N

This equation can be rearranged:

e 2Pn

to give the inbreeding effective population size for the coalescent model over one generation. The effective size of a haploid population is defined identically except that the population contains — instead of 2— gene copies. (The reasoning used is parallel to that used to arrive at the inbreeding effective population size in equations 3.53 and 3.54 based on the probability of identity by descent in a finite population.)

Interact box 3.5 Simulating gene genealogies in populations with different effective sizes

(see Fig. 3.24). In a diploid population with two sexes and a 1:1 breeding sex ratio, half of the gene copies reside in females and half reside in males. A coalescence event requires that two gene copies descend from a single ancestor, either a single male or single female individual. The total probability that two randomly sampled gene copies coalesce in the previous generation, PC, is the sum of the probability that the coalescence was in the population of females and the probability that the coalescence was in the population of males, or

PC = PC(female population) + PC(male population)

Simulating lineage branching events forward in time for populations of different sizes is a direct way to understand how sampling leads to patterns of lineage coalescence. Use the link on the text web page to reach a simulation to model lineage branching events in finite populations.

Start by simulating populations of N : 4 and N : 10. Before pressing the run button (the arrow that points to the right), determine the expected time for all of the gene copies in the present to find a single most recent common ancestor. Use your answer to intelligently set the number of generations (the G: entry field) to run the simulation to be able to see most of the lineage coalescence events (note that the maximum number of generations is 30). Run 25 simulations for each population size and tabulate the number of generations for all lineages in the present to find their most recent common ancestor. What does the distribution of coalescence times look like?

Use 100% for a Speed value to carry out replicate runs rapidly. The "untangle" button (the leftmost button at the bottom) rearranges the lineages so that the branches do not overlap and is very useful when tracing individual genealogies to visualize common ancestors.

We can apply the coalescent definition of the effective population size to the case of the breeding sex ratio in a population. For the coalescent model, it helps to think of the two sexes as two separate populations where gene copies can find their ancestors

Regardless of whether or not the two gene copies come from the male or female population, the probability of coalescence is the chance of sampling the same gene copy twice in the previous generation, or 1

2—~. To take the populations of the two sexes into e account, notice that the probability that a gene copy is sampled from either the female or male population is 1/2. The probability that both gene copies are sampled from the population of the same sex is therefore (1/2)(1/2) = 1/4. Putting these probabilities together into equation 3.78 gives

ef et

where the effective size of the female and male populations sum to the total effective size (N'e = Nf + Ne'm). Based on the definition of the effective population size as the probability of coalescence in equation 3.63,

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