Download Classical Model of Selection at a Single Locus

CLASSICAL MODEL OF SELECTION AT A SINGLE LOCUS
THE MODEL: Same conditions as Hardy-Weinberg, but with selection included.
Genetic system:
1) diploid population
2) sexual reproduction
3) random mating
Selection
1) identical selection in both sexes
2) viability selection
3) constant selection on each genotype
Other factors
1) non-overlapping generations
2) infinite population size
3) no migration (gene flow)
4) no mutation
Single locus with 2 alleles, A1 and A2, in frequencies p and q.
RELATIVE FITNESS
GENOTYPES
A1 A1
A1 A2
A2 A2
FREQUENCY
p2
2pq
q2
FITNESS
w11
w12
w22
Fitnesses are denoted as relative fitnesses: highest fitness is standardized to 1, others
are denoted as a fraction less than 1 (see Table 3.5)
Example: deleterious recessive
GENOTYPES
A1 A1
A1 A2,
A2 A2
FITNESS
1
1
1-s
Average fitness in the population:
w
_ = p2w11 + 2pqw12 + q2w22
CHANGE IN GENE FREQUENCIES
Remember that the frequencies of the alleles in the population are (relative to genotype
frequencies): p = P + 2H = p2 + pq, q = Q + 2H = q2 + pq
To get frequencies of the alleles in the next generation, multiply each genotype frequen
cy by its relative fitness and divide by the mean fitness:
p'
= [p2w11 + pqw12]/w
_
similarly:
q'
= [q2w22 + pqw12]/w
_
We can denote the change in gene frequency in terms of the change in gene frequency,
w - p
= [p2w11 + pqw12]/_
∆p = p' - p
= [p2w11 + pqw12]/_
w - p_
w/_
w
(multiplied out)
= {[p2w11 + pqw12] - p[p2w11 + 2pqw12 + q2w22]}/_
w
= [p2w11 + pqw12 - p3w11 - 2p2qw12 - pq2w22]/_
w
(gather "w" terms)
= [p2w11(1 - p) + pqw12(1 - 2p) - pq2w22]/_
w
(sub. q = 1 - p)
= [p2qw11 + pqw12(q - p) - pq2w22]/_
w
(factor out pq)
= pq[pw 11 - pw12 + qw12 - qw22]/_
w
THE BOTTOM LINE:
∆p = p' - p
_
= pq[p(w11 - w12) + q( w12 - w22)]/w
Note that ∆p is proportional to the gene frequencies, but also to the relative
fitnesses of the genotypes.
SHORTCUT
Deleterious recessive alleles represent a special case, if we substitute w11, w12 = 1, w22
= 1 - s into the equations above:
Mean fitness (_
w) = p2w11 + 2pqw12 + q2w22 = p2 + 2pq + q2 - sq2 = 1 - sq2
∆p = p' - p =
pq[p(w 11 - w12) + q(w12 - w22)]/_
w
2
=
pq[q(1 - (1 - s))]/(1 - sq )
=
spq2/(1 - sq2)