Section: New Results

Direct and inverse Problems in Structured-population equations

Modelling Polymerization Processes

In 2017, we evidenced the presence of several polymeric species by using data assimilation methods to fit experimental data from H. Rezaei's lab  [64]; new experimental evidence reinforced these findings [19], [35]. The challenges are now to propose mathematical models capable of tracking such diversity while keeping sufficient simplicity to be tractable to analysis.

In collaboration with Klemens Fellner from the university of Graz, we propose a new model, variant of the Becker-Döring system but containing two monomeric species, capable of displaying sustained though damped oscillations as is experimentally observed. We also proposed a statistical test to validate or invalidate the presence of oscillations in experimental highly nonstationary signals [55].

Asymptotic behaviour of structured-population equations

Pierre Gabriel and Hugo Martin studied the mathematical properties of a model of cell division structured by two variables – the size and the size increment – in the case of a linear growth rate and a self-similar fragmentation kernel [16]. They first show that one can construct a solution to the related two dimensional eigenproblem associated to the eigenvalue 1 from a solution of a certain one dimensional fixed point problem. Then they prove the existence and uniqueness of this fixed point in the appropriate $L^1$ weighted space under general hypotheses on the division rate. Knowing such an eigenfunction proves useful as a first step in studying the long time asymptotic behaviour of the Cauchy problem.

Etienne Bernard, Marie Doumic and Pierre Gabriel proved in [9] that for the growth-fragmentation equation with fission into two equal parts and linear growth rate, under fairly general assumptions on the division rate, the solution converges towards an oscillatory function, explicitely given by the projection of the initial state on the space generated by the countable set of the dominant eigenvectors of the operator. Despite the lack of hypo-coercivity of the operator, the proof relies on a general relative entropy argument in a convenient weighted $L^2$ space, where well-posedness is obtained via semigroup analysis. They also propose a non-dissipative numerical scheme, able to capture the oscillations.

Pierre Gabriel and Hugo Martin then extended this asymptotic result in the framework of measure solutions [50]. To do so they adopt a duality approach, which is also well suited for proving the well-posedness when the division rate is unbounded. The main difficulty for characterizing the asymptotic behavior is to define the projection onto the subspace of periodic (rescaled) solutions. They achieve this by using the generalized relative entropy structure of the dual problem.

Estimating the division rate from indirect measurements of single cells

Marie Doumic and Adélaïde Olivier

Is it possible to estimate the dependence of a growing and dividing population on a given trait in the case where this trait is not directly accessible by experimental measurements, but making use of measurements of another variable? The article [46] adresses this general question for a very recent and popular model describing bacterial growth, the so-called incremental or adder model - the model studied by Hugo Martin and Pierre Gabriel in [16]. In this model, the division rate depends on the increment of size between birth and division, whereas the most accessible trait is the size itself. We prove that estimating the division rate from size measurements is possible, we state a reconstruction formula in a deterministic and then in a statistical setting, and solve numerically the problem on simulated and experimental data. Though this represents a severely ill-posed inverse problem, our numerical results prove to be satisfactory.