Evolution__3rd_Edition

106 PART 2 / Evolutionary Genetics
The model predicts the rate of
change in gene frequency as the
superior gene is fixed
Mean fitness = p 2 + 2pq + q 2 (1 − s) = 1 − sq 2 (5.3)
Dividing by mean fitness in the algebraic case is the same as dividing by the population
size after selection in the numerical example. Notice that now the adult genotype frequencies
are not in Hardy–Weinberg ratios. If we tried to predict the proportion of aa
from q 2 , as in the MN blood group (Section 5.4), we should fail. The frequency of aa is
q 2 (1 − s)/1 − sq 2 , not q 2 .
What is the relation between p′ and p? Remember that the frequency of the gene A at
any time is equal to the frequency of AA plus half the frequency of Aa. We have just
listed those frequencies in the adults after selection:
p2+ pq p
p′
= =
1−sq 1−sq
2 2
(5.4)
(remember p + q = 1, and therefore p 2 + pq = p(p + q) = p.) The denominator 1 − sq 2 is
less than 1, because s is positive, so p′ is greater than p: selection is increasing the frequency
of the A gene. We can now derive a result for ∆p, the change in gene frequency
in one generation. The algebra looks like this.
p
∆p = p′ − p =
1 − sq
2
p − p + spq
=
1 − sq2
spq2
=
1 − sq
2
− p
2
(5.5)
For example, if p = q = 0.5 and aa individuals have fitness 0.9 compared with AA and Aa
individuals (s = 0.1) then the change in gene frequency to the next generation will be
(0.1 × 0.5 × (0.5) 2 )/(1 − 0.1 × (0.5) 2 ) = 0.0128; the frequency of A will therefore increase
to 0.5128.
We can use this result to calculate the change in gene frequency between successive
generations for any selection coefficient (s) and any gene frequency. The result in this
simple case is that the A gene will increase in frequency until it is eventually fixed (that
is, has a frequency of 1). Table 5.4 illustrates how gene frequencies change when selection
acts against a recessive allele, for each of two selection coefficients. There are two
points to notice in the table. One is the obvious one that with a higher selection
coefficient against the aa genotype, the A gene increases in frequency more rapidly. The
other is the more interesting observation that the increase in the frequency of A slows
down when it becomes common, and it would take a long time finally to eliminate the a
gene. This is because the a gene is recessive. When a is rare it is almost always found in
Aa individuals, who are selectively equivalent to AA individuals: selection can no
longer “see” the a gene, and it becomes more and more difficult to eliminate them.
Logically, selection cannot eliminate the one final a gene from the population, because
if there is only one copy of the gene it must be in a heterozygote.
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