Page 194 - Applied Probability

TABLE 9.1. Allele Vectors for the Components of the Founder Tree Graph
Legal Allele Vectors
Descent Trees in Component
All singleton allele vectors
{B}
(1, 2, 1) and (2, 1, 2)
{A, C, E}
(1, 2, 3, 4)
{D, F, G, H} 9. Descent Graph Methods 179
of a single node. In a nonconsanguineous pedigree, a descent tree forms
a singleton component if it passes through no one typed at its associated
locus. In this case, all alleles can be legally assigned to the founder gene at
the top of the descent tree. In a consanguineous pedigree, a descent tree
can form a singleton component even if it descends through a typed child.
However, the descent tree must descend to the child via both of its parents.
If the typed child has homozygous genotype a i /a i , then a i is the only allele
permitted for the founder gene. If the typed child is heterozygous, then no
legal allele exists for the founder gene.
The situation for a multinode (or nonsingleton) component of the founder
tree graph is equally simple. If we label the nodes of the component as
t 1 ,...,t k , then the founder gene of node t 1 is transmitted to some typed
person who is either homozygous or heterozygous. If the person is homozy-
gous, then there is only one legal choice for the founder gene of t 1 . Because
this founder gene is connected to another founder gene through the current
typed person or another typed person, the connected founder gene is also
completely determined. This second founder gene is in turn connected to
a third founder gene through some typed person. Hence, the third founder
gene is also uniquely determined. In general, a cascade of connecting edges
completely determines the permissible alleles for each of the founder genes
of the component, unless, of course, an inconsistency is encountered at
some step. If descent tree t 1 passes through a typed heterozygote, then
the founder gene of t 1 may be either observed allele. Once one of these
two alleles is chosen for t 1 , then the alleles of all other founder genes in
the component are determined by the argument just given. Thus, we can
summarize the situation for a multinode component t 1 ,... ,t k by noting
) can be legally
that either two, one, or no allele vectors a =(a t 1 ,... ,a t k
assigned to the founder genes of the component. Table 9.1 displays all legal
allele vectors for each component of the founder tree graph shown in part
(b) of Figure 9.2 based on the genotypes shown in part (a) of the same
figure.

To simplify Prior(G), label the connected components of the
G à 0 G∩M
founder tree graph C 1 ,...,C m , and let G ø G∩M be a consistent descent
0
state. As just noted, there is an allele vector a i with constituent alleles a ij
assigned to each component C i of the descent state G. Under conditions of