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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.Signi?cancebacterialgrowthlawsjautocatalysisjtranscriptionjtranslationjBacterialcellscontainvariousautocatalyticcycles,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.Contrarytoribosomemerases?rsttranscribegenestoformmessengerRNA(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—asigni?cantbiologicalfunctionofwell-knownfeedbackmechanisms,suchaschangeisobservedinthegrowthrate,butthischangeisnotthestringentresponse(7,8)orproductfeedbackinhibitioninaccompaniedbyachangeintheribosomefraction,asexpectedmetabolism(9),andinribosomeassembly(10),arerequiredforfromtheribosomegrowthlawpresentedinref.3.Explainingthisoptimizingthecoupling,e.g.,forgrowthrateoref?ciency,ratherdeviationrequiresamoregeneralapproach.thanforgrowthcoordination.Here,wetakesuchageneralapproachbyconsideringOurmodelingapproachoffersasimplewaytorecognize“l(fā)im-bothtranscriptionandtranslationonanequalfooting,anditationregimes,”characterizedbyacompletelistofcatalystsandwederivegrowthlawsthatarebasedontheautocataly-substratesthatlocallylimitthereactionsinwhichtheypartici-sisofthetranscription–translationmachinery.Furthermore,pate.Thenumberoflimitationregimescombinatoriallyexplodesweshowthatbothtranscriptionandtranslationcouplesallwiththenumberofreactionsaccountedforbythemodel.Manyotherautocatalyticcyclesinthebacterialcell,leadingtobal-limitationregimescanbefurtheraggregatedtoform“growthancedexponentialgrowthofallcomponents,atthesameregimes,”whicharecharacterizedbyhavingacommonlimitinggrowthrate,withoutrequiringcomplexfeedbackmechanismsautocatalyticcycle.Asmentionedabove,eachlimitingcyclegives(Fig.1).risetotwotypesofgrowthlaws.WedemonstratethateachautocatalyticcycleleadstotwoNotably,despitetheirelegantsimplicityandexperimentalsuc-typesofgrowthlaws.The?rsttype,whichwerefertoasthecess(3,4,11),bacterialgrowthlawsdonotuniquelyde?netherelativeabundancegrowthlaw,relatesthegrowthratetotherel-cellularstateorelucidatethecomplexevolutionarilyshapedcon-ativeabundancesofthecatalyststhatdrivethecycleandtotheirtrolmechanismsthatdrivethecelltoaparticularlimitationcatalysisrates.Thesecondtype,whichwerefertoastheclosed-andgrowthregime.Findingsuchanevolutionarydesignlogiccyclegrowthlaws,relatethegrowthratetoallthecatalysisratesremainsaninterestingopenchallenge(11).andallocationparameterswithinagivenautocatalyticcycle.InordertounderstandourmathematicalderivationsandAnallocationparameteristhefractionofcatalystsallocatedresults,readersmay?nditusefultostartwithMethods,wheretowardaparticulartask,e.g.,thefractionofribosomesallocatedweexplainourformalismusingasimpli?edtoyexample.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.Inthemain?gure,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;and