Page 23 - Applied Probability

1. Basic Principles of Population Genetics
6
1
1

2
2
2p 1 p 2 + p
=2p 1(p 1 + p 2 )p 2 (p 1 + p 2 )
2 p + 2p 1 p 2
1
2
2
2
=2p 1p 2
1
2

2

p 2 (p 1 + p 2 )
=
2p 1 p 2 + p
2
2
2
= p . 2
2
Thus, after a single round of random mating, genotype frequencies stabilize
at the Hardy-Weinberg proportions.
We may deduce the same result by considering the gamete population.
A 1 gametes have frequency p 1 and A 2 gametes frequency p 2 . Since random
union of gametes is equivalent to random mating, A 1 /A 1 is present in the
2
next generation with frequency p , A 1 /A 2 with frequency 2p 1p 2 , and A 2 /A 2
1
2
with frequency p . In the gamete pool from this new generation, A 1 again
2
2
occurs with frequency p + p 1 p 2 = p 1 (p 1 + p 2 )= p 1 and A 2 with frequency
1
p 2 . In other words, stability is attained in a single generation. This random
union of gametes argument generalizes easily to more than two alleles.
Hardy-Weinberg equilibrium is a bit more subtle for X-linked loci. Con-
sider a locus on the X chromosome and any allele at that locus. At genera-
tion n let the frequency of the given allele in females be q n and in males be
r n . Under our stated assumptions for Hardy-Weinberg equilibrium, one can
2
1
show that q n and r n converge quickly to the value p = q 0 + r 0 . Twice as
3 3
much weight is attached to the initial female frequency since females have
two X chromosomes while males have only one.
Because a male always gets his X chromosome from his mother, and his
mother precedes him by one generation,
= q n−1 . (1.1)
r n
Likewise, the frequency in females is the average frequency for the two sexes
from the preceding generation; in symbols,
1 1
= q n−1 + r n−1 . (1.2)
q n
2 2
Equations (1.1) and (1.2) together imply
2 1 2 1 1 1

q n + r n = q n−1 + r n−1 + q n−1
3 3 3 2 2 3
2 1
= q n−1 + r n−1 . (1.3)
3 3
2
1
It follows that the weighted average q n + r n = p for all n.
3 3
From equations (1.2) and (1.3), we deduce that
3 1
q n − p = q n − p + p
2 2