Analytic derivation of bacterial growth laws from a simple model of in

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TheoryBiosci.(2016)135:121130DOI10.1007/s12064-016-0227-9ORIGINALPAPERAnalyticderivationofbacterialgrowthlawsfromasimplemodelofintracellularchemicaldynamics11,2,3ParthPratimPandey•SanjayJainReceived:13March2016/Accepted:20April2016/Publishedonline:11May2016ÓTheAuthor(s)2016.ThisarticleispublishedwithopenaccessatSpringerlink.comAbstractExperimentshavefoundthatthegrowthrateandKeywordsBacterialgrowthlaws
Growthratecertainothermacroscopicpropertiesofbacterialcellsinoptimization
Cellulareconomy
Chemicaldynamics
steady-stateculturesdependuponthemediuminasurpris-Mathematicalmodelinginglysimplemanner;thesedependenciesarereferredtoasgrowthlaws.HereweconstructadynamicalmodelofinteractingintracellularpopulationstounderstandsomeoftheIntroductiongrowthlaws.Themodelhasonlythreepopulationvariables:anaminoacidpool,apoolofenzymesthattransportanBacterialcellscontainthousandsofmolecularspeciesandexternalnutrientandproducetheaminoacids,andribosomesareexceedinglycomplex,yettheyexhibitcertainremark-thatcatalyzetheirownandtheenzymesproductionfromtheableregularitiesatthesystemlevelwhichhavebeenaminoacids.Weassumethatthecellallocatesitsresourcesquantifiedexperimentally.Theregularitiesofconcerninbetweentheenzymesectorandtheribosomalsectortomax-thispaperareasubsetoftheso-calledbacterialgrowthimizeitsgrowthrate.Weshowthattheempiricalgrowthlawslaws(Monod1949;Schaechteretal.1958;MaaloeandfollowfromthisassumptionandderiveanalyticexpressionsKjeldgaard1966;Maaløe1979;BremerandDennis1996;forthephenomenologicalparametersintermsofthemoreScottetal.2010)whichhighlighttherelationshipsbetweenbasicmodelparameters.Interestingly,themaximizationofthemacroscopicallymeasuredquantitiessuchascellcompo-growthrateofthecellasawholeimpliesthatthecellallocatessition,size,growthrateandtheenvironmentormediuminresourcestotheenzymeandribosomalsectorsininversewhichthecellgrows.Theempiricalrelationshipsareproportiontotheirrespectiveefficiencies.Theworkintro-summarizedintermsofphenomenologicalequations.Inducesamathematicalschemeinwhichthecellulargrowthratethispaperweattempttodeducethesephenomenologicalcanbeexplicitlydeterminedandshowsthattwolargerelationshipsfromamathematicalmodelofacellcon-parameters,thenumberofaminoacidresiduesperenzymetainingafewinteracting(poolsof)molecularspecies.Theandperribosome,areusefulformakingapproximations.populationdynamicsofthesemolecularspeciesbasedonstandardchemicalkinetics,togetherwithanoptimizationprinciple,givesrisetothegrowthlaws.ThisarticleformspartofaspecialissueofTheoryinBiosciencesincommemorationofOlafBreidbach.Whengeneticallyidenticalbacterialcellsdrawnfromanovernightcultureareintroducedintoavesselcontaininga&SanjayJainmediumwithacertainconcentrationofnutrients,temper-jain@physics.du.ac.inature,etc.,theyexhibitseveralphasesofgrowth(Monod1DepartmentofPhysicsandAstrophysics,UniversityofDelhi,1949).Theseinclude,insequence,alagphasewherethereDelhi110007,Indiaisverylittlegrowthinthenumberofcells,anacceleration2TheSimonsCenterforSystemsBiology,Instituteofphasewheregrowthpicksup,anexponentialphaseinAdvancedStudy,Princeton,NJ08540,USAwhichthepopulationofcellsgrowsexponentiallywith3SantaFeInstitute,1399,HydeParkRoad,SantaFe,time(ataconstantgrowthrate),adecelerationphasewithNM87501,USAdeclininggrowthratethatsetsinwhenthefoodbeginsto123 122TheoryBiosci.(2016)135:121130runoutandastationaryphasewherethepopulationisconstants(Scottetal.2010).Thesimplicityanduniversalityconstant,followedbyaneventualpopulationdecline.ofthesephenomenologicallawsaresurprisinggiventheRegularitiesaremostapparentintheexponentialphasecomplexityanddiversityofbacteria.Inadditiontotheabovewhichisoftenreferredtoasasteadystate.Inthisphasethegrowthlaws,thesizeofbacterialcellsalsoexhibitsaveragesanddistributions(acrossthepopulationofcells)remarkablepropertieswhicharenotthesubjectofthispaper.ofcelldoublingtime,cellsizeatbirth,intracellularcon-Therehavebeenseveralrecentworkswhichhavecentrationofribosome,totalproteinandmetabolitesattemptedtounderstandthegrowthlawstheoretically,becomeconstantintime(foraslongastheexponentialthroughmathematicalmodeling(Molenaaretal.2009;phaselasts).TheseconstantaveragevaluesdependuponScottetal.2010,2014;MaitraandDill2015;Weißeetal.thestrainofbacteriaandonthemedium(itsconcentration2015;Bosdrieszetal.2015).Scottetal.(2010,2014)haveofnutrients,temperature,etc.).Repeatedexperimentswithrelatedthephenomenologicalconstantstomolecularthesamestrainandmediumbutwithdifferentinitialparametersofthecell.TakingforwardanideaduetoMaaløeconditions(correspondingtodifferentovernightcultures)(1979),theyhavearguedthatthegrowthlawsreflectregu-yieldthesamegrowthrateinthesteadystateandthesamelatorymechanismsinthecellthatoptimizeitsgrowthrateinvaluesoftheseaverages.Thegrowthlawsarestatementsofanygivenmedium.Theyandotherauthors(MaitraandDillhowthegrowthrateandtheseaveragesdependuponthe2015;Weißeetal.2015;Bosdrieszetal.2015)havecon-environmentandcellularparameters.Thefirstofthese,duestructedmodelsforthemolecularregulatorymechanismstoMonod(1949),isthehyperbolicdependenceoftheinsidethecellthatcanproducetheabovegrowthlaws.steady-stategrowthratelupontheconcentration[F]ofaInthispaperweadoptadifferentapproachthatisclosergrowth-limitingnutrient(orfoodmolecule)inthemedium:inspirittotheworkofMolenaaretal.(2009).Molenaaretal.consideredanonlineardynamicalmodelofacellwith½Fl¼l1:ð1Þafewclassesofmetabolitesandenzymesaswellasribo-C1þ½Fsomesandshowedthroughcomputersimulationsthatl1isthemaximumvalueofthegrowthratepossibleinthemaximizationofthecellulargrowthratequalitativelymediumandC1thevalueof[F]atwhichthegrowthrateisreproducedsomeofthegrowthlawsandotherobservedhalfitsmaximumvalue.propertiesofcells.HereweconsiderasimplernonlinearInthecell,theribosomewhichassemblesaminoacidstodynamicalmodelofthecellcontainingonlythreemolec-produceproteinsfromamessengerRNAtemplateisanularpopulations:onemetabolitepool,oneenzymepoolandimportantcatalystofcellgrowth.Theamountofcellularribosomes.Weareabletoobtainanexplicitformulafortheinvestmentinribosomesisfoundtodependuponthegrowthgrowthrateofthecellasafunctionofcellularandmediumrateinacharacteristicmanner.Inparticular,theratioofparameters,whichhassofarbeenlackinginexistingribosomalproteininthecelltototalproteininthecell(bymodels.Maximizingthegrowthratewithrespecttooneofweight),referredtoastheribosomalfractionUR,isfoundtheparameters,thefractionofribosomesmakingribo-tobealinearincreasingfunctionoflwhenlisincreasedbysomes,wederiveallthethreegrowthlawsanalytically.Theimprovingthenutritionalqualityofthemedium(Schaechtermethodproducesanalyticexpressionsforthephenomeno-etal.1958;Maaløe1979;BremerandDennis1996):logicalparametersintermsofthemolecularparametersinminlthemodel.TheseexpressionsaregeneralizationsoftheUR¼URþ;ð2ÞjtonesobtainedbyScottetal.andreducetotheirresultswhencertainprocessesareignored.Weshowthattheoptimiza-whereUminandjareconstants.However,whenlisalteredRttionofgrowthrateleadstoasimpleprincipleofcellularbychangingthecatalyticefficiencyofribosomes(e.g.,byeconomy.Theworkprovidesadirectconnectionbetweenproducingmutantswithdifferentcatalyticefficienciesorbygrowthrateoptimizationandthegrowthlaws.addingantibioticsinthemediumthatparticularlyaffecttheAtamethodologicallevelweidentifynaturallargecatalyticefficiency)keepingthenutritionalqualityoftheparametersinthecellthatareusefulinmakingapproxi-mediumthesame,thenURisfoundtobealineardecreasingmations.Thismightproveusefulinmorecomplexcellularfunctionofl(Scottetal.2010):modelsandinmodelingothercellularphenomenaaswell.maxlUR¼UR;ð3ÞPrecursor-Transporter-Ribosome(PTR)cell:jnacoarsegrainedmodelwhereUmaxandjareconstants.TheabovethreeequationsRncanbeconsideredtobephenomenologicalequationsConsiderasimplemathematicalmodelofagrowingcelldescribingbacterialgrowthsteadystates,withthesixcon-consistingofthreetypesofmolecules;precursors,trans-stantsl;C;Umin;Umax;j;jasphenomenological11RRtnportersandribosomes.Werefertothismodelasthe123 TheoryBiosci.(2016)135:121130123fTkfRkKT¼;KR¼;fTþfR¼1;ð5ÞmTmRwherekrepresentsribosomalcatalyticefficiencyandistherateatwhichasingleribosomeconsumesPmolecules,perunitconcentrationofP,fortheproductionofproteins.ThisaccountsforthetermkRP/VintheP_equation,thetotalrateofconsumptionofP.AfractionfToftheribosomesmakestheTproteinsandtheremainingfractionfRtheribosomalFig.1ThePTRcell.Precursormolecules(P)areproducedbytheproteins.Thus,ofthePconsumptionfluxapartfTkRP=Vcatalyticactionofthemetabolicproteins(T)ontheexternalfoodgoestoproduceTandtheremainingpartfRkRP=Vgoestomolecules(F).Metabolicproteinsandribosomalproteins(R)areproduceR.EachTmolecule(ribosome)containsmT(mR)synthesizedfromthePmoleculesinreactionscatalysedbyRaminoacidresidues;hencetherateofproductionofTisfTkRP=VmTandthatofRisfRkRP=VmR.ThisexplainsthePrecursor-Transporter-Ribosome(PTR)model.ThesystemassumedformsofKTandKR.dTanddRarethedegradationhasthefollowingthreereactions(Fig.1):ratesofTandR,respectively,intoawasteproduct;weassumeanegligibledegradationrateforP.T1.F!,whereexternalfoodmolecules(F)aretrans-VistheinstantaneousvolumeoftheinteriorofthecellPportedintothecellbytheactionoftransporterproteinsandweassumethatitisalinearfunctionofthemolecular(T)andconvertedintoprecursormolecules(P)repre-populations.Sincemolecularpopulationsinthebulkaresentingaminoacids;PandR,wecantakeittobeproportionaltoPþR.OurRresultsdonotdependuponthisparticularchoiceandfor2.P!,wherePmoleculesareconvertedintoTbytheTgeneralityweassumecatalyticactionofribosomes(R),andRV¼vPPþvTTþvRR;ð6Þ3.P!,whereRcatalysestheproductionofitselfusingRwherev;v;vareconstants.NotethatEqs.(4a)(4c)doPTRP.notcontainatermproportionaltoV_=Vontheright-handsidebecausetheyrefertopopulationsinsteadofAllthemoleculesareproducedintheinteriorofthecell.concentrations.Themembraneconsistssolelyoftransportermolecules,whichareassumedtomigrateimmediatelyuponformationtothecellboundary.TheinteriorofthecellconsistsofSteady-statesolutionofthePTRcellprecursormoleculesandribosomes.Themodelisdescri-bedbythefollowingsetofdifferentialequations:ThesteadystateofabacterialculturecorrespondstocellsdPRPgrowingexponentiallywithaconstantrate.Welookforan¼KPTk;ð4aÞdtVexponentialsolutionforthechemicalpopulations:PðtÞ¼Pelt;TðtÞ¼Telt;RðtÞ¼Relt,wherel,acon-dTRP000¼KTdTT;ð4bÞstant,isthegrowthrateofthePTRcell.SubstitutingthisdtVansatzintoEq.(4),wegetdRRP¼KRdRR;ð4cÞR0P0dtVlP0¼KPT0k;ð7aÞV0wherePrepresentsthenumberofprecursormoleculesinR0P0thecell(aminoacidpool),TthenumberofallmetabolicðlþdTÞT0¼KT;ð7bÞproteinmoleculesthattransportfoodintothecellandV0convertitintoprecursorandRisnumberofribosomesinR0P0ðlþdRÞR0¼KR;ð7cÞthecell.TherateconstantKPrepresentstheefficiencyofV0metabolisminmakingPfromexternalfood.ItisanwhereV0¼vPP0þvTT0þvRR0.Henceforthwedroptheincreasingfunctionoftheexternalfoodconcentration[F]subscript0astheequationsarevalidforthetime-depen-(explicitformstobediscussedlater)andcanalsoencap-dentquantitiesP(t),T(t),R(t)aswell.Thelastofthesesulatethequalityofthefoodsource(e.g.,thenumberofPequationsimmediatelygivesmoleculesproducedperfoodmoleculetransportedin).Theotherproductionrateconstantsareparametrizedasfollows:P=V¼ðlþdRÞ=KR:ð8Þ123 124TheoryBiosci.(2016)135:121130Substituting(8)in(7b)givestheratioT/R:bechosensuchthatl[0.Whenparametervaluesweresuchthatl,anexponentialdeclineofTmRfTðlþdRÞ¼;ð9Þpopulationswasobservedinsteadofgrowth.)RmTfRðlþdTÞ2.OnecanexaminethetwolimitsfR!0andfR!1.andsubstituting(8)and(9)in(7a)givestheratioP/R:WhendT¼dR¼0,inboththeselimitslmustgotozero.Physically,whenfR!0,thenKR!0andPmRðlþdRÞKPfT¼1:ð10ÞEq.(4c)impliesthatribosomesarenotproduced;RfRlmTðlþdTÞhenceRisaconstant,orl¼0.WhenfR!1,thenThustheratiosofthepopulationsandtheconcentrationsofKT!0,andTisnotproduced;henceagainl¼0.Itthethreechemicalsatsteadystatecanbeexpressediniseasytoseethatlgoestozeroinboththeselimitstermsoflandtheparametersofthemodel.Inordertoandnotlþ.solvetheproblemfully,weneedtofindlintermsofthe3.WehaveverifiedanalyticallyfromEq.(7)thatwhenparameters.dT¼dR¼0andmT¼mR1,lþgivesrisetoGrowthrateTheEq.(8)givesl¼KRP=VdR.NotenegativepopulationswhilelgivesrisetopositivethatVcanbewrittenasV¼vPð1þvTTþvRRÞ¼populations.PvPPvPPvPð1þ½vTTþvRRÞ.ThusP/ViscompletelyexpressedinWeremarkherethatithasbeenpossibletoobtainanPvPRvPPtermsoftheratiosT/RandP/Rwhichareknownasexplicitsolutionforlbecausewehaveexpressedthecellfunctionsoflandtheparameters[Eqs.(9)and(10)].volumeasafunctionofthepopulationsandfurtherTherefore,theequationl¼KRP=VdRbecomesanassumedthatitisalinearfunctionofthepopulations,(6).equationthatcontainsonlylandtheparameters.Simpli-Thisassumption(a)makestheexponentialansatzasolu-fyingit,wegetaquadraticequationinlwithcoefficientstionof(4),and(b)causestheabsolutepopulationstobedependingontheparameters:eliminatedfrom(8),leavinganequationconnectinglandtheparameters.Inourviewthevolumeassumptionisaal2blþc¼0ð11Þcrucialonethathasbeenmissingfrompreviousmodels.withRibosomalfraction(UR)TheratioofribosomalproteinmRRa¼11;b¼aþbþ2;c¼ab;ð12aÞtototalprotein(byweight)isgivenbyUR¼mTþmR.TRa¼mfTdT;b¼qfRdR;ð12bÞUsingEq.(9)URbecomes01fTfRðdTdRÞm¼KP=mT¼nutritionalefficiency;ð12cÞUR¼¼fRþ:fTðlþdRÞlþfTdRþfRdTð14Þ1þq¼k=ðmvÞ¼ribosomalefficiency0;ð12dÞfðlþdÞRPRT1vT1vRNoticethatthisexpressionforURisanonlinearfunctionof1¼fTþfR;ð12eÞmTvPmRvPlifdT6¼dRandaconstantindependentoflifdT¼dR.Thisisquitedifferentfromtheobservedlineargrowthlaws1vT1vR2¼fTdRþfRdT:ð12fÞ(2)and(3).ThusthePTRmodeldoesnotreproducethemTvPmRvPobservedgrowthlaws.ThemodelasitstandsismissinganEquation(11)hastwosolutions:importantingredientregulationthatwenowturnto.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2bb4acl¼:ð13Þ2aThePTRmodelwith‘regulation’andbacterialThelsolutionisthephysicallyrelevantone,inwhichthegrowthlawssquare-rootisalwaystakenwiththenegativesign.Thereareseveralwaystoseethis:UptonowwehavetreatedfTandfR,thefractionofribo-1.TheEqs.(4a)(4c)canbesimulatednumericallyforasomescatalysingtheproductionoftransportersandribo-fixedsetofparametervaluesandinitialconditions.somalprotein,respectively,asconstantparametersoftheThiswasdoneforseveralparametersetsandinitialmodel.However,itisawell-knownfactthatregulatoryconditions.WefoundthatatlargetimesP,TandmechanismsexistinbacteriathatregulatehowmuchRalwaysgrewexponentiallywithtimeandtheirrateribosomeisengagedinproducingribosomalproteinandofexponentialgrowthwasgivenbylandnotlþ.howmuchinproducingmetabolicprotein.InthecontextofthePTRmodelthesemechanismswouldmodulatetheFurther,theobservedasymptoticratiosweregivenbyEqs.(9),(10)withl¼l.(ParametervalueshadtovalueofthefRparameter(andhencefT¼1fR).The123 TheoryBiosci.(2016)135:121130125absenceofthismechanisminthePTRmodelasdescribedHowever,intheinterestofmathematicalsimplicityweaboveisthereasonthatitdoesnotreproducetheobservedtakeanalternativeapproachinvolvingoptimization,growthlaws.employedearlierbyMolenaaretal.(2009)foradifferentTrade-offbetweenmetabolicandribosomalproteinmodel.WeassumethatforanyfixedmediumandotherproductionSincelisafunctionofthecellularandcellularparametersadditionalregulatorymechanismsmediumparameters[Eq.(13)],wefirstaskhowitvariesasexistinginthecellacttomodifyfR(e.g.,bychangingthefRisincreasedkeepingthemediumandallothercellularproportionofmessengerRNAmoleculescorrespondingtoparametersfixed.Numericalanalysisofthesteady-stateofRandT)suchthatthecellulargrowthrateisoptimized,thePTRmodelshowsthatwhenallotherparametersarei.e.,foragivenmediumandothercellularparameters,thefixed,lisanon-monotonicfunctionoffRasshowninregulationadjustsfRtofmax.ThisisinspiritsimilartotheFig.2a.Thisreflectsatrade-offbetweenproductionofoptimalityassumptionmadeinfluxbalanceanalysisofmetabolicproteinsandribosomalproteinsinthemodel.metabolicnetworks(Orthetal.2010).Inotherwords,weThereisadistinctvalueoffR(fmax)wherelisamaximumassumethatthesteadystatereachedwhentheseother(lmax).fmax,lmaxdependupontheotherparametersandin(unspecified)regulatorydynamicsareincludedisapprox-particular,fmaxincreasesasKPisincreased(keepingtheimatedbythesteadystateofthePTRmodelwithothersconstant).ForconvenienceweherewriteKP¼qkpf¼f;ð15ÞRmaxwhereqequalsthenumberofPmoleculesproducedperfoodmoleculeconsumed(qualityofthemedium),andkPwherefmaxisthevalueoffRthatmaximizesl[Eq.(13)]dependsuponexternalfoodconcentration.Weobserveinkeepingallotherparametersfixed.WecallthissteadystateFig.2athatasthequalityofmediumisincreased,fmax(whenfRissetequaltofmax)astheoptimizedsteady-state.increases.Thesetwoproperties,namelynon-monotonicityAchangeinmedium,ingeneral,leadstoadifferentfmaxoflwithrespecttofRandtheincreaseoffmaxwithmediumsincelisafunctionofallthemedium-dependentparam-qualityhavealsobeennotedinScottetal.(2014)usingaetersparametersandfR.differentapproach.OptimizedsteadystateofthePTRcellreproducesIncorporatingtheeffectofregulatorymechanismsqualitativefeaturesofobservedgrowthlawsFigure2b,cthroughanoptimizationassumptionInordertobringinshowsthattheoptimizedsteadystateofthePTRcellregulatorymechanismswecanmaketherateconstantsqualitativelysatisfiesthegrowthlawssummarizedindependentonmolecularconcentrationsreflectingfeedbackEqs.(1)(3).InFig.2btoincreasethegrowthrateforthemechanismsorintroduceothermolecularspecies(thePTRcellweonlyincreasethemediumqualityq(keepingregulators)intothemodel(Scottetal.2014;MaitraandkP;k;mT;mR;dT;dR;vP;vT;vRconstant).ForeachmediumDill2015;Weißeetal.2015;Bosdrieszetal.2015).qualityqwenumericallyobtainfmaxðqÞusingEq.(13),i.e.,acbFig.2ThePTRmodelintheoptimizedsteadystatequalitativelycURversuslmaxfordifferentvaluesofqandk(otherparametersreproducestheobservedgrowthlaws.aTrade-offbetweenproductionsameasbefore).Thecolouredlines(positiveslope)correspondtoofribosomalandmetabolicproteins:lasafunctionoffRfordifferentEq.(2)(changingmediumqualityatfixedtranslationalefficiency).valuesofq(KP¼qkP,kP¼250hr1,k¼5104hr1lm3,dT¼Thegreylines(negativeslope)correspondto(3)(changingtransla-0:1hr1,d¼0hr1,m¼104,m¼5102,v¼v¼v¼tionalefficiencyatfixedmediumquality)RRTPTR108lm3).blasafunctionofq.Otherparameterssameasin(a).max123 126TheoryBiosci.(2016)135:121130thevalueoffRthatgivesthelargestlforthegivenq.WeSubstitutingfR¼fmaxandl¼lmaxinEqs.(9)and(14)denotethisoptimizedlaslmaxðqÞsinceitdependsonq.givesWeplotthedependenceoflmaxonqandfindthequali-TmRqtativebehavioursimilartotheMonodcurve(1).¼;ð18ÞRmTmNextweshowthedependenceofribosomefractionURU¼monlmaxintheoptimizedsteadystatewhenlmaxisRmþq:ð19Þincreasedbyincreasingq.Foreachmediumqualityq,wealreadyhavefmaxðqÞandlmaxðqÞ.ToobtainURweusetheThisexpressestheribosomalfractionattheoptimizedrelationEq.(14)withfR¼fmaxandl¼lmax.Figure2csteadystateofthePTRcellintermsofmediumandcel-showstheplotofURversuslmaxasthequalityofthelularparameters.Thegrowthlawsinthestandardformmediumisincreased.Thelineswithpositiveslopein(2),(3)followfrom(19)and(17).Forexample,toFig.2ccorrespondtothisvariation.NoticethelinearunderstandthedependenceofURonlwhenthemediumbehaviourofthecurvesashasbeenobservedinexperi-qualityisvaried,onecaneliminateminfavouroflinments,Eq.(2).Eq.(17)andsubstitutethatinEq.(19).Thisyields(2)withForasmallervalueofk(smallerribosomalefficiency),mindTtheURversuslmaxcurveremainslinearbutwithalargerUR¼;jt¼qþdTdR:ð20ÞqþdTdRslope(colouredlinesin2c)ashasbeenobservedinexperiments(Scottetal.2010).Figure2cqualitativelySimilarly,onecaneliminateqinfavouroflfromEq.(17)reproducestheobservedbehaviourofUR[Eqs.(2),(3)]andsubstituteinEq.(19)togetEq.(3),withwhenthegrowthrateisvaried,bothbyincreasingmediummaxmdTqualityandbydecreasingribosomalefficiency.UR¼;jn¼mdTþdR:ð21ÞmdTþdRAnalyticderivationofthegrowthlawsforthePTRmodelthelargemT;mRapproximationTheaboveresultsThisreproducestheequationsofthegrowthlawsandobtainednumericallyandillustratedinFig.2canbeexpressestheconstantsappearinginthoseequationsinderivedanalytically.Theexpressionsturnouttobeverytermsofthemediumandcellularparameters.Equation(2)simplewhenmTandmRaremuchgreaterthanunity,whichwithparametersgivenby(20)describesthepositiveslopeweassumeinthefollowing(mTbeingthenumberofaminolinesinFig.2candEq.(3)withparametersgivenin(21)acidmoleculesneededtomakeanenzymeis300,anddescribesthenegativeslopelines.mR,thenumberofaminoacidsinallribosomalproteinperribosomeis7000BremerandDennis1996).Wealsoneedtoassumethattheparametersmandqdefinedin(12)DiscussionareindependentofmTandmR;inotherwords,KP,therateofPproductionperunitTmoleculeinthecell,andk=vP,NutritionalandribosomalefficiencyWenowdiscussthetherateatwhicharibosomeaddsaminoacidstoaprotein,meaningoftheformulaeobtained.Theformulaearearesufficientlylarge,infact,respectively,ofordermT;mR.expressedintermsoftwoquantitiesmandqanditisusefulmandqwillturnouttobethetwonaturaltimescalesthattointerpretthesequantitiesfirst.WefollowScottetal.determinethesystemlevelpropertiesofthecell.Thetime(2014)incallingmthenutritionalefficiencyofthePTRscalesdT;dR,andthevolumevPwillalsobeassumedtobecellinthegivenmedium.NotethattheproductionterminP_independentofmT;mR.vT;vRmaybeindependentoronlyisKPT¼mmTT.SincemTTisthetotalnumberPmoleculesweaklydependentonmT;mR,respectively(sublinearlockedupinT,misthenumberofaminoacidmoleculesdependence).Withtheseassumptions,11andproducedinthecellperunittimeperaminoacidresidue2aþb.lockedupinthemetabolicenzymes.m,beingtherateofPThen,asshownintheAppendix,productionperunitPinvestedinmetabolicenzymes,isappropriatelythemetabolicefficiencyornutritionalf¼mþdRdTmaxmþq;ð16Þefficiencyofthecellinthegivenenvironment.Inordertoseethemeaningofqitisconvenienttoconsiderthesitu-andtheoptimizedsteady-stategrowthrateofthePTRcellationwheretheconcentrationofPishighenoughsothatitsisgivenbyavailabilityisnolongeralimitingfactorforribosomalactivity.InthemodelthelargestvalueofP/Vis1=vP,whichl¼l¼qðmdTÞmdRmaxmþq:ð17ÞariseswhenthecontributionofPtothevolumedominatesoverthecontributionfromTandR,i.e.,VvPP.Then(4c)ThisleadstotheMonodcurveaswillbediscussedlater.becomesR_¼ðqfRdRÞR.ThenRbyitselfformsan123 TheoryBiosci.(2016)135:121130127autocatalyticset(ACS)withgrowthrateqfRdR.qistheTheMonodcurveWeturntoadiscussionofthemaximalgrowthrateofthisACS(whendR¼0andfR¼1),analyticexpressionforl,Eq.(17).FirstwediscusstheortherateatwhichRcanmakecopiesofitselfifitwassituationwhendT¼dR¼0.Thenfrom(20),(21),jt¼qsolelyfocusedondoingthat(thatis,iffR¼1).q,beingtheandjn¼m,andourresultsforlandalltheotherquantitiesmaximalrateofRproductionperunitRpresent,willbereproduceexactlytheresultsofScottetal.(2010,2014).referredtoastheribosomalefficiencyofthecell.TheThegrowthratereducestofactork=vPinq¼k=ðvPmRÞistherateatwhicharibosomeqml¼:ð24Þcanaddanaminoacidtoaproteinwhenthereisnolimi-qþmtationofPandthefactorofmRaccountsforthenumberofPThisisthesameastheexpressionl¼ðUmaxrequiredtomakearibosome.InScottetal.(2014)qisRUminÞqm=ðqþmÞderivedinScottetal.(2010,2014),whenreferredtoasthetranslationalefficiencyofthecell.R(20)and(21)areusedtosetUmin¼0;Umax¼1.TomakeOptimizationasaprincipleofcellulareconomyAsRRmentionedearlier,thegrowthlaws(2)and(3)followfromcontactwiththeMonodequation(1),onehastosayhowm(19).ThelatterisamorebasicequationasitexpressesURdependsupontheconcentration[F]oftheexternalnutrient.directlyintermsoftheparameterswithoutreferencetotheAsmentionedbelow(4)KPandhencemisanincreasinggrowthrate,anditencapsulatestheconsequenceofgrowthfunctionof[F].IfonesubstitutesthesimplestfunctionratemaximizationwhenmT;mR1.(19)orequivalentlym¼k1½F,wherek1isaconstant,into(24),oneobtains(1)(18)canberecastaswithl1¼qandC1¼q=k1.Alternatively,ifthetransportðmTTÞm¼ðmRRÞq:ð22ÞlimitedMichelis-Mentenformoffooduptakem¼m0½F=ðKþ½FÞ,wherem0andKareconstants,isWecaninterpretmTTastheallocationorinvestmentofthesubstitutedin(24),onegets(1)withl1¼qm0=ðqþm0Þcellinthemetabolicsector(measuredinunitsofP)andandC1¼K=½1þðm0=qÞ(Scottetal.2014).mRRastheinvestmentintheribosomesector.WedefineThedifferencebetweenourderivationof(24)andthattheoutputofeachsectorastheinvestmenttimeseffi-ofScottetal.isthatthelatterusesthegrowthlaws(2),(3)ciencyofthesector.Thentheinvestmentstrategyoftheasthestartingpointandobtainstheabovementionedcell,namely(22),canbestatedasexpressionforl.ItdoesnotrequireanyfurtherassumptionOutputofmetabolicsector¼Outputofribosomalsector:ofgrowthrateoptimalityinderivingthatexpressionas(2),(3)alreadyincorporateoptimality.Ontheotherhand,ð23Þourderivationstartswithequations(4)describingtheEquivalently,(22)canbestatedasthefollowingprincipledynamicsofthethreepools,obtainslinthesteadystateofcellulareconomy:theresourcesallocatedtotheenzymebeforeoptimizationandthenusestheoptimalityassump-andribosomalsectorsareinverselyproportionaltotheirtiontoderive(2),(3)aswellastheoptimizedl.Thisrespectiveefficiencies.Inotherwords,thePTRcellfollowscrisplyestablishestherelationshipbetweenoptimalityandthedictum:Fromeachsectoraccordingtoitsability,toeachthegrowthlaws.sectoraccordingtoitsneed.HereabilityofasectoristheItmaybehelpfultomakeafewremarksabout(24).Thesameasitsefficiency,definedearlier,andneedistheright-handsideisasymmetricfunctionofmandq,whichallocationorinvestmentinthesectorthatwouldmakeitsdefinethetwonaturaltimescalesintheproblem.(1)Foroutputequaltothatoftheothersector.Thisprinciplefixedqasafunctionofm,itsaturatesatamaximumvaluefollowsfromtheoptimizationofthegrowthrateofthePTRl¼q.ThesaturationisnotaconsequenceofaMichelis-cellasawholeinthelargemT;mRapproximation.NotethatMententypesaturationkineticsassumedinthemodelefficiencyishardwiredintothecellularandmedium[Eq.(4)hasnoMichelis-MentenorHilltypeterms],butisparameterswhiletheallocation,inthecontextofthepresentaconsequenceoftheexistenceofthesetwotimescalesinmodel,isamatterofcellularchoice(though,ofcourse,incellulardynamics.Whenmq,fRin(16)approaches1;practice,eventhatishardwiredintotheregulatorymecha-thusthecoreautocatalyticsetthatdrivesthePTRcellnismsthatdynamicallyimplementthechoice.)ribosomeproducingmoreribosomeisfocusedlargelyonWeremarkthat(22)isnotarequirementforthesystemproducingitself.Eventhen,weknowthatthemaximalratetohaveasteadystate.Indeed,steadystatesareachievedinofRself-reproductionproductioncanonlybeq,whichthemodelevenwhenfRisnotatitsoptimalvaluegivenbyexplainsthesaturation.(2)Interestingly,notonlyisthe(16),asdiscussedearlier.WhenfR6¼fmax,wecanstillhavesaturationvalueoflequaltoq,thevalueofmatwhichlisasteadystatewithconstantconcentrationssatisfyingthehalfitsmaximumvalueisalsoq.ThishasasimpleEqs.(8)(14),but(22)doesnothold.(22)istheconditionexplanation.If,toachievethemaximumgrowthrate,thethatthesteadystatehasthemaximalpossiblevalueoflribosomepoolisfocusedsolelyonmakingribosome,thengiventhatallparametersotherthanfRarefixed.athalfthemaximalrateonlyhalfthepoolisfocusedon123 128TheoryBiosci.(2016)135:121130makingribosome.TheotherhalfisthenfocusedonmakingfTþfRþfQ¼1,andin(6),whereatermvQQisaddedtoT,andthisequalinvestmentinbothsectorsmeansthedefinitionofV.URisnowdefinedbymTT¼mRR.ButfromtheprincipleofcellulareconomymRR=ðmTTþmRRþmQQÞ.Intheoptimization,fQisthetwosectorialoutputsareequal;therefore,mmustbetreatedasafixednumber;fRcanrangebetween0and1fQequaltoq.Analternativewayofsayingthisistoobserveandischosentomaximizethegrowthrate.Doingthefrom(16)thatfR¼1=2atm¼q.(3)ThesymmetryanalysisasforthePTRmodel,onereproducesthegrowthbetweenmandqimpliesthatifmisheldfixedandqislaws(2)and(3)inwhichUmin;jarethesameasfortheRtincreased,lwillsaturateatavaluem,andthevalueofqatPTRmodel,andUmax¼1f,j¼mð1fÞ.Further,RQnQhalf-saturationisalsom.fR¼mð1fQÞ=ðmþqÞ,and,asbefore,l¼qfR,UR¼fR.DissipationtermsEquation(17)isageneralizationofAsimplerderivationoftheresultsfromalinear(24)whendT;dRarenonzero.NotethatevenwiththemodelAbove,wehavepresentedadetailedderivationofadditionaltermsthereisasymmetrybetweenthetwofR,landURfromthePTRmodelassumingmT;mR1.Itsectors:lisunchangedunderthesimultaneousinterchangeisworthmentioningthatthesameresultsfollowfromam$q,dT$dR.WhendR¼0;dT[0,thefactormmuchsimplerheuristicargument.SupposingweassumedT/KPmTdTinthenumeratorreflectsthatthemeta-thatthedominantcontributiontoVisvPP,i.e.,weignorebolicefficiencyhastobe[dTtosustainanonzerogrowththecontributionofTandRtoV.(Thisdoesnotmeanthatrate.ThisisbecauseforeveryKPmoleculesofPproducedthecontributionofTandRtothemassofthecellismuchbyeachmoleculeofTperunittime,anumbermTdTislostsmallerthanthatofP.IfmT;mR1,thecontributionofTthroughthedTTterm.Similarly,whendT¼0,dR[0,andRtothemassofthecellcouldbelarge,evenlargerthefactorqdRinthenumeratormeansthattheriboso-thanthecontributionofP,whiletheircontributiontothemalefficiencyhastobegreaterthandRfortheribosomalvolumeismuchsmallerthanthatofP,aslongasvT;vRareACStogetofftheground.AnonzerodRrequiresagreaterindependentof(orsufficientlyweaklydependenton)fractionofribosomestobemakingribosomes,andamT;mR).Then(4)reducestoasetoflinearequationsX_¼nonzerodTrequiresagreaterfractiontobemakingT[seeAXwithEq.(16)].However,therelativeinvestmentbythecellin0101thetwosectorsasmeasuredbyT/RorURisindependentofP0KPk=vPBCBCdT;dR[seeEqs.(18),(19)].TheequalityoffRandURhasX¼@TA;A¼@0dTKT=vPA:ð25ÞbeencommenteduponbyScottetal.(2010).TheyhaveR00bconsideredmodelsinwhichthedegradationtermsarezero.ThelargesteigenvalueofAisb;hencethegrowthrateofInthepresentmodelalsofR¼URwhendT¼dR.Butwhenthecellisl¼b.TheeigenvectorcorrespondingtobhasdT6¼dR,thetwoarenotequal.WenotethatinthemodelthephenomenologicalT¼KTR=½vPðbþdTÞ,P¼½k=ðvpbÞ½mfT=ðbþdTÞ1R.minmaxSinceP0wehavemfT=ðbþdTÞ10orab.WeparameterURiszeroifdT¼0,andtheUR¼1ifdR¼nowaskthefollowing:whatisthelargestvalueofl0[Eqs.(20),(21)].InbacterialcellsdTmaybeoftheorder1possible,andforwhatvalueoffRdoesthatoccur?Sinceof0.1h(Dressaireetal.2009;MaitraandDill2015),l¼b¼qfRdR,onemaynaivelythinkthatthelargestwhiledRmaybemuchlower(Zundeletal.2009).ThisminpossiblevalueoffR,namelyfR¼1willgivethelargestl.predictsavalueofURabout23timessmallerthantheHowever,wealsohavetheinequalityba;therefore,theobservedvaluegiveninScottetal.(2010).Equation(20)largestvalueofloccurswhenb¼a.ThisisthesamepredictsthatwhendT[dR,jtasafunctionofqislinearconclusionasreachedintheAppendixforthefullPTRwithapositiveintercept.ThisfeatureisseeninthedatamodelunderthelargemT;mRapproximation.Thecondi-(Scottetal.2010).However,againthevalueoftheinter-tiona¼bimmediatelyyieldsfR¼fmaxwithfmaxgivenbyceptpredictedby(20)issmallerthanthevaluefromthe(16),andl¼lmaxgivenby(17).Furthertheabovedata.ThissuggeststhatothercontributionstoUminandj,Rteigenvectoralsoreproduces(18)forT/R.Inthelinearizednotdescribedbythepresentmodel,aresignificant.equationP_¼lP¼KPTðk=vPÞR¼ðmTTÞmðmRRÞq,TheconstantfractionsectorScottetal.(2010)intro-wecanrecognizethetwotermsastheoutputsoftheducedanothersectorofproteinsQinadditiontoTandRmetabolicandribosomalsectors.whichtakesupafixedfractionoftheproteinmassUQ,toTheaboveapproximationisreasonableforfRfmax.ItisaccountforthefactthatUmaxwasobservedinexperimentsRmeaninglessforfR[fmaxbecausePturnsnegativeinthattobelessthanunity.Inthepresentmodelthissectorcanberegimeunderthisapproximation,thoughthefullmodelhasaddedasfollows(weconsiderthecasedT¼dR¼0):to(4),aperfectlyreasonablebehaviourevenforfR[fmax.AsaddanotherequationQ_¼KQRP=V,whereKQ¼fQk=mQ.seenearlier,thisapproximationisalsogoodfordeducingTheotherchangesarein(5),wherewenowhavefmax;lmaxandT/RasthesetendtofinitelimitswhenfR123 TheoryBiosci.(2016)135:121130129approachesfmaxfrombelow.Itisnotusefulforestimatingcreativecommons.org/licenses/by/4.0/),whichpermitsunrestrictedP/RnearfR¼fmaxwhichapproacheszerointhisapproxi-use,distribution,andreproductioninanymedium,providedyougiveappropriatecredittotheoriginalauthor(s)andthesource,provideamation.OnecanseefromthefullmodelthatP/RreceiveslinktotheCreativeCommonslicense,andindicateifchangeswerecorrectionsinasmallrangeoffRofsize1=mRaroundmade.fmax,inwhichrangeitgoesfromavalue1toasmallervalue.InthefullmodelP/RdoesnotgotozeroatfR¼fmax.AppendixUnderthelargemT;mRassumptionsdescribedinthemainConclusiontext,weignore1comparedto1and2comparedtoaþbin(12).ThusInthispaperwehaveconstructedasimpledynamicala1;baþb;c¼ab;ð26Þsystemdescribingacellintermsofitsthreecoarse-grainedmolecularpoolsandshownthattheoptimizationoftheItfollowsthatb24acisaperfectsquare;steady-stategrowthrateofthecellwithrespecttoa222parameterthatcanbetunedbyintracellularregulationb4ac¼ðaþbÞ4ab¼ðabÞ:ð27Þleadstothegrowthlaws(1),(2)and(3).Wehaverepro-FromEq.(13)wehaveducedandextendedexistingformulaeforthegrowthratepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiandotherphysiologicalparameters.Thisdeepensour2bb4ac1l¼¼½ðaþbÞjabj;understandingofthemacroscopicphysiologicalvariables2a2ð28Þintermsofmicroscopicparameters.Weexpectthatthis¼bifab;kindofmodelcanbeextendedtoincludeothermolecular¼aifab:sectorsinthecell(Huietal.2015).pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAtamethodologicallevelwehaveintroducedaIntheequationaboveb24achasbeenreplacedbyschemethatallowsanexplicitcomputationofthesteady-jabj,because,asdiscussedearlier,itisthissolutionstategrowthrateofthecellintermsofthecellularandmediumparameters.Inthisschemeakeyassumptionisthatthevolumeofthecellisdeterminedbyitsmolecularpopulations.Wehavealsoputtousetwonaturallargeparametersinthecell,mTandmR,tosetthescaleofcertainotherparametersandtomakeapproximations.Thishasallowedustogetanalyticresultsforthenonlinearsystemleveldynamics.Ourmodelusesanoptimizationprincipletofixaninternalparameter,fR,thefractionofribosomesmakingribosomes.Themodelissilentonthedynamicalmecha-nismsinsidethecellthatimplementthisoptimization.Thesemechanismshavebeenthesubjectofseveralrecentworks(Scottetal.2014;MaitraandDill2015;Weißeetal.2015;Bosdrieszetal.2015).Wehopethatcombin-ingsomeofthemethodsintroducedherewiththemecha-nismsdiscussedintheseworkswillproducemodelsthataremoresatisfactorythanthepresentone.AcknowledgmentsWethankPoojaSharmaandHarshantSinghfordiscussions,andtheInternationalCentreforTheoreticalSciences,BangaloreandAbdusSalamInternationalCentreforTheoreticalPhysics,Trieste,forhospitality,wherepartsofthisworkweredone.SJacknowledgesgrantsfromtheDepartmentofBiotechnology,Fig.3lvsfRinthePTRmodel.ThefigureisillustratedforGovernmentofIndia,andaResearchandDevelopmentgrantfromdR¼0;dT¼d.ThetwosolidlinesshowlasanincreasingandthentheUniversityofDelhi.PPPwouldliketothanktheUniversitydecreasingfunctionoffR,withamaximumlmaxatfR¼fmax.TheGrantsCommissionforaJuniorResearchFellowship.solidlineofslopemisshownforagenericvalueofm.TheotherthreedottedlineswithspecificvaluesofmrepresentwhatthissolidOpenAccessThisarticleisdistributedunderthetermsofthelinewouldhavebeenforthosevaluesofm.AsmincreasesfromdtoCreativeCommonsAttribution4.0InternationalLicense(http://qþ2dto1,lmaxincreasesfrom0toq=2toq123 130TheoryBiosci.(2016)135:121130thathasthecorrectphysicalbehaviour.Thepointa¼bHuiS,SilvermanJM,ChenSS,EricksonDW,BasanM,WangJ,correspondstomfTdT¼qfRdR,orHwaT,WilliamsonJR(2015)QuantitativeproteomicanalysisrevealsasimplestrategyofglobalresourceallocationinmþdRdTbacteria.MolSystBiol11:784fR¼¼f0:ð29ÞMaaløeO(1979)Regulationoftheprotein-synthesizingmachinerymþqribosomes,tRNA,factors,andsoon.In:GoldbergerRF(ed)Theregiona[bcorrespondstofRf0andabtoBiologicalregulationanddevelopment,vol1.PlenumPress,NewYork,pp487582fR[f0.ThuswehaveMaaloeO,KjeldgaardNO(1966)ControlofMacromolecularl¼qfRdRforfRf0;Synthesis.BenjaminPress,NewYorkð30ÞMaitraA,DillKA(2015)Bacterialgrowthlawsreflectthe¼mdTfRmforfRf0:evolutionaryimportanceofenergyefficiency.ProcNatlAcadSciUSA112:406ThuslasafunctionoffRisgivenbythetwostraightMolenaarD,vanBerloR,deRidderD,TeusinkB(2009)ShiftsinlinesofslopeqandmasshowninFig.3(thesolidlines).growthstrategiesreflecttradeoffsincellulareconomics.MolItisevidentthatthemaximumvalueoflisobtainedwhereSystBiol5:323MonodJ(1949)Thegrowthofbacterialcultures.AnnuRevthetwolinesmeet,whichisatfR¼f0.Using(29)thisMicrobiol3:371proves(16)inthemaintext.Further,fromthefirstofOrthJD,ThieleI,PalssonBØ(2010)Whatisfluxbalanceanalysis?Eqs.(30)itfollowsthatlmax¼qfmaxdR¼½qðmdTÞNatBiotechnol28:245mdR=ðmþqÞ.Thisproves(17).SchaechterM,MaaloeO,KjeldgaardN(1958)DependencyonmediumandtemperatureofcellsizeandchemicalcompositionduringbalancedgrowthofSalmonellatyphimurium.JGenMicrobiol19:592ScottM,GundersonCW,MateescuEM,ZhangZ,HwaT(2010)ReferencesInterdependenceofcellgrowthandgeneexpression:originsandconsequences.Science330:1099ScottM,KlumppS,MateescuEM,HwaT(2014)EmergenceofBosdrieszE,MolenaarD,TeusinkB,BruggemanFJ(2015)Howfast-robustgrowthlawsfromoptimalregulationofribosomegrowingbacteriarobustlytunetheirribosomeconcentrationtosynthesis.MolSystBiol10:747approximategrowth-ratemaximization.FEBSJ282:2029WeißeAY,Oyarzu´nDA,DanosV,SwainPS(2015)MechanisticBremerH,DennisPP(1996)Modulationofchemicalcompositionlinksbetweencellulartrade-offs,geneexpression,andgrowth.andotherparametersofthecellbygrowthrate.In:NeidhardtFCProcNatlAcadSciUSA112:E1038(ed)EscherichiacoliandSalmonella.ASMPress,WashingtonZundelMA,BastureaGN,DeutscherMP(2009)InitiationofDC,pp15531569ribosomedegradationduringstarvationinEscherichiacoli.DressaireC,GittonC,Loubie`reP,MonnetV,QueinnecI,Cocaign-RNA15:977BousquetM(2009)TranscriptomeandproteomeexplorationtomodeltranslationefficiencyandproteinstabilityinLactococcuslactis.PLoSComputBiol5:e1000606123