Stochastic Models for the Inference of Life Evolution


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.


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.


You can reach us by email (amaury.lambert - at - or (smile - at -

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Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations

The theoretical investigation of how spatial structure affects the evolution of social behavior has mostly been done under the assumption that parent-offspring strategy transmission is perfect, ie, for genetically transmitted traits, that mutation is very weak or absent. Here, we investigate the evolution of social behavior in structured populations under arbitrary mutation probabilities. We consider populations of fixed size N, structured such that in the absence of selection, all individuals have the same probability of reproducing or dying (neutral reproductive values are the all same). Two types of individuals, A and B, corresponding to two types of social behavior, are competiting; the fidelity of strategy transmission from parent to offspring is tuned by a parameter μ. Social interactions have a direct effect on individual fecundities. Under the assumption of small phenotypic differences (weak selection), we provide a formula for the expected frequency of type A individuals in the population, and deduce conditions for the long-term success of one strategy against another. We then illustrate this result with three common life-cycles (Wright-Fisher, Moran Birth-Death and Moran Death-Birth), and specific population structures (graph-structured populations). Qualitatively, we find that some life-cycles (Moran Birth-Death, Wright-Fisher) prevent the evolution of altruistic behavior, confirming previous results obtained with perfect strategy transmission. We also show that computing the expected frequency of altruists on a regular graph may require knowing more than just the graph{\textquoteright}s size and degree.



Time Reversal Dualities for some Random Forests

We consider a random forest \$$\mathcal{F}^*\$$, defined as a sequence of i.i.d. birth-death (BD) trees, each started at time 0 from a single ancestor, stopped at the first tree having survived up to a fixed time \$$T\$$. We denote by \$$\left(\xi^*_t, 0\leq t \leq T \right)\$$ the population size process associated to this forest, and we prove that if the BD trees are supercritical, then the time-reversed process \$$\left(\xi^*_{T-t}, 0 \leq t \leq T\right)\$$, has the same distribution as \$$\left(\widetilde\xi^*_t, 0 \leq t \leq T\right)\$$, the corresponding population size process of an equally defined forest \$$\widetilde{\mathcal{F}}^*\$$, but where the underlying BD trees are subcritical, obtained by swapping birth and death rates or equivalently, conditioning on ultimate extinction. We generalize this result to splitting trees (i.e. life durations of individuals are not necessarily exponential), provided that the i.i.d. lifetimes of the ancestors have a specific explicit distribution, different from that of their descendants. The results are based on an identity between the contour of these random forests truncated up to \$$T\$$ and the duality property of L\'evy processes. This identity allows us to also derive other useful properties such as the distribution of the population size process conditional on the reconstructed tree of individuals alive at \$$T\$$, which has potential applications in epidemiology.



Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach

Kingman's representation theorem (Kingman 1978) states that any exchangeable partition of \$$\mathbb{N}\$$ can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e.\ ordered partition of \$$\mathbb{N}\$$) can be represented as a paintbox based on an interval-partition (Gnedin 1997. Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a paintbox based on a random non-decreasing process valued in interval-partitions, called nested interval-partition, generalizing the notion of comb metric space introduced by Lambert & Uribe Bravo (2017) to represent compact ultrametric spaces. As a special case, we show that any \$$\Lambda\$$-coalescent can be obtained from a paintbox based on a unique random nested interval partition called \$$\Lambda\$$-comb, which is Markovian with explicit semi-group. This nested interval-partition directly relates to the flow of bridges of Bertoin & Le~Gall (2003). We also display a particularly simple description of the so-called evolving coalescent by a comb-valued Markov process. Next, we prove that any measured ultrametric space \$$U\$$, under mild measure-theoretic assumptions on \$$U\$$, is the leaf set of a tree composed of a separable subtree called the backbone, on which are grafted additional subtrees, which act as star-trees from the standpoint of sampling. Displaying this so-called weak isometry requires us to extend the Gromov-weak topology, that was initially designed for separable metric spaces, to non-separable ultrametric spaces. It allows us to show that for any such ultrametric space \$$U\$$, there is a nested interval-partition which is 1) indistinguishable from \$$U\$$ in the Gromov-weak topology; 2) weakly isometric to \$$U\$$ if \$$U\$$ has complete backbone; 3) isometric to \$$U\$$ if \$$U\$$ is complete and separable.

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