SMILE

Stochastic Models for the Inference of Life Evolution

SMILE | Stochastic Models for the Inference of Life Evolution | Collège de France

Presentation

SMILE is an interdisciplinary research group gathering mathematicians, bio-informaticians and biologists.
SMILE is affiliated to the Institut de Biologie de l'ENS, in Paris.
SMILE is hosted within the CIRB (Center for Interdisciplinary Research in Biology) at Collège de France.
SMILE is supported by Collège de France and CNRS.
Visit also our homepage at CIRB.

Directions

SMILE is hosted at Collège de France in the Latin Quarter of Paris. To reach us, go to 11 place Marcelin Berthelot (stations Luxembourg or Saint-Michel on RER B).
Our working spaces are rooms 107, 121 and 122 on first floor of building B1 (ask us for the code). Building B1 is facing you upon exiting the traversing hall behind Champollion's statue.

Contact

You can reach us by email (amaury.lambert - at - college-de-france.fr) ; (guillaume.achaz - at - college-de-france.fr) or (smile - at - listes.upmc.fr).

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Publication

2020

Kingman's coalescent with erosion

Consider the Markov process taking values in the partitions of \$$\mathbb{N}\$$ such that each pair of blocks merges at rate one, and each integer is eroded, i.e., becomes a singleton block, at rate \$$d\$$. This is a special case of exchangeable fragmentation-coalescence process, called Kingman's coalescent with erosion. We provide a new construction of the stationary distribution of this process as a sample from a standard flow of bridges. This allows us to give a representation of the asymptotic frequencies of this stationary distribution in terms of a sequence of hierarchically independent diffusions. Moreover, we introduce a new process called Kingman's coalescent with immigration, where pairs of blocks coalesce at rate one, and new blocks of size one immigrate at rate \$$d\$$. By coupling Kingman's coalescents with erosion and with immigration, we are able to show that the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman's coalescent with erosion to \$$\{1, \dots, n\}\$$ converges to the total progeny of a critical binary branching process.

Publication

2017

Renewal Sequences and Record Chains Related to Multiple Zeta Sums

For the random interval partition of \([0,1]\) generated by the uniform stick-breaking scheme known as GEM\((1)\), let \(u_k\) be the probability that the first \(k\) intervals created by the stick-breaking scheme are also the first \(k\) intervals to be discovered in a process of uniform random sampling of points from \([0,1]\). Then \(u_k\) is a renewal sequence. We prove that \(u_k\) is a rational linear combination of the real numbers \(1, \zeta(2), \ldots, \zeta(k)\) where \(\zeta\) is the Riemann zeta function, and show that \(u_k\) has limit \(1/3\) as \(k \to \infty\). Related results provide probabilistic interpretations of some multiple zeta values in terms of a Markov chain derived from the interval partition. This Markov chain has the structure of a weak record chain. Similar results are given for the GEM\((\theta)\) model, with beta\((1,\theta)\) instead of uniform stick-breaking factors, and for another more algebraic derivation of renewal sequences from the Riemann zeta function.

Sharing

June 4, 2018

Entretien avec Hélène et Amaury

(Movie in french) Entretien avec Hélène Morlon et Amaury Lambert. Mathématicienne de formation, Hélène Morlon est responsable d'une équipe de recherche (Modélisation de la Biodiversité) à l'Institut de Biologie de l'ENS Paris. Amaury Lambert est mathématicien, chercheur au Centre Interdisciplinaire de Recherche en Biologie (Collège de France)

Publication

2015

The reconstructed tree in the lineage-based model of protracted speciation

A popular line of research in evolutionary biology is the use of time-calibrated phylogenies for the inference of diversification processes. This requires computing the likelihood of a given ultrametric tree as the reconstructed tree produced by a given model of diversification. Etienne and Rosindell in Syst Biol 61(2):204–213, (2012) proposed a lineage-based model of diversification, called protracted speciation, where species remain incipient during a random duration before turning good species, and showed that this can explain the slowdown in lineage accumulation observed in real phylogenies. However, they were unable to provide a general likelihood formula. Here, we present a likelihood formula for protracted speciation models, where rates at which species turn good or become extinct can depend both on their age and on time. Our only restrictive assumption is that speciation rate does not depend on species status. Our likelihood formula utilizes a new technique, based on the contour of the phylogenetic tree and first developed by Lambert in Ann Probab 38(1):348–395, (2010). We consider the reconstructed trees spanned by all extant species, by all good extant species, or by all representative species, which are either good extant species or incipient species representative of some good extinct species. Specifically, we prove that each of these trees is a coalescent point process, that is, a planar, ultrametric tree where the coalescence times between two consecutive tips are independent, identically distributed random variables. We characterize the common distribution of these coalescence times in some, biologically meaningful, special cases for which the likelihood reduces to an elegant analytical formula or becomes numerically tractable.

Publication

2016

Testing for Independence between Evolutionary Processes

Evolutionary events co-occurring along phylogenetic trees usually point to complex adaptive phenomena, sometimes implicating epistasis. While a number of methods have been developed to account for co-occurrence of events on the same internal or external branch of an evolutionary tree, there is a need to account for the larger diversity of possible relative positions of events in a tree. Here we propose a method to quantify to what extent two or more evolutionary events are associated on a phylogenetic tree. The method is applicable to any discrete character, like substitutions within a coding sequence or gains/losses of a biological function. Our method uses a general approach to statistically test for significant associations between events along the tree, which encompasses both events inseparable on the same branch, and events genealogically ordered on different branches. It assumes that the phylogeny and themapping of branches is known without errors. We address this problem from the statistical viewpoint by a linear algebra representation of the localization of the evolutionary events on the tree.We compute the full probability distribution of the number of paired events occurring in the same branch or in different branches of the tree, under a null model of independence where each type of event occurs at a constant rate uniformly inthephylogenetic tree. The strengths and weaknesses of themethodare assessed via simulations; we then apply the method to explore the loss of cell motility in intracellular pathogens.

Publication

2017

The genealogical decomposition of a matrix population model with applications to the aggregation of stages

Matrix projection models are a central tool in many areas of population biology. In most applications, one starts from the projection matrix to quantify the asymptotic growth rate of the population (the dominant eigenvalue), the stable stage distribution, and the reproductive values (the dominant right and left eigenvectors, respectively). Any primitive projection matrix also has an associated ergodic Markov chain that contains information about the genealogy of the population. In this paper, we show that these facts can be used to specify any matrix population model as a triple consisting of the ergodic Markov matrix, the dominant eigenvalue and one of the corresponding eigenvectors. This decomposition of the projection matrix separates properties associated with lineages from those associated with individuals. It also clarifies the relationships between many quantities commonly used to describe such models, including the relationship between eigenvalue sensitivities and elasticities. We illustrate the utility of such a decomposition by introducing a new method for aggregating classes in a matrix population models to produce a simpler model with a smaller number of classes. Unlike the standard method, our method has the advantage of preserving reproductive values and elasticities. It also has conceptually satisfying properties such as commuting with changes of units.

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