Section: New Results

Mathematical analysis of structured branching populations

The investigation of cellular populations at the single-cell level has led to the discovery of important phenomena, such as the co-occurrence of different phenotypes in an isogenic population. Novel experimental techniques, such as time-lapse fluorescence microscopy combined with the use of microfluidic devices (Section 3.2), enable one to take the investigation further by providing time-course profiles of the dynamics of individual cells over entire lineage trees. The development of models that take into account the genealogy of individual cells is an important step in the study of inheritance in bacterial population. As a prerequisite, the efficient analysis of single-cell data relies on the mathematical analysis of those models.

Structured branching processes allow for the study of populations, where the lifecycle of each cell is governed by a given characteristic or trait, such as the concentration of a specific protein inside the cell. The dependence of bacterial phenotypes like cell division times or ageing on such characteristics has been investigated by Aline Marguet using mathematical analysis of the underlying processes. To understand the long-time behavior of structured branching populations, the process describing the trait of a typical individual along its ancestral lineage, called auxiliary process [21] and its asymptotic behavior play a key role. In a publication in ESAIM: Probability and Statistics that appeared this year [20], we proved that the empirical measure of the structured branching process converges to the mean value of this auxiliary process. The approach relies on ergodicity arguments for the time-inhomogeneous auxiliary Markov process. The novelty compared to existing spectral methods is that our method allows to consider processes with time-varying rates for the modeling of changing environments. For example, we studied the case of a size-structured population in a varying environment and proved the convergence of the empirical measure in this specific case.

In collaboration with Charline Smadi (IRSTEA Grenoble), Aline Marguet also investigated the long-time behavior of a general class of branching Markov processes. This work, which has been submitted for publication [27], aims at understanding the link between the dynamic of the trait and the dynamic of the population. In the case of a trait modelling the proliferation of a parasite infection in a cellular population, we exhibit conditions on the dynamics of the parasites to survive in the population, despite the cellular divisions that dilute the number of parasites in each cell.

The study of the asymptotic behavior of general non-conservative semigroups is important for several aspects of branching processes, especially to prove the efficiency of statistical procedures. Vincent Bansaye from École Polytechnique, Bertrand Cloez from INRA Montpellier, Pierre Gabriel from Université Versailles Saint-Quentin, and Aline Marguet obtained necessary and sufficient conditions for uniform exponential contraction in weighted total variation norm of non-conservative semigroups. It ensures the existence of Perron eigenelements and provides quantitative estimates of spectral gaps, complementing Krein-Rutman theorems and generalizing recent results relying on probabilistic approaches. This work was submitted for publication this year [26].