Pi = Gi Ei

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Pi Gi Ei
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Pi Gi Ei
What is
Some algebra
Pi 2 (Gi Ei)2
variance of phenotype
Sum over all individuals
Take the mean
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Variance P Variance G Variance E
something else
Pi Gi Ei
true for individuals
VP VG VE 2 Cov (G,E) true
for populations
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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)
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Look at G only for a while
What is this phenotype?
Let freq A1 0.5
pi .5
What is ?
0.5
pi 0
pi -.5
What is this phenotype?
pi -0.5 0
0.5
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What will offspring of A1A1 look like?
1/2 will be A1A2 and 1/2 will be A1A1
Offspring of random mating are 1/2 as far from
mean as their parents were
For offspring of A2A2
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Additive
Genotype score gives us the phenotype Offspring
are 1/2 as deviant as parents Well-behaved
inheritance
P G A
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Dominance
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.
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We want P G A D
What will offspring of A1A1 look like?
What will offspring of A2A2 look like?
What will offspring of A1A2 look like?
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We want P G A D
Additive Offspring are 1/2 as deviant as
parents Well-behaved inheritance
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(No Transcript)
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Now we have a component, the Additive component
that predicts what the offspring will be
However we still want P G A D
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What is D?
Now we have P A D
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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.)
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Covariance
Each point is (xi, yi)
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VP VG VE 2 Cov (G,E)
What is Cov (G,E)? Say Gene-Environment
Covariance G by E
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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?
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What about parents and offspring?
When offspring resemble parents their phenotypes
covary
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Let us ignore E for a moment
Ai Di
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
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Cov(Offspring,Parents) 1/2 VA
Heritability, h2 is defined as
The proportion of phenotypic variance that is
additive
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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
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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
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h2 0.7-0.8
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h2 0 h2 0.5 h2 1
offspring mean
h2
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Parents and offspring share more than just genes,
they share environments
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Pi Ai Di Ei
VP VA VD VE some other stuff
(covariances)
What is parental phenotype? Pi Ai Di
EiP
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
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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
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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
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R h2 s
Often h2 R / s
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s -- Selection Differential
With a single gene the change in phenotype is the
change in allele frequency
With a quantitative trait R h2 s
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How big are selection differentials?
R h2 s
Selection differentials
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How much heritability is there? Why is that
important? How do traits differ?
R h2 s Or resemblance among relatives