The MAMBA (Modelling and Analysis in Medical and Biological Applications) team is the continuation of the BANG (Biophysics, Numerical Analysis and Geophysics) team, which itself was a continuation of the former project-team M3N. Historically, the BANG team, headed by Benoît Perthame during 11 years (2003-2013), has developed models, simulations and numerical algorithms for problems involving dynamics of Partial Differential Equations (PDEs).

The dynamics of complex physical or biophysical phenomena involves many agents, e.g. proteins or cells. The latter can be seen as active agents. Mathematically, agents can be represented either explicitly as individuals with their dynamics modelled e.g. through branching trees and piecewise deterministic Markov processes (PDMP), or as deterministic or stochastic differential equations, or under certain conditions be grouped or locally averaged, in which case their dynamics is mimicked by Ordinary or Partial Differential Equations (ODEs/PDEs).

Biology and medicine presently face the difficulty to make sense of the data newly available by means of recent signal acquisition methods and to take appropriate actions through possible treatment pathways. Modeling through agent-based or continuous models is a unique way to explain (model) experimental or clinical observations and then compute, control and predict the consequences of the mechanisms under study. These are the overall goals of Mamba.

Data and image analysis, statistical, ODEs, PDEs, and agent-based approaches are used either individually or in combination, with a strong focus on PDE analysis and agent-based approaches. Mamba was created in January 2014. It aims at developing models, simulations, numerical and control algorithms to solve questions from life sciences involving dynamics of phenomena encountered in biological systems such as protein intra-cellular spatio-temporal dynamics, cell motion, early embryonic development, multicellular growth, wound healing and liver regeneration, cancer evolution, healthy and tumor growth control by pharmaceuticals, protein polymerization occurring in neurodegenerative disorders, control of dengue epidemics, etc.

Another guideline of our project is to remain close to the most recent questions of experimental biology or medicine. In this context, we develop many close and fruitful collaborations with biologists and physicians.

We focus mainly on the creation, investigation and transfer of new mathematical models, methods of analysis and control, and numerical algorithms, but in selected cases software development as that of CellSys and TiQuant by D. Drasdo and S. Hoehme is performed. More frequently, the team develops “proof of concept” numerical codes in order to test the adequacy of our models to experimental biology.

We have organized the presentation of our research program in three methodological axes (Subsections , and ) and two application axes (Subsections and ). Evolving along their own logic in close interaction with the methodological axes, the application axes are considered as application-driven research axes in themselves. The methodological research axes are the following.

Axis 1
is devoted to work in physiologically-based design, analysis and control of population dynamics. It encompasses populations of bacteria, of yeasts, of cancer cells, of neurons, of aggregating proteins, etc. whose dynamics are represented by partial differential equations (PDEs), structured in evolving physiological traits, such as age, size, size-increment, time elapsed since last firing (neurons).

Axis 2
is devoted to reaction equations and motion equations of agents in living systems. It aims at describing biological phenomena such as tumor growth, chemotaxis and wound healing.

Axis 3
tackles questions of model and parameter identification, combining stochastic and deterministic approaches and inverse problem methods in nonlocal and multi-scale models.

Population dynamics is a field with varied and wide applications, many of them being in the core of MAMBA interests - cancer, bacterial growth, protein aggregation. Their theoretical study also brings a qualitative understanding on the interplay between individual growth, propagation and reproduction in such populations. In the past decades, many results where obtained in the BANG team on the asymptotic and qualitative behavior of such structured population equations, see e.g. , , , . Other Inria teams interested by this domain are Mycenae, Numed and Dracula, with which we are in close contacts. Among the leaders of the domain abroad, we can cite among others our colleagues Graeme Wake (New Zealand), Glenn Webb (USA), Jacek Banasiak (South Africa), Odo Diekmann (Netherlands), with whom we are also in regular contact. Most remarkably and recently, connections have also been made with probabilists working on Piecewise Deterministic Markov Processes (F. Malrieu at the university of Rennes, Jean Bertoin at the ETH in Zurich, Vincent Bansaye at Ecole Polytechnique, Julien Berestycki at Cambridge, Amaury Lambert at College de France, M. Hoffmann at Paris Dauphine, Alex Watson in UCL, London and J. Bertoin in Zurich), leading to a better understanding of the links between both types of results – see also the Methodological axis 3.

We divide this research axis, which relies on the study of structured population equations, according to four different applications, bringing their own mathematical questions, e.g., stability, control, or blow-up.

Time asymptotics for nucleation, growth and division equations

Following the many results obtained in the BANG team on the asymptotic and qualitative behavior of structured population equation, we put our effort on the investigation of limit cases, where the trend to a steady state or to a steady exponential growth described by the first eigenvector fails to happen. In , the case of equal mitosis (division into two equally-sized offspring) with linear growth rate was studied, and strangely enough, it appeared that the general relative entropy method could also be adapted to such a non-dissipative case. Many discussions and common workshops with probabilists, especially through the ANR project PIECE coordinated by F. Malrieu, have led both communities to work closer.

We also enriched the models by taking into account a nucleation term, modeling the spontaneous formation of large polymers out of monomers . We investigated the interplay between four processes: nucleation, polymerization, depolymerization and fragmentation.

New perspectives are now to consider not only one species but several interacting ones, which may exhibit complex interplays which may lead to damped oscillations or to infinite growth; these are in collaboration with C. Schmeiser and within the Vienna associated team MaMoCeMa (J. Delacour's Ph.D) and with K. Fellner from Graz (M. Mezache's Ph.D).

Cell population dynamics and its control

One of the important incentives for such model design, source of many theoretical works, is the challenging question of drug-induced drug resistance in cancer cell populations, described in more detail below in the Applicative axis 1, Cancer. The adaptive dynamics setting used consists of phenotype-structured integro-differential [or reaction-diffusion, when phenotype instability is added under the form of a Laplacian] equations describing the dynamic behavior of different cell populations interacting in a Lotka-Volterra-like manner that represents common growth limitation due to scarcity of expansion space and nutrients. The phenotype structure allows us to analyse the evolution in phenotypic traits of the populations under study and its asymptotics for two populations , , , . Space may be added as a complementary structure variable provided that something is known of the (Cartesian) geometry of the population , which is seldom the case. A recent review of mathematical methods aiming at improving cancer treatments has been published in Physics of Life Reviews .

Modelling Mendelian and non-Mendelian inheritances in density-dependent population dynamics

Classical strategies for controlling mosquitoes responsible of vector-borne disease are based on mechanical methods, such as elimination of oviposition sites; and chemical methods, such as insecticide spraying.
Long term usage of the latter generates resistance , , transmitted to progeny according to Mendelian inheritance (in which each parent contributes randomly one of two possible alleles for a trait).
New control strategies involve biological methods such as genetic control, which may either reduces mosquito population in a specific area or decreases the mosquito vector competence , , .
Among the latter, infection of wild populations by the bacterium Wolbachia appears promising (see also Applicative axis 2 below).
Being maternally-transmitted, the latter obeys non-Mendelian inheritance law.
Motivated by the effects of the (possibly unwanted) interaction of these two types of treatment, we initiated the study of modelling of Mendelian and non-Mendelian inheritances in density-dependent population dynamics.
First results are shown in .

Control and macroscopic limits of collective dynamics

The term self-organization is used to describe the emergence of complex organizational patterns from simple interaction rules in collective dynamics systems. Such systems are valuable tools to model various biological systems or opinion dynamics, whether it be the collective movement of animal groups, the organization of cells in an organism or the evolution of opinions in a large crowd.
A special case of self-organization is given by consensus, i.e. the situation in which all agents' state variables converge. Another phenomenon is that of clustering, when the group is split into clusters that each converge to a different state.
A natural question in this framework is that of control: can the system be guided to a desired predetermined configuration? In the case when self-organization is not achieved naturally by the system, can it be driven to it? On the contrary, in the case where consensus and clustering are situations to be avoided (for example in crowd dynamics), can we design control strategies to keep the system away from clustering?

Another natural question is that of the large population limit. When the number of agents tends to infinity, the previous system of equations becomes unmanageable, a problem well-known as the curse of dimensionality. A common answer to this issue consists of studying the macroscopic limit of the system. It is then crucial to understand whether the limit system retains the properties of the microscopic one.

Models of neural network

Mean field limits have been proposed by biophysicists in order to describe neural networks based on physiological models. The various resulting equations are called integrate-and-fire, time elapsed models, voltage-conductance models. Their specific nonlinearities and the blow-up phenomena make their originality which has led to develop specific mathematical analysis , followed by , , , . This field also yields a beautiful illustration for the capacity of the team to combine and compare stochastic and PDE modelling (see Methodological axis 3), in .

Models of interacting particle systems

The organisation of biological tissues during development is accompanied by the formation of sharp borders between distinct cell populations. The maintenance of this cell segregation is key in adult tissue homeostatis, and its disruption can lead tumor cells to spread and form metastasis. This segregation is challenged during tissue growth and morphogenesis due to the high mobility of many cells that can lead to intermingling. Therefore, understanding the mechanisms involved in the generation and maintain of cell segregation is of tremendous importance in tissue morphogenesis, homeostasis, and in the development of various invasive diseases such as tumors. In this research axis, we aim to provide a mathematical framework which enables to quantitatively link the segregation and border sharpening ability of the tissue to these cell-cell interaction phenomena of interest . As agent-based models do not enable precise mathematical analysis of their solutions due to the lack of theoretical results, we turn towards continuous -macroscopic- models and aim to provide a rigorous link between the different models .

Models of population dynamics structured in phenotype

The collaboration of Jean Clairambault with Emmanuel Trélat and Camille Pouchol (from September last year assistant professor at MAP5 Paris-Descartes, University of Paris), together now with Nastassia Pouradier Duteil, has been continued and presently leads us to a possible quantitative biological identification of the structuring phenotypes of the model developed in , through a beginning collaboration with an Indian systems biologist (Mohit Kumar Jolly, IIS Bangalore). Our motivation in this collaboration is to couple a physiologically based system of 6 ODEs developed by our Indian collaborator with our phenotype-structured cell population dynamics model , .

In the framework of the HTE project EcoAML 2016-2020, Thanh Nam Nguyen, Jean Clairambault, Delphine Salort and Benoît Perthame, in collaboration with Thierry Jaffredo at IBPS-SU, have designed a phenotype-structured integrodifferential model of interactions between haematopoietic stem cells (healthy or leukaemic) and their supporting stromal cells . In this model, without diffusion, to our relative astonishment, our postdoctoral fellow T.N. Nguyen predicts in particular that under special circumstances, a coexistence between healthy and leukaemic stem cell subpopulations is possible. The explanation of such possible theoretical coexistence still remains to be explained.

The idea of cooperation between cell subpopulations in a tumour is also studied using phenoytype-structured models of cell populations by Frank Ernesto Alvarez Borges, PhD student of Stéphane Mischler (Paris-Dauphine University), Mariano Rodríguez Ricard (University of Havana, Cuba) and Jean Clairambault, in collaboration with José Antonio Carrillo (Oxford). A feature of these models, in as much as conflicting continuous phenotypes (e.g., adhesivity vs. motility, or fecundity vs. viability, or fecundity vs. motility ) are supposed to structure a unique cell population, is that they can also represent the emergence of multicellularity in such a cell population, when two subpopulations of the same population, i.e., endowed with the same genome and represented w.r.t. relevant heterogeneity in the cell population by such conflicting phenotypes, are determined by two different choices of the 2-d phenotype. This has been this year the object of a submitted article associating Frank Ernesto Alvarez Borges, José antonio Carrillo (Oxford) and Jean Clairambault , and this collaboration will be continued. In a simplified representation when the two phenotypes are just extreme values of a 1-d continuous phenotype (e.g., 0 for total adhesivity and no motility, 1 for no adhesivity and complete motility) this situation may be related to the previously described case, developed in , in which two extreme values of a convex function linked to proliferation are occupied by the two extreme phenotype values (0 and 1), leading to the coexistence of two cell subpopulations.

Collaborations

The Mamba team had initiated and is a leader on the works developed in this
research axis. It is a part of a consortium of several mathematicians
in France through the ANR Blanc project Kibord, which involves
in particular members from others INRIA team (DRACULA, COMMEDIA).
Finally, we mention that from Sept. 2017 on, Mamba benefited from the ERC Advanced Grant ADORA (Asymptotic approach to spatial and dynamical organizations) of Benoît Perthame.

We divide this research axis, which relies on the study of partial differential equations for space and time organisation of biological populations, according to various applications using the same type of mathematical formalisms and methodologies: asymptotic analysis, weak solutions, numerical algorithms.

Aggregation equation

In the mathematical study of collective behavior, an important class of models is given by the aggregation equation. In the presence of a non-smooth interaction potential, solutions of such systems may blow up in finite time. To overcome this difficulty, we have defined weak measure-valued solutions in the sense of duality and its equivalence with gradient flows and entropy solutions in one dimension . The extension to higher dimensions has been studied in . An interesting consequence of this approach is the possibility to use the traditional finite volume approach to design numerical schemes able to capture the good behavior of such weak measure-valued solutions , .

Identification of the mechanisms of single cell motion

In this research axis, we aim to study the mechanisms of single cell adhesion-based and adhesion free motion. This work is done in the frame of the recently created associated team MaMoCeMa (see Section 9) with the WPI, Vienna. In a first direction with N. Sfakianakis (Heidelberg University), we extended the live-cell motility Filament Based Lamellipodium Model to incorporate the forces exerted on the lamellipodium of the cells due to cell-cell collision and cadherin induced cell-cell adhesion. We took into account the nature of these forces via physical and biological constraints and modelling assumptions. We investigated the effect these new components had in the migration and morphology of the cells through particular experiments. We exhibit moreover the similarities between our simulated cells and HeLa cancer cells.

In a second work done in collaboration with the group of biologist at IST (led by Michael Sixt Austria), we developed and analyzed a two-dimensional mathematical model for cells migrating without adhesion capabilities . Cells are represented by their cortex, which is modelled as an elastic curve, subject to an internal pressure force. Net polymerization or depolymerization in the cortex is modelled via local addition or removal of material, driving a cortical flow. The model takes the form of a fully nonlinear degenerate parabolic system. An existence analysis is carried out by adapting ideas from the theory of gradient flows. Numerical simulations show that these simple rules can account for the behavior observed in experiments, suggesting a possible mechanical mechanism for adhesion-independent motility.

Free boundary problems for tumor growth

Fluid dynamic equations are now commonly used to describe tumor growth with two main classes of models: those which describe tumor growth through the dynamics of the density of tumoral cells subjected to a mechanical stress; those describing the tumor through the dynamics of its geometrical domain thanks to a Hele-Shaw-type free boundary model. The first link between these two classes of models has been rigorously obtained thanks to an incompressible limit in for a simple model. This result has motivated the use of another strategy based on viscosity solutions, leading to similar results, in .

Since more realistic systems are used in the analysis of medical images, we have extended these studies to include active motion of cells in , viscosity in and proved regularity results in . The limiting Hele-Shaw free boundary model has been used to describe mathematically the invasion capacity of a tumour by looking for travelling wave solutions, in , see also Methodological axis 3. It is a fundamental but difficult issue to explain rigorously the emergence of instabilities in the direction transversal to the wave propagation. For a simplified model, a complete explanation is obtained in .

Coupling of diffusion and growth

The growth of an organism is triggered by signaling molecules called morphogens that diffuse in the organism during its development. Meanwhile, the diffusion of the morphogens is itself affected by the changes in shape and size of the organism. In other words, there is a complete coupling between the diffusion of the morphogens and the evolution of the shapes.
We are working on the elaboration of a mathematical framework for diffusion equations on time-evolving manifolds, both theoretically and in collaboration with developmental biologists, for the special case of the diffusion of Gurken during the oogenesis of Drosophila.

Migration of cells in extracellular matrix

A single cell based model has been developed that reproduces a large set of experimental observations of cells migrating in extracellular matrix based on physical mechanisms with mimimal internal cell dynamics. This includes individually migrating cells in micro-channels of different size, and their collective dynamics in case of many cells, as well as the impact of cell division and growth. The model explicitly mimics the extracellular matrix as the cells as deformable objects with explicit filopodia.

Collaborations

Direct parameter identification is a great challenge particularly in living systems in which part of parameters at a certain level are under control of processes at smaller scales. Mamba developed and addressed model and parameter identification methods and strategies in a number of mathematical and computational model applications including growth and fragmentation processes emerging in bacterial growth and protein misfolding, in liver regeneration , TRAIL treatment of HeLa cells , growth of multicellular spheroids , blood detoxification after drug-induced liver damage , .

This naturally leads to increasingly combine methods from various fields: image analysis, statistics, probability, numerical analysis, PDEs, ODEs, agent-based modeling methods, involving inverse methods as well as direct model and model parameter identification in biological and biomedical applications. Model types comprise agent-based simulations for which Mamba is among the leading international groups, and Pharmocokinetic (PK) simulations that have recently combined in integrated models (PhD theses Géraldine Cellière, Noémie Boissier). The challenges related with the methodological variability has led to very fruitful collaborations with internationally renowned specialists of these fields, e.g. for bacterial growth and protein misfolding with Marc Hoffmann (Paris Dauphine) and Patricia Reynaud-Bouret (University of Nice) in statistics, with Philippe Moireau (Inria M3DISIM) in inverse problems and data assimilation, and with numerous experimentalists.

Estimation methods for growing and dividing populations

In this domain, all originated in two papers in collaboration with J.P. Zubelli in 2007 , , whose central idea was to used the asymptotic steady distribution of the individuals to estimate the division rate. A series of papers improved and extended these first results while keeping the deterministic viewpoint, lastly . The last developments now tackle the still more involved problem of estimating not only the division rate but also the fragmentation kernel (i.e., how the sizes of the offspring are related to the size of the dividing individual) . In parallel, in a long-run collaboration with statisticians, we studied the Piecewise Deterministic Markov Process (PDMP) underlying the equation, and estimated the division rate directly on sample observations of the process, thus making a bridge between the PDE and the PDMP approach in , a work which inspired also very recently other groups in statistics and probability , and was the basis for Adélaïde Olivier's Ph.D thesis , and of more recent work (see also axis 5).

Data assimilation and stochastic modeling for protein aggregation

Estimating reaction rates and size distributions of protein polymers is an important step for understanding the mechanisms of protein misfolding and aggregation (see also axis 5). In , we settled a framework problem when the experimental measurements consist in the time-dynamics of a moment of the population.

To model the intrinsic variability among experimental curves in aggregation kinetics - an important and poorly understood phenomenon - Sarah Eugène's Ph.D, co-supervised by P. Robert , was devoted to the stochastic modeling and analysis of protein aggregation, compared both with the deterministic approach traditionally developed in Mamba and with experiments.

Parameter identification in multi-level and multi-scale models of liver

Several projects are pursued on multiscale, multilevel modeling of liver regeneration and its consequences with integration of an increasingly amount of data. So far the most promising strategy working was for every additional data set, first testing whether the model would be able to simulate it without any modifications, and to modify the model if necessary by inclusion of further biological mechanisms or information. A key unsolved problem is that biological data seem often not perfectly reproducable, and measurements at different times may differ from each other. This can result from slightly different experimental settings or conditions, or different measurement methods. While for testing of qualitative mechanisms this is usually sufficient, the quantitative difference is sometimes of the order of the effect which makes a quantitative modeling very challenging. For ammonia detoxification during fibrosis, extensive simulations have been performed varying multiple clinically relevant parameters. The basis model needed to integrate multiple data sets and could only be modelled if modifications in tissue microarchitecture, adaptations of intracellular enzyme activities, and possible aging effects were taken into account (ongoing project close to finalization).

Collaborations

The team has three main application-driven research axes. Applicative axis 1 focuses on cancer, an application on which almost all team members work, with various approaches. A main focus of the team is to study cancer as a Darwinian evolutionary phenomenon in phenotype-structured cell populations. Optimal control methods take into account the two main pitfalls of clinical cancer therapeutics, namely unwanted toxic side effects in healthy cell populations and drug resistance in cancer cell populations. Other studies concern telomere shortening, and multi-scale models. Applicative axis 2 is devoted to growth, evolution and regeneration in populations and tissues. It involves protein aggregation and fragmentation models for neurodegenerative diseases (prion, Alzheimer), organ modeling, mainly of the liver, its damages induced by toxic molecules, and its regeneration after toxic insult. Applicative axis 3 is new and encompasses works related to epidemiology, both for infectious and vector-borne diseases.

The MAMBA team designs and analyses mathematical models of tumor growth and therapy, at the cell population level, using agent-based or partial differential equations, with special interest in methodologies for therapeutic optimization using combined anticancer drug treatments. Rather than, or not only, modeling the effect of drugs on molecular targets, we represent these effects by their functional consequences on the fate of healthy and cancer cell populations: proliferation (velocity of the cell division cycle, decreasing it, e.g., by antagonizing growth factor receptors), apoptosis, cell death or senescence. Our goal in doing this is to circumvent the two main issues of anticancer therapy in the clinic, namely unwanted toxic side effects in populations of healthy cells and emergence of drug-induced drug resistance in cancer cell populations. This point of view leads us to take into account phenomena of transient and reversible resistance, observed in many cancer cell populations, by designing and analyzing models of cell populations structured in continuous phenotypes, relevant for the description of the behavior of cell populations exposed to drugs: either degree of resistance to a given drug, or potential of resistance to drug-induced stress, proliferation potential, and plasticity. Such modeling options naturally lead us to to take into account in a continuous way (i.e., by continuous-valued phenotype or relevant gene expression) the wide phenotypic heterogeneity of cancer cell populations. They also lead us to adopt the point of view of adaptive dynamics according to which characteristic traits of cell populations evolve with tumor environmental pressure (drugs, cytokines or metabolic conditions, mechanical stress and spatial conditions), in particular from drug sensitivity to resistance. This position is original on the international scene of teams dealing with drug resistance in cancer. Recently, we have also extended our approach to the study of Tumour-Immune System interac

Modeling Acute Myeloid Leukemia (AML) and its control by anticancer drugs by PDEs and Delay Differential equations

In collaboration with Catherine Bonnet (Inria DISCO, Saclay) and François Delhommeau (St Antoine hospital in Paris), together with DISCO PhD students José Luis Avila Alonso and Walid Djema, this theme has led to common published proceedings of conferences: IFAC, ACC, CDC, MTNS , , , , , . These works study the stability of the haematopoietic system and its possible restabilization by combinations of anticancer drugs with functional targets on cell populations: proliferation, apoptosis, differentiation.

Adaptive dynamics setting to model and circumvent evolution towards drug resistance in cancer by optimal control

We tackle the problem to represent and inhibit - using optimal control algorithms, in collaboration with Emmanuel Trélat, proposed Inria team CAGE - drug-induced drug resistance in cancer cell populations. This theme, presently at the core of our works on cancer modeling with a evolutionary perspective on tumor heterogeneity, is documented in a series of articles , , , , . Taking into account the two main pitfalls of cancer therapy, unwanted side effects on healthy cells and evolution towards resistance in cancer cells, it has attracted to our team the interest of several teams of biologists, with whom we have undertaken common collaborative works, funded by laureate answers to national calls (see ITMO Cancer HTE call).

This theme is also at the origin of methodological developments (see Research axis 1). In collaboration with Shensi Shen from Institut Gustave Roussy and Francois Vallette from Université de Nantes, we aim to develop simple non-spatial models to understand the mechanisms of drug resistance acquisition -and loss- in melanoma and glioblastoma. The models are systematically compared with in vitro and in vivo data generated by our collaborators and treated via image processing techniques developed in the team.

Senescence modeling by telomere shortening

In many animals, aging tissues accumulate senescent cells, a process which is beneficial to protect from cancer in the young organism. In collaboration with Teresa Teixeira and Zhou Xu from IBCP, we proposed a mathematical model based on the molecular mechanisms of telomere replication and shortening and fitted it on individual lineages of senescent Saccharomyces cerevisiae cells, in order to decipher the causes of heterogeneity in replicative senescence .

Biomechanically mediated growth control of cancer cells other cell types

Model simulations indicate that the response of growing cell populations on mechanical stress follows a simple universal functional relationship and is predictable over different cell lines and growth conditions despite the response curves look largely different. We developed a hybrid model strategy in which cells were represented by coarse-grained individual units calibrated in a high resolution cell model and parameterized each model cell by measurable biophysical and cell-biological parameters. Cell cycle progression in our model is controlled by volumetric strain, the latter being derived from a bio-mechanical relation between applied pressure and cell compressibility. After parameter calibration from experiments with mouse colon carcinoma cells growing against the resistance of an elastic alginate capsule, the model adequately predicts the growth curve in i) soft and rigid capsules, ii) in different experimental conditions where the mechanical stress is generated by osmosis via a high molecular weight dextran solution, and iii) for other cell types with different growth kinetics. Our model simulation results suggest that the growth response of cell population upon externally applied mechanical stress is the same, as it can be quantitatively predicted using the same growth progression function . This model has now been extended to compare the efficiency of different culturing methods, monolayer growth, multicellular spheroids growth and growth within elastic capsules. The methodology of culturing is relevant in terms of cell yield and cell homogeneity.

Bio-mechanical models of tissue growth

The degenerate Cahn-Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn-Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose in [hal-02274417] a relaxation system formed of two second order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system; global existence and convergence of the relaxed system to the degenerate Cahn-Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.

Free boundary multiphase models of tumor growth

Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous media equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a segregation result.
Several major mathematical difficulties arise in the two species case which are addressed in . Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson-Bénilan estimates cannot be established in our context. We are lead, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an

Philosophy of cancer

The quite natural idea that cancer is a disease of the control of coherent multicellularity, expressed when cohesion of tissues and coherence of (unknown, except maybe for the case of a centralised circadian clock) synchronising signals fail to ensure it, by a regression towards unicellularity, stopping in this “reverse evolution path” at a coarse, incoherent multicellularity state continues to be developed and popularised by Jean Clairambault in seminars and workshops, and published in review articles , and conference proceedings . This view, and the investigation of the immune system in the design of such coherence of all multicellular organisms is naturally inscribed in a philosophy of cancer perspective, and from a mathematical viewpoint, to multicellularity genes - and links between them and unicellularity genes - seen as a hyperstructure above structures consisting of the genes of unicellularity, i.e., those that make a single cell a coherent living system, such hyperstructure being failed in cancer; this view is presently under development with colleagues from universities of the Paris region, together with Nils Baas at NTNU, Trondheim, Norway). This perspective, that makes use of category theory as a structuring point of view to apprehend multicellularity and cancer, is also meant to endow us with an innovative methodology to apply topological data analysis (TDA) to investigate cancer genome data.

Modelling of TMZ induced drug resistance

Temozolomide (TMZ) is a standard chemotherapy treatment in patients with glioblastoma. Resistance to this drug is correlated to the presence of a specific enzyme, which activity in cancer cells creates a drug-induced cell death resistant phenotype. Understanding the transition of cancer cells to a resistant phenotype is still a topic of research where multiple hypothesis have been studied: From an adaptive process to an inherent resistance to treatment. It has been recently shown that if TMZ treatment does not significantly induce cell death in glioblastoma, it still generates a response in terms of the spatial arrangement of cell aggregates. Moreover, the coupling of TMZ with irradiation has been shown to generate a better response in patients compared with using irradiation alone. Therefore, understanding the mechanisms of glioblastoma reaction to TMZ treatment could open new therapeutic avenues. In the frame of the post-doctorate of Gissell Estrada Rodriguez, we developed a 2D mathematical model in , suggesting a new possible mechanism for TMZ induced rearrangement of cancer cells (see section new results).

Modelling of the Epithelial-Mesenchymal Transition (EMT)

Understanding cell-fate decisions remains a major research challenge in developmental biology. In particular, the forward and backward epithelial-mesenchymal cellular transitions (EMT-MET) play a crucial role in embryonal development, tissue repair and cancer metastasis. The epithelial cell phenotype (E) is characterized by strong cell-to-cell adhesion, while the mesenchymal phenotype (M) is characterized by a strong cellular motility. Recent research has shown that there even exists a third hybrid phenotype (E/M) with mixed characteristics, that enables collective cell migration. EMT and MET play a crucial role in cancer metastasis, for instance when cancer cells from a primary tumor gain the ability to migrate through the bloodstream or lymph system to distant organs and then recover their adhesion to form secondary tumors. Thus, understanding the dynamics of MET and EMT is crucial for decoding metastasis and for designing effective therapeutics.

Collaborations

Adaptive dynamics to model drug resistance and optimal control to circumvent it:

Alexandre Escargueil, Michèle Sabbah (1 PhD thesis in common), St Antoine Hospital, Paris

Emmanuel Trélat (1 PhD thesis in common) at Inria team CAGE and Laboratoire Jacques-Louis Lions at Sorbonne Université.

Frédéric Thomas at CREEC, Montpellier.

Tommaso Lorenzi (Univ. of St Andrews).

The applications in this category span very different subjects from amyloid diseases, wound healing, liver regeneration and toxicity, up to bacterial growth and development of organisms. As the applications, the methods span a wide range. Those concerning identification of models and parameters with regard to data have partially been outlined in axis 3. Focus in this axis is on the model contribution to the biologically and/or medically relevant insights and aspects.

Liver-related modelling is partially performed within the INRIA team MIMESIS (Strasbourg) with the focus on real-time, patient-specific biomechanical liver models to guide surgery and surgeons. Internationally, spatial temporal liver related models are developed in Fraunhofer MEVIS (Bremen), by T. Ricken (TU Dortmund), and P. Segers group (Leuven).

Different from these, Mamba has a strong focus on spatial-temporal modeling on the histological scale, integration of molecular processes in each individual cell, and single-cell (agent) based models . Works by Schliess , have been highlighted in editorials.

Mathematical modeling of protein aggregation is a relatively recent domain, only a few other groups have emerged yet; among them we can cite the Inria team Dracula, with whom we are in close contact, and e.g., the work by Jean-Michel Coron (Sorbonne Université) and Monique Chyba (Hawaii, USA) in control, and Suzanne Sindi (USA) for the modeling of the yeast prion. We have interactions with all these groups and organized a workshop in June 2017, gathering both the biophysics and applied mathematics communities.

Amyloid disease

Application to protein aggregation in amyloid diseases is a long-standing interest of Mamba, dating back to 2010 , and developed through the collaboration with n rHuman Rezaei's team at Inra. More recently, with Wei-Feng Xue in Canterbury, we investigated the intrinsic variability among identical experiments of nucleation , , Sarah Eugène's Ph.D subject (co-supervised by Philippe Robert) .

In collaboration with Tom Banks first , and then Philippe Moireau, we developed quantitative comparisons between model and data. Through data assimilation and statistical methods , we proposed new models and mechanisms.

Wound healing: adipose tissues

After injury, if regeneration can be observed in hydra, planaria and some vertebrates, regeneration is rare in mammals and particularly in humans. In this research axis, we investigated the mechanisms by which biological tissues recover after injury. We explored this question on adipose tissue, using the mathematical framework recently developed in . Our assumption is that simple mechanical cues between the Extra-Cellular Matrix (ECM) and differentiated cells can explain adipose tissue morphogenesis and that regeneration requires after injury the same mechanisms. We validated this hypothesis by means of a two-dimensional Individual Based Model (IBM) of interacting adipocytes and ECM fiber elements . The model successfully generated regeneration or scar formation as functions of few key parameters, and seemed to indicate that the fate of injury outcome could be mainly due to ECM rigidity.

Following these encouraging results, the team is currently taking a step further in the model validation and confrontation to experimental data. The first direction concerns the development of a 3D framework to validate the mechanisms observed in 2D, in the frame of the PhD of P. Chassonnery, co-directed by D. Peurichard and L. Casteilla (RESTORE, Toulouse).

Influence of cell mechanics in embryonic bile duct lument formation: insight from quantitative modeling

In vitro construction of hepatic tissue for regenerative therapy consists in recapitulating mechanisms of embryonic development. However, implementing those mechanisms in a spatially and temporally coordinated way remains difficult. Specifically, the construction of bile ducts and in particular the controlled formation of luminal structures formed by cholangiocytes is a challenge. The team works on a high resolution individual-based computational model which can help in unravelling the mechanisms of initial bile duct lumen formation. Guided by the quantification of morphological features and expression of genes in developing bile ducts from embryonic mouse liver, hypotheses for the mechanisms of biliary lumen formation were generated and tested with the model. Our simulations with a hybrid simulation technology as developed in ref. suggest that successful bile duct lumen formation primarily requires the simultaneous contribution of several mechanisms discussed in the literature.

Mathematical modelling of axolotl regeneration

Tissue response after injury/amputation induces one or two alternatives: scar formation versus regeneration (complete recovery of tissue shape and functions). In most mammals, regeneration is considered largely impaired for the benefit of a fibrotic scar after injury automatically associated with dysfunctions, but complete regeneration has been largely described and investigated in animal models such as zebra fish, salamander, or axolotl. Despite several processes regulating regeneration have been identified at different scales -from diffusing molecules and cellular gene expression patterns up to tissue mechanics-, how these mechanisms individually or collectively play a role in the regulation of regenerative processes remains poorly understood. In order to give insights into the mechanisms of tissue regeneration, Valeria Caliaro started an Inria PhD project in october 2019, in collaboration with Osvaldo Chara, internationally recognized group leader of SysBio in Argentina. This project focuses on the role of cell proliferation in space and time along the two first phases of regeneration after injury: (i) initiation of a regeneration response, (ii) tissue patterning during regenerate growth. The first part of the project aims at building an agent-based model featuring few key mechanisms regulating cell proliferation after injury. By introducing heuristic rules which rely on Prof O. Chara expertise, we propose a 2D-ABM using methodologies borrowed from socio-dynamics and collective behavior studies (based on many interacting agent systems). While the focus is made on proliferation-based mechanisms, other mechanisms responsible for collective behavior such as volume exclusion, diffusion or aggregation are taken into account. The resulting model will provide a synthetic tissue model which will serve to investigate regeneration in cellular systems, focusing on cell proliferation properties. The second part of the PhD will be devoted to the derivation of continuous models from the agent-based formalism. This will provide a large scale ‘synthetic tissue’ model to explore the role of large scale effects in general tissue models.

Quantitative cell-based model predicts mechanical stress response of growing tumor spheroids

Model simulations indicate that the response of growing cell populations on mechanical stress follows the same functional relationship and is predictable over different cell lines and growth conditions despite experimental response curves look largely different. We developed a hybrid model strategy in which cells are represented by coarse-grained individual units calibrated with a high resolution cell model and parameterized by measurable biophysical and cell-biological parameters. Cell cycle progression in our model is controlled by volumetric strain, the latter being derived from a bio-mechanical relation between applied pressure and cell compressibility. After parameter calibration from experiments with mouse colon carcinoma cells growing against the resistance of an elastic alginate capsule, the model adequately predicts the growth curve in i) soft and rigid capsules, ii) in different experimental conditions where the mechanical stress is generated by osmosis via a high molecular weight dextran solution, and iii) for other cell types with different growth kinetics from the growth kinetics in absence of external stress. Our model simulation results suggest a generic, even quantitatively same, growth response of cell populations upon externally applied mechanical stress, as it can be quantitatively predicted using the same growth progression function (.

Bacterial population growth

We exploited all the methods developed to estimate the division rate of a population (see axis 3) to address a seminal question of biology: is it a size-sensing or a timing mechanism which triggers bacterial growth? In , we showed that a sizer model is robust and fits the data well. Several studies from other groups came at the same time, showing a renewed interest on a question dated back to Jacques Monod's PhD thesis (1941). Of special interest is the “adder” model, for which we are currently developing new estimation methods .

A quantitative high resolution computational mechanics cell model for growing and regenerating tissues

Mathematical models are increasingly designed to guide experiments in biology, biotechnology, as well as to assist in medical decision making. They are in particular important to understand emergent collective cell behavior. For this purpose, the models, despite still abstractions of reality, need to be quantitative in all aspects relevant for the question of interest. Considered was as showcase example the regeneration of liver after drug-induced depletion of hepatocytes, in which the surviving and dividing hepatocytes must squeeze in between the blood vessels of a network to refill the emerged lesions. Here, the cells’ response to mechanical stress might significantly impact the regeneration process. We present a 3D high-resolution cell-based model integrating information from measurements in order to obtain a refined and quantitative understanding of the impact of cell-biomechanical effects on the closure of drug-induced lesions in liver. Our model represents each cell individually and is constructed by a discrete, physically scalable network of viscoelastic elements, capable of mimicking realistic cell deformation and supplying information at subcellular scales. The cells have the capability to migrate, grow, and divide, and the nature and parameters of their mechanical elements can be inferred from comparisons with optical stretcher experiments. Due to triangulation of the cell surface, interactions of cells with arbitrarily shaped (triangulated) structures such as blood vessels can be captured naturally. Comparing our simulations with those of so-called center-based models, in which cells have a largely rigid shape and forces are exerted between cell centers, we find that the migration forces a cell needs to exert on its environment to close a tissue lesion, is much smaller than predicted by center-based models. To stress generality of the approach, the liver simulations were complemented by monolayer and multicellular spheroid growth simulations. In summary, our model can give quantitative insight in many tissue organization processes, permits hypothesis testing in silico, and guide experiments in situations in which cell mechanics is considered important .

Liver regeneration and disease: towards a full virtual liver model at histological scale

In our work towards a full virtual liver model at histological level, a number of steps were performed. The models under points (1)-(4) focus on either a single or a few liver lobules. A liver lobule is the smallest repetitive functional and anatomical building block of liver, while (5) addresses a much larger organisational building block of the liver, a liver lobe that consists of thousands to hundreds of thousands of lobules depending on the species. A second strand (6), (7) addresses image analysis, which in most cases forms the entrance to modeling as it provides the data necessary to generate model hypotheses and to parameterize a model.

(2) Liver disease: Degenerative liver diseases such as liver fibrosis and cirrhosis develop out of a disturbed balance of degenerative and regenerative processes.
The model under (1) has thereby been extended by the formation of extracellular matrix, mimicked as fiber networks, to capture the disease process leading to liver fibrosis. In that process characteristic streets form that modify the mechanics, perfusion behavior and detoxification capacity of the liver. The model is now used to simulate disease pathways emerging from different administration schemes of drugs that are knowning to long-term lead to hepatocellular cancer.

(3) Consequence of liver fibrosis: Whole-slide scans from fibrotic liver in a mouse model has been analysed at different time points after emergence of the disease with regard to the degree of excess matrix to mimic the possible consequences of fibrotic inclusions on perfusion and function of liver within a multiscale model that considers ammonia detoxification in each individual hepatocyte as well as blood flow and transport processes in the liver lobule. This model has now be confronted on multimodal data in healthy liver, liver after a toxic dose of a drug, and fibrosis. The requirement to explain simultaneously all data sets in the same model imposes significant challenges for which solutions are currently explored.

(4) Bile flux: Bile flux has been for decades believed to be controlled by convection at the level of liver lobules as well as at the level of the entire organ.
By a methodology based on correlative imaging for quantitative intravital flux analysis no directed advection was detectable in bile canaliculi at the resolution limit. Instead, after active transport across hepatocyte membranes bile salts within the liver lobules are transported in the canaliculi by a diffusion-dominated process. Only in the interlobular ducts i.e., at super-lobular level, diffusion is augmented by advection. In silico simulations of bile transport in real 3D bile network microarchitectures can quantitatively explain the data assuming diffusive transport as sole mechanism.

(5) Liver regeneration after partial hepatectomy (partial organ removal): Partial hepatectomy is an adequate therapy in case of diseases or events that destructed only part of the liver. A typical case is a primary tumor or a metastasis affecting only a single liver lobe. Within an biophysical agent-based model capturing many aspects of the cell mechanics we studied regrowth of liver after partial organ removal in mouse calibrated with multivariate experimental data. Our model predicts characteristic proliferation pattern that change from small animals (as mouse) to large animals (as pig).

(6) Bile duct ligation: Bile duct ligation (BDL) is an experimental procedure that mimics obstructive cholestatic disease.
One of the early consequences of BDL in rodents is the appearance of so-called bile infarcts that correspond to Charcot-Gombault necrosis in human cholestasis. The mechanisms causing bile infarcts and their pathophysiological relevance are unclear.
Therefore, intravital two photon–based imaging of BDL mice was performed with fluorescent bile salts (BS)
and non-BS organic anion analogues. Key findings were followed up by matrix-assisted laser desorption ionization
imaging, clinical chemistry, immunostaining, and gene expression analyses.
Our group performed analysis of intravital imaging.
The key finding is that bile microinfarcts occur in the acute phase after BDL in a limited number of dispersed hepatocytes followed by larger infarcts involving neighboring hepatocytes, and they allow leakage of bile from the BS-overloaded biliary tract into blood, thereby protecting the liver from BS toxicity; in the chronic phase after BDL, reduced sinusoidal BS uptake is a dominant protective factor, and the kidney contributes to the elimination of BS until cholemic nephropathy sets in .

(7) Periportalisation during liver fibrosis formation: Within a liver lobule, the function of hepatocytes is zonated i.e., certain functions are only executed by either hepatocytes close to the center (pericentral region) or hepatocytes in the periphery of the lobule (periportal region).
Little is known about how liver fibrosis influences lobular zonation. To address this
question, three mouse models of liver fibrosis were used, CCl4 administration repeated for 2, 6 and 12
months to induce pericentral damage, as well as bile duct ligation (21 days) and a particular mdr2-mouse model to
study periportal fibrosis. Analyses were performed by RNA-sequencing, immunostaining of
zonated proteins and image analysis. Image analysis was performed by our group.
The key result was that liver fibrosis leads to strong alterations of lobular zonation, where the pericentral region adopts periportal features. Beside adverse consequences, periportalization supports adaptation to repeated doses of hepatotoxic compounds .

Toxicity extrapolation from in vitro to in vivo

In vivo toxicity prediction from in vitro data is a major objective in toxicology as it permits bypassing animal experiments, and as the predictive power of animal experiments for human is limited. Objective was the prediction of paracetamol (acetaminophen)-induced hepatotoxicity from in vitro experiments. For this purpose, numerous iterations between in vitro experiments, in vivo experiments and simulations were performed for mouse. Using a recent thesis (Géraldine Cellière'ns PhD thesis ) as a start point, two candidate mechanisms could be identified both explaining the in vivo data after calibration of the in silico model with in vitro toxicity data.

Relating imaging on microscopic scales with imaging on macroscopic scales: From Diffusion-Weighted MRI Calibrated With Histological Data: an Example
From Lung Cancer

Diffusion-weighted magnetic resonance imaging (DWI) is a key non-invasive imaging technique for cancer diagnosis and tumor treatment assessment, reflecting Brownian movement of water molecules in tissues. Since densely packed cells restrict molecule mobility, tumor tissues produce usually higher signal (less attenuated signal) on isotropic maps compared with normal tissues. However, no general quantitative relation between DWI data and the cell density has been established. In order to link low-resolution clinical cross-sectional data with high resolution histological information, we developed an image processing and analysis chain, which was used to study the correlation between the diffusion coefficient (D value) estimated from DWI and tumor cellularity from serial histological slides of a resected non-small cell lung cancer tumor. Color deconvolution followed by cell nuclei segmentation was performed on digitized histological images to determine local and cell-type specific 2d (two-dimensional) densities. From these, the 3d cell density was inferred by a model-based sampling technique, which is necessary for the calculation of local and global 3d tumor cell count. Next, DWI sequence information was overlaid with high resolution CT data and the resected histology using prominent anatomical hallmarks for co-registration of histology tissue blocks and non-invasive imaging modalities’ data. The integration of cell numbers information and DWI data derived from different tumor areas revealed a clear negative correlation between cell density and D value. Importantly, spatial tumor cell density can be calculated based on DWI data. In summary, our results demonstrate that tumor cell count and heterogeneity can be predicted from DWI data, which may open new opportunities for personalized diagnosis and therapy optimization . The work of that paper has been further advanced to adapt the procedures for clinical use (in preparation).

Collaborations

This axis is new and encompasses different works related to epidemiology, both for infectious and vector-borne diseases. The team was working since several years on the modeling, analysis and control of the propagation of vector-borne diseases such as dengue fever. Ordinary or partial differential equations of reaction-diffusion are used, and various (optimal or not) control strategies. In parallel and with the acknowledged opportunity of the onset and spreading of the Covid-19 pandemic, we expanded our interest to issues related to infectious diseases, using similar evolution systems.

Biological control of arboviroses

A related technique is based on the infection by Wolbachia.
This symbiotic bacterium is maternally transmitted from infected females to their offspring, but induces cytoplasmic incompatibility, : mating between infected males and uninfected females gives no offspring.
Releases of Wolbachia infected males alone is thus comparable to classical SIT.

On the other hand, releasing both infected males and females in sufficient quantity may result in infection of the wild population.
This gives rise to an interesting new control principle, as Wolbachia has been shown to severely reduce the insect vectorial ability to transmit dengue, zika or chikungunya, indirectly by lifespan and fertility reduction, and directly by reducing the ability of the viruses to proliferate within the organism .

We proposed new insights on the practical and theoretical issues raised by the implementation of the previous methods. Concerning the SIT, we obtained control synthesis results through impulsive periodic release of controlled amplitude , and through optimal control approach . Concerning Wolbachia technique, we investigated general control principles capable of spreading the infection.

We also considered the effects of hindrances to these strategies .

Moreover, while continuing our work on the problem of controling the mosquito population by itself, we have extended our studies to systems that consider the dynamics or the arboviroses in both the human and mosquito populations. This should allow us to confirm the pertinence of our previous and future works concerning the control of the mosquito population, but also motivate new criteria that take directly into account the dynamicsz of the epidemics in the human population.

Mathematical epidemiology of infectious diseases

The current outbreak of Covid-19 resulted in the appearance of many novel experiences at individual and collective, biological and social, national and international levels, making this pandemic a full epistemological experience as well. Motivated by the great number of questions raised by this global event, some members of the team devoted part of their time to exploring more or less closely related scientific issues. One should notice however that this evolution constitutes indeed the continuation of a movement already initiated previously, and only accelerated by the current events.

The issues raised by the effective implementation of the social distancing measures largely implemented on the Earth’s surface during the whole year 2020, have been the focus of intense reflection. We contributed to this debate by studying optimal control policies aiming at reducing the total number of infected people during the whole epidemic outbreak, the so-called epidemic final size. In another research line, we established the equation fulfilled by the epidemic final size for a fully general SEIR model in a heterogenous population characterized by some trait in a discrete or continuous subset, and studied the uniqueness of its solution. This allowed to extend the use and meaningfulness of the classical concept of next-generation operator introduced by O. Diekmann et al. in 1990 . Last, in cooperation with the Inria team NeCS (Inria Grenoble-Rhône-Alpes), we studied in a control theory perspective the effects of the testing policies in the dynamics and in the control of the epidemic.

Collaborations

All members of the team decided to carefully review his or her trip policy (especially by air), in order to reduce carbon footprint.

Dirk Drasdo, one of the former members of MAMBA and its precursor BANG, left the team and co-founded the new team SIMBIOTX at INRIA Center Saclay-Ile de France together with Irene Vignon-Clementel (formerly team REO) starting 03/2021. The team SIMBIOTX works on SImulations in Medicine, BIOtechnology and ToXicology of multicellular systems, see details in nd

Gaëtan Vignoud received with three co-authors the award "Interne innovant" of “Trophées APinnov 2021” of the AP-HP for their contribution "Approche informatique de Deep Learning pour l’analyse vidéo du mouvement dans le cadre des syndromes parkinsoniens", see .

The many results obtained during the last years have oriented us towards new research directions in the wide field of structured population equations: the study of the direct and inverse problem in the newly-proposed "adder model" ; oscillatory behaviors of such equations; the study of models for heterogeneous aggregation, i.e. where the aggregates are formed out of several monomeric species.

Modelling, analysing and inferring triggering mechanisms in population reproduction is fundamental in many biological applications. It is also an active and growing research domain in mathematical biology. In the book chapter , we review the main results developed over the last decade for the estimation of the division rate in growing and dividing populations in a steady environment. These methods combine tools borrowed from PDE's and stochastic processes, with a certain view that emerges from mathematical statistics. A focus on the application to the bacterial cell division cycle provides a concrete presentation, and may help the reader to identify major new challenges in the field.

To date, there exists very few studies of heterogeneous aggregation, i.e. aggregates formation out of several monomeric species. Last year, we proposed a bimonomeric model of Becker-Döring type, capable of explaining damped oscillations observed in prion fibrils aggregates ; however, in this study, we kept the standard formalism where a given aggregate is characterised by its size, i.e. by the number of monomers it contains, irrespective of the monomeric species.

In a different and still more complex direction, there is the case where each aggregate is formed out of two or more monomeric species, arranged in a particular way. This is typically the case of the aggregation of ubiquitinated cargo by oligomers of the protein p62. This is an important preparatory step in cellular autophagy, which has been Julia Delacour's Ph.D subject, defended in December 2020 , and co-supervised by M. Doumic and C. Schmeiser of the associated team MaMoCeMa. The dynamics of protein aggregation has been studied by mathematical modelling for several decades, but most models consider the aggregation of only one type of protein, which gives rise to models belonging to the class of nucleation-coagulation-fragmentation equations.Contrary to these studies, Julia Delacour's Ph.D thesis studied aggregates composed of two different types of particles with varying mixing ratios, which drastically increases the complexity of the problem. This phenomenon appears in autophagy, a natural mechanism of the cell which degrades unnecessary material.

Aggregation of ubiquitinated cargo by oligomers of the protein p62 is an important preparatory step in cellular autophagy. In a first study , a mathematical model for the dynamics of these heterogeneous aggregates in the form of a system of ordinary differential equations is derived and analyzed. Three different parameter regimes are identified, where either aggregates are unstable, or their size saturates at a finite value, or their size grows indefinitely as long as free particles are abundant. The boundaries of these regimes as well as the finite size in the second case can be computed explicitly. The growth in the third case (quadratic in time) can also be made explicit by formal asymptotic methods. The qualitative results are illustrated by numerical simulations. A comparison with recent experimental results permits a partial parametrization of the model.

The dynamics by which polymeric protein filaments divide can be described by the universal mathematical equations of 'pure fragmentation'. The rates of fragmentation reactions reflect the stability of the protein filaments towards breakage, which is of importance in biology and biomedicine for instance in governing the creation of amyloid seeds and the propagation of prions. In the numerical study , we devised from mathematical theory inversion formulae - analysed in their own right in previous studies - to recover the division rates and division kernel information from time dependent experimental measurements of filament size distribution. The numerical approach to systematically analyze the behaviour of pure fragmentation trajectories was also developed. We illustrate how these formulae can be used, provide some insights on their robustness, and show how they inform the design of experiments to measure fibril fragmentation dynamics. These advances are made possible by our central theoretical result on how the length distribution profile of the solution to the pure fragmentation equation aligns with a steady distribution profile for large times.

Thanks to the thorough numerical investigation carried out in , we realized the limitation of the previous approaches. Whereas analysing the long-time dynamics allowed us to infer in a very efficient way the division rate characteristics, this does not provide enough information do infer the fragmentation kernel, i.e. the distribution of offspring sizes out of the breakage of a polymer. We thus developed a totally different approach in , where we considered short-time instead of large-time behaviour, which revealed strikingly informative. We first provided a new representation of the solution to the fragmentation equation as a power series in the Banach space of Radon measures endowed with the total variation norm. This representation is used to justify how the fragmentation kernel, which is one of the two key parameters of the fragmentation equation, can be recovered from short-time experimental measurements of the particle size distributions when the initial condition is a delta function. A new stability result for this equation is also provided using a Wasserstein-type norm. We exploit this stability to prove the robustness of our reconstruction formula with respect to noise and initial data.

Background Telomerase-negative cells have limited proliferation potential. In these cells, telomeres shorten until they reach a critical length and induce a permanently arrested state. This process called replicative senescence is associated with genomic instability and participates in tissue and organismal ageing. Experimental data using single-cell approaches in the budding yeast model organism show that telomerase-negative cells often experience abnormally long cell cycles, which can be followed by cell cycles of normal duration, before reaching the terminal senescent state. These series of non-terminal cell cycle arrests contribute to the heterogeneity of senescence, as we already established in previous studies , and likely magnify its genomic instability. Due to their apparent stochastic nature, investigating the dynamics and the molecular origins of these arrests has been difficult. In particular, whether the non-terminal arrests series stem from a mechanism similar to the one that triggers terminal senescence is not known.

Replicative senescence is a complex process that affects cell divisions earlier than anticipated, as exemplified by the frequent occurrence of non-terminal arrests early after telomerase inactivation. The present work unravels two kinetically and mechanistically distinct generation-dependent processes underlying non-terminal and terminal senescence arrests. We suggest that these two processes are responsible for two consequences of senescence at the population level, the increase of genome instability on the one hand, and the limitation of proliferation capacity on the other hand.

In neuroscience, learning and memory are usually associated to long-term changes of neuronal connectivity.
In this context, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight.
Spike-Timing Dependent Plasticity (STDP) is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons.

If numerous models of neuronal cells have been proposed in the mathematical literature, few of them include a variable for the time-varying strength of the connection.
A new, general, mathematical framework is introduced to study synaptic plasticity associated to different STDP rules.
The system composed of two neurons connected by a single synapse is investigated and a stochastic process describing its dynamical behavior is presented and analyzed.
The notion of plasticity kernel is introduced as a key component of plastic neural networks models, generalizing a notion used for pair-based models.
We show that a large number of STDP rules from neuroscience and physics can be represented by this formalism.
Several aspects of these models are discussed and compared to canonical models of computational neuroscience.
An important sub-class of plasticity kernels with a Markovian formulation is also defined and investigated.
In these models, the time evolution of cellular processes such as the neuronal membrane potential and the concentrations of chemical components created/suppressed by spiking activity has the Markov property.

In neuroscience, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. A Spike-Timing Dependent Plasticity (STDP) rule is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. A general mathematical framework has been introduced in .

In this paper we develop and investigate a scaling approach of these models based on several biological assumptions. Experiments show that long-term synaptic plasticity evolves on a much slower timescale than the cellular mechanisms driving the activity of neuronal cells, like their spiking activity or the concentration of various chemical components created/suppressed by this spiking activity. For this reason, a scaled version of the stochastic model of in is introduced and a limit theorem, an averaging principle, is stated for a large class of plasticity kernels. A companion paper is entirely devoted to the tightness properties used to prove these convergence results.

These averaging principles are used to study two important STDP models: pair-based rules and calcium-based rules. Our results are compared with the approximations of neuroscience STDP models. A class of discrete models of STDP rules is also investigated for the analytical tractability of its limiting dynamical system.

In these papers we consider a stochastic system with two connected nodes, whose unidirectional connection is variable and depends on point processes associated to each node.
The input node is represented by an homogeneous Poisson process, whereas the output node jumps with an intensity that depends on the jumps of the input nodes and the connection intensity.
We study a scaling regime when the rate of both point processes is large compared to the dynamics of the connection.
In neuroscience, this system corresponds to a neural network composed by two neurons, connected by a single synapse.
The strength of this synapse depends on the past activity of both neurons, the notion of synaptic plasticity refers to the associated mechanism.
A general class of such stochastic models has been introduced in and to describe most of the models of long-term synaptic plasticity investigated in the literature.
The scaling regime corresponds to a classical assumption in computational neuroscience that cellular processes evolve much more rapidly than the synaptic strength.

The central result of the paper is an averaging principle for the time evolution of the connection intensity. Mathematically, the key variable is the point process, associated to the output node, whose intensity depends on the past activity of the system. The proof of the result involves a detailed analysis of several of its unbounded additive functionals in the slow-fast limit, and technical results on interacting shot-noise processes.

In

we investigate spike-timing dependent plasticity (STPD) in the case of a synapse connecting two neuronal cells. We develop a theoretical analysis of several STDP rules using Markovian theory. In this context there are two different timescales, fast neuronal activity and slower synaptic weight updates. Exploiting this timescale separation, we derive the long-time limits of a single synaptic weight subject to STDP. We show that the pairing model of presynaptic and postsynaptic spikes controls the synaptic weight dynamics for small external input, on an excitatory synapse. This result implies in particular that mean-field analysis of plasticity may miss some important properties of STDP. Anti-Hebbian STDP favors the emergence of a stable synaptic weight, but only for high external input. In the case of an inhibitory synapse the pairing schemes matter less, and we observe convergence of the synaptic weight to a non-null value only for Hebbian STDP. We extensively study different asymptotic regimes for STDP rules, raising interesting questions for future works on adaptive neuronal networks and, more generally, on adaptive systems.

In this project, stationary non-linear Hawkes processes are revisited by formulating the Hawkes dependence as a Markovian property on the space of non-negative sequences. A characterization of the associated point process is obtained in terms of the solution of stochastic differential equations. When the influence of past jumps decreases exponentially over time, the Palm measure of the associated stationary point process is expressed with the distribution of the stationary version of a one-dimensional Harris ergodic Markov chain. Finally a scaling result for some Hawkes processes exhibiting a blow-up phenomenon is derived.

We develop a simple numerical model of the striatum, integrating cortical spiking inputs to study the role of anti-Hebbian STDP in sequence learning. In this model, cortical neurons are seen as binary input neurons and one striatal MSN is modeled as a leaky-integrate-and-fire neuron. Patterns, composed by sequences of cortical spikes, are presented to the MSN whose spiking binary pattern models the output of the circuit. Combined information from the output, reward and timing between the different spikes modify the intensity of each connection, through two mechanisms: anti-Hebbian STDP and reward signaling, using a simple additive three-factor learning rule. The network learning capacity is measured by a score, based on the prediction of rewarded and non-rewarded patterns. The learning dynamics and efficiency are compared between different settings (number of neurons, intensity of the plasticity, types of STDP, tolerance to random noise). Two important properties of the striatal networks, spiking latency and collateral inhibition have subsequently been added to the model and lead to an improvement of the global accuracy.

Altogether, these results suggest that anti-Hebbian STDP favors the learning of complete sequence of spikes, such as is needed in the striatum.

Bradykinesia is defined as a motor slowness and is associated with decrement of the amplitude and/or the speed of movement. Bradykinesia is a key parkinsonian feature yet subjectively assessed by the MDS-UPDRS score, making reproducible measurements and follow-up challenging. Using a deep learning inspired approach, we developed a tool to compute an objective score of bradykinesia. A large database of videos showing parkinsonian patients performing MDS-UPDRS protocols has been acquired in a Movement Disorder unit. Several deep learning algorithms, including DeepLabCut®, were applied to detect a 2D skeleton of the hand composed of 21 predefined points, and transpose it into a 3D network. A two- and three-dimensional semi-automated analysis tool was then developed to study the evolution of several key parameters during the protocol repetitions.

The general goal of this PhD work, started in September 2020, is of developing a scaling approach to analyze stochastic models of chemical networks.

A chemical network is defined with three components

As an example, if

corresponds to an edge of the graph for the transformation of the
complex

if

Historically, Horn, Jackson and Feinberg developed in the 70's, in a deterministic framework associated to a polynomial differential system, several results based on the geometry of the graph. We can mention the famous deficiency zero theorem, that give sufficient conditions on the structure of the graph of the reaction network to have a stable equilibrium point. Most of the stochastic stability results are essentially based on these results.

The goal of this study is to find alternative ways to show results of stability and convergence in distribution. The general approach is to use some scaling arguments on the Markov process associated to the Chemical Reaction Network studied. Scaling in time and/or in space gets us some converging results, avoiding the usual search for a Lyapunov function. We also used averaging results in specific border cases. We are currently analyzing simple networks, binary (i.e. with complexes composed only of two molecules), and weakly reversible (i.e. with a graph weakly reversible).

The information flow from DNA genes to proteins is a fundamental process that is common to all living organisms. We analyze it in the context of a bacterial cell. The production of proteins is the most important process that takes place inside a bacterium consuming around 90% of its resources.

Bacterium adapts very quickly to changes in the environment. It succeeds in making a balance between supply allocating the resources available in the environment for its growth. All of this is achieved via regulatory mechanisms in the bacterium.

Basically the bacterium has two different regimes. The exponential phase during which resources are abundant, enabling the doubling of the cell on a fast pace. The Stationary phase during which resources are scarce, a regulatory mechanisms known by the "sequestration by the 6S RNA" comes into play. The 6S RNA is a macromolecule which is a regulator of transcription by preventing the polymerase from binding to the gene promoter.

Our main objectives are to estimate the efficiency of the 6S RNA regulatory mechanisms via the analysis of the time evolution of the concentration of free, sequestered and in transcription polymerases according to the growth rate. Due the disorganized medium, stochastic models are used for such analysis.

We worked on a model in which we have the same constant number

such that

We have also analyzed the fluctuations of

where

where

In this model we take into consideration: the transcription of mRNA by a polymerase, the sequestration by the 6S RNA, as well as the production (at rate

Our main result in this model is this convergence in distribution:

where

For this model the number of sequestered polymerases is of the order of

We are currently studying the fluctuations of

We proposed different models to serve as a basis for the design of control strategies relying on releases of sterile male mosquitoes (Aedes spp) and aiming at elimination of wild vector population.
Different types of releases were considered (constant, periodic or impulsive) and sufficient conditions to reach elimination were provided in each case
We also estimated sufficient and minimal treatment times.
A feedback approach was introduced, in which the impulse amplitude is chosen as a function of the actual wild population .

We modelled and designed optimal release control strategy with the help of a least square problem. In a nutshell, one wants to minimize the number of uninfected mosquitoes at a given time horizon, under relevant biological constraints. We derived properties of optimal controls and studied a limit problem providing useful asymptotic properties of optimal controls , , .

The main subject of this study is concerned with the effects of the migration of mosquitoes on the spread and sustainability of Wolbachia invasion. Recent works focused on the time dynamical systems or consider the spatial dynamics in an unbounded domain, typically the real line. Based on a previous work by M. Strugarek and N. Vauchelet, we studied more realistic situations by considering the spatial dynamics in a bounded domain, with Robin boundary conditions taking into account migration term. Existence and stability of the equilibrium points have been considered.

Understanding mosquitoes life cycle is of great interest presently because of the increasing impact of vector borne diseases. Observations yields evidence of oscillations in these populations independent of seasonality, still unexplained. We proposed a simple mathematical model of egg hatching enhancement by larvae which produces such oscillations that conveys a possible explanation.

On the other hand, population oscillations may be induced by seasonal changes. We considered a biological population whose environment varies periodically in time, exhibiting two very different “seasons”, favorable and unfavorable. We addressed the following question: the system's period being fixed, under what conditions does there exist a critical duration above which the population cannot sustain and extincts, and below which the system converges to a unique periodic and positive solution? We obtained , sufficient conditions for such a property to occur for monotone differential models with concave nonlinearities, and applied the obtained criterion to a two-dimensional model featuring juvenile and adult insect populations.

The issue of effective scheduling of the releases of Wolbachia-infected mosquitoes is an interesting problem for Control theory.
Having in mind the important uncertainties present in the dynamics of the two populations in interaction, we attempted to identify general ideas for building release strategies, which should apply to several models and situations .
These principles were exemplified by two interval observer-based feedback control laws whose stabilizing properties were demonstrated when applied to a model retrieved from .

In order to tackle the issue of mosquito population control in presence of insecticide, we developed a class of fast-slow models for adaptive resistance evolution . This allowed to model altogether the Mendelian inheritance of the resistance insecticide, and the maternal inheritance of Wolbachia .

The sterile insect technique consists in massive release of sterilized males in the aim to reduce the size of mosquitoes population or even eradicate it. In this work, we investigate the feasability of using the sterile insect technique as a barrier against invasion of a mosquito-free zone (e.g. an urban area) by the mosquitoes coming from outside. In and we were able to show (both analytical and numerical results) that performing the sterile insect technique on a sufficiently large band we may stop invasion. In we were also able to extend this study to the feasability of implementing a Rolling Carpet strategy to increase the size of the mosquito-free area thanks to performing sterile male releases on a band that is moved towards the mosquito infected area at an appropriate speed depending on its size and on the number of sterile mosquitoes released.

We introduce a mathematical description of the impact of contacts in the spread of infectious diseases by integrating an epidemiological dynamics with a kinetic modeling of population-based contacts. The kinetic description leads to study the evolution over time of Boltzmann-type equations describing the number densities of social contacts of susceptible, infected and recovered individuals, whose proportions are driven by a classical SIR-type compartmental model in epidemiology. Explicit calculations show that the spread of the disease is closely related to moments of the contact distribution. Furthermore, the kinetic model allows to clarify how a selective control can be assumed to achieve a minimal lockdown strategy by only reducing individuals undergoing a very large number of daily contacts.

The current outbreak of Covid-19 and the entailed implementation of social distancing on an unprecedented scale, led to a renewed interest in modelling and analysis of the nonpharmaceutical intervention strategies to control infectious diseases. The term ‘‘social distancing” (including, but not limited to, “physical distancing”) refers to attempts to directly reduce the infecting contacts within the population. In absence of vaccine or therapy, such containment strategies constitute probably the only mid-term option. An issue of interest is to understand how one can minimize the epidemic final size, or equivalently the total number of individuals infected during the outbreak, given maximal social distancing duration and intensity. Voluntarily ignoring many features important in the effective handling of a human epidemic, we investigated this question on a simple SIR model. A complete answer was given in for optimal control on an interval with prescribed starting date, and in in the case of free starting date.

Testing for the infected cases is one of the most important mechanisms to control an epidemic. It enables to isolate the detected infected individuals, thereby limiting the disease transmission to the susceptible population. We presented in , , an epidemic model that incorporates the testing rate as a control input. The proposed model differentiates the undetected infected from the detected infected cases, who are assumed to be removed from the disease spreading process in the population. The model has been estimated and validated for Covid-19 data in France, and two testing policies were proposed and evaluated by predicting the number of active intensive care unit (ICU) cases and the cumulative number of deaths.

The phenomenon of reinfection, and particularly the counting of the number of reinfections, have been little studied so far. We considered here an infinite-dimensional system of ordinary differential equations (pretty much in the spirit of Becker-Döring system ) extending a usual SEIRS model and capable to count explicitly the number of reinfections of the individuals of the population. Having disentangled this underlying structure allows to express mean numbers of reinfections in the population. Also, we have demonstrated that measuring the number of infected individuals and the number of primo-infected individuals allows to collect more information on the system – more precisely, this may render observable and identifiable a simple SIS model which has none of this property when only one measurement is achieved.

Modelling the movements and mixing of populations in complex urban environments is an important issue to attempt to reproduce epidemic dynamics. To begin the exploration of this question, we study in this methodological work the effects of various modelling options on the stability of the disease-free equilibrium of simple systems.

Several class of models have been devised to describe the macroscopic dynamics of living tissues (including growing tumors), depending on the mechanical behaviour of the tissue. The team has progressed on several aspects: analysis of models and asymptotic analysis towards free boundary problem, numerical methods compatible with energy properties.

Collective motion of cells plays an important role and the case of self-propelled particles confined between two parallel plates has attracted a lot of attention. In , we considered cells moving with a constant speed while their direction changes by rotational diffusion. The probability distribution of such micro-organisms in confined environment is singular because particles accumulate at the boundaries. This leads us to distinguish between the probability distribution densities in the bulk and in the boundaries. They satisfy a degenerate Fokker-Planck system and we propose boundary conditions that take into account the switching between free-moving and boundary-contacting particles. Relative entropy property, a priori estimates and the convergence to an unique steady state are established. The steady states of both the PDE and individual based stochastic models are compared numerically.

The research topic “Evolution and cancer”, designed in the framework
of adaptive dynamics to represent and overcome acquired drug
resistance in cancer, initiated in , and later continued
in , , has been summarised
in , presented in more detail in , and has been the object of the PhD
thesis work of Camille Pouchol, see above “Cell population dynamics
and its control” . In collaboration with F. Vallette's INSERM team in Nantes, it gave rise to the publication of the article . It is now oriented, thanks to work underway by
Frank Ernesto Alvarez Borges, Jean Clairambault, and Stéphane Mischler , in particular towards the mathematical representation of bet hedging in cancer, namely a supposed optimal strategy consisting
for cancer cell populations under life-threatening cell stress in
diversifying their phenotypes according to several resistance
mechanisms, such as overexpression of ABC transporters (P-glycoprotein
and many others), of DNA repair enzymes or of intracellular
detoxication processes. According to different deadly insults the
cancer cell population is exposed to, some phenotypes may be selected,
any such successful subpopulation being able to store the cell
population genome (or subclones of it if the cell population is
already genetically heterogeneous) and make it amenable to survival
and renewed replication.

In the setting of Emma Leschiera's thesis, we have studied the interaction between the tumor and the immune system. In particular, a first paper concerning the role of Intra-tumour heterogeneity (ITH) was recently accepted for publication in J. Theor. Biol. . Several expoerimental papers that inspired this work, have shown that ITH has a strong impact on the efficacy of the immune response against solid tumours. In this work we considered that ITH can be measured both through the number of sub-populations of cancer cells expressing different antigens and through the percentage of immunogenic cells in a tumour. We proposed and implemented a spatially explicit stochastic individual-based model of the interaction dynamics between tumour cells and CD8+ T cells, which makes it possible to evaluate the contribution of these two aspects of ITH on anti-tumour immune response and which are able to retrieve several immunosurveillance scenarios (both successful and unsuccessful) identified in the experimental works.

We are now working on trying to model T-cell infiltration in solid tumours and to retrieve the notion of immunoscore which is a widely used tool to evaluate the chances of success of immunotherapy and an important information to guide the choice of a particular treatment or combination of treatments.

In the frame of the HTE project MoGlImaging and the post-doctorate of Gissell Estrada Rodriguez, we developed a 2D mathematical model to study and analyse the evolution of a population of glioblastoma cells that are exposed to TMZ . Based on the experimental data generated by our partner team led by F. Valette (Inserm Nantes), we proposed a Keller-Segel type model where tumour aggregate formation is obtained as the result of nutrient-limited cell proliferation coupled with chemotaxis-based cell movement. The introduction of a chemotherapeutic treatment is supposed to induce mechanical changes at the cell level, with cells undergoing a transition from rigid bodies to semi-elastic entities. We analysed the influence of these individual mechanical changes on the properties of the aggregates obtained at the population level by introducing a nonlinear volume-filling chemotactic system of partial differential equations. The elastic properties of the cells were taken into account through the so-called squeezing probability, which allowed us to change the packing capacity of the aggregates, depending on the concentration of the treatment in the extracellular microenvironment. By confronting the model results to experimental data, we showed that the changes observed in cellular structures under a non-cytotoxic drug could be due to this mechanotransduction phenomenon. This study suggests a new mechanism which, if experimentally validated, opens interesting therapeutic avenues.

Following these works, we studied how the volume-filling Keller-Segel model (VFKS) could be obtained as the diffusion limit of a kinetic ‘velocity jump’ model, giving insights into how the individual mechanisms at the cell level can lead to volume-filling effects at the population level. In the framework of Xinran Ruan post-doctorate (upcoming paper), we were able to successfully derive the VFKS model from a kinetic 'run-and-tumble' model, and showed that density effects had to be included both in the transport term and in the turning operator. We developed an asymptotic preserving numerical method to show numerically the convergence of the kinetic model to the PDE system.

From a biological point of view, adaptive dynamics and its asymptotics rely on the so-called plasticity of cancer cell populations, i.e., their ability to easily change their phenotypes, thanks to their poor differentiation, to adapt to a changing environment, in particular to develop resistance to cancer treatments. This point of view has been reviewed, from a biological, mathematical and `philosophy of cancer' point of view in , . In these articles, and in the invited conference paper , is particularly developed the idea according to which cancer is characterized, not so much as a default of control on cell proliferation, but at least equally as a default of control on cell differentiations. This idea is not new (in particular it has been put forward in Marta Bertolaso's book of 2016 “Philosophy of cancer”, Springer Publ.), nevertheless it could lead to modeling developments that should complement the classical models based on sheer proliferation of cell populations, and possibly open the way to new therapeutic tracks, provided that can be found actual means of control and reestablishment of physiological cell differentiation, that so far exist for very few cancer diseases (e.g., for acute promyelocytic leukemia).

Of note, philosophy of cancer is thus a point of convergence between mathematics, biology and social and human sciences, that may help biologists and mathematicians to bring new insight to understanding this old disease.

We have developed a model for collective dynamics with weights, in which each agent is described not only by its position, but also by a positive “weight of influence”. The weights allow us to model a social hierarchy within the group, where the most influencial agents (the ones with the largest weights) have a larger impact on the behavior of the group. Moreover, the weights of influence are susceptible to evolve in time, which models the changing social hierarchy.

As often in models of collective dynamics, we focused on the question of its continuum limit when the number of agents tends to infinity. However, for this particular model, the classical mean-field limit can fail when the weight dynamics do not preserve indistinguishability of the particles: indeed, the essense of the mean-field approach is to describe the population by its density, hence requiring all particles to be indistinguishable. This led us to another approach: the graph limit method.

In 2014, Medvedev used techniques from the recent theory of graph limit to derive rigorously the continuum limit of dynamical models on deterministic graphs. In our paper , we extended this idea to our collective dynamics model with time-varying weights, adopting the graph point of view. The central point of this approach consists of describing the infinite population by two functions

In a collaboration between MAMBA and the Inria team CAGE, centered around the post-doctoral position of Benoit Bonnet, we studied the large-time behavior of interacting particle systems modeled by graph dynamics.

Although the case of finite-dimensional Hegselmann-Krause systems is well-known in the literature, the question of understanding the formation of consensus in large interacting systems where the number of agents goes to infinity is less understood. Indeed, as explained in the previous paragraph, when the interaction kernels depend explicitly on the agent labels, the standard mean-field approach is not applicable. The latter can only account for systems whose dynamics are invariant under permutations, which greatly limits the admissible interaction models. Motivated by this observation, several recent contributions have been aiming at studying macroscopic approximations of collective dynamics which allow for more general and possibly asymmetric interaction functions. The corresponding approaches are based on the theory of graph limits introduced by Lovász and Segedy, and popularised by Medvedev.

In

, we modeled and studied the patterns created through the interaction of collectively moving self-propelled particles (SPPs) and elastically tethered obstacles. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivated the derivation of a macroscopic partial differential equations model for the interactions between the self-propelled particles and the obstacles, for which we assumed large tether stiffness. The result is a coupled system of non-linear, non-local partial differential equations. By performing a linear stability analysis, we showed that patterning was expected if the interactions are strong enough and allowed for the predictions of pattern size from model parameters. The macroscopic equations revealed that the obstacle interactions induce short-ranged SPP aggregation, irrespective of whether obstacles and SPPs are attractive or repulsive.

To model the morphogenesis of rod-shaped bacterial micro-colony, several individual-based models have been proposed in the biophysical literature. When studying the shape of micro-colonies, most models present interaction forces such as attraction or filial link. In the article , we propose a model where the bacteria interact only through non-overlapping constraints. We consider the asymmetry of the bacteria, and its influence on the friction with the substrate. Besides, we consider asymmetry in the mass distribution of the bacteria along their length, and the division follows the so-called "adder model" (see Section ). These new modelling assumptions allow us to retrieve mechanical behaviours of micro-colony growth without the need of interaction such as attraction. We compare our model to various sets of experiments, discuss our results, and propose several quantifiers to compare model to data in a systematic way. We now aim at deriving a space-and-size structured population equation as the macroscopic limit of a simplified version of this model.

In , we propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and intersticial fluid flows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. This work offers a new paradigm for capillary network creation by placing the flow of blood at the central place in the process. The model proposed in provides a proof of concept of this approach and elaborates a road map by which the model can be gradually improved towards a fully fledged simulator of blood capillary network formation. Such simulator would have huge potential for biological or clinical applications in cancer, wound healing, tissue engineering and regeneration.

Cell-fate transition can be modeled by ordinary differential equations which describe the behavior of several molecules in interaction, and for which each stable equilibrium corresponds to a possible phenotype (or 'biological trait'). In the paper , we focused on simple ODE systems modeling two molecules which each negatively (or positively) regulate the other. It is well-known that such models may lead to monostability or multistability, depending on the selected parameters. However, extensive numerical simulations have led systems biologists to conjecture that in the vast majority of cases, there cannot be more than two stable points. Our main result is a proof of this conjecture. More specifically, we provided a criterion ensuring at most bistability, which is indeed satisfied by most commonly used functions. This includes Hill functions, but also a wide family of convex and sigmoid functions. We also determined which parameters lead to monostability, and which lead to bistability, by developing a more general framework encompassing all our results.

Computer-Vision and Deep Learning applied to Safran Electronics Defense objectives.Survey of multiple-instance learning and few-shot learning algorithms applied to DRI (detection, recognition, identification).

Simulation of growth efficiency and cell yield in multiple in vitro experimental settings to better understand the impact of the chosen culturing method and to guide potential improvements of the outcome.

Gaëtan Vignoud has obtained a Fulbright Fellowship, as a visiting student researcher at Brandeis University, from October 2021 to March 2022. He will work with Pr. Jonathan D. Touboul on the switch between habitual and goal-directed behaviors in models of the striatum.

Mamba (Marie Doumic and Philippe Robert) participates to the GDR "MeDyna" (mechanisms and dynamics of assemblies of peptides and proteins), coordinated by Stéphane Bressanelli from IBPC.

, headed by Vincent Andrieu, univ. of Lyon.

Telomere dynamics, headed by Teresa Teixeira (IBPC, Paris).

Khê HOANG-XUAN, Hôpital Universitaire La Pitié Salpêtrière, Paris. Mathematical modeling at micro and macroscopic level of primary central nervous system lymphomas (PCNSL).

Early leukaemogenesis in Acute Myelogenous Leukaemia (AML), 8 teams headed by François Delhommeau (CDR St Antoine, Paris).

Treatment-induced treatment resistance and heterogeneity in glioblastoma, 8 teams headed by Elizabeth Moyal (INSERM, Toulouse).

Pierre-Alexandre Bliman participates in the project HealthyMobility (Optimal Policies for Human Mobility to Control CoVID19-Epidemic Spread under Health and Economics Constraints), in cooperation with the Necs-Post team (CNRS, Gipsa, UGA, Inria), in the framework of Inria Covid-19 mission.

This project establishes liver systems medicine approaches to understand disease pathways and consequences of liver disease on liver function. The project is a large network projects linking many partners all over Germany.

MATHREGEN 2021-2023: MAthematical modeling of Tissue Homeostasis and REGENeration - Project Emergence Alliance Sorbonne University awarded to Diane Peurichard in collaboration with RESTORE, Toulouse

Summary: The goal of MATHREGEN is to open a therapeutic breakthrough in one of the great societal challenges that is regeneration of whole organs. To this aim, MATHREGEN proposes to study the mechanisms of tissue homeostasis and repair, via the development of innovative tools combining mathematical agent-based modelling, software development and sophisticated image processing techniques for calibration. This highly interdisciplinary project proposes a functional and simple view of complex biological systems, relying on the use of mathematical models to identify the main parameters involved in tissue repair coupled with state-of-the-art model calibration and model-data coupling methodologies. The experimental data provided by our biological partners will permit to test, validate and improve the underlying hypotheses of the models. If successful, this project will provide a list of key determinants that induce tissue reprogramming, suggesting new avenues for antiscarring therapies and tissue regeneration.

NoMicMac 2021-2023: Novel Microscopic-to-Macroscopic limit
of short-range repulsion models - Project Emergence Alliance Sorbonne University awarded to Nastassia Pouradier Duteil

Summary:
Large systems of interacting agents have long been the focus of a wide community of both mathematicians
and applied scientists, captivated by the following question: how do globally organized patterns
emerge in a system driven only by local interactions? Such behavior is ubiquitous in many systems,
and understanding the emergence of patterns has many applications in crowd dynamics, animal group
behavior (swarming,
flocking, migration), and cells' organization in tissues.
The classical approach to link microscopic and macroscopic models is a limit process called “mean-field limit”. However, this approach has a main drawback: it does not take into account the intrinsic
volume of the individuals, since they are approximated by their centers of mass. As a result, in many
cases the limiting PDE fails to reproduce the behavior of the microscopic system, in particular when
modeling congestion effects due to size constraints.
The aim of this project is to address this issue by deriving rigorously a macroscopic limit for a microscopic
model of a population subject to non-overlapping constraints, and to reveal the link between distance and density constraints.

Pierre-Alexandre Bliman was the scientific chair and organizer of the seminary of the STIC AmSud project NEMBICA, from September 20–24, 2021.

Diane Peurichard and Nastassia Pouradier Duteil co-organized, together with Chloé Audebert, a Mittag-Leffler summer school entitled “Multi-scale modeling for pattern formation in biological systems”. The summer school gathered the participants around partial differential equations and numerical analysis applied to biological systems. It was composed of two courses of five hours each, one on partial differential equations and modeling of biological systems (taught by Marie-Therese Wolfram), and one on numerical analysis (taught by Magali Ribot). The rest of the school was organized around themed mini-symposia by all participants. Due to the pandemic, the school took place entirely online.

Marie Doumic and Diane Peurichard organized a minisymposium entitled 'Exploring the processes of bacteria self-organization using mathematical modelling and experimental studies' at the online SMB meeting, from June 13–17 2021.

Marie Doumic was a member of the SMAI conference scientific committee and of the workshop "nonlocal models arising from mathematical biology"

Philippe Robert is co-chair of the Sigmetrics/Performance 2022 conference.

Pierre-Alexandre Bliman was associate editor of the 2022 European Control Conference.

Pierre-Alexandre Bliman was reviewer for IEEE Conference on Decision and Control.

Jean Clairambault is Associate Editor of the journal “Frontiers in Applied Mathematics and Statistics", Section “Frontiers in mathematical biology", and Invited Editor of the journal “Mathematical Modelling of Natural Phenomena".

Marie Doumic is Editor in Chief of ESAIM Proc. and Associate Editor of "Kinetic and Related Models", Bulletin des Sciences Mathématiques" and the Journal of Mathematical Biology.

Philippe Robert is Associate Editor of the journal “Stochastic Models”.

Nastassia Pouradier Duteil is Guest Editor for the OCAM (Optimal Control Applications and Methods) special issue on “Optimal control in therapeutics and epidemiology”.

Nastassia Pouradier Duteil was a reviewer for the following journals: Communications in Mathematical Sciences, Journal of Mathematical Biology, SIAM Journal on Mathematical Analysis.

Diane Peurichard was a reviewer for the following journals: Mathematics, SIAM Journal on Mathematical analysis, JCOMP, European Journal of Applied Mathematics, Symmetry.

Pierre-Alexandre Bliman was reviewer for the following journals: Automatica, Communications in Nonlinear Science and Numerical Simulation, Communication Physics, Discrete and Continuous Dynamical Systems Series B, Journal of the Franklin Institute, IEEE Transactions on Automatic Control, Journal of Mathematical Biology, Journal of THeoretical Biology, Mathematical Biosciences, Mathematical Modelling of Natural Phenomena, Nonlinear Analysis: Real World Applications, International Journal of Robust and Nonlinear Control.

Jean Clairambault gave the following talks:

Philippe Robert gave a talk “Stochastic Models of Neural Synaptic Plasticity” at the seminar “Stochastic Networks, Applied Probability, and Performance” on Oct 04, 2021.

Gaëtan Vignoud gave a talk “On the Spontaneous Dynamics of Synaptic Weights in Stochastic Models” at the 2021 International Conference on Mathematical Neuroscience - Digital Edition on 29th of June 2021. He has presented a poster “Movement disorders analysis using a deep learning approach” at IDAI Summer School 2021.

Nastassia Pouradier Duteil gave a seminar at the Online IntComSin Colloquium of Erlangen-Regensburg (Germany), “Continuum limits of collective dynamics with time-varying weights” on June 18, 2021. She also gave a presentation at the workshop “Numerical aspects of hyperbolic balance laws and related problems – Young Researchers Conference” in Verona (Italy) on Dec. 16, 2021.

Marie Doumic gave online Colloquiums for the university of Bonn (Germany) on February 2nd and for ECMTB on June 25th. She gave two talks as a two-week visiting fellow in the COLIBRI network of the university of Graz (Austria) from October 20th to November 2nd.

Diane Peurichard gave an (online) talk at the 15th International Conference on Free Boundary Problems on september 13th-17th and a talk at the Young Researcher Meeting for the conference "Numerical Aspects of Hyperbolic Balance Laws and Related Problems”, held from December 15 to December 17, 2021, at the University of Verona, Italy.

Pierre-Alexandre Bliman gave invited talks in the International Conference “Dynamical Systems Applied to Biology and Natural Sciences – DSABNS" (February 2021) and in the “Encontro nacional de modelagem matemática da Covid-19 (May 2021), presented communications at the conferences 2021 European Control Conference (June 2021) and at the seminaries Séminaire AFRIMath de modélisation mathématique (June 2021), GIPSA-Lab in Grenoble (July 2021), “Epidemics: modeling, identification, control" (October 2021), and in Universidade de Lisboa (October 2021).

Pierre-Alexandre Bliman was member of the Comités d’évaluation scientifique for the two ANR Calls for projects RA-Covid-19 and Résilience Covid-19, and expert for the ANR Appel spécifique - Action Liban.

Marie Doumic is a member of the Scientific Council of INSMI (mathematical institute of the CNRS) and member of the interdisciplinary committee CID 51 of CNRS. She also represents Inria at CNFM (Comité National Français des Mathématiques).

Philippe Robert is teaching the master M2 course `Modèles Stochastiques de la Biologie Moléculaire`” at Sorbonne Université.

Nastassia Pouradier Duteil is teaching a course on Mathematics in the Sorbonne University program RESPE “Retour aux Etudes Supérieures des Personnes Exilées”. She also gave a two-hour graduate-level course at the University of Erlangen-Nuremberg (online), entitled “Introduction to mean-field limits”.

Marie Doumic is a part-time "professeuse chargée de cours" in Ecole Polytechnique (67 h/year), teaches probability (L3) and variationnal analysis of PDE (M1).

Diane Peurichard is teaching the module 'Fondement des methodes numeriques' (M1, 48h) and 'Approximation des équations aux dérivées partielles' (M1,24h) at Sorbonne University. She gave a 4h course for the 'Master de physique cellulaire', Strasbourg University on january 21th, 2022 entitled 'Mathematical modelling in biological systems'

Jean Clairambault is a co-advisor of Frank Ernesto Alvarez Borges at Paris-Dauphine University

Philippe Robert is the PhD advisor of

Luis Almeida and Pierre-Alexandre Bliman supervised, together with Nicolas Vauchelet, the master M2 internship of Thi Quynh Nga Nguyen, now PhD student in the team with the same co-advisors.

Pierre-Alexandre Bliman was also the supervisor of Marcel Fang, presently PhD student in MAMBA team with the same advisor. Pierre-Alexandre Bliman is also co-supervisor of Assane Savadogo, together with the Professor Boureima Sangaré from Université Nazi Boni, Bobo-Dioulasso, in the framework of the IRD Programme Doctoral International “Modélisation des Systèmes Complexes".

Nastassia Pouradier Duteil and Diane Peurichard co-advised the M1 master thesis of Agathe Yvinec-Tolmer. Nastassia Pouradier Duteil advised the L2 internship of Maxime Guil. Nastassia Pouradier Duteil is currently co-advising with Camille Pouchol the PhD thesis of Jules Guilberteau.

Marie Doumic and Marc Hoffmann co-supervise, together with Lydia Robert from INRAE, Guillaume Garnier's Ph.D thesis, begun in October 2021.

Marie Doumic co-supervises Anaïs Rat's Ph.D with Magali Tournus from Ecole Centrale Marseille.

Diane Peurichard co-supervises the Phd of Pauline Chassonnery (ED Biologie-Santé-Biotechnologie, Toulouse) with Louis Casteilla (RESTORE, Toulouse)

Diane Peurichard co-supervises the PhD of Valeria Caliaro (Inria Paris) with Osvaldo Chara (Sysbio, Argentina and Technische Universität Dresden, Dresden, Germany)

Diane Peurichard and Luis Almeida supervised together the post-doctorate of Xinran Ruan (now professor at Capital Normal University, Beijin, China)

Jean Clairambault has been a reviewer of Giada Fiandaca's PhD thesis “Tumour phenotypic heterogeneity in a niche-construction evolutionary perspective: investigating the impact of trade-offs via a mathematical approach”, Politecnico di Torino, defended on January 28, 2022.

Philippe Robert has been the reviewer of Pascal Helson's PhD document “Plasticity in networks of spiking neurons in interaction”, Université Côte d'Azur, defended on March 29, 2021.

Pierre-Alexandre Bliman has been reviewer of Amidou Traore's PhD “Modélisation et contrôle d’un insecte ravageur", Université Joseph Ki-Zerbo, Ouagadougou and Université de Bordeaux, defended on February 19, 2021. He has also been reviewer of Souâd Yacheur's PhD “Modélisation et étude mathématique de la propagation d’une maladie vectorielle (paludisme) au sein d’une population", Université Aboubekr Belkaid, Tlemcen and Université de Lorraine, Metz, defended on December 16, 2021. last, he has been member of the panel of Muhammad Umar B. Niazi “Aggregated Monitoring of Large-scale Network Systems and Control of Epidemics", Université de Grenoble Alpes, defended on July 13, 2021.

Diane Peurichard is a member of the selection commitee for the CRCN/ISFP positions 2022, Inria Saclay

Diane Peurichard is a member of the selection commitee for the CRCN positions at INRAE 2022 "Modèles dynamiques pour la modélisation en écologie microbienne"

Diane Peurichard is a member of the selection commitee for the MCF positions 2022, Université de Montpellier (section 26, EDPs appliquées à la dynamique des populations)

Marie Doumic gave a conference at the Mathpark seminar of Institut Henri Poincaré on December 11.

Diane Peurichard is member of the CDT (Commission de Développement Technologique) at Inria Paris

Diane Peurichard is member of the CSD (Comité de Suivi Doctoral) at Inria Paris

Diane Peurichard is member of the pôle ecoute at Sorbonne Université (instance in charge of support of researchers regarding communication problems, discomfort, harrassement issues etc)