Section: Research Program

PDE analysis and simulation

PDEs arise at several levels of our models. Parabolic equations can be used for large cell populations and also for intracellular spatio-temporal dynamics of proteins and their messenger RNAs in gene regulatory networks, transport equations are used for protein aggregation / fragmentation models and for the cell division cycle in age-structured models of proliferating cell populations. Existence, uniqueness and asymptotic behaviour of solutions have been studied (B. Perthame, Transport equations in biology, Springer, 2007) [67] [65] . Other equations, of the integro-differential type, dedicated to describing the Darwinian evolution of a cell population according to a phenotypic trait, allowing exchanges with the environment, genetic mutations and reversible epigenetic modifications, are also used [82] , [83] , [81] [23] . Through multiscale analysis, they can be related to stochastic and free boundary models used in cancer modelling.

Inverse problems

When studying biological populations (usually cells or big molecules) using PDE models, identification of the functions and parameters that govern the dynamics of a model may be achieved to a certain extent by statistics performed on individuals to reconstruct the probability distribution of their relevant characteristics in the population they constitute, but quantitative observations at the individual level (e.g., fluorescence in single cells [63] or size/age tracking  [89] ) require sophisticated techniques and are most often difficult to obtain. Relying on the accuracy of a PDE model to describe the population dynamics, inverse problem methods offer a tractable alternative in model identification, and they are presently an active theme of research in MAMBA. Following previous studies  [70] , [71] , some combining statistical and deterministic approaches  [69] with application to raw experimental data [13] , we plan to develop our methods to new structured-population models (or stochastic fragmentation processes as in  [68] ), useful for other types of data or populations (e.g. size/age tracking, polymer length distribution, fluorescence in single cells).