Page 149 - Applied Probability

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7. Computation of Mendelian Likelihoods
Inspection of the location score curve shows that it rises above the mag-
ical level of 3 and that the episodic ataxia gene probably resides on the
interval from D12S372 to pY2/1. Where the marker S372 is uninformative
for linkage, other markers fill the information gap. Thus, location scores
make better use of scarce disease pedigrees than lod scores do. In Chapter
9 we will revisit this problem and demonstrate how the haplotypes dis-
played in Figure 7.3 are reconstructed and how one can compute location
scores using an almost arbitrary number of markers.
7.7 Problems
1. Under Haldane’s model of independent recombination on disjoint in-
tervals, it is possible to compute the recombination fraction θ ij be-
tween two loci i< j by Trow’s formula

j−1

1 − 2θ ij = (1 − 2θ k,k+1 ), (7.10)
k=i
where the loci occur in numerical order along the chromosome, and
where θ k,k+1 is the recombination fraction between the adjacent loci
k and k + 1. Verify Trow’s formula first for three loci (i = 1 and
j = 3) and then by induction for an arbitrary number of loci. (Hint:
For three loci, recombination occurs between loci 1 and 3 if and only
if it occurs between loci 1 and 2 and not between loci 2 and 3, or vice
versa.)

2. Consider the partially typed, inbred pedigree depicted in Figure 7.6.
The phenotypes displayed in the figure are unordered genotypes at a
single codominant locus with three alleles. Show that the genotype
elimination algorithm fails to eliminate some superfluous genotypes
in this pedigree.

3. The sum of array products

··· A(G 1 ,G 2 ,G 3 ,G 4 )B(G 4 ,G 5 )
G 1 ∈S 1 G 9 ∈S 9
× C(G 5 ,G 6 )D(G 6 ,G 7 ,G 8 ,G 9 )

can be evaluated as an iterated sum by the greedy algorithm. If all
range sets S i have the same number of elements m> 2, then show
that one greedy summation sequence is (5, 1, 2, 3, 4, 7, 8, 9, 6). Prove
that the alternative nongreedy sequence (1, 2, 3, 4, 5, 7, 8, 9, 6) requires
fewer arithmetic operations (additions plus multiplications) [18].