Page 25 - Applied Probability

1. Basic Principles of Population Genetics
8
Example 1.3.1 Hardy-Weinberg Equilibrium for the Xg(a) Locus
The red cell antigen Xg(a) is an X-linked dominant with a frequency
in Caucasians of approximately p = .65. Thus, about .65 of all Caucasian
males and about p +2p(1 − p)= .88 of all Caucasian females carry the
antigen. 2
1.4 Linkage Equilibrium
Loci on nonhomologous chromosomes show independent segregation at
meiosis. In contrast, genes at two physically close loci on the same chromo-
some tend to stick together during the formation of gametes. The recombi-
nation fraction θ between two loci is a monotone, nonlinear function of the
physical distance separating them. In family studies in man or in breeding
studies in other species, θ is the observable rather than physical distance.
1
In Chapter 12 we show that 0 ≤ θ ≤ . The upper bound of 1 is attained
2 2
by two loci on nonhomologous chromosomes.
The population genetics law of linkage equilibrium is of fundamental
importance in theoretical calculations. Convergence to linkage equilibrium
can be proved under the same assumptions used to prove Hardy-Weinberg
equilibrium. Suppose that allele A i at locus A has frequency p i and allele B j
at locus B has frequency q j . Let P n (A i B j ) be the frequency of chromosomes
with alleles A i and B j among those gametes produced at generation n.
Since recombination fractions almost invariably differ between the sexes,
let θ f and θ m be the female and male recombination fractions, respectively,
between the two loci. The average θ =(θ f + θ m )/2 governs the rate of
approach to linkage equilibrium.
We can express P n (A i B j ) by conditioning on whether a gamete is an egg
or a sperm and on whether nonrecombination or recombination occurs. If
recombination occurs, then the gamete carries the two alleles A i and B j
with equilibrium probability p i q j . Thus, the appropriate recurrence relation
is
1
P n (A i B j )= (1 − θ f )P n−1 (A i B j )+ θ f p i q j
2
1
+ (1 − θ m )P n−1 (A i B j )+ θ m p i q j
2
=(1 − θ)P n−1 (A i B j )+ θp i q j .
Note that this recurrence relation is valid when the two loci occur on non-
homologous chromosomes provided θ = 1 and we interpret P n (A i B j )as
2
the probability that someone at generation n receives a gamete bearing the
two alleles A i and B j . Subtracting p i q j from both sides of the recurrence
relation gives

P n (A i B j ) − p i q j =(1 − θ)[P n−1 (A i B j ) − p i q j ]