Page 188 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

or in matrix notation the model is:
x +
y = b Zu + å W v + e
j
j
j
The terms are as defined in Eqn 10.17. The vector v contains the effects of the paternal
j
and maternal MQTL alleles at each locus. The summation is over chromosome segments
bounded by markers. The variance of u and v are as defined in Eqn 10.19, such that:
j
var(v ) = G J 2
j vj vj
Assuming j = 2, the BLUP equations for the above model are:
′
′
′
′
⎛ X X X Z XW 1 XW 2 ⎞ ⎛ ˆ ⎞
b
⎜ ′ Z Z + −1 ′ ′ ⎟ ⎜ ⎟
′
⎜ ZX A a 1 Z W 1 Z W 2 ⎟ ⎜ u ˆ ⎟
⎜ WX W Z W W′ + − 1 W′ W ⎟ v ˆ ⎜ ⎜ ⎟
′
′
′
⎜ 1 1 1 1 G a 2 1 1 2 ⎟ ⎜ 1 ⎟
v
1
⎜ ⎝ W X W Z WW 1 WW + G a 3 ⎟ ⎠ ⎝ v ˆ ⎠
−
′
′
′
′
1
2
2
2
2
v
2
2
2
⎛ Xy′ ⎞
⎜ Zy ⎟
′
= ⎜ ⎟ (10.24)
⎜ Wy′ 1 ⎟
⎜ ⎝ Wy′ 2 ⎟ ⎠
where:
a = s /s , 2 2 and 2 2
2
2
e
3
e
1 e u a = s /s v1 a = s /s v2
2
10.9.2 Calculating the covariance matrix, G
Consider a single MQTL bounded by two marker loci with marker distances as follows:
M 1 Q M 2
p q = 1)
pr qr (+
r
With the assumption of no crossover, the recombination rates are (Haldane, 1919) between:
−2r
M and M = a = 0.5(1 − e )
1 2
M and Q = b = 0.5(1 − e −2pr )
1
Q and M = c = 0.5(1 − e −2qr )
2
Similar to the situation with a single marker, the variance of v depends on the
relationship among the v terms. The MQTL alleles in the progeny can be expressed
in terms of parental MQTL. Thus given, for instance, that the genotype of the sire is:
1 v 1
s 11
2 v s 22 2
172 Chapter 10