Page 102 - Applied Probability

5. Genetic Identity Coefficients
86
A more complex example is afforded by the two offspring of the brother–
sister mating in Figure 5.1. Straightforward but tedious calculations show
1
1
1
1
7
1
,∆ 2 =
that ∆ 1 =
,∆ 3 = ,∆ 4 =
16
32
8
1
5
, and ∆ 9 =
∆ 8 =
in this case. Because inbreeding is rare, the three
16
16
coefficients ∆ 7 ,∆ 8 , and ∆ 9 originally introduced by Cotterman [1] suffice
for most practical purposes. 1 8 32 ,∆ 5 = ,∆ 6 = 32 ,∆ 7 = 32 ,
5.4 Generalized Kinship Coefficients
We generalize classical kinship coefficients by randomly sampling one gene
from each person on an ordered list of n people rather than one gene from
each of two people [7, 8, 9, 17]. If a person is repeated in the list, then
sampling is done with replacement. The ordered sequence of n sampled
genes G 1 ,... ,G n can be partitioned into nonoverlapping blocks whose con-
stituent genes are i.b.d. A generalized kinship coefficient gives the proba-
bility that a particular partition occurs.
This simple verbal description necessarily involves complex notation. For
instance, with four individuals i, j, k, and l, there are 15 partitions of the
four sampled genes G i , G j , G k , and G l . Again these range from the single-
block partition {G i ,G j ,G k ,G l }, where all of the genes are i.b.d., to the
four-block partition {G i }, {G j }, {G k }, {G l }, where no genes are i.b.d. We
will denote the probability of any such partition by enclosing its blocks
within parentheses preceded by Φ, bearing in mind that this probability
depends neither on the order of the blocks nor on the order of the sampled
genes within a block. Thus, the two partitions mentioned above have prob-
abilities Φ({G i ,G j ,G k ,G l }) and Φ({G i }, {G j }, {G k }, {G l }), respectively.
The classical kinship coefficient between i and j becomes Φ({G i ,G j })in
this notation, and its complementary probability becomes Φ({G i }, {G j }).
5.5 From Kinship to Identity Coefficients
What is the relationship between generalized kinship coefficients and con-
densed identity coefficients? Suppose we randomly sample two genes G 1 i
2
1
2
and G from i and two genes G and G from j. Now consider one of the
i j j
1
2
∗
detailed identity states S of Figure 5.2. If we imagine G and G occupying
k i i
1
2
the upper two gene positions and G and G occupying the lower two gene
j j
positions, then S defines a detailed identity state of the four randomly
∗
k
sampled genes. Corresponding to the 15 random detailed identity states
are 9 random condensed identity states as shown in Figure 5.3. Denote the
probability of a random condensed identity state S k by Ψ k . The probability
Ψ k is an integer multiple of a generalized kinship coefficient. For example,
2
2
1
1
Ψ 1 =Φ({G ,G ,G ,G }),
i
i
j
j