Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Similarity among Relatives Response to Selection Multivariate Selection Introduction to Quantitative Genetics Michael Morrissey August 2015 Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Similarity among Relatives Response to Selection Multivariate Selection 1 Gene and Genotype Frequencies (population genetics) 2 Fundamentals of Quantitative Genetics 3 Similarity among Relatives 4 Response to Selection 5 Multivariate Selection Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Similarity among Relatives Response to Selection Multivariate Selection Introductory remarks - my interests My interests: nuts and bolts of evolution in the wild do populations contain genetic variation for ecologically-important traits? how are di↵erent traits selected? do we expect contemporary evolution, if so, why, if not, why not? theoretical and empirical approaches Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Similarity among Relatives Response to Selection Multivariate Selection Introductory remarks - genetical background Will assume knowledge of diploidy and Mendel’s laws chromosomes meiosis Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Similarity among Relatives Response to Selection Multivariate Selection Introductory remarks - statistical background Will assume some knowledge of relationships among correlation, variance, covariance, and regression variance of X : VAR (X )=σ2(X )=E[(X X¯)2] − covariance of X and Y : COV (X , Y )=σ(X , Y )=E[(X X¯)(Y Y¯)] − − COV (X ,Y ) regression of Y on X : bY ,X = VAR (X ) COV (X ,Y ) 2 2 correlation of Y on X : σ(X )σ(Y ) or bY ,X if σ (X )=σ (Y ) variance in Y arising from b : σ2(X ) b2 Y ,X · Y ,X Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Similarity among Relatives Response to Selection Multivariate Selection Introductory remarks - goals of these lectures There is a massive volume of QG material out there Foundational statistical genetics of Wright and Fisher Long traditions of statistical approaches to animal breeding in the UK and USA Evolutionary quantitative genetics Not possible to cover this comprehensively! Goal is to generate sufficient familiarity with core concepts as to make independent study productive. Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Gene and Genotype Frequencies (population genetics) Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Random mating 1 In a diploid population allele type A occurs at frequency p allele type a occurs at frequency q q =1 p − individuals mate randomly What are the frequencies of genotypes AA, Aa,andaa? Michael Morrissey, Intro to QG So summing the two ways of getting a heterozygote, the expected genotypic proportions at a locus under random mating are AA Aa aa p2 2pq q2 These are called “Hardy-Weinberg proportions”, and will be very useful! Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Random mating 2 p = freq(A), q = freq(a), q =1 p − male gamete female gamete probability AAp2 Aapq aApq aaq2 Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Random mating 2 p = freq(A), q = freq(a), q =1 p − male gamete female gamete probability AAp2 Aapq aApq aaq2 So summing the two ways of getting a heterozygote, the expected genotypic proportions at a locus under random mating are AA Aa aa p2 2pq q2 These are called “Hardy-Weinberg proportions”, and will be very useful! Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Hardy-Weinberg proportions 1.0 0.8 P H Q 0.6 P=p(AA) H=p(Aa) 0.4 genotype frequency Q=p(aa) 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 p Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Genotypic fitnesses di↵erent genotypes may have di↵erent fitness this may result in allele frequency change Can we construct a general model? fitnesses of the three genotypes AA, Aa and aa are WAa, WAa,and Waa, frequencies of A and a are p and q what, then, are the allele frequencies in the next generation? Michael Morrissey, Intro to QG 2 WAA 1 WAB p0 = p + 2pq W¯ 2 W¯ 2 2 W¯ = p WAA +2pqWAB + q WBB Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Allele frequency change 1 2 WAA P(AA)⇤ = p W¯ WAB P(AB)⇤ =2pq W¯ 2 WBB P(BB)⇤ = q W¯ Michael Morrissey, Intro to QG 2 2 W¯ = p WAA +2pqWAB + q WBB Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Allele frequency change 1 2 WAA P(AA)⇤ = p W¯ WAB P(AB)⇤ =2pq W¯ 2 WBB P(BB)⇤ = q W¯ 2 WAA 1 WAB p0 = p + 2pq W¯ 2 W¯ Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Allele frequency change 1 2 WAA P(AA)⇤ = p W¯ WAB P(AB)⇤ =2pq W¯ 2 WBB P(BB)⇤ = q W¯ 2 WAA 1 WAB p0 = p + 2pq W¯ 2 W¯ 2 2 W¯ = p WAA +2pqWAB + q WBB Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Allele frequency change iterator delta_p <- function(W_AA,W_AB,W_BB,p){ Wbar<-p^2*W_AA+ 2*p*(1-p)*W_AB+(1-p)^2*W_BB p*(p*W_AA+(1-p)*W_AB-Wbar)/(Wbar) } Tmax<-100 p0<-0.005 W_AA<-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,pt,ylim=c(0,1),xlab="gen",ylab="p",type=’l’) Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Dominance and allele frequency change Question: Which goes to fixation fastest - a dominant, recessive, or additive mutant? We can use the allele frequency iterator to find out. Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Dominance and allele frequency change Dominant Recessive 0.8 0.8 0.4 0.4 0.0 0.0 0 20 40 60 80 100 0 20 40 60 80 100 p Overdominant Additive 0.8 0.8 0.4 0.4 0.0 0.0 0 20 40 60 80 100 0 20 40 60 80 100 generation Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Drift simulator N<-100 p0<-0.5 Tmax<-100 plot(-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,]) Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Drift 1.0 1.0 0.8 0.8 0.6 0.6 frequency 0.4 0.4 0.2 0.2 mean frequency across simulations 0.0 0.0 0 20 40 60 80 100 0 20 40 60 80 100 generation generation Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Facts about drift undirected !! The variance of di↵erences between generations is pq σ2(∆p)= N The probability that A is fixed at generation t is given by 1 t P(fixed) = p 3p q 1 . t 0 − 0 0 − N ✓ ◆ surprisingly simple implication when t !1 Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Similarity