Problem 16.1
A set of demes all receive ten migrants per generation from a mainland source that is fixed for an allele. These migrants arrive just before breeding and then the number of breeding adults is counted. Initially, all the demes lack this allele. What will the expected allele frequency be after 20 generations, in the following cases?

*i) 
A deme with 100 breeding adults. HINT 16A

*ii) 
A deme with 1000 breeding adults.

**iii) 
A deme that fluctuates in size, with 20 breeding adults for one generation and 1000 for the following nine. HINT 16B

Problem 16.2
Six demes are arranged in a line; each exchanges 6% of its individuals with the neighbor to the left and 6% with the neighbor to the right. (This is known as the steppingstone model, with a total rate of gene flow into each deme of m = 12%.) The leftmost and rightmost demes are very large, so that allele frequencies there can be taken as constant. Initially, the leftmost three demes are fixed for one allele, and the rightmost three for another allele (p = 0, 1, respectively).

**i) 
What equilibrium will be reached? HINT 16C

**ii) 
Suppose now that there is a barrier to gene exchange at the center: The rate of exchange between demes 3 and 4 is reduced to 1% in either direction. What is the equilibrium now? HINT 16D

Problem 16.3
A steppingstone model is set up as in the previous question, but now with 21 demes in total, and m = 50% (i.e., 1/4 move to the left, 1/4 to the right, and 1/2 stay in the same deme). Demes are 100 meters apart.

*i) 
What is the rate of diffusion, σ^{2}? HINT 16E

*ii) 
Use the diffusion approximation to find the width of the cline after 20 generations. HINT 16F

*iii) 
What will be the final cline width?

Problem 16.4

*i) 
An allele has frequencies p = 0.1, 0.2, 0, 0.3, 0.4, 0.4, 0.5, 0.1, 0.7, 0.3 across ten demes. What is F_{ST}?

**ii) 
What would your estimate of F_{ST} be if these were the allele frequencies in samples of ten individuals taken from much larger demes? HINT 16G

**iii) 
If the species has total size 10^{6}, what would you estimate the effective size to be? What assumptions do you need to make to answer this question?

Problem 16.5
A population of mammals is subdivided into very many demes, each containing N = 20 diploids; these exchange genes according to the island model. Throughout, assume that both males and females produce a Poisson number of offspring so that for autosomal genes, N_{e} = N. Males and females are equally frequent, but males disperse more: A fraction m_{M} = 0.1 of males are replaced by immigrants in every generation, but only m_{F} = 0.05 of females. What is F_{st} for genes on the following?

**i) 
An autosome.

*ii) 
The Y chromosome.

**iii) 
The X chromosome. HINT 16H

*iv) 
The mitochondrial DNA.

Problem 16.6**
A survey of allozyme markers shows that F_{ST} = 0.15. A quantitative trait has narrowsense heritability h^{2} = 0.3, measured within demes. What is the variance of the trait mean between demes? HINT 16I

Problem 16.7
Suppose that there is negligible migration among the demes of a metapopulation. Assume that the number of demes, n, is very large.

**i) 
Demes become extinct at a rate λ per deme per generation and are immediately recolonized by migrants from one other deme. Once the metapopulation has settled to a steady state, what is the expected proportion of heterozygotes within a deme? What is F_{ST} ? HINT 16J

***ii) 
Assuming that the allele frequency over the whole population is p, what is the increase in variance of allele frequency per unit time? What is the effective size of the whole population? HINT 16K

***iii) 
Now, suppose that empty patches are recolonized by individuals drawn from the whole population. Each deme has N diploid individuals and reproduces according to the Wright–Fisher model. What is F_{ST}? HINT 16L

Problem 16.8
Samples of sequences are taken from multiple demes. The average number of differences between sequences from within the same deme (i.e., the withindeme pairwise diversity) is π = 0.002. The average number for sequences from different demes is π^{*} = 0.003. The mutation rate is estimated from divergence between species as μ = 2.5 × 10^{–9} per site per generation.

**i) 
Extimate the total number of individuals in the whole population. State any assumptions that you make about the nature of gene flow. HINT 16M

*ii) 
Estimate F_{ST}.

Problem 16.9
The butterfly Heliconius erato is divided into many different geographic races, which differ in color pattern. These patterns are maintained by Müllerian mimicry, by which species converge to a common pattern that advertises their distastefulness (see p. 475). Two of these pattern races meet near Tarapoto, in Peru, and change in a cline approximately w ~ 10 km wide. Their differences are largely determined by three unlinked genes, with the alleles indicated by uppercase letters at each locus fixed in one race and the recessive alleles in the other. 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. (See Fig. P16.1.)
Table P16.1 gives the numbers of each genotype from a sample of 84 butterflies taken where the two pattern races meet.

*i) 
Calculate the allele frequencies at the three loci. HINT 16N

**ii) 
With random union of gametes, the frequency of the doublerecessive homozygote (e.g., cr/cr sd/sd) is expected to be the frequency of the gamete carrying the two recessive alleles. For each pair of loci, find the frequency of this doublerecessive gamete and hence estimate the pairwise linkage disequilibrium. HINT 16O

**iii) 
At an equilibrium between gene flow and recombination, the linkage disequilibrium is approximately D ~ σ^{2}/(rw^{2}). Estimate the rate of gene flow, σ^{2}. NOTE 16A

Problem 16.10 *C28*
A sample of 35 sika deer is taken from an area where red deer are also found. They are scored for 25 microsatellite marker loci, which carry different alleles in red and sika deer. Many deer that appear phenotypically as sika in the area of overlap carry several alleles derived from the other species. Table P16.2 gives the numbers of the 35 apparently sika deer that carry 0, 1, 2, ... “red” alleles. (For example, an individual heterozygous for “red” alleles at three loci, or homozygous at one and heterozygous at another would be classed as carrying three “red” alleles. In fact, almost all “red” alleles are found in heterozygotes.)

**i) 
Assuming that red alleles have the same frequency at all 25 loci, how many would you expect in each class if the population were at linkage equilibrium? HINT 16P

**ii) 
If individuals with multiple “red” alleles are due to recent backcrossing, what generation of backcross are they likely to be? HINT 16Q

**iii) 
Make a rough estimate of the rate of hybridization from these data. HINT 16R

*iv) 
These deer have been hybridizing for about 90 years, or 25 generations. What frequency of introgressed alleles would you expect after that time? NOTE 16B
