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

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

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

4. 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)

5. Look at G only for a while
Genotype
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 =

6. A2A2 A1A2 A1A1 0.5 1.0 Offspring of random mating are 1/2 as far
What will offspring of A1A1
look like?
Genotype
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

7. A2A2 A1A2 A1A1 0.5 1.0 Additive P = G = A 0 1 2 Phenotype
Genotype
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

8. Dominance
Genotype
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.

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

10. We want: P = G = A + D A A2A2 A1A2 A1A1 0.5 1.0 0 1 2
Genotype
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

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

12. However we still want: P = G = A + D
Genotype
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

13. What is D? A2A2 A1A2 A1A1 0.5 1.0 Now we have P = A + D 0 1 2
Genotype
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

14. 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.)

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

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

17. 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?

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

19. 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

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

21. 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 =

22. 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 =

23. h2 =

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

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

26. 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

27. 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

28. 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

29. 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

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

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

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