If:     variation exists for some trait, and
              fitness difference is correlated with that trait, and
              trait is to some degree heritable (determined by genetics),
      Then: trait distribution will change
                over life history within single generation, and
                between generations.

      Process of change called "adaptation"

      Or, "Natural Selection" is a process in which
            "adaptation" occurs such that "fitness" increases
         
      Under certain conditions, this results in Descent with Modification (evolution)
                                               & The Origin of Species

Evolution & Natural Selection can be modeled genetically

Natural Selection results in change of allele frequency (q) [read "delta q"]
    in consequence of differences in relative fitness (W)
    of phenotypes to which alleles contribute.

Fitness is a phenotype of individual organisms (Darwinian fitness)
    Fitness determined genetically (at least in part): follows Mendelian rules
    Fitness related to success at Survival & Reproduction
    Fitness can be measured & quantified : Analysis of survivorship & fecundity schedule
          relative fitness of phenotypes (genotypes) assigned numerical values

Consequences of natural selection depend on Dominance of Fitness:
          Are "fitter" phenotypes due to dominant or recessive allele ?

Then, allele frequency change over time predicted by General Selection Model [see derivation]

q = [pq] [(q)(W₂ - W₁) + (p)(W₁ - W₀)] /

            where W₀, W₁, & W₂ are phenotypic measures of fitness
            of AA, AB, & BB genotypes, respectively,
            read as "W bar" = Mean fitness 

      genotype:   AA     AB      BB     (A & B are alternative alleles at same locus)
      phenotype: W₀  =  W₁    W₂    (AA & AB phenotypes identical: A dominant to B)

      Model simplifies to q  pq²(W₂ - W₁)     (since W₁ - W₀ = 0)
                                                                             (read   as 'proportional to')

            If 'B' phenotype more fit than 'A' phenotype,
                  W₂ > W₁   & q > 0    so   q increases

            If 'B' phenotype less fit than 'A' phenotype,
                  W₂ < W₁    & q < 0    so   q decreases

            then q  (W₂ - W₁) : greater difference in fitness,
                                               greater intensity of selection
                                               more rapid change

A numerical example of Selection:
       Tay-Sachs Disease (TSD) arises from deficiency of Hexosaminidase-A
              The alleles are
                       rare            (q =  0.001)
                       recessive   (W₀ = W₁ = 1)
                       lethal         (W₂ = 0)

        Then q = pq²(W₂ - W₁) = -pq²  -q²   (if q << p, then p ~ 1)

        Natural Selection reduces frequency q of such an allele by
            ~ one part in a million (0.001²) per generation
              q' = 0.001000 - 0.000001 = 0.000999

         s = 1 - W

        Selection Coefficient (s) = difference in fitness
            of phenotype relative to 'standard' phenotype with fitness  W = 1
            Math simpler because only one variable used

      (1) Complete dominance

      genotype:   AA      AB     BB
      phenotype:  W₀ =  W₁   W₂    (AA & AB phenotypes identical, as before)
                or        1   =   1     1 - s

         if  0 < s < 1 : 'B' is deleterious (at a selective disadvantage)
         if  s < 0       : 'B' is advantageous

        then     q = -spq² / (1 - sq²)      [see derivation]
 

      (2) Incomplete dominance

      genotype:    AA     AB        BB
      phenotype:  W₀    W₁     W₂    (all phenotypes different)
         or           1 - s₁    1     1 - s₂

      if 0 < s₁ & s₂ < 1 : heterozygote advantage ( "overdominance" of fitness)
      Population has optimal fitness when both alleles retained:
           q reaches an equilibrium where q = 0
                   0 <  < 1   (read as, "q hat")

           then     = (s₁) / (s₁ + s₂)          [see derivation]

Other alternatives

       genotype:   AA      AB      BB
      phenotype:    _{1     1 - hs   1 - s}

        _{1 - s = fitness difference between homozygotes
        h scales relative fitness of heterozygote wrt homozygotes
                if h = 1         then (1 - hs) = (1 - s)                 fitness of AB = BB}
               if h = 0      then (1 - hs) = 1                      fitness of AB = AA
               if h = 0.5   then (1 - hs) = (1 - (0.5)(s))     fitness of AB intermediate bx AA & BB
                                semi-dominance: each allele contributes equally to heterozygote fitness

        HOMEWORK: What situation is described by 0 < h < 1 and h  ≠ 0.5 ?

      Direction of allele frequency change due to fitness difference of alleles
            (whether effect of allele on phenotype deleterious or advantageous).
      Ultimate consequences depend on Dominance of Fitness
            (whether allele dominant, semi-dominant, or recessive).
      Rate of change an interplay of both factors (see MATLAB exercise),

      AA   AB     BB    Consequence of natural selection   [ let q = change in f(B) ]

      W₀ =  W₁ =  W₂    No selection (neither allele has selective advantage):
                         then     q = 0,  H-W proportions remain constant

      W₀ =  W₁ >  W₂    deleterious recessive (= advantageous dominant):
                         then     q < 0,     q  0.00  (loss): how fast? Does it get there?

      W₀ =  W₁ <  W₂    advantageous recessive (= deleterious dominant):
                          then     q > 0,     q  1.00  (fixation): how fast?

      W₀ <  W₁ >  W₂    heterozygote superiority     [special case of incomplete dominance]:
                                          AKA "overdominance" [SR2019 4.7, 4.6]
                                    q   , where q = 0

                                    Ex.: Balancing selection for Hemoglobin S & A alleles
                                            See National Public Radio story on societal aspects of Sickle-Cell Anemia
                                            See National Public Radio story on use of CRISPR to treat Sickle-Cell Anemia