2
population.
Further, understanding the variance in the population will allow us to
understand how it will evolve, for example, in response to selection.
Call the Phenotypic value of an individual
P
.
Until now we've considered the phenotype of an individual as being entirely genetically
determined, so
P = G
, which we can break down into
P = A + D
, i.e., an individual's
phenotype is composed of its breeding value plus its dominance deviation.
Let us
further consider a contribution of the environment to the phenotype, call it
E
, so
P = A + D + E
(assuming no epistasis)
Recall the variance =
∑
−
i
i
i
x
x
p
)
(
2
, where
p
i
is the frequency of the
i
th type
Variance of a sum Z = X + Y is Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)
Covariance is a measure of association between two variables; if they are independent
the covariance is zero.
Cov(X,Y) =
)
)(
(
y
y
x
x
p
j
i
j
ij
i
−
−
∑
∑
IF covariance is zero, then Var(X+Y) + Var(X) + Var(Y)
Thus, the phenotypic variance in a population, V
P
= V
A
+ V
D
+ V
E
IF
the covariances
between the terms are zero.
i)
It Can Be Shown that the covariance between the breeding value and the dominance
deviation is always zero. Proof
**
Below
ii)
We ASSUME the covariance between the genotypic component of variance V
G
= V
A
+ V
D
and the environmental component of variance is zero, i.e., Cov(G,E) = 0.
So, the phenotypic variance present in a population is composed of a component due to
variation in breeding values, V
A
, variation in dominance deviations, V
D
, and variation
due to the random effects of the environment, V
E
.
Q: