SMILE

Stochastic Models for the Inference of Life Evolution

Presentation

SMILE is an interdisciplinary research group gathering probabilists, statisticians, bio-informaticians and biologists.
SMILE is affiliated to the Stochastics and Biology group of LPSM (Lab of Probability, Statistics and Modeling) at Sorbonne Université (ex Université Pierre et Marie Curie Paris 06).
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.

Recent contributions of the SMILE group related to SARS-Cov2 and COVID-19.

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 - upmc.fr) or (smile - at - listes.upmc.fr).

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Publication

2018

Coagulation-transport equations and the nested coalescents

The nested Kingman coalescent describes the dynamics of particles (called genes) contained in larger components (called species), where pairs of species coalesce at constant rate and pairs of genes coalesce at constant rate provided they lie within the same species. We prove that starting from \$$rn\$$ species, the empirical distribution of species masses (numbers of genes\$$/n\$$) at time \$$t/n\$$ converges as \$$n\to\infty\$$ to a solution of the deterministic coagulation-transport equation $$ \partial_t d \ = \ \partial_x ( \psi d ) \ + \ a(t)\left(d\star d - d \right), $$ where \$$\psi(x) = cx^2\$$, \$$\star\$$ denotes convolution and \$$a(t)= 1/(t+\delta)\$$ with \$$\delta=2/r\$$. The most interesting case when \$$\delta =0\$$ corresponds to an infinite initial number of species. This equation describes the evolution of the distribution of species of mass \$$x\$$, where pairs of species can coalesce and each species' mass evolves like \$$\dot x = -\psi(x)\$$. We provide two natural probabilistic solutions of the latter IPDE and address in detail the case when \$$\delta=0\$$. The first solution is expressed in terms of a branching particle system where particles carry masses behaving as independent continuous-state branching processes. The second one is the law of the solution to the following McKean-Vlasov equation $$ dx_t \ = \ - \psi(x_t) \,dt \ + \ v_t\,\Delta J_t $$ where \$$J\$$ is an inhomogeneous Poisson process with rate \$$1/(t+\delta)\$$ and \$$(v_t; t\geq0)\$$ is a sequence of independent rvs such that \$$\mathcal L(v_t) = \mathcal L(x_t)\$$. We show that there is a unique solution to this equation and we construct this solution with the help of a marked Brownian coalescent point process. When \$$\psi(x)=x^\gamma\$$, we show the existence of a self-similar solution for the PDE which relates when \$$\gamma=2\$$ to the speed of coming down from infinity of the nested Kingman coalescent.

Publication

2019

The species problem from the modeler’s point of view

How to define and delineate species is a long-standing question sometimes called the species problem. In modern systematics, species should be groups of individuals sharing characteristics inherited from a common ancestor which distinguish them from other such groups. A good species definition should thus satisfy the following three desirable properties: (A) Heterotypy between species, (B) Homotypy within species and (E) Exclusivity, or monophyly, of each species. In practice, systematists seek to discover the very traits for which these properties are satisfied, without the a priori knowledge of the traits which have been responsible for differentiation and speciation nor of the true ancestral relationships between individuals. Here to the contrary, we focus on individual-based models of macro-evolution, where both the differentiation process and the population genealogies are explicitly modeled, and we ask: How and when is it possible, with this significant information, to delineate species in a way satisfying most or all of the three desirable properties (A), (B) and (E)? Surprisingly, despite the popularity of this modeling approach in the last two decades, there has been little progress or agreement on answers to this question. We prove that the three desirable properties are not in general satisfied simultaneously, but that any two of them can. We show mathematically the existence of two natural species partitions: the finest partition satisfying (A) and (E) and the coarsest partition satisfying (B) and (E). For each of them, we propose a simple algorithm to build the associated phylogeny. We stress that these two procedures can readily be used at a higher level, namely to cluster species into monophyletic genera. The ways we propose to phrase the species problem and to solve it should further refine models and our understanding of macro-evolution.

Upcoming seminars

Resources

Planning des salles du Collège de France.
Intranet du Collège de France.