Friday, November 19, 2010

The Coalescent in a Continuous, Finite, Linear Population

An interesting paper describes the process of gene flow that we see in the Fertile Crescent between the West Asian component and the Southwest Asian component:

The Coalescent in a Continuous, Finite, Linear Population, Jon F Wilkins and John Wakeley, Genetics Society of America, March 4, 2002.

Link

From the Abstract:

"In this article we present a model for analyzing patterns of genetic diversity in a continuous, finite, linear habitat with restricted gene flow. The distribution of coalescent times and locations is derived for a pair of sequences sampled from arbitrary locations along the habitat. The results for mean time to coalescence are compared to simulated data. As expected, mean time to common ancestry increases with the distance separating the two sequences. Additionally, this mean time is greater near the center of the habitat than near the ends. In the distant past, lineages that have not undergone coalescence are more likely to have been at opposite ends of the population range, whereas coalescent events in the distant past are biased toward the center. All of these effects are more pronounced when gene flow is more limited. The pattern of pairwise nucleotide differences predicted by the model is compared to data collected from sardine populations. The sardine data are used to illustrate how demographic parameters can be estimated using the model."

The model postulates the normal distribution for the expansion of populations and also suggests that migration is conservative.

Thus, the location of a parent is normally distributed around the location of its offspring, with variance 2 sigma squared.

The boundaries of the habitat are reflecting, so a gamete that would otherwise land outside the habitat range is reflected back an equal distance within it. Each individual thus has the same expected number of offspring regardless of its location. This means that migration is conservative, so migration alone is sufficient to maintain the relative population densities at all locations in the habitat (Nagylaki 1980). Nonreflecting boundaries would correspond to the case where those gametes dispersing outside the habitat range are lost. In such a system, individuals near the edges of the habitat would have a reduced effective fecundity relative to those nearest the center.

As Felsenstein (1975) pointed out, most continuous space models in population genetics assume a uniform population density that would not actually be maintained by the proposed reproductive scheme. A normal distribution of gametes without severe density regulation generates a population that is clumped together at certain locations and sparsely populated at others. With its absolute density regulation at all locations, the model of reproduction proposed here will immediately generate and maintain a population that is uniformly distributed across its habitat range.

What's striking about this paper is that it exactly describes the diffusion process that has been occuring in the Fertile Crescent.


In the previous post, I've looked at two other diffusion models, in addition to the first order approximation. Here, we add the Normal Distribution Model described by Wilson and Wakeley:
 
We can see that the First Order Approximation is in good agreement with the Normal Distribution Model for populations that are uniformly distributed.

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