Download Phenotype-Genotype covariances, statistical background

What matters for evolution is the additive
genetic variance VA
V P = VA + VD + VI + VE
Why?
The additive genetic variance component is not
affected by interactions within and between loci,
and therefore the genetic part that predictably
passed on from parent to offspring.
Some Definitions
• Genotypic value: the phenotypic value a genotype would
produce in the absence of environmental effects. A property of
the diploid genotype. Not (fully) passed on to next generation.
OR (working definition): The average phenotype of a genotype
across all environments.
• Average effect: the mean deviation from the population mean
of individuals which received that gene from one parent, the
gene received from the other parent having come at random
from the population (Falconer & Mackay 1996). Not easily
measurable.
• Breeding value: mean genotypic value of an individual‘s
offspring (determined by the average effect of the gene). Also
termed „additive genotype“, or „additive genetic value“. A
property of the haploid genotype. Passed on to next generation!
The „variance in breeding values“ determines the response to
selection and hence evolutionary change.
1
From Values to Means and Variances: The
Normal (Gaussian Distribution)
A: small mean, large variance
B: small mean, small variance
C: large mean, large variance
Density = (2πVP ) e
0.5
⎡ ( P − P )2 ⎤
⎢− i
⎥
2VP ⎥⎦
⎢⎣
The standard normal distribution is defined by two parameters, the mean and
the variance
From Values to Means and Variances:
The Normal Distribution
Phenotypic Mean
1
P=
N
Phenotypic Variance
∑ Pi
i =1
∑ (P − P )
N
N
Vp =
i =1
The
The standardized
standardized normal
normal distribution:
distribution:
Values
Values minus
minus mean,
mean, devided
devided by
by standard
standard deviation
deviation
→
→ mean
mean of
of distribution
distribution == 0,
0, Variance
Variance == 1;
1;
→
→ N(0,1).
N(0,1).
2
i
N −1
P −P
Pˆi = i
Vp
xˆi =
xi − x
sd x
2
From Gene-Frequencies to Population Means
Genotype
Frequency
Genotypic Value
Mean value per genotype
AA
p2
2pq
q2
+a
d
-a
p2a
2pqd
Aa
aa
Population mean
genotypic/phenotypic
value (sum of products)
-q2a
G = P = p 2 a + 2 pqd − q 2 a
= a( p − q )( p + q ) + 2 pqd
= a( p − q ) + 2 pqd
From Gene-Frequencies to Population Means
m Loci (no epistasis):
m
m
j =1
j =1
P = G = ∑ a j ( p j − q j ) + 2 ∑ d j p jq j
j: index for locus identity
3
From Gene-Frequencies to Population Variances
Additive Variance (variance of breeding values)
VA = 2 pq[a + d (q − p )]
2
Note: The additive genetic variance partially contains
variance due to dominance effects!
Dominance Variance (Variance of dominance deviations)
VD = (2 pqd )
2
Note: The dominance genetic variance is purely based on variation
from dominance effects!
From Gene-Frequencies to Population Variances
VA = 2 pq[a + d (q − p )]
2
VD = (2 pqd )
2
Pure Additivity (d=0)
Complete Dominance (a=d)
Genotypic Variance (VG)
Additive Variance (VA)
Complete Overdominance (a=0)
Dominance Variance (VA)
4
From Gene-Frequencies to Population Variances
Level: Population
Variance components: VP = VA + VD + VI + VE
Phenotypic
Additive
Dominance
Interaction
(epistatic)
Environmental
Genotypic
Summary
We built now a distribution of phenotypes in a population
by simulating genetic and environmental effects on the
phenotypes. We used a “bottom-up” approach, starting
with assumed gene action patterns and predicting the
phenotypic distribution as determined by genotype
frequencies and gene action effects.
In applied quantitative genetics, we usually do the
opposite – measuring the distribution of phenotypes in a
population and designing experiments to separate
statistically the different genetic and environmental
components of variation. This can be said to be a “topdown” approach to inferring genetic components
involved in generating heritable trait variation.
5
Heritability
Non-additive Variance
Broadsense
V
VA + VD + VI
h2 = G =
VP VA + VD + VI + VE
Broad-sense heritability: „degree of genetic determination“
Narrowsense
V
VA
h2 = A =
VP VA + VD + VI + VE
= regression of breeding value on phenotypic value
= phenotype – additive genotype covariance
Heritability
6
The importance of heritability
The „Breeders Equation“
R = h2S
Va
=
S
Vp
R
R: response to selection
S: selection differential; the difference between the mean
trait value of selected individuals and the whole population
before selection; can also be estimated by the covariance of
fitness with the trait.
h2
1
S
Empirical Estimation of
Variance Components
1.
2.
3.
Statistical background
Experimental design
Statistical models
7
1. Statistical background
Variance / Covariance
Definitions
Y
-
+
Covariance(X,X) = Variance
Covariance(X,Y) = Covariance
Y
-
+
0
∑ (X
n
0
X
Cov ( X , Y ) =
i =1
i
− X )(Yi − Y )
n −1
X
n: sample size, number of replicates
8
Regression
Regression
y = a + bx
Y
a = y − bx
b=
b
Cov( X , Y )
Var ( X )
a: intercept
b: slope
a
0
0
Correlation
X
r ( X , Y )=
The regression line is the line through a cloud
of datapoints that minimizes the summ of
squared deviations of Y from a fitted line.
Cov( X , Y )
Var ( X )Var (Y )
Check
Check out
out Appendix
Appendix in
in Conner
Conner and
and Hartl!
Hartl!
4-offspring mean
pistil length
2-offspring mean
pistil length
Heritability, Regression and Correlation
Slope=h2=0.79(0.12)
R2=0.47
Slope=h2=0.74(0.09)
R2=0.59
Midparent pistil length
Slope of
regression
stays quite
stable,
Correlation
becomes
stronger
Pistil: Stempel
9
Y
ANOVA
(Analysis of Variance)
Y1
Y2
Comparison of
two groups
••
••
••
••
••
V
: Variance between
Vmeans
means: Variance between
group
group means
means
n:
n: number
number of
of groups
groups
SS:
SS: sum
sum of
of squared
squared
deviations
deviations
d.f.:
d.f.: degrees
degrees of
of freedom
freedom
n-1:
n-1: because
because 11 d.f.
d.f. is
is
„used
„used up“
up“ for
for estimating
estimating
the
the grand
grand mean
mean
ANOVA
(Analysis of Variance)
Comparison
of multiple
groups
••
••
••
MS:
MS: Mean
Mean squares
squares
F:
F: F-statistics,
F-statistics, MS
MS among
among
devided
devided by
by MS
MS within
within
P:
P: statistical
statistical significance,
significance,
calculated
calculated based
based on
on FFstatistics
statistics
10
ANOVA
(Example)
11