Universal tree structures in directed polymers and models of evolving populations

Brunet, Éric; Derrida, Bernard; Simon, Damien

doi:10.1103/physreve.78.061102

Cited by 23 publications

(34 citation statements)

References 49 publications

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“…A particular -coalescent, usually called the Bolthausen-Sznitman coalescent, is thought to be an important object for describing the conjectured universal ultrametric structure of numerous mean-field spin glass models including the Sherrington-Kirkpatrick model (see [12,13,38]). The same coalescent has also been recently linked in [15] to scaling limits of directed polymers.…”

Section: Motivation and Main Resultsmentioning

confidence: 87%

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Global divergence of spatial coalescents

Angel

Berestycki

Limic

2010

Probab. Theory Relat. Fields

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We study several fundamental properties of a class of stochastic processes called spatial -coalescents. In these models, a number of particles perform independent random walks on some underlying graph G. In addition, particles on the same vertex merge randomly according to a given coalescing mechanism. A remarkable property of mean-field coalescent processes is that they may come down from infinity, meaning that, starting with an infinite number of particles, only a finite number remains after any positive amount of time, almost surely. We show here however that, in the spatial setting, on any infinite and bounded-degree graph, the total number of particles will always remain infinite at all times, almost surely. Moreover, if G = Z d , and the coalescing mechanism is Kingman's coalescent, then starting with N particles at the origin, the total number of particles remaining is of order (log * N ) d at any fixed positive time (where log * is the inverse tower function). At sufficiently large times the total number of particles is of order (log * N ) d−2 , when d > 2. We provide parallel results in the recurrent case d = 2. The spatial Beta-coalescents behave similarly, where log log N is replacing log * N .

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Section: Motivation and Main Resultsmentioning

confidence: 87%

“…Recall the martingales (15). On the event As in (25), Doob's maximal inequality yields that for large enough n…”

Section: Lemma 52 For Any T > 0 We Have That Z (T) − → ∞ Almost Surementioning

confidence: 99%

Global divergence of spatial coalescents

Angel

Berestycki

Limic

2010

Probab. Theory Relat. Fields

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“…These rates satisfy a self-consistency relation l b;k ¼ l bþ1;k þ l bþ1;kþ1 (see Pitman 1999;Sagitov 1999;Schweinsberg 2000;Brunet et al 2008). For the Moran model, we have for all b $ 2;…”

Section: B Coalescence Ratesmentioning

confidence: 99%

Correlated Mutations and Homologous Recombination Within Bacterial Populations

Lin

Kussell

2017

Genetics

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Inferring the rate of homologous recombination within a bacterial population remains a key challenge in quantifying the basic parameters of bacterial evolution. Due to the high sequence similarity within a clonal population, and unique aspects of bacterial DNA transfer processes, detecting recombination events based on phylogenetic reconstruction is often difficult, and estimating recombination rates using coalescent model-based methods is computationally expensive, and often infeasible for large sequencing data sets. Here, we present an efficient solution by introducing a set of mutational correlation functions computed using pairwise sequence comparison, which characterize various facets of bacterial recombination. We provide analytical expressions for these functions, which precisely recapitulate simulation results of neutral and adapting populations under different coalescent models. We used these to fit correlation functions measured at synonymous substitutions using whole-genome data on Escherichia coli and Streptococcus pneumoniae populations. We calculated and corrected for the effect of sample selection bias, i.e., the uneven sampling of individuals from natural microbial populations that exists in most datasets. Our method is fast and efficient, and does not employ phylogenetic inference or other computationally intensive numerics. By simply fitting analytical forms to measurements from sequence data, we show that recombination rates can be inferred, and the relative ages of different samples can be estimated. Our approach, which is based on population genetic modeling, is broadly applicable to a wide variety of data, and its computational efficiency makes it particularly attractive for use in the analysis of large sequencing datasets.KEYWORDS bacteria; homologous recombination; population diversity; sample selection bias; sample ages; adapting populations; Bolthausen-Sznitman coalescent B ACTERIA can receive DNA fragments from their environment by different mechanisms, and integrate them into their genome in a set of processes collectively known as horizontal gene transfer (HGT) (Thomas and Nielsen 2005). While the importance of HGT in bacterial evolution is increasingly appreciated (Koonin and Wolf 2008;Shapiro et al. 2012;Oren et al. 2014;Ravenhall et al. 2015;Rosen et al. 2015), quantifying its impact across bacterial genomes, and in diverse environmental samples, remains a key challenge (Maynard Smith 1991;Soucy et al. 2015). In particular, a prevalent form of HGT involves homologous recombination of fragments that bear a high degree of sequence similarity to the recipient genome (Andam and Gogarten 2011;Williams et al. 2012). Such transfer events are particularly difficult to detect, since they leave no obvious marks, and are indistinguishable based on nucleotide composition, yet they are likely to represent the majority of HGT events in bacteria (Fraser et al. 2007).Three major mechanisms of HGT-transformation, conjugation, and transduction-mediate the passage of external DNA into bacte...

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“…1. BBM is a prototypical model of evolution, but has also been extensively used as a simple model for reaction-diffusion systems, disordered systems, nuclear reactions, cosmic ray showers, epidemic spreads amongst others [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. In one dimension, the position of the existing particles at time t constitute a set of strongly correlated variables that are naturally ordered according to their positions on the line with x 1 (t) > x 2 (t) > x 3 (t) .…”

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confidence: 99%

Universal Order and Gap Statistics of Critical Branching Brownian Motion

Ramola¹,

Majumdar²,

Schehr³

2014

Phys. Rev. Lett.

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We study the order statistics of one dimensional branching Brownian motion in which particles either diffuse (with diffusion constant D), die (with rate d) or split into two particles (with rate b). At the critical point b = d which we focus on, we show that, at large time t, the particles are collectively bunched together. We find indeed that there are two length scales in the system: (i) the diffusive length scale ∼ √ Dt which controls the collective fluctuations of the whole bunch and (ii) the length scale of the gap between the bunched particles ∼ D/b. We compute the probability distribution function P (g k , t|n) of the kth gap g k = x k −x k+1 between the kth and (k +1)th particles given that the system contains exactly n > k particles at time t. We show that at large t, it converges to a stationary distribution The statistics of the global maximum of a set of random variables finds applications in several fields including physics, engineering, finance and geology [1] and the study of such extreme value statistics (EVS) has been growing in prominence in recent years [2][3][4][5][6][7]. In many real world examples where EVS is important, the maximum is not independent of the rest of the set and there are strong correlations between near-extreme values. Examples can be found in meteorology where extreme temperatures are usually part of a heat or cold wave [8] and in earthquakes and financial crashes where extreme fluctuations are accompanied by foreshocks and aftershocks [9][10][11][12]. Nearextreme statistics also play a vital role in the physics of disordered systems where energy levels near the ground state become important at low but finite temperature [4]. In this context, the distribution of the kth maximum x k of an ordered set {x 1 > x 2 > x 3 ...} (order statistics [13]) and the gap between successive maxima g k = x k − x k+1 provides valuable information about the statistics near the extreme value. Such near-extreme distributions have recently been of interest in statistics [14] and physics [15,[17][18][19]. Although the order and gap statistics of independent identically distributed (i.i.d.) variables are fully understood [13], very few exact analytical results exist for strongly correlated random variables. In this context, random walks and Brownian motion offer a fertile arena where near-extreme distributions for correlated variables can be computed analytically [16][17][18][19].Another interesting system where order statistics plays an important role is the branching Brownian motion (BBM). In BBM, a single particle starts initially at the origin. Subsequently, in a small time interval dt, the particle splits into two independent offsprings with probability b dt, dies with probability d dt and with the remaining probability (1 − (b + d) dt) it diffuses with diffusion constant D. A typical realization of this process is shown in Fig. 1. BBM is a prototypical model of evolution, but has also been extensively used as a simple model for reaction-diffusion systems, disordered systems, nuclear reactio...

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Universal tree structures in directed polymers and models of evolving populations

Abstract: By measuring or calculating coalescence times for several models of coalescence or evolution, with and without selection, we show that the ratios of these coalescence times become universal in the large size limit and we identify a few universality classes.

Cited by 23 publications

References 49 publications

Global divergence of spatial coalescents

Global divergence of spatial coalescents

Correlated Mutations and Homologous Recombination Within Bacterial Populations

Universal Order and Gap Statistics of Critical Branching Brownian Motion

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