The Mathematical Theory of Natural Selection

  Charles Darwin described Natural Selection as an evolutionary process

      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
                    [Discourse on SVO structure of English language: 'adaptation' as noun & verb]

      Under certain conditions, this results in Descent with Modification (evolution)
                                               & The Origin of Species

The General Selection Model

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).
    Fitness related to success at Survival AND 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 measure of fitness phenotypes
            of AA, AB, & BB genotypes, respectively,
            read as "W bar" = Mean fitness 

Consider simplest case:  Complete Dominance

      genotype:   AA     AB      BB
      phenotype: W0  =  W  W2    (AA & AB phenotypes identical)

      Model simplifies to  pq2(W2 - W1)     (since W1 - W0 = 0)
  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 (W2 - W1) : greater difference in fitness,
                                              greater intensity of selection
                                              more rapid change

A numerical example of Selection:
       Tay-Sachs Disease is caused by a series of alleles that are
                       rare            (q =  0.001)
                       recessive   (W0 = W1 = 1)
                       lethal         (W2 = 0)

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

        That is, Natural Selection reduces frequency of Tay-Sachs allele
            ~ one part in a million (1 / 0.0012) per generation
              q' = 0.001000 - 0.000001 = 0.000999

General cases of Directional Selection
        Dependence on Dominance model [SR2019 4.3]
        Rate of vs q [SR2019 4.2]

Alternate notation with selection coefficients 

         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 =  W  W2    (AA and AB have identical phenotypes)
                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:  W  W    W2    (all phenotypes different)
         or           1 - s   1     1 - s2

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

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

Other alternatives (Gillespie, 1968) [see SR2019 Table 4.4]

       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
: each allele contributes equally to heterozygote fitness

        HOMEWORK: what if 0 < h < 1 ?        

The General Selection Model: Summary

      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 [SR2019 4.2]
            (whether allele dominant, semi-dominant, or recessive).
      Rate of change: interplay of both factors (see MATLAB exercise),
                                 q0 at start, if  q < or << 0.1) at start
[SR2019 4.3]

      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]
                                      , where q = 0

                                    Ex.: Balancing selection for Hemoglobin S & A alleles [NS_07-Box7 smc] HOMEWORK
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

Stabilizing, Disruptive, & Directional selection [SR2019 4.1 modified]

Alternative models of Natural Selection

Natural Selection in natural populations

Text material © 2022 by Steven M. Carr