Introduction to Quantitative Genetics - Sinica

1 Gene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionIntroduction to Quantitative GeneticsMichael MorrisseyAugust 2015 Michael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate Selection1 Gene and Genotype Frequencies (population Genetics )2 Fundamentals of Quantitative Genetics3 Similarity among Relatives4 Response to Selection5 Multivariate SelectionMichael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionIntroductory remarks - my interestsMy interests: nuts and bolts of evolution in thewilddo populations contain genetic variationfor ecologically-important traits?

2 How are di erent traits selected?do we expect contemporary evolution, ifso, why, if not, why not?theoretical and empirical approachesMichael Morrissey, Intro to @SoaySheepGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionIntroductory remarks - genetical backgroundWill assume knowledge ofdiploidy and Mendel s lawschromosomesmeiosisMichael Morrissey, Intro to QGDon t need much background though. If you understand that youresemble your maternal grandfather, because your mother s eggcontained chromosomes that were a mix of those she inherited from himand from her mother, then we ll be and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionIntroductory remarks - statistical backgroundWill assume some knowledge of relationships among correlation, variance,covariance, and regressionvariance ofX:VA R(X)= 2(X)=E[(X X)2]covariance ofXandY:COV(X,Y)= (X,Y)=E[(X X)(Y Y)]regression ofYonX:bY,X=COV(X,Y)VA R(X)correlation ofYonX:COV(X,Y) (X) (Y)orbY,Xif 2(X)= 2(Y)variance inYarising frombY,X.

3 2(X) b2Y,XMichael Morrissey, Intro to QGregression relations given are for simple, not multiple, regressionGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionIntroductory remarks - goals of these lecturesThere is a massive volume of QG materialout thereFoundational statistical Genetics ofWright and FisherLong traditions of statistical approachesto animal breeding in the UK and USAE volutionary Quantitative geneticsNot possible to cover thiscomprehensively!Goal is to generate su cient familiaritywith core concepts as to makeindependent study Morrissey, Intro to QGThese books Falconer and Mackay. 1996. Introduction to QuantitativeGenetics. Longman Press. andM. Lynch and B. Walsh. 1998. Genetics and Analysis of QuantitativeTraits.

4 First is the standard introductory text to Quantitative Genetics . Thesecond is the bible of Quantitative and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationGene and Genotype Frequencies (population Genetics )Michael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationRandom mating 1In a diploid populationallele typeAoccurs at frequencypallele typeaoccurs at frequencyqq=1 pindividuals mate randomlyWhat are the frequencies of genotypesAA,Aa,andaa?Michael Morrissey, Intro to QGF&M, page and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationRandom mating 2p=freq(A),q=freq(a),q=1 pmale gamete female gamete probabilityAAp2 AapqaApqaaq2So summing the two ways of getting a heterozygote, the expectedgenotypic proportions at a locus under random mating areAAAaaap22pqq2 These are called Hardy-Weinberg proportions , and will be very useful!

5 Michael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationHardy-W einberg frequencyPHQP=p(AA)H=p(Aa)Q=p(aa)Michael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationGenotyp ic fitnessesdi erent genotypes may have di erent fitnessthis may result in allele frequency changeCan we construct a general model?fitnesses of the three genotypesAA,AaandaaareWAa,WAa,andWaa,fre quencies ofAandaarepandqwhat, then, are the allele frequencies in the next generation?Michael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationAllele frequency change 1P(AA) =p2 WAA WP(AB) =2pqWAB WP(BB) =q2 WBB Wp0=p2 WAA W+122pqWAB W W=p2 WAA+2pqWAB+q2 WBBM ichael Morrissey, Intro to QGThe parts of the first three expressions that goWx Wcan be thought of asweighting by fitness, where division by the mean makes the weights addup to the frequency after selection, which is the frequency withwhichAwill be represented in the next generationGene and Genotype Frequencies (population Genetics )

6 Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationAllele frequency change iteratordelta_p <- function(W_AA,W_AB,W_BB,p){Wbar<-p^2*W_A A+ 2*p*(1-p)*W_AB+(1-p)^2*W_BBp*(p*W_AA+(1- p)*W_AB-Wbar)/(Wbar)}Tmax<-100p0< <-2; W_AB<-2; W_BB<-1;pt<-array(dim=Tmax); pt[1]<-p0;for(t in 1:(Tmax-1)){pt[t+1] = pt[t]+delta_p(2,2,1,pt[t])}plot(1:Tmax,p t,ylim=c(0,1),xlab="gen",ylab="p",type= l )Michael Morrissey, Intro to QGThe first four lines represent the declaration of a custom function. Itaccepts the four arguments (genotypic fitnesses andp)andreturnsthechange and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationDominan ce and allele frequency changeQuestion:Which goes to fixation fastest - a dominant, recessive, or additive mutant?

7 We can use the allele frequency iterator to find Morrissey, Intro to QGThis can be answered by playing with the genotypic fitnesses in the allelefrequency and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationDominan ce and allele frequency changeMichael Morrissey, Intro to QGNote that the figure is suppressed from the and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationDrift simulatorN<-100p0< <-100plot(-100,-100,xlim=c(0,Tmax),ylim= c(0,1),xlab="generation",ylab="frequency ")pt<-array(dim=Tmax); pt[1]<-p0;for(t in 1:(Tmax-1)){pt[t+1] <- rbinom(1,N,pt[t])/N}lines(1:T,pt[s,])Mic hael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationDriftMi chael Morrissey, Intro to QGFigure suppressed - please play with the drift simulator!

8 Gene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationFacts about driftundirected !!The variance of di erences between generations is 2( p)=pqNThe probability thatAis fixed at generationtis given byP(fixed)t=p0 3p0q0 1 1N simple implication whent!1 Michael Morrissey, Intro to QGPlease forgive the quick run through this. I want to give a flavour for thefact that a lot of useful theory about drift has been worked out. SeeF&M p. 51-63 for a fuller and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationPopulat ion structure and migrationif demes (local populations) di er in allele frequencies, migrationmight change allele frequenciesHow much will populations di er in allele frequencies?

9 HS: expected heteroaygosity in demesHT: expected heteroaygosity in the population as a wholeFST=1=HSHT=VA R(p)p(1 p)under very simple assumptions:FST=14Nm+1 Michael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationSelecti on and drift togetherN<-100p0< <-100plot(-100,-100,xlim=c(0,Tmax),ylim= c(0,1),xlab="generation",ylab="frequency ")pt<-array(dim=Tmax); pt[1]<-p0;W_AA<-3; W_Aa<-2; W_aa<-1;for(t in 1:(Tmax-1)){pt_prime<-pt[t]+delta_p(W_AA ,W_Aa,W_aa,pt[t])pt[t+1] <- rbinom(1,N,pt_prime)/N}lines(1:Tmax,pt)M ichael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationFixatio n of a beneficial mutant in a finite populationRunning the previous simulation many times suggests that beneficialmutations are often lost.

10 For an additive mutant, that increases the fitnessof the heterozygote by s %, the probability of fixation is2s(1 e 4Ns)Some thoughts:Note thate 4 Nswill typically be very to probability of fixation of neutral a 1% fitness advantage seem bit?Michael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationMutatio nWe will not focus too much on of our problems will deal with existing genetic much variation can be maintained by mutation at anecologically-relevant locus?Michael Morrissey, Intro to QGGene and Genotype Frequencies (population Genetics )Fundamentals of Quantitative GeneticsSimilarity among RelativesResponse to SelectionMultivariate SelectionHardy-Weinberg equilibriumSelectionDriftMutationMutatio n-selection balancepandqas throughout,WAA=1 s,WAa=1andWaa=1uandvmutation rates ofa!