Evolution in random fitness landscapes the infinit

arXiv:0711.1989v2[q-bio.PE]18Dec2007Evolutioninrandomfitnesslandscapes:theinfinitesitesmodelSu-ChanParkandJoachimKrugInstitutf¨urTheoretischePhysik,Universit¨atzuK¨oln,Z¨ulpicherStr.77,50937K¨oln,GermanyE-mail:[e-mail]@thp.uni-koeln.deAbstract.Weconsidertheevolutionofanasexuallyreproducingpopulationinanuncorrelatedrandomfitnesslandscapeinthelimitofinfinitegenomesize,whichimpliesthateachmutationgeneratesanewfitnessvaluedrawnfromaprobabilitydistributiong(w).ThisisthefinitepopulationversionofKingman’shouseofcardsmodel[J.F.C.Kingman,J.Appl.Probab.15,1(1978)].IncontrasttoKingman’swork,thefocushereisonunboundeddistributionsg(w)whichleadtoanindefinitegrowthofthepopulationfitness.ThemodelissolvedanalyticallyinthelimitofinfinitepopulationsizeN→∞andsimulatednumericallyforfiniteN.Whenthegenome-widemutationprobabilityUissmall,thelongtimebehaviorofthemodelreducestoapointprocessoffixationevents,whichisreferredtoasadilutedrecordprocess(DRP).TheDRPissimilartothestandardrecordprocessexceptthatanewrecordcandidate(anumberthatexceedsallpreviousentriesinthesequence)isacceptedonlywithacertainprobabilitythatdependsonthevaluesofthecurrentrecordandthecandidate.WedevelopasystematicanalyticapproximationschemefortheDRP.AtfiniteUthefitnessfrequencydistributionofthepopulationdecomposesintoastationarypartduetomutationsandatravelingwavecomponentduetoselection,whichisshowntoimplyareductionofthemeanfitnessbyafactorof1−UcomparedtotheU→0limit.Evolutioninrandomfitnesslandscapes:theinfinitesitesmodel21.IntroductionAfruitfulexchangeofconceptsandmethodshastakenplacebetweenevolutionarypopulationbiologyandthestatisticalphysicsofdisorderedsystemsoverthepastseveraldecades[1,2,3].Onthemostbasiclevel,oneimaginesthatabiologicalpopulationevolvesbysearchingahigh-dimensionallandscapeforfitnesspeaks,inmuchthesamewayasadisorderedsystemrelaxestowardsitslowenergyconfigurations[4,5].Notsurprisingly,extremalstatisticsargumentsplayaprominentroleinbothcontexts[5,6,7,8,9,10,11].Tomaketheanalogymoreprecise,wenotethattheinheritablecharactersofanindividual(itsgenotype)areencodedinageneticsequence(consistingofnucleotidelettersortheallelesofgenes),whichformanypurposescanbereducedtoabinarysequenceσ=(σ1,...,σL)offixedlengthL.Forastatisticalphysicistitisverynaturaltoassignthevaluesσi=±1totheletters,andtotreatthesequenceas,e.g.,arowofspinsinthetwo-dimensionalIsingmodel[12]oraconfigurationofaquantumspinchain[13].Afitnesslandscapeisthenareal-valuedfunctionW(σ)ontheL-dimensionalsequencespace,analogoustothe(negative)energyofthespinsystem.Thenotionofafitnesslandscapeisavenerableandpersistentimageinevolutionarybiology[14,15],butithasalwaysbeenplaguedbyacertainelusiveness,inthesensethatverylittleisknownaboutthefitnesslandscapesinwhichrealorganismsevolve.Thissituationmayeventuallychange,astheexperimentalmappingofgenotypicfitnessbecomesfeasibleforsimplemicrobialsystems[16,17].MeanwhileitisreasonabletohandleourignoranceofrealfitnesslandscapesbytreatingW(σ)asarealizationofasuitablychosenensembleofrandomfunctions.ThisapproachwaspioneeredbyKauffmanandcoworkers[7,8],whointroducedtheNKfamilyofrandomfitnesslandscapeswhicharecloselyanalogoustoDerrida’sp-spinmodelofspinglassesdevelopedafewyearsearlier[6,18].Twolimitingcasesofthemodelareofinteresthere:Therandomenergymodel(REM),inwhichfitness(orenergy)isassignedrandomlywithoutcorrelationstothegenotypes(orspinconfigurations),andthecaseinwhichthelettersσicontributeindependently(multiplicativelyfordiscretetimedynamics[15])tothefitness.Inthelattercasethereisalwaysasinglefitnessmaximum,whichexplainswhythisisalsoreferredtoastheFujiyamalandscape.Intheevolutionarycontextdeviationsfrommultiplicativefitnessareassociatedwithepistasis[16].WithintheNKfamily,theREMlandscapeismaximallyepistatic[19].InpreviousworktheevolutionaryprocessintheREMlandscapehasbeenstudiedmostlyinthelimitofinfinitepopulationsize,wherefluctuationsduetosamplingnoise(alsoknownasgeneticdriftinpopulationgenetics)areignored[15].Whilethisallowsonetoderivearathercompletepictureofbothstationary[20,21]andtime-dependent[22,23,24,25]propertiesofthemodel,theassumptionofaninfinitepopulationisunrealistic,becausethenumberofpossiblegenotypes2LexceedsanyconceivablepopulationsizeNalreadyformoderatevaluesofL.Ontheotherhand,individual-basedsimulationstudieswhichexplicitlyfollowthepopulationthroughsequencespacearerestrictedtorathershortsequences[26,27].Evolutioninrandomfitnesslandscapes:theinfinitesitesmodel3InthiscontributionwethereforeproposetoperformthelimitofinfinitesequencelengthatfinitepopulationsizeN.Thiskindoflimitiswellknowninpopulationgenetics,whereitisviewedalternativelyasalimitonthenumberofgeneticlociorsites(thesequencelengthLinoursetting)orasalimitonthenumberofalleles(thenumberofpossiblevaluesthatthevariablesσicantake).Althoughmathematicallydistinct,thetwovariantsareequivalentforourpurposes.TheimplementationoftheinfinitesiteslimitfortheREMlandscapeisstraightforward:ForL→∞everymutationleadstoanewgenotype,whosefitnesscanberandomlygeneratedwithoutneedtokeeptrackoftheneighborhoodrelationsofthesequencespace.Inthiswayl