# What is the difference between nucleotide polymorphism (θ) and nucleotide diversity (π)?

In my population genetics book (see reference at bottom) they define them as:

• Nucleotide polymorphism (θ): proportion of nucleotide sites that are expected to be polymorphic in any sample of size 5 from this region. Of the genome. $\hat{θ}$ equals the proportion of nucleotide polymorphism observes in the sample (S) divided by $a_1 = \sum_{i=1}^{n-1} \frac{1}{I}$. If n= 5, $a_1 = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}=2.083$
• Nucleotide diversity is the average proportion of nucleotide differences between all possible pairs of sequences in the sample. In R, I came up with that code which is in accordance with what is in the book.

   data # Only polymorphisms
total.snp # This is the total number of sites that were looked (e.g. 16 might be polymorphic over the 500 sites that we've looked at. So 484 sites are monomorphic)

n = nrow(data) # Number of samples
pwcomp = n*(n-1)/2 # Number of pairwise comparisons
for(i in 1:n.col){ # Compute the number of differences in the samples that are polymorphic
t.v = as.vector(table(data[,i]))
z = outer(t.v,t.v,'*')
temp= c(temp,sum(z[lower.tri(z)]))
}

pi.hat = sum(temp)/(pwcomp*total.snp) # Nb of different pairwise comparison/(nb all pairwise comparison * total nb of loci (polymorphic or not))


In the book, they say

On the other hand, there is a theoretical relation between θ and π that is expected under simplifying assumption that the alleles are invisible to natural selection.

In summary θ = π with this assumption and large sample sizes.

1. So what is the difference in the
2. Why are we calculating both (what could they tell us)?

Hartl, D. L., & Clark, A. G. (1997). Principles of Population Genetics (3rd ed.). Sinauer Associates Incorporated.

You are also confusing definition of these terms with their expected values (typically computed from coalescent theory). Let's look at an example.

Consider a haploid population of 6 individuals. We will represent these individuals with 5 sites that can take values 0 or 1.

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Polymorphism

In this population there are $S=2$ polymorphic sites. In other words, there are two sites for which different individuals take different values in the population. Over these 5 sites, the fraction of polymorphic sites is $\frac{2}{5}$. You use $\theta$ for this quantity but to me, $\theta$ is typically used as a short hand for $2 N \mu$ (or $4 N \mu$ for diploid populations).

Under an infinite allele model, in a single panmictic haploid population (without selection and other assumptions), $E[S] = 2 N \mu a$, where $a = \sum_{i=1}^{k-1} \frac{1}{i}$, where $k$ is the sample size.

Genetic diversity

Genetic diversity is also known as expected heterozygosity. While it is most often called $\pi$, I will call it $H$ to avoid confusion. Imagine, you randomly sample two individuals in the population, what is the probability that there have different alleles at a given locus?

• At locus 1, $H = 0$
• At locus 2, $H = 0$
• At locus 3, $H = 0$
• At locus 4, $H = 2 \frac{2}{5} \frac{3}{5} = \frac{12}{25}$
• At locus 5, $H = 2 \frac{1}{5} \frac{4}{5} = \frac{8}{25}$

You can average over all loci if you want.

The expected heterozygosity in panmictic population (without selection and other assumptions), $E[H] = \frac{4 N \mu}{1 + 4 N \mu}$. You should notice a relationship between $H$ and $F_{ST}$ here!

Nucleotide diversity

The nucleotide diversity is the equivalent of $D$ from Nei and Li (1979) and is highly related to $D_{XY}$ (Nei, 1987). Once you've managed to demonstrate that the expected coalesence time is $N$ (I won't do the demonstration here), then under an infinite site model, the number of mismatch between two individuals is logically $E[D] = 2 N \mu$ (it is the number of mutations that has happened in both lineages since the the coalescence event). For diploid population, multiply everything by 2.

Relatedness between $D$ and $S$

Under a neutral model $\frac{S}{a} = D$. Actually the standardize difference between $\frac{S}{a}$ and $D$ is what is used in Tajima's $D$ tests.

Source of information

Hartl and Clarck is good IMO but I would also like to recommend reading Population Genetics: A Concise Guide by Gillespie who explain all of that wonderfully.

• Thanks! Actually, $S = nb.polymorphic.sites / total.nb.loci$. In my example θ = $S/a_1$. This is what you say is $S/a =D$. So in this case, it's not exactly $4Nμ$. And I guess here $μ$ is the mutation rate. Thanks for the book! I'm going to read this carefully. Dec 29 '17 at 16:03