A_unifying_autocatalytic_network-based_framework_f

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Aunifyingautocatalyticnetwork-basedframeworkforbacterialgrowthlawsaaa,b,1AnjanRoy,DotanGoberman,andRamiPugatchaDepartmentofIndustrialEngineeringandManagement,Ben-GurionUniversityoftheNegev,BeerSheva8410501,Israel;andbQuantitativeLifeScienceSection,TheAbdusSalamInternationalCenterforTheoreticalPhysics,Trieste34014,ItalyEditedbyPeterSchuster,UniversitatWien,Vienna,Austria,andapprovedJuly8,2021(receivedforreviewApril25,2021)Recentlydiscoveredsimplequantitativerelations,knownasbac-teinsubunitsthatcomprisetheRNApolymerase.Eachoftheseterialgrowthlaws,hintattheexistenceofsimpleunderlyingtwocyclesalsoinvolvesaself-assemblystep.principlesattheheartofbacterialgrowth.Inthiswork,weBoththeRNApolymeraseandtheribosomeautocatalyticprovideaunifyingpictureofhowtheseknownrelations,ascyclesalsorelyonotherautocatalyticcyclesthatareintegralwellasrelationsthatwederive,stemfromauniversalauto-partsofthetranscription–translationmachinery.Theseaddi-catalyticnetworkcommontoallbacteria,facilitatingbalancedtionalautocatalyticcyclesareresponsibleforchargingtransferexponentialgrowthofindividualcells.WeshowthatthecoreofRNA(tRNA)withaminoacidsandinassistingtheribosomesthecellularautocatalyticnetworkisthetranscription–translationtoinitiate,translocate,andterminatethetranslationprocess.machinery—initselfanautocatalyticnetworkcomprisingsev-Theautocatalyticnatureofthesecyclesislessfamiliaranderalcoupledautocatalyticcycles,includingtheribosome,RNAbecomesmoreevidentwhenweconsidereachofitselements,polymerase,andtransferRNA(tRNA)chargingcycles.Wederivee.g.,tRNAascatalyzingitselfwiththehelpofRNApolymerasestwotypesofgrowthlawsperautocatalyticcycle,onerelatingandribosomes,asweexplainbelow.growthratetotherelativefractionofthecatalystanditscatal-Alltheautocatalyticcyclesmentionedaboveareintertwinedysisrateandtheotherrelatinggrowthratetoallthetimescalesandrequireeachothertoperformautocatalysis;removinganyinthecycle.Thestructureoftheautocatalyticnetworkgener-keycatalystfromanyoneofthesecyclesbreaksautocatalysisinatesnumerousregimesinstatespace,determinedbythelimitingallthecycles.components,whilethenumberofgrowthlawscanbemuchRecently,the“ribo-centric”view,whichfocusesontheribo-smaller.WealsoderiveagrowthlawthataccountsfortheRNAsomeautocatalyticcycle(“ribosomesmakeribosomes”),hasledSYSTEMSBIOLOGYpolymeraseautocatalyticcycle,whichweusetoexplainhowtothediscoveryofabacterialgrowthlawthatquantitativelygrowthratedependsontheinducibleexpressionoftherpoBandrelatesbacterialgrowthratetotheribosomalproteinfractionrpoCgenes,whichcodefortheRpoBandCproteinsubunitsofandtheribosometranslationrate(3,4).ArecentstudyfocusedRNApolymerase,andhowtheconcentrationofrifampicin,whichontherelationshipbetweengrowthrate,translationrate,andthetargetsRNApolymerase,affectsgrowthratewithoutchangingtranscriptionofrRNA(5).theRNA-to-proteinratio.WederivegrowthlawsfortRNAsyn-Despiteitssuccesses,theribo-centricapproachalsohasshort-thesisandchargingandpredicthowgrowthratedependsoncomings,asitdisregardsbothtranscription—animportantpillartemperature,perturbationtoribosomeassembly,andmembraneBIOPHYSICSANDCOMPUTATIONALBIOLOGYsynthesis.SignificancebacterialgrowthlawsjautocatalysisjtranscriptionjtranslationjBacterialcellscontainvariousautocatalyticcycles,e.g.,theself-replicationribosomecycle,whereribosomestranslateribosomalpro-teinsthatsubsequentlyself-assembletoformnewribosomes.hetranscription–translationmachineryisauniversalsetofHere,weshowthatthetranscription–translationmachin-Tmolecularmachinesatthecoreofallknownself-reproducingerycouplesallcellularautocatalyticcycles,resultinginbal-single-cellorganisms.Itcanbeconsideredasanembodimentancedexponentialgrowth.EachautocatalyticcyclegeneratesofvonNeuman’sconceptofauniversalconstructor—amachinetwotypesofgrowthlaws.WederivetheRNApoly-capableofmakingothermachines,self-included,byreadinganmerase(RNAP)growthlawbasedontheRNAPautocat-instructionsetandconsumingrawmaterials(1,2).alyticcycle,whereRNAPstranscribemessengerRNAs(mRNAs)Thetranscription–translationmachineryiscomposedoftwoofitsconstituentRpoproteinsubunits.Beforedegrading,keymolecularmachines,RNApolymeraseandtheribosome.thesemRNAscatalyzeRpoproteinsemployingribosomes.Accordingtothecentraldogma,allcellularproteinsaresynthe-TheRpoproteinssubsequentlyself-assemble,formingnewsizedbythiscoremachineryinatwo-stepprocess:RNApoly-RNAPs,thuscompletingthecycle.ContrarytoribosomemerasesfirsttranscribegenestoformmessengerRNA(mRNA)growthlaw,areductioningrowthrateduetoshortagein“instructionsets,”whicharethentranslatedbyribosomestoRNAPsoccurswithoutaffectingtheribosomalproteinmassformproteins.fraction.Toqualifyasa“universalconstructor,”thetranscription–Authorcontributions:R.P.conceptualizedresearch;A.R.andR.P.designedresearch;A.R.,translationmachinerymustalsobecapableofreplicatingitself.D.G.,andR.P.performedresearch;R.P.supervisedresearch;A.R.,D.G.,andR.P.developedTheself-replicationofthetranscription–translationmachineryismethodologyandcontributedanalyticaltools;A.R.andR.P.analyzeddata;andA.R.andacomplexprocess,whichis,nevertheless,universaltoallsingle-R.P.wrotethepaper.ycellorganismscapableofself-replication.ItproceedsviatwoTheauthorsdeclarenocompetinginterest.yprominentcoupledautocatalyticcycles,theRNApolymeraseThisarticleisaPNASDirectSubmission.yautocatalyticcycleandtheribosomeautocatalyticcycle.ThetwoThisopenaccessarticleisdistributedunderCreativeCommonsAttribution-NonCommercial-cyclesarecoupledbecausethedenovosynthesisofnewribo-NoDerivativesLicense4.0(CCBY-NC-ND).ysomescannottakeplacewithoutRNApolymerasetranscribing1Towhomcorrespondencemaybeaddressed.Email:rpugatch@bgu.ac.il.yribosomalRNA(rRNA),whilethedenovosynthesisofRNAThisarticlecontainssupportinginformationonlineathttps://www.pnas.org/lookup/suppl/polymerasecannottakeplacewithoutribosomestranslatingthedoi:10.1073/pnas.2107829118/-/DCSupplemental.ymRNAsofrpogenes,whichcodefortheRNApolymerasepro-PublishedAugust13,2021.PNAS2021Vol.118No.33e2107829118https://doi.org/10.1073/pnas.2107829118j1of12DownloadedbyguestonAugust13,2021

1inthecentraldogma—andotherautocatalyticcyclesinthecell.Thefactthatglobalcoupling,byitself,canguaranteebal-Incertaincasesdiscussedbelow—e.g.,whenthetemperatureancedgrowthwithacommongrowthrateimpliesthatthechangesmildlyorwhentranscriptionisperturbed—asignificantbiologicalfunctionofwell-knownfeedbackmechanisms,suchaschangeisobservedinthegrowthrate,butthischangeisnotthestringentresponse(7,8)orproductfeedbackinhibitioninaccompaniedbyachangeintheribosomefraction,asexpectedmetabolism(9),andinribosomeassembly(10),arerequiredforfromtheribosomegrowthlawpresentedinref.3.Explainingthisoptimizingthecoupling,e.g.,forgrowthrateorefficiency,ratherdeviationrequiresamoregeneralapproach.thanforgrowthcoordination.Here,wetakesuchageneralapproachbyconsideringOurmodelingapproachoffersasimplewaytorecognize“lim-bothtranscriptionandtranslationonanequalfooting,anditationregimes,”characterizedbyacompletelistofcatalystsandwederivegrowthlawsthatarebasedontheautocataly-substratesthatlocallylimitthereactionsinwhichtheypartici-sisofthetranscription–translationmachinery.Furthermore,pate.Thenumberoflimitationregimescombinatoriallyexplodesweshowthatbothtranscriptionandtranslationcouplesallwiththenumberofreactionsaccountedforbythemodel.Manyotherautocatalyticcyclesinthebacterialcell,leadingtobal-limitationregimescanbefurtheraggregatedtoform“growthancedexponentialgrowthofallcomponents,atthesameregimes,”whicharecharacterizedbyhavingacommonlimitinggrowthrate,withoutrequiringcomplexfeedbackmechanismsautocatalyticcycle.Asmentionedabove,eachlimitingcyclegives(Fig.1).risetotwotypesofgrowthlaws.WedemonstratethateachautocatalyticcycleleadstotwoNotably,despitetheirelegantsimplicityandexperimentalsuc-typesofgrowthlaws.Thefirsttype,whichwerefertoasthecess(3,4,11),bacterialgrowthlawsdonotuniquelydefinetherelativeabundancegrowthlaw,relatesthegrowthratetotherel-cellularstateorelucidatethecomplexevolutionarilyshapedcon-ativeabundancesofthecatalyststhatdrivethecycleandtotheirtrolmechanismsthatdrivethecelltoaparticularlimitationcatalysisrates.Thesecondtype,whichwerefertoastheclosed-andgrowthregime.Findingsuchanevolutionarydesignlogiccyclegrowthlaws,relatethegrowthratetoallthecatalysisratesremainsaninterestingopenchallenge(11).andallocationparameterswithinagivenautocatalyticcycle.InordertounderstandourmathematicalderivationsandAnallocationparameteristhefractionofcatalystsallocatedresults,readersmayfinditusefultostartwithMethods,wheretowardaparticulartask,e.g.,thefractionofribosomesallocatedweexplainourformalismusingasimplifiedtoyexample.Further-tomakeribosomalproteins.Usingthisformalism,werederivemore,wenotethat,inordertoapplyourmethod,detailedknowl-existinggrowthlawsandalsoderiveanddiscussothergrowthedgeofautocatalyticprocessesinbacterialcells,theircoupling,laws,thusdemonstratingthemeritsofaholisticpictureofbacte-andtheallocationofcatalyststodifferentcyclesisrequired.rialcellulargrowth.Weshowthattheuniversalcouplinginducedbythetranscription–translationmachineryisresponsibleforResultslockingallcyclestothesameexponentialgrowthrate,irrespec-TheTranscription–TranslationMachinerySelf-ReplicatesUsingSev-tiveofthenatureofthecoupling,whichcanbenonoptimaleralCoupledAutocatalyticCycles.Thetranscription–translation(Methods).machineryself-replicatesusingthreemaincoupledautocatalyticFig.1.Schematicdiagramofabacterialautocatalyticnetwork,showcasingdifferentautocatalyticcyclescoarselygrained.Thetopleftcornershowsanexplanationofthegraphicalnotation,followingref.6.Areactionnodeismarkedbyasquare.Substratesconsumedbythereactionaredepictedinsideopenboxes.Catalyststhatdriveareaction,butarenotconsumedbyit,aredepictedinsidedashedcurvedarrows.Eacharrowemanatingfromareactionnodepointstoproductofthesynthesisreaction.Inanautocatalyticreaction,thesynthesizedproductsthemselvesserveasthecatalyststhatdrivethereaction.InMethods,weexplainhowthisgraphicalnotationistranslatedtoasetofcoupledODEs,fromwhichwederivethegrowthlawsbysolvingforthesteadygrowthcondition.Inthemainfigure,wepresentaschematicautocatalyticreactionnetworkforanentirecell.Thetranscription–translationmachineryconsumesrawmaterialsandenergy(notshown)andproducescopiesofalltheproteins,includingcopiesofitself.DNAisreplicatedbythereplisomemachinery,usingtheexistingDNAasatemplate.Metabolicproteinsimportandconvertexternalmetabolitesintonucleotides,aminoacids,fattyacids,andothermetabolites.Themembraneissynthesizedbythemembranesynthesisproteins.Thus,fourmajorautocatalyticcyclesareshowninthiscoarse-grainedpicture—theautocatalysisofthetranscription–translationmachinery,oftheDNA,ofmetabolism,andofthemembrane.Alltheseautocatalyticcyclesarecoupledbythetranscription–translationmachinery.2of12jPNASRoyetal.https://doi.org/10.1073/pnas.2107829118Aunifyingautocatalyticnetwork-basedframeworkforbacterialgrowthlawsDownloadedbyguestonAugust13,2021

2cycles:1)theribosomecycle,2)theRNApolymerase1RPjRb(SA(R)+1)(pool(RPj)+1)+=,[1]cycle,and3)thetRNA-chargingcycle;othertranslation-life(R)RPjRfacilitatingcyclesarenotdiscussedhere.InFig.2,thesethreeautocatalyticcyclesareschematicallydepictedandwhereisthegrowthrate;SA(R)istheribosomeassemblytime;explained.pool(RPj)isthedurationthattheribosomalproteinspendinitsTheribosomeautocatalyticcycle.Tosynthesizeribosomesdeassemblyprecursorpool;Risthetotalnumberofribosomes;Rbnovo,existingribosomesmustcreatemorethan50differentribo-isthetotalnumberofactiveribosomes,life(R)isthelifetimeofsomalproteinsubunits.ThemRNAsfortheribosomalproteinstheribosome;RPisthefractionofribosomesallocatedtosyn-jaretranscribedbyRNApolymerases.Asubgroupoftheriboso-thesizeribosomalproteinRPj;RP=LRPaaisthetranslationjjmalproteinsdirectlybindtorRNA,whichisalsotranscribedbydurationofribosomalproteinRPj,whoselengthisLRPjaminoRNApolymerases.Subsequently,otherribosomalproteinsbindacids;and1istheribosomeelongationrate.aatothesubassembledribosome,inapredefinedpartialorder(12),Ifwefurtherassumethatboththeribosomeassemblytimeandelucidatedbythewell-knownsmallandlargeribosomesubunitthedurationthatfree-floatingribosomalproteinsspendintheirassemblymaps(13,14).precursorpoolsarenegligiblecomparedwiththedoublingtime,Wederivetheautocatalyticcycleofribosomalproteinsbyi.e.,1=SA(R)and1=pool(RPj),weobtainassumingthatrRNAisabundantandbyfocusingonthefractionRPofribosomesthatareallocatedtosynthesizethesepro-RPjRP+=RPb,[2]teins.Theribosomalproteinsspendsometimefreefloatingandjjlife(R)eventuallyentertheribosomeassemblyline,wheretheyspendsometimeintheassemblyprocessandthenexit,embeddedinwhereRbb=isthefractionofactiveribosomes.Rthesmallorlargesubunitoftheribosome.Thenewlysynthe-ThetermRPbinEq.2isthefractionofactiveribosomesjsizedsmallandlargeribosomesubunitsspendsometimefreethatareallocatedtotranslateribosomalproteins.Thus,thesec-floatingbeforebindingtomRNAandbecomingengagedintheondtermontheleft-handsidestandsfortheallocationofactivetranslationprocess.ribosomestothetranslationofribosomalproteinsatzerogrowthWederiveboththeabundanceandtheclosed-cyclegrowth=0.lawsbywritingasetofcoupledordinarydifferentialequationsEq.2isthusequivalenttothewell-knownbacterialgrowthlaw,(ODEs)fortherateofchangeintheabundanceofribosomal+0=R,presentedandexperimentallytestedinrefs.3and4proteins,therateofconversiontoanactiveassemblingstate,undertheconditionsSA(R)1andpooli1.Inthisapprox-SYSTEMSBIOLOGYtherateofassemblyofnewribosomes,andtherateofinter-imation,themassfractionoftheribosomalproteins(whichisconversionbetweenrestingandactivestates(SIAppendix).The2=3oftheRNA-to-proteinratio)isequivalenttotheallocationresultingclosed-cyclegrowthlawisparameteroftheribosomes,namely,thefractionofribosomesBIOPHYSICSANDCOMPUTATIONALBIOLOGYFig.2.Thetranscription–translationautocatalyticnetwork.Inthisschematicdiagram,wecoarselyshowhowthetranscription–translationmachineryself-replicatesviathreemaincoupledautocatalyticcycles:1)Theribosomeautocatalyticcycle,whichcomprisestworeactions:oneforribosomesthatsynthesizeribosomalproteinsandoneforRNApolymerasethatsynthesizesrRNA.TheribosomalproteinsandtherRNAsmergeinaself-assemblyreactiontoformnewribosomes.2)TheRNApolymeraseautocatalyticcycle,inwhichRNApolymerasestranscribethemRNAsthatcatalyzetheproductionoftheRpoproteinsubunits,which,inturn,self-assembletoformnewRNApolymerases.3)ThetRNA-chargingreactionwhere,e.g.,aa-tRNA-synt.catalyzethechargingoftRNAwithaminoacids,which,inturn,transfertheaminoacidstoribosomesthattranslatemRNAs,includingthemRNAsofaa-tRNA-synt.Anysubstratethatisnotconsumedbythereactioncanbeconsideredasacatalyst;forexample,mRNAcanbeviewedasacatalystforproteinsynthesis,butitscatalysisrateisnitimeshigherthanthatofasingleribosome,whereniistheaveragenumberofribosomescotranslatingthismRNA.Importantly,intheabsenceofanytypeofmaterialinputs—eithercatalystsliketRNAs,mRNAs,ribosomes,RNApolymerases,aa-tRNA-synt.,orsubstrateslikeaminoacidsorrRNAs—willbringtheautocatalysisoftheentirenetworktoahalt.Royetal.PNASj3of12Aunifyingautocatalyticnetwork-basedframeworkforbacterialgrowthlawshttps://doi.org/10.1073/pnas.2107829118DownloadedbyguestonAugust13,2021

3allocatedtotranslateribosomalproteins(SIAppendix).Signif-Theclosed-cycleRNApolymerasegrowthlaw(Fig.3)isicantly,thevalidityoftheseassumptionsandtheagreementobtainedbywritingasetofcoupledODEsfortherateofchangebetweenribosomeallocationandribosomemassfractioncaninthemRNAstranscribedfromtherpogenes,whileaccount-betestedexperimentallybecauseribosomeallocationcanbeingfortheirfinitelifetimes,therateofchangeinRpoproteinmeasureddirectlythroughribosome-profilingexperiments,whilesubunits,therateofproductionofnewRNApolymerasesfromribosomalproteinmassfractioncanbemeasuredbyusingRNA-theRposubunitsafterassembly,andthefractionofactiveRNAsequencingcombinedwithmassspectrometry(4).Indeed,inapolymerasesthatare(re)allocatedtotranscribetherpogenes,20-mindoublingtime,themeasuredribosomalmassfractionwasthussustainingtheexponentialgrowthofRNApolymerasesinmeasuredtobe30%(3),whilearibosome-profilingexperimentthecell.TheseODEsyieldsthefollowingclosed-cyclegrowthfoundthat28:5%oftheactiveribosomes,or32%ofthetotallaw:numberofribosomes,areengagedintheprocessoftranslatingribosomalproteins(15).Nevertheless,deviationsfromthecorre-spondencebetweenmassfractionandtheallocationparameter4life(Rpol)rpojRm(Rpoj)life(m(Rpoj))~i=1(1+i)=b,[5]arepredictedbyourmodelwhenoneormoreoftheafore-Rpojrpojmentionedassumptionsbreaksdown,e.g.,whentheribosomeassemblytimeincreases,asinref.16.Thesepredicteddeviationscanbeexperimentallytested.where1=SA(Rpol)istheRNApolymeraseassemblydura-Sinceribosomesaresynthesizedbytheself-assemblyofrRNAtion,2=life(m(Rpoj))isthelifetimeofthemRNAofRpoj,andribosomalproteins,wecanalsowritearibosomegrowth3=pool(Rpoj)isthedurationthatRpojspendsinitsprecursorlawthatisbasedontheproductionofrRNAsbyRNApoly-pool,andlife(Rpol)istheRNApolymeraselifetime.Additionally,merases.Theresultingabundanceandclosed-cyclegrowthlawsrpoisthefractionofactiveRNApolymerasestranscribingthejareobtainedbywritingasetofcoupledODEs,fortherateofmRNAsoftheRNApolymerasesubunitRpoj,Rmistheaver-changeintheabundanceofRNApolymerasesubunits,therateagenumberofribosomestranslatingthismRNA,and~Rpolbb=Rpolofconversiontoanactivelytranscribingstate,therateoftran-isthefractionofactiveRNApolymerases.AsthegrowthratescriptionofnewrRNAbyRNApolymerases,andtherateofdecreasestowardzero,Eq.5predictsthatthecellwillstillcon-productionofnewribosomesbytheserRNAsafterassembly(SItainafinitefractionofactiveRNApolymerases,aswasthecaseAppendix).Weobtainthefollowingclosed-cyclegrowthlaw,intheribosomegrowthlaw.TodemonstratethemeritofourRNApolymerasegrowthlaw,weuseddatafromtworecentexperiments(20,21).IntheR(SA(Rpol)+1)(SA(R)+1)(pool(Rpoj)+1)R+firstexperiment,theresearchersdevelopedareversiblegrowthlife(R)switchinEscherichiacolibyremovingrpoBCgenesfromitsRpolrRNAjRpojgenomeandplacingthemonaplasmidwithaninduciblepro-Rpol+=RbRpolb,[3]moterandafluorescentreporter(20).TheexpressionofrpoBClife(Rpol)rRNAjRpojgeneswasthuscontrolledviatheexternalconcentrationoftheinducer,isopropyl-D-1-thiogalactopyranoside(IPTG).WhentheIPTGconcentrationwashigh,nominalgrowthrateswerewhererRNAisthetranscriptiontimeofthejthrRNA;Rpojjobserved.However,whentheIPTGconcentrationdropped,aisthetranslationtimeofthejthRNApolymerasesubunit;rapiddecreaseingrowthratewasobserved.SA(Rpol)andlife(Rpol)aretheassemblytimeandlifetimeortheToexplaintherapiddecreaseingrowthrate,weemployourRNApolymerase,respectively;RpoisthefractionofactivejRNApolymerasegrowthlaw,Eq.5,withRpoBasthelimitingribosomestranslatingtheRpojproteinsubunitoftheRNApoly-factorintheassemblyprocess.AslongasthelevelsofRpoBmerase;rRNAisthefractionofactiveRNApolymerasestran-jundersteadygrowthconditionsarenonzero,thegrowthratescribingrRNAj;andRpolbisthenumberofRNApolymerasesdoesnotchange.However,assoonasthefreeRpoBpoolvan-thatareactivelytranscribing.ishesduetotheinducedreductioninitsexpression,theassemblyIfweassumefurtherthat1=SA(R),1=SA(Rpol),durationofnewRNApolymerasesstartstoincrease.Toaccount1=pool(Rpoj),andthatthelifetimesofribosomesandRNAforthisscenario,weassumethattheincreaseintheassemblypolymerasesarelongerthanthedoublingtime,weobtaintimeequalsthedelayinthedeliveryofRpoB.Themeasuredfluorescenceofthereporterproteinisnotpro-2rRNAjRpojRbRpolbportionaltothesizeofthefreeRpopool,becausetheRpoB=,[4]proteinsareconsumedbytheassemblyreactionatafasterraterRNARpoRRpoljjthanthedecayrateofthefluorescence.Therefore,weexpectthefluorescenceleveltomonotonicallyincreasewiththeexpres-sionlevel.TheequationthatweusefortheassemblydurationwhichisequivalenttotherRNAautocatalysisgrowthlawhFpresentedinref.5.is^SA(Rpol)=SA(Rpol)Fh+Kh,whereSA(Rpol)istheassemblydurationinthenominal(highIPTGorwild-type)case.TheRNAPolymeraseAutocatalyticCycle.Inbacteria,RNApoly-Forthethreeenvironmentstestedintheexperiment,wefittedmerasecomprisesfourcoreproteinsubunits—rpoA(),rpoBtheRNApolymeraseallocationparametertoyieldthenominal0(),rpoC(),andrpoZ(!)—andaninterchangeablefactor.growthratewithoutlimitingtheexpressionoftherpoBandrpoCConsiderthe“RNApolymerasemakesRNApolymerase”auto-genes.Allotherparametersweretakenfromknownmeasure-catalyticcycle,whichoperatesasfollows.Afractionoftheactivements(18,19);seeSIAppendixformoredetails.Next,wefittedRNApolymerases,rpo,isallocatedtotranscribingmRNAsfromthevaluesofKandhtothegrowthrateasafunctionoftheflu-therpogenes,whileafractionoftheactiveribosomes,Rpo,orescenceintheM9+glucosemedium.WefoundthatthefittedisallocatedtotranslatingthesemRNAsandsynthesizingthevaluesofbothKandh(K=500,h=4)remainedvalidfortheRNApolymeraseproteinsubunits.Translationfromaspecificothertwoenvironments,withoutfurtherfitting,supportingthemRNAcontinuesuntilthemRNAdegrades,asithasafinitehypothesisthatourmodelisconsistent(Fig.3C).lifetimeof3min(17).TheRpoproteinsubunitssubsequentlyInthesecondexperiment(21),sublethaldosagesofself-assembletoformnewRNApolymerases.rifampicin—adrugthattargetsDNA-boundRNApolymerases4of12jPNASRoyetal.https://doi.org/10.1073/pnas.2107829118Aunifyingautocatalyticnetwork-basedframeworkforbacterialgrowthlawsDownloadedbyguestonAugust13,2021

4ABCDFig.3.TheRNApolymerasegrowthlaw.(A)IllustrationoftheRNApolymeraseautocatalyticcycle.AmongtheactiveRNApolymerases,afractionrpoAD=Zisallocatedtotranscribe(transc.)therpoA-DandrpoZgenes(rpoDnotshown).ThetranscribedmRNAsproducetheRpoA-DandRpoZproteinsubunits1duringtheirlifetime.TherateofproteinsynthesisbymRNAisequaltoRm,whereRmistheaveragenumberofribosomesonanmRNAoftypeiitransl:i1andtransl:=Liaaisthedurationforasingleribosometotranslate(transl.)mRNAoftypei,whoselengthisLibase-pairtriplets,andaaistheribosomeSYSTEMSBIOLOGYelongationrate.TheRpojproteins,j2fA,B,C,D,Zgself-assembletoformnewRNApolymerases,whichjointhecollectivepoolofRNApolymerases.ThefractionofactiveRNApolymerasesbistakenfromrefs.18and19.(B)TheRNApolymerasegrowthlaw.lifetimeistherpoBmRNAlifetime,takentobe3min(17),transc:isthedurationoftranscription,calculatedbasedonthetranscriptionratefromref.18andthelengthoftherpoBgene.ThefractionofactiveRNApolymerasesandtheRNApolymeraseassemblydurationaremodulatedbyusingseparateHillfunctions,inaccordancewiththeexperimentunderconsideration(seeCandDfordetails).(C)Thegrowthrateasafunctionofthefluorescentreporterprotein,inducedsubsequentlytotheinductionoftherpoBandrpoCgenesintheexperimentdetailedinref.20.SinceRpoBprecedesRpoCintheassemblyofRNApolymerase,uponthedepletionoftheRpoBpool,thesynthesisrateofRpoBgovernstheRNApolymeraseassemblyduration.ThetheoreticalfitswereproducedbyusingtheRNApolymerasegrowthlaw(B),byfittingonlyoncetheKandhparameters(curlybracketsinB).Tofacilitatethecomparisonbetweenthetheoreticalfitsandthedata,weartificiallyshiftedtheM9+glucoseby500arbitrary(arb.)unitsoffluorescenceandtheM9+casaminoacidsby1,000arbitraryunitsoffluorescence.(D)BIOPHYSICSANDCOMPUTATIONALBIOLOGYTheeffectofrifampicinonthegrowthrateofE.coli;dataweretakenfromref.21.Asrifampicinlevelsincrease,thefractionofactivelytranslatingRNApolymerases,^b,decreases.Wefindthat,whentheconcentrationofrifampicincisc17g=mL,thegrowthrateisreducedbyhalfcomparedwiththenominal(c=0)case.ThemeasuredRNA-to-proteinratio(whichisproportionaltotheribosomalproteinmassfraction)remainsconstant,indicatingthatribosomesdonotlimitthegrowthrateinthisexperiment.ThisisbecauseRNAtranscriptionbecomeslimiting,whichequallyattenuatesbothribogenesisandproteinsynthesisduetoaglobalshortageinallformsofmRNA,asexplainedinTheRNAPolymeraseAutocatalyticCycle.thatarejustbeginningtotranscribeRNA—wasadministeredtoInagrowingcell,mostRNAtranscriptionisofrRNA.There-E.coli.ThedependenceofthegrowthrateandtheRNA-to-fore,itisnaturaltoaskwhetherrifampicinreducesgrowthbyproteinsratioasafunctionoftheconcentrationofrifampicinpreventingrRNAtranscriptionand,thereby,ribogenesis.Usingwerebothmeasured.Thegrowthratewasfoundtodecreaseasourmodel,wecancalculatethegrowthrateandRNA-to-proteintheconcentrationofrifampicinincreased.TheRNA-to-proteinratioforvariouslimitationregimes.Weconsiderthreerelevantratio,however,remainedconstant(Fig.3D),inmarkeddiffer-limitationregimes:1)rRNAislimitingbutmRNAisnot;2)bothencetotheribosomegrowthlaw,wheretheRNA-to-proteinrRNAandmRNAarelimiting;and3)rRNAisnotlimiting,butratiowasfoundtochangelinearlywiththegrowthrate(3).mRNAis.WeexplainthisdiscrepancybyusingourRNApolymeraseInthefirstlimitationregime,rRNAcanlimitthesynthe-growthlaw(Eq.5;alsoseeFig.3D).OurmodelnaturallysisofnewribosomesbecauseRNApolymerasesarelimited,accountsfortheseobservationsbyassumingthatrifampicinbutmRNAdoesnotlimittranslation.ThereductioninrRNAturnstheRNApolymeraseautocatalyticcycletothelimit-synthesisisaccompaniedbyareductioninribosomalproteiningcyclebyreducingthenumberofactiveRNApolymerases.translation.ThisisduetoaremarkablemechanismdiscoveredAstheconcentrationofrifampicinincreases,thefractionofbyNomuraetal.(10),namely,thatribosomalproteinsthatactiveRNApolymerasesdecreases,and,accordingly,thegrowthareprimaryrRNAbinderscandown-regulatetheirowntrans-rateofRNApolymerasesdecreases.Moreover,thedecreaselation,aswellasthetranslationofotherribosomalproteinsinthenumberofactiveRNApolymerasesgloballydecreasesonthesameoperon,iftheyfailtofindtheirtargetrRNARNAtranscriptioninthecell,thusreducingthemRNAlev-sequence.Thistranslationalfeedbackmechanismisduetoanelsofallproteins,ribosomalandnonribosomalalike.Thisaffinityofribosomalproteins,whichareprimarybinders(14),processexplainswhytheRNA-to-proteinratioremainscon-tobindtoaregionontheirmRNA,whichissimilartotheirstant.WealsoderivedthisresultbyrewritingtheribosomerRNAbindingsite.BindingoftheseproteinstotheirownmRNAgrowthlawundertheassumptionthatmRNAislimiting(SIpreventsfurthertranslationfromthesemRNAs.TogetherwithAppendix).WenotethattheRNA-to-proteinratiocouldpoten-othermechanisms,thistranslationalfeedbackkeepsthelevelstiallyincreaseifproteinsynthesisisaffectedmoreseverelythanofribosomalproteinprecursorpoolsinsyncwithrRNAtran-ribogenesis.scription,which,inturn,isgovernedbythemodulationofRNARoyetal.PNASj5of12Aunifyingautocatalyticnetwork-basedframeworkforbacterialgrowthlawshttps://doi.org/10.1073/pnas.2107829118DownloadedbyguestonAugust13,2021

5polymerasetranscriptionviathestringentresponse(8).Thus,cyclelimiting.However,inthiscase,wewouldexpectthealimitationonrRNAtranscriptionwithoutanaccompaniedstringentresponsetoreducetheproductionofrRNA,which,limitationonmRNAisexpectedtoreduceribosomalproteininturn,wouldreducethetranslationofribosomalproteins.translationand,thereby,increasethenumberofribosomesthatThefreedribosomeswouldbedivertedtothetranslationofareallocatedtotranslateotherproteins.Therefore,theregimeproteinsbelongingtothelimitingmetaboliccycle.Thus,ifaofrRNAlimitationwithoutmRNAlimitationisexpectedtometabolicautocatalyticcyclewerelimiting,wewouldexpectreducetheRNA-to-proteinratio,incontrasttotheobservationtheRNA-to-proteinratiotodecreasewithincreasingrifampicinthatitremainedconstant.concentrations.ThesecondlimitationregimetoconsideristheregimeinThetRNA-synthetaseandtRNAautocatalyticcycles.Totrans-whichbothrRNAandmRNAsimultaneouslylimitribogenesislatemRNAs,ribosomesrelyonaseriesofessentialproteinsandtranslation.ThisregimeisconsistentwithaconstantRNA-thatfacilitateribosomebindingtomRNA,tRNAchargingandto-proteinratio,asthedecreaseintranslationiscommontotransferintotheribosome,translocatingtheribosomealongallproteinsectors.Inparticular,theribosomesthatarefreedthemRNA,andreleasingtheribosomesfromthemRNAuponfrommakingribosomalproteinscannotsynthesizeotherproteinscompletingthetranslationprocess.insteadbecausemRNAisinshortage.However,aglobalshort-Inallautocatalyticcycles,anyofthecatalyststhatarepartofageinmRNAalsomeansthattheRNApolymeraseautocatalyticthecyclecanbeseenasthepivotcatalysts,aroundwhichthecycleislimiting,astheshortageinmRNAiseventuallytheresultautocatalyticcycleisconstructed.Todemonstratethisnotion,weofashortageinRNApolymerase.Wethusconcludethat,ifbothpresenttwogrowthlaws—theaminoacyl-tRNAsynthetase(aa-rRNAandmRNAarelimitingbecauseRNApolymeraseislim-tRNA-synt.)growthlawandthetRNAgrowthlaw(Fig.4).iting,growthratewillbedeterminedbytheRNApolymeraseConsider,first,theaa-tRNA-synt.(aaS)foraparticularaminoautocatalyticcycle,andtheRNA-to-proteinratiowillremainacidi.TheaaSicatalyzestheloadingofanaminoacidoftypeconstant.iontoitscorrespondingtRNA.TheloadedtRNAthenbindstoThethirdlimitationregimetoconsideristhatinwhichrRNAEF-Tu—themostabundantproteininE.coli—andproceedstoisnotlimitingandmRNAislimiting.Thisregimeisalsoconsis-enteraribosomeanddeposittheaminoacidtotheelongatingtentwithaconstantRNA-to-proteinratio,astheglobalshortagepeptidechain,which,subsequently,foldstoformthenewpro-inmRNAimpliesareductioninproteinsynthesis,includingoftein.AfractionoftheproteinsformedwillbeaaSiproteins,thusribosomalproteins.IfonlymRNAislimitingbecauseRNApoly-closingthecycle.meraseislimiting,growthratewillbedeterminedbytheRNASimilarly,considertRNAsynthesisbyRNApolymerases.polymeraseautocatalyticcycle,andtheRNA-to-proteinratioAftermaturation,thetranscribedtRNAsarechargedwithaminowillremainconstant.acidsandtransferredtoribosomes.AfractionofthesetRNAsApossiblealternativeexplanationtothedecreaseingrowthwillcontributetheiraminoacidstoRpoproteins,which,inturn,rateasafunctionofrifampicinconcentrationisthattheself-assembletoformnewRNApolymerases,someofwhichmRNAshortagemakesaparticularmetabolicautocatalyticareallocatedtomaketRNAs,therebyclosingthecycle.MoreABFig.4.tRNAandaa-tRNA-synt.(aaS)autocatalyticcyclesandgrowthlaws.(A)ThetRNAautocatalyticcycle.tRNAsaretranscribedbyRNApolymerases(poly.).Afterthematurationprocess(notshown),eachtRNAischargedwithanaminoacidandloadedontoEF-Tu.TheEF-Tu-tRNA-aaisalsochargedwithGTP(notshown)andsubsequentlydeliverstheaminoacidtotheribosome.ThiscyclerepeatsuntilthetRNAdegradesinatimescalethatweassumetobemuchlongerthanthedoublingtime.SomeofthedeliveredaminoacidsareembeddedinRpoproteins,whichself-assembletoformnewRNApolymerases,someofwhichareallocatedtotranscribetRNAs,thuscompletingthecycle.(B)Theaa-tRNA-synt.autocatalyticcycle.Anaa-tRNA-synt.proteinofaspecifiedtypechargestRNAwithitscorrespondingaminoacid.AfractionofthechargedtRNAscontributestheaminoacidstoformnewaa-tRNA-synt.ofthesametype,thuscompletingthecycle.Inthemiddleofthefigure,weshowtheratiobetweentRNAoftypeianditsassociatedaa-tRNA-synt.Neglectingcross-charging,thisratioequals,atbalancedgrowth,theratioofthechargingcycledurationtothechargingduration.6of12jPNASRoyetal.https://doi.org/10.1073/pnas.2107829118Aunifyingautocatalyticnetwork-basedframeworkforbacterialgrowthlawsDownloadedbyguestonAugust13,2021

6generally,usingthismethodallowsustoconnectalmostanytworate,allunderthesameconditionsand,preferably,onthesameprocessesinthecell,e.g.,transcription,translation,tRNAcharg-experiment.ing,metabolicrate,DNAsynthesisrate,membranesynthesis,InFig.4,wepresentthesetwoautocatalyticcyclesandtheandassemblytimesofproteincomplexes.resultinggrowthlaws.InthecurrentcontextoftRNAcharging,wefindaconnec-tionbetweenthenumberoftRNAsandthenumberofRNAFurtherApplicationsoftheAutocatalyticGrowthLaws.polymerasesrequiredtosustaingrowthatagivenrate,givenGrowthratedependenceontemperature.OurstartingpointistheallocationofRNApolymerasetowardthetranscriptionofthebacterialgrowthlaw=transl:(R0),whichwasexperi-tRNAandrpogenes.Wefindthatthegrowthrateisequalmentallymeasuredat37C.Werecallafewexperimentalfacts.tothefractionofactiveRNApolymerasestranscribingmRNAsThefirstobservationistheexistenceofanArrheniusregime,dividedbythetranscriptiondurationmultipliedbytheactivewhichisarangeoftemperaturesbetweenTAc=20CandRNApolymerasetotRNAratio,=tRNARpolb.TAh=40CinE.coli,wheretheRNA-to-proteinratiodoestRNAtRNAnotchange,whiletheribosomeelongationratechangeswithTraditionally,theratiooftRNAtoribosomeswasstudied
GsincetRNAtransportsaminoacidstotheribosomes.Addi-anArrheniustemperaturedependence,i.e.,^=ekBT.elong:elong:tionally,theRNApolymerasetoribosomesratiowasstudiedThefactthatgrowthratescaleswithtemperaturewithansinceRNApolymerasewritesmRNAinstructionsforribosomesArrhenius-typedependence,whichistypicallyrelevanttoasin-totranslate.However,evidently,tRNAalsoservesRNApoly-glechemicalreaction,mightseemsurprising;however,ifthemerases,albeitindirectly,becauseRNApolymerasesaremaderibosomeautocatalyticcycleisthelimitingcycle,allotherauto-ofproteins,andtRNAmustdelivertheaminoacidsrequiredcatalyticcycleslockstoitsgrowthrate,andthescalingbecomesformakingtheseproteinstotheribosomesthataresynthe-anaturalconsequenceoftheincreaseintheelongationratewithsizingthem.Thus,italsomakessensetoinquireaboutthetemperature.Furthermore,thefactthattheribosomefractionratiobetweentRNAsandRNApolymerases,atbalancedgrowthremainsconstantwithintheArrheniusregimecomesasanat-conditions.uralconsequencetothefactthatthiscycleistheleadingcycleTheaa-tRNA-synt.growthlawthatwederiveisgivenbyacrosstheentireArrheniusregime.WhathappensbeyondtheArrheniusregime?Bothaboveandbelowthisregime,thecellaaSBIOLOGY=i,[6]cannotsustaingrowth,becauseanincreasingnumberofproteinsLifusage(i)charging(i)denature,ifT>TAh,ormisfold,ifT<
(TTAc)+
,T:
(TTAh)+
,T>TAh,active,i.e.,that~b=1(otherwise,theright-handsideofEq.8(TAhThd)shouldbemultipliedby~b).WealsofindthattheratiobetweenthetRNAanditsaa-where
=R0,RisthemeasuredribosomefractionintRNA-synt.isinverselyproportionaltotherateoftRNAaminothemediumat37C,and0istheribosomefractionwhentheacidtransfercycletotherateofaa-tRNA-syntcharging,growthrateiszero,takentobeaconstantindependentofthemedia,asmeasuredinref.22.TheelongationrateinthemediumtRNAitransfer(i)atT=37Cisgivenbyelong:and
G~=
G(T)
G(T==:[9]aaSicharging(i)37C).Followingref.23,wecalculate
G(T)byusingtheformulaThus,thefasterthechargingofaparticularaa-tRNA-synt.,thedevelopedinref.24,namely,fewertRNAsperaa-tRNA-synt.arerequired.Testingthepre-dictionarisingfromthesetRNAgrowthlawsrequiresaccurateT
G(T)=
H+
CpTT^T
ST
Cpln,measurementsofthefractionofactiveRNApolymerasetran-385scribingeachgeneandofthetRNAabundancesandthegrowth[10]Royetal.PNASj7of12Aunifyingautocatalyticnetwork-basedframeworkforbacterialgrowthlawshttps://doi.org/10.1073/pnas.2107829118DownloadedbyguestonAugust13,2021

7first-ordertermisidenticaltothebacterialgrowthlawwhenset-where
H=4N+143,1000
S=13:27N+448,
Cp=RPiSAting0=0.For1,thesecond-ordercorrectiontothe0:048N+0:85,andT^=49C.WecalculatetheaverageRPiexpressedproteinsizeN=364aminoacidsfromtheribo-RPiSAgrowthrateisnegligible.However,when1,thegrowthRPisomeprofilingexperimentpresentedinref.15,forrichdefinedratewilldecreasetowardzeroastheassemblydurationincreases.medium(MOPS,asinref.15)at37C(doublingtimeof21Recently,theanticonvulsiondruglamotriginewasfoundtomin).InFig.5,weplotthepredictedgrowthrateasafunctionofadverselyaffecttheribosomeassemblyprocessinE.coli(26).thetemperature.Weobtainagoodfittothemeasuredtemper-Usingthegeneralformoftheribosomegrowthlawthataccountsaturedependenceofthegrowthrate(18)withoutusingfittingforassemblytime,wecanfittheobserveddependenceoftheparameters.growthrateontheconcentrationoflamotrigine.ThisisdoneEffectoflamotrigineonribosomeassemblydurationinE.coli.As1presentedinEq.1,theribosomegrowthlawcanbeusedtoyieldbyequating^SA=SA1+(c)handthenfittingcHMandh.WecHMtherelationshipbetweengrowthrateandallthetimescalesinfindthatcHM=0:0385ng/mL,whichisinaccordwiththeempir-thecycle,namely,theribosomerestingduration0,thetransla-icalconcentrationatwhichthegrowthrateisreducedbyhalf.tiondurationofallribosomalproteinsR,theassemblydurationWealsofindthath=1,indicatinganoncooperativeeffectofSA,andtheallocationparameterR,whichrepresentstheper-lamotrigineontheribosomeself-assemblyprocess.InFig.6,wecentageofribosomesthatactivelytranslateribosomalproteins.presenttheresultingfit.AmoredetailedtheoryisrequiredinUpontheapproximationthat01,weobtainasecond-orderordertoobtainamechanisticunderstandingoftheeffectoflam-equationforthegrowthrate,whichisreadilysolved,otrigineontheassemblyprocess,whichweleaveforafutureqstudy.1+RP4SA1Effectoftriclosanoncell-wallsynthesisinE.coli.CylindricallyiRPishapedbacteria,suchasE.coli,areknowntoelongateatan=,[11]2SAexponentialrate(27).Ifthewidthofthebacteriadoesnotchangesignificantlyoverthedoublingtime,thisalsoimpliesexponentialwhere,asabove,SAistheribosomeassemblyduration,RPigrowthofthevolumeandsurfacearea.However,theprocessofisthetranslationdurationofaribosomalproteinthatisapri-synthesizingnewmembraneislinear:ThemembraneisinsertedmaryrRNAbinder,andRPisthefractionofactiveribosomesithroughaninsertionsiteinaprocesshypothesizedtobecoor-allocatedtotranslatingthisribosomalprotein.dinatedbyproteinsbelongingtothe“elongasome,”includingByusingaTaylorexpansionofthenumeratorinEq.11,weMreBandpenicillin-bindingproteinsthatservekeyroles,whichobtain2arestillsubjecttoresearch(28,29).RPiRPiSAAmechanismthatcancoarselyaccountfortheobservation+


,[12]2RPiRPiofexponentialelongationisasynchronousthresholdinitiationwhichtofirstorderis1ofnewinsertionsitesbyconstitutivemembranesynthesispro-RP,i.e.,tofirstorder,theiRPiteins.Underbalancedgrowthconditions,theseproteinsareribosomeassemblytimedoesnotaffectthegrowthrate.Thisexpectedtogrowexponentiallyduetotheinherentcouplingwiththetranscription–translationmachinerythatsynthesizesthem.Ifeithertheinitiationisasynchronousorthereisaninhomogeneity0.04intherateofinsertionbetweendifferentinsertionsites,asmooth0.035ColdshockArrheniusregimeHeatshockexponentialgrowthofthemembraneensues.Wedevisedasimplemathematicaldescriptionofthiscoupling]andobtainedthefollowingrelativeabundancegrowthlaw,which-10.03relatesthegrowthratetothesurfacedensityofinsertionsitesS,thewidthoftheinsertionWin,andthespeedofinsertionin0.025unitsoflengthovertime1:m0.02SWin=:[13]0.015mGrowthrate[min0.01Thisequationisidenticaltotheequationusedinref.29toobtainthewidthofinsertionsiteasafunctionofthegrowthrate.0.005RNA/Protein=const.Membrane-boundvolumeandmembranesurfaceareaareTTTcdTAcAhhdalsophysicalresourcesthatareessentialtoallcellularpro-0cessessimplybecauseallcellularconstituentsoccupysome1020304050volume.ItisalsorequiredinordertomaintainthereactionoTemperature[C]ratesandpreventdilutionbydiffusion.Furthermore,surfaceareaisrequiredbymetabolisminordertoexchangemetabolitesFig.5.Growthratedependenceontemperature.Fourtemperatures,andheatwiththesurroundingenvironment(30,31).Obtainingderivedfromexperiments,arerequiredtoaccountfortheobservations:theaclosecyclegrowthlawis,however,moreinvolvedthanthecoldtemperatureTcd=8C,atwhichgrowthrateceases;thetemperaturerange[T,T]=[20,40]C,whichdefinestheArrheniusregime,whereothercycleswealreadydiscussedandgoesbeyondthescopeofAcAhtheRNA-to-proteinratioismeasuredtobeconstant(const.)(25);thehotthiswork.Whathappenswhenmembranesynthesisisdisrupted?Con-temperatureThd=49C,atwhichgrowthceases,i.e.,=0;andtheribo-someelongationratedependenceonthetemperature,whichisassumedsidertheantibacterialagenttriclosan,whichtargetsfattyacid
G~biosynthesis.TriclosandisruptstheproteinFabI,whichcatalyzestochangebytheArrheniusfactor(T)=optekBT,where
G~iscal-elong:anessentialstepinthebiosynthesisoffattyacids,necessaryforculatedbasedonamodelpresentedinref.24,asexplainedinthemaintext.BeyondtheArrheniusregime,weassumetheRNA-to-proteinratiobuildingthebacterialmembrane.Toexplaintheeffectoftri-decreaseslinearlyfromitsnominalvalueatbothsidesoftheArrheniusclosanonthegrowthrate,wemodulatetherateofinsertionregimetoitsminimalvalue0=0:035,bothatTcdandatThd.ofnewmembranesurfaceareainEq.13byaHillfunction,8of12jPNASRoyetal.https://doi.org/10.1073/pnas.2107829118Aunifyingautocatalyticnetwork-basedframeworkforbacterialgrowthlawsDownloadedbyguestonAugust13,2021

8Fig.6.Effectoftwoantibacterialagents—lamotrigineandtriclosan—onE.coligrowthrate.(Upper)Theeffectoflamotrigine,ananticonvulsivedrugthatSYSTEMSBIOLOGYwasrecentlyfoundtoadverselyaffectribosomeassemblyinE.coli(26).ThegrowthlawsthatweusetofitthedataarebasedonEq.1.Weneglectboththeribosomalproteinpooltimeandthefreeribosomeidlingtimetoobtainasimplequadraticequation.WhentheassemblydurationbecomescomparableRPorlongerthanthebaredoublingtimei,nonnegligiblecorrectionstothestandardbacterialgrowthlawarerequired.Inthelamotrigineexperiment,ithisoccursatconcentrationsabove0:03ng=mL.(Lower)TheeffectoftriclosanonthegrowthrateofE.coli.Dataweretakenfromref.20.TriclosandisruptstheproteinFabI,whichcatalyzesanessentialstepinthebiosynthesisoffattyacidsrequiredforbuildingthebacterialmembrane.Toincorporatethiseffect,wedecreasetheutilizationofthemembranesynthesisproteins(depictedinblackintheillustrationabovethegraph).Thisisasimplificationbecause1thereareseveralsuchproteinsthatcollectivelysynthesizethemembrane.Themembranesynthesisrateisunaffectedbytriclosanatconcentrationsmc

9tochangesinexternalmetabolitecomposition(7)andproductallautocatalyticcycles,andthesecondstepistoinsertmoreinternalstatesfeedbackinhibition,e.g.,inmetabolism,whichenablethecelltoperautocatalyticcycletoaccountforidlingvs.activecatalysts,self-assemblyoptimallytunegeneexpressionandproteincontent(9).steps,andprecursorpools,asweexplainfurtherbelow.Thesuccessofbacterialgrowthlawsindescribingtherela-WerecalltheproverbattributedtoG.Box,“Allmodelsarewrong,buttionshipbetweenthegrowthrateanddifferentphysiologicalsomeareuseful.”Wearguethattheusefulnessofourmodelingapproachliesinitsmodularityandinitsabilitytounravelbothsimplegrowthlawsparametersstandsontwopillars.Thefirstpillaristhefactandthecomplexcircumstancesthatrenderthemvalid.thattypicallyautocatalyticnetworkyieldsbalancedexponentialConsiderthenthedrasticallysimplifiedmodelshowninFig.7A,whichgrowth(11,35).Thesecondpillaristheuniversalstructurewetermthe“UPFmodel.”Inthismodel,afractionofmachinesoftypeUofthetranscription–translationautocatalyticnetworkandthecatalyzethemselves.TheremainingUmachinessynthesizeasecondtypeofuniversalmannerinwhichitcouplestoothercellularauto-machinesP.ThePmachinesconvertanexternalsubstrateftoaninternalcatalyticcycles,alsoleadingtobalancedexponentialgrowthsubstrateF,whichisusedbyUtomakebothmorecopiesofitselfandnewandmoreovertomanybiologicalconstraintsbetweendifferentPs.TheUmachineshavealifetimeL,whilethePmachineshaveaninfinitephysiologicalparameters.lifetime.TocatalyzeanewUmachine,FUunitsofFarerequired.TocatalyzeIntheclashbetweenthephysics-inspiredstriveforsimpleanewPmachine,FPunitsofFarerequired.TherateatwhichUcatalyzeseitherPorUisequaltotheincorporationunderlyinglawsofbacterialphysiologyandthebiologicalhard-111rateofasingleF,aa,timesFPorFUrespectively,assumingthatFiswonunderstandingoftheintricaciesoflife,weendinamiddleabundant.TherateatwhichPconvertsanexternalsubstrateftoaninternalground.Ononehand,wehavefoundvalidandsimplegrowthsubstrateFis1,assumingthatfisabundant.Theminimumfunction,Flaws.Ontheotherhand,wedemonstratedthatthevalidityofmin(
),intheequationspresentedinFig.7A,allowsustouncoverthefouragivengrowthlawdoesnotfullyrevealthephysiologicalstatedifferentlimitationregimesofthismodel.Inregime1,fPandFU,ofthecell.UnderstandinghowthecellularstateisdeterminedbothPsandUscatalyzeattheirfastestrate,becauseatanygivenmomentinresponsetointernalandexternalcues,andhowevolutionaryneitherofthemarestarvedforsubstrates.Inregime2,fPandFU,Psarestarved,butUsarenot.Finally,inregimeMethods4,f0.theword—i.e.,materialsthatarerequiredtofacilitateareaction,butareTofindthegrowthrateasafunctionofthetime-scalesinthemodel,notconsumedbyit—andsubstrates—i.e.,materialsthatrequiredbyareac-itisinstructivetoformulatetheequationsbyusingmatrixalgebra.Definetionandareconsumedbyit—acquireequalfootingintheequations.Thisthecolumnstatevector~S=(U,P,F,f)y.Then,thedynamicsperlimitationfacilitatesastraightforwardcharacterizationofwhichcycleislimitingandregimeindexedbyi=1,:::,4canbedescribedasd~S=M~S.Forexample,whichisnot,underdifferentcircumstances.dtithematrixforregime1,M,isdefinedas(M)=1,(M)=Webeginbyexplaininghowwetranslatethegraphicalnotationpre-1111FUaaL1211,(M)=1,(M)=1,(M)=1,and(M)=D,whileallsentedinFigs.1and2intoasetofcoupledODEs,whicharepiecewiseFPaa131aa132F142F144linear,andhowwederivevariousgrowthlawsfromthemthatarevalidforotherelementsofM1arezero.Weuseabsorbingboundaryconditionsfordifferentlimitationregimes—whichwealsodefine.allthestatevariables,toensuretheirnonnegativity:~S0.Forthesakeofclarity,weuseasimplifiedtoymodeltoelucidatethetwo-SinceforeachlimitationregimethematrixMiisconstant,thecou-stepprocessthatweemploytodevisethemorecomplexmodels,whichbearpledODEsbecomelinearand,thus,supportexponentialgrowth.AtsteadytbiologicalrelevanceandarepresentedinResults.Thefirststepistoidentifygrowthconditions,thesolutionwillbe~S(t)=~Sie,whereisthelargestABFig.7.Two-stepmodelbuildup.(A)Couplingoftwoautocatalyticcycles(comparewithref.32).Twominimumfunctionsimplyfourdifferentlimitationregimes.(B)Addingidleandactivestatesandanintermediateself-assemblystep.Theequationsprovideaquantitativedescriptionofthegraphicalnotation.10of12jPNASRoyetal.https://doi.org/10.1073/pnas.2107829118Aunifyingautocatalyticnetwork-basedframeworkforbacterialgrowthlawsDownloadedbyguestonAugust13,2021

10eigenvalueofM,and~Sisthecorrespondingeigenvector,i.e.,M~S=~S.1,with0=iiiiithegrowthlawfortheregimeF>U:=0aaaa=,andaaLInthismodel,therearetwocoupledautocatalyticcycles,“UsmakeUs”and~F=,where~FandU~aretheeigenvectorcomponentsofthematrixinthe“PsmakeFs,FsmakePs.”Evidently,allcomponentsofbothcycleswillgrowU~~Fatthesamegrowthrate,irrespectiveofthevalueoftheallocationparame-growthregimeFU,Udoesnotwait,and=1.Extend-Bothregime1andregime3,forexample,sharethesamegrowthlaw,ingthisideatotheribosomegrowthlaw(Eqs.1and2),wecanexpectthatwhichcanbeanalyticallycalculatedtobe=1.WedefinethewheneverribosomesarestarvedforchargedtRNA,theribosomegrowthFUaaLlawwillremainvalid,aslongastheelongationrateoftheribosomesusedunionofthesetwolimitationregimesasagrowthregime.Moregener-intheformulaisthemeasuredelongationrateratherthanthemaximalally,agrowthregimeisdefinedastheunionofalllimitationregimesthatelongationrate.Aslongastheribosomesaretranslatingatthemaximalshareacommonlimitingautocatalyticcycle.Inourcase,thecommonlimit-rate,theyareinonegrowthregime,equivalenttotheF>Uregime.WheningautocatalyticcycleistheUsmakeUsautocatalyticcycle.KnowledgethatribosomearestarvedforchargedtRNAs,theelongationratedecreases,andacertaingrowthlawdescribesagivenexperimentalcondition,therefore,istheribosomegrowthlawremainsvalidbyusingthemeasuredelongationnotenoughforspecifyingwhichlimitationregimethecellwasin.Ifwerate,whichislessthanthemaximal(<1).canmeasuretheparametersofourmodel,namely,thereactionratesandThegrowthlawobtainedfromsolvingthecharacteristicpolynomialistheallocationparameters,wecaninferthelimitationregimebysolvingfortheclosed-cyclegrowthlaw,agrowthlawthatdependsonallthetimescaleseigenvectorscorrespondingtothelargesteigenvalue,forallthelimitationandallocationparametersinthecycle.TherelativeabundancegrowthlawisregimematricesMiandcheckingfortheirconsistency.Consistencycheckingderiveddirectlyfromtheeigenvalueequation.Forexample,assumingthatsucheigenvectorofagivenmatrixMirequirescheckingiftheeigenvector’selementsobeythesameinequalitiesasthelimitationregimethatdefinethelimitationregimeiisconsistent,weknowthatMi~Si=~Si.DividingbythematrixMi.Ifalltheinequalitiesaresatisfied,thelimitationregimeiiscon-kthcomponentof~Si,namely,by(~Si)k,weobtainthat,forthelthcompo-sistentwiththemeasuredparameters.Inthiscase,alltheotherlimitationP(~Si)j(~Si)jnent,=j(Mi)lj.Thisisusefuliftherelativeabundancescanberegimeswillbeinconsistent.Forexample,inthecurrentsimplifiedmodel,(~Si)k(~Si)kmeasuredexperimentally,asinthecaseoftheribosomalproteinmassfrac-settingFU=FP=1,aa=F=6min,if=0:2wecanshowthatregimes1tion.Evidently,becauseallthecomponentsgrowexponentiallyatthesameand3areconsistent,while2and4areinconsistent.Ifwefurtherknowthat,rate,manymorerelativeabundancegrowthlawscanbederived,involvinge.g.,D>max,thenonlyregime1willbeconsistent.otherrelativeabundanceratios,e.g.,proteintomRNAorRNApolymeraseToyieldapositivegrowthrate,UsmustautocatalyzeataratethatistomRNA.fasterthantheirdecayrate1.Thisimpliesaminimalvalueforthealloca-LBeforemovingtoexplainFig.7B,wenotethatusingODEsthatswitchBIOLOGYtionparameter,abovewhichexponentialgrowthbegins.Intheregimesbetweenregimesisawell-knownpracticeincontroltheoryunderthenamewhereUsarestarved,thesecondautocatalyticcycle—thePsmakeFs,Fspiecewise-smoothdynamicalsystemorswitchingdynamicalsystems(33),makesPscycle—determinesthegrowthrate.Inthisregime,thelargestbutalsoinclassicalphysics,e.g.,whenconsideringthebouncingofabasket-SYSTEMSeigenvaluecanbeagaincalculatedanalyticallyandisgivenasthepositiveqballonthefloor.Inparticular,theuseofaminimumfunctionisconsideredrootofthequadraticcharacteristicequation,=1(1+4(1)aa1).2aaFPFstandardininput–outputeconomy,whereitiscalledLeontif’sproductionAssumingf>P,thesimplegrowthlaw=1willbevalidasfunction(34).FUaaLSofar,weexplainedhowwedeviseamodelthatcouplestwoautocat-longasopt—theoptimalallocationthatleadstomaximalgrowth.optcanbefoundbyequatingthetwogrowthlaws=1andalyticcycles,usingtheminimumfunctiontodefinethevariousregimesthatqFUaaLsupportbalancedexponentialgrowth.Next,weexplainhowweaddmoreAND=1(1+4(1)aa1)andsolvingfor.When>,limitation2aaFPFoptbiologicaldetailstothemodel,soastofacilitateitsuseforderivingbiolog-regime1eventuallybecomesinconsistent.icallyrelevantgrowthlaws.WeturntoFig.7B,whichdepictsthenextstepInactualexperiments,whereaparameterequivalenttotheparameterinbuildingourmodel.Forthesakeofclarity,wesimplifythemodelfur-BIOPHYSICSCOMPUTATIONALBIOLOGYaaismeasured,thereisnoaprioriwayofknowinginwhichgrowthregimetherbytaking=1andassumingFU.Thissimplifiedmodelcomprisesthemeasurementwasperformed.Hence,whatisactuallybeingmeasuredonlyoneautocatalyticcycle;however,weaddnewgenericstatesinordermaynotbethe“bare”aa,i.e.,thefastestelongationduration,but,rather,tomakeitmorerealistic.First,wedifferentiatebetweentwostatesoftheU0aa,whichcanbedifferentduetoahiddenutilizationfactor2[0,1]thatmachines,idlingandactive.ThenumberofidlingU’sisdenotedbyU0.Themeasurestheaveragepercentageoftimeacatalystiswaitingforsubstrate,idlingstateischaracterizedbyatimescale0,whichistheaverageidling0aaatsteadygrowthconditions,i.e.,aa=.Usingthisnotionofutilizationduration.WhenU’sarenotidling,theyareactive.ThenumberofactiveU’sparameters,,wecanextendthevalidityofgrowthlawsbeyondthelimi-isdenotedbyUb.Thetimescaleforbeingactiveisacomplexfunctionoftationregimethatdefinesthem.Forexample,inthegrowthregimewheretheallocationtowardallthesynthesistasksthatUisallocatedto,andofFF,i.e.,whentheUmachinesarestarvedforFsubstrate.(B)Theallocationparameterasafunctionofthegrowthrate,usingthetwogrowthlawsderivedinMethodsforthetwogrowthregimes.Royetal.PNASj11of12Aunifyingautocatalyticnetwork-basedframeworkforbacterialgrowthlawshttps://doi.org/10.1073/pnas.2107829118DownloadedbyguestonAugust13,2021

11thereisonlyonetaskthatUisallocatedtoperform—whichistoreplicateperiods;ribosomeandRNApolymeraseself-assemblysteps;andthefiniteitself.Hence,thetimescaleforbeingactiveisthetimescaleformakingnewlifetimeofmRNAs,ribosomes,RNApolymerases,andproteins.Theresult-aaU’s—thatis,U=FU.ingmodelisalsopiecewiselinear,albeitmuchlarger,andwederivefromitFinally,weaddaself-assemblystepbyfurtherbreakingtheprocessofthevariousgrowthlawsthatwepresentabove,byalgebraicallycalculatingmakingnewU’stotheprocessofsynthesizinga1,a2,anda3subunitsofU,thecharacteristicpolynomialroots(closed-cyclegrowthlaw)orbyusingtheandtotheprocessofassemblingthemtoformanewU.Thisself-assemblyeigenvectorequation(relativeabundancegrowthlaw)fortheappropriate1processproceedsatarate.Inourmodel,weusewhatweduba“Tetris”limitationregimes.SAmodel,whichassumesthatwheneverastoichiometricseriesofasubunits—thatisa1=a2=a3=1—isformedforthefirsttime,theassemblyofanewUDataAvailability.Previouslypublisheddata(3,4,15,17–27)wereusedforisinitiatedwiththesesubunits.Theassemblywillbecompleted,onaverage,thiswork.SAtimeunitsaftertheassemblyisinitiated.InResults,wecombinetheautocatalyticcyclesofthetranscription–ACKNOWLEDGMENTS.WethankTerryHwaandSuckjoonJunforcriticaltranslationmachinerywithothercycles,whileaccountingforthecouplingsremarksattheearlystagesofthiswork.WethankMatteoMarsili,Yin-betweenthedifferentcycles;theexistenceofdifferentallocationparame-nonM.Bar-On,andYitzhakFishovforusefulcomments.ThisresearchwastersforRNApolymerasesandribosomes;theexistenceofidlingandactivesupportedbytheIsraelScienceFoundation(Grant776/19).1.J.vonNeumann,A.W.Burks,TheoryofSelf-ReproducingAutomata(Universityof18.R.Milo,R.Phillips,CellBiologybytheNumbers(TaylorandFrancisInc,Abingdon,UK,IllinoisPress,Champaign,IL,1966).2015).2.R.Pugatch,Greedyschedulingofcellularself-replicationleadstooptimaldoubling19.H.Bremer,P.P.Dennis,Modulationofchemicalcompositionandotherparametersoftimeswithalog-Frechetdistribution.Proc.Natl.Acad.Sci.U.S.A.112,2611–2616thecellatdifferentexponentialgrowthrates.EcosalPlus,10.1128/ecosal.5.2.3(2008).(2015).20.J.Izardetal.,AsyntheticgrowthswitchbasedoncontrolledexpressionofRNA3.M.Scott,C.W.Gunderson,E.M.Mateescu,Z.Zhang,T.Hwa,Interdependenceofpolymerase.Mol.Syst.Biol.11,840(2015).cellgrowthandgeneexpression:Originsandconsequences.Science330,1099–110221.F.Sietal.,InvarianceofinitiationmassandpredictabilityofcellsizeinEscherichia(2010).coli.Curr.Biol.27,1278–1287(2017).4.S.Huietal.,Quantitativeproteomicanalysisrevealsasimplestrategyofglobal22.X.Daietal.,ReductionoftranslatingribosomesenablesEscherichiacolitomaintainresourceallocationinbacteria.Mol.Syst.Biol.11,784(2015).elongationratesduringslowgrowth.Nat.Microbiol.2,16231(2016).5.S.Kostinski,S.Reuveni,RibosomecompositionmaximizescellulargrowthratesinE.23.K.Chenetal.,Thermosensitivityofgrowthisdeterminedbychaperone-coli.Phys.Rev.Lett.125,028103(2020).mediatedproteomereallocation.Proc.Natl.Acad.Sci.U.S.A.114,11548–115536.W.Hordijk,S.A.Kauffman,M.Steel,Requiredlevelsofcatalysisforemergenceof(2017).autocatalyticsetsinmodelsofchemicalreactionsystems.Int.J.Mol.Sci.12,3085–24.K.A.Dill,K.Ghosh,J.D.Schmit,Physicallimitsofcellsandproteomes.Proc.Natl.3101(2011).Acad.Sci.U.S.A.108,17876–17882(2011).7.V.Chubukov,L.Gerosa,K.Kochanowski,U.Sauer,Coordinationofmicrobial25.J.Ryals,R.Little,H.Bremer,TemperaturedependenceofRNAsynthesisparametersmetabolism.Nat.Rev.Microbiol.12,327–340(2014).inEscherichiacoli.J.Bacteriol.151,879–887(1982).8.M.F.Traxleretal.,Theglobal,ppGpp-mediatedstringentresponsetoaminoacid26.J.M.Stokes,J.H.Davis,C.S.Mangat,J.R.Williamson,E.D.Brown,DiscoverystarvationinEscherichiacoli.Mol.Microbiol.68,1128–1148(2008).ofasmallmoleculethatinhibitsbacterialribosomebiogenesis.eLife3,e035749.S.Goyal,J.Yuan,T.Chen,J.D.Rabinowitz,N.S.Wingreen,Achievingoptimalgrowth(2014).throughproductfeedbackinhibitioninmetabolism.PLoSComput.Biol.6,e100080227.P.Wangetal.,RobustgrowthofEscherichiacoli.Curr.Biol.20,1099–1103(2010).(2010).28.T.denBlaauwen,M.A.dePedro,M.Nguyen-Disteche,J.A.Ayala,Morphogenesisof`10.M.Nomura,J.L.Yates,D.Dean,L.E.Post,Feedbackregulationofribosomalpro-rod-shapedsacculi.FEMSMicrobiol.Rev.32,321–344(2008).teingeneexpressioninEscherichiacoli:StructuralhomologyofribosomalRNAand29.C.Billaudeauetal.,Contrastingmechanismsofgrowthintwomodelrod-shapedribosomalproteinMRNA.Proc.Natl.Acad.Sci.U.S.A.77,7084–7088(1980).bacteria.Nat.Commun.8,15370(2017).11.S.Jun,F.Si,R.Pugatch,M.Scott,Fundamentalprinciplesinbacterialphysiology-30.S.Klumpp,M.Scott,S.Pedersen,T.Hwa,Molecularcrowdinglimitstrans-history,recentprogress,andthefuturewithfocusoncellsizecontrol:Areview.Rep.lationandcellgrowth.Proc.Natl.Acad.Sci.U.S.A.110,16754–16759Prog.Phys.81,056601(2018).(2013).12.B.Schroder,OrderedSets:AnIntroductionwithConnectionsfromCombinatoricsto31.M.Szenk,K.A.Dill,A.M.R.deGraff,Whydofast-growingbacteriaenterover-Topology(Birkhauser,Basel,Switzerland,2016).flowmetabolism?Testingthemembranerealestatehypothesis.CellSyst.5,95–10413.Z.Shajani,M.T.Sykes,J.R.Williamson,A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