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_
Pi =
Gi
+
Pi = pi - p
Ei
_
Gi = gi - g
_
Phenotype
Ei = ei - e
0
1
Genotype
2
Pi =
Gi
+
Ei
What is:
Some algebra:
Pi 2 =
(Gi +
1
1
2
Pi   ( pi  p ) 2

n i
n
Ei)2
= variance of phenotype
Sum over all individuals:
P
i
2

 (G
i
+
Ei )2
i
Take the mean:
1
1
2
Pi   (G i +

n i
n
Ei ) 2
1
1
2
Pi   (G i +

n i
n
1
1
1
2
2
Pi   G i +

n i
n
n
Ei )

2
1
2
Pi = variance P

n i
1
Ei  2  Gi Ei
n
2
Variance P = Variance G + Variance E + something else
Pi = Gi
VP
=
+
Ei
true for individuals
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
What is this phenotype?
1.0
Let freq A1 = 0.5
Phenotype
pi = .5
_
What is p ? =0.5
0.5
What is phenotype? pi = 0
0
What is this phenotype? pi = -.5
0
1
2
Genotype
A2A2
pi = -0.5
A1A2
A1A1
0
0.5
What will offspring of A1A1
look like?
1/2 will be A1A2
and 1/2 will be A1A1
Phenotype
1.0
on average phenotype = 0.75
0.5
= .25
0
0
1
2
Genotype
A2A2
For offspring of A2A2
A1A2
A1A1
Offspring of random
mating are 1/2 as far
from mean as their
parents were
Additive
Phenotype
1.0
Genotype score gives us the phenotype
Offspring are 1/2 as ‘deviant’ as parents
Well-behaved inheritance
0.5
0
P=G =A
0
1
2
Genotype
A2A2
A1A2
A1A1
Dominance
What is the mean phenotype?
= 0.75
Phenotype
1.0
0.5
Now the genotype score does
not give us the phenotype.
0
P A
0
1
2
Genotype
A2A2
A1A2
A1A1
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
What will offspring of A1A1
look like?
Phenotype
1.0
p=1
0.5
p = .5
What will offspring of A2A2
look like?
0
0
1
2
Genotype
A2A2
A1A2
A1A1
What will offspring of A1A2
look like?
We want:
P=G =A+D
A
Phenotype
1.0
If offspring are here
where should the parents
be?
0.5
Additive:
Offspring are 1/2 as ‘deviant’ as parents
Well-behaved inheritance
0
0
1
2
Genotype
A2A2
A1A2
A1A1
Phenotype
1.0
0.5
If offspring are here
where should the parents
be?
A
0
0
1
2
Genotype
A2A2
A1A2
A1A1
Phenotype
1.0
0.5
Now we have a component, the
Additive component that predicts
what the offspring will be
0
0
1
2
Genotype
A2A2
A1A2
A1A1
However we still want: P = G = A + D
What is D?
Phenotype
1.0
0.5
A = 0, D = .25
0
A = .5, D = -.25
A = -.5, D = -.25
0
1
2
Genotype
A2A2
A1A2
A1A1
Now we have P = A + D
Pi = Gi
VP
=
+
Ei
VG + VE
Pi = Ai + Di +
VP
Ei
= 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.)
Mean of y
Covariance
Each point is (xi, yi)
Mean of x
1
( xi  x )( yi  y )  Cov( x, y)

n i
1
1
2
Pi   (G i +

n i
n
1
1
1
2
2
Pi   G i +

n i
n
n
VP
=
Ei ) 2

1
Ei  2  Gi Ei
n
2
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?
Offspring
What about parents and offspring?
Parents
When offspring resemble parents their phenotypes covary
Offspring
Let us ignore E for a moment
Parents
1
cov( P, O)   ( pi  p )(oi  o )
n
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
Offspring
Cov(Offspring,Parents)
= 1/2 VA
Parents
Heritability,
h2
is defined as:
The proportion of phenotypic
variance that is additive
VA
 h2
VP
Offspring
Suppose we want to predict the
phenotype of the offspring
from the phenotype of the
parents.
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 =
cov XY
VA
 (1 / 2)
 (1 / 2)h 2
Var x
VP
Offspring
Offspring have two parents
Which to choose?
Use the average, call it the
Midparent
Parents
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 =
cov OM VA

 h2
VarM
VP
h2 = 0.7-0.8
h2 ~ 0.5
offspring
mean
h2 ~ 0
= h2
h2 ~ 1
Parents and offspring
share more than just genes,
they share environments
Pi = Ai + Di +
= VA + VD + VE + some other stuff (covariances)
What is parental phenotype?
Pi = Ai + Di + EiP
Offspring
VP
Ei
What is offspring phenotype?
Oi = 1/2 Ai + EiO
Parents
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
Sibling
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
Offspring
How does a population
respond to selection?
On average,
Offspring = h2 Parents
Mid-Parent
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
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:
 sq 2 (1  q)
q 
1  sq 2
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