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1、DynamicsandbifurcationsinasimplequasispeciesmodeloftumorigenesisVanessaCastillo1,J.Tom´asL´azaro1,∗JosepSardany´es2,3,∗,1.UniversitatPolit`ecnicadeCatalunya,Barcelona,Spain2.ICREA-ComplexSystemsLab,DepartmentofExperimentalandHealthSciences,UniversitatPompeuFabra
2、,Dr.Aiguader88,08003Barcelona,Spain3.InstitutdeBiologiaEvolutiva(CSIC-UniversitatPompeuFabra),PasseigMaritimdelaBarceloneta37,08003Barcelona,Spain∗Correspondingauthors.E-mail:jose.tomas.lazaro@upc.edu;josep.sardanes@upf.edu.AbstractCancerisacomplexdiseaseandthus
3、iscomplicatedtomodel.However,simplemodelsthatdescribethemainprocessesinvolvedintumoraldynamics,e.g.,competitionandmutation,cangiveuscluesaboutcancerbehaviour,atleastqualitatively,alsoallowingustomakepredictions.Hereweanalyzeasimplifiedquasispeciesmathematicalmode
4、lgivenbydifferentialequationsdescribingthetimebehaviouroftumorcellspopulationswithdifferentlevelsofgenomicinstability.Wefindtheequilibriumpoints,alsocharacterizingtheirstabilityandbifurcationsfocusingonreplicationandmutationrates.Weidentifyatranscriticalbifurcati
5、onatincreasingmutationratesofthetumorcellspopulation.Suchabifurcationinvolvesanscenariowithdominanceofhealthycellsandimpairmentoftumorpopulations.Finally,wecharacterizethetransienttimesforthisscenario,showingthataslightincreasebeyondthecriticalmutationratemaybee
6、noughtohaveafastresponsetowardsthedesiredstate(i.e.,lowtumorpopulations)duringdirectedmutagenictherapies.1IntroductionCancerprogressioniscommonlyviewedasacellularmicroevolutionaryprocess[1,2].Genomicinstability,whichseemstobeacommontraitinmosttypesofcancer[3],is
7、akeyfactorresponsiblefortumorprogressionsinceallowsaDarwinianexploratoryprocessrequiredtoovercomeselectionbarriers.Bydisplayingeitherhighlevelsofmutationorchromosomalaberrations,cancercellscangenerateaprogenyofhighlydiversephenotypesabletoevadesuchbarriers[4].Ge
8、nomicinstabilityreferstoanincreasedtendencyofalterationsinthegenomeduringthelifecycleofcells.Normalcellsdisplayaverylowrateofmutation(1.4×10−10changespernucleotideand
