Page 173 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

p
m
its paternal (p) and its maternal (m) parents. Also, let Q and Q denote alleles at
i
i
m
p
the quantitative trait loci linked to M and M as illustrated below:
i i
M p i Q p i
_____________________
| |
_____________________
| |
m m
M i Q i
p
m
m
p
Let v and v be the genetic additive effects of Q and Q , respectively, and u the
i i i i i
genetic additive effects of the remaining quantitative trait loci not linked to the ML.
Then the additive genetic value (a ) of individual i is:
i
a = v + v + u (10.1)
m
p
i i i i
Given only phenotypic information, the usual BLUP equation for additive genetic
effects (Section 3.2) is:
y = x b + a + e (10.2)
i i i i
Replacing a above by Eqn 10.1 gives:
i
p
y = x b + v + v + u + e (10.3)
m
i i i i i i
From Section 2.2, the covariance matrix for u , A, is the usual relationship matrix
i
(Henderson, 1976) but the covariance for v , G , depends on both the relationship
i v
matrix and marker information. Thus given A and G , the BLUP of v and u can be
v i i
obtained using the usual MME. The calculation of A and its inverse has been covered
in Chapter 2. The calculation of G and its inverse are covered in the next section.
v
10.3 Calculating the Covariance Matrix (G )
v
for MQTL Effects
2
The matrix G s represents the covariance between the additive effects of the MQTL
v v
alleles. For simplicity, consider only maternal MQTL. Assume two arbitrary individu-
m
m
m
m
als b and b′ inherit MQTL alleles Q and Q with additive effects v and v from
b b′ b b ′
m
m
dams d and d′, respectively. The covariance between the additive effects v and v for
b b′
the maternal MQTL in b and b′ is:
m
m
m
m
m
m
m
m
cov(v , v ) = cov(v , v | Q ≡ Q ) · P(Q ≡ Q )
b b′ b b′ b b′ b b′
m
m
m
= var(v ) · P(Q ≡ Q ) (10.4)
b b b′
2
= s G
v v(b,b′)
m
2
m
m
where var(v ) = s is the variance of the MQTL allele, P(Q ≡ Q ) is the proba-
b v b b′
m
m
bility that Q is identical by descent (IBD) to Q and the matrix G is the
b b′ v(b,b′)
covariance matrix for the MQTL between b and b′. Given that b is not a direct
m
descendant of b′, Q can only be identical by descent to Q in two mutually exclusive
m
b b′
manners: (i) if Q is IBD to Q , the paternal MQTL allele of the dam of b′, and
p
m
b d′
m
b′ has inherited Q ; or (ii) Q is IBD to Q , the maternal MQTL allele of the dam
p
m
d′
d′
b
of b′, and b′ has inherited Q . This is akin to calculating A where the relation-
m
d′
ship, say, between b and b′ is evaluated through the relationship of b with the
parents of b′.
Use of Genetic Markers in Breeding Value Prediction 157