Principles of Population Genetics
Various aspects of "Population"

     gene pool (a genetic unit):
          all the alleles at a (single) locus
     deme (an ecological unit):
          all the conspecific individuals in an area
     panmictic unit (a reproductive unit):
          a group of randomly interbreeding individuals
     sample (a numerical unit):
          a statistical subset of size 'N'
Theory of allele frequencies: p's & q's

Genetic variation in populations can be described by genotype and allele frequencies.
            (not "gene" frequencies)

Consider a diploid autosomal locus with two alleles and no dominance
      (=> semi-dominance: AA , Aa , aa  phenotypes are 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)=  p2 + 2pq + q2  =  1

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

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

        q is usually used for the
                  rarer, recessive, or deleterious (disadvantageous) allele;

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



The Hardy-Weinberg Theorem

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

Consider a population of monoecious organisms reproducing by random union of
      gametes ("tide pool" model)...

      (1) Determine the expectations
            of parental alleles coming together in various genotype combinations.
            [expectation: the anticipated value of a variable  probability]
            The  probability , binomial expansion , Punnet Square methods
            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 (A' & 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



The Hardy - Weinberg Theorem (1908):
     In the absence of other genetic or evolutionary factors,
        allele frequencies are invariant between generations, and
            constant genotype frequencies are reached in one generation.

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


      The Hardy-Weinberg Theorem holds 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.

              Hobs = f(Aa) = observed heterozygosity
              Hexp = 2pq   = expected heterozygosity (for two alleles)

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

                                n
                  He = 1 - (qi)2      for n alleles
                               i=1

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

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

            (3) dioecious organisms
                    sexes are separate
                    H-W is produced by random mating of individuals (random union of genotypes).
                        expand (p2 'AA' + 2pq 'AB' + q2 'BB')2 :
                               nine possible 'matings' among genotypes
                              (See derivation)
                    No selfing (self-fertilization not possible)

  [Read "Suggestions for using the Website" for comments on the math in this course]



Application of Hardy-Weinberg to population biology

Genotype proportions in natural populations can be tested for H-W conditions
     Ho(null hypothesis): no outside factors are acting.

    Ex.: MN blood groups in Homo

      Among North American whites:
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:
 
#
expected
observed
(obs-exp)
d2/exp
MM
(0.539)2(6129)
1781
1787
6
0.020
MN
(2)(0.539)(0.461)(6129)
3046
3039
-7
0.012
NN
(0.461)2(6129)
1302
1303
1
0.000
 
     
2
0.032ns
                  (cf. critical value p.05[1 d.f.] = 3.84)                              ( p >> 0.05)
note: there is only one degree of freedom, because there are only two alleles



     But (you ask) won't "expected" always more or less equal "observed",
            cuz that's where "expected" comes from?
            Consider an artificial data set :
 
 
MM
MN
NN
Sum
f(M)
f(N)
 Navaho (US)
305
52
4
361
0.92
0.08
Koori (Aus.)
22
216
492
730
0.18
0.82
Combined
327
268
496
1091
0.42
0.58
    (Homework: show that Navaho & Koori populations exhibit H-W proportions)

         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 Hardy-Weinberg proportions,
            will not show expected Hardy-Weinberg proportions
            if the allele frequencies are different in the separate populations.

           Wahlund Effect: an artificial mixture of populations will have a deficiency of heterozygotes


Evolutionary Genetics:
    modification of Hardy-Weinberg conditions

The Hardy-Weinberg conditions are the 'null hypothesis':
      What are the consequences of other genetic / evolutionary phenomena?

     Five major factors:

      1. Natural selection
            Change of allele frequencies (q) [read as 'delta q']
                  occurs due to differential effects of alleles on 'fitness'
            Consequences depend on dominance of fitness (see Lab #1)
            Natural Selection is the principle concern of evolutionary theory
               (& first half of this course)

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

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

      4. Population structure
           Inbreeding: preferential mating of relatives at some rate F (see Homework).
           Non-random reproduction: variable sex ratio, offspring number, population size

      5. Statistical sampling error
            Chance fluctuations occur in finite populations, especially those with small size N.
            Genetic drift: random change of allele frequencies
                                 over time & among populations (see Homework)


All text material © 2010 by Steven M. Carr