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Asymptotic approach to spatial and dynamical organizations

Periodic Reporting for period 4 - ADORA (Asymptotic approach to spatial and dynamical organizations)

Reporting period: 2022-03-01 to 2023-02-28

Despite the immense progresses made over the last decades in mathematical biology, the multifaceted nature of biological processes still represents an enormous challenge for mathematical modeling.
Technological advances lead to many new experimental observations, which can then be used to improve the accuracy of modeling. As a result, sophisticated mathematical methods have become crucial for addressing the key questions and paradigms in diverse biological systems, for making predictions of the effects of system perturbations and for their control. Examples are numerous and Adora focusses on population of bacteria (as the famous E. coli of medical interest), the mathematical modeling of tissue growth and tumor development, networks of neurones.

Adora focusses on those systems because spatial, social and dynamical organization of large numbers of agents plays an important role for their understanding and control. More precisely, we aim to develop mathematical models, under the form of Partial Differential Equations, able to explain the experimental observations, the qualitative behavior of solutions and to quantify their properties.

Being given the inter-disciplinary dimension with biology of ADORA, the recent COVID-19 pandemic has motivated new research directions in the area of mathematical epidemiology.
A core of mathematical models for biological questions has been developed which address many areas of biology, namely
- Population based Darwinian evolution
- Bacterial colonies self-organization and movement by run and tumble
- Mechanical models of living tissues
- Models for neural assemblies based voltage distribution or elapsed times distribution between spikes

Our approach to Darwinian evolution of large populations is based on integro-differential equations and the selection of the fittest is interpreted as a Dirac concentration of solutions in the trait space. From this perspective, it encompasses a large class of problems which include nonlocal parabolic equations. A remarkable outcome is the use of the Effective Hamiltonian to define the fitness, in particular in the case of the classical problem of the evolution of dispersal. Another new application has been to the selection of the recovery time for cells under stress.

The description of run and tumble movement of bacteria uses the kinetic formalism where the population is described by space, velocity and internal state of the cells. The main questions covered by Adora concern multiscale analysis and the relations between various levels of modeling (macroscopic, kinetic, kinetic with internal states). The emergence of anomalous diffusions is a fascinating issue.

From a mechanical point of view, living tissues can be seen as a porous media flow of cells through the extra-cellular matrix. Departing from a number of models from the literature, our aim has been to unify them and understand how they are connected. Here, the incompressible limit plays an important role and a major discovery is that a free boundary problem describes the tumor edge. The major difficulty is to handle systems when nutrients or chemotaxis are included. Compactness methods based on Aronson-Benilan estimates are used and allow to establish optimal regularity and the so called 'complementarity relation'.

Models for neural assemblies also lead to nonlinear integro-differential equations for the population of neurones structured by voltage, conductances, time elapsed between spikes or a combination of those. In particular voltage-conductance models are degenerate parabolic with specific boundary conditions, and regularity has been established. Also voltage equations have been interpreted as 'macroscopic limits' of voltage-conductance models in the terminology of kinetic equations.

While the main goal was the mathematical understanding of these problems, it is remarkable that they also opened the route for discussions and collaborations with biologists.
Among the most remarkable outcomes, let us mention
- the development of a new class of kinetic model for processing (chemical) information in bacterial movement and their macroscopic limit
- a successful collaboration with biologist on tumor cell movement and initiation of metastases
- new types of estimates for the porous medium equation including Aronson-Benilan estimates in complex situations and a new L4 estimate for the porous media equations.

Unexpectedly, these problems have also lead to develop methods based on the Monge-Kantorovich distance, in particular with the approach of variable doubling. They cover a large class of structured equations from biology including the elapsed time model or the multi-time renewal equation.

These methods and results have always been developed in an international context and they have motivated several teams around the world. They contribute to place biology as one of the main front for applied mathematics nowadays.
Natural network formation along with Cai-Hu model
Spheroid formation of cell aggregates by chemotaxis