2. Here is a neat result due to Mather (1938). Note each chiasma in
meiosis involves two of the four chromatids, and hence has probability
1/2 of being in a resulting gamete. We say there is no
chromatid interference if each chiasma has probability
1/2 of being in a resulting gamete, INDEPENDENTLY of the others.
(a) If a fair coin is tossed 1,2 or 3 times, what is the probability of an
odd number of heads?
(b) Show by induction (or elsehow if you prefer), that if a fair coin
is tossed any number of N of times (N>0), the probability of an odd
number of heads is 1/2.
(c) Deduce Mather's formula, that the recombination probability is
(1/2) P(N>0), where N is the random
number of chiasmata between the two loci in a meiosis.
(This means that under no chromatid interference, recombination
probabilities must be an increasing function of genetic distance.)
3. (a)
J. H. Edwards has referred to the "half-life" of a two-locus
haplotype as the number of generations (or meioses)
until there is probability 1/2 that
recombination has occurred between the two loci. Show that the half-life
is - log(2)/log(1-r) at recombination frequency r. (What base logs?)
(b) In the ancestry of a sample of disease alleles, suppose there are M
meioses in the ancestry of the sample, back to most recent common ancestor
of the sample. Show that the probability that all the sample are IBD at a
linked marker loci is at least (1-r)M.
(c) Suppose the allele frequencies at the linked
marker locus are p1,
..., pk, and remain constant over time. Show that the
probability that all the current sampled disease haplotypes in (b) have the
same allele at the linked marker is at least
sumi pi (1 - r (1-pi))M
4. These data come from a study by Dr. Arno Motulsky and coworkers, and
are published in Thompson et al. (1998; Am.J.Hum.Genet, 42, 113-124).
There were three population samples (all from around Seattle),
(Caucasian, African American, and
Japanese American), and three tightly linked diallelic loci,
designated M, P and S.
(a) In the Caucasian sample of 205 individuals typed at the
P and M loci, the counts of the two-locus phenotypes were
143, 35, 3, 17, 5, 0, 2, 0, and 0, respectively, for the nine types
P1P1,M1M1;
P1P1,M1M2;
P1P1,M2M2;
P1P2,M1M1;
P1P2,M1M2;
P1P2,M2M2;
P2P2,M1M1;
P2P2,M1M2;
P2P2,M2M2.
Use the EM algorithm to estimate the 4 haplotype frequencies.
In the Japanese subsample, the same 68 individuals (136 chromsomes)
were scored from all three loci. As in (a), there were few
double-heterozygotes for any pair of the loci. The following three
parts all refer to this Japanese sample.
(b) For loci S and M, the estimated haplotype frequencies are
0.551, 0.082, 0.008, 0.360.
Is there evidence for disequilibrium between loci S and M?
(c) For loci P and M, the estimated haplotype frequencies are
0.514, 0.398, 0.045, 0.043.
Is there evidence for disequilibrium between loci P and M?
(d)
For loci S and P, the estimated haplotype frequencies are
0.592, 0.041, 0.320 0.047.
Is there evidence for disequilibrium between loci S and P?
(You can use a 2-by-2 contingency table chi-squared to test for significantly
non-zero associations.)
(The interesting fact is that all three loci are known to be very tightly
linked, and locus P is between the other two.)
5. In a simple backcross experiment between two inbred lines, hybrid
AB/ab individuals are crossed with ab/ab individuals, and the numbers of
recombinant offspring are counted.
Among a total of
120 offspring in which the hybrid individual was female, 54 were of
the recombinant types.
Among 90 offspring in which the hybrid individual was male, 36 were of
the recombinant types.
(a) Is there evidence for linkage, using only the data on the offspring of
hybrid males?
(b) Is there evidence for linkage, using only the data on the offspring of
hybrid females?
(c) Assuming male and female recombinantion frequencies are the same,
is there evidence for linkage?
(d) Is there evidence from this expecriment that male and female
recombination frequencies differ?