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ARTICLEhttps://doi.org/10.1038/s41467-020-20472-xOPENExploringtheeffectofnetworktopology,mRNAandproteindynamicsongeneregulatorynetworkstabilityYipeiGuo1,2&ArielAmir1✉Homeostasisofproteinconcentrationsincellsiscrucialfortheirproperfunctioning,requiringsteady-stateconcentrationstobestabletofluctuations.Sincegeneexpressionisregulated1234567890():,;byproteinssuchastranscriptionfactors(TFs),thefullsetofproteinswithinthecellcon-stitutesalargesystemofinteractingcomponents,whichcanbecomeunstable.WeexplorefactorsaffectingstabilitybycouplingthedynamicsofmRNAsandproteinsinagrowingcell.WefindthatmRNAdegradationratedoesnotaffectstability,contrarytopreviousclaims.However,globalstructuralfeaturesofthenetworkcandramaticallyenhancestability.Importantly,anetworkresemblingabipartitegraphwithalowerfractionofinteractionsthattargetTFshasahigherchanceofbeingstable.ScramblingtheE.colitranscriptionnetwork,wefindthatthebiologicalnetworkissignificantlymorestablethanitsrandomizedcoun-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.DespiteproteinsingrowingcellsandfindthatwhilethemRNAdegra-theimportanceofmaintainingdesiredproteinconcentrationsdationratecanaffectrelaxationratebacktosteady-statelevels,itwithincells,factorsaffectingthestabilityoftheseconcentrationsdoesnotaffectwhetherthesystemisstable.Instead,stabilitycantoperturbationshavereceivedlittleattention.dependstronglyontheglobalstructuralfeaturesoftheinterac-Oneapproachofstudyingthestabilityofsuchsystemswithationnetwork.Inparticular,giventhesamenumberofproteins,largenumberofinteractingcomponentswasintroducedbyMayTFs,numberofinteractions,andregulationstrengths,anetworkinthe1970sinthecontextofcomplexecologicalcommunities2.withalowerfractionofinteractionsthattargetTFshasahigherTheideaisthatinan-speciescommunity,thedynamicsofthechanceofbeingstable.InthelimitwheretherearenoTF–TFabundancesNiofeachspeciesmayingeneralbedescribedbyainteractionsi.e.allTFsregulateproteinsthatarenotTFs,itissetofordinarydifferentialequations:possibleforthesystemtoremainstableforarbitrarilylargesys-temsizes,unlikerandomnetworkswhichbecomeunstablewhendNi¼fiðN1;N2;:::NnÞð1Þsystemsizebecomestoolarge.ByscramblingtheE.coli.tran-dtscriptionnetwork,wefindthatthetopologyofrealnetworkscanfori=1,2,...,n,withcorrespondingsteady-statesolutionNssstabilizethesystemsincetherandomizednetworkwiththesamei!ssnumberofregulatoryinteractionsisoftenunstable.Thesefind-suchthatfðNÞ¼0∀i.Thedynamicsofsmallperturbationsiaboutthissteady-statexðtÞ¼NðtÞNss,whenlinearizedaboutingssuggestthatconstraintsimposedbysystemstabilitymayiiiNss,hastheform:haveplayedasignificantroleinshapingtheexistingregulatoryinetworkduringtheevolutionaryprocess.Bycarryingoutthe!dx!analysisfordifferentphysiologicalstatesthecellcanbein(cor-¼Ax;ð2Þdtrespondingtodifferentsetsofdynamicalequations)andwithssdifferentchoicesofparameterdistributions,wealsoshowthatwhereAistheJacobianmatrixwithelementsA¼∂fi.Ifallij∂NjourmainresultsandconclusionsarerobusttothedetailsoftheeigenvaluesofAhaveanegativerealpart,thesystemrelaxesthemodel.backtothesteady-stateuponperturbationsandthesteady-stateissaidtobestable;ifanyoftheeigenvalueshaveapositiverealpart,Resultsthesteady-stateisunstableasthesystemwillmoveawayfromitThemodel.Geneexpressioninvolvestwomajorsteps:tran-(exponentiallyfast)wheninfinitesimallyperturbed.Toconstructscriptionandtranslation(Fig.1a).TranscriptionistheprocessinA,onewouldneedtopreciselyknowthefunctionsfi,whichiswhichmRNAissynthesizedbyRNApolymeraseusingDNAasaoftenhardtoobtain.May’sapproachwastomodelAasaran-template.Thetranscriptionrateofageneithereforedependsondommatrixwithindependent,identicallydistributedoff-diagonalthenumberofRNApolymerasesnanditseffectivegenecopyelements(withmean0,standarddeviationσ,andfractionofnon-numbergiwhichtakesintoaccountbothitscopynumberandzeroelementsC)andconstantdiagonalelements—a.InthehowstronglyRNApolymerasecanbindtothepromoterofthatcontextofecology,σreflectstheaverageinteractionstrengthgene11.DuetothepresenceofTFs,gð!cÞcaningeneraldependibetweenspecies,Cisthedensityofinteractionsortheprobability!onthesetofproteinconcentrationsc(Fig.1a).Weassumethatthatanytwospeciesinteract,whileaistheself-regulationtermmultipleTFsactingonthesamegeneactindependently,withwhichsetstherelaxationtime-scaleofthesystemiftherewerenotheireffectsstackingmultiplicatively.ThisallowsforbothOR-otherpairwiseinteractions.Fromrandommatrixtheory(RMT)12andAND-gate-likecombinatorialeffects,andcanemergefromandinparticularthecircularlawformatrixeigenvaluedis-pffiffiffiffiffiffiathermodynamicmodelofTFbinding(SupplementaryNote1).tributions3,4,thissystemisstableifandonlyifa>σnC.ThisTherefore,weadoptthefollowingformfortranscriptionalreg-impliesthatthesystembecomesunstableabovesomecriticalsize,ulationthroughoutthepaper:andthatincreasingastabilizesthesystemandallowsforstrongerY!interactionsbetweenspecies.giðcÞ¼gi0ð1þγijfijðcjÞÞ;ð3ÞThisapproachhasalsobeenusedtoanalyzeotherlargejinteractingsystems.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>0ðupregulatingÞresourcemodel),evenwhentheinteractions(i.e.,preferencesofγij¼11ifγ<0ðdownregulatingÞð4ÞΩijeachspeciesforthedifferentresources)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)TranscriponalEffecvegenecopynumber: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,Γmisproportionaltothegeneallocationfractionϕi=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!1¼kmϕiðcÞcncmikpcrþ;ð9Þsimulationsintherestofthissectionarecarriedoutinphasedtτ1,ourfindingsandconclusionsalsoapplytotheotherphases.dccpffiffiffiffii¼kcmic;ð10ÞStabilityofthesystemscaleswithNforrandomregulatorypridtcmTnetworks.WestartbyexploringthestabilityoffullyrandomwherecmT=∑icmiisthetotalconcentrationofallmRNAsandregulatorynetworks,whichwetaketobeournullmodel.1¼11isthedifferencebetweenmRNAandproteinSincethemaximumeigenvalueofarandommatrixdependsonττmτpthestandarddeviationofitselements,wefirstcarryoutanaivedegradationrates(whichcanbepositiveornegative).AsummaryestimateofhowtheelementsofMscalewithN.Withgi(c)givenofthelistofmodelparameterscanbefoundinSupplementarybyEq.(3),Table1.Whiletheseequationsgovernthedynamicsofaverageconcentrationsandhencedonotcapturestochasticeffects∂loggiγij∂fij¼:ð16Þinherentingeneexpressionandinthebinomialsamplingof∂cj1þγfðcjÞ∂cjijijmoleculesduringcelldivision,thesefluctuationsdonotaffecttheaveragesteady-stateconcentrationsifthenumberofmoleculesislarge(seeSupplementaryNote2,SupplementaryFig.1).Infact,Biologically,TFconcentrationsareoftencomparabletothevaluesthesefluctuationscanbeconsideredasperturbationsaboutofdissociationconstantsKdforDNAbinding26.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,wefirstconsiderthepffiffiffiffimaxregimewherethelifetimeofmRNAsismuchshorterthanthatofapproximatelyasNforrandominteractionnetworks.λM;rmaxproteins,whichistypicallytrueforwild-typecells26.Inthislimitalsoincreaseswiththestrengthoftheinteractionsγ,implyingoffastmRNAdegradation,therelaxationdynamicsofmRNAisthatthesystemwillbecomeunstableeitherwhenNexceedsamuchfasterthanthatofproteinssuchthatdcmi0atalltimes.criticalnumberortheregulationstrengthbecomestoohigh.dtEliminatingthefastprocess(bysubstitutingthesteady-stateHowever,thisargumentneglectscorrelationsbetweenthemRNAconcentrationsc¼kmcnϕð!cÞobtainedfromEq.(9)elementsofM,whichcouldpotentiallyberelevant.Infact,wemikcþ1iprτwillseeinthelatersectionsthatthestructureofM(Eq.(13))intoEq.(10)),thedynamicsofproteinconcentrationscanbeplaysanimportantroleininfluencingthestabilityofthesystem.writtenasasetofNODEs:Therefore,totestifthisscalingrelationholds,weconstructeddcnetworksofaspecifiedinteractiondensityρbyrandomlyikcϕð!cÞc:ð11Þ2priiselectingρNinteractionsfromtheN(N−1)possibilities(wheredtwehaveassumedthatribosomescannotactasTFs),andchooseThestabilityofthesystemthereforedependsonlyontheeigen-halfoftheinteractionstobeupregulatingwiththeremaininghalfvaluesoftheN×NJacobianmatrixA¼kcssðMIÞ,whereweprbeingdownregulating.definetheinteractionmatrixBytakingtheensembleaverageovertherandomlydrawnpffiffiffiffi∂ϕnetworks,weindeedrecovertheNscaling(Fig.2a),whichisiMij¼j!!ss;ð12Þalsorobusttothefractionofup-anddownregulatoryinteractions∂cc¼cj(seeSupplementaryNote4,SupplementaryFig.2a)andthewiththesteady-stateproteinconcentrationsgivenbycss¼distributionoffold-changesP(Ω)(seeSupplementaryNote5,i!ssSupplementaryFig.3).ForsufficientlylargeNorΩ,wecannoϕiðcÞ(fromEq.(11)).maxDenotingλMastheeigenvaluesofM,thesystemisstableaslongerfindthefixedpointofthesystem.Nevertheless,bylongasthemaximalrealpartoftheseeigenvaluesλM;rissimulatingthedynamics,wefindthatforinteractionnetworksofmaxagivenNandρ,wegetoscillatory,followedbychaoticbehaviorsmallerthan1(suchthatalleigenvaluesofAhaveanegativerealpart).ItisthereforeusefultounderstandthestructureofMbyasΩmaxisincreased(Fig.2b).Similarphenomenahavealsobeendescribedandanalyzedinmodelsofneuralnetworks27andbreakingitintotwopartsusingEq.(6):ecologicalsystems28.WhilecertainbiochemicalcircuitshavebeenM¼cssðMMÞ;ð13Þiji1;ij2;ijknowntogenerateoscillationssuchasinthecellcycleandthecircadianclock,theoscillatorydynamicsobservedhereisofawheredifferentnature—itdoesnotcomeaboutfromanyspecificfine-∂loggituningofthenetworkbut,rather,emergesfromhavingalargeM1;ij¼ð14Þ∂cjnumberofrandomlyandstronglyinteractinggenes.However,transcriptionalregulatorynetworksaretypicallynotcapturesthedirectinteractionsbetweenproteins,whilerandom.Instead,theyareenrichedfordistinctstructuralfeatures∂loggXss∂loggsuchasthefollowingmotifs:feedforwardloops(FFL),single-M¼T¼ck2;ijkð15Þinputmodule(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,theyareaspecificsubsetofDAGsinwhichnoneoftheTFsarethemselvesregulated.Thisisalsoakeydifferencebetweenthesenetworksandbipartite,mutualisticnetworkscommonlystudiedinecologicalmodels9,10.Inthissubsection,weinvestigatethestabilityofsuchnetworks.Fig.2Stabilityofrandominteractionnetworks.aForrandominteractionTostudythisproblem,wefirstgroupproteinsintotwonetworks(redmarkers,`rand'),themaximalrealpartoftheeigenvaluesofcategories:qTFsandN−qnon-TFs,suchthatforanygeneralpffiffiffiffitheinteractionmatrixλM;rscaleswithN.Surprisingly,forrandomnetworkthecomponentsoftheJacobianmatrixhavethemaxdirectedacyclicnetworks(bluemarkers,`DAG'),λM;ralsoscalesfollowingstructure:pffiffiffiffimaxapproximatelywithN.Inbothofthesecases,increasingtheinteractionT01strengthfromΩmax¼1:5(circles)toΩmax¼2(triangles)increasesλM;rmax.M1¼ð17ÞTheseresultssuggestthatthesystemwillbecomeunstable(i.e.,R10log10ðλM;rmaxÞexceeds0,indicatedbytheblackdashedline)whenNorT0Ωbecomestoolarge.Eachdatapointisobtainedfromanaverageof102maxM2¼;ð18Þrandomlydrawnnetworks,witherrorbarsindicatingtheinterquartilerange.R20EachrandomnetworkisconstructedbyrandomlyselectingρN2whereT1(T2)isaq×qmatrixrepresentingthedirect(indirect)interactionsfromN(N−1)possibilities,withhalfoftheinteractionschoseneffectofTFsonTFswhileR1(R2)isa(N−q)×qmatrixtobeupregulatingandtheremaininghalftobedownregulating.Therepresentingthedirect(indirect)effectofTFsonnon-TFs,withconstructionofDAGsisdescribedin(c).Foreachregulatoryinteraction,theirelementsdefinedpreviously(Eqs.(13)–(15)).Thenon-zerofoldchangeischosenuniformlybetween1andΩmax.[Otherparameters:eigenvaluesofMarethereforetheeigenvaluesofthesub-matrixρ=0.01,h=1].bWhensystemsgooutofstability,dynamicsofproteinQwithelements:concentrationscexhibitoscillatory(left,Ωmax¼20)followedbychaoticQ¼cssðTTÞ:ð19Þbehavior(right,Ωmax¼200)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.andweexpectthemaximaleigenvaluetobesmallerthanthatofInsystemswheretheJacobianmatrixreflectsthepresenceofdirectrandomnetworksandDAGs.Moreover,sinceinthiscaseQisofinteractionsbetweencomponents,theelementsoftheJacobianrank-1,ithasauniquerealeigenvalueλQ,bwhichcanbeshowntomatrixAijis0ifjdoesnotinfluenceorregulatei.Insuchcases,ifbe(seeSupplementaryNote6):therearenointeractionloopsinvolving2ormorecomponents(e.g.,Xq∂loggEregulatesFwhichalsoregulatesE),Acanbewrittenasatriangularλ¼cT;ð20ÞQ;bimatrixforsuchaDAGandtheeigenvaluesarethediagonalelementsi¼1∂ciofthematrix,i.e.,theself-regulationloops.Thesystemistherefore∂loggPN∂loggjwhereT¼casdefinedinEq.(15)aretheelementsstableiftherearenoauto-activationamongthecomponents,i.e.,∂cij¼1j∂citherearenopositiveelementsalongthediagonalofA.oftheM2matrix(andthereforesmallwhentheinteractionInourcase,thepresenceofindirectinteractionscapturedbythedensityislow).ThemaximumeigenvalueoftheinteractionadditionalM2matrix(Eq.(13))impliesthateveniftheregulationmatrixMisthengivenbyλM;b¼maxðλQ;b;0Þ,since0isalsoannetworkisaDAG,thestabilityofthesystemisnotdeterminedeigenvalueofM(seeEqs.(17,18)).solelybytheself-regulationloops.Instead,wefindthatifwedrawThisexpression(Eq.(20))impliesthatunlikeforfullyrandomDAGsrandomly(constructedbyaddingaconnectiononlyifthenetworksandrandomDAGs,thestabilityofbipartitenetworksresultantnetworkisstillacyclic,Fig.2c),eveniftherearenocandependstronglyontheratioofup-anddownregulatingselfinteractions,thelargesteigenvaluestillscalesapproximatelypffiffiffiffiinteractions(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;rmax¼0andabovewhichλM;rmax>0.IntheregimewhereλM;rmax¼0(whichcanbeconsideredtobedeeplystablesinceitisfarfromthepointλM;rmax¼1wherethesystembecomesunstable),thisvalueofλM;rmaxstaysthesameevenwhenthenumberofdifferentproteinsN(starmarkersvs.circles)orinteractionstrengthsΩmax(starmarkersvs.squares)areincreased.cWhenthereisanequalfractionofup/downregulatoryinteractionsPneg=0.5,λM;rmaxisindependentofbothNandΩmaxforbipartitenetworks(greenmarkers).Thisisincontrasttofullyrandomnetworks(‘Random’,redmarkers)andrandomdirectedacyclicgraphs(‘DAG’,bluemarkers)wherethesystemapproachestheinstabilitylimit(λM;rmax¼1)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,werandomlychoosebepositivewhenthefractionofdownregulationsissufficientlyupoftheinteractionstobeupregulating(γij>0)andtheresttobehigh.Thistendencyforinhibitory(activating)interactionstodownregulating(γij<0).Wedrawthefold-changeΩijofeachdestabilize(stabilize)thesystemcomesfromtheindirecteffectregulatoryinteractionfromauniformdistributionbetween1andthataregulatorhasonitself:aslightincreaseintheconcentrationΩmax¼1000.ThischoiceofΩmaxismotivatedbythefactthatofaninhibitorfromitssteady-statevaluewillreducethegeneTFshavebeenshownexperimentallytochangetargetproteincopynumberandhencemRNAlevelsoftheregulatedgene.Thelevelsby100–1000fold13.mRNAsoftheinhibitorthereforemakeupalargerfractionoftheWefindthatwiththerealnetwork,thesystemalwaystotalmRNAinthecell.SinceallmRNAscompeteforthesharedconvergestoastablefixed-pointregardlessoftheregulationpoolofribosomes,thisinturncausestheinhibitorconcentrationsstrengths(Fig.4a).Incontrast,fortherandomlyconstructedtoincreasefurther.Thispositivefeedbackalsoexistsintheothernetworks(bothwithandwithoutkeepingqfixed),theprobabilityphases,althoughitsphysicaloriginmaybedifferent(seeofthesystembecomingunstabledrasticallyincreaseswhentheSupplementaryNote4,SupplementaryFig.2b).interactionsbecometoostrong(Fig.4a).ThislossofastablefixedIndeed,bynumericallyconstructingmultipleinstancesofapointcangiverisetoeitheranoscillatory(Fig.4b)orchaoticbipartitenetworkandvaryingthefractionofinhibitoryinterac-behavior(Fig.4c).ThissuggeststhatfortypicalregulationtionsPneg,wefindthatλM,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(ofwhichinteractionsρ¼u0:0011,andratioofpositive(activating)to2944areactivating)(Fig.5a).Incomparison,thescramblingN2negative(inhibitory)regulation.methodthatmaintainedboththenumberofTFsandthetotalWefirstexploredtwodifferentwaysofscramblingtheoriginalnumberofinteractionsgivesasmallernumberofself-interactionsnetwork:(1)randomlychoosingudirectedconnectionsoutofthe(〈us〉=2.5)andalargernumberofdirectTF-otherTFN(N−1)possibleconnections,and(2)fixingthenumberofTFsinteractions(〈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)ofthesystemhavingastablefixedpointdecreasesforscramblednetworksofthesameinteractiondensityρ=0.0011,regardlessofwhetherthenumberoftranscriptionfactors(TFs)q=211iskeptfixed(yellowcircles)ornot(redsquares).However,scramblingthenetworkwhilemaintainingthesamenumberofTF-otherTF,TF-nonTF,andselfinteractionscansignificantlyenhancetheprobabilityofthesystemisstable(greentriangles).Eachofthedatapointsrepresentsanaverageover15setsof10regulatorynetworks,witherrorbarsindicatingtheinterquartilerange.[Otherparameters:h=2].bAtypicalexampleofoscillatorydynamicsinproteinconcentrationscwhenthesystemnolongerhasastablefixedpoint.[Parameters:Ωmax¼1585,h=2].cAnexampleofthesystemgoingunstableandexhibitingchaoticbehaviorwhentherealnetworkisscrambledattimet=5×106markedbythedashedverticalline.[Parameters:Ωmax¼1000,h=5].Inboth(b)and(c),timetisinunitsof1/kp.downregulationstobeequallylikely,arandomnetworkisalmost(a)(b)1alwaysstablewhenthedensityofTF-otherTFinteractionsρ¼=134(42+)qutissufficientlylow(Fig.5b).Abovethisthresholdvalueofρ,=373qðq1Þq0.8(201+)theprobabilityofthesystemnotexhibitingastablesteady-stateTFs0.6increaseswithρq(Fig.5b).Thiseffectisobservedregardlessofthe=5148numberofself-interactionsorwhetheruniskeptfixed(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-nonTFmorefine-tuned.interactions(ofwhich2944ofthemareactivating).ThetotalnumberofThephenomenonthatasmallρqpromotesstabilityisinteractionsisgivenbyu.bArandomlyconstructednetworkisalmostconsistentwiththestabilityofbipartitenetworks(ρq=0)andalwaysstablewhenρqissufficientlylow.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,wefindnumericallyN=2274,q=211,h=2,Ωmax¼1000.].thatthepresenceofT2canaffectλM;rmax(SupplementaryNote7,SupplementaryFig.4),suggestingthattheindirectcouplingbetweenproteinscanalsoplayaroleininfluencingthestabilityofthesystem.scramblingmethodwiththecompositionoftheinteractionskeptfixed.Inparticular,aftersettingthefirstq=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.Wefindthatthisscramblingprocedure,affectwhetherthesystemisstable.Therelaxationratesarealsowhichfixesthecompositionofregulatoryinteractions(inindependentofτmandτp,withtherelaxationrateintheabsenceadditiontoN,q,andρ),significantlyincreasestheprobabilityofinteractionsgivenby(fromEq.(11)):ofthenetworkhavingastablefixedpoint(Fig.4a).ssβ0¼kpcr:ð21ÞDirectinteractionsamongTFscaneitherbeauto-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=0.4̃)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~J¼J:ð22Þspectrumfordifferentdegradationratesτ.WhenmRNAandproteinβ0degradationratesarecomparable,alleigenvaluesfallwithinacircularregion(red).Ontheotherhand,whenτm≪τp,theeigenvaluespectrumForanarbitraryregulatorynetworkwithacorrespondingapproximatelyresemblestwocircularregions,onecorrespondingtotheinteractionmatrixM(Eq.(12)),wefindthattheeigenvalues~λof~JdynamicsofmRNAsandoneforthatofproteins.Inthislimit,increasingaregivenby(seeSupplementaryNote3):mRNAdegradationrateonlyshiftstheeigenvaluesforthemRNAsectorto1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimorenegativevalues,leavingthemaximalrealpartoftheeigenvalues~λ¼ω±ω2þ4λMð1þωÞ1;ð23Þ2approximatelyunchanged,ω=0.5(green)vsω=0.4(blue).whereλMaretheeigenvaluesofMasbefore,andωisaTheexpressionfor~λ(Eq.(23))impliesthatwhenthesystemisdimensionlessquantitygivenby:stable(λM;r<1),therateatwhichthesystemrelaxestosteady-1maxω¼;ð24Þstateinitiallyincreasesasωincreasesfrom−1,buteventuallyτβ0plateauoff−intheω→∞limit(whereτm≪τp),~λ!λM1whichreflectsthedifferencebetweenmRNAandprotein(Eq.(23),Fig.6a).Thisimpliesthatthereissomebenefitto111havingfastmRNAdegradationintermsofresponsetimes,butdegradationrates¼.ττmτponcemRNAdegradesmuchfasterthanproteins,furtherSinceonaveragecellvolumeincreasesexponentiallywithrateincreasingmRNAdegradationratenolongeraffectstheresponse(seeEq.(8)):timeofthesystem.Theeigenvaluespectruminthisτm≪τplimit1appearstoconsistoftwocircularregions,oneforthedynamicsofμ¼kpϕr;ð25ÞτmRNAsandtheotherforthatofproteins(Fig.6b),reminiscentpoftheRMT’scircularlaw.Increasingτmonlyshiftstheagrowingcellhastosatisfythecondition1<1.Therefore,τpkpϕreigenvaluescorrespondingtothemRNAsectorandhencedoessinceτm≥0,wehaveω≥−1.Theexpressionfor~λ(Eq.(23))notaffect~λrmax.Thisisconsistentwiththefactthatwhenτm≪τp,thereforeimpliesthatthesystemisstableifandonlyifλM;r≤1,thedynamicsoftheoverallsystemisgovernedonlybythemaxregardlessofthevalueofτmandτp(Fig.6a).Wefindthatdespiteproteinsector(Eq.(11)).Therefore,theslowestrelaxationratedifferencesinthedetailsofthemodel,thisconclusionstillholdsbacktosteady-statelevelsdependsonlyonMandincreasingintheotherphases(seeSupplementaryNote3).mRNAdegradationratenolongerimprovestheresponsetime.Therefore,unlikewhathasbeenarguedintheliteratureandwhatonemightexpectfromRMT,changingmRNAnorproteinDiscussiondegradationrateshasnoeffectonwhethertheoverallsystemisInsystemswithalargenumberofinteractingcomponents,thestable.Ifsteady-stateproteinconcentrationsareunstablebecausequestionofstabilityisoftenanimportantone,asresultsfromλM;ristoolarge(e.g.,wheninteractionsaretoostrong),maxRMTpredictinstabilitywhenthesystemsizeNbecomestoolargeincreasingmRNAorproteindegradationratescanneverhelptoorinteractionsbecometoostrong.Inthecontextofgenestabilizethesystem.expression,transcriptionalregulationiscrucialforcellstoadaptImportantly,thisfindingalsoimpliesthatourresultsforhowtodifferentenvironmentalconditionsbychangingtheirgenestructuralfeaturesofthetranscriptionnetworkaffectsstabilityexpressionlevels.ItisthereforeimportantfortranscriptionalholdsoutsidetheregimeoffastmRNAdegradation,sinceregulatorynetworks(TRNs)tobeabletoaccommodatealargestabilityonlydependsonM.numberofregulatoryinteractionswithoutthesystemgoingunstable.However,wepfindherethatsimilartotheintuitionffiffiffiffiIncreasingmRNAdegradationratecanimproveresponsetimes,providedbyRMT,λNforafullyrandomregulationnet-butonlyuptosomelimit.Besidessystemstability,anotherwork,suggestingthatthesystemwillgounstableasthenumberquantityofbiologicalinterestistheresponsetimeofthesystemtoofgenesexceedsathreshold.Infact,basedontypicalvaluesforperturbations,whichisespeciallyrelevantforcellsexperiencingthedensityofactualregulatorynetworksandinteractionchangesinnutrientconditions32,33.Sincethisrelaxationtimescalestrengths,wefindthatthesystemhasahighprobabilityofbeingisdeterminedbytheslowesteigenvalueoftheJacobianmatrix,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.Whilethestabilityofthesystemmayhaveplayedaroleinshapingcurrentpropertiesofthesespecificmotifshavebeenwidelystudiedandexistingregulatorynetworksthroughtheevolutionaryshowntobeimportantforspecificfunctionssuchasadaptation,process.Ourapproachcanthereforeprovideinsightsintotherobustness,andfastresponsetoenvironmentalchanges1,29,30,designandevolutionaryconstraintsforafunctionalregulatoryhowtheycontributetotheoverallstabilityofthenetworkislessnetwork,whichmaypotentiallybeusefulforguidingthecon-clear.Wefindherethatglobalstructuralfeaturesofthenetwork,structionofsyntheticgeneticcircuits36–38.Inthefuture,thewhicharefundamentallyshapedbymanyofthesemotifs,canabilitytoexperimentallyengineeralarge,randomregulatoryplayahugeroleindeterminingthestabilityofthesystem.Incircuitwithincellscouldalsoallowtestingoftheresultswehaveparticular,giventhesamenumberofproteins,TFs,interactiondescribed.density,andregulationstrengths,anetworkthatresemblesaInadditiontotranscriptionalregulation,geneexpressionisbipartitegraphwithalowerdensityofTF-otherTFinteractionsρqalsoregulatedatthepost-transcriptional(e.g.,throughsmall-hasahigherchanceofbeingstable.Thesignificanceofρqfun-RNAsormicro-RNAs)andpost-translational(e.g.,throughdamentallyarisesbecauseoftwomainfactors:(i)theeigenvaluespost-translationalmodifications)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,thepresenceoftheseoutgoinghubshasbeenshowntosignificantlyincreasetheprobabilityofthesystemreachingaReportingsummary.FurtherinformationonresearchdesignisavailableintheNaturestabletargetphenotypewhentheinteractionstrengthsareResearchReportingSummarylinkedtothisarticle.allowedtovarywhilethenetworktopologyiskeptfixed34.Here,wefindthathavingalowρqcanalreadysignificantlystabilizetheDataavailabilitysystemwithouttheneedtocontrolthedegreedistributions.TheE.coli.transcriptionalregulatorynetworkdatathatsupportthefindingsofthisstudyNevertheless,havingjustafewmasterregulatorsmaycontributeisavailableinthesupplementaryfilesofthepaper(ref.31):https://doi.org/10.1073/pnas.1702581114.ThisdatausedforanalysisisalsoavailableinaMATLABdatafileontothenetworkhavingalowρqifforinstancemostofthereg-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,wefindthatthisisnotthecase.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).actionsmediatedthroughcompet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