## N

Figure 5.12 Haploid reproduction in the context of coalescent and mutation events. In a haploid population the probability of coalescence is (solid lines) while the probability that the two lineages do not have a common ancestor in the previous generation is 1 (dashed lines). The process of mutation can also occur simultaneously (stars), changing the state of alleles (filled circles to unfilled circles). Compare with Figure 3.23.

Building a coalescent model with mutation is as simple as adding another type of possible event that can occur between the present and some time in the past (Fig. 5.12). We will assume that both coalescent and mutation events are rare (or that Ne is large and the rate of mutation is small), so that when an event does occur going back in time it is either coalescence or mutation. In other words, we will assume that mutation and coalescence events are mutually exclusive.

Every generation there is the chance a mutation occurs. The rate of mutation, ||, can be thought of as the chance that each lineage experiences a mutation each generation. The chance that a lineage does not experience a mutation is therefore 1 -1 each generation. The chance that t generations pass before a mutation event occurs is then the product of the chances of t - 1 generations of no mutation followed by a mutation, or

P(T, mutation

This equation has an identical form to the chance that a coalescent event occurs after t generations given in Chapter 3. Like the probability of coalescence, the probability of a mutation through time is a geometric series that can be approximated by the exponential distribution (see Math box 3.2).

To obtain the exponential expression or the exponent of e that describes the frequency of mutation events, we need to determine the rate at which mutations are expected to occur. When time is measured on a continuous scale with t = ——, one unit of

2N e time is equivalent to 2Ne generations. If 2Ne generations elapse and | is the rate of mutation per generation, then 2Ne| mutations are expected during one unit of continuous time. If we define 0 = 4Ne| then 0/2 is equivalent to 2Ne|. This leads to the exponential approximation for the chance that a mutation occurs in a single lineage at generation t:

on a continuous time scale. When there is more than one lineage, each lineage has an independent chance of experiencing a mutation but only one lineage can experience a mutation. When events are independent but mutually exclusive, the probability of each event is added over all possible events to obtain the total chance that an event occurs. Adding

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