Pi = Gi + Ei Pi = pi - p Gi = gi - g Ei = ei - e _ _ _ 0 1 2 Phenotype Genotype 0 1 2 Phenotype Ei = ei - e _

Pi = Gi + Ei Pi 2 = (Gi + Ei)2 What is: Some algebra: = variance of phenotype Sum over all individuals: Take the mean:

= variance P Variance P = Variance G + Variance E + something else Pi = Gi + Ei true for individuals VP = VG + VE + 2 Cov (G,E) true for populations

Pi = Gi + Ei VP = VG + VE When the Phenotype is the sum of the Genotype and the Environment, then the Variance in Phenotypes is the sum of the Variances in Genotype and Environment. (plus the covariance term, which we will come back to later)

Look at G only for a while Genotype 0 1 2 Phenotype A2A2 A1A2 A1A1 0.5 1.0 What is this phenotype? Let freq A1 = 0.5 pi = .5 _ p What is ? =0.5 What is phenotype? pi = 0 What is this phenotype? pi = -.5 pi = -0.5 0 0.5

A2A2 A1A2 A1A1 0.5 1.0 Offspring of random mating are 1/2 as far What will offspring of A1A1 look like? Genotype 0 1 2 Phenotype A2A2 A1A2 A1A1 0.5 1.0 1/2 will be A1A2 and 1/2 will be A1A1 on average phenotype = 0.75 = .25 Offspring of random mating are 1/2 as far from mean as their parents were For offspring of A2A2

A2A2 A1A2 A1A1 0.5 1.0 Additive P = G = A 0 1 2 Phenotype Genotype 0 1 2 Phenotype A2A2 A1A2 A1A1 0.5 1.0 Additive Genotype score gives us the phenotype Offspring are 1/2 as ‘deviant’ as parents Well-behaved inheritance P = G = A

Dominance Genotype 0 1 2 Phenotype A2A2 A1A2 A1A1 0.5 1.0 What is the mean phenotype? = 0.75 Now the genotype score does not give us the phenotype. P  A Offspring still resemble their parents and it would still be very useful to be able to determine what is inherited.

We want: P = G = A + D A2A2 A1A2 A1A1 0.5 1.0 0 1 2 What will offspring of A1A1 look like? Genotype 0 1 2 Phenotype A2A2 A1A2 A1A1 0.5 1.0 p = 1 p = .5 What will offspring of A2A2 look like? What will offspring of A1A2 look like?

We want: P = G = A + D A A2A2 A1A2 A1A1 0.5 1.0 0 1 2 Genotype 0 1 2 Phenotype A2A2 A1A2 A1A1 0.5 1.0 If offspring are here where should the parents be? Additive: Offspring are 1/2 as ‘deviant’ as parents Well-behaved inheritance

A2A2 A1A2 A1A1 0.5 1.0 0 1 2 Phenotype If offspring are here Genotype 0 1 2 Phenotype A2A2 A1A2 A1A1 0.5 1.0 A If offspring are here where should the parents be?

However we still want: P = G = A + D Genotype 0 1 2 Phenotype A2A2 A1A2 A1A1 0.5 1.0 Now we have a component, the Additive component that predicts what the offspring will be However we still want: P = G = A + D

What is D? A2A2 A1A2 A1A1 0.5 1.0 Now we have P = A + D 0 1 2 Genotype 0 1 2 Phenotype A2A2 A1A2 A1A1 0.5 1.0 A = .5, D = -.25 A = -.5, D = -.25 A = 0, D = .25 Now we have P = A + D

VP = VA + VD + VE + some other stuff (covariances) VP = VG + VE Pi = Gi + Ei Pi = Ai + Di + Ei VP = VA + VD + VE + some other stuff (covariances) Additive and Dominance components are both genetic However, dominance is not inherited--it is a relationship between alleles only one of which is inherited. Additive component is inherited. When an individual reproduces only half of its genes (sexual reproduction) are transmitted. The additive component is relevant for natural selection. (It is what is inherited.)

Covariance Mean of x Mean of y Each point is (xi, yi)

VP = VG + VE + 2 Cov (G,E) What is Cov (G,E)? Say: “Gene-Environment Covariance” “G by E”

What happens to the Phenotypic Variance if there is a substantial G by E term? Suppose genetic variation for size in plants and two types of environment--one that makes them large and one that makes them small. If plants with genes for small size are found in places that make plants small (and vice versa) what is the sign of the G by E term? What about the converse? What sort of experimental techniques eliminate G by E?

What about parents and offspring? When offspring resemble parents their phenotypes covary

The product = 1/2 Ai2 + 1/2 Ai Di Parents Offspring Let us ignore E for a moment What is (pi - ) ? p = Ai + Di What is (oi - ) ? o = 1/2 Ai The product = 1/2 Ai2 + 1/2 Ai Di The mean over all pairs = 1/2 VA + 1/2 Cov (A,D) = 1/2 VA

Cov(Offspring,Parents) = 1/2 VA Heritability, h2 is defined as: The proportion of phenotypic variance that is additive

Regression want to predict y from x Parents Offspring Suppose we want to predict the phenotype of the offspring from the phenotype of the parents. Regression want to predict y from x We produce a regression coefficient or slope for a line The line goes through the mean x and mean y Regression coefficient =

Offspring have two parents Which to choose? Use the average, call it the Midparent Variance of midparents is less than variance of one parent (the variance of an average is always less) VMidparent = 1/2 VP, therefore regression of offspring on Midparent =

h2 = 0.7-0.8

h2 ~ 0 h2 ~ 0.5 h2 ~ 1 offspring mean = h2

Parents and offspring share more than just genes, they share environments

VP = VA + VD + VE + some other stuff (covariances) Pi = Ai + Di + Ei VP = VA + VD + VE + some other stuff (covariances) What is parental phenotype? Pi = Ai + Di + EiP Parents Offspring What is offspring phenotype? Oi = 1/2 Ai + EiO CovO,P = 1/2 VA + 1/2 Cov (A,D) + 1/2 Cov (A,EP ) + Cov (A,EO ) + Cov (D,EO ) + Cov (EP,EO ) CovO,P = 1/2 VA + “G by E terms” + covariance in environment

Sibling Siblings have the same parents They have resemblance through to parents---AND it is possible for both to get the same alleles. In that case their phenotypes will be influenced by Dominance in the same way. Covsiblings = 1/2 VA + 1/4 VD

R = h2 s How does a population respond to selection? On average, Mid-Parent Offspring How does a population respond to selection? On average, Offspring = h2 Parents If we only allow some parents to breed (e.g. above the mean) Then the offspring will be larger. By how much? offspring = h2 Parents R = h2 s

R = h2 s Often: h2 = R / s R Mean of Parents Threshold for Survival s Mean of Surviving Parents Mean of Offspring R R = h2 s Often: h2 = R / s

s -- Selection Differential With a single gene the change in phenotype is the change in allele frequency: With a quantitative trait: R = h2 s

How big are selection differentials? R = h2 s Selection differentials

How much heritability is there? Why is that important? How do traits differ? R = h2 s Or resemblance among relatives