Section: New Results

Modelling and analysis of cellular trait dynamics over lineage trees

The investigation of cellular populations at a single-cell level has already led to the discovery of important phenomena, such as the occurrence of different phenotypes in an isogenic population. Nowadays, several experimental techniques, such as microscopy combined with the use of microfluidic devices, enable one to take investigation further by providing time-profiles of the dynamics of individual cells over entire lineage trees. The difficulty, and at the same time the opportunity, in exploiting these data and inferring mathematical models from them is the fact that the behavior of different cells is correlated because of inheritance.

From the modelling point of view, lineage trees are well described by structured branching population models where the life cycle of each cell depends on individual characteristics, such as size and internal protein dynamics, which play a key role in the mechanisms of cell division. One important aspect in the analysis of these population models consists in the investigation of biases arising from the sampling of a finite set of observed individuals. In order to characterize bias, we studied the dynamics of a structured branching population where the trait of each individual evolves in accordance with a Markov process. We assumed that the rate of division of each individual is a function of its trait and when a branching event occurs, the trait of the descendants at birth depends on their number and on the trait of the mother. We explicitly described the Markov process, named auxiliary process, corresponding to the dynamics of the trait of a "typical" individual by deriving its associated infinitesimal generator. In particular, we proved that this process characterizes exactly the process of the trait of a uniformly sampled individual in a large population approximation. This work, carried out by Aline Marguet, has been accepted for publication in the journal Bernoulli [19].

We also investigated the long-time behavior of the population and proved that a typical individual in the population asymptotically behaves like the auxiliary process previously introduced. These results have been submitted for publication [22]. Structured branching processes and their analysis also provide the basis for identification tools for lineage-tree data. In particular, in the context of a bifurcating Markov chain, where each individual is characterized by a trait evolving in accordance with a scalar diffusion, we proved that the maximum-likelihood estimator of the division rate is asymptotically efficient and demonstrate the method on simulated data. This work, in collaboration with M. Hoffmann at Univ Paris-Dauphine, has also been submitted for publication [21].

Along the same lines, modelling and identification of gene expression models with mother-daughter inheritance are being investigated in the context of the ANR project MEMIP. Starting from an earlier work of the group [7], with reference to an application on osmotic shock response by yeast, the key question is to what extent leveraging an inheritance model improves inference of individual cell dynamics as well as of inheritance dynamics themselves, relative to state-of-art approaches where inheritance is not accounted for at a modelling stage.