2025‌Activity reportProject-TeamMERGE‌​‌

From stochastic processes to​ constrained Hamilton-Jacobi (HJ) equations.​‌

Permanent members: Sylvie​​ Méléard, Gaël Raoul

Most​​​‌ models, for instance for​ the "normal" bacterial division​‌ cycle 83, consider​​ asexual populations with clonal​​ reproduction and vertical inheritance.​​​‌ We want to consider‌ here a more general‌​‌ model with a transfer​​ term, justified by biological​​​‌ considerations in the case‌ of bacterial transfer 87‌​‌ (see also the application​​ section 4.1). The​​​‌ individual-based population process is‌ given for any $\mathrm{K‌‌}$ by the jump point-measure​​ Markov process ${\left({\mathrm{\mu ‌}}_t^K\right)}_{\mathrm{t‌}}$ on a trait subset‌​‌ of ${ℝ}^d$ weighted​​ by $1/\mathrm{K‌}$. An individual with‌ trait $x$ gives birth‌​‌ to a new individual​​ with rate $b(‌x)$. With‌ probability $1-{\mathrm{p‌‌}}_K$, the new​​ individual carries the trait​​​‌ $x$ and with probability‌ $p_K$, it‌​‌ carries a mutant trait​​ $z$ chosen according to​​​‌ the distribution $m(‌x,z)‌‌dz$. An​​ individual with trait $\mathrm{x‌}$ in the population ${\mathrm{\mu ‌}}^K$ dies with death‌​‌ rate $d\left(\mathrm{x}\right)+C\langle ‌{\mu }^K\left(\mathrm{t‌}\right),1\rangle ‌‌$. Further an individual​​ with trait $x$ chooses​​​‌ a partner with trait‌ $y$ at rate ${\mathrm{h‌‌}}_K(x,y,{\mu }^{\mathrm{K‌}})=\tau (‌x,y)‌‌/K\langle {\mathrm{\mu }}^K,1\rangle ‌$ and after transfer, the‌ couple $(x,‌‌y)$ becomes $(x,x)‌$. Then for any‌ “good” test function $\mathrm{\phi ‌‌}$, we have

where​​​‌ $M^{K,\mathrm{\phi ‌}}$ is a square integrable‌​‌ martingale whose quadratic variation​​ can be easily made​​​‌ explicit. By letting $\mathrm{K‌}$ go to infinity, and‌​‌ $p_K$ go to​​ $p$, we can​​​‌ derive an integro-differential equation‌ with non local non‌​‌ linearities due to both​​ competition and transfer. Uniqueness​​​‌ of its solution is‌ obvious but its long-time‌​‌ behaviour is unknown, as​​ well as the existence​​​‌ of stationary solutions. Formally,‌ applying a limiting procedure‌​‌ for small mutations and​​ time rescaling usually leads​​​‌ to a HJ type‌ equation with constraints, in‌​‌ the formalism introduced in​​ 75, successfully developed​​​‌ in 51, 118‌, 50, and‌​‌ in many extensions so​​ far. Concentrations in such​​​‌ equations are too fast‌ for realistic evolution 119‌​‌. Indeed the evolutionary​​​‌ dynamics strongly depend on​ the positivity of the​‌ density although it is​​ exponentially small for some​​​‌ traits. Different papers 119​, 95, 111​‌ proposed to slow down​​ the concentration speed by​​​‌ the addition of artificial​ terms. With Nicolas Champagnat,​‌ Sylvie Méléard introduced another​​ point of view. The​​​‌ rare mutation assumption introduced​ in 61 allowed to​‌ obtain a time scale​​ separation between demography and​​​‌ mutation. Under this assumption,​ they were able to​‌ characterize rigorously a general​​ evolutionary jump process describing​​​‌ the successive evolutionary population​ states 63. This​‌ approach was very fruitful​​ and allowed to quantify​​​‌ the complete scheme from​ individuals to macroscopic behaviors​‌ as suggested in 108​​ and 74. Nevertheless,​​​‌ the assumptions imposed very​ small mutation rates considered​‌ too slow to explain​​ evolution (especially for microorganisms),​​​‌ but also too slow​ to capture the concentration​‌ effects of the HJ​​ equations. At this point​​​‌ one may recall the​ usual critics by some​‌ biologists 136, of​​ unrealistic evolutionary time scale,​​​‌ at least for certain​ species.

The first task​‌ in this study is​​ to integrate fast mutation​​​‌ time scales and to​ show how stochastic models​‌ based on logarithmic scales​​ can capture small populations​​​‌ in large approximations and​ explain deterministic concentration phenomena.​‌ In particular we aim​​ to obtain new singular​​​‌ and constrained HJ equations​ taking into account the​‌ local population extinctions. We​​ hope that these new​​​‌ scales will provide an​ intermediate approach consistent with​‌ biological observations.

The second​​ task is to characterize​​​‌ the different asymptotic behaviors​ in the Hamilton-Jacobi equation​‌ and to understand the​​ role of the trade-off​​​‌ between demography and transfer.​

Space-and-trait structured models

Permanent​‌ members: Vincent Bansaye, Marie​​ Doumic, Gaël Raoul

Effects​​​‌ of spatial heterogeneity on​ structured population dynamics need​‌ to be studied for​​ many applications, in ecology​​​‌ as well as in​ microbiology. Here again, relating​‌ macroscopic to individual-based models​​ is of key importance​​​‌ for a correct interpretation​ of macroscopically observed phenomena​‌ such as morphogenesis or​​ front propagation. Let us​​​‌ develop two examples: size-and-space​ structured models and phenotypic​‌ trait-and-space structured models.

Non-markovian interactions: from‌ local interaction model to‌​‌ the parabolic-parabolic Keller-Segel system​​

Permanent members: Milica Tomašević​​​‌

An important mathematical challenge‌ is to derive mean-field‌​‌ limits for non-Markovian interaction,​​ i.e., when the past​​​‌ also needs to be‌ taken into account. Such‌​‌ models appear for instance​​ in neuroscience 64,​​​‌ 65 and chemotaxis. To‌ model chemotaxis, the parabolic-parabolic‌​‌ Keller-Segel model has been​​ stated phenomenologically, but to​​​‌ interpret it we need‌ to introduce interaction memory,‌​‌ which provides tremendous analysis​​ difficulties since particles are​​​‌ now non-markovian both in‌ time and space. New‌​‌ methods have been proposed​​ by Milica Tomašević, with​​​‌ a stochastic representation of‌ the mild formulation of‌​‌ the equation and a​​ particle approximation 125,​​​‌ 130, 127.‌ The equations obtained have‌​‌ been little studied before​​ Milica Tomašević's PhD thesis,​​​‌ so that many questions‌ remain open. Concerning the‌​‌ convergence of the particle​​ systems towards the Keller-Segel​​​‌ model, an important problem‌ is the obtention of‌​‌ explicit convergence rates, when​​ the number of particles​​​‌ tends to infinity, for‌ the propagation of chaos‌​‌ of the particle system​​ in 1D. A possible​​​‌ way is to extend‌ techniques developed by Jabin‌​‌ and Wang 96 for​​ the quantitative study of​​​‌ the mean-field boundaries of‌ particle systems in non-regular‌​‌ Markovian interaction. The aim​​ is to control the​​​‌ relative entropy between the‌ joint law of the‌​‌ particles and the law​​ of $N$ independent copies​​​‌ of the Keller-Segel system.‌ By exploiting the results‌​‌ on the Keller-Segel nonlinearity​​ in 1 dimension and​​​‌ on the Sobolev type‌ estimation on the densities‌​‌ of the system (chapter​​ 4 in 129),​​​‌ a regularization of the‌ interaction kernel of the‌​‌ particles allows to obtain​​ a first convergence rate​​​‌ for the marginal laws‌ in time of an‌​‌ arbitrary particle, explicit but​​ suboptimal (due to the​​​‌ regularization procedure). To obtain‌ the optimal convergence rate,‌​‌ we think to develop​​ an essentially probabilistic approach​​​‌ suggested by the recent‌ works of Veretennikov 133‌​‌ and Lacker 102 as​​ well as by the​​​‌ partial Girsanov transformations introduced‌ in 97.

Diffusion-growth-fragmentation​​​‌ processes and equations

Permanent‌ members: Marie Doumic, Sylvie‌​‌ Méléard

To model the​​ growth of a bacterial​​​‌ population in a chemostat,‌ a new model of‌​‌ growth and fragmentation, coupled​​ to a differential equation​​​‌ for the resource, was‌ proposed by Josué Tchouanti‌​‌ in his thesis 126​​. Using a combination​​​‌ of probabilistic and analytical‌ methods, he proved the‌​‌ existence, uniqueness and regularity​​ of solutions, as well​​​‌ as the convergence in‌ large populations of the‌​‌ individual-based model. This model​​ also present similiarities with​​​‌ the proliferation of parasites‌ in dividing cells studied‌​‌ by Vincent Bansaye 48​​, 47.

One​​​‌ of the very interesting‌ novelties of this model‌​‌ is to consider the​​ growth not as a​​​‌ purely deterministic process, leading‌ to a transport term‌​‌ in the size structured​​​‌ equation $\frac{\partial }{\partial \mathrm{x}}(\tau \left(\mathrm{x‌}\right)u(\mathrm{t},x))‌$, but to take​ into account the intrinsic​‌ stochasticity in growth, so​​ that a diffusion-type term​​​‌ $\frac{{\partial }^2}{\partial {\mathrm{x}}^2}\left(D(‌x)u(t,x)‌\right)$ is added, which​ degenerates at the boundary​‌ $x=0.$ We thus want to​​​‌ study further this equation,​ its long-time dynamics with​‌ and without interaction (​​i.e. in the linear​​​‌ case as well as​ with nonlinear couplings), how​‌ it differs from the​​ much more studied growth-fragmentation​​​‌ equation, and which model​ seems more relevant in​‌ which applicative case. We​​ also want to adapt​​​‌ the model to metabolic​ heterogeneity cases, i.e. when​‌ we model the capacity​​ for bacteria to feed​​​‌ on two distinct nutrients,​ which leads to distinguish​‌ two populations competing for​​ two resources.

Ergodicity analysis​​​‌ and exponential convergence for​ multi-dimensional growth-fragmentation processes and​‌ equations

Permanent members: Vincent​​ Bansaye, Sylvie Méléard, Milica​​​‌ Tomašević

Based on data​ from the Edinburgh's lab​‌ of Meriem El Karoui,​​ Ignacio Madrid Canales introduced​​​‌ during his thesis an​ adder growth-fragmentation stochastic process​‌ modelling the growth of​​ bacteria. He studied its​​​‌ long time behaviour and​ proved that conveniently renormalized,​‌ the associated semigroup converges​​ exponentially to a well​​​‌ defined measure. The aim​ is now to generalize​‌ this result in higher​​ dimensions, motivated by the​​​‌ different growth strategies that​ bacteria can have under​‌ stress. Mathematically the question​​ is also largely open.​​​‌

Understanding the links between​ genealogical and population behaviours​‌

Permanent members: Vincent Bansaye,​​ Marie Doumic and Sylvie​​​‌ Méléard

Microfluidic experiments allow​ one to follow a​‌ genealogical lineage of cells,​​ whereas most previous experiments​​​‌ as well as "natural"​ conditions consist in observing​‌ a full population dynamics.​​ The natural question then​​​‌ comes to relate the​ two models, and to​‌ understand how certain phenomena​​ may be observed in​​​‌ one setting but not​ in the other -​‌ for instance, how a​​ few individuals may finally​​​‌ invade the whole population;​ how survivor cells may​‌ emerge from a senescent​​ population; or yet, how​​​‌ to find the "signature"​ of a phenomenon that​‌ happened in the past​​ from the observation of​​​‌ a population at a​ fixed time.

Estimate growth, division,​​​‌ interaction features in structured​ populations

The estimation of​‌ the division rate in​​ non-interacting populations has been​​​‌ developed in a series​ of papers over the​‌ last decade 82.​​ The question we want​​​‌ to address now is​ whether growth and division​‌ rates are modified by​​ cell-to-cell interaction (or yet​​​‌ by antibiotic resistance or​ by competition), and reciprocally,​‌ how distributed growth and​​ division rates may have​​​‌ an influence on the​ morphogenesis of the bacterial​‌ microcolony. In this task,​​ we aim to provide​​​‌ answers based on more​ realistic individual-based models. We​‌ plan the following steps:​​

• Develop parametric and non-parametric​​​‌ inference of the interaction​ function from single individual​‌ tracking. A similar study​​ has been carried out​​​‌ by Laetitia Della Maestra​ and Marc Hoffmann 73​‌ for Mc-Kean-Vlasov equation; we​​ would like to add​​​‌ a size structure and​ a non-constant number of​‌ individuals. We will first​​ assume that the growth​​​‌ and division rates do​ not depend on the​‌ interaction between cells, so​​ that prior to this​​ step we have used​​​‌ the methods already developed‌ to infer these functional‌​‌ parameters. We may also​​ build upon biophysical studies​​​‌ such as in 138‌.
• Develop statistical hypothesis‌​‌ testing to accept or​​ reject the assumption made​​​‌ in the previous step‌ that division and growth‌​‌ are not influenced by​​ the interaction inferred. Reciprocally,​​​‌ test whether different division‌ or growth rates would‌​‌ give rise to different​​ morphogenesis.
• Generalise the methods​​​‌ and adapt them to‌ new problems, in particular‌​‌ the mycelial networks 76​​.

Estimate mutation or​​​‌ fragmentation kernel density

The‌ question of estimating the‌​‌ fragmentation kernel in polymer​​ breakage experiments 79 surprisingly​​​‌ rejoins the question of‌ estimating the so-called Distribution‌​‌ of Fitness Effects (DFE)​​ which characterizes the accumulation​​​‌ of mutations in bacteria‌ 123. As shown‌​‌ in 79, these​​ are so-called severely ill-posed​​​‌ inverse problems, for which‌ we aim at developing‌​‌ new approaches, two in​​ particular: rely on short-time​​​‌ instead of long-term behaviour,‌ adapt statistical methods developed‌​‌ for decoumpounding Poisson processes​​ and deconvolution 85.​​​‌

State estimation and observation‌ inequalities for depolymerisation models‌​‌

In depolymerisation experiments, prior​​ to parameter estimation, we​​​‌ began to address the‌ question of state estimation,‌​‌ i.e. how to infer​​ the initial condition out​​​‌ of measurements of moments‌ time dynamics. Whereas it‌​‌ is relatively straightforward if​​ we approximate the discrete​​​‌ system by a backward‌ transport equation 39,‌​‌ we address the question​​ of estimating it from​​​‌ the next order approximation,‌ namely a transport-diffusion equation;‌​‌ this new problem is​​ closer to the experimental​​​‌ system but gives rise‌ to a severely ill-posed‌​‌ inverse problem, for which​​ we want to find​​​‌ an observation inequality thanks‌ to Carleman estimates 71‌​‌, 70.

Calibrating​​ the mycelial network model​​​‌

The model developed in‌ 128 paves the way‌​‌ to new parametric calibration​​ methods that we wish​​​‌ to confront with the‌ real observations made by‌​‌ mycological colleagues of the​​ LIED laboratory (Paris Diderot​​​‌ University), as well as‌ with their empirical results.‌​‌

The parametric calibration based​​ on the solutions of​​​‌ the spectral problem can‌ lead to new simple‌​‌ descriptors that characterize the​​ growth of the fungus.​​​‌

The first objective is‌ to see how values‌​‌ obtained in 76 for​​ the exponential growth rates​​​‌ compare with the one‌ obtained in 128 as‌​‌ a solution of the​​ spectral problem related to​​​‌ the corresponding growth and‌ fragmentation equation. For the‌​‌ latter, there is an​​ interpretation through the main​​​‌ characteristics of the network‌ (ratio between the number‌​‌ of external nodes and​​ the total length of​​​‌ the network at a‌ sufficiently large time $\mathrm{t‌‌}$).

Then, we could​​ test how these descriptors​​​‌ change in different growth‌ environments. This will allow‌​‌ us to quantify the​​ impact of various forms​​​‌ of stress (nutrient depletion,‌ pH, ...).

From a‌​‌ theoretical point of view,​​ we would have to​​​‌ justify this empirical approach‌ and demonstrate a "many‌​‌ to one" formula to​​ be able to correctly​​​‌ sample our model. It‌ should also be proved‌​‌ that the estimators thus​​​‌ constructed are consistent and​ converge, when $t\to ‌\infty$, to the​​ quantities they are supposed​​​‌ to approximate.

The bacterial​ cell cycle

Coordination between​‌ cell growth and division​​ is often carried out​​​‌ by ‘size control’ mechanisms,​ where the cell size​‌ has to reach a​​ certain threshold to trigger​​​‌ some event of the​ cell cycle, such as​‌ DNA replication or cell​​ division. Concerning bacteria, recent​​​‌ articles 38, 124​ stated the excellent adequacy​‌ of the so-called "incremental​​ model", where the structuring​​​‌ variable which triggers division​ is the size increase​‌ of the bacteria since​​ birth, to experimental data.​​​‌ This opens up new​ questions to refine and​‌ analyse this model, test​​ its validity in extreme​​​‌ growth conditions such as​ antibiotic treatments, and understand​‌ its links with intracellular​​ mechanisms. Main research axis:​​​‌ 3, and the CellDiv​ platform.

Antibiotic response and​‌ resistance emergence

To address​​ the emergence of antibiotic-resistant​​​‌ strains of bacteria, it​ is essential to understand​‌ quantitatively the response of​​ bacteria to antibiotic treatments.​​​‌ Under the action of​ an antibiotic that causes​‌ damage to cellular DNA,​​ bacteria change their growth​​​‌ strategy and do not​ respond homogeneously to this​‌ stress. Of particular importance​​ is the so-called SOS​​​‌ response: in response to​ DNA damage induced by​‌ antibiotic treatments, the cell​​ cycle is arrested and​​​‌ DNA repair and mutagenesis​ are induced (cf. 41​‌). Cells with high​​ SOS response will grow​​​‌ for an abnormal duration,​ producing long filaments that​‌ are impervious to antibiotics.​​ Understanding the distribution of​​​‌ sizes in the population​ of bacteria will allow​‌ a better quanitfication of​​ antibiotics effects. On this​​​‌ subject, we work with​ Meriem El Karoui who​‌ carries out microfluidic experiments​​ in Edinburg university. Main​​ research axis: 2.

Microcolony​​​‌ morphogenesis

When bacterial microcolonies‌ grow, they can aggregate‌​‌ to one another and​​ form a biofilm. How​​​‌ do they interact? How‌ do their growth and‌​‌ division characteristics translate into​​ the shape of the​​​‌ colony? Inside the gut,‌ it has been proved‌​‌ that the immune response​​ acts not by killing​​​‌ bacteria but by making‌ them aggregate after division;‌​‌ how do these aggregates​​ form and break is​​​‌ another question tackled by‌ Claude Loverdo at Lab.‌​‌ Jean Perrin (Sorbonne). Main​​ research axis: 1 (short​​​‌ term in collaboration with‌ Diane Peurichard and Sophie‌​‌ Hecht).

Bacterial growth in​​ a chemostat; the gut​​​‌ as a chemostat

A‌ chemostat is a specific‌​‌ experiment, where the number​​ of bacteria is let​​​‌ constant by a permanent‌ influx and outflux. The‌​‌ functional mechanism of the​​ very gut could be​​​‌ modeled as a chemostat.‌ Main research axis: 2‌​‌ (mid to long term​​ / only first contacts​​​‌ made).

Mutations

The pace‌ of evolution and possible‌​‌ trajectories depend on the​​ dynamics of mutation incidence​​​‌ and the effects of‌ mutations on fitness. Mutation‌​‌ dynamics has been for​​ the first time analyzed​​​‌ directly by Lydia Robert‌ and co-authors 123,‌​‌ using two different microfluidic​​ experiments which led them​​​‌ to the conclusion of‌ a Poissonnian appearance of‌​‌ bacterial mutations, and to​​ a first parametric estimation​​​‌ of the so-called "distribution‌ of fitness effects" (DFE)‌​‌ of mutations. How to​​ assess better the shape​​​‌ of the DFE, and‌ apply the method not‌​‌ only to deleterious or​​ neutral but also to​​​‌ possibly beneficial mutations, is‌ one of our goals.‌​‌ Main research axis: 3,​​ short term (Guillaume Garnier's​​​‌ ongoing PhD).

Horizontal gene‌ transfer

Microorganisms such as‌​‌ bacteria tend to exhibit​​ a relatively large “evolution​​​‌ speed”. They have also‌ the particularity to exchange‌​‌ genes by direct cell-to-cell​​ contact. We are particularly​​​‌ interested in plasmids horizontal‌ gene transfer (HGT): plasmids‌​‌ carry pathogens or genes​​ coding for antibiotics resistances,​​​‌ and plasmid exchange is‌ considered by biologists as‌​‌ the primary reason for​​ antibiotics resistance. Main research​​​‌ axis: 1, both short‌ and long-term research, included‌​‌ in the ERC project​​ SINGER.

Leukemic mutations and‌ hematopoeisis

Hematopoiesis is the‌​‌ process of producing blood​​ cells from stem cells​​​‌ and progenitors. These highly‌ regulated mechanisms keep at‌​‌ equilibrium the number of​​ blood cells such as​​​‌ red blood cells, white‌ blood cells and platelets‌​‌ (mature cells). We want​​ to understand the emergence​​​‌ of leukaemia or resistance‌ to chemotherapy through the‌​‌ mechanisms of erythropoiesis (production​​​‌ of red blood cells)​ and leukopoiesis (white blood​‌ cell formation). This application​​ also rejoins the application​​​‌ 3.1.

Senescence by telomere​ shortening

Telomeres cap the​‌ ends of linear chromosomes,​​ and help maintain genome​​​‌ integrity by preventing the​ ends being recognized and​‌ processed as accidental chromosomal​​ breaks. When telomeres fall​​​‌ below a critical length,​ cells enter replicative senescence.​‌ However, the exact structure(s)​​ of the short or​​​‌ dysfunctional telomeres either triggering​ permanent replicative senescence or​‌ promoting genome instability remains​​ to be defined; this​​​‌ is the main focus​ of Teresa Teixeira's lab​‌ at IBPC, which has​​ developed microfluidic as well​​​‌ as population experiments to​ follow senescence triggering in​‌ yeast cells. Main research​​ axis: 1 and 2.​​​‌ This application is both​ a long-term goal, in​‌ a long-lasting collaboration with​​ Teresa Teixeira and Zhou​​​‌ Xu, and has short​ and mid-terms objectives, through​‌ Anaïs Rat's finishing D​​ and Jules Olayé forthcoming​​​‌ PhD (co-supervised by Milica​ Tomašević and Marie Doumic).​‌

Ageing in drosophyla

Ageing’s​​ sensitivity to natural selection​​​‌ has long been discussed​ because of its apparent​‌ negative effect on an​​ individual’s fitness. In the​​​‌ recent years, a new​ 2-phases model of ageing​‌ has been proposed by​​ Hervé Tricoire and Michael​​​‌ Rera 72, 132​, describing the ageing​‌ process not as being​​ continuous but as made​​​‌ of at least 2​ consecutive phases separated by​‌ a dramatic transition. It​​ was first observed in​​​‌ drosophila, and then shown​ to be evolutionary conserved;​‌ this raises the question​​ of an active selection​​​‌ of the underlying mechanisms​ throughout evolution. Main research​‌ axis: 2 and 3.​​

Protein polymerisation: amyloid formation​ and autophagy

Protein polymerisation​‌ occurs in many different​​ situations, from functional situations​​​‌ (actin filaments, autophagy) to​ toxic ones (amyloid diseases).​‌ It involves complex reaction​​ networks, making it a​​​‌ challenge to identify the​ key mechanisms, for instance​‌ which mechanisms lead to​​ the initial formation of​​​‌ polymers during the first​ reaction steps (nucleation), how​‌ and where the polymers​​ break, or yet the​​​‌ aggregates formation, out of​ (at least) two different​‌ proteins, in autophagy. With​​ our biologist collaborators, our​​​‌ aim in these applications​ is to isolate the​‌ most meaningful reactions, study​​ their behaviour(Research axis​​​‌ 2), and compare​ them - qualitatively and,​‌ if possible, quantitatively -​​ with experimental data.

Mycelial​​​‌ network

Filamentous fungi are​ complex expanding organisms that​‌ are omnipresent in nature.​​ They form filamentous structures,​​​‌ growing and branching to​ create huge networks called​‌ mycelia. We aim​​ at modelling, understanding and​​​‌ estimating the main mechanisms​ of mycelial formation. We​‌ have already studied a​​ first model without interactions​​​‌ and we will now​ study the impact of​‌ fusion of filaments on​​ the growth of the​​​‌ network. Main research axes:​ 2 and 3.

Emergence‌​‌ of bacterial resistance in​​ heterogeneous environments

When an​​​‌ antibiotic treatment is applied‌ to a population, bacteria‌​‌ resistant to the treatment​​ have an opportunity to​​​‌ develop. If several treatments‌ are used, life threatening‌​‌ multi-resistant bacteria can appear.​​ Understand the dynamics of​​​‌ bacterial populations in such‌ heterogeneous environments would provide‌​‌ interesting perspectives to improve​​ treatments and keep antibiotic​​​‌ resistance in control. On‌ this topic, we will‌​‌ collaborate with S. Gandon​​ lab at CEFE, that​​​‌ tackles this problem with‌ a combination of theory‌​‌ and experiments. This also​​ rejoins the application domain​​​‌ 3.1., and the main‌ research axis is axis‌​‌ 2.

Dynamics of species​​ submitted to climate change​​​‌

The impact of climate‌ change on natural species‌​‌ is a complicated matter.​​ An important research effort​​​‌ has been made on‌ the modification of species'‌​‌ niche in coming years,​​ but this is only​​​‌ a partial clue for‌ the future of species.‌​‌ In collaboration with Ophélie​​ Ronce at ISEM, we​​​‌ will investigate how the‌ local adaptation of species‌​‌ will is shook by​​ global changes. With François​​​‌ Massol in CIIL and‌ Nicolas Loeuille in IEES,‌​‌ we will focus on​​ the impact of interspecies​​​‌ effects: predation, parasitism, cooperation,‌ etc. Main research axis:‌​‌ 1.

7.1.1 telomeres​​

• Name:
  Simulation of cell​​​‌ populations and lineages during​ replicative senescence.
• Keywords:
  Stochastic​‌ models, Statistical inference, Monte-Carlo​​ methods, Branching system, Population​​​‌ approach, Computational biology
• Functional​ Description:

  Code associated with​‌ "Mathematical model linking telomeres​​ to senescence in Saccharomyces​​​‌ cerevisiae reveals cell lineage​ versus population dynamics".

  Preprint​‌ version of the associated​​ article: https://www.biorxiv.orgcontent/10.1101/2023.11.22.568287v1 See also​​​‌ Chapter 3 of the​ PhD thesis: https://theses.hal.science/tel-04250492

  The​‌ telomeres package contains all​​ the necessary auxiliary code.​​​‌ This is where the​ mathematical model is encoded,​‌ with its default parameters​​ (parameters.py). More generally, it​​​‌ contains all the functions​ allowing to

  Posttreat the​‌ raw data (make_*.py) Simulate​​ the model (simulation.py) Plot​​ the simulated and experimental​​​‌ data, the laws of‌ the model... (plot.py)

  The‌​‌ scripts in this folder​​ are not intended to​​​‌ be modified (unless you‌ find errors, in which‌​‌ case please let me​​ know) or used directly​​​‌ to run simulations.

  The‌ makeFiles folder contains scripts‌​‌ to run to generate​​ the data/processed directory, that​​​‌ contains the posstreated data.‌

  The main folder contains‌​‌ the scripts that should​​ be run to perform​​​‌ the simulations and plot‌ their results.

• URL:
  https://­zenodo.­org/­records/­14525443‌​‌
• Contact:
  Anais Rat

8.1.1 Hematopoiesis‌ as a continuum: from‌​‌ stochastic compartmental model to​​ hydrodynamic limit

Participants: Vincent​​​‌ Bansaye, Ana Fernandez‌ Baranda, Stephane Giraudier‌​‌, Sylvie Méléard.​​

This work is now​​​‌ submitted 20.

We‌ consider a multiscale stochastic‌​‌ compartmental model with three​​ types of cells (stem​​​‌ cells, immature cells and‌ mature cells) which combines‌​‌ cell proliferation and cell​​ differentiation. We derive a​​​‌ hydrodynamic limit when the‌ number of immature compartments‌​‌ goes to infinity obtaining​​ a PDE system with​​​‌ boundary conditions, modelling hematopoiesis‌ as a continuum. We‌​‌ assume that proliferation and​​ differentiation are regulated and​​​‌ let the corresponding rates‌ depend on the number‌​‌ of mature cells. This​​ leads us to model​​​‌ the dynamics of the‌ population by a Markov‌​‌ process in continuous time​​ and discrete space, which​​​‌ does not satisfy the‌ branching property. We prove‌​‌ the convergence in law​​ of the stem and​​​‌ mature cells population size‌ processes and of the‌​‌ empirical measures of the​​ immature cells dynamics, conveniently​​​‌ rescaled, to the unique‌ triplet involving coupled functions‌​‌ and measure, solutions of​​ a deterministic measured valued​​​‌ equation with boundary dynamics.‌ The cell differentiation induces‌​‌ a transport term in​​ space and the main​​​‌ difficulty comes from the‌ boundary effects coming from‌​‌ stem and mature cells.​​ We also prove that​​​‌ the limiting measure admits‌ at each time a‌​‌ density with respect to​​ Lebesgue measure and can​​​‌ be characterized by as‌ solution of a partial‌​‌ differential equation.

8.1.2 Convergence​​ of individual-based models of​​​‌ population to Hamilton-Jacobi equations‌

Participants: Anouar Jeddi.‌​‌

Anouar Jeddi's PhD work​​ is co-supervised by Nicolas​​​‌ Champagnat, Sepideh Mirrahimi and‌ Sylvie Méléard. He has‌​‌ carried out several studies​​ to link individual-based models​​​‌ to Hamilton-Jacobi type equations.‌

8.1.3 Functional Limit Theorems​​ for the range of​​​‌ stable random walks

Participants:​ Maxence Baccara.

Maxence​‌ Baccara carries out his​​ PhD work under Vincent​​​‌ Bansaye and Jean-René Chazottes'​ co-supervision. His work complements​‌ the fluctuations obtained at​​ fixed time and the​​​‌ functional limit Theorems obtained​ in the strongly transient​‌ regime. The techniques involve​​ original ideas of Le​​​‌ Gall and Rosen for​ fluctuations and allow to​‌ show tightness in some​​ Hölder space, thus also​​​‌ providing sharp regularity results​ about the limiting processes.​‌ The original motivation of​​ this work is the​​​‌ description of functionals appearing​ in spatial ecology for​‌ consumption of resources induced​​ by random motion. It​​​‌ is applied our to​ estimate the large fluctuations​‌ of energy and mortality​​ for a simple prey​​​‌ predator model 17.​

8.1.4 Interacting populations

8.1.5 Dynamics of​​​‌ a kinetic model describing​ protein transfers in a​‌ cell population

Participants: Pierre​​ Magal$†$, Gaël​​​‌ Raoul.

In 9​, we consider a​‌ cell population structured by​​ a positive real number​​​‌ which represents the number​ of P-glycoproteins carried by​‌ the cell. In this​​ article, we introduce a​​​‌ kinetic model to describe​ the dynamics of the​‌ cell population, and consider​​ an asymptotic limit of​​​‌ this equation: if transfers​ are frequent, the population​‌ can be described through​​ a system of two​​​‌ coupled ordinary differential equations.​ The main idea of​‌ this manuscript is to​​ combine Wasserstein distance estimates​​​‌ on the kinetic operator​ to more classical estimates​‌ on the macroscopic quantities.​​

The model described above​​​‌ can leads to the​ formation of singular solutions:​‌ in a population of​​ cells, a positive fraction​​​‌ of the individuals could​ have depleted its P-glycoproteins,​‌ creating a Dirac mass​​ at the origin. We​​​‌ believe this unusual property​ could be representative of​‌ a biological reality, and​​ have therefore proposed an​​​‌ existence and uniqueness framework​ for measure-valued solutions in​‌ 34.

8.1.6 Macroscopic​​ limit from structured population​​​‌ models to simpler models​

Participants: Sirine Boucenna,​‌ Vasilis Dakos, Gaël​​ Raoul.

In 12​​​‌, we consider an​ ecology model in which​‌ the population is structured​​ by a spatial variable​​​‌ and a phenotypic trait.​ The model combines a​‌ parabolic operator on the​​ spatial variable with a​​​‌ kinetic operator on the​ trait variable. We combine​‌ a contraction argument based​​ on Wasserstein estimates on​​​‌ the phenotypic variable with​ parabolic estimates controlling the​‌ spatial regularity of solutions​​ to prove the convergence​​​‌ of the population size​ and the mean phenotypic​‌ trait to solutions of​​ the Kirkpatrick-Barton model, which​​​‌ is a well-established model​ in evolutionary ecology.

We​‌ have used a similar​​ macroscopic argument to understand​​​‌ the effect of plastic​ traits on tree phenology:​‌ In tropical regions (French​​ Guyana in particular), trees​​​‌ can enter a summer​ dormancy when temperatures exceed​‌ a certain trigger temperature,​​ which protects the individual​​​‌ from water stress. In​ the study 22,​‌ we propose a model​​ to study the effect​​ of this plasticity in​​​‌ the context of an‌ environmental shifts (higher temperatures,‌​‌ lower precipitations).

8.2.1 Long-time behaviours

Long-time‌ behaviour of a multidimensional‌​‌ age-dependent branching process with​​ a singular jump kernel.​​​‌

Participants: Jules Olayé,‌ Milica Tomašević.

Jules‌​‌ Olayé and Milica Tomašević​​ published the article “Long-time​​​‌ behaviour of a multidimensional‌ age-dependent branching process with‌​‌ a singular jump kernel​​ modelling telomere shortening” in​​​‌ “Electronic Journal of Probability”‌ 1. In this‌​‌ work, the authors study​​ a quite complex model​​​‌ representing the biological phenomenon‌ of telomere shortening with‌​‌ an age structure, and​​ aim to obtain the​​​‌ convergence of the telomere‌ lengths and age distribution‌​‌ towards a stationary profile.​​ Due to the fact​​​‌ that the kernel for‌ updating telomere lengths at‌​‌ each cell division is​​ irregular with respect to​​​‌ the Lebesgue measure, and‌ to the age structure‌​‌ (implying a semi-Markovian setting),​​ the proof of this​​​‌ convergence is quite technical.‌ The authors manage these‌​‌ difficulties by exploiting a​​ criterion developed by Aurélien​​​‌ Velleret corresponding to a‌ weak form of a‌​‌ Harnack inequality, and by​​ using results from the​​​‌ renewal theory.

Quantitative approximation‌ of a Keller–Segel PDE‌​‌ by a branching moderately​​ interacting particle system and​​​‌ suppression of blow-up

Participants:‌ Thomas Cavalazzi, Alexandre‌​‌ Richard, Milica Tomašević​​.

The Keller–Segel PDE​​​‌ is a model for‌ chemotaxis known to exhibit‌​‌ possible finite-time blow-up. Following​​ a seminal work by​​​‌ Tello and Winkler, a‌ logistic damping term is‌​‌ added in this PDE​​ and local well-posedness of​​​‌ mild solutions is proven.‌ When the space dimension‌​‌ is 2 or when​​ the damping is strong​​​‌ enough, the solution is‌ global in time. In‌​‌ the second part of​​ this work, a microscopic​​​‌ description of this model‌ is introduced in terms‌​‌ of a system of​​ stochastic moderately interacting particles.​​​‌ This system features two‌ main characteristics: the interaction‌​‌ between particles happens through​​ a singular (Coulomb-type) kernel​​​‌ which is attractive; and‌ the particles are subject‌​‌ to demographic events, birth​​ and death due to​​​‌ local competition with other‌ particles. The latter induces‌​‌ a branching structure of​​ the particle system. Then​​​‌ the main result of‌ this work 24 is‌​‌ the convergence of the​​ empirical measure of the​​​‌ particle system towards the‌ Keller–Segel PDE with logistic‌​‌ damping, with a rate​​ of order $N^{-‌1/\left(\mathrm{2‌}\right(d+\mathrm{1‌‌}\left)\right)}$.

Long-time​​ behaviour of a degenerate​​​‌ stochastic system modeling the‌ response of a population‌​‌ to its environmental perception.​​

Participants: Pierre Collet,​​​‌ Claire Ecotière, Sylvie‌ Méléard.

In 4‌​‌, accepted for publication​​ in Electronic Communications in​​​‌ Probability, we study the‌ asymptotics of a two-dimensional‌​‌ stochastic differential system with​​ a degenerate diffusion matrix.​​​‌ This system describes the‌ dynamics of a population‌​‌ where individuals contribute to​​ the degradation of their​​​‌ environment through two differentbehaviors,‌ responding more or less‌​‌ intensively to their environmental​​​‌ perception. We exploit the​ almost one-dimensional form of​‌ the dynamical system to​​ compute explicitly the Freidlin-Wentzell​​​‌ action functional. This allows​ us to give conditions​‌ under which the small​​ noise regime of the​​​‌ invariant measure is concentrated​ around the equilibria of​‌ the dynamical system having​​ the smallest diffusion coefficient.​​​‌

Long time behavior of​ Feynman-Kac semigroups.

Participants: Pierre​‌ Collet, Sylvie Méléard​​, Jaime San Martin​​​‌.

The article 69​, accepted for publication​‌ in the Ann. Fac.​​ Sc. Toulouse, studies the​​​‌ long time behavior of​ Feynman-Kac semigroups by means​‌ of spectral properties. We​​ adapt the ideas developed​​​‌ in Cattiaux et al.​ 2009 for a non​‌ conservative semigroup. We consider​​ a situation where the​​​‌ underlying diffusion process doesn't​ come down rapidly from​‌ infinity but the compactness​​ properties follow from the​​​‌ divergence of the potential​ at infinity. We establish​‌ the complete spectral decomposition​​ for the Feynman-Kac semigroup.​​​‌ An interesting consequence is​ the identification of the​‌ law of the time​​ reversed spinal process issued​​​‌ from the unique quasi-stationary​ distribution (q.s.d.) with the​‌ $Q$-process of the​​ Feynman-Kac semigroup.

Long time​​​‌ behavior and Yaglom limit​ for real trait-structured Birth​‌ and Death Processes.

Participants:​​ Pierre Collet, Sylvie​​​‌ Méléard, Jaime San​ Martin.

In this​‌ article 26, submitted​​ to Probability Theory and​​​‌ Related Fields, we study​ the long time behaviour​‌ of measure-valued birth and​​ death processes in continuous​​​‌ time, where the dynamics​ between jumps are one-dimensional​‌ Markov processes including diffusion​​ and jumps. We consider​​​‌ the three regimes, critical,​ subcritical and supercritical. Under​‌ suitable hypotheses on the​​ Feynman-Kac semigroup, we prove​​​‌ a new recurrence for​ the moments and the​‌ extinction probability, their time​​ asymptotics and the convergence​​​‌ in law for the​ measure-valued birth and death​‌ process conditioned to non​​ extinction, leading to the​​​‌ existence of $Q$-process​ and Yaglom limit (in​‌ this infinite dimensional setting).​​ We develop three classes​​​‌ of natural examples where​ our results apply.

Ancestral​‌ lineages for horizontal gene​​ transfer modelling

Participants: Mateo​​​‌ Deangeli Bravo.

Mateo​ Deangeli Bravo numerically observed​‌ the convergence towards a​​ stationary profile, but he​​​‌ also observed a non-damped​ cyclic profile. The aim​‌ is to study ancestral​​ lineages, that is, to​​​‌ trace the evolution of​ genetic traits back to​‌ the initial time, starting​​ from an individual sampled​​​‌ at time $T>0$. He numerically​‌ proved that the lineage​​ of a randomly sampled​​​‌ individual takes values, over​ a long period, in​‌ a neighborhood of the​​ best-adapted traits, although he​​​‌ found other sets of​ parameters for which this​‌ observation does not hold.​​ He calculated the generator​​​‌ of the time-reversed process​ characterizing this evolution. To​‌ do this, he developed​​ an approach generalizing previous​​​‌ results, notably in the​ absence of stationary densities​‌ and in the presence​​ of asymmetric jumps. An​​​‌ in-depth study of the​ large population limit (​‌2) (existence and​​ uniqueness of a stationary​​​‌ solution, convergence) remains to​ be conducted. A parallel​‌ avenue is to deepen​​ the approach for diffusions​​ other than drifted Brownian​​​‌ motion. Finally, he is‌ planning to study horizontal‌​‌ transfer from the perspective​​ of Mutualism, following discussions​​​‌ initiated on regulation models‌ not depending on a‌​‌ quadratic competition term.

8.2.2​​ Random Models in Biology,​​​‌ Ecology and Evolution

Participants:‌ Sylvie Méléard.

This‌​‌ book accepted by Springer​​ is intended for master​​​‌ students in applied mathematics‌ or theoretical biologists who‌​‌ wish to expand their​​ knowledge of probabilistic modeling​​​‌ tools. It originated from‌ a course given to‌​‌ third-year students at the​​ École polytechnique (France), but​​​‌ over the years it‌ has grown to far‌​‌ exceed the scope of​​ the course. Its aim​​​‌ is to provide the‌ reader with rigorous tools‌​‌ for modeling biological phenomena​​ subject to random fluctuations.​​​‌ It focuses on stochastic‌ models built from individual‌​‌ behaviors.

8.2.3 Laws of​​ large numbers for cross-diffusion​​​‌ models and supercritical branching‌ processes

Participants: Vincent Bansaye‌​‌, Tresnia Berah,​​ Alexandre Bertolino, Bertrand​​​‌ Cloez, Ayman Moussa‌.

In a submitted‌​‌ work 19, Vincent​​ Bansaye, Ayman Moussa (Sorbonne​​​‌ Université) and Alexandre Bertolino‌ (Ecole polytechnique and Sorbonne‌​‌ Université) study the stability​​ of non-conservative deterministic cross​​​‌ diffusion models and prove‌ that they are approximated‌​‌ by stochastic population models​​ when the populations become​​​‌ locally large. In this‌ model, the individuals of‌​‌ two species move, reproduce​​ and die with rates​​​‌ sensitive to the local‌ densities of the two‌​‌ species. Quantitative estimates are​​ given and convergence is​​​‌ obtained as soon as‌ the population per site‌​‌ and the number of​​ sites go to infinity.​​​‌ The proofs rely on‌ the extension of stability‌​‌ estimates via duality approach​​ under a smallness condition​​​‌ and the development of‌ large deviation estimates for‌​‌ structured population models, which​​ are of independent interest.​​​‌ The proofs also involve‌ martingale estimates in ${\mathrm{H‌‌}}^{-1}$ and improve​​ the approximation results in​​​‌ the conservative case as‌ well.

In the preprint‌​‌ 18, On the​​ strong law of large​​​‌ numbers and $L-‌logL$ condition for‌​‌ supercritical general branching processes​​, Vincent Bansaye, Tresnia​​​‌ Berah (Imperial College) and‌ Bertrand Cloez (MISTEA) consider‌​‌ branching processes for structured​​ populations: each individual is​​​‌ characterized by a type‌ or trait which belongs‌​‌ to a general measurable​​ state space. We focus​​​‌ on the supercritical recurrent‌ case, where the population‌​‌ may survive and grow​​ and the trait distribution​​​‌ converges. The branching process‌ is then expected to‌​‌ be driven by the​​ positive triplet of first​​​‌ eigenvalue problem of the‌ first moment semigroup. Under‌​‌ the assumption of convergence​​ of the renormalized semigroup​​​‌ in weighted total variation‌ norm, we prove strong‌​‌ convergence of the normalized​​ empirical measure and non-degeneracy​​​‌ of the limiting martingale.‌ Convergence is obtained under‌​‌ an $L-logL$ condition which provides​​​‌ a Kesten-Stigum result in‌ infinite dimension and relaxes‌​‌ the uniform convergence assumption​​ of the renormalized first​​​‌ moment semigroup required in‌ the work of Asmussen‌​‌ and Hering in 1976.​​ The techniques of proofs​​​‌ combine families of martingales‌ and contraction of semigroups‌​‌ and the truncation procedure​​​‌ of Asmussen and Hering.​ We also obtain ${\mathrm{L‌}}^1$ convergence of the​​ renormalized empirical measure and​​​‌ contribute to unifying different​ results in the literature.​‌ These results greatly extend​​ the class of examples​​​‌ where a law of​ large numbers applies, as​‌ we illustrate it with​​ absorbed branching diffusion, the​​​‌ house of cards model​ and some growth-fragmentation processes.​‌

Finally, Alexandre Bertolino, PhD​​ student co-supervised by Vincent​​​‌ Bansaye and Ayman Moussa,​ is currently working on​‌ the existence and uniqueness​​ theory and explosion criteria​​​‌ for regular solutions (​$H^s$, $\mathrm{s‌}>d/\mathrm{2}$) of the triangular​​​‌ SKT system, i.e. the​ Shigesada–Kawasaki–Teramoto cross-diffusion system.

8.2.4​‌ Evolutionary dynamics - stochastic​​ and deterministic mutation models​​​‌

Evolution in ecosystem dynamics​

Participants: Sirine Boucenna,​‌ Vasilis Dakos, Gaël​​ Raoul.

The article​​​‌ 57 is now published​ in the Journal of​‌ Theoretical Biology. Shallow lakes​​ ecosystems may experience abrupt​​​‌ shifts (ie tipping points)​ from one state to​‌ a contrasting degraded alternative​​ state as a result​​​‌ of gradual environmental changes.​ It is crucial to​‌ elucidate how eco-evolutionary feedbacks​​ affect abrupt ecological transitions​​​‌ in shallow lakes. We​ explore the eco-evolutionary dynamics​‌ of submerged and floating​​ macrophytes in a shallow​​​‌ lake ecosystem under asymmetric​ competition for nutrients and​‌ light. We show how​​ rapid trait evolution can​​​‌ result in complex dynamics​ including evolutionary oscillations, extensive​‌ diversification and evolutionary suicide.​​ Overall, this study shows​​​‌ that evolution can have​ strong effects in the​‌ ecological dynamics of bistable​​ ecosystems.

Is heterogeneity beneficial​​​‌ or detrimental?

Participants: Marie​ Doumic, Anaïs Rat​‌, Magali Tournus.​​

Is there an advantage​​​‌ of displaying heterogeneity in​ a population where the​‌ individuals grow and divide​​ by fission? This is​​​‌ a wide-ranging question, for​ which a universal answer​‌ cannot be easily provided.​​ In 30, we​​​‌ aim at providing a​ quantitative answer in the​‌ specific context of growth​​ rate heterogeneity by comparing​​​‌ the fitness of homogeneous​ versus heterogeneous populations. We​‌ focus on a size-structured​​ population, where an individual's​​​‌ growth rate is chosen​ at its birth through​‌ heredity and/or random mutations.​​ We use the long-term​​​‌ behaviour to define the​ Malthus parameter of such​‌ a population, and compare​​ it to the ones​​​‌ of averaged homogeneous populations.​ We obtain analytical formulae​‌ in two paradigmatic cases:​​ first, constant rates for​​​‌ growth and division, second,​ linear growth rates and​‌ uniform fragmentation. Surprisingly, these​​ two cases happen to​​​‌ display similar analytical formulae​ linking effective and individual​‌ fitness. They allow us​​ to investigate quantitatively the​​​‌ crossed influence of heredity​ and heterogeneity, and revisit​‌ previous results stating that​​ heterogeneity is beneficial in​​​‌ the case of strong​ heredity.

8.2.5 Temporal dynamics​‌ of an oscillatory polymerisation​​ model

Participants: Marie Doumic​​​‌, Klemens Fellner,​ Mathieu Mezache, Juan​‌ Velazquez.

To provide​​ a mechanistic explanation of​​​‌ sustained then damped oscillations​ observed in a depolymerisation​‌ experiment, a bi-monomeric variant​​ of the seminal Becker-Döring​​​‌ system has been proposed​ in 80. When​‌ all reaction rates are​​ constant, the equations are​​ the following:

where $u$ and​​ $w$ are two distinct​​​‌ unit species, and ${\mathrm{c‌}}_i$ represents the concentration‌​‌ of clusters containing $\mathrm{i}$ units. We study in​​​‌ detail the mechanisms leading‌ to such oscillations and‌​‌ characterise the different phases​​ of the dynamics, from​​​‌ the initial high-amplitude oscillations‌ to the progressive damping‌​‌ leading to the convergence​​ towards the unique positive​​​‌ stationary solution. We give‌ quantitative approximations for the‌​‌ main quantities of interest:​​ period of the oscillations,​​​‌ size of the damping‌ (corresponding to a loss‌​‌ of energy), number of​​ oscillations characterising each phase.​​​‌ We illustrate these results‌ by numerical simulation, in‌​‌ line with the theoretical​​ results, and provide numerical​​​‌ methods to solve the‌ system 5.

8.3.1 The​​ mycelial network

Participants: Lena​​​‌ Kuwata, Thibault Chassereau‌, Florence Chapeland-Leclerc,‌​‌ Pascal David, Eric​​ Herbert, Gwenaël Ruprich-Robert​​​‌, Milica Tomašević,‌ Amandine Véber.

Quantifying‌​‌ the impact of different​​ forms of stress on​​​‌ fungal growth: an inference‌ method based on high-resolution‌​‌ pictures of the mycelial​​ network

In a previous​​​‌ work 76, a‌ complete methodology for monitoring‌​‌ the growth of a​​ filamentous fungus was introduced,​​​‌ covering all aspects of‌ this complex task going‌​‌ from the multi-scale imaging​​ of the network of​​​‌ filaments to the automated‌ extraction of the graph‌​‌ structure and its key​​ statistics at regular time​​​‌ points. This methodology was‌ applied to the fungus‌​‌ Podospora anserina grown in​​ the lab under various​​​‌ conditions. In parallel, a‌ stochastic growth-fragmentation model for‌​‌ the dynamics of such​​ mycelial networks was introduced​​​‌ and studied in 128‌. This simple model‌​‌ depends on three parameters​​ only: the elongation speed​​​‌ $v$ of a single‌ filament, the branching rate‌​‌ $b_1$ of a​​ filament at its open​​​‌ end, and the per‌ unit length rate ${\mathrm{b‌‌}}_2$ at which a​​ budding event happens, resulting​​​‌ in a new filament‌ branching off from an‌​‌ existing one. In this​​ work, we develop a​​​‌ statistical inference method based‌ on the large-time behaviour‌​‌ of the growth-fragmentation model​​ shown in 128,​​​‌ and on the high-resolution‌ pictures of the mycelial‌​‌ network obtained using the​​ methodology described in 76​​​‌, to reconstruct the‌ parameters $v,{\mathrm{b‌‌}}_1$ and $b_{\mathrm{2}}$ from experimental data. In​​​‌ 33, we use‌ this method to analyse‌​‌ the growth of P.​​ anserina observed under standard​​​‌ conditions and when several‌ forms of stress are‌​‌ applied, in order to​​​‌ quantify the effect of​ these stresses on the​‌ different mechanisms of fungal​​ growth. By comparing with​​​‌ the parameter estimates obtained​ from the dynamical tracking​‌ of individual filaments, we​​ show that reliable estimates​​​‌ of the individual elongation​ speed and branching rates​‌ can be computed from​​ the easily accessible data​​​‌ consisting in a single​ panorama of the filament​‌ network pictured after several​​ hours of growth and​​​‌ an empirical measure of​ the exponential growth rate​‌ of the number of​​ branch points and free​​​‌ extremities.

8.3.2 Telomere shortening​ and senescence: modelling and​‌ estimation

Individual cell fate​​ and population dynamics revealed​​​‌ by a mathematical model​ linking telomere length and​‌ replicative senescence.

Participants: Prisca​​ Berardi, Anaïs Rat​​​‌, Marie Doumic,​ Veronica Martinez Fernandez,​‌ Teresa Teixeira, Zhou​​ Xu.

Progressive shortening​​​‌ of telomeres ultimately causes​ replicative senescence and is​‌ linked with aging and​​ tumor suppression. Studying the​​​‌ intricate link between telomere​ shortening and senescence at​‌ the molecular level and​​ its population-scale effects over​​​‌ time is challenging with​ current approaches but crucial​‌ for understanding behavior at​​ the organ or tissue​​​‌ level. In the article​ 2, published in​‌ Nature Communications, we developed​​ a mathematical model for​​​‌ telomere shortening and the​ onset of replicative senescence​‌ using data from Saccharomyces​​ cerevisiae without telomerase. Our​​​‌ model tracks individual cell​ states, their telomere length​‌ dynamics, and lifespan over​​ time, revealing selection forces​​​‌ within a population. We​ discovered that both cell​‌ genealogy and global telomere​​ length distribution are key​​​‌ to determine the population​ proliferation capacity. We also​‌ discovered that cell growth​​ defects unrelated to telomeres​​​‌ also affect subsequent proliferation​ and may act as​‌ confounding variables in replicative​​ senescence assays. Overall, while​​​‌ there is a deterministic​ limit for the shortest​‌ telomere length, the stochastic​​ occurrence of non-terminal arrests​​​‌ drive cells into a​ totally different regime, which​‌ may promote genome instability​​ and senescence escape. Our​​​‌ results offer a comprehensive​ framework for investigating the​‌ implications of telomere length​​ on human diseases. Our​​​‌ model has also been​ used further in another​‌ experimental device, where one​​ telomere is cut at​​​‌ a given very short​ length thanks to CrisPr-Cas9​‌ technique 21. Alongside​​ this modelling and simulation​​​‌ approach, Jules Olayé's PhD​ work focused on several​‌ interesting problems raised by​​ this collaboration, see Sections​​​‌ 8.2.1, 8.4 and​ 35, 1.​‌

Recovering an initial distribution​​ of telomere lengths from​​​‌ measurements of senescence times​

Participant: Jules Olayé.​‌

Jules Olayé submitted the​​ pre-publication “An inverse problem​​​‌ in cell dynamics: Recovering​ an initial distribution of​‌ telomere lengths from measurements​​ of senescence times”. In​​​‌ this work, the author​ studies a deterministic model​‌ for telomere shortening, corresponding​​ to an integro-differential equation.​​​‌ His aim is to​ know if this is​‌ possible to retrieve the​​ initial distribution of telomere​​​‌ lenghts from the senescence​ times density. To solve​‌ this inverse problem, the​​ author first obtains an​​​‌ approximation of the system​ of integro-differential equations he​‌ study by a system​​ of partial differential equations,​​ in which there is​​​‌ a transport equation that‌ allows to simplify the‌​‌ computations. Then, he developed​​ an estimator on this​​​‌ approximated model, and solve‌ dthe inverse problem by‌​‌ using the equivalent of​​ this estimator on the​​​‌ original model. The inference‌ method he developed is‌​‌ then applied on numerical​​ and experimental data.

8.3.3​​​‌ Hematopoiesis and myelo-proliferative neoplasms‌ (MPN) modelling

Participants: Ana‌​‌ Fernandez Baranda, Vincent​​ Bansaye, Nadia Belmabrouk​​​‌, Celine Bonnet,‌ Stéphane Giraudier, Simon‌​‌ Girel, Panhong Gou​​, Emelyne Lauret,​​​‌ Duanya Liu, Sylvie‌ Méléard.

Mathematical modeling‌​‌ offers the opportunity to​​ test hypothesis concerning Myeloproliferative​​​‌ emergence and development.

Hematopoiesis‌ mimics stress hematopoiesis and‌​‌ induces the clonal advantage​​ at the stem cell​​​‌ level of Myelo-Proliferative Neoplasm‌ cells.

Participants: Vincent Bansaye‌​‌, Celine Bonnet,​​ Stéphane Giraudier, Simon​​​‌ Girel, Panhong Gou‌, Emelyne Lauret,‌​‌ Duanya Liu, Sylvie​​ Méléard.

Joint analysis​​​‌ of experimental data and‌ simulations shows that JAK2-V617F‌​‌ mice evolve in a​​ regime corresponding to a​​​‌ state of chronic stress,‌ modeled by a persistent‌​‌ change in kinetic parameters.​​ In this regime, the​​​‌ system's response to additional‌ acute stress is greatly‌​‌ attenuated, reflecting resistance to​​ acute stress. From a​​​‌ dynamic perspective, the mutant‌ hematopoietic system already appears‌​‌ to be displaced far​​ from its basal state,​​​‌ limiting its ability to‌ transiently adjust its cell‌​‌ flows in response to​​ external disturbance. When normal​​​‌ cells and JAK2-V617F cells‌ are integrated into a‌​‌ comparative framework and subjected​​ to stress (competitive transplantation​​​‌ of normal and JAK2V617F-mutated‌ cells, replicating the situation‌​‌ in humans where both​​ cell types exist in​​​‌ subjects with myeloproliferative neoplasia),‌ the model predicts and‌​‌ the data confirm the​​ emergence of a quantifiable​​​‌ proliferative advantage in favor‌ of mutant cells. Stress‌​‌ then acts as a​​ selection mechanism, amplifying differences​​​‌ in kinetic parameters and‌ leading to a clonal‌​‌ advantage for JAK2-V617F cells.​​ Overall, this work demonstrates​​​‌ formally, quantitatively, and mathematically‌ coherently that JAK2-V617F myeloproliferative‌​‌ syndromes correspond to a​​ state of chronic stress​​​‌ in the hematopoietic system.‌ This state partly explains‌​‌ the proliferative and clonal​​ advantage observed. This work​​​‌ will be submitted soon.‌

Effects of secondary mutations‌​‌ on growth and treatment​​ response in MPNs

Participants:​​​‌ Ana Fernandez Baranda,‌ Vincent Bansaye, Nadia‌​‌ Belmabrouk, Stéphane Giraudier​​, N Maslah,​​​‌ Sylvie Méléard.

Using‌ clinical data, our aim‌​‌ is to better understand​​ the evolution of JAK2V617F​​​‌ Myeloproliferative Neoplasms depending on‌ the mutational profile of‌​‌ a patient. We work​​ with a cohort of​​​‌ $n=291$ patients‌ with multiple blood samples‌​‌ through time. We present​​ a mixed effect model​​​‌ that integrates a logistic‌ growth for the percentage‌​‌ of JAK2V617F cells, the​​ effect that treatment or​​​‌ treatments have on the‌ mutant population and the‌​‌ effect of additional mutations​​ both on the growth​​​‌ and the treatment. The‌ question of study is‌​‌ if and to what​​ extent each additional mutation​​​‌ promotes or limits the‌ JAK2V617F growth and makes‌​‌ the mutant cells more​​​‌ or less responsive to​ treatment. We will estimate​‌ the parameters of the​​ model using the software​​​‌ Monolix and study their​ significance to answer this​‌ question. This is a​​ work in progress.

8.3.4​​​‌ Deciphering the Replication-Division Coordination​ in E. coli: A​‌ Unified Mathematical framework for​​ Systematic Model Comparison

Participants:​​​‌ Alexandre Perrin, Marie​ Doumic, Meriem El​‌ Karoui, Sylvie Méléard​​.

Despite extensive research,​​​‌ the quantitative principles that​ govern the coordination between​‌ DNA replication and cell​​ division in bacteria remain​​​‌ debated. Multiple theoretical models​ have been proposed, some​‌ postulating that a single​​ regulatory process is sufficient​​​‌ to ensure replication–division coordination,​ while others argue that​‌ two concurrent processes are​​ required for robust control.​​​‌ To enable the comparison​ of these approaches, we​‌ developed a unifying mathematical​​ framework within which models​​​‌ can be consistently formulated​ and quantitatively compared 36​‌. Through theoretical analysis,​​ we establish the necessary​​​‌ and sufficient conditions under​ which single-process models can​‌ reproduce physiological cell behaviours.​​ Beyond the correlation-based analyses​​​‌ extensively used to date,​ we further demonstrate within​‌ a comprehensive statistical framework​​ that double-process models more​​​‌ accurately recapitulate experimental data​ across all growth conditions.​‌ Finally, we developed a​​ unified model that robustly​​​‌ captures the replication-division coordination​ in every growth regime,​‌ thereby providing a foundation​​ for future mechanistic studies.​​​‌

8.3.5 Estimation of the​ lifetime distribution from fluctuations​‌ in Bellman-Harris processes

Participants:​​ Jules Olayé, Hala​​​‌ Bouzidi, Andrey Aristov​, Antoine Barizien,​‌ Salomé Gutiérrez Ramos,​​ Charles Baroud, Vincent​​​‌ Bansaye.

The growth​ of a population is​‌ often modeled as branching​​ process where each individual​​​‌ at the end of​ its life is replaced​‌ by a certain number​​ of offspring. We are​​​‌ interested in the estimation​ of the parameters of​‌ the Bellman-Harris model, motivated​​ by the estimation of​​​‌ cell division time. Lifetimes​ are distributed according a​‌ Gamma distribution and we​​ follow a population that​​​‌ starts from a small​ number of individuals by​‌ performing time-resolved measurements of​​ the population size. The​​​‌ exponential growth of the​ population size at the​‌ beginning offers an easy​​ estimation of the mean​​​‌ of the lifetime. Using​ fine and recent results​‌ on these fluctuations, we​​ describe two time-asymptotic regimes​​​‌ and explain how to​ estimate the variance. Then,​‌ we both consider simulations​​ and biological data to​​​‌ validate and discuss our​ method. The results described​‌ here provide a method​​ to determine single-cell parameters​​​‌ from time-resolved measurements of​ populations without the need​‌ to track each individual​​ or to know the​​​‌ details of the initial​ condition. This work is​‌ now published 10.​​

8.3.6 Asymptotic inverse problems​​​‌ for depolymerisation models

Participants:​ Marie Doumic, Philippe​‌ Moireau.

In this​​ study, we focused on​​​‌ depolymerisation reactions, which constitute​ frequent experiments, for instance​‌ in biochemistry for the​​ study of amyloid fibrils.​​​‌ The quantities experimentally observed​ are related to the​‌ time dynamics of a​​ quantity averaged over all​​​‌ polymer sizes, such as​ the total polymerised mass​‌ or the mean size​​ of particles. The question​​ analysed here is to​​​‌ link this measurement to‌ the initial size distribution.‌​‌ To do so, we​​ first derive, from the​​​‌ initial reaction system two‌ asymptotic models: at first‌​‌ order, a backward transport​​ equation, and at second​​​‌ order, an advection-diffusion/Fokker-Planck equation‌ complemented with a mixed‌​‌ boundary condition at x​​ = 0. We estimate​​​‌ their distance to the‌ original system solution. We‌​‌ then turn to the​​ inverse problem, i.e., how​​​‌ to estimate the initial‌ size distribution from the‌​‌ time measurement of an​​ average quantity, given by​​​‌ a moment of the‌ solution. This question has‌​‌ been already studied for​​ the first order asymptotic​​​‌ model, and we analyse‌ here the second order‌​‌ asymptotic. Thanks to Carleman​​ inequalities and to log-convexity​​​‌ estimates, we prove observability‌ results and error estimates‌​‌ for a Tikhonov regularization.​​ We then develop a​​​‌ Kalman-based observer approach, and‌ implement it on simulated‌​‌ observations. Despite its severely​​ ill-posed character, the second​​​‌ order approach appears numerically‌ more accurate than the‌​‌ first-order one 7.​​

8.3.7 Estimating the hazard​​​‌ rate with associated kernels;‌ application to a two-phase‌​‌ aging model

Participants: Luce​​ Breuil, Sarah Kaakai​​​‌.

Luce Breuil's PhD‌ focuses on modeling aging‌​‌ as a 2-phase process,​​ based on the biological​​​‌ discovery by Michaël Rera‌ 122, 132 of‌​‌ two consecutive phases in​​ the aging of drosophila.​​​‌ She is supervised by‌ Marie Doumic, Sarah Kaakaï‌​‌ and works in collaboration​​ with Michaël Rera. She​​​‌ and S. Kaakaï submitted‌ a preprint 23 on‌​‌ the convergence of the​​ hazard rate kernel estimator​​​‌ for a very general‌ class of kernels called‌​‌ associated kernels, for which​​ the dependence of the​​​‌ kernel on the bandwidth‌ and the point of‌​‌ estimation is not explicit.​​ In this preprint, they​​​‌ also prove an oracle‌ type inequality for both‌​‌ a local (pointwise) and​​ global minimax bandwidth selection​​​‌ procedure. A second work‌ she pursued in 2025‌​‌ is the study a​​ deterministic PDE model of​​​‌ 2-phase aging for a‌ wild population, for which‌​‌ she proved general existence,​​ uniqueness and positivity results​​​‌ of the solutions as‌ well as stability and‌​‌ convergence results for simplified​​ systems.

8.3.8 Non-Asymptotic Convergence​​​‌ of Discrete Diffusion Models:‌ Masked and Random Walk‌​‌ dynamics

Participants: Giovanni Conforti​​, Alain Durmus,​​​‌ Le-Tuyet-Nhi Pham, Gaël‌ Raoul.

Diffusion models‌​‌ for continuous state spaces​​ based on Gaussian noising​​​‌ processes are now relatively‌ well understood, as many‌​‌ works have focused on​​ their theoretical analysis.

In​​​‌ contrast, results for diffusion‌ models on discrete state‌​‌ spaces remain limited and​​ pose significant challenges, particularly​​​‌ due to their combinatorial‌ structure and their more‌​‌ recent introduction in generative​​ modelling.

In this work​​​‌ 28, we establish‌ new and sharp convergence‌​‌ guarantees for three popular​​ discrete diffusion models (DDMs).​​​‌

Two of these models‌ are designed for finite‌​‌ state spaces and are​​ based respectively on the​​​‌ random walk and the‌ masking process.

The third‌​‌ DDM we consider is​​ defined on the countably​​​‌ infinite space ${\mathbb{N}}^{\mathrm{d‌}}$ and uses a drifted‌​‌ random walk as its​​​‌ forward process.

For each​ of these models, the​‌ backward process can be​​ characterized by a discrete​​​‌ score function that can,​ in principle, be estimated.​‌ However, even with perfect​​ access to these scores,​​​‌ simulating the exact backward​ process is infeasible, and​‌ one must rely on​​ approximations.

In this work,​​​‌ we study Euler-type approximations​ and establish convergence bounds​‌ in both Kullback-Leibler divergence​​ and total variation distance​​​‌ for the resulting models,​ under minimal assumptions on​‌ the data distribution. In​​ particular, we show that​​​‌ the computational complexity of​ each method scales linearly​‌ in the dimension, up​​ to logarithmic factors.

Furthermore,​​​‌ to the best of​ our knowledge, this study​‌ provides the first non-asymptotic​​ convergence guarantees for these​​​‌ noising processes that do​ not rely on boundedness​‌ assumptions on the estimated​​ score.

9.1.1 Participation​ in other International Programs​‌

International Emerging Actions.

CNRS​​ project with University of​​​‌ Bath, 2024-2025, coordinator: Milica​ Tomašević and A. Mayorcas​‌

9.3.1 Other​​​‌ european programs/initiatives

ERC SINGER​ 101054787 ADG, PI: Sylvie​‌ Méléard

10.1.1 Scientific events: organisation​​​‌

Marie Doumic and Vincent‌ Bansaye co-organised an FMJH‌​‌ day for the programs​​ "math for life sciences"​​​‌ and "scientific computing" (November).‌

Viviana Gavilanes organised the‌​‌ first event of the​​ Research Group in Statistics,​​​‌ Probability, and Data Analysis‌ (GIEPAD). The event‌​‌ was announced on 02​​ October 2025 and took​​​‌ place on 18 October‌ 2025 in a virtual‌​‌ format.

Jules Olayé co-organized​​ the seminar of PhD​​​‌ students.

10.1.2 Scientific events: selection​​​‌

10.1.3 Journal​

10.1.4 Invited​​​‌ talks

Marie Doumic: Branching​ processes conference (Orsay, January);​‌ "Taming Complexity in Partial​​ Differential Systems" workshop (Vienna,​​​‌ February); "Young Women in​ Mathematical Biology" minicourse (Bonn,​‌ March); Collège de France​​ seminar (Paris, December).

Vincent​​​‌ Bansaye : Probability seminar​ in Villetaneuse seminar at​‌ the conference “Genealogies for​​ interacting populations” in Vienna​​​‌ (September); presentation of the​ CMAP Hôpital–Saint Louis collaborations​‌ during the IHU visit;​​ talk and mathematics workshop​​​‌ in middle school (G.​ Philippe and Diderot middle​‌ schools in Massy)

S.​​ Méléard: Plenary speakers in​​​‌ Conferences in Mexico, Merida,​ Hong-Kong, Scientific lectures in​‌ Santiago (Chile), Zurich, French​​ Academy of Sciences (for​​​‌ the Award Femme scientifique​ de l’année), Université Paris-Cité​‌ (twice) Research Lectures at​​ Université de Dschang, Cameroun.​​​‌

Gaël Raoul: June 2025​ "Measure-valued solutions for a​‌ structured population with transfers",​​ Le Havre University, France.​​​‌ August 2025 "Propagation of​ pathogens in a heterogeneous​‌ environment and emergence of​​ resistance strains", at the​​​‌ Summer School on Mathematical​ Biology at VIASM, Vietnam.​‌ October 2025 "From structured​​ population models to simplified​​​‌ limits: insights into tree​ population dynamics", University of​‌ French Guyana.

Milica Tomasevic:​​ Conference "Dynamics of Collective​​​‌ (Bio-)Systems: Mathematical Modelling and​ Applications", LPSM, October 2025.​‌

10.1.5 Participation at conferences​​

Maxence Baccara: Presentation :​​​‌ Doctoral Seminar (Sorbonne University)​

Alexandre Bertolino: Presentation :​‌ Journée Internes du Laboratoire​​ Jacques-Louis Lions: "Approximation of​​​‌ parabolic problems by particle​ systems"

Luce Breuil: Participation​‌ in the activities of​​ Chaire MMB (Aussois summer​​​‌ school and conference days​ held by the Chaire​‌ throughout the year), the​​ Young Women in Mathematical​​​‌ Biology conference (Bonn) and​ New trends in mathematical​‌ models for Biology (Paris)​​ ; talks at the​​​‌ PhD student seminar of​ MAP5 (Université Paris Cité)​‌ - March 2025 ;​​ Young Women in Mathematical​​​‌ Biology, Hausdorff Center (Bonn​ University) - April 2025;​‌ and Journées Maths Bio​​ Santé in Montpellier -​​​‌ October 2025.

Nicoleta Cazacu:​ Poster for the conference​‌ " New trends of​​ stochastic nonlinear systems :​​​‌ well-posedness, dynamics and numerics",​ CIRM, Marseille, October 2025.​‌ Short talk at Journées​​ de Probabilités, Marseille, June​​​‌ 2025. Talk at Séminaire​ des Doctorants, CMAP, Palaiseau,​‌ June 2025. Short talk​​ for the day of​​​‌ PhD Students of FdM​ , CentraleuSupélec, Gif-sur-Yvette, June​‌ 2025. Poster for the​​ visit of Insmi representatives​​​‌ at Centrale-Supélec, Gif-sur-Yvette, May​ 2025.

Mateo Deangeli Bravo​‌ : Talk at the​​ PhD Students Seminar of​​​‌ CMAP, participation in the​ activities of Chaire MMB​‌ (Aussois summer school and​​ conference days held by​​ the Chaire throughout the​​​‌ year)

Ana Fernandez-Baranda :‌ Speaker at Mathematical Biology‌​‌ Modelling days of Besancon,​​ 5-7 november 2025, Besancon​​​‌ (France), Journées Maths-Bio-Santé, 5-7‌ november 2025, Montpellier (France)‌​‌

Viviana Gavilanes: Poster presentation​​ at the 32nd International​​​‌ Conference on Yeast Genetics‌ and Molecular Biology (2025).‌​‌

Anouar Jeddi : Summer​​ school, Aussois. Colloque Probabilistes​​​‌ et Statisticiens, Oléron, Septembre‌ 2025.

Le Tuyet Nhi‌​‌ PHAM: June 12, 2025​​ "Bit-Level Discrete Diffusion with​​​‌ Markov Probabilistic Models: An‌ Improved Framework with Sharp‌​‌ Convergence Bounds under Minimal​​ Assumptions", SIMPA retreat, France.​​​‌ September 23, 2025 "Bit-Level‌ Discrete Diffusion with Markov‌​‌ Probabilistic Models: An Improved​​ Framework with Sharp Convergence​​​‌ Bounds under Minimal Assumptions",‌ workshop Generative Models in‌​‌ Science and Machine Learning​​ in Heidelberg, Germany.

Jules​​​‌ Olayé gave a talk‌ at the conference “Les‌​‌ probabilités de demain” in​​ Paris and at the​​​‌ seminar of the team‌ MUSCA at INRIA. He‌​‌ also gave an online​​ talk at the workshop​​​‌ “Grupo de Estadística, Probabilidad‌ y Análisis de Datos”‌​‌ organized by Viviana Gavilanes,​​ an other member of​​​‌ the team. Finally, he‌ presented a poster at‌​‌ the workshop “Emerging Connections​​ between Reaction-Diffusion, Branching Processes,​​​‌ and Biology” in Canada.‌

Alexandre Perrin : Seminar‌​‌ at Laboratoire d'Optique et​​ Biosciences. Poster at JMBS​​​‌ 2025

10.1.6 Leadership within‌ the scientific community

S.‌​‌ Méléard : Senior Fellow​​ of the Institute of​​​‌ Advanced Studies, CityU Hong‌ Kong since August 2025.‌​‌

10.1.7 Research administration

Vincent​​ Bansaye is vice-president of​​​‌ the Applied Math Department‌ of Ecole Polytechnique and‌​‌ of Fondation Mathématique Jacques​​ Hadamard (FMJH).

Marie Doumic,​​​‌ Sylvie Méléard and Vincent‌ Bansaye are members of‌​‌ the steering committee of​​ the MMB chair.

Marie​​​‌ Doumic is a member‌ of the Scientific Committee‌​‌ of Inria Paris-Saclay ;​​ of the board of​​​‌ the ESMTB (European Society‌ for Mathematical and Theoretical‌​‌ Biology) and of the​​ Committee for Applications and​​​‌ Interdisciplinary Relations (CAIR) of‌ the European Mathematical Society‌​‌ (EMS).

Sylvie Méléard is​​ a member of the​​​‌ Scientific Advertisory Board of‌ HIM (Hausdorff Research Institute‌​‌ for Mathematics, Bonn, Germany),​​ CMM (Center for Mathematical​​​‌ Modeling, Santiago, Chili) and‌ of CRM(Centre de recherches‌​‌ mathématiques,Montréal, Canada).

Milica Tomašević​​ is responsable of the​​​‌ thematic group MABIOME of‌ SMAI (with Y. Mameri),‌​‌ she is a member​​ of the Scientific Council​​​‌ of the Thematic Network‌ Maths Bio Santé and‌​‌ a member of the​​ Department Council of DMAP​​​‌ at Ecole polytechniqie.

10.2.1 Teaching​​​‌

Vincent Bansaye : 3‌ courses

Mateo Deangeli Bravo‌​‌ : Enseignement, encadrement d’un​​ stagiaire

Marie Doumic: master​​​‌ 2 course ; tutorials‌ for Bachelor 3; juries‌​‌ for MSV master

Ana​​ Fernandez : Tutorial classes​​​‌ to two groups of‌ around 20 students each‌​‌ for the course Discrete​​ Mathematics for first year​​​‌ students of the Bachelor‌ program at Ecole Polytechnique‌​‌ during 13 weeks. Supervision​​ and correction of the​​​‌ midterm and final exam.‌

Viviana Gavilanes: Bachelor 1‌​‌ level, How to Write​​ Mathematics, problem sessions​​​‌ and lectures ; Summer‌ course (Ecuador, undergraduate level):‌​‌Introduction to Markov Chains​​​‌ and Ergodic Theory,​ 10 hours, 4-8 August​‌ 2025.

Anouar Jeddi :​​ -Encadrement de séances de​​​‌ TD, TP et de​ remise à niveau pour​‌ les élèves polytechniciens et​​ les étudiants du Bachelor.​​​‌

Sylvie Méléard: head of​ the master "Mathematics for​‌ living sciences" till August​​ 2025.

Jules Olayé gave​​​‌ the course “How to​ write mathematics” for the​‌ Bachelor 1 students of​​ Ecole Polytechnique, which corresponds​​​‌ to a course of​ 64 hours.

Milica Tomasevic:​‌ PCC at Ecole Polytechnique​​ with "démi-décharge" due to​​​‌ maternity leave

10.2.2 Supervision​

Vincent Bansaye: 4 PhD​‌ students (co-supervised), 1 postdoc,​​ 1 gap-year internship in​​​‌ 2025 (co-supervised)

S. Méléard​ : Head of the​‌ Master Mathematics for Life​​ Sciences. Courses in the​​​‌ Master. Co-supervision of 4​ PhD Students (A. Fernandez-Baranda,​‌ A. Jeddi, A. Perrin,​​ M. Deangeli Bravo). Supervision​​​‌ of two post-doc (N.​ Belmabrouk, now ATER in​‌ University Paris Ouest-Nanterre), M.​​ Esser since May 2025).​​​‌

Gaël Raoul: 2 PhD​ students (co-supervised) Milica Tomasevic:​‌ Supervision of thesis of​​ N. Cazacu (with A.​​​‌ Richard) and A. Le​ Meur (with P. Monmarché).​‌ Supervision of postdoc of​​ Shyam Popat (with A.​​​‌ Richard).

10.2.3 Juries

Vincent​ Bansaye: 1 HDR jury,​‌ 1 PhD jury

Marie​​ Doumic: member of the​​​‌ recruitment committees for a​ professor in Evry ;​‌ research directors at INRAE;​​ an assistant professor in​​​‌ Toulouse ; a Junior​ Chair Professor in Nice​‌ and a junior chair​​ professor at CNRS. HDR​​​‌ juries: Jimmy Garnier, Coralie​ Fritsch (chairwoman). PhD thesie​‌ juries: Guillaume Garnier (co-supervisor),​​ Jules Olayé (co-supervisor), Matthias​​​‌ Rakotomalala (chairwoman), Maxime Payan,​ Virgile Brodu (chairwoman).

Sylvie​‌ Méléard : PhD thesis​​ juries : J. Olayé,​​​‌ N. Blassel Member of​ the Jury IUF Junior,​‌ member of the Jury​​ for the Award Blaise​​​‌ Pascal (European Academy of​ Sciences), member of the​‌ Jury for the Award​​ Dargelos (Association anciens Ecole​​​‌ polytechnique)

Gaël Raoul: Thesis​ Jury of Nathanaël Boutillon,​‌ Aix-Marseille Université.

Milica Tomasevic:​​ Thesis Jury of Lena​​​‌ Kuwata, MAP 5, October​ 2025. Thesis defense of​‌ J. Olayé (co-supervised with​​ Marie Doumic).

10.2.4 Educational​​​‌ and pedagogical outreach

Marie​ Doumic participated in several​‌ "matinées des métiers" and​​ gave talks for highschool​​​‌ children.

Vianney De la​ Salle : Participation in​‌ a “Maths in Jeans”​​ Workshop.

Viviana Gavilanes:

• AMARUN​​​‌ Association: authored and published​ two interviews (in French)​‌ with researchers:
  • Interview with​​ Marie Doumic: link​​​‌
  • Interview with Sylvie Méléard​: link
• co-organized a​‌ 4-month mentoring initiative (October​​ 2025-January 2026) Mentoring program​​​‌ for women in mathematics​ (undergraduate level): matching Ecuadorian​‌ women mathematicians (academia and​​ industry) with undergraduate students​​​‌ across Ecuador. The program​ offered introductory activities, networking​‌ and discussions, and workshops​​ to raise women’s awareness​​​‌ of academic and industry​ careers.

International journals

• 3​​​‌ articleC.Charles Bertucci‌, M.Matthias Rakotomalala‌​‌ and M.Milica Tomašević​​. Curvature in chemotaxis:​​​‌ A model for ant‌ trail pattern formation.‌​‌Mathematical Models and Methods​​ in Applied Sciences35​​​‌12September 2025,‌ 2695-2740HALDOIback‌​‌ to text
• 4 article​​P.Pierre Collet,​​​‌ C.Claire Ecotière and‌ S.Sylvie Méléard.‌​‌ Long time behavior of​​ a degenerate stochastic system​​​‌ modeling the response of‌ a population face to‌​‌ environmental impacts.Electronic​​ Communications in Probability30​​​‌noneJanuary 2025HAL‌DOIback to text‌​‌
• 5 articleM.Marie​​ Doumic, K.Klemens​​​‌ Fellner, M.Mathieu‌ Mezache and J. J.‌​‌Juan J L Velázquez​​. Asymptotic Analysis of​​​‌ a bi-monomeric nonlinear Becker-Döring‌ system.NonlinearityMay‌​‌ 2025HALDOIback​​ to text
• 6 article​​​‌M.Marie Doumic,‌ S.Sophie Hecht,‌​‌ M.Marc Hoffmann and​​ D.Diane Peurichard.​​​‌ Scaling limits for a‌ population model with growth,‌​‌ division and cross-diffusion.​​Mathematical Models and Methods​​​‌ in Applied Sciences35‌12November 2025,‌​‌ 2611-2660HALDOIback​​ to text
• 7 article​​​‌M.Marie Doumic and‌ P.Philippe Moireau.‌​‌ Asymptotic approaches in inverse​​ problems for depolymerization estimation​​​‌.Inverse Problems and‌ Imaging 20February 2026‌​‌, 105-155HALDOI​​back to text
• 8​​​‌ articleM.Maxime Ligonnière‌. Ergodic behavior of‌​‌ products of random positive​​ operators.ALEA :​​​‌ Latin American Journal of‌ Probability and Mathematical Statistics‌​‌XXII2025, 93-129​​HALDOIback to​​​‌ text
• 9 articleP.‌Pierre Magal and G.‌​‌Gaël Raoul. Dynamics​​ of a kinetic model​​​‌ describing protein exchanges in‌ a cell population.‌​‌Journal of Mathematical Biology​​9162025,​​​‌ 76HALDOIback‌ to text
• 10 article‌​‌J.Jules Olayé,​​ H.Hala Bouzidi,​​​‌ A.Andrey Aristov,‌ A.Antoine Barizien,‌​‌ S. G.Salomé Gutiérrez​​ Ramos, C.Charles​​​‌ Baroud and V.Vincent‌ Bansaye. Estimation of‌​‌ the lifetime distribution from​​ fluctuations in Bellman-Harris processes​​​‌.Journal of Mathematical‌ Biology90May 2025‌​‌, 56HALDOI​​back to text
• 11​​​‌ articleJ.Jules Olayé‌ and M.Milica Tomasevic‌​‌. Long-time behaviour of​​ a multidimensional age-dependent branching​​​‌ process with a singular‌ jump kernel modelling telomere‌​‌ shortening.Electronic Journal​​ of Probability31January​​​‌ 2026HALDOI
• 12‌ articleG.Gaël Raoul‌​‌. Macroscopic limit from​​ a structured population model​​​‌ to the Kirkpatrick-Barton model‌.Bulletin des Sciences‌​‌ Mathématiques2052025,​​ 103697HALDOIback​​​‌ to text
• 13 article‌A.Anaïs Rat,‌​‌ V.Veronica Martinez Fernandez​​​‌, M.Marie Doumic​, M. T.Maria​‌ Teresa Teixeira and Z.​​Zhou Xu. Individual​​​‌ cell fate and population​ dynamics revealed by a​‌ mathematical model linking telomere​​ length and replicative senescence​​​‌.Nature Communications16​1January 2025,​‌ 1024HALDOI

Doctoral​​ dissertations and habilitation theses​​​‌

• 14 thesisG.Guillaume​ Garnier. Effects of​‌ a mutation on the​​ fitness of a bacterium​​​‌ : theoretical estimation and​ numerical implementation.Sorbonne​‌ UniversitéJune 2025HAL​​back to text
• 15​​​‌ thesisM.Maxime Ligonnière​. Products of random​‌ matrices and infinite dimensional​​ multitype branching processes.​​​‌Université de toursApril​ 2025HALback to​‌ text
• 16 thesisJ.​​Jules Olayé. Stochastic​​​‌ and deterministic approaches for​ studying telomere shortening and​‌ other cell dynamics.​​Institut Polytechnique de Paris​​​‌July 2025HALback​ to text

Reports &​‌ preprints

• 17 miscM.​​Maxence Baccara. Functional​​​‌ Limit Theorems for the​ range of stable random​‌ walks.December 2025​​HALback to text​​​‌
• 18 miscV.Vincent​ Bansaye, T.Tresnia​‌ Berah and B.Bertrand​​ Cloez. On the​​​‌ strong law of large​ numbers and Llog L​‌ condition for supercritical general​​ branching processes.March​​​‌ 2025HALback to​ text
• 19 miscV.​‌Vincent Bansaye, A.​​Alexandre Bertolino and A.​​​‌Ayman Moussa. Stability​ of non-conservative cross diffusion​‌ model and approximation by​​ stochastic particle systems.​​​‌March 2025HALback​ to text
• 20 misc​‌V.Vincent Bansaye,​​ A.Ana Fernández Baranda​​​‌, S.Stéphane Giraudier​ and S.Sylvie Méléard​‌. Hematopoiesis as a​​ continuum: from stochastic compartmental​​​‌ model to hydrodynamic limit​.January 2026HAL​‌back to text
• 21​​ miscP.Prisca Berardi​​​‌, V.Veronica Martinez​ Fernandez, A.Anaïs​‌ Rat, F.Fernando​​ Rosas Bringas, P.​​​‌Pascale Jolivet, R.​Rachel Langston, S.​‌Stefano Mattarocci, A.​​Alexandre Maes, T.​​​‌Théo Aspert, B.​Bechara Zeinoun, K.​‌Karine Casier, H.​​Hinke Kazemier, G.​​​‌Gilles Charvin, M.​Marie Doumic, M.​‌Michael Chang and M.​​ T.Maria Teresa Teixeira​​​‌. The shortest telomere​ in telomerase-negative cells triggers​‌ replicative senescence at a​​ critical threshold length and​​​‌ also fuels genomic instability​.January 2025HAL​‌DOIback to text​​back to textback​​​‌ to text
• 22 misc​S.Sirine Boucenna,​‌ V.Vasilis Dakos and​​ G.Gaël Raoul.​​​‌ A model for a​ population of trees structured​‌ by phenological traits.​​January 2026HALback​​​‌ to text
• 23 misc​L.Luce Breuil and​‌ S.Sarah Kaakai.​​ Nonparametric hazard rate estimation​​​‌ with associated kernels and​ minimax bandwidth choice.​‌October 2025HALDOI​​back to text
• 24​​​‌ miscT.Thomas Cavallazzi​, A.Alexandre Richard​‌ and M.Milica Tomasevic​​. Quantitative approximation of​​​‌ a Keller–Segel PDE by​ a branching moderately interacting​‌ particle system and suppression​​ of blow-up.February​​​‌ 2026HALback to​ text
• 25 miscN.​‌Nicoleta Cazacu. Stochastic​​ numerical approximation for nonlinear​​ Fokker-Planck equations with singular​​​‌ kernels.April 2025‌HALback to text‌​‌
• 26 miscP.Pierre​​ Collet, S.Sylvie​​​‌ Méléard and J.Jaime‌ San. Long time‌​‌ behavior and Yaglom limit​​ for real trait-structured Birth​​​‌ and Death Processes.‌August 2025HALback‌​‌ to text
• 27 report​​M.Maxime Colomb,​​​‌ D.Denis Talay,‌ A.Aline Carneiro Viana‌​‌, Q.Quentin Cormier​​, J.Josselin Garnier​​​‌, N.Nicolas Gilet‌, C.Carl Graham‌​‌, L.Laura Grigori​​, J.Julien Perret​​​‌, R.Raphael Porcher‌, P.Philippe Ravaud‌​‌, R.Razvan Stanica​​, M.Milica Tomasevic​​​‌ and V.-T.Viet-Thi Tran‌. Validation du simulateur‌​‌ ICI de propagation d'épidémies​​ à l'aide des données​​​‌ publiques recueillies pendant la‌ première vague de la‌​‌ pandémie de COVID-19.​​InriaFebruary 2026HAL​​​‌
• 28 miscG.Giovanni‌ Conforti, A.Alain‌​‌ Durmus, L.-T.Le-Tuyet-Nhi​​ Pham and G.Gael​​​‌ Raoul. Non-Asymptotic Convergence‌ of Discrete Diffusion Models:‌​‌ Masked and Random Walk​​ dynamics.2025HAL​​​‌back to text
• 29‌ miscM.Mete Demircigil‌​‌ and M.Milica Tomasevic​​. Convergence and Wave​​​‌ Propagation for a System‌ of Branching Rank-Based Interacting‌​‌ Brownian Particles.May​​ 2025HALback to​​​‌ text
• 30 miscM.‌Marie Doumic, A.‌​‌Anaïs Rat and M.​​Magali Tournus. Comparison​​​‌ Between Effective and Individual‌ Growth Rates in a‌​‌ Heterogeneous Population.May​​ 2025HALback to​​​‌ text
• 31 miscA.‌Anouar Jeddi. Asymptotic‌​‌ behavior of some stochastic​​ models in population dynamics:​​​‌ a Hamilton-Jacobi approach.‌February 2026HAL
• 32‌​‌ miscM.Madeleine Kubasch​​. Empirical distribution of​​​‌ ancestral lineages in populations‌ with density-dependent interactions.‌​‌March 2025HAL
• 33​​ miscL.Lena Kuwata​​​‌, T.Thibault Chassereau‌, F.Florence Chapeland-Leclerc‌​‌, P.Pascal David​​, E.Eric Herbert​​​‌, G.Gwenaël Ruprich-Robert‌, M.Milica Tomašević‌​‌ and A.Amandine Véber​​. Quantifying the impact​​​‌ of different forms of‌ stress on fungal growth:‌​‌ an inference method based​​ on high-resolution pictures of​​​‌ the mycelial network.‌July 2025HALback‌​‌ to text
• 34 misc​​P.Pierre Magal and​​​‌ G.Gaël Raoul.‌ On measure-valued solutions for‌​‌ a structured population model​​ with transfers.June​​​‌ 2025, 76HAL‌back to text
• 35‌​‌ miscJ.Jules Olayé​​. An inverse problem​​​‌ in cell dynamics: Recovering‌ an initial distribution of‌​‌ telomere lengths from measurements​​ of senescence times.​​​‌February 2025HALback‌ to textback to‌​‌ text
• 36 miscA.​​Alexandre Perrin, M.​​​‌Marie Doumic, M.‌Meriem El Karoui and‌​‌ S.Sylvie Méléard.​​ Deciphering the Replication-Division Coordination​​​‌ in E. coli: A‌ Unified Mathematical framework for‌​‌ Systematic Model Comparison.​​July 2025HALDOI​​​‌back to textback‌ to text