THE GENETiCAl THEoRy of NATuRAl SElECTioN 109
dark A 2 A 2 homozygotes are well camouflaged, and all of them survive. Selection
has increased the frequency of A 2 to 0.58. When the survivors reproduce, this new
allele frequency is passed to the next generation.
The rate at which an allele’s frequency changes—that is, the speed of evolu-
tion—is determined by the relative fitness advantage of the favored allele. The
mathematical theory behind that fact is explained in BOX 5A.
FIGURE 5.7 shows examples of selective sweeps, the situation in which a ben-
eficial mutation spreads through a population. The population is initially fixed for
allele A 1 , which means that the allele is at a frequency of 1.0. We will use the A 1 A 1
homozygote for the fitness reference. Mutation then produces a new allele A 2 thatThis box explores the evolutionary change that results
from selection on a single locus with two alleles, A 1 and A 2.
We will start with the simplest case, in which the heterozy-
gote has intermediate fitness, as is often the case, and we
will use the A 1 A 1 genotype as our fitness reference. The
A 2 allele increases survival, and for each copy of A 2 that an
individual carries, its fitness increases by a proportion s.
The frequency of A 2 is p, and we assume the population is
in Hardy-Weinberg equilibrium. The relative fitnesses and
frequencies of the three genotypes are therefore:
Genotype A 1 A 1 A 1 A 2 A 2 A 2
Relative fitness 1 1 + s 1 + 2s
Frequency at birth (1 – p)^2 2 p(1 – p) p^2
Our goal is find the frequency of A 2 in the next genera-
tion. The first step is to calculate the frequencies of the
three genotypes among surviving adults after selection
has acted. Those are:Genotype A 1 A 1 A 1 A 2 A 2 A 2Frequency in adultsIn each numerator, we see the product of the genotype’s
frequency at birth and its relative fitness. In the denomi-
nators is w—, a factor that is needed to ensure that all the
frequencies sum to 1 (which is required by the definition
of frequencies). That number has an important interpreta-
tion: it is the mean fitness of the population. It is given by
the sum of the three numerators:w—
= (1 – p)^2 + 2p(1 – p)(1 + s) + p^2 (1 + 2s)
= 1 + 2sp (A1)We now have what we need to find the allele frequen-
cies at the start of the next generation. Using the logic
sketched in Figure 4.6, the frequency of allele A 2 is givenby the frequency among surviving adults of A 2 A 2 individu-
als plus half the frequency of A 1 A 2 individuals. After a bit of
algebra, we find the answer is:
pʹ
= [1 + s(1 + p)] p/ w—
(A2)We can learn more by asking how much allele frequen-
cies change from one generation to the next. Using our re-
sults from above, we find that the change in the frequency
of allele A 2 caused by one generation of selection is∆p
= pʹ – p
= sp( 1 – p) / w—
≈ sp( 1 – p) (A3)The first two expressions for ∆p are exact. The last one is
an approximation that is very accurate when s is less than
0.1. It is useful because it is so simple.
Equation A3 has two key implications. The rate that the
allele frequency changes—that is, the rate of evolution—
caused by selection is proportional to two quantities. The
first is s, the selection coefficient. Logically enough, if s = 0,
then all genotypes have the same fitness, there is no selec-
tion, and therefore no evolutionary change. The second
factor that governs the rate of evolution is the quantity
p(1 – p), which represents the genetic variation at the locus.
Notice that p(1 – p) is 0 if either p = 0 or (1 – p) = 0. In either
case, there is only one allele in the population and therefore
no genetic variation. Variation is maximized when both al-
leles are equally frequent (see Figure 5.8).
We’ve now seen how to calculate the change in allele
frequencies for the particular case of the relative fitnesses
shown above. Fitnesses that follow that pattern are com-
mon for alleles with small fitness effects. For other situa-
tions, including cases in which heterozygotes have a fit-
ness higher or lower than both homozygotes, we calculate
the allele frequency change using the same logic used
here. The results, however, are a bit more complicated
than Equation A3. For more details, see [18].BOX 5A
Evolution by Selection on a Single Locus
w(1 – p)^22 p(1 – p)(1 + s)p^2 (1 + 2s)www(1 – p)^22 p(1 – p)(1 + s)p^2 (1 + 2s)www(1 – p)^22 p(1 – p)(1 + s)p^2 (1 + 2s)ww05_EVOL4E_CH05.indd 109 3/23/17 9:01 AM