# What fraction of sites are expected to be polymorphic?

Question

Consider a very long (eventually infinite) DNA sequence of neutral sites. Consider a panmictic population of constant size $N$ with a per site mutation rate of $\mu$ where all individuals have the exact same fitness.

What is the fraction of sites that we'd expect to be polymorphic in the population (SNPs)?

Motivation behind this question

I am asking this question to verify the results of simulations I run. For example, I run a simulation with $x$ ($x$ will be varying below) neutral sites, with a per-site mutation rate $\mu = 10^{-9}$ and a population size of $N=100$. I run the simulations for 10,000 generations. There is no recombination. When the number of sites:

• $x=10^3$ I get 0 SNP
• $x=10^4$ I get 1 SNP
• $x=10^5$ I get 3 SNPs
• $x=10^6$ I get 25 SNPs
• $x=10^7$ I get 238 SNPs

Is there a bug in my model or is it what we'd expect given the parameters?

In the human genome, 1 out of 300 sites are polymorphic (SNPs) (ref.). This is a frequency of SNPs that is 100 times greater than what I observe in my simulations. Note however, that the assumption of neutrality and out demographic assumptions would not perfectly hold and this result could pretty far off neutral expectation. My goal is not to reproduce something that look like the human genome but only to reproduce the neutral expectations for the moment.

• Have a look at https://en.wikipedia.org/wiki/Tajima's_D. It provides an estimate for the number of segregation sites for a population under a neutral mutation model. – putnampp Jul 8 '15 at 19:29
• @putnampp Sounds interesting indeed. Does it mean that the expected number of SNP's in a sample of size $n=N$ (I sample the whole population) is $E[S] = 4N\mu\sum_{i=1}^{N-1}\frac{1}{i}$? Therefore, in my case $N=100, \mu = 10^{-9}, E[S] ≈ 1/481939$. Therefore, out of $10^7$ sites I'd expect to have $20.75$ SNPs. Is that right? This is 10 times less than what I observe. – Remi.b Jul 8 '15 at 20:23
• I would agree with your comment. If you were to use the entire population as your sample size you would expect to find roughly the number you suggest. I've written some example simulation software that is capable of performing such evolutionary scenarios (Clotho). You could also check your numbers against MS. – putnampp Jul 8 '15 at 20:54
• Ok, sounds good. I'll have a look at clotho. If you want an make a short answer out of your comments by repeating the calculation I made. And you'll hopefully get enough reputation to comment in the future thanks to this answer! I'd welcome you link the paper describing clotho as well. Thanks! I've got to understand why I have so much polymorphism now! – Remi.b Jul 8 '15 at 21:11

Reiterating the above comments. Have a look at Tajima's D. It provides an estimate for the number of segregation sites for a population under a neutral mutation model.

The general form of the estimation for a diploid population is $E[S]=4N\mu\sum_{i=0}^{n-1} \frac{1}{i}$. Here the mutation rate of is per-genome not per-site, so $\mu=L * 10^{-9}$ where $L$ is the genome size. Estimating the segregation sites of an entire population of $n=N=100$ with genome size of $L=10^{7}$ where each site has a per genome mutation rate of $\mu=10^{-2}$ one would expect that $E[S] \approx 20.75$. So, your numbers seem higher than expected.

I've written some example simulation software that is capable of performing such evolutionary scenarios (Clotho manuscript). Similarly, you can check your numbers against a population generated using MS.

The fraction of polymorphic sites that exist in a population is dependent on the biology of the organism. For instance, you would expect to find different rates of polymorphism in related plants that have different breeding systems, e.g. in Silene [1]. Past bottlenecks are also expected to decrease polymorphisms [2]. So, the answer to your question would depend on the exact species and population that you are looking at.

• Thanks for your answer. Yes, I am well aware that pattern of selection (frequency-dependent, purifying selection, spatially or temporally heterogenous environment, LD, etc..) as well as demographic patterns (bottleneck, population structure, range expansion, etc..) affect these polymorphism. I am interested in a theoretical predictions assuming a whole bunch of things. I made those assumptions a bit more obvious in my post. – Remi.b Jul 4 '15 at 21:24

we included a script to calculate this in supplemental material

http://onlinelibrary.wiley.com/doi/10.1111/mec.13034/full

....single segregating site per locus or up to a maximum of four SNPs, as is expected for short-read genomic data (see attached R script for estimation).

• Could you tell a bit more about the method? It's more of a comment now. – AliceD Jul 15 '15 at 9:01