# Coalescence time: Is it different for haploids and diploids in population genetics?

I'm trying to model Cyanobacteria cells divergence in 2 populations with mutation rate $-\mu$ and I need to verify my model with a valid theory. I don't have much biology background and all the theories I can find are either valid for haploid population or mating populations. All I need to know is a theory for coalescence time/divergence for a haploid population without mating. So far I have this:

$$T = 2\times N \times d\\ D = 2 \times \mu \times T$$ in which N is the number of diploid individuals, d is the number of subpopulations (in this case 2), T is the coalescence time and D is divergence. Can I apply this to my case even though it's defined for diploid population?

Thanks

• Welcome to Biology.SE. The question is a little unclear to me for the moment. $D$ is divergence. Can you tell us more about what you call divergence? You write $T=2 \cdot Ku \cdot T$, therefore, $\frac{1}{2}=Ku$. This is probably not what you wanted to write. $T$ the coalescent time is in unit of generations I suppose. You might want to let us know where this equation comes from. It is always good to follow conventions; the mutation rate is usually $\mu$, not Ku. – Remi.b Aug 13 '15 at 17:16
• You should write your equations in MathJax (LaTex). Here is a quick tutorial. You might want to have a deeper look at coalescent theory. There is a wikipedia entry. Coalescent Theory: an introduction is a very good book on the subject. – Remi.b Aug 13 '15 at 17:17
• Out of curiosity, what is your background (or what are you studying) if not biology? – Remi.b Aug 14 '15 at 2:26
• Can you provide a background for the assumed model (the equation) and the details of the model that you want to build? Details would include assumptions, parameters to be considered etc, phenomena to be accounted for etc. If you do not include these details then your question would be too broad. There can be several ways to model a system. – WYSIWYG Aug 14 '15 at 4:22
• Need background for those equations and need a definition for "divergence". – Remi.b Aug 20 '15 at 19:22

Coalescence time: Is it different for haploids and diploids in population genetics?

The coalescent time is twice as high in a diploid population than in a haploid population

Imagine you sample at random one chromosome. Then, you sample a second one and you ask the question, what is the probability that they coalesce in the previous generation. This probability is $\frac{1}{PN}$, where $N$ is the population size and $P$ is the ploidy number (1 for haploids, 2 for diploids). The probability that they do not coalesce is therefore $1-\frac{1}{PN}$. The probability that the two chromosomes coalesce $t$ generations ago is the probability that they don't coalesce for $t-1$ generations and then coalesce. That is the probability that the two chromosomes coalesce $t$ generations ago is

$$P(t) = \left(1-\frac{1}{PN}\right)^{t-1}\frac{1}{PN}$$

You might recognize here a geometric distribution, where the probability of success is $\frac{1}{PN}$. In order to simplify the math, let's assume that the population size is very large. In such case the above formulation is well approximated by

$$P(t) = \frac{1}{PN} e^{-t\frac{1}{PN}}$$

You might now recognize the exponential distribution with rate $\frac{1}{PN}$. We can now compare the mean expected time to coalescent of a diploid population and a haploid population.

The mean of the above exponential distribution is simply $PN$. Therefore, the expected time to coalescent is $2N$ in diploids and only $N$ in haploids.

In your post you talk about structured populations. The calculations are a little more complicated in structured populations and I don't remember on the top of my head how they work.

You talk about divergence $D$. I don't know what you mean by divergence. From coalescent results you can compute the expected heterozygosity, the number of pairwise differences, and even the whole site frequency spectrum. You can as well make those calculations for the structured coalescent model.

Yes you can, as the model speaks of allele coalescence as states the wiki article on that.

It comprises a probabilistic assessment of variation in time to common ancestry of alleles in a relatively small sample of individuals, from a much larger population.

What you are trying to check is how many generations have passed since any number of alleles were a common ancestral copy. For that we have:

## $t=\frac{4N}{n·(n-1)}$

This was developed by JFC Kingman (based on the Wright-Fisher previous model) for a constat population size of $N$ where $t$ is the coalescence time of $n$ alleles. In this equation the numerator is $4N$ because $2N$ is the total allele population size. You have to multiply it again by $2$ because the total time two alleles have been differing is twice the time it has been separating from the most common recent ancestor. Therefore for haploid populations you should use $2N$ in the numerator.

There are versions of this model that include mutation rate and non-constant $N$, but maybe it would be easier to simply look a good model to find your $N_e$ (effective size of a population, or the N that in a constant size population has the same genetic drift as your actual population) as it can be used in this formula when $N$ is not constant.

• Can you please provide some equations to show how it is affected by the "only difference" you cite? You might want to add a reference for your claim as well especially if you don't provide a complete explanation. Given that $N$ is the number of individuals, I would expect the coalescence time to be $T = Nd$ for haploids (assuming $T=2Nd$ is correct for diploids). – Remi.b Aug 20 '15 at 19:26
• It's T=2Nd because there are two paths by which the alleles have been changing, not because there are two copies im each genome. Anyway, let me rewrite it propperly. – Athe Aug 20 '15 at 19:29