Jochen Blath

Computing likelihoods for coalescents with multiple collisions in the infinitely many sites model.

One of the central problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of the classical Fleming–Viot process, such as the Wright–Fisher- or the Moran model, then the standard means to describe its genealogy is Kingman’s coalescent. For this coalescent process, powerful inference methods are well-established. An important feature of the above class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size, and hence all ancestral collisions are binary only. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised Λ-Fleming–Viot processes, as well as their (dual) genealogies, given by the so-called Λ-coalescents, which allow multiple collisions. Moreover, Eldon and Wakeley (Genetics 172:2621–2633, 2006) provide evidence that such more general coalescents might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier and Griffiths (Ann Probab 15(2):515–545, 1987) and Griffiths and Tavaré (Theor Pop Biol 46:131–159, 1994a, Stat Sci 9:307–319, 1994b, Philos Trans Roy Soc Lond Ser B 344:403–410, 1994c, Math Biosci 12:77–98, 1995) to obtain a likelihood based inference method for general Λ-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of Λ-coalescents, the parametrisable sub-family of Beta(2 − α, α)-coalescents, where α ∈ (1, 2], are of particular relevance. We illustrate our method using simulated datasets, thus obtaining maximum-likelihood estimators of mutation and demographic parameters.

Can the Site-Frequency Spectrum Distinguish Exponential Population Growth from Multiple-Merger Coalescents?

The ability of the site-frequency spectrum (SFS) to reflect the particularities of gene genealogies exhibiting multiple mergers of ancestral lines as opposed to those obtained in the presence of population growth is our focus. An excess of singletons is a well-known characteristic of both population growth and multiple mergers. Other aspects of the SFS, in particular, the weight of the right tail, are, however, affected in specific ways by the two model classes. Using an approximate likelihood method and minimum-distance statistics, our estimates of statistical power indicate that exponential and algebraic growth can indeed be distinguished from multiple-merger coalescents, even for moderate sample sizes, if the number of segregating sites is high enough. A normalized version of the SFS (nSFS) is also used as a summary statistic in an approximate Bayesian computation (ABC) approach. The results give further positive evidence as to the general eligibility of the SFS to distinguish between the different histories.

Statistical Properties of the Site-Frequency Spectrum Associated with Lambda-Coalescents

Statistical properties of the site-frequency spectrum associated with Λ-coalescents are our objects of study. In particular, we derive recursions for the expected value, variance, and covariance of the spectrum, extending earlier results of Fu (1995) for the classical Kingman coalescent. Estimating coalescent parameters introduced by certain Λ-coalescents for data sets too large for full-likelihood methods is our focus. The recursions for the expected values we obtain can be used to find the parameter values that give the best fit to the observed frequency spectrum. The expected values are also used to approximate the probability a (derived) mutation arises on a branch subtending a given number of leaves (DNA sequences), allowing us to apply a pseudolikelihood inference to estimate coalescence parameters associated with certain subclasses of Λ-coalescents. The properties of the pseudolikelihood approach are investigated on simulated as well as real mtDNA data sets for the high-fecundity Atlantic cod (Gadus morhua). Our results for two subclasses of Λ-coalescents show that one can distinguish these subclasses from the Kingman coalescent, as well as between the Λ-subclasses, even for a moderate (maybe a few hundred) sample size.

An Ancestral Recombination Graph for Diploid Populations with Skewed Offspring Distribution

A large offspring-number diploid biparental multilocus population model of Moran type is our object of study. At each time step, a pair of diploid individuals drawn uniformly at random contributes offspring to the population. The number of offspring can be large relative to the total population size. Similar “heavily skewed” reproduction mechanisms have been recently considered by various authors (cf. e.g., Eldon and Wakeley 2006, 2008) and reviewed by Hedgecock and Pudovkin (2011). Each diploid parental individual contributes exactly one chromosome to each diploid offspring, and hence ancestral lineages can coalesce only when in distinct individuals. A separation-of-timescales phenomenon is thus observed. A result of Möhle (1998) is extended to obtain convergence of the ancestral process to an ancestral recombination graph necessarily admitting simultaneous multiple mergers of ancestral lineages. The usual ancestral recombination graph is obtained as a special case of our model when the parents contribute only one offspring to the population each time. Due to diploidy and large offspring numbers, novel effects appear. For example, the marginal genealogy at each locus admits simultaneous multiple mergers in up to four groups, and different loci remain substantially correlated even as the recombination rate grows large. Thus, genealogies for loci far apart on the same chromosome remain correlated. Correlation in coalescence times for two loci is derived and shown to be a function of the coalescence parameters of our model. Extending the observations by Eldon and Wakeley (2008), predictions of linkage disequilibrium are shown to be functions of the reproduction parameters of our model, in addition to the recombination rate. Correlations in ratios of coalescence times between loci can be high, even when the recombination rate is high and sample size is large, in large offspring-number populations, as suggested by simulations, hinting at how to distinguish between different population models.

Coexistence in locally regulated competing populations and survival of branching annihilating random walk

We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which do not, at present, incorporate all of the competitive strategies that a population might adopt. The second is a simplification of the first, in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching annihilating random walk. For each model, using a comparison with oriented percolation, we show that for certain parameter values, both populations will coexist for all time with positive probability. As a corollary, we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates. We also present a number of conjectures relating to the r\^{o}le of space in the survival probabilities for the two populations.

Trends in Stochastic Analysis

Preface Part I. Foundations and techniques in stochastic analysis: 1. Random variables - without basic space Gotz Kersting 2. Chaining techniques and their application to stochastic flows Michael Scheutzow 3. Ergodic properties of a class of non-Markovian processes Martin Hairer 4. Why study multifractal spectra? Peter Morters Part II. Construction, simulation, discretisation of stochastic processes: 5, Construction of surface measures for Brownian motion Nadia Sidorova and Olaf Wittich 6. Sampling conditioned diffusions Martin Hairer, Andrew Stuart and Jochen Voss 7. Coding and convex optimization problems Steffen Dereich Part III. Stochastic analysis in mathematical physics: 8. Intermittency on catalysts Jurgen Gartner, Frank den Hollander and Gregory Maillard 9. Stochastic dynamical systems in infinite dimensions Salah-Eldin A. Mohammed 10. Feynman formulae for evolutionary equations Oleg G.Smolyanov 11. Deformation quantization in infinite dimensional analysis Remi Leandre Part IV. Stochastic analysis in mathematical biology: 12. Measure-valued diffusions, coalescents and genetic inference Matthias Birkner and Jochen Blath 13. How often does the ratchet click? Facts, heuristics, asymptotics Alison M. Etheridge, Peter Pfaffelhuber and Anton Wakolbinger.

On the moments and the interface of the symbiotic branching model

In this paper we introduce a critical curve separating the asymptotic behavior of the moments of the symbiotic branching model, introduced by Etheridge and Fleischmann [ Stochastic Process. App. 114 (2004) 127-160] into two regimes. Using arguments based on two different dualities and a classical result of Spitzer [ Trans. Amer. Math. Soc. 87 (1958) 187-197] on the exit-time of a planar Brownian motion from a wedge, we prove that the parameter governing the model provides regimes of bounded and exponentially growing moments separated by subexponential growth. The moments turn out to be closely linked to the limiting distribution as time tends to infinity. The limiting distribution can be derived by a self-duality argument extending a result of Dawson and Perkins [ Ann. Probab. 26 (1998) 1088-1138] for the mutually catalytic branching model. As an application, we show how a bound on the 35th moment improves the result of Etheridge and Fleischmann [ Stochastic Process. Appl. 114 (2004) 127-160] on the speed of the propagation of the the interface of the symbiotic branching model.

Genetic Variability Under the Seedbank Coalescent

We analyze patterns of genetic variability of populations in the presence of a large seedbank with the help of a new coalescent structure called the seedbank coalescent. This ancestral process appears naturally as a scaling limit of the genealogy of large populations that sustain seedbanks, if the seedbank size and individual dormancy times are of the same order as those of the active population. Mutations appear as Poisson processes on the active lineages and potentially at reduced rate also on the dormant lineages. The presence of “dormant” lineages leads to qualitatively altered times to the most recent common ancestor and nonclassical patterns of genetic diversity. To illustrate this we provide a Wright–Fisher model with a seedbank component and mutation, motivated from recent models of microbial dormancy, whose genealogy can be described by the seedbank coalescent. Based on our coalescent model, we derive recursions for the expectation and variance of the time to most recent common ancestor, number of segregating sites, pairwise differences, and singletons. Estimates (obtained by simulations) of the distributions of commonly employed distance statistics, in the presence and absence of a seedbank, are compared. The effect of a seedbank on the expected site-frequency spectrum is also investigated using simulations. Our results indicate that the presence of a large seedbank considerably alters the distribution of some distance statistics, as well as the site-frequency spectrum. Thus, one should be able to detect from genetic data the presence of a large seedbank in natural populations.

Surveys in Stochastic Processes

We discuss problems posed by the quantitative study of time inhomogeneous Markov chains. The two main notions for our purpose are merging and stability. Merging (also called weak ergodicity) occurs when the chain asymptotically forgets where it started. It is a loss of memory property. Stability relates to the question of whether or not, despite temporary variations, there is a rough shape describing the long time behavior of the chain. For instance, we will discuss an example where the long time behavior is roughly described by a binomial, with temporal variations.Based on an intuitive approach to the Ray-Knight representation of Feller’s branching diffusion in terms of Brownian excursions we survey a few recent developments around exploration and mass excursions. One of these is Bertoin’s “tree of alleles with rare mutations” [6], seen as a tree of excursions of Feller’s branching diffusion. Another one is a model of a population with individual reproduction, pairwise fights and emigration to ever new colonies, conceived as a tree of excursions of Feller’s branching diffusion with logistic growth [14]. Finally, we report on a Ray-Knight representation of Feller’s branching diffusion with logistic growth in terms of a reflected Brownian motion whose drift depends on the local time accumulated at its current level [19, 27]. 2010 Mathematics Subject Classification. Primary 60J70; Secondary 60J80, 60J55.The parabolic Anderson model is the Cauchy problem for the heat equation with random potential. It offers a case study for the effects that a random, or irregular, environment can have on a diffusion process. The main focus in the present survey is on phenomena that are due to a highly irregular potential, which we model by a spatially independent, identically distributed random field with heavy tails. Among the effects we discuss are random fluctuations in the growth of the total mass, localisation in the weak and almost sure sense, and ageing.Simple random walk is well understood. However, if we condition a random walk not to intersect itself, so that it is a self-avoiding walk, then it is much more difficult to analyse and many of the important mathematical problems remain unsolved. This paper provides an overview of some of what is known about the critical behaviour of the self-avoiding walk, including some old and some more recent results, using methods that touch on combinatorics, probability, and statistical mechanics. 2010 Mathematics Subject Classification. Primary 60K35, 82B41.

Genealogy of a Wright Fisher model with strong seed bank component

We investigate the behaviour of the genealogy of a Wright-Fisher population model under the influence of a strong seedbank effect. More precisely, we consider a simple seedbank age distribution with two atoms, leading to either classical or long genealogical jumps (the latter modeling the effect of seed-dormancy). We assume that the length of these long jumps scales like a power N β of the population size N, thus giving rise to a ‘strong’ seedbank effect. For a certain range of β, we prove that the ancestral process of a sample of n individuals converges under a non-classical time-scaling to Kingman’s n−coalescent. Further, for a wider range of parameters, we analyze the time to the most recent common ancestor of two individuals analytically and by simulation.