Page 189 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The sire will produce the following four types of gametes on the basis of marker
haplotypes:
1 v 1 1 v 2 2 v 1 2 v 2
o11 o12 o21 o22

Assuming no double recombination between markers, the frequency, means and
approximate means for the four gametes (Goddard, 1992) are:

Haplotype Frequency Mean Approximate mean
1 1 1 (1 − a) [(1 − b)(1 − c)/1 − a]v + [bc/1 − a]v v
2 s11 s22 s11
1 2 1 a [(1 − b)c/a]v + [b(1 − c)/a]v qv + pv
2 s11 s22 s11 s22
2 1 1 a [b(1 − c)/a]v + [(1 − b)c/a]v pv + qv
2 s11 s22 s11 s 22
2 2 1 (1 − a) [bc/1 − a]v + [(1 − b)(1 − c)/1 − a]v v
2 s11 s 22 s 22

Given, for instance, that r = 0.2, p = 0.8 and q = 0.2, then a, b and c are 0.1649,
0.1370 and 0.0385, respectively. The means for the haplotypes are 0.99v and 0.01v
s11 s11
for (1 1), 0.2v and 0.8v for (1 2), 0.8v and 0.2v for (2 1) and 0.01(v )
s11 s11 s11 s11 s11
and 0.99(v ) for (2 2). The approximate means are very similar to these estimates.
s11
The maximum errors associated with the above approximate means are when p = q = 0.5
for haplotypes (1 1) and (2 2) (Goddard, 1992). Using the approximate means, the value
of the MQTL in each gamete can be written in terms of the parental MQTL as:
æ v o11 ö æ 10 ö 11 ö
ç ÷ ç ÷ ç æe ÷
÷
ç v o12 ÷ = ç q p æ v s11 ö + ç e 12 ÷
ç v ÷ ç p q ÷ ç è v s22 ø ÷ ÷ (10.25)
ç ç o21 ÷ ÷ ç ÷ 2 ç ç e 21 ÷ ÷
ç
è v o22 ø è 01 ø è e 22 ø
where e is the deviation of each gamete from the mean of the haplotype. Since v
ij o11
is identical to v and v to v with the approximate means, then e = e = 0.
s11 o22 s22 11 22
Eqn 10.25 may be expressed as:
v = Pv + e
where P is as defined in Section 10.5 and has at most two non-zero elements, which
sum to unity. Thus:
v = (I − P) e
−1
Therefore:
G = var(v) = (I − P) var(e)((I − P) )′
−1
−1
and:
−1
G = (I − P)′H (I − P) (10.26)
−1
2
where Hs = var(e) and H is a diagonal matrix. Since e = e = 0, var(e ) = var(e ) = 0.
v 11 22 11 22
The main interest therefore is in calculating var(e ) and var(e ). The calculation of
12 21
Use of Genetic Markers in Breeding Value Prediction 173