Page 110 - Applied Probability

5. Genetic Identity Coefficients
94
f n+1 = φ n and that
1
1
=
φ n+1
4
2
4
1
1
=
φ n + φ n−1 + .
4
2 φ n + f n + 1 1
4
From this second-order difference equation, deduce that
* √ + n * √ + n

1 1 1+ 5 1 1 1 − 5
φ n =1 − + √ − − √ .
2 5 4 2 5 4
Thus, lim n→∞ φ n = lim n→∞ f n = 1, and one random allele is fixed
at each locus.
10. Wright proposed a path formula for computing inbreeding coefficients
that can be generalized to computing kinship coefficients [15]. The
pedigree formula is
n(p ij )
1
Φ ij = [1 + f a(p ij ) ],
2
p ij
where the sum extends over all pairs p ij of nonintersecting paths
descending from a common ancestor a(p ij )of i and j to i and j,
respectively, and where n(p ij ) is the number of people counted along
the two paths. The common ancestor is counted only once. If i = j,
there is only the degenerate pair of paths that start and end at i but
possess no arcs connecting a parent to a child. In this case, the formula
1
reduces to the fact Φ ii = (1 + f i ). In general, a path is composed of
2
arcs connecting parents to their children. Two paths intersect when
they share a common arc. To get a feel for Wright’s formula, verify it
for the case of siblings of unrelated parents. Next prove it in general
by induction. Note that although founders are allowed to be inbred,
no two of them can be related. (Hint: Consider first the founders of
a pedigree and then, recursively, each child of parents already taken
into account.)
11. The definition of a generalized X-linked kinship coefficient exactly
parallels the definition of a generalized kinship coefficient except that
genes are sampled from a generic X-linked locus rather than a generic
autosomal locus. Adapt the algorithm of Section 5.6 to the X-linked
case by showing how to revise the boundary conditions and recurrence
relations [18].
5.8 References
[1] Cotterman CW (1940) A Calculus for Statistico-Genetics. Ph.D. the-
sis, Ohio State University. in Genetics and Social Structure: Mathemat-