# How many generations are required for a specific neutral mutation to reach fixation?

In population genetics, the term “time to fixation” is defined as the time it takes for a specific mutation to appear in a population, plus the time required for this mutation to spread throughout this population. My question is, how many generations are required for a specific neutral mutation to reach fixation? I need a general formula for number of generations as a function of mutation rate and population size, valid for all kinds of biological entities. Note that a neutral mutation is not fixed by selection, it is fixed by genetic drift.

• 1) The only definition of fixation (in genetics) I've seen involves one allele completely replacing all other variants of that gene. 2) I doubt that there can be a "universal" law for all biological entities. Some examples: many organisms are not sexual, ploidy varies particularly within plants, and dispersal rates within a population are going to be different. All these factors seem likely to affect how long fixation takes. – tyersome Jan 10 at 19:16

How many generations are required for a specific neutral mutation to reach fixation?

Kimura and Ohta (1968) showed that the expected time for a neutral allele to reach fixation is

$$\bar t(p_0)=-4N\left(\frac{1-p_0}{p_0}\right)\ln(1-p_0),$$

where $$p_0$$ is the initial frequency and $$N$$ is the population size. The model assumes a Wright-Fisher population (panmixia, constant population size, exclusively sexually reproducing hermaphroditic individuals, ...) and negligible mutation rate at the locus of interest.

how do i calculate po in terms of known quantities, such as mutation rate?

You can compute the expected allele frequency at mutation - drift balance (e.g. see here or any decent intro textbook to population genetics). In short, let $$\theta = 4N\mu$$, where $$\mu$$ is the mutation rate, the expected heterozygosity is

$$H = \frac{\theta}{\theta + 1}$$

From that you can compute the allele frequencies $$p$$ and $$q=1-p$$ by solving $$H = 2pq$$.

If needed, you can get the entire distribution of allele frequency given $$N$$ and $$\mu$$ (and even $$s$$ and $$h$$ if needed) in Wright (1937).

• I think your answer would be more generally useful if you provided some examples of actual numbers. – David Jan 10 at 17:01
• Maybe worth noting what assumptions are behind this model? I could easily be wrong, but I would expect that how fast individuals, propagules, and gametes disperse through a population must have some effect. Also, is this true for species that aren't obligately sexual (e.g. many plants can reproduce vegetatively) ... – tyersome Jan 10 at 19:06
• @tyersome The model assumes a Wright-Fisher population. Information added to the post. Thank you – Remi.b Jan 10 at 21:37
• thanks for the formula. Since neutral mutations are often phenotypically silent, it can be difficult to detect their presence in wild populations, which means it can be difficult to assign an accurate value to po. So, I would be very grateful if you rewrite the formula in terms of the mutation rate, which is known for a broad range of organisms. I apologize that my math skills are too low for the job, otherwise I would do it myself. Actually, I found a paper (Lanfear et al, 2014:36) that gives the rate of neutral evolution as a function of mutation rate, but I can’t see how it reduces to your – igor brznovskaya Jan 10 at 22:21
• Lanfear, R. et al, (2014) Trends in Ecology & Evolution 29:33-­42 dx.doi.org/10.1016/j.tree.2013.09.009 – igor brznovskaya Jan 10 at 22:29