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1Distinguishingdi?erentmodesofgrowthusing2single-celldataPrathithaKar1,2,SriramTiruvadi-Krishnan3,JaanaM?nnik3,JaanM?nnik3,3andArielAmiry141SchoolofEngineeringandAppliedSciences,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,we23?ndthegrowthoflengthinE.colitobenon-exponential.Ouranalysisshowsthatinthe24laterstagesofthecellcyclethegrowthrateisfasterthanexponential.2
2251Introduction26Thelastdecadehasseenatremendousincreaseintheavailabilityofhigh-qualitylarge27datasetsinbiology,inparticularinthecontextofsingle-celllevelmeasurements.Such28dataarecomplementaryto“bulk”measurementsmadeoverapopulationofcells.They29haveledtonewbiologicalparadigmsandmotivatedthedevelopmentofquantitativemodels30[1–7].Nevertheless,theyhavealsoledtonewchallengesindataanalysis,andherewe31willpointoutsomeofthepitfallsthatexistinhandlingsuchdata.Inparticular,wewill32showthatthecommonlyusedprocedureofbinningdataandlinearregressionmayhint33atspeci?cfunctionalrelationsbetweenthetwovariablesplottedthatareinconsistentwith34thetruefunctionalrelations.Asweshallshow,thismaycomeaboutduetothe“hidden”35noisesourcesthata?ectthebinningprocedureandthephenomenonof“inspectionbias”36wherecertainbinshavebiasedcontributions.Oneofourmaintakehomemessagesisthe37signi?canceofhavinganunderlyingmodel(ormodels)toguide/test/validatedataanalysis38methods.Theunderlyingmodelisreferredtoasagenerativemodelinthesensethat39itleadstosimilardatatothatobservedintheexperiments.Theimportanceofaso-40calledgenerativemodelhasbeenbeautifullyadvocatedinthecontextofastrophysicaldata41analysis[8],yetbiologybringsinaplethoraofexcitingdi?erences:whileinphysicsnoisefrom42measurementinstrumentsoftendominates,inthebiologicalexampleswewilldwellonhereit43istheintrinsicbiologicalnoisethatcanobscurethemathematicalrelationbetweenvariables44whennothandledproperly.Inthefollowing,wewillillustratethisratherphilosophical45introductiononaconcreteandfundamentalexample,albeitepluribusunum.Wewillfocus46ontheanalysisoftheEscherichiacoligrowthcurvesobtainedviahighthroughputoptical47microscopy.Neverthelessweanticipatetheconceptualpointsmadehere–anddemonstrated48onaparticularexampleofinterest–willtranslatetoothertypesofmeasurements,which49makeuseofmicroscopybutalsobeyond.3
350Binningcorrespondstogroupingdatabasedonthevalueofthex-axisvariable,and?nd-51ingthemeanofthe?uctuatingy-axisvariableforthisgroup.Byremovingthe?uctuations52ofthey-variable,thebinningprocessoftenaimstoexposethe“true”functionalrelation53betweenthetwovariableswhichcanbeusedtoinfertheunderlyingbiologicalmechanism.54Whilebinningmayprovideasmoothnon-linearrelationbetweenvariables,linearregression55isusedto?ndalinearrelationshipbetweenthevariables.Inadditiontobinning,weuse56theordinaryleastsquaresregressionwheretheslopeandtheinterceptofthebestlinear?t57lineareobtainedbyminimizingthesquaredsumofthedi?erencebetweenthedependent58variablerawdataandthepredictedvalue.Here,thebest?t/thebestlinear?tisobtained59usingtherawdataandnotthebinneddata.Similartobinning,theassumptionunderlying60linearregressionisthatourknowledgeofx-axisvariableisprecisewhilethenoiseisinthe61y-axisvariable.62Itisimportanttodiscussthesourcesof?uctuationsinthey-axisvariablebeforewe63proceed.Inbiology,?uctuationsinthevariablesariseinevitablyfromtheintrinsicvariability64withinacellpopulation.Cellsgrowinginthesamemediumandenvironmenthavedi?erent65characteristics(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
476biologicalmodelknownastheaddermodelasshowninFigure1Binwhichthelengthat77divisionissetbyadditionoflengthLfrombirth[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.,dueto94itspotentiale?ectsoncellsizehomeostasis.Exponentiallygrowingcellscannotemploya95mechanismwheretheycontroldivisionbytimingaconstantdurationfrombirthbutsuch96amechanismispossibleincaseoflineargrowth[3,13,29].Linearregressionperformedonln(Ld)vshiTplot,wherehiisthemeangrowthrate,wasusedtoinferthemode97Ldb98ofgrowthinthearchaeonH.salinarum[16],andinthebacteriaM.smegmatis[30]and5
599C.glutamicum[31],forexample.Ifthebestlinear?tfollowsthey=xtrend,theresulting100functionalrelationmightpointtogrowthbeingexponential.Acorollarytothisisthe101rejectionofexponentialgrowthwhentheslopeandinterceptofthebestlinear?tdeviatefrom102oneandzerorespectively[31].Thus,binningandlinearregressionappliedonsingle-celldata103appeartoprovideinformationabouttheunderlyingbiology,inthiscase,themodeofcellular104growth.WewilltestthevalidityofsuchinferencebyanalyzingsyntheticdatageneratedLd)vs105usinggenerativemodels.We?ndthatlinearregressionperformedontheplotln(Lb106hiTd,surprisingly,doesnotprovideinformationaboutthemodeofgrowth.Nonetheless,107weshowthatothermethodsofstatisticalanalysissuchasbinninggrowthratevsageplots108areadequateinaddressingtheproblem.Usingthesevalidatedmethodsonexperimental109data,we?ndthatE.coligrowsnon-exponentially.Inlaterstagesofthecellcycle,the110growthrateishigherthanthatinearlystages.1112Statisticalmethodslikebinningandlinearregression112shouldbeinterpretedbasedonamodel.113Toillustratethepitfallsassociatedwithbinning,weusedatafromrecentexperimentsonE.114coliwherethelengthatbirth,thelengthatdivisionandthegenerationtimewereobtained115formultiplecells(seeSection5.1and[32]).Phase-contrastmicroscopywasusedtoobtain116celllengthatequalintervalsoftime.Notethatweconsiderlengthtore?ectcellsizein117thispaperratherthanothercellgeometrycharacteristicssuchassurfaceareaandvolume.118Thelengthgrowthratethatweelucidateinthepapercanbedi?erentfromthecellvolume119growthrateasshowninAppendix1assumingasimplecellmorphologyandexponential120growth.Usingthesamecellmorphology,wealso?ndthelengthgrowthratetobeidentical121tocellsurfacegrowthrate.Toinvestigateifthecellgrowthwasexponential,weplottedLd)vshiTforcellsgrowinginM9alanineminimalmediumat28C(hTi=214min).122ln(Lddb6
6123Thelinearregressionofthesedatayieldsaslopeof0.3andaninterceptof0.4asshownin124Figure2A.Thebinneddataandthebestlinear?tdeviatesigni?cantlyfromthey=xline125(seeTableS2).Additionally,thebinneddatafollowsanon-lineartrendand?attensout126atlongergenerationtimes.Wealsofoundsimilardeviationsinthebinneddataandbest127linear?tinglycerolmedium(hTdi=164min)showninFigure2-?guresupplement1A,and128glucose-casmedium(hTdi=65min)showninFigure2-?guresupplement1B.Qualitatively129similarresultshavebeenrecentlyobtainedforanotherbacterium,C.glutamicum,inRef.130[31].Theseresultsmightpointtogrowthbeingnon-exponential.131Nextwewillapproachthesameproblembutwithagenerativemodel.Wewill?rstLd)vshiTbinnedplotcouldnotdistinguishexponentialgrowthfrom132showthattheln(Ldb133non-exponentialgrowth.Forthatpurpose,weuseapreviouslystudiedmodel[16]which134considersgrowthtobeexponentialwiththegrowthratedistributednormallyandindepen-135dentlybetweencellcycleswithmeangrowthratehiandstandarddeviationCVhi.CV136isthusthecoe?cientofvariation(CV)ofthegrowthrateandisassumedtobesmall.To137maintainanarrowdistributionofcellsize,cellsmustemployregulatorymechanisms.In138ourmodel,weassumethat,barringthenoiseduetostochasticbiochemicalreactions,cells139attempttodivideataparticularsizeLdgivensizeatbirthLb.Keepingthemodelasgeneric140aspossible,wecanwriteLdasafunctionofLb,f(Lb)whichcanbethoughtofasacoarse-141grainedmodelfortheregulatorymechanism.Ref.[13]providesaframeworktocapturetheregulatorymechanismsbychoosingf(L)=2L1
