Derivation of the General Selection Equation
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Genotype
AA
AB
BB
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(1) Frequency p2
+
2pq
+ q2
=
1
before selection
(2) Fitness
W0
W1
W2
(3)
Relative p2W0
+ 2pqW1
+ q2W2
=
Contribution
(4) Frequency p2W0/ + 2pqW1/ +
q2W2/ = / = 1
after selection
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(1) Genotype distributions before selection follow Hardy-Weinberg
expectations.
(2) Each genotype AA, AB, and BB
has a distinct phenotype: W0, W1,
& W2, respectively.
W is the expectation
that an individual with a particular genotype will survive
& reproduce
(3) Each genotypic class makes a relative
contribution to the next generation,
which is proportional
to its initial frequency, weighted by its fitness.
[e.g., if the AA genotype has a
frequency of 0.25 and 80% survive to reproduce,
the relative contribution of AA to the next
generation is (0.25)(0.8)=0.20]
The sum of the relative contributions
of all three genotypes is
(read as, "W bar") = mean population fitness
In
this
simple model,< 1,
because not all individuals on Line (1) survive.
(4) Because
< 1, the surviving genotypic contributions have to be "normalized
":
Dividing the proportion of each genotype
by returns the
sum to unity,
& the final values are the relative genotype
frequencies after selection.
To derive the allele
frequencies after selection,
take Line (4) above and
recall q = f(BB) + (1/2) f(AB)
so q' = q2W2/ + (1/2) 2pqW1/ = q(qW2
+ pW1)/
then q
= q' - q = qafter
- qbefore
= q(qW2 + pW1)/ - q/
= [(q)(qW2 + pW1) - (q)(p2W0 + 2pqW1 + q2W2)] / [Note 1]
= [(q)(qW2 + pW1
- p2W0 - 2pqW1
- q2W2)]
/
[Note 2]
= [(q)(pqW2 + W1p(1-2q) - pW0p)]
/
[Note 3]
= [(pq)(qW2 + W1(1-2q)
- W0p)] /
[Note 4]
= [(pq)(W2q + W1(p-q) - W0p)]
/
[Note 5]
= [(pq)(W2q + W1p - W1q
- W0p)/
[Note 6]
q = [pq] [(q)(W2 - W1)
+ (p)(W1 - W0)] / []
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Notes:
[1] Expand
in numerator, from Line 4 of Table
[2] Combine terms by factoring out q from
[3] trick: Combine
W2 terms by noting (q - q2) = (q)(1 - q) = pq
[4] Factor out p from W0,
W1, & W2
terms
[5] trick: (1 - 2q)
= (1 - q) - q = (p - q)
[6] Expand W1 term, gather p
& q terms
Homework
: Repeat the derivation of the model for in
terms of the other allele, p = p' - p