2025Activity​​ reportProject-TeamMUSICS

6.1 Latest software​‌ developments

7.1​‌ Multiscale modeling of the​​ spatial structure of stem​​ cells in neuroblastoma patient-derived​​​‌ tumoroids reveals a critical‌ role for a short-range‌​‌ diffusive process

Participants: Thi​​ Nhu Thao Nguyen,​​​‌ Samuel Bernard, Olivier‌ Gandrillon.

Neuroblastomas are‌​‌ heterogeneous pediatric tumors of​​ the sympathetic nervous system​​​‌ for which treatments are‌ still limited. Fundamental and‌​‌ applied approaches have been​​ enabled thanks to the​​​‌ generation of patient-derived tumoroids‌ (PDTs), ex vivo 3D‌​‌ structures used as avatars​​ of the original tumor.​​​‌ In 32, we‌ generated neuroblastoma PDTs and‌​‌ quantified the spatial distribution​​ of CD133 + cancer​​​‌ stem cells using immunohistochemistry.‌ We observed that those‌​‌ cells tend to aggregate​​ in the PDT. In​​​‌ order to better understand‌ the set of rules‌​‌ needed for generating such​​ structures, we implemented a​​​‌ multiscale agent-based neuroblastoma tumoroid‌ model. Model rules specify‌​‌ single cell's fate based​​ on its intracellular content,​​​‌ which dynamically evolves according‌ to a stochastic gene‌​‌ regulatory network. The state​​ of this network can​​​‌ be modulated by cell-to-cell‌ signalling through neighbor cells‌​‌ fate decisions and, possibly,​​ spatial location. We first​​​‌ observed that in the‌ absence of any spatial‌​‌ rules for inter-cellular interactions,​​ no spatial structure emerged.​​​‌ The addition of simple‌ rules (signalling by cell-to-cell‌​‌ contact or differential cell​​ adhesion) only marginally improved​​​‌ the quantitative agreement to‌ the experimental dataset. In‌​‌ sharp contrast, the addition​​ of short-range pro-stem cell​​​‌ diffusive signalling among stem‌ cells produced very realistic‌​‌ 3D PDT-like structures. This​​ works highlights the power​​​‌ of our multiscale approach‌ to discard too simplistic‌​‌ rules and to propose​​ a minimal set of​​​‌ hypotheses required to reproduce‌ qualitatively and quantitatively experimentally‌​‌ observed spatial structures. In​​ the case of neuroblastomas-derived​​​‌ PDTs, short-range spatial diffusion‌ of stem-to-stem cell signalling‌​‌ proved to play a​​ key role in successfully​​​‌ reconstructing the spatial structure.‌

7.2 Inferring and simulating‌​‌ a gene regulatory network​​ for the sympathoadrenal differentiation​​​‌ from single-cell transcriptomics in‌ human.

Participants: Olivier Gandrillon‌​‌.

Neuroblastoma is a​​ malignant childhood cancer with​​​‌ significant interand intrapatient heterogeneity‌ arising from the abnormal‌​‌ differentiation of neural crest​​ cells into sympathetic neurons.​​​‌ The lack of actionable‌ mutations limits therapeutic options,‌​‌ highlighting the need to​​ better understand the molecular​​​‌ mechanisms that drive this‌ differentiation. Although RNA velocity‌​‌ has provided some insights,​​ modeling regulatory relationships is​​​‌ limited.

Methods: To address‌ this, we applied in‌​‌ 13 our integrated gene​​ regulatory network (GRNs) inference​​​‌ (CARDAMOM) and simulation (HARISSA)‌ tools using a published‌​‌ single-cell RNAseq dataset from​​ human sympathoadrenal differentiation

Results:​​​‌ Our analysis identified a‌ 97-gene GRN that drives‌​‌ the transition from Schwann​​ cell precursors to chromaffin​​​‌ cells and sympathoblasts, highlighting‌ dynamic interactions such as‌​‌ self-reinforcing loops and toggle​​ switches. The simulation of​​​‌ that GRN was able‌ to reproduce very satisfactorily‌​‌ the experimentally observed gene​​ expression distributions

7.3 A​​​‌ refractory density approach to‌ a multi-scale SEIRS epidemic‌​‌ model

Participants: Laurent Pujo-Menjouet​​.

In 5,​​​‌ we propose a novel‌ multi-scale modeling framework for‌​‌ infectious disease spreading, borrowing​​ ideas and modeling tools​​​‌ from the so-called Refractory‌ Density (RD) approach. We‌​‌ introduce a microscopic model​​​‌ that describes the probability​ of infection for a​‌ single individual and the​​ evolution of the disease​​​‌ within their body. From​ the individual-level description, we​‌ then present the corresponding​​ population-level model of epidemic​​​‌ spreading on the mesoscopic​ and macroscopic scale. We​‌ conclude with numerical illustrations,​​ taking into account either​​​‌ a white Gaussian noise​ or an escape noise​‌ to showcase the potential​​ of our approach in​​​‌ producing both transient and​ asymptotic complex dynamics as​‌ well as finite-size fluctuations​​ consistently across multiple scales.​​​‌ A comparison with the​ epidemiology of coronaviruses is​‌ also given to corroborate​​ the qualitative relevance of​​​‌ our new approach.

7.4​ Mast cells act as​‌ pro-angiogenic and pro-tumorigenic players​​ in pituitary gonadotroph tumors​​​‌

Participants: Olivier Gandrillon.​

Background – The tumor​‌ microenvironment (TME) represents a​​ promising avenue to understand​​​‌ gonadotroph tumors and develop​ therapeutic tools. In 14​‌ , we aimed to​​ gain insight into the​​​‌ tumorigenesis mechanisms driven by​ the gonadotoph TME. Methods​‌ Single-cell and spatial-omics were​​ combined with histological analysis.​​​‌ Mice engrafted with tumor​ cells were used for​‌ functional validation.

Results –​​ Using single-cell and spatial​​​‌ transcriptomic data from gonadotroph​ tumors and normal tissues,​‌ we identified mast cells​​ in the microenvironment of​​​‌ gonadotroph tumors and confirmed​ their physical and functional​‌ interaction with endothelial cells.​​ Quantification of mast cells​​​‌ in 40 patients suggested​ their pro-tumoral role as​‌ tumors relapsing after surgery​​ harbored more mast cells.​​​‌ More interestingly, the distribution​ of mast cells was​‌ associated with the presence​​ of a higher number​​​‌ of blood vessels, with​ an increased microvessel density​‌ (MVD), and with blood​​ vessels with thicker walls.​​​‌ Ligand-receptor network analysis highlighted​ VEGFA as a modulator​‌ of mast/endothelial cell communication,​​ a result confirmed by​​​‌ the identification of intratumoral​ mast cells expressing VEGFA​‌ in mouse and human​​ gonadotroph tumors. Finally, using​​​‌ mice engrafted with gonadotroph​ tumor cells, we demonstrated​‌ that the depletion of​​ mast cells reduces tumor​​​‌ volume through increased apoptosis.​ These observations were associated​‌ with increased hemorrhagic areas​​ and a significant reduction​​​‌ of the number of​ blood vessels and MVD​‌ as evidenced in human​​ gonadotroph tumors.

Conclusion –​​​‌ we demonstrate that mast​ cells represent a new​‌ actor of the gonadotroph​​ TME, and highlight their​​​‌ pro-angiogenic and pro-tumorigenic roles​ as potential targets for​‌ the therapeutic treatment of​​ gonadotroph tumors.

7.5 Mathematical​​​‌ modeling of the feather​ follicle morphogenetic wave in​‌ birds

Participants: Maxime Estavoyer​​, Thomas Lepoutre.​​​‌

During the development of​ the avian skin, feather​‌ follicles are produced in​​ a medio-lateral morphogenetic wave​​​‌ that results in their​ spatial arrangement in typical​‌ patterns. This wave involves​​ the timely acquisition of​​​‌ pattern-forming competence followed by​ a row-by-row produc tion​‌ of feather follicles. While​​ several mathematical models combining​​​‌ self-organizing systems accurately reproduced​ dynamics of feather follicle​‌ pattern formation, the events​​ that control timely parameters​​​‌ of wave propagation remain​ poorly understood. In 10​‌, we built on​​ previous modeling work to​​​‌ theoretically calculate the speed​ at which tissue competence​‌ progresses. Using a weakly​​ non-linear analysis, we calculated​​ the speed at which​​​‌ follicles emerge once competence‌ is attained. We produced‌​‌ numerical simulations of our​​ model to predict the​​​‌ respective influences of competence‌ acquisition and follicle emergence‌​‌ on each other and​​ on wave propagation. Our​​​‌ results show that the‌ theoretical speed of follicle‌​‌ emergence is limited by​​ competence acquisition, but that,​​​‌ in turn, competence acquisition‌ is not constrained by‌​‌ follicle emergence. This modeling​​ work provides an approximation​​​‌ of the timely parameters‌ of the morphogenetic wave,‌​‌ and sheds light on​​ the interplay between competence​​​‌ and patterning events in‌ the developing skin.

7.6‌​‌ Multi-serotype nested immuno-epidemiological model​​ for dengue hemorrhagic fever​​​‌ involving backward bifurcation and‌ serotype invasion

Participants: Mostafa‌​‌ Adimy, Charlotte Dugourd-Camus​​, Ruben Taieb.​​​‌

Reinfection with the same‌ dengue serotype is generally‌​‌ benign, as individuals develop​​ protective immunity. On the​​​‌ other hand, in the‌ case of reinfection with‌​‌ a different serotype, pre-existing​​ antibodies can increase the​​​‌ risk of developing Dengue‌ Hemorrhagic Fever (DHF), by‌​‌ inducing Antibody-Dependent Enhancement (ADE).​​ To model this dynamic,​​​‌ we introduce in 24‌ a multi-scale immuno-epidemiological system.‌​‌ The immunological part is​​ described by a system​​​‌ of ODEs representing the‌ interaction between two antibodies‌​‌ (from previous and current​​ infection) and the virus.​​​‌ The epidemiological part is‌ represented by an infection-age‌​‌ structured SIRS system (for​​ both the primary and​​​‌ secondary infections) and a‌ recovery-age structured equation (for‌​‌ the first infection). A​​ detailed mathematical analysis of​​​‌ the equilibrium points of‌ the multi-scale reinfection model,‌​‌ including disease-free, mono-endemic and​​ bi-endemic states, is performed.​​​‌ We establish necessary and‌ sufficient conditions for the‌​‌ existence of backward bifurcations​​ and derive an expression​​​‌ for the invasion reproduction‌ number, which shows that‌​‌ the second serotype can​​ invade the population after​​​‌ a mono-endemic first serotype.‌ We also investigate the‌​‌ dependence of the basic​​ and invasion reproduction numbers​​​‌ on the immunological parameters‌ of the first and‌​‌ second infections. This gives​​ us a better understanding​​​‌ of the relationship between‌ DHF and ADE during‌​‌ secondary infection.

7.7 Uncovering​​ candidate Nanog-Helper genes in​​​‌ early mouse embryo differentiation‌ using differential entropy and‌​‌ network inference

Participants: Olivier​​ Gandrillon.

In the​​​‌ preimplantation mammalian embryo, stochastic‌ cell-to-cell expression heterogeneity is‌​‌ followed by signal reinforcement​​ to initiate the specification​​​‌ of Inner Cell Mass‌ (ICM) cells into Epiblast‌​‌ (Epi). The expression of​​ NANOG, the key transcription​​​‌ factor for the Epi‌ fate, is necessary but‌​‌ not sufficient: coincident expression​​ of other factors is​​​‌ required. To identify possible‌ Nanog-helper genes, we analyzed‌​‌ in 17 gene expression​​ variability in five time-stamped​​​‌ single-cell transcriptomic datasets using‌ differential entropy, a quantitative‌​‌ measure of cell-to-cell heterogeneity.​​ The entropy of Nanog​​​‌ displays a peak-shaped temporal‌ pattern from the 16-cell‌​‌ to the 64-cell stage,​​ consistent with its key​​​‌ role in Epi specification.‌ By estimating the entropy‌​‌ profiles of the 21​​ genes common to all​​​‌ five datasets, we identified‌ three genes - Pecam1,‌​‌ Sox2, and Hnf4a -​​ whose variability in expression​​​‌ patterns mirrors that of‌ Nanog. We further performed‌​‌ gene regulatory network inference​​​‌ using CARDAMOM, an algorithm​ that exploits temporal dynamics​‌ and transcriptional bursting. The​​ results revealed that these​​​‌ three genes exhibit reciprocal​ activation with Nanog at​‌ the 32-cell stage. This​​ regulatory motif reinforces fate-switching​​​‌ decisions and co-expression states.​ Our innovative analysis of​‌ single-cell transcriptomic data thus​​ uncovers a likely role​​​‌ for Pecam1, Sox2, and​ Hnf4a as key genes​‌ that, when coincidentally expressed​​ with Nanog, initiate ICM​​​‌ differentiation.

7.8 Modelling of​ anti-inflammatory treatment in the​‌ Alzheimer disease: optimal regimen​​ and outcome

Participants: Laurent​​​‌ Pujo-Menjouet, Léon Tine​.

The application of​‌ non-steroidal anti-inflammatory drugs (NSAIDs)​​ for Alzheimer’s disease is​​​‌ considered to be a​ promising therapeutic approach. Epidemiological​‌ studies suggest potential benefits​​ of NSAIDs; however, these​​​‌ findings are not consistently​ supported by clinical trials.​‌ This long-standing discrepancy has​​ persisted for decades and​​​‌ remains a significant barrier​ to developing effective treatment​‌ strategies. To assess the​​ efficacy of NSAIDs in​​​‌ Alzheimer’s disease, we have​ developed in 7 a​‌ mathematical model based on​​ a system of ordinary​​​‌ differential equations. The model​ captures the dynamics of​‌ key players in disease​​ progression, including Aβ-monomers, oligomers,​​​‌ proinflammatory mediators (M1 microglial​ cells and pro-inflammatory cytokines),​‌ and anti-inflammatory mediators (M2​​ microglial cells and anti-inflammatory​​​‌ cytokines). The effects of​ NSAIDs are modeled through​‌ a reduction in the​​ production rate of inflammatory​​​‌ cytokines (IC). While a​ single NSAID administration temporarily​‌ reduces IC levels, their​​ concentration eventually returns to​​​‌ baseline due to drug​ elimination. The return time​‌ depends on the drug​​ dose, resulting in a​​​‌ patient-specific return time function.​ By analyzing this function,​‌ we propose an optimal​​ treatment regimen and identify​​​‌ conditions under which NSAID​ treatment is most effective​‌ in reducing IC levels.​​ Our results suggest that​​​‌ NSAID efficacy in Alzheimer’s​ disease is influenced by​‌ the stage of the​​ disease (with earlier intervention​​​‌ being more effective), patient-specific​ parameters, and the treatment​‌ regimen. The approach developed​​ here can also be​​​‌ generalized to evaluate the​ efficacy of anti-inflammatory treatments​‌ for other diseases.

7.9​​ A temporary challenge by​​​‌ tumor cells can lead​ to a permanent partial-impairment​‌ of memory CD8 T​​ cell function

Participants: Olivier​​​‌ Gandrillon.

Memory CD8​ T cells typically exhibit​‌ improved effector functions compared​​ to their naive counterparts.​​​‌ However, under certain activation​ conditions, such as chronic​‌ viral infections or cancer,​​ these cells may develop​​​‌ functional defects. In 29​, we compared the​‌ functional quality of memory​​ CD8 T cells generated​​​‌ following tumor rejection with​ those arising from an​‌ acute viral infection. We​​ found that tumor-induced (Tum-CD8)​​​‌ memory cells exhibited a​ distinct phenotype and transcriptomic​‌ profile compared with viral-induced​​ (Vir-CD8) memory cells. These​​​‌ memory cells are characterized​ by the expression of​‌ inhibitory receptors and displayed​​ altered functions including reduced​​​‌ IFN$\gamma$ and TNF​$\alpha$ production as well​‌ as changes in integrin​​ expression. Additionally, the protective​​​‌ capacity of Tum-CD8 memory​ cells was flawed relative​‌ to that of Vir-CD8​​ memory cells. Importantly, the​​​‌ functional defects of Tum-CD8​ persisted upon viral recall.​‌ Together, these findings indicate​​ that transient tumoral stimulation​​ can imprint a stable​​​‌ partial exhaustion-like program on‌ memory CD8 T cells.‌​‌

7.10 Spatial pattern analysis​​ of A Aβ-monomer model​​​‌ with inflammation processes for‌ Alzheimer’s disease

Participants: Maxime‌​‌ Estavoyer, Laurent Pujo-Menjouet​​.

We study the​​​‌ emergence of spatial patterns‌ for a system of‌​‌ reaction-diffusion equations, modeling the​​ progression of Alzheimer’s disease​​​‌ through the interaction of‌ Aβ-monomers, oligomers, microglial cells,‌​‌ and interleukins with neurons.​​ In our work, these​​​‌ spatial patterns stand for‌ inert amyloid plaques, which‌​‌ are extracellular deposits of​​ Aβ-proteins and a characteristic​​​‌ feature of this neurodegenerative‌ disease. Using linear analysis‌​‌ and numerical simulations, we​​ show in 11 the​​​‌ existence of spatially heterogeneous‌ solutions and exhibit a‌​‌ wide variety of possible​​ spatially-dependent solutions: time-oscillating, low-amplitude,​​​‌ and high-amplitude patterns. Moreover,‌ we carry out an‌​‌ extensive analysis of high-amplitude​​ patterns in the one-​​​‌ and two-dimensional domains. In‌ particular, we study the‌​‌ stability of branches of​​ heterogeneous steady states through​​​‌ bifurcation diagrams and their‌ selection. From this numerical‌​‌ bifurcation analysis, we develop​​ some conjectures concerning the​​​‌ influence of inflammation and‌ microglial cells in the‌​‌ formation of amyloid plaques.​​ These findings offer insights​​​‌ into potential anti-inflammatory treatments‌ that might be used‌​‌ to mitigate the progression​​ of Alzheimer’s disease and​​​‌ the emergence of inert‌ amyloid plaques.

7.11 Velocity‌​‌ Trapping in the Lifted​​ Totally Asymmetric Simple Exclusion​​​‌ Process and the True‌ Self-Avoiding Random Walk

Participants:‌​‌ Brune Massoulié, Clément​​ Erignoux.

We discuss​​​‌ in 15 nonreversible Markov-chain‌ Monte Carlo algorithms that,‌​‌ for particle systems, rigorously​​ sample the positional Boltzmann​​​‌ distribution and that have‌ faster than physical dynamics.‌​‌ These algorithms all feature​​ a nonthermal velocity distribution.​​​‌ They are exemplified by‌ the lifted totally asymmetric‌​‌ simple exclusion process (lifted​​ TASEP), a one-dimensional lattice​​​‌ reduction of event-chain Monte‌ Carlo. We analyze its‌​‌ dynamics in terms of​​ a velocity trapping that​​​‌ arises from correlations between‌ the local density and‌​‌ the particle velocities. This​​ allows us to formulate​​​‌ a conjecture for its‌ out-of-equilibrium mixing timescale, and‌​‌ to rationalize its equilibrium​​ superdiffusive timescale. Both scales​​​‌ are faster than for‌ the (unlifted) TASEP. They‌​‌ are further justified by​​ our analysis of the​​​‌ lifted TASEP in terms‌ of many-particle realizations of‌​‌ true self-avoiding random walks.​​ We discuss velocity trapping​​​‌ beyond the case of‌ one-dimensional lattice models and‌​‌ in more than one​​ physical dimensions. Possible applications​​​‌ beyond physics are pointed‌ out.

7.12 Cell Trajectory‌​‌ Inference based on Schrödinger​​ Problem and a Mechanistic​​​‌ Model of Stochastic Gene‌ Expression

Participants: Clémence Fournié‌​‌, Aymeric Baradat,​​ Olivier Gandrillon.

Cellular​​​‌ differentiation is the biological‌ process that leads a‌​‌ cell to opt for​​ a particular cellular identity.​​​‌ Recently, single-cell RNA-sequencing has‌ enabled the simultaneous measurement‌​‌ of gene expression levels​​ at specific times for​​​‌ a large number of‌ individual cells and a‌​‌ large number of genes.​​ Repeating such measurements at​​​‌ different time points gives‌ then access to the‌​‌ temporal variation, or transport,​​ of a distribution on​​​‌ a gene expression space.‌ The whole temporal trajectory‌​‌ of distributions thus characterizes​​​‌ the differentiation process at​ population level, but trajectories​‌ of individual cells are​​ still out of reach​​​‌ since most measurement techniques​ are destructive. The optimal​‌ transport theory that has​​ been used so far​​​‌ to infer cellular differentiation​ trajectories from time-stamped single-cell​‌ RNA-seq data involves solving​​ the so-called Schrödinger problem​​​‌ in its most common​ version. This implies assuming​‌ that cells move, in​​ the gene expression space,​​​‌ by diffusion. Yet, real​ gene dynamics are much​‌ more complex. In 28​​, we assume that​​​‌ mRNA dynamics are characterized​ by brief and important​‌ production of RNA, with​​ long periods of inactivity​​​‌ in between, and consider​ the so-called Bursty model​‌ of gene dynamics. We​​ use this model to​​​‌ define a reference process​ for the Schrödinger problem.​‌ By comparing the solutions​​ of the Schrödinger problems​​​‌ with a Diffusive and​ a Bursty reference process,​‌ under different conditions, we​​ show that the Bursty​​​‌ model provides a better​ approximation of the underlying​‌ gene dynamics than the​​ standard Diffusive process when​​​‌ inferring cell trajectories.

7.13​ A mathematical model to​‌ the melanoma dynamics involving​​ CAR T-cells

Participants: Mostafa​​​‌ Adimy.

Melanoma is​ one of the most​‌ aggressive types of cancer.​​ Although it has a​​​‌ low percentage of incidence​ in the population, a​‌ high degree of lethality​​ is observed due to​​​‌ its rapid metastasis. As​ melanoma is a highly​‌ immunogenic cancer, it has​​ been used as an​​​‌ experimental model in several​ studies aimed at developing​‌ therapies, such as immunotherapy​​ with Chimeric Antigen Receptor​​​‌ (CAR) T-cells. We propose​ in 18 a mathematical​‌ model of three ordinary​​ differential equations to describe​​​‌ the dynamics of melanoma​ in the presence of​‌ Tumor-Associated Macrophages (TAMs) and​​ CAR T-cell therapy, to​​​‌ assess the role of​ TAMs cells in the​‌ failure of this melanoma​​ therapy. We examine the​​​‌ existence and asymptotic stability​ of equilibrium points of​‌ this system, giving a​​ biological interpretation to each​​​‌ of them. Based on​ our theoretical and numerical​‌ results, we conclude that​​ immunosuppression has a negative​​​‌ impact on CAR T-cell​ immunotherapy and that increasing​‌ the immunotherapy dose can​​ improve tumor control. Furthermore,​​​‌ an increase in the​ action of the TAMs​‌ population on tumor proliferation​​ can induce oscillations that​​​‌ eventually become periodic.

7.14​ Steady state large deviations​‌ for one-dimensional, symmetric exclusion​​ processes in weak contact​​​‌ with reservoirs

Participants: Clément​ Erignoux.

Consider the​‌ symmetric exclusion process evolving​​ on an interval and​​​‌ weakly interacting at the​ end-points with reservoirs. Denote​‌ by $I_{[\mathrm{0},T]}(‌·)$ its dynamical​ large deviations functional and​‌ by $V(·)$ the associated quasi-potential,​​​‌ defined as $V(\gamma )={inf‌}_{T>0}{inf}_u{I}_{[\mathrm{0‌},T]}(u)$, where​‌ the infimum is carried​​ over all trajectories $\mathrm{u‌}$ such that $u(0)=\overline{\mathrm{\rho ‌}}$, $u(T)=\mathrm{\gamma ‌}$, and $\stackrel{¯}{\rho }$ is the stationary density​‌ profile. We derive in​​ 4 the partial differential​​ equation which describes the​​​‌ evolution of the optimal‌ trajectory, and deduce from‌​‌ this result the formula​​ obtained by Derrida, Hirschberg​​​‌ and Sadhu 33 for‌ the quasi-potential through the‌​‌ representation of the steady​​ state as a product​​​‌ of matrices.

7.15 Scaling‌ relations for the CLG's‌​‌ critical exponents

Participants: Clément​​ Erignoux.

In 26​​​‌ we consider, in any‌ dimension, the constrained lattice‌​‌ gas introduced by Rossi​​ et al., which is​​​‌ an exclusion process on‌ a d-dimensional lattice following‌​‌ the additional constraint that​​ only particles with at​​​‌ least one occupied neighbour‌ can jump. In dimension‌​‌ d=2, this model features​​ self-organized criticality at some​​​‌ critical density of particles.‌ Numerical simulations predict the‌​‌ existence of scaling exponents​​ close to criticality, and​​​‌ several relations can be‌ derived between these exponents.‌​‌ The goal of this​​ article is to give​​​‌ a mathematical framework for‌ these relations, which have‌​‌ been numerically established in​​ a companion article.

7.16​​​‌ Quantifying uncertainty and sensitivity‌ in an Alzheimer's disease‌​‌ model: A mathematical approach​​

Participants: Laurent Pujo-Menjouet.​​​‌

To understand the dynamics‌ of Alzheimer's disease, we‌​‌ formulate in 31 a​​ generalized mathematical model based​​​‌ on three events: aggregation‌ of disease-related proteins, activation‌​‌ of immune cells, and​​ initiation of inflammation. We​​​‌ incorporate functional forms in‌ the model to represent‌​‌ the complex biological interactions​​ between components related to​​​‌ Alzheimer's disease. We take‌ explicit forms depending on‌​‌ the properties of functions​​ in the model. We​​​‌ describe the system dynamics‌ by locating biologically feasible‌​‌ steady states, determining stability​​ properties, and identifying the​​​‌ effective parameters. Parameters are‌ estimated using two methods:‌​‌ biological literature and data​​ fitting. We perform sensitivity​​​‌ and uncertainty analyses to‌ identify the most influential‌​‌ parameters. Partial Rank Correlation​​ Coefficient and scatter plots​​​‌ are used to visualize‌ global sensitivity. Our results‌​‌ reveal that lower activation​​ rate and higher proliferation​​​‌ rate of microglia may‌ contribute to a reduction‌​‌ in toxic protein aggregate​​ levels, thus slowing the​​​‌ disease progression.

7.17 A‌ reaction telegraph model reveals‌​‌ synergy between motility strategies​​ in Myxococcus xanthus predation​​​‌

Participants: Maxime Estavoyer.‌

The predatory bacterium Myxococcus‌​‌ xanthus can invade prey​​ bacteria using two distinct​​​‌ motility apparatuses. It is‌ commonly acknowledged that adventurous‌​‌ motility is used for​​ isolated bacteria, while social​​​‌ motility corresponds to bacterial‌ clusters. Inspired by recent‌​‌ biological findings, we propose​​ in 27 a simple​​​‌ model of predatory invasion‌ focusing on the co-occurrence‌​‌ of these two mechanisms​​ and their possible synergistic​​​‌ effects. At microscopic scale,‌ cell motion is persistent;‌​‌ therefore, we opt for​​ a transport-reaction model, extending​​​‌ previous reaction-diffusion models. Another‌ specificity is the structuration‌​‌ of the bacterial population​​ into clusters with varying​​​‌ speeds and persistence times.‌ In the linear regime,‌​‌ we find a transition​​ from normal speed to​​​‌ anomalous speed, consistent with‌ reaction-diffusion theory but with‌​‌ specificities due to the​​ hyperbolic nature of the​​​‌ model. For the nonlinear‌ regime, we numerically observe‌​‌ and study the existence​​ of transitions between pulled​​​‌ and pushed fronts. Finally,‌ we reproduced biological experiments‌​‌ with mutants lacking each​​​‌ of the motility apparatuses​ based on relevant modifications​‌ of the model. Moreover,​​ we propose a rational​​​‌ basis for the reported​ synergistic effects. Our work​‌ paves the way for​​ a better understanding of​​​‌ the complex waves of​ bacterial population advance, which​‌ are precursors to biofilm​​ formation.

7.18 Mixed precision​​​‌ implicit numerical schemes for​ systems of ordinary differential​‌ equations

Participants: Mouhamad Al​​ Sayed Ali, Samuel​​​‌ Bernard, Arsène Marzorati​.

Ordinary differential equations​‌ (ODEs) are widely used​​ to model complex systems​​​‌ in biology, which result​ from the interactions of​‌ a large number of​​ cells or organisms. This​​​‌ can lead to a​ substantial system size. These​‌ complex interactions can quickly​​ alter their behavior, and​​​‌ some biological systems are​ stiff when represented as​‌ ODEs. Therefore, these stiff​​ ODEs could benefit from​​​‌ implicit numerical schemes. However,​ each iteration of these​‌ schemes involves solving a​​ large nonlinear system, typically​​​‌ using the Newton method.​ To guarantee the global​‌ convergence of this method,​​ we use line search​​​‌ (LS) and trust region​ (TR) algorithms. In 2​‌, we introduce a​​ new approach to accelerate​​​‌ the computation of the​ implicit schemes by using​‌ mixed precision arithmetic, combining​​ float and double precision,​​​‌ within the LS and​ TR algorithms, as well​‌ as in the Newton​​ method. This approach aims​​​‌ at balancing the performance​ of lower precision arithmetic​‌ with the accuracy of​​ higher precision arithmetic. We​​​‌ give theoretical results that​ show the efficiency of​‌ our approach. Numerical experiments​​ show that our approach,​​​‌ running in either sequential​ or in parallel with​‌ MPI, is up to​​ twice as fast as​​​‌ the double precision approach​ with the same level​‌ of accuracy. These experiments​​ also show that increasing​​​‌ the size of the​ ODEs does not impact​‌ the quality of our​​ mixed precision solution.

7.19​​​‌ Spatial pattern analysis of​ an Aβ-monomer model with​‌ inflammation processes for Alzheimer's​​ disease

Participants: Maxime Estavoyer​​​‌, Laurent Pujo-Menjouet.​

We study in 12​‌ the emergence of spatial​​ patterns for a system​​​‌ of reaction-diffusion equations, modeling​ the progression of Alzheimer's​‌ disease through the interaction​​ of Aβ-monomers, oligomers, microglial​​​‌ cells, and interleukins with​ neurons. In our work,​‌ these spatial patterns stand​​ for inert amyloid plaques,​​​‌ which are extracellular deposits​ of Aβ-proteins and a​‌ characteristic feature of this​​ neurodegenerative disease. Using linear​​​‌ analysis and numerical simulations,​ we show the existence​‌ of spatially heterogeneous solutions​​ and exhibit a wide​​​‌ variety of possible spatiallydependent​ solutions: time-oscillating, low-amplitude, and​‌ high-amplitude patterns. Moreover, we​​ carry out an extensive​​​‌ analysis of high-amplitude patterns​ in the one-and two-dimensional​‌ domains. In particular, we​​ study the stability of​​​‌ branches of heterogeneous steady​ states through bifurcation diagrams​‌ and their selection. From​​ these numerical simulations, we​​​‌ develop some conjectures concerning​ the influence of inflammation​‌ and microglial cells in​​ the formation of amyloid​​​‌ plaques. These findings offer​ insights into potential anti-inflammatory​‌ treatments that might be​​ used to mitigate the​​​‌ progression of Alzheimer's disease​ and the emergence of​‌ inert amyloid plaques.

7.20​​ Analysis of a polarization​​ model with exchange at​​​‌ the boundary

Participants: Thomas‌ Lepoutre.

We analyze‌​‌ in 30 a drift-diffusion​​ model on the half-line.​​​‌ The drift is linked‌ to exchanges at 0.‌​‌ This corresponds to the​​ modeling of filament generation​​​‌ following protein exchanges between‌ the membrane and the‌​‌ cytoplasm of a cell.​​ This model has a​​​‌ critical mass and the‌ asymptotic behaviors can be‌​‌ divided into three categories.​​ In subcritical mass regime,​​​‌ diffusive behavior dominates and‌ convergence in self-similar variables‌​‌ can be obtain with​​ parabolic scaling. In supercritical​​​‌ mass regime, drift compensates‌ for diffusion and convergence‌​‌ towards equilibrium distributions occurs.​​ In the critical case,​​​‌ convergence towards 0 for‌ the drift occurs over‌​‌ a long period of​​ time, the behavior exhibits​​​‌ a distinct relevant scaling.‌ We quantify the convergences‌​‌ in the relevant variables​​ in the three regimes.​​​‌

7.21 Global stability and‌ periodicity in a delay‌​‌ differential model

Participants: Mostafa​​ Adimy, Laurent Pujo-Menjouet​​​‌.

Simple form delay‌ differential equation (DDE) is‌​‌ considered in 23 as​​ a mathematical model of​​​‌ several biological processes. The‌ problems of the global‌​‌ asymptotic stability (GAS) of​​ the unique positive equilibrium​​​‌ and the existence of‌ periodic solutions slowly oscillating‌​‌ about this equilibrium are​​ studied. Sufficient conditions for​​​‌ the GAS are derived‌ in terms of the‌​‌ global attractivity of the​​ unique fixed point of​​​‌ induced interval maps, one‌ set being delay independent‌​‌ conditions and the other​​ one dependent on the​​​‌ size of the delay.‌ Slowly oscillating periodic solutions‌​‌ always exist when the​​ linearized about the equilibrium​​​‌ DDE is unstable. The‌ theoretical results are demonstrated‌​‌ by extensive numerical simulations.​​

7.22 Regional Control Strategies​​​‌ for a Spatiotemporal SQEIAR‌ Epidemic Model: Application to‌​‌ COVID-19

Participants: Mohammed Elghandouri​​.

In 8,​​​‌ we develop a spatial‌ SEIAR-type epidemic model considering‌​‌ a quarantined population (denoted​​ as Q), which we​​​‌ call the SQEIAR model.‌ The dynamics of the‌​‌ SQEIAR model are described​​ by six Partial Differential​​​‌ Equations (PDEs) that represent‌ the changes in the‌​‌ susceptible, quarantined, exposed, asymptomatic,​​ infected, and recovered populations.​​​‌ Our goal is to‌ reduce the number of‌​‌ susceptible, exposed, asymptomatic, and​​ infected individuals while accounting​​​‌ for the environment, which‌ plays a critical role‌​‌ in the spread of​​ epidemics. We then propose​​​‌ a novel strategy for‌ epidemic control, incorporating two‌​‌ key control measures: regional​​ quarantine for the susceptible​​​‌ population and treatment for‌ the infected. This ap-‌​‌ proach serves as an​​ alternative to widespread quarantine,​​​‌ minimizing the economic, social,‌ and other potential impacts.‌​‌ Additionally, we consider the​​ possibility of re-infection among​​​‌ recovered individuals, a common‌ occurrence in many diseases.‌​‌ To demonstrate the practical​​ utility of our results,​​​‌ a numerical example centered‌ on COVID-19 is presented.‌​‌

7.23 Exploring well-posedness and​​ asymptotic behavior in an​​​‌ Advection Diffusion Reaction (ADR)‌ model

Participants: Mohammed Elghandouri‌​‌.

In 9,​​ the existence, uniqueness, and​​​‌ positivity of solutions, as‌ well as the asymptotic‌​‌ behavior through a finite​​ fractal dimensional global attractor​​​‌ for a general Advection-Diffusion-Reaction‌ (ADR) equation, are investigated.‌​‌ Our findings are innovative,​​​‌ as we employ semigroups​ and global attractors theories​‌ to achieve these results.​​ Also, an analytical solution​​​‌ of a two-dimensional Advection-Diffusion​ Equation is presented. And​‌ finally, two Explicit Finite​​ Difference schemes are used​​​‌ to simulate solutions in​ the two- and three-dimensional​‌ cases. The numerical simulations​​ are conducted with predefined​​​‌ initial and Dirichlet boundary​ conditions.

7.24 Optimal control​‌ of an impulsive VS-EIAR​​ epidemic model with applications​​​‌ to COVID-19

Participants: Mohammed​ Elghandouri.

In 6​‌, we investigate a​​ VS-EIAR epidemiological model that​​​‌ incorporates vaccinated individuals ${\mathrm{V}}_i:i=‌1,\cdots ,n$, where $\mathrm{n‌}\in {\mathbb{N}}^{*}$.​ The dynamics of the​‌ VS-EIAR model are governed​​ by a system of​​​‌ ordinary differential equations describing​ the evolution of vaccinated,​‌ susceptible, exposed, infected, asymptomatic,​​ and deceased population groups.​​​‌ Our primary objective is​ to minimize the number​‌ of susceptible, exposed, infected,​​ and asymptomatic individuals by​​​‌ administering vaccination doses to​ susceptible individuals and providing​‌ treatment to the infected​​ population. To achieve this,​​​‌ we employ optimal control​ theory to regulate the​‌ epidemic dynamics within an​​ optimal terminal time ${\mathrm{\tau ‌}}^{*}$. Using Pontryagin’s​ Maximum Principle (PMP), we​‌ establish the existence of​​ an optimal control pair​​​‌ $(v^{*}(t),{\mathrm{u‌}}^{*}(t))$. Additionally, we​​​‌ extend the model to​ an impulsive VS-EIAR framework,​‌ with particular emphasis on​​ the impact of immigration​​​‌ and population movement. Finally,​ we present numerical simulations​‌ to validate the theoretical​​ results and demonstrate their​​​‌ practical applicability.

7.25 Analytical​ derivation of delayed prey-predator​‌ model with hunting-and-resting delay​​

Participants: Mostafa Adimy.​​​‌

We investigate in 1​ a prey-predator model based​‌ on a general Gause​​ type system. We take​​​‌ for the predator two​ phases into account, the​‌ hunting phase and the​​ resting one. We suppose​​​‌ that the predators stop​ hunting after they catch​‌ the prey. Then they​​ enter the resting phase​​​‌ where they stay for​ a fixed limited time.​‌ The resulting mathematical model​​ is a system of​​​‌ two age-structured partial differential​ equations. By integrating this​‌ system over age and​​ using the characteristics method,​​​‌ we reduce it to​ a delay differential system,​‌ and we investigate the​​ existence and stability of​​​‌ the steady states. In​ particular, we have shown​‌ that the introduction of​​ the delay (the duration​​​‌ of the resting phase)​ stabilizes the coexistence equilibrium.​‌

7.26 Traveling Waves in​​ a Hybrid Reaction‐Diffusion‐Difference SEIR​​​‌ Epidemic Model With Nonlocal​ Transmission and Protection Phase​‌ With Delay

Participants: Mostafa​​ Adimy.

In 16​​​‌ we study the existence​ and non‐existence of traveling​‌ waves for a SEIR​​ epidemic model with diffusion.​​​‌ This system is coupled​ to an age‐structured protection​‌ equation that takes into​​ account a proportion of​​​‌ susceptible individuals who protect​ themselves from the disease​‌ through medical treatment or​​ vaccination with temporary immunity,​​​‌ for example. The model​ includes a compartment of​‌ exposed individuals to represent​​ diseases with an incubation​​​‌ period. In addition, the​ model contains a nonlinear​‌ saturated incidence rate to​​ describe the interaction between​​ susceptible and infected individuals.​​​‌ By solving the equation‌ associated with the exposed‌​‌ compartment, we obtain a​​ nonlocal integral representation of​​​‌ the exposed population as‌ a function of the‌​‌ nonlinear incidence rate. This​​ reformulation reduces the model​​​‌ to an hybrid system‌ of reaction‐diffusion equations coupled‌​‌ to a continuous difference​​ equation that includes a​​​‌ delay and a nonlocal‌ term, reflecting the movement‌​‌ of individuals during the​​ protection and exposure phases.​​​‌ For wave speeds above‌ a certain threshold, we‌​‌ construct lower and upper​​ solutions for the resulting​​​‌ hybrid system. Finally, we‌ apply Schauder's fixed point‌​‌ theorem in an appropriate​​ function space to establish​​​‌ the existence of progressive‌ wave solutions.

7.27 Three-State‌​‌ Gene Expression Model Parameterized​​ for Single-Cell Multi-Omics Data​​​‌

Participants: Thomas Lepoutre.‌

We present in 19‌​‌ a novel three-state gene​​ expression model designed to​​​‌ elucidate the underlying mechanisms‌ of mRNA transcription and‌​‌ its regulation. Our model​​ incorporates gene regulatory processes​​​‌ by explicitly including a‌ transcription factor-bound state, thereby‌​‌ capturing the dynamic interplay​​ between transcription activation and​​​‌ chromatin dynamics. We fit‌ the model to paired‌​‌ single-cell ATAC-seq and single-cell​​ RNA-seq data, as these​​​‌ data give us simultaneous‌ information on a gene’s‌​‌ transcriptional state and its​​ accompanying chromatin state. Working​​​‌ at the pseudo-bulk level,‌ we extract biologically meaningful‌​‌ high-level descriptors from homogeneous​​ cell (sub)populations, such as​​​‌ the mean and variance‌ of gene expression as‌​‌ well as the fraction​​ of accessible chromatin. Crucial​​​‌ to the computational feasibility‌ of our approach, these‌​‌ descriptors can be analytically​​ related to our model​​​‌ parameters. Despite the increased‌ complexity needed to capture‌​‌ regulatory processes in our​​ model, it remains sufficiently​​​‌ parsimonious to infer parameters‌ reliably from experimental data.‌​‌ Each parameter has a​​ clear biological interpretation, reflecting​​​‌ properties such as burst‌ frequency, chromatin opening and‌​‌ closing dynamics, and basal​​ or regulated expression. Fitting​​​‌ the model to a‌ large collection of genes‌​‌ allows us to analyze​​ the parameters and distinguish​​​‌ so-called gene expression strategies.‌ The model parameters reveal‌​‌ a small number of​​ distinct expression strategies among​​​‌ gene clusters, providing data-driven‌ novel insight into context-dependent‌​‌ regulation of gene expression.​​

7.28 Subdiffusive fractional limit​​​‌ of a jump-renewal equation‌

Participants: Thomas Lepoutre.‌​‌

We present in 25​​ an age-structured jump model​​​‌ that arises as a‌ description of continuous time‌​‌ random walks with infinite​​ mean waiting time between​​​‌ jumps. We prove that‌ under a suitable rescaling,‌​‌ this equation converges in​​ the long time large​​​‌ scale limit to a‌ time fractional subdiffusion equation.‌​‌

8.1 International initiatives

8.2​​ International research visitors

8.3 National initiatives

8.4 Regional‌ initiatives

9.1 Promoting scientific activities​​​‌

9.2 Teaching​​ - Supervision - Juries​​​‌ - Educational and pedagogical​ outreach

9.3 Popularization

10.1​​​‌ Publications of the year​

10.2​​​‌ Cited publications

• 33 article‌B.Bernard Derrida,‌​‌ O.Ori Hirschberg and​​ T.Tridib Sadhu.​​​‌ Large deviations in the‌ symmetric simple exclusion process‌​‌ with slow boundaries.​​Journal of Statistical Physics​​​‌18212021,‌ 15back to text‌​‌
• 34 articleM. B.​​M. B. Elowitz,​​​‌ A. J.A. J.‌ Levine, E. D.‌​‌E. D. Siggia and​​ P. S.P. S.​​​‌ Swain. Stochastic gene‌ expression in a single‌​‌ cell.Science297​​55842002, 1183-1186​​​‌back to text
• 35‌ articleO.O. Gandrillon‌​‌ and M. P.M.​​ P. H. Stumpf.​​​‌ Editorial overview: ‘Theoretical approaches‌ to analyze single-cell data’‌​‌ (April 2021) within the​​ theme ‘Mathematical modelling’.​​​‌Current Opinion in Systems‌ Biology282021,‌​‌ 100382back to text​​
• 36 articleU.U.​​​‌ Herbach. Gene regulatory‌ network inference from single-cell‌​‌ data using a self-consistent​​ proteomic field.arXiv​​​‌2109.148882022back to‌ text
• 37 articleJ.‌​‌ M.Jeffrey M Levsky​​ and R. H.Robert​​​‌ H Singer. Gene‌ expression and the myth‌​‌ of the average cell​​.Trends Cell Biol​​​‌1312003,‌ 4--6back to text‌​‌
• 38 articleR.R.​​ Noble, K.K.​​​‌ Tasaki, P. J.‌P. J. Noble and‌​‌ D.D. Noble.​​ Biological Relativity Requires Circular​​​‌ Causality but Not Symmetry‌ of Causation: So, Where,‌​‌ What and When Are​​ the Boundaries?Front Physiol​​​‌102019, 827‌back to textback‌​‌ to text
• 39 article​​M.M. Smart and​​​‌ A.A. Zilman.‌ Emergent properties of collective‌​‌ gene-expression patterns in multicellular​​ systems.Cell Reports​​​‌ Physical Science42‌2023, 101247back‌​‌ to text
• 40 article​​D. M.D. M.​​​‌ Suter, N.N.‌ Molina, D.D.‌​‌ Gatfield, K.K.​​ Schneider, U.U.​​​‌ Schibler and F.F.‌ Naef. Mammalian genes‌​‌ are transcribed with widely​​ different bursting kinetics.​​​‌Science33260282011‌, 472-4URL: http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=21415320‌​‌DOIback to text​​
• 41 articleA.A.​​​‌ Wilbrey-Clark, K.K.‌ Roberts and S. A.‌​‌S. A. Teichmann.​​ Cell Atlas technologies and​​​‌ insights into tissue architecture‌.Biochem J477‌​‌82020, 1427-1442​​back to text