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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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.



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.



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.

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