0
$\begingroup$

I am trying to find an equation that relates the variables of probability of fixation and generations. Or, how does number of generations affect the probability a gene will fix, if population size doesn't significantly decrease or increase.

$\endgroup$
  • 1
    $\begingroup$ You may want to have a look at the notion of Moran process, that's pretty much what these models are designed for. $\endgroup$ – Mowgli Dec 6 '18 at 4:21
2
$\begingroup$

From Hartl and Clark's 'Principles of Population Genetics' Third Edition chapter 7: Kimura and Ohta (1969) showed that the mean time in generations until the allele of frequency p is fixed (ignoring cases where the allele is lost) is:

$$t_1(p) = -4N[(1-p)log(1-p)]/p$$

The mean time to loss of the allele is:

$$t_0(p) = -4N[p \cdot log(p)]/(1-p)$$

Thus the mean persistence time of an allele is:

$$t(p) = -4N[p \cdot log(p)+(1-p)log(1-p)]$$

Hopefully I didn't make any typos there, but I reccommend this book to learn all the ins and outs of popgen. There is a PDF of the third edition online in an archive. For more book recommendations, please have a look at the post Books on population or evolutionary genetics?

You can also find these equations in the original article: Kimura and Otha (1968)

$\endgroup$
  • $\begingroup$ By wrapping your equations in dollar signs, you can use LaTex type of formatting. For example $\frac{1}{\sqrt{2}}$ becomes $\frac{1}{\sqrt{2}}$. See your answer edited for more info. I also added the original article and a link to more book recommendations. $\endgroup$ – Remi.b Dec 6 '18 at 21:14
1
$\begingroup$

It seems like answer to your question can be related to effective population size and idealized population. One of the models to to consider is Wright-Fisher model: $$\frac{(2N)!}{k!(2N-k)!)}p^{k}q(2N-k))$$ where N is individuals, so there are 2N copies of each gene, k is number of copies of an allele that had frequency p.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.