Genetic variation in populations can be described by genotype and allele frequencies.
            (not "gene" frequencies) [NS 01-01]

Consider a diploid autosomal locus with two alleles and no dominance
      (=> semi-dominance: AA , Aa , aa  phenotypes distinguishable)

      # AA = x    # Aa = y    # aa = z    x + y + z = N (sample size)

      f(AA) = x / N       f(Aa) = y / N       f(aa) = z / N

      f(A) = (2x + y) / 2N          f(a) = (2z + y) / 2N

            or    f(A) = f(AA) + 1/2 f(Aa)      f(a) = f(aa) + 1/2 f(Aa)

            let p = f(A), q = f(a)    p & q are allele frequencies 

      Properties of p & q

        p + q = 1     p = 1 - q    q = 1 - p

            (p + q)²  =  p² + 2pq + q²  =  1

            (1 - q)² + 2(1 - q)(q) + q² = 1

        p & q interchangeable wrt [read, "with respect to"] A & a;

        q usually used for
                  rarer, recessive, deleterious (disadvantageous), or "interesting" allele;

              BUT   'common' & 'rare' are statistical properties
                         'dominant' & 'recessive' are genotypic properties
                         'advantageous' & 'deleterious' are phenotypic properties
                  *** any combination of these properties is possible ***

What happens to p & q in one generation of random mating?

For a population of monoecious organisms that reproduce by random union of gametes
       ("tide pool" model)...

      (1) Determine the expectation
            of parental alleles coming together in various genotype combinations.
            expectation: the anticipated value of a variable
                                   not quite the same as probability [NS 03-Box1]
            Proofs by the probability, binomial expansion, & Punnet Square methods [SR2019 3.1]
            all show that expectation of f(AA) = p²
                                 expectation of f(Aa) = 2pq
                                 expectation of f(aa) = q²

     (2) Re-describe allele frequencies among offspring (f(A') & f(a')).

       f(A') = f(AA) + 1/2 f(Aa)
                    = p² + (1/2)(2pq) = p² + pq = p(p+q) = p' = p
 

       f(a') = f(aa) + 1/2 f(Aa)
                    = q² + (1/2)(2pq) = q² + pq = q(p+q) = q' = q

     p² : 2pq : q² are Hardy-Weinberg proportions (cf. Mendelian ratios 1 : 2 : 1 )

     Not an "equilibrium": proportions shift within & between generations during evolution

      Hardy-Weinberg Proportions (HWP) obtained under more realistic conditions:

            (1) multiple alleles / locus

                  p + q + r = 1
                  (p + q + r)² = p² + 2pq + q² + 2qr + r² + 2pr = 1

                  The proportion of heterozygotes (H = 'heterozygosity')
                        is a measure of genetic variation at a locus.

              H_obs = f(Aa) = observed heterozygosity
              H_exp = 2pq   = expected heterozygosity (for two alleles)

              H_e = 2pq + 2pr + 2qr = 1 - (p² + q² + r²)    for three alleles

                                n
                  H_e = 1 -  (q_i)²      for n alleles
                               i=1

                        where q_i = freq. of i th allele of n alleles at a locus
 
             Ex.: if q₁ = 0.5, q₂ = 0.3, & q₃ = 0.2
                            then H_e = 1 - (0.5² + 0.3² + 0.2²) = 0.62

            HOMEWORK: Calculate H_e for a locus with 10 alleles, all at equal frequency

            (2) sex-linked loci
                    iff [read: "if and only if"] allele frequencies in males and females identical
                    If frequencies initially unequal, they converge over several generations.

            (3) dioecious organisms [NS 01-Box2]
                    sexes separate
                    HWP produced by random mating of individuals
                        expand (p² 'AA' + 2pq 'AB' + q² 'BB')² :
                               nine possible 'matings' among genotypes
                    selfing (self-fertilization) remains possible

Genotype proportions in natural populations can be tested for Hardy-Weinberg Proportions (HWP)
     H_o (null hypothesis): no outside factors acting
     Note:  HWP often called a HW equilibrium, BUT
                HWP observed only at birth of any single generation
                         change between newborns & adults
                         allele frequencies in population change due to outside factors:
                HWP not an "equilibrium"
    See Excel spreadsheets for Chi-Square calculations   

    Ex.: MN blood groups in Homo

      Among Euro-Americans:

MM	MN	NN	Sum
1787	3039	1303	6129

        f(M) = [(2)(1787) + 3039] / (2)(6129) = 0.539

        f(N) = [(2)(1303) + 3039] / (2)(6129)  = 0.461    = 1.0 - 0.539

     Chi-square (²) test (NS 01-Box 3):

	N genotypes	#	expected	observed	(obs-exp)	d2/exp
MM	p2N	(0.539)2(6129)	1781	1787	6	0.020
MN	2pqN	(2)(0.539)(0.461)(6129)	3046	3039	-7	0.012
NN	q2N	(0.461)2(6129)	1302	1303	1	0.000
			6129	6129	2 =	0.032ns

 
                (cf. critical value p_{.05[1 d.f.]} = 3.84)                              ( p >> 0.05)
      Note: only one degree of freedom, because there are only two alleles

      HOMEWORK: S&R Table 3.1 & Eqn 3.2 are wrong: explain the error, correct the calculation
            See notes on Chi-Square calculations for some hints

         Chi-square test on combined data:

	exp	obs	d=(o-e)	d2/exp
MM	192	327	135	94.9
MN	532	268	-264	131.0
NN	367	496	129	45.3
			2 =	271.2**

                                                                                  (p << 0.01)

      *=> A mixture of populations, each of which shows HWP,
            will not show expected HWP
            if the allele frequencies are different in the separate populations.

      Wahlund Effect: artificial mixture of populations has deficiency of heterozygotes [NS 01-02]
                    (Relate this to F statistics and population structure, later on)

Hardy-Weinberg Proportions a 'null hypothesis':
      What are consequences of other genetic / evolutionary phenomena?

     Five major factors:

      1. Natural selection
            Change of allele frequencies (q) [read 'delta q']
                  occurs due to differential effects of alleles on 'fitness'
            Consequences depend on dominance of fitness (see NatSel MATLAB exercise)
            Natural Selection is the principle concern of evolutionary theory

      2. Mutation
             A and A' inter-converted at some rate µ
             If µ(AA')    µ'(AA'), net change in one direction.

      3. Gene flow
            Net movement of alleles between populations at some rate m
            (Im)migration introduces new alleles, changes frequency of existing alleles.

      4. Statistical sampling error
            Chance fluctuations occur in finite populations, especially those with small N
            Genetic drift: random change of allele frequencies
                                     over time and (or) space, within and (or) among populations
            Non-random reproduction: variable sex ratio, offspring number, population size, etc.

      5. Population structure
           Inbreeding: preferential mating of relatives at some rate F
                Inbreeding modifies genotype proportions but not allele frequencies
          Assortative Mating: differential mating of phenotypes and (or) genotypes
           Metapopulation structure (SR2019 3.8): sub-populations of total population differ