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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 48 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Interpreting Λcoalescent speed of coming down from infinity via particle representation of superprocesses
 In preparation
, 2008
"... Consider a Λcoalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number Nt of blocks at any positive time t> 0). We exhibit a deterministic function v: (0,∞) → (0,∞), such that Nt/v(t) → 1, almost ..."
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Cited by 29 (9 self)
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Consider a Λcoalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number Nt of blocks at any positive time t> 0). We exhibit a deterministic function v: (0,∞) → (0,∞), such that Nt/v(t) → 1, almost surely and in Lp for any p ≥ 1, as t → 0. Our approach relies on a novel martingale technique.
A coalescent model for the effect of advantageous mutations on the genealogy of a population
, 2008
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Random partitions approximating the coalescence of lineages during a selective sweep
, 2005
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The stepping stone model, II: Genealogies and the infinite sites model, submitted
, 2005
"... This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the twodimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman’s coalescent. With DNA sequence data in mind, we inves ..."
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Cited by 17 (3 self)
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This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the twodimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman’s coalescent. With DNA sequence data in mind, we investigate mutation patterns under the infinite sites model, which assumes that each mutation occurs at a new site. Our results suggest that the spatial structure of the human population contributes to the haplotype structure and a slower than expected decay of genetic correlation with distance revealed by recent studies of the human genome. 1. Introduction. Sequencing
Splitting trees with neutral Poissonian mutations II: Large or old families
, 2010
"... We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a split ..."
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Cited by 16 (3 self)
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We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree [8], and the population counting process (Nt; t ≥ 0) is a homogeneous, binary Crump–Mode–Jagers process. We assume that individuals independently experience mutations at constant rate θ during their lifetimes, under the infinitealleles assumption: each mutation instantaneously confers a brand new type, called allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e., the number A(t) of distinct alleles represented in the population at time t, and more specifically, the numbers A(k, t) of alleles represented by k individuals at time t, k = 1, 2,..., Nt. We mainly use two classes of tools: coalescent point processes, as defined in [14], and branching processes counted by random characteristics, as defined in [10, 11]. We provide explicit formulae for the expectation of A(k, t) conditional on population size in a coalescent point process, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k, t)/Nt and of A(t)/Nt thanks to random characteristics, in the same vein as in [18]. Last, we separately compute the expected homozygosity by applying a method introduced in [13], characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations.
The allelic partition for coalescent point processes
, 2008
"... Assume that individuals alive at time t in some population can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d. The ranked sequence of these branches is called a coalescent point process. We have shown in a previous work [14] that splitting trees are impo ..."
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Assume that individuals alive at time t in some population can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d. The ranked sequence of these branches is called a coalescent point process. We have shown in a previous work [14] that splitting trees are important instances of such populations. Here, individuals are given DNA sequences, and for a sample of n DNA sequences belonging to distinct individuals, we consider the number Sn of polymorphic sites (sites at which at least two sequences differ), and the number An of distinct haplotypes (sequences differing at one site at least). It is standard to assume that mutations arrive at constant rate (on germ lines), and never hit the same site on the DNA sequence. We study the mutation pattern associated with coalescent point processes under this assumption. Here, Sn and An grow linearly as n grows, with explicit rate. However, when the branch lengths have infinite expectation, Sn grows more rapidly, e.g. as n ln(n) for critical birth–death processes. Then, we study the frequency spectrum of the sample, that is, the numbers of polymorphic sites/haplotypes carried by k individuals in the sample. These numbers are shown to grow also linearly with sample size, and we provide simple explicit formulae for mutation frequencies and haplotype frequencies. For critical birth–death processes, mutation frequencies are given by the harmonic series and haplotype frequencies by Fisher’s logarithmic series. Running head. The allelic partition for coalescent point processes.
Deterministic and stochastic regimes of asexual evolution . . .
, 2008
"... We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length L. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is determin ..."
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Cited by 15 (6 self)
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We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length L. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance deff upto which the population can spread in genotype space. For deff = L, the deterministic quasispecies theory operates while for deff < 1, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasispecies theory below a crossover time T × while above T×, the population gets trapped at a local fitness peak and manages to find a better peak either via stochastic tunneling or double mutations. In the stochastic regime deff < 1, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interepretation of evolution experiments with microbial populations.
Biological applications of the theory of birthanddeath processes
 Briefings Bioinfo
, 2005
"... In this review, we discuss applications of the theory of birthanddeath processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described. Birthanddeath processes, with some straightforward additions suc ..."
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In this review, we discuss applications of the theory of birthanddeath processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described. Birthanddeath processes, with some straightforward additions such as innovation, are a simple, natural and formal framework for modeling a vast variety of biological processes such as population dynamics, speciation, genome evolution, including growth of paralogous gene families and horizontal gene transfer and somatic evolution of cancers. We further describe how empirical data, e.g. distributions of paralogous gene family size, can be used to choose the model that best reflects the actual course of evolution among different versions of birthdeathandinnovation models. We conclude that birthanddeath processes, thanks to their mathematical transparency, flexibility and relevance to fundamental biological processes, are going to be an indispensable mathematical tool for the burgeoning field of systems biology.