It is a classical result that the expected time for a neutral mutation to occur and to get fixed is $2 N \mu \frac{1}{2N} = \mu$, where $N$ is the population size and $\mu$ is the neutral mutation rate (wiki).

I am not sure I understand what this actually mean. What is the probability of (at least) one fixation event to occur in $\frac{1}{\mu}$ generations? Is it 0.5? Could we calculate the probability that (at least) one fixation event occurs in $n$ generations? For example after $n=\frac{\mu}{10}$ generations, is the probability of observing (at least) one fixation event equal to $0.1$ or is it more complicated?


Following @DanielWeissman comments:

Let's consider we are starting with an initially monomorphic population (population of clones). In addition to that, let's consider that the population is relatively small $N\mu < 1$

  • 1
    $\begingroup$ You should probably specify whether you're considering an initially monomorphic population, or a steady-state population (with infinite alleles), or something else. $\endgroup$ Mar 12 '15 at 17:16
  • $\begingroup$ I didn't realize that would matter. Let's consider an initially monomorphic population. (see update) $\endgroup$
    – Remi.b
    Mar 12 '15 at 17:57

First of all, the $\mu$ is not expected time for a mutation to occur and get fixed; it is the rate at which mutations are fixed in the population. The basic result states that if neutral mutations arise at a locus at rate $\mu$ within individuals, mutations at this locus will be fixed in the population at rate $\mu$ as well.

The expected time for a given neutral mutation to fix after arising, given that it is not lost from the population, is approximately $4N_e$, where $N_e$ is the effective population size. Kimura and Ohta (1969) derive this result using a diffusion approximation; Kingman (1982) develops coalescent theory and in the process obtains what in my opinion is a more elegant derivation of the same approximation.

Now what is the probability that some mutation goes to fixation in the population as a function of time? Even though the fixation rate in the population is $\mu$, this does not mean that with probability 0.5 some mutation fixes within $1/\mu$ time units. That probability will depend on the distribution of fixation events in the population. Here's a good way to think about it. If fixation events were evenly spaced and occurred at rate $\mu$, then with probability 1 you would have a mutation within an interval of length $1/\mu$. If fixation events were highly clumped, then you would have a very low probability of having a fixation event within an interval of length $1/\mu$ -- but if you did have one event, you would probably have many. If fixation events are distributed as a Poisson process--which I suspect they would be in a purely neutral model--the probability of having a fixation event within a time interval $1/\mu$ would be given by a (negative) exponential distribution and thus the probability of having at least one fixation event in a time interval of length $1/\mu$ would be $1-\frac{1}{e}$.

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    $\begingroup$ This final answer isn't quite correct -- it neglects the drift time mentioned in the second paragraph. If $N\mu \ll 1$ it's a good approximation though. (If $N\mu \gg 1$, the process of fixation is almost deterministic.) $\endgroup$ Mar 12 '15 at 17:14
  • $\begingroup$ @DanielWeissman Thank you for the informative correction. $\endgroup$
    – Corvus
    Mar 13 '15 at 3:31

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