Page 107 - Applied Probability

5. Genetic Identity Coefficients
1
1
2
1
1
1
2
=
1
2
1
1
2
1
8
8
1
1
2
1
Φ({G }{G }{G })
=
2
1
1
4
1
.
= 1 Φ({G }{G }{G })+ Φ({G }{G }{G }) 91
8
Although here we replace 3 by the grandparents 1 and 2 before we replace
4 by the same pair, this is permitted because 4 is not a descendant of 3. In
2
1
2
1
like manner, B =Φ({G ,G }{G }{G }) can be reduced to
3 4 4 3
1 1 2 1 1 1 1 2 1 1
B = Φ({G ,G }{G }{G })+ Φ({G ,G }{G }{G })
4
4
2
4
1
2
4
1
4 4
1 1 2 1 1
= Φ({G ,G }{G }{G })
1
2
4
4
2
1 1 2 2 1 1 1 2 2 1
= Φ({G ,G }{G }{G })+ Φ({G ,G }{G }{G })
2
2
2
1
2
1
1
1
8 8
1 1 1
= × + × 0
8 4 8
1
= .
32
1
These reductions yield 1 Ψ 8 = 1 A + B = 5 for this particular sibling
4 8 8 256
pair. The condensed identity coefficient ∆ 8 =4Ψ 8 = 5 for the pair follows
16
directly from equation (5.3).
5.7 Problems
1. Consider two non-inbred relatives i and j with parents k and l and
m and n, respectively. Show that
∆ 7 =Φ km Φ ln +Φ kn Φ lm
∆ 8 =4Φ ij − 2∆ 7
∆ 9 =1 − ∆ 7 − ∆ 8 ,
where the condensed identity coefficients all pertain to the pair i and
j. Thus, in the absence of inbreeding, all nonzero condensed identity
coefficients can be expressed in terms of ordinary kinship coefficients.
2. Given the assumptions and notation of Problem 1 above, show that
2
4∆ 7 ∆ 9 ≤ ∆ [15]. This inequality puts an additional constraint on
8
∆ 7 ,∆ 8 , and ∆ 9 besides the obvious nonnegativity requirements and
the sum requirement ∆ 7 +∆ 8 +∆ 9 = 1. (Hints: Note first that
1 1
Φ ij = ∆ 7 + ∆ 8
2 4
1 1 1 1
= Φ km + Φ kn + Φ lm + Φ ln .
4 4 4 4