Page 134 - Applied Probability

7. Computation of Mendelian Likelihoods
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7.3 Genotype Elimination and Allele Consolidation
As hinted above, systematic genotype elimination is a powerful technique
for accelerating likelihood evaluation. This preprocessing step involves more
than just using those genotypes compatible with a person’s observed phe-
notype. The phenotypes of the person’s relatives also impose rigid com-
patibility constraints on his or her possible genotypes. Although this fact
has long been known informally, it is helpful to state a formal algorithm
that mimics how geneticists reason. In doing so, we will focus on ordered
genotypes at a single autosomal locus. Because ordered genotypes carry
phase information, applying the algorithm separately to several linked loci
automatically eliminates superfluous phases among the loci as well as su-
perfluous genotypes within each locus.
Here then is the algorithm [18, 19]:
(A) For each pedigree member, list only those ordered genotypes compat-
ible with his or her phenotype.

(B) For each nuclear family:
(1) Consider each mother–father genotype pair.
(a) Determine which zygotes can arise from the genotype pair.
(b) If each child in the nuclear family has one or more of these
zygote genotypes among his or her current list of genotypes,
then save the parental genotypes and any child genotype
matching one of the created zygote genotypes.
(c) If any child has none of these zygote genotypes among his
or her current list of genotypes—in other words, is incom-
patible with the current parental pair of genotypes—then
take no action to save any genotypes.
(2) For each person in the nuclear family, exclude any genotypes not
saved during step (1) above.

(C) Repeat part (B) until no more genotypes can be excluded.

As an illustration of the algorithm, consider the pedigree of Figure 7.1
at the ABO locus, and suppose individuals 2, 3, and 5 alone are typed at
this locus. Then applying part (A) of the algorithm leads to the genotype
sets displayed in column 2 of Table 7.1. Applying (B) to the nuclear family
{3, 4, 5} gives column 3, and finally, applying (B) to the nuclear family
{1, 2, 3} gives column 4. Recall that the maternal allele is listed to the left
of the paternal allele in an ordered genotype. This convention shows up in
the genotype set for person 3. No further genotypes can be eliminated by
repeated use of (B), and column 4 provides the minimal genotype sets. In
more extensive pedigrees, genotype eliminations can ripple up and down