Fitness landscapes and evolution

arXiv:cond-mat/9505003v12May1995FITNESSLANDSCAPESANDEVOLUTION∗LUCAPELITIDipartimentodiScienzeFisicheandUnit`aINFMUniversit`a“FedericoII”,Mostrad’Oltremare,Pad.19I-80125Napoli(Italy)†February1,2008AbstractTheconceptoffitnessisintroduced,andasimplederivationoftheFundamentalTheoremofNaturalSelection(whichstatesthattheaveragefitnessofapopulationincreasesifitsvarianceisnonzero)isgiven.Afterashortdiscussionoftheadaptativewalkmodel,ashortreviewisgivenofthequasispeciesapproachtomolecularevolutionandtotheerrorthreshold.Therelevanceofflatfitnesslandscapestomolecularevolutionisstressed.Finallyafewexampleswhichinvolvewiderconceptsoffitness,andinparticulartwo-levelselection,areshortlyreviewed.1FitnessandthefundamentaltheoremofnaturalselectionTheterm“fitness”derivesfromthephrase“survivalofthefittest”thatthephilosopherHerbertSpencersuggestedtouseinsteadof“naturalse-lection”.Inthestrugglemadebytheevolutionarytheoriststoavoidthe∗LecturesgivenattheNATOASIonPhysicsofBiomaterials:Fluctuations,Self-Assembly,andEvolution,Geilo(Norway),March27-April6,1995.†AssociatoINFN,SezionediNapoli.E-mail:[e-mail]1tautologylurkinginthephrase,thetermhasbeentwistedtoseveralmean-ings.R.Dawkins[6]distinguishesnolessthanfivedifferentmeaningstothewordintheevolutionaryliterature.Fromthepointofviewofmodelbuild-ing,themostconvenientmeaning—andtheoneweshalladopt—ishoweverthefollowing:Thefitnessofanindividualisproportionaltotheaveragenumberofoffspringitmayhaveinthegivenenvironment.Inthisdefinition,fitnessisassignedtoindividualsratherthattogenesortogroupsofindividuals.Itisfurtherassumedthatreproductiontakesplaceviaastochasticprocess,andthat,inagivenpopulation,theaveragenumbersof(immediate)offspringoftwoindividualshavethesameratioastheirfitnesses:therefore,onlyratiosoffitnesseshaveawell-definedmeaning,andnottheirabsolutevalue.Letusconsiderapopulationformedbyacertainnumberofindividuals,whoseinheritablecharacteristics(genotype)aresummarizedbythevariableσ.Letusfurtherassumethatthepopulationreproducesasexually,thattheoffspringofanindividualhavethesamegenotypeastheparent,andfinallythatthenumberofoffspringisexactlyproportionaltothefitnessoftheparent:briefly,letusneglectmutationsinthegenotypeandfluctuationsinthenumberofoffspring.Wecanthuswritedownanequationexpressingthenumbernt(σ)ofindi-vidualscarryingthegenotypeσatgenerationt+1,giventhesamequantityatgenerationt,assumingthatthefitnessA(σ)ofanindividualisafunctiononlyofitsgenotype:nt+1(σ)=1ZtA(σ)nt(σ),(1)whereZtisaproportionalityconstant.Inordertosimplifytheargumentwehavealsoassumedthatthegenerationsarenonoverlapping,i.e.,thatalindividuals,oncereproduced,die.ThetotalnumberNtofindividualsinthepopulationatgenerationtisgivenbyNt=Xσnt(σ).(2)2WedefinethepopulationaveragehQitatgenerationtofaquantityQ(σ),whichdependsonlyonthegenotypeσ,inthefollowingway:hQit=1NtXσQ(σ)nt(σ).(3)WecannowprovethattheaveragefitnesshAialwaysincreases,unlessallindividualshavethesamefitness.Wehaveinfact:hAit+1=1Nt+1XσA(σ)nt+1(σ)=1Nt+1ZtXσA2(σ)nt(σ).(4)Ontheotherhand,onehasNt+1=1ZtXσA(σ)nt(σ)=NtZthAit.(5)ThereforehAit+1hAit=DA2Et≥(hAit)2,(6)andtheequalityholdsonlyifallindividualsinthepopulationhavethesamefitness.Infact,thelargerthevarianceinthefitness,thefasteritsaveragegrows.ThisresultisasimplifiedversionoftheFundamentalTheoremofNaturalSelectionduetoR.Fisher[16,p.22ff].Someauthorshaveconsidereditasthekeypointofdifferencebetweenthelivingandtheinorganicworld.AsK.Sigmundputsit[32,p.108]:Sowesee,inphysics,disordergrowinginexorablyinsystemsiso-latedfromtheirsurroundings;andinbiology,fitnessincreasingsteadilyinpopulationsstrugglingforlife.Ascenthereanddegra-dationthere—almosttoogoodtobetrue.Infact,theresultdependsonmanyunrealisticassumptions.Letalonethecomplicationsintroducedbysex,whichleadtomaddeninglycomplexbehavior,letusfocusontheeffectsofmutation:onthatsetofcauseswhichmakesoffspringdifferentfromtheirparent,evenamongbacteria.WeallknowthatgeneticinformationiscarriedbytheDNA,intheformofasequenceofnucleotidebases,whichbelongtofourdifferenttypes:AadenineandGguanine(purines);TthymineandCcytosine(pyrimidines).3InthedoublehelixofDNAtheyarefoundinmatchingpairs:A-TandG-C.Duringthereplication,itmayhappenthatthereplicationmechanism,whichassociatesoneofthe“old”strandstothe“new”ones,stumblesinsomeerrors.Theseerrorscanbedividedinafewclasses:Pointmutations:Substitutionofonenucleotidebasetoanother.Theycanbedividedintotwoclasses:Transitions(themostcommon):substitutionofonepurinebytheother,orofonepyrimidinebytheother;Transversions,inwhichapurineisreplacedbyapyrimidineandvicev-ersa.Insertionsanddeletions:Theycorrespondtotheintroductionofnewbasesinthestrandorintheiromissionrespectively.Inthecaseofsequencescodingforaprotein,thesemutationsareoftenfatal,sincetheyentailaframeshiftinthetranslationintoproteins,unlesstheyoccurbythrees.Majorrearrangements:Inthisclassoneconsiderstheinsertions(ordele-tion)ofcomparativelylongsequences.Thisisthecase,e.g.,ofthetransposableelementswhichareknowntomoveeasilyfromoneplacetoanotherinthegenotype.Asubclassofspecialinterestisgenedou-bling.Theseprocessesdonothavethesam