# What is the probability distribution of a neutral mutation's allele frequency after k-generations? (in asexually reproducing organisms)

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.

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.