Evolution Chapter 16 Answers

In each generation, a fraction m = 10/100 = 0.1 come from outside. The new allele frequency is therefore p_t = mp_S+(1 – m)p_t–1, where the allele frequency in the source is p_S = 1. It is easier to work with the frequency of the native allele, q_t = 1 – p_t, since q_S = 0, and so q_t = (1 – m)q_t–1. After 20 generations, therefore, p₂₀ = 1 – q₂₀ = 1 – 0.9²⁰ = 0.878.

Similarly, p₂₀ = 1 – 0.99²⁰ = 0.182.

The rate of immigration, m, now varies from generation to generation: it is m = 10/20 for one generation in ten, and 10/1000 otherwise. Hence,

Gene flow mostly occurs during the brief times when the native population is small. NOTE 16C

The equilibrium will be a linear cline. Using the same method as that in Box 16.1, we know that allele frequency changes according to

where p_{i – 1,t} is the ith deme in the line. So, at equilibrium, we must have

which means that the difference in allele frequency between each pair must stay the same. Because the ends are fixed at p₁ = 0, p₆ = 1, we have p₂ = 0.2, p₃ = 0.4, p₄ = 0.6, p₅ = 0.8. This is independent of the rate of gene flow, m.

The same argument as above holds for the demes on either side. At the center, we must have

p_3,t = 0.06 p_2,t + (1 – 0.06 – 0.01)p_3,t + 0.01p_4,t ,

so that

0 = –0.06 (p_3,t – p_2,t) + 0.01(p_4,t – p_3,t).

Thus, a reduction in gene flow by a factor of 6 makes the gradient in allele frequency six times steeper (i.e., (p_4,t – p_3,t) = 6(p_3,t – p_2,t) = 6d, say). So, there are four intervals of size d and one of size 6d. These sum to 1, and so d = 0.1. The solution is that p₂ = 0.1, p₃ = 0.2, p₄ = 0.8, p₅ = 0.9.

The rate of diffusion is the variance in distance between parent and offspring. m = 0.5 move ±100 meters, and the other half do not move. Therefore, σ² = (1/2)(100 meters)² per generation = 5,000 m² per generation.

With neutral mixing, cline width after t generations is (see Fig. 16.6). After 20 generations, this is 793 meters. The diffusion approximation, ignoring the fixed endpoints, is very accurate up to 20 generations (see Fig. P16.2).

Ultimately, the cline will settle to a linear form, with width 20 × 100 m = 2 km.

The mean allele frequency is 0.3, and the variance is 0.0444. Therefore, F_ST = var(p)/ = 0.212.

The observed variance in allele frequency will be increased by the sampling variance, which is /2n for a sample of n diploid individuals from a population with actual allele frequencies p,q. So, we expect the observed variance to be F_ST + E[pq]/2n, where var(p) = F_ST is the true variance in the actual allele frequencies. Because E[pq] = – var(p) = (1 – F_ST, we have an observed variance of (F_ST + (1 – F_ST)/2n). This leads to a revised estimate of F_ST = 0.170. (It would be a good approximation just to subtract the scaled sampling error, 1/2n = 0.05, from the estimate in i).)

Assuming conservative migration—so that gene flow does not directly alter allele frequencies—the effective size of the population as a whole is inflated by a factor 1/(1 – F_ST) = 1.205. However, fluctuations in population size will tend to reduce the overall N_e.

From Box 16.2, 1/(1 + 4Nm). The appropriate migration rate is the average of that in males and females, (m_M + m_F)/2 = 0.075, because an autosomal gene spends half its time in males and half in females. Thus, F_ST = 0.143.

Now, the number of copies of the gene is N/2 rather than 2N, and migration is through males only. Therefore, F_ST = 1/(1 + Nm_M) = 0.333.

From Box 15.2, the effective population size for an X-linked gene is

An X-linked gene spends 2/3 of its time in females and 1/3 in males, and so m = (m_M/3 + 2m_F/3). Thus, F_ST = 1/[1 + N(m_M + 2m_F)] = 0.2.

Following ii), but replacing males by females, F_ST = 1/(1 + Nm_F) = 0.5.

The measure Q_ST = var()/(var() + 2V_A) is expected to equal F_ST, on the assumption that both allozyme frequencies and the trait mean are in a balance between gene flow and random drift. Also, h² = 0.3 = V_A/V_P. Therefore, var() = 0.106V_P, corresponding to a standard deviation of 0.326 phenotypic standard deviations.

Every deme traces its ancestry back through a single deme. There is no mixing, and so demes must fix at 0 or 1. Overall, a fraction q of demes is fixed at 0 and p at 1. There are no heterozygotes (E[2pq] = 0), and var(p) = , so that F_ST = 1.

Extinctions and recolonizations occur at a rate λn. Whenever one occurs, the new deme has frequency 0 or 1 with probability q or p. If it is 0, then the overall mean allele frequency has decreased by –p/n, whereas if it is 1, the mean has increased by + q/n. The variance of this change is q(–p/n)² + p(–q/n)² = pq/n². Therefore, the rate of increase in variance of allele frequency is λpq/n. Defining this as 1/2N_E, we have N_E = n/λ. The faster the rate of extinction and recolonization, the faster the rate of drift, and the smaller the overall effective size.

The effect of extinction and recolonization is now quite different. If empty patches are colonized from the population as a whole, they will immediately return to the overall average allele frequency, p. Thus, extinction and recolonization now increase diversity within demes and reduce differences between them. Conversely, random drift at a rate 1/2N_e will reduce diversity within demes, and increase variation between them.

In a small time interval δt, λnδt demes will return to the overall average. Thus, differences in allele frequency will decrease by a factor (1 – λnδt), and the variance V of these differences will decrease by a factor (1 – λnδt)² ~ 1 – 2λnδt. The decrease in variance per unit time is therefore –2λnV; this is balanced by random drift that increases V by

The net change in variance is

so that at equilibrium

Provided gene flow is conservative (i.e., provided it does not change allele frequencies directly), then for any population structure, the mean coalescence time for two genes from the same deme is T_W = 2nN_e (see p. 449). Assuming infinite-sites mutation, we therefore expect π = 2μT_W = 4nN_eμ. Hence, nN_e = 200,000.

(assuming that there are a large number of demes so that ~ T_B). (See p. 449.)

Assuming Hardy–Weinberg proportions, the frequencies of the recessive sd and cr alleles are estimated as the square root of the sd/sd and cr/cr homozygotes. The frequency of the d allele is estimated directly, as half the D/d heterozygote frequency plus the d/d homozygote frequency. So, p_sd = 0.512, p_cr = 0.476, p_d = 0.518. Because the population is formed by mixing populations fixed for different alleles and because each locus is under similar selection against unusual patterns, these allele frequencies are similar.

The frequency of the double-recessive homozygote (sd/sd cr/cr say) is expected to be the square of the frequency of the double-recessive gamete (p_{sd cr}², say). Subtracting the frequency expected at linkage equilibrium gives an estimate of the coefficient of linkage disequilibrium (D = p_{sd cr} – p_sdp_cr, say). Thus, D_{sd cr} ~ 0.0453, D_{cr d} ~ 0.0623, and D_{sd d} ~ 0.0623.

The average D is 0.0566, which can be equated to the approximation σ²/(rw²), with r = 1/2 for unlinked loci, and w = 1 km. Hence, σ² ~ 2.8 km² per generation, a standard deviation of σ ~ 1.7 km. (See Mallet et al. [1990] for a detailed analysis.)

There are 66 red alleles among 2 × 25 × 35 genes, and so the average allele frequency is p = 0.0377. The chance that an individual has no “red” alleles among 50 genes (2 copies at each of 25 loci) is therefore q⁵⁰ = 0.9623⁵⁰ = 0.146, and the expected number in the zero class is 35 × 0.9623⁵⁰ = 5.12. The number with one red allele is 50pq⁴⁹, the number with 2 is (50 × 49/2)p²q⁴⁸, and so on, following a binomial distribution.

The individual with six red alleles is unlikely to have arisen by chance in a panmictic population, and the individual with ten “red” alleles is vanishingly unlikely.

The F₁ has 25 heterozygous loci, the first generation backcross (Bx₁) has 12.5 on average, and so on. Thus, the individual in class 10 is most likely to be a first-generation Bx₁, and the individual with 6 is most likely to be a Bx₂. However, the actual number of introgressed alleles varies widely, according to a binomial distribution, so these inferences are uncertain.

Of 35 deer, 2 are hybrids, most likely one Bx₁ and one Bx₂. Later backcrosses would be undetectable against the background of randomly combining alleles. Because hybrids could be detected over three generations (as F₁, Bx₁, or Bx₂), we can estimate the rate at which F₁s are produced as 2/(35 × 3) ~ 0.020 per generation. Because only half the genes of an F₁ are from red, this corresponds to m = 0.01 per generation.

Assuming one-way immigration (the simplest assumption), then following the first problem, we expect allele frequency to be 1 – 0.99²⁵ = 0.22. In fact, it is much lower. This may be because the species have met more recently than 25 generations or have only just begun to hybridize, because they overlap in a limited area, or because selection acts against introgressing alleles, or simply because by chance, there is more introgression than average in this sample. (See Goodman et al. [1999] for a full analysis.)