Biology Reference
In-Depth Information
q
0
by the time of reproduction. Thus, the number of A alleles at that
time will be 2p
0
N
to N
a
þ
2p
0
q
0
N, and the number of a alleles will be
q
0
N
2
2p
0
q
0
N. Denoting the proportions of the A and a alleles in the
first generation by p
1
and q
1
, we now calculate that:
a
þ
q
0
N
2
þ
2p
0
q
0
N
a
q
1
¼
q
0
N
2p
0
N
2
þ
2p
0
q
0
N
þ
þ
2p
0
q
0
N
a
0
1
(3-5)
q
0
þ
p
0
q
0
q
0
þ
p
0
a
a
@
A
:
¼
p
0
¼
q
0
q
0
þ
q
0
þ
p
0
2p
0
q
0
þ
2p
0
q
0
þ
a
a
The proportion p
1
could be calculated similarly, but we assumed that
A and a are the only alleles present in the population, so it is easier to
state that p
1
¼
q
1
. Similarly, if p
n
and q
n
denote the proportions of the
A and a alleles in the nth generation, then because of weaker fitness
of the aa genotype, these frequencies will change in the (n
1
þ
1)-st
generation to:
q
n
þ
1
q
n
þ
p
n
a
¼
q
n
and p
n
þ
1
¼
1
q
n
þ
1
:
(3-6)
a
q
n
þ
2p
n
q
n
þ
p
n
Notice how different this situation is compared with the Hardy-
Weinberg case discussed previously. The allelic frequencies of A and
a now change from generation to generation, and, when we know the
frequencies for any given generation, Eq. (3-6) allows us to compute
their values for the following generation. Formulas such as
Eq. (3-6) are called recursive formulas.
Notice that because:
q
n
þ
p
n
¼ð
a
2p
n
q
n
þ
q
n
þ
p
n
Þ
q
n
þ
p
n
ð
q
n
þ
p
n
Þ¼ð
a
q
n
þ
p
n
Þ
q
n
þ
p
n
;
a
Eq. (3-6) could be rewritten as:
q
n
þ
p
n
a
q
n
þ
1
¼
q
n
p
n
:
(3-7)
ð
a
q
n
þ
p
n
Þ
q
n
þ
Now, because
< 1
;
a
p
n
<p
n
and because p
n
¼
1
q
n
, we have
a
a
ð
1
q
n
Þ
<p
n
;
which is the same as
<
q
n
þ
p
n
. This implies that
a
a
a
q
n
þ
p
n
< 1 and, together with Eq. (3-7), proves that q
n
þ
1
<q
n
.
ð
a
q
n
þ
p
n
Þ
q
n
þ
p
n
Thus, the allelic frequency of the harmful allele a does indeed decrease
from generation to generation.
Our next goal is to establish what happens to the harmful allele a in the
long run. Will it disappear from the gene pool or stabilize at a nonzero
value? More importantly, are we sure the limit for the sequence q
0
, q
1
,
q
2
,
, q
n
,
exists?
...
...
The last question has an easy answer. Because the sequence q
0
,q
1
,q
2
,
,
...
q
n
,
is a decreasing sequence of numbers bounded below by 0, the
...
