Synchronization_of_a_genetic_oscillator_with_the_c

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NewJ.Phys.24(2022)033050https://doi.org/10.1088/1367-2630/ac5c16PAPERSynchronizationofageneticoscillatorwiththecelldivisionOPENACCESScycleRECEIVED23December20211,∗1,21,∗GabrielKnotz,UlrichParlitzandStefanKlumppREVISED1InstitutefortheDynamicsofComplexSystems,UniversityofGottingen,Friedrich-Hund-Platz1,37077G¨ottingen,Germany¨23February20222MaxPlanckInstituteforDynamicsandSelf-Organization,AmFaßberg17,37077Gottingen,Germany¨ACCEPTEDFORPUBLICATION∗Authorstowhomanycorrespondenceshouldbeaddressed.9March2022E-mail:g.knotz@theorie.physik.uni-goettingen.deandstefan.klumpp@phys.uni-goettingen.dePUBLISHED31March2022Keywords:geneticcircuits,repressilator,synchronization,cellcycleOriginalcontentfromthisworkmaybeusedunderthetermsoftheAbstractCreativeCommonsAttribution4.0licence.GeneticcircuitsthatcontrolspecificcellularfunctionsareneverfullyinsulatedagainstinfluencesAnyfurtherdistributionofotherpartsofthecell.Forexample,theyaresubjecttoperiodicmodulationbythecellcycleofthisworkmustmaintainattributiontothroughvolumegrowthandgenedoubling.Toinvestigatepossibleeffectsofthecellcycleontheauthor(s)andtheoscillatorygenecircuitsdynamics,wemodelledasimplesyntheticgeneticoscillator,thetitleofthework,journalcitationandDOI.repressilator,andstudiedhallmarksoftheresultingnonlineardynamics.WefoundthattherepressilatorcoupledtothecellcycleshowstypicalquasiperiodicmotionwithdiscreteFourierspectraandwindowsinparameterspacewithsynchronizationofthetwooscillators,withadevil’sstaircaseindicatingtheArnoldtonguesofsynchronization.Inthecaseofidenticalparametersforthethreegenesoftherepressilatorandsimultaneousgeneduplication,weidentifytwoclassesofsynchronizationwindows,symmetricandasymmetric,dependingonwhetherthetrajectoriessatisfyadiscretethree-foldrotationsymmetry,correspondingtocyclicpermutationofthethreegenes.UnexpectedlychangingthegenedoublingtimerevealedthatthewidthoftheArnoldtonguesisconnectedtothatthree-foldsymmetryofthesynchronizationtrajectories:non-simultaneousgeneduplicationincreasesthewidthofasymmetricsynchronizationregions,forsomeofthembyanorderofmagnitude.Bycontrast,thereisonlyasmallorevenanegativeeffectonthewindowsizeforsymmetricsynchronization.Thisobservationpointstoacontrolmechanismofsynchronizationviathelocationofthegenesonthechromosome.1.IntroductionGeneticcircuitscontrolmanyaspectsofthebehaviorofacellsuchastheadaptationofmetabolismtoavailablenutrientsandtheresponsetoexternalstressesaswellasdevelopmentalprograms[1,2].Inmanycases,thesecircuits,whicharebasedonproductsofgenescontrollingtheactivitiesofothergenes,canbeunderstoodasdeterministicorstochasticdynamicalsystems[3].Particularlysimpledynamicalsystemsdisplaying,forexample,bistability[4]oroscillations[5–7]havebeendesignedinsyntheticcircuits,butoftensimilarcircuitdesignsalsoformthecoreofmorecomplexnativecircuits.However,onecomplicationinunderstandinggeneticcircuitsassimpledynamicalsystemsisthattheyarenevertrulyisolatedsystems,butrathercoupledtoprocessesgoingoninthebackgroundofthecell.Thus,acellisnotaninert‘chassis’forageneticcircuit,butitsphysiologicalstatemayaffectthedynamicsofthatcircuitinunexpectedways.Onecasewherethiscouplinghasbeenstudiedinsomedetailisthegrowthratedependenceofgeneexpressioninbacteria[8],asinmanycasesthephysiologicalstateofabacteriumcanbecharacterizedbythegrowthrate[9–12].Mostinterestingly,growtheffectscanmediatefeedbacks,whichmayresultinbistabilitythatisabsentwithoutsuchcoupling[8,13,14].Anotherbackgroundprocesstowhichgeneticregulatorydynamicsisunavoidablycoupledisthecelldivisioncycle[15–17].Again,thisisparticularlysimpleinbacteria,wherethecellcycleaffectstheconcentrationofanyproteinviadilutionbyvolumegrowthandviathedoublingofthegeneencodingthatprotein.Thesetwoeffectsresultinaperiodicmodulationoftheprotein’sconcentration,whichmaybe©2022TheAuthor(s).PublishedbyIOPPublishingLtdonbehalfoftheInstituteofPhysicsandDeutschePhysikalischeGesellschaft

1NewJ.Phys.24(2022)033050GKnotzetalamplifiedbygeneregulation[18].Aperiodicmodulationoftheconcentrationbythecellcyclehasbeenobservedinyeast[19]andEscherichiacoli[20].Ifgenesarereplicatedatdifferenttimesinthedivisioncycle,theresultingimbalanceinstoichiometryofthecorrespondingproteinscantriggersignaling[17]asproposedfortheinductionofsporulationinBacillussubtilis[21].Sincethedivisioncycleisaperiodicperturbationofthedynamicsofgeneticcircuits,itsinfluenceoncircuitswithoscillatorydynamicsisofparticularinterest,asinteractionoftwooscillatorsmayresultinsynchronization[22,23].Ifanoscillatingcircuitsynchronizestothecellcycle,thiscouldbeconsideredastrongperturbation,specificallyiftheoscillator’speriodisrelevantforitsfunction,raisingthequestionwhetheroscillationsarerobustagainsttheperturbationbythedivisioncycle.Ontheotherhand,insomesituationssynchronizationmaybefunctionallydesirable,thereareforexamplereportsthatthemammaliancircadianclockandthecelldivisioncyclearesynchronized[24].Thesimplestgeneticoscillatoristherepressilator,asyntheticbacterialgeneticcircuitofthreetranscriptionfactorswithcyclicrepression[5,7].Inanearlymodelforcellcycleeffectsitwasfoundthattherepressilatorisonlymildlyaffectedbythedivisioncyclewithamodulationoftheamplitude[18].Paijmansetal[25]latershowedthatsynchronizationoftherepressilatorwiththedivisioncyclecanbeobserved,howeveronlyinsmallparameterwindows.Thesesynchronizationwindowsbecamebiggerwhenthegenesareduplicatedatdifferenttimes[25].Moreover,anotherdesignofageneticoscillator[6]showedmorepronouncedsynchronizationwiththedivisioncycle[25].Anexperimentaltestofsynchronizationbetweenageneticoscillatorandthebacterialcelldivisioncycledidnotshowsynchronization,unlessabackcouplingwasintroducedviaacontrolofcellcycleproteinbytheoscillatortoobtainmutualentrainment[26].Herewerevisitsynchronizationoftherepressilatorwiththecelldivisioncyclebyanalyzingthesynchronizationwindowsinmoredetail.Wefindthatinthesymmetriccasewherethethreeproteinsarecharacterizedbythesameparametersandthesamegeneduplicationtimes,therearetwotypesofsynchronizationwindowsthatreflectthecyclicsymmetryofthesystemindifferentways.Moreover,whenthesymmetrybetweenthethreeproteinsisbrokenbyduplicatingthegenesatdifferenttimesinthecelldivisioncycle,thetwotypesofsynchronizationwindowsshowdifferentbehavior:largeincreasesofthewidthofthesynchronizationwindowsareonlyseeninthecaseofasymmetricsynchronization.Thisobservationsuggestsarelationbetweenadiscretesymmetryofthetrajectoriesandtheefficiencyofsynchronizationasreflectedinthewidthofthesynchronizationregions.Fromthepointofviewofoptimizingsynchronization,maximizingthewidthofthesynchronizationwindowisoneofseveralpossibletargetfeatures[27](othersincludethespeedofentrainment[28]orphaseresetting[29,30]).Ourresultspointtowardsanintriguingconnectionwiththesymmetrypropertiesofthetrajectoriesandsuggestthatthetimingofgeneduplicationcanplayaroleinsuchoptimizationforgeneticoscillators.2.Cellcycledependentrepressilator2.1.RepressilatorwithoutexplicitcouplingtocellcycleTherepressilatorisasyntheticgenenetworkofthreetranscriptionfactorswithcircularrepressionasshowninfigure1(a).Westartfromadescriptionoftheproductionanddegradationofasingleproteinexpressedfromanunregulatedgene.ThecopynumberofthatproteinpercellisdenotedbyP.ItsdynamicsisgivenbyP˙=α·g−β·P(1)withthesynthesisanddegradationratesαandβ,respectively,andthegenecopynumberg.ThisequationcanbederivedfromasystemofequationsdescribingthecopynumbersoftheproteinandthecorrespondingmRNA(alsousedinearlierstudiesoftherepressilator[5,25])bytakingthemRNAdynamicstobeinitssteadystate,asmRNAdynamicsistypicallyfasterthanproteindynamicsduetotheshortlifetimeofmRNA[2,15].Atcelldivision,whichwewillconsiderbelow,theproteincopynumberishalved.Nextweneedtomodeltherepressionofproteinproduction.AcommonapproachistodescriberepressionbyaHillfunction1R([P])=n,(2)1+[P]Kwhichmultiplicativelymodulatesthesynthesisrate.R([P])describestheactivityoftherepressedgeneasafunctionoftherepressorconcentration[P]inrelationtoareferenceconcentrationKandtheHillcoefficientn.BecausetheHillfunctiondependsontheconcentrationoftherepressor,[P]=P/Vwithcell2

2NewJ.Phys.24(2022)033050GKnotzetalFigure1.(a)Schematicdiagramoftherepressilator.ThethreeproteinsA,BandCrepresseachotherinacyclicmanner.(b)Stabilitydiagramofthefreerepressilator.Inthestableregimethefixedpointoftherepressilatorisstable.Byincreasingtheα[g]Hillcoefficientnorthedimensionlessquantityc=thefixedpointbecomesunstableandtrajectoriesconvergetowardsaβeffKstablelimitcycle.volumeV,werewriteequation(1)asdPP˙V˙P[P˙]==−.(3)dtVVVVThisexpressionshowsthatequationsforconcentrationshavethesameformasthecorrespondingequationforproteincopynumberwithtwomodifications:thegenecopynumbergisreplacedbythegeneconcentration[g]=g/V(alternatively,afactorV−1isoftenabsorbedintoα,whichthenbecomesasynthesisratepervolume)andthedegradationrateβisincreasedbythedilutiontermV˙/V.Importantly,concentrationsarenotaffectedbycelldivision,asincelldivisionbothvolumeandproteincopynumbersarehalved,sotheconcentrationiscontinuousatthetimesofcelldivision[15].Ifthecellcycleisnotconsideredexplicitly,onecanapproximatethevolumeastimeindependentandtakecellcycleeffectsintoaccountinanimplicitfashionthroughthetwomodificationsoftheequationdiscussedabove.Thishasbeendoneinmanystudiesofgeneregulatorysystemsandwillbeusedasareferencesystemhere(freerepressilatorwithimplicitcelldivision).Inthatcase,thedynamicsoftherepressilatorwithitscyclicrepressionofthreegenescanbeexpressedbythefollowingsystemofcoupleddifferentialequations[P˙1]=α1·R([P3])·g−βeff,1[P1][P˙2]=α2·R([P1])·g−βeff,2[P2](4)[P˙3]=α3·R([P2])·g−βeff,3[P3]withtheconstantaveragegeneconcentration[g]andeffectivedegradationratesβeff,i=βi+V˙/V(fori=1,2,3)thatincludethedilutiontermwiththeaveragevolumegrowthrateV˙/V(whichbelowwewillspecifyforexponentialgrowthasV˙/V=ln2/T).Forsimplicitywewillconsiderthesymmetricsystemwithα1=α2=α3=αandβ1=β2=β3=β(and,thus,βeff,i=βeff=β+V˙/V)andonlyintroduceanasymmetryinthegenereplicationtimestowardstheendofthepaper.Therepressilatorasdefinedbyequation(4)isadissipativesystemwithasinglefixedpoint.Inthesymmetriccasethatfixedpointisgivenby[P1]=[P2]=[P3]=pKwherep>0isdeterminedbytheimplicitequationp=c/(1+pn)withthedimensionlessconstantc=α[g],whichdescribestheratiooftheβeffKconcentrationobtainedbyunregulatedgeneexpression(α[g]/βeff)andthethresholdconcentrationforrepression,K.Standardlinearstabilityanalysisshows[31]thatthefixedpointisstableforsmallcandbecomesunstablethroughaHopfbifurcationatacriticalvalueofc,whichdependsontheHillcoefficientnandforlargenis≈1.Intheunstableregime,therepressilatordevelopsstablerelaxationoscillations.Amapofastableandunstableregimeoftheoscillatorisshowninfigure1(b).Typicaltrajectoriesofthethreeproteinsintheoscillatingregimeareshowninfigure2andwillbecomparedtothecase,wheretherepressilatorisexplicitlycoupledtothecellcyclebelow.3

3NewJ.Phys.24(2022)033050GKnotzetalFigure2.Trajectoriesoftherepressilator(a)and(c)withoutand(b)and(d)withexplicitcouplingtothecellcyclewithT=60min.Thegreyverticallinesin(d)indicatethetimesofcelldivisionand,thus,theperiodicdrivingoftherepressilator.In(a)and(c),cellcycleeffectsareincludedimplicitlythroughaconstantaveragegeneconcentrationg=1√1andaneffectiveV02ln2ln2degradationrateβeff=β+thatincludesdilutionduetovolumegrowth.In(a)and(b)fordifferentβandin(c)and(d)Twithβ=0.2min−1.Thefreerepressilatorintheoscillatoryregimedevelopsstableoscillationswhereasthetrajectoriesofthedrivenrepressilatortypicallycoversabroaderregioninthephasespace,forexampleforβ=0.2and0.02min−1(blueandorangein(b).However,onecangetstableperiodictrajectoriesaswell,e.g.forβ=0.01min−1(green).2.2.CellcycleeffectsThecellcycleprovidesaperiodicmodulationofparametersoftherepressilatorandthusanexternaldrivingforcefortheoscillator.Toimplementthecellcycleintoourmodelexplicitly,weintroducetwoeffects.Thefirstoneisthevolumegrowthanddivisionofthecell.Thecellvolumegrowsexponentially[32]andafteraperiodofthecellcycleoftimeT,thecellistwiceasbigasatthebeginningofthecellcycle.Atthattime,thecelldividesandthevolumeissetbacktoitsinitialvalue.Asasideremark,wenotethatthedifferencebetweenlinearandexponentialvolumegrowthisrathersmallforsuchamodelforproteinsynthesis[15].Tosimplifytheexpressionsforcellcycle-dependentparameters,weintroduceasecondtimevariableτ=tmodT,theageofthecell,whichmeasuresthetimesincethelastdivision.DivisionsaretakentooccurattimesnTwithintegern,correspondingtoτ=0.Thesecondeffectisthegenedoubling.Attimesτ=td,allthreegenesoftherepressilatorareduplicatedsimultaneously.Thisisagoodassumptionifthegenesareclosetogetheronthechromosome,buttheremaybesignificantimpactifthatisnotthecase,asshowninreference[25]anddiscussedfurtherinsection3.3.Atcelldivision,τ=0,thecopynumberofallgenesisreducedagaintoasinglecopy.Likeforproteins,theconcentrationsofthegenesarecontinuousattimesofdivision.Allinallwegetthefollowingequationsforthecellcycleeffectsτln2V(t)=V0exp(5)T10τ

4NewJ.Phys.24(2022)033050GKnotzetalwhereboththecell’svolumeVandthegenecopynumbergdependontimeviathecell’sageτ=tmodT.Asaconsequence,theequationforthedynamicsoftheconcentrationofanunregulatedproteintakestheformln2[P˙]=αg(t)−β+[P].(7)TImportantly,thegeneconcentration⎧⎪⎪1τln2⎨exp−0τ

5NewJ.Phys.24(2022)033050GKnotzetalFigure3.(a)Analyticsignalfortherepressilatorwithβ=0.011min−1,T=60min.(b)AverageperiodoftherepressilatordeterminedviatheHilberttransformwithβ=0.0215min−1formultiplecelldivisiontimesT.Onenoticesthatthereareregionswheretheperiodfollowsstraightlinesstartingfromtheorigin.Figure3(b)showstheperiodmeasuredinthiswayasafunctionofthecellcycletimeT.Superimposedonanoverallincrease,weobtaincharacteristicwindows,inwhichthedependenceonTislinear,indicatingsynchronizationoftherepressilatorwiththecellcycle,forexamplearoundT=400minandT=150min,inagreementwithearlierresults[25].Themostprominentonesamongthoselinearsegmentshavesimpleintegerorrationalslopesasindicatedbythedashedlinesinfigure3(b).WenotethattheHilberttransformisnottheonlyoptionfordeterminingthephase.Forexample,aphasecanalsobeextractedbyprojectingthedynamicsinthethree-dimensionalspaceofthethreeconcentrationsontoasuitabletwo-dimensionalsubspaceinwhichagainacircle-likeorbitisobtained(weusedtheprojectiony1=−2[P1]+[P2]+[P3],y2=[P2]−[P3]andobtainedthesameresults).TheadvantageoftheHilberttransformisthatitonlyneedsthetimeseriesofoneofthevariables,soitcanalsobeappliedtoexperimentaldataifonlyoneoftheconcentrationsisobservedwithafluorescencereporter.WefurthernotethattheperioddeterminedviatheHilberttransformagreeswiththedominantperiodobtainedfromthepowerspectrum(seebelow),whiletheaveragepeak-to-peaktime,whichwasusedinreference[25],systematicallyunderestimatestheperiod,asshownintheappendixA.Finally,todetermineonlytheperiodwithoutthephase,onecanalsousethezerocrossingsofashiftedtimeseriesandgetthesameresultsaswiththeHilberttransform(seeappendixA).Figure4(a)showsafullpowerspectrumoftheoscillation.Forcomparison,thepowerspectrumofthefreerepressilatorisalsoincluded.Someprominentpeaksofthedrivenspectrumareseentocoincidebetweenthetwospectra.Thepositionofthedominantpeakscanbeapproximatedbyω≈|n·ωcell+m·ωfree|(12)withintegersn,masindicatedinfigure4(a)bythedashedverticallines.Whilethisexpressionisnotexact,itprovidesaratheraccurateapproximationofthepositionofthepeaks.Thetwofrequenciesinthisapproximation,ωcellandωfree,correspondtothetwouncoupledoscillators,i.e.thecellcycleandthefreerepressilator(withimplicitcelldivision).Moreprecisely,ωfreeisthefrequencyofanfreerepressilatorwithTtheaveragegeneconcentrationg=1g=1√1.SuchabehaviouristypicalforT0V02log2quasi-periodicmotion[22].Thepowerspectrumalsoprovidesaquantitativedescriptionofthemodulatedoscillationsobservedinearlierstudies[18].Thehighestpeakofthespectrumisseentobetheonewiththelowestfrequencyamongthosepeakswithoutacontributionfromthecellcyclefrequency,i.e.thepeakwithn=0,m=1inequation(12).Itsfrequency(whichisapproximately,butnotexactlythatofthefreerepressilator)agreeswiththefrequencyobtainedfromtheHilberttransform.Figure4(a)comparesthepowerspectraofthedrivenandfreerepressilator.Thedominantpeaksofthetwospectracoincide.Thisobservationsuggestsalreadythattheinfluenceofthecellcycleonthefrequencyoftherepressilatorissmallenoughthatitcanoftenbeneglected.Indeed,ifwedirectlymeasuretheperiodofthedrivenandfreerepressilatorandcomparethem(figure4(b)),thedifferenceisseentobesmall,apartfromcertainareasinparameterspace,wherethedrivenrepressilatorspeedsupandslowsdownquiterapidly.6

6NewJ.Phys.24(2022)033050GKnotzetalFigure4.(a)Apowerspectrumofarepressilatorwithβ=0.011min−1,T=60min.Additionallythedashedverticallinescorrespondtofrequencies|n2π+m2π|withTtheperiodofacorrespondingfreerepressilator.InthegraphtheyarelabeledTTfreefreeas[n,m].Thereddashedlineisthepowerspectrumofthecorrespondingfreerepressilator.(b)Thedifferencebetweentheperioddeterminedforthedrivenandthefreerepressilator.Overallthedifferencebetweenbothoscillatorsisrathersmall,apartfromareaswherethedrivenrepressilatorspeedsupandslowsdownagainrapidly.Afterthatthedifferenceconvergesforincreasingβtoasmallervalue.3.2.TwotypesofsynchronizationTofurtheranalyzethedynamicsinthewindowsinwhichtheobservedperiodshowsalineardependenceonthedivisiontimeT,weplottheperiod(figure5(a))togetherwithanorbitdiagraminfigure5(b).Inthisdiagram,wedisplaythevaluestakenbytheconcentration[P1(nT)]atintegermultiplesofthecelldivisiontimeasafunctionofthedivisiontimeT.SynchronizationwiththecelldivisioncycleatagivenT(andthusperiodicdynamicsofthefullsystem)isseenasadiscretesetofvaluesthatrepeatafterafinitenumberofcycles,whilewithoutsynchronization(quasiperiodicdynamicsofthefullsystem)acontinuousdistributionofvaluesisobtained.Thecomparisonrevealsthatthetrajectoriesoftheoscillatorbecomeperiodicinthewindowswiththelineardependenceoftheperiodonthedivisiontime.Fortheparametersusedhere(witharealisticallysmalldegradationrateβ=0.01min−1),thebiggestintervalwithperiodicbehaviorisseenaroundT=352–375min,whichcorrespondstoveryslowcellgrowth,buttwosmallerwindowsaroundT=60minandT=130minareintherangeaccessibleforthegrowthofE.coliwithcommongrowthmedia.Asobservedabove,inthewindowswithlineardependence,theperiodoftherepressilatorislockedtorationalmultiplesofthecelldivisiontimeT.Forthethreementionedwindows,thedependenciesare3T,6Tand9T,respectively.AnothersmallwindownearT=215minshowsa3Tdependence(insetin2figure5(a)).TheobservedlockingoftherepressilatorperiodtothecellcycletogetherwiththeperiodicorbitsseeninthePoincaremapsindicatethesynchronizationoftherepressilatorwiththecellcycle.Anotherhallmarkofsynchronizationcanbeobtainedfromthephaseφ(t)extractedfromtheHilberttransform.Bycalculatingthewindingnumberφ(n·T)−φ(0)Ω=lim(13)n→∞noneobtainsareasofsynchronizationsinparameterspace,alsocalledArnoldtongues,asareasofconstantwindingnumber.Intheplotofthewindingnumber(figure5(c)),theseareasappearasintervalsofconstantwindingnumberthatcorrespondtothewindowsoflineardependenceinfigure5(a)andthewindowsofperiodicdynamicsintheorbitdiagram(figure5(b)).Thevaluesofthewindingnumberintheseintervalsaregivenbytheinverseoftheslopeofthelineardependenceoftheperiodonthedivisiontime.Thewindingnumber,whichindicatestheratioofthedrivenrepressilatorfrequencytothedrivingfrequencyofthecellcycle,allowsforageometricinterpretationofsynchronization[22,33]:itindicatesthenumberoffullturnsofoneoscillatorinoneperiodoftheotheroscillator.Anintegerorrationalvalueasinthesynchronizationwindowsthusdistinguishesperiodicfromquasiperiodicmotionofthecoupledsystem.Astheorbitoftwooscillatorsislocatedonatorus,theformercaseischaracterizedbyaclosedorbitonthattorus,whileinthelattercase,theorbit‘fills’thewholetorus[22,33].Figure6showstrajectoriesinthesynchronizationwindowsandalsoindicatesthetimesofgenedoublingandcelldivision.Basedonthesetrajectories,wedistinguishtwotypesofsynchronization,towhichwereferassymmetricandasymmetric.Forthesymmetriccase,thestabletrajectoryisinvariantundercyclicpermutationoftheproteins,so(A,B,C)→(B,C,A)→(C,A,B)→(A,B,C).(14)7

7NewJ.Phys.24(2022)033050GKnotzetalFigure5.Multiplegraphsforarepressilatorwithβ=0.01min−1forchangingcelldivisiontimesT.In(a)theperioddeterminedviatheHilberttransform,in(b)anorbitdiagramwiththecurrentproteinconcentrationattimesn·Twithn∈Nandin(c)thewindingnumbercalculatedfromthephaseoftheHilberttransform.Theinsetsin(b)showaperiod9(blue)andperiod6(red)window.Notethatsomevaluesarebarelydistinguishableinthisvisualization.Themostimportantfeatureisthatareasofsynchronizationsin(a)correspondtoperiodicwindowsin(b)andtoplateausin(c).ThevalueofthewindingnumberinArnoldtonguesisthesameastheinverseslopeofthesynchronizationstraightsin(a).Thisisequivalenttoathree-folddiscreterotationsymmetryofthetrajectoryaroundthe(1,1,1)axis.Reflectingthissymmetry,inthecorrespondingsynchronizationwindowsthelinearrelationbetweentheperiodoftherepressilatorandthecelldivisiontimetakestheformT=3nTwithcoprimepositiverepmintegersn,m.Thelargestsychronizationwindowsdiscussedabove(andindicatedinfigure5)areofthistype.Allsynchronizationwindowsthatdonotshowthispermutationordiscreterotationsymmetryarereferredtoasasymmetric.AnexampleisthewindowwithTrep=1T(figure6(a)).3.3.Synchronizationfornon-simultaneousduplicationofthethreegenesPajimansetalobservedthatsynchronizationoftherepressilatorwiththecellcyclebecomesmorepronouncedwhenthegenesarenotduplicatedatthesametime[25].Duplicationatdifferenttimesinthedivisioncycleisobtainedifthethreegenesareindifferentlocationsonthechromosome.Indeed,ifweshiftthegenedoublingtimeofproteinArelativetotheothertwoproteinsBandC,someofthemostpronouncedsynchronizationwindowsgetbiggerandmanysmallwindowsbecomemoreapparentintheorbitdiagrambasedonthePoincar´emap(figure7(a),varyingT)aswellasintheplotofthefrequencyasafunctionofthedegradationrateβ(figure7(b)).Thisisshowninfigure7(b),wherewecomparethecasewhereallthreegenesareduplicatedatτ=td=T/2withthecasewheregeneAisduplicated20minearlier.Hereandinthefollowingfiguresweplotthefrequencyoftherepressilatorasafunctionofthedegradationrateβ(whichmodulatesthefrequencyofthefreerepressilator)ratherthanasafunctionof8

8NewJ.Phys.24(2022)033050GKnotzetalFigure6.Exampletrajectoriesforasymmetric(a),(c)and(e)andsymmetricsynchronizationorbits(b),(d)and(f).Thesynchronization1T(a)and2T(c)arenotsymmetricundercyclicpermutationoftheproteincoordinates,whichcaneasilybeseenbythepositionofthegenedoubling(redcross)orthecelldivision(bluedot)inthesynchronizationtrajectory.Forthe3T9(b)andT(d)orbitthisconditionisfulfilled,eventhoughthismightbenotsoobviousforthelatter.(e)and(f)Showtime4representationsfortheasymmetricorbit2Tandthesymmetricorbit3T,respectively.Here,theverticaldashedlinesindicatethetimesofcelldivision.thecelldivisiontimetoavoidcomplicationswhentheearlierduplicationtimeapproachesthebeginningofthecellcycle.Themostprominentexamplesforsynchronizationbecomingmorepronouncedarethe1Tand2Twindows.However,notallsynchronizationwindowsgetbigger,the3Twindow,forexample,actuallygetssmaller.Thedifferentresponsesofthesynchronizationwindowstotheshiftofthegenedoublingtimeofoneproteinareillustratedinfigure7,whereweplottwoofthesynchronizationwindowsasfunctionoftheshiftintheduplicationtimeforshiftsrangingbetweenthetwoextremesshowninfigures7(c)and(d).The3Twindowisseentodecreaseinsize,whilethe2Twindowincreases.Thetwocasescorrespondtoasymmetricandanasymmetricsynchronizationscenario,respectively,asshownbythetrajectoriesinfigure6.9

9NewJ.Phys.24(2022)033050GKnotzetalFigure7.(a)Thegenedoublingtimetd,AofproteinAisshiftedΔtd=−20minbeforetheothertwoproteins.Theorbitdiagraminthissetupshowsaincreasednumberofvisiblewindowsandthefrequencyofthedrivenrepressilator(b)showsforTmostwindowsincreasedArnoldtonguesaswell.Bluecurve:allthreeproteinsdoubledatthesametimetd=2.Redcurve:proteinA20minbeforetheothertwoproteins.Howevernotallwindowsincreaseinsizewhichcanbeseenin(c)and(d)wherethesizeofthe3Tandthe2Twindowinβspace(indicatedbythearrowsin(b))aredrawnfordifferentshiftsΔtontheproteindA.Forincreasing|Δt|thesizeofthe2Twindowincreasessignificantlyandthe3Twindowdecreases.T=60minin(b)–(d).dTotestwhetherthereisageneralrelationbetweenthetypeofthesynchronizationandtheresponsetotheshiftintheduplicationtime(whichbreaksthesymmetrybetweenthethreeproteins),weconsiderthesizesofthe28synchronizationwindowsthatcanbeidentifiedinfigure7.Wedeterminethewindowsize(inthespaceofdegradationratesβandforT=60min)forsynchronousgenedoublingatt=Tandforad2shiftof20minappliedtothegenedoublingtimeoftheAprotein.Theresultsareshowninfigure8,whereweplotthewindowsizewiththeshiftagainstthatwithouttheshift.Windowsofasymmetricsynchronizationareshownascircles,thosecorrespondingtosymmetricsynchronizationascrosses.Thedashedlineindicatesunchangedwindowsize.Strikingly,allwindowscorrespondingtoasymmetricsynchronizationareabovethedashedlineandthusgetbiggerupontheshift,someofthemincreasebymorethanafactor10.Bycontrast,thewindowscorrespondingtosymmetricsynchronizationlieverynearthedashedlineorevenbelowit.Theseresultsthussuggestthatasynchronousgenedoublingcanindeedmakesynchronizationmorepronounced,butonlyforcasesofasymmetricsynchronization,whiletheeffect10

10NewJ.Phys.24(2022)033050GKnotzetalFigure8.Eachmarkerrepresentsasynchronizationwindowwhichcanbefoundinfigure7(b).Onthex-axiswehavethesizeofthesynchronizationwindowinβspacewithsynchronousgenedoublingatTandonthey-axisthewindowsizewithashifton2thegenedoublingtimeofproteinAbyΔtd=−20min(a)orwithgenedoublingtimesτA=10min,τB=30minandτC=50min(b).Everysynchronizationwindowabovethegreydottedlinedoesincreasewiththeshiftandeverywindowbelowthelinedecreases.Thecrossescorrespondtosymmetricsynchronizationsandthedotstoasymmetricsynchronizations.Thecolorofeachdotindicatestheperiodofthewindow,thatcanbefoundinanorbitdiagram.onsymmetricsynchronizationissmallorevennegative.Totestwhetherthereisadependenceontheperiodofthesynchronizedorbits,weindicatedtheperiodbythecolorofthesymbols,butnoobviouspatternisseen.Asafurthertestofthishypothesis,weshiftedthedoublingnotonlyofonegene,butoftwoawayfromtd=T/2andrepeatedtheanalysis.Figure8(b)showsthesizeofthesynchronizationwindowsforthecasewheregeneAisduplicated20minearlierandgeneC20minlaterthangeneB.Theoverallpictureisthesame:breakingthesymmetrybyshiftedgeneduplicationtimesincreasesthesizeofsynchronizationwindowsforasymmetricsynchronization,withbigincreasesforsomeofthem,buthasonlyasmallpositiveornegativeeffectforsymmetricsynchronization.4.DiscussionGeneticcircuitsaregenericallycoupledtoaperiodicforcingbythecelldivisioncycleviavolumegrowthandgeneduplication.Herewehavestudiedthespecificcaseofhowthiscouplingaffectsageneticoscillator,therepressilator.Inagreementwithearlierstudies[18,25],weseethattheperturbationsoftherepressilatoroscillationsaremostlymoderate.However,therearedistinctwindowsintheparameterspaceinwhichthetwooscillatorsaresynchronized.Someofthesewindowsoccurinaccessibleparameterrangesandshouldthusbeexperimentallyobservable.Importantly,however,thecouplingtothecelldivisioncycleasstudiedhereappliestothecasewheretherepressilatorgenesarelocatedonthechromosome.Iftherepressilatorisencodedonaplasmid,thecouplingwillbedifferentandshouldtypicallybeweakerduetoasynchronousduplicationofmultiplecopiesoftheplasmid.Inthecontextofsyntheticcircuits,thissuggeststhatdependingonwhetheroneisinterestedintheperiodicforcingbythecellcycleasafeatureorwhetheroneconsidersitratherasaperturbation,implementationstrategiesusingthechromosomemayormaynotbepreferableoverstrategiesbasedonplasmids.Wenotethatanumberofstudieshaveemphasizedthecompetitionbetweengenesandallocationofgeneexpressionmachinery[11,34,35],includingrecentworkoncellcycleeffects[16].Inthedescriptionusedhere,wehavenotincludedsucheffects,astheycanbeneglectedifthegenesconsideredaccountonlyforasmallfractionofthetotaltranscriptomeandproteome.Specifically,LinandAmir[16]haveproposedthatinsteadofthegenecopynumbergofagenei,thegenefraction,g/g=g/(g+g),whereiijjiij=ijthesumrunsoverallexpressedgenes,shouldbeusedinthetranscriptionrateandlikewiseformRNAfractionsinthetranslationrate.Ifthegeneofinterestonlycontributesasmallamounttothesesums,however,thedifferencebetweenthetwodescriptionsissmall.Specifically,transcriptioncanbeexpectedtobeproportionaltothegenecopynumber.Forthespecialcasewherethethreeproteinsoftherepressilatorarecharacterizedbythesameparameters(synthesisanddegradationrates,parametersoftheHillfunction),wehaveshownthatthesynchronizationwindowscanbegroupedintotwoclasses:inoneclass,theperiodictrajectoriesexhibitathree-folddiscretesymmetryundercyclicpermutationoftheproteinsandthenumberofcelldivision11

11NewJ.Phys.24(2022)033050GKnotzetalcyclesinoneperiodisanintegermultipleof3.Intheotherclass,thereisnosuchsymmetryandinsteadthreeperiodicsolutionsexistthatareshiftedrelativetoeachother.Unexpectedly,wefoundthatthethesetwotypesofsynchronizationshowdifferentbehaviorwhenthesymmetrybetweentheparametersofthethreeproteinsisbrokenbynon-synchronousgeneduplication:thesynchronizationwindowswithoutthepermutationsymmetrybecomewider,whilethosewiththatsymmetrygetsmallerorremainunchanged.Ingeneral,thewidthofsynchronizationwindowscanbeconsideredasameasureoftheefficiencyofentrainmentandcanbeoptimizedbymodulatingthewaveformoftheperiodicforcing[27],atleastinthelimitofweakforcing.Ourobservationindicatesarelationbetweensuchoptimalforcingandadiscretesymmetryoftheoscillation.Onlysynchronizationwindowscorrespondingtoasymmetricsynchronizationshowapotentialforconsiderableimprovementoftheefficiencyofentrainment.Whethersucharelationholdsmoregeneralthanforthespecificsystemstudiedhereandhowthesymmetryofthetrajectoriesinfluencesthewidthofthesynchronizationregionsremaintobeinvestigated.Fromthebiologicalpointofview,onecanaskwhethersynchronizationofageneticoscillatorwiththecellcycleprovidesabenefitorratheraperturbationoftheoscillatorsdynamics.Indeed,bothscenariosarepossible.Botheukaryoticandcyanobacterialcircadianclockshavebeenshowntobecoupledtothecellcycle,howeverthecouplingistypicallysuchthatthetwooscillatorsinfluenceeachother[24,36].Synchronizationinthatcasecanstabilizetheoscillationsandcoordinatecellularfunctionswiththeday–nightcycle.Bycontrast,ifanoscillatorhasapreciselytunedfunctionrequiringadefinedperiod,thecouplingtothecellcycleamountstoanunwanteddisturbanceandinsulatingtheoscillationsagainsttheseperturbationsmaybecrucialtotheirfunction.Ithasindeedbearguedthatsomefeaturesofthecyanobacterialcircadianclockserveexactlythatfunction,forexamplemultiple,independentlyreplicatedcopiesoftheclockgenes[37].Insummary,ourresultsshowthatthegenericcouplingofgeneticoscillatorstothecellcycleduetodiscretegeneduplicationeventsprovidesbothasourceofperturbationsandanadditionallayerofcontrolofthebehaviorofsuchgeneticcircuits.Synchronizationwiththedivisioncycleoccursintypicallysmallparameterregions,whichmaybeoptimizedviathegenelocationonthechromosome,withanintriguingrelationbetweensynchronizationefficiencyandthesymmetryoftheoscillator’strajectoriesinphasespace.AcknowledgmentsWeacknowledgesupportbytheOpenAccessPublicationFundsoftheUniversityofGottingen.¨DataavailabilitystatementThedatathatsupportthefindingsofthisstudyareavailableuponreasonablerequestfromtheauthors.AppendixA.DeterminingfrequenciesApreviousstudyofsynchronizationofgeneticoscillatorswiththecellcycle[25]usedthemeanpeak-to-peaktime,thetimebetweenlocalmaximaofthetrajectory,todeterminethefrequencyoftheoscillator.However,thisapproachisnotgenerallyapplicableandmaysystematicallyunderestimatetheperiodofthedrivenrepressilator:theincreasedproteinproductionaftergenedoublingcanleadtoarapidincreaseofproteinconcentration.IfthisincreasehappensnearamaximumoftheoscillationsthiscanresultintwocloselyspacedlocalmaximaasshowninfigureA1(a).Thislowerstheestimatedperiodoftheoscillations.TheperioddeterminedwiththepeaktopeakmethodisplottedasafunctionofthecellcycledurationTinfigureA1(b)(greenline),togetherwiththeperiodobtainviatheHilberttransform(blueline).ThepeaktopeaktimecanindeedbeseentobesystematicallysmalleroratmostequaltotheperioddeterminedviatheHilberttransform.Moreover,themeanpeaktopeaktimeappearstobeveryunstable(greenline)andfluctuatesstrongly.Inaddition,weplotthestandarddeviationofthepeak-to-peaktime(greenarea),whichtendstoberatherlargeandindicatesthebroaddistributionoftheindividualpeak-to-peaktimes.However,inthesynchronizationwindow,seehereasintervalswherethemeasuredperiodincreasesinalinearfashion,thestandarddeviationisnegligiblysmallandthepeak-to-peaktimeisinagreementwiththeperiodcalculatedviatheHilberttransform.Incaseofthesinusoidal-likeoscillationsoftheproteinconcentrations(seefigures6(e)and(f))analternativemethodtodetermineanaverageperiodistousethezeroscrossingsofashiftedtimeseries.Thismethodalsoavoidstheproblemsofincreasedproteinproductionaftergeneduplication.Forthisweshift12

12NewJ.Phys.24(2022)033050GKnotzetalFigureA1.(a)Exampletrajectoryoftherepressilatorwith(β=0.011min−1,T=60min).Thelocalmaximacanbeveryclosetogetherwhichleadstoasystematicunderestimationoftheperiod.(b)Periodoftherepressilatordeterminedbythepeaktopeakmethodforβ=0.01min−1.ThegreenareadisplaysthestandarddeviationforthemeanofthelocalmaximaandbluelineistheperiodweobtainedbytheHilberttransformation.WhentherepressilatorsynchronizesthepeaktopeakmethodandtheHilberttransformationgivethesameresult.(c)Perioddeterminedbyzerocrossingsofthetimeseriesfromwhichtheaverageofthemaximalandtheminimalvaluehasbeensubtracted,againforβ=0.01min−1.Thestandarddeviation(orangearea)showsasimilarbehaviourasthepeaktopeakmethod,butismuchsmaller.TheaverageitselfgivesthesameresultsastheHilberttransform.thetrajectorybythemedianvalueofthemaximumandtheminimumoftheproteinconcentrationandmeasurethedistancebetweentwozerocrossingsfromnegativetopositiveasperiod.Becausethisdistanceisnotconstantforquasiperiodicoscillations,weaverageovermanycrossingstogetanaverageperiod.TheresultsfordifferentcellcyclesTareshowninfigureA1(c).Withthismethodtheaverageismuchsmoother(orangeline)andagreeswellwithresultsobtainedviatheHilberttransform(blueline).Inaddition,wecanagainobservehowthestandarddeviation(orangearea)isreducedintheregionscorrespondingtoaperiodicorbit.ORCIDiDsGabrielKnotzhttps://orcid.org/0000-0003-1485-9230UlrichParlitzhttps://orcid.org/0000-0003-3058-1435StefanKlumpphttps://orcid.org/0000-0003-0584-2146References[1]PtashneM1986AGeneticSwitch:PhageLambdaandHigherOrganisms2ndedn(Oxford:Blackwell)[2]AlonU2007AnIntroductiontoSystemsBiology:DesignPrinciplesofBiologicalCircuits(BocaRaton,FL:CRCPress)[3]JeffH,DavidM,FarrenIandCollinsJJ2001Nat.Rev.Genet.2268–79[4]GardnerTS,CantorCRandCollinsJJ2000Nature403339–42[5]ElowitzMBandLeiblerS2000Nature403335–8[6]StrickerJ,CooksonS,BennettMR,MatherWH,TsimringLSandHastyJ2008Nature456516–9[7]Potvin-TrottierL,LordND,VinnicombeGandPaulssonJ2016Nature538514–7[8]KlumppS,ZhangZandHwaT2009Cell1391366–75[9]SchaechterM,MaaløeOandKjeldgaardNO1958J.Gen.Microbiol.19592–606[10]BremerHandDennisP1996ModulationofchemicalcompositionandotherparametersofthecellbygrowthrateEscherichiaColiandSalmonellaTyphimurium:CellularandMolecularBiologyvol2edFNeidhard(WashingtonDC:AmericanSocietyforMicrobiology)pp1553–69[11]ScottM,GundersonCW,MateescuEM,ZhangZandHwaT2010Science3301099–10213

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