# Probability of Extinction under Genetic Drift

Here is the Wright-Fisher model of genetic drift:

$$\frac{(2N)!}{k!(2N-k)!}p^kq^{2N-k} \Leftrightarrow \binom{2N}{k}p^kq^{2N-k}$$

where $\binom{2N}{k}$ is the binomial coefficient.

This formula gives the probability of obtaining $k$ copies of an allele at generation $t+1$ given that there are $p$ copies of this allele at generation $t$. $N$ is the population size and $2N$ is the number of copies of each gene (this model applies to diploid population only).

From this formula, how can we calculate the probability of extinction of an allele in say 120 generations starting at a given frequency, let's say 0.2?

and

How can we calculate the probability of extinction rather than fixation of an allele present at frequency $p$ if we wait an infinite amount of time?

• See answer there. – Did Dec 5 '13 at 6:54

## 2 Answers

update

The answer is here!

Original comment/answer

Kimura and Ohta (1969) showed that assuming an initial frequency of $p$, the mean time to fixation $\bar t_1(p)$ is:

$$\bar t_1(p)=-4N\left(\frac{1-p}{p}\right)ln(1-p)$$

similarly they showed that the mean time to loss $\bar t_0(p)$ is

$$\bar t_0(p)=-4N\left(\frac{p}{1-p}\right)ln(p)$$

Combining the two, they found that the mean persistence time of an allele $\bar t(p)$ is given by $\bar t(p) = (1-p)\bar t_0(p) + p\bar t_1(p)$ which equals

$$\bar t(p)=-4N\cdot \left((1-p)\cdot ln(1-p)+p\cdot ln(p)\right)$$

This does not answer any of the two questions!

This answer gives...

• the average persistence time

but not...

• the probability of fixation rather than extinction if we wait an infinite amount of time

neither...

• the probability that the allele get extinct over a period of say 120 generations.

Can someone improve this answer?

• I was also thinking about Kimura's papers, but have seen some more recent stuff as well. I will look at it later when I have the time. – fileunderwater Oct 29 '13 at 8:56
• The probability of fixation given infinite time is just $p$ (the frequency of the allele at that time) - I just ran a simulation in R with initial frequency of 0.2 (1000 replicates over 5000 generations $N$ =10000 and $p$ =0.2) and the average fixation rate was 0.201 (though simulations with smaller populations are more sensitive to stochasticity thus vary more around $p$ but are still very close, $N$ =10 had a fixation of 0.184) ... I'm sure there must be a way to calculate the probability after a given number of generations... – rg255 Nov 29 '13 at 10:48
• ps. does that answer your second point in your question? @Remi.b – rg255 Nov 29 '13 at 11:07
• @GriffinEvo Nice! Thank you. You answer to "what is the probability of fixation rather than extinction if we wait an infinite amount of time" via simulation (rather than mathematical modeling). But that's already a very informative point. – Remi.b Nov 29 '13 at 11:11
• @remi.b mathematically speaking it's just simple probability - if something is randomly sampled it, on average, will be sampled at the same frequency it is found in the population it is being sampled from. (Probability of showing heads when tossing a coin is 0.5 because the frequency of head-sides on coins is 0.5). The simulation I did was just to support this (and because I like playing in R) :) – rg255 Nov 29 '13 at 11:18

Here is a simple proof that the probability of fixation given an infinite time is indeed p (in a finite population, otherwise there will be no fixation): Let's look at all 2N gametes in the population. Eventualy, according to the Wright-Fisher model, only one of them will be represented in the population. The probability for this gamete to be of an allele with initial frequency p is just p. Therefore the probability of fixation is p. Simple.