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A short introduction to population genetics modeling.
In this chapter we define the important notions of population genetics, that we are going to work with, and the evolution forces that act on popu-lations.
Alleles and genotypes.
A locus is a fixed position on the genome composed of one or several DNA bases. This can be a gene, but does not have to be. If several variants of this locus exist, we call them alleles. In diploid populations the loci are associated by pairs in the individuals, one inherited from the father and one from the mother. Here, since our focus is at the population level, we are going to see the population as a genetic pool and the individuals as transitory genetic carriers.
Considering a locus that has two possible alleles, A and a, a diploid individual can have three possible genotypes: AA, Aa, aa. The amount of possible genotypes increase very much with the number of alleles and makes calculations more complicated. So, in the following we will assume only two alleles per locus in the populations, unless otherwise specified.
To describe the genetic composition at a locus in a population, there are two different measures: the allele frequencies and genotype frequencies. The first one forgets that the loci are paired, so is the frequency of allele A (or allele a). The second one takes the pairs of alleles into account, i.e. the frequencies of AA, Aa, aa. If we know the genotype frequencies, it is direct to calculate the allele frequencies, as follows:
The Hardy-Weinberg Equilibrium
The model developed by Hardy and Weinberg is a purely theoretical model, but it illustrates very well how the genetic structure is transmitted from one generation to another and it is useful as an initial model.
The hypothesis of the model are:
H1: • diploid population
• autosomal chromosomes
• generations do not overlap
• panmixia
• same allele frequencies in gametes of both sexes H2: • infinite population
• lack of mutation
• lack of migration from other populations
• lack of natural selection on any gene.
The model: As we have an infinite population, in the gametic phase, we are going to have an infinite number of gametes, some of them are A and some a. We can imagine that we have an urn with two types of “balls”, whose frequencies are p and q and that to build the next generation we choose balls randomly from this urn. As the parental population is infi-nite, we choose with replacement, with a probability p of choosing A, and probability q of choosing a. As the next generation is also infinite, applying the large numbers law, we have f req(A) = p and f req(a) = q in the next generation.
The Hardy-Weinberg equilibrium (HWE): In an isolated population of infinite size, which is not under selection or mutation, the allele frequencies are constant across generations. Besides, if we suppose that the population is panmictic, i.e., its mating regime is at random, and the allele frequencies do not change between maternal and parental gametes, then computing the genotype frequencies from the allele frequencies is possible according to the following table:
Note that when we sample from a population, the calculated frequencies from the sample are unbiased estimators of the population’s frequencies.
Although the infinite size hypothesis is unrealistic, in populations with really large sizes and in short periods of time this equilibrium holds. Be-sides, if a population has non equally distributed frequencies between sexes, the population will reach the equilibrium in one generation. So, whatever the composition of the parental population, random mating will produce a genotypic distribution approximately stationary.
When the frequencies do not remain constant we say that the population evolves, this evolution can be due mainly to four different forces, which we will describe in the next section.
The four evolutionary forces.
Genetic drift
As real populations have a finite size, every time a new generation is born, gametes from the parental population are only a finite sample. Although we can apply the large numbers law and the expected frequency does not vary over generations, the frequencies are going to change due to this sampling.
Table of contents :
1 A short introduction to population genetics modeling.
1.1 Alleles and genotypes
1.2 The Hardy-Weinberg Equilibrium
1.3 The four evolutionary forces
1.3.1 Genetic drift
1.3.2 Mutation
1.3.3 Migration
1.3.4 Selection
1.3.5 Evolutionary forces considered in our model
1.4 Introduction to multilocus models
2 Theoretical framework
2.1 The Lewontin and Krakauer test and extensions
2.1.1 LK-test
2.1.2 F-LK-test
2.1.3 Multiallelic versions
2.1.4 Estimation of the kinship matrix
2.2 A multilocus model for linkage disequilibrium
2.2.1 A cluster model
3 A new haplotype-based test for detecting signatures of se-lection.
3.1 The test implementation
3.2 Simulations
3.3 Results
3.4 Perspectives-Future Work
3.5 Conclusion
