YEP VI 2009 (Young European Probabilists) workshops
"Fragmentation, coalescence and probabilistic genetics"
March 23-27, 2009
EURANDOM, Eindhoven, The Netherlands
Joint work with Viet Chi Tran
Continuous time model for parasite infection with cell division
We are interested by the proliferation of parasites in dividing cells in continuous time. The quantity of parasites in a cell follows a Feller diffusion. When the cell divides, a random fraction of them goes in the first daughter cell, whereas the rest goes in the other one. We consider the case when the rate of division is constant but also when it increases when the quantity of parasites increases. We are interested by the asymptotic behaviours of the quantity of parasites in a random cell line and the proportion of infected cells. This model can be seen as a fragmentation where the mass evolves as a Feller diffusion and splits in two fragments at a rate which may depend on its mass.
Joint work with Christina Goldschmidt)
The allele frequency spectrum associated with the Bolthausen-Sznitman coalescent
In this talk, we investigate the distribution of number of individuals in an haploid population carrying the same genotype. The genealogy of such a population can be seen as a coalescent process. We assume that individuals are subject to neutral mutations at some constant rate. The allelic partition is then defined by grouping together individuals with the same genotype. We describe here some asymptotics of this partition when the size of the population tends to infinity and when the underlying coalescent process is the Bolthausen-Sznitman coalescent. In particular, we consider, for any integer , the number of blocks of size of this partition and prove that, for this model, singletons scale differently from larger blocks.
Coalescent processes and population genetics
Coalescent theory is the study
of random processes where particles may join each other to form clusters as time
evolves. This course provides
an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and (time permitting) 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.
Joint work with Matthias Steinrücken
Lambda-coalescents and population genetic inference
We discuss applications of multiple merger coalescents as models for the genealogies in natural populations, ensuing statistical questions and some computational answers, illustrated by datasets from Atlantic cod.
(Joint work with R. Neininger, M. Krawczak, U. Rösler)
The length of external branches in coalescence trees
We study the asymptotic distribution of the length of a randomly chosen external branch of a Kingman coalescent. Let denote the length of such a branch. Based on a recursion for the distribution of , we show that converges in distribution, as , to a non-negative random variable with Lebesgue density
This asymptotic distribution allows the study of the time to the most recent common ancestor of a randomly chosen individual and its closest relative in the population. It therefore is a measure of the diversity within a population.
Modelling Protein Translocation: A Brownian Ratchet
Motivated by protein translocation models we consider a Brownian ratchet. It is defined as a reflecting Brownian motion with moving reflection boundary given by a non-decreasing jump process . At rate proportional to a new boundary is chosen uniformly in the interval . In the talk we outline the proof of the law of large numbers and the central limit theorem for the Brownian ratchet.
On the number of allelic types for samples taken from coalescents with mutation
Let denote the number of types of a sample of size taken from an exchangeable coalescent process (-Coalescent) with mutation rate . It is shown that fulfills a distributional recursion. If the coalescent does not have proper frequencies, then in probability, where, for , is the fraction of singletons present in the coalescent at time .
1. Regenerative partitions: the two-parameter family and beyond
Regenerative partitions appear naturally in various contexts, including data structures and the decomposition of fragmentation trees. An exchangeable partition of the infinite set of integers is regenerative if its restrictions on the finite sets are consistent with respect to some operation of deleting a block. This property, introduced in joint work with Jim Pitman, generalises the species non-interference property of Ewens’ partitions, in which case the deleted block is the size-biased pick. Partitions of the larger two-parameter family (with positive values of the parameters) also possess the regeneration property, albeit with respect to less obvious deletion algorithms. In general, the regeneration property allows to introduce an intrinsic order on the collection of blocks and to relate the partition, in essentially unique way, to a random discrete distribution obtained by the exponential transformation of some increasing Lévy process.
2. A coupling method for exchangeable coalescents
A class of -coalescents with the characteristic measure having finite moment of order is considered. The majority of collisions in the process starting with particles involve some of these primary particles, while relatively few collisions merge only secondary particles. The evolution of primary particles is relatively simple, and can be analysed by the methods developed for regenerative compositions, in some cases by means of the renewal theory. An explicit coupling with compositions is constructed and exploited to approximate the number of collisions and the absorption time of coalescents.
Measures for the exceptionality of gene order in conserved genomic regions
The goal of this work is to find "good" measures for quantifying the exceptionality of the order of the orthologs in conserved genomic regions between two different species. Here "good" means both biologically relevant and computationally accessible. We propose three measures based on the transposition distance in the permutation group for measuring the gene order conservation in orthologous gene clusters found by the reference region approach. We obtain analytic expressions for their distribution in the case of a random uniform permutation, i.e. under the null hypothesis of random gene order. Our results can be used to increase the power of the significance tests for gene clusters which take into account only the proximity of the orthologous genes, and not their order.
Self-similar fragmentations and random real trees
This mini-course will start with an introduction to the theory of self-similar fragmentation processes, as developed by Bertoin in a series of papers in 2001-2003. These models are used to describe the evolution of systems of particles that undergo splitting, so that each particle evolves independently of the others, with a splitting rate proportional to a power of its mass. There is a natural tree-like structure behind such fragmentation processes, which can be formalized rigorously using the concept of random real trees.
Random real trees have been extensively studied in the last fifteen years, starting at the beginning of the nineties with the work of Aldous on the Brownian continuum random tree. We will introduce these objects in a general framework and then focus on random real trees coding the genealogy of self-similar fragmentation processes. We will, in particular, study some fractal and symmetry properties of these fragmentation trees.
Last, we will show that a large class of fragmentation trees arises as scaling limits of natural families of finite trees possessing a branching property.
Nabin Kumar Jana
Large deviation in random energy models: some new problems
The branching process as well as the coalescent process play a vital role in Derrida's generalized random energy model. In this talk, we discuss how the large deviation techniques brings some more problems in this model.
Strong law of large numbers for fragmentation processes with immigration
In the spirit of a classical result for Crump-Mode-Jagers processes, we prove a strong law of large numbers for fragmentation processes. Specifically, for self-similar fragmentation processes we prove the almost sure convergence of an empirical measure associated with the stopping line corresponding to first fragments of size strictly smaller than for . Moreover, we extend this result to fragmentation processes with immigration.
Entropic repulsion for a Gaussian membrane model
We consider a model for a random interface, or semiflexible membrane, which is defined as the centered Gaussian field on the d-dimensional integer lattice, with covariances given by the Green’s function of the discrete Bilaplacian. Our main aim is to discuss the effect of a "forbidden region" on the interface. We will show that the constraint for the field to take only positive values in a domain of side-length N results in an effect of "entropic repulsion", which means that the average height of the interface above the reference hyperplane is of order in the supercritical dimensions, and of order in the critical dimension. To obtain these results, a detailed analysis of the decay of the covariances of the field is necessary, which uses a random walk representation. In the critical dimension (), the model displays a certain hierarchical structure, which explains the difference between critical and supercritical dimensions concerning the repulsion height.
Evolution of the Ancestral Recombination Graph along the genome in case of selective sweep
We consider the genome of a sample of n individuals taken at the end of a selective sweep, which is the fixation of an advantageous allele in the population. When the selective advantage is high, the genealogy at a locus under selective sweep can be approximated by a comb with n teeth. However, because of recombinations that occur during the selective sweep, the effect of selection decreases as the distance from the selected site increases, so that far from this locus, the tree is a Kingman coalescent tree, as in the neutral case. We first give the distribution of the tree at a given locus. Then we focus on the evolution of this tree along the genome. Since this tree-valued process is not Markovian, we study the Ancestral Recombination Graph. We finally obtain its evolution along the genome in case of selective sweep.
Joint work with Peter Moerters.
The minimal supporting tree for the free energy of a random polymer model
We consider the model of directed polymers in random environment, where a polymer modelled as the path of a simple random walk on a lattice interacts with a random environment given by independent, identically distributed random variables. Under certain conditions on the spatial dimension and the distribution of the environment, this model exhibits a phase transition. In the weak disorder regime, we can identify the minimal set of paths that supports the free energy. We compare the situation on the lattice with a mean field model defined on a regular tree, where we can also show that in the strong disorder regime a single path suffices to support the free energy.
Random walk on discrete point process
We will describe a model of random walk on random environment on a random subset of , with uniform transition probabilities on 2d points, two "nearest neighbors" in each of the d coordinate directions. We will than show that the velocity of the walk is almost surely 0, and will discuss transience and recurrence classifications. Finally we will present a conjecture concerning CLT theorem for such random walks.
On the representation of Fleming-Viot models from a Bayesian perspective
We review some recent contributions on particle processes for measure-valued diffusions of Fleming-Viot type which shed new light on the links between population genetics and Bayesian nonparametric theory. Special emphasis will be given to the construction of time-dependent random probability measures by means of generalised Polya urn schemes and Gibbs sampling procedures. This approach yields intuitive representations of Fleming-Viot processes yet allowing explicit specification of various evolutionary mechanisms for the individual dynamics.
Branching random walks : selection, survival and genealogies
I will give an overview of the results physicists can have about branching random walks. More precisely, I will consider the effect of absorbing boundaries on the survival of a population of branching random walks and study the phase transition to the absorbing state. In a second part, I will consider other types of selection and characterize the genealogies of the population for each of them. Links will be made with other problems of statistical mechanics such as directed polymers in random media.
The two oldest families in population genetics models
Some of population genetics models (as well as Moran model) allow us to divide the actual population in two families, generated by the two children of the most recent common ancestor. We study the evolution in time of frequencies of these two families, infinite (Moran model) or infinite (obtained as rescaled Wright-Fisher models) populations. Results about stationarity will be given.
Spatial Lambda-Fleming-Viot process and associated genealogies
A spatial Lambda-Fleming-Viot process models a population distributed on ( in practice) which experiences local partial extinctions, followed by the recolonization of the affected area by the offspring of an individual presentin this region. After introducing the model and some of its properties, we will consider the genealogy of a sample of individuals when the population lives on a two-dimensional torus of size . In particular, we will see that this ancestral process converges as grows to infinity towards a coalescent with multiple mergers, whose characteristics depends on the range of the extinction areas (joint work with Alison Etheridge).
On the number of collisions in certain -coalescents with heavy mass around zero.
The results of Gnedin and Yakubovich (2007) about the number of collisions in the -coalescent are extended to some family of -coalescents. Namely we consider the case when only simultaneous collisions are possible and impose further restrictions on the measure supposing that and are -independent. Then under certain regularity conditions we show that the number of simultaneous multiple collisions in a proper scaling approaches stable limit.
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