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Introduction to Quantitative Genetics 1 Gene and Genotype Frequencies (Population Genetics) Gene and Genotype Frequencies (population genetics) Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Fundamentals of Quantitative Genetics Similarity among Relatives Similarity among Relatives Response to Selection Response to Selection Multivariate Selection Multivariate Selection Introduction to Quantitative Genetics 1 Gene and Genotype Frequencies (population genetics) Michael Morrissey 2 Fundamentals of Quantitative Genetics 3 Similarity among Relatives August 2015 4 Response to Selection 5 Multivariate Selection Michael Morrissey, Intro to QG Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics http://synergy.st-andrews.ac.uk/megaloceros/ Similarity among Relatives Response to Selection Multivariate Selection http://soaysheep.biology.ed.ac.uk Introductory remarks - my interests @SoaySheep 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 Don’t need much background though. If you understand that you Similarity among Relatives Response to Selection resemble your maternal grandfather, because your mother’s egg Multivariate Selection contained chromosomes that were a mix of those she inherited from him Introductory remarks - genetical background and from her mother, then we’ll be OK. 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 regression relations given are for simple, not multiple, regression 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 These books are: Similarity among Relatives Response to Selection Multivariate Selection D.S. Falconer and T.F.C. Mackay. 1996. Introduction to Quantitative Genetics. Longman Press. and Introductory remarks - goals of these lectures M. Lynch and B. Walsh. 1998. Genetics and Analysis of Quantitative Traits. Sinauer. There is a massive volume of QG material out there The first is the standard introductory text to quantitative genetics. The Foundational statistical genetics of second is the bible of quantitative genetics. 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 F&M, page 5. 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 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 Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Wx Fundamentals of Quantitative Genetics The parts of the first three expressions that go can be thought of as Selection W¯ Similarity among Relatives Drift Response to Selection weighting by fitness, where division by the mean makes the weights add Mutation Multivariate Selection up to one. Allele frequency change 1 p0 indicates the frequency after selection, which is the frequency with which A will be represented in the next generation 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 The first four lines represent the declaration of a custom function. It Similarity among Relatives Drift Response to Selection Mutation accepts the four arguments (genotypic fitnesses and p)andreturnsthe Multivariate Selection change in p. 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 This can be answered by playing with the genotypic fitnesses in the allele Similarity among Relatives Drift Response to Selection Mutation frequency simulator. 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 Note that the figure is suppressed from the handouts. Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Dominance and allele frequency change 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 Figure suppressed - please play with the drift simulator! Similarity among Relatives Drift Response to Selection Mutation Multivariate Selection Drift Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Hardy-Weinberg equilibrium Fundamentals of Quantitative Genetics Selection Please forgive the quick run through this. I want to give a flavour for the Similarity among Relatives Drift Response to Selection Mutation fact that a lot of useful theory about drift has been worked out. See Multivariate Selection F&M p. 51-63 for a fuller introduction. 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 among Relatives Drift Response to Selection Mutation Multivariate Selection Population structure and migration if demes (local
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