Evolutionandspeciationonholeyadaptivelandscapes

Fitness landscapesSergey GavriletsDepartments of Ecology and Evolutionary Biology and Mathematics,University of Tennessee,KnoxvilleTable of contentslGeneral notion of fitness landscapeslFitness landscapes in simple population genetic modelslRugged landscapeslSingle-peak landscapeslFlat landscapeslHoley landscapesSewall Wright(1889-1988)lA founder of theoretical population genetics(with Fisher and Haldane)lIntroduced the notion of“fitness landscapes”(a.k.a.adaptive landscapes,adaptive topographies,surfaces of selective values)in 1931lHis last publication on fitness landscapes was published in 1988Papers on fitness landscapesTitle onlyTitle,keywords,abstract1980-1989 8 no data1990-1999 59 2122000-34 181Some of the journals that publish these papers:JOURNAL OF THEORETICAL BIOLOGY,PROTEIN ENGINEERING,PHYSICAL REVIEW E,CANCER RESEARCH,EVOLUTION,JOURNAL OF MATHEMATICAL BIOLOGY,LECTURE NOTES IN COMPUTER SCIENCE,CURRENT OPINION IN BIOTECHNOLOGY,MARINE ECOLOGY-PROGRESS SERIES,INTEGRATED COMPUTER-AIDED ENGINEERING,PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS,BIOLOGY&PHILOSOPHY,INTERNATIONAL JOURNAL OF TECHNOLOGY MANAGEMENT,BIOSYSTEMS,JOURNAL OF GENERAL VIROLOGY,ECOLOGY LETTERS,RESEARCH POLICY,SYSTEMS RESEARCH AND BEHAVIORAL SCIENCE,ANNALS OF APPLIED PROBABILITY,BIOPOLYMERS Working example:one-locus two-allele model of viability selectionlTwo allele at a single locus:A and alAllele frequencies:p and 1-plThree diploid genotypes:AA,Aa and aalGenotype frequencies:lViabilities:lAverage fitness of the population:AAaAaawww,22)1(),1(2,pppp22)1()1(2pwppwpwwaaAaAAFitness landscape as fitness of gene combinationsFitness landscape as the average fitness of populationsdpwdwppp2)1(Genotype spaceL=2,A=3 caseDimensionality:D=L(A-1)for haploids and D=2L(A-1)for diploidsOne-locus multi-allele model of stepwise mutationFitness landscape in a two-locus two-allele modelDimensionality of the population state spacelGeneral case:lRandomly mating population under constant viability selection:12LAD1LADAverage fitness of the population in a 2-locus 2-allele model with additive fitnessesD=2(because of linkage equilibrium)Fitness landscapes for mating pairs:fertilityFitness landscapes for mating pairs:mating preferenceDrosophila silvestris,and hybridsFitness landscapes for quantitative characterslRelationship between a set of Q quantitative characters that an individual has and its fitness;dimensionality of phenotype space is QlRelationship between the average fitness of the population and its genetic structure;dimensionality is equal to the number of phenotypic moments affecting the average fitnessFitness landscape with two quantitatie charactersMating preference function as fitness landscape)2)(exp(),(2VyxyxAverage fitness of the population under stabilizing selection)2exp(2sVzwzwVzGlnMetaphor of fitness landscapeslTwo or three dimensional visualization of certain features of multidimensional fitness landscapes Wright 1932Rugged fitness landscapeHill climbing on a rugged fitness landscape(Kauffman and Levin 1987)lL diallelic haploid locilFitnesses are assigned randomlylThe walk starts on a randomly chosen genotypelAt each time step,the walk samples one of the L one-step neighbors.If the neighbor has higher fitness,the walk moves there.Otherwise,no change happens.The walk stops when it reaches a local fitness peak,so that all L neighbors have smaller fitnessSample of Kauffman and Levins resultslExpected number of local peaks islExpected fraction of fitter neighbors dwindles by on each improvement steplAverage number of steps till a local peak is lRatio of accepted to tried mutations scales aslFrom most starting points,a walk can climb only to an extremely small fraction of the local peaks.Any one local peak can be reached only from an extremely small fraction of starting points.l“Complexity catastrophe”:as L increases,the heights of accessible peaks fall towards the average fitness)1(log2L)1/(2LLkk/lnSingle-peak fitness landscape Ronald Fisher(1890-1962)Fishers geometric model of adaptationlEach organism is characterized by Q continuous variableslThere is a single optimum phenotype O and fitness decreases monotonically with increasing(Euclidean)distance from the optimumlLet d/2 be the current distance to the optimum OlEach mutation is advantageous if it moves the organism closer to O.lLet r be the mutation size(i.e.distance between the current state and the mutant)lFor large Q,the probability that a mutation is advantageous is P(r)=1-F(r)where F is the cumulative distribution function of a standard normal distribution,and dQrx/Mutations of small size are the most important in evolutionCorrections to the Fisher modellKimura(1983):the probability that an advantageous mutation with effect s is fixed is 2s.Therefore,the rate of adaptive substitutions is 2x(1-F(x).Thus,mutation of intermediate size are most important.lOrr(1998):distance to the optimum continuously decreases.The distribution of factors fixed during adaptation is exponential.“Error threshold”(Manfred Eigen)lAssume that there is a single optimum genotype(“master sequence”)that has fitness 1;all other genotypes have fitness 1-s.Let n be the mutation rate per sequence per generationlThen,if ns,the master sequence is not maintained in the populationFlat fitness landscape(of the neutral theory of molecular evolution)Motoo Kimura(1924-1994)Evolution of flat landscapeslRandom walk on a hypercubeEquilibrium distribution:equal probability to be at any vertex;time to reach the equilibrium distribution is order stepsTransient dynamics of the distance to the initial stateThe index of dispersion(i.e.var(x)/E(x),where x is the number of steps per unit of time)is equal to 1.LLlog)2exp(1 2tLdtEvolution of flat landscapes(cont.)lIn a population of N alleles,any two alleles can be traced back to a common ancestor about N generations ago(under the Fisher-Wright binomial scheme for random genetic drift)lThe average number of mutations fixed per generation is equal to the mutation rate nlThe average genetic distance between two organisms is 2NnlPopulation can be clustered into 2(2Nn)/d clusters such that the average distance within the same cluster is d.How many dimensions do real fitness landscapes have?lThe world as we perceive it is three dimensionallSuperstring theory:10 to 12 dimensions are required to explain physical worldlBiological evolution takes place in a space with millions dimensions (3/27/03)SuperKingdom#of species#of sequences range(in million base pairs)Archae16161.5-5.8Bacteria1011300.4-9.1Eukaryotes11110.2-282Extremely high dimensionality ofthe genotype space results in:redundancy in the genotype-fitness map a possibility that high-fitness genotypes form networks that extend throughout the genotype space(=substantial genetic divergence without going through adaptive valleys)increased importance of chance and contingency in evolutionary dynamics (=mutational order as a major source of stochasticity)Russian roulette model Genotype is viable with probability p and is inviable otherwise:There exists a giant cluster of viable genotypes if p0.5973(percolation in two dimensions)Percolation on a hypercubeIn the L-dimensional hypercube(e.g.if there are L diallelic loci),viable genotypes form a percolating neutral network if p1/L(assuming that L is very large).Each genotype has L“neighbors.”Uniformly rugged landscapeThe nearly neutral network of genotypes with fitnesses between w1 and w2 percolates if w2-w11/L.Fitness w is drawn from a distribution on(0,1):Metaphor of holey fitness landscapes disregards fitness differences between different genotypes belonging to the network of high-fitness genotypes and treats all other genotypes as holesMicroevolution and local adaptation climbing from a“hole”macroevolution movement along the holey landscapespeciation takes place when populations come to be on opposite sides of a hole in the landscape The origin of the idealVerbal argumentsBateson(1909)Dobzhansky(1937)Muller(1940,1942)Maynard Smith(1970,1983)Nei(1976)Barton and Charleswoth(1984)Kondrashov and Mina(1986)lFormal modelsNei (1976)Wills(1977)Nei et al (1983)Bengtsson and Christiansen(1983)Bengtsson(1985)Barton and Bengtsson(1986)Dobzhansky model(1937)“This scheme may appear fanciful,but it is worth considering further since it is supported by some well-established facts and contradicted by none.”(Dobzhansky,1937,p.282)(1900-1975)Maynard Smith(1970):“It follows that if evolution by natural selection is to occur,functional proteins must form a continuous network which can be traversed by unit mutational steps without passing through nonfunctional intermediates”(p.564)TerminologylA neutral network is a contiguous set of genotypes(sequences)possessing the same fitness.lA nearly neutral network is a contiguous set of genotypes possessing approximately the same fitness.lA holey fitness landscape is a fitness landscape in which relatively infrequent high-fitness genotypes form a contiguous set that expands throughout the genotype space.Conclusions from modelsThe existence of percolating nearly-neutral networks of high-fitness combinations of genes which allow for“nearly-neutral”divergence is a general property of fitness landscapes with a very large number of dimensions.Experimental evidencelDirect analyses of relationships between genotype and fitness in plants,Drosophila,mammals and mothslRing species and hybrid zones lArtificial selection experimentslNatural hybridization in plants and animalslIntermediate forms in the fossil recordlProperties of RNA and proteinslPatterns of molecular evolutionlArtificial lifeApplicationslSpeciationlHybrid zoneslMorphological macroevolutionlRNA and proteinslAdaptationlMolecular evolutionlGene and genome duplicationlCanalization of development