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How to Model Microevolution Evolution is a change over time in the frequency of alleles or allele combinations in the gene pool, so any model of evolution must include at the minimum the passing of genetic material from one generation to the next. Hence, our fundamental time unit will be the transition between two consecutive generations at comparable stages. All such trans-generational models of microevolution have to make assumptions about three major mechanisms: • • • Mechanisms of producing gametes Mechanisms of uniting gametes Mechanisms of developing phenotypes. How to Model Microevolution In order to specify how gametes are produced, we have to specify the genetic architecture. Genetic architecture refers to the number of loci and their genomic positions, the number of alleles per locus, the mutation rates, and the mode and rules of inheritance of the genetic elements. For example, the first model we will develop assumes a single autosomal locus with two alleles with no mutation. Under this genetic architecture, we need only to use Mendel’s first law of inheritance to specify how genotypes produce gametes. 1 Demes and Gene Pools • Meiosis Interconnects the Deme to the Gene Pool • Therefore, Given Mendel’s Laws and Normal Meiosis, You Can Always Calculate the Allele Frequencies in the Gene Pool From the Genotype Frequencies in the Deme • Can You Predict the Deme (Genotype Frequencies) from the Gene Pool (Allele Frequencies)? Demes AA 1/ 2 aa 1/ 2 1 AA 1/ 4 1 A 1/ 2 a 1/ 2 Aa 1/ 2 1/ 1 A 1/ 2 2 aa 1/ 4 1/ 1 2 a 1/ 2 Gene Pools 2 Hardy (and Weinberg) Solution FERTILIZATION To Go From Gene Pool (Gametes) to Deme (Initially Zygotes),Need to Specify The Rules by Which Gametes Unite (Fertilization) Population Structure Population Structure is the rules at the level of the deme by which gametes are united in fertilization, thereby defining the transition from haploidy to diploidy. 3 Models in Population Genetics Minimally Specify How To Go From One Generation To The Next Deme of Adult Diploid Individuals Meiosis Gene Pool of Haploid Gametes Need to Specify Genotype Frequencies, And Therefore Genetic Architecture (Number of Loci, Alleles per Locus, Linkage, Rules of Inheritance, etc.). Need to Specify Population Structure. Fertilization Deme of Adult Diploid Individuals Need to Specify How Individuals Develop Phenotypes. Assumptions of Hardy-Weinberg Mechanisms of Producing Gametes (Genetic Architecture) One Autosomal Locus Two Alleles No Mutation Mendel’s First Law (50:50 Segregation in heterozygotes) Mechanisms of Uniting Gametes (Population Structure) System of Mating: Random Size of Population: Infinite Genetic Exchange: None (One Isolated Population) Age Structure: None (Discrete Generations) Mechanisms of Developing Phenotypes All Genotypes Have Identical Phenotypes With Respect to Their Ability for Replicating Their DNA 4 Random Mating Random Mating occurs when both of the gametes united in a zygote are drawn at random and independently from the gene pool. This means that the probability of a gamete bearing a specific allele = the frequency of that allele in the gene pool, and this is true for all gametes involved in fertilization. Random Mating A p a q = 1-p Gene Pool Paternal Gamete Maternal Gamete A p a q A p AA p×p=p2 Aa pq a q aA qp aa q×q=q2 5 Hardy-Weinberg Genotype Frequencies AA p2 Aa 2pq aa q2 Mendelian Probabilities of Offspring (Zygotes) Weinberg’s Derivation Mating Pair Frequency of Mating Pair AA Aa aa AA × AA GAA×GAA = GAA2 1 0 0 AA × Aa GAA×GAa = GAAGAa Aa × AA GAa×GAA = GAAGAa AA × aa GAA×Gaa = GAAGaa 0 1 0 aa × AA Gaa×GAA = GAAGaa 0 1 0 0 0 1 G’AA G’Aa G’aa 0 0 2 Aa × Aa GAa×GAa = GAa Aa × aa GAa×Gaa = GAaGaa 0 aa × Aa Gaa×GAa = GAaGaa 0 aa × aa Gaa×Gaa = Gaa 2 Total Offspring Summing Zygotes Over All Mating Types: G’AA=GAA2 + [2GAAGAa] + GAa2 = [GAA+ GAa]2 = p2 G’Aa= [ 2GAAGAa]+ 2GAAGaa+ GAa2+ [ GAaGaa]= 2[GAA+ GAa][Gaa+ GAa] = 2 p q G’aa= GAa2 + [ 2GAaGaa] + Gaa2 = [Gaa+ GAa]2 = q2 6 The Life Cycle for a Population AA GAA Deme of Diploid Individuals Meiosis Random Mating Fertilization 1/ 1 Mendelian Probabilities Gene Pool of Haploid Gametes Aa GAa 1/ 2 1 2 A a p=GAA+ 1/2GAa q=Gaa+ 1/2GAa p p AA p2 Deme of Diploid Individuals aa Gaa p q q Aa 2pq q aa q2 Testing for Hardy-Weinberg Genotype Frequencies. E.g., a Population of Pueblo Indians Scored for the MN Blood Group Type Blood Type M MN N Genotype MM MN NN Number 83 46 11 140 (0.24)2 = 0.06 1 H.-W. Freq. (0.76)2 = 2(0.76)(0.24) 0.58 = 0.36 Exp. Number 0.58(140) = 81.2 0.36(140) = 50.4 (Obs.-Exp.)2 Exp. (83-81.2)2 81.2 (46-50.4)2 50.4 Sum 0.06(140) 140 = 8.4 (11-8.4)2 8.4 1.23 Degrees of Freedom = 3 Categories - 1 - 1 estimated parameter = 1 7 Random Mating Is Locus Specific Although the Pueblo Indians are randomly mating for the MN Blood Group Locus, They Are Not Randomly Mating For All Loci, e.g., the X and Y Chromosomes Hardy-Weinberg Frequencies Represent An Equilibrium No Assumption is AA Made About These Genotype Frequencies; GAA They May or May Mendelian 1 Not Be in Probabilities Hardy-Weinberg Random Mating One Generation of Random Mating Insures These Are Hardy-Weinberg Genotype Frequencies Aa GAa 1/ aa Gaa 1/ 2 1 2 A a p=GAA+ 1/2GAa q=Gaa+ 1/2GAa p p AA p2 p q Aa 2pq q q aa q2 8 Hardy-Weinberg Frequencies Represent An Equilibrium The Frequency of The A Allele in the Next Generation’s Gene Pool Is: p’= p2 + 1/22pq = p2 + pq = p(p + q) =p Therefore, the Gene Pool Is Unchanged There Is NO Evolution Under The HardyWeinberg Model A p=GAA+ Random Mating p a 1/ p 2 GAa p AA p2 q q Aa 2pq 1/ 1 Mendelian Probabilities q=Gaa+ 1/2GAa aa q2 1/ 2 a 22pq =p(p+q) = p q’=q2+ 1/ 22pq =q(q+p)=q A a p=GAA+ 1/2GAa q=Gaa+ 1/2GAa p p p AA p2 Mendelian Probabilities 1 2 A p’=p2+ 1/ Random Mating q q q Aa 2pq 1/ 1 2 A p’=p2+ 1/ q aa q2 1/ 1 2 a 22pq =p(p+q) = p q’=q2+ 1/ 22pq =q(q+p)=q 9 Importance of Hardy-Weinberg • Acceptance of Mendelian Genetics (Punnett’s dilemma) • Resurrection of Natural Selection (Jenkin’s critique) • A useful null model of evolutionary stasis. • A valuable springboard for the investigation of many forces of evolutionary change by relaxing its assumptions. The First Assumption That We Will Relax Is the Assumption of a Genetic Architecture of OneLocus With Two Alleles. What Happens When We Have Two Loci, Each With Two Alleles? 10 Two Locus Hardy Weinberg Gene Pool AB Ab aB ab gAB gAb gaB gab Mechanisms of Uniting Gametes (Random Mating) Zygotic/Adult Population AB/AB AB/Ab AB/aB AB/ab gAB2 2g AB gAb 2g ABgaB 2g ABgab Ab/Ab gAb 2 Ab/aB Ab/ab aB/aB aB/ab 2g Ab gaB 2g Ab gab gaB 2 2g aB gab ab/ab gab2 Mechanisms of Producing Gametes (Mendel's First Law & Recombination) Gene Pool of Next Generation AB Ab aB ab g' AB g'Ab g' aB g' ab Recombination Occurs in All Genotypes, But Can Change The State of the Parental Gametes Only in Double Heterozygotes. 11 Two Locus Hardy Weinberg Gene Pool AB Ab aB ab gAB gAb gaB gab Mechanisms of Uniting Gametes (Random Mating) Zygotic/Adult Population AB/AB AB/Ab AB/aB AB/ab gAB2 2g AB gAb 2g ABgaB 2g ABgab Ab/Ab gAb 2 Ab/aB Ab/ab aB/aB aB/ab 2g Ab gaB 2g Ab gab gaB 2 2g aB gab ab/ab gab2 Mechanisms of Producing Gametes (Mendel's First Law & Recombination) Gene Pool of Next Generation AB Ab aB ab g' AB g'Ab g' aB g' ab Double heterozygotes can produce all four gamete types. Two Locus Hardy Weinberg 2 g'AB = 1" gAB + 12 (2gAB gAb )+ 12 (2gAB gaB )+ 12 (1# r )(2gAB gab )+ 12 r(2gAb gaB ) = gAB [gAB + gAb + gaB + (1# r )gab ] + rgAb gaB = gAB [gAB + gAb + gaB + gab ] + rgAb gaB # rgAB gab = gAB + r(gAb gaB # gAB gab ) = gAB # rD ! Where D = gAB gab - gAb gaB D is called Linkage Disequilibrium or Gametic Phase Imbalance 12 Two Locus Hardy Weinberg Similarly, can show: g’AB = gAB - rD g’Ab = gAb + rD g’aB = gaB + rD g’ab = gab - rD Where D = gAB gab - gAb gaB D, “linkage disequilibrium” • It measures the degree of association at the population level between the two sites/loci • D is created by many evolutionary forces and historical events, including the very act of mutation because the new mutant variant initially exists on only one chromosomal background. 13 Two Locus Hardy Weinberg That is, Evolution Occurs! Two Locus Hardy Weinberg D1 The two locus equilibrium is approached gradually, at a rate determined by r. Historical information is encoded in D (and other multi-locus/site measures) that decays gradually with time! This information persists for long periods of time for tightly linked sites. 14 Theoretical Decay of LD in a Random-Mating Population In a genomic region with no recombination, the LD created by mutation never dissipates. Two Locus Hardy Weinberg Equilibrium pA = gAB + gAb pB = gAB + gaB so pA pB = ( gAB + gAb )( gAB + gaB ) 2 = gAB + gAB gaB + gAB gAb + gAb gaB = gAB ( gAB + gaB + gAb) + gAb gaB = gAB (1" gab ) + gAb gaB ! = gAB " gAB gab + gAb gaB = gAB " D As t goes to infinity, D goes to 0 (the equilibrium), so at the two-locus equilibrium, gAB=pApB, and similarly for the other ! gamete frequencies. 15 Two Locus Hardy Weinberg At equilibrium the two loci associate at random (proportional to their allele frequencies) in the Population’s Gene Pool: B k b m A p AB pk=gAB Ab pm=gAb a q aB qk=gaB ab qm=gab D = gAB gab - gAb gaB=pkqm-pmqk=0 D≠ 0 measures the degree of non-random association at the population gene pool level between the two sites/loci Many Factors Create Disequilibrium, Including the Very Act Of Mutation Once created, disequilibrium decays at a rate determined in part by recombination, and in part by population structure (as we will see later). 16 Linkage disequilibrium is created when mutation creates new variation D = gAB gab - gAb gaB = gAB0 - 0gaB =0 Initial Gene Pool: A B a B a B Mutation At A Second Site Produces Three Gamete Types: Gene Pool After Mutation: A B a B a b D = gAB gab - gAb gaB = gAB gab - 0gaB = gAB gab ≠ 0 Can see the effects of mutation on Linkage disequilibrium more clearly through D’ " D ,D<0 $ min p p ,p p ( A B a b) $ D '= # $ D ,D>0 $ % min( pA pb ,pa pB ) D’ varies between -1 and +1, and when mutation first creates the third gamete type, D’=-1 or +1, so mutation creates maximal linkage disequilibrium. ! 17 D (or D’) decays with recombination: A B A B a B a B D’=D=0 Mutation At A Second Site Produces Three Gamete Types: A B A D = gAB gab D’=1 (pA=gAB, pb=gab) A B B a Recombination Produces Four Gamete Types A b a B a b B a b D = gAB gab - gAb gaB< gAB gab, D’<1 (pA=gAB+gAb) In regions of little to no recombination, the pattern of disequilibrium is determined primarily by the historical conditions that existed at the time of mutation, resulting in little to no correlation of D with physical distance Indel Taq I Pst I Sst I Pvu II Xmn I Apo AI Apo CIII Apo AIV Significant linkage disequilibrium 18 On larger physical scales, |D| is negatively correlated with physical distance 0.9 0.8 0.7 |D’| 0.6 0.5 0.4 0.3 0.2 0.1 0 5 10 20 40 80 kb AllYor Distance Utah Swed 160 S U (kb) YorBot YorTop Reich et al. (2001 Nature 411:199-204) Disequilibrium and Historical Effects Create Both Opportunities and Difficulties for the Analysis of Population Genetic Data, As We Shall See 19 Some lessons from 1 vs. 2-locus HW: A seemingly slight change in the model can create qualitative differences (e.g., no evolution in 1 locus HW vs. evolution in 2-locus HW; instantaneous equilibrium in 1 locus HW vs. gradual or no equibrium in 2 locus HW) Scale matters (e.g., the relationship between D’ and physical distance on different scales of physical distance). The inferences made from a model are often very sensitive to the assumptions of that model. Generalize with care! 20