Counting unique molecular identifiers in sequencing using a multi-type branching process with immigration.

Detection of extremely rare variant alleles, such as tumor DNA, within a complex mixture of DNA molecules is experimentally challenging due to sequencing errors. Barcoding of target DNA molecules in library construction for next-generation sequencing provides a way to identify and bioinformatically remove polymerase induced errors. During the barcoding procedure involving t consecutive PCR cycles, the DNA molecules become barcoded by Unique Molecular Identifiers (UMIs). Different library construction protocols utilize different values of t. The effect of a larger t and imperfect PCR amplifications in relation to UMI cluster sizes is poorly described. This paper proposes a branching process with growing immigration as a model describing the random outcome of t cycles of PCR barcoding. Our model discriminates between five different amplification rates r1, r2, r3, r4, r for different types of molecules associated with the PCR barcoding procedure. We study this model by focussing on Ct, the number of clusters of molecules sharing the same UMI, as well as Ct(m), the number of UMI clusters of size m. Our main finding is a remarkable asymptotic pattern valid for moderately large t. It turns out that E(Ct(m))/E(Ct)≈2-m for m=1,2,…, regardless of the underlying parameters (r1,r2,r3,r4,r). The knowledge of the quantities Ct and Ct(m) as functions of the experimental parameters t and (r1,r2,r3,r4,r) will help the users to draw more adequate conclusions from the outcomes of different sequencing protocols.

A consistent estimator of the evolutionary rate.

We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate λ, conditioned on the observed number of particles to be equal to n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance σ(2). In this paper we treat n particles as a sample of related species. The spatial Brownian motion of a particle describes the development of a trait value of interest (e.g. log-body-size). We propose an unbiased estimator Rn(2) of the evolutionary rate ρ(2)=σ(2)/λ. The estimator Rn(2) is proportional to the sample variance Sn(2) computed from n trait values. We find an approximate formula for the standard error of Rn(2) based on a neat asymptotic relation for the variance of Sn(2).

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).

Time to a single hybridization event in a group of species with unknown ancestral history.

We consider a stochastic process for the generation of species which combines a Yule process with a simple model for hybridization between pairs of co-existent species. We assume that the origin of the process, when there was one species, occurred at an unknown time in the past, and we condition the process on producing n species via the Yule process and a single hybridization event. We prove results about the distribution of the time of the hybridization event. In particular we calculate a formula for all moments and show that under various conditions, the distribution tends to an exponential with rate twice that of the birth rate for the Yule process.

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.

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.

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.

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.

Coalescent patterns in diploid exchangeable population models.

A class of two-sex population models is considered with N females and equal number N of males constituting each generation. Reproduction is assumed to undergo three stages: 1) random mating, 2) exchangeable reproduction, 3) random sex assignment. Treating individuals as pairs of genes at a certain locus we introduce the diploid ancestral process (the past genealogical tree) for n such genes sampled in the current generation. Neither mutation nor selection are assumed. A convergence criterium for the diploid ancestral process is proved as N goes to infinity while n remains unchanged. Conditions are specified when the limiting process (coalescent) is the Kingman coalescent and situations are discussed when the coalescent allows for multiple mergers of ancestral lines.