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Here is a quote from Wagner (2008)

A second line of evidence [against neutralism] comes from the relationship between the mean number of polymorphic differences between alleles within a species, $\pi$, and the number of fixed differences between genes in two species, $d$. For neutral mutations, a positive association between $\pi$ and $d$ should exist, because the neutral theory predicts that both quantities are linearly proportional to the rate at which neutral mutations arise. Recent genome-scale data shows instead that this association is in fact negative.

What do "the mean number of polymorphic differences between alleles within a species" and "the number of fixed differences between genes in two species" have to do with each other? Why should there be any relationship at all? And by "two species", are they talking about Eastern Yellowback Whooping Finches versus Western Yellowback Whooping Finches, or any arbitrary two species, like E. Coli versus Muskrats?

I found this related question, but it doesn't say anything about different species.

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  • $\begingroup$ Good question +1. I edited your post to get rid of what I thought was unnecessary information. Feel free to roll back if you don't like the edit. $\endgroup$ – Remi.b May 8 '16 at 19:26
  • $\begingroup$ Assuming everyone else here has the necessary background, it's fine. I'm not a biologist, so the neutralism/selectionism theories were news to me. $\endgroup$ – alexw May 8 '16 at 19:50
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Metrics of interest

The two metrics you are interested in are

  • $\pi$ - the mean number of differences between two randomly sampled (with replacement) alleles in a population
  • $d$ - the mean number of differences between two randomly sampled (with replacement) alleles coming from two different species

Consider two sequences

  • ATCGTCAAT
  • ATAGTTAAT

There are 2 pairwise differences between these two sequences (positions 3 and 6).

The whole point here is to understand that two individuals in the same population coalesce at a given time in the past just like two individuals coming from two different species. The number of pairwise differences is just equal to the rate at which mutations accumulate multiplied by the coalescence time. Let me develop this idea with a few equations below.

Neutral Expectations

Let's do the math! We will do two important assumptions below.

  1. Every mutation makes a new allele (it is an infinite allele model)
  2. All mutations are substitutions (no indels, no gene duplication, etc...)

Intro to Coalescence Theory

Let's first make sure you understand the concept of coalescence. Imagine you were to look at an evolving population backward in time. A coalescent event, is an event by which two lineages become one (looking backward in time). For simplicity consider a case of an asexual population (but future calculations don't do this assumption). In an asexual population, two siblings coalesced the previous generation, as in the previous generation the genes they are carrying were in a single individual (their parent).

We will see below how to calculate coalescence time between two randomly drawn individual from a population and how we can make inference about genetic diversity from this coalescent time.

Rate of accumulation of neutral mutations

The rate of accumulation of neutral mutation is also called the fixation rate of neutral mutations. Let $fix$ be this fixation rate. Consider a panmictic diploid population of constant size $N$. At a given locus of interest, the mutation rate is $\mu$. The number of mutations occurring every generation in this population is $2N\mu$ (The $2$ comes from the fact that in a diploid population, there are two homologous copies of every gene in each cell). If the mutation is neutral, then every single individual in the population has the same probability to end up being the parent of the whole population in a long time in the future. In other words, starting to look at this problem from the future and looking backward in time, any individual in a distance far enough in the future descends from a single individual in the present. As a consequence, the probability of a new mutation to reach fixation is $\frac{1}{2N}$. Multiplying the probability of fixation by the rate at which mutations occur in the whole population we obtain $\frac{1}{2N} 2N\mu = \mu$. In other words the rate of fixation of new neutral mutations is simply equal to the rate at which mutation occur.

Key equation:

$$fix=\frac{1}{2N} 2N\mu = \mu$$

$d$ - number of pairwise differences between randomly sampled (with replacement) alleles coming from different species

Consider two species which common ancestor lived $T$ generations ago. Each lineage accumulated mutations at rate $\mu$ (and explained above) and therefore the expected (average) total number of pairwise differences is $\bar d=2\mu T$. The probability of having exactly $d$ pairwise differences comes from a Poisson distribution with rate $2\mu T$ ($P(D=d) = Poisson(2\mu T)$).

Key equation: $$\bar d=2\mu T$$

$\pi$ - number of pairwise differences between randomly sampled (with replacement) alleles coming from the same population

Here the calculations follows the same logic as above $\pi = 2 \mu T$. The whole issue is that $T$ (the coalescence time) is unknown for the moment and we need to calculate it.

Let $P(T)$ be the probability that two randomly sampled (with replacement) individuals from a diploid panmictic population coalesce exactly $T$ generations ago. The probability of coalescing in a given generation is simply the probability for drawing the same individual, that is $\frac{1}{2N}$ and the probability of not coalescing in any subsequent generation is $1-\frac{1}{2N}$. Iterating over the generations we obtain

$$P(T) = \frac{1}{2N} \left( 1-\frac{1}{2N} \right)^T \approx \frac{1}{2N}e^{-\frac{T}{2N}}$$

, where $e \approx 2.7$ is Euler's number. The above approximation is accurate for large $N$ (say larger than 100). In case you are interest, you will find on this Math.SE post, an explanation for this approximation.

The expected value (average) $\bar T$ of this distribution is $\bar T=2N$ (and the variance is $var(T) = 4N^2$). As a consequence the expected number of pairwise differences between these two individuals is $\pi = 2 \mu 2 N = 4N\mu$

Quickly speaking, these calculations can be extended to calculate the coalescent time between $k$ randomly sampled alleles. The expected coalescent time is then $\bar T = \frac{4N}{k(k-1)}$ and therefore some algebra shows that the total expected time (along all branches) is $4 N \sum_{i=1}^{k-1}\frac{1}{i}$ and the expected number of segregating sites is $4 N \mu \sum_{i=1}^{k-1}\frac{1}{i}$.

Key equation: $$\pi = 4N\mu$$

Conclusion

It is clear from the above that $\pi = 4N\mu$ and $d=2\mu T$ are correlated as they are both linearly related to the mutation rate. As it is said in your quote

For neutral mutations, a positive association between $\pi$ and $d$ should exist, because the neutral theory predicts that both quantities are linearly proportional to the rate at which neutral mutations arise

The quote goes one saying

Recent genome-scale data shows instead that this association is in fact negative.

Such negative relationship between $d$ and $\pi$ cannot be caused by neutral processes. Therefore selection must be involved somehow.

Imagine for example that two species are both selected at different optima (opposite selection pressures). In such case, you would see purifying selection within species reducing the probability of a new mutation to reach fixation to a value lower than $\frac{1}{2N}$ and therefore reducing the rate of fixation $fix$ so that $fix<\mu$. However, such opposite selection pressure among species will yield the species to diverge more than expected by random processes that is $\pi > 4N\mu$. Such selective pattern could explain the observed negative relationship.

More information

This post offers book recommendation on the subject (population genetics).

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  • $\begingroup$ when you say "mean number of differences", how is this quantified? Number of differing base pairs? $\endgroup$ – alexw May 8 '16 at 19:52
  • $\begingroup$ Yes exactly. There are two assumptions (I forgot about in my answer but I will add them) 1. All mutations are substitutions (no indel, no whole chromosome duplication etc..) and 2. every mutation makes a new allele (infinite-allele model) $\endgroup$ – Remi.b May 8 '16 at 19:57
  • $\begingroup$ Are those assumptions specific to neutral theory? I guess this is supposed to be a proof by contradiction, so I want to see which specific premises are leading us to the contradiction. $\endgroup$ – alexw May 8 '16 at 20:19
  • $\begingroup$ Those are very common assumptions found in the neutral theory literature and more globally in the literature on genetic drift. The concept of proof by contradiction applies to math but not to empirical sciences (isn't it?). We just have a null prediction (model without selection) and if the data don't match these then something else must be going on. (Thanks for the edit) $\endgroup$ – Remi.b May 8 '16 at 20:30

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