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I've had trouble finding info on how to model this problem, so figured I'd ask it here. I'm trying to figure out what the probability distribution is of the frequency of a mutation in a population with no affect on fitness as k generations after occurring (assuming stable population size N). Obviously there are other parameters such as lifespan, but the simplest version would be that each generation, every cell (it's a single cell organism) either undergoes cell division, leaving two daughter cells, or it dies, with probability 0.5.

E.g., a mutation that occurs at t = 0 has probability 0.5 of going extinct and the same probability of being present in 2 cells (thus frequency x = 2/N) after one generation; then at t=2 it p(x = 0) = 62.5; p(x=2/N) = 0.25; p(x=4) = 0.25. How do I generalize this to get a distribution for after k generations? Thanks.

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I think that you are interested in the original work about neutral mutations from Kimura (1954, 1962). This work established that the probability of fixation of a neutral mutation is equal to its current allele frequency.

The take-home message is that the neutral mutations are taking a random walk through allele frequency space, i.e. a diffusion process, which makes it tough to come up with well-behaved distributions.

You can model this with the normal Wright-Fisher model in the absence of selection (page 19 here). It's a bit of a headache because it involves following a Markov chain for $k$ generations. Possibly the most straightforward thing to do is to just simulate a million trajectories using the Wright-Fisher model and see what they look like for any given $k$ and a new mutation. For an example of how to do this look here, or alternatively you could use an existing program like SLiM.

If you are looking for something a bit more closed form, what Kimura (1954) did was to apply a diffusion model approximation of the Wright-Fisher model (equation 1), with some further development.

He writes:

If $\phi(x,t)dx$ is the probability that the gene frequency lies between x and x + dx in the t-th generation, it can be proved that $\phi(x,t)$ satisfies the partial differential equation,

$\frac{\partial \phi (x,t)}{\partial t} = \frac{\partial ^2}{\partial x^2} [\frac{V_{\delta\_{x}}}{2} \phi(x,t)] - \frac{\partial}{\partial x}[M_{\delta_{x}} \phi(x,t)]$

This is where my math fails me in population genetics and I am less able to help you applying and untangling how to come up with a distribution. But I would suggest reading through that Kimura paper and using it as a starting point to do some research and see who cites it to give some kind of solution that's more useful for you.

A treatment of this in slides for a class is given here.

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