Serik Sagitov

Statistical Inference of Allopolyploid Species Networks in the Presence of Incomplete Lineage Sorting.

Polyploidy is an important speciation mechanism, particularly in land plants. Allopolyploid species are formed after hybridization between otherwise intersterile parental species. Recent theoretical progress has led to successful implementation of species tree models that take population genetic parameters into account. However, these models have not included allopolyploid hybridization and the special problems imposed when species trees of allopolyploids are inferred. Here, 2 new models for the statistical inference of the evolutionary history of allopolyploids are evaluated using simulations and demonstrated on 2 empirical data sets. It is assumed that there has been a single hybridization event between 2 diploid species resulting in a genomic allotetraploid. The evolutionary history can be represented as a species network or as a multilabeled species tree, in which some pairs of tips are labeled with the same species. In one of the models (AlloppMUL), the multilabeled species tree is inferred directly. This is the simplest model and the most widely applicable, since fewer assumptions are made. The second model (AlloppNET) incorporates the hybridization event explicitly which means that fewer parameters need to be estimated. Both models are implemented in the BEAST framework. Simulations show that both models are useful and that AlloppNET is more accurate if the assumptions it is based on are valid. The models are demonstrated on previously analyzed data from the genera Pachycladon (Brassicaceae) and Silene (Caryophyllaceae).

On the path to extinction

Populations can die out in many ways. We investigate one basic form of extinction, stable or intrinsic extinction, caused by individuals on the average not being able to replace themselves through reproduction. The archetypical such population is a subcritical branching process, i.e., a population of independent, asexually reproducing individuals, for which the expected number of progeny per individual is less than one. The main purpose is to uncover a fundamental pattern of nature. Mathematically, this emerges in large systems, in our case subcritical populations, starting from a large number, x, of individuals. First we describe the behavior of the time to extinction T: as x grows to infinity, it behaves like the logarithm of x, divided by r, where r is the absolute value of the Malthusian parameter. We give a more precise description in terms of extreme value distributions. Then we study population size partway (or u-way) to extinction, i.e., at times uT, for 0 < u < 1, e.g., u = 1/2 gives halfway to extinction. (Note that mathematically this is no stopping time.) If the population starts from x individuals, then for large x, the proper scaling for the population size at time uT is x into the power u − 1. Normed by this factor, the population u-way to extinction approaches a process, which involves constants that are determined by life span and reproduction distributions, and a random variable that follows the classical Gumbel distribution in the continuous time case. In the Markov case, where an explicit representation can be deduced, we also find a description of the behavior immediately before extinction.

Multitype Bienayme-Galton-Watson processes escaping extinction.

In the framework of a multitype Bienaymé–Galton–Watson (BGW) process, the event that the daughters type differs from the mothers type can be viewed as a mutation event. Assuming that mutations are rare, we study a situation where all types except one produce on average less than one offspring. We establish a neat asymptotic structure for the BGW process escaping extinction due to a sequence of mutations toward the supercritical type. Our asymptotic analysis is performed by letting mutation probabilities tend to 0. The limit process, conditional on escaping extinction, is another BGW process with an enriched set of types, allowing us to delineate a stem lineage of particles that leads toward the escape event. The stem lineage can be described by a simple Markov chain on the set of particle types. The total time to escape becomes a sum of a random number of independent, geometrically distributed times spent at intermediate types.

Interspecies correlation for neutrally evolving traits.

A simple way to model phenotypic evolution is to assume that after splitting, the trait values of the sister species diverge as independent Brownian motions. Relying only on a prior distribution for the underlying species tree (conditioned on the number, n, of extant species) we study the random vector (X(1),…,X(n)) of the observed trait values. In this paper we derive compact formulae for the variance of the sample mean and the mean of the sample variance for the vector (X(1),…,X(n)). The key ingredient of these formulae is the correlation coefficient between two trait values randomly chosen from (X(1),…,X(n)). This interspecies correlation coefficient takes into account not only variation due to the random sampling of two species out of n and the stochastic nature of Brownian motion but also the uncertainty in the phylogenetic tree. The latter is modeled by a (supercritical or critical) conditioned branching process. In the critical case we modify the Aldous-Popovic model by assuming a proper prior for the time of origin.

Stochasticity in the adaptive dynamics of evolution: the bare bones

First a population model with one single type of individuals is considered. Individuals reproduce asexually by splitting into two, with a population-size-dependent probability. Population extinction, growth and persistence are studied. Subsequently the results are extended to such a population with two competing morphs and are applied to a simple model, where morphs arise through mutation. The movement in the trait space of a monomorphic population and its possible branching into polymorphism are discussed. This is a first report. It purports to display the basic conceptual structure of a simple exact probabilistic formulation of adaptive dynamics.

The coalescent effective size of age-structured populations.

We establish convergence to the Kingman coalescent for a class of age-structured population models with time-constant population size. Time is discrete with unit called a year. Offspring numbers in a year may depend on mothers age.

An Accurate Model for Genetic Hitchhiking

We suggest a simple deterministic approximation for the growth of the favored-allele frequency during a selective sweep. Using this approximation we introduce an accurate model for genetic hitchhiking. Only when Ns < 10 (N is the population size and s denotes the selection coefficient) are discrepancies between our approximation and direct numerical simulations of a Moran model notable. Our model describes the gene genealogies of a contiguous segment of neutral loci close to the selected one, and it does not assume that the selective sweep happens instantaneously. This enables us to compute SNP distributions on the neutral segment without bias.

Limit Processes for Age-Dependent Branching Particle Systems

We consider systems of spatially distributed branching particles in Rd. The particle lifelengths are of general form, hence the time propagation of the system is typically not Markov. A natural time-space-mass scaling is applied to a sequence of particle systems and we derive limit results for the corresponding sequence of measure-valued processes. The limit is identified as the projection on Rd of a superprocess in R+×Rd. The additive functional characterizing the superprocess is the scaling limit of certain point processes, which count generations along a line of descent for the branching particles.

Linkage Disequilibrium Under Recurrent Bottlenecks

To model deviations from selectively neutral genetic variation caused by different forms of selection, it is necessary to first understand patterns of neutral variation. Best understood is neutral genetic variation at a single locus. But, as is well known, additional insights can be gained by investigating multiple loci. The resulting patterns reflect the degree of association (linkage) between loci and provide information about the underlying multilocus gene genealogies. The statistical properties of two-locus gene genealogies have been intensively studied for populations of constant size, as well as for simple demographic histories such as exponential population growth and single bottlenecks. By contrast, the combined effect of recombination and sustained demographic fluctuations is poorly understood. Addressing this issue, we study a two-locus Wright–Fisher model of a population subject to recurrent bottlenecks. We derive coalescent approximations for the covariance of the times to the most recent common ancestor at two loci in samples of two chromosomes. This covariance reflects the degree of association and thus linkage disequilibrium between these loci. We find, first, that an effective population-size approximation describes the numerically observed association between two loci provided that recombination occurs either much faster or much more slowly than the population-size fluctuations. Second, when recombination occurs frequently between but rarely within bottlenecks, we observe that the association of gene histories becomes independent of physical distance over a certain range of distances. Third, we show that in this case, a commonly used measure of linkage disequilibrium, σd2 (closely related to r^2), fails to capture the long-range association between two loci. The reason is that constituent terms, each reflecting the long-range association, cancel. Fourth, we analyze a limiting case in which the long-range association can be described in terms of a Xi coalescent allowing for simultaneous multiple mergers of ancestral lines.

Markovian Paths to Extinction

Subcritical Markov branching processes {Z t } die out sooner or later, say at time T < ∞. We give results for the path to extinction {Z uT , 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of x u−1 Z uT , as Z 0=x → ∞. The limit reflects an interplay of branching and extreme value theory. Then we consider the population on the verge of extinction, as modelled by Z T-u , u > 0, and show that as Z 0= x → ∞ this process converges to a Markov process {Y u }, which we describe completely. Emphasis is on continuous time processes, those in discrete time displaying a more complex behaviour, related to Martin boundary theory.