Distinguishing_different_modes_of_growth_using_sin

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1Distinguishingdifferentmodesofgrowthusing2single-celldataPrathithaKar1,2,SriramTiruvadi-Krishnan3,JaanaMännik3,JaanMännik3,3andArielAmiry141SchoolofEngineeringandAppliedSciences,HarvardUniversity,Cambridge,56MA02134,USA2DepartmentofChemistryandChemicalBiology,HarvardUniversity,78Cambridge,MA02138,USA3DepartmentofPhysicsandAstronomy,UniversityofTennessee,Knoxville,910TN37996,USACorrespondingauthor-email:jmannik@utk.edu;phone:+1(865)9746018yCorrespondingauthor-email:arielamir@seas.harvard.edu;phone:+1(617)49558181

111Abstract12Collectionofhigh-throughputdatahasbecomeprevalentinbiology.Largedatasetsallow13theuseofstatisticalconstructssuchasbinningandlinearregressiontoquantifyrelationships14betweenvariablesandhypothesizeunderlyingbiologicalmechanismsbasedonit.Wediscuss15severalsuchexamplesinrelationtosingle-celldataandcellulargrowth.Inparticular,we16showinstanceswherewhatappearstobeordinaryuseofthesestatisticalmethodsleads17toincorrectconclusionssuchasgrowthbeingnon-exponentialasopposedtoexponential18andviceversa.Weproposethatthedataanalysisanditsinterpretationshouldbedonein19thecontextofagenerativemodel,ifpossible.Inthisway,thestatisticalmethodscanbe20validatedeitheranalyticallyoragainstsyntheticdatageneratedviatheuseofthemodel,21leadingtoaconsistentmethodforinferringbiologicalmechanismsfromdata.Onapplying22thevalidatedmethodsofdataanalysistoinfercellulargrowthonourexperimentaldata,we23findthegrowthoflengthinE.colitobenon-exponential.Ouranalysisshowsthatinthe24laterstagesofthecellcyclethegrowthrateisfasterthanexponential.2

2251Introduction26Thelastdecadehasseenatremendousincreaseintheavailabilityofhigh-qualitylarge27datasetsinbiology,inparticularinthecontextofsingle-celllevelmeasurements.Such28dataarecomplementaryto“bulk”measurementsmadeoverapopulationofcells.They29haveledtonewbiologicalparadigmsandmotivatedthedevelopmentofquantitativemodels30[1–7].Nevertheless,theyhavealsoledtonewchallengesindataanalysis,andherewe31willpointoutsomeofthepitfallsthatexistinhandlingsuchdata.Inparticular,wewill32showthatthecommonlyusedprocedureofbinningdataandlinearregressionmayhint33atspecificfunctionalrelationsbetweenthetwovariablesplottedthatareinconsistentwith34thetruefunctionalrelations.Asweshallshow,thismaycomeaboutduetothe“hidden”35noisesourcesthataffectthebinningprocedureandthephenomenonof“inspectionbias”36wherecertainbinshavebiasedcontributions.Oneofourmaintakehomemessagesisthe37significanceofhavinganunderlyingmodel(ormodels)toguide/test/validatedataanalysis38methods.Theunderlyingmodelisreferredtoasagenerativemodelinthesensethat39itleadstosimilardatatothatobservedintheexperiments.Theimportanceofaso-40calledgenerativemodelhasbeenbeautifullyadvocatedinthecontextofastrophysicaldata41analysis[8],yetbiologybringsinaplethoraofexcitingdifferences:whileinphysicsnoisefrom42measurementinstrumentsoftendominates,inthebiologicalexampleswewilldwellonhereit43istheintrinsicbiologicalnoisethatcanobscurethemathematicalrelationbetweenvariables44whennothandledproperly.Inthefollowing,wewillillustratethisratherphilosophical45introductiononaconcreteandfundamentalexample,albeitepluribusunum.Wewillfocus46ontheanalysisoftheEscherichiacoligrowthcurvesobtainedviahighthroughputoptical47microscopy.Neverthelessweanticipatetheconceptualpointsmadehere–anddemonstrated48onaparticularexampleofinterest–willtranslatetoothertypesofmeasurements,which49makeuseofmicroscopybutalsobeyond.3

350Binningcorrespondstogroupingdatabasedonthevalueofthex-axisvariable,andfind-51ingthemeanofthefluctuatingy-axisvariableforthisgroup.Byremovingthefluctuations52ofthey-variable,thebinningprocessoftenaimstoexposethe“true”functionalrelation53betweenthetwovariableswhichcanbeusedtoinfertheunderlyingbiologicalmechanism.54Whilebinningmayprovideasmoothnon-linearrelationbetweenvariables,linearregression55isusedtofindalinearrelationshipbetweenthevariables.Inadditiontobinning,weuse56theordinaryleastsquaresregressionwheretheslopeandtheinterceptofthebestlinearfit57lineareobtainedbyminimizingthesquaredsumofthedifferencebetweenthedependent58variablerawdataandthepredictedvalue.Here,thebestfit/thebestlinearfitisobtained59usingtherawdataandnotthebinneddata.Similartobinning,theassumptionunderlying60linearregressionisthatourknowledgeofx-axisvariableisprecisewhilethenoiseisinthe61y-axisvariable.62Itisimportanttodiscussthesourcesoffluctuationsinthey-axisvariablebeforewe63proceed.Inbiology,fluctuationsinthevariablesariseinevitablyfromtheintrinsicvariability64withinacellpopulation.Cellsgrowinginthesamemediumandenvironmenthavedifferent65characteristics(e.g.,growthrate)duetothestochasticnatureofbiochemicalreactionsin66thecell[9].Forexample,thedivisioneventiscontrolledbystochasticreactions,whose67variabilityleadstocelldividingatasizesmallerorlargerthanthemean.Inthispaper,68whenmodelingthedata,wewillconsidertheintrinsicnoiseastheonlysourceofvariability69andassumethatthemeasurementerrorismuchsmallerthantheintrinsicvariationinthe70population.71OneexampleoftheuseofbinningandlinearregressionisshowninFigure1Awheresize72atdivision(Ld)vssizeatbirth(Lb)isplottedusingexperimentaldataobtainedbyTanouchietal.forE.coligrowingat25C[10].InFigure1A,thefunctionalrelationbetweenlengthat7374divisionandlengthatbirthforE.coliisobservedtobelinearandclosetoLd=Lb+
L(see75Section5.11.1fordetails).Therelationobtainedallowsustohypothesizeacoarse-grained4

476biologicalmodelknownastheaddermodelasshowninFigure1Binwhichthelengthat77divisionissetbyadditionoflength
Lfrombirth[4,11–16].Thispreviouslydiscussed78exampledemonstratesandreiteratestheuseofstatisticalanalysisonsingle-celldatato79understandtheunderlyingcellregulationmechanisms.Usingstatisticalmethodssuchas80binningandlinearregression,otherphenomenologicalmodelsapartfromadderhavealso81beenproposedinE.coliwherethedivisionlength(Ld)isnotdirectly“set”bythatatbirth82[17–19].Thephenomenologicalmodels,inturn,canberelatedtomechanistic(molecular-83level)modelsofcellsizeandcellcycleregulation[20].Recentworkhasshedlightonthe84subtletiesinvolvedininterpretingthelinearregressionresultsfortheLdvsLbplotwhere85seeminglyadderbehaviorinlengthcanbeobtainedfromasizermodel(divisionoccurring86onreachingacriticalsize)duetotheinterplayofmultiplesourcesofvariability[21].This87issueissimilarinspirittothosewehighlighthere.88Thevolumegrowthofsinglebacterialcellshasbeentypicallyassumedtobeexponential89[4,14,22–25].Assumingribosomestobethelimitingcomponentintranslation,growthis90predictedtobeexponentialandgrowthratedependsontheactiveribosomecontentinthe91cell[26–28].Undertheassumptionofexponentialgrowth,thesizeatbirth(Lb),thesizeat92division(Ld),andthegenerationtime(Td)arerelatedtoeachotherby,Ldln()=Td;(1)Lb93whereisthegrowthrate.Understandingthemodeofgrowthisimportante.g.,dueto94itspotentialeffectsoncellsizehomeostasis.Exponentiallygrowingcellscannotemploya95mechanismwheretheycontroldivisionbytimingaconstantdurationfrombirthbutsuch96amechanismispossibleincaseoflineargrowth[3,13,29].Linearregressionperformedonln(Ld)vshiTplot,wherehiisthemeangrowthrate,wasusedtoinferthemode97Ldb98ofgrowthinthearchaeonH.salinarum[16],andinthebacteriaM.smegmatis[30]and5

599C.glutamicum[31],forexample.Ifthebestlinearfitfollowsthey=xtrend,theresulting100functionalrelationmightpointtogrowthbeingexponential.Acorollarytothisisthe101rejectionofexponentialgrowthwhentheslopeandinterceptofthebestlinearfitdeviatefrom102oneandzerorespectively[31].Thus,binningandlinearregressionappliedonsingle-celldata103appeartoprovideinformationabouttheunderlyingbiology,inthiscase,themodeofcellular104growth.WewilltestthevalidityofsuchinferencebyanalyzingsyntheticdatageneratedLd)vs105usinggenerativemodels.Wefindthatlinearregressionperformedontheplotln(Lb106hiTd,surprisingly,doesnotprovideinformationaboutthemodeofgrowth.Nonetheless,107weshowthatothermethodsofstatisticalanalysissuchasbinninggrowthratevsageplots108areadequateinaddressingtheproblem.Usingthesevalidatedmethodsonexperimental109data,wefindthatE.coligrowsnon-exponentially.Inlaterstagesofthecellcycle,the110growthrateishigherthanthatinearlystages.1112Statisticalmethodslikebinningandlinearregression112shouldbeinterpretedbasedonamodel.113Toillustratethepitfallsassociatedwithbinning,weusedatafromrecentexperimentsonE.114coliwherethelengthatbirth,thelengthatdivisionandthegenerationtimewereobtained115formultiplecells(seeSection5.1and[32]).Phase-contrastmicroscopywasusedtoobtain116celllengthatequalintervalsoftime.Notethatweconsiderlengthtoreflectcellsizein117thispaperratherthanothercellgeometrycharacteristicssuchassurfaceareaandvolume.118Thelengthgrowthratethatweelucidateinthepapercanbedifferentfromthecellvolume119growthrateasshowninAppendix1assumingasimplecellmorphologyandexponential120growth.Usingthesamecellmorphology,wealsofindthelengthgrowthratetobeidentical121tocellsurfacegrowthrate.Toinvestigateifthecellgrowthwasexponential,weplottedLd)vshiTforcellsgrowinginM9alanineminimalmediumat28C(hTi=214min).122ln(Lddb6

6123Thelinearregressionofthesedatayieldsaslopeof0.3andaninterceptof0.4asshownin124Figure2A.Thebinneddataandthebestlinearfitdeviatesignificantlyfromthey=xline125(seeTableS2).Additionally,thebinneddatafollowsanon-lineartrendandflattensout126atlongergenerationtimes.Wealsofoundsimilardeviationsinthebinneddataandbest127linearfitinglycerolmedium(hTdi=164min)showninFigure2-figuresupplement1A,and128glucose-casmedium(hTdi=65min)showninFigure2-figuresupplement1B.Qualitatively129similarresultshavebeenrecentlyobtainedforanotherbacterium,C.glutamicum,inRef.130[31].Theseresultsmightpointtogrowthbeingnon-exponential.131Nextwewillapproachthesameproblembutwithagenerativemodel.WewillfirstLd)vshiTbinnedplotcouldnotdistinguishexponentialgrowthfrom132showthattheln(Ldb133non-exponentialgrowth.Forthatpurpose,weuseapreviouslystudiedmodel[16]which134considersgrowthtobeexponentialwiththegrowthratedistributednormallyandindepen-135dentlybetweencellcycleswithmeangrowthratehiandstandarddeviationCVhi.CV136isthusthecoefficientofvariation(CV)ofthegrowthrateandisassumedtobesmall.To137maintainanarrowdistributionofcellsize,cellsmustemployregulatorymechanisms.In138ourmodel,weassumethat,barringthenoiseduetostochasticbiochemicalreactions,cells139attempttodivideataparticularsizeLdgivensizeatbirthLb.Keepingthemodelasgeneric140aspossible,wecanwriteLdasafunctionofLb,f(Lb)whichcanbethoughtofasacoarse-141grainedmodelfortheregulatorymechanism.Ref.[13]providesaframeworktocapturetheregulatorymechanismsbychoosingf(L)=2L1L.Listhetypicalsizeatbirthand,142bb00143whichcantakevaluesbetween0and2,reflectsthestrengthofregulationstrategy.=0144correspondstothetimermodelwheredivisionoccursonaverageafteraconstanttimefrom145birth,and=1isthesizermodelwhereacelldividesuponreachingacriticalsize.=1461/2canbeshowntobeequivalenttotheaddermodelwheredivisioniscontrolledbyaddi-147tionofconstantsizefrombirth[13].Inadditiontothedeterministicfunction(f)specifying148division,thesizeatdivisionisaffectedbynoise()indivisiontiming.Weassumeithashi7

7nandthatitisindepen-149aGaussiandistributionwithmeanzeroandstandarddeviationhi150dentofthegrowthrate.Thus,thegenerationtime(Td)canbemathematicallywrittenasT=1ln(f(Lb))+andisinfluencedbygrowthratenoiseanddivisiontimingnoise.Note151dLhib152thatreplacingthetimeadditivedivisiontimingnoisewithasizeadditivedivisiontiming153noisewillnotaffecttheresultsqualitatively(seeSections5.2and5.3fordetailsandTable154S1forvariabledefinitions).155Forperfectlysymmetricallydividingcellswhosesizesarenarrowlydistributed,wefindLd)vshiTplottobe(seeSection5.4),156thetrendinthebinneddataforln(Ldb011x@1+ln(2)A:(2)y=x221+n2CV2ln2(2)157FixingCV=n=0.15,weshowusingsimulationsinFigure2Cthenon-lineartrendinthe158binneddataeventhoughweassumedexponentialgrowth.Similarly,onperforminglinearLd)vshiTplot,wefindthattheslopeofthebestlinear159regressionontherawdataofln(Ldb160fitisnotequaltooneandtheinterceptisnon-zero(seeEqs.27and28andFigure2C).161Eq.2showsthatthetrendinthebinneddatadependsontheratioofgrowthratenoise162anddivisiontimingnoise.Theslopeisequaltooneandinterceptiszeroonlyifthenoise163ingrowthrateisnegligibleascomparedtothedivisiontimingnoise.Inexperimentsthatis164rarelythecase,hence,thebinneddatatrendandthebestlinearfitdeviatefromthey=x165lineeventhoughgrowthmightbeexponential.Thus,wecannotruleoutexponentialgrowth166intheE.coliexperimentsdespitethebinneddatatrendbeingnon-linearandthebest-fit167linedeviatingfromthey=xline.Ld)vshiTarise168Whydoesanon-linearrelationshipinthebinneddatafortheplotln(Ldb169evenforexponentialgrowth?Accordingtothemodel,Ldisdeterminedbyadeterministic170strategy,f(Lb)andatime/sizeadditivedivisiontimingnoise.ThenoisecomponentwhichLd)isthusthenoiseindivisiontimingandnot171affectsLdandsubsequentlythequantityln(Lb8

8172thegrowthrate.Thegenerationtime(Td)plottedonthex-axisisinfluencedbythenoisein173divisiontimingaswellasthenoiseingrowthrate.Binningassumesthatforafixedvalueof174thex-axisvariable,thenoisefromothersourcesaffectsonlythey-axisvariable(thebinned175variable).Similarlyforlinearregression,theunderlyingassumptionisthattheindependent176variableonx-axisispreciselyknownwhilethedependentvariableonthey-axisisinfluenced177bytheindependentvariableandfromexternalfactorsotherthantheindependentvariable.178Inthiscase,onlyhiTdplottedonx-axisisinfluencedbygrowthratenoisewhilebothhiTdLd)areinfluencedbynoiseindivisiontime.Thisdoesnotfittheassumptionfor179andln(Lbbinningandlinearregressionandhence,thebestlinearfitforln(Ld)vshiTplotmight180Ldb181deviatefromthey=xlineeveninthecaseofexponentialgrowth.182Anotherwayofexplainingthedeviationfromthelineary=xtrendisbyinspectionbias,183whichariseswhencertaindataisover-represented[33].Cellswhichhavealongergeneration184timethanthemeanwillmostlikelyhaveaslowergrowthrate.Thus,inFigure2Aand185Figure2C,atlargervaluesofhiTdorTd,thebinaveragesarebiasedbyslowergrowingLd)orTtobelowerthanexpected.Thisprovidesanexplanation186cells,thusmakingln(Ldb187fortheflatteningofthetrend.Itfollowsfromthepreviousdiscussionthatifonebinsdatabyln(Ld)thentheassumption188Lbforbinningismet.BothofthevariableshiTandln(Ld)areinfluencedbythenoisein189dLb190divisiontimebuthiTdplottedonthey-axisisalsoinfluencedbythegrowthratenoise.Ld),andanexternal191Thus,they-axisvariable,hiTdisdeterminedbythex-axisvariable,ln(Lb192sourceofnoise,inthiscase,thegrowthratenoise.Thus,basedonourmodel,weexpect193thetrendinbinneddataandlinearregressionperformedontheinterchangedaxestofollow194they=xtrendforexponentiallygrowingcells(seeSection5.4).Indeed,oninterchangingtheLd)forsyntheticdata,wefindthatthetrendinthebinned195axisandplottinghiTdvsln(Lb196dataandthebestlinearfitcloselyfollowsthey=xline(Figure2D).Wealsofindthatthe197bestlinearfitfollowsthey=xlineinthecaseofalanine(Figure2B),glycerol(Figure2-9

9198figuresupplement1A)andglucose-cas(Figure2-figuresupplement1B).Achangefrom199non-linearbehaviortothatoflinearoninterchangingtheaxesisalsoobservedinarelated1)areconsidered(Figure2-200problemwheregrowthrate()andinversegenerationtime(Td201figuresupplement2andSection5.10).202Thusfar,weshowedforarangeofmodelswherebirthcontrolsdivisionthatthebinnedLd)asfunctionofhiTisnon-linearanddependentonthenoiseration203datatrendforln(LdCVb204inthecaseofexponentialgrowth.Oninterchangingtheaxesthebinneddatatrendagrees205withthey=xlineindependentofthegrowthrateanddivisiontimenoise.However,wewill206shownextthatthisagreementwiththey=xtrendcannotbeusedasa“smokinggun”for207inferringexponentialgrowthfromthedata.208Toinvestigatethisfurther,letusconsiderlineargrowth,whichhasalsobeensuggested209tobefollowedbyE.colicells[34,35].Theunderlyingequationforlineargrowthis,0LdLb=Td;(3)where0isthetheelongationspeedi.e.,dL.Forcellsgrowinglinearly,thebestlinearfit210dtfortheplothiTvsln(Ld)isexpectedtodeviatefromthey=xline.Asbefore,wefixhi211dLbtobethemeanof1ln(Ld),agnosticofthelinearmodeofgrowth.Surprisingly,wefound212TdLb213thatfortheclassofmodelswherebirthcontrolsdivisionbyastrategyf(Lb)andcellsgrowLd)agreescloselywiththey=xtrend.Oncarrying214linearly,thebestlinearfitforhiTdvsln(Lb215outanalyticalcalculationsbasedonthismodel,weobtaintheslopeandtheinterceptofthehiTvsln(Ld)plottobe3ln(2)1.04and-0.03respectively,whichisveryclosetothat216dL2b217forexponentialgrowth(seeSection5.6).Thisisshownforsimulationsoflineargrowthwith218cellsfollowinganaddermodelinFigure3A.Givennoinformationabouttheunderlying219model,Figure3Acouldbeinterpretedascellsundergoingexponentialgrowthcontraryto220theassumptionoflineargrowthinsimulations.Thus,whenhandlingexperimentaldata,10

10221cellsundergoingeitherexponentialorlineargrowthmightseemtoagreecloselywiththeLd)222y=xtrend.Deforetetal.[36]usedthelinearbinneddatatrendincaseofhiTdvsln(Lb223plottoinferexponentialgrowthbutasweshowedinthissection,thelineartrenddoesnot224ruleoutlineargrowth.Thisagainreiteratesourmessageofhavingagenerativemodelto225guidethedataanalysismethodssuchasbinningandlinearregression.Forcompleteness,weLd)vshTianditsinterchangedaxesplotstoelucidatethemode226alsotesttheutilityofln(Ldb227ofgrowth(Appendix2).Wefindthatbinningandlinearregressionappliedontheseplots228cannotdifferentiatebetweenexponentialandlineargrowth.229Toconcludethediscussionoflineargrowth,wenotethatthenaturalplotforthisgrowth230regimeishliniTdvsldlbandtheplotobtainedoninterchangingtheaxes(seeSection5.5231andFigure3-figuresupplements1A,1B).Herelb,ldandlinaredefinedtobequantitiesL,Land0,respectively,normalizedbythemeanlengthatbirth.Forcellsgrowing232bd233exponentially,thebestlinearfitforthehliniTdvsldlbplotisexpectedtodeviatefromthe234y=xline.ThisisindeedwhatisobservedinFigure3-figuresupplement1Cwheresimulations235ofexponentiallygrowingcellsfollowingtheaddermodelarepresented(seeSection5.6for236extendeddiscussion).237Inallofthecasesabove,theproblemathanddealswithdistillingthebiologicallyrelevant238functionalrelationbetweentwovariables.However,thedataisassumedtobesubjectedto239fluctuationsofvarioussources,anditisimportanttoensurethatthestatisticalconstructwe240areusing(e.g.binning)isrobusttothese.Howcanweknowaprioriwhetherthestatistical241methodisappropriateanda"smokinggun"forthefunctionalrelationweareconjecturing?242Theexamplesshownabovesuggestthatperformingstatisticaltestsonsyntheticdataob-243tainedusingagenerativemodelisaconvenientandpowerfulapproach.Notethatincases244suchastheonesstudiedherewhereanalyticalcalculationsmaybeperformed,onemaynot245evenneedtoperformanynumericalsimulationstotestthevalidityofthemethods.11

112463Growthratevsageplotsareconsistentwiththeun-247derlyinggrowthmode.Ld)vshiTandhiTvsln(Ld)arenot248Inthelastsection,weshowedthattheplotsln(LddLbb249decisiveinidentifyingthemodeofgrowth.RecentworksonB.subtilis[37]andfissionyeast[38]haveuseddifferentialmethodsofquantifyinggrowthnamelygrowthrate(=1dL)vs250LdtdL)vsageplotstoprobethemodeofgrowthwithina251ageplotsandelongationspeed(=dt252cellcycle.Here,Ldenotesthesizeofthecellaftertimetfrombirthinthecellcycleand253agedenotestheratiooftimettoTdwithinacellcycle(henceitrangesfrom0to1by254constructionwithinacellcycle).Inthissection,usingvariousmodelsofcellgrowthand255cellcycle,wetestthegrowthratevsagemethod.Notethatthegrowthratevsageand256theelongationspeedvsageplotsarenotdimensionlessunlikethepreviousplots.Usingthe257growthratevsageandelongationspeedvsageplots,weaimtoquantifythegrowthrate258changeswithinacellcycle.Forcellsassumedtobegrowingexponentially,growthrateis259constantthroughoutthecellcycle.Onaveragingovermultiplecellcycles,thetrendofbinned260dataisexpectedtobeahorizontallinewithvalueequaltomeangrowthratewhichisindeed261whatwefindinthenumericalsimulationsoftheadderandtheadderperoriginmodel[17],262asshowninFigure3B.Thebinneddatatrendineachofthemodelsmatchesthetheoretical263predictionsofgrowthrate(shownasdottedlines).Incontrast,forlinearlygrowingcells,the264elongationspeedisexpectedtoremainconstant.Weshowthisconstancyusingnumerical265simulationsoflinearlygrowingcellsfollowingtheaddermodel(Figure3-figuresupplement2663A).Inaccordancewiththisresult,thegrowthrateisexpectedtodecreasewithcellageas1.ThisisverifiedinFigure3Bbyagainusingthenumericalsimulationsoflinear267/1+age268growthwithcellsfollowingtheaddermodel.Thebinneddatatrendforlineargrowth(greensquares)matchesthetheoreticalpredictionsof/1(greendottedline).2691+age270Thus,thetwogrowthmodes(exponentialandlinear)couldbedifferentiatedusingthe12

12271growthratevsageplot(fordetailsseeSection5.7).However,thegrowthratevsageplots272canbeusedtoinferthemodeofgrowthbeyondthetwodiscussedabove.Weshowthisby273usingsimulationsofcellsfollowingtheaddermodelandundergoingfasterthanexponential274orsuper-exponentialgrowth(seeSection5.11.2fordetails).Insuchacase,thegrowthrateis275expectedtoincrease.ThisincreaseingrowthrateisshowninFigure3Busingsimulations.276Thebinneddatatrend(redtriangles)againmatchesthegrowthratemodeusedinthe277simulations(reddottedline).Thus,thegrowthratevsageplotsareaconsistentmethodto278distinguishlinearfromexponentialandsuper-exponentialgrowths.279Usingthevalidatedgrowthratevsageplots,weobtainedthegrowthratetrendfor280experimentaldataonE.coliforthethreegrowthconditionsstudiedinthispaper(Figures2814A-4C).Wefoundanincreaseingrowthrateinallgrowthconditionsduringthecourseof282thecellcycle.OnemaywonderwhethersuchanincreasemaybeexplainedbytheE.coli283morphologyalone,duetothepresenceofhemisphericalpoles.Forexponentiallygrowingcell284volumeandconsideringageometryofE.coliwithsphericalcapsatthepoles,thepercentage285increaseinthegrowthrateoflengthoveracellcycleisaround3%whichissignificantly286smallerthanthatobservedinourexperimentaldata.Consideringcellsizetrajectories(cell287size,Lattime,tdata)wherecelllengthsweretrackedbeyondthecelldivisionevent(by288consideringcellsizeinbothdaughtercells),wealsofoundthatthegrowthratedecreasesclose289todivision(age1)andreturnstoavaluenearlyequaltothatobservedatthebeginning290ofcellcycle(age0)asshowninFigure4-figuresupplements1A-1C(seeSection5.7for291extendeddiscussion).292Theabovequestionofmodeofgrowthwithinacellcyclecanalsobeanalyzedinrelation293toaspecificevent.Severalstudieshavepointedtoachangeingrowthrateattheonsetof294constriction[39,40].Thischangeingrowthratecanbeprobedusinggrowthratevstime295plotswheretimeistakenrelativetotheonsetofconstrictionasshowninFigure4-figure296supplement2.Theseplotsshowadecreaseingrowthratesatthetwoextremesoftheplot.13

13297Thesedecreasesareduetoinspectionbias,wherethegrowthratetrendisaffectedbythe298biasedcontributionofcellswithahigherthanaveragegenerationtimeorequivalentlyslower299growthrate(seeSection5.8forextendeddiscussion).Inspectionbiasisalsoobservedwhen300timingisconsideredrelativetoothercelleventssuchascellbirth(seeSection5.8andFigure3013-figuresupplements2C,2D).302Itmightnotalwaysbepossibletoobtaingrowthratetrajectoriesasafunctionoftime/celldM)asa303age.Godinetal.insteadobtainedtheinstantaneousbiomassgrowthspeed(dt304functionofitsbuoyantmass(M)[22].Onapplyinglinearregressionforinstantaneous305massgrowthspeedvsmass,weexpecttheslopeofthebestlinearfitobtainedtoprovidetheaveragegrowthrate(h1dMi)undertheassumptionofexponentialgrowthwhilefor306Mdt307lineargrowththeinterceptprovidestheaveragegrowthspeed.Usingthismethod,biomass308wassuggestedtobegrowingexponentially.Thismethodcanbeappliedtostudythelength309growthratewithinthecellcyclebyplottingelongationspeedasafunctionoflength[41].We310findthatthebinneddatatrendandthebestlinearfitofthisplotfollowtheexpectedtrend311forlinearandexponentialgrowthasshowninFigure3-figuresupplement3BandFigure3-312figuresupplement3D,respectively,foracellcyclemodelwheredivisioniscontrolledviaan313addermechanismfrombirth.However,thetrendobtainedappearstobemodel-dependent314asshowninFigure3-figuresupplement3Fwheretheunderlyingcellcyclemodelusedin315thesimulationsistheadderperoriginmodel.Forthismodel,thebinneddatatrendis316foundtobenon-linearwiththegrowthratespeedingupatlargesizes,despitethesynthetic317databeinggeneratedforperfectlyexponentialgrowth.Thisnon-lineartrendcanleadto318growthratebeingmisinterpretedasnon-exponentialwithinthecellcycle(seeSection5.9319fordetails).Thus,ananalysisusingtheelongationspeedvssizeplotmustbeaccompanied320withanunderlyingcellcyclemodel.321Insummary,wefoundthatthegrowthratevsageplotwasaconsistentmethodto322determinethechangesingrowthratewithinacellcycle.Unlikethegrowthratevsage14

14323plots,theinferencefromthegrowthratevssizeplotswasfoundtobemodel-dependent.324Usingthegrowthratevsageplots,weshowthatthelengthgrowthofE.colicanbefaster325thanexponential.3264Discussion327Statisticalmethodssuchasbinningandlinearregressionareusefulforinterpretingdataand328generatinghypothesesforbiologicalmodels.However,weshowinthispaperthatpredicting329therelationshipsbetweenexperimentallymeasuredquantitiesbasedonthesemethodsmight330leadtomisinterpretations.Constructingagenericmodelandverifyingthestatisticalanalysis331onthesyntheticdatageneratedbythismodelprovidesamorerigorouswaytomitigatethese332risks.Ld)vshiTandhiTvsln(Ld)plotsfail333Inthepaper,weprovideexamplesinwhichln(LddLbb334asamethodtoinferthemodeofgrowth.Thebinneddatatrendandthebestlinearfitfortheln(Ld)vshiTplotwasfoundtobedependentuponthenoiseparametersintheclass335Ldbofmodelswherebirthcontrolleddivision(Equation2).WealsoshowthathiTvsln(Ld)336dLb337plotcouldnotdifferentiatebetweenexponentialandlinearmodesofgrowth(Figures2D,3383A).Thus,weconcludethatthebestlinearfitfortheaboveplotsmightnotbeasuitable339methodtoinferthemodeofgrowthbuttheyarejustoneofthemanycorrelationswhich340thecorrectcellcyclemodelshouldbeabletopredict.341Wefoundgrowthratevsageandelongationspeedvsageplotstobeconsistentmethods342toprobegrowthwithinacellcycle.Themethodwasvalidatedusingsimulationsofvarious343cellcyclemodels(suchastheadder,andadderperoriginmodel,whereinthelatter,control344overdivisioniscoupledtoDNAreplication)andthebinnedgrowthratetrendagreedclosely345withtheunderlyingmodeofgrowthforthewiderangeofmodelsconsidered(Figure3B).In346thecaseofgrowthratevstimeplots,itwasimportanttotakeintoconsiderationtheeffects15

15347ofinspectionbias.Weusedcellcyclemodelstoshowthetimeregimeswhereinspectionbias348couldbeobserved(Figure3-figuresupplement2).Intheregimewithnegligibleinspection349bias,wecouldreconcilethegrowthratetrendobtainedusinggrowthratevsage(Figures4A-3504C)andgrowthratevstimeplots(Figure4-figuresupplement2).TheauthorsinRef.[31]351circumventinspectionbiasintheelongationspeedvstimefrombirthplotsbyfocusingtheir352analysisonthetimeperiodfromcellbirthtothegenerationtimeofthefastestdividingcell.353TheauthorsofRef.[42],whileinvestigatingthedivisionbehaviorinthecellsundergoing354nutrientshiftwithintheircellcycle,usebothmodelsandexperimentaldatafromsteady-355stateconditionstoidentifyinspectionbias.Theseserveasgoodexamplesofusingmodels356toaiddataanalysis.357StatisticsobtainedfromlinearregressionsuchasinFigure1Ahelpnarrowdownthe358landscapeofcellcyclemodels,butmanyhavepotentialpitfallslurkingwhichmightleadto359misinterpretations(Figure2C,Figure3A).Thereareadditionalissuesbeyondthoseconcern-360inglinearregressionandbinningdiscussedhere.Forexample,Ref.[43]discussesSimpson’s361paradox[44]wheredistinctcellularsub-populationsmightleadtoerroneousinterpretation362ofcellcyclemechanisms.Examplesofsuchdistinctsub-populationsarefoundinasymmet-363ricallydividingbacteriasuchasM.smegmatis[30,45].Anothersourceofmisinterpretation364couldarisefrompresenceofmeasurementerrors.Throughoutthiswork,wedealwithin-365trinsicnoiseandneglectmeasurementerror.However,whenmeasurementnoiseaffectsboth366x-axisandy-axisvariables,theslopeofthebestlinearfitisbiasedtowardszero.Thiscan367leadtopotentiallyrelatedvariablesbeingmisinterpretedasuncorrelated.Measurementer-368rorscanhoweverbehandledbasedonamodel.Usingamodelwhichincludesmeasurement369errorasasourceofnoise,wecanguidethebinninganalysis.Usingthismethodology,we370verifiedthattypicalmeasurementerrors(0:02Lb)[31,46]havenegligibleeffectsonthe371growthratetrendsobtainedfromtheexperimentaldatausedinourwork.372SinglecellsizeinE.colihasbeenreportedtogrowexponentially[4,14,22–25],linearly16

16373[34],bilinearly[47]ortrilinearly[39].TheseareinconsistentwithourobservationsinFigures3744A-4Cwherewefindthatgrowthcanbesuper-exponential.Thenon-monotonicbehaviorin375thefastest-growthconditionisreminiscentoftheresultsreportedinRef.[37]forB.subtilis.376TheauthorsofRef.[37]attributetheincreaseingrowthratetoamultitudeofcellcycle377processessuchasinitiationofDNAreplication,divisomeassembly,septumformation.In378thetwoslowergrowthconditions(Figures4A-4B),wefindthatthegrowthrateincrease379startsbeforethetimewhentheseptalcellwallsynthesisstartsi.e.,theconstrictionevent.380However,inthefastestgrowthcondition(Figure4C),thetimingofgrowthrateincrease381seemstocoincidewiththeonsetofconstrictionwhichisinagreementwithpreviousfindings382[39,40].383Itisimportanttodistinguishbetweenlengthgrowthandbiomassgrowth.Ref.[48]384measuresbiomassandcellvolumeandfindsthemass-densityvariationswithinthecell-cycle385tobesmall.Inthispaper,sinceweobservethelengthgrowthtobenon-exponential(Figure3864),itremainstobeseenwhetherbiomassgrowthalsofollowsasimilarnon-exponential387behaviororifitisexponentialaspreviouslysuggested[22,48].388Inconclusion,thepaperdrawstheattentionofthereaderstothecarefuluseofstatistical389methodssuchaslinearregressionandbinning.Althoughshowninrelationtocellgrowth,390thisapproachtodataanalysisseemsubiquitous.Thegeneralframeworkofcarryingoutdata391analysisispresentedinFigure5.Itproposestheconstructionofagenerativemodelbasedon392theexperimentaldatacollected.Ofcourse,wedonotalwaysknowwhetherthemodelused393isanadequatedescriptionofthesystem.Whatisthefateofthemethodologydescribedhere394insuchcases?First,weshouldberemindedofBox’sfamousquote“allmodelsarewrong,395someareuseful”.Thegoalofamodelisnottoprovideasaccurateadescriptionofasystem396aspossible,butrathertocapturetheessenceofthephenomenaweareinterestedinand397stimulatefurtherideasandunderstanding.Inourcontext,thegoalofthemodelistoprovide398arigorousframeworkinwhichdataanalysistoolscanbecriticallytested.Ifverifiedwithin17

17399themodel,itisbynomeansproofofthesuccessofthemodelandthemethoditself,and400furthercomparisonswiththedatamayfalsifyitleadingtotheusual(andproductive)cycle401ofmodelrejectionandimprovementviacomparisonwithexperiments.However,ifthebest402modelwehaveathandshowsthatthedataanalysismethodisnon-informative,aswehave403shownhereonseveralmethodsusedtoidentifythemodeofgrowth,thenclearlyweshould404revisetheanalysisasitprovidesuswithanon-consistentframework,whereourmodelingis405atoddswithourdataanalysis.Furthermore,testingthemethodsonasimplifiedmodelis406stilladvantageouscomparedwiththeoptionofusingthemethodswithoutanyvalidation.407Tomitigatetheriskofusingirrelevantmodels,insomecasesitmaybedesirabletotestthe408analysismethodsonasbroadaclassofmodelsaspossibleaswehavedoneinthepaper,for409examplebyouruseofageneralvalueoftodescribethesize-controlstrategywithinour410models.Thus,guidedbythemodel,thedataanalysismethodscanbeultimatelyappliedto411experimentaldataandunderlyingfunctionalrelationshipscanbeinferred.Reiteratingthe412messageoftheauthorsinRef.[8],thedataanalysisusingthisframeworkaimstojustify413themethodsbeingused,thus,reducingarbitrarinessandpromotingconsensusamongthe414scientistsworkinginthefield.4155Methods4165.1Experimentalmethods417Strainengineering:STK13strain(
ftsN::frt-Ypet-FtsN,
dnaN::frt-mCherry-dnaN)is418derivativeofE.coliK12BW27783(CGSC#:12119)constructedby-Redengineering[49]419andbyP1transduction[50].ForchromosomalreplacementofftsNwithfluorescencederiva-420tive,weusedprimerscarrying40nttailswithidenticalsequencetotheftsNchromosomal421locusandaplasmidcarryingacopyofypetprecededbyakanamycinresistancecassetteflankedbyfrtsites(frt-kanR-frt-Ypet-linker)asPCRtemplate(akindgiftfromR.Reyes-42218

18423LamotheMcGillUniversity,Canada;[51]).TheresultingPCRproductwastransformedby424electroporationintoastraincarryingthe-Red-expressingplasmidpKD46.Colonieswere425selectedbykanamycinresistance,verifiedbyfluorescencemicroscopyandbyPCRusing426primersannealingtoregionsflankingftsNgene.Afterremovalofkanamycinresistanceby427expressingtheFlprecombinasefromplasmidpCP20[52],wetransferredthemCherry-dnaN428genefusion(BN1682strain;akindgiftfromNynkeDekkerfromTUDelft,TheNether-429lands,[53])intothestrainbyP1transduction.Tominimizetheeffectoftheinsertionon430theexpressionlevelsofthegeneweremovedthekanamycincassetteusingFlprecombinase431expressingplasmidpCP20.432Cellsgrowth,preparation,andculturingE.coliinmothermachinemicroflu-433idicdevices:AllcellsweregrownandimagedinM9minimalmedium(Teknova)supple-mentedwith2mMmagnesiumsulfate(Sigma)andcorrespondingcarbonsourcesat28C.434435Threedifferentcarbonsourceswereused:0.5%glucosesupplementedby0.2%casamino436acids(Cas)(Sigma),0.3%glycerol(Fisher)and0.3%alanine(Fisher)supplementedwith1x437traceelements(Teknova).438Formicroscopy,weusedmothermachinemicrofluidicdevicesmadeofPDMS(poly-439dimethylsiloxane).Thesewerefabricatedfollowingtopreviouslydescribedprocedure[54].440Togrowandimagecellsinmicrofluidicdevice,wepipetted2-3lofresuspendedconcen-441tratedovernightcultureofOD6000.1intomainflowchannelofthedeviceandletcellsto442populatethedead-endchannels.Oncethesechannelsweresufficientlypopulated(about1443hr),tubingwasconnectedtothedevice,andtheflowoffreshM9mediumwithBSA(0.75444g/ml)wasstarted.Theflowwasmaintainedat5l/minduringtheentireexperimentby445anNE-1000SyringePump(NewEraPumpSystems,NY).Toensuresteady-stategrowth,446thecellswerelefttogrowinchannelsforatleast14hrbeforeimagingstarted.447Microscopy:ANikonTi-Einvertedepifluorescencemicroscope(NikonInstruments,448Japan)witha100X(NA=1.45)oilimmersionphasecontrastobjective(NikonInstru-19

19449ments,Japan),wasusedforimagingthebacteria.ImageswerecapturedonaniXonDU897450EMCCDcamera(AndorTechnology,Ireland)andrecordedusingNIS-Elementssoftware451(NikonInstruments,Japan).Fluorophoreswereexcitedbya200WHglampthroughan452ND8neutraldensityfilter.AChroma41004filtercubewasusedforcapturingmCherryim-453ages,andaChroma41001(ChromaTechnologyCorp.,VT)forYpetimages.Amotorized454stageandaperfectfocussystemwereutilizedthroughouttime-lapseimaging.Imagesinall455growthconditionswereobtainedat4minframerate.456Imageanalysis:ImageanalysiswascarriedoutusingMatlab(MathWorks,MA)scripts457basedonMatlabImageAnalysisToolbox,OptimizationToolbox,andDipImageToolbox458(https://www.diplib.org/).Celllengthsweredeterminedbasedonsegmentedphasecontrast459images.DissociationofYpet-FtsNlabelfromcellmiddlewasusedtodeterminetheexact460timingofcelldivisions.461FurtherexperimentaldetailscanalsobefoundinRef.[32].4625.2Model463Consideramodelofcellcyclecharacterizedbytwoevents:cellbirthanddivision.Inour464model,weassumethat,barringthenoise,cellstendtodivideataparticularsizevdgiven465sizeatbirthvb,viasomeregulatorymechanism.Hence,wecanwritevdasafunctionof466vb,f(vb).Ref.[13]providesaframeworktocapturetheregulatorymechanismsbychoosing1v.visthetypicalsizeatbirthandcapturesthestrengthofregulation467f(vb)=2vb00468strategy.=0correspondstothetimermodelwheredivisionoccursafteraconstanttime469frombirth,and=1isthesizerwhereacelldividesonreachingacriticalsize.=1/2can470beshowntobeequivalenttoanadderwheredivisioniscontrolledbyadditionofconstant471sizefrombirth[13].Fromhereon,wewouldbeusingthelengthofthecell(Lb,Ld,etc.)as472aproxyforsize(vb,vd,etc.).Toreiterate,thelengthgrowthisnotthesameascellvolume473growthasshowninAppendix1.AllofthevariabledefinitionsaresummarizedinTableS1.20

20Lbandl=Ld.Usingthis,wecanwritethedivisionstrategyf(l)474Wealsodefinelb=hLidhLibbb1475tobeld=f(lb)=2lb.Thetotaldivisionsizeobtainedwillbeacombinationoff(lb)and476noiseinthedivisiontiming,thesourceofwhichcouldbethestochasticityinbiochemical477reactionscontrollingdivision.Wewillassumethatdivisionisperfectlysymmetrici.e.,sizeatbirthinthe(n+1)th478generation(ln+1)ishalfofsizeatdivisioninthenthgeneration(ln).Usingthesizeadditive479bd480divisiontimingnoise(s(0;bd))andf(lb)specifiedabove,weobtain,s(0;bd)xn+1=(1)xn+ln1+2(1+x)1;(4)nn).Sizeatbirth(L)isnarrowlydistributed,hencel1andwecanwrite481wherexn=ln(lbbb482x=ln(lb)=ln(1+)whereisasmallnumber.Weobtainx1and,x=lb1:(5)483Thesizeadditivenoise,s(0;bd)isassumedtobesmallandhasanormaldistributionwith484mean0andstandarddeviationbd.Notethatbdisadimensionlessquantity.Sinces(0;bd)485isassumedtobesmallandxn1,wecanTaylorexpandthelasttermofEquation4to486firstorder,s(0;bd)xn+1(1)xn+:(6)2487Equation6showsarecursiverelationforcellsizeanditisagnosticofthemodeofgrowth.488Wewillshowlaterforexponentialgrowththatreplacingthesizeadditivenoisewithtime489additivenoisedoesnotchangethestructureofEquation6.21

214905.3Exponentialgrowth491Next,wewilltrytoobtainthegenerationtime(Td)inthecaseofexponentiallygrowing492cells.Forexponentialgrowth,thetimeatdivisionTdisgivenby,1LdTd=ln():(7)Lb493Forsimplicity,wewillassumeaconstantgrowthrate()withinthecell-cycle.Growthrate494isfixedatthestartofthecell-cycleandisgivenby=hi+hi(0;CV),wherehiis495themeangrowthrateand(0;CV)isassumedtobesmallwithanormaldistributionthat496hasmean0andstandarddeviationCV.CVdenotesthecoefficientofvariation(CV)of497thegrowthrate.Thiscapturesthevariabilityingrowthratewithincellsarisingfromthe498stochasticnatureofbiochemicalreactionsoccurringwithinthecell.4995.3.1Sizeadditivenoise500Herewewillcalculatethegenerationtimeusingthedivisionstrategyf(lb)andasizeadditive501divisiontimingnoise(s(0;bd))asdescribedpreviously.OnsubstitutingLd=(f(lb)+502s)hLbiintoEquation7weobtain,112lb+s(0;bd)Td=ln();(8)hi+hi(0;CV)lb503wherethesizeadditivenoise(s(0;bd))isGaussianwithmean0andstandarddeviation504bd.505Thenoises(0;bd)isassumedtobesmall,andweobtaintofirstorder,1s(0;bd)Tdln(2)xn+:(9)2(1+xn)122

221tofirstorder,506Sincexn0,onTaylorexpanding(1+xn)11s(0;bd)Tdln(2)xn+(1+(1)xn):(10)2507Assumingnoiseingrowthratetobesmallandexpandingtofirstorder,weobtain,1s(0;bd)Tdln(2)xnln(2)(0;CV)+:(11)hi2508Equation11givesthegenerationtimefortheclassofmodelswherebirthcontrolsdivision509undertheassumptionthatgrowthisexponential.5105.3.2Timeadditivenoise511Next,weensurethattherecursiverelationforsizeatbirthandtheexpressionforthe512generationtimegivenbyEquations6and11,respectively,arerobusttothenatureofnoise513assumed.Inthissection,thegenerationtimeisobtainedusingthedivisionstrategyf(lb)as514describedpreviouslyalongwithatimeadditivedivisiontimingnoise().Insuchacase,hi515Tdisobtainedtobe,1(0;n)Td=(ln(2)xn)+:(12)hi(0;n)516Thetimeadditivenoise,,isassumedtobesmallandhasanormaldistributionwithhin.Notethatisadimensionlessquantity.517mean0andstandarddeviationhin518Assumingnoiseingrowthratetobesmall,wefindTdtofirstordertobe,1Td(ln(2)xnln(2)(0;CV)+(0;n)):(13)hi519Equation13issameasEquation11,ifthetimeadditivenoiseterm,(0;n),inEquation23

2312isreplacedby(0;)=2.UsingEquation13,thevarianceinT(2)is,520sbddt122222nt=2ln(2)CV+:(14)hi2521Forexponentialgrowth,wealsofind,Ldln()=xn+1xn+ln(2)=Td:(15)Lb522OnsubstitutingEquation12intoEquation15weobtaintofirstorder,xn+1(1)xn+(0;n):(16)523Onreplacingthetimeadditivenoiseterm,(0;n),inEquation16withs(0;bd)=2,we524recovertherecursiverelationforsizeatbirthobtainedinthecaseofsizeadditivenoise525showninEquation6.Hence,themodelisinsensitivetonoisebeingsizeadditiveortime526additivewithasimplemappingforgoingfromonenoisetypetoanotherinthesmallnoise527limit.Atsteadystate,xhasanormaldistributionwithmean0andvariance2whosevalueis528x529givenby,22nx=:(17)(2)530WenotethatsomeofthederivationsabovehavealsobeenpresentedinRef.[16],butare531providedhereforcompleteness.24

245.4Predictingtheresultsofstatisticalconstructsappliedonln(Ld)532LbvshiTandhiTvsln(Ld)533ddLb5345.4.1ObtainingthebestlinearfitNext,wecalculatetheequationforthebestlinearfitforthechoiceofln(Ld)asy-axisand535Lb536hiTdasx-axisandviceversa.Forsimplicity,inthissection,wewillconsidertimeadditive537divisiontimingnoise.However,theresultsobtainedherewillholdforsizeadditivenoiseas538wellbecausethemodelisrobusttothetypeofnoiseaddedasshownintheprevioussection.First,wecalculatethecorrelationcoefficient()forln(Ld)andtimeofdivisionT,539expLdbh(ln(Ld)hln(Ld)i)(ThTi)iLbLbddexp=;(18)ltLd).UsingEquations15and16weobtain,540wherelisthestandarddeviationinln(LbLdln()ln(2)xn+(0;n):(19)LbSubstitutingEquations13and19intothenumeratorofEquation18,LdLdh(ln()hln()i)(TdhTdi)iLbLb(xnln(2)(0;CV)+(0;n))=h(xn+(0;n))i:(20)hi541Astheterms(0;n),(0;CV)andxnareindependentofeachother,h(0;CV)(0;n)i=5420,h(0;CV)xni=0andhxn(0;n)i=0.Equation20simplifiesto,LdLd2221h(ln()hln()i)(TdhTdi)i=(x+n):(21)LbLbhi25

25Ld)obtainedusingEquation19is,543Thevarianceofln(Lb222222nl=x+n=:(22)2544InsertingEquations14,21and22intoEquation18,weget,vuu1exp=t(1)ln2(2)CV2:(23)1+22n545Theslopeofalinearregressionlineisgivenby,ym=;(24)x546wherex,yandarethestandarddeviationofthex-variable,thestandarddeviationof547they-variableandthecorrelationcoefficientofthe(x,y)pair,respectively.Theinterceptis,c=hyimhxi:(25)Onthex-axis,weplothiTandthey-axisischosenasln(Ld).Theslopeforthischoice548dLb549(mtl)canbecalculatedby,lmtl=exp:(26)thi550Onsubstitutingthevaluesweget,1mtl=(1)ln2(2)CV2:(27)1+22n551OnlyforCVnwewouldexpectaslopecloseto1.26

26Ld)vshiTplotisgivenby,552Theintercept(ctl)fortheln(Ldb01Ld1ctl=hln()imtlhhiTdi=ln(2)@1(1)ln2(2)CV2A:(28)Lb1+22nHowever,ifwechoosethex-axisasln(Ld)andthey-axisischosenashiT,weobtainthe553Ldb554slopemlt,thimlt=exp:(29)l555Onsubstitutingthevaluesweobtainmlt=1independentofthenoiseparametersandfind556thattheinterceptiszero.5575.4.2Non-linearityinbinneddataIntheMaintext,fortheplotln(Ld)vshiT,wefindthebinneddatatobenon-linear(see558Ldb559Figure2CoftheMaintext).Inthissection,weexplainthenon-linearityobservedusingthe560modeldevelopedintheprevioussections.561Binningdatabasedonthex-axismeanstakinganaverageofthey-variableconditioned562onthevalueofthex-variable.Mathematically,thisamountstocalculatingE[yjx]i.e.,563theconditionalexpectationofthey-variablegiventhatxisfixed.Inourcase,weneedtoLd)jhiT].ln(Ld)=Tbydefinitionofexponentialgrowth,hence,564calculateE[ln(LdLdbbLdE[ln()jhiTd]=E[TdjhiTd]:(30)Lb565SinceTdisfixed,thisisequivalenttocalculatingE[jTd].UsingEquation13,R1R1R1ln(2)xln(2)p(x;;)(Td(+)dxdd111hihihihiE[jTd]=R1R1R1ln(2)xln(2):(31)p(x;;)(Td(+)dxdd111hihihihip(x;;)isthejointprobabilitydistributionofxandnoiseparametersand.Since,they27

27areindependentofeachother,thejointdistributionisproductoftheindividualdistributionsf1(x),f2()andf3(),thedistributionsbeingGaussianwithmean0andstandarddeviationx,CVandn,respectively.x,narerelatedbyEquation17.Sincex,,andarenarrowlydistributedaroundzero,thecontributionfromlargepositiveornegativevaluesisextremelysmall.ThisensuresthatTdisalsoclosetoitsmeanandnon-negativedespitethelimitsoftheintegralbeing1to1.Using=hi+hi(0;CV)inEquation31,E[jTd]R1R1R1ln(2)xln(2)!111f1(x)f2()f3()(Td(hihihi+hi))dxdd=hi1+RRR:111f(x)f()f()(T(ln(2)xln(2)+))dxdd111123dhihihihi(32)566Onevaluatingtheintegrals,weobtain,01hiTd1ln(2)E[jTd]=hi@1+22A:(33)1+2n1+2n2CV2ln2(2)2CV2ln2(2)567Thus,thetrendofbinneddataisfoundtobe,01hiTdLd@1+1ln(2)A:(34)E[ln()jhiTd]=hiTd2222Lb1+n1+n2CV2ln2(2)2CV2ln2(2)568IntheregimeCVn,thelasttwotermsontheRHSofEquation34vanishandthe569binneddatafollowsthetrendy=x.ForthehiTvsln(Ld)plot,weneedtocalculateE[hiTjln(Ld)].UsingEquations13570dLdLbb571and19,weobtain,LdhiTd=ln()ln(2)(0;CV):(35)Lb28

28ln(Ld)isindependentof(0;CV).Usingthis,wecanwriteE[hiTjln(Ld)]as,LbdLbLdE[hiTdjln()]LbRR11(hiT)f()f(ln(Ld))hiT(ln(Ld)ln(2))d(hiT)d11d24LbdLbd=:(36)f(ln(Ld))4Lb572NotethattheintegraloverhiTdgoesfrom1to1althoughhiTdcannotbenegative.573Asbefore,thisisnotanissuebecauseweassumehiTdtobetightlyregulatedaroundln(2)andthecontributiontotheintegralfrom1to0isnegligible.f(ln(Ld))denotesthe5744Lbprobabilitydistributionforln(Ld),thedistributionbeingGaussianwithmeanln(2),and575Lb576standarddeviationlwhichiscalculatedinEquation22.PuttingtheGaussianformof577f2()intotheintegralandsimplifyingweget,LdLdE[hiTdjln()]=ln():(37)LbLbLd)]=ln(Ld).Thisis578ThetrendofbinneddatatofirstorderinnoiseandxisE[hiTdjln(LLbb579showninFigure2DoftheMaintextwherethebinneddatafollowsthey=xline.5805.5Lineargrowth581Inthissection,wewillfocusonfindingtheequationofthebestlinearfitforrelevantplots582inthecaseoflineargrowth.Thetimeatdivisionforlineargrowthisgivenby,LdLbTd=:(38)0Notethat0hasunitsof[length/time]andisdefinedastheelongationspeed.Thisis583584differentfromtheexponentialgrowthratewhichhasunits[1/time].Here,wewillworkwith29

29585thenormalizedlengthatbirth(lb)anddivision(ld),ldlbTd=:(39)lin586Considerthenormalizedelongationspeedtobelin=hlini+hlinilin(0;CV;lin),where587hliniisthemeannormalizedelongationspeedforalineageofcellsandlin(0;CV;lin)is588normallydistributedwithmean0andstandarddeviationCV;lin.Thus,theCVofelongation589speedisCV;lin.Theregulationstrategywhichthecellundertakesisequivalenttothatin590previoussectionsandisgivenbyg(lb)=2+2(1)(lb1).Notethatwecanobtaing(lb)591byTaylorexpandingf(lb)aroundlb=1.Usingtheregulationstrategyg(lb)andaddinga592sizeadditivenoises(0;bd)whichisindependentoflb,wefind,2+2(1)(ln1)+(0;)lnT=bsbdb:(40)dhlini(1+lin(0;CV;lin))593Notethatwechosesizeadditivedivisiontimingnoise(s(0;bd))forconvenienceinthis594section.However,itcanbeshownasdonepreviouslythatthemodelisrobusttothenoise595indivisiontimingbeingsizeadditiveortimeadditive.Assumingthatthenoiseterms596lin(0;CV;lin)ands(0;bd)aresmall,weobtaintofirstorder,(12)(lb1)+1+s(0;bd)lin(0;CV;lin)Td:(41)hlini597Thetermslb,s(0;bd)andlin(0;CV;lin)areindependentofeachother.Thestandard598deviationofTd(t)canbecalculatedtobe,(12)22+2+CV22bbd;lint=:(42)hlini230

30Assumingperfectlysymmetricdivisionandusingln=g(ln)+(0;),wefindtherecursive599dbsbdrelationforlntobe,600bnnn+1nnldlb=2lblb=(12)lb+2+s(0;bd):(43)NotethatEquation43isthesameasEquation6undertheapproximationx=ln1.At601nb602steadystate,thestandarddeviationoflbisdenotedbybandusingEquation43itsvalue603isobtainedtobe,22bdb=:(44)4(2)604Similarly,thestandarddeviationofld-lb,orequivalentlylinTd,denotedbyl;lin,iscalculated605tobe,24+12l;lin=bd:(45)4(2)Forlineargrowth,anaturalplotisl-lvshiT(reminiscentoftheln(Ld)vshiTplot606dblindLdb607forexponentialgrowth).Tocalculatetheslopeofthebestlinearfit,wehavetocalculate608thecorrelationcoefficientlingivenby,h(ldlbhldlbi)(hliniTdhhliniTdi)ilin=:(46)hlinil;lint609Againusingtheindependenceoftermslb,s(0;bd)andlin(0;CV;lin)fromeachother,we610get,(12)22+2=bbd=l;lin:(47)linhlinil;linthlinit611TheslopeofbestlinearfitfortheplotldlbvshliniTdisgivenby,l;lin1mtl;lin=lin=2:(48)CV4(2)hlinit1+;lin2(4+1)bd31

31612Theinterceptctl;linisfoundtobe,1ctl;lin=hldlbimtl;linhhliniTdi=1CV24(2):(49)1+;lin2(4+1)bd613Onflippingtheaxis,theslope(mlt;lin)fortheplothliniTdvsldlbisobtainedtobe,hlinitmlt;lin=lin=1:(50)l;lin614Theinterceptclt;linisfoundtobe,clt;lin=hhliniTdimlt;linhldlbi=0:(51)615ThebestlinearfitforthehliniTdvsldlbplotfollowsthetrendy=x.616Simulationsoftheaddermodelforlinearlygrowingcellswerecarriedout.Thedeviation617ofthebestlinearfitfortheldlbvshliniTdplotfromthey=xlineisshowninFigure3-618figuresupplement1A,whileinFigure3-figuresupplement1B,thebestlinearfitfortheplot619hliniTdvsldlbisshowntoagreewiththey=xline.6205.6Differentiatinglinearfromexponentialgrowth621Inthissection,weexploretheequationforthebestlinearfitofhliniTdvsldlbplotinthecaseofexponentialgrowthandhiTvsln(Ld)plotforlineargrowth.Intuitively,we622dLb623expectthebestlinearfitinbothcasestodeviatefromthey=xline.Inthissection,wewill624calculatethebestlinearfitexplicitly.Surprisingly,wewillfindthat,inthecaseoflinearLd)plotfollowsthey=xlineclosely.625growth,thebestlinearfitforthehiTdvsln(Lb626Letusbeginwithexponentialgrowthwithgrowthrate,=hi+hi(0;CV)as627definedpreviously.Again,(0;CV)hasanormaldistributionwithmean0andstandard32

32628deviationCV,itbeingtheCVofthegrowthrate.ThetimeatdivisionisgivenbyEquationln(2)ln(2)6297.Theaveragegrowthratehi=hi.Forexponentialgrowth,wewillplotTdhTdi630hliniTdvsldlb.Aspreviouslydefined,hliniisthemeannormalizedelongationspeedandhi=h1i1.hiisrelatedtohiby,631linThTilinddhihlini=:(52)ln(2)632ldlbcanbecalculatedbyusingtheregulationstrategyf(lb)introducedinSection5.2and633anormallydistributedsizeadditivenoises(0;bd).Notethatwehavechosenthenoisein634divisiontimingtobesizeadditive.However,themodelisrobusttothechoiceoftypeof635noiseasweshowedinSection5.3.UsingEquations5and6weobtain,nnldlb1+(12)xn+s(0;bd):(53)636UsingEquation11,hliniTdisobtainedtobe,xs(0;bd)hliniTd=1(0;CV)+:(54)ln(2)2ln(2)637Tocalculatetheexpressionformlt;lin,theslopeofthebestlinearfitforhliniTdvsldlbplot,638wefirstcalculatelingivenbyEquation46.Theexpressionforl;lin(standarddeviationof639ldlb)andt(standarddeviationofTd)arefoundtobe,2222l;lin=(12)x+bd;(55)64021x22bd2t=2()+CV+():(56)hliniln(2)2ln(2)bd.Using641xisrelatedtonviaEquation17.InSection5.3,wealsoshowedthatn=233

33642these,wecanwrite,22bdx=:(57)4(2)643Nowusingtheexpressionsfort,l;linandthefactthatx,(0;CV)ands(0;bd)are644independentofeachother,weget,(21)22x+bdln(2)2ln(2)lin=:(58)hlinil;lint645FortheplothliniTdvsldlb,theslopemlt;linisgivenby,(21)22x+bdthliniln(2)2ln(2)mlt;lin=lin=2:(59)l;linl;linInsertingEquation55intoEquation59andsubstituting2givenbyEquation57,weobtain,646x13mlt;lin=:(60)ln(2)4+1647Theinterceptclt;linisfoundtobe,13clt;lin=hhliniTdimlt;linhldlbi=1:(61)ln(2)4+1Fortheaddermodel(=1),wegetthevalueofslopem=10:7213andintercept6482lin;lt2ln(2)c=110:279.Thisisdifferentfromthebestlinearfitobtainedforsame649lin;lt2ln(2)650regulatorymechanismcontrollingdivisioninlinearlygrowingcellswherewefoundthatthe651bestlinearfitfollowsthey=xline.Intuitively,weexpectthebestlinearfitofhliniTdvs652ldlbplottodeviatefromy=xlineinthecaseofexponentialgrowth.Weshowedanalytically653thatforaclassofmodelswherebirthcontrolsdivision,itisindeedthecase.Thisisalso654shownusingsimulationsoftheaddermodelinFigure3-figuresupplement1C.34

34Ld)plottofollowthey=x655InSection5.4.1,wefoundthebestlinearfitforhiTdvsln(Lb656lineforexponentiallygrowingcellswheredivisionisregulatedbybirtheventviaregulationstrategyf(l).Next,wecalculatetheequationforthebestlinearfitofhiTvsln(Ld)657bdLb658plotgivengrowthislinear.ThemodelfordivisioncontrolwillbesameasthatinSection6595.5i.e.,theregulationstrategyfordivisionisgivenbyg(lb)=2+2(1)(lb1)which660isalsoequivalenttof(lb).Thelinearlygrowingcellsgrowwithelongationspeedlin=661hlini(1+lin(0;CV;lin)).Asdiscussedbefore,lin(0;CV;lin)hasanormaldistributionwith662mean0andstandarddeviationCV;lin,itbeingtheCVoftheelongationspeed.Using663Equations5and6,weget,Ldns(0;bd)ln()=ln(2)x+:(62)Lb2664UsingEquations5and52,weobtainfromEquation41,hiTd=ln(2)+(12)ln(2)x+ln(2)s(0;bd)ln(2)lin(0;CV;lin):(63)Sincex,(0;CV)and(0;)areuncorrelated,thestandarddeviationofln(Ld)and665lin;linsbdLb666Tddenotedbylandtrespectivelyarecalculatedtobe,2222bdl=x+;(64)466722ln(2)2222t=2((12)x+bd+CV;lin):(65)hiWecalculatethecorrelationcoefficientforthepair(ln(Ld),hiT).Sincethecorrelation668Ldb669coefficientisunaffectedbymultiplyingoneofthevariableswithapositiveconstant,wecancalculatethecorrelationcoefficientforthepair(ln(Ld),T)orasgivenbyEquation18.670Ldexpb35

35671Usingtheindependenceoftermsx,lin(0;CV;lin)ands(0;bd),2ln(2)(2(21)+bd)x2exp=:(66)hiltFortheplothiTvsln(Ld),theslopemofthebestlinearfitisgivenby,672dLltb2hiln(2)(2(21)+bd)tx2mlt=exp=2:(67)ll673InsertingEquation64intoEquation67andusingEquation57,weget,3mlt=ln(2)1:0397:(68)2Similarlytheintercept(c)fortheplothiTvsln(Ld)isfoundtobe,674ltdLbLd3clt=hhiTdimlthln()i=ln(2)(1ln(2))0:0275:(69)Lb2675Thisisveryclosetoy=xtrendobtainedforthesameregulatorymechanismcontrolling676divisioninexponentiallygrowingcells(Figure3A).6775.7Growthratevsageandelongationspeedvsageplots.Intheprevioussections,wefoundthatbinningandlinearregressionontheplotln(Ld)vs678Lb679hiTd,andtheplotobtainedbyinterchangingtheaxes,wereinadequatetoidentifythemode680ofgrowth.Inthissection,wetrytovalidatethegrowthratevsageplotasamethodto681elucidatethemodeofgrowth.682Inadditiontocellsizeatbirthanddivisionandthegenerationtime,cellsizetrajectories683(cellsize,Lvstimefrombirth,t)wereobtainedformultiplecellcycles.Inourcase,thecell684sizetrajectorieswerecollectedeitherviasimulations(inFigure3B)orfromexperiments(for36

36685Figures4A-4C)atintervalsof4min.Notethatifthemeasurementsweretobecarriedout686atequallengthintervalsinsteadoftime,theresultsdiscussedinthepaperwouldstillremainunchanged.Foreachtrajectory,growthrateattimetoragetiscalculatedas1L(t+
t)L(t)687TdL(t)
t688where
tisthetimebetweenconsecutivemeasurements.ToobtainelongationspeedvsL(t+
t)L(t)689ageplots,theformulabeforeneedstobereplacedwith.Thegrowthrateis
t690interpolatedtocontain200pointsatequalintervalsoftimeforeachcelltrajectory.The691growthratetrendsappeartoberobustwithregardstoadifferentnumberofinterpolated692points(from100to500points).Toobtainthegrowthratetrendasafunctionofcellage,we693usethemethodpreviouslyappliedinRef.[37].Inthismethod,growthrateisbinnedbased694onageforeachindividualtrajectory(50bins)andtheaveragegrowthrateisobtainedin695eachofthebins.Thebinneddatatrendforgrowthratevsageisthenfoundbytakingthe696averageofthegrowthrateineachbinoveralltrajectories.Binningthegrowthrateforeach697trajectoryensuresthateachtrajectoryhasanequalcontributiontothefinalgrowthrate698trendsoastoavoidinspectionbias.Thisstepisespeciallyimportantwhendatacollected699atequalintervalsoftimeisanalyzed.Insuchacase,cellswithlargergenerationtimes700haveagreaternumberofmeasurementsthancellswithsmallergenerationtimes.Obtaining701thegrowthratetrendwithoutbinninggrowthrateforeachtrajectorywouldhavebiased702thebinneddatatrendforthegrowthratevsageplottoasmallervaluebecauseofover-703representationbyslower-growingcells(orequivalentlycellswithlongergenerationtime).704Thisbiastowardslowergrowthratevaluesinthegrowthratevsageplotsisaninstanceof705inspectionbias.706InFigures4A-4C,wefindthegrowthrateobtainedfromE.coliexperimentstochange707withinthecellcycle.Inthetwoslowergrowthmedia(Figures4A,4B),thegrowthrateis708foundtoincreasewithcellagewhileforthefastestgrowthmedia(Figure4C)thegrowth709ratefollowsanon-monotonicbehavioursimilartothatobservedinRef.[37]forB.subtilis.710AbruptchangesingrowthratearereportedatconstrictioninRefs.[39,40].Wefindthatthe37

37711growthratechangesstartbeforeconstrictioninthetwoslowergrowthconditionsconsidered.712Onepossibilityisthatthisincreaseisduetopreseptalcellwallsynthesis[55].Preseptalcell713wallsynthesisdoesnotrequireactivityofPBP3(FtsI)butinsteadreliesonbifunctional714glycosyltransferasesPBP1AandPBP1BthatlinktoFtsZviaZipA.Onehypothesisthat715canbetestedinfutureworksisthatattheonsetofconstriction,activityfromPBP1A716andPBP1BstartstograduallyshifttothePBP3/FtsWcomplexandthereforenoabrupt717changeingrowthrateisobserved.Inthefastestgrowthcondition(glucose-casmedium),we718findthattheincreaseingrowthrateapproximatelycoincideswithonsetofconstriction,in719agreementwiththepreviousfindings[39,40].720InFigures4A-4C,thegrowthratetrendsarenotobtainedforageclosetoone.This1L(Td+
t)L(Td)andthisrequiresknowing721isbecausegrowthrateatage=1isgivenbyL(Td)
t722thecelllengthsbeyondthedivisionevent(L(Td+
t)).Toestimategrowthratesatage723closetoone,weapproximateL(Td+
t)tobethesumofcellsizesofthetwodaughter724cells.Inordertominimizeinspectionbias,weconsideredonlythosecellsizetrajectories725whichhadL(t)datafor12minafterdivision(correspondingtoanageofapproximately7261.1).However,thegrowthratetrendsinallthreegrowthmediawererobustwithregardsto727adifferenttimeforwhichL(t)wasconsidered(4minto20minafterdivision).Weusethe728binningprocedurediscussedbeforeinthissection.Tovalidatethismethod,weappliedit729onsyntheticdataobtainedfromthesimulationsofexponentiallygrowingcellsfollowingthe730adderandtheadderperoriginmodel.Cellswereassumedtodivideinaperfectlysymmetric731mannerandbothofthedaughtercellswereassumedtogrowwiththesamegrowthrate,732independentofthegrowthrateinthemothercell.Thegrowthratetrendsforthetwo733modelsconsidered(adderandadderperorigin)areexpectedtobeconstantevenforcellage734>1.Wefoundthatthegrowthratetrendswereindeedapproximatelyconstantasshownin735Figure4-figuresupplement1D.Wealsoconsideredlineargrowthwithdivisioncontrolledvia736anaddermodel.Thedaughtercellswereassumedtogrowwiththesameelongationspeed,38

38737independentoftheelongationspeedinthemothercell.Inthiscase,weexpecttheelongation738speedtrendtobeconstantforcellage>1.Thisisindeedwhatweobservedasshowninthe739insetofFigure4-figuresupplement1D.WeusedthismethodonE.coliexperimentaldata740andfoundthatthegrowthratetrendsobtainedforthethreegrowthconditions(Figure4-741figuresupplements1A-1C)wereconsistentwiththatshowninFigures4A-4Cintherelevant742ageranges.Forcellageclosetoone,wefoundthatthegrowthratedecreasedtoavalue743closetothegrowthratenearcellbirth(age0)forallthreegrowthconditionsconsidered.744Insummary,wefindthatthegrowthratevsageplotsareaconsistentmethodtoprobe745themodeofcellgrowthwithinacellcycle.7465.8Growthratevstimefromspecificeventplotsareaffectedby747inspectionbias748Toprobethegrowthratetrendinrelationtoaspecificcellcycleevent,forexamplecellbirth,749growthratevstimefrombirthplotsareobtainedforsimulationsofexponentiallygrowing750cellsfollowingtheaddermodel.Inthegrowthratevstimefrombirthplot,therateisfound751tostayconstantandthendecreaseatlongertimes(Figure3-figuresupplement2C)even752thoughcellsareexponentiallygrowing.Becauseofinspectionbias(orsurvivorbias),atlater753times,onlythecellswithlargergenerationtimes(orslowergrowthrates)“survive”.The754averagegenerationtimeofthecellsaverageduponineachbinofFigure3-figuresupplement7552CisshowninFigure3-figuresupplement2D.ThedecreaseingrowthrateinFigure3-756figuresupplement2Coccursaroundthesametimewhenanincreaseingenerationtimeis757observedinFigure3-figuresupplement2D.Thus,thetrendingrowthrateisbiasedtowards758lowervaluesatlongertimes.Theproblemmightbecircumventedbyrestrictingthetimeon759thex-axistothesmallestgenerationtimeofallthecellcyclesconsidered[31].760Tocheckforgrowthratechangesatconstriction,weusedplotsofgrowthratevstime39

39761fromconstriction(tTn).GrowthratetrendsobtainedfromE.coliexperimentaldatashow762adecreaseattheedgesoftheplots(Figure4-figuresupplements2A,2C,and2E).These763deviatefromthetrendsobtainedusingthegrowthratevsageplots(Figures4A-4C).To764investigatethisdiscrepancy,weuseamodelwhichtakesintoaccounttheconstrictionand765thedivisionevent.Currentlyitisunknownhowconstrictionisrelatedtodivision.Forthe766purposeofmethodsvalidation,weuseamodelwherecellsgrowexponentially,constriction767occursafteraconstantsizeadditionfrombirth,anddivisionoccursafteraconstantsize768additionfromconstriction.Notethatothermodelswhereconstrictionoccursafteraconstant769sizeadditionfrombirthwhiledivisionoccursafteraconstanttimefromconstriction,aswell770asamixedtimer-addermodelproposedinRef.[40],leadtosimilarresults.Weexpectthe771growthratetrendtobeconstantforexponentiallygrowingcells.However,wefindusing772numericalsimulationsthatitdecreasesattheplotedgesbothbeforeandaftertheconstriction773event(Figure3-figuresupplement2A).Thisdecreasecanbeattributedtoinspectionbias.774Theaveragegrowthrateintimebinsattheextremesarebiasedbycellswithsmallergrowth775rates.ThisisshowninFigure3-figuresupplement2Bwheretheaveragegenerationtime776forthecellscontributingineachofthebinsofFigure3-figuresupplement2Aisplotted.777Thetimeatwhichthegrowthratedecreasesonbothsidesoftheconstrictioneventisclose778tothetimeatwhichtheaveragegenerationtimeincreases.Forexample,inalaninemedium,779thegenerationtimeforeachofthebinsisplottedinFigure4-figuresupplement2B.The780averagegenerationtimeforthecellscontributingtoeachofthebinsisalmostconstantfor781thetimingsbetween-80minto20min.Thus,forthistimerangethechangesingrowthrate782arenotbecauseofinspectionbiasbutarearealbiologicaleffect.Thebehaviorofgrowth783ratewithinthistimerangeinFigure4-figuresupplement2Aisinagreementwiththetrend784ingrowthratevsageplotofFigure4A.Onaccountingforinspectionbias,thegrowthrate785vsageplotsagreewiththegrowthratevstimefromconstrictionplotsinothergrowthmedia786aswell(Figure4-figuresupplement2C,Figure4-figuresupplement2E).Thus,growthrate40

40787vstimeplotsarealsoaconsistentmethodtoprobegrowthratemodulationinthetimerange788whenavoidingtheregimespronetoinspectionbias.7895.9Resultsofelongationspeedvssizeplotsaremodel-dependent.790Cellsassumedtoundergoexponentialgrowthhaveelongationspeedproportionaltotheir791size.Inthecaseofexponentialgrowth,thebinneddatatrendoftheplotelongationspeedvs792sizeisexpectedtobelinearwiththeslopeofthebestlinearfitprovidingthevalueofgrowth793rateandinterceptbeingzero.Inthissection,weusethesimulationstotestifbinningand794linearregressionontheelongationspeedvssizeplotsaresuitablemethodstodifferentiate795exponentialgrowthfromlineargrowth[41].796Totestthemethod,wegeneratecellsizetrajectoriesusingsimulationsoftheaddermodel797withasizeadditivedivisiontimingnoiseandassumingexponentialgrowth.ElongationspeedL(t+
t)L(t)798atsizeL(t)iscalculatedforeachtrajectoryaswhere
tisthetimebetween
t799consecutivemeasurements(=4mininourcase).Eachtrajectoryisbinnedinto10equally800sizedbinsbasedontheircellsizesandtheaverageelongationspeedisobtainedforeachbin.801Thefinaltrendofelongationspeedasafunctionofsizeisthenobtainedbybinning(based802onsize)thepooledaverageelongationspeeddataofallthecellcycles.803Wefindthatthebinneddatatrendislinearwiththeslopeofthebestlinearfitclosetothe804averagegrowthrateconsideredinthesimulations(Figure3-figuresupplement3D).Thisis805inagreementwithourexpectationsforexponentialgrowth.Inordertocheckifthismethod806coulddifferentiatebetweenexponentialgrowthandlineargrowth,weusedsimulationsof807theaddermodelundergoinglineargrowthtogeneratecellsizetrajectoriesformultiplecell808cycles.Forlineargrowth,elongationspeedisexpectedtobeconstant,independentofits809cellsize.Thebinneddatatrendfortheelongationspeedvssizeplotisalsoobtainedtobe810constantforthesimulationsoflinearlygrowingcells(Figure3-figuresupplement3B).The811interceptofthebestlinearfitobtainedisclosetotheaverageelongationspeedconsideredin41

41812thesimulations.Thebinneddatatrendforlinearandexponentialgrowthareclearlydifferent813asshowninFigure3-figuresupplement3BandFigure3-figuresupplement3D,respectively,814andthisresultholdsforabroadclassofmodelswherethedivisioneventiscontrolledby815birthandthegrowthrate(forexponentialgrowth)/elongationspeed(forlineargrowth)is816distributednormallyandindependentlybetweencell-cycles.817Next,weconsidertheadderperorigincellcyclemodelforexponentiallygrowingcells818[17].Inthismodelspace,thecellinitiatesDNAreplicationbyaddingaconstantsizeper819originfromthepreviousinitiationsize.Thedivisionoccursonaverageafteraconstanttime820frominitiation.Forexponentiallygrowingcells,thebinneddatatrendisstillexpectedtobe821linearasbefore.Instead,wefindusingsimulationsthatthetrendisnon-linearanditmight822bemisinterpretedasnon-exponentialgrowth(Figure3-figuresupplement3F).823Thus,theresultsofbinningandlinearregressionfortheplotelongationspeedvssizeis824model-dependent.8255.10Interchangingaxesingrowthratevsinversegenerationtime826plotmightleadtodifferentinterpretations.827Sofar,ourdiscussionwasfocusedonthequestionofmodeofsingle-cellgrowth.Arelated1).828problemregardstherelationbetweengrowthrate()andtheinversegenerationtime(Td829Onapopulationlevel,thetwoareclearlyproportionaltoeachother.However,single-cell830studiesbasedonbinningshowedanintriguingnon-lineardependencebetweenthetwo,with831thetwovariablesbecominguncorrelatedinthefaster-growthmedia.[25,56].Withinthe1flattenedoutforfasterdividing832samemedium,thebinneddatacurvefortheplotvsTd833cells.Thetrendinthebinneddatawasdifferentfromthetrendofy=ln(2)xlineasobserved834forthepopulationmeans.Apriorionemightspeculatethattheflatteninginfasterdividing835cellscouldbebecausethefasterdividingcellsmighthavelesstimetoadapttheirdivision42

42836ratetotransientfluctuationsintheenvironment.Kennardetal.[56]insightfullyalsoplotted1vsandfoundacollapseofthebinneddataforallgrowthconditionsontothey=1x837Tdln(2)line.Theseresultsarereminiscentofwhatwepreviouslyshowedfortherelationofln(Ld)838Lb839andhiTd.840Inthefollowing,wewillelucidatewhythisoccursinthiscaseusinganunderlyingmodel841andpredictingthetrendbasedonit.Weusesimulationsoftheaddermodelundergoing842exponentialgrowth.Theparametersforsizeaddedinacellcycleandmeangrowthrates843areextractedfromtheexperimentaldata.CVofgrowthrateisassumedlowerinfaster-844growthmediaasobservedbyKennardetal.Usingthismodel,wecouldobtainthesame845patternofflatteningatfaster-growthconditionsthatisobservedintheexperiments(Figure1followstheexpectedy=ln(2)x8462-figuresupplement2A).ThepopulationmeanforandTd847equation(shownasblackdashedline)aswasthecaseinexperiments.Intuitively,sucha848departurefromtheexpectedy=ln(2)xlineforthesinglecelldatacanagainbeexplainedby849determiningtheeffectofnoiseonvariablesplottedonbothaxes.AspreviouslystatedTdis850affectedbybothgrowthratenoiseandnoiseindivisiontimingwhilegrowthratefluctuates851independentlyofothersourcesofnoise.Thisdoesnotagreewiththeassumptionforbinning852asnoiseindivisiontimingaffectsthex-axisvariableratherthanthey-axisvariable.Insuch853acase,thetrendinthebinneddatamightnotfollowtheexpectedy=ln(2)xline.However,854oninterchangingtheaxes,wewouldexpecttheassumptionsofbinningtobemetandthe1xline(Figure2-figuresupplement2B).855trendtofollowthey=ln(2)8565.11Dataandsimulations8575.11.1Experimentaldata858ExperimentaldataobtainedbyTanouchietal.[10]wasusedtoplotLdvsLbshowninFigure1A.E.colicellsweregrownat25Cinamothermachinedeviceandthelengthat85943

43860birthanddivisionwerecollectedformultiplecellcycles.LdvsLbplotwasobtainedusing861thesecellsandlinearregressionperformedonitprovidedabestlinearfit.862DatafromrecentmothermachineexperimentsonE.coliwasusedtomakeallother863plots.DetailsareprovidedinSection5.1andRef.[32].Theexperimentswereconductedat28Cinthreedifferentgrowthconditions-alanine,glycerolandglucose-cas(alsoseeSection8648655.1).Cellsizetrajectorieswerecollectedformultiplecellcyclesandallofthedatacollected866wereconsideredwhilemakingtheplotsinthepaper.8675.11.2Simulations868MATLABR2021awasusedforsimulations.Simulationsoftheaddermodelforexponentially869growingcellswerecarriedoutoverasinglelineageof2500generations(Figures2C,2D,870Figure3-figuresupplement1C).Themeanlengthaddedbetweenbirthanddivisionwas871setto1.73minlinewiththeexperimentalresultsforalaninemedium.Growthratewas872variableandsampledfromanormaldistributionatthestartofeachcellcycle.Themeanln(2)873growthratewassettohTdi,wherehTdi=212minandcoefficientofvariation(CV)=CV874=0.15.Thenoiseindivisiontimingwasassumedtobetimeadditivewithmean0andn,where=0.15.Thebinningdatatrendsandthebestlinearfits875standarddeviationhin876obtainedusingthesesimulationscouldbecomparedwiththeanalyticalresultsobtainedin877Sections5.4.2and5.6.878Forsimulationsoflineargrowth(Figures3A-3B,Figure3-figuresupplements1A,1B,3A,hLdLbi,withthevalues8793B,Figure4-figuresupplement1D),themeangrowthratewassettohTdi880ofhLdLbiandhTdiusedasmentionedpreviously.Thenoiseindivisiontimingwassize881additivewithstandarddeviation=0.15hLbi.Noisewasalsoconsideredtobesizeadditive882withthesamestandarddeviationforthesimulationsofexponentiallygrowingcellsshown883inFigure3B,Figure3-figuresupplements2C,3C,3D,andFigure4-figuresupplement1D.884Inthesimulationsofsuper-exponentialgrowthcarriedoverasinglelineageof2500gen-44

44885erations(Figure3B),thecellsinitiallygrewexponentiallybutinthelaterstagesofthecell886cycle,thegrowthrateincreasedas,d=2k(ttc);(70)dtwherekwasfixedtobe2andtwasthetimefrombirthatwhichthegrowthratechanged887T3cd888fromexponentialtosuper-exponentialgrowth.tcwasfixedtobehalfofthegenerationtime889ofthecellorequivalentlyanageof0.5.Thedivisionsizewassetbytheaddermodelwitha890timeadditivenoisewithsimilarparametersasbeforeforexponentialgrowthsimulations.The891exponentialgrowthrateatthestartofeachcellcyclewasdrawnfromanormaldistributionln(2)min1andCV=0.15.892withmeansetto242893ForFigure3B,Figure3-figuresupplements3E,3F,Figure4-figuresupplement1D,894simulationswerecarriedoutoveralineageof2500generationsforexponentiallygrowingcells895followingtheadderperoriginmodel.Inthesimulations,thetimeincrementis0.01min.896TheinitialconditionforthesimulationsisthatcellsarebornandinitiateDNAreplication897attimet=0buttheresultsareindependentofinitialconditions.Thenumberoforiginsis898alsotrackedthroughoutthesimulationsbeginningwithaninitialvalueof2.Cellsdivide899intotwodaughtercellsinaperfectlysymmetricalmanner(nonoiseindivisionratio),and900oneofthedaughtercellsisdiscardedforthenextcellcycle.Insimulations,thegrowthrate901wasfixedwithinacellcyclebutvariedbetweendifferentcellcycles.Ondivision,thegrowth902rateforthatcellcyclewasdrawnfromanormaldistributionwithmeanhiandcoefficientof903variation(CV)whosevalueswerefixedusingtheexperimentaldatafromalaninemedium.904Thetotallengthatwhichthenextinitiationhappensisdeterminedby,tot;nextLi=Li+O
ii;(71)905where
iiisthelengthaddedperoriginandOisthenumberoforigins.Todetermine45

45tot;next906Li,
iiwasdrawnonreachinginitiationlengthfromanormaldistribution.Themean907andCVof
iiwasobtainedfromexperimentsdoneinalaninemedium.Intheadderper908originmodel,divisionhappensafteraC+Dtimefrominitiation.Thedivisionlength(Ld)909isobtainedtobe,(C+D)Ld=Lie:(72)910Inthesimulations,oncetheinitiationlengthwasreached,thecorrespondingdivisionoc-911curredatimeC+Dafterinitiation.C+Dtimingsforeachinitiationeventwereagaindrawn912fromanormaldistributionwiththesamemeanandCVasthatoftheexperimentsinalanine913medium.914ForFigure3-figuresupplement2A,cellswereassumedtogrowexponentiallyinthe915simulations.Theconstrictionlength(Ln)wassettobe,Ln=Lb+
bn:(73)916Thelengthadded(
bn)wasassumedtohaveanormaldistributionwiththemeanlength917addedbetweenbirthandconstrictionsetto1.18mandtheCV=0.23,inlinewiththe918experimentalresultsforalaninemedium.Thelengthatdivisionwassetas,Ld=Ln+
nd:(74)919Thelengthadded(
nd)wasalsoassumedtohaveanormaldistributionwiththemean920lengthaddedsetto0.53mandtheCV=0.26,againinlinewiththeexperimentalresults921foralaninemedium.922ForFigure3B,Figure3-figuresupplements2A-2D,3A-3F,Figure4-figuresupplement9231D,thecellsizesarerecordedwithinthecellcycleatequalintervalsof4min,similarto924thatintheE.coliexperimentsofRef.[32].46

46925ForsimulationsshowninFigure4-figuresupplement1D,thecellsizetrajectoriesare926obtainedatintervalsof4minbeyondthecurrentcell-cycle.Thesizeafterthedivisionevent927issaidtobethesumofthesizesofthedaughtercells.Itisalsofurtherassumedthat928thedaughtercellsareequalinsize(perfectlysymmetricdivision)andtheybothgrowwith929thesamegrowthrate(forexponentialgrowth)orelongationspeed(forlineargrowth).The930growthrates/elongationspeedsforthedaughtercellsaresampledfromanormaldistribution931withameanandCVasdiscussedbefore.Thecellsizetrajectoriesarerecordedfor80min932afterthedivisioneventinthecurrentcellcycle.933InFigure2-figuresupplement2,simulationsoftheaddermodelforexponentiallygrowing934cellswerecarriedoutuntilapopulationof5000cellswasreached.Theparametersforsize935addedinacellcycleandmeangrowthrateswereextractedfromtheexperimentaldata[56].936Thevalueofnusedinallgrowthconditionswas0.17whileCVdecreasedinfastergrowth937conditions(0.2inthethreeslowestgrowthconditions,0.12and0.07inthesecondfastest938andfastestgrowthconditionsrespectively).9396Acknowledgements940TheauthorsthankEthanLevien,JieLinforusefuldiscussions,JaneKondev,XiliLiu,and941MarcoCosentinoLagomarsinofortheirusefulfeedbackonthemanuscript,DaYangand942ScottRettererforhelpinmicrofluidicchipmaking,andRodrigoReyes-Lamotheforakind943giftofstrain.AuthorsacknowledgetechnicalassistanceandmaterialsupportfromtheCenter944forEnvironmentalBiotechnologyattheUniversityofTennessee.Apartofthisresearchwas945conductedattheCenterforNanophaseMaterialsSciences,whichissponsoredatOakRidge946NationalLaboratorybytheScientificUserFacilitiesDivision,OfficeofBasicEnergySciences,947U.S.DepartmentofEnergy.ThisworkhasbeensupportedbytheUS-IsraelBSFresearch948grant2017004(JM),theNationalInstitutesofHealthawardunderR01GM127413(JM),47

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5410991100Figure1:Utilityofbinningandlinearregression:A.Lengthatdivision(Ld)vslength1101atbirth(Lb)isplottedusingdataobtainedbyTanouchietal.[10].Rawdataisshownas1102bluedots.Wefindthetrendinbinneddata(red)tobelinearwiththeunderlyingbest1103linearfit(yellow)followingtheequation,Ld=1:09Lb+2:24m.Thisisclosetotheadder1104behaviorwithanunderlyingequationgivenbyLd=Lb+
L,where
Listhemeansize1105addedbetweenbirthanddivision(shownasblackdashedline).B.Aschematicoftheadder1106mechanismisshownwherethecellgrowsoveritsgenerationtime(Td)anddividesafter11071108additionoflength
Lfrombirth.Thisensurescellsizehomeostasisinsinglecells.55

5511091110Figure2:Plotsthatcouldpotentiallyleadtomisinterpretingexponentialgrowth:1111A,B.DataisobtainedfromexperimentsinM9alaninemedium(hTdi=214min,N=816cells).A.ln(Ld)vshiTplotisshown.Thebluedotsaretherawdata,theredcorrespond1112Ldb1113tothebinneddatatrend,theyellowlineisthebestlinearfitobtainedbyperforminglinear1114regressionontherawdataandtheblackdashedlineisthey=xline.Apriori,non-lineartrendinbinneddatamightpointtogrowthbeingnon-exponential.B.hiTvsln(Ld)1115dLb1116plotisshownforthesameexperiments.C,D.Simulationsofexponentiallygrowingcells1117followingtheaddermodelarecarriedoutforN=2500cells.TheparametersusedareprovidedinSection5.11.2.C.ln(Ld)vshiTplotisshown.Thetrendinbinneddata1118Ldb1119showninredisnon-linearandthebestlinearfitofrawdata(yellow)deviatesfromthey=x1120line(blackdashedline).Theblackdottedlineistheexpectedtrendobtainedfromtheory1121(Equation2).Forparametersusedinthesimulationshere,theblackdottedlinefollowsLd2Ld1122ln(L)=1:26hiTd0:38(hiTd).D.hiTdvsln(L)plotisshownwithbinneddatainbb1123redandthebestlinearfitonrawdatainyellowcloselyfollowingtheexpectedtrendofy=x1124line(blackdashedline).Thetheoreticalbinneddatatrend(blackdottedline)isexpected1125tofollowthey=xtrend.Inalloftheseplots,thebinneddataisshownonlyforthosebins5611261127withmorethan15datapointsinthem.

561128Figure3:Differentiatinglineargrowthfromexponentialgrowth:A.hiTvsln(Ld)1129dLb1130plotisshownforsimulationsoflinearlygrowingcellsfollowingtheaddermodelforN=25001131cellcycles.Thebinneddata(red)andthebestlinearfitonrawdata(yellow)closelyfollows1132they=xtrend(blackdashedline)whichcouldbeincorrectlyinterpretedascellsundergoing1133exponentialgrowth.B.Thebinneddatatrendforgrowthratevsageplotisshownas1134purplecirclesforsimulationsofN=2500cellcyclesofexponentiallygrowingcellsfollowing1135theaddermodel.Weobservethetrendtobenearlyconstantasexpectedforexponential1136growth(purpledottedline).Sincethegrowthrateisfixedatthebeginningofeachcellcycle1137intheabovesimulations,wedonotshowerrorbarsforeachbinwithinthecellcycle.Also1138shownasgreensquaresisthegrowthratevsageplotforsimulationsofN=2500cellcyclesof1139linearlygrowingcellsfollowingtheaddermodel.Asexpectedforlineargrowth,thebinnedgrowthratedecreaseswithageas/1(greendottedline).Thebinnedgrowthrate11401+age1141trend(shownasmagentadiamonds)isalsofoundtobenearlyconstantasexpected(shown1142asmagentadottedline)forthesimulationsofexponentiallygrowingcellsfollowingtheadder1143peroriginmodel.Wealsoshowthatthebinnedgrowthratetrend(redtriangles)increases1144forsimulationsoftheaddermodelwiththecellsundergoingfasterthanexponentialgrowth.1145Thetrendisinagreementwiththeunderlyinggrowthratefunction(shownasreddotted1146line)usedinthesimulationsofsuper-exponentialgrowth.Thus,theplotgrowthratevs1147ageprovidesaconsistentmethodtoidentifythemodeofgrowth.Parametersusedinthe1148abovesimulationsofexponential,linearandsuper-exponentialgrowtharederivedfromthe11491150experimentaldatainalaninemedium.DetailsareprovidedintheSection5.11.2.57

5711511152Figure4:Growthratevsageobtainedfromexperiments:Growthratevsageplots1153areshownforE.coliexperimentaldata.Thereddotscorrespondtothebinneddatatrends1154showingthevariationingrowthrate.Themediuminwhichtheexperimentswereconducted1155areA.Alanine(hTdi=214min)B.Glycerol(hTdi=164min)C.Glucose-cas(hTdi=651156min).Theerrorbarsshowthestandarddeviationofthegrowthrateineachbinscaledbyp1,whereNisthenumberofcellsinthatbin.Thedashedverticallinesmarktheageat1157N1158initiationofDNAreplication(leftline)andthestartofseptumformation(rightline).In11591160caseofglucose-cas,theinitiationageisnotmarkedasitoccursinthemothercell.58

5811611162Figure5:Aflowchartofthegeneralframeworkproposedinthepapertocarryoutdata11631164analysis.59

5911658Appendix1:Comparinglength,surfaceareaandvol-1166umegrowthrate1167Inthepaper,weusecelllengthtorepresentcellsize.However,othercellsizecharacteristics1168suchascellsurfaceareaandcellvolumecouldalsobeusedtodenotecellsize.Howdoes1169thegrowthratevarywithourchoiceofcelllength,cellsurfacearea,orcellvolumetobethe1170cellsize?1171Tostudythis,weassumeacellmorphologyasshowninFigure1A-Appendix1.We1172assumethatE.colicellsarecylindricalwithhemisphericalpoles.Thetotallengthofthe1173cellisLwitharadiusR.Thecellvolume(V)isthen,223V=RLR:(1-A1)31174ThemorphologyofthecellafterconstrictionisalsoshowninFigure1A-Appendix1.The1175volumeinthiscaseis,2432223V=RLR+2Rh2hR+h:(2-A1)331176Ifwemaketheassumptionthatcellbiomassgrowsexponentiallyandthetotalcellsurface1177areaiscoupledtothebiomass[48],thencellsurfaceareagrowsexponentiallywithtime.1178UsingthemorphologyinFigure1A-Appendix1,thetotalsurfacearea(S)beforeandafter1179constrictionis,S=2RL:(3-A1)1180Surprisingly,thisisindependentofh.Sincethesurfaceareaisproportionaltothecell1181length(Equation3-A1),thelengthgrowthisalsoexponentialwithanidenticalgrowthrate1182assurfaceareagrowth,assumingthewidthofthecellisconstant.Theexponentialgrowth60

601183ofcelllengthisshowninFigure1B-Appendix1usingsimulationswherethecellsurfaceis1184assumedtogrowexponentially.So,forthismodelofcellgrowthandmorphology,thelength1185andthesurfacegrowthratesarefoundtobeidentical.Figure1-Appendix1:Lengthgrowthratevsvolumeandsurfaceareagrowthrate:A.CellmorphologyofE.coliusedinthemodelisshown.TheE.colicellsareassumedtobecylindricalwithhemisphericalendcaps.Beforeconstriction,thecellelongateswithconstantwidth(2R).However,afteronsetofconstriction,theseptumstartsformingatthemid-cell.B.Lengthgrowthrateasafunctionofageassumingthatthetotalcellsurfaceareagrowthisexponential,andtheradiusisconstant(R=0.35m).C.Lengthgrowthrateasafunctionofageassumingthatthevolumegrowthisexponential,radiusisconstant(R=0.35m)andseptumsurfacegrowsataconstantrate.1186Next,wecomparelengthgrowthratetovolumegrowthrateconsideringthesamecell1187morphologyasthatinFigure1A-Appendix1.Inthismodel,thevolumegrowthisassumedto1188beexponential.ThevolumebeforeandaftertheonsetofconstrictionaregivenbyEquations11891-A1and2-A1,respectively.1190Beforeconstriction,thevolumegrowsonlybyanincreaseinlengthofthecylindrical1191partofthecellwhilethewidthstaysconstant.However,aftertheconstrictionatmid-cell61

611192starts,thevolumegrowsbyanincreaseinlengthaswellasbyaddingaseptumsurfaceat1193themid-cell.Weassumethattheseptumwallsurfacegrowsataconstantrate(c1)[39].We1194canobtainc1intermsofcellmorphologyvariablestobe,dhc1=4R:(4-A1)dt1195Wecansolveforh(t)usingthefollowingboundaryconditions,h(t=Tn)=R;h(t=Td)=0;(5-A1)1196whereTnisthetimefrombirthatwhichconstrictionstarts.UsingEquations4-A1and11975-A1,wecanobtainc1intermsofcellcyclevariablesR,TnandTd,4R2c1=(6-A1)TdTn1198Undertheseassumptions,forexponentialvolumegrowth,weobtainthelengthgrowthvia1199simulations.ThelengthgrowthrateisshowninFigure1C-Appendix1.Thegrowthrate,1200thelengthatbirth,thetimeatconstrictionfrombirthandthegenerationtimeparameters1201usedinthesimulationsareobtainedfromexperimentaldatainalaninegrowthmedium.The1202widthofthecellsisassumedtobe0.35m.Wefindthatbeforeconstriction,thelength1203growthrateincreasestoasmallextent(6%).However,afterconstrictionthereisarapid1204increaseinlengthgrowthrate.Themodeofgrowthinlengthandvolumearenotidentical.62

629Appendix2:Linearregressiononln(Ld)vshTiplot1205dLb1206anditsinterchangedaxesplotLd)vshiTand1207InSection2,wefoundthatbinningandlinearregressionontheplotsln(Ldb1208itsinterchangedaxeswerenotasuitablemethodtoidentifytheunderlyingmodeofgrowth.Inthissection,weexplorebinningandlinearregressiononsimilarplotsln(Ld)vshTi1209Ldb1210plotanditsinterchangedaxes.Wetesttheusabilityoftheseplotstoelucidatethemodeof1211growthusingthemethodologyproposedinthepaper.Assumingexponentialgrowth,foracellcyclecanbecalculatedas1ln(Ld).Onplotting1212TdLbLd)vshTi(Figures1A-Appendix2-1C-Appendix2)andhTivsln(Ld)(Figures1F-1213ln(LddLbb1214Appendix2-1H-Appendix2)fortheexperimentaldata,weobtaintheslopeofthebestlinear1215fittobeclosetozero(valuesshowninTable1-Appendix2).Next,usingthemethodology1216ofthepaper,weinterprettheseresultsusinganunderlyingmodel.Weconsideramodelin1217whichcellsgrowexponentiallywiththedivisiondeterminedbybirth.Inthemodel,growth1218rateisfixedatthebeginningofeachcellcycleandisindependentofsizeatbirth.Themodelpredictsthatln(Ld)willbeindependentofthegrowthrate(Equation19inmain1219Lbtext).Thus,wewouldexpecttheslopetobezeroforbothoftheplotsln(Ld)vshTi1220LdbLd).ThisisalsoshownusingsimulationsoftheaddermodelinFigures1221andhTdivsln(Lb12221D-Appendix2and1I-Appendix2wheretheslopeoftheplotsisclosetozero.Inorder1223todifferentiatebetweenexponentialgrowthandlineargrowth,thebestlinearfitincaseof1224lineargrowthfortheseplotsmustdeviatefromy=constantline.However,wefindforthe1225simulationsoftheaddermodelwherecellsgrowlinearlythattheslopeofthebestlinearfit1226forbothoftheaboveplotsisstillzero(Figures1E-Appendix2and1J-Appendix2).Note1ln(Ld).Aslopeofzeroincaseof1227thatinthecaseoflineargrowthisstillcalculatedasTdLb1228lineargrowthcanbeexplainedusingEquation62ofthemaintext.Usingtheequation,weLd)isindependentoftheunderlyinggrowthrateforlineargrowth.Thus,the1229findthatln(Lb63

63Figure1-Appendix2:ln(Ld)vshTianditsflippedaxesplots:A-E.ln(Ld)vshTiLbdLbdareshownforA.Experimentaldatainalaninemedium.B.Experimentaldatainglycerolmedium.C.Experimentaldatainglucose-casmedium.D.Simulationsoftheaddermodelwherecellsgrowexponentially,carriedoutforN=2500cells.E.Simulationsoftheaddermodelwherecellsgrowlinearly,carriedoutforN=2500cells.F-J.Forthesameorderoftheaboveexperimentalconditionsandsimulations,hTivsln(Ld)plotsareshown.InalldLboftheplots,bluerepresentstherawdata,redrepresentsthebinneddata,andtheyellowlinerepresentsthebestlinearfitobtainedbyapplyinglinearregressionontherawdata.Inalloftheplots,theslopeofthebestlinearfitisclosetozero.Thus,wefindthattheseplotsarenotasuitablemethodtodifferentiatebetweenlinearandexponentialgrowthastheyprovideasimilarbestlinearfit.64

641230bestlinearfitforbothplotshaveaslopeofzerointhecaseoflineargrowth.ThisindicatesLd)vshTianditsinterchangedaxesplotsare1231thatbinningandlinearregressionontheln(Ldb1232unsuitableforelucidatingthemodeofgrowth.Table1-Appendix2:Theslopeandtheinterceptofthebestlinearfitalongwiththeir95%confidenceintervals(CI)obtainedonperforminglinearregressiononexperimentaldata.ThedataiscollectedforcellsgrowinginM9alanine,glycerolandglucose-casmedia[Srirametal.(2021)].ln(Ld)vshTiplothTivsln(Ld)plotLbddLbMediaNo.ofTdcells(min)Slope(withInterceptSlope(withIntercept95%CI)(with95%95%CI)(with95%CI)CI)Alanine8162140.04(-0.01,0.65(0.62,0.05(-0.01,0.67(0.63,0.09)0.69)0.12)0.72)Glycerol648164-0.12(-0.75(0.71,-0.19(-0.83(0.78,0.16,-0.07)0.79)0.27,-0.11)0.89)Glucose-737650.11(0.06,0.55(0.52,0.16(0.09,0.56(0.51,cas0.16)0.58)0.23)0.61)65

65123310SupplementaryFiguresandTablesTableS1:Variabledefinitions.VariablesDescriptionLbLengthofthecellatbirthandalsoaproxyforsizeatbirthLdLengthofthecellatdivisionandalsoaproxyforsizeatdivisionlLb,wherehLiismeansizeatbirthbhLbiblLd,wherehLiismeansizeatbirthdhLbibf(lb)Mathematicalfunctionwhichcapturestheregulationstrategy1determiningdivisiongivensizeatbirth.f(lb)=2lbTdGenerationtimetStandarddeviationofgenerationtimexorxx=ln(ln).Sincel1,xln1nnbbnbxStandarddeviationofxnf1(xn)Gaussiandescribingthedistributionofxn.f1(xn)=1x2pexpn222x2xhiMeangrowthrateCVCoefficientofvariationofgrowthrate(0;CV)Normallydistributedgrowthratenoise.Growthrateisde-finedas=hi+hi(0;CV)f2()Gaussiandescribingthedistributionofrandomvariable12(0;CV).f2()=p2exp2CV22CV(0;n)Normallydistributedtimeadditivedivisiontimingnoisewithhimean0andstandarddeviationnhi66

66f3()Gaussiandescribingthedistributionofrandomvariable12(0;n).f3()=p2exp222nns(0;bd)Normallydistributedsizeadditivedivisiontimingnoisewithmean0andstandarddeviationbdStandarddeviationofln(Ld)lLbfln(Ld)Gaussiandescribingthedistributionofln(Ld).fln(Ld)4Lb!Lb4Lb2ln(Ld)ln(2)p1Lb=exp2222llCorrelationcoefficientofthepair(ln(Ld),hiT)expLdbmSlopeofthebestlinearfitforln(Ld)vshiTplottlLdbcInterceptofthebestlinearfitforln(Ld)vshiTplottlLdbmSlopeofthebestlinearfitforhiTvsln(Ld)plotltdLbcInterceptofthebestlinearfitforhiTvsln(Ld)plotltdLbhliniMeannormalizedelongationspeedCV;linCoefficientofvariationofnormalizedelongationspeedlin(0;CV;lin)Normallydistributednormalizedelongationspeednoise.Nor-malizedelongationspeedisdefinedaslin=hlini+hlinilin(0;CV;lin)l;linStandarddeviationofldlblinCorrelationcoefficientofthepair(ldlb,hliniTd)mtl;linSlopeofthebestlinearfitforldlbvshliniTdplotctl;linInterceptofthebestlinearfitforldlbvshliniTdplotmlt;linSlopeofthebestlinearfitforhliniTdvsldlbplotclt;linInterceptofthebestlinearfitforhliniTdvsldlbplotLiCellsizeatthestartofDNAreplication(initiation)67

67tot;nextLiTotalcellsizeofthedaughtercellsatthestartofDNArepli-cation
iiSizeaddedperoriginbetweeninitiationsONumberoforiginsjustafterinitiationC+DTimebetweeninitiationanddivisionTnTimingofstartofseptumformation/onsetofconstrictionLnCellsizeattimeTnTableS2:Theslopeandtheinterceptofthebestlinearfitalongwiththeir95%confidenceintervals(CI)obtainedonperforminglinearregressiononexperimentaldata.ThedataiscollectedforcellsgrowinginM9alanine,glycerolandglucose-casmedia[32].ln(Ld)vshiTplothiTvsln(Ld)plotLbddLbMediaNo.ofTdcells(min)Slope(withInterceptSlope(withIntercept95%CI)(with95%95%CI)(with95%CI)CI)Alanine8162140.34(0.31,0.44(0.42,1.06(0.98,-0.01(-0.07,0.36)0.46)1.14)0.04)Glycerol6481640.34(0.32,0.43(0.41,1.26(1.16,-0.13(-0.20,0.37)0.44)1.35)-0.07)Glucose-737650.31(0.28,0.42(0.40,0.91(0.83,0.09(0.03,cas0.34)0.44)1.00)0.15)68

681234Figure2-figuresupplement1:Experimentaldata:ln(Ld)vshiT(left)andhiTvs1235Lbddln(Ld)plot(right)isshownfor,A.Cellsgrowinginglycerolmedium(hTi=164min,N=1236Ldb1237648cells).B.Cellsgrowinginglucose-casmedium(hTdi=65min,N=737cells).Binned1238data(red),andthebestlinearfit(yellow)obtainedbyperforminglinearregressionontheLd1239rawdatadeviatefromthey=xline(blackdashedline)inthecaseofln(L)vshiTdplotsinb1240bothmedia.However,bothbinneddataandthebestlinearfitareincloseagreementwith1241they=xline(blackdashedline)oninterchangingtheaxes.Inalloftheseplots,thebinned12421243dataisshownonlyforthosebinswithmorethan15datapointsinthem.69

6912441245Figure2-figuresupplement2:Binneddatatrendingrowthrate()andinversegenerationtime(1)plots:A-B.Simulationsoftheaddermodelforexponentially1246Td1247growingcellswerecarriedoutatmultiplegrowthratesforN=2500cells.Thesizeadded1248betweenbirthanddivisionandthemeangrowthrateswereextractedfromKennardetal.,1249[56].TheCVofgrowthrateswasgreaterforcellsgrowinginslower-growthmedia.SeeSection5.11.2fortheparametervalues.Forthesesimulations,weshowA.vs1plot.B.1250Td1vsplot.Thesmallercirclesshowthetrendinbinneddatawithinagrowthmedium.1251Td1252Differentcolorscorrespondtodifferentgrowthmedia.Populationmeansareshownaslarger1253markers.Thepopulationmeansagreewiththeexpectedy=ln(2)xline(blackdashedline)1254inFigure2-figuresupplement2Abutthetrendwithinasinglegrowthmediumisnon-linear1255anddeviatesfromthey=ln(2)xline.However,inFigure2-figuresupplement2B,population1256meansacrossgrowthconditionsandthetrendinbinneddatawithinasinglegrowthmediumfollowtheexpectedy=1xline(blackdottedline).12571258ln(2)70

7012591260Figure3-figuresupplement1:Predictingstatisticsbasedonamodeloflinear1261growth:A-B.Simulationsoflinearlygrowingcellsfollowingtheaddermodelarecar-1262riedoutforN=2500cellcycles.A.ldlbvshliniTdplotisshown.Therawdataisshown1263asbluedots.Thebinneddata(inred)andthebestlinearfitonrawdata(inyellow)deviate1264fromthey=xline(blackdashedline).Suchadeviationcanbepredictedbasedonamodel1265asdiscussedindetailinSection5.5.B.hliniTdvsldlbplotisshown.Thebinneddata(in1266red)andthebestlinearfitonrawdata(inyellow)agreewiththey=xline(inblack).C.1267SimulationsofexponentiallygrowingcellsfollowingtheaddermodelarecarriedoutforN=12682500cellcycles.hliniTdvsldlbplotisshown.Thebinneddata(inred)andthebestlinear1269fitonrawdata(inyellow)deviatefromthey=xline(inblack)asexpectedforexponential12701271growth.ParametersusedinthesimulationsaboveareprovidedinSection5.11.2.71

7112721273Figure3-figuresupplement2:Inspectionbiasinthegrowthratevstimeplots1274obtainedfromsimulations:A.Thebinnedgrowthratetrendasafunctionoftime1275fromtheonsetofconstriction(t-Tn)isshowninred.Timet-Tn=0correspondstoonsetof1276constriction.Theplotisshownforsimulationsofexponentiallygrowingcellscarriedoutover1277N=2500cellcycles.Constrictionlengthisdeterminedbyaconstantlengthadditionfrom1278birthanddivisionoccursafteraconstantlengthadditionfromconstriction.B.Theaverage1279generationtimeforthecellspresentineachbinofFigure3-figuresupplement2Aisshown.1280C.Forsimulationsofexponentiallygrowingcellsfollowingtheaddermodel(N=2500),the1281binnedgrowthrate(inred)vstimefrombirthplotisshown.D.Theaveragegeneration1282timeforthecellspresentineachbinofFigure3-figuresupplement2Cisshown.Thevertical1283dashedlinesshowthetimerangeinwhichthegenerationtimesareapproximatelyconstant1284andhence,theeffectsofinspectionbiasarenegligible.Withinthattimerange,thegrowth12851286ratetrendisfoundtobeconstant,consistentwiththeassumptionofexponentialgrowth.72

72128773

73Figure3-figuresupplement3:Differentialmethodsofquantifyinggrowth:A-B.SimulationsoflinearlygrowingcellsfollowingtheaddermodelarecarriedoutforN=2500cellcycles.Cellsize(L)dataisrecordedasafunctionoftimewithinthecellcycle.A.Thereddotsshowthebinneddataforelongationspeedasafunctionofage.Thetrendisalmostconstantinagreementwiththelineargrowthassumption.B.Elongationspeedisalsoconstantwithcellsizeasexpectedforlineargrowth.Theinterceptvalueofthebestlinearfitonrawdata(inyellow)providestheaverageelongationspeed.C-D.SimulationsofexponentiallygrowingcellsfollowingtheaddermodelarecarriedoutforN=2500cellcycles.C.Elongationspeedtrend(inred)increaseswithageinagreementwiththeexponentialgrowthassumption.D.Elongationspeedtrend(inred)increaseslinearlywithsize.Theslopeofthebestlinearfitonrawdata(inyellow)isequaltotheaveragegrowthrate.E-F.SimulationsofexponentiallygrowingcellsfollowingtheadderperoriginmodelarecarriedoutforN=2500cellcycles.E.Again,theelongationspeedtrend(inred)increaseswithageinagreementwiththeexponentialgrowthassumption.F.Elongationspeedtrend(inred)andthebestlinearfitonrawdata(inyellow)deviatesfromtheexpectedlineartrend(blackdashedline).Thiscouldbemisinterpretedasnon-exponentialgrowth.Thus,wefindthatthebinneddatatrendfortheplotelongationspeedvssizeismodel-dependent.74

741288128975

75Figure4-figuresupplement1:Growthratevsagecurvesextendedbeyondthedivisionevent:A,B,C.ThebinnedgrowthratetrendisshowninredasafunctionofageforE.coliexperimentaldata.Thetrendsareobtainedusingthecellsizetrajectoriesextendingbeyondthedivisionevent(age>1).TheplotsareshownforA.Alaninemedium(N=720cells)B.Glycerolmedium(N=594cells).C.Glucose-casmedium(N=664cells).Theerrorbarsinallthreeplotsrepresentthestandarddeviationofthegrowthrateineachbinscaledbyp1,whereNisthenumberofcellsinthatbin.ThegrowthratetrendNappearstobeperiodicineachofthegrowthmediai.e.,atage1isclosetoatage0.ThesetrendsagreewiththatofFigure4intheappropriateageranges.D.SimulationsarecarriedoutforN=2500cellcycles.Thecellsizetrajectoriesarecollectedbeyondthedivisionevent(age>1).Thebinneddatatrendforgrowthratevsageplotisshownaspurplecirclesforexponentiallygrowingcellsfollowingtheaddermodel.Weobservethetrendtobenearlyconstantasexpectedforexponentialgrowth.Thebinnedgrowthratetrendisalsofoundtobenearlyconstantforthesimulationsofexponentialgrowingcellsfollowingtheadderperoriginmodel(shownasmagentadiamonds).(Inset)ShownasgreensquaresistheelongationspeedvsageplotforsimulationsofN=2500cellcyclesoflinearlygrowingcellsfollowingtheaddermodel.Asexpectedforlineargrowth,thebinnedelongationspeedtrendremainsapproximatelyconstantwithage.ThegrowthratetrendsforthemodelswithexponentialgrowthagreewiththatofFigure3B.Theelongationspeedtrend(inset)alsoagreeswiththetrendinFigure3-figuresupplement3A.76

76129077

77Figure4-figuresupplement2:Inspectionbiasinthegrowthratevstimefromconstrictionplotsobtainedfromexperiments:A,C,E.Thebinnedgrowthratetrendisshowninredasafunctionoftimefromtheonsetofconstriction(t-Tn).Timet-Tn=0correspondstotheonsetofconstrictionforallcellsconsidered.TheplotsareshownforA.Alaninemedium.C.Glycerolmedium.E.Glucose-casmedium.Theerrorbarsinallthreeplotsrepresentthestandarddeviationofthegrowthrateineachbinscaledbyp1,whereNNisthenumberofcellsinthatbin.B,D,F.TheaveragegenerationtimeforthecellspresentineachbinofB.Alaninemedium(Figure4-figuresupplement2A)D.Glycerolmedium(Figure4-figuresupplement2C)F.Glucose-casmedium(Figure4-figuresupplement2E)areshown.Theverticaldashedlinesrepresentthetimerangewithinwhichtheaveragegenerationtimeremainsapproximatelyconstant.ThegrowthratetrendswithinthistimerangeareconsistentwiththatinFigure4fortherespectivegrowthconditionasthereisnegligibleinspectionbias.78

78Figure1

79Figure2

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82Figure3

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86Figure4

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89Figure5

90Appendix1--figure1

91Appendix2-figure1