Biology Reference
In-Depth Information
limit lim
n
!1
q
n
must exist. Let q
¼
lim
n
!1
q
n
. Then p
¼
lim
n
!1
p
n
¼
lim
n
!1
ð
q. Because we began with q
0
<1
and proved the values of the
a
allele frequencies decrease from generation to
generation, it is clear the limit value q will be such that
q < 1. We now show that q
1
q
n
Þ¼
1
lim
n
!1
q
n
¼
1
¼
0.
Taking limits as n
!1
of both sides in Eq. (3-7), we obtain:
þ
a
q
n
p
n
lim
n
q
n
þ
1
¼
lim
n
q
n
ð
a
q
n
þ
p
n
Þ
q
n
þ
p
n
!1
!1
Because q
¼
lim
n
!1
q
n
¼
lim
n
!1
q
n
þ
1
and p
¼
1
q, this implies that:
q
þ
p
q
þð
1
q
Þ
q
q
þ
1
q
ð
a
1
Þ
q
þ
1
a
a
a
q
¼
q
p
¼
q
Þ
¼
q
1
¼
1
:
q
2
þ
pq
þ
q
2
þð
1
q
Þ
q
þð
1
q
q
2
q
2
þ
ð
a
1
Þ
q
2
þ
a
a
a
Thus, the limit value q satisfies the following algebraic equation:
q
ð
a
1
Þ
q
þ
1
q
¼
1
:
(3-8)
ð
a
Þ
q
2
þ
1
It is easy to see the value q
0
is the only solution of Eq. (3-8) with q<1. Assume Eq. (3-8) has another
solution; that is, assume we can find a value q
¼
0 satisfies Eq. (3-8). We next show that q
¼
6¼
0 that satisfies Eq. (3-8).
To solve for q, because we assumed q
6¼
0, we can divide both sides of
Eq. (3-8) by q to obtain:
¼
ð
a
1
Þ
q
þ
1
1
1
:
ð
a
1
Þ
q
2
þ
Then
q
2
ð
a
1
Þ
þ
1
¼ð
a
1
Þ
q
þ
1
;
and so, because
a
6¼
1,
q
2
¼
q
:
This equation has two solutions: q
1
is the only nonzero solution of the algebraic Eq. (3-8). As we know that
q
¼
0 and q
¼
1, establishing that q
¼
¼
1 cannot be the limit of the decreasing sequence of the allelic
frequencies q
n
< 1, this implies that Eq. (3-8) does not have a nonzero
solution q<1. Then q
¼
0 is the only solution of Eq. (3-8), and this
implies lim
n
!1
q
n
¼
0.
We have therefore established the following fact: Under the assumption
that the recessive homozygous genotype
aa
is less fit than the homozygous
dominant
AA
and the heterozygous
Aa
genotypes, the frequency of the harmful
allele
a
will diminish from generation to generation, and allele
a
will be
eventually eliminated from the gene pool.
An interesting question here is how many generations it takes for the
maladaptive allele to reach virtually negligible frequency levels in the
gene pool. The exact answer depends on the survival level
—the lower
the survival level, the faster the harmful allele is eliminated from the
a
