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Fromepigeneticlandscapetophenotypicfitnesslandscape:evolutionaryeffectofpathogensonhosttraitsMarkJaysonV.Cortez1,JomarF.Rabajante1,*,JerroldM.Tubay1,ArielL.Babierra11InstituteofMathematicalSciencesandPhysics,UniversityofthePhilippinesLosBaños,College,Laguna4031,Philippines*Correspondingauthor:jfrabajante@up.edu.phAbstractTheepigeneticlandscapeillustrateshowcellsdifferentiateintodifferenttypesthroughthecontrolofgeneregulatorynetworks.Numerousstudieshaveinvestigatedepigeneticgeneregulationbuttherearelimitedstudiesonhowtheepigeneticlandscapeandthepresenceofpathogensinfluencetheevolutionofhosttraits.Hereweformulateamultistabledecision-switchmodelinvolvingmanypossiblephenotypeswiththeantagonisticinfluenceofparasitism.Asexpected,pathogenscandrivedominant(common)phenotypestobecomeinferior,suchasthroughnegativefrequency-dependentselection.Furthermore,novelpredictionsofourmodelshowthatparasitismcansteerthedynamicsofphenotypespecificationfrommultistableequilibriumconvergencetooscillations.Thisoscillatorybehaviorcouldexplainpathogen-mediatedepimutationsandexcessivephenotypicplasticity.TheRedQueendynamicsalsooccurincertainparameterspaceofthemodel,whichdemonstrateswinnerlesscyclicphenotype-switchinginhostsandinpathogens.Theresultsofoursimulationselucidatehowepigeneticlandscapeisassociatedwiththephenotypicfitnesslandscapeandhowparasitismfacilitatesnon-geneticphenotypicdiversity.Keywords:parasitism,disease,plasticity,inclusiveinheritance,biodiversity,mathematicalmodel

.CC-BY-NC-ND 4.0 International licensecertified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprint (which was notthis version posted May 5, 2016. . https://doi.org/10.1101/051904doi: bioRxiv preprint

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IntroductionThemechanismsofepigeneticsaremultifacetedandcomplex[Danchinetal.2011;Gomez-Diazetal.2012;Huang2013;Duncanetal.2014;Kilvitisetal.2014;Skinner2015].Inspiteofnumerousexperimentalandtheoreticalstudiesontheepigeneticlandscapeofphenotypespecification[Richards2008;Rohlfetal.2012;Huang2013;Kilvitisetal.2014;Rabajanteetal.2015],itsbehaviorisnotfullyunderstood,especiallyinthepresenceofpathogens(e.g.,parasites).Parasitism-mediatedalterationsinthedynamicsoftheepigeneticlandscapehavenotgainedhighattentionincomputationalsystemsbiology[Poulin&Thomas2008;Boyko&Kovalchuk2011;Bierneetal.2012;Gomez-Diaz2012].Particularly,mathematicalmodelsneedtoadvancetocapturetheimportantqualitativebehavioroftheepigeneticlandscapethatinvolvesmanyfactors,suchasdiseasesthataffectphenotypespecificationandfitnessofspecies[Raghavanetal.2010;Geoghegan&Spencer2011;Huang2012;Rabajante&Babierra2015;Rabajanteetal.2015].Theoreticalpredictionsthatemployepigeneticperspectivecanguideecological,agricultural,epidemiologicalandbiomedicalstudiesininvestigatingtheimplicationsofhost-pathogeninteractioninphenotypevariationandtraitheritability[Woolhouseetal.2002;Mallard&Wilkie2007;Poulin&Thomas2008;Kasuga&Gijzen2013;Rabajanteetal.2015].Itisplausibletoincorporateepigeneticstothestudyofparasite-inducedevolutionsincetheepigeneticpattern(epigenotype)ingeneexpressionisanon-geneticfactorthatcouldbepassed-onfromparentstooffspring[Pal&Miklos1999;Bond&Finnegan2007;Bonduriansky&Day2009;Petronis2010;Danchinetal.2011;Kilvitisetal.2014;Englishetal.2015].Epigeneticattractors.Mathematicalmodelsofepigeneticlandscapedescribecell-fatedetermination/specificationasaprocessconvergingtocellular‘attractors’(seeindividualepigeneticlandscapesinFig.1)[Cinquin&Demongeot2005;Huang2012;Furusawa&Kaneko2012;Huang2013;Rabajante&Babierra2015;Rabajante&Talaue2015].Stemcelldifferentiationisillustratedasabranchingprogressionfromtotipotencytovariouscelllineagestodifferentterminallyspecializedcelltypes[Furusawa&Kaneko2012;Huang2013;Rabajante&Babierra2015].Throughtheregulationofgeneinteraction,cells‘decide’wheretoconvergefromthemultipleattractorspresentintheepigeneticlandscape.Anattractorcanbeanyfateofthecell,suchasmetastable,terminallyspecialized,cancer,quiescent,senescenceandapoptoticstates[Furusawa&Kaneko2012;Huang2013;Li&Wang2014;Marcoetal.2014;Rabajante&Babierra2015;Rabajanteetal.2015].Metastablestatescouldbetotipotent,pluripotent,multipotentorprogenitor.Theattractorsthatcharacterizeterminallyspecializedcellsrepresentdifferentphenotypicfates(e.g.,muscle,skin,blood,neuron,bone,fat).Moreover,cellsmaynottrackaone-directionallinearpathwayintheepigeneticlandscape,thatis,thepathwaycanbemultidirectionalorcircular[Enveretal.2009;Furusawa&Kaneko2012;Rabajante&Babierra2015].Cellsaresaidtobeplasticandcanbedriventoundergodedifferentiation(cellsregresstoearlierstate,e.g.,fromspecializedcellsbacktopluripotentstate)andtransdifferentiation(cellstransfertootherlineages,


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e.g.,frommesenchymaltoneurallineage)[Joplingetal.2011;Xuetal.2014].Considerabledegreeofbiologicalnoiseandinductionbyexternalstimulimayalsoaffectthedirectionofthedevelopingcells[Rabajante&Babierra2015;Rabajante&Talaue2015].

Theconceptofcellularattractorisevolvingandhasvariousinterpretations.However,ageneralruleforbeinganattractoristhatthetrajectoryofdevelopingcellsintheepigeneticlandscapefromaneighborhoodofinitialconditions(basinofattraction)shouldconvergetothisattractorstate[Huang2012].Anyperturbationwithinthebasinofattractionshouldnotresultininstability,thatis,thestateisresilientandhomeostatic[Huang2013].Mathematically,theattractorsarerepresentedbystableequilibriumpoints[Huang2013;Rabajante&Babierra2015],stablelimitcycles[Rabajante&Babierra2015],strangechaoticattractors[Furusawa&Kaneko2012]ornoisyattractors[Huang2009].Eachattractorhasitsownbasinofattraction,whichcanbeofdifferentsizes.Moreover,theparametervaluesinthemathematicalmodelsaredictatedbythegenesandtheirinteraction.Geneinteractionisnotstaticbutratherdynamic[Rabajante&Babierra2015].Modificationintheparametervaluesmayormaynotpreservethestabilityofanattractor.Sometimesthemodificationsresultinbifurcationthattransformsthetopographyoftheepigeneticlandscapetoformanewattractororobliterateanexistingattractor[Huang2013;Rabajante&Talaue2015].Dynamicallychangingaparametervaluecouldleadcellstochangefatesbymovingtheirtrajectoryfromonebasinofattractiontoanotherbasinofattraction[Rabajante&Talaue2015].Host-parasitecoevolution.Theinteractionandcoevolutionbetweenhostsandparasitesarewidelyinvestigated.Inevolutionarybiology,armsracecompetitionandtheRedQueendynamicshavebeenhypothesizedtocausecoadaptation/coevolutioninhostsandpathogens[Avranietal.2012;Brockhurstetal.2014;Rabergetal.2014;Rabajanteetal.2015;Vojeetal.2015].Forthehosttosurviveparasitism,itincreasesitsdefensetraits;butforthepathogenthatreliesheavilyonthehost,itneedstocounteractthehostdefenseandincreaseitspathogenicity.Thismayresultinawinnerlessarmsracecompetition[Brockhurstetal.2014;Rabajanteetal.2015].Inseveralcaseswheretherearemultipletypesofhosts,parasitismcandecreasetheabundanceofthecommonhosttypepermittingararetypetobecomethenewdominant(negativefrequency-dependentselection).Commonhosttypes,especiallymonocultures,aremoresusceptibletotheattackofpathogens.Thisphenomenonisakintothe‘killingthewinner’hypothesis[Avranietal.2012;Rabajanteetal.2015].Host-pathogeninteractionisoneoftheexplanationsfortheextinctionanddiversityofspecies[Miuraetal.2006;Brockhurstetal.2014].

Ifnegativefrequency-dependentselectionpersists,itcouldresultinfluctuatingRedQueendynamicswhichisillustratedbycyclicphenotype-switching[Brockhurstetal.2014;Rabajanteetal.2015].ThefluctuatingRedQueendynamicsisacandidatemodelofrecurrentpunctuatedequilibriumprocessthatisdrivenbybioticinteraction[Rabajanteetal.2015].Thisrecurrentpunctuatedequilibriumprocesshasout-of-phase‘heteroclinic’cyclesofrapidnegativefrequency-


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dependentselection,followedbystasis,thennegativefrequency-dependentselectionagain.SimulationsshowthattheRedQueendynamicsarerobustagainstcertainlevelofnoisebuttheorderingofcyclicphenotype-switchingcouldvary,resultinginadiversifiedpossibilitiesofevolutionarypatterns[Rabajanteetal.2015].

Bridgingthegap.Phenotypicvariationisaffectedbyinheritedandnon-inheritedbiologicalinformation[Danchinetal.2011;Day&Bonduriansky2011;Englishetal.2015;Skinner2015;Soubry2015].Inheritedinformation,whichisinfluencedbygeneticandnon-genetic(e.g.,epigenetic)factors,isimportantinevolution[Danchinetal.2011;Richardsetal.2012;Klironomosetal.2013;Smith&Ritchie2013;Nishikawa&Kinjo2014;Santosetal.2015].Thatis,genotype-by-epigenotype-by-environmentregulatestransmissionofphenotypesfromonegenerationtoanother(inclusiveheritability)[Danchinetal.2011;Gervasietal.2015].Hereweaimtocontributeinbridgingagapbetweenthestudyofepigeneticsandevolutionarybiologyusingamathematicalmodel,specificallybylinkingtheepigeneticlandscapeattheindividualleveltothephenotypicfitnesslandscapeatthepopulationlevel(Fig.1).Thereareearlydiscussionsabouttheconnectionbetweenthedynamicsoftheepigeneticlandscapeandfitnesslandscape,suchasinthecaseoftumorprogression[Huang2013],buthereweconsidertheeffectofparasitismasamajordriverofphenotypicevolution.InFigure2,thegenotype-by-epigenotypeinteractionisrepresentedbyaphenotypedecision-switchnetworkwhereparasitismactsastheinfluencingenvironmentalfactor.Inourmathematicalmodel(seeMethods),weassumethatphenotypespecificationisdeterminedexclusivelybyepigeneticmechanismswheregenotypesarestable,i.e.,thestructureofgeneregulatorynetworkisfixed.Thisassumptionallowsustodecoupletheeffectofgeneticandepigeneticfactorsinthegenotype-by-epigenotypeinteraction,andfocusourobservationontheeffectsofparasitism-modifiedepigenotypesonhosttraits.ResultsandDiscussionAtthecellularlevel,cellsdecidefrommultiplechoicesofphenotypes(epigeneticlandscapeinFig.1).Inthepresenceofintenseevolutionarypressures,suchaspresenceofpathogens,geneticornon-geneticselectionmayoccur[Danchinetal.2011;Gomez-Diazetal.2012;Duncanetal.2014;Kilvitisetal.2014;Skinner2015;Rabajanteetal.2015].Ourdiscussionfocusesonnon-geneticselectiondrivenbyepigeneticmechanismswiththeinfluenceofpathogens.Ifaphenotypeisadvantageousandthereisadequateepigeneticmemory[Pal&Miklos1999;Bond&Finnegan2007;Geoghegan&Spencer2012;Gomez-Diazetal.2012;D’Urso&Brickner2014;Englishetal.2015],apopulationofspeciesmayselectthisphenotyperesultinginhigherphenotypicfitness(Fig.1).However,thisselectionprocessandaphenotypebecomingprevalentwithinapopulationarenotstraightforwardandstrictconditionsarenecessary[Takahashietal.2010;Vermeij&Roopnarine2013;Rabajanteetal.2015].Forexample,parasitismresultinginnegativefrequency-dependentselectionrequiresspecificstructureoffunctionalresponseandcombinationofparametervalues(e.g.,Figs.3and4)[Rabajanteetal.


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2015].Thecharacteristicsofthehosts(e.g.,growthrateandcompetitiveability)andpathogens(e.g.,pathogenicity,specificityanddeathrate)dictatethephenotypicoutcome.

Questionsariseonwhetheritistheepigeneticsstructurethatcausespathogenesisorwhetherthepathogensaretheonescausingepigeneticchanges[Gomez-Diazetal.2012].Wearguethatthesetwoquestionsarevalidsincetheepigenomeofhostsandthemechanismsofpathogensarecomponentsofaninteractingsystemwhichispossiblyinfluencedbyfeedbackloops(Fig.2).Particularparametersoftheepigeneticmodel(Eq.1inMethods)mayormaynotallowdiseasetoaffecttheoutcomeofgeneregulation.Onanotherside,theparameters(e.g.,pathogenicityandspecificity)andfunctionalresponseintheparasitismmodel(Eq.2inMethods)dictatetheseverityofdiseasethatcanalterthetopographyoftheepigeneticlandscape.Equilibriumdynamics.Asexpectedinmanycases,coexistenceofallphenotypes(sometimeswithequalfrequencies)andcompetitiveexclusionoccur(Fig.4aand4b).Coexistencerepresentedbyequilibriumpointwithallcomponentshavingequalvaluesmayhavelimitedrangeofinitialvalueandparameterspace,thatis,someofthephenotypescouldbecomeinferiorwhenconditionsarechanged.Incompetitiveexclusion,therearedominantphenotypesandrareorextinctphenotypesinthelongrun.Competitiveexclusionillustratessilencingorknockdownofsomegenes,andactivationorsometimesoverexpressionoftheothergenes.Insomecases,hostsarenotabletoregaingrowth,resultingintotalsilencing/extinction(Fig.4c).Thecoexistence,competitiveexclusionandtotalextinctionofphenotypesareexpectedinawiderangeofparameterspacebecausethefundamentalepigeneticmodel(Eq.1inMethods)describesamultistablesystem[Mostowyetal.2012;Rabajante&Talaue2015].

Multistabilitypertainstotheexistenceofmanystableequilibriumpoints.

Heremultistabilityimpliesnon-geneticvariation.Themeaningofnon-geneticvariationdrivenbyepigeneticmechanismsisthattwocellshaveidenticalDNAsequences(samesetofgenes)buttheydifferonwhatgenesareexpressed(e.g.,activatedorsilencedduringtranscription)resultingindifferentphenotypes.Forexample,skincellsandmusclecellsinanorganismhavesimilarsetofgenesbecausetheycamefromonemothercellbuttheyhavedifferentexpressedgenesthatdefinetheproteinsspecificforskincellsandspecificformusclecells.Thisideacanbeextendedtopopulationepigeneticswherenon-geneticvariationimpliesphenotypicheterogeneitynotexplainedbygenetics[Schlichting&Smith2002;Johnson&Tricker2010;Hughes2012;Duncanetal.2014;Kilvitisetal.2014].Non-geneticphenotypicheterogeneityisoneofthesurvivalstrategiesofspeciesagainstenvironmentalstress[Donaldson-Matascietal.2008;Soen2014].Forexample,someintroducedspecieswithlowgeneticvariationcanstillbecomeinvasivebecausetheycanhavehighphenotypicdiversityduetoepigeneticvariation[Kilvitisetal.2014].


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Oscillatorydynamics.Oscillationsinphenotypefrequenciesarealsopossibletoariseandtheymaycharacterizenon-equilibriumpolymorphism[Takahashietal.2010;Mostowyetal.2012]andnon-equilibriumphenotypicdiversity(Figs.4dand4e).Infact,oscillatorybehaviorisacharacteristicofstemcellsandprogenitorcells,whichimpliesthatoscillationshavehighpotentialtogeneratedifferentcelltypes[Furusawa&Kaneko2012;Rabajante&Babierra2015].Inpopulationbiologypoint-of-view,iftheminimumvalueoftheoscillationsarefarfromtheedgeofextinction,thenoscillationsexhibitpermanentcoexistence[Huisman&Weissing2001].Iftheminimumvalueoftheoscillationsareneartheedgeofextinction,thenoscillations(Fig.4e;e.g.,theRedQueendynamics)mayleadtoimpermanentcoexistence[Huisman&Weissing2001].Itiscalledimpermanentcoexistencebecauserandomperturbationscoulddrivethepopulationfrequencyofaspeciestozero(guaranteedextinction);however,therearecaseswhereRedQueendynamicsarerobustagainstrandomnoise[Rabajanteetal.2015].

Inoursimulation,fluctuatingnegativefrequency-dependentselection,specificallytheRedQueendynamics,happensinthemodelwhereeachhostphenotypehasanassociatedspecificpathogen(Fig.4e).Thishighpathogenspecificitycharacterizesahost-pathogeninteractionwithtightmatchingofhostandpathogentraits,asinthematching-allelesmodel[Weitzetal.2012;Rabajanteetal.2015].IntheRedQueendynamicsoccurringinourmodel(Fig.4e),theoccurrenceofcyclesthatareout-of-phaseandhaveidenticalamplitudeisconsistentwiththedefinitionofthecanonicalRedQueendynamics[Brockhurstetal.2014;Rabajanteetal.2015;Rabajanteetal.2016].ThecanonicalRedQueendynamicsstatethattheaveragefitnessofspeciesstaysthesame(representedbyidenticalamplitude)eventhoughthespeciesundergocontinuousevolution(representedbyout-of-phasecycles).Theout-of-phasecyclesimplythatonlyonespeciestypeisdominantforacertainperiodoftimeandtherestarerare;butaftersometime,thedominanttypeisreplacedbyarareone(cyclicphenotype-switching).NoticethattheRedQueendynamicsdonotonlyincludecyclicphenotype-switchinginhostsbutalsocyclicphenotype-switchinginpathogens(Fig.4e).SuchRedQueendynamicsisoneofthemanymanifestationsoftheRedQueenhypothesis[Avranietal.2012;Brockhurstetal.2014].ThereareseveralempiricalevidencesbasedonexperimentsandfossilrecordssupportingtheRedQueenhypothesis[Brockhurstetal.2014;Rabergetal.2014;Vojeetal.2015].Investigatingcycliccoevolutionarydynamicscouldalsohelpindesigningstrategiestomanagedrugresistance(forexample,seeFig.17in[Miraetal.2015]).

Inthepoint-of-viewofcellanddevelopmentalbiology,oscillationsliketheRedQueendynamicsmayrepresentexcessivetemporalplasticityofcells.Thisexcessiveplasticityisoneofthecharacteristicsofamutator(epimutator)phenotypethatmayleadtophenotypicdiversityortotumorheterogeneityincancer[Shackletonetal.2009;Marusyk&Polyak2010;Loeb2011;Peltomaki2012;Swanton2012].Oscillationsthatcharacterizeamutatorphenotypeoccurduetoaberrationinthegeneregulatorynetwork[Loeb2011;Rabajante&Babierra2015].Herewehaveshownthatpathogens(e.g.,oncovirus)canalsocausetheoccurrence


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ofmutatorphenotype[Liao2006],especiallywhenthereiswinnerlesscoevolutionbetweenhostimmunesystemandthepathogens(Fig.4e).Abnormaldampeningoftheseoscillationscouldleadtoirreversiblegenemisexpression.Inpopulationbiologypoint-of-view,theRedQueendynamicsshowpathogen-mediatedtemporaldiversityinhostandpathogenspecies.However,thenever-endingcyclesintheRedQueendynamicsmaynotproceedindefinitelyinecologicalsystems.Thisisbecausehost-pathogeninteractionsarefiniteactivities[Avranietal.2012;Brockhurstetal.2014;Vojeetal.2015].Soonerorlater,somehostandparasitetypesmayescapethearmsracecompetition.Severalspeciesmaylikewisebecomelosersinthearmsracecompetition(e.g.,speciesextinction)becausetheywerenotabletocopeupwiththecoevolutionaryprocess.Epigeneticinheritance.Oneoftheimportantcomponentsforbringingtheoutcomesofepigeneticlandscapetotheleveloffitnesslandscapeisepigeneticmemory,whichischaracterizedbyepigeneticmarks.Epigeneticmemoryisrequiredinnon-geneticinheritanceofphenotypesfromcelltocellorfromorganismtoorganism(transgenerational)[Pal&Miklos1999;Bond&Finnegan2007;Pilu2011;Geoghegan&Spencer2012;D’Urso&Brickner2014;Englishetal.2015].Muchoftheepigeneticmemoriesareresetinthegermlinebutsomeareconservedandimpartedtooffspring[Rakyanetal.2001;Molinieretal.2006;Bonduriansky&Day2009;Robertson&Richards2015].Transgenerationalinheritancehappenindifferenttimescales[Rando&Verstrepen2007;Rabajante&Babierra2015].Insomecases,epigeneticchangescouldleadtorapidphenotypevariationinpopulationsbutepigeneticmarkscouldalsodecayinfewgenerations[Rando&Verstrepen2007;Geoghegan&Spencer2012].

Inourdecision-switchmodel,self-stimulation(positivefeedback;Fig.2)and

multistabilitysupportepigeneticmemory[Cinquin&Demongeot2005;Kaufmannetal.2007;Lim&vanOudenaarden2007;Chengetal.2008;Smitsetal.2008;Hughes2012].Self-stimulationallowstheexpressionofaphenotypetobefitterthroughtime.Inamultistablesystem,eachphenotypeattractorhastheirownbasinofattraction.Theprobabilityofaphenotypebeingheritableishigherif(i)thephenotype’sbasinofattractionislarge,thatis,itisstableforawiderangeofinitialconditions,and(ii)thephenotypeattractorisstructurallystable,thatis,itisstableforawiderangeofparametervalues.Iftheinitialconditionandparametervaluesareresetduringmeiosis,epigeneticmarkscanescaperesettingifthenewinitialconditionandparametervaluesstillfallwithinthestabilityrangeofthephenotypeattractor.Largebasinofattractionandstructuralstabilityalsoproviderobustnessagainstenvironmentalnoise.Nevertheless,furtherstudiesareneededtomaptheextentofopportunitiesandlimitationsofepigeneticinheritance,especiallyonhowitaffectsthegeneticsandevolutionofpopulations[Crispetal.2016].Futureprospects.Empiricalinvestigationsontheinterfacebetweenpathogens,epigeneticsandevolutionareinprogress[vanNhieu&Arbibe2009;Cosseauetal.2010;Thomasetal.2011;Gomez-Diazetal.2012;Brockhurst&Koskella2013].Diversityofperspectivesisusefulbutaunifiedperspectiveinvolvingdifferent


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fields,suchasepigenetics,parasitology,pathology,epidemiologyandevolutionarybiology,willhelpbetterunderstandandsolveproblemsaboutdiseases[Gomez-Diazetal.2012;Noble2015;Skinner2015].Ourtheoreticalpredictionscouldaidongoingandfuturestudiesonidentifyingthenon-geneticeffectandmechanismsofdiseasesfromthecellulartopopulationlevel.Bacteria-phageinteractioninmarineecosystems,plant-pathogeninteractioninagriculturalsystems,drugdynamicsinsynthetic/artificialorganisms,andepigenetic-baseddiseasesininsectsaresomeoftheplausiblesubjectsforexperimentalvalidation[Takken&Rep2010;Koskella&Brockhurst2014;Williams2013;Gijzenetal.2014;Mukherjeeetal.2015;Cinietal.2015;Trozzet&Carbonell2015].Inthispaper,wehaveemployedaminimalmodeltostudytheexpressionofphenotypesincellsandpopulations.Otherscenarioscanbeexploredfurther,suchasthedynamicsalongthecontinuumbetweenmatching-allelesandgene-for-genesystems[Agrawal&Lively2002;Weitzetal.2012;Rabajanteetal.2015],theeffectofthenumberofhostsandpathogens[Woolhouseetal.2002;Rabajanteetal.2015],theeffectofdifferentinfectionnetworks(e.g.,random,modular,nested)[Weitzetal.2012],andtheinterplayamongmanygenetic,epigeneticandenvironmentalfactors[Priceetal.2003;Mostowy&Engelstadter2011;Soen2014].Theepigeneticfactorscouldincludeindetailthedifferentmechanismsthataltergeneexpression,suchasDNAmethylation,chromatinremodeling,histonemodificationsandRNAinterference[Rakyanetal.2001;Bonduriansky&Day2009;Rohlfetal.2012;Duncanetal.2014;Kilvitisetal.2014;Brazel&Vernimmen2016].MethodsDefinitionofvariablesandparameters.Supposewehavemvarietiesofhostphenotypesandntypesofpathogens/parasites.Therearetwostatevariablesinthemodel,HiandPj.WecancomputeforF(Hi)thatdenotesthepopulationfrequencyofhostphenotypei,whereF(Hi)=Hi/NH,NH=∑kHk(k=1,2,…,m).F(Pj)denotesthepopulationfrequencyofpathogenstrainj,whereF(Pj)=Pj/NP,NP=∑lPl(l=1,2,…,n).Populationfrequencymayalsobeinterpretedastheprobabilitythataphenotypewillbeexpressed.Forthedefinitionofparameters,seeTable1.Allstatevariablesandparametersarenon-negative.Cellulardecision-makingmodel.Thefollowingepigeneticmodel(Eq.1)describesthequalitativedynamicsofmultistabledecisionswitchesincell-fatedetermination,basedonthegeneregulatorynetworkinFig.2.EachnodeinFig.2representsasetofgeneregulatoryfactorsinvolvedinexpressingaspecificphenotype.Thenodeshavemutualrepressionbecauseitispresumedthatamaturespecializedcellexpressesauniquephenotypeandinhibitstheexpressionoftheotherphenotypes.Forfurtherdetailsaboutthismodelanddefinitionofvariablesandparametersinepigeneticspoint-of-view,referto[Cinquin&Demongeot2005;Macarthuretal.2008;Andrecut2011;Rabajante&Babierra2015;Rabajante&Talaue2015](notethattherearesomedifferencesinthesymbolsused).


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iiii

ik

ckik

cii

ciii gHK

HHHr

dtdH

iki

i

+−++

=∑≠

ργθ

(1)

i,k=1,2,…,m. Frequency-dependentselection.Withproperdefinitionofvariablesandreparametrization,weassumesimilardecision-makingmechanism(Eq.1)happensduringevolutionaryselectionofaphenotype.Ifaphenotypeisadvantageous,apopulationofspeciesmayselectthisphenotyperesultinginhigherphenotypicfitness(characterizedbyahigherphenotypefrequency)comparedtotheotherphenotypes.ThismodelissimilartotheLotka-Volterracompetitionmodelbutinvolvinganon-polynomialfunctionthatdescribesmultistabilityandself-stimulation(auto-activation/auto-catalysis)[Rabajante&Talaue2015].Multistabilityandself-stimulationsupportsthepreservationofepigeneticmemory.

Thebattleforphenotypedominanceisakintointer-hostcompetition.Anincreaseininter-hostcompetition( γ ikHk

cik

k≠i∑ inthedenominatorofequation(1))

reducestheneteffectofthehostgrowthratecoefficientri.Thevalue(riKi+gi)/ρiisanupperboundoftheequilibriumvalueofHi.TheparameterKiaffectstheupperbound,whichwecaninterpretasafactorinfluencingthecarryingcapacityallocatedtoeachhosttype.Hereweassumetheallocationisfixed.ThetotalcarryingcapacityisK=∑k(rkKk+gk)/ρk.Modelofhost-pathogeninteraction.Tomodelhost-pathogeninteraction,wecoupleequation1withpathogendynamics.Weassumethatthepathogendynamicsfollowaparasitismmodel,asfollows[Smoutetal.2010;Joveretal.2013;Rabajanteetal.2015;Rabajanteetal.2016]:

iij

jijii

ik

ckik

cii

1ciii gHPfK

HHHr

dtdH

iki

i

+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−++

= ∑∑≠

−

ργθ

(2)

dPjdt

= ϕij fijHi − dji∑#

$%

&

'(Pj (3)

where

fij =αijHi

qi−1

δ φi + βikjHkqi

k∑

#

$%

&

'(

(4)

i,k=1,2,…,m ; j=1,2,…,n.


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ThepopulationfrequencyofhostphenotypeisdecreasedbyparasitismquantifiedbythefunctionalresponsefijHi.Thegrowthofpathogensdependsonhostutilization(numericalresponse ϕij fijHi

i∑ )withdeathratedj.Equation(4)defines

thepathogenfunctionalresponse.Ifqi=1andδ =1

φi + βikjHkqi

k∑

thenthefunctional

responsecurveisoftype-I(linear).Ifqi=1andδ=1thenthefunctionalresponsecurveisoftype-II(hyperbolic).Ifqi=2andδ=1thenthefunctionalresponsecurveisoftype-III(sigmoidal).

Inoursimulations,weareprimarilyconcernedwiththedynamicsofthe

epigeneticlandscape.Weassumeafixedsetofgenotypes(fixedstructureofgeneregulatorynetwork).Thepredictedphenotypevariationduetoepigeneticfactorsundertheinfluenceofparasitismcouldbeamplifiedifmodificationsingeneticfactorsareincorporatedinourmodel.References1. AgrawalA,LivelyCM.2002Infectiongenetics:gene-for-geneversus

matchingallelesmodelsandallpointsinbetween.Evol.Ecol.Res.4,79–90.

2. AndrecutM.2011Monte-Carlosimulationofamulti-dimensionalswitch-likemodelofstemcelldifferentiation.InApplicationsofMonteCarloMethodsinBiology,MedicineandOtherFieldsofScience(edCJMode),pp.25-40.Rijeka,Croatia:InTech.(doi:10.5772/634)

3. AvraniS,SchwartzDA,LindellD.2012Virus-hostswingingpartyintheoceans:

incorporatingbiologicalcomplexityintoparadigmsofantagonisticcoexistence.MobileGeneticElements2(2),88-95.(doi:10.4161/mge.20031)

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5. BondDM,FinneganEJ.2007Passingthemessageon:inheritanceofepigenetictraits.TrendsPlantSci.12(5),211-216.(doi:10.1016/j.tplants.2007.03.010)

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AcknowledgmentWewouldliketothankJuanchoA.ColleraofUPBaguioandEdithaC.JoseofUPLBfordiscussionaboutthemodelanditsanalysis.AuthorcontributionsMJVCranthesimulationsandproducedtheresults.JFRconceivedthestudyandbuiltthemodel.JMTcreatedthefigures.JFR,ALB,MJVCandJMTwrotethemanuscript.CompetingfinancialinterestsTheauthorsdeclarenocompetingfinancialinterests.FiguresFig.1.Thelinkbetweentheepigeneticlandscapeandfitnesslandscape.Inourcase,naturalselection(e.g.,negativefrequency-dependentselection)isinducedbyenvironmental/externalfactors,suchasparasitism[Poulin&Thomas2008;Bonduriansky&Day2009;Boyko&Kovalchuk2011;Gomez-Diazetal.2012;Kasuga&Gijzen2013;Soen2014].Weassumethatthecollectivedynamicsoftheindividualepigeneticlandscapesarereflectedinthephenotypicfitnesslandscapeatthepopulationlevelthroughepigeneticinheritance(becauseofepigeneticmemory)andparasitism-inducedselection.Ifaphenotypeisadvantageousandthereisadequateepigeneticmemory,apopulationofspeciesmayselectthisphenotyperesultinginhigherphenotypicfitness.


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Fig.2.Thephenotypedecision-switchnetwork.Thequalitativedynamicsofthegeneregulatorynetworkthatdeterminestheselectedphenotypesfollowthisdecision-switchnetwork[Cinquin&Demongeot2005;Macarthuretal.2008;Rabajante&Babierra2015].Supposethereare5phenotypes(nodes).Attheindividualepigeneticslevel,thenodeshavemutualrepressionbecauseitispresumedthataspecializedcellexpressesauniquephenotypeandinhibitstheexpressionoftheotherphenotypes.Theredpositivefeedbackarrowrepresentsself-stimulation.Forsimplicity,weassumethatthestructureofgeneregulatorynetworkisfixed,i.e.,fixedparametervalues(seeearlystudies[Rabajante&Babierra2015;Rabajante&Talaue2015]forthesimulationwithdynamicgeneregulatorynetworkbutwithoutparasitism).Correspondingly,weassumethattheaveragecollectivedynamicsatthepopulationlevelalsoadheretosuchdecision-switchnetwork.Notethatthefrequencyofphenotypeisacontinuumfrom0to1.Thefrequencyofaphenotypeisnegativelyinfluencedbyparasitism.Fig.3.Effectofparasitismonhostpopulation.(a)Hostpopulationdynamicswithoutparasitism.(b-d)Hostpopulationdynamicswithparasitismusingdifferentfunctionalresponse.HerehostpopulationdensityisdefinedasHi/K,whereKisthetotalcarryingcapacity.Forcomparison,wealsoassumethatparasitepopulationdensityisdefinedasPj/K.Fig.4.Differentparasitism-inducedpopulationdynamics.HerehostpopulationdensityisdefinedasHi/K,whereKisthetotalcarryingcapacity.Forcomparison,wealsoassumethatparasitepopulationdensityisdefinedasPj/K.(a)Equilibrium-convergenceshowingcoexistenceofallhostandparasitetypes.(b)Equilibrium-convergenceshowingcompetitiveexclusion.(c)Extinctionofallhostandparasitetypes.(d)Parasitism-inducedoscillations.(e)RedQueendynamics,aspecialtypeofoscillatorydynamics.TheRedQueendynamicsisdescribedbyatimeseriesofever-changingdominantphenotypefrequenciescharacterizedbyout-of-phasecyclesandidenticalamplitude[Rabajanteetal.2015;Rabajanteetal.2016].Thecyclesarepathogen-induced,thatis,acommonhosttypewithhighfrequencyisinfectedbypathogensresultingintheeventualdecreaseinfrequency.Asthefrequencyofthecommonhosttypedecreases,ararehosttypeisgiventheopportunitytoincreaseitsfrequencyandbecomethenewdominanttyperesultinginwinnerlesscycles.Thewinnerlesscyclesalsohappeninthepathogentypesbecausepathogenstracktheirspecifichosts.


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TableTable1.Definitionofparameters.Parameter Definition

ri coefficientassociatedwiththegrowthofhosttypei,ortheefficiencyinexpressingphenotypei

Ki factorinfluencingthecarryingcapacityallocatedtohosttypei

ρi deathrateofhosttypeinotduetopathogens,orthedecayrateoftheexpressionofphenotypei

ciexponentaffectingthecurveofself-stimulationofhostphenotypei;hyperbolicifci=1,sigmoidalifci>1.Forsimplicity,assumeciisinteger-valued.

cikexponentaffectingthenonlinearrepressionbehaviorofhostphenotypek≠iagainsthostphenotypei.Forsimplicity,wecanassumecik=ciforalli,k.

θinon-zerothresholdconstant;ifallHk≠i=0,then riHi

ci

θi +HiciKi =

riKi

2when

Hi =θi1ci

ϒik interactioncoefficientassociatedwiththerepressionofhostphenotypeibyhostphenotypek≠i

gibasalorconstitutivegrowthrate(sometimescalledthe‘leak’term),i.e.,theexpressionofhostphenotypeistillincreasesatthisrateatHi=0

αij strengthofpathogenicityorinfectivityofpathogenstrainjagainsthosttypei

φij hosti-to-pathogenjconversionratethatdefinesthegrowthrateofpathogenpopulation

dj deathrateofpathogenstrainj

qiexponentdefiningtheshapeofthefunctionalresponsecurve.Forsimplicity,wecanassumeallhost-pathogenrelationshipshavethesamefunctionalresponsetype.

ϕi non-zerothresholdconstantofthefunctionalresponsecurve

βikj

coefficientrepresentingtheenergyallotmentofpathogenstrainjtodifferenthosttypes;ifβikj=0,i≠k,thenthepopulationofhosttypekisnegligibleinalteringthefunctionalresponsecurveforinfectinghosttypei

SupplementarymaterialParametervaluesusedinthesimulations


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Fig.1

Fig.2


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Fig.3


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Fig.4a


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Fig.4b


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Fig.4c


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Fig.4d


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Fig.4e


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