Introduction to Quantitative Genetics

Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Similarity among Relatives Response to Selection Multivariate Selection

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

Will assume knowledge of diploidy and Mendel’s laws chromosomes meiosis

Will assume some knowledge of relationships among correlation, variance, covariance, and regression variance of X : VAR (X )=2(X )=E[(X X¯)2] covariance of X and Y : COV (X , Y )=(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 : (X )(Y ) or bY ,X if (X )= (Y ) variance in Y arising from b : 2(X ) b2 Y ,X · Y ,X

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 sucient 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)

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

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!

genotype frequency Q=p(aa) 0.2 0.0

0.0 0.2 0.4 0.6 0.8 1.0

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

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

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 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

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’)

Question:

Which goes to ﬁxation fastest - a dominant, recessive, or additive mutant?

We can use the allele frequency iterator to ﬁnd out.

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

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,])

0 20 40 60 80 100 0 20 40 60 80 100

generation generation

undirected !! The variance of di↵erences between generations is pq 2(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

if demes (local populations) di↵er in allele frequencies, migration might change allele frequencies How much will populations di↵er in allele frequencies?

HS : expected heteroaygosity in demes HT : expected heteroaygosity in the population as a whole HS VAR (p) FST =1= H = p(1 p) T 1 under very simple assumptions: FST = 4Nm+1

N<-100 p0<-0.005 Tmax<-100 plot(-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)

Running the previous simulation many times suggests that beneficial mutations are often lost. For an additive mutant, that increases the fitness of the heterozygote by s %, the probability of fixation is

4Ns 2s(1 e ) Some thoughts: 4Ns Note that e will typically be very small. Compare to probability of ﬁxation of neutral allele. Does a 1% ﬁtness advantage seem bit?

We will not focus too much on mutation. Most of our problems will deal with existing genetic variation. How much variation can be maintained by mutation at an ecologically-relevant locus?

p and q as throughout, W =1 s, W =1andW =1 AA Aa aa u and v mutation rates of a A and A a,respectively ! ! Selection pW + qW p(1 s)+q p2qs p = p AA Aa = p = W¯ p2(1 s)+2pq + q2 1 p2s Mutation uq vp

Set e↵ects of mutation and selection to be equal, and solve for equilibrium

p2qs uq vp = 1 p2s vp will be small, p2 will be small, q will be large

u uq = p2s, p = s r Serious recessive conditions will have s close to one, so p will be on the order of the mutation rate. However, consider consequences if many such loci are relevant

Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Parent-o↵spring similarity Similarity among Relatives Many loci Response to Selection Fisher’s regression Multivariate Selection

Core principles of Quantitative Genetics

74 ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● 72 ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● from: (Sir) Francis Galton. 70 ● ● ● ● ● ● ● ● ● ● 1886. Regression towards ● ● ● ● ● ● ● ● ● 68 mediocrity in hereditary ● ● ● ● ● ● ● ● ● stature. The Journal of the ● ● ● ● ● ● ● ● ● 66 Anthropological Institute of ● ● ● ● ● ● ● ● ● offspring height (inches) ● ● ● ● ● ● ● Great Britain and Ireland. 15: 64 ● ● ● ● ● ● ● ● 246-263.

● ● ●

62 ● ● ● ● ●

64 66 68 70 72

mid−parent height (inches) Michael Morrissey, Intro to QG solution to toy model: Those o↵spring mate (with their generally white population-mates) Their o↵spring are less dark-skinned Eventually the population goes back to being light-skinned Can you spot the mistake?

Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Parent-o↵spring similarity Similarity among Relatives Many loci Response to Selection Fisher’s regression Multivariate Selection Challenge to natural selection?

Michael Morrissey, Intro to QG Can you spot the mistake?

toy model from Fleeming Jenkin, University of Edinburgh, 1867 (original was shockingly racist, I have changed it): African man comes to Edinburgh Ingratiates himself with locals through physical and social superiority African gentleman’s > average number of children are somewhat dark-skinned solution to toy model: Those o↵spring mate (with their generally white population-mates) Their o↵spring are less dark-skinned Eventually the population goes back to being light-skinned

1 t 2(t)=2(0) 2 ✓ ◆ blending inheritance destroys variation adaptive evolution by natural selection requires variation. blending inheritance therefore would limit the ecacy of selection What maintains variation????

What if many loci contribute to a trait? Let’s simulate...

k<-100 # number of loci n<-1000 # parent-offspring-child groups p<-0.5 # allele frequency g<-c(0,1,2) # genotypic values

data<-as.data.frame(list(midPar=array(dim=n), offVal=array(dim=n)))

for(i in 1:n){ #genotypes dad<-rbinom(k,2,p) mum<-rbinom(k,2,p) kid<-rbinom(k,1,dad/2)+rbinom(k,1,mum/2)

#phenotypes z_dad<-rnorm(1,sum(g[dad+1]),sqrt(50)) z_mum<-rnorm(1,sum(g[mum+1]),sqrt(50)) z_kid<-rnorm(1,sum(g[kid+1]),sqrt(50)) data$midPar[i]<-mean(c(z_dad,z_mum)) data$offVal[i]<-sum(g[kid+1]) }

● 74 ● ● ● ● 120 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● 72 ● ● ●●●● ● ● ●● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ●● ●● ●● ●●● ● ● ● ● ● ● ● ●● ●●● ●●●● ● ● ● ● ●● ●●●●●●●●● ● 110 ● ● ● ● ●●●●● ●●●● ● ●● ● ● ●● ● ●●●●●●●●● ● ● ● ● ●●●●●● ●● ● ● ● ● ● ● ● ● ● ●●●●●● ●●●●● ●●●● ● ● ● ● ● ● ● ● ● ●●● ● ●●●●● ●●●●●● ● ●●●● 70 ● ●● ●● ● ● ●●● ●● ● ● ● ● ● ● ● ● ●● ●●●● ●●● ●●●●●● ●● ● ●●●● ● ●●●●● ●●●●●● ●●●●●● ●● ●●●●● ●● ● ● ● ● ● ● ● ●●●●●● ●●●● ●●●●●●● ●●● ● ● ● ● ●● ●●●●●●●●●● ●● ●●● ●●● ● ● ● ● ● ●● ●●●●●●●●●●●● ●●●●● ● ●●● ● ● ● ●●● ●● ●●●●●●●●● ●●●●●●●● ●●●●● ●● ● ●● ● ●●● ● ●●●●● ●●● ●● ● ● ● ● ●●●●● ●● ●●●●●●●●●● ●●● ● ● ● ● ● ● ● ●● ● ●●● ●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ● ● ●●●● ●●●●● ●●●●●●●● ●● ● 68 100 ●●● ●●● ●●●●●●●●●●●● ●●●●● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●●●●●●●●● ●●●●●●●●● ● ●● ●● ● ● ● ● ●● ● ● ●●●● ●●●●●●● ●● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●● ● ● ● ● ●●●●●● ● ●●●●●● ●●●● ●● ●●●● ● ● ● ●● ●●● ● ●●●● ●●● ●●●● ●● ●●● ●●●●●●●● ●●●●● ● ● ● ●● ● ● ●● ●●●●● ● ●●●● ●● ●●●● ● ● ● ● ● ●● ● ●●●● ●●●●●●●●●●●●● ● ● ● ● ● ● ● ●●●●●●● ●●● ●●●● ● 66 ● ● ●●●●● ● ●● ● ● ● ● ●●●●●● ● ●●●● ● ●●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●●●●●●●●● ● ●●● ● ● ● ● ● 90 ●● ● ● ● ●● ●●● ● ● ● ● ●● ● ● offspring height (inches) Simulated offspringSimulated value ● ● ●● ● ● ● ●● ●● ●●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●●●●●● ● ● ● ● ● 64 ●●● ●● ●● ●● ● ● ● ● ●● ●●● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● 80 ● ● ● 62 ● ● ● ● ●

80 90 100 110 120 64 66 68 70 72

Simulated mid−parent value mid−parent height (inches)

>summary(lm(offVal~midPar,data=data))

●

120 Call: ● ● ● ● ● ● ● ● ● ● ●● ●● ● lm(formula = offVal ~ midPar, data = data) ● ● ●● ●● ● ● ●● ● ● ● ● ●●●● ● ● ●● ● ● ● ● ●●● ●● ● ● ● ●● ●● ●● ●●● ● ● ● ● ●● ●●● ●●●● ● ● ● ● ●● ●●●●●●●●● ● 110 ● ● ● ●● ● ●●●● ● ●● ● ● ● ●●● ●●●●●●●●●●●● ● ● ●● ●●● ●●●●● ● ●● ●● ● ●● ●● Residuals: ●●●● ●●●● ● ● ● ● ● ● ● ●●● ●● ●●●●● ●●●● ● ●●●●●● ● ● ● ●● ●● ●●●● ●●● ●● ● ● ● ●●● ● ● ● ● ●●●●●●●●●● ● ●●●●●●● ●● ●●●●● ● ● ●●●●●●●●●●●●●●●● ●●●●● ●●●● ● ● Min 1Q Median 3Q Max ●●●● ●● ●●●●● ●● ●●●●●●● ●● ● ●● ● ● ●●● ●●●●●●●●● ●● ●● ● ●●● ● ● ● ●●● ●● ●●●●●●●●● ●●●●●●●●●●●●●●●●● ●● ● ●● ● ●●● ● ●●●●● ●●●●● ● ● ● ●●●●● ●● ●●●●●●●●●● ●●● ● ● ● ●● ● ●●● ●●●●●● ●●●●●●●●●●●●● ●● -22.9329 -4.4264 -0.2072 4.6185 17.9158 ● ● ●●●● ●●●●● ●●●●●●●● ●● ● 100 ● ● ● ●●●● ● ● ●●●● ●● ●●●●●●●●●●●●● ● ● ● ● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ●●●●● ●●●●●●●● ● ●● ●● ●● ● ● ●●● ●●●●●●●● ●● ● ● ●●●●●●●●●●●●●●●● ●●● ● ● ●●●●●●●● ● ●●●● ●●●●●●●●● ●●●● ● ●● ● ●● ● ● ● ●●●● ●● ●● ●● ● ● ● ●● ●●●●●●● ●●●● ●●●●●● ●●● ●● ● ● ● ●● ● ● ● ●●●●●●●●●●●●●● ● ●●● ●●● ●● ●●● ●● ● Coefficients: ● ●● ●●●●●●●● ●●●●● ●●● ● ● ● ●●●● ● ●●● ● ●●● ● ● ●●● ● ● ●●● ● ● ●● ●● ● ●●●●●●●●● ● ●●● ● ●

90 ● ● ● ●● ●● ●●● ● ● Estimate Std. Error t value Pr(>|t|) ● ●● ●● ● Simulated offspringSimulated value ● ● ●● ● ● ● ●● ●● ●●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●●●●●● ● ●●● ●● ●● ●● ● ● ● (Intercept) 46.2453 4.2966 10.76 <2e-16 *** ● ●● ●●● ● ● ● ●● ● ●●● ● midPar 0.5343 0.0429 12.45 <2e-16 ***

80 ● ● ● --- Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 80 90 100 110 120

Simulated mid−parent value Residual standard error: 6.522 on 498 degrees of freedom Multiple R-squared: 0.2375, Adjusted R-squared: 0.2359 F-statistic: 155.1 on 1 and 498 DF, p-value: < 2.2e-16

● d ● a 1.0

0.5 The mean: 0.0 m = ap2 +2dpq + aq2 − 0.5 = a(p q)+2dpq genotypic or breeding value ● −a − 1.0 − 1.5

0 1 2

number of a alleles

● d ● a The mean: 1.0

2 2 0.5 m = ap +2dpq + aq

0.0 = a(p q)+2dpq

− 0.5 Breeding values: genotypic or breeding value ● −a

− 1.0 predictions from least-squares regression − 1.5

0 1 2

number of a alleles

M+2qα ● d ● a 1.0 Breeding values: M+(q−p)α predictions from least-squares 0.5 regression M−2pα 0.0 ↵ = a + d(p q) is average e↵ect of a

− 0.5 gene substitution ● genotypic value

genotypic or breeding value breeding value ● −a Variance of breeding values: − 1.0 2 Va =2pq↵ − 1.5

0 1 2

number of a alleles

1.5 Variance of breeding values: M+2qα ● d ● a 1.0 2 Va =2pq↵ M+(q−p)α 0.5

M−2pα Dominance variance: 0.0 V =(2pqd)2 − 0.5 d ● genotypic value

genotypic or breeding value breeding value ● −a − 1.0 Total genetic variance: − 1.5 0 1 2 Vg = Va + Vd

number of a alleles

Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Heritability Similarity among Relatives Co-ancestry, Relatedness, IBD, and Fraternity Response to Selection Multivariate Selection

Similarity among Relatives

Michael Morrissey, Intro to QG End of the last lecture: additive genetic variance genotypic (or total genetic) variance Heritability: V h2 = a Vp a.k.a. narrow-sense heritability V H2 = g Vp a.k.a. broad-sense heritability

Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Heritability Similarity among Relatives Co-ancestry, Relatedness, IBD, and Fraternity Response to Selection Multivariate Selection Heritability

Initially, easily misunderstood concept: does not refer to the genetic basis of a trait refers to genetic basis of variation in a trait

Initially, easily misunderstood concept: does not refer to the genetic basis of a trait refers to genetic basis of variation in a trait End of the last lecture: additive genetic variance genotypic (or total genetic) variance Heritability: V h2 = a Vp a.k.a. narrow-sense heritability V H2 = g Vp a.k.a. broad-sense heritability Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics)116 SEWALL WRIGHT Fundamentals of Quantitative Genetics Heritability Similarity among Relatives represents theCo-ancestry, constitution Relatedness,of the fertilized IBD, egg and and Fraternity hence all that is received Response to Selection Multivariate Selectionby the individual by inheritance. The environmental factors are sepa- rated into two elements, tangible environment (E) and the intangible 2 factors (D)which are not common even to litter mates, and yet appear to Where does the term h become responsible from? for much variation in early development. The path coefficients Po.,, Po., and p,., are represented by h, e and d respectively. From equation (1) we have the following equation which is of use in calculating ...and why is it squared? the relative importance of heredity and environment: h2 + d2 + e2 = 1

FIGURE2.-A diagram illustrating the relationsFigure between from: two Wright mated (1921) individuals Genetics and their progeny. H, HI, H" and 11"' are the genetic constitutions of the four individuals. G, G', G" and G"' are four gem-cells.Michael Morrissey,E and D represent Intro totangible QG external conditions and chance irregu- larities as factors in development. C represents chance at segregation as a factor in determining the composition of the germ-cells. Path coefficients are represented by small letters.

THE GRADING OF GAMETIC AND ZYGOTIC FORMULAE In applying the methods of correlation and of path coefficients to Men- delian inheritance, we must adopt scales of measurement for gametic and zygotic formulae, as well as for the physical characters in question. This would be easy to do if the effects of factors were always combined simply Gene and Genotype Frequencies (population genetics)116 SEWALL WRIGHT Fundamentals of Quantitative Genetics Heritability Similarity among Relativesrepresents the constitution of the fertilized egg and hence all that is received Co-ancestry, Relatedness, IBD, and Fraternity Response to Selectionby the individual by inheritance. The environmental factors are sepa- Multivariate Selectionrated into two elements, tangible environment (E) and the intangible factors (D)which are not common even to litter mates, and yet appear to be responsible for much variation in early development. The path coeffi- Parent-o↵spring covariancecients Po.,, Po., and p,., are represented by h, e and d respectively. From equation (1) we have the following equation which is of use in calculating the relative importance of heredity and environment: h2 + d2 + e2 = 1

If X Y with path coecient ! a,thenvarianceinY due to X is a2

Wright worked out: a2 = b2 = ab =0.5

P-0 correlation & 1 regression: 2 h2 o↵spring on mid-parent regression: h2 1 P-0 covariance: 2 Va

Mid-parent variance: em" 2 2 1 2 2 1 2 1" mp =2 e (1 2 ) + a (1 2 ) 1 2 ·2 · zm" 1 = + 2 2 e a 1" a mp mid-parent - o↵spring covariance: m" " 2 1 1 1 2 1 =2 1 = 2 ao ,mp a · · 2 · 2 2 a ef" 1 Regression of o↵spring on mid-parent e 2 f" 1 2 1" 2 a 2 zf" b = = h o,mp 1 (2+2) 1 e a zf" ao" 2 2

1" Is bo,mp a↵ected by covariance of af" parents? cs" no - it contributes equally to mp and ao,andsocancels

Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics)116 SEWALL WRIGHT Fundamentals of Quantitative Genetics Heritability Similarity among Relativesrepresents the constitution of the fertilized egg and hence all that is received Co-ancestry, Relatedness, IBD, and Fraternity Response to Selectionby the individual by inheritance. The environmental factors are sepa- Multivariate Selectionrated into two elements, tangible environment (E) and the intangible factors (D)which are not common even to litter mates, and yet appear to be responsible for much variation in early development. The path coeffi- Parent-o↵spring covariancecients - Po.,, non-random Po., and p,., are represented mating by h, e and d respectively. From equation (1) we have the following equation which is of use in calculating the relative importance of heredity and environment: h2 + d2 + e2 = 1

What if m is non-zero? P-O regression: ab + m ab = 1+m · 2 So, if there is covariance among parents, 1+m b = h2 PO 2 2b h2 = PO 1+m

2 1 1 Through Mum: h 2 2

So:h2 =2COV (full sibs)(if e2 = 0)

If Mum and Dad covary:

2 1 1 1 1 2 1 h 2( 2 2 + m 2 2 ) h 2(1 + m) 4 2 1 h 2 (1 + m)

Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics 116 SEWALL WRIGHT Heritability Similarity among Relatives Co-ancestry, Relatedness, IBD, and Fraternity represents the constitution of the fertilizedResponse egg and hence to Selection all that is received Multivariate Selection by the individual by inheritance. The environmental factors are sepa- rated into two elements, tangible environment (E) and the intangible factors (D)which are not common even to litter mates, and yet appear to Covariancebe responsible for much ofvariation full in early sibs development. The path coefficients Po.,, Po., and p,., are represented by h, e and d respectively. From equation (1) we have the following equation which is of use in calculating the relative importance of heredity and environment: h2 + d2 + e2 = 1

2 1 1 1 1 2 1 h 2( 2 2 + m 2 2 ) h 2(1 + m) 4 2 1 h 2 (1 + m)

2 1 1 Through Mum: h 2 2

So:h2 =2COV (full sibs)(if e2 = 0)

FIGURE2.-A diagram illustrating the relations between two mated individuals and their progeny. H, HI, H" and 11"' are the genetic constitutions of the four individuals. G, G', G" and G"' are four gem-cells. E and D represent tangible external conditions and chance irregu- larities as factors in development. C represents chance at segregation as a factor in determining the composition of the germ-cells. Path coefficients are represented by small letters. Michael Morrissey, Intro to QG THE GRADING OF GAMETIC AND ZYGOTIC FORMULAE In applying the methods of correlation and of path coefficients to Men- delian inheritance, we must adopt scales of measurement for gametic and zygotic formulae, as well as for the physical characters in question. This would be easy to do if the effects of factors were always combined simply Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics 116 SEWALL WRIGHT Heritability Similarity among Relatives Co-ancestry, Relatedness, IBD, and Fraternity represents the constitution of the fertilizedResponse egg and hence to Selection all that is received Multivariate Selection by the individual by inheritance. The environmental factors are sepa- rated into two elements, tangible environment (E) and the intangible factors (D)which are not common even to litter mates, and yet appear to Covariancebe responsible for much ofvariation full in early sibs development. The path coefficients Po.,, Po., and p,., are represented by h, e and d respectively. From equation (1) we have the following equation which is of use in calculating the relative importance of heredity and environment: h2 + d2 + e2 = 1 2 1 1 Through Dad: h 2 2

2 1 1 Through Mum: h 2 2

So:h2 =2COV (full sibs)(if e2 = 0)

If Mum and Dad covary:

2 1 1 1 1 2 1 h 2( 2 2 + m 2 2 ) h 2(1 + m) 4 2 1 h 2 (1 + m) FIGURE2.-A diagram illustrating the relations between two mated individuals and their progeny. H, HI, H" and 11"' are the genetic constitutions of the four individuals. G, G', G" and G"' are four gem-cells. E and D represent tangible external conditions and chance irregu- larities as factors in development. C represents chance at segregation as a factor in determining the composition of the germ-cells. Path coefficients are represented by small letters. Michael Morrissey, Intro to QG THE GRADING OF GAMETIC AND ZYGOTIC FORMULAE In applying the methods of correlation and of path coefficients to Men- delian inheritance, we must adopt scales of measurement for gametic and zygotic formulae, as well as for the physical characters in question. This would be easy to do if the effects of factors were always combined simply 2 2bPO 2 0.47 o↵spring on parent: h = = · 0.71 1+m 1+0.33 ⇡

Boag and Grant (1978 Nature 274:793-794) reported the following regressions for bill depth in one of Darwin’s ﬁnches (Geospiza fortis):

O↵spring on midparent 0.82 0.15 ± O↵spring on father 0.47 0.17 ± O↵spring on mother 0.48 0.13 ± Correlation of parents 0.33 What interpretations can we make about heritability? o↵spring-midparent regression = h2

Boag and Grant (1978 Nature 274:793-794) reported the following regressions for bill depth in one of Darwin’s ﬁnches (Geospiza fortis):

Michael Morrissey, Intro to QG seems esoteric but actually important: co-ancestry with self is 1 2 if not inbred, 1 2 (1 + f )ifinbred f is co-ancestry of parents

Coecient of co-ancestry (⇥); the probability that two alleles, each chosen at random from di↵erent relatives, are descended from a common ancestor, i.e., are Identical By Descent, or IBD

1 1 1 2 2 2

seems esoteric but actually m" f" m" important: co-ancestry with self is 1 2 if not inbred, 1 m" f" 2 (1 + f )ifinbred f is co-ancestry of parents

1 (1+ f ) Θmf 1 2 (1+ f ) 2

m" f" parent-o↵spring

x" y"

1 (1+ f ) Θmf 1 2 (1+ f ) 2

m" f" full-sib

x" y"

Coecient of fraternity (); the probability that genotypes of two individuals are identical, due to identity by descent of both alleles.

(compare to ⇥, which was about individual alleles, rather than genotypes)

1 (1+ f ) Θmf 1 2 (1+ f ) 2 2 x,y =⇥m⇥f +⇥mf m" f" for non-inbred, non-inbreeding parents, 1 x,y = 4 x" y"

The (de)composition if individual phenotypes of relatives x and y

x x x G(x)=µg + ↵i + ↵j + ij + ... y y y G(y)=µg + ↵i + ↵j + ij + ...

x x y y covariance due to additive e↵ects: E[(↵i + ↵j )(↵i + ↵j )] x y ↵i and ↵i contribute to similarity if they are the same 2 Va They are IBD with probability ⇥xy and e↵ect E(↵i )= 2 There are four combinations so covariance due to additive genetic e↵ects is 2⇥xy Va

VG = 2⇥xy Va +xy Vd ...

2 2 2 relationship A D AA 1 1 parent-o↵spring 2 4 1 1 grandparent-grandchild 4 16 1 1 half sibs 4 16 1 1 1 full sibs, dizygotic twins 2 4 4 1 1 avuncular 4 16 1 1 ﬁrst cousins 8 64 1 1 1 double ﬁrst cousins 4 16 16 1 1 second cousins 32 1024 monozygotic clones (clonemates) 1 1 1

Lynch and Walsh 1998 Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Heritability Similarity among Relatives Co-ancestry, Relatedness, IBD, and Fraternity Response to Selection Multivariate Selection Segregation What is wrong with this simulation?

n<-5000 # mum-dad-kid groups Va<- 2.5 # additive genetic variance

dad_a<-rnorm(n,0,sqrt(Va)) mum_a<-rnorm(n,0,sqrt(Va)) kid_a<-(dad_a+mum_a)/2

> c(var(dad_a),var(mum_a),var(kid_a)) [1] 2.456265 2.501833 1.235058

How would we correctly simulate o↵spring breeding values?

kid_a<-rnorm(n,(dad_a+mum_a)/2,sqrt(Va/2))

> c(var(dad_a),var(mum_a),var(kid_a)) [1] 2.456265 2.501833 2.412038

1 If a parents are inbred, then segregational variance is Va (2 fm ff ) 4 Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Heritability Similarity among Relatives Co-ancestry, Relatedness, IBD, and Fraternity Response to Selection Multivariate Selection Advanced: Epistasis

Hill et al. 2008’s perspective

Genotypic values: AA Aa aa BB 2a a 0 Bb a a a bb 0 a 2a

p1: frequency of A p2: frequency of B

Proportion Va

0.8

0.8 0.4 Hill et al. 2008’s perspective

0.4 0.2 Genotypic values: 0.6 1 AA Aa aa 0.0

2 0.0 0.4 0.8 BB 2a a 0 p Proportion Vaa

Bb a a a 0.4 0.8 bb 0 a 2a 0.8 0.4 p1: frequency of A 0.6 p2: frequency of B 0.2 0.0

0.0 0.4 0.8 p1

Non−equilibrium scenario: Equilibrium scenario: z1 z1 −4 −2 0 2 4 −4 −2 0 2 4

10 10 10 10

8 8

6 6 E(z2|z1) 5 E(z2|z1) 5 E(z2|z1) 4 4

2 2

0 0 0 0

0.00 0.02 0.04 0.06 0.00 0.02 0.04 0.06 p(z2) p(z2)

0.4 0.4

0.3 0.3 p(z1) 0.2 p(z1) 0.2

0.1 0.1

0.0 0.0

−4 −2 0 2 4 −4 −2 0 2 4 z1 z1

Wright’s 1935 perspective 0.8 2 z 0.6 Non−equilibrium scenario: Equilibrium scenario: z1 z1 0.4 −4 −2 0 2 4 −4 −2 0 2 4 10 10 10 10 0.2 8 8 proportion Va for for proportion Va

6 6 E(z2|z1) 5 E(z2|z1) 5 E(z2|z1) 0.0 4 4

2 2 −2 −1 0 1 2

0 0 0 0 1.0 0.00 0.02 0.04 0.06 0.00 0.02 0.04 0.06 2 p(z2) p(z2) z 0.8

0.4 0.4 0.6

0.3 0.3 0.4 p(z1) 0.2 p(z1) 0.2 proportion Vaa for for proportion Vaa 0.1 0.1 0.2

0.0 0.0 −2 −1 0 1 2 −4 −2 0 2 4 −4 −2 0 2 4 mean of z1 z1 z1

Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics The breeder’s equation Similarity among Relatives Theorems of selection Response to Selection Multivariate Selection

The response to selection

n<-1000 a<-rnorm(n,0,sqrt(0.5)) z<-rnorm(n,a,sqrt(0.5))

# truncation selection W<-(z>1)+0 plot(a,z,col=c("red","blue")[W+1])

● 3 ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ●● ●●● ● ● ●●● ●●●● ● ● ● ● ● ● ●●●●●●●●●●● ●● ●●● ●●●●●● ● ● ● ●● ●●●● ● ● ●●●●● ● ● ● ●●● ●●●●●●●●●●● ● ●● ●●● ●●●●●●●●●●●●●●● ●●● ●● 1 ●● ● ● ● ● ● ●● ●●●●●●●●●●●●●●●●● ●●●●● ● ● ●● ●●●●●●●●●●●● ●● ●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ● ● ●●●●● ●●●●●●●●●●●●●●●● ● ● ● ● ● ● ●● ●●●●●●●●●● ●●●●●● ●●●●●● ●● ●●● ● ●●● ●●●●● ●●●●●●●●●●●●●●●●● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● 0 ●●●●●●●●● ●●●●● ●●●●● ● z ●●●●●● ●●●●●●● ●● ●●●●● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ● ● ● ● ●● ●●●●●●●●●●●●●● ● ● ● ● ● ●●●●●● ●●●●●●●●●●●●●●●● ● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● ●●●●●●●●● ●●●●●●●●●●●●● ●●● ● ●●●●● ●●●● ●●●●●●●●● ●●● ● ● ● ●●●●●●● ●●● ●●●● ●● ●● ● − 1 ● ● ● ●● ●● ● ●● ●●●●●●●●●●●●●● ●●●●● ● ● ●●●●●● ● ●●●●● ● ●● ● ●●● ● ●●●● ● ● ● ●●●●●●●●● ● ● ● ●● ● ● ● ● ●● ●●● ● ●● ● ●●●● ●● − 2 ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● − 3 ● ●

−2 −1 0 1 2

Let’s ﬁll in the table

Scenario V V h2 S (¯z z¯)z¯ a p ⇤ strong truncation 0.5 1.0 0.5 weak truncation 0.5 1.0 0.5 blurry selection 0.5 1.0 0.5 strong truncation weak truncation blurry selection

Michael Morrissey, Intro to QG Note that expected mid-parent value isz ¯ z¯ ⇤ 2 2 E[¯z z¯]=h (¯z⇤ z¯)=h S 0 R = h2S

Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics The breeder’s equation Similarity among Relatives Theorems of selection Response to Selection Multivariate Selection The breeder’s equation

Hopefully, we concluded that z¯ = h2S

Recall that h2 is the regression of o↵spring on mid-parent value

z + z z =¯z + h2 m f z¯ o p 2 p ✓ ◆

Michael Morrissey, Intro to QG R = h2S

Hopefully, we concluded that z¯ = h2S

Recall that h2 is the regression of o↵spring on mid-parent value

z + z z =¯z + h2 m f z¯ o p 2 p ✓ ◆ Note that expected mid-parent value isz ¯ z¯ ⇤ 2 2 E[¯z z¯]=h (¯z⇤ z¯)=h S 0

Hopefully, we concluded that z¯ = h2S

Recall that h2 is the regression of o↵spring on mid-parent value

z + z z =¯z + h2 m f z¯ o p 2 p ✓ ◆ Note that expected mid-parent value isz ¯ z¯ ⇤ 2 2 E[¯z z¯]=h (¯z⇤ z¯)=h S 0 R = h2S

Michael Morrissey, Intro to QG Clones: R = H2S - H2 drastically reduced each generation

Two sexes: S + S S = m f 2 - can often select in only one sex (unless multiple generations)

Clones: R = H2S - H2 drastically reduced each generation

ing the last 233 generations of Strictly, Rselection= h showedS holds continued at progress for oil in IHO, RHO, a speciﬁcRLO, time, and SHO and (Dudley set and of environmentalLambert 2004) conditions and for protein in IHP, RHP, and RLP. Va a functionMeans are an of important allele measure of progress and an frequenciesimportant statistical tool in V aall function the work which of has the been e done to help understand the environmentgenetic control of and oil and protein. Mean values for indi- interactionvidual ears were with the units genes of selection. Means have been In practice,used the in conjunction breeder’s with other quantitative genetic equationtools can in work hold following well that for Figure 1. Plot of mean oil concentration against generation for Illinois High Oil (IHO), Reverse of Smith (1908). T us, fur- High Oil (RHO), Switchback High Oil (SHO), Illinois Low Oil (ILO), and Reverse Low Oil (RLO). several generationsther discussion of means will be included with discussion of variances or other quantitative genetic tools. strains tested (Dudley and Lambert, 2004) although signif - cant genetic variance was found in all three strains. Variances Dudley (2007) Crop Science Winter (1929) was the f rst to report the ef ect of divergentMichael Random Morrissey, Mating Intro to QG selection on standard errors and coef cients of variation Moreno-Gonzalez et al. (1975) reported estimates of (CVs). He showed that selection in the high direction for genetic variance obtained from a Design III study of the

either oil or protein decreased the CV among ears within F2 and random-mated F6 generations of the cross of gen- strains while selection in the low direction increased it. erations 70 of IHO × ILO. In accordance with quantita- Conversely, selection in the high direction increased the tive genetic theory, estimates of additive genetic variance

among ears within strains standard deviation while selec- in the F6 were signif cantly lower than in the F2 indicating tion in the low direction decreased it. Based on these the build-up of coupling phase linkages for alleles for high results, Winter (1929) concluded that continued progress oil in IHO and for alleles for low oil in ILO. Estimates of

in the high direction should be possible for both oil and dominance variance were similar in the F2 and F6 suggest- protein but that a selection limit was being approached ing a lack of ef ect of linkage on dominance variance.

in the low strains. Woodworth et al. (1952) noted the In a similar Design III study of the F2 and F6 genera- same trends in standard errors and CVs within strains as tions of the cross of generations 70 of IHP × ILP (Dudley reported by Winter (1929). 1994), additive genetic variance for protein was signif -

T e f rst estimates of genetic variance in the selected cantly less in the F6 than in the F2 suggesting the build- strains were from a half-sib mating design in generation up of coupling phase linkages for high protein alleles in 65 of IHO, ILO, IHP, and ILP (Dudley and Lambert, 1969). IHP and for low protein alleles in ILP. T e measurement

Signif cant genetic variance was found for both oil and of marker-QTL associations in the F2 and in subsequent protein in all four strains. Genetic variance in IHO was random-mated generations was proposed as a method of larger than in ILO for both oil and protein with the dif er- identifying tight marker-QTL associations. ence being larger for oil. Similarly the genetic variance in IHP was larger for both oil and protein than in ILP. Pre- Regression dicted gain for one cycle of 20% selection for the selected Dudley et al. (1974) f rst reported use of regression of means trait in all four strains was similar to the average gain per on cycles of selection to measure rate of progress through generation for the preceding 65 generations. generation 70. T e experiment was divided into segments

Genetic variances among S1 lines were estimated for the based on changes in breeding procedure and the selection earliest (generation 69) and latest (generation 98) cycles avail- environment. Response was measured by regression of able of IHP, IHO, and SHO (Dudley and Lambert, 2004). No means on cycles within segments. Realized heritabilities signif cant dif erences in genetic variances between genera- were calculated for each segment as regression of means on tions 69 and 98 were found for the selected trait in any of the cumulative selection dif erentials. Results of this process

INTERNATIONAL PLANT BREEDING SYMPOSIUM • DECEMBER 2007 S-23 Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics The breeder’s equation Similarity among Relatives Theorems of selection Response to Selection Multivariate Selection Cumulative response to selection

I I I I"! I I I I I I I I 1 I I I I I I I I 1 I I I I M I I 1 I I 1 I I 1 I I II I I [ I I I I I 1 I I I I I I I I I I40

24 46

Selection relaxed

10 10 I I I I I 1 I I I I 1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 I I I I 1 I I I I I I I I I I I 0 10 20 30 40 50 60 70 80 90 100 110 120 Generations Fig. 1. Female mean bristle numbers plotted against generationsYoo. (1980) Gen Res Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics The breeder’s equation Similarity among Relatives Theorems of selection Response to Selection Multivariate Selection Selection intensity - 1

The selection differential is a function of both the phenotypic variance and the fraction selected

20% selected

Vp = 1, S = 1.4 50% selected 20% selected V = 4, S = 1.6 p Vp = 4, S = 2.8 (C)

(A) (B)

S S S

Michael Morrissey, Intro to QG

The Selection Intensity, i As the previous example shows, populations with the same selection differential (S) may experience very different amounts of selection

The selection intensity i provides a suitable measure for comparisons between populations,

14 Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics The breeder’s equation Similarity among Relatives Theorems of selection Response to Selection Multivariate Selection Selection intensity - 2 S depends on ﬁtness-determining rules phenotypic variance

The selection differential is a function of both the phenotypic variance and the fraction selected A more general measure can be: 20% selected

Vp = 1, S = 1.4 50% selected 20% selected V = 4, S = 1.6 S p Vp = 4, S = 2.8 i = (C) SD(z) (A) (B) so, S S S 2 13 SD(a) R = iSD(z)=ihSD(a) SD(z)2

The Selection Intensity, i or As the previous example shows, populations with the 2 same selection differential (S) may experience very R = h i different amounts of selection Michael Morrissey, Intro to QG The selection intensity i provides a suitable measure for comparisons between populations,

Was all that fuss about i worthwhile? Consider that h2 is the correlation of breeding value with phenotype, i.e., it relates to how predictive the phenotype is of o↵spring value.

R = iSD(a)ru,a

Is there a better u than z? the mean of progeny weighted mean values of other relatives mean values of relatives grown in a variety of environments

S has been the change in the mean due to selection Intuitive for viability selection More generally, S is the change in the mean, weighted by ﬁtness

S = COV (z, w)

where w is relative ﬁtness, w = Wi i W¯

Fisher (1930) The Genetical Theory of Natural Selection The rate of increase in ﬁtness of any organism at any time is equal to its genetic variance in ﬁtness at that time

Modern interpretation The increase in relative fitness, within a generation, due to selection, is equal to the additive genetic variance in relative fitness at that time If the parent-o↵spring regression for fitness is linear, this will be the rate of adaptation across generations

The change in mean breeding value is equal to the covariance of a trait with relative ﬁtness.

also applies within generations predicts evolutionary change if parent-o↵spring regression is linear very useful in simulation - checking your own ideas leads to a measure of the adequacy of speciﬁc models of selection, genetics, and evolution

Michael Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Genetic and phenotypic correlations Similarity among Relatives Measuring multivariate selection (in the wild) Response to Selection Multivariate Selection

Multivariate Selection

Just as variance in a trait can be partitioned, so can covariance

P = G + E

2(z ) (z , z ) 2(a ) (a , a ) 2(e ) (e , e ) 1 1 2 = 1 1 2 + 1 1 2 (z , z ) 2(z ) (a , a ) 2(a ) (e , e ) 2(e )  1 2 2  1 2 2  1 2 2 Further decomposition possible.

Michael Morrissey, Intro to QG slope = 1 COVa(z1,z2) 2 Vp(z1) V (z , z )=2 slope V (z ) a 1 2 · ⇤ p 1

Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Genetic and phenotypic correlations Similarity among Relatives Measuring multivariate selection (in the wild) Response to Selection Multivariate Selection Estimating genetic correlations

● 1.5 ● ● ●

● 0.5

● ●

in offspring ● − 0.5 1 z ● − 1.5 −1.5 −0.5 0.5 1.5

● z1 in parent

slope = 1 Va(z1) 2 Vp(z1) V (z )=2 slope V a 1 · ⇤ p

● 1.5 1.5 ● ● ● ● ● ● ● 0.5 0.5 ●● ● ● ● ●

in offspring ● in offspring − 0.5 − 0.5 1 2 ● z z

● ● − 1.5 − 1.5 −1.5 −0.5 0.5 1.5 −1.5 −0.5 0.5 1.5 ●

● z1 in parent z1 in parent

slope = 1 Va(z1) slope = 1 COVa(z1,z2) 2 Vp(z1) 2 Vp(z1) V (z )=2 slope V V (z , z )=2 slope V (z ) a 1 · ⇤ p a 1 2 · ⇤ p 1

n<-1000 G<-matrix(c(0.5,-0.4,-0.4,0.5),2,2) a<-rmvnorm(n,c(0,0),G) E<-matrix(c(0.5,0,0,0.5),2,2) e<-rmvnorm(n,c(0,0),E) z<-a+e W<-(z[,1]>1)+0 plot(z[,1],z[,2],col=c("red","blue")[W+1]) lines(c(0,0),c(-10,10),lty="dashed") lines(c(-10,10),c(0,0),lty="dashed") lines(rep(weighted.mean(z[,1],W),2),c(-10,10)) lines(c(-10,10),rep(weighted.mean(z[,2],W),2))

● ● ● 3

● ● ● ● ●●●● ● ● ● ● ● ● ● ●● ● ● ●● 2 ●● ● ●● ● ● ● ● ● ● ●● ● ● ●●●● ●● ● ● ● ● ● ●●● ●● ● ●● ●● ●● ●●● ●●●● ● ●● ● ●●● ●● ●●●●● ●●● ●●●●●●● ● ● ● ● ●●●●●● ●● ●● ●●●● ● ●●● ●●●●●● ●● ● ●● ● ●● ● ● ● ● ●●●●● ●●● ●●●●● ●●● ● ● 1 ● ● ● ● ● ●●●●●● ●●●●●●●●●●●●● ●● ●●●●● ● ●● ● ●●●●●●● ●● ●●●●●●● ● ●●●● ●● ●●● ● ● ●●●● ●●●●●●●●●●●●●● ● ●●●●● ● ● ● ● ● ●●● ●● ●●●● ●●●●●●●●●●●●●●●●● ●● ● ●● ●●●●● ●●●●●●●●● ●●● ●● ●● ● ● ● ● ●●●●●●●●●●●●●●●● ●●●●●●●●● ● ● ● ●●●●●● ●●●●●●●●●●●●●●●●●●●●● ● ●●● ● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●● 0 ● ● ● ● ● ● ●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●● ●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●● ● ● ● ● ● ●●●●●●●●●●●● ●●●●●●● ●●●●●●● ● ● z[, 2] ● ● ●● ●●●●●●●●●●●●●●●●●● ●●●●●●● ● ● ●●● ●● ●●●●●● ●●●●●●●●●●● ●● ● ● ● ●● ●●● ●● ●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●● ●● ●● ●●●● ● ● ●●●● ●●●●●●●●●● ●●●● ●● ● ●● ● ●●●●● ●● ●●●●●●● ●● ● ● − 1 ● ●● ●●●●●●● ●●●● ●● ● ●●● ● ● ● ● ●●●● ●● ●●●●● ● ● ● ●●●● ●●●●●●● ● ●●● ● ● ● ●● ●● ●● ●●●● ● ● ●● ●●● ●●●● ● ●● ● ● ●● ● ● ● ●●● ● ● ●● ●● ●●● ●● ● ●● ● ● ●● − 2 ● ● ● ● ●● ● ● ● ●●

− 3 ● ● ●

−3 −2 −1 0 1 2 3

z[, Michael1] Morrissey, Intro to QG Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Genetic and phenotypic correlations Similarity among Relatives Measuring multivariate selection (in the wild) Response to Selection Multivariate Selection Direct and indirect selection - 3

phenotypic correlation induced a within-generation change in the unselected trait

● ● ● 3

● ● ● ● ●●●● ● ● ● ● ● ● ● ●● ● ● ●● 2 ●● ● ●● ● > weighted.mean(z[,1],W) ● ● ● ● ● ●● ● ● ●●●● ●● ● ● ● ● ● ●●● ●● ● ●● ●● ●● ●●● ●●●● ● ●● ● ●●● ●● ●●●●● ●●● ●●●●●●● ● ● ● ● ●●●●●● ●● ●● ●●●● ● ●●● ●●●●●● ●● ● ●● ● ●● ● ● ● ● ●●●●● ●●● ●●●●● ●●● ● ● 1 ● ●●●● ● ● ● ● ● ●●● ●● ● ● ●●●● ●●●●●●●●●●●●●●●●●●●●●●● ● ● ●● ● ●●●●●●● ●●●●●●●●● ●●●●● ● ● ● [1] 1.520319 ●● ● ●●● ●●●●●●● ●●●● ● ●● ●● ● ● ● ● ●●● ●● ●●●● ●●●●●●●●●●●●●●●●● ●● ● ●● ●●●●● ●●●●●●●●● ●●● ●● ●● ● ● ● ● ●●●●●●●●●●●●●●●● ●●●●●●●●● ● ● ● ●●●●●● ●●●●●●●●●●●●●●●●●●●●● ● ●●● ● ● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●● 0 ● ● ● ● ● ● ●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●● ●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●● ● ● ● ● ● ●●●●●●●●●●●● ●●●●●●● ●●●●●●● ● ● z[, 2] ● ●● ● ● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ● > weighted.mean(z[,2],W) ● ●● ● ●●●●● ●●●●●●● ● ● ● ● ●● ●●● ●● ●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●● ●● ●● ●●●● ● ● ●●●● ●●●●●●●●●● ●●●● ●● ● ●● ● ●●●●● ●● ●●●●●●● ●● ● ● − 1 ● ●● ●●●●●●● ●●●● ●● ● ●●● ● ● ● ● ●●●● ●● ●●●●● ● ● ● ●●●● ●●●●●●● ● ●●● ● ● ● ●● ●● ●● ●●●● ● ● ●● ●●● ● ● ●● ● [1] -0.5974898 ●●● ● ●● ● ● ● ●●● ● ● ●● ●● ●●● ●● ● ●● ● ● ●● − 2 ● ● ● ● ●● ● ● ● ●● > − 3 ● ● ● > weighted.mean(a[,1],W) −3 −2 −1 0 1 2 3

z[, 1] [1] 0.7561953 > weighted.mean(a[,2],W) [1] -0.5992512

Michael Morrissey, Intro to QG S(z1)=1.52, and S(z )= 0.60 2 Traits had heritabilities of 0.5 R(z )=0.5 1.52 = 0.76, and 1 · R(z )=0.5 0.60 = 0.30 2 ·

> weighted.mean(z[,1],W) Both traits had non-zero [1] 1.520319 selection di↵erentials: > weighted.mean(z[,2],W) [1] -0.5974898 > > weighted.mean(a[,1],W) [1] 0.7561953 > weighted.mean(a[,2],W) [1] -0.5992512

Michael Morrissey, Intro to QG Traits had heritabilities of 0.5 R(z )=0.5 1.52 = 0.76, and 1 · R(z )=0.5 0.60 = 0.30 2 ·

> weighted.mean(z[,1],W) Both traits had non-zero [1] 1.520319 selection di↵erentials: > weighted.mean(z[,2],W) S(z1)=1.52, and [1] -0.5974898 S(z2)= 0.60 > > weighted.mean(a[,1],W) [1] 0.7561953 > weighted.mean(a[,2],W) [1] -0.5992512

> weighted.mean(z[,1],W) Both traits had non-zero [1] 1.520319 selection di↵erentials: > weighted.mean(z[,2],W) S(z1)=1.52, and [1] -0.5974898 S(z2)= 0.60 > > weighted.mean(a[,1],W) Traits had heritabilities of 0.5 [1] 0.7561953 R(z1)=0.5 1.52 = 0.76, and · > weighted.mean(a[,2],W) R(z2)=0.5 0.60 = 0.30 · [1] -0.5992512

Michael Morrissey, Intro to QG genetic regression of z2 on z1: b = COVa(z1,z2) a2,a1 VARa(z2)

z¯ = COVa(z1,z2) z¯ 2 VARa(z2) 1 = 0.8 0.76 = 0.60 ·

The univariate breeder’s equation didn’t work so well...

Michael Morrissey, Intro to QG z¯ = COVa(z1,z2) z¯ 2 VARa(z2) 1 = 0.8 0.76 = 0.60 ·

The univariate breeder’s equation didn’t work so well...

Worked for z1 R(z )=0.5 1.52 = 0.76 > weighted.mean(z[,1],W) 1 · [1] 1.520319 Remember, only z1 was > weighted.mean(z[,2],W) selected. How does evolution [1] -0.5974898 of z1 predict z2? genetic > regression of z2 on z1: b = COVa(z1,z2) > weighted.mean(a[,1],W) a2,a1 VARa(z2) [1] 0.7561953 > weighted.mean(a[,2],W) [1] -0.5992512

The univariate breeder’s equation didn’t work so well...

Michael Morrissey, Intro to QG We could simply have applied the multivariate equivalent of all the terms in the breeder’s equation R = h2S 1 R = GP S

We just did two things:

1 we ignored S(z2) 2 we used information in G calculate the indirect response. (1) requires either knowledge of the mechanism of selection, or information about phenotypic correlations (2) requires information about genetic correlations We could simply have applied the multivariate equivalent of all the terms in the breeder’s equation R = h2S 1 R = GP S

Michael Morrissey, Intro to QG > G %*% solve(P) %*% S [,1] [1,] 0.7591917 [2,] -0.6044011 >

>G [,1] [,2] [1,] 0.5 -0.4 [2,] -0.4 0.5 > > P<-G+E >P [,1] [,2] [1,] 1.0 -0.4 [2,] -0.4 1.0 > > S<-c(weighted.mean(z[,1],W)-mean(z[,1]), + weighted.mean(z[,2],W)-mean(z[,2])) >S [1] 1.5249438 -0.5962006 >

>G > G %*% solve(P) %*% S [,1] [,2] [,1] [1,] 0.5 -0.4 [1,] 0.7591917 [2,] -0.4 0.5 [2,] -0.6044011 > > > P<-G+E >P [,1] [,2] [1,] 1.0 -0.4 [2,] -0.4 1.0 > > S<-c(weighted.mean(z[,1],W)-mean(z[,1]), + weighted.mean(z[,2],W)-mean(z[,2])) >S [1] 1.5249438 -0.5962006 >

What if we did not know that (artiﬁcial, truncation) selection was applied only to z1?

1 = P S

> solve(P) %*% S [,1] [1,] 1.53150418 [2,] 0.01640106 >

What if we did not know that (artiﬁcial, truncation) selection was applied only to z1?

1 = P S

> solve(P) %*% S [,1] [1,] 1.53150418 [2,] 0.01640106 >

if 1 R = GP S and 1 = P S then R = G This is no mere re-statement.

Michael Morrissey, Intro to QG selection smallness darkness of 33 for 37

selection smallness darkness of 33 for 37

is the direction of maximally-increasing ﬁtness.

For our example: As vectors, G means this:

z1 > 1 gives survival z2 despite covariance of z2 with fitness, increasing z2 would not increase fitness β z1 increasing z1 increases fitness R

is a regression coecient As a least-squares regression of Y on X

1 ˆ = XT X XY ⇣ ⌘ The response is relative ﬁtness w,andthepredictorsaretraits. Consequently, we can estimate by multiple regression of relative ﬁtness on multivariate phenotype.

wi =1+1z1,i + 2z2,i ... + ei

Michael Morrissey, Intro to QG 248 The American Naturalist

Table 4: Number of estimates of quadratic selection in the database as a function of taxon, trait type, and ﬁtness component Taxon Trait Fitness component

a Estimates of quadratic selection gradients, gii/gij Invertebrates 215/44 Morphology 358 Mating success 139/26 Plants 147/59 Life history/phenology 77 Survival 112/19 Vertebrates 103/6 Principal component 9 Fecundity 199/64 … … Behavior 15 Total fitness 0/0 … … Interaction 109 Net reproductive rate 0/0 … … Other 6 Other 0/0 Estimates of quadratic selection differentialsb Invertebrates 56 Morphology 183 Mating success 52 Plants 69 Life history/phenology 28 Survival 110 Vertebrates 104 Principal component 6 Fecundity 65 … … Behavior 3 Total fitness 2 … … Interaction NR Net reproductive rate 0 … … Other 3 Other 0 Note: NR p not recorded. a N p 465/109 total estimates. b N p 229 total estimates. ably remain and another set of researchers might code the overall “average” strength of selection is unlikely to these same studies somewhat differently, this process be very informative. We focus our analyses on the distri- helped to ensure that our codings are internally consistent. butions of selection strengths and on how methodological (e.g., type of study, sample size) or biological (e.g., trait GeneAnalysis and Genotype Frequencies (population genetics) type, fitness component, taxon) characteristics may con- Fundamentals of Quantitative Genetics Genetic and phenotypic correlations Similarity among Relatives The database includes a highly heterogeneous set of studies tributeMeasuring to multivariate the heterogeneity selection (in the wild) of selection strengths. We ex- Response to Selection and study systems with disparate biological characteristics:Multivariate Selection plore these issues graphically. We did not conduct formal Distribution of selection gradient (estimates) in nature

Kingsolver et al. (2001) Am Nat Michael Morrissey, Intro to QG

Figure 1: Linear selection gradient estimates (b) as a function of sample size (log10 scale;N p 993 estimates). The statistical significance (at the P p .05 level) of each estimate is given: filled circles indicate significantly different from 0; open circles indicate not significant; x’s indicate significance of the estimate not available.

This content downloaded from 138.251.156.7 on Mon, 6 May 2013 10:28:09 AM All use subject to JSTOR Terms and Conditions Gene and Genotype Frequencies (population genetics) Fundamentals of Quantitative Genetics Genetic and phenotypic correlations Similarity among Relatives Measuring multivariate selection (in the wild) Response to Selection Multivariate Selection Missing variables - 1

n<-1000 E<-rnorm(n,0,1) a<-rnorm(n,0,sqrt(0.5)) z<-a+0.5*E+rnorm(n,0,sqrt(0.25)) W<-(E>0)+0

interpret the simulation ecologically, esp. w.r.t E is there selection of the trait, z?

Michael Morrissey, Intro to QG > summary(lm(w~z))$coefficients Estimate Std. Error t value Pr(>|t|) (Intercept) 0.9972185 0.02996689 33.27734 5.827234e-164 z 0.4435020 0.03002950 14.76888 8.637279e-45

> cov(a,w) [1] 0.02048253

> weighted.mean(z,W)-mean(z) [1] 0.4416373 > w<-W/mean(W) > cov(z,w) [1] 0.4420794

Michael Morrissey, Intro to QG > cov(a,w) [1] 0.02048253

> weighted.mean(z,W)-mean(z) [1] 0.4416373 > w<-W/mean(W) > cov(z,w) [1] 0.4420794

> summary(lm(w~z))$coefficients Estimate Std. Error t value Pr(>|t|) (Intercept) 0.9972185 0.02996689 33.27734 5.827234e-164 z 0.4435020 0.03002950 14.76888 8.637279e-45

> weighted.mean(z,W)-mean(z) [1] 0.4416373 > w<-W/mean(W) > cov(z,w) [1] 0.4420794

> summary(lm(w~z))$coefficients Estimate Std. Error t value Pr(>|t|) (Intercept) 0.9972185 0.02996689 33.27734 5.827234e-164 z 0.4435020 0.03002950 14.76888 8.637279e-45

> cov(a,w) [1] 0.02048253

Michael Morrissey, Intro to QG > cov(z_1,W/mean(W)) [1] 0.4314736 > summary(lm(I(W/mean(W))~z_1+z_2))$coefficients Estimate Std. Error t value Pr(>|t|) (Intercept) 0.98449160 0.01819543 54.106531 6.502900e-299 z_1 0.03597539 0.02045771 1.758525 7.896497e-02 z_2 0.75251185 0.02133128 35.277387 1.369164e-177

=0; ecologically irrelevant! n<-1000 z_1 <- rnorm(n,0,1) z_2 <- 0.5*z_1 + rnorm(n,0,sqrt(0.75)) W <- (z_2>0)+0

interpret the simulation ecologically, esp. w.r.t z_1 is there selection of the trait, z_1?

=0; ecologically irrelevant! n<-1000 z_1 <- rnorm(n,0,1) z_2 <- 0.5*z_1 + rnorm(n,0,sqrt(0.75)) W <- (z_2>0)+0

interpret the simulation ecologically, esp. w.r.t z_1 is there selection of the trait, z_1? > cov(z_1,W/mean(W)) [1] 0.4314736 > summary(lm(I(W/mean(W))~z_1+z_2))$coefficients Estimate Std. Error t value Pr(>|t|) (Intercept) 0.98449160 0.01819543 54.106531 6.502900e-299 z_1 0.03597539 0.02045771 1.758525 7.896497e-02 z_2 0.75251185 0.02133128 35.277387 1.369164e-177

Michael Morrissey, Intro to QG