Stochastic Models for the Inference of Life Evolution

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


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.


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 - ; (guillaume.achaz - at - or (smile - at -

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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.



From individual-based epidemic models to McKendrick-von Foerster PDEs: A guide to modeling and inferring COVID-19 dynamics

We present a unifying, tractable approach for studying the spread of viruses causing complex diseases, requiring to be modeled with a large number of types (infective stage, clinical state, risk factor class...). We show that recording for each infected individual her infection age, i.e., the time elapsed since she was infected,
1. The age distribution \$$n(t,a)\$$ of the population at time \$$t\$$ is simply described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick--von Foerster equation;
2. The frequency of type \$$i\$$ at time \$$t\$$ is simply obtained by integrating the probability \$$p(a,i)\$$ of being in state \$$i\$$ at age \$$a\$$ against the age distribution \$$n(t,a)\$$.
The advantage of this approach is three-fold. First, regardless of the number of types, macroscopic observables (e.g., incidence or prevalence of each type) only rely on a one-dimensional PDE ``decorated'' with types. This representation induces a simple methodology based on the McKendrick-von Foerster PDE with Poisson sampling to infer and forecast the epidemic. This technique is illustrated with French data of the COVID-19 epidemic.
Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of ODEs to account for disease complexity. We show that such models can always be rewritten in our framework, thus providing a low-dimensional yet equivalent representation of these complex models.
Third, beyond the simplicity of the approach and its computational advantages, we show that our population model naturally appears as a universal scaling limit of a large class of fully stochastic individual-based epidemic models,
where the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.



Bottlenecks can constrain and channel evolutionary paths

Population bottlenecks are commonplace in experimental evolution, specifically in serial pas- saging experiments where microbial populations alternate between growth and dilution. Natural populations also experience such fluctuations caused by seasonality, resource limitation, or host- to-host transmission for pathogens. Yet, how unlimited growth with periodic bottlenecks influence the adaptation of populations is not fully understood. Here we study theoretically the effects of bottlenecks on the accessibility of evolutionary paths and on the rate of evolution. We model an asexual population evolving on a minimal fitness landscape consisting of two types of beneficial mutations with the empirically supported trade-off between mutation rate and fitness advantage. In the limit of large population sizes and small mutation rates, we show the existence of a unique most likely evolutionary scenario, determined by the size of the wild-type population at the be- ginning and at the end of each cycle. These two key demographic parameters determine which adaptive paths may be taken by the evolving population by controlling the supply of mutants during growth and the loss of mutants at the bottleneck. We do not only show that bottlenecks act as a deterministic control of evolutionary paths but also that each possible evolutionary sce- nario can be forced to occur by tuning demographic parameters. This work unveils the effects of demography on adaptation of periodically bottlenecked populations and can guide the design of evolution experiments.

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