Page 30 - Applied Probability

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1. Basic Principles of Population Genetics
1.7 Problems
1. In blood transfusions, compatibility at the ABO and Rh loci is im-
portant. These autosomal loci are unlinked. At the Rh locus, the +
allele codes for the presence of a red cell antigen and therefore is
dominant to the − allele, which codes for the absence of the antigen.
Suppose that the frequencies of the two Rh alleles are q + and q − .
Type O− people are universal donors, and type AB+ people are uni-
versal recipients. Under genetic equilibrium, what are the population
frequencies of these two types of people? (Reference [2] discusses these
genetic systems and gives allele frequencies for some representative
populations.)
2. Suppose that in the Hardy-Weinberg model for an autosomal locus
the genotype frequencies for the two sexes differ. What is the ultimate
frequency of a given allele? How long does it take genotype frequencies
to stabilize at their Hardy-Weinberg values?
3. Consider an autosomal locus with m alleles in Hardy-Weinberg equi-
librium. If allele A i has frequency p i , then show that a random non-
m 2
inbred person is heterozygous with probability 1 − p . What is
i=1 i
the maximum of this probability, and for what allele frequencies is
this maximum attained?
4. In forensic applications of genetics, loci with high exclusion probabil-
ities are typed. For a codominant locus with n alleles, show that the
probability of two random people having different genotypes is

n−1 n n
2 2
e = 2p i p j (1 − 2p i p j )+ p (1 − p )
i
i
i=1 j=i+1 i=1
under Hardy-Weinberg equilibrium [8]. Simplify this expression to
n 2 n
2 4
e =1 − 2 p i + p .
i
i=1 i=1
2 1
Prove rigorously that e attains its maximum e max =1 − n 2 + n 3
√
1
when all p i = . For two independent loci with n alleles each, verify
n
that the maximum exclusion probability based on exclusion at either
4 4 1
locus is 1 − 2 + 3 . How does this compare to the maximum
n n 5/2 − n
exclusion probability for a single locus with n equally frequent alleles
when n = 16? What do you conclude about the information content of
two loci versus one locus? (Hint: To prove the claim about e max, note
that, without loss of generality, one can assume p 1 ≤ p 2 ≤· · · ≤ p n .If
p i <p i+1 , then e can be increased by replacing p i and p i+1 by p i + x
and p i+1 − x for x positive and sufficiently small.)