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The Influence of Tissue Architecture on Somatic Tissue Evolution, Homeostasis, Aging, and Cancer

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Title:
The Influence of Tissue Architecture on Somatic Tissue Evolution, Homeostasis, Aging, and Cancer
Creator:
Cannataro, Vincent L
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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Language:
english
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1 online resource (102 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Zoology
Biology
Committee Chair:
ST MARY,COLETTE MARIE
Committee Co-Chair:
MCKINLEY,SCOTT
Committee Members:
BAER,CHARLES
LU,JIANRONG J
Graduation Date:
4/30/2016

Subjects

Subjects / Keywords:
Cells ( jstor )
Crypts ( jstor )
Genetic mutation ( jstor )
Homeostasis ( jstor )
Intestinal crypts ( jstor )
Population dynamics ( jstor )
Population estimates ( jstor )
Population size ( jstor )
Stem cells ( jstor )
Tumors ( jstor )
Biology -- Dissertations, Academic -- UF
aging -- cancer -- evolution -- mutation -- somatic -- stemcell -- tumorigenesis
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Zoology thesis, Ph.D.

Notes

Abstract:
The vast majority of the cells within a mammalian body are in a continual flux. The cells lost from the body are replaced through the constant division and differentiation of stem cells located in stem cell niches distributed throughout tissues. Gene replication is prone to errors, and the perpetual division of stem cells presents a tremendous risk of mutation accumulation. This problem is exacerbated by the fact that mutations, even those that confer a deleterious effect to cellular fitness, may drift to fixation in small populations. Within this dissertation I examine the influence of mutation accumulation in the intestinal epithelium on the processes of tissue homeostasis, aging, tumor formation, and tumor growth in light of small stem cell niche population size. First, I examine the implications of mutational effects along the entire continuous distribution of fitness effects (DFE), and examine if DFE commonly measured in whole organisms are sufficient to explain known patterns of tumorigenesis. I find that the expected effect of mutation accumulation is a decrease in stem cell population fitness, and that whole organism DFE are capable of accounting for tumor incidence in humans. Next, I calculate the expected effect mutation accumulation within stem cell niches has on whole-tissue maintenance. I find that whole-tissue population size is expected to decrease, and stem cell niches exist at a population size that minimizes the probability of initiating tumorigenesis, at the expense of accumulating deleterious mutations due to drift. Finally, I examine the influence of small stem cell population sizes on the modular nature of tumor growth. I find that the high non-synonymous to synonymous nucleotide replacement rate and the high heterogeneity found in whole-genome sequencing efforts of tumors, both of which are commonly assumed to be indicative of the lack of selection or the presence of pervasive positive selection, are naturally emergent properties of tumor population architecture. Overall, the research presented here highlights the importance of considering population structure when investigating evolutionary dynamics within somatic tissue, and emphasizes the strength and implications of genetic drift within compartmentalized populations. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2016.
Local:
Adviser: ST MARY,COLETTE MARIE.
Local:
Co-adviser: MCKINLEY,SCOTT.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-05-31
Statement of Responsibility:
by Vincent L Cannataro.

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Source Institution:
UFRGP
Rights Management:
Copyright Vincent L Cannataro. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
5/31/2017
Classification:
LD1780 2016 ( lcc )

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THEINFLUENCEOFTISSUEARCHITECTUREONSOMATICTISSUEEVOLUTION,HOMEOSTASIS,AGING,ANDCANCERByVINCENTL.CANNATAROADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2016

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c2016VincentL.Cannataro

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Idedicatethistoalloftheteachersthroughoutmylife.Thankyou.

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ACKNOWLEDGMENTSManythankstomyfamily.Tomyparents,RichandElaine,whostartedmeonthisamazingjourneywithscience.Andtomywife,Begum,whobrightenstheway,everyday.IoerveryheartfeltandsincerethankstoDr.ColetteM.St.MaryandDr.ScottA.McKinley.ThankyouforgivingmethetoolsIneedtopursuemydreams.ThankyouDr.CharlieBaerandDr.JianrongLu,foralwayshavinganopendoor.ThankyoutoeveryoneintheSt.MaryLab,Pop.BioSeminar,theIGERTgroup,andalltheotheroutletsthatfosteredthesharingofresearchovertheyears.Andthankyouforreading. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................... 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1BIOLOGICALINTRODUCTION ........................... 11 1.1StemCellsAreMaintainedWithinTheirNiche ................. 11 1.2StemCellsInitiateandMaintainTumorGrowth ................ 14 2THEIMPLICATIONSOFSMALLSTEMCELLNICHESIZESANDTHEDISTRIBUTIONOFFITNESSEFFECTSOFNEWMUTATIONSINAGINGANDTUMORIGENESIS ................................... 17 2.1IntroductoryRemarks .............................. 17 2.1.1EvolutioninTheStemCellNiche .................... 17 2.1.2DistributionofFitnessEects ...................... 17 2.2MaterialsandMethods ............................. 20 2.2.1DescriptionoftheModel ......................... 20 2.2.2ParameterChoices ............................ 24 2.3Results ...................................... 27 2.3.1MutationsResultinBothAgingandTumorigenesisWithintheIntestine 27 2.3.2PredictedIncidenceCurvesinMiceandHumansUsingDFEDerivedfromaWholeOrganism ......................... 29 2.3.3AlteringtheExpectedBenecialFitnessEectsandtheMutationRateProvidesBetterFitsforBothExponentialandPower-lawDerivedIncidenceCurves .................................. 31 2.3.4MutationsThatAltertheDierentiationRateofStemCellsResultinRapidAgingandTumorigenesis ..................... 31 2.4Discussion .................................... 34 2.4.1WholeOrganismDFEAreSucienttoExplainTumorigenesis ..... 34 2.4.2SmallPopulationsandGeneticDriftLeadtoAging ........... 38 2.4.3MutationsThatOnlyAectDierentiationRateDoNotMatchIncidenceDataCurves ............................... 39 2.4.4TheInuenceofOrganismSpecicFactorsonSomaticEvolution ... 41 3THEEXPECTEDEFFECTOFSOMATICEVOLUTIONONTISSUEHOMEOSTASIS 43 3.1IntroductoryRemarks .............................. 43 3.2Methods/Modeling ................................ 44 5

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3.2.1ModelingtheCryptSystem ....................... 44 3.2.2ModelParameterization ......................... 48 3.2.3ModelingEvolutionWithinSomaticTissue ............... 51 3.2.4LinearApproximation ........................... 54 3.2.5ExploringtheEvolutionaryTrade-oBetweenStemCellNicheSize,Aging,andTumorigenesis ........................ 56 3.3Results ...................................... 56 3.3.1TheExpectedFitnessofStemCellsDecreaseswithFixedMutations .. 56 3.3.2MouseandHumanStemCellFitnessisExpectedtoDecreasewithAge,ReducingTissueRenewal ......................... 59 3.3.3ThereIsanEvolutionaryTrade-oBetweenTumorigenesisandAgingMediatedbyStemCellNicheSize .................... 60 3.4Discussion .................................... 62 4HIGHdN/dSRATIOSANDHETEROGENEITYAREANEXPECTEDOUTCOMEOFTUMORPOPULATIONARCHITECTURE .................... 66 4.1IntroductoryRemarks .............................. 66 4.2MethodsandModel ............................... 67 4.2.1TheIntestinalTumorGland ....................... 68 4.2.2ParameterChoices ............................ 69 4.3Results ...................................... 70 4.3.1TheMajorityofMutationsinaGrowingTumorHaveRecentlyArisen 70 4.3.2WholeTumordN/dSIsHigh ....................... 72 4.3.3dN/dSRatiosofCommonVs.RareMutations ............. 74 4.4Discussion .................................... 75 4.4.1ALargeProportionofNon-synonymousMutationsandHighHeterogeneityAretheDirectResultoftheModularNatureofTumorGrowth ..... 75 4.4.2ModelingConsiderations,SamplingTechniques,andResults ...... 79 5SUMMARYANDCONCLUSIONS .......................... 80 APPENDIX:DESCRIPTIONOFTHEMATHEMATICALMETHODOLOGY ....... 82 REFERENCES ........................................ 92 BIOGRAPHICALSKETCH ................................. 102 6

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LISTOFTABLES Table page 1-1Stemcelltissuerenewalinsomecommonlystudiedtissues .............. 16 2-1EvolutionandCryptParameters ............................ 29 3-1Approximateparametervaluesforthemouseintestine ................ 49 4-1Parametervaluesforatumorglandinourmodel ................... 71 7

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LISTOFFIGURES Figure page 2-1Arepresentationofourmodel ............................. 24 2-2PosteriorDensitiesOfStemCellDivisionRate .................... 28 2-3TumorigenesisincidenceinmiceandhumansusingwholeorganismDFEparameters. 30 2-4LeastSquaresAnalysisHeatmapFortheExponentialScenario ............ 32 2-5LeastSquaresAnalysisHeatmapForthePower-lawScenario ............. 33 2-6Tumorigenesiscurvesresultingfromleastsquarestting. ............... 34 2-7Theaccumulationofprobabilitydensitiesdescribingdierentiationrate. ....... 35 2-8Tumorigenesisresultingfrommutationaleectsondierentiationrate. ....... 36 2-9LeastSquaresAnalysisHeatmapForthePower-lawScenario ............. 37 3-1Thegeneralarchitectureofacryptsystem ...................... 47 3-2Expectedvalueoftheprobabilitydensitydescribingdivisionrate ........... 57 3-3Theexpectedvalueofmutationvs.theexpecteddeleterioustnesseect. ..... 58 3-4Expectedvalueoftheprobabilitydensitydescribingdierentiationrate ....... 59 3-5Theeectofmutationaccumulationintheintestinesofamouse .......... 60 3-6Theeectofmutationaccumulationinthelargeintestinesofahuman ....... 61 3-7AnEvolutionaryTradeowithNicheSize ....................... 62 4-1Flowchartofthesimulation. .............................. 69 4-2Intratumorheterogeneitywithinasingletumor .................... 72 4-3Lineagefrequencywithinasinglesimulatedtumor .................. 73 4-4dN/dSof30tumorsimulations ............................ 75 4-5Thedistributionoftnesseectsfromallsimulatedtumors ............. 76 4-6dN/dSratiosforthemostcommonlineagesinthetumors .............. 77 4-7DistributionsofFitnessEectsfromthemostcommonlineagesinasingletumor .. 78 A-1CryptEvolutionaryDynamicsSimulation ....................... 88 A-2Mutationprolesofatumorattheonsetoftumorigenesis. ............. 91 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTHEINFLUENCEOFTISSUEARCHITECTUREONSOMATICTISSUEEVOLUTION,HOMEOSTASIS,AGING,ANDCANCERByVincentL.CannataroMay2016Chair:ColetteM.St.MaryCochair:ScottA.McKinleyMajor:ZoologyThevastmajorityofthecellswithinamammalianbodyareinacontinualux.Thecellslostfromthebodyarereplacedthroughtheconstantdivisionanddierentiationofstemcellslocatedinstemcellnichesdistributedthroughouttissues.Genereplicationispronetoerrors,andtheperpetualdivisionofstemcellspresentsatremendousriskofmutationaccumulation.Thisproblemisexacerbatedbythefactthatmutations,eventhosethatconferadeleteriouseecttocellulartness,maydrifttoxationinsmallpopulations.WithinthisdissertationIexaminetheinuenceofmutationaccumulationintheintestinalepitheliumontheprocessesoftissuehomeostasis,aging,tumorformation,andtumorgrowthinlightofsmallstemcellnichepopulationsize.First,Iexaminetheimplicationsofmutationaleectsalongtheentirecontinuousdistributionoftnesseects(DFE),andexamineifDFEcommonlymeasuredinwholeorganismsaresucienttoexplainknownpatternsoftumorigenesis.Indthattheexpectedeectofmutationaccumulationisadecreaseinstemcellpopulationtness,andthatwholeorganismDFEarecapableofaccountingfortumorincidenceinhumans.Next,Icalculatetheexpectedeectmutationaccumulationwithinstemcellnicheshasonwhole-tissuemaintenance.Indthatwhole-tissuepopulationsizeisexpectedtodecrease,andstemcellnichesexistatapopulationsizethatminimizestheprobabilityofinitiatingtumorigenesis,attheexpenseofaccumulatingdeleteriousmutationsduetodrift.Finally,Iexaminetheinuenceofsmallstemcellpopulationsizesonthemodularnatureoftumorgrowth.Indthatthehigh 9

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non-synonymoustosynonymousnucleotidereplacementrateandthehighheterogeneityfoundinwhole-genomesequencingeortsoftumors,bothofwhicharecommonlyassumedtobeindicativeofthelackofselectionorthepresenceofpervasivepositiveselection,arenaturallyemergentpropertiesoftumorpopulationarchitecture.Overall,theresearchpresentedherehighlightstheimportanceofconsideringpopulationstructurewheninvestigatingevolutionarydynamicswithinsomatictissue,andemphasizesthestrengthandimplicationsofgeneticdriftwithincompartmentalizedpopulations. 10

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CHAPTER1BIOLOGICALINTRODUCTIONTheresearchpresentedinthisdissertationusesmathematicalmodelstoexaminetheexpectedeectsofmutationaccumulationwithinhealthysomatictissueandgrowingtumors.Here,Iintroducethemainmechanismoftissuehomeostasis:tissuereplenishmentthroughthecontinualdivisionanddierentiationofsomaticstemcells.Ialsointroducerecentevidencesuggestingthatmanytumorsarebothinitiatedbyandmaintainedthroughthedivisionofstem-likecellsexistingatsmallpopulationsizes. 1.1StemCellsAreMaintainedWithinTheirNicheBillionsofcellswithinanadulthumanundergoregulatedcelldeatheveryday( ElliottandRavichandran 2010 ).Thiscontinualdeathiscompensatedbypluripotentstemcells,whichdivideanddierentiateintotissue-speciccelltypes( BlanpainandSimons 2013 ).Stemcellsoccupyasmallproportionofthetotalcellsinatissue(Table 1-1 )andaredenedbothbytheirabilitytogeneratemultiplecelltypesandself-renewindenitely( SimonsandClevers 2011 ).Thestemcellpopulationsarelocatedandregulatedthroughouttissuesinstemcellniches,or,localmicroenvironmentscapableofhousingandmaintainingoneormorestemcellsthroughbothintrinsicandextrinsicsignals( MorrisonandSpradling 2008 ).Sincestemcellsandtheirnichesarethesourceoftissuehomeostasistheyarealsoresponsibleforconditionsanddiseasesrelatedtotheperturbationofthishomeostasis,suchasaging( Rossietal. 2008 )andcancer( Barkeretal. 2009 ).Importantly,thecauseofstemcellandtissuedisregulationisoftenaheritablemutationwithinastemcelllineageand,thus,evolutionarydynamicscontrolthefateofmutationsandtheirassociateddiseaseswithinsomatictissue.Understandingthepopulationstructureoftissues,andthedynamicswithintheirstemcellniches,isessentialtounderstandingthecause,andprevention,ofagingandcancer.TheideaofaspecializedenvironmenttopreservestemcellproliferativepotentialandblockdierentiationofstemcellswasrstproposedbyR.Schoeld( Schofield 1977 ; 11

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MooreandLemischka 2006 )whenitwasdiscoveredthathematopoieticcellsfrombonemarrowhadmoreproliferativepotentialthansimilarcellsfromthespleen.EarlyexperimentstodiscoverputativestemcellsandtheirnichesinvolvedlabelingcellularDNAwithbromo-deoxyuridine(BrdU)orintroducinguorescentlylabeledhistoneH2BduringDNAsynthesis( BlanpainandSimons 2013 ).TheDNAlabelbecomesdilutedwithinadividinglineageasnewDNAissynthesizedduringreplication,andthusslowcyclingstemcellswillretaintheirlabellongerthanthequickcyclingtransientamplifyingcells( VoogandJones 2010 ).Inaddition,itwasalsopostulatedthatstemcellsalwaysdivideasymmetricallyintoanotherstemcellandadaughtercellboundtodierentiate,andthecellthatmaintainsthestemcellphenotypealwaysretainsthesameDNAstrand(the`immortal'strandhypothesis),andthusalwaysmaintainsthelabel( Pottenetal. 2002 ).Theselabelingexperimentsandtheviewthatanindividualstemcellfunctionedaloneandalwaysdividedasymmetricallyledtovarioushypothesesconcerningstemcellnichedynamics.Onesuchhypothesiswasthe`epidermalproliferativeunit'intheskin,whereonestemcellindependentlygivesrisetoaportionofthedermis( PottenandMorris 1988 ).Anotherconcernedintestinalepitheliamaintenance,whereitwasbelievedthatasmallnumberofmainlyquiescentstemcellsindependentlygaverisetolineagesofepidermalcellsintheintestinalcrypt( PottenandMorris 1988 ).However,modernlabelingtechniquesusingdenitivemarkershavedemonstratedthatthepropertiesdescribedaboveunreliablycharacterizethetruestemcellniche( MorrisonandSpradling 2008 ).Importantly,theselabelingtechniquestracespeciclineagesandsubclonesofstemcells,andrevealtheclonaldynamicswithinthestemcellniche.Oneoftherstlineagetracingexperimentsinvestigatedtheclonaldynamicsofthemouseepidermisusinganimalsthatcouldbeinducedtoexpressenhancedyellowuorescentproteinwithinasmallfractionofepidermalcells( Claytonetal. 2007 ).Attwodayspost-induction,theresearcherssawsinglylabeledcellsthroughouttheepidermisatafrequencyof1in600cellsand,sincetheexpressionoftheuorescentproteinisheritable,anycells 12

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subsequentlyexpressingtheproteinatlatertimepointswouldbeclonesoftheserstcells.Invariantasymmetricdivisionofstemcells,acrucialelementoftheepidermalproliferativeunithypothesisproposedmorethan40yearspriortothisexperiment,wouldresultinlabeledclonesgrowingtoaxedsizeasthedermisdirectlysurroundinglabeledstemcellsalsoexpressedtheuorescentprotein.However,theresearchersdiscoveredthatthenumberofsurvivingclonesdiminished,whilethesizeoftheclonesthatsurvivedcontinuedtoexpand.Infact,theextinctionofclonesthroughdierentiationandnaturalepithelialturnoverwasperfectlycompensatedbytheexpansionofsurvivingclones,suchthatthetotalnumberoflabeledcellsremainedconstantovertime.This`scaling'behaviorisahallmarkoftheneutralcompetitionofequipotentstemcellsdividingthroughpopulationasymmetry( BlanpainandSimons 2013 ).Inthissystem,stemcells1dividebothsymmetricallyintootherstemcellsandasymmetricallyintocellsthatdierentiate,andthefateofanyonestemcellinthenicheisnotintrinsicallydierentthananyotherstemcell,meaningthatstemcellsarelostandreplacedinastochasticmanner.Stochasticstemcellfateandneutralcompetitionwithinthestemcellnichehassincebeendiscoveredinmanyothertissues(Table 1-1 )andmaybeauniversalparadigmfortissuerenewal( KleinandSimons 2011 ).ThelistprovidedinTable 1-1 doesnotrepresentalltissuetypesthatarerenewedthroughstemcelldivision,butrathersomeofthetissuetypesmostcommonlystudied.Inaddition,lineagetracingexperimentshavenotbeenconductedonallstemcellniches,sotheclonaldynamicswithinmanystemcellnichesremainunknown.Withinastemcellnicheunderneutralcompetition,anyonecloneorlineageinthestemcellnichewiththesameintrinsicdivisionordierentiationrateastheotherlineagesinthenichehasanequalchanceofreachingmonoclonalityintheniche(i.e.,allotherclonesleave 1Afterthediscoverythattheinterfollicularcellsdidnotdivideinasolelyasymmetricfashiontheauthorsreferredtothemas\progenitorcells",andthedesignationepidermalstemcellisusedforthequiescent\reserve"populationmobilizedafterwoundingfoundinthebulgeofhairfollicles( DoupeandJones 2013 ).Theprogenitorcellshaveindeniteself-renewalanddierentiateintothevariouscelltypesoftheepidermis,andmaintaintheirownniche( Limetal. 2013 ),andthusstillfunctionasthestemcellsoftheinterfollicularepidermis. 13

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thesystem),aprocesscommonlyreferredtoasneutraldrift.Inasystemwherethestemcellnicheislimitedbyspace,andstemcellspushothersoutofthenicheupondivision,asintheintestinalcrypt( Ritsmaetal. 2014 ),mutationsthatchangethecellulardivisionratewillchangetheprobabilitythatthelineageharboringthatmutationreachesmonoclonality.Atsmallpopulationsizes,lineageswithadisadvantageintheirdivisionratemayreachmonoclonalityinthenichethroughthestochasticnatureofcellulardivision(i.e.viageneticdrift).Mutationsthatincreasethedivisionrateofstemcellshaveanincreasedprobabilityofreachingmonoclonalityintheintestinalcryptstemcellniche( Vermeulenetal. 2013 ; Snippertetal. 2014 ).Mutationsthataccumulateinthestemcellnicheandchangetherateofstemcelldynamicswillaectthehomeostasisofthetissueasawhole.InthenextsectionIelaborateontheconceptofstemcellsascells-of-originintumorinitiationandtheroleofstemcellsincancergrowth. 1.2StemCellsInitiateandMaintainTumorGrowthEarlyworksuggestingthattumorsarisefromandaremaintainedbyrarepopulationsofcellswithstemcellproperties,i.e.cancerstemcells( SneddonandWerb 2007 ),camefromexperimentsdemonstratingthatonlyasmallsubpopulationoftumorcellsarecapableofforminganewtumorwhentransplantedintoanewhost,andthesecellsexpressstemcellmarkers( Lapidotetal. 1994 ; Al-Hajjetal. 2003 ).Thesestudiesshowedthatthemajorityofatumorconsistsofcellsthataredierentiatedandhavenotumorigenicpotentialiftransplantedintoanotheranimal.Muchoftheworkinvestigatingthecancerstemcellhypothesisandstemcellsasthecells-of-originincancerhasbeendoneintheintestinalepithelium,asthestemcellmarkersintheintestinearewellcharacterizedandthearchitectureoftheepitheliumisbrokenintoeasilyidentiablediscretepopulations.Oneoftherstexperimentsinvestigatingthecell-of-originintheintestinalepitheliumdeletedtheApcgene,oneofthemostcommonlymutatedgeneincoloncancer,fromcellsthroughouttheintestinalepitheliuminmice( Barkeretal. 2009 ).TheyfoundthatmostoftheApc-decientcellswerelostwithinafewdays,correspondingtothenaturaltransient 14

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amplication,dierentiation,anddeathofnon-stemcells.Afterover100days,asmallnumberofadenomasweredetectable,correlatingwiththeirmethod'sinfrequentdeletionofApcinstemcells.Laterworkutilizinglineagetracingdemonstratedthatasubpopulationofcellsexpressingthemarkersforintestinalstemcellsfueledthegrowthofestablishedadenomas( Schepersetal. 2012 ).Thisstudyalsodemonstratedthatthesameproportionofcellswithintheadenomasexpressedthestemcellmarkersasnormaltissue,reinforcingthehypothesisthatthegrowingtumorretainedthearchitectureofhealthytissueandthehierarchyofcelltypesderivedfromstemcellnichesfoundinhealthytissue.Asimilarstudyinvestigatingepidermaltumorigenesisusingalineagetracingstrategyfoundthatasimilarproportionofthetumorconsistedofstemcellsanddierentiatedtissueasthenormaltissue( Driessensetal. 2012 ).Inaddition,althoughintestinaltumorsaredrivenbytheexpansionofthestemcellpopulation,thesizeofthestemcellpopulationoccupyingeachnicheremainsextremelysmall( Kozaretal. 2013 ).Theresearchpresentedinthefollowingchaptersinvestigatestheevolutionarydynamicsthatresultfrompopulationsofstemcellsunderneutralcompetition.Itisestimatedthatthecompartmentalarchitectureoftheintestinalepitheliumleadstoboththeaccumulationofdeleteriousmutationsbygeneticdriftthroughouttheepithelium,andtheaccumulationofbenecialmutationswithinasinglepopulationthatcaninitiatetumorgrowth.Theexpectedeectofmutationaccumulationthroughgeneticdriftontissueself-renewalisestimated,anditisshownthatthesizeofthestemcellnichesandnumberofcryptspresentwithinthemouseintestinalepitheliumminimizestheprobabilityoftumorigenesis,attheexpenseofaccumulatingdeleteriousmutationsthatdiminishtissueself-renewal.Finally,tumorsgrowingunderthedynamicsdescribedwithinthischapteraresimulated,andcommonmeasurementsoftumorheterogeneityandnon-synonymoustosynonymousmutationratiosareshowntobetheproductofthepopulationstructureofagrowingtumor. 15

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Table1-1. Stemcelltissuerenewalinsomecommonlystudiedtissues. TissueApproximatepercentofcellsthatarestemcellsEvidenceforaniche?Evidenceforstochasticstemcellfate? Blood0.01%a( Pangetal. 2011 )Yes( Zhangetal. 2003 )Yes( Lenaertsetal. 2010 ; DingliandPacheco 2010 )Bone0.16%( TomasettiandVo-gelstein 2015 )Yes( Chanetal. 2015 )n/abEpidermis0.01%( Schneideretal. 2003 )Yes( Blanpainetal. 2004 )Yes( Claytonetal. 2007 ; KleinandSimons 2011 )Esophagus0.4%( Doupeetal. 2012 )Likely( De-Wardetal. 2014 )Yes( Doupeetal. 2012 )Intestine-Large1%( Nicolasetal. 2007 )Yes( Satoetal. 2011 )Yes( Lopez-Garciaetal. 2010 )Intestine-Small1%( Kozaretal. 2013 )Yes( Satoetal. 2011 )Yes( Lopez-Garciaetal. 2010 )Liver1%( Turneretal. 2011 )Yes( Turneretal. 2011 )n/aLung0.4%( Kimetal. 2005 )Yes( Kajsturaetal. 2011 )Yes( Teixeiraetal. 2013 )Mammary1%( Clarke 2005 )Yes( Villadsenetal. 2007 )Potential( BlanpainandSimons 2013 )Pancreas2.5%( SangiorgiandCapec-chi 2009 )Yes( Balicetal. 2012 )n/aProstate0.2%( Leongetal. 2008 )Yes( Tsu-jimuraetal. 2002 )Potential( Oussetetal. 2012 ; Blan-painandSimons 2013 )Testis0.03%( TagelenboschanddeRooij 1993 )Yes( LiandXie 2005 )Yes( Kleinetal. 2010 )Trachea0.1%( Schochetal. 2004 )Yes( Borth-wicketal. 2001 )n/a a:6%to10%( Pangetal. 2011 )oftheheterogeneousCD34+populationcomprising0.18%ofallcells( KatoandRadbruch 1993 ).b:Skeletalstemcellswerejustrecentlyidentied,andmoreresearchisnecessarytoinvestigatetheclonaldynamicswithintheskeletalstemcellniche( BiancoandRobey 2015 ). 16

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CHAPTER2THEIMPLICATIONSOFSMALLSTEMCELLNICHESIZESANDTHEDISTRIBUTIONOFFITNESSEFFECTSOFNEWMUTATIONSINAGINGANDTUMORIGENESIS 2.1IntroductoryRemarks 2.1.1EvolutioninTheStemCellNicheTheepithelialtissueswithinmanyanimalsarecontinuallyreplenishedbypopulationsofstemcellsthatdividethroughouttheorganism'slifetime.Forinstance,theepithelialliningoftheintestinaltractisreplacedweeklybymillionsofindependentpopulationsofstemcellslocatedinintestinalcrypts(reviewedin Barker ( 2014 )).Thiscontinualdivisionprovidesanopportunityformutation,resultingintheaccumulationofmutantlineagesandsomaticevolution( Lynch 2010 ).Stemcelllineageswithdecreasedtness,oradiminishedabilitytodivideandsurvive,willrepresentafailureinthistissuerenewalprocessandtheagingoftissuesandmulticellularorganismsasawhole( Lopez-Otnetal. 2013 ; Moskalevetal. 2013 ).Lineageswithincreasedtness,orfasterdivisionratesandanincreasedpropensitytosurvive,willresultintheaccumulationofcellsandneoplasia( Merloetal. 2006 ).Althoughconsideredpremalignantattheonset,theaccumulationofcellsintoapolyp,inwhichcellscontinuallydivideandaccumulatesubsequentmutations,candevelopacancerousphenotypeovertime( Winawer 1999 ). 2.1.2DistributionofFitnessEectsTheeectthatanewmutationwillhaveonanindividual'stnesscanbecharacterizedbyadistributionoftnesseects(DFE).TheDFEofseveralorganismshavebeenexperimentallyestimatedusingmutationaccumulationexperimentsordirectedmutagenesisexperimentsinthelaboratory( Eyre-WalkerandKeightley 2007 ; HalliganandKeightley 2009 ).Themajorityofrandommutationstoagenomethataecttnesshaveadeleteriouseectontness,whileasmallsubsetincreasetness( Eyre-WalkerandKeightley 2007 ).Additionally,manymutationsthataecttnesshaveasmalleect,whilefewhavealargeeect.Ingeneral,bothbenecial( ImhofandSchlotterer 2001 ; KassenandBataillon 2006 ; Orr 2003 )anddeleterious( Elenaetal. 1998 )mutationaltness 17

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eectscanbedescribedwellusinganexponentialdistribution.WenotethatcertainbenecialDFEmaynotbeexponentiallydistributed,andarebetterclassiedashavingeitheraboundedorheavier-than-exponentialtail( Rokytaetal. 2008 ; Banketal. 2014 ; Levyetal. 2015 ),andcompounddistributionsordistributionswithmoreparametersmaybettertempiricalmeasuresofDFE( Sanjuanetal. 2004 ).ByunderstandingthemutationalDFEinsomatictissuewecanpredicttheevolutionarytrajectoriesoftissueswithinmulticellularorganismsastheyage.Absolutetnessistypicallymeasuredasthereproductivesuccessofageneticallyidenticallineage,whichcanbemeasuredempiricallyasthegrowthrateofapopulationandinterpretedecologicallyasthedeathrateofindividualsinapopulationsubtractedfromthebirthrate.Withinstemcellpopulationsthisisanalogoustothedierentiationrateofthestemcelllineagesubtractedfromthedivisionrate.However,thetotalgrowthrateofthehealthystemcellpopulationisnecessarilyzerotoinsuretissuehomeostasis.Aswedescribeinthenextsection,thestemcellsexistintwopopulations:astaticnichepopulationwithcellsthatareundergoingdivisionandmigratingintothesecondpopulation,containingcellsthatareundergoingdivisionanddierentiation.Therefore,althoughonlymutationstostemcelldivisionratewouldconferachangeinselectionpressurewithinthestemcellniche,xedmutationstobothdivisionanddierentiationratewillaltertherateofgrowthofthetotalstemcelllineage,theexpectedsizeofthestemcellpopulationasawhole,andcontributetotheprobabilityofatumorigenesisevent.Hence,mutationstodivisionanddierentiationrateaectthereproductivesuccessofstemcelllineages,i.e.tness,andweconsiderdistributionsofmutationaleectsonthesetworatesinthiswork.Althoughtherehasbeennodirectmeasurementofthedistributionoftnesseectsinsomatictissue(butsee Vermeulenetal. ( 2013 ); Snippertetal. ( 2014 )forestimationsoftheselectiveadvantageforsomeknowncancerdrivers),theevolutionofcancerprogressionhasbeenpreviouslymodeledusingdiscrete( Beerenwinkeletal. 2007 ; Bozicetal. 2010 ; McFarlandetal. 2013 )andcontinuous( Fooetal. 2011 )tnesseects.Here,wedierfromthesepreviousmodelsbyinvestigatingdierentmutationaleectframeworksusing 18

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parametersderivedfromwholeorganismstoexploremutationaccumulationincryptsinitializedattheirmeasuredhealthysizeinmiceandhumansandquantifybothagingandtumorigenesis.Whenquantifyingtumorincidenceweareconcernedwiththemomentthattheregulatoryregimeintheintestinalcryptbreaksdown:whenthestemcelldivisionrateexceedsitsdierentiationrate.Wecallthispointthetumorigenesisthreshold.Theresultingpopulationwillaccumulatestemcellswithoutbound,whichisthoughttobethecauseofcryptssionandthemainmechanismofpolyporadenomagrowth( Wongetal. 2002 ; LoefflerandGrossmann 1991 ).Weinvestigatethefullspectrumofdeleteriousandbenecialmutationaleectsontheprogressionofahealthycrypttotumorinitiationusingempiricallymeasuredratesofdivision.Theevolutionofmulticellularityhasnecessitatedtheevolutionofregulatorysystemsthatholdsomaticstemcellsatarelativelylowtness(whencomparedtotheirmaximumpotential)inordertoensurethecooperationofthedierentcellularsystemsconstitutingawholeorganism.Assuch,inadditiontothebenecialmutationsthatwouldbeaccountedforbyatypicalDFEforwholeorganisms(whicharecommonlyassumedtoalreadybehighlyt( Orr 2010 )),weexpectmutationsoflargeeectinsomatictissueasregulatoryprocessesbecomedysfunctional,suchasthedeactivationoftumorsuppressorgenesortheactivationofoncogenes.Itisreasonabletohypothesizethataheavy-taileddistributioncouldbetterclassifymutationaleectsthathaveabenecialeectinsomaticstemcellsbycapturingboththemutationsofsmalleectandalsohavinganon-trivialprobabilityofcapturingthemutationsoflargeeectoftenassociatedwithcancer.WeevaluatewhetherornottheDFEestimatedinwholeorganismscanexplainknowntumorincidenceintheintestine.Further,weexplorewhetherornottumorincidenceisbetterexplainedbyaheavy-taileddistributionformutationsbenecialtotness.Thus,wecreateamodelofanevolvingintestinalstemcellpoolandimplementalternateDFEandcomparetheresultantincidencecurvestoknowntumorincidencecurves. 19

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2.2MaterialsandMethods 2.2.1DescriptionoftheModelCryptPopulationStructure.Thebaseofeachintestinalcryptharborsapopulationofsymmetricallydividingcellsexpressingmarkersassociatedwiththestemcellphenotype( Lopez-Garciaetal. 2010 ; Snippertetal. 2010 ).Withinthispopulationthereexistsasubpopulationnichethatisresponsibleformaintainingtissuehomeostasis( Kozaretal. 2013 ; Vermeulenetal. 2013 ).Wemodelthestemcellsoftheintestinalcryptastwopopulationsofcells,therstbeingthisstemcellniche,whichconsistsofaxedpopulationofstemcells,N,andthesecondconsistingofthestemcellsdisplacedfromthisnichebutnotyetcommittedtodierentiation.Thesumofthesepopulationsrepresentthetotalnumberofstemcellswithinthecrypt,NT.Inourmodel,cellswithinthenichedivideatrateanddisplacetheirneighborsthroughovercrowding,asproposedby Lopez-Garciaetal. ( 2010 )andrevealedbyinvivoliveimagingby Ritsmaetal. ( 2014 ).Thispopulationofcellsexperiencesgeneticdriftandselection;cellsthathaveahigherdivisionratearemorelikelytopushtheirneighborsoutoftheniche(asdemonstratedby Snippertetal. ( 2014 ))andcellswithlowerdivisionratesaremorelikelytobedisplaced.Mutationsmayoccuratdivisionwithmutationrateandresultineitheralineagewithanewdivisionrateoralineagewithanewrateofcommittingtodierentiation.Displacedstemcellsdivideattherateoftheirprogenitorcellsinthenicheandcommittodierentiationatrate,hereafterreferredtoasthedierentiationrate.Weassumethatoncealineagecommitstodierentiationitisdestinedtobeexpelledfromthecrypt.Wedenetumorigenesisinthecryptasthemomentalineageofstemcellswithadivisionrategreaterthanitsdierentiationratehasbecomexedintheniche,resultinginexponentialpopulationgrowth.Wenotethat,althoughastemcell'spropensitytocommittodierentiationinhealthytissueispartiallydependentonexternalsignalingqueues,suchasWntsignalsfromPanethcellsinthesmallintestinalcryptstemcellniche( Clevers 2013 ),theabilityofastemcelltointerpretandrespondto,orevengainindependencefrom,externalsignalsisanintrinsicandheritablepropertyofthestemcell( ReyaandClevers 2005 ).The 20

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parameters,N,andNThavebeenpreviouslyestimated( Kozaretal. 2013 ; Vermeulenetal. 2013 ),andwecalculateaccordingtoequation4inthe APPENDIXDESCRIPTIONOFTHEMATHEMATICALMETHODOLOGY ,where=1+N y(t)andyistheaveragenumberofstemcellsoutsideoftheniche.DistributionofFitnessEects.Werstdescribeourmodelofmutationsthataectthedivisionrateofstemcellsandaddressmutationsthataectdierentiationratelaterinsection\ MutationsThatAltertheDierentiationRateofStemCellsResultinRapidAgingandTumorigenesis "WhenmutationsoccurthenewdivisionrateisgreaterthanthepreviousratewithprobabilityPB,andthemeanpositivechangeofrateiss+.WeconsiderpositiveandnegativechangesthatareexponentiallydistributedfordeleteriouseectsandexponentiallyorParetodistributedforbenecialeects,seethe APPENDIXDESCRIPTIONOFTHEMATHEMATICALMETHODOLOGY .Themeannegativechangeiss)]TJ /F1 11.955 Tf 7.08 1.79 Td[(.WedenetheexponentialDFEinEquation 3{3 andthepower-lawDFEinEquation 2{2 .m(;0)exp=8>><>>:(1)]TJ /F9 11.955 Tf 11.96 0 Td[(PB) 0e)]TJ /F14 7.97 Tf 6.59 0 Td[((1)]TJ /F16 5.978 Tf 9.69 3.26 Td[( 0)<0PB 0e)]TJ /F14 7.97 Tf 6.58 0 Td[(( 0)]TJ /F12 7.97 Tf 6.58 0 Td[(1)>0 (2{1)m(;0)Pareto=8>><>>:(1)]TJ /F9 11.955 Tf 11.96 0 Td[(PB) 0e)]TJ /F14 7.97 Tf 6.59 0 Td[((1)]TJ /F16 5.978 Tf 9.69 3.26 Td[( 0)<0PB)]TJ /F12 7.97 Tf 6.59 0 Td[(1 0( 0))]TJ /F14 7.97 Tf 6.59 0 Td[(>0 (2{2)Thepowerlawdistributioniswelldenedif>1andisconsideredtobeheavy-tailed(havinginnitevariance)if1<<3.SelectionAssumptions.Weareconcernedwiththemutationsthatariseandreachxationwithinthestemcellniche.Duetodrift,allstemcellswiththesamedivisionrateasthebackgroundpopulationhaveanequalprobabilityofreachingxation,commonlyreferredtoasneutraldriftdynamics( Lopez-Garciaetal. 2010 ; Snippertetal. 2010 ).Following WodarzandKomarova ( 2005 )weuseaMoranmodeltoestimatetheprobabilitythata 21

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mutantlineagexesinthestemcellniche: px(;old)=1)]TJ /F10 11.955 Tf 11.96 0 Td[(old= 1)]TJ /F7 11.955 Tf 11.95 0 Td[((old=)N(2{3)whereNisthenumberofcellsintheniche.Themutationrateislowrelativetothedivisionrate,soweassumethatthereareatmosttwocompetingdivisionratesatanygiventime.Usingtheaboveformula( 2{3 ),wecanuseBayes'Theoremtocomputetheprobabilitydensity(jold)ofanewxeddivisionrategiventhatthepreviousdivisionrateisold: (jold)=px(;old)m(;old) R10px(`;old)m(`;old)d`.(2{4)Asdescribedabove,tumorigenesisoccurswhenthedivisionrateisgreaterthandierentiationrateandwedenethepointatwhichthishappenstobethetumorigenesisthreshold.Inourmodelingframework,eachnewxedmutationpresentsanewpossibilitythatthedivisionrateexceedsthethresholdfortumorigenesis.From( 3{5 )wecaniterativelyderivethesequenceoffunctionsffngthatrepresentthedensityofthedistributionofthestemcelldivisionratesconditionedthatnmutationshavexedinthestemcellnicheandtumorigenesishasnotoccurredasofmutationn)]TJ /F7 11.955 Tf 12.8 0 Td[(1.Ifwelet0denotetheinitialstemcelldivisionrate,thenf1()=(jold=0).Foreachnletpndenotetheprobabilitythattumorigenesisoccursduetothenthmutation(giventhatnmutationshaveoccurred).Thenp1=R1f1()d.Fromthere,wecanwritetherecursiveformulae fn+1()=1 1)]TJ /F9 11.955 Tf 11.96 0 Td[(pnZ0(jold=`)fn(`)d`andpn+1=Z1fn+1()d.(2{5)Fromthis,wehavearecursiveformulafortheprobabilities,fqng,thattumorigenesishasnotoccurredgivennxedmutations:q1=1)]TJ /F9 11.955 Tf 11.96 0 Td[(p1and qn+1=(1)]TJ /F9 11.955 Tf 11.96 0 Td[(pn+1)qn.(2{6) 22

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Totranslatethisresulttoanindividual'slifetime,wemodelthetime-dependentarrivalofnewmutationsasaPoissonprocesswithxedrateparametermutationspercelldivision.Wekeeptrackofthetime-dependentnumberM(t)ofmutationsthatxinthestemcellnichesbytimet.Then,usingTtodenotethetimethattumorigenesisoccursinagivencrypt,wecanwritetheprobabilitythattumorigenesishasnotoccurredasoftimetbytheequation P(T>t)=1Xn=1qnP(M(t)=n)(2{7)Dependingonthespecies,anindividualhashundredsofthousandsorevenmillionsofcrypts.Theprobabilitythatanindividualhasatleastonecryptthathasundergonetumorigenesiscanbecalculatedbyconsideringthedistributionofxedmutationsthathaveaccumulatedamongtheindividual'scryptsandtheprobabilitythatthesemutationsresultintumorigenesis.Thiscanthenbeextrapolatedtotheincidencerateoftumorsamongapopulationofindividuals(Fig. 2-1 ).LetTrepresentthetimethattumorigenesisrstoccursinanyofanindividual'scrypts.WeusethefollowingestimatetocalculatetumorigenesisincidencedatareportedintheResultssection.IntheSupportingInformationwedescribethefullcalculationandafewsimplifyingassumptionswemaketodevelopacomputationallytractablemodel.P(Notumorigenesisattimet)^nYn=1P(Nocrypttumorigenesisattimetjnmutations)E(#ofcryptswithnmutations)i.e., P(T>t)nmaxYn=1(qn)C(^t)ne)]TJ /F15 5.978 Tf 6.53 0 Td[(^t=n!.(2{8)Intheabove,Cisthenumberofcryptsinthelengthofintestinebeinginvestigated,nmaxisthemaximumnumberofmutationssimulatedand^=N0R10px(;0)m(;0)d, 23

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where,asabove,Nisthenumberofcellsinthestemcellniche,isthemutationratepercelldivisionandpxisdenedbyEq. 2{3 Figure2-1. A:Acrosssectionofanintestinalcrypt,bluecirclesatthebaseofthecryptrepresentstemcellswhileyellowcirclesrepresentcellsthathavecommittedtodierentiation.Theovalcrosssectionatthebaseencompassesthestemcellniche,whilestemcellsabovethisnichearedestinedtocommittodierentiation.Takingatop-downlookattheoval,largecirclesrepresentacrosssectionoftheintestinalcryptbase,whichhousestheintestinalstemcells,representedbysmallerblueandredcircles.Mutationsmayoccurtoasinglecellinthestemcellniche.Thesemutationsalterthetnessofthecellaccordingtoaspecieddistributionoftnesseects.Giventhenewtness,themutatedlineagehasacertainprobability,px(;old),ofreachingxationwithinthestemcellniche.B:Here,therectanglesrepresentacrosssectionoftheintestinalepitheliumwiththenumbersrepresentingthelocationsofindividualcryptsanddescribingthenumberofxedmutationsforeachcrypt.Anorganismaccumulatesxedmutationsoveritslifetime. 2.2.2ParameterChoicesSomeestimatesofcryptdynamicsparametershaveshiftedovertime,forexamplethestemcelldivisionrateinthemousewasformerlythoughttobeonceevery1-1.5days( Lopez-Garciaetal. 2010 ),butmorerecentestimatesindicatetheydivideonceevery3-10days( Kozaretal. 2013 ). Kozaretal. ( 2013 )demonstratedthatthedivisionrateofstemcells 24

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inthestemcellnicheofmicevariedfromapproximately0.1to0.2to0.3divisionsperdayalongtheproximalsmallintestine,distalsmallintestine,andcolon,respectively.Likewisetheestimatednumberofstemcellswithinthemousestemcellnichevariesfromapproximatelyvetosixtoseven,respectively.Thetotalnumberofcellsincryptsexpressingstemcellmarkershasbeenreportedtobe14-16inmice( Snippertetal. 2010 ; Clevers 2013 ; Lopez-Garciaetal. 2010 ).Fortheanalysisofourmousemodelwechosethemiddlevalueoftheseparameterranges,acryptwith15totalcellsexpressingstemcellmarkers,with6ofthecellsconstitutingthestemcellnichedividing0.20timesperstemcellperday.Toestimatethedierentiationrateofstemcellsoutsidethestemcellniche,weusedacontinuoustimeMarkovchain,describedinthe APPENDIXDESCRIPTIONOFTHEMATHEMATICALMETHODOLOGY .Accordingtothismodel,inorderforthetotalstemcellsinthecryptofamousetostayataconstantpopulationsizethedierentiationrateofstemcellsoutsideofthestemcellnichemustbe0.333perstemcellperday.Theparametersassociatedwithcryptdynamicsinmicehavebeenwelldescribed,however,wewereunabletoobtainanydataonpopulationincidenceofintestinalpolypsortumorsinwildtypemice.Ontheotherhand,whilecryptdynamicsinhumanshavenotbeenaswell-studied,thereexistsincidencedataforlargeintestinepolyps( Chapman 1963 ).Toparameterizethehumancoloncryptsystemweconsideredafewsources. Nicolasetal. ( 2007 )analyzedthemethylationpatternswithinthehumancoloncryptandtheirBayesiananalysissuggestsaposteriordensitymodebetween15and20stemcellsmaintaininghomeostasisandconstitutingthestemcellnichewithinthecrypt.Theirdistributionisskewedtotheleftsowechose20asaninitialvalueforthenumberofstemcellswithinthestemcellniche. BravoandAxelrod ( 2013 )reportanaverageof35.7quiescentstemcellswithinthehumancoloncryptthroughastainingexperiment,soweassumethereare36totalstemcellswithinthehumancoloncrypt.Humancolonstemcellsdivideaboutonceeverysevendays 25

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( Pottenetal. 2003 ),whichwouldmeantheywouldhavetodierentiateatarateofabout0.321perdayinordertomaintainhomeostasisattheassumedinitialparameters.WeparameterizedtheinitialDFEbasedonthosemeasuredinwholeorganismstoevaluatewhethertheycanaccountforknowntumorigenesisincidence.Thedistributionoftnesseectshasbeenestimatedinmutationaccumulationexperimentsanddirectedmutagenesisexperiments.WeconsidertheDFEproposedby JosephandHall ( 2004 )inamutationaccumulationstudybecausetheyreporttheexpectedeectsizeofdeleteriousandbenecialmutations,aswellasthemutationrateandtheproportionofmutationsthatwerebenecialinadiploideukaryoticsystem(Saccharomycescerevisiae).Theyreportanaveragebenecialheterozygoustnesseectof0.061,whichisslightlylowerbutwithinanorderofmagnitudeoftheeectofaveragebenecialmutationmeasuredforvesicularstomatitisvirusof0.07( Sanjuanetal. 2004 )andE.coliof0.087( KassenandBataillon 2006 ).Theyfoundthat5.75%ofaccumulatedmutationswerebenecialandthattheoverallmutationratetoallelesthataltertnesswas6.310)]TJ /F12 7.97 Tf 6.58 0 Td[(5mutationsperhaploidgenomepergeneration.Thiswouldresultinadiploidbenecialmutationrateof26.310)]TJ /F12 7.97 Tf 6.59 0 Td[(50.0575=7.24510)]TJ /F12 7.97 Tf 6.58 0 Td[(6.ThisiswithinanorderofmagnitudeofthebenecialmutationratereportedforE.coli( Wiseretal. 2013 ).Mutationaccumulationexperimentsmaynotcapturethetruedistributionoftnesseectsbecausetheyrelyonobservingthemutationsoflineagesthatsurviveandpersistinapopulation.Becauseofthis,theyarebiasedagainstmutationsoflargedeleteriouseect.Additionally,therandompassagingofindividualstorepopulatenewgenerationsmayresultindrasticallydierentestimatesofaveragemutationaleectsizeforthesamespecies.Forinstance,averagedeleteriouseectofmutationsinSaccharomycescerevisiaehasbeenestimatedtobe0.061( JosephandHall 2004 ),0.086( Wlochetal. 2001 ),and0.217( ZeylandDeVisser 2001 ).DirectedmutagenesisofrandomgenometargetsinanRNAvirusrevealedanaveragenon-lethaldeleterioustnesseectof0.244( Sanjuanetal. 2004 ).Itislikelythattheinherentaverageeectsizeofamutationofdeleteriouseectwouldbe 26

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betterreectedbythelargerestimatessincemutationsoflargedeleteriouseectmaybelostinmutationaccumulationexperiments.AfterbuildingourmodelwithDFEparametersestimatedfromwholeorganismswedescribeoverallpatternsofmutationaccumulationandriskoftumorigenesisandthenweutilizedleastsquaresanalysistoexplorethebesttamongaseriesofplausiblechoicesofandtheexpectedvalueofs+forthehumanincidencecurvesandcomparedtodatafrom Chapman ( 1963 )(besttguresavailableasSupportingFigures 2-4 2-5 ,and 2-9 ).Forthedivisionratescenario,bothfortheexponentialandpower-lawDFE,wevarytheexpectedvalueofs+from0.041to0.07andfrom2.510)]TJ /F12 7.97 Tf 6.58 0 Td[(5to710)]TJ /F12 7.97 Tf 6.59 0 Td[(4.Forthedierentiationratescenariowevarys+from0.041to0.07andfrom2.510)]TJ /F12 7.97 Tf 6.58 0 Td[(7to710)]TJ /F12 7.97 Tf 6.59 0 Td[(4.ThemodeldescribedabovewasexecutedusingRversion3.1.1.Rscriptsdevelopedforthisstudyareavailableat https://github.com/vcannataro/Somatic Evo DFE 2.3Results 2.3.1MutationsResultinBothAgingandTumorigenesisWithintheIntestineBecausestemcellnichepopulationsaresmall,itispossibleformutantlineageswithatnessdisadvantagetoxintheniche.This,coupledwiththefactthatthevastmajorityofmutationsthatoccurwillhaveadeleteriouseectonstemcelltness,resultsintheexpectedvalueoftheprobabilitydensitydescribingthenewdivisionratestomoveawayfromthetumorigenesisthresholdwithsubsequentxedmutations(Fig. 2-2 A,C,E).Ingeneral,theaccumulationofxedmutationswithincryptsresultsinimpairedstemcellmaintenanceandlowerstemcellproduction,contributingtotheagingofthetissueandorganism.Theprobabilitythataparticularxedmutationwillresultintumorigenesisinthecrypt,pn(Eq. 3{6 ),isequaltotheareaunderthesedensitiesthatcrossesthetumorigenesisthreshold(Fig. 2-2 B,D).Fortheinitialparameterizationinmiceandhumansthisincreasesatrst,butthendecreaseswithsubsequentxedmutationsastheprobabilitydensitiesdescribingdivisionratemoveawayfromthetumorigenesisthreshold. 27

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Figure2-2. Theaccumulationofprobabilitydensitiesdescribingstemcelldivisionrate.A:ExponentiallydistributedtnesseectsondivisionrateusingtheparametersinTable 2-1 forthemouse.Therstdensityisagreendashedline.Eachprobabilitydensityrepresentsthedivisionrateofaxedlineageafternxedmutations,withnindicatedbyanarrow.B:Zoominginonthetumorigenesisthreshold,weseethattheareaofthedivisionratedensitythatisoverthetumorigenesisthresholdincreasesatrstandthendecreaseswithsubsequentmutation.Thereisachangeinslopeofthedensitiesatthetumorigenesisthresholdbecausesubsequentdensitiesarecalculatedfromthepreviousdensitywhichhashadtheareatotherightofthetumorigenesisthresholdremovedandtheareatotheleftrenormalizedto1.C,DarethesameasAandB,respectively,butareforthehumanscenario.Thelargerpopulationsizedecreasesthestrengthofdrift.OrderofmutationsinCproceedsasinA,andproceedsfrom1through8frombottomtotopinD.E:TheexpectedvaluesoftheprobabilitydensitiesinAandBdividedbytheiroriginalvaluesoversubsequentxedmutations 28

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Table2-1. Initialmodelparameters,combiningwholeorganismDFEwithorganismalcryptparameters.Seetextaboveforreasoningbehindinitialparameterchoices. ParameterDescriptionValueinmouse[Ref]Valueinhuman[Ref] PBPercentofmutationswithabenecialeect0.0575( JosephandHall 2004 )0.0575( JosephandHall 2004 )s+Eectsizeofamutationofbenecialeect0.061( JosephandHall 2004 )0.061( JosephandHall 2004 )s)]TJ /F1 11.955 Tf 38.02 1.79 Td[(Eectsizeofamutationofdeleteriouseect0.217( ZeylandDeVisser 2001 )0.217( ZeylandDeVisser 2001 )Mutationratepergenesinuencingtnessperdivision26.310)]TJ /F12 7.97 Tf 6.58 0 Td[(5( JosephandHall 2004 )26.310)]TJ /F12 7.97 Tf 6.59 0 Td[(5( JosephandHall 2004 )0Normalstemcelldivisionrateperday0.2( Kozaretal. 2013 )0.143( Pottenetal. 2003 )0Normalstemcelldierentiationrateperday0.333(thisstudy)0.321(thisstudy)NNumberofstemcellsinthestemcellnicheatthebaseofthecrypt6( Kozaretal. 2013 )20( Nicolasetal. 2007 )NTTotalnumberofstemcellsexpressingstemcellmarkersinthecrypt15( Clevers 2013 )36( BravoandAxelrod 2013 )CryptsNumberofcryptsinthesmallandlargeintestine,respectively7.5105,4.5105( Pottenetal. 2003 )5107,2107( Pottenetal. 2003 ) 2.3.2PredictedIncidenceCurvesinMiceandHumansUsingDFEDerivedfromaWholeOrganismUsingthemodeldescribedin\ DescriptionoftheModel ",wedeterminedthecumulativeprobabilitydistributionoftumorigenesiswithinapopulationofcryptsinanindividualorganism.Formice,usingtheinitialparametersinTable 2-1 andexponentiallydistributedbenecialtnesseects,wendthattheincidenceoftumorigenesisispredictedtoincreaselinearlywithage,withclosetoninepercentofmiceexperiencingtumorigenesisatthreeyearsofage(Fig. 2-3 A).Humantumorigenesisincidenceinthelargeintestineispredictedtobeapproximately 29

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36percentat80yearsofage(Fig. 2-3 B),usinganexponentiallydistributedbenecialtnesseectsandtheinitialnullparametersfromTable 2-1 .Theonlyincidencedataforearlytumorsorpolypswasfoundforthelargeintestineinhumans.ThepredictedincidencecurvederivedfromanexponentiallydistributedDFEfollowsthesamequalitativedynamicsasthetumorincidencedata.Incidencecurvesthatarederivedfromapower-lawdistributionusingtheinitialparametersinTable 2-1 predictnearly100percenttumorigenesisby80yearsofageanddonotfollowtheincidencedatadynamics.Hence,weperformedaleastsquaresanalysis,varyingparametersthathavenotbeencharacterizedforhumansomatictissue,tondtheparametersetinourexploratoryspacewiththebestttotheobservedincidencecurvetothedata. Figure2-3. TumorigenesisincidenceinmiceandhumansusingwholeorganismDFEparameters.A:Thepopulationincidenceoftumorigenesisthroughouttheentireintestinaltractofthemouse.B:Thepopulationincidenceoftumorigenesisthroughoutthelargeintestineinhumans.TheblackdashedlinesaregeneratedfromthespeciesspecicparameterslistedinTable 2-1 .Thesolidredlineconnectslargeintestinepolypincidencedatafoundduringautopsy( Chapman 1963 ). 30

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2.3.3AlteringtheExpectedBenecialFitnessEectsandtheMutationRateProvidesBetterFitsforBothExponentialandPower-lawDerivedIncidenceCurvesTheexpectedmeantnesseects(s+,s)]TJ /F1 11.955 Tf 7.08 1.79 Td[()oftheDFEandthemutationrate()perdivisionofamutationwhichaltersthestemcelltnesswereinferredfromwholeorganismsasaninitialparameterchoice(Table 2-1 ).Aparameterspacearoundtheinitialchoiceswasexploredandaleastsquaresanalysiswasperformedtondabetterttothedata(additionalinformationinthe APPENDIXDESCRIPTIONOFTHEMATHEMATICALMETHODOLOGY andFigures 2-4 2-5 ,and 2-9 ).Justthemutationrate()andexpectedbenecialtnesseect(s+)arepresentedbecausechangestotheexpecteddeleterioustnesseect(s)]TJ /F1 11.955 Tf 7.08 1.79 Td[()hadlittleeectontheresultanttumorigenesisincidencecurves.Wefoundthatboththeexponentialandpower-lawscenarioscanprovidesimilarlygoodtstothedata,howeverwithdistinctlydierentparameters.TheexponentialDFEderivedcurveprovidedthebesttwiththesamemutationrateasourinitialchoice(Table 2-1 ),withaslightlylargerexpectedbenecialtnesseect(E[s+]=0.064,Fig. 2-6 A,reddashedline).Interestingly,assumingthesameexpectedbenecialtnesseectasinTable 2-1 andvaryingthemutationrateprovidesareasonabletwithaslightlylargermutationrate(=1.7510)]TJ /F12 7.97 Tf 6.59 0 Td[(4, 2-6 A,bluedashedline).Thepower-lawDFEprovidedasimilarlygoodttotheincidencecurve,butforaparameterspacethatassumesamuchsmallerexpectedbenecialtnesseectandalargemutationrate(E[s+]=0.044,=510)]TJ /F12 7.97 Tf 6.59 0 Td[(4,Fig. 2-6 Breddashedline). 2.3.4MutationsThatAltertheDierentiationRateofStemCellsResultinRapidAgingandTumorigenesisMutationsaectingdierentiationrateinuencethelifetimeofastemcelllineage.Mutationsthatincreasedierentiationratewilldecreasethetnessofthelineage,whilemutationsthatdecreasedierentiationrateincreasetness.Mutationsaectingdierentiationrateneutrallydrifttoxationinthestemcellnichebecausethedierentiationphenotypeisnotexpressedintheniche,henceallcellsdivideatthesamerate.Thus,theprobability 31

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Figure2-4. HeatmapdepictingvaluesfortheleastsquaresanalysisofpredictedtumorincidenceandhumantumorincidencedatafortheexponentialbenecialDFEondivisionratescenario.Parametercombinationswithredcolorshavethesmallestsumofsquaredresiduals,whilebluecolorshavethelargest. ofxationofmutationstodierentiationrateis(1=N),regardlessofmutationaleect.Weonlyconsideredanexponentialmutationaleectdistributionbecausethedistinctionbetweenexponentialandpowerlawdistributionsisonlysignicantinprevalenceoflargedeviationsfromthemean,andbenecialmutationaleectsinthisscenarioexistbetween0andzero.Becauseallmutationsthatsolelyaectdierentiationratedriftneutrally,andthemajorityofmutationsdecreasetness(byincreasingdierentiationrate),themajorityofxedmutationsmovestemcellpoolsawayfromthetumorigenesisthreshold.(Fig. 2-7 ). 32

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Figure2-5. Heatmapdepictingvaluesfortheleastsquaresanalysisofpredictedtumorincidenceandhumantumorincidencedataforthepower-lawbenecialDFEondivisionratescenario.Parametercombinationswithredcolorshavethesmallestsumofsquaredresiduals,whilebluecolorshavethelargest. Mutationaleectsaretypicallydescribedasaproportionofthephenotypetheyareaecting,andassuchthesameDFEappliedtoalargerratewillhavealargerabsoluteexpectedeect.Thedierentiationrateofstemcellsdisplacedfromthenicheisnecessarilylargerthantheintrinsicdivisionratesinceonlyasubpopulationoftheentirestemcellpopulationisexposedtocommittingtodierentiation,howeverallcellsaredividing( Ritsmaetal. 2014 ),andthestemcellpopulationismaintainedatasteady-stateequilibrium.Thus,mutationsaectingdierentiationrateinourmodelhavealargerabsoluteeectforthesameproportionalchangeinratewhencomparedtothepreviousanalysisonmutationsto 33

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Figure2-6. A:IncidencecurvesderivedfromtheassumptionofanexponentialbenecialDFE.Thebestttothedataoutoftheexploredparameterspacehasthesameastheyeastreportedin 2-1 andE[s+]=0.064(reddashedline).BlackdashedlinesderivedfromE[s+]=0.064and,frombottomtotop,=7.510)]TJ /F12 7.97 Tf 6.59 0 Td[(5to1.7510)]TJ /F12 7.97 Tf 6.58 0 Td[(4by2.510)]TJ /F12 7.97 Tf 6.58 0 Td[(5.BluedashedlineisthepredictedincidencecurvewiththebesttwithE[s+]=0.061(initialDFEderivedfromyeastreportedin 2-1 ),whichhad=1.7510)]TJ /F12 7.97 Tf 6.59 0 Td[(4.B:Incidencecurvesderivedfromtheassumptionofapower-lawbenecialDFE.AllparametersarethesameasinTable 2-1 ,exceptE[s+]=0.044foreachcurveand,rangingfromtoptobottom,rangesfrom4.510)]TJ /F12 7.97 Tf 6.59 0 Td[(4to5.510)]TJ /F12 7.97 Tf 6.58 0 Td[(4by2.510)]TJ /F12 7.97 Tf 6.58 0 Td[(5,with510)]TJ /F12 7.97 Tf 6.59 0 Td[(4providingthebestt. divisionrate.Hence,givenaxedmutation,weseeahighincidenceoftumorigenesiswhenthemutationaectsdierentiationrate(Fig. 2-8 A,B).Fittinganalysesalongarangeofplausibleparameterspacerevealedapoorertwhencomparedtomutationsthatalterdivisionratebecausemutationsthatalterdierentiationratewillalwaysresultinlargetumorincidenceatearlyage. 2.4Discussion 2.4.1WholeOrganismDFEAreSucienttoExplainTumorigenesisWehypothesizedthatmutationsinsomatictissueswoulddierintheirdistribution,comparedtounicellularwholeorganisms,becauseoftheregulatoryprocessesthatcontrolcelldivisionanddierentiationratesinmulticellularorganisms.However,wefoundthatwholeorganismDFEweresucienttoaccountforpatternsoftumorigenesisintheintestines.This 34

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Figure2-7. A:ExponentiallydistributedtnesseectsondierentiationrateusingtheparametersinTable 2-1 forthemouse.Therstdensityisagreendashedline.Eachprobabilitydensityrepresentsthedierentiationrateofaxedlineageafternxedmutations,withsubsequentmutationstravelingawayfromtheoriginaldierentiationrate.B:Zoominginonthetumorigenesisthreshold,weseethattheareaofthedierentiationratedensitythatisoverthetumorigenesisthresholddecreaseswithsubsequentmutation.Thereisachangeinslopeofthedensitiesatthetumorigenesisthresholdbecausesubsequentdensitiesarecalculatedfromthepreviousdensitywhichhashadtheareatotheleftofthetumorigenesisthresholdremovedandtheareatotherightrenormalizedto1.C,DarethesameasAandB,respectively,butareforthehumanscenario.OrderofmutationsinCproceedsasinA.E:TheexpectedvaluesoftheprobabilitydensitiesinAandBdividedbytheiroriginalvaluesoversubsequentxedmutations 35

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Figure2-8. CalculationspresentedfortumorincidenceinA:mice,B:humans,andC:Besttincidencecurveinred;expectedbenecialtnesseectof0.057andmutationrateof2.510)]TJ /F12 7.97 Tf 6.58 0 Td[(6.Theothercurveshavethesamemutationratebutvaryaroundtheexpectedbenecialtnesseectbyincrementsof0.001. suggeststhatsomaticevolutionisnotunique;butinsteadisbasedonthesamepatternsofmutationthatweseeinwholeorganisms.Hence,thedierencesinevolutionarypatternsbetweensomatictissuesandwholeorganisms,suchasthetendencyoftissuestoageviatheaccumulationofdeleteriousmutationswhilepopulationsofwholeorganismsinsteadevolvetogreatermeantnessinbenignenvironments,ariseasaconsequenceofthesmallpopulationsofstemcellswithinmulti-cellularorganismsandtheasexualnatureofcelldivision.SomaticagingviamutationisthusakintotheactionofMuller'sratchet,theaccumulationofdeleterious 36

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Figure2-9. Heatmapdepictingvaluesfortheleastsquaresanalysisofpredictedtumorincidenceandhumantumorincidencedataforthemutationsaectingdierentiationratescenario.Parametercombinationswithwhitecolorshavethesmallestsumofsquaredresiduals,whilebluecolorshavethelargest. mutationsinorganismsthatcannoteliminatethemviarecombination.Indeed,theratchetactsmorestronglythaninpopulationsoforganismsasaresultoftherelativeimportanceofdriftversusselectioninverysmallstemcellpopulations(i.e.,niches).Thisraisestheinterestingquestionofwhysomatictissuesareorganizedinthiswayandwhethersmallstemcellpoolspredominatetominimizetumorigenesisattheexpenseofaging,ashasbeensuggestedby Michoretal. ( 2003a ).Giventheroleofwell-knownlargeeectmutationsincancer,itistempting,fromamathematicalmodelingpoint-of-view,toadoptaheavy-tailed(innitevariance)distributionfortheDFE.IncontrasttotheDFEemployedformodelingpopulationsofwholeorganisms(e.g.anexponentialdistribution),whichtendtoexhibitsmallincrementalchanges,aheavy-tailedregimeenablesasignicantcontributionfrom\one-shot"largemutations.Toprobethispossibility,weincludedinoursimulationsapower-law(Pareto)distributionwhich,throughitsshapeparameter,canbeeitherheavy-tailed(1<3)ornot(>3).Itisnoteworthy,then,thatthebesttparameterswereveryfarfromtheheavy-tailedregime(16).The 37

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prevalenceoflargeoutliermutationsforsuchadistributioniscomparabletowhatwouldbeseenfromanexponentialdistribution,meaningthattheheavy-tailedregimeisnotanappropriatemodelingframeworktoexplainthedata. 2.4.2SmallPopulationsandGeneticDriftLeadtoAgingOneofourprimaryndingsisthatmutationeectsdrivecryptagingasmuch,ifnotmoreso,thantumorigenesis.Tumorformationandagingaretwomanifestationsoftheaccumulationofcellulargeneticdamage.Thisdamageisespeciallyrelevanttotheagingprocesswhenitaectsthefunctionalcompetenceofstemcellsandcompromisestheirabilitytoreplenishthevariouscellpopulationsoftheirconstituenttissue( Lopez-Otnetal. 2013 ).Mutationstostemcellsthatresultinaginghavebeenassociatedwithdiminishingthestemcell'spotentialtoproliferate( Rossietal. 2007 ),competitivelyexcludehealthystemcells( Nijniketal. 2007 ),andself-renewordierentiate( Moskalevetal. 2013 ; JonesandRando 2011 ).Theseeectsonstemcelldynamicswoulddecreasethenumberoffunctionalstemandnon-stemcellsintissues,thusresultingintissueaging,asdenedinagingreviewsandexperimentalwork(above)andpreviousmathematicalmodels( Wodarz 2007 ).Asmutationsbecomexedintheintestinalstemcellnichetheexpectedvalueoftheprobabilitydensitydescribingnewstemcelllineagedivisionratesdecreaseswhenweconsidermutationsthataectdivisionrate,andtheexpectedvaluefordierentiationrateincreaseswhenweconsidermutationsthataectdierentiationrate,andthuscryptsarepredominatelyaging.Theintestinalstemcellnicheismaintainedatapopulationsizesmallerthantheeectivepopulationsizesofwholeorganismsandourndingsderivedirectlyfromthispopulationstructure.Ourstudy,whichemphasizessmallhealthycryptpopulations,contrastswithpreviousstudiesthathaveinvestigatedtheaccumulationofdeleteriousmutationsinsomatictissue.Thesestudieshavelookedatlargerinitialpopulationsizesandineectmodelhyperplasiaorgrowingtumors.Forinstance, McFarlandetal. ( 2013 )modeledpopulationswithaninitialpopulationsizeofapproximately1000cellsbasedonestimatesfromhyperplasia 38

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inmicetwoweeksafterAPCdeletion.Similarly, McFarlandetal. ( 2014 )investigatedmutationaccumulationinmodelsofhyperplasiaandgrowingcancers. Dattaetal. ( 2013 )modeleddeleteriousmutationsinhousekeepinggenesinanexponentiallygrowingtumorinitializedat1106cells. BeckmanandLoeb ( 2005 )assumedtheirpopulationofcellswassucientlylargetoensureadeleteriousmutationofanystrengthcouldnotbecomexed.Theseapproachesareusefultodescribetumorgrowthininitiatedtumorsbutfailtocapturetherelativeimportanceofdriftinevolvingstemcellnichesandtheprocessoftumorigenesisfromhealthystemcellniches,whichexistasverysmallpopulations.Wendthatcryptswithxedmutationsaredistributedalongarangeofbothagingandtumorformation.Theexpectedvalueofthedivisionratedensitymovingawayfromthetumorigenesisthresholdcausestheprobabilityoftumorigenesisperxedmutationtoeventuallydecrease.Forexample,inmice,thereisasmallerprobabilitythatthefourthxedmutationinastemcellnichewillresultintumorigenesiswhencomparedtothethirdmutationintheexponentialbenecialDFEscenario.Thehumanintestinalcryptstemcellnicheconsistsofalargernumberofstemcellssodriftplaysasmallerroleintheevolutionarytrajectoryofthesecrypts.Nonetheless,themodeofthedistributionofdivisionratestillmovesawayfromthetumorigenesisthreshold,albeitataslowerrate.Althoughourmodelassumesthatthesizeofthestemcellniche(N)remainsconstantandmutationsonlychangethedivisionrateordierentiationrateoflineages,it'spossiblethatmutationscouldalterthenichesize.Ifmutationsalteredthesizeofacrypt'sstemcellnichetheywouldchangetheprobabilityofxationofsubsequentmutations. 2.4.3MutationsThatOnlyAectDierentiationRateDoNotMatchIncidenceDataCurvesAnalysesofcoloncancergenomesfromdierentindividualsrevealsthatasmallnumberofgenes,associatedwithlargetnessadvantage,arecommonlymutatedamongcancers( Woodetal. 2007 ).Forinstance,manycoloncancerscontaincellsthathavemutationsingenesinvolvedintheWntsignalingcascaderesponsibleformaintaining\stemness"( Cleversand 39

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Nusse 2012 ).Astudyby Smithetal. ( 2002 )foundthat56%of106sequencedtumorshadmutationsintheAPCgene,which,whennonfunctional,resultsintheactivationoftheWntcascade( ReyaandClevers 2005 ).Additionally,cancersthathaveamutationintheAPCgenetendtohavethemutationdistributedthroughoutthetumor,suggestingthemutationsoccurredearlyintumorgrowth( Sottorivaetal. 2015 ).BecausetheWntsignalingcascadeisinvolvedwithmaintainingastemcellphenotypemutationsinthiscascadewouldinuencethepropensityforstemcellstodierentiate.Additionally,Smithetal.(2002)alsofoundthat61.3%ofcolorectalcancershadmutationsinp53,involvedinregulatingapoptosis,and27.4%ofcolorectalcancershadmutationsinK-ras,thoughttodrivecancergrowthbyacceleratingstemcelldivisionandleadingtoenhancedcryptssion( Snippertetal. 2014 ).Ourmodelingscenarioofmutationsonlyhavinganeectonthedierentiationrateofstemcellsandhavingeectsizesequaltothosemeasuredinwholeorganismsresultsinrapidtumorigenesis,withnearly100%ofhumanindividualshavingapolypintheirlargeintestineatyoungage.Indeed,individualswithfamilialadenomatouspolyposis(FAP),whoalreadyhaveagermlinemutationinonecopyoftheirAPCgeneandonlyneedonemutationalhitontheothertoformanadenoma,regularlydevelopadenomasasteenagers( Bozicetal. 2010 ).Inourmodelthelargetumorigenesisincidenceassociatedwithmutationssolelyaectingdierentiationrateisduetoboththemutationshavingalargerabsoluteeecttowardsthetumorigenesisthresholdandtherebeingahigheroverallprobabilityofxationofnewmutationsamongallthecryptsduetothemutationsxingthroughneutraldrift.Evenwhenwedecreasetheexpectedbenecialmutationaleectsizeanddecreasethemutationrateinanattempttobettertthetumorigenesisincidencedata,wendthatmutationsonlyaectingdierentiationratestillresultinmoretumorigenesisthanpredictedatyoungage.However,thedatawasderivedfromautopsiesonindividualsgreaterthan10yearsofage,sodatafortumorigenesisislackinginthisagegroup.Additionally,wemodeledscenarioswheremutationsonlyaectdivisionordierentiation,natureiscertainlymorecomplexandmutationinbothdierentiationanddivisionratesarelikelytoco-occurwithinacryptpopulation. 40

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Indeed,theAPCproteindiscussedabovecontributesdirectlyorindirectlytocellulardivision,dierentiation,migration,cellorientation,andapoptosis( McCartneyandNathke 2008 ; Dikovskayaetal. 2007 ).Wemodelstemcelldynamicsandmutationaleectsonthosedynamicsasapropertythatiscontrolledbyanindividualstemcell'sgenome,i.e.astemcell'sheritableabilitytoproduceorrespondtointernalsignals,orrespondtoexternalsignals,todivideordierentiate.Theexternalsignalsregulatingstemcellphenotype,suchasWntsignalsproducedbyPanethcellsinthesmallintestine( Clevers 2013 ),areproducedbycellsdierentiatedfromstemcells.Mutationstothestemcellgenomemayeventuallyinuencetheproductionofthesesignalsindaughtercells.Thesemutationswoulddriftneutrallyintheniche,astheyarenotexpresseduntilafterstemcelldierentiation,and,unlessthelineageharboringthemutationreachesxationintheniche,wouldeventuallybelostfromthecryptsincePanethcellsdieinapproximately20days( Bryetal. 1994 ).Thus,mutationsthatresultindierentialsignalingoutputbydaughtercellscanbemodeledasneutrallyxedmutationsactingintrinsicallyinthestemcells. 2.4.4TheInuenceofOrganismSpecicFactorsonSomaticEvolutionWendlesstumorincidenceinmicethanhumansthroughouttheirrespectivelifetimesusingthesameDFEparameters.Miceonlyliveafewyearsandhaveanorderofmagnitudefewercryptsintheirentireintestinethanhumanshaveinjusttheirlargeintestine( Pottenetal. 2003 ).Theyalsohavesmallernumbersofstemcellswithintheircrypts,althoughthosestemcellsaredividingatafasterratethanhumanstemcells.Overall,thisresultsinalowerchanceofmutantlineagesreachingxationwithincryptsduringtheshortermouselifetime,andthereforeareductionintheoverallnumberofcryptswithxedmutationsislower.Forinstance,usingthedistributionofxedmutationsderivedinthe APPENDIXDESCRIPTIONOFTHEMATHEMATICALMETHODOLOGY ,attwoyearsoldamouseisexpectedtohaveabout75cryptswithtwomutations,andonlyabout28percentofmicewillhaveasinglecryptwiththreemutations.At85yearsold,ahumanisexpected 41

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tohaveabout44cryptswithvemutations,onecryptwithsixmutations,andaboutfourpercentofhumansat85yearsoldwillhaveacryptwithsevenxedmutations.Ashumansagetheyexperiencemorexedmutations,eachofwhichconfersahigherprobabilityoftumorigenesisthantheprevious,whereasmiceareexpectedtoexperiencetheaccumulationoffewermutations,possiblyexplainingthenearlinearityofthemouseincidencecurveandtheupwardscurvatureofthehumanincidencecurve.Ofnote,giventhatatumorigenesiseventhasoccurred,itislikelytheproductofonemutationinthemousemodel,whereasmultiplemutationsmaycontributetotheinitiationofatumorinthehumanmodel(the APPENDIXDESCRIPTIONOFTHEMATHEMATICALMETHODOLOGY A-2 ).Theincidenceofpolypsatautopsyreportedby Chapman ( 1963 )wasbasedonvisualobservationsofdiscernibleelevationsofthemucosaintheentirelargeintestineduringautopsy.Itwouldtaketimeforaninitiatedtumortogrowtoavisiblemass,sothetruetumorigenesisincidencecurvemaylieinfrontofthedatarecordedinthisstudy,withalagtimeofgrowthbeforethetumorisvisible.Thislagtimewouldbeafunctionoftheindividualmutationalspectrumoftheinitiatedtumorandthetumor'senvironment.Overall,wehaveshownthatsmallhomeostaticpopulationsofstemcells,typicalofsomatictissuesinmulticellularorganisms,accumulatemutationsthataectcellulartness,contributingbothtoagingandtumorigenesisoveranorganism'slifetime.WeshowthattheevolutionofintestinalstemcellpopulationsundertheassumptionofanorganismalDFE,asopposedtotheassumptionofaheavy-tailedbenecialDFE,bestpredictedearlytumorformation.However,aging,ratherthantumorigenesis,predominatedamongcryptsintheintestine.Ourmodelingapproachemphasizestumorigenesisinthecontextofaging,andviceversa,anddemonstratestheimportanceofmutationalprocesseswithinverysmallpopulationsinboththesephenomena. 42

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CHAPTER3THEEXPECTEDEFFECTOFSOMATICEVOLUTIONONTISSUEHOMEOSTASIS 3.1IntroductoryRemarksTheconstantonslaughtofmutation,coupledwiththeimperfectnatureofdamagerecognitionandrepairmechanisms,resultsinthecontinualaccumulationofgenomicalterationswithinthecellsconstitutingmulticellularorganisms( Lynch 2008 ).Thefullextentofmutationaccumulation,andtheburdenthesemutationsplaceonthesomatictissueswithinmulticellularorganisms,isnotknown.Here,weutilizetherecentlyelucidatedpopulationdynamicsofintestinalstemcells,alongwiththedynamicsofstemcellprogeny,tomodeltheevolutionarytendencyofstemcellpopulationsandtheexpectedburdenthatmutationaccumulationplacesthroughouttheintestinaltissue.Theparadigmofstemcellturnoverwithintheintestineisnotuniquetothattissue( KleinandSimons 2011 ),andthustheworkpresentedheremaybeapplicabletoothersystemsundergoingcontinualself-renewalthroughthedivisionofstemcells.FitnessandAging.Themajorityofmutationsthataectanindividual'stnesswilldecreasetness( Eyre-WalkerandKeightley 2007 ).Inpopulationsofwholeorganisms,especiallythoseinwhichtnesseectsarestudiedinthelaboratory,thesemutationsrarelybecomexedduetopurifyingselectionandlargepopulationsizes(see Levyetal. ( 2015 )forarecentexample).However,theindependentlyevolvingpopulationsofadultstemcellsthatcontinuallydividetoreplenishtissuesareusuallymaintainedatverysmallpopulationsizes( MorrisonandSpradling 2008 ),andarethusmoresusceptibletogeneticdriftandthexationofdeleteriousmutations.Inaddition,duetotheasexualnatureofmitoticdivisionandstemcellmaintenance,stemcellpopulationsareaprimeexampleofMuller'sratchet,or,theirreversibleaccumulationofmutationsandresultantdecreaseinapopulation'smeantnessintheabsenceofrecombination( Muller 1964 ).Wehavepreviouslydemonstratedthatcryptsarepredominantlyaccumulatingdeleteriousmutationsoverthelifetimeofthehostorganism( Cannataroetal. 2016 ).Here,wequantifytheexpectedchangeinwhole-tissueequilibrium 43

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populationsizeasmutationsaccumulateinthestemcellpopulationsatthebaseofintestinalcryptsoveranorganism'slifetime.Theaccumulationofdamagecausingthelossofcellulartnessisahallmarkofaging,andisespeciallyrelevantwhenDNAdamageoccursinstemcells,compromisingtheirroleintissuerenewal( Lopez-Otnetal. 2013 ).Indeed,severalmousemodelswiththediminishedabilitytomaintaincellulargenomeintegritysuccumbtoacceleratedage-relatedphenotypesthroughthelossoftissuehomeostasiscausedbystemandprogenitorcellattrition( Ruzankinaetal. 2008 ).Justasstemcellmutationsconferringabenecialtnesseectwillincreasecellproduction,mutationsconferringadeleterioustnesseectwillleadtodecreasedcellproductionandthediminishedmaintenanceofhealthytissue.Stemcellsatthebaseoftheintestinalcryptdierentiateintoallotherintestinalcellpopulations( Barker 2014 ).Hence,mutationsaectingtheratesofstemcelldynamicswillpropagatethroughotherpopulations,aectingtheirsteadystateequilibriumpopulationsizes.Wemodelthevariouscellpopulationsoftheintestinalcryptandepitheliumtocalculatehowmutationsoccurringinstemcelllineagesgovernpopulationdynamicsthroughoutthetissue. 3.2Methods/ModelingWeareinterestedinhowtheratesofstemcelldivisionanddierentiationscaleuptowholetissuedynamics.Withinthissectionwerstdescribethegeneralarchitectureofcryptsystems.Then,wedetailourparameterizationofthemodelsinlightofrecentexperimentsinmice.Finally,wedescribehowwemodeltheevolutionaryprocessesoccurringwithincryptsandthroughoutthetissue. 3.2.1ModelingtheCryptSystem Intestinalcryptsarecomposedofthevarioustypesofcellpopulationsderivedfromstemcells..MousecryptscontainapopulationofLGR5+putativestemcellsattheirbase.Withinthispopulation,thereexistsafunctionalsubpopulationresponsibleformaintaininghomeostasiswithinthecrypt( Kozaretal. 2013 ; Bakeretal. 2014 ).WecallthissubpopulationresponsibleforcrypthomeostasisX1.Thestemcellsinthestem 44

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cellnichedividesymmetricallyandundergoneutraldrift,whereanylineagewiththesamedivisionrateastheotherlineagesinthecrypthasequalprobabilitytoreachmonoclonalitybydisplacingallotherlineagesthroughdivision( Ritsmaetal. 2014 ).Cellswithgreaterdivisionratehaveanincreasedprobabilityofdisplacingtheirneighbors;thosewithlowerdivisionratearemorelikelytobedisplaced.Inourmodel,bothstemcellswithinthenicheanddisplacedstemcellsdividesymmetricallyatrate,anddisplacedstemcellscommittodierentiationatrateandjointheTransient-Amplifying(TA)compartment,Y1,locatedabovethestemcellcompartment.ATAcellrapidlydividesatrateanumberofrounds,R,joiningsubsequentTApools(Y1,Y2,...,YR).Afterthelastroundthecelldividesanaltimeandjoinstheterminally-dierentiatedpost-mitoticcellpoolZ( Potten 1998 ).Cellswithinthepost-mitoticcellpoolexistuntiltheyundergoapoptosisatrateeitheratthevillustiporlumenalsurfaceinthesmallintestineandlargeintestine,respectively( Grossmannetal. 2002 ).Theterminallydierentiatedcellsmaintainthefunctionalityoftheintestinaltissue,withmanyexistingatthetopofthecrypt,ontheepithelialsurfaceliningthelumen,and,inthecaseofthesmallintestine,alongthevilli.ThedynamicsdescribedabovearedepictedinFigure 3-1 .Thesedynamicsarerepresentedbythetransitionrates 45

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(Z)!(Z)]TJ /F7 11.955 Tf 11.95 0 Td[(1)atrateZ,(YR,Z)!(YR)]TJ /F7 11.955 Tf 11.95 0 Td[(1,Z+2)atrateYR,...(Y2,Y3)!(Y2)]TJ /F7 11.955 Tf 11.96 0 Td[(1,Y3+2)atrateY2,(Y1,Y2)!(Y1)]TJ /F7 11.955 Tf 11.96 0 Td[(1,Y2+2)atrateY1,(X2,Y1)!(X2)]TJ /F7 11.955 Tf 11.96 0 Td[(1,Y1+1)atrateX2,(X2)!(X2+1)atrateX2,(X1,X2)!(X1,X2+1)atrateX1.Thetime-dependentmeansofthissystemsatisfytheordinarydierentialequations:d dtz(t)=2yR(t))]TJ /F10 11.955 Tf 11.95 0 Td[(z(t),d dtyR(t)=2yR)]TJ /F12 7.97 Tf 6.59 0 Td[(1(t))]TJ /F10 11.955 Tf 11.95 0 Td[(yR(t,)...d dty2(t)=2y1(t))]TJ /F10 11.955 Tf 11.96 0 Td[(y2(t,)d dty1(t)=x2(t))]TJ /F10 11.955 Tf 11.96 0 Td[(y1(t),d dtx2(t)=1x1(t)+2x2(t))]TJ /F10 11.955 Tf 11.96 0 Td[(x2(t),d dtx1(t)=0,wemodelX1asaxedpopulationsize. (3{1)Settingtheleft-handsidesforeachequationinthesystem 3{1 tozero,wecansolveforthesteadystatemeanoftheterminallydierentiatedpopulation,Z?.WendthatZ?canbeexpressedintermsofthesystem'srateparametersandthestemcellnichepopulationsize 46

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Figure3-1. Thegeneralarchitectureofacryptsystem.Populationnamesarewithintheboxesandtheratesatwhichcellsaccumulatewithinoraretransferredbetweenpopulationsarenexttothearrowportrayingtheirtransition. X1.Z?(Eq. 3{2 )isafunctionofalloftherateparametersinthesystemexcepttheTAcelldivisionrate: Z?=2RX?1 ()]TJ /F10 11.955 Tf 11.95 0 Td[().(3{2)Thisfunctionisusedinsubsequentanalysestocalculatehowmutationstothevariousrateparameterschangethetissuepopulationsize.Ofnote,apredictionofthismodelisthatamutationtoeitherdivisionrateordierentiationratewillresultinanampliedeectontheproportionofchangeinsteadystatepost-mitoticcells.Thatis,if0ismutatedto1,theproportionaldierenceinthepost-mitoticcellpopulationisZ?1 Z?0=1 0()]TJ /F14 7.97 Tf 6.59 0 Td[(0) ()]TJ /F14 7.97 Tf 6.59 0 Td[(1). 47

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3.2.2ModelParameterizationParameterizingthecryptmodelusingmousesmallintestinedata.Themoststudiedcryptsystemisinthemousesmallintestine.Thereare14-16Lgr5+putativestemcellsinthebaseofthemousesmallintestinalcryptsanditisbelievedthattheydividesymmetricallyunderneutralcompetition( Snippertetal. 2010 ; Lopez-Garciaetal. 2010 ). Kozaretal. ( 2013 )foundthatthereexistsasubsetofthispopulationthatmaintainscrypthomeostasis,andthatthissubpopulationconsistsofapproximately5,6,and7cellsdividingapproximately0.1,0.2,and0.3timesperdayalongtheproximalsmallintestine,distalsmallintestine,andcolon,respectively.Whenestimatingtheexpectedeectsofmutationaccumulationalongtheentireintestineinthemousewetakethemedianvalueofcellnumberanddivisionratefoundby Kozaretal. ( 2013 )tobethenumberofcellsintheniche,X1,andthestemcellsymmetricdivisionrate,,withtheremainderofthestemcellpopulationconstitutingtheX2population.InordertomaintainthenumberofLgr5+stemcellsatasteadystateof15wecalculatearateofcommittingtodierentiation,,of0.333perday.TAcellsdividetwiceaday( PottenandLoeffler 1987 ; Potten 1998 )andundergoapproximatelysixgenerationsoftransientamplication,witheachstemcelleventuallycontributingabout64cellstothecryptsystem( Marshmanetal. 2002 ).Theselineagesendattheirterminallydierentiatedcellstage,andlivefor2-3daysbeforedyingandleavingtheintestine( Snippertetal. 2010 ).TheparametersdescribedaboveareprovidedinTable 3-1 .ParameterizingourstructuredmodelwiththevaluesfromTable 3-1 themodelgeneratesobservedcellpoolpopulationsizes.UsingthemodelwecalculatethesteadystatemeanpopulationsizesofthevariousTApopulationstobe(Y?1,Y?2,Y?3,Y?4,Y?5,Y?6)=(1.5,3,6,12,24,48).ThisresultsinthetotalrapidlydividingTApopulationbeingY?total95cells,slightlyunderestimatingestimatesfromtheliteratureofthenumberofcellswithinthiscompartmentwhicharearound120( Marshmanetal. 2002 ).Thesedynamicsresultinasteadystatemeanoftheterminallydierentiatedcellpopulationsize,Z?,toequal576cells.Thecryptsconsistofapproximately250cells( PottenandLoeffler 1990 ),meaningthat(accounting 48

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Table3-1. Approximateparametervaluesforthemouseintestine ParameterValueNoteCitation X16CellsExpectednumberofstemcellsinthesmallintestinestemcellniche( Kozaretal. 2013 )X1+X?215CellsThetotalnumberofLgr5+inthebaseofthecrypt( Snippertetal. 2010 ; Clevers 2013 )0.2perdayThedivisionrateofthesubpopulationofthestemcellsconstitutingtheniche( Kozaretal. 2013 )0.333perdayTherateatwhichcellspassivelyadoptaTAfatethisstudy2perdayThedivisionrateofTAcells( PottenandLoef-fler 1987 )0.333perdayThedeathrateofterminallydierentiatedcells( Snippertetal. 2010 )R6RoundsofdivisionofrapidlydividingTAcells( Marshmanetal. 2002 ) forthe95TAcells,the15stemcells,andthe10Panethcellsnotmodeledhere( Clevers 2013 ))theterminallydierentiatedcellpopulationexistsas130cellswithinthecryptand466cellsoutsideofthecrypt.Thevilliinthesmallintestinearesupportedby6-10cryptsandconsistofapproximately3500cells( PottenandLoeffler 1990 ).Thetotalsteadystatemeanpopulationcontributedby8cryptstoavillususingtheparametersinTable 3-1 is8446=3568cells.Weareinterestedinestimatingtheeectsofmutationaccumulationthroughoutthewholeintestinaltractinthemouse,hencewetakethemedianvalueofcertainparametersthatvaryfromproximalsmallintestinetocolon(asdescribedabove).Wenotethatifonewasinterestedintheeectsofsteadystateterminallydierentiatedpopulationsizeinjusttheproximalsmallintestineorcolonthisanalysismayoverestimateorunderestimatetheeects,respectively,sincetherearemorestemcellsdividingfasterinthecolonthantheproximalsmallintestine.Additionally,thelargeintestineTAcellsmayundergomoreroundsofdivisionthanthesmallintestineTAcells( Potten 1998 ). 49

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Intrinsicmutationaleects.Theentiredistributionoftnesseectsisunknownforsomatictissues.Additionally,althoughthemutationratehasbeenestimatedonapernucleotidebasis( Jonesetal. 2008 ),themutationaltargetsizeofallmutationsthataectcryptcelltnessisunknown.Thus,weexploretheimplicationsofawiderangeofintrinsicmutationalparameterswithinsomatictissue.However,despitethedierencesbetweenthegenomesofspecies,mutationaccumulationexperimentsandanalysisofDNAsequencedatahaverevealedgeneralprinciplesregardingdistributionsofintrinsicmutationaleects.Namely,mutationsaremuchmorelikelytoconferadeleterioustnesseectthanabenecialeect,andthatthedeleteriouseectwillhavealargerexpectedvalue( Eyre-WalkerandKeightley 2007 ; Banketal. 2014 ).Deleterioustnesseectsareextremelydiculttocharacterizeintypicallaboratoryexperimentsbecauseselectionagainstdeleteriousmutationsiseectiveinevenmoderatelysizedpopulations( Estesetal. 2004 ; Keight-leyandHalligan 2009 ).Directedmutagenesisexperimentsareonewaytoclassifythedistributionandaverageeectofmutationsdeleterioustotness.Forinstance, Sanjuanetal. ( 2004 )performedsite-specicsingle-nucleotidesubstitutionsinanRNAvirusandfoundthattheaveragedeleteriousnon-lethaltnesseectdecreasedtnessapproximately20%.Similarly,whengrowninpermissiveconditionsthatallowspontaneousdeleteriousmutationstoaccumulatethroughgeneticdrift, KeightleyandCaballero ( 1997 )founda21%deleterioustnesseectpermutationinCaenorhabditiselegansand ZeylandDeVisser ( 2001 )founda21.7%tnessdeclineperxedmutationindiploidstrainsofthesinglecelledeukaryoteSaccharomycescerevisiae.Othermutationaccumulationstudieshavefoundmoremodestdeleteriouseects,suchastheaveragetnessdeclineinhaploidSaccharomycescere-visiaepermutationof8.6%foundby Wlochetal. ( 2001 ).AnothermutationaccumulationexperimentinSaccharomycescerevisiaefoundtheexpectedbenecialincreaseoftnesspermutationtobe6.1%,therateofmutationthataectstnesspermutationtobe1.2610)]TJ /F12 7.97 Tf 6.59 0 Td[(4,andthepercentoftnesseectsthatarebenecialtobe5.75%( JosephandHall 2004 ).Whenouranalysisrequiresspecicparameterchoices,asinSection 3.3.3 whenwe 50

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juxtaposethedynamicsofmutationsthatxneutrallywiththoseunderselection,weutilizetheSaccharomycescerevisiaeparametersdescribedhere,butnotethatweareinterestedincharacterizingthedynamicsoftumorigenesisandaging,andwearenotmakingconclusionsabouttheabsolutemagnitudeofeithergivenourlimitedknowledgeofmutationaleectsinsomatictissue. 3.2.3ModelingEvolutionWithinSomaticTissueModelingtheexpectedmutationaleectofasinglemutationwithinacrypt.Inordertoquantifytheexpectedeectontissuehomeostasisofmutationsinepithelialtissueitisnecessarytounderstandtheprocessesofmutationaccumulationandxationwithinthestemcellnichepopulationatthebaseoftheintestinalcrypts.Mutationsinthenichecanbeplacedintotwodierentcategories:mutationsthatdirectlyaectthestemcellphenotypeassociatedwithcellulartness,i.e.divisionrate,withinthestemcellniche,andmutationsthatdonotaectthetnessofstemcellswithintheniche.Mutationsthataectthedivisionrateofstemcellswillconferatnessadvantageordisadvantagebecausestemcellswithinthenichearesymmetricallydividingandreplacingtheirneighbors.Forinstance,certainmutationstoKRASincreasestemcelldivisionrateandtheprobabilitythismutantlineagereachesxation( Vermeulenetal. 2013 ; Snippertetal. 2014 ).Mutationsthatdonotdirectlyaectstemcelldivisionratewillnotalterstemcelltnesswithinthenicheandwillxneutrally.Wemodelthedistributionofmutationaleectsandmutationaccumulationsimilarlyasin Cannataroetal. ( 2016 ),whereweprovideadetailedmathematicalmethodology.Briey,mutationaleectsaredistributedexponentially,withexpecteddeleteriouseects)]TJ /F1 11.955 Tf 7.09 1.8 Td[(,expectedbenecialeects+,andPBpercentofmutationsconferringabenecialeect.Thus,mutationaleectsontheinitialstemcelldivisionrate,0,aredistributedaccordingtoEq. 3{3 ,where 51

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m(;0)exp=8>><>>:(1)]TJ /F9 11.955 Tf 11.96 0 Td[(PB) 0e)]TJ /F14 7.97 Tf 6.58 0 Td[((1)]TJ /F16 5.978 Tf 9.69 3.26 Td[( 0)<0PB 0e)]TJ /F14 7.97 Tf 6.58 0 Td[(( 0)]TJ /F12 7.97 Tf 6.59 0 Td[(1)>0, (3{3)where=1 s)]TJ /F1 11.955 Tf 11.35 5.2 Td[(and=1 s+.Distributionsthataremoreleptokurticthanexponential,and/orbimodal,mayprovidebettercharacterizationofdeleteriousandbenecialmutations( Eyre-WalkerandKeightley 2007 ; Levyetal. 2015 ),however,modelingtheDFEasanexponentialdistributionminimizesthenumberofparametersinourassumptionswhilestillcapturingadistributionthathasprovidedreasonablygoodtstobothdeleterious( Elenaetal. 1998 )andbenecial( KassenandBataillon 2006 )tnesseectsduringexperiments.Anewlineagewithadivisionraterelativetothebackgrounddivisionrate 0hasprobabilityofeventuallyreplacingtheoriginallineage px 0=px(r)=1)]TJ /F9 11.955 Tf 11.95 0 Td[(r)]TJ /F12 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 11.955 Tf 11.96 0 Td[(r)]TJ /F8 7.97 Tf 6.58 0 Td[(X1.(3{4)WeuseBayes'theoremtocalculatetheprobabilitydensityofthedivisionrategiventhemutantlineagexed, (j0)=px(;0)m(;0) R10px(`;0)m(`;0)d`,(3{5)and,redeningEq. 3{5 suchthatf1()isequaltothedensitygiventherstxationofamutantlineage,wecalculatetheexpectedvalueofthedivisionrateofthisnewlineage:E[f1()]=1=R10f1()d.NotethatwecanextendthisanalysistotheaccumulationofmutationsthatdonotalterthetnessofstemcellswithinthenichebychangingEq. 3{4 tobeequalto1 X1,theprobabilityofxationofaneutralmutation.Additionally,mutationsthataectdierentiationratecanalsobemodeledusingEq. 3{3 byswitchingthedirectionofeectsuchthatbenecialmutationsnowdecreasethedierentiationrate(i.e.increasethelifetimeofthecells). 52

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Calculatingtheexpectedeectofmultiplemutationswithinacrypt,andtheaccumulationofmutationsthroughoutallcrypts.Wecalculateprobabilitydensitiesdescribingthedivisionrateofmsubsequentxedmutations,andtheprobabilityoftumorigenesistheyconfer,accordingtotherecursiveformula fn+1()=Z10(j0=`)fn(`)d`.(3{6)FromEq. 3{6 wecancalculatetheexpectedvalueofdivisionrategivenmmutations,whichistheexpectedvalueoftheseprobabilitydensities:E[fm()]=m=R10fm()d.Wemodelthearrivalrateofxedlineagesinthecryptsasbeingconstantovertimewithrate^=^p0X1,where^pisthetotalprobabilityofxationR10px(;0)m(;0)dandisthemutationrate,asin Cannataroetal. ( 2016 ).Here,thenumberofxedmutationswithinacrypt,m,areapproximatedbythePoissondistributionwithmean^t,andwecanestimatethenumberofncryptswithmmutationsbymultiplyingthisdistributionbythenumberofcryptsinthesystem,C: nmCe)]TJ /F12 7.97 Tf 7.15 0 Td[(^t(^t)m=m!.(3{7)Thus,wecanestimatethetotalnumberofcryptswithmmutationsastheorganismages.Calculatingtheprobabilitythatxedmutationsinitiatetumorigenesis.Forpartofouranalysiswejuxtaposetheexpectedmagnitudeoftissuechangewiththeriskoftumorigenesisforagivenstemcellnichesize.Wedenetheinitiationoftumorigenesisasthemomentwhenthedivisionrateofstemcells,,becomesgreaterthanthedierentiationrateofstemcells,,thusinitiatingexponentialgrowthwithinthetissue.Wecalculatethisbyintegratingtheprobabilitydensitiesdescribingthechangeinthestemcellratesfromtoinnityforscenarioswheremutationsaectdivisionrate,andtozeroforscenarioswheremutationsaectdierentiationrate,withbothcalculationsdeterminingtheprobabilitythatacertainmutationresultedinatnesschangethatinitiatedtumorigenesis.Theseprobabilitiesaresummedoverallmutationsinallcrypts,givingthetotalprobabilityoftumorigenesis. 53

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Calculatingtheexpectedeectofmutationaccumulationontissuemainte-nance.Usingourcalculationofthenumberofcryptswithmmutations(Eq. 3{7 ),theexpecteddivisionrateofstemcellswithinnicheswithmmutations,andthesteadystatepost-mitoticcellpopulationgiventhisexpecteddivisionrate(Eq. 3{2 ),wecanestimatetheexpectedpost-mitoticepithelialpopulationsizeoverthetimesinceadulthood,t,oftheindividual, Z?(t)=Z?Total)]TJ /F12 7.97 Tf 18.77 14.94 Td[(^mXm=1nm(Z?Normal)]TJ /F9 11.955 Tf 11.95 0 Td[(Z?m),(3{8)whereZ?Totalisequaltothetotalnumberofcryptstimesthenumberofpost-mitoticcellsproducedbyahealthycrypt,Z?Normal.Mutationsthatdonotaectstemcelldivisionratexneutrallyinthestemcellniche,however,thesemutationsmaystillalterthesteadystatemeanofthepost-mitoticterminallydierentiatedpopulationbychangingpopulationdynamicsratesthatareimportantfortissuemaintenance.Forinstancemutationsthatstrictlydealwiththerateforalineagetodierentiate()wouldstillchangeZ?. 3.2.4LinearApproximationPlotsofZ?(t)=Z?Totalasafunctionoforganismallifetimeareapproximatelylinear(seeSection 3.3.2 ).Hence,weutilizeanasymptoticanalysistoapproximatethedynamicsdenedbyinEq. 3{8 .Whenonlyconsideringtherelativeaectoftherstxedmutationwithinstemcellniches,andthearrivalofthisrstxedmutationamongtheniches,wecansimplifyEq. 3{8 to Z?(t) Z?Total1)]TJ /F11 11.955 Tf 11.96 16.85 Td[(1)]TJ /F9 11.955 Tf 13.15 8.08 Td[(Z?1 Z?00^pX1t,(3{9)whereZ?1 Z?0isthesteadystatepopulationsizeofthepost-mitoticcellpopulationafteronexedmutationdividedbythehealthypopulationsizewithzeromutations(Eq. 3{2 ).WhenmutationsalterthedivisionratethefractionZ?1 Z?0simpliesto1X?1 ()]TJ /F14 7.97 Tf 6.59 0 Td[(1)()]TJ /F14 7.97 Tf 6.59 0 Td[(0) 0X?1,orr)]TJ /F12 7.97 Tf 6.59 0 Td[(1 )]TJ /F8 7.97 Tf 6.58 0 Td[(r, 54

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wherer=1 0and= 0.Thus,Eq. 3{9 simpliestoZ?t Z?Total=1)]TJ /F11 11.955 Tf 12.1 13.27 Td[(1)]TJ /F9 11.955 Tf 11.95 0 Td[(r)]TJ /F12 7.97 Tf 6.58 0 Td[(1 )]TJ /F8 7.97 Tf 6.59 0 Td[(r0X1^pt,whosetimederivativeis d dtZ?(t) Z?Total=)]TJ /F10 11.955 Tf 9.29 0 Td[(0^p(r)]TJ /F7 11.955 Tf 11.96 0 Td[(1)X1 r)]TJ /F10 11.955 Tf 11.95 0 Td[(.(3{10)Whenmutationsaectthedierentiationrateofstemcells,thefractionZ?1 Z?0simpliestor1)]TJ /F14 7.97 Tf 6.58 0 Td[()]TJ /F15 5.978 Tf 5.75 0 Td[(1 r)]TJ /F14 7.97 Tf 6.59 0 Td[()]TJ /F15 5.978 Tf 5.76 0 Td[(1,wherer=1 0.InthiscaseEq. 3{9 simpliestoZ?t Z?Total=1)]TJ /F11 11.955 Tf 12.22 13.27 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(r1)]TJ /F14 7.97 Tf 6.58 0 Td[()]TJ /F15 5.978 Tf 5.76 0 Td[(1 r)]TJ /F14 7.97 Tf 6.59 0 Td[()]TJ /F15 5.978 Tf 5.75 0 Td[(10t,andthederivativeofthisfunctionwithrespecttotimeis: d dtZ?(t) Z?Total=)]TJ /F10 11.955 Tf 9.3 0 Td[(0(r)]TJ /F7 11.955 Tf 11.96 0 Td[(1) r)]TJ /F7 11.955 Tf 11.96 0 Td[(1.(3{11)NotehowEq. 3{11 isindependentofX1and^p,sincemutationsxneutrallyandthesetermsaretheinverseofoneanother.Equations 3{10 and 3{11 arecloseapproximationsoftherateoftissuesizechangeperday.Importantly,theyareindependentofmanyoftheparametersinthecryptmodel( 3-1 ),suchasthedivisionrateandnumberofgenerationsofTAcellsandtheapoptosisrateofpost-mitoticcells.Theseparametersarenoteasilymeasuredinhumans,andthusEquations 3{10 and 3{11 enableustoextendourinferencetohumanscenarios.Parameterizationforthehumancolon.Thedivisionrateofstemcellswithinhumancoloncryptshasbeenhasbeenreportedasclosetoonceaweek( Pottenetal. 2003 ; Nicolasetal. 2007 )andthetotalnumberofstemcellsmaintainingthecryptpopulation,i.e.stemcellsintheniche,hasbeenestimatedtobebetween6( Bakeretal. 2014 )and15to20cells( Nicolasetal. 2007 ),withBayesiananalysisfromthelatterstudyshowingmoresupportforlargercryptpopulationsizes.Assumingthehumanstemcellnicheinthecoloncrypthouses6cells,andthetotalmeannumberofpotentialstemcellsinthecoloncryptisapproximately36cells( BravoandAxelrod 2013 ),weuseEquation 3{1 tocalculatethatthedierentiationrateofstemcellsisapproximately0.172perday.Alternatively,ifthehumanstemcellnichehouses20cells,andthereareatotalof36putativestemcellswithinthecrypt, 55

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wecalculatethedierentiationrateofstemcellstobe0.321perday.Thus,wehavealltheparametersnecessarytocalculatetheexpectedeectoftheaccumulationofmutationsinstemcellnichesontissuehomeostasisinhumanintestines,andcancomparetheexpectedeectsgiventhetwodierentestimatedsizesofthestemcellniche. 3.2.5ExploringtheEvolutionaryTrade-oBetweenStemCellNicheSize,Aging,andTumorigenesisGiventhatsmallpopulationsizesarepronetomutationalmeltdownviatheaccumulationofdeleteriousmutationsthroughgeneticdrift,weexploretheselectivepressuresthatmayhaveinuencedtheevolutionofsmallstemcellnichepopulationsize.Specically,wejuxtaposethemagnitudeofepitheliumtissuepopulationchangeandthetotalriskoftumorigenesisthroughouttheintestinalepitheliumfordierentstemcellnichesizes.Throughoutthisanalysisweassumethathalfoftheputativestemcellsinthecryptresideintheniche,asisapproximatelythecaseforbothmiceandhumans(seeabove).Additionally,weemphasizethattheabsolutemagnitudesoftissuechangeandtumorincidencemaynotreectthetruevalues,astheparametersassociatedwiththetrueexpectedmutationaleectsareunknown,howevertheemergentselectivedynamicsfromouranalysisarerelativelyindependentofexactparameterspecications. 3.3Results 3.3.1TheExpectedFitnessofStemCellsDecreaseswithFixedMutationsWhenmutationsaectdivisionrate.Thesmallpopulationsizeofthestemcellnicheatthebaseofthecryptspromotesweakselectionandpervasivedriftthroughouttheintestine.Foreachcrypt,alongdistributionoftnesseectparametersconsistentwiththosemeasuredinwholeorganismsandstemcellnichesizesconsistentwiththosemeasuredinmiceandhumans,theexpectedvalueoftherstxedmutationhasalowertnessthanthepreviouslineagewhenmutationsaectdivisionrate(Figure 3-2 ).Aspopulationsize,theexpectedeectofabenecialmutation,ortheprobabilityarandommutationresultsinabenecial 56

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eectincreases,thestrengthofdriftdecreasesandthestrengthofselectionincreases,resultingintheexpectedvalueforxedmutationsaectingdivisionratetoincrease. Figure3-2. Expectedvalueoftheprobabilitydensitydescribingdivisionrate,,afteronexedmutation,dividedbytheoriginaldivisionrate.Withinheatmaps,theXaxisofeachindividualheatmapisthestemcellnichesize,from1to40cellsandtheYaxisistheprobability,givenamutationevent,themutationisbenecialvs.deleterioustotness,andthecellsoftheheatmaprepresentthevalueoftheexpecteddivisionrateafteronemutationrelativetotheoriginaldivisionrate.Thenineheatmapsrepresentthecombinationsalongthreeexpectedeectsofdeleteriousandbenecialmutationaldistributions. Thepowerofgeneticdriftatsmallpopulationsisexhibitedbyconsideringtheexpectedeectofaxedmutationasafunctionoftheexpectedeectofdeleterioustnesseectsattwopopulationsizes(Figure 3-3 ).Forsmallerpopulations,suchaswhenX1=6,evenmutationsoflargedeleteriouseectmayxthroughgeneticdrift,andthentheexpected 57

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valueoftherstxedmutationcontinuestodecreaseastheexpectedvalueofadeleteriousmutationincreases.Alternatively,largerstemcellnichepopulationsizesarelesspronetodeleteriousmutationsxingthroughdrift,andastheexpectedeectofdeleteriousmutationsincrease,thereisapointatwhichdeleteriousmutationsbecomelessandlesslikelytox.Givenaxationevent,itislikelythemutationconferredabenecialeect. Figure3-3. Theexpectedvalueofdivisionratedividedbytheoriginaldivisionrate,r,versuss)]TJ /F1 11.955 Tf 7.08 1.8 Td[(,theexpectedmutationaleectgivenadeleteriousmutation,forthestemcellnichesizesof20and6.Here,s+is0.061andPBis0.0575. Whenmutationsaectdierentiationrate.Sincethedierentiationratephenotypeisnotexpressedintheniche,allmutationstothisratexneutrally,andtheexpectedvalueofgivenaxedmutationdoesnotdependonnichepopulationsize(Figure 3-4 ).Noteherethat,becausethevastmajorityofmutationsaredeleterious,theexpectedvalueofmutationsisalwaysdeleteriousintheabsenceofselection.Additionally,nowmutationsdeleterioustotnessaregreaterthantheoriginalrate,ashigherratesofdierentiationdecreasestemcelllifetime. 58

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Figure3-4. Expectedvalueoftheprobabilitydensitydescribingdierentiationrate,,afteronexedmutation,dividedbytheoriginaldierentiationratewithvepercentofmutationsconferringabenecialmutationaleect,i.e.decreasingstemcelldierentiationrate. 3.3.2MouseandHumanStemCellFitnessisExpectedtoDecreasewithAge,ReducingTissueRenewalAsdemonstratedinsection 3.3.1 ,thetnessofstemcellsisexpectedtodecreaseasmutationsaccumulateinstemcellniches.Thisresultsindiminishedwhole-tissuepopulationsizesasmutationsaccumulatethroughoutthecryptswithinthetissueinmice(Figure 3-5 )andhumans(Figure 3-6 ).Wendthatthelinearapproximationdescribedinsection 3.2.4 providesagoodapproximationtothesimulatedcurvesandwethusemploythisapproximationinestimatingtissuesizechangecurvesforhumans.Remarkably,despitehavingdierentsizedstemcellpopulationsandlivingvastlydierentamountsoftime,thestrengthofdriftisbalancedbythelongevityofeachorganism,andhencebothspeciesarepredictedtoexperiencesimilarcontributionstowardstissuesizechange(mutationsaectdivisionrateandthehumanstemcellnichecontains20cells). 59

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Figure3-5. Theeectofmutationaccumulationintheintestinesofamouse 3.3.3ThereIsanEvolutionaryTrade-oBetweenTumorigenesisandAgingMediatedbyStemCellNicheSizeWevaryinitialstemcellpopulationsize,alongwiththetotalnumberofcryptsinthesystem,suchthatthetotaloutputofcrypts,i.e.,healthytissue,remainsconstant.ThisanalysiswasconductedusingthecryptparametersdescribedforthemouseinTable 3-1 andtheyeastmutationparametersdescribedinsection 3.2.2 ,namelymutationsthataecttnessoccuratrate1.26104,benecialtnesseectshaveanexpectedvalueof0.061,deleteriouseectshaveanexpectedvalueof0.217,and5.75percentofmutationsconferabenecialtnesseect.Formutationsthataectdivisionrate,thereexistsanoptimalintermediatecryptsizetominimizetheprobabilityoftumorigenesis(Figure 3-7 A).However,at 60

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Figure3-6. Theeectofmutationaccumulationinthelargeintestinesofahuman thiscryptsize,theexpectedvalueoftheepitheliumtissuesizeisexpectedtodecreaseduetotheaccumulationofdeleteriousmutationsinstemcellniches(Figure 3-7 B).Furthermore,theprobabilityoftumorigenesisisminimizedforlargestemcellnichesizes(Figure 3-7 C)andtheexpectedeectontissuesizeisinvarianttostemcellnichesize(Figure 3-7 D)whenmutationsaectdierentiationrateandxneutrally.Asthetruemutationalparametersgoverningsomatictissueevolutionareunknown,wedonotwishtoanalyzethepredictedmagnitudeoftumorigenesisandagingpresentedhere,butratherthetrade-othatexistsbetweennichepopulationsize,drift,andselectionamongmutationsthatconferaselectiveadvantageandthosethatxneutrally.ThenatureofthedynamicspresentedinFigure 3-7 holdstrueforalargerangeofmutationparameters. 61

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Figure3-7. AnEvolutionaryTradeowithNicheSize.Forscenarioswheremutationsaectdivisionrate,tumorigenesisisminimizedatanintermediatepopulationsize(A),attheexpenseofxingdeleteriousmutationsanddecreasingtissuerenewal(C).Forscenarioswheremutationsaectdierentiationrate,tumorigenesisisminimizedforlargepopulationsizes(B),andtheeectofmutationaccumulationontheepitheliumisindependentofnichesize(D).Theredverticaldottedlineisplacedat6,themediannichesizemeasuredinthemouse. 3.4DiscussionDierencesinselectivepressuresonoptimalstemcellnichesizeformutationsthatconferaselectiveadvantagevs.xneutrally.Whenmutationsconferaselectiveadvantageordisadvantagewithintheniche,thereexistsanintermediatecryptsizethatminimizestheprobabilitythatanycryptaccumulatesthelargebenecialmutationsnecessarytoinitiateatumor.Thisresultreinforcespreviousresultsby Michoretal. ( 2003b ),who 62

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alsofoundanintermediatenichesizeexiststhatdecreasestherateoftumorinitiationformutationsunderselection.Atsmallstemcellnichesizesthereexistsalargenumberofcryptstomaintainhomeostasis,andahigherprobabilitythatanyonecryptwillobtainararemutationoflargeeectthatwouldresultintumorigenesis.Asstemcellnichesizeincreases,thenumberofcryptsneededtomaintainthesameamountofepitheliumdecreases,andsodoestheprobabilityofxingmutationswithinthecrypts,andthereforethechanceofxingararemutationoflargeeect.However,forlargervaluesofstemcellnichesize,thestrengthofselectionincreases,thusincreasingthechancethatmutationsconferringabenecialtnesseectxwithinaniche,leadingtohigherchancesoftumorigenesis.Here,giventhespeciedparameters,wendthattheminimumprobabilityoftumorigenesisexistsat6cellsinthemouse,whichisthemedianstemcellnumberfoundby Kozaretal. ( 2013 )formiceintestinestemcellniches.Atthispopulationsizethewholetissuesizeisexpectedtodeclinewithageasdeleteriousmutationsaccumulateinstemcellniches,andifselectivepressuresagainsttumorigenesishaveselectedforstemcellnichepopulationsizesatthisintermediatethenithasbeenattheexpenseofincreasingepithelialattrition.Whenmutationsaectdierentiationrate,andthusxneutrallyinthestemcellniche,largerstemcellnichesizesresultindecreasingtheprobabilityoftumorigenesis.However,therateatwhichtheepitheliumchangestotalsizeisindependentofstemcellnichesize.Atsmallnichepopulationsizes,thenumberofxedmutationsintheentiresystemismaximizedasboththenumberofcryptsandprobabilityofxationofeverymutationwithineachcryptismaximized,thusleadingtothemaximumprobabilitythatanyonexedmutationresultedintumorigenesis.Becausethesemutationsxneutrally,theexpectedvalueofaxedmutationisindependentofstemcellnichepopulationsize.Furthermore,theprobabilityofxationofamutationinanicheistheinverseofthecontributionofthatnichetotissuehomeostasis.Forinstance,amutationthatariseswithinanichethatistentimesthesizeofanothernichehastentimessmallerprobabilityofxation,buttentimeshigherinuenceonthetotalepithelium,meaningthattheexpectedinuenceofthetotalamountofaccumulatedmutationsinsystems 63

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withdierentstemcellpopulationsizesbutconsistenttotalepitheliumsizesisinvariant.Itisprobablethatthemutationaltargetsizeformutationsthataectthepropensityforstemcellstocommittodierentiationissmallerthanthatformutationsthatmightaecttheoveralldivisionrate,andthereforeselectionmaynotactasstronglytooptimizenichesizeinlightonminimizingtumorigenesiscausedbyfailuretodierentiate.Theintestinalepitheliumpopulationisexpectedtodeclinewithagethroughstemcellattrition.Whenemployingdistributionsofmutationaleectscommonlyfoundinexperimentsonwholeorganismswendthatthetotalintestinalepitheliumsizeisexpectedtodecreasewithage.Thisattritionismodestinthemouse,withamousethreeyearsintoadulthoodhavinganintestinalepitheliumapproximately0.3%-0.5%smallerthanamousethatjustenteredadulthood.Thisattritionispotentiallymuchmoresubstantialforhumans,giventheirlongerlifetime.Whenmutationsaectdivisionrateandforthescenariowherethestemcellnichesizeexistsas20cells,wendremarkablysimilartotalepithelialpopulationchangetothemousescenariobecause,despitethelongerlifetime,thedivisionrateofstemcellsisslowerandthepopulationsizeislarger,decreasingthestrengthofdriftandtheaccumulationofdeleteriousmutations.Whenmutationsxneutrallyforthesamepopulationsize,andaectdierentiationrate,wecalculatethatthetotalpost-mitoticepitheliumwilldecreaseapproximately5%at75yearspost-adulthood.Ifthetruestemcellnichesizeinhumansisclosertothelowerestimateof6cells,wecalculateamuchhigherrateoftissueattrition,withtheepitheliumdecreasingover10andover20percentinthedivisionanddierentiationscenarios,respectively.Thisisduetothemuchstrongerinuenceofdeleteriousmutationsdriftingtoxationatsmallerpopulationsizes.Mechanismsofintestinalhomeostasisandevolutionarytradeos.Althoughmiceandhumanshaveanintrinsicrateofcryptdivision,thiseventisexceedinglyrare( Snip-pertetal. 2014 ; Bakeretal. 2014 )andlikelydoesnotplayaroleinmaintainingtissuehomeostasis.Thus,populationequilibriumduringnormaltissuemaintenanceisprimarilycontrolledthroughthecontinualdivisionanddierentiationofstemcells.Here,weshow 64

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that,duetothesmallpopulationsizesoftheindependentlyevolvingpopulationsofstemcellsconstitutingtheintestinalepithelium,thetotaltissuesizeisexpectedtodecreaseoveranorganism'slifetimeduetotheaccumulationofdeleteriousmutationsinstemcellcompartments.Thesesmallcompartments,whileatriskformutationalmeltdown,minimizetheoverallprobabilitythatanymutationintheintestinesresultsinatumorigenesiseventwhenmutationsconferaselectiveadvantageintheniche,andisthusaprimeexampleofanevolutionarytradeo. 65

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CHAPTER4HIGHDN/DSRATIOSANDHETEROGENEITYAREANEXPECTEDOUTCOMEOFTUMORPOPULATIONARCHITECTURE 4.1IntroductoryRemarksRecentevidencesuggeststhatbothnormaltissue( Martincorenaetal. 2015 )andgrowingtumors( WooandLi 2012 ; Ostrowetal. 2014 ; Williamsetal. 2016 )haveahighrateofnon-synonymoustosynonymousDNAmutations(dN/dS).Thishasbeeninterpretedasbeingeitherindicativeofthepresenceofpervasivepositiveselection( Mar-tincorenaetal. 2015 ; Ostrowetal. 2014 )ortheproductofweakorabsentpurifyingselection( WooandLi 2012 ; Lingetal. 2015 ; Williamsetal. 2016 ),aspurifyingselectionwouldprohibittheaccumulationofnon-synonymousmutations.Furthermore,growingtumorsharboranextremeamountofgeneticdiversity,furthersuggestingthatselectivepressuresareweakwithincancers( Lingetal. 2015 ; Sottorivaetal. 2015 ; Williamsetal. 2016 ).Thesepreviousinterpretationsofselectivepressureswithintumorsdidnotaccountforthearchitectureofthevariouscellpopulationswithinagrowingtumor.Here,weinvestigatetheexpecteddN/dSratioandmagnitudeofgeneticdiversitywithinagrowingtumorinlightofempiricallydescribedpopulationdynamics.Wehypothesizethattheseconictinginterpretationswillbereconciledifweconsiderpopulationstructure.WeaddressthishypothesisbysimulatingtumorgrowthunderempiricallymeasuredparametersanddescribedpopulationstructureandsampletheresultantheterogeneityanddN/dSratios.Manyformsofbothhealthy( MorrisonandSpradling 2008 )andhyperplastic( Clevers 2011 )tissuesareconstitutedbyindependentlyevolvingpopulationsofstemcells.Thesepopulationsareoftenmaintainedatsmallsize,andareatconstantriskofaccumulatingmutationsduetothestrengthofgeneticdrift(seeChapter 2 ).Additionally,giventhehighlikelihoodthatanygivenmutationhasadeleteriouseecttocellulartness,deleteriousmutationsmayxandcontributetohighnon-synonymoustosynonymousreplacementratios.Here,weexaminetheexpectedevolutionarydynamicsofagrowingtumorpopulationthatisconstructedofdividingsubpopulations,eachwithitsownevolutionarydynamics.Thissystem 66

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isamodelofanintestinaltumorgrowingviastemcellexpansionandcryptssion(i.e.,thebirthofnewcrypts),theproposedmethodofcancergrowthwithintheintestines( Schepersetal. 2012 ; Garciaetal. 1999 ).Wemodelagrowingintestinaltumoraccordingtothemechanismsandrateparametersdescribedempiricallyanddemonstratethatatumorgrowingviathedivisionofconstituentstemcellpopulationsexhibitsbothhighnon-synonymoustosynonymousnucleotidereplacementsandhighintratumorheterogeneity,andthatthesearenotaproductoftheabsenceofselectionorthepresenceofpervasivepositiveselection,butratherthexationofpredominantlydeleteriousmutationsviageneticdrift.Furthermore,wedemonstratethathighheterogeneityistheexpectedresultofrapidlydividingtransientamplifyingcellpopulations.Thehealthyintestinalepitheliumiscomposedofmillionsofcompartments,namedcrypts,andthebaseofeachcrypthousesasmallpopulationofstemcellsthatcontinuallydividesymmetricallyintomorestemcellsthroughoutanorganism'slife.Thesestemcellsdisplacetheirneighbors,leadingtolineagesdriftingneutrallytoxation( Kozaretal. 2013 ; Ritsmaetal. 2014 ).Intestinaltumorsgrowthroughthessionofthesecrypts( Wasanetal. 1998 ; Garciaetal. 1999 )(oftenreferredtoasglandsinthecontextofagrowingtumor).Eachglandharborsasmallpopulationofstemcellsatthebase( Bakeretal. 2015 ),andthesestemcellsstilldisplacetheirneighborsinamechanismconsistentwithneutraldrift( Kozaretal. 2013 ; Bakeretal. 2014 ),andareresponsibleforgeneratingallotheradenomacelltypes( Schepersetal. 2012 ).Theproportionofstemcellsanddierentiatedcellswithinagrowingtumorremainsconstant( Schepersetal. 2012 ),furthersuggestingthatthepopulationarchitectureofnormaltissueisretainedaftertumorigenesis. 4.2MethodsandModelThemurineintestinalandassociatedcarcinogenesispathwaysareoneofthebeststudiedandparameterizedepithelialtissuesystems.Wesimulatemouseintestinaltumorglandsundergoingssionafterthexationofamutationthatincreasedstemcelldivisionratesuchthatthestemcellpopulationiscontinuingtoexpand,fuelingtumorgrowth.Thesimulation 67

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utilizestheGillespiealgorithm,andassumesthereareindependentexponentialwaitingtimesbetweenalleventsinthesystem. 4.2.1TheIntestinalTumorGlandWemodeltheintestinalcryptsasinChapter 3 .Briey,stemcellsexistatthebaseofthecryptinanicheanddividesymmetricallyatrate,displacingtheirneighbors.Stemcellsdisplacedfromthisnichemayeitherdividesymmetricallyatrateorcommittodierentiationatrate.Oncecommittedtodierentiation,eachlineageundergoesRgenerationsoftransientamplication,dividingatrate,andeventuallyterminallydierentiateintoapost-mitoticpopulation.Thesecellsleavethesystematrate.Glandsundergossionifthestemcellnumberexceedsaspeciedthreshold( Garciaetal. 1999 )andthestemcellniche,X1,isfull.Atssion,halfofthestemcellsinthenicheleavetheircurrentnicheandformanewgland.Thenewglandthengrowsfromthestemcellnichecells,asobservedby Snippertetal. ( 2014 ).Theglandpositionsarearrangedonatwodimensionalsurface,andglandsexpandoutwardatthetimeofdivisionbypickingarandomspaceoutsidethetumorandpushingallneighborstowardsthatposition.Wenotethat,althoughclonalexpansionsmaybeginasatwo-dimensionaleldcancerizationevent( Snippertetal. 2014 ),theyeventuallymayformthree-dimensionalpolypsandtumors,andthistwodimensionalsurfaceisonlymeanttorepresentrelativedistancebetweenlineages.Thisspatialarrangementisusefulbecausecancergenomesequencesareoftentakenfromtumorresections,meaningthatheterogeneitypresentinasampleisonlyasmallrepresentationofthetotalheterogeneityinthesystem.ThesesimulationswererunonRversion3.1.1.Wesimulated30tumorsstartingatthesameinitialconditions.Duetothehighcomputationalworkrequiredtosimulateeveryeventinthesystemallsimulationswereterminatedwhenthepopulationreached200,000cells,anaverageof66.4simulateddaysaftertumorinitiation.Allcellsaresampledandthecompletemutationalhistoryofalllineagesarequantiedsowecancalculateboththemagnitude 68

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ofheterogeneityinthetumorandtheratioofsurvivingnon-synonymoustosynonymousmutations. Figure4-1. Diagram(A)andowchart(B)describingourmodel.(A):Withinglands,stemcellsdividesymmetricallyanddierentiate.Thecellsthenundergoseveralroundsoftransientamplicationbeforereachingtheirpost-mitoticstage,afterwhichtheydieandleavethesystem.(B):Mutationsmayoccurtoanycellatdivisionevents.Ifanewlineageappears,thetnessofthatlineagemaybechangedaccordingtoanintrinsicdistributionoftnesseects.Ifthecrypthasmorestemcellsthanthestemcellthresholditundergoesssion. 4.2.2ParameterChoicesWithin-glandparametersarelistedinTable 4-1 Snippertetal. ( 2014 )foundthathealthyglandsinamouseundergossioneventsapproximatelyonceevery5600days.Inourmodelthistimeisconsistentwithastemcellpopulationssionthresholdofapproximately39cells.Wechoosetomodelthegrowingtumorafterthexationofamutationinthestemcellnichethatcausedanincreaseinstemcelldivisionrate(fromapproximately0.2perday( Kozaretal. 2013 )to0.35perday),increasingthepopulationofstemcellsinthe 69

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systemandtherateatwhichthestemcellpopulationpassesthessionthreshold,resultingintumorglanddivisionandtumorgrowthfueledbytheexpansionofthestemcellcompartment( Schepersetal. 2012 ; Snippertetal. 2014 ).Mutationsinthemodelthatarenon-synonymouschangethelineage'sdivisionrateaccordingtoaspecieddistributionoftnesseects(Figure 4-1 ).Thedistributionofmutationaltnesseectsisheavilybiasedtowardsmutationsthanhaveadeleterioustnesseect( Eyre-WalkerandKeightley 2007 ).Thedistributionofmutationaleectsongrowthrateofnewmutationswithinsomatictissueisunknown;however,ithasbeencharacterizedforseveralmodelorganisms.Inthismodel,weassumeadistributionofmutationaleectssimilartothatmeasuredinthesinglecelledeukaryoteSaccharomycescere-visiae,forwhichtheexpectedvalueofdeleteriousmutationshasbeenmeasuredtohaveanexpectedeectof0.217( ZeylandDeVisser 2001 ),with0.0575proportionofmutationsthatarebenecialtogrowthandanexpectedbenecialeectof0.061( JosephandHall 2004 ).Divisionratecannotbecomenegative,soinourmodelanylargedeleteriousmutationsthatdrawsfromtheDFElowerthanzeroisxedatzero.Weparameterizeourmodelsuchthatnon-synonymousmutationsarethreetimesaslikelyassynonymousmutations,avalueclosetothatcalculatedby Ostrowetal. ( 2014 ),althoughwenotethatthedN/dSrationormalizestheoccurrenceofeachtypeofmutationbythelikelihooditoccurred.Weparameterizemutationrateasin Lingetal. ( 2015 ),whobasedtheirmutationratefromsinglenucleotidemutationrateandtheexpectedtargetsizeoftheentirecodingregionofthegenome,butnotethatthetruetargetsizeformutationsthataectsomaticcelltnessisunknown. 4.3Results 4.3.1TheMajorityofMutationsinaGrowingTumorHaveRecentlyArisenThevastmajorityofmutationsinourmodelatthepointofsamplingexistinonlyafewcells.Figure 4-2 depictsthedistributionofheterogeneitywithinatumoratthetimeofsamplingforarepresentativetumor.Mostsurvivinglineagesatthetimeofsampling(>93%) 70

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Table4-1. Parametervaluesforatumorglandinourmodel ParameterValueNoteCitation X19CellsExpectednumberofstemcellsinthetumorstemcellniche( Kozaretal. 2013 )0.35perdayThedivisionrateofstemcellsinourmodelafterthexationoftheinitialmutationtriggeringtumorigenesisthisstudy0.333perdayTherateatwhichcellspassivelyadoptaTAfatethisstudy2perdayThedivisionrateofTAcells( PottenandLoef-fler 1987 )0.333perdayThedeathrateofterminallydierentiatedcells( Snippertetal. 2010 )R6RoundsofdivisionofrapidlydividingTAcells( Marshmanetal. 2002 )SCT39StemCellThreshold,abovewhichcryptsundergossionthisstudyand( Snip-pertetal. 2014 ) aroseinthetransientamplifyingpopulation,asthispopulationhasthelargestproportionofdividingcellsinthemodel.Thesemutationsareallveryrare,astheywerejustborn,butaswewilldemonstrateinSection 4.3.2 ,theyarethemaincontributorstotheresultantdN/dSratioiffullknowledgeoftumorheterogeneityisconsideredintheanalysis.Thenumberofmutationscarriedbymanycells(16+,Figure 4-2 )ishigherthanthosecarriedbyfewcellsformutationsthataroseinthestemcellpool,aslineagespresentinstemcellsmaycontinuetoexpandinthesystem,andthoseinthenichecanbetransferredtonewcrypts,seedingnewpopulationscarryingthatmutation.Themostcommonmutatedallelesinthesimulatedtumoratthetimeofsampling,i.e.thealleleswiththelowestinverseallelefrequency,originatedwithinthestemcellniche(Figure 4-3 ).Theallelesthatarisewithinthetransientamplifyingpopulationaredestinedtoleavethesystem,unlesstheyhaveadivisionrateofzero,andthereforedonotspreadthroughoutthetumor.Interestingly, Williamsetal. ( 2016 )usedalinearrelationshiponalog-logplotofmutationnumbervs.inverseallelicfrequencytoindicatethearrivalofmutationsintheabsenceofselection.Here,wendthatallelesthathavearisenwithinthetransientamplifying 71

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Figure4-2. Intratumorheterogeneitywithinarepresentativetumorafterthepopulationreached200,000cells.Thedierentcolorbarscorrespondtothepopulationfromwhichthelineageoriginallyarose. populationfollowalinearrelationshipwhenplottedinthisfashion,despitethefactthatinourmodelweknowthatmostmutationshaveadirectaectonthedivisionrateandselectionoftheselineages.Theallelesthatxwithinthestemcellnichemayspreadthroughoutthetumorthroughcryptssion,andarespreadthroughoutallotherprogenycellsinthepopulation,andthereforehavethehighestfrequency. 4.3.2WholeTumordN/dSIsHighAttheendofthesimulationwemeasureddN/dSsamplingallsurvivinglineages(mean=16518survivinglineagespertumor)withinallcellpopulationsthroughouttheentiretumor.Theratiowasmeasuredbynormalizingthenumberofnon-synonymousandsynonymousmutationspresentbytheirintrinsicprobabilityofoccurringgivenamutationevent,andthendividingthenormalizednumberofnon-synonymousmutationsbythenormalizednumberofsynonymousmutations.ThetumordN/dSratiosrangedfrom1.011to1.069(Figure 4-4 ).TherangeofdN/dSvaluesobtainedismuchhigherthanexpectedgivenevolutionunder 72

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Figure4-3. Lineagefrequencywithinasinglesimulatedtumorafterthepopulationreached200,000cells.Inverseallelicfrequencyistheinverseofthefrequencyatwhichallelesarepresentamongallcellsinthesimulation,suchthat5101representsanalleleatfrequency1/50.Thedierentcolorsrepresentthepopulationthatthemajorityofthelineagesatthatdatapointarosefrom,andblackpointsindicatethattherewasnomajority,i.e.themutationsarosefromanequalnumberofdierentpopulations.Thelinearredlineisttothereddatapoints,andR2=0.99 73

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strictpurifyingselection,forinstance,thedN/dSratioforgenesamonghumansmeasuredfromthegermlineis0.28( Ostrowetal. 2014 ).ThishighdN/dSislargelyreectiveoftheintrinsicdistributionofmutationaleects(Figure 4-5 (A)),aseachtumorcontainsalargeamountofrecently-bornlineagesinthetransientamplifyingpool,andtheirlargemutationalheterogeneityreectstheintrinsicdistributionoftnesseects.Thus,thedN/dSratioofallpresentmutationsiscloseto1,asmostmutationshavejustbeenbornandhavenotbeensubjecttoselection.WecanvisualizetheselectivepressureonlineagesbysubtractinglineagesrandomlygenerateddirectlyfromtheintrinsicDFE(withallhavingthesamesurvivalpotentialunderneutralevolution)fromthesurvivinglineagesinoursimulation(Figure 4-5 (B)).Weseethatlineageswithextremelylowdivisionratespreferentiallysurviveinoursimulations,astheydonotreadilydividethroughtheirtransientamplicationstages,andarethusselectedtopersist.ThepersistenceoftheseslowlydividingcellsresultsinourdN/dSratiotobeslightlyover1,asnon-synonymousmutationsconferredtheslowdivisionphenotype. 4.3.3dN/dSRatiosofCommonVs.RareMutationsThemostcommonallelesinthesimulatedtumorarethosethatwerebornwithinthestemcellniche,subsequentlyxed,andspreadthroughcryptssion(Section 4.3.1 ).DuetothenatureofDNAsequencingtechniques,itismorelikelythatallelespresentathighfrequencyaredetectedthanallelesatlowfrequency.Hence,weperformthesameanalysisasinSection 4.3.2 ,butonlydetectthemostcommonlineages,specicallythe50,75,and100mostcommonlineages.WendthatthedN/dSratioislowerthanthefulltumorscenarioforallthreedetectionscenarios,andthedN/dSratioincreasesasweincreasedetectionability(Figure 4-6 ).ThemediandN/dSvalueforthe50,75,and100mostcommonlineagescenariosare0.5,0.56,and0.62.Ofnote,thedN/dSratiomeasuredfromTheCancerGenomeAtlasforcoloncancersampleresectionsby( Ostrowetal. 2014 )is0.6.AlthoughthisdN/dSratioisgreaterthanwhatiscommonlyseenunderpurifyingselection,itdoesnotimplypervasivepositiveselection,asthedistributionoftnesseectsdrawnfromthelineagesinasinglesimulation(Figure 4-7 )revealthatthemajorityofnon-synonymousmutationshavealower 74

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Figure4-4. dN/dSof30tumorsimulations.Eachtumorgrewuntilitreached200,000totalcells.Tumorsreachedthissizeonaverage66.4simulateddaysfromthemomentoftumorinitiation.Thereddottedlinerepresentsthe0.28germlinedN/dSratioforhumans( Ostrowetal. 2014 ). divisionratewithrespecttotheirparentlineages.Theselineageswithalowertnessxthroughgeneticdriftinthestemcellniche.Thus,themostcommonallelesinthesystem,thosewhichresultinahighdN/dSatthetimeofsampling,areonaveragedeleterioustocellulardivisionanddrifttoxationinsmallstemcellniches. 4.4Discussion 4.4.1ALargeProportionofNon-synonymousMutationsandHighHeterogeneityAretheDirectResultoftheModularNatureofTumorGrowthWedemonstratethatattributespreviouslyreasonedtobesuggestiveofthelackofselectionorthepresenceofpervasivepositiveselection,i.e.relativelyhighdN/dSratiosandhighheterogeneity,areproductsoftheaccumulationofmutationsdeleterioustocellulardivisionrate.Mutationsdeleterioustocellulardivisionrateaccumulateandpersistinourmodelfortwomainreasons.Firstly,thestrengthofgeneticdriftislargewithinourstemcell 75

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Figure4-5. (A)Thedistributionoftnesseectssampledfromalltumors(black)andtheintrinsicdistributionoftnesseectsoverlay(transparentred).TheintrinsicDFEdatawasgeneratedbydrawingfromthedistributionspeciedatthebeginningofthesimulationanumberoftimesequaltothesurvivinglineagesattheendofthesimulation.(B)TheintrinsicDFEsubtractedfromthesampledDFEfromsurvivinglineages,dividedbytheintrinsicDFE.Positivevaluescorrespondtomoresurvivinglineagesthananeutralscenario(positiveselection),whilenegativevaluessignifynegativeselection.Theextremelyhighdivisionratesontherightsideof(B)areveryrare,andthussinglemutationsdisplaylargeeects. 76

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Figure4-6. dN/dSratiosforthemostcommonlineagesinthetumorsafterthepopulationsreached200,000cells.BlueverticallinesareplacedatthemediandNdSforthe30simulations. nicheduetothestochasticfateofstemcellsatsmallpopulationsize( Kozaretal. 2013 )ascellsdisplacetheirneighborsandcompeteforspacewithintheniche( Ritsmaetal. 2014 ).Secondly,sincetheamountoftimeanysinglelineagepersistsinthetransientamplifyingpooliscontingentonthenumberofdivisioneventsthelineagehasbeenthrough,mutationsthatdecreasedivisionrateresultinlineagesthatpersistlongerinthesystemandhaveahigherprobabilityofbeingdetectedatthetimeofsampling.Furthermore,deleteriousmutationsareintrinsicallymuchmorelikelythanbenecialmutations,increasingthelikelihoodofthesemutationsbeingprevalentintherapidlydividingtransientamplifyingpopulation.Thus,ahighdN/dSratiointhisstructuredpopulationistheresultofbothdriftinthestemcellpopulationandextremeheterogeneityinthetransientamplifyingpopulation,andreectstheaccumulationofdeleteriousnon-synonymousmutations. 77

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Figure4-7. DistributionsofFitnessEectsfromthemostcommonlineagesinasingletumorafterthepopulationsreached200,000cells.Thedivisionratesarerelativetotheirparentlineage. Wealsodemonstratedthatatumorgrowingviatheexpansionofstemcellswhileretainingthepopulationarchitectureofhealthysomatictissueexhibitsextremeintratumorheterogeneity.Mostofthemutationsinourmodelexistinonlyafewcells.Thisresultisconsistentwiththeextensivesequencingeortsof Lingetal. ( 2015 ),whofoundthatapproximately99%ofmutationsexistinfewerthan100cells,aswellasresearchby Sottorivaetal. ( 2015 ),whofounduniformlyhighintratumorheterogeneityincolonadenomas.However,whereasthishighheterogeneityhasbeenpreviouslyinterpretedastheproductoftheabsenceofselectioninanunstructuredpopulation,ourmodelsuggeststhisextremeheterogeneitycanbeexplainedbythepopulationarchitectureofatumorgrowingviacancerstemcells. 78

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4.4.2ModelingConsiderations,SamplingTechniques,andResultsWetreatsomatictissueandagrowingtumorasahaploidsystem.Somatictissueisdiploid,andgrowingtumorsmayreachhigherploidylevels,meaningthatthefrequencyofallelespercellspresentinthetumormayreachhigherlevelsthaninoursimulation.However,wechosenottoincorporatethisadditionalcomplexityintoourmodel,becausethemechanismsofploidyincreaseandtranslatingtheaectofmultipleallelecopiestophenotypearenotinthescopeofthiswork.Wehaveacompletehistoryoftumorgrowthandfullknowledgeoflineagefrequencyfromoursimulations.Itisnotclearhowwhole-exomesequencingofrandomresectionsofatumorcantranslateintotheprobabilityofobtaininganallelevariantinananalysis.However,itislikelythatonewouldonlyobtainvariantsofhighfrequency,limitingtheestimationofthetruedN/dSratioandmagnitudeofheterogeneitywithinthetumor.Mutationsathighfrequencywithinagrowingtumormaynothavebeenselectedforandmaynotbefuelingtumorgrowth,butinsteadmayhavebeenxedthroughgeneticdriftwithinastemcellnicheandpropagatedthroughnichession.Thus,todetectmutationsresponsiblefortumorgrowthandsurvivalitisnecessarytosamplecommonmutationswithindierenttumorsindierentindividuals.Commonlyrecurrentmutationsathighfrequencywithindierenttumorswouldlikelyindicateconsistentselection. 79

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CHAPTER5SUMMARYANDCONCLUSIONSWithinthisdissertationIhavemodeledmutationaccumulationwithinhealthysomatictissueandgrowingtumorsandanalyzedhowtheresultantevolutionarydynamicscontributetoaging,tumorformation,andtumorgrowth.Importantly,thisisthersttimetheseprocesseshavebeeninvestigatedinlightoftheextremelysmallpopulationsofstemcellsthatgoverntissuerenewalandtumorgrowth.Ihavedemonstratedthat,duetothecompartmentalarchitectureoftheintestinalepitheliumandthesmallpopulationsizeofthestemcellnicheatthebaseofeveryintestinalcrypt,mutationsdeleterioustocellulartnessareexpectedtoaccumulatethroughouttheintestinalepithelium.Asmutationsaecttherateofstemcellturnover,tissuerenewaliscompromised,andtheexpectedeectisadecreaseintissuerenewalcapabilitywithage.Furthermore,raremutationsofbenecialtnesseecthaveachanceofxinginanyonestemcellniche,initiatingtumorformation,anddistributionsofmutationaleectsconsistentwiththosemeasuredinwholeorganismsaresucienttoaccountforthisprocess.Thesizeofthestemcellnicheandthenumberofintestinalcryptsinthemouseintestineminimizesthisprobabilityoftumorigenesisattheexpenseofaccumulatingdeleteriousmutationsthroughgeneticdrift.Finally,Ihavedemonstratedthatdierentselectivepressuresexistindierentpopulationswithinagrowingtumor;mutationsdeleterioustodivisionrateareselectedforinthetransientamplifyingpopulationwhilemutationsbenecialtodivisionrateareselectedforinthestemcellniche,althoughmostlydeleteriousmutationsaccumulateinthenicheviageneticdrift.TheseselectivepressuresandthepopulationarchitecturewithinagrowingtumordrivehighdN/dSratiosandextremeintratumorheterogeneity.Theresultspresentedwithinthisdissertationhighlighttheimportanceofconsideringpopulationstructurewheninvestigatingevolutioninseeminglylargepopulations,suchaswholetissuesortumors,asthecompartmentalarchitectureofpopulationsresultsinpervasivegeneticdriftandtheaccumulationofmutationsdeleterioustotness.Theresultsofthemodelspresentedheremaybetestedempirically.Forinstance,themagnitudeofgeneticheterogeneity 80

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withinandamongnormalcryptsinagedindividualscanbequantied,andcorrelatedwithmeasurementsofself-renewalpotentialsuchaspopulationsizeanddivisionrate.Furthermore,theindividualpopulationsofcellswithindierentglandsofagrowingtumorcanbesequencedtodeterminethedistributionofintratumorheterogeneityamongpopulations.Lastly,thisworkemphasizesthatunderstandingtheclonaldynamicswithinstemcellnichesiscrucialinpredictingtheevolutionarydynamicsofatissue,andthus,morelineagetracingexperimentsarewarrantedwithindierenttissuetypes. 81

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APPENDIXDESCRIPTIONOFTHEMATHEMATICALMETHODOLOGYInthissectionwedescribethemathematicalmodelunderlyingourresearchapproach.Itisamulti-scalemodel,includingthedynamicsofthestemcellniche;theconsequencesforthelargerstemcellpopulationandacrypt;thedynamicsinapopulationofcryptsthatcompriseanindividual'scolon;andthedynamicsoftumorigenesisamongmanyindividualsinapopulation.Duetotheverylargenumberofcryptsinthecolonandthedesiretoanalyzepopulationlevelincidencecurves,weusedseveralprinciplesofrare-eventanalysisinournumericalcomputationsandalsointroducedafewapproximationstomakecomputationstractable.Populationdynamicswithinthestemcellniche.First,wedevelopamodelforthepopulationdynamicsofacryptimmediatelyafteramutationhasoccurred.SupposethatthereareNcellsinthestemcellnicheandletX(t)representthenumberofcellsthataredescendedfromtheoriginalmutatedcellattimet.WemodelX(t)asacontinuoustimeMarkovchain(CTMC)thattakesitsvaluesinthesetf0,...,NgwithX(0)=1.InaccordancewiththeMarkovprocessassumption,thetimebetweendivisionsofagivenstemcellareindependentofallothercellsandexponentiallydistributedwithrateparametersoldorfortheoldandthenewlineages,respectively.Whenacelldivides,weassumethatthereiscrowdinginthestemcellnicheandanoldcellisforcedout.(Infact,theactualorderofeventsremainsunclear.Ithasalsobeenhypothesizedthatacellmayleavethenicherst,thentriggeringacelldivisiontoreplaceit( Lopez-Garciaetal. 2010 ).)WhetherornotthevalueoftheprocessX(t)changesdependsonwhetherthecellthathasbeenforcedoutisfromthesamelineageastheonethatdivided.Theassumptionthatleadstothesimplestmathematicalmodelisnearestneighbordisplacement.Therearetwocases:1)thedividingcellisofthesamelineageasbothofitsneighbors;and2)thedividingcellisadjacenttoacelloftheopposinglineage.Intherstcase,thevalueofX(t)doesnotchangeasaresultofthecell 82

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division.Inthelattercase,thereisaone-halfprobabilitythatacelloftheopposinglineagewillbedisplaced.Assuch,forX(t)2f1,2,...,N)]TJ /F7 11.955 Tf 11.96 0 Td[(1g,theMarkovtransitionratesaregivenby Nearestneighbordisplacement:8><>:X(t)!X(t)+1atrateX(t)!X(t))]TJ /F7 11.955 Tf 11.95 0 Td[(1atrate0.(A{1)(Therateofoneofthetwobordercellsdividingis2forthenewlineageand2oldfortheoldlineageandtheneachismultipliedbytheone-halfprobabilityofdisplacinganopposinglineagecell.)Analternatehypothesisisthatafterdivision,anyothercellinthecryptmightbedisplaced.Thecorrespondingtransitionrateswouldbe Nonlocaldisplacement:8><>:X(t)!X(t)+1atrate1 NX(t)(N)]TJ /F9 11.955 Tf 11.95 0 Td[(X(t))X(t)!X(t))]TJ /F7 11.955 Tf 11.95 0 Td[(1atrate1 NX(t)(N)]TJ /F9 11.955 Tf 11.95 0 Td[(X(t))old(A{2)Theprobabilityofxationisactuallythesameforbothmodels(thoughtheexpectedtimeuntilxationwilldier).Letft1,t2,...gbethesequenceoftimeswhenX(t)changesvalues.Disregardingtheroleoftimeintheprocess,wetrackthevalueswiththeprocessfXngn0denedbyXn:=X(tn).TheprobabilityofatransitionX!X+1istherateatwhichthesizeofthemutantlineageincreasesdividedbythetotalrateofchangeinlineagecount.Forbothmodelsthisprobabilityofanincreaseinthemutantlineagesizeisp==(+old).Usingtheclassicaltheoryofhittingprobabilitiesforbiasedrandomwalks( WodarzandKomarova 2005 ),onecanreadilyderivetheprobabilitypx(;old)recordedinEq.( 2{3 )inthemaintext.Theintervalsbetweenmutationsthatxinthestemcellniche.TheDFEsusedinthisworkarebothconsideredintermsofpercentageincreaseordecrease,ratherthanintermsofabsolutequantitiesofchange.Inmathematicalterms,thismeansthatthedensitiescanbeexpressedintermsoftheratio=old.Aremarkableconsequenceofthisassumptionisthattheprobabilityofanewlineagexinginthenicheisindependentoftheprevailingdivisionrateold.Toseethis,considertheprobabilityofthatanewlineagexesafteramutationdrawn 83

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fromtheexponentialDFE.RecallingEquation( 2{3 ),wenotethattheprobabilityofxationformulacanbewrittenintermsoftheratioofthenewtotheolddivisionrate,r==old,px(;old)=px(r)=1)]TJ /F9 11.955 Tf 11.95 0 Td[(r)]TJ /F12 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 11.955 Tf 11.96 0 Td[(r)]TJ /F8 7.97 Tf 6.58 0 Td[(N.Wecanthenwrite^p=PfFixationjoldg=Z10px(;old)m(;old)d=Zold0px old(1)]TJ /F9 11.955 Tf 11.96 0 Td[(PB) olde)]TJ /F14 7.97 Tf 6.59 0 Td[((1)]TJ /F16 5.978 Tf 12.02 3.26 Td[( old)d+Z1oldpx oldPB olde)]TJ /F14 7.97 Tf 6.59 0 Td[((1)]TJ /F16 5.978 Tf 12.01 3.26 Td[( old)d=Z10px(r)(1)]TJ /F9 11.955 Tf 11.95 0 Td[(PB)e)]TJ /F14 7.97 Tf 6.59 0 Td[((1)]TJ /F8 7.97 Tf 6.59 0 Td[(r)dr+Z11px(r)PBe)]TJ /F14 7.97 Tf 6.59 0 Td[((r)]TJ /F12 7.97 Tf 6.58 0 Td[(1)dr,whichisindependentofthechoiceofvalueold.AsimilarresultholdsforthepowerlawDFE.Generally,thispropertyholdsforanyDFEthatcanbeexpressedintermsoftheratio=old.ItfollowsthatthenumberofmutationsthatmustoccurinorderforanewdivisionratetoxisdistributedGeometricallywithsuccessprobability^p.BystandardpropertiesofCTMC,wecanthensaythatthetimebetweenthearrivalsof\successful"mutationsisExponentiallydistributedwithrateparameter^poldN.Populationdynamicsoutsidethestemcellniche.Onceoutsidetheniche,astemcellcaneitherdivide(atrate),oritcandierentiateintotransientamplifyingcells(atrate).Forthepurposesofthismodel,weconsiderdierentiatedcellstobedead.Thereisachancethatthelineageofastemcelloutsidethenichecanundergosucientlymanymutationstocausetumorigenesis,butwefoundbywayofnumericalinvestigationsthatthisdoesnotsignicantlycontributetooverallincidenceofthesecancers.Assuch,letY(t)denotethenumberofstemcellsoutsidethenichethathavenotyetdierentiated.Assuming,forthe 84

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moment,thatallmembersofthestemcellnichehaveadivisionrate,theCTMCY(t)isdenedbythetransitionratesY(t)!Y(t)+1atrate(N+Y(t))Y(t)!Y(t))]TJ /F7 11.955 Tf 11.96 0 Td[(1atrateY(t).TheformoftherateofincreasefollowsfromtheobservationthatY(t)increasesanytimeastemcelldivides,whetherthatstemcellisinthecryptornot.Ontheotherhand,sincestemcellsinthenicheareassumedtonotdierentiate,thetotalrateofdecreaseisproportionaltothenumberofstemcellsoutsidetheniche.Becausethepopulationsizeissosmall,thereishighvariabilityandwenotethatY(t)canregularlyhitthevaluezero.Becausethenicheisprotectedbyunrelatedbiologicalprocesses,thisdoesnotconstituteextinctionofthefullstemcellpopulation.Assoonasanotherstemcellinthenichedivides,thepopulationoutsidethenicheisrenewedagain.AtypicaltraceforY(t)canbeseeninFigure A-1 .ThelawofthisCTMC,yn(t)=PfY(t)=ng,satisesthesystemofmasterequations d dtyn(t)=(N+(n)]TJ /F7 11.955 Tf 11.63 0 Td[(1))yn)]TJ /F12 7.97 Tf 6.58 0 Td[(1(t)1n1(n)+(n+1)yn+1(t))]TJ /F11 11.955 Tf 11.63 9.68 Td[()]TJ /F7 11.955 Tf 5.48 -9.68 Td[((N+n)+nyn(t).(A{3)Onecanthenshowthatthemeany(t)=P1n=1nyn(t)satisestheODE d dty(t)=N+()]TJ /F10 11.955 Tf 11.95 0 Td[()y(t).(A{4)If<,thisODEconvergestoasteady-statevalueN=()]TJ /F10 11.955 Tf 12.9 0 Td[().Otherwisethemeandivergestoinnitywithexponentialgrowth.Forthisreason,weconsiderthisthresholdtobetheinitiationoftumorigenesis.Analternatewaytoviewthedynamicsistonotethateachtimeastemcellinthenichedividesitcreatesanewindependentlineageoutsidethecrypt.LetYj(t)bethenumberoflivingstemcellsoutsidethecryptthataredescendedfrom(andinclude)theproductofthejthstemcelldivisionintheniche.AssuchY(t)=P1j=1Yj(t).Each 85

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processYj(t)canbeunderstoodasabranchingprocess,withanospringdistributionthatisGeometricallydistributedwith\successprobability"q==(+).(Thenumberofospringisdeterminedbythenumberoftimesthecelldividesbeforedierentiating.Thisisasequenceofindependenttrialswheretheprobabilityofhavinganotherospring,ratherthandierentiating,is=(+).)Aslongasthemeanofthisospringdistributionislessthanorequaltoone,theselineageswilleventuallygoextinct.Therefore,thecriticalstemcelldivisionratecorrespondstowhenthemeanoftheospringdistribution(whichcanbeshowntobe(1)]TJ /F9 11.955 Tf 10.55 0 Td[(q)=q==)islessthanone.Inotherwords,thecriticaldivisionrateissimply=.Populationdynamicsinthecrypt.Ofcourse,tumorigenesisinagivencryptisexceedinglyunlikely,evenoverthelifetimeofanindividual.WemodelthecolonasacollectionofC107individualcryptsthataremathematicallyidenticalandindependent.ThenumberofxedmutationsintheithcryptattimetisdenotedMi(t),andletfi0,i1...gdenotethesequenceofdivisionratesthatbecomexedintheithcryptattimesf0,i1,i2...grespectively.Itfollowsthattheinter-arrivaltimesareindependentanddistributedas ik+1)]TJ /F10 11.955 Tf 11.95 0 Td[(ikExp(^pikN).(A{5)Whethertumorigenesishasoccurredintheithcryptwillbetrackedbythefunctioni(t),denedbyi(t)=n1,ifmax)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(ik:kMi(t)0,otherwise.Thatistosay,i(t)=1iftumorigenesishasoccurredbeforetimet.Itfollowsthatthetimeofrsttumorigenesisinthecolonisgivenbythetime T:=inft>0:CXi=1i(t)1.(A{6) 86

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ThepercapitapopulationincidencecurvesarethenjustthecumulativedistributionfunctionoftherandomvariableT,whichcanbeexpressedintermsoftheindividualcryptdynamicsasfollows:PfT>tg=Pi(t)=0foralli2f1,...,Cg.Toprepareforournumericalapproximationofthisquantityweintroduceonelastbitofnotation,fNm(t)g1m=0,whichrepresentsthenumberofcryptsthathaveseenthearrivalofmnewxedlineagesasoftimet.Then PfT>tjNm(t)=nmforallmg=1Ym=0Pf(t)=0jM(t)=mgnm.(A{7)ThesedynamicscanbesimulatedbyGillespie'smethod( Gillesple 1977 ),butsuchanapproachiscomputationallyintensive.Forthisreasonweintroducedafewsimplifyingassumptions.Forexample,wemodelthearrivalratesofnewxedlineagesinthecryptsasbeingconstantovertime(havingxedrate^=^p0N,ratherthanasequenceofratesgiveninEq.( A{5 )).ThisallowsustoassumethatthenumberofmutationsineachcryptattimetisPoissondistributedwithmean^t.Withsuchatremendouslylargenumberofcryptsinthecolon,itisinturnreasonabletoassumethatthenumberofcryptstakestheformnmCPfM(t)=mgCe)]TJ /F12 7.97 Tf 7.15 0 Td[(^t(^t)m=m!.TocompletethederivationofEq.( 2{8 )inthemaintext,wetruncatetheinniteproductinEq.( A{7 )andnotethatinthenotationofthemaintext,Pf(t)=0jM(t)=mg=qm.DFEequations.Todenetheparametersofthesystem,wespeciedtheprobabilityPBofabenecial(versusdeleterious)mutationandtherespectivemeanss+ands)]TJ /F1 11.955 Tf 10.99 1.8 Td[(oftheDFEconditionedontheeventthatthemutationisbenecialordeleterious.TheformofthemeanoftheDFEdependsonwhetheritisexponentialorheavy-tailed.TheexponentialformDFEhasmean 87

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FigureA-1. Thesimulatedstemcelldynamicswithinahumancryptforboththedisplacedstemcells(A,B)andthestemcellniche(C,D)showingaxationeventofamutantlineage.Originaldivisionrate0=0.143,newxeddivisionrate1=0.141.A:Thedisplacedstemcellpopulationsizeuctuatesstochasticallyasstemcellsenterthepopulationbybeingdisplacedfromthenicheandleavebycommittingtodierentiation.B:Zoominginatthemomentanewstemcelllineagebeginsbeingdisplacedfromtheniche,weseetheoriginal(representedinblack)lineagegoingextinctasthenew(representedinred)lineageeventuallydominatesthepopulation.C:Thespatiallyexplicitstemcellnicheisarrangedinacirclewithcellsdisplacingtheirneighborsthroughdivision.Herethatcircleisopened(cellpositions1-20representedontheY-axisandshownthroughthesametimeseriesasB.Eachtimepointcorrespondstoanyeventoccurringintheentiresimulation(divisioninthestemcellniche,divisionanddierentiationoccurringindisplacedcells).Anewmutantlineagearisesthrougherrorscausedbydivisionattime23994.13daysand,bystochasticallydividinganddisplacingneighbors,reachesxationinthepopulationbyday24536.87.ThebeginningandenddynamicsofthisnewlineageareshownininsetD 88

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E(newjold)exp=old(1+PB )]TJ /F7 11.955 Tf 13.15 8.09 Td[((1)]TJ /F9 11.955 Tf 11.95 0 Td[(PB) )whiletheheavy-tailDFEhasmeanE(newjold)Pareto=PB()]TJ /F7 11.955 Tf 11.96 0 Td[(1 )]TJ /F7 11.955 Tf 11.96 0 Td[(2)old+(1)]TJ /F9 11.955 Tf 11.95 0 Td[(PB)(old)]TJ /F10 11.955 Tf 13.15 8.09 Td[(old ).Theconditionalmeanshavetheforms+:=E(newjold,benecial)exp=old(1+1 )s)]TJ /F7 11.955 Tf 10.4 1.79 Td[(:=E(newjold,deleterious)exp=old(1)]TJ /F7 11.955 Tf 13.64 8.09 Td[(1 )fortheexponentialDFE,ands+:=E(newjold,benecial)Pareto=old()]TJ /F7 11.955 Tf 11.96 0 Td[(1 )]TJ /F7 11.955 Tf 11.96 0 Td[(2)s)]TJ /F7 11.955 Tf 10.4 1.79 Td[(:=E(newjold,deleterious)Pareto=old(1)]TJ /F7 11.955 Tf 13.64 8.09 Td[(1 )fortheheavy-tailDFE.Leastsquaresanalysis.Wegeneratedtumorincidencecurvesandusedaleastsquaresanalysistodeterminewhichsetoftheseparametersbesttthetumorincidencedatadescribedin Chapman ( 1963 ).ThebestthasthesmallestsumofsquaredresidualsoftheparameterspaceexploredinFigures 2-4 2-5 ,and 2-9 .TumormutationalproleSupportingInformationFigureS4containstheprobabilitiesthateachindividualmutationalprolewastheculpritintumorigenesisgiventhattumorigenesisoccurredinasinglecrypt.TheywerecalculatedbyusingBayes'Theoremtocomputetheprobabilitythatacertainmutationalloadxedinthecryptgiventhatatumorigenesiseventhappened, 89

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P(M(T)=njT=t)=fT(tjM(T)=n)P(M(T)=n) P^nj=1fT(tjM(T)=j)P(M(T)=j)wherewerecallthatM(t)isthenumberofxedmutationsasoftimet,TistheprecisetimethattumorigenesisoccursinthecryptandfTreferstodensityoftherandomvariableT.ThequantityP(M(T)=n)istheprobabilitythattumorigenesisoccursexactlyonthenthmutation,aquantitywedenedearlieraspnandgavearecursiveformulaforinEquation5inthemaintext.TocomputethequantityfT(tjM(T)=n),notethatsincethearrivaltimeofthenthmutationisindependentoftheeventthatitcausestumorigenesis,wehavethatfT(tjM(T)=n)=fn(t),wherenisthearrivaltimeofthenthmutation.Byhypothesis,nisPoissondistributedwithmean^twith^beingdenedafterEquation8inthemaintext.Thedistributionsofmutationalprolesgiventhetumorigenesiseventoccurredatacertainpointintimethroughoutanorganism'slifetimearegiveninFigure A-2 90

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FigureA-2. Probabilitiesdescribingthedistributionsofmutationsthatcausedaninitiatedtumorthroughoutthelifetimeofamouse(A,B)andhuman(C,D)underthemutationsaectingdivisionrate(A,C)anddierentiationrate(B,D)modelingscenarios. 91

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BIOGRAPHICALSKETCHVincentL.CannatarowasbornonLongIsland,NewYork,andgraduatedfromEastIslipHighSchoolinMay2006.DuringhissenioryearofhighschoolhereceivedtheScienceTeacherAssociationofNYOutstandingHighSchoolScienceStudentAwardforSuolkCounty,andtheIslipTownBoardOutstandingAchievementAwardintheCategoryofScience.InMay2010hegraduatedfromtheStateUniversityofNewYorkatGeneseowithaBachelorofSciencewithhonorsinbiochemistry.HishonorsthesiswasunderthedirectionofDr.ElisabethCox-PaulsonandwastitledTMD-1/tropomodulinregulatesintestinalshapeandvolumeduringdevelopmentinC.elegans,andhewontheDr.MarkDiamondMemorialBiologyResearchAwardforthemostoutstandingbiologyresearchpresentationoftheacademicyear.VincentjoinedthegraduateprogramintheDepartmentofBiologyattheUniversityofFloridainAugust2010.HewasawardedtheNationalScienceFoundationQuantitativeSpatialEcology,Evolution,andEnvironmentIntegrativeGraduateResearchTraineeshipFellowshipandwonthenationalchoiceawardforthe2013NSFIGERTscienceoutreachvideocompetitiononresearchcompletedunderthisfellowship.VincentservedasvicepresidentoftheBiologyGraduateStudentAssociation,andreceivedtheBestGraduateStudentTeacherawardfortheUFDepartmentofBiology.Hisdissertationresearchinvestigatedtheevolutionarydynamicswithinsomatictissueandexploredhowthisevolutionmanifestsclinicallybycontributingtoagingandcancer.HewasadvisedbyDr.ColetteM.St.MaryandDr.ScottA.McKinley. 102