Not sure?  Look it up! EEB/MCB 182: Spring 2010.
Diploid Genetics.
 (Much of this Material Covered in Lab)

                                        (1)

  • Post-selection genotypic frequencies are given by the formulae in the table above.

  • From the post-selection genotypic frequencies, we can calculate gene frequencies for the next generation. To do this, we note that
    1. Each AA homozygote contributes 2 A genes to the next generation;
    2. Each aa homozygote contributes 2 a genes to the next generation;
    3. Each heterozygote contributes 1 A and 1 a gene to the next generation;
    4. The total number of genes in the next generation is 2W.

  • Then, if pt is the current frequency of A, and pt+1 the frequency next generation,
  • pt+1 = [2pt2wAA + 2pt(1-pt)wAa] / 2W                                                                    (2)

  • Moreover, the change, Dp, in gene frequency from one generation to the next, is given by
  • Dpt = pt+1 -  pt = [pt(1-pt) / 2W] (dW/dp)                                                              (3)

    where

    (dW/dp) = pt[(wAa- wAA) + (wAa- waa)] - (wAa- waa)                                                     (4)


    1. Introduction.

      1. Most eukaryotes have a diploid stage in which there are 2n chromosomes and therefore 2 copies of each autosomal gene.
        1. Dependiding on the organism, the diploid stage can be infrequent and transitory - e.g., a zygote which immediately undergoes meiosis - or it can be the dominant part of the life cycle.
        2. In plants, one observes an alternation of generations with multicellular haploid (gametophyte) and diploid (sporophyte) phases.

         
      2. Important evolutionary consequences of diploidy include
        1. Facilitates DNA repair;
        2. Creates new genotypes via recombination;
        3. Allows for partial or complete phenotypic masking of deleterious mutants (dominance);
        4. Allows for heterosis (hybrid vigor) in which heterozygous individuals are more "fit" than either homozygote.

       
      Click to enlarge.

      Figure 1. In large, panmictic populations, genotypic frequencies in generation iare determined by gene frequencies in generation i-1 according to formulae independently discovered by Hardy and Weinberg in the early years of the 20th century.

    2. Hardy-Weinberg Equilibrium.

      1. H-W formulae relate genotypic frequencies in the present geneteration, pij, to gene frequencies, pi, the preceding generation.

      2.  
      3. In large, panmictic populations, and in the absence of complicating factors such as
        1. Migration,
        2. Mutation and
        3. Selection,
        H-W frequencies are established in a single generation.
         
      4. Panmictic means "well mixed," i.e., that mating takes is independent of genotype.

      5.  
      6. The observation of genotypic frequencies that depart from H-W expectations is evidence that such forces are at work.

      7.  
      8. For two alleles, A and a,

                 pAA = pA2          pAa = 2 pApa          paa = pa2

      9. Note that both genic and genotypic frequencies must sum to 1 i.e.,

      10.  
        1. pA + pa = 1
        2. pAA + pAa + paa = 1

      11. Often, the frequencies are written as p and q, or as p and 1-p.
        1. In the first case, the genotypic frequencies are p2, 2pq and q2;
        2. In the second, p2, 2p(1-p) and (1-p)2.

      12. You can view H-W genotypic frequencies as deriving from two tosses of a weighted coin.

       
    3. Selection I.
      1. We assign fitnesses to each genotype. Call them wAA, wAa and waa.

      2. Outcomes:
        1. If wAA> wAa > waa, gene A prevails, i.e., p --> 1.
        2. If wAA< wAa < waa, gene a prevails, i.e., p --> 0.
        3. If wAA< wAa > waa, (heterozygote advantage), both genes are mainatined in the population, i.e., p --> p*, where 0 < p* < 1.
        4. If wAA> wAa < waa, (heterozygote infereiority), either gene A or gene a prevails depending on the initial frequencies, i.e., p --> 0 or p --> 1.

    4. Selection II.
    5. Genotype AA Aa aa
      Pre-Selection
      Frequency
      p2 2p(1-p) (1-p)2
      Fitness wAA wAa waa
      Post-Selection
      Frequency
      p2wAA/W 2p(1-p)wAa/W (1-p)2waa/W
      1. Now asign fitnesses, wAA, wAa and waa to the three genotypes.
        1. Think of fitnesses as relative numbers of offspring produced by each genotype.
        2. Then the total number of offspring is given by what is sometimes called the "average" or "population" fitness,
        3. W = p2wAA + 2p(1-p)wAa + (1-p)2waa

    Click for additional cases
    Click to enlarge

    Figure 2. When the average fitness, W, passes through a maximum on p = (0,1), the interior equilibrium, p* is stable. This is the case that obtains when heterozygotes have greater fitness than both homozygotes. For other cases, click the figure.

  • Equation (4) implies that in the presence of selection, the gene frequency, p, changes so as to maximize W, i.e., Dp takes its sign from dW/dp.

  •  
  • Sometimes, there is an interior equilibrium, i.e., a value of p, call it p*, between 0 and 1, such that Dp = 0. Then
  • p* = (wAa- waa) / [(wAa- wAA) + (wAa- waa)]           (5a)

    and

    q* = 1 - p*                                                                      
     
    = (wAa- wAA) / [(wAa- wAA) + (wAa- waa)]        (5b)

    1. At this value, dW/dp = 0, and W passes through a local maximum or minimum.
      1. If W passes through a maximum, p* is stable as shown in Figure 2.
      2. If W passes through a minimum, p* is unstable as shown here.
    2. Stability of p* results when heterozygotes have greater fitness than homozygotes (both types) - in which case we say that non-zero frequencies of the two alleles are maintained by balancing selection or (alternatively) that there is a balanced polymorphism.
    3. Instability of p* results when heterozygotes are inferior to both homozygotes - in which case, we say that disruptive selection leads to the fixation of one allele or the other.

  • p = 0 and p = 1 are also equilibria.
    1. If wAA ≥ wAa > waa (AA homozygote superior), ptà 1
    2. If wAA < wAa ≤ waa (aa   homozygote superior), ptà 0
    3. Click here for plots of Dp and W vs. p for these cases.

     
    Click to enlarge.

    Figure 3. Characteristic collapse or "sickling" of an RBC drawn from an individual with sickle cell anemia.

  • Sickle-Cell Anemia.

  •  
    1. Arguably the most famous example of a balanced polymorphism involves a disease called sickle-cell anemia (Figure 3).

    2.  
    3. Caused by a point mutation that results in the substitution of valine for glutamic acid at a single amino position in the b-globin polypeptides in the hemoglobin molecules that transport oxygen in your blood.
      1. This mutation is called HbS or, more simply, s.
      2. As such, it is one of many hemoglobin variants, of which three produce clinical abnormalities.
      3. Recall that
        1. Hemoglobin is comprised of 4 subunits - two a and two b chains, each of which is organized about a heme group that binds oxygen.
        2. Hemoglobin is found in red blood cells (RBCs, also called erythrocytes) anucleate (in mammals) cells produced by the bone marrow.

    4. Individuals diagnosed with sickle-cell anemia have two copies of the sickling gene. More precisely,
      1. Hemoglobin picks up oxygen in the lungs (oxygenated state) and releases oxygen to the tissues (deoxygenated state).
      2. The substitution of valine, which is hydrophobic, for glutamic acid, which is hydrophilic, in the b-globin of HbS individuals causes the b-chains of deoxygenated hemoglobin-S to polymerize, forming rigid fibers that collapse the cells.
      3. Although, HbS molecules de-polymerize when the blood is re-oxygenated, repeated cycling between polymerized de-polymerized states makes for abnormal cell rigidity and causes the cells themselves to aggregate into fibrous threads.
      4. The tendancy of HbS cells to aggregate, together with their abnormal shape, obstructs small blood vessels, thereby leading to oxygen deprivation and damage to tissues and organs.
      5. Result is life long morbidity, i.e., much pain and suffering plus diminished life expectancy (about 45 years in the US).

      Click to enlarge
      Click to enlarge

      Figure 4. Geographic distributions of the malaria parasite, Plasmodium falciparum, and the sickling gene, HbS.

    5. One copy of HbS results in a condition called sickle-cell trait, a condition which is not nearly so serious as the full-blown anemia.

    6.  
    7. In the presence of the malaria parasite, Plasmodium falciparum, heterozygous individuals actually have a selective advantage.
      1. The malaria plasmodium attacks red blood cells
      2. In Hb/HbS individuals, infected RBCs undergo sickling and are preferentially removed by the spleen.
      3. Sickling also interferes directly with the parasite's life cycle within the RBC.

    8. Although estimated fitness values for the three genotypes, Hb/Hb, Hb/HbS and HbS/HbS in the presence of malaria vary, the following numbers are probably reasonable:
    9. wHb/Hb = 0.88;      wHb/HbS = 1.0      wHbS/HbS = 0.14.      

    10. These allow us to calculate from Equations (5) the equilibrium frequencies two alleles. Thus,
    11. pHbS* = 1 - {(wHb/HbS - wHbS/HbS) / [(wHb/HbS - wHb/Hb) + (wHb/HbS - wHbS/HbS)]}                

      = 1 - {(1.0 - .14) / [(1.0-.88) + (1.0-.14)]} = 1 - 0.87 @ 0.13.                                               

    12. This estimate squares reasonably well with
      1. Geographical distribution of malaria and observed frequencies of HbS in the Old World Tropics (Figure 4);
      2. pHbS = .04 among Afro-Americans, a figure which reflects selection against HbS in North America.

      Click to enlarge

      Figure 5. Life cycle of the blood parasite, Plasmodium falciparum, one of four species of apicomplexans (see Purves et al., pp. 485-486) that cause malaria in man.

    13. Malaria is a blood infection spread by mosquitos.
      1. Caused by one of four protozoan species, Plasmodium spp., of which P. falciparum is the most dangerous.
      2. Life cycle alternates between anopheline mosquitos and man (Figure 5).
        1. Female mosquito injests haploid gametocytes when it sucks blood from an infected individual.
        2. Fertilization takes place in the mosquito's gut.
        3. Resulting diploid ookinete
          1. Penetrates wall of the gut;
          2. Develops (sporogeny) into an oocyst, which
          3. Ruptures liberating sporozoites that make their way to the insect's salivary gland.
        4. Sporozoites
          1. Transferred to human host when mosquito injects saliva as it bites.
          2. Develop into merozoites that enter the blood stream and penetrate red blood cells (RBCs).
        5. Merozoites
          1. Reproduce in the RBCs, lyzing them in the process - causes the characteristic bouts of fever and chills;
          2. Re-invade more RBCs;
          3. Produce gametocytes that are injested by another mosquito, thereby perpetuating the cycle.

    14. To summarize, P. falciparum is an extracellular parasite in mosquitos and an intracellular parasite in man.

    15.  
      Click to enlarge

      Figure 6. Rates of malaria-induced mortality in the 20th century. In recent decades, mortality has increased, principally in sub-Saharan Africa,


      Click to enlarge

      Figure 7. Mosquito netting of the sort sewn and donated by American church groups to help fight the spread of malaria. Often, the netting is treated with insecticide.

    16. Malaria infects on the order of 600 million souls worldwide, and kills 1-3 million of them annually.

    17.  
    18. Among the factors often cited to account for increasing malaria's increasing prevalence - especially in Africa - are the following:
      1. Deforestation and development which result in stagnant bodies of water in which mosquitos breed.
      2. Evolution of plasmodium resistance to anti-malarial drugs.
      3. Evolution of mosquito resistance to insecticides.
      4. Climate warming, which has been predicted to spread the mosquitoes' geographic range - but for which there is currently little supporting evidence.

    19. Equally important has been the WHO's policy of emphasizing the treatment of infected individuals as opposed to killing mosquitoes.
      1. Analogous to discouraging use of flu vaccine and telling people to seek treatment after they get sick.
      2. Unworkable in areas of the world where significant fractions of the populace don't see a doctor from one year to the next.
      3. Spraying the interiors of houses once or twice a year (so-called "residual spraying),
        1. Is far safer than exposing their occupants to malaria.
        2. Repells mosquitos, even if the insecticide no longer kills them in poplulations that have evolved resistance.
        3. Is more effective than encouraging people to sleep under mosquito netting.
      4. However, use of insecticide treated nets (ITNs), which WHO does emphasize, kills mosquitoes and therefore reduces infection rates even in households not using them.

       
    20. Spraying homes with fungi that kill mosquitoes may substitute for spraying with insecticides.

    21.  
    22. Genetically modified (GM) mosquitoes, in which the parasite cannot complete its life cycle, and an effective malaria vaccine still for the future.