Problem 19.1  A large outbred population of Drosophila melanogaster is polymorphic at a gene coding for a metabolic enzyme. There are two alleles (Q, P) at equal frequencies (q = p = 1/2). To determine whether this polymorphism is maintained by balancing selection, ten replicate cages are established, each with 100 PQ individuals, and 400 QQ individuals, taken from the base population. These are then propagated under the same conditions as the base population. The aim is to determine whether the allele frequency increases consistently, as expected if it is maintained at an equilibrium of 50% by balancing selection.

After 20 generations, samples of 200 flies are taken from each cage. The frequency of the P allele is {0.185, 0.17, 0.1825, 0.235, 0.305, 0.185, 0.1625, 0.1775, 0.155, 0.23}.

Problem 19.2  A study aims to measure the selection on the F allele of alcohol dehydrogenase in D. melanogaster, which is polymorphic in natural populations. A cage is set up, initially containing F₁ individuals from a cross between inbred lines that are fixed for S and fixed for F. The cage is maintained under constant conditions and at a large population size for 80 generations. Samples of 100 flies are taken every five generations. The frequency of the F allele in these samples is shown in Figure P19.1 and Table P19.1.

Problem 19.3  On February 1, 1898, New England suffered a severe snowstorm. In a classic study of natural selection, Hermon Bumpus collected 136 sparrows (Passer domesticus) that had been grounded by the storm; 72 of these survived, and the rest died. Bumpus measured nine morphological traits. Table P19.2 shows data for humerus length (in millimeters) in males and females.

Problem 19.4  The butterfly Heliconius erato carries a conspicuous color pattern, which advertises its distastefulness to birds. Two distinct patterns meet near Tarapoto, in Peru (see Fig. 18.17). They remain separated by a set of narrow clines, which are maintained by selection that favors whichever pattern is most common in an area (see pp. 475–476). In effect, rare alleles are at a disadvantage, because they tend to be found in butterflies with unusual patterns, which are not recognized by birds as being distasteful, and so are eaten.

The differences between the two patterns are mainly determined by three unlinked autosomal genes. The alleles d, D determine the pattern of rays on the hindwing; heterozygotes can be distinguished from both homozygotes. The alleles cr, Cr determine the size of the hindwing bar, and sd, Sd determine the size of the forewing bar. Cr is dominant over cr, and Sd over sd (Fig. P19.2).

Table P19.3 shows the position of each site along a linear transect, X, and the number of butterflies sampled, N. The numbers of butterflies homozygous for cr and homozygous for sd are shown; the last three columns show the numbers with the three genotypes d/d, D/d and D/D. (These numbers were determined simply by examining the wing patterns.)

Problem 19.5  The α and β globins diverged following a gene duplication event about 500 million years ago (Mya), near the origin of the vertebrates. Table P19.4 shows four comparisons between α and β globins, involving different pairs of species that have had largely independent evolutionary histories. For each pair, the sequences of the two proteins are aligned, and sites are classified according to whether they have the same, or a different, amino acid. NOTE 19D

Problem 19.6  Figure 19.11 gives the number of amino acid differences between the β globins of each pair of six mammalian species. The total length of the protein is 146 amino acids. In this problem, we investigate whether these proteins differ more than we would expect under the simplest model of molecular evolution. We assume that these mammals all diverged at about the same time, so that their phylogeny is a star. To make the analysis, we have to make two corrections: First, to allow for multiple substitutions, and second, to find how the observed distribution of differences between two species is related to the underlying distribution of the number of changes along each lineage.

Problem 19.7  The neutral theory can be tested by comparison with the distribution of allele frequencies within populations. One of the first tests of this kind was devised by Ewens and Watterson (see p. 534, and Web Notes).

Suppose that we see k alleles in a sample of n genes. The numbers of each allele are {n₁, n₂, ..., n_k). Under the neutral theory, the chance of seeing this configuration is

where S_n(θ) ≡ θ(θ + 1) ··· (θ + n – 1), |S_n^(k)| is the coefficient of θ^k in S_n (θ), and θ = 4 N_eμ.

This formula has two terms. The first gives the probability of seeing k alleles, and depends on the parameter θ = 4N_eμ. (For example, with n = 4 genes, S₄(θ) = θ(θ + 1)(θ + 2)(θ + 3) = 6θ + 11θ² + 6θ³ + θ⁴. The probability of there being 1, 2, 3, 4 alleles in the sample is just proportional to the four terms in this series.) The second term is the probability of seeing the numbers {n₁, n₂, ...}, given that we have seen k alleles in a sample of n. (For example, with four genes, the chance of seeing two alleles in numbers {3,1} is 4!/2! × 11 × (3 × 1) = 4/11. This does not depend on θ, and so can be used to test whether the numbers we see are likely if the neutral theory is correct.

In a sample of 12 genes, four alleles are found. There are n₁ = 8 copies of the first, n₂ = 2 of another, and n₃ = n₄ = 1 of the third and fourth.

Problem 19.8  This question looks at some data used to estimate the fraction of amino acid substitutions that were established by positive selection (see Box 19.1). Table P19.5 shows data for 25 genes, from Drosophila simulans and Drosophila yakuba. These two species shared a common ancestor about 6 Mya. Each row shows the gene; the number of nucleotides in the coding sequence; the divergence between D. simulans and D. yakuba, corrected for multiple substitutions, for nonsynonymous (D_n) and synonymous (D_s) changes; and the number of polymorphic sites within D. simulans, again classified into nonsynonymous (P_n) and synonymous (P_s) polymorphisms. NOTE 19G

Problem 19.9  The region around a gene has lower genetic diversity in three Drosophila species: D. melanogaster, D. yakuba, and D. simulans. This reduction is thought to be due to recent selective sweeps; this is a rapidly evolving gene that is involved in the defense against viruses. This problem shows how we can estimate the strength of selection on favorable alleles.

Problem 19.10  Figure 19.21A shows that the synonymous diversity is lower in regions of the D. melanogaster genome that have lower rates of recombination per base pair. Background selection is predicted to reduce neutral diversity by exp(–U/R), where U is the total rate of deleterious mutation per diploid genome and R is the total rate of recombination. (See Web Notes.)

Problem 19.11  Balancing selection has maintained a stable polymorphism for a very long time, with two alleles at frequencies 0.3:0.7. Over the genome as a whole, nucleotide diversity at synonymous sites averages π = 0.02.

Problem 19.12  This question shows how selection to maintain pairing in RNA structures slows down the rate of molecular evolution at the sites involved. Figure 19.28 shows the secondary structure of the bicoid mRNA that is required for this molecule to be correctly localized in the developing egg. This structure is maintained by pairing between complementary bases; a total of 71 bp, or 142 nucleotides, are involved. Nevertheless, paired regions do evolve: Typically, a change at one site is followed by a change at the complementary site, so that pairing is maintained. NOTE 19I The estimated rate of substitution per nucleotide site in these paired regions is estimated to be one-fifth that in adjacent unpaired regions.

Assume that the mutation rate per site per generation, μ, is low relative to the rate of random drift (N_eμ << 1), so that each species is usually fixed: Polymorphism can be neglected.

Problem 19.13  Suppose that 1000 polymorphisms are maintained by heterozygote advantage: At each locus, there are two alleles, and heterozygotes have a fitness advantage s = 1% over both homozygotes.

Problem 19.14 *C28*  After a sudden change in environment, a rare allele P becomes favored: The fitnesses of genotypes QQ, PQ and PP are 1– 2s:1 – s:1, respectively. The frequency of P increases from a low value p₀ to fixation, but although it does so, the population is less fit than it might be.

Problem 19.15 *C28*  Two neutral alleles (A^U, A^V) at locus A segregate at frequencies u, v (u + v = 1); initially, u₀ = 0.2. A new mutation, B^P, which increases relative fitness by s = 1%, occurs at a linked locus, B, and increases at the expense of the original allele, B^Q. The population size is N = 10⁷ diploid individuals. The rate of recombination between A and B is c = 0.1%. The new mutation happens to arise on a genome that carried A^U, and, initially, there is a complete association between A^U and B^P. This problem shows how we can find the overall increase in A^U that is caused by the substitution of B^P for B^Q. NOTE 19K