Local fluctuations of genetic processes defined on two time scales, with applications to effective size estimation.

In this paper we develop a general framework for how the genetic composition of a structured population with strong migration between its subunits, evolves over time. The dynamics is described in terms of a vector-valued Markov process of allele, genotype or haplotype frequencies that varies on two time scales. The more rapid changes are random fluctuations in terms of a multivariate autoregressive process, around a quasi equilibrium fix point, whereas the fix point itself varies more slowly according to a diffusion process, along a lower-dimensional subspace. Under mild regularity conditions, the fluctuations have a magnitude inversely proportional to the square root of the population size N, and hence can be used to estimate a broad class of genetically effective population sizes Ne, with genetic data from one time point only. In this way we are able to unify a number of existing notions of effective size, as well as proposing new ones, for instance one that quantifies the extent to which genotype frequencies fluctuate around Hardy-Weinberg equilibrium.

Exact Markov chain and approximate diffusion solution for haploid genetic drift with one-way mutation.

The classical Kimura solution of the diffusion equation is investigated for a haploid random mating (Wright-Fisher) model, with one-way mutations and initial-value specified by the founder population. The validity of the transient diffusion solution is checked by exact Markov chain computations, using a Jordan decomposition of the transition matrix. The conclusion is that the one-way diffusion model mostly works well, although the rate of convergence depends on the initial allele frequency and the mutation rate. The diffusion approximation is poor for mutation rates so low that the non-fixation boundary is regular. When this happens we perturb the diffusion solution around the non-fixation boundary and obtain a more accurate approximation that takes quasi-fixation of the mutant allele into account. The main application is to quantify how fast a specific genetic variant of the infinite alleles model is lost. We also discuss extensions of the quasi-fixation approach to other models with small mutation rates.

A monoecious and diploid Moran model of random mating.

An exact Markov chain is developed for a Moran model of random mating for monoecious diploid individuals with a given probability of self-fertilization. The model captures the dynamics of genetic variation at a biallelic locus. We compare the model with the corresponding diploid Wright-Fisher (WF) model. We also develop a novel diffusion approximation of both models, where the genotype frequency distribution dynamics is described by two partial differential equations, on different time scales. The first equation captures the more slowly varying allele frequencies, and it is the same for the Moran and WF models. The other equation captures departures of the fraction of heterozygous genotypes from a large population equilibrium curve that equals Hardy-Weinberg proportions in the absence of selfing. It is the distribution of a continuous time Ornstein-Uhlenbeck process for the Moran model and a discrete time autoregressive process for the WF model. One application of our results is to capture dynamics of the degree of non-random mating of both models, in terms of the fixation index fIS. Although fIS has a stable fixed point that only depends on the degree of selfing, the normally distributed oscillations around this fixed point are stochastically larger for the Moran than for the WF model.

Wright-Fisher model of social insects with haploid males and diploid females.

An inhomogeneous discrete Markov model is formulated for sexual random mating in finite populations of haploid male and diploid female individuals. This is a Wright-Fisher type of model for social insects. The generations are non-overlapping and of given finite sizes. Bottlenecks are included, allowing different sizes to change from generation to generation. Mutations and selection are included in this exact model for the stochastic process. Computations of the exact Markov model are presented, focussing on the sexually asymmetric genetic drift caused by haplodiploidy.

Exact Markov chains versus diffusion theory for haploid random mating.

Exact discrete Markov chains are applied to the Wright-Fisher model and the Moran model of haploid random mating. Selection and mutations are neglected. At each discrete value of time t there is a given number n of diploid monoecious organisms. The evolution of the population distribution is given in diffusion variables, to compare the two models of random mating with their common diffusion limit. Only the Moran model converges uniformly to the diffusion limit near the boundary. The Wright-Fisher model allows the population size to change with the generations. Diffusion theory tends to under-predict the loss of genetic information when a population enters a bottleneck.

A sexually neutral discrete Markov model for given sum males + females.

An inhomogeneous discrete Markov model is formulated for sexual random mating with diploid male and female individuals. The generations are nonoverlapping and of given sizes. The genetic variation is in a sexually neutral allele with two varieties, giving three different genotypes. Taking sex as a marker, the Markov model works with six genotypes. The sex of each offspring is random. This implies a probability of extinction, giving the model an algorithmic nature. We compute expected genotype frequencies, their standard deviations and fixation probabilities.

Markov model of haploid random mating with given distribution of population size.

An exact Markov chain model is formulated and computed for random mating in a haploid gamete pool. There are two versions of the gamete, and there is a finite number of diploid monoecious organisms. The founder population is given, and the subsequent generations allow a prescribed statistical distribution over different population sizes. The non-homogeneous Markov chain works on the haploid gamete level provided the probability of self-fertilization is 1/n, where n is the number of diploid individuals. Standard deviations of gamete frequencies and fixation probabilities are calculated. Effective population sizes for different population size distributions are estimated, including periodic bottlenecks.