Modeling Evolution of Heterogeneous Populations: Theory and Applications PDF
Preview Modeling Evolution of Heterogeneous Populations: Theory and Applications
MODELING EVOLUTION OF HETEROGENEOUS POPULATIONS MATHEMATICS IN SCIENCE AND ENGINEERING MODELING EVOLUTION OF HETEROGENEOUS POPULATIONS Theory and applications I K RINA AREVA G K EORGY AREV AcademicPressisanimprintofElsevier 125LondonWall,LondonEC2Y5AS,UnitedKingdom 525BStreet,Suite1650,SanDiego,CA92101,UnitedStates 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom ©2020ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicormechanical, includingphotocopying,recording,oranyinformationstorageandretrievalsystem,withoutpermissioninwritingfromthe publisher.Detailsonhowtoseekpermission,furtherinformationaboutthePublisher’spermissionspoliciesandour arrangementswithorganizationssuchastheCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefound atourwebsite:www.elsevier.com/permissions. ThisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythePublisher(otherthanas maybenotedherein). Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperiencebroadenour understanding,changesinresearchmethods,professionalpractices,ormedicaltreatmentmaybecomenecessary. Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluatingandusingany information,methods,compounds,orexperimentsdescribedherein.Inusingsuchinformationormethodstheyshouldbe mindfuloftheirownsafetyandthesafetyofothers,includingpartiesforwhomtheyhaveaprofessionalresponsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assumeanyliabilityforany injuryand/ordamagetopersonsorpropertyasamatterofproductsliability,negligenceorotherwise,orfromanyuse oroperationofanymethods,products,instructions,orideascontainedinthematerialherein. LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-814368-1 ForinformationonallAcademicPresspublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Theviewspresentedinthisbookaretheauthors’personalviewsanddonotnecessarilyrepresenttheviewsofany otherorganization. Publisher:CandiceJanco AcquisitionEditor:ScottJBentley EditorialProjectManager:SusanIkeda ProductionProjectManager:DebasishGhosh CoverDesigner:GregHarris TypesetbySPiGlobal,India Dedication To Faina Berezovsky, wife,mother, friend, collaborator, and anextraordinary mathematician C H A P T E R 1 Using mathematical modeling to ask meaningful biological questions Abstract Classicalapproachestoanalyzingdynamicalsystems,includingbifurcationanalysis,canprovideinvaluable insightsintotheunderlyingstructureofamathematicalmodelandthespectrumofallpossibledynamical behaviors.However,thesemodelsfrequentlyfailtotakeintoaccountpopulationheterogeneity.Whilehetero- geneityiscriticallyimportanttounderstandingandpredictingthebehaviorofanyevolvingsystem,thischar- acteristiciscommonlyomittedwhenanalyzingmanymathematicalmodelsofecologicalsystems.Attemptsto include population heterogeneity frequently result in expanding system dimensionality, effectively preventingqualitativeanalysis.However,Reductiontheorem,orhiddenkeystonevariable(HKV)method, allowsincorporatingpopulationheterogeneitywhilestillpermittingtheuseofclassicalbifurcationanalysis. Acombinationofthesemethodsallowsvisualizationofevolutionarytrajectoriesandpermitsmakingmean- ingfulpredictionsaboutdynamicsovertimeofevolvingpopulations. 1.1 Introduction Heterogeneityisamajordrivingforcebehindthedynamicsofevolvingsystems.Whenit is heritable and when it affects fitness, heterogeneity is what makes evolution possible (Bell,2008;Darwin,1880;Johnson,1976;Page,2010).Thiscomesfromthefactthattheenvi- ronmentinwhichtheindividualsinteractiscomposednotonlyoftheoutsideworld(suchas the resources necessary for survival, or members of other species) but also of individuals themselves. Therefore, selective pressures that are imposed on the individuals come both fromtheenvironmentandfromoneanother.Furthermore,selectivepressuresthatindivid- uals experience from one another will be imposed and perceived differently depending on population composition, which in turn may be changing as a result of these selective pres- sures.ThisselectionprocessunderliesDobzhansky’sfamousthesis:nothinginbiologymakes sense except in light ofevolution (Dobzhansky, 1973). Inamajorityofconceptualandoftenevendescriptivemathematicalmodelsofpopulation dynamics—whetheritbemodelsofpredator-preyinteractions,spreadofinfectiousdiseases or tumor growth—population homogeneity is the first simplification that is made. The ModelingEvolutionofHeterogeneousPopulations 1 #2020ElsevierInc.Allrightsreserved. https://doi.org/10.1016/B978-0-12-814368-1.00001-1 2 1. Usingmathematicalmodelingtoaskmeaningfulbiologicalquestions population is not treated as homogeneous per se; rather, one assumes an average rate of growth, death, or infectiousness as a reasonable enough approximation if the system has reachedaquasi-stablestateofevolutionarydevelopment.However,byignoringpopulation heterogeneityinsuchaway,oneendsupeitherignoringnaturalselectionorassumingthatit has already “done its work.” This assumption is often incorrect within the context of such models however, since natural selection may in fact be a key driver behind the dynamics of systems that are of most interest andimportance. The process of evolution of heterogeneous populations is typically modeled using replicator equations that capture the “basic tenet of Darwinism” (Hofbauer and Sigmund, 1998;NowakandSigmund,2004).Theseequationscapturetheprocessofselectioninhetero- geneous populations as they demonstrate how when an individual’s value of a heritable characteristicisaboveaverage,thatindividualstaysinthepopulation,butwhentheheritable characteristic is below average, the individual is expunged. However, such models have a major drawback: modeling high levels of heterogeneity is accompanied by an inevitable increase of system dimensionality, which makes obtaining any kind of qualitative under- standing of the system nearly impossible. Assuming population homogeneity makes systems of equations computationally and sometimes even analytically manageable, but at the cost of losing many of the system dynamics caused by intraspecies interactions and natural selection. Despitetheirshortcomings,parametricallyhomogeneoussystemscanstillprovideexcep- tionallyvaluableinformationaboutthestructureofthesystemthroughtheuseofextensively developedanalyticaltechniques,suchasbifurcationanalysis(Kuznetsov,2013).Askillfully constructedbifurcationdiagramcanrevealvariouspossibledynamicalregimesofasystem thatresultfromvariationsinparametervaluesandinitialconditions,andalsoprovideana- lytical boundaries as functions of system parameters. This information can then be used to construct a theoretical framework for understanding a biological system that could never have been obtained experimentally. Inthisbook,wewilldescribeindetailamethodtointroducepopulationheterogeneityback intoequation-basedmodelsusingtheReductiontheorem.Alsoknownasparameterdistribu- tion technique or hidden keystone variable (HKV) method, Reduction theorem can use and buildoninsightsobtainedfrombifurcationanalysis,whileincorporatingpopulationheteroge- neity.Reductiontheoremallowsthedynamicsofanevolvingsystemtobeinvestigatedmore fully,whileovercomingtheproblemofimmensesystemdimensionalityinawideclassofmath- ematicalmodels. 1.2 General strategy ThekeystepsforusingtheHKVmethodareasfollows.Assumeapopulationofindivid- ualsiscomposedattimetofclonesx(t,a).Eachindividualclonex(t,a)ischaracterizedbypa- rametervaluea2A,andeachparametercorrespondstoameasureofsomeintrinsicheritable trait,suchasbirtPhrate,deathrate,resourceconsumptionrate,eÐtc.Thetotalpopulationsizeis givenbyNðtÞ¼ a2Axðt,aÞifthesystemisdiscrete,andN(t)¼ Ax(t,a)daifthesystemiscon- tinuous. Then, since different clones can grow and die at different rates, the distribution of 3 1.2 Generalstrategy clones within the population Pðt,aÞ¼xðt,aÞ can change over time due to system dynamics. NðtÞ Consequently the mean value of the parameter Et[a] now becomes a function of time and changes over time as well. Intheupcomingchaptersweshowhowtoanalyzeaparametricallyheterogeneoussystem usingthe following steps: 1. Analyzetheautonomousparametricallyhomogeneoussystemtotheextentpossibleusing well-developed analytical tools, such as bifurcation analysis. 2. Replace parametera with its meanvalue Et[a],whichisa function of time. 3. Introduce anauxiliary system of differential equationsto define keystonevariables that determine the actual dynamics of the system. (Note:the term “keystone” isused here to parallelthe function of keystonespeciesin ecology. Just like keystone species have disproportionatelylargeeffectontheirenvironmentrelativetotheirabundance,keystone variablesdeterminethedirectioninwhichthesystemwillevolve,withoutbeingexplicitly present in theoriginal system.) 4. Express the distribution of thedistributed parameter through keystone variables. This transformation allows finding allstatistical characteristics of interest,including parameter’smeanandvariance,whichnowchangeovertimeduetosystemdynamics.The mean of the parameter can now “travel” through the different domainsof the phase- parameter portrait ofthe original parametrically homogeneous system. 5. Calculate numerical solutions. ExactformulationoftheReductiontheoremandthetheoryunderlyingtheHKVmethod willbefoundlaterinthebook.Asummarydefinitionsandassociatednotationareprovided in Table 1. TABLE1 DefinitionsandnotationusedintheapplicationoftheHKVmethod. Definition Notationandexplanation Selectionsystem Amathematicalmodelofaninhomogeneouspopulationinwhichevery individualischaracterizedbyavector-parametera¼(a1,…,an)thattakeson valuesfromset Clonex(t,a) Setofallindividualsthatarecharacterizedbyafixedvalueofparametera R TotalpopulationsizeN(t) NðtÞ¼ xðt,aÞda Growthrateofaclonex(t,a) dxðt,aÞ dt Fitnessofanindividualwithinthe dxðdtt,aÞ=xðt,aÞ population Distributionofcloneswithinthe Pðt,aÞ¼xNðtð,taÞÞ population R Expectedvalueofafunctionon Et½f(cid:2)¼ fðaÞxðt,aÞda distributedparameter NðtÞ 4 1. Usingmathematicalmodelingtoaskmeaningfulbiologicalquestions 1.3 Advantages and drawbacks of the Reduction theorem Oneofthemostimportantpropertiesofthismethodisthatitallowsreducinganotherwise many- oreven infinitely-dimensional system to low dimensionality. However,aswithanymethod,thismethodisnotuniversal.Mostimportantly,thetrans- formationcanbedone(withsomegeneralizations)onlytoKolmogorovtypeequationsofthe form x(t)’¼x(t)F(t,Et[f(a)]), where: (cid:129) x(t)isa vector, (cid:129) aisaparameteroravectorofparametersthatcharacterizeindividualheterogeneitywithin the population, (cid:129) Et[f(a)] is ofsystem-specific form. Reductiontheoremcanalsoincreasethedimensionalityoftheoriginalparametricallyho- mogeneoussystematapossiblecostofauxiliarykeystoneequations(althoughthesewould typicallybeuptoonlyoneortwoextraequations,dependingontheoriginalsystem).Finally, the resultingsystemistypically non-autonomous,soonecannotperform standardbifurca- tion analysis. Whenstudyingnumericalsolutionsofsuchparametricallyheterogeneoussystems,trajec- toriescanbeobservedthatcouldnotpreviouslyhavebeenseeninparametrically homoge- neoussystems.Thisphenomenonarisesfromtheexpectedvalueoftheparameter“traveling” throughthephaseparameter portrait,andthesystemundergoescorresponding qualitative phase transition as the parameter’s expected value crosses the bifurcation boundaries. Furthermore, if there exists a complete bifurcation diagram for the specific parametrically homogeneous model, the boundaries crossed during system evolution can be identified analytically. Classic techniques for analyzing dynamical systems, such as bifurcation theory (Kuznetsov, 2013), can provide critical insights into the possible dynamical regimes that a system can realize. Unfortunately, doing full bifurcation analysis is labor intensive and is not always possible due to increasing complexities of many models. However, a very rich body of literature exists of fully analyzed parametrically homogeneous models in many fields, including ecology (Bazykin, 1998; Berezovskaya et al., 2005), epidemiology (Brauer andCastillo-Chavez,2001),amongothers.Astheexamplespresentedthroughoutthisbook will demonstrate, even relatively simple two-dimensional systems can reveal rich, unex- pectedandmeaningfulbehaviors.ApplicationoftheHKV-methodtointroducepopulation heterogeneityinameaningfulwayandutilizingpreviouslyperformedanalysiscanreveala new layer of understandingof many existing modelsthat was not accessiblebefore. InChapter1ofthisbook,weintroducetheHKVmethodformodelingpopulationhetero- geneity. Chapter 2 shows how applications of these methods to some classical models can reveal new and unexpected dynamical behaviors. Chapter 3 demonstrates how the HKV- method can be applied to more complex biological systems, including models of world demography,microbialresistancetoantibiotics,andthedynamicsoftreestandself-thinning. In Chapter4,we go over morein-depth theory. Chapter5 features numerous examplesex- ploringsuchtopicsasevolutionofaltruism,competitionbetweentwoinhomogeneousequa- tions,aputsanewspinontheclassicalLotka-Volterrapredator-preymodel.Wealsodiscuss 5 1.3 AdvantagesanddrawbacksoftheReductiontheorem theFisher-Haldane-Wrightequation,theHaldaneprincipleforselectionsystems,andfinally Fisher’s fundamental theorem—the latter three topics we will return to again and again throughout the book. In Chapter 6, we discuss models of frequency versus density-dependent population growth,andhighlightsomekeydifferencesbetweenthem.Chapter7divesintothediscus- sion of inhomogeneous logistic and Gompertzian growth; we look at dynamics of distribu- tions in inhomogeneous models and discuss various types of Darwinian (“survival of the fittest”) and non-Darwinian (“survival of the common” and “survival of everybody”) selection. InChapter8,weintroducethePrincipleofminimalinformationgain,whereweshowthat it can be derived from system dynamics rather than being postulated a priori. Chapter 9 discussessub-exponentialsystemdynamicsandthePrincipleofminimalTsallisinformation gain.ThemainresultofthisChapteristhatthePrincipleofminimalinformationgainisthe underlying variationalprinciple that governs replicator dynamics. In Chapter 10, we discuss some philosophical issues on time perception and propose several hypotheses on how a model of inhomogeneous population extinction can be applied to time perception in a dying brain. Chapter 11 explores several seemingly similar models of population growth—logistic, Gompertz and Verhulst, among others—and shows that intrinsic population composition mayinfactbeverydifferentdependingonwhichmodeldescribeddatabest.Wethenapply this developed theory to cancer cell growth. InChapter12,weshowhowtheHKV-methodcanbeappliedtopreviouslyanalyzedpara- metricallyhomogeneoussystemstorevealthephenomenonoftheexpectedvalueofthedis- tributed parameter “traveling” through the phase-parameter portrait. This analysis can revealnew,complexandsometimesunexpecteddynamicalbehaviorsthathelpanswermany interestingandimportantquestions.Weshowcaseseveralexamplesofthisphenomenon,in- cludingthetragedyofthecommons,naturalselectionbetweenresourceallocationstrategies, and applicationof oncolyticvirustherapy to apopulation of heterogeneous cancer cells. Chapter13appliestheHKV-methodtogametheoryandlooksatdynamicsofselectionof strategieswithinasinglegame.InChapter14,welookatselectionbetweengames,andthen discusshowsomeoftheseinsightscanbeappliedtounderstandingthecomplexdynamicsof cancer cells in tumors. Finally, Chapter 15 (which can be read a stand-alone chapter), demonstrates how the HKV-method can beapplied to selection systems with discretetime(maps). Let usbegin.
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