-- Coalescence

-- Mutation -- Coalescence





-- Coalescence Present

Figure 5.13 A genealogy constructed under the simultaneous processes of coalescence in a single finite population and mutation. Working backward in time from the present, both mutation and coalescence events can occur. The blue dots represent mutation events, each assigned at random to a lineage present when the event occurred. Mutation events alter the state of a lineage, causing divergence from the ancestral state of the most recent common ancestor (MRCA) of all the lineages in the present.

have more signs. Applying this logic to genealogies like that in Fig. 5.13 tells us that more mutations are expected during the long average waiting time for coalescence with two lineages (k = 2) than are expected to occur when there are six lineages that can coalesce (k = 6). Another example would be the pattern of mutations expected for lineages in two demes with different levels of migration (see Fig. 4.17). With very limited migration, multiple mutations are expected on the long branches before the single lineages within each deme coalesce. Alternatively, when migration rates are high then many fewer mutations are expected between migration events. In the former case mutations cause lineages to diverge substantially between the two demes, while in the latter case the lineages in the different demes have less opportunity to accumulate differences.

A genealogy with generic mutations like that shown in Fig. 5.13 is an abstraction until it is joined with a mutation model. Figure 5.14 shows the same genealogy under the assumptions that the most recent common ancestor has an allelic state of A and mutational changes follow the infinite alleles

Interact box 5.5 Build your own coalescent genealogies with mutation

Again building a few coalescent trees can help you to better understand the evolution of genealogies when both the processes of mutation and coalescence are operating. You can use an expanded version of the Microsoft Excel spreadsheet used to build coalescent trees in Chapter 3 that now models waiting times for both mutation and coalescence. The spreadsheet contains the cumulative exponential distributions used to determine the time until a coalescent or mutation event (see equation 5.44) for up to six lineages. To determine the time that an event occurs for a given number of lineages k and mutation rate, a random number between zero and one is picked and then compared to the cumulative exponential distribution. The time interval on the distribution that matches the random number is taken as the event time. The next step is to determine whether the event was a mutation or a coalescence, again accomplished by comparing a random number to the chances of each type of event (equations 5.45 and 5.46).

Step 1 Open the spreadsheet and click on cells to view the formulas used, especially the cumulative probability of coalescence for each k. This will help you understand how the equations in this section of the chapter are put into practice. You can compare the cumulative probability distributions graphed for k = 6 and k = 2. Step 2 Look at the section of the spreadsheet under the heading "Event times:" on the right side of the sheet. This section gives the waiting times until an event occurs and then determines if the event was a coalescence or a mutation. Press the recalculate key(s) to generate new sets of random numbers (see Excel help if necessary). Watch the times to an event change.

Step 3 Now draw a genealogical tree with the possibility of mutations (do not recalculate again until Step 6 is complete). Along the bottom of a blank sheet of paper, draw six evenly-spaced dots to represent six lineages. Step 4 Start at the first "Decide event time:" panel to determine how much time passes (going backward in time) until either a mutation or a coalescence occurs. Then use the entries under "Decide what type of event:" to determine if the event was a coalescence or a mutation. If the event is a mutation go to Step 5, otherwise go to Step 6. Step 5 If the event is a mutation, draw the lines for all lineages back in time by a length equal to the waiting time (e.g. if the time is 0.5, draw lines that are 0.5 cm). Use the random number table to pick one lineage and draw an X on the lineage at the event time to indicate a mutation occurred. If a mutation occurred the number of lineages (k) remains the same. Move down to the next "Decide event time:" panel and obtain the next event time for the same value of k. Repeat Step 5 until the event is a coalescent event. Step 6 Using the random number table, pick two lineages that will experience coalescence. Label the two left-most dots with these lineage numbers. Then, using a ruler, draw two parallel vertical lines that start at the end of the last event and extend as long as the time to coalescence in continuous time (e.g. if the time is 0.5, draw lines that are 0.5 cm). Connect these vertical lines with a horizontal line. Assign the lineage number of one of the coalesced lineages to the pair's single ancestor at the horizontal line. Record the other lineage number on a list of lineages no longer present in the population (skip over these numbers if they appear in the random number table). There are now k - 1 lineages. Step 7 Return to Step 4 until all lineages have coalesced (k = 1).

You should obtain a coalescence tree with mutation events on the branches like that in Fig. 5.13. Your trees will be different because the random coalescence and mutation times vary around their averages, but the overall shape of your trees (e.g. shorter branches when k is large) and frequencies of mutations (for a given mutation rate) will be similar.

Allelic state of MRCA A

Allelic state of MRCA

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