclassesofdynamicspossessthefollowingexperimentallytestableproperties (a) Duringtheapproachtobalancedgrowth,theperipheralenzymeandribosomelevelspassthroughamaximum,buttheenzymelevelreachesamaximumbeforetheribosomelevel. (b) Duringtheapproachtothenalsteadystate,thecelldensitypassesthroughamaximumandtheribosomelevelpassesthroughaminimum.Thecelldensityreachesitsmaximumbeforetheribosomelevelreachesitsminimum. Wecanexpresstheseobservationsmoreconciselyintheparlanceofnonlineardy-namics.Wehaveshown,ineect,thatthechemostatdisplaysexcitabledynamics.Inotherwords,therearetwo\slow"submanifolds(surfaces)embeddedwithinthespaceofallvariablesinthemodel.Themotionintoandoutofthesesubmanifoldsisveryfast.Thedynamicsonthetimescaleofinterest(hours)occuronlywhenthevariableslieinoneofthesubmanifolds.ThemotiononthesetwosubmanifoldscorrespondstoPhase1/Phase2andPhase3/Phase4dynamics,respectively.

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Wehavecharacterizedthesluggishresponseinthechemostattotheenvi-ronmentalperturbationintwocategories.Firstly,thelagswereobservedduetosubstrateuptakeratelimitation,whichwereassignedtotheinsucienttrans-portenzymelevel.Theroleoftransportenzymewasstudiedinchapter2.Itwasdemonstratedthatthemodelaccountingfortransportenzymecapturestheexperimentaldataquantitatively(Figure 1{5 ).Secondly,inthecasewherethegrowthwaslimitedbyproteinsynthesis,thephenomenawasencompassedusingtheautocatalyticsynthesisofRibosomeorRNA.Theroleofribosomeswasstudiedinchapter3.WearguedthatalowerconcentrationofRNAatalowdilutionratewouldlimitthespecicgrowthrate.Weassumedthatthebiosynthesiswaslimitedbyribosomes,andfurtherexplainedthesedynamicsqualitatively. Thisstudyismotivatedbythemetabolicregulationofthespecicgrowthrate.Whenthecellpicksupthesubstrate,thedierentenzymesinsidethecellbrakesdowntheincomingsubstrateintoprecursormoleculesi.e.glucose-6-phosphate,fructose-6-phosphate,pyruvateetc.Theprecursormoleculeshaveanopportunitytogoeithertobiosynthesis,respiration,storageproducts,orexcretion.Thequestionnowarisesthathowthecelldecideswheretosendtheincomingsubstrate?Tosolvethisdiscrepancy,wehaveusedenergychargedparticle,adenosinetriphosphate(ATP),asanenergysourceandasamajorregulatoryfactorincontrollingthemetabolicuxdistribution.Theenergychargedparticle(ATP)convertsintoenergyunchargedparticle,adenosinediphosphate(ADP),inenergyconsumingprocessesandtheenergyunchargedparticle(ADP)convertsbacktoenergychargedparticle(ATP)inenergygeneratingprocesses.Forthis 68

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purpose,wehavedenedenergychargeastheratioofthechargedadenosinenucleotidestototaladenosinenucleotidesandwillbeusedtoshowhowtheratioofATPandATPplusADPregulatesthemetabolicuxes.WewishtoextendourpreviousmodeltoaccountfortheseobservationsbyincludingATPinthemodel.WehaveincorporatedtheconcentrationofATPandADPindierentkineticexpressionstoexplainthemetabolicregulationinvarioustransientdynamicsinthechemostatincludingthestationaryphaseinthebatchculture. ThegoalofthisworkistostudyanextendedmodeltakingdueaccountofATPtocaptureandexplainthemetabolicuxdistribution,andadditionaldatadiscussedbelowwhilekeepingthevariousdynamicsofpreviousmodelintact.Webeginbyformulatingtheextendedmodel(Section 4.1 ).Wethensimulatethemodeltoshowthatityieldsresultsinqualitativeagreementwiththedata(Section 4.2 ).Finally,wecomparethekeyresultsofthemodeltothosesummarizedinreviewsoftheexperimentalliterature. 4.1 TheModel Figure 4{1 showsthekineticschemeofthemodel.Here,Sdenotesthegrowth-limitingcarbonsource,EdenotestheperipheralenzymesthatcatalyzethetransportandperipheralcatabolismofthecarbonsourceandXdenotestheinternalizedformofthesubstratethatinducesthesynthesisofE;PdenotesthepoolofprecursorsproducedbycatabolismofX;RdenotesribosomesorrRNA;Cdenotesproteins;Eadenotestheenzymesrequiredtosynthesizeadenosinediphosphate(ADP)

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Figure4{1: KineticschemeoftheATPmodel.Here,Sdenotesthesubstrate,Edenotestheinducibleenzyme(s)catalyzingtheuptakeandperipheralcatabolismofS,XdenotestheinducerforE,PdenotesthebiosyntheticprecursorsderivedfromcatabolismofX,RdenotesRNA,Cdenotesproteins,A3denotesthethechargednucleotides,A2denotestheunchargednucleotides,Eadenotestheunchargednu-cleotidessynthesisenzymeandCdenotestheentirecellconsistingofE,X,P,R,A3,A2,EaandC. roleofadenosinemonophosphate(AMP)hasbeenomittedtoavoidcomplexityinthemodel.AsAMPisnotapartofthismodel,ATPandADParedenedasenergychargedandenergyunchargednucleotides.Thusenergycharge,Ec,isdenedastheconcentrationofA3=At.TheentirecellconsistingofE,X,P,R,C,Ea,A3,andA2isdenotedbyC.ThetermbiosynthesiswillbeusedtoreferthesynthesisofRNA(R),andproteins(C),whilegrowthwillreferthesynthesisofallintracellularcomponents. Throughoutthiswork,theinstantaneousconcentrationsofthevariablesaredenotedbythecorrespondinglower-caseletterss;e,x,p,r,c,ea,a3,a2,atandc,whilesteadystateandquasisteadystateconcentrationsaredenotedbyoverlayingtheletterswithand,respectively(forinstance,~xandx).Theconcentrationsoftheenvironmentalvariables,sandc,arebasedonthevolumeofchemostat(g/Landgdw/Lrespectively),theconcentrationsofthephysiologicalvariables,e,x,p,r,a,c,ea,a3,a2,andatarebasedonthedryweightofbiomass(g/gdw).

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Wemakethefollowingassumptionsregardingthekineticsoftheprocesses. 1. Thespecicsubstrateuptakerate,denotedrs,satisesthekineticlawrsVsa3es Ks+s1 1+p Ki2 1+p Ki2denotestheinhibitionofsubstrateuptakebyprecursors. 2. ThespecicrateofbreakdownofXintoenergyandprecursorsP,denotedrx,isgivenbyrxkxa2x Thespecicrateofrespirationisrco2kco2a2p Thespecicrateofenzymesynthesis,denotedr+e,isr+eVer ke+r1+K2x2 5. Thespecicrateofexcretionisrpxkpxa2p3 ThespecicrateofrRNAsynthesisisr+rk+ra23pr Thespecicrateofprotein(C)synthesisisr+cVca3rp Kc+p Thespecicrateofnucleotidesynthesisenzymeisr+eak+eapr Thespecicrateofdenovonucleotidesynthesisisr+a2k+a2a3pea

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10. Thespecicratesofperipheralenzyme,nucleotide,rRNA,nucleotidesynthesisenzyme,andproteindenotedre,ra2,rr,reaandrcrespectively,aregivenbyrekee;ra2ka2a2;rrkrr;reakeaea;rckcc Thesynthesisofperipheralenzymes,unchargednucleotides,nucleotidesynthesisenzyme,andproteinsdepletesthepoolofprecursorsandtheirdegradationreplenishesthepoolofprecursors. 12. ThesynthesisofRNAdepletesthepoolofchargednucleotidesandtheirdegradationreplenishesthepoolofunchargednucleotides. Amassbalanceonthestatevariablesyieldsds dt=D(sfs)rsc dtc dt=r+ereD+1 dte dtea dt=r+rrrD+1 dtr dta2 dta3 dt=rsrxD+1 dtx dt=rxrco2rpxXi=a2;c;e;ea(r+iri)D+1 dtp wheresfdenotestheconcentrationofSinthefeed,andDdenotesthedilutionrate.sdenotestheenergyrequiredtotransportthesubstrateinsidethe

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cell,xdenotesenergygeneratedintheprocessofproducingprecursors,+iandidenotetheenergyrequiredtogeneratethedierentcellularcomponentsandenergygeneratedwhiledegradingthecellularcomponentsrespectively,andco2denotesenergygeneratedfromutilizingtheprecursorsintheTCAcycle. Toderivethereducedequationsfromtheoriginalequations( 4.20 { 4.28 ),weappealtothefollowingfacts 1. Themassfractionofallintracellularentitiesequalsunity,i.e., Then,addingequations( 4.2 { 4.9 )yields dt=rsrco2rpx(4.12) Wedene wherergdenotesthespecicgrowthrateofthecells.Itfollowsfrom( 4.12 { 4.13 )thatthelastterminequations( 4.2 { 4.9 )representsthedilutionoftheintracellularvariablesduetocellgrowth,and( 4.12 )canberewritteninthemorefamiliarform dt=(rgD)c(4.14) Toderivethereducedequations,wesubstituteequation( 4.2 )withequa-tion( 4.14 ),andreplacetheexpressionD+(1=c)(dc=dt)inequations( 4.2 { 4.9 )withrg. 2. Althoughbothvariables,a2anda3,changerapidly.Thetotalconcentrationatevolvesslowerthana2anda3.ata2+a3: 4.6 )and( 4.7 )yieldsthefollowingequation Toderivethereducedequations,wereplace( 4.6 )with( 4.15 ).

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3. SinceA3,At,XandParepresentinsmallconcentrations,butparticipateinrapidreactions,theyachievequasi-steadystatealmostinstantly.Thusda3 (4.16)dat (4.17)dx dt=0)rxrs dt=0)rxrco2rpx(r+crc)+(r+a2ra2) (4.19) wherewehaveassumedthatthedilutionratesofA3,X,PandAtarenegligiblysmallcomparedtotheratesatwhichtheseentitiesarepro-duced/consumedbyvariouscatabolicandanabolicreactions. Toderivethereducedequations,wereplacethedierentialequations( 4.7 { 4.9 )withthealgebraicrelations( 4.16 { 4.19 ). 4. Furthersimplicationoftheexpressionobtainsbyobservingthat( 4.17 { 4.19 )implytherelationrgrsrco2rpx(r+crc)+(r+rrr) wherewehaveassumedthatPi=e;ea(r+iri)(r+crc)Thisexpressionisconsistentwithourintuitionthatsincex;p;e;a2;a3r;c,thespecicgrowthrateiseectivelyequaltothesumofthenetspecicratesofRNAandproteinsynthesisrates. 5. Finally,notethatsincee;x;p;a2;a31,equation( 4.11 )impliesthatc+r1)c1r 4.1 { 4.10 )canbeapproximatedbythereducedequations dt=D(sfs)Vsa3es Ks+sc dt=(rgD)c dt=Ver ke+r1+K2x2 dt=k+rra23pkrrrgr

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0(xs)rs+co2rco2Xi=c;r;a2(+ir+iiri)r+r 0r+a2ra2r+r+rr 0Vsa3es Ks+skxa2x 0kxa2xkco2a2pkpxa2p3rg whererg,thespecicgrowthrate,isgivenby Kc+pkcc+k+rra3pkrr 1c+r ItiseectivelyequaltothespecicrateofnetrRNAandprotein. 4.2 Simulations ThesimulationsweredonewiththeparametervaluesinTable 4{1 . Table4{1: ParametervaluesusedintheATPmodelsimulations. ghrKs=102g LKi=0:02g gdwkx=106gdw ghrVe=1104g gdwhrK2=1011gdw g2K3=104ke=0:011 hrke=0:1g gdwkco2=105g ghrk+r=3106gdw3 hrVc=103gdw ghrkc=0:031 hrKc=0:002g gdwkpx=5107k+ea=0:01gdw ghrkea=0:061 hrk+a2=3107gdw2 gdwx=10g gdwpx=0g gdwa2=20g gdwc=40g gdwco2=23g gdwr=8g gdw SteadyStates Theequations( 4.20 { 4.29 )permittwotypesofsteadystates,namely,thepersistencesteadystate(~c6=0)andthewashoutsteadystate(~c=0).Foragivenmicrobialspeciesandgrowth-limitingsubstrate,thesesteadystatescandependontwoparameters|thedilutionrateandthefeedconcentration.

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(b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) VariationofthesteadystateswithDatsf=1:5g/L.Stableandun-stablesteadystatesaredenotedbyfullanddashedlines,respectively.

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Underthiscondition,thereexistswashoutsteadystateforalldilutionrateandcriticalwashoutdilutionrateatDw=0:811/hrwherethemaximumgrowthrateisunabletobalancethedilutionrate,consequentlythecellwashesout.Thewashoutsteadystateisstableabovethecriticalwashoutdilutionrateandunstablebelowthecriticalwashoutdilutionrate.Thereexistsauniquepersistencesteadystateforalldilutionrates.Thepersistencesteadystateisstablebelowthewashoutdilutionrateandisunstableabovethewashoutdilutionrate.Thesteadystategures(Figure 4{2 )areformallysimilartotheresultsobtainedinexperiments(Figure 1{3 ).Thesubstrateconcentrationandribosomelevelsincreasemonotonically(Figure 4{2 a,d).Thecelldensityandtransportenzymelevelspassthroughamaximum(Figure 4{2 b,c).Atlowdilutionrate,thecellproducessmallconcentrationofexcretoryproductscomparetohighdilutionrate(Figure 4{2 f).TheconcentrationsofATP,andtotalnucleotidesshowahyperbolicproleandenergychargestaysapproximatelyconstantforalldilutionrates(Figure 4{2 i,j,l).TheconcentrationofADPpassesthroughmaximaatDw=0:21/hr(Figure 4{2 k). Notethatthesteadystateconcentrationsofthesubstrateandthephysiolog-icalvariablesareindependentofthefeedconcentration,sf.ThisisbecausethesteadystateconcentrationsofSandtheeightphysiologicalvariables(E,Ea,R,X,P,Ea,A3andAt)arecompletelydeterminedbytheeightequations( 4.21 { 4.28 ).Theseequationsdependonlyonthedilutionrate;andareindependentofthefeedconcentration. 4.2.2 Transients Thecalculationofenergychargeraisesthequestionofitspotentialutility?Theanswerliesinthestructureofthemodel.Theabsoluteconcentrationsofchargedandunchargednucleotidesdeneonlytherateofallthereactions.Themetabolicdistributionoftheincominguxisdenedbytheenergychargeasthegrowthrateisonlyinitiatedbytheconcentrationofchargednucleotides,ATP,and

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therateofcarbondioxideevolutionandexcretoryproductsareonlydictatedbytheconcentrationofunchargednucleotides,ADP.TheratioofchargednucleotidestototalnucleotideshelpsthecelltomakethedecisionsregardingthedistributionoftheincomingsubstrateeitherforcreatingbiomassorforenergygenerationinTCAcycledependingontheconcentrationofribosomesandtheavailabilityofthesubstrate.Whentheenergychargeishigh,impliestheATPconcentrationishigh,thecelldivertstheincomingsubstratetowardsenergyconsumingpathways,i.e.,proteinsynthesisandribosomesynthesis.Conversely,whentheenergychargeislow,thecelldivertsthesubstrateuxtorespirationtogenerateenergy.Thisregulationisoccurringbecauseofinterconversionbetweenchargednucleotides,ATP,andunchargednucleotidesADP,hencethetimeperiodofthisregulationliesinseconds.Thus,thecellutilizestheconcentrationofribosomesfortheregulationofthecellgrowthinaslowtimescaleandtheconcentrationofchargedandunchargednucleotidesinafasttimescale. 4.2.2.1 Substratepulse Insubstratepulseexperiment,thereactorwasmaintainedatsteadystateatxeddilutionrateandasmallperturbationwasgiventotheglucoseconcentrationbyinjectingapulseofextraglucose.ThetransientconcentrationofATPwasmeasuredfor180seconds(seeFigure 1{9 )[ 51 ].ThefastdropintheconcentrationofATPinsecondstimescaleimmediatelyaftertheglucosepulsewasattributedtotheresponseoftheATPconsumptioninphosphorylationofthesubstrate.Toreplicatethisexperiment,thesimulationwasstartedwiththesteadystatevaluesatxeddilutionD=0:21/hr.Att=0,thesubstrateconcentrationinthereactorwasshiftedtoahighervaluethanthesubstrateconcentrationatsteadystate.Inthissimulation,thesubstrateconcentrationhasbeenincreasedto0:1g/L.Assoonasthecellperceivesanexcesssubstrateintheenvironment,theunsaturatedtransportenzymebecomessaturatedwithsubstrate,hence,thesubstrateuptake

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(b) (c) Transientresponsetosubstratepulseinsecondstimescale.Att<0,theculturewasatsteadystatecorrespondingtodilutionrated=0:21/hr.Attimet=0,thesubstrateconcentration,s,inthereactorisincreasedtos=0:1g/lwhilefeedconcentrationanddilutionratewereheldconstant. ratereachestomaximallevel.Thecellphosphorylatestheincomingsubstrateintheprocessofsubstratetransport.Thephosphorylationofthesubstrateusesadenosinetriphosphate,ATP,andconvertsintoadenosinediphosphate,ADP.Asthetotalnucleotideconcentrationisreallylowinsidethecell,thistransportprocesscausestheconcentrationofchargednucleotides,ATP,todropinsecondstimescale.ThedropinATPconcentrationrecoversbackonlywhenthesubstratestartsaccumulatingastheinducermoleculeinsidethecellandcommencesenergygenerationfromglycolysisintheprocessofproducingprecursors(seeFigure 4{3 ). Thetimeperiodofthistransientissosmallthattheincomingsubstratedoesnotreachbiosynthesisespeciallythesynthesisofribosomeandinuencesthetotalnucleotideconcentration,hencethetotalnucleotideconcentrationstaysconstant.Theconservationoftotalnucleotideshelpsinexplainingthesimulation

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results.Theinterconversionbetweenthechargednucleotide,ATP,andunchargednucleotide,ADP,intheprocessofsubstratetransportandglycolysisprojectsalightnotonlyfortheobservedminimainATPconcentration,butalsotheobservedmaximainADPconcentrationandminimainenergycharge.Energychargeisanincreasingfunctionofthechargednucleotide,ATP,anddecreasingfunctionoftheunchargednucleotide,ADP.AsATPconvertstoADPintransport,thisprocessdecreasestheATPconcentrationandincreasesADPconcentration,hence,decreasesenergycharge.WhenconcentrationofATPrecoversbackintheprocessofglycolysis,itincreasestheconcentrationofATPanddecreasestheconcentrationofADP,henceincreasesenergycharge. Whentheincomingsubstrateinuencesthebiomasssynthesis,theslowdropintheconcentrationofATPwithin30minutewasobserved[ 17 ].ThisfurtherdropinATPconcentrationwasexplainedbyanincreasedutilizationofATPforthesynthesisofbiomassespeciallythesynthesisofribosomeorRNA.Toreplicatethisexperimenttheabovesimulationwascontinuedtorunfor30minutes.Whenthesubstrateconvertsintotheprecursormolecules,theslowribosomalkineticsdiverttheincomingsubstratetorespirationorexcretionresultinginanincreaseintheconcentrationofchargednucleotides,henceenergychargeincreasesabovethesteadystatevalue(Figure 4{4 f).AsthesynthesisofRNAdependsonthehigherorderofthechargednucleotides,a23,thanthedenovosynthesisofunchargednucleotidesonthechargednucleotides,a3,theincreaseinthechargednucleotidecausesthehigherjumpinRNAsynthesisratethantheunchargednucleotides.ThisdierencebetweenthesynthesisofRNAandunchargednucleotidesresultsinthedecreaseintheconcentrationoftotalnucleotidesandconsequentlytheslowdropinthechargednucleotides(seeFigure 4{4 d,e).Ifthesubstratepulseissucientenoughtosustaintheperturbationforawhile,thenitincreasestheconcentrationofribosomesabovethesteadystatelevel.Thehigherlevelsribosomeconcentration

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(b) (c) (d) (e) (f) Transientresponsetosubstratepulseinminutetimescale.Att<0,theculturewasatthesteadystatecorrespondingtothedilutionrate,D0=0:21/hr,andfeedconcentration,sf=1:5g/L.Att=0,thesubstrateconcentrationinthereactorwasshifteduptos=0:1g/L,whilethedilutionratewasheldconstant.

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canalsoproduceadropinchargednucleotidesaftertheexhaustionofthesubstrate(simulationsarenotshown). 4.2.2.2 Continuous-to-batchshifts Incontinuous-to-batchshifts,att<0,thecultureisinsteadystateatsomedilutionrate,D.Attimet=0,asmallsampleofthecultureisrapidlytransferredtothebatchreactorcontainingahighsubstrateconcentration.Theinitialratesofbiochemicalprocessesaremeasuredwithin10{15minutesofexposingtheculturetoexcesssubstrateconditions(seeFigure 1{6 ).Tosimulatetheseexperimentswiththemodel,weobservethatinthisshorttimeperiod,theconcentrationsoftheperipheralenzymes,ribosomes,andnucleotidessynthesisenzymelevelsessentiallyremainunchangedbecauseoftheslowkinetics.Theinducer,precursor,ATPandtotalnucleotidesconcentrationsrapidlymovetonewquasisteadystatescorrespondingtosubstrate-excessconditions.Thisnewquasisteadystateisobtainedbylettings=(Ks+s)1in( 4.25 { 4.28 ),whiletheperipheralenzyme,nucleotidesynthesisenzymeandribosomelevelsareheldattheirinitialsteadystatevalues,e0=~e(D),r0=~r(D)andea0=~ea(D). Figure 4{5 showsthatsimulationresultsareconsistentwiththeexperiments(Figure 1{6 ).Assoonasthecellsareexposedtoexcesssubstrateconditions,thesubstrateuptakerategoestoitsmaximumvaluedependingupontheenzymeconcentrationatthatdilutionrate(Figure 4{5 a).Thisincreaseinrsincreasestheinducerconcentration,hencetheprecursorconcentrationwhichfurtherresultsintheincreaseoftherespirationrate,rco2andexcretionrate,rpx(Figure 4{5 b,e,h).Theincreaseintherespirationrate,rco2increasestheenergycharge(Figure 4{5 l),hencedivertstheincomingsubstrateuxtothebiomasssynthesis.Butthegenerationofnucleotide,r+a2,andtheconsumptionofnucleotidesinRNAsynthesis,r+r,maintainstheconstantconcentrationofATP,a3,whichmakestheADPconcentration,a2,togodown(Figure 4{5 i,j).

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(b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Initialresponsefollowingacontinuous-to-batchshift.Thedashedlineshowstheinitialsteadystatevalue.Thefulllineshowsthevalueafterthecellshavebeenexposedtosubstrate-excessconditions:(a)Specicsubstrateuptakerate(b)Specicrespirationrate(c)Specicgrowthrate(d)Yieldofbiomass(e)Spe-cicrespirationrate(f)Netspecicproteinsynthesisrate(rcr+crc)(g)NetspecicRNAsynthesisrate(rrr+rrr)(h)Precursorconcentration(i)ATPconcentration(j)ADPconcentration(k)Totalnucleotideconcentration(l)Energycharge.

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4.2.2.3 Substrateswitch Thetransientresponsetoasubstrateswitchcouldbecapturedusingourearliermodel(seeFigure 1{5 ).Simulationsoftheextendedmodelpreservethesetransients.Thistransientdependsontheinduciblenatureofthetransportenzymesynthesis.Whenthesubstrateisnotpresentintheenvironment,thecelldoesnotsynthesizethetransportenzymecorrespondingtothatsubstrate.Intheabsenceofthesubstrate,thesynthesisofthecorrespondingtransportenzymecontinuesonlyatreallysmallconstitutivelevel.Afterthesubstrateswitch,thetraceleveloftransportenzymecorrespondingtoanewsubstrate,limitsthespecicsubstrateuptakerate,rs,atvanishinglysmalllevelsforapproximatelytherst30hours(Figure 4{6 ).Duringthistimeperiod,theintercellularvariablesshowtheresponsesimilartothestarvationphase(Figure 4{8 ).Theribosomeconcentrationdecreasesgradually(Figure 4{8 e).Similarly,theATPconcentration,a3,showsabiphasicstarvationresponse.Initially,theATPconcentrationdropsinstantaneousbecauseoftheconsumptionofATPinRNAandproteinsynthesis,andshowsagradualdecreaseafterthatbecauseofthegradualdegradationofthetotalnucleotide,at(Figure 4{8 f,g).Theslowsynthesisoftransportenzymetakeslongtimetobuildupitsconcentrationtothedesirelevel(Figure 4{8 c),hence,thetransientresponseshowsmassivelossofthecellsandaccumulationofsubstrateconcentrationinthereactor(Figure 4{6 a,b).Theutilizationofchargednucleotidesbythesubstratetransportenzymesmakesthesubstrateuptakeratekineticsevenmorenonlinearthanthepreviousmodels[ 41 ]andenhancesthelagperiod.ThesesimulationsareingoodagreementwiththedatashowninFigure 1{5 . 4.2.2.4 Dilutionrateshift-up Theresponsetodilutionrateshift-upsisalsopreservedforbothcases,thetransportenzymedecientculture[ 41 ]andRNAdecientculture[ 65 ].Inthismodel,thespecicgrowthratecanbelimitedbyeitherthetransportenzyme

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(b) (c) (d) (e) (f) (g) (h) TransientresponseforaswitchintheidentityofthesubstrateatD=0:11/hr.Beforetheswitch,thecultureisinasteadystatecorrespondingtotheabsenceofthesubstrate.

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(b) (c) (d) (e) (f) (g) Transientresponsetoadilutionrateshift-up.Att<0,thecultureisatthesteadystatecorrespondingtothedilutionrate,D0=0:011/hr(solidline)andD0=0:11/hr(dashline),andfeedconcentration,sf=1:5g/L.Att=0,thedilutionrateisshifteduptoD=0:41/hrandD=0:71/hrrespectively,whilethefeedconcentrationisheldconstant.

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concentrationorribosomeconcentration.Figure 4{7 showsthesimulationsofthedilutionrateshift-upforbothlimitingcases. ThedilutionrateshiftupfromD0=0:011/hrtoD=0:41/hr(solidline)isanexampleoflimitationcausedbythetransportenzyme.Thelowinitialconcentrationoftransportenzymelimitstheinstantaneousincreaseinthespecicsubstrateuptakerateandconsequentlytheinstantaneousincreaseinthespecicgrowthrate.Whentheinstantaneousincreaseinthespecicgrowthratedoesnotmatchwiththeincreaseinthedilutionrate,thesubstrateconcentrationincreasesandthecelldensitydecreasesinthereactor.Theslowkineticsofthetransportenzymetakesseveralhourstobuilduptothedesiredlevelcausingthelonglaginthereactor. ThedilutionrateshiftupfromD0=0:11/hrtoD=0:71/hr(dashline)isanexampleoftheribosomeslimitation.Thehighconcentrationoftransportenzymeletthesubstrateuptakerategoesup,butthelowconcentrationofribosomeslimitstheincreaseininstantaneousspecicgrowthrate.Theribosomelimitationresultsinaccumulationofprecursorsinsidethecellhence,increaseinthespecicrespirationrateandspecicexcretionrate.Thehighconcentrationofprecursorsalsoinhibitsthespecicsubstrateuptakerateandthusresultsintheaccumulationofthesubstrateinthereactor. 4.2.2.5 Starvation Thestarvationisdeneasthestatewhenthecellexhauststhegrowthlimitingsubstrateandstopsthegrowth.Inthistransientexperiment,thecultureisgrowninsinglegrowthlimitingsubstrate.Theconcentrationsofphysiologicalvariablesaremeasuredforthegrowthphaseandevenaftertheexhaustionofthesubstrateinthebatchreactor.ItwasobservedthatcelldensitystaysconstantforhourswhereasATPconcentrationandenergychargesuddenlydroptothelowerlevelandstaythere[ 66 ].Toreplicatethisexperiment,thesimulationiscomputedeven

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(b) (c) (d) (e) (f) Simulationresultsofthebatchcultureduringexhaustionofthesub-strate.

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afterthesubstrateconcentrationgoestozero.Thesimulationresults(Figure 4{8 )qualitativelymatchestheexperimentaldata. Severalpapers[ 67 ]suggestthatthemaintenanceofconstantcelldensityinthestationaryphaseisduetotheconsumptionofexcretorybyproducts(i.eacetate).Sincethismodeldoesnotincorporatetheconsumptionofexcretoryproducts;asthesubstrateconcentrationreacheszero,thedecayinproteinandRNAmakethespecicgrowthratenegative,andthecelldensitydecreasesintheculture(Figure 4{8 b).InordertoaccountforthedecreaseintheconcentrationofATPandenergycharge,refertoFigure 4{1 ,whenthecellpicksupthesubstrate,therearetwowaystogenerateprecursormoleculesandenergy,eitherfromtheuptakeofsubstrateorfromthedegradationofthebiomass.Thecellgeneratestheenergyinglycolysisandrespiration,andthetotalgeneratedenergyisusedintheproductionofthebiomass.Attheexhaustionofthegrowthlimitingsubstrate,precursorsaregeneratedonlybythedegradationofthebiomass.Thegeneratedprecursorsredistributethemselveseitherbygoingintotherespirationtogenerateenergyorbypolymerizingbackintothebiomassusingtheenergygeneratedintherespiration.Thisprocessresultsinthenetdecayofthebiomassknownasfutilecycle.Thus,thisfutilecycledegradesbiomass.Sincethecellnolongerpicksupthesubstrate,itlackstheenergygeneratedthroughglycolysis.Thislosesofenergyresultsinthesuddendecreaseintheconcentrationofchargednucleotidesandconsequentlyenergycharge.Theadvantageoffutilecycleisinmaintainingthethelowconcentrationofchargednucleotide,hencetheviabilityofthecell. 4.2.2.6 Resumptionofthegrowthinthestarvationstate Thisexperimentisanextensionofstarvationexperiment.Inthisexperiment,thecellsweregrowninabatchreactor.Attheexhaustionofthesubstrate,thecul-turewasallowedtoenterintothestarvationphaseandglucosewasresuppliedafter45minutesofstarvation(Figure 4{10 )[ 66 ].Theenergychargeofexponentially

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(b) (c) (d) (e) (f) Simulationresultsofthebatchculturewithresupplyofthesubstrateduringstarvation.

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Figure4{10: Transientresponseoftheglucoselimitedbatchculturewithsupplyofglucoseafter45minutesofstarvation. growingcellswasnear0:8.Attheexhaustionofglucose,thecessationofgrowthcausesenergychargetodecreaseto0:6.Whenglucosewasresuppliedtothesamecultureafter45minutesofstarvation,thevalueoftheenergychargejumpstothelevelcharacteristicofthegrowingculture.Toreplicatethisexperiment,thesimula-tionwasperformedforanexponentialgrowthinbatchcultureandcontinuedevenaftertheexhaustionofthesubstrate.Atthestarvationphase,thecellstopsthegrowthandtheenergychargecomesdownsuddenly(seeFigure 4{9 b)becauseofthereasonasmentionedinSection 4.2.2.5 .Thesubstrateconcentrationwasshiftedtosomepositivevalueafterapproximately2hoursofstarvation.Theconcentrationoftransportenzymeandribosomestayhighbecauseoftheslowdegradationduringallofthisperiod.Theenergylevelwasmaintainedinthecellbythedegradationofthebiomass.Thus,assoonasthesubstrateisaddedtothereactor,thesubstrateuptakerategoesup(seeFigure 4{9 e).Initially,theincomingsubstratedivertstorespirationbecauseoflowenergycharge,consequently,theconcentrationofchargednucleotidesandenergychargegoesup.Thesuddenincreaseintheconcentrationofchargednucleotidesandavailabilityofhighconcentrationofribosomesputsthegrowthratetoahigherlevel(seeFigure 4{9 f).

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(b) (c) (d) Simulationresultsofthebatchculturewiththeslowsupplyofthesubstrateduringstarvation. 4.2.2.7 Slowsupplyofthesubstrateduringstarvation Thestarvationexperimentsindicatethatenergychargecanbemaintainedbetween0:8and0:9aslongasthereissubstrateintheenvironment.Assoonasthesubstrateisexhausted,energychargegoesdown.Peoplewereabletoextendthismaintenanceofenergychargeat0:8fordaysbycontinuousslowsupplyofglucosetoaglucoselimitedcultureinthestarvationstate(Figure 4{12 )[ 67 ].Aftercessationofthegrowth,glucosewasaddedattherateof0:15moleofglucose/mgofproteinperh.Whenglucosewasaddedathigherrate;ahigherenergychargewasmaintained,similarly,atlowerglucoseadditionrates;alowerenergychargewasmaintained.Toreplicatetheseresults,itisassumedthatthevolumetricowrateoftheslowsupplyofglucoseusedintheexperimentissosmallthatthevolumeoftheculturedoesnotchange.Thisimpliesthattheroleofslowglucosesupplyistomaintainaconstantsmalllevelofsubstrateconcentrationinthe

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Figure4{12: Transientresponseofaglucoselimitedbatchculturewiththeslowsupplyofglucoseduringstarvationtomaintainenergycharge. reactor.Simulationswereperformedatconstantsubstrateconcentrationwithoutworryingthefactthatreactordilutionoccursbecauseofowintothereactor.Theresultingenergychargeprole(Figure 4{11 )wassimilartotheresultsobtainedintheexperiment(Figure 4{12 ).Theconcentrationsofthecells,chargednucleotides,andtotalnucleotidesstayconstantorcomedowndependinguponthesubstratesupplyrate.Theexplanationbehindthisdynamicisthatifthecellsareexposedtosomeconstantsubstrateconcentrationtheywillmaintainaconstantenergychargeinsidethecell.Accordingtothemodelconstruction,theenergychargeisanincreasingfunctionofsubstrateconcentrationboundedbythevaluescorrespondingtostarvationandsubstrateabundanceconditions.Forthissimulation,energychargeliesbetween0:4and1respectively.Theenergychargeprolesaturateoutataverysmallvalueofsubstrateconcentrationintheenvironment.Thesimulationusesdierentvaluesofthesubstrateconcentrationbetweenzeroandsubstrateconcentrationcorrespondingtosaturationlevelofenergychargetogeneratedierentenergycharges. 4.3 Discussion Themodelaccountingadeninenucleotides,ATPandADP,capturesawidevarietyofexperimentaldata.Thesimulationresultsqualitativelymatcheswith

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theexperimentaldata.TheincorporationofATPinourpreviousmodelnotonlypreservesthedynamicsoftransportenzymeandribosomemodelbutalsoincludesthefastdynamicsofATPinasubstratepulseexperimentandtheslowevolutionofATPinvariousstarvationexperiments.ThefastdynamicsofATPwasattributedtoitslowconcentrationinvolvedinthefastreactionkinetics,hence,thefastturnovertime.However,theslowdynamicsofATPisaresultantofthetransitionofATPintoquasisteadystatemanifoldofslowvariables.ThismodelalsocontradictsthehypothesisthatcellalwaystriestomaintainATPandADPconcentrationsuchthatitmaintainstheconstantenergycharge.Accordingtoourformulation,themaintenanceofconstantlevelofATPconcentrationispurelybalancebetweendenovosynthesisofnucleotideandconsumptionofnucleotidesinRNAsynthesis.Thehomeostasisofenergychargeismaintainedbecauseofthebalancebetweenenergygenerationprocessesi.e.glycolysis,respirationandenergyconsumptionprocessesi.e.synthesisofproteinandRNA.

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Wehaveformulatedamodelofmicrobialgrowthaccountingfortransportenzymesynthesis,ribosome-mediatedproteinsynthesisandadeninenucleotideasenergymolecule.Basedontheexperimentalobservation,thefundamentalassumptionwasmadethatthesynthesisoftransportenzymeandribosomeisautocatalytic.Boththesteadystatesandthetransientsofthemodelshowremarkableagreementwithawidevarietyofexperimentaldata.Themodelyieldssimpleexplanationsofseeminglycomplexbehavior. 1. Steadystates (a) Thecelldensitydecreasesatlowdilutionrates.Thisbehaviordoesnotstemfromanadhocmaintenance.ItfallsoutnaturallyasaconsequenceofthefutilecyclingofRNAatlowdilutionrates. (b) Theperipheralenzymelevelpassesthroughamaximum.Thisreectsthecompetingeectsofenzymeinductionanddilution.Atlowdilutionrates,inductiondominates,sothattheperipheralenzymelevelisanincreasingfunctionofD.Athighdilutionrates,dilutiondominates,andtheperipheralenzymelevelisadecreasingfunctionofD. (c) Theribosomelevelsapproachanonzerolevelatsmalldilutionrates.Accordingtothemodel,thisistheresultofthefutilecyclingofproteins. (d) ThenonzerolevelofATPwasalsomaintainedbythefutilecycleasthedilutionrateapproacheszero. (e) Athighdilutionrate(D>0:1),theconstantlevelofATPinthecellisaconsequenceofthebalancebetweendenovosynthesisofnucleotideandRNA.Thehomeostasisofenergychargeisasimplebalancebetweenthenetenergygenerationinglycolysisandrespirationandthenetenergyconsumptioninbiosynthesis. 95

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2. Transients (a) Themodelshowsgoodagreementwithawidevarietyoftransientsincludingcontinuous-tobatchshifts,feedswitches,dilutionrateshift-ups,dilutionrateshift-downsandstarvationphase. (b) Feedswitchsandlowdilutionrateshift-upstransientsaregovernedbytransportenzyme. (c) Thelowlevelofribosomesproducesthetransientobservedincontinuoustobatchshiftsandhighdilutionrateshiftups. (d) Thedistributionoftotalnucleotidebetweenchargedandunchargednucleotidedenesenergycharge,whichhelpsinthemetabolicdivisionofincomingsubstrateintorespirationandbiosynthesis. (e) Themaintenanceoflowlevelofenergychargeistheresultofthefutilecycleduringstarvationphaseinthebatchreactor. (f) Thetransientsobservedincontinuousreactorareinfactthecombi-nationsoftwocanonicaldynamics,namely,theapproachtobalancedgrowthundersupersaturatingsubstrateconcentrations,andtheap-proachtosteadystateundersubsaturatingsubstrateconcentrations.

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[1] M.H.Saier,Jr.,\Multiplemechanismscontrollingcarbonmetabolisminbacteria,"Biotechnol.Bioeng.,vol.58,pp.170{174,1998. [2] O.MaaloeandN.O.Kjeldgaard,Controlofmacromolecularsynthesis:AstudyofDNA,RNA,andproteinsynthesis,W.A.Benjamin,NewYork,1966. [3] M.Nomura,\Thecontrolofribosomesynthesis,"ScienticAmerican,vol.250,pp.102{114,1984. [4] X.Zhang,P.Dennis,M.Ehrenberg,andH.Bremer,\KineticpropertiesofrrnpromotersinEscherichiacoli,"Biochimie,vol.84,pp.981{996,2002. [5] K.F.JensenandS.Pedersen,\MetabolicgrowthratecontrolinEscherichiacolimaybeaconsequenceofsubsaturationofthemacromolecularbiosyntheticapparatuswithsubstratesandcatalyticcomponents,"Microbiol.Rev.,vol.54,pp.89{100,1990. [6] E.A.Dawes,\Storagepolymersinprokaryotes,"inProkaryoticstructureandfunction,S.Mohan,C.Dow,andJ.A.Coles,Eds.,NewYork,1991,SocietyforGeneralMicrobiology,SymposiaoftheSocietyforGeneralMicrobiology,pp.81{122,CambridgeUniversityPress. [7] E.A.DawesandP.J.Senior,\Theroleandregulationofenergyreservepolymersinmicroorganisms,"Adv.Microb.Physiol.,vol.10,pp.135{266,1973. [8] W.H.Holms,\ThecentralmetabolicpathwaysofEscherichiacoli:Rela-tionshipbetweenuxandcontrolatabranchpoint,eciencyofconversiontobiomass,andexcretionofacetate,"Curr.Top.Cell.Regul.,vol.28,pp.69{105,1986. [9] D.W.TempestandO.M.Neijssel,\Thephysiologyofmetaboliteoverpro-duction,"inMicrobialtechnology:Currentstate,futureprospects,vol.29ofSymposiaoftheSocietyforGeneralMicrobiology,pp.53{82.CambridgeUniversityPress,NewYork,1979. [10] T.Suzuki,\PhosphotransacetylaseofEscherichiacoliB:ActivationbypyruvateandinhibitionbyNADHandcertainnucleotides,"Biochem.Biophys.Acta,vol.191,pp.559{569,1969. 97

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ShaktiGuptawasborninIndia,onJune1,1976.HereceivedaBachelorofSciencefromIndiaInstituteofTechnology,Kanpur,India.AftercompletinghisbachelorsdegreehejoinedtheDepartmentofChemicalEngineering,UniversityofFlorida,in2000.Hisresearchinterestsaremodelingofbiologicalsystems. 103