Academic journal article Genetics

Stepping-Stone Spatial Structure Causes Slow Decay of Linkage Disequilibrium and Shifts the Site Frequency Spectrum

Academic journal article Genetics

Stepping-Stone Spatial Structure Causes Slow Decay of Linkage Disequilibrium and Shifts the Site Frequency Spectrum

Article excerpt


The symmetric island model with D demes and equal migration rates is often chosen for the investigation of the consequences of population subdivision. Here we show that a stepping-stone model has a more pronounced effect on the genealogy of a sample. For samples from a small geographical region commonly used in genetic studies of humans and Drosophila, there is a shift of the frequency spectrum that decreases the number of low-frequency-derived alleles and skews the distribution of statistics of Tajima, Fu and Li, and Fay and Wu. Stepping-stone spatial structure also changes the two-locus sampling distribution and increases both linkage disequilibrium and the probability that two sites are perfectly correlated. This may cause a false prediction of cold spots of recombination and may confuse haplotype tests that compute probabilities on the basis of a homogeneously mixing population.

(ProQuest-CSA LLC: ... denotes formulae omitted.)

HOMOGENEOUSLY mixing populations of constant size are a convenient setting to develop the theory of population genetics. However, when one wants to understand patterns observed in data, one must consider the effects of population growth, bottlenecks, and population subdivision. When the consequences of population subdivision are investigated, the symmetric island model with D demes and equal migration rates is the usual choice, and the case of two demes is especially popular. The island model is easy to analyze mathematically due to the fact that if two lineages are not in the same deme, then their relative location is not important. However, this has the unrealistic consequence that after one migration event, the lineage is distributed uniformly over the species range.

An alternative approach tomodeling spatial structure that does not suffer from this defect is the steppingstone model. In this article, we investigate the consequences of modeling space as a two-dimensional stepping-stone model in which there is an L × L grid of colonies and migration only to neighboring colonies. The migration scheme is very simple; however, it results in what Wright called isolation by distance. In other words, it takes a number of migration events for the lineages to spread across the system. As we will see, this feature, which is certainly present in Drosophila and early human populations, causes a dramatic change in the coalescence structure of lineages.

The reason for this is intuitively clear. At small times the lineages have not had a chance to spread across the population, so the effectivepopulation size is reduced.The coalescence rate is increased, reducing the number of lowfrequency- derived alleles, skewing the site frequency spectrum, and increasing linkage disequilibrium. These effects occur in the island model as well; two lineages sampled fromone deme have an increased coalescence rate until one of them migrates, at which point they behave like a random sample from the overall population. In contrast, as we later explain, in the stepping-stone model the effective population size increases roughly linearly in time.

The main point of this article is to argue that spatial structure in the form of the stepping-stone model has a different effect from the symmetric island model and can have a much greater impact on genealogies, so it should also be considered when one wants to assess the impact of spatial structure on estimation procedures or statistical tests.We begin by reviewing theoretical results of Cox and Durrett (2002) and Zähle et al. (2005) for the coalescence time of a sample of size n and contrast these results with the corresponding facts about the symmetric island model. The strange nonlinear time scaling needed to reduce genealogies in the steppingstone model to Kingman's coalescent indicates that there is a strong effect on commonly used statistics, but the exact nature of the changes is difficult to analyze mathematically. Because of this, we turn to simulations to demonstrate the effect of stepping-stone spatial structure on the decay of linkage disequilibrium, the site frequency spectrum, and the distribution of test statistics based on the site frequency spectrum. …

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