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ARTICLEhttps://doi.org/10.1038/s41467-020-20472-xOPENExploringtheeffectofnetworktopology,mRNAandproteindynamicsongeneregulatorynetworkstabilityYipeiGuo1,2&ArielAmir1?Homeostasisofproteinconcentrationsincellsiscrucialfortheirproperfunctioning,requiringsteady-stateconcentrationstobestableto?uctuations.Sincegeneexpressionisregulated1234567890():,;byproteinssuchastranscriptionfactors(TFs),thefullsetofproteinswithinthecellcon-stitutesalargesystemofinteractingcomponents,whichcanbecomeunstable.WeexplorefactorsaffectingstabilitybycouplingthedynamicsofmRNAsandproteinsinagrowingcell.We?ndthatmRNAdegradationratedoesnotaffectstability,contrarytopreviousclaims.However,globalstructuralfeaturesofthenetworkcandramaticallyenhancestability.Importantly,anetworkresemblingabipartitegraphwithalowerfractionofinteractionsthattargetTFshasahigherchanceofbeingstable.ScramblingtheE.colitranscriptionnetwork,we?ndthatthebiologicalnetworkissigni?cantlymorestablethanitsrandomizedcoun-terpart,suggestingthatstabilityconstraintsmayhaveshapednetworkstructureduringthecourseofevolution.1JohnA.PaulsonSchoolofEngineeringandAppliedSciences,HarvardUniversity,Cambridge,MA,USA.2PrograminBiophysics,HarvardUniversity,Boston,MA02115,USA.?email:arielamir@seas.harvard.eduNATURECOMMUNICATIONS|(2021)12:130|https://doi.org/10.1038/s41467-020-20472-x|www.nature.com/naturecommunications1ContentcourtesyofSpringerNature,termsofuseapply.Rightsreserved
1ARTICLENATURECOMMUNICATIONS|https://doi.org/10.1038/s41467-020-20472-xellsrequiredifferentproteinlevelstosurviveindifferentmodels7–10.However,howthesefeaturesaffectthestabilityofexternalenvironments.Theexpressionoftheseproteinsgeneregulatorynetworkshasnotbeenexplored.Cwithinthecellarethereforehighlyregulated.Animpor-Here,byanalyzingamodelthattakesintoaccountthetran-tantregulatorymechanisminvolvestranscriptionfactors(TFs),scriptionofmRNAsfromgenes,translationofmRNAsintowhicharethemselvesproteinsthatcaneitherupordownregulateproteins,andtranscriptionalregulationbyproteins,weinvesti-thetranscriptionofmRNAscodingforotherproteinsbybindinggatethestabilityofthislargesystemofcoupledmRNAsandtoenhancerorpromoterregionsoftheregulatedgene1.Despiteproteinsingrowingcellsand?ndthatwhilethemRNAdegra-theimportanceofmaintainingdesiredproteinconcentrationsdationratecanaffectrelaxationratebacktosteady-statelevels,itwithincells,factorsaffectingthestabilityoftheseconcentrationsdoesnotaffectwhetherthesystemisstable.Instead,stabilitycantoperturbationshavereceivedlittleattention.dependstronglyontheglobalstructuralfeaturesoftheinterac-Oneapproachofstudyingthestabilityofsuchsystemswithationnetwork.Inparticular,giventhesamenumberofproteins,largenumberofinteractingcomponentswasintroducedbyMayTFs,numberofinteractions,andregulationstrengths,anetworkinthe1970sinthecontextofcomplexecologicalcommunities2.withalowerfractionofinteractionsthattargetTFshasahigherTheideaisthatinan-speciescommunity,thedynamicsofthechanceofbeingstable.InthelimitwheretherearenoTF–TFabundancesNiofeachspeciesmayingeneralbedescribedbyainteractionsi.e.allTFsregulateproteinsthatarenotTFs,itissetofordinarydifferentialequations:possibleforthesystemtoremainstableforarbitrarilylargesys-temsizes,unlikerandomnetworkswhichbecomeunstablewhendNi?fieN1;N2;:::NnTe1Tsystemsizebecomestoolarge.ByscramblingtheE.coli.tran-dtscriptionnetwork,we?ndthatthetopologyofrealnetworkscanfori=1,2,...,n,withcorrespondingsteady-statesolutionNssstabilizethesystemsincetherandomizednetworkwiththesamei!ssnumberofregulatoryinteractionsisoftenunstable.These?nd-suchthatfeNT?0?i.Thedynamicsofsmallperturbationsiaboutthissteady-statexetT?NetTNss,whenlinearizedaboutingssuggestthatconstraintsimposedbysystemstabilitymayiiiNss,hastheform:haveplayedasigni?cantroleinshapingtheexistingregulatoryinetworkduringtheevolutionaryprocess.Bycarryingoutthe!dx!analysisfordifferentphysiologicalstatesthecellcanbein(cor-?Ax;e2Tdtrespondingtodifferentsetsofdynamicalequations)andwithssdifferentchoicesofparameterdistributions,wealsoshowthatwhereAistheJacobianmatrixwithelementsA??fi.Ifallij?NjourmainresultsandconclusionsarerobusttothedetailsoftheeigenvaluesofAhaveanegativerealpart,thesystemrelaxesthemodel.backtothesteady-stateuponperturbationsandthesteady-stateissaidtobestable;ifanyoftheeigenvalueshaveapositiverealpart,Resultsthesteady-stateisunstableasthesystemwillmoveawayfromitThemodel.Geneexpressioninvolvestwomajorsteps:tran-(exponentiallyfast)whenin?nitesimallyperturbed.Toconstructscriptionandtranslation(Fig.1a).TranscriptionistheprocessinA,onewouldneedtopreciselyknowthefunctionsfi,whichiswhichmRNAissynthesizedbyRNApolymeraseusingDNAasaoftenhardtoobtain.May’sapproachwastomodelAasaran-template.Thetranscriptionrateofageneithereforedependsondommatrixwithindependent,identicallydistributedoff-diagonalthenumberofRNApolymerasesnanditseffectivegenecopyelements(withmean0,standarddeviationσ,andfractionofnon-numbergiwhichtakesintoaccountbothitscopynumberandzeroelementsC)andconstantdiagonalelements—a.InthehowstronglyRNApolymerasecanbindtothepromoterofthatcontextofecology,σre?ectstheaverageinteractionstrengthgene11.DuetothepresenceofTFs,ge!cTcaningeneraldependibetweenspecies,Cisthedensityofinteractionsortheprobability!onthesetofproteinconcentrationsc(Fig.1a).Weassumethatthatanytwospeciesinteract,whileaistheself-regulationtermmultipleTFsactingonthesamegeneactindependently,withwhichsetstherelaxationtime-scaleofthesystemiftherewerenotheireffectsstackingmultiplicatively.ThisallowsforbothOR-otherpairwiseinteractions.Fromrandommatrixtheory(RMT)12andAND-gate-likecombinatorialeffects,andcanemergefromandinparticularthecircularlawformatrixeigenvaluedis-p??????athermodynamicmodelofTFbinding(SupplementaryNote1).tributions3,4,thissystemisstableifandonlyifa>σnC.ThisTherefore,weadoptthefollowingformfortranscriptionalreg-impliesthatthesystembecomesunstableabovesomecriticalsize,ulationthroughoutthepaper:andthatincreasingastabilizesthesystemandallowsforstrongerY!interactionsbetweenspecies.giecT?gi0e1tγijfijecjTT;e3TThisapproachhasalsobeenusedtoanalyzeotherlargejinteractingsystems.Inparticular,ithasbeenusedtoarguewhywheregi0istheeffectivegenecopynumberofiifitwereunre-weakrepressionsbymicroRNAs,thoughtofaseffectivelygulated(randomlydrawnfromauniformdistribution),andγijincreasingthedegradationrateofmRNAs,conferstabilitytocontrolsthetypeandstrengthofregulation,i.e.,howmuchgenegeneregulatorynetworks5,6.However,suchaframeworkdoesnotexpressionofichangesinthepresenceoftheTFj.Inparticular,takeintoaccountthefunctionalformoffiandinparticularthatγij>0ifjupregulatesiand?1≤γij<0ifjdownregulatesi.Forthematrixelementsoftendependonthesteady-statesolutionseachregulatoryinteraction,weassumethatthefold-changeΩijisthemselves.Thesedetailsofthemodelcanbeimportant—fordrawnfromauniformdistributionbetween1andΩmax,suchthatexample,whencompetitionforresourcesbetweenecological(speciesareexplicitlymodeled(usingMacArthur’sconsumer-Ωij1ifγij>0eupregulatingTresourcemodel),evenwhentheinteractions(i.e.,preferencesofγij?11ifγ<0edownregulatingTe4TΩijeachspeciesforthedifferentresources)arecompletelyrandom,ijthespectrumoftheJacobianmatrixthatrepresentseffectivesincethiswouldallowgi(cj)toincrease(ifjupregulatesi)orpairwiseinteractionbetweenspeciesisnolongercircular(butdecrease(ifjdownregulatesi)byafactorofΩijinthelimitofrather,followstheMarchenko-Pasturdistribution)7.Further-highcj.InSupplementaryNote5,weshowthatthemainresultsmore,transcriptionalregulatorynetworksarenotrandombutdonotdependontheparticulardistributionP(Ω)used.insteadhavedistinctstructuralfeatures.Thestructureofinter-MotivatedbyexperimentalmeasurementsoftherelationshipactionnetworkshasbeenknowntoaffectstabilityinotherbetweenTFinputandgeneexpressionoutputshowinga2NATURECOMMUNICATIONS|(2021)12:130|https://doi.org/10.1038/s41467-020-20472-x|www.nature.com/naturecommunicationsContentcourtesyofSpringerNature,termsofuseapply.Rightsreserved
2NATURECOMMUNICATIONS|https://doi.org/10.1038/s41467-020-20472-xARTICLE(a)TranscriponalE?ecvegenecopynumber:regulaonbyTFs=∏1+TranscriponTranslaon(rateΓ)(rateΓ)(b)(c)RNApolymerasesRibosomesareareliming(<)[phases1,4]liming(<)[phases1,2]Γ=Γ=∑∑mRNAproteinGenecopynumbersmRNAsareareliming(≥)liming(≥)Γ=Γ=[phases2,3][phases3,4]Fig.1Schematicillustrationofthegeneexpressionmodel.aThedynamicsofproteinandmRNAconcentrationsarecoupledthroughtranscriptionalregulation,wheresomeoftheproteins(e.g.,transcriptionfactors)modulatetheeffectivegenecopynumbersgiandhencethetranscriptionrateofothergenes.bIfRNApolymeraseisinexcess,transcriptionrateΓmofgeneiisproportionaltoitseffectivegenecopynumbergi.IfinsteadRNApolymeraseislimiting,Γmisproportionaltothegeneallocationfraction?i=gi/∑jgj.cTranslationrateΓpisproportionaltomRNAnumbermiifmRNAsarelimiting,andproportionaltothemRNAfractionmi/∑jmjifribosomesarelimiting.TherearefourdifferentphasesofthemodeldependingonwhetherRNApolymerasesandribosomesarelimiting.sigmoidalfunctionalformoffij(cj)13,14,wetakeittobeaHillpolymerasesandpN=rcorrespondingtoribosomes,arethere-functionforegivenby:h8mpcj 3ARTICLENATURECOMMUNICATIONS|https://doi.org/10.1038/s41467-020-20472-xassumeforsimplicitythateachproteinhasthesamemassandsetItcanbeshownthatboththestructureofM(Eq.(13))andthethecelldensitytobe1,suchthatV=∑ipi.ThedynamicsforfactthatstabilityonlydependsonMstillholdintheotherphases,concentrationsinphase1arethengivenby:despitetheexactequationsforproteindynamicsbeingdifferent(seeSupplementaryNote3).Therefore,eventhoughthedcmi!1?km?iecTcncmikpcrt;e9Tsimulationsintherestofthissectionarecarriedoutinphasedtτ1,our?ndingsandconclusionsalsoapplytotheotherphases.dccp????i?kcmic;e10TStabilityofthesystemscaleswithNforrandomregulatorypridtcmTnetworks.WestartbyexploringthestabilityoffullyrandomwherecmT=∑icmiisthetotalconcentrationofallmRNAsandregulatorynetworks,whichwetaketobeournullmodel.1?11isthedifferencebetweenmRNAandproteinSincethemaximumeigenvalueofarandommatrixdependsonττmτpthestandarddeviationofitselements,we?rstcarryoutanaivedegradationrates(whichcanbepositiveornegative).AsummaryestimateofhowtheelementsofMscalewithN.Withgi(c)givenofthelistofmodelparameterscanbefoundinSupplementarybyEq.(3),Table1.Whiletheseequationsgovernthedynamicsofaverageconcentrationsandhencedonotcapturestochasticeffects?loggiγij?fij?:e16Tinherentingeneexpressionandinthebinomialsamplingof?cj1tγfecjT?cjijijmoleculesduringcelldivision,these?uctuationsdonotaffecttheaveragesteady-stateconcentrationsifthenumberofmoleculesislarge(seeSupplementaryNote2,SupplementaryFig.1).Infact,Biologically,TFconcentrationsareoftencomparabletothevaluesthese?uctuationscanbeconsideredasperturbationsaboutofdissociationconstantsKdforDNAbinding26.Therefore,sincesteady-statevalues,andweinvestigatethestabilityofthesystemcj~1/N,wealsochooseKij~1/N(Eq.(5)),whichwouldallowtosuchperturbationsintherestofthepaper.cellstomaintainthefullrangeofgeneexpressionresponse.From?fijEq.(5),thisimpliesthatfij~O(1)and?cN,andhenceM1andEffectsofnetworkfeaturesandtopologyonstabilityofthejsystem.TostudyhowpropertiesofthetranscriptionalregulatoryM2alsoscalewithN(Eqs.(14),(15)).WethereforeexpectMij~O(1)(Eq.(13)),andhence(fromRMT),forλM;rtoscalenetworkaffectthestabilityofthesystem,we?rstconsiderthep????maxregimewherethelifetimeofmRNAsismuchshorterthanthatofapproximatelyasNforrandominteractionnetworks.λM;rmaxproteins,whichistypicallytrueforwild-typecells26.Inthislimitalsoincreaseswiththestrengthoftheinteractionsγ,implyingoffastmRNAdegradation,therelaxationdynamicsofmRNAisthatthesystemwillbecomeunstableeitherwhenNexceedsamuchfasterthanthatofproteinssuchthatdcmi0atalltimes.criticalnumberortheregulationstrengthbecomestoohigh.dtEliminatingthefastprocess(bysubstitutingthesteady-stateHowever,thisargumentneglectscorrelationsbetweenthemRNAconcentrationsc?kmcn?e!cTobtainedfromEq.(9)elementsofM,whichcouldpotentiallyberelevant.Infact,wemikct1iprτwillseeinthelatersectionsthatthestructureofM(Eq.(13))intoEq.(10)),thedynamicsofproteinconcentrationscanbeplaysanimportantroleinin?uencingthestabilityofthesystem.writtenasasetofNODEs:Therefore,totestifthisscalingrelationholds,weconstructeddcnetworksofaspeci?edinteractiondensityρbyrandomlyikc?e!cTc:e11T2priiselectingρNinteractionsfromtheN(N?1)possibilities(wheredtwehaveassumedthatribosomescannotactasTFs),andchooseThestabilityofthesystemthereforedependsonlyontheeigen-halfoftheinteractionstobeupregulatingwiththeremaininghalfvaluesoftheN×NJacobianmatrixA?kcsseMIT,whereweprbeingdownregulating.de?netheinteractionmatrixBytakingtheensembleaverageovertherandomlydrawnp??????networks,weindeedrecovertheNscaling(Fig.2a),whichisiMij?j!!ss;e12Talsorobusttothefractionofup-anddownregulatoryinteractions?cc?cj(seeSupplementaryNote4,SupplementaryFig.2a)andthewiththesteady-stateproteinconcentrationsgivenbycss?distributionoffold-changesP(Ω)(seeSupplementaryNote5,i!ssSupplementaryFig.3).Forsuf?cientlylargeNorΩ,wecanno?iecT(fromEq.(11)).maxDenotingλMastheeigenvaluesofM,thesystemisstableaslonger?ndthe?xedpointofthesystem.Nevertheless,bylongasthemaximalrealpartoftheseeigenvaluesλM;rissimulatingthedynamics,we?ndthatforinteractionnetworksofmaxagivenNandρ,wegetoscillatory,followedbychaoticbehaviorsmallerthan1(suchthatalleigenvaluesofAhaveanegativerealpart).ItisthereforeusefultounderstandthestructureofMbyasΩmaxisincreased(Fig.2b).Similarphenomenahavealsobeendescribedandanalyzedinmodelsofneuralnetworks27andbreakingitintotwopartsusingEq.(6):ecologicalsystems28.WhilecertainbiochemicalcircuitshavebeenM?csseMMT;e13Tiji1;ij2;ijknowntogenerateoscillationssuchasinthecellcycleandthecircadianclock,theoscillatorydynamicsobservedhereisofawheredifferentnature—itdoesnotcomeaboutfromanyspeci?c?ne-?loggituningofthenetworkbut,rather,emergesfromhavingalargeM1;ij?e14T?cjnumberofrandomlyandstronglyinteractinggenes.However,transcriptionalregulatorynetworksaretypicallynotcapturesthedirectinteractionsbetweenproteins,whilerandom.Instead,theyareenrichedfordistinctstructuralfeatures?loggXss?loggsuchasthefollowingmotifs:feedforwardloops(FFL),single-M?T?ck2;ijke15Tinputmodule(SIM),anddenseoverlappingregulons(DOR)?cjk?cj1,29whichdonotcontainanyloopsbesidesautoregulatoryones.isarank-1matrixthatcapturestheindirectinteractionsarisingInthenextfewsubsections,wethereforeexploretheeffectsoffromcompetitionforribosomes.networktopologyonsystemstability.4NATURECOMMUNICATIONS|(2021)12:130|https://doi.org/10.1038/s41467-020-20472-x|www.nature.com/naturecommunicationsContentcourtesyofSpringerNature,termsofuseapply.Rightsreserved 4NATURECOMMUNICATIONS|https://doi.org/10.1038/s41467-020-20472-xARTICLE(a)(c)unstable.Nevertheless,thereisanegativeoffsetinλM;rmaxcomparedDAGtothefullyrandomcase(Fig.2a),implyingthatthelackofloopsdoeshelptostabilizethesystem.0.5Bipartitestructurecanmaintainstabilityoflargenetworks.A,commonlyfoundmotifintheEscherichiacolitranscriptionnet-workisthedense-DORswhichconsistofasetofregulatorsthat10combinatoriallycontrolasetofoutputgenes1,29,30.Therearerand(Ω=1.5)logrand(Ω=2)severaloftheseDORsinE.coli,eachwithhundredsofoutputDAG(Ω=1.5)DAG(Ω=2)genes,andtheyappeartooccurinasinglelayer,i.e.,thereisnoDORattheoutputofanotherDOR.Suchastructurecanbelog10thoughtofasabipartitegraphinwhichtherearetwotypesof(b)nodesrepresentingTFsandnon-transcriptionfactors(non-TFs),andeverydirectededgegofromaTFtoanon-TF.Sincesuchgraphsdonotcontainanyregulatoryloops(andarethereforealsoDAGs),weexpectthemtobemorestablethanrandomnetworks.However,theyareaspeci?csubsetofDAGsinwhichnoneoftheTFsarethemselvesregulated.Thisisalsoakeydifferencebetweenthesenetworksandbipartite,mutualisticnetworkscommonlystudiedinecologicalmodels9,10.Inthissubsection,weinvestigatethestabilityofsuchnetworks.Fig.2Stabilityofrandominteractionnetworks.aForrandominteractionTostudythisproblem,we?rstgroupproteinsintotwonetworks(redmarkers,`rand'),themaximalrealpartoftheeigenvaluesofcategories:qTFsandN?qnon-TFs,suchthatforanygeneralp????theinteractionmatrixλM;rscaleswithN.Surprisingly,forrandomnetworkthecomponentsoftheJacobianmatrixhavethemaxdirectedacyclicnetworks(bluemarkers,`DAG'),λM;ralsoscalesfollowingstructure:p????maxapproximatelywithN.Inbothofthesecases,increasingtheinteractionT01strengthfromΩmax?1:5(circles)toΩmax?2(triangles)increasesλM;rmax.M1?e17TTheseresultssuggestthatthesystemwillbecomeunstable(i.e.,R10log10eλM;rmaxTexceeds0,indicatedbytheblackdashedline)whenNorT0Ωbecomestoolarge.Eachdatapointisobtainedfromanaverageof102maxM2?;e18Trandomlydrawnnetworks,witherrorbarsindicatingtheinterquartilerange.R20EachrandomnetworkisconstructedbyrandomlyselectingρN2whereT1(T2)isaq×qmatrixrepresentingthedirect(indirect)interactionsfromN(N?1)possibilities,withhalfoftheinteractionschoseneffectofTFsonTFswhileR1(R2)isa(N?q)×qmatrixtobeupregulatingandtheremaininghalftobedownregulating.Therepresentingthedirect(indirect)effectofTFsonnon-TFs,withconstructionofDAGsisdescribedin(c).Foreachregulatoryinteraction,theirelementsde?nedpreviously(Eqs.(13)–(15)).Thenon-zerofoldchangeischosenuniformlybetween1andΩmax.[Otherparameters:eigenvaluesofMarethereforetheeigenvaluesofthesub-matrixρ=0.01,h=1].bWhensystemsgooutofstability,dynamicsofproteinQwithelements:concentrationscexhibitoscillatory(left,Ωmax?20)followedbychaoticQ?csseTTT:e19Tbehavior(right,Ωmax?200)asinteractionstrengthsareincreased.[Otheriji1;ij2;ijparameters:N=200,ρ=0.2,h=1,fullyrandomnetwork,timetisinunitsWhenthenetworkissparse,eachTFonlyregulatesasmallof1/kp.]cRandomdirectedacyclicnetworksareconstructedbyrandomlyssfractionofthetotalnumberofgenes.Sincec~1/N,thestrengthdrawingconnectionsbetweenproteins(redcirclesrepresentTFs,blueofindirectinteractionsarethereforetypicallymuchweakerthancirclesrepresentnon-TFs).Ifadrawnconnectioncreatesaloop(e.g.,thethatofdirectinteractions(i.e.,thenon-zeroelementsofM2aregrayarrowwithacrossonit),itisrejected.muchsmallerinmagnitudethanthatofM1,Eqs.(14),(15)).Whenconstructingrandombipartitenetworks,weonlyallowRandomdirectedacyclicnetworkscanalsobeunstable.SinceTFstoregulatenon-TFs(Fig.3a),implyingthatT1=0.ThetranscriptionnetworksasawholeresembledirectedacyclicmatrixQthereforeonlyconsistsofweakindirectinteractions,graphs(DAGs)1,29,weexplorethestabilityofsuchnetworks.andweexpectthemaximaleigenvaluetobesmallerthanthatofInsystemswheretheJacobianmatrixre?ectsthepresenceofdirectrandomnetworksandDAGs.Moreover,sinceinthiscaseQisofinteractionsbetweencomponents,theelementsoftheJacobianrank-1,ithasauniquerealeigenvalueλQ,bwhichcanbeshowntomatrixAijis0ifjdoesnotin?uenceorregulatei.Insuchcases,ifbe(seeSupplementaryNote6):therearenointeractionloopsinvolving2ormorecomponents(e.g.,Xq?loggEregulatesFwhichalsoregulatesE),Acanbewrittenasatriangularλ?cT;e20TQ;bimatrixforsuchaDAGandtheeigenvaluesarethediagonalelementsi?1?ciofthematrix,i.e.,theself-regulationloops.Thesystemistherefore?loggPN?loggjwhereT?casde?nedinEq.(15)aretheelementsstableiftherearenoauto-activationamongthecomponents,i.e.,?cij?1j?citherearenopositiveelementsalongthediagonalofA.oftheM2matrix(andthereforesmallwhentheinteractionInourcase,thepresenceofindirectinteractionscapturedbythedensityislow).ThemaximumeigenvalueoftheinteractionadditionalM2matrix(Eq.(13))impliesthateveniftheregulationmatrixMisthengivenbyλM;b?maxeλQ;b;0T,since0isalsoannetworkisaDAG,thestabilityofthesystemisnotdeterminedeigenvalueofM(seeEqs.(17,18)).solelybytheself-regulationloops.Instead,we?ndthatifwedrawThisexpression(Eq.(20))impliesthatunlikeforfullyrandomDAGsrandomly(constructedbyaddingaconnectiononlyifthenetworksandrandomDAGs,thestabilityofbipartitenetworksresultantnetworkisstillacyclic,Fig.2c),eveniftherearenocandependstronglyontheratioofup-anddownregulatingselfinteractions,thelargesteigenvaluestillscalesapproximatelyp????interactions(seeSupplementaryNote4).Inparticular,thereisawithN,suggestingthatitisstillpossibleforsuchanetworktogolimitonthetotalstrengthofdown-regulation(relativetothatofNATURECOMMUNICATIONS|(2021)12:130|https://doi.org/10.1038/s41467-020-20472-x|www.nature.com/naturecommunications5ContentcourtesyofSpringerNature,termsofuseapply.Rightsreserved 5ARTICLENATURECOMMUNICATIONS|https://doi.org/10.1038/s41467-020-20472-x(a)(b)(c)BiparteBiparteTFsDAG,,Random1?Ω=1.5Ω=2non-TFsFig.3Stabilityofbipartitenetworks.aWhenconstructingabipartiteinteractionnetwork,wegrouptheproteinsintotranscriptionfactors(TFs,redcircles)andnon-TFs(bluecircles),andonlyallowdirectedregulatoryinteractionstogofromaTFtoanon-TF.bForbipartitenetworks,thereisacriticalvalueforthefractionofinhibitoryinteractionsPneg(thatisslightly>0.5)belowwhichthemaximalrealpartoftheeigenvaluesoftheinteractionmatrixλM;rmax?0andabovewhichλM;rmax>0.IntheregimewhereλM;rmax?0(whichcanbeconsideredtobedeeplystablesinceitisfarfromthepointλM;rmax?1wherethesystembecomesunstable),thisvalueofλM;rmaxstaysthesameevenwhenthenumberofdifferentproteinsN(starmarkersvs.circles)orinteractionstrengthsΩmax(starmarkersvs.squares)areincreased.cWhenthereisanequalfractionofup/downregulatoryinteractionsPneg=0.5,λM;rmaxisindependentofbothNandΩmaxforbipartitenetworks(greenmarkers).Thisisincontrasttofullyrandomnetworks(‘Random’,redmarkers)andrandomdirectedacyclicgraphs(‘DAG’,bluemarkers)wherethesystemapproachestheinstabilitylimit(λM;rmax?1)asNorΩmax(circlestotriangles)isincreased.ThisimpliesthatabipartitenetworkstructurecanmaintainandenhancethestabilityofthesystemasNorΩmaxisincreased.Inboth(b)and(c),eachdatapointisobtainedfromanaverageof10randomlydrawnnetworks,witherrorbarsindicatingtheinterquartilerange.[Otherparameters:h=1,ρ=0.01forfullyrandomandrandomDAGs,numberofTFsforbipartitenetworksq=0.1N].up-regulation)forthesystemtobestable.Forexample,ifthethefactthatq?Nandthestabilityofthesystemisgovernedmajorityoftheinteractionsareupregulating,λQ,bshouldbesolelybytheq×qmatrixQrepresentinghowTFsaffectTFs(Eq.negativeandhenceλM,bmustbe0.Ontheotherhand,λM,bmust(19)).Foreachdrawninteractionnetwork,werandomlychoosebepositivewhenthefractionofdownregulationsissuf?cientlyupoftheinteractionstobeupregulating(γij>0)andtheresttobehigh.Thistendencyforinhibitory(activating)interactionstodownregulating(γij<0).Wedrawthefold-changeΩijofeachdestabilize(stabilize)thesystemcomesfromtheindirecteffectregulatoryinteractionfromauniformdistributionbetween1andthataregulatorhasonitself:aslightincreaseintheconcentrationΩmax?1000.ThischoiceofΩmaxismotivatedbythefactthatofaninhibitorfromitssteady-statevaluewillreducethegeneTFshavebeenshownexperimentallytochangetargetproteincopynumberandhencemRNAlevelsoftheregulatedgene.Thelevelsby100–1000fold13.mRNAsoftheinhibitorthereforemakeupalargerfractionoftheWe?ndthatwiththerealnetwork,thesystemalwaystotalmRNAinthecell.SinceallmRNAscompeteforthesharedconvergestoastable?xed-pointregardlessoftheregulationpoolofribosomes,thisinturncausestheinhibitorconcentrationsstrengths(Fig.4a).Incontrast,fortherandomlyconstructedtoincreasefurther.Thispositivefeedbackalsoexistsintheothernetworks(bothwithandwithoutkeepingq?xed),theprobabilityphases,althoughitsphysicaloriginmaybedifferent(seeofthesystembecomingunstabledrasticallyincreaseswhentheSupplementaryNote4,SupplementaryFig.2b).interactionsbecometoostrong(Fig.4a).Thislossofastable?xedIndeed,bynumericallyconstructingmultipleinstancesofapointcangiverisetoeitheranoscillatory(Fig.4b)orchaoticbipartitenetworkandvaryingthefractionofinhibitoryinterac-behavior(Fig.4c).ThissuggeststhatfortypicalregulationtionsPneg,we?ndthatλM,b=0whenPnegisbelowacriticalvaluestrengthsanddensity,theinteractionnetworkcannotberandom,thatisapproximately(butslightlygreaterthan)0.5(Fig.3b).andthatcertainstructuralfeaturesofrealnetworksareimportantImportantly,withinthisregime,thevalueofλM,b=0isforstability.independentofbothNandthestrengthofinteractionsΩmax(Fig.3b,c).ThissuggeststhatsuchabipartitenetworkstructureNetworkstabilitydependsonthedensityofTF–TFinteractions.canhelptomaintainandenhancethestabilityofthesystem,Sinceitisthemaximaleigenvalueoftheq×qsub-matrixQ(Eq.especiallyforlargeNandΩmax.(19))thatdeterminesthestabilityofthesystem,anddirectreg-ulatoryinteractionsaretypicallystrongerthantheindirectScramblingtheinteractionsofE.colitranscriptionalregulatorybackgroundeffects,weexpectahigherdensityofdirectinterac-networkcandestabilizethesystem.Realtranscriptionnetworks,tionsamongTFstodestabilizethesystem.Thissuggeststhathowever,arenotstrictlybipartitegraphs—thereareauto-whatmattersforstabilityisnotonlythenumberofTFsandtheregulatoryelementsaswellasTFsthatregulateotherTFs.Tototalnumberofregulatoryinteractions,butalsothefractionofinvestigatehowrelevantnetworkstabilityistobiologicalnet-thoseinteractionsthattargetTFs.works,weobtainedtheE.colitranscriptionalregulatorynetworkWethereforeanalyzedthecompositionofregulatoryinterac-fromref.31.Thenetworkconsistsofu=5654regulatoryinter-tionsintheE.colitranscriptionnetwork,andfoundthatthereareactions(ofwhichup=3187areupregulating),withq=211TFs(i)us=134self-regulations(ofwhich42areactivating),(ii)ut=regulatingN=2274genes.Wecompareditsstabilitywiththatof373TF-otherTFinteractions(ofwhich201areactivating),andrandomlyconstructednetworkswiththesameN,densityof(iii)un=u?us?ut=5148TF-nonTFinteractions(ofwhichinteractionsρ?u0:0011,andratioofpositive(activating)to2944areactivating)(Fig.5a).Incomparison,thescramblingN2negative(inhibitory)regulation.methodthatmaintainedboththenumberofTFsandthetotalWe?rstexploredtwodifferentwaysofscramblingtheoriginalnumberofinteractionsgivesasmallernumberofself-interactionsnetwork:(1)randomlychoosingudirectedconnectionsoutofthe(?us?=2.5)andalargernumberofdirectTF-otherTFN(N?1)possibleconnections,and(2)?xingthenumberofTFsinteractions(?ut?=522)thanintherealnetwork.qandrandomlychoosingudirectedconnectionsoutofqNToinvestigateifthiscouldbetheoriginoftheenhancedpossibilities.ThesecondmethodofscramblingismotivatedbystabilityoftheE.coliregulatorynetwork,wetriedanother6NATURECOMMUNICATIONS|(2021)12:130|https://doi.org/10.1038/s41467-020-20472-x|www.nature.com/naturecommunicationsContentcourtesyofSpringerNature,termsofuseapply.Rightsreserved 6NATURECOMMUNICATIONS|https://doi.org/10.1038/s41467-020-20472-xARTICLEP(stable))Fig.4ComparingtheE.colitranscriptionalregulatorynetworkwithrandomnetworksofthesamedensity.aTheactualE.colinetworkdoesnotbecomeunstableevenwhenthemaximumregulationstrengthΩmaxisincreased(bluestars).Incontrast,asΩmaxincreases,theprobabilityP(stable)ofthesystemhavingastable?xedpointdecreasesforscramblednetworksofthesameinteractiondensityρ=0.0011,regardlessofwhetherthenumberoftranscriptionfactors(TFs)q=211iskept?xed(yellowcircles)ornot(redsquares).However,scramblingthenetworkwhilemaintainingthesamenumberofTF-otherTF,TF-nonTF,andselfinteractionscansigni?cantlyenhancetheprobabilityofthesystemisstable(greentriangles).Eachofthedatapointsrepresentsanaverageover15setsof10regulatorynetworks,witherrorbarsindicatingtheinterquartilerange.[Otherparameters:h=2].bAtypicalexampleofoscillatorydynamicsinproteinconcentrationscwhenthesystemnolongerhasastable?xedpoint.[Parameters:Ωmax?1585,h=2].cAnexampleofthesystemgoingunstableandexhibitingchaoticbehaviorwhentherealnetworkisscrambledattimet=5×106markedbythedashedverticalline.[Parameters:Ωmax?1000,h=5].Inboth(b)and(c),timetisinunitsof1/kp.downregulationstobeequallylikely,arandomnetworkisalmost(a)(b)1alwaysstablewhenthedensityofTF-otherTFinteractionsρ?=134(42+)qutissuf?cientlylow(Fig.5b).Abovethisthresholdvalueofρ,=373qeq1Tq0.8(201+)theprobabilityofthesystemnotexhibitingastablesteady-stateTFs0.6increaseswithρq(Fig.5b).Thiseffectisobservedregardlessofthe=5148numberofself-interactionsorwhetheruniskept?xed(Fig.5b).(2944+)P(stable)0.4WhilethisimpliesthatsystemswithasmallnumberofTF–TF=5655,=134interactionsarealmostalwaysstable,itdoesnotmeanthathaving0.2=5000,=134ahighdensityofTF–TFinteractionswillnecessarilyleadtoan=5000,=00unstablesystem.Thiscanbeseenfromthefacttheprobabilityof-3-2-1thesystemisstabledoesnotdropsharplywithρ(Fig.5b)—thereqarestillsystemswitharelativelyhighdensityofTF–TFinteractionsthatarestillstable.ThissuggeststhatinthehighFig.5Effectofdensityρqoftranscriptionfactor(TF)-otherTFρqregime,thedetailsoftheinteractionsbecomeimportant.Forinteractionsonstability.aIntherealnetworkanalyzed,thereareus=134suchanetworkwithalargenumberofTF–TFinteractionstobeself-regulations(ofwhich42ofthemareactivating),ut=373TF-otherTFstable,thetypeandstrengthofthoseinteractionswillneedtobeinteractions(ofwhich201ofthemareactivating),andun=5148TF-nonTFmore?ne-tuned.interactions(ofwhich2944ofthemareactivating).ThetotalnumberofThephenomenonthatasmallρqpromotesstabilityisinteractionsisgivenbyu.bArandomlyconstructednetworkisalmostconsistentwiththestabilityofbipartitenetworks(ρq=0)andalwaysstablewhenρqissuf?cientlylow.Aboveathresholdvalue,thethefactthatdirectregulatoryinteractionsaretypicallymuchprobabilityofbeingstable(P(stable))decreaseswithρq.Thisistruewithstrongerthantheindirectbackgroundinteractions.Nevertheless,(redandgreencircles)orwithout(bluecircles)self-interactions,andregardlessofwhetheritisthetotalnumberofinteractionsu(redcircles)orsinceQ(whichhascontributionsfrombothT1andT2,Eq.(19))isnotasparsematrixevenwhenρqissmall,wedonotexpectthethenumberofTF-nonTFinteractionsun(greenandbluecircles)thatiskeptconstant.Eachdatapointisanaverageover15setsof10regulatorymaximaleigenvalueλM;rmaxtoscalewithρqthewayitdoesforanetworks,witherrorbarsindicatingtheinterquartilerange.[Parameters:q×qrandommatrixwithdensityρq.Indeed,we?ndnumericallyN=2274,q=211,h=2,Ωmax?1000.].thatthepresenceofT2canaffectλM;rmax(SupplementaryNote7,SupplementaryFig.4),suggestingthattheindirectcouplingbetweenproteinscanalsoplayaroleinin?uencingthestabilityofthesystem.scramblingmethodwiththecompositionoftheinteractionskept?xed.Inparticular,aftersettingthe?rstq=211(outofN=2274)proteinstobeTFs,werandomlydrawthenumbersofEffectofdegradationratesonproteinlevelstability.Sofar,weinteractionpairswithinthethreecategories(self,TF-otherTF,havebeenworkinginthelimitoffastmRNAdegradation,whereandTF-nonTF)bychoosingeachTFanditstargetseparately.thestabilityofthesystemisgovernedonlybytheinteractionThesignoftheinteractionsarethenrandomlyassignedwhilematrixM(Eq.(12)).Inthisregime,sinceMisindependentofmaintainingthefractionofpositive/negativeinteractionswithindegradationrates1/τmand1/τp(seeEqs.(12,6,3)),thesedonoteachofthesecategories.We?ndthatthisscramblingprocedure,affectwhetherthesystemisstable.Therelaxationratesarealsowhich?xesthecompositionofregulatoryinteractions(inindependentofτmandτp,withtherelaxationrateintheabsenceadditiontoN,q,andρ),signi?cantlyincreasestheprobabilityofinteractionsgivenby(fromEq.(11)):ofthenetworkhavingastable?xedpoint(Fig.4a).ssβ0?kpcr:e21TDirectinteractionsamongTFscaneitherbeauto-regulatoryloopsorTFsregulatingotherTFs.WeexploredtheeffectsofHowever,itisnotclearifthisinsensitivity(ofbothstabilityandbothofthesefactors,andfoundthatassumingup-andrelaxationrates)toτmandτpstillholdsoutsideoftheτm?τpNATURECOMMUNICATIONS|(2021)12:130|https://doi.org/10.1038/s41467-020-20472-x|www.nature.com/naturecommunications7ContentcourtesyofSpringerNature,termsofuseapply.Rightsreserved 7ARTICLENATURECOMMUNICATIONS|https://doi.org/10.1038/s41467-020-20472-xregime.WithintheframeworkofRMT,amorenegativeself-?10=10regulationtermtypicallyincreasestherelaxationrateandhencehasastabilizingeffect2.Here,weaskifthisisthecasebyinvestigatinghowmRNAandproteindegradationratesaffectthe=0.4?)stabilityofthesystemanditsrelaxationtimescale.Inparticular,canfastermRNAdegradationrateshelptostabilizeasystemthatIm(wouldotherwisebeunstableifmRNAsdegradetooslowly??=0.5ValuesofmRNAandproteindegradationratesdonotaffectwhetherthesystemisstable.ToinvestigatehowthedegradationratesofproteinsandmRNAsaffectthestabilityofthesystemwhenτmisnottoosmall,hereweconsiderthefullsetof2NRe()equations(Eqs.(9,10))andstudyhowtheeigenvaluesofthe(2N×2N)JacobianmatrixJvarieswithτmandτp.Fig.6Effectofdegradationratesonstability.aThesystemisstableifandTocomparetherelaxationratesofthefullsystemwiththeonlyifthemaximalrealpartoftheeigenvaluesoftheinteractionmatrixλM;r1,regardlessofthevalueofωwhichincreaseswithmRNAproteinrelaxationrateswhentherearenointeractions,weworkmaxwiththetransformedJacobianmatrix:degradationrates(Eq.(24)).Thescaledeigenvalues~λ!λM1inthelimitoffastmRNAdegradationrateω→∞(Eq.(23)).bEigenvalue1~J?J:e22Tspectrumfordifferentdegradationratesτ.WhenmRNAandproteinβ0degradationratesarecomparable,alleigenvaluesfallwithinacircularregion(red).Ontheotherhand,whenτm?τp,theeigenvaluespectrumForanarbitraryregulatorynetworkwithacorrespondingapproximatelyresemblestwocircularregions,onecorrespondingtotheinteractionmatrixM(Eq.(12)),we?ndthattheeigenvalues~λof~JdynamicsofmRNAsandoneforthatofproteins.Inthislimit,increasingaregivenby(seeSupplementaryNote3):mRNAdegradationrateonlyshiftstheeigenvaluesforthemRNAsectorto1p??????????????????????????????????morenegativevalues,leavingthemaximalrealpartoftheeigenvalues~λ?ω±ω2t4λMe1tωT1;e23T2approximatelyunchanged,ω=0.5(green)vsω=0.4(blue).whereλMaretheeigenvaluesofMasbefore,andωisaTheexpressionfor~λ(Eq.(23))impliesthatwhenthesystemisdimensionlessquantitygivenby:stable(λM;r<1),therateatwhichthesystemrelaxestosteady-1maxω?;e24Tstateinitiallyincreasesasωincreasesfrom?1,buteventuallyτβ0plateauoff?intheω→∞limit(whereτm?τp),~λ!λM1whichre?ectsthedifferencebetweenmRNAandprotein(Eq.(23),Fig.6a).Thisimpliesthatthereissomebene?tto111havingfastmRNAdegradationintermsofresponsetimes,butdegradationrates?.ττmτponcemRNAdegradesmuchfasterthanproteins,furtherSinceonaveragecellvolumeincreasesexponentiallywithrateincreasingmRNAdegradationratenolongeraffectstheresponse(seeEq.(8)):timeofthesystem.Theeigenvaluespectruminthisτm?τplimit1appearstoconsistoftwocircularregions,oneforthedynamicsofμ?kp?r;e25TτmRNAsandtheotherforthatofproteins(Fig.6b),reminiscentpoftheRMT’scircularlaw.Increasingτmonlyshiftstheagrowingcellhastosatisfythecondition1<1.Therefore,τpkp?reigenvaluescorrespondingtothemRNAsectorandhencedoessinceτm≥0,wehaveω≥?1.Theexpressionfor~λ(Eq.(23))notaffect~λrmax.Thisisconsistentwiththefactthatwhenτm?τp,thereforeimpliesthatthesystemisstableifandonlyifλM;r≤1,thedynamicsoftheoverallsystemisgovernedonlybythemaxregardlessofthevalueofτmandτp(Fig.6a).We?ndthatdespiteproteinsector(Eq.(11)).Therefore,theslowestrelaxationratedifferencesinthedetailsofthemodel,thisconclusionstillholdsbacktosteady-statelevelsdependsonlyonMandincreasingintheotherphases(seeSupplementaryNote3).mRNAdegradationratenolongerimprovestheresponsetime.Therefore,unlikewhathasbeenarguedintheliteratureandwhatonemightexpectfromRMT,changingmRNAnorproteinDiscussiondegradationrateshasnoeffectonwhethertheoverallsystemisInsystemswithalargenumberofinteractingcomponents,thestable.Ifsteady-stateproteinconcentrationsareunstablebecausequestionofstabilityisoftenanimportantone,asresultsfromλM;ristoolarge(e.g.,wheninteractionsaretoostrong),maxRMTpredictinstabilitywhenthesystemsizeNbecomestoolargeincreasingmRNAorproteindegradationratescanneverhelptoorinteractionsbecometoostrong.Inthecontextofgenestabilizethesystem.expression,transcriptionalregulationiscrucialforcellstoadaptImportantly,this?ndingalsoimpliesthatourresultsforhowtodifferentenvironmentalconditionsbychangingtheirgenestructuralfeaturesofthetranscriptionnetworkaffectsstabilityexpressionlevels.ItisthereforeimportantfortranscriptionalholdsoutsidetheregimeoffastmRNAdegradation,sinceregulatorynetworks(TRNs)tobeabletoaccommodatealargestabilityonlydependsonM.numberofregulatoryinteractionswithoutthesystemgoingunstable.However,wep?ndherethatsimilartotheintuition????IncreasingmRNAdegradationratecanimproveresponsetimes,providedbyRMT,λNforafullyrandomregulationnet-butonlyuptosomelimit.Besidessystemstability,anotherwork,suggestingthatthesystemwillgounstableasthenumberquantityofbiologicalinterestistheresponsetimeofthesystemtoofgenesexceedsathreshold.Infact,basedontypicalvaluesforperturbations,whichisespeciallyrelevantforcellsexperiencingthedensityofactualregulatorynetworksandinteractionchangesinnutrientconditions32,33.Sincethisrelaxationtimescalestrengths,we?ndthatthesystemhasahighprobabilityofbeingisdeterminedbytheslowesteigenvalueoftheJacobianmatrix,unstableiftheTRNisrandomlyconstructed.herewediscusshowthemaximalrealpartoftheeigenvalues~λrBesidesthenumberofgenes,andthedensityandstrengthsofmaxchangeswithτ.interactions,thereareotherfactorsthatcanaffectthestabilityof8NATURECOMMUNICATIONS|(2021)12:130|https://doi.org/10.1038/s41467-020-20472-x|www.nature.com/naturecommunicationsContentcourtesyofSpringerNature,termsofuseapply.Rightsreserved 8NATURECOMMUNICATIONS|https://doi.org/10.1038/s41467-020-20472-xARTICLEthesystem,oneofwhichisthenetworktopology.Thisaspectisnutrientconditions,buttheonlyonesthatcansurvivearethoseparticularlyrelevantinthissystemsinceTRNsarefarfrombeingthatalsomaintainthestabilityofthesystem.Inotherwords,therandombutinsteadconsistofrecurringmotifs.Whilethestabilityofthesystemmayhaveplayedaroleinshapingcurrentpropertiesofthesespeci?cmotifshavebeenwidelystudiedandexistingregulatorynetworksthroughtheevolutionaryshowntobeimportantforspeci?cfunctionssuchasadaptation,process.Ourapproachcanthereforeprovideinsightsintotherobustness,andfastresponsetoenvironmentalchanges1,29,30,designandevolutionaryconstraintsforafunctionalregulatoryhowtheycontributetotheoverallstabilityofthenetworkislessnetwork,whichmaypotentiallybeusefulforguidingthecon-clear.We?ndherethatglobalstructuralfeaturesofthenetwork,structionofsyntheticgeneticcircuits36–38.Inthefuture,thewhicharefundamentallyshapedbymanyofthesemotifs,canabilitytoexperimentallyengineeralarge,randomregulatoryplayahugeroleindeterminingthestabilityofthesystem.Incircuitwithincellscouldalsoallowtestingoftheresultswehaveparticular,giventhesamenumberofproteins,TFs,interactiondescribed.density,andregulationstrengths,anetworkthatresemblesaInadditiontotranscriptionalregulation,geneexpressionisbipartitegraphwithalowerdensityofTF-otherTFinteractionsρqalsoregulatedatthepost-transcriptional(e.g.,throughsmall-hasahigherchanceofbeingstable.Thesigni?canceofρqfun-RNAsormicro-RNAs)andpost-translational(e.g.,throughdamentallyarisesbecauseoftwomainfactors:(i)theeigenvaluespost-translationalmodi?cations)level.OurframeworkcanbeoftheJacobianmatrixandhencethestabilityofthesystemaboutextendedtotakeintoaccounttheseeffects(seeSupplementaryitssteady-statearegovernedonlybytheTFsector(i.e.,howNote8foranexample).HowthestabilityofthesystemisperturbationsinTFconcentrationsaffectTFs),and(ii)foraaffectedbythecouplingbetweenthesedifferentformsofreg-sparseregulatorynetwork,theindirectbackgroundinteractionsulationwithpotentiallydifferentnetworkstructuresisanarisingfromcompetitionforribosomesbetweendifferentgenesinterestingquestionthatweleaveforfuturework.Besidesaretypicallymuchweakerthanthedirectregulatoryinteractions.stability(determinedbytheeigenvaluesofJ),inthefuture,itTRNsarealsoknowntobescale-free,havingapower-lawout-couldalsobeinstructivetoinvestigatethespreadofperturba-degreedistribution.ThisisconsistentwiththefactthatmostTFstionswithintheregulatorynetwork(i.e.,theeigenvectorsofJ).onlyregulateasmallnumberofgenes,butthereareTFs(oftenThisisanalogoustothestudyofhowconcentrationpertur-referredtoasmasterregulators)thatregulateaverylargenumberbationspropagateinprotein–proteininteractionnetworksofgenes.Withinamoreabstractmodelofgeneregulatorywithinthecell39.dynamics,thepresenceoftheseoutgoinghubshasbeenshowntosigni?cantlyincreasetheprobabilityofthesystemreachingaReportingsummary.FurtherinformationonresearchdesignisavailableintheNaturestabletargetphenotypewhentheinteractionstrengthsareResearchReportingSummarylinkedtothisarticle.allowedtovarywhilethenetworktopologyiskept?xed34.Here,we?ndthathavingalowρqcanalreadysigni?cantlystabilizetheDataavailabilitysystemwithouttheneedtocontrolthedegreedistributions.TheE.coli.transcriptionalregulatorynetworkdatathatsupportthe?ndingsofthisstudyNevertheless,havingjustafewmasterregulatorsmaycontributeisavailableinthesupplementary?lesofthepaper(ref.31):https://doi.org/10.1073/pnas.1702581114.ThisdatausedforanalysisisalsoavailableinaMATLABdata?leontothenetworkhavingalowρqifforinstancemostofthereg-GitHubrepository40:https://github.com/yipeiguo/TRNstability.ulationsonTFsarecarriedoutbythemasterregulators(andnon-masterregulatorspredominantlyregulatenon-TFs).CodeavailabilityBesidesthestructuralfeaturesofthenetwork,anotherfactorAllsimulationsanddataanalysisarecarriedoutusingcodeswritteninMATLABthatcouldaffectstabilityisthedegradationratesofmRNAandR2019a.ThesecanbefoundonGitHubrepository40:https://github.com/yipeiguo/proteins.BasedonRMT,onemayexpectfasterdegradationtoTRNstability.stabilizethesystem.Thishasinfactbeenarguedtobethecase5,6.However,bytakingintoaccountthedynamicsofproteincon-Received:29May2020;Accepted:3December2020;centrationsandhowitcouplestothedynamicsofmRNAlevels,we?ndthatthisisnotthecase.Instead,thestabilityofthesystemdependssolelyontheregulatorynetworkandthestrengthsofthoseregulations—ifthesystemisunstable,itwillbeunstableregardlessofhowfastmRNAorproteindegrades.ThishighlightsReferencestheimportanceoftakingintoaccountkeyaspectsoftheinter-1.Alon,U.AnIntroductiontoSystemsBiology:DesignPrinciplesofBiologicalactions(throughtheformofthedynamicalequations)whenCircuits.(CRCpress,2019).analyzingthestabilityoflargecoupledsystems,similarinspiritto2.May,R.M.Willalargecomplexsystembestable?Nature238,413(1972).studiesofecologicalmodelswhereexplicitlyconsideringinter-3.Ginibre,J.Statisticalensemblesofcomplex,quaternion,andrealmatrices.J.Math.Phys.6,440–449(1965).actionsmediatedthroughcompetitionfornutrientscangive4.Girko,V.L.Circularlaw.TheoryProbab.Appl.29,694–706(1985).drasticallydifferentresultsfromassumingrandompairwise5.Chen,Y.,Shen,Y.,Lin,P.,Tong,D.&Zhao,Y.etal.Generegulatorynetworkinteractionsbetweenspecies7.Thispredictioncanalsopotentiallystabilizedbypervasiveweakrepressions-micrornafunctionsrevealedbythebetestedinthelabbyvaryingthedegradationratesofmRNAsmay-wignertheory.NatlSci.Rev.6,1176–1188(2019).(e.g.,byusinggeneticallymodi?edRNases)orproteins(e.g.,by6.Zhao,Y.,Shen,X.,Tang,T.&Wu,C.-I.Weakregulationofmanytargetsiscumulativelypowerful-?anevolutionaryperspectiveonmicrornausinggeneticallymodi?edproteases)inthecellandobservingthefunctionality.Mol.Biol.Evolution34,3041–3046(2017).dynamicsofproteinconcentrations.7.Cui,W.,Marsland,W.III&Mehta,P.DiversecommunitiesbehavelikeFromanevolutionaryperspective,therearemanypossibletypicalrandomecosystems.Preprintathttps://www.biorxiv.org/content/factors(suchastherangeofgeneexpressionlevels,environ-10.1101/596551v2.full(2019).mentalconditions,responsetime32,33,levelofunwantedcross-8.Allesina,S.&Tang,S.Stabilitycriteriaforcomplexecosystems.Nature483,35205(2012).talk,etc.)thatdrivetheadditionorremovalofregulatory9.Thébault,E.&Fontaine,C.Stabilityofecologicalcommunitiesandtheconnections.Our?ndingssuggestthatinadditiontothesecon-architectureofmutualisticandtrophicnetworks.Science329,853–856(2010).siderations,anotherfundamentalfactoristhestabilityofthe10.Okuyama,T.&Holland,J.N.Networkstructuralpropertiesmediatetheoverallnetwork.Forexample,therecouldbemanywaysofstabilityofmutualisticcommunities.Ecol.Lett.11,208–216(2008).achievingacertaintasksuchasallowingthecelltoswitch11.Lin,J.&Amir,A.Homeostasisofproteinandmrnaconcentrationsingrowingcells.Nat.Commun.9,4496(2018).betweentwodesiredgeneexpressionlevelsintwodifferentNATURECOMMUNICATIONS|(2021)12:130|https://doi.org/10.1038/s41467-020-20472-x|www.nature.com/naturecommunications9ContentcourtesyofSpringerNature,termsofuseapply.Rightsreserved 9ARTICLENATURECOMMUNICATIONS|https://doi.org/10.1038/s41467-020-20472-x12.Buchler,N.E.,Gerland,U.&Hwa,T.Onschemesofcombinatorial36.Adamala,K.P.,Martin-Alarcon,D.A.,Guthrie-Honea,K.R.&Boyden,E.S.transcriptionlogic.Proc.NatlAcad.Sci.USA100,5136–5141(2003).Engineeringgeneticcircuitinteractionswithinandbetweensyntheticminimal13.Kuhlman,T.,Zhang,Z.,Saier,M.H.&Hwa,T.Combinatorialtranscriptionalcells.Nat.Chem.9,431(2017).controlofthelactoseoperonofescherichiacoli.Proc.NatlAcad.Sci.USA104,37.Ellis,T.,Wang,X.&Collins,J.J.Diversity-based,model-guidedconstruction6043–6048(2007).ofsyntheticgenenetworkswithpredictedfunctions.Nat.Biotechnol.27,14.Kim,H.D.&Oashea,E.K.Aquantitativemodeloftranscriptionfactor-465–471(2009).activatedgeneexpression.Nat.Struct.Mol.Biol.15,1192(2008).38.Noireaux,V.,Maeda,Y.T.&Libchaber,A.Developmentofanarti?cialcell,15.Paulsson,J.Modelsofstochasticgeneexpression.Phys.LifeRev.2,157–175fromself-organizationtocomputationandself-reproduction.Proc.NatlAcad.(2005).Sci.USA108,3473–3480(2011).16.Shahrezaei,V.&Swain,P.S.Analyticaldistributionsforstochasticgene39.Maslov,S.,Sneppen,K.&Ispolatov,I.Spreadingoutofperturbationsinexpression.Proc.NatlAcad.Sci.USA105,17256–17261(2008).reversiblereactionnetworks.N.J.Phys.9,273(2007).17.Thattai,M.&Oudenaarden,A.VanIntrinsicnoiseingeneregulatory40.Guo,Y.&Amir,A.Exploringtheeffectofnetworktopology,mRNAandnetworks.Proc.NatlAcad.Sci.USA98,8614–8619(2001).proteindynamicsongeneregulatorynetworkstability.GitHubrepository,18.Zhurinsky,J.etal.Acoordinatedglobalcontrolovercellulartranscription.DOI:10.5281/zenodo.4294914,(2020).CurrentBiol.20,2010–2015(2010).19.Schmidt,E.E.&Schibler,U.Cellsizeregulation,amechanismthatcontrolscellularrnaaccumulation:consequencesonregulationoftheubiquitousAcknowledgementstranscriptionfactorsoct1andnf-yandtheliver-enrichedtranscriptionfactorWethankRuiFang,JieLin,HaimSompolinsky,GraceZhang,DavidNelson,Naamadbp.J.CellBiol.128,467–483(1995).Brenner,andGuyBuninforusefuldiscussionsandfeedback.Thisresearchwassup-20.Neurohr,G.E.etal.ExcessivecellgrowthcausescytoplasmdilutionandportedbytheNationalScienceFoundationthroughMRSECDMR14-20570,theKavlicontributestosenescence.Cell176,1083–1097(2019).Foundation,andtheNSFCAREER1752024.21.Reuveni,S.,Ehrenberg,M.&Paulsson,J.Ribosomesareoptimizedforautocatalyticproduction.Nature547,293(2017).Authorcontributions22.Scott,M.,Gunderson,C.W.,Mateescu,E.M.,Zhang,Z.&Hwa,T.Y.G.andA.A.designedresearch,performedresearch,andwrotethepaper.Interdependenceofcellgrowthandgeneexpression:originsandconsequences.Science330,1099–1102(2010).23.Kubitschek,H.E.,Baldwin,W.W.,Schroeter,S.J.&Graetzer,R.CompetinginterestsIndependenceofbuoyantcelldensityandgrowthrateinescherichiacoli.J.Allauthorsdeclarenocompetinginterests.Bacteriol.158,296–299(1984).24.Basan,M.etal.In?atingbacterialcellsbyincreasedproteinsynthesis.Mol.AdditionalinformationSyst.Biol.11,836(2015).Supplementaryinformationisavailableforthispaperathttps://doi.org/10.1038/s41467-25.Crissman,H.A.&Steinkamp,J.A.Rapid,simultaneousmeasurementofdna,020-20472-x.protein,andcellvolumeinsinglecellsfromlargemammaliancellpopulations.J.CellBiol.59,766(1973).CorrespondenceandrequestsformaterialsshouldbeaddressedtoA.A.26.Milo,R.&Phillips,R.CellBiologybytheNumbers.(GarlandScience,2015).27.Sompolinsky,H.,Crisanti,A.&Sommers,H.-J.ChaosinrandomneuralPeerreviewinformationNatureCommunicationsthanksTamarFriedlanderandthenetworks.Phys.Rev.Lett.61,259(1988).other,anonymous,reviewer(s)fortheircontributiontothepeerreviewofthiswork.28.Roy,F.,Biroli,G.,Bunin,G.&Cammarota,C.Numericalimplementationofdynamicalmean?eldtheoryfordisorderedsystems:applicationtotheReprintsandpermissioninformationisavailableathttp://www.nature.com/reprintslotka-volterramodelofecosystems.J.Phys.A:Math.Theor.52,484001(2019).Publisher’snoteSpringerNatureremainsneutralwithregardtojurisdictionalclaimsin29.Shen-Orr,S.S.,Milo,R.,Mangan,S.&Alon,U.Networkmotifsinthepublishedmapsandinstitutionalaf?liations.transcriptionalregulationnetworkofescherichiacoli.Nat.Genet.31,64(2002).30.Alon,U.Networkmotifs:theoryandexperimentalapproaches.Nat.Rev.Genet.8,450–461(2007).OpenAccessThisarticleislicensedunderaCreativeCommons31.Fang,X.,Sastry,A.,Mih,N.,Kim,D.&Tan,J.etal.GlobaltranscriptionalAttribution4.0InternationalLicense,whichpermitsuse,sharing,regulatorynetworkforEscherichiacolirobustlyconnectsgeneexpressiontoadaptation,distributionandreproductioninanymediumorformat,aslongasyougivetranscriptionfactoractivities.Proc.NatlAcad.Sci.USA114,10286–10291appropriatecredittotheoriginalauthor(s)andthesource,providealinktotheCreative(2017).Commonslicense,andindicateifchangesweremade.Theimagesorotherthirdparty32.vanHeerden,J.H.etal.Lostintransition:start-upofglycolysisyieldsmaterialinthisarticleareincludedinthearticle’sCreativeCommonslicense,unlesssubpopulationsofnongrowingcells.Science343,1245114(2014).indicatedotherwiseinacreditlinetothematerial.Ifmaterialisnotincludedinthe33.Erickson,D.W.etal.Aglobalresourceallocationstrategygovernsgrowtharticle’sCreativeCommonslicenseandyourintendeduseisnotpermittedbystatutorytransitionkineticsofEscherichiacoli.Nature551,119–123(2017).regulationorexceedsthepermitteduse,youwillneedtoobtainpermissiondirectlyfrom34.Schreier,H.I.,Soen,Y.&Brenner,N.Exploratoryadaptationinlargerandomthecopyrightholder.Toviewacopyofthislicense,visithttp://creativecommons.org/networks.Nat.Commun.8,1–9(2017).licenses/by/4.0/.35.Friedlander,T.,Prizak,R.,Guet,C.C.,Barton,N.H.&Tka?ik,G.Intrinsiclimitstogeneregulationbyglobalcrosstalk.Nat.Commun.7,12307(2016).?TheAuthor(s)2021,correctedpublication202110NATURECOMMUNICATIONS|(2021)12:130|https://doi.org/10.1038/s41467-020-20472-x|www.nature.com/naturecommunicationsContentcourtesyofSpringerNature,termsofuseapply.Rightsreserved 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