Page 21 - Applied Probability

1. Basic Principles of Population Genetics
4
gametes. This forces geneticists to rely on indirect statistical arguments to
overcome the problem of missing information. The experimental situation
is analogous to medical imaging, where partial tomographic information is
available, but the full details of transmission or emission events must be
reconstructed. Part of the missing information in pedigree data has to do
with phase. Alleles O and A 2 are in phase in individual 3 of Figure 1.1. In
general, a gamete’s sequence of alleles along a chromosome constitutes a
haplotype. The alleles appearing in the haplotype are said to be in phase.
Two such haplotypes together determine a multilocus genotype (or simply
a genotype when the context is clear).
Recombination or linkage studies are conducted with loci called traits
and markers. Trait loci typically determine genetic diseases or interesting
biochemical or physiological differences between individuals. Marker loci,
which need not be genetic loci in the traditional sense at all, are signposts
along the chromosomes. A marker locus is simply a place on a chromosome
showing detectable population differences. These differences, or alleles, per-
mit recombination to be measured between the trait and marker loci. In
practice, recombination between two loci can be observed only when the
parent contributing a gamete is heterozygous at both loci. In linkage analy-
sis it is therefore advantageous for a locus to have several common alleles.
Such loci are said to be polymorphic.
The number of haplotypes possible for a given set of loci is the product
of the numbers of alleles possible at each locus. In the ABO-AK1 example,
2
there are k =3 × 2 = 6 possible haplotypes. These can form k genotypes
based on ordered haplotypes or k + k(k−1) = k(k+1) genotypes based on
2 2
unordered haplotypes.
To compute the population frequencies of random haplotypes, one can
invoke linkage equilibrium. This rule stipulates that a haplotype fre-
quency is the product of the underlying allele frequencies. For instance,
are the
the frequency of an OA 1 haplotype is p O p A 1 , where p O and p A 1
population frequencies of the alleles O and A 1 , respectively. To compute
the frequency of a multilocus genotype, one can view it as the union of two
random gametes in imitation of the Hardy-Weinberg law. For example,
2
) ,
the genotype of person 2 in Figure 1.1 has population frequency (p O p A 2
being the union of two OA 2 haplotypes. Exceptions to the rule of linkage
equilibrium often occur for tightly linked loci.

1.3 Hardy-Weinberg Equilibrium

Let us now consider a formal mathematical model for the establishment
of Hardy-Weinberg equilibrium. This model relies on the seven following
explicit assumptions: (a) infinite population size, (b) discrete generations,
(c) random mating, (d) no selection, (e) no migration, (f) no mutation, and