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

Text material © 2024 by Steven M. Carr