MUSCA is intrinsically interdisciplinary and brings together applied mathematicians and experimental biologists. We address crucial questions arising from biological processes from a mathematical perspective. Our main research line is grounded on deterministic and stochastic population dynamics, in finite or infinite dimension. We study open methodological issues raised by the modeling, analysis and simulation of multiscale in time and/or space dynamics in the field of physiology, with a special focus on developmental and reproductive biology, and digestive ecophysiology.

The formalism at the heart of our research program is that of structured population dynamics, both in a deterministic and stochastic version. Such a formalism can be used to design multiscale representations (say at the meso and macro levels), possibly embedding two-way (bottom-up and top-down) interactions from one level to another. We intend to couple structured population dynamics with dynamics operating on the microscopic level -typically large biochemical networks (signaling, metabolism, gene expression)-, whose outputs can be fed into the higher level models (see section 3.4). To do so, model reduction approaches have to be designed and implemented to properly formulate the “entry points” of the micro dynamics into the meso/macro formalism (e.g. formulation of velocity terms in transport equations, choice of intensities for stochastic processes) and to enable one to traceback as much as possible the variables and parameters from one scale to another. This approach is common to EPC MUSCA's two main applications in reproductive/developmental biology on one side, and microbiota/holobiont biology on the other side, while being applied to different levels of living organisms. Schematically, the meso level corresponds to the cells of a multi-cellular organism in the former case, and to the individual actors of a microbial community for the latter case.

Our general multiscale framework will be deployed on the study of direct problems as well as inverse problems. In some situations these studies will be accompanied with a post-processing layer of experimental data, which may be necessary to make the observations compatible with the model state variables, and will be based on dedicated statistical tools. Even if our approach may use classical modeling bricks, it is worth highlighting that the design of de novo models, specifically suited for addressing dedicated physiological questions, is a central part of our activity. Due to their intrinsic multiscale nature (in time and/or space), infinite dimensional formulation (PDE and/or measure-valued stochastic processes) and nonlinear interactions (across scales), such models raise most of the time open questions as far as their mathematical analysis, numerical simulation, and/or parameter calibration. We intend to cope with the resulting methodological issues, possibly in collaboration with external experts when needed to tackle open questions.

We will deal with models representing dynamic networks, whether in a biochemical or ecological context.
The mathematical formulation of these models involve Ordinary Differential Equations (ODE), Piecewise Deterministic Markov Processes (PDMP), or Continuous Time Markov Chains (CTMC). A prototypical example is the (mass-action) Chemical Reaction Network (CRN) 54, defined by a set of

Most of EPC MUSCA's models based on CRNs deal with (unstructured) population dynamics (complex microbial communities, neutral models in ecology, cell dynamics in developmental processes, macromolecule assemblies), biochemical kinetics and chemical reaction networks (signaling, gene, and metabolic networks), coagulation-fragmentation models (in particular Becker-Döring model). Notwithstanding the diversity of our modeling applications, we have to face common methodological issues to study such models, ranging from the theoretical analysis of model behavior to parameter inference.

In the case of autonomous systems (with no explicit dependency on time), the main theoretical challenge is the prediction of the long time dynamics, given the algebraic complexity associated with putative stationary states in high dimension. In physiological systems, the intracellular reaction networks are not under a static or constant input stimulation but rather subject to complex and highly dynamic signals such as (neuro-)hormones 21 or metabolites. These systems are thus non-autonomous in nature. Understanding to what extent reaction network motifs are able to encode or decode the dynamic properties of a time-dependent signal is a particularly challenging theoretical question, which has yet been scarcely addressed, either in simplified case-studies 71,11 or in the framework of “pulse-modulated systems” 52.

The high dimension of realistic networks calls for methods enabling to perform model reduction. Our strategy for model reduction combines several tools, that can be applied separately or sequentially to the initial model. Both in stochastic biochemical systems and population dynamics, large species abundance calls in general for the functional law of large number and central limit theorems, for which powerful results are now established in standard settings of finite dimension models 59. However, in more and more biological applications, the very large spectrum of orders of magnitude in reaction rates (or birth and death rates) leads naturally to consider simultaneously large species abundance with timescale separation, which generally results in either algebraic-differential reduced models, or to hybrid reduced models with both deterministic and stochastic dynamics. We will apply the generic methodology provided by the singular perturbation theory of Fenichel-Tikhonov in deterministic systems, and Kurtz's averaging results in stochastic systems, which, in the context of high dimensional reaction networks or population dynamics, are still the matter of active research both in the deterministic 60, 53 and stochastic context 42, 58, 70.

Other reduction approaches of deterministic systems will consist in combining regular perturbation expansion with standard linear model order reduction (MOR) techniques. We will continue our previous work 14, 13 on the derivation of convergence and truncation error bounds for the regular perturbation series expansion (also known as Volterra series expansion) of trajectories of a wide class of weakly nonlinear systems, in the neighborhood of stable hyperbolic equilibria. The challenge will be to obtain biologically interpretable reduced models with appropriate features such as for instance positivity and stability. Finding a general approach for the reduction of strongly nonlinear systems is still an open question, yet it is sometimes possible to propose ad-hoc reduced models in specific cases, using graph-based decomposition of the model 74, combined with the reduction of weakly nonlinear subsystems.

Once again, a key challenge in parameter estimation is due to the high dimension of the state space and/or parameter space. We will develop several strategies to face this challenge. Efficient Maximum likelihood or pseudo-likelihood methods will be developed and put in practice 128, using either existing state-of-the art deterministic derivative-based optimization 75 or global stochastic optimization 50. In any case, we pay particular attention to model predictivity (quantification of the model ability to reproduce experimental data that were not used for the model calibration) and parameter identifiability (statistical assessment of the uncertainty on parameter values). A particularly challenging and stimulating research direction of interest concerning both model reduction and statistical inference is given by identifiability and inference-based model reduction 62. Another strategy for parameter inference in complex, nonlinear models with fully observed state, but scarce and noisy observations, is to couple curve clustering, which allows reducing the system state dimension, with robust network structure and parameter estimation. We are currently investigating this option, by combining curve clustering 56 based on similarity criteria adapted to the problem under consideration, and an original inference method inspired by the Generalized Smoothing (GS) method proposed in 73, which we call Modified Generalized Smoothing (MGS). MGS is performed using a penalized criterion, where the log-likelihood of the measurement error (noisy data) is penalized by a model error for which no statistical model is given. Moreover, the system state is projected onto a functional basis (we mainly use spline basis), and the inference simultaneously estimates the model parameters and the spline coefficients.

The mathematical formulation of structured population models involves Partial Differential Equation (PDE) and measure-valued stochastic processes (sometimes referred as Individual-Based Models–IBM). A typical deterministic instance is the McKendrick-Von Foerster model, a paragon of (nonlinear) conservation laws. Such a formalism rules the changes in a population density structured in time and (possibly abstract) space variable(s). The transport velocity represents the time evolution of the structured variable for each “individual” in the population, and might depend on the whole population (or a part of it) in the case of nonlinear interactions (for instance by introducing nonlocal terms through moment integrals or convolutions). The source term models the demographic evolution of the population, controlled by birth or death events. One originality of our multiscale approach is that the formulation of velocities and/or source terms may arise, directly or indirectly, from an underlying finite-dimension model as presented in section 3.2. According to the nature of the structuring variable, diffusion operators may arise and lead to consider second-order parabolic PDEs. For finite population dynamics, the stochastic version of these models can be represented using the formalism of Poisson Measure-driven stochastic differential equations.

From the modeling viewpoint, the first challenge to be faced with this class of models yields in the model formulation itself. Obtaining a well-posed and mathematically tractable formulation, that yet faithfully accounts for the “behavioral law” underlying the multiscale dynamics, is not an obvious task.

On one side, stochastic models are suited for situations where relatively few individuals are involved, and they are often easier to formulate intuitively. On the other side, the theoretical analysis of deterministic models is generally more tractable, and provides one with more immediate insight into the population behavior. Hence, the ideal situation is when one can benefit from both the representation richness allowed by stochastic models and the power of analysis applicable to their deterministic counterparts. Such a situation is actually quite rare, due to the technical difficulties associated with obtaining the deterministic limit (except in some linear or weakly nonlinear cases), hence compromises have to be found. The mathematical framework exposed above is directly amenable to multiscale modeling. As such, it is central to the biomathematical bases of MUSCA and transverse to its biological pillars. We develop and/or analyze models for structured cell population dynamics involved in developmental or tissue-homeostasis processes, structured microbial populations involved in eco-physiological systems and molecule assemblies.

As in the case of finite dimension models, the study of these various models involve common methodological issues.

The theoretical challenges associated with the analysis of structured population models are numerous, due to the lack of a unified methodological framework. The analysis of the well-posedness 19 and long-time behavior 7, and the design of appropriate numerical schemes 1, 3 often rely on more or less generic techniques 69, 64 that we need to adapt in a case-by-case, model-dependent way: general relative entropy 65, 47, measure solution framework 57, 44, 51, martingale techniques 45, finite-volume numerical schemes 61, just to name a few.

Due to their strong biological anchorage, the formulation of our models often leads to new mathematical objects, which raises open mathematical questions. Specific difficulties generally arise, for instance from the introduction of nonlocal terms at an “unusual place” (namely in the velocities rather than boundary conditions 19), or the formulation of particularly tricky boundary conditions 9. When needed, we call to external collaborators to try to overcome these difficulties.

Even if the use of a structured population formalism leads to models that can be considered as compact, compared to the high-dimensional ODE systems introduced in section 3.2, it can be useful to derive reduced versions of the models, for sake of computational costs, and also and above all, for parameter calibration purposes.

To proceed to such a reduction, we intend to combine several techniques, including moment equations 68, dimensional reduction 6, timescale reduction 4, spatial homogenization 4010, discrete to continuous reduction 9 and stochastic to deterministic limit theorems 15.

Once again, all these techniques need to be applied on a case-by-case basis, and they should be handled carefully to obtain rigorous results (appropriate choice of metric topology, a priori estimates).

The calibration of structured population models is challenging, due to both the infinite-dimensional setting and the difficulty to obtain rich enough data in our application domains. Our strategy is rather empirical. We proceed to a sequence of preliminary studies before using the experimental available data. Sensitivity analyses 55, 46, and theoretical studies of the inverse problems associated with the models 5 intend to preclude unidentifiable situations and ill-posed optimization problems. The generation and use of synthetic data (possibly noised simulation outputs) allow us to test the efficiency of optimization algorithms and to delimit an initial guess for the parameters. When reduced or simplified versions of the models are available (or derived specifically for calibration purposes) 2, these steps are implemented on the increasingly complex versions of the model. In situations where PDEs are or can be interpreted as limits of stochastic processes, it is sometimes possible to estimate parameters on the stochastic process trajectories, or to switch from one formalism to the other.

A major challenge for multiscale systems biology is to rigorously couple intracellular biochemical networks with physiological models (tissue and organic functions) 72, 41, 76, 63. Meeting this challenge requires reconciling very different mathematical formalisms and integrating heterogeneous biological knowledge in order to represent in a common framework biological processes described on very contrasting spatial and temporal scales. On a generic ground, there are numerous methodological challenges associated with this issue (such as model or graph reduction, theoretical and computational connection between different modeling formalisms, integration of heterogenous data, or exploration of the whole parameter space), which are far from being overcome at the moment.

Our strategy is not to face frontally these bottlenecks, but rather to investigate in parallel the two facets of the question, through (i) the modeling of the topology and dynamics of infra-individual networks or dynamics, accounting for individual variability and local spatialization or compartmentalization at the individual level, as encountered for instance in cell signaling; and (ii) the stochastic and/or deterministic multiscale modeling of populations, establishing rigorous link between the individual and population levels. To bridge the gap, the key point is to understand how intracellular (resp. infra-individual) networks produce outputs which can then be fed up in a multicellular (resp. microbial population) framework, in the formulation of terms entering the multiscale master equations. A typical example of such outputs in individual cell modeling is the translation of different (hormonal or metabolic) signaling cues into biological outcomes (such as proliferation, differentiation, apoptosis, or migration). In turn, the dynamics emerging on the whole cell population level feedback onto the individual cell level by tuning the signal inputs qualitatively and quantitatively.

The multiscale modeling approach described in section 3 is deployed on biological questions arising from developmental and reproductive biology, as well as digestive ecophysiology.

Our main developmental and reproductive thematics are related to gametogenesis, and gonad differentiation and physiology. In females, the gametogenic process of oogenesis (production and maturation of egg cells) is intrinsically coupled with the growth and development of somatic structures called ovarian follicles. Ovarian folliculogenesis is a long-lasting developmental and reproductive process characterized by well documented anatomical and functional stages. The proper morphogenesis sequence, as well as the transit times from one stage to another, are finely tuned by signaling cues emanating from the ovaries (especially during early folliculogenesis) and from the hypothalamo-pituitary axis (especially during late folliculogenesis). The ovarian follicles themselves are involved in either the production or regulation of these signals, so that follicle development is controlled by direct or indirect interactions within the follicle population. We have been having a longstanding interest in the multiscale modeling of follicle development, which we have tackled from a “middle-out”, cell dynamics-based viewpoint 2, completed progressively with morphogenesis processes 17.

On the intracellular level, we are interested in understanding the endocrine dialogue within the hypothalamo-pituitary-gonadal (HPG) axis controling the ovarian function. In multicellular organisms, communication between cells is critical to ensure the proper coordination needed for each physiological function. Cells of glandular organs are able to secrete hormones, which are messengers conveying information through circulatory systems to specific, possibly remote target cells endowed with the proper decoders (hormone receptors). We have settled a systems biology approach combining experimental and computational studies, to study signaling networks, and especially GPCR (G Protein-Coupled Receptor) signaling networks 12. In the HPG axis, we focus on the pituitary hormones FSH (Follicle-Stimulating Hormone) and LH (Luteinizing Hormone) – also called gonadotropins-, which support the double, gametogenic and endocrine functions of the gonads (testes and ovaries). FSH and LH signal onto gonadal cells through GPCRs, FSH-R and LH-R, anchored in the membrane of their target cells, and trigger intracellular biochemical cascades tuning the cell enzymatic activity, and ultimately controlling gene expression and mRNA translation. Any of these steps can be targeted by pharmacological agents, so that the mechanistic understanding of signaling networks is useful for new drug development.

Our main thematics in digestive ecophysiology are related to the interactions between the host and its microbiota. The gut microbiota, mainly located in the colon, is engaged in a complex dialogue with the large intestinal epithelium of its host, through which important regulatory processes for the host's health and well-being take place. Through successive projects, we have developed an integrative model of the gut microbiota at the organ scale, based on the explicit coupling of a population dynamics model of microbial populations involved in fiber degradation with a fluid dynamics model of the luminal content. This modeling framework accounts for the main drivers of the spatial structure of the microbiota, specially focusing on the dietary fiber flow, the epithelial motility, the microbial active swimming and viscosity gradients in the digestive track 16.

Beyond its scientific interest, the ambitious objective of understanding mechanistically the multiscale functioning of physiological systems could also help on the long term to take up societal challenges.

In digestive ecophysiology, microbial communities are fundamental for human and animal wellbeing and ecologic equilibrium. In the gut, robust interactions generate a barrier against pathogens and equilibrated microbiota are crucial for immune balance. Imbalances in the gut microbial populations are associated with chronic inflammation and diseases such as inflammatory bowel disease or obesity. Emergent properties of the interaction network are likely determinant drivers for health and microbiome equilibrium. To use the microbiota as a control lever, we require causal multiscale models to understand how microbial interactions translate into productive, healthy dynamics 20.

In reproductive physiology, there is currently a spectacular revival of experimental investigations (see e.g. 66, 78), which are driven by the major societal challenges associated with maintaining the reproductive capital of individuals, and especially female individuals, whether in a clinical (early ovarian failure of idiopathic or iatrogenic origin in connection with anticancer drugs in young adults and children), breeding (recovery of reproductive longevity and dissemination of genetic progress by the female route), or ecological (conservation of germinal or somatic tissues of endangered species or strains) context. Understanding the intricate (possibly long range and long term) interactions brought to play between the main cell types involved in the gonadal function (germ cells, somatic cells in the gonads, pituitary gland and hypothalamus) also requires a multiscale modeling approach.

Given our positioning in comparative physiology, future outcomes of MUSCA's basic research can be expected in the fields of Medicine, Agronomy (breeding) and Ecophysiology, in a One Health logic.
For instance, a deep understanding of female gametogenesis can be instrumental for the clinical management of ovarian aging, the development of sustainable breeding practices, and the monitoring of micro-pollutant effects on wild species (typically on fish populations). These issues will be especially investigated in the framework of the OVOPAUSE project and they are also implemented as part of our collaboration with INERIS (GinFiz project). In the same spirit, we intend to design methodological and sofware tools for the model-assisted validation of alternatives to hormone use in reproduction control (ovarian stimulation, contraception). This line is driven by the Contrabody project, which has stimulated associated actions such as that dedicated to the automatic assessment of the reproductive status from ovary imaging.
In the same spirit, our mechanistic view of the interactions between the host and gut microbiota leads to new approaches of the antibioresistance phenomenon, which is the topic of the PARTHAGE project and has already been the matter of a translational project (COOPERATE).
Finally, our systems biology and computational biology approaches dedicated to cell signaling and structural biology clearly target pharmacological design and screening, and, on the long term, have the potential to accelerate and improve drug discovery in the field of reproduction and beyond. Such approaches have proven particularly fruitful with the MabSilico start-up (a spin-off of the BIOS group), which continues to interact with BIOS and MUSCA on antibody-related projects (SELMAT and Contrabody for example).

In the framework of Guillaume Ballif's PhD, we have introduced an ODE-based compartmental model of ovarian follicle development all along lifespan 43. The model monitors the changes in the follicle numbers in different maturation stages with aging. Ovarian follicles may either move forward to the next compartment (unidirectional migration) or degenerate and disappear (death). The migration from the first follicle compartment corresponds to the activation of quiescent follicles, which is responsible for the progressive exhaustion of the follicle reserve (ovarian aging) until cessation of reproductive activity. The model consists of a data- driven layer embedded into a more comprehensive, knowledge-driven layer encompassing the earliest events in follicle development. The data-driven layer is designed according to the most densely sampled experimental dataset available on follicle numbers in the mouse. Its salient feature is the nonlinear formulation of the activation rate, whose formulation includes a feedback term from growing follicles. The knowledge-based, coating layer accounts for cutting-edge studies on the initiation of follicle development around birth. Its salient feature is the co-existence of two follicle subpopulations of different embryonic origins. We have then setup a complete estimation strategy, including (i) the study of structural identifiability based on differential elimination, using the Structural identifiability Julia package, (ii) a sensitivity analysis based on the elementary effect method of Morris, (iii) the elaboration of a relevant optimization criterion combining different sources of data (the initial dataset on follicle numbers, together with data in conditions of perturbed activation, and data discriminating the subpopulations) with appropriate error models, and a model selection step. We have finally illustrated the model potential for experimental design (suggestion of targeted new data acquisition) and in silico experiments.

We have designed and analyzed a stochastic model of embryonic neurogenesis in the mouse cerebral cortex, within the framework of compound Poisson processes, with time-varying, probabilistic fate decisions, and possibly stochastic cell cycle durations 33. The core of the model is the stochastic counterpart of our former deterministic compartmental model based on transport equations 18. The model accounts for the dynamics of different progenitor cell types and neurons. The expectation and variance of the cell number of each type are derived analytically and illustrated through numerical simulations. The effects of stochastic transition rates between cell types, and stochastic duration of the cell division cycle have been investigated sequentially. The model does not only predict the number of neurons, but also their spatial distribution into deeper and upper cortical layers. The model outputs are consistent with experimental data providing the number of neurons and intermediate progenitors according to embryonic age in control and mutant situations.

In the framework of the COMPARTIMENTAGE exploratory action, we have initiated a new thematics on the compartmentalization of cell signaling, with a special focus on the compartimentalization of G Protein-Coupled Receptors

During the CEMRACS 2022 summer school, Romain Yvinec, Erwan Hingant and Juan Carlo supervised a project dedicated to the modeling of compartmentalization within intracellular signaling pathways. Together with Claire Alamichel, Nathan Quiblier, and Saoussen Latrach, they have introduced a new modeling approach for the signaling systems of G protein-coupled receptors, taking into account the compartmentalization of receptors and their effectors, both at the plasma membrane and in dynamic intra-cellular vesicles called endosomes 31. The first building block of the model is about compartment dynamics. It takes into account creation of de novo endosomes, i.e. endocytosis, recycling of endosomes back to the plasma membrane, degradation through transfer into lysosomes, as well as endosome fusion through coagulation dynamics. The second building block corresponds to the biochemical reactions arising in each compartment and to the transfer of molecules between the dynamical compartments. They have proven sufficient conditions to obtain exponentially the ergodicity for the size distribution of intracellular compartments. In parallel, they have designed a finite volume scheme to simulate the model and illustrated two application cases for receptor trafficking and spatially biased second effector signaling.

In the framework of Leo Darrigade's post-doc, we have then designed a piecewise deterministic Markov process of intracellular GPCR trafficking and cAMP production. The stochastic part of the model accounts for the formation, coagulation, fragmentation and recycling of intracellular vesicles carrying the receptors, while the deterministic part of the model represents the chemical reactions mediating the response to the activated receptor. Assuming that the different stochastic jump rates are constant, and that the deterministic flow associated with chemical reactions is exponentially contractive, we have proven that this process converges exponentially to a unique stationary measure. In parallel, we have developped a simplified ODE-based model of receptor signaling and trafficking to analyze experimental time series of cAMP concentration. The goal is to estimate kinetic parameters of receptor trafficking and signaling activity in different compartments. Special care has been devoted to describe rigorously the metadata (e.g. type of ligand, dose, pharmacological perturbations) related to each dataset.

In the framework of the master internship of Alice Fohr (M2 Mathématiques pour les Sciences du Vivant, Université Paris-Saclay), we have initiated a new thematics on mathematical modeling for the understanding of X chromosome inactivation. In mammals, females are endowed with two X chromosomes, which could lead to an over-transcription of X-linked genes compared to males. Early during embryonic development, a compensation mechanism settles, which ends up by silencing either the father-inherited or the mother-inherited X chromosome, in a random manner. We have studied the qualitative behavior of an ODE-based toggle-switch model proposed in 67. Using the theory of bifurcation analysis, we have confirmed the numerical results obtained in 67 on the stationary states, from which one can select different configurations of small-size gene networks ensuring the initiation and maintenance of a single X chromosome inactivation. We have then derived the deterministic model as the large-size limit of a continuous-time Markov process representing the unitary events associated with transcription. Finally, we have started investigating the clonal propagation of the X-chromosome inactivation status along cell lineages in the framework of branching processes.

Oogenesis is the process of production and maturation of female gametes (oocytes), which ends up in fish with spawning. This process is critical to the survival of species, and particularly sensitive to environmental alterations (e.g. temperature, pollutants). In the framework of Louis Fostier's PhD, we have developed a model representing the oocyte population dynamics, from the earliest phases to egg laying, and taking into account the key stages of physiological and environmental controls. The model formulation is based both on knowledge available in two model fish species, the zebrafish and medaka, and on mathematical models that we have previously developed for mammalian oogenesis. The evolution of the oocyte population is governed by a size-structured population dynamics model, formalized in the form of a transport partial differential equation, with nonlocal nonlinearities on the velocity term and boundary conditions, capturing the effect of interactions between oocytes on the recruitment of new oocytes and on the growth rate. We have shown the well-posedness of the model in its generic formulation, and we have studied the associated stationary problem. Under certain additional hypotheses, concerning the growth rate term, we have determined the long-time behavior of the model, and in particular the local stability of the stationary solutions, by linearization methods.

Fat cells, called adipocytes, are designed to regulate energy homeostasis by storing energy in the form of lipids. The adipocyte size distribution is assumed to play a role in the development of obesity-related diseases. The population of adipocytes is characterized by a bimodal size distribution. We have proposed a model based on a partial differential equations to describe the adipocyte size distribution 35. The model includes a description of the lipid fluxes and cell size fluctuations. From the formulation of a stationary solution we can obtain a fast computation of bimodal distributions. We have investigated the parameter identifiability and estimated parameter values with the CMA-ES algorithm. We have first validated the procedure on synthetic data, then estimated parameter values with experimental data of thirty-two rats. We have discussed the estimated parameter values and their variability within the population, as well as the relation between estimated values and their biological significance. Finally, a sensitivity analysis has been performed to specify the influence of parameters on the cell size distribution and explain the differences between the model and measurements. The proposed framework enables the characterization of adipocyte size distribution with four parameters and can be easily adapted to measurements of cell size distribution in different health conditions.

Biological data show that the size distribution of adipocytes follows a bimodal distribution. In 39, we have introduced a Lifshitz-Slyozov type model, based on a transport partial differential equation, for the dynamics of the size distribution of adipocytes. We have proven a new convergence result from the related Becker-Döring model, a system composed of several ordinary differential equations, toward mild solutions of the Lifshitz-Slyozov model using distribution tail techniques. This result allowed us to propose a new advective-diffusive model, the second-order diffusive Lifshitz-Slyozov model, which is expected to better fit the experimental data. Numerical simulations of the solutions to the diffusive Lifshitz-Slyozov model have been performed using a well-balanced scheme and the model outputs have been compared to solutions to the transport model. The simulations show that both bimodal and unimodal profiles can be reached asymptotically, depending on several parameters. We put in evidence that the asymptotic profile for the second-order system does not depend on initial conditions, unlike for the transport Lifshitz-Slyozov model.

We have studied the Lifshitz-Slyozov model with inflow boundary conditions of nucleation type 23. We have shown that, for a collection of representative rate functions, the size distributions approach degenerate states concentrated at zero size for sufficiently large times. The proof relies on monotonicity properties of some quantities associated with an entropy functional. Moreover, we have given numerical evidence on the fact that the convergence rate to the goal state is algebraic in time. Besides their mathematical interest, these results can be relevant for the interpretation of experimental data.

The Lifshitz-Slyozov model is a nonlocal transport equation that can describe certain types of phase transitions in terms of the temporal evolution of a mixture of monomers and aggregates. Most applications of this model so far do not require boundary conditions. However, there is a recent interest in situations where a boundary condition might be needed-e.g. in the context of protein polymerization phenomena. Actually, the boundary condition may change dynamically in time, depending on an activation threshold for the monomer concentration. This new setting raises a number of mathematical difficulties for which the existing literature is scarce. In 32, we have constructed examples of solutions for which the boundary condition becomes activated (resp. deactivated) dynamically in time. We also discussed how to approach the well-posedness problem for such situations.

The health and well-being of a host are deeply influenced by the interactions with its gut microbiota. Diet, especially the amount of fiber intake, plays a pivotal role in modulating these interactions impacting microbiota composition and functionality. We have introduced a novel mathematical model 37, designed to delve into these interactions, by integrating dynamics of the colonic epithelial crypt, bacterial metabolic functions and sensitivity to inflammation as well as colon flows in a transverse colon section. Unique features of our model include accounting for metabolic shifts in epithelial cells based on butyrate and hydrogen sulfide concentrations, representing the effect of innate immune pattern recognition receptors activation in epithelial cells, capturing bacterial oxygen tolerance based on data analysis, and considering the effect of antimicrobial peptides on the microbiota. Using our model, we show a proof-of-concept that a high-protein, low-fiber diet intensifies dysbiosis and compromises symbiotic resilience. Our simulation results highlight the critical role of adequate butyrate concentrations in maintaining mature epithelial crypts. Through differential simulations focused on varying fiber and protein inputs, our study offers insights into the system's resilience following the onset of dysbiosis. The present model, while having room for enhancement, offers essential understanding of elements such as oxygen levels, the breakdown of fiber and protein, and the basic mechanisms of innate immunity within the colon environment.

Mathematical models of biological tissues are a promising tool for multiscale data integration, computational experiments and system biology approaches. While some data and insights are rooted at the cell level, macroscopic mechanisms emerge and are observed at the tissue scale, rendering tissue modeling an inherently multiscale process. As a consequence, tissue models can be broadly categorized as either individual-based or continuous population-based. In 34, we have introduced a generic individual-based model of epithelial tissue including the main regulation processes such as cell division, differentiation, migration and death, together with cell-cell mechanical interactions. We have also considered the coupling with diffusing molecules. The model is a measure-valued piecewise-deterministic Markov process, coupled with reaction-diffusion PDEs. The well-posedness of the model is assessed, and the large population deterministic limit is rigorously derived. Finally, numerical experiments are conducted: the model is applied to the context of epithelial tissues in the intestinal crypt and the convergence towards the deterministic model is illustrated numerically.

We have considered the study of an inverse problem for an intestinal crypt model 38. The original model is based on the interaction of epithelial cells with microbiota-derived chemicals diffusing in the crypt from the gut lumen. The five types of cells considered in the original model were reduced in this work to three types of cells for simplifications of the inverse problem. The inverse problem consists in determining the shape of the secretory cells of the deep crypt from observations of the stem cells and progenitor cells at a fixed time. The method used is the calculation of the adjoint state associated with the second-order BGK numerical scheme, which allows calculating the critical points of the Lagrangian associated with the inverse problem, and applying a gradient method in order to minimize the cost function. The algorithm is described, and some numerical examples are given.

Deciphering the complex interactions between the gut microbiome and host requires evolved analysis methods focusing on the microbial ecosystem functions. We have integrated a priori knowledge on anaerobic microbiology with statistical learning to design synthetic profiles of fiber degradation from metagenomic analyses 26. We have identified four distinct functional profiles related to diet, dysbiosis, inflammation and disease. We have used non-negative matrix factorization to mine metagenomic datasets, after selecting manually 91 KEGG orthologies and 33 glycoside hydrolases, further aggregated in 101 functional descriptors. The profiles were identified from a training set of 1153 samples and thoroughly validated on a large database of 2571 unseen samples from 5 external metagenomic cohorts. Profiles 1 and 2 are the main contributors to the fiber-degradation-related metagenome. Profile 1 takes over Profile 2 in healthy samples, and the unbalance of these profiles characterizes dysbiotic samples. Profile 3 takes over Profile 2 during Crohn’s disease, inducing functional reorientations towards unusual metabolism such as fucose and H2S degradation or propionate, acetone and butanediol production. Profile 4 gathers under-represented functions, like methanogenesis. Two taxonomic makes up of the profiles were investigated, using either the covariation of 203 prevalent genomes or metagenomic species, both providing consistent results with their functional characteristics. It appeared that Profiles 1 and 2 were respectively mainly composed of bacteria from the phyla Bacteroidetes and Firmicutes, while Profile 3 is representative of Proteobacteria and Profile 4 of Methanogens.

Monoclonal antibodies are biopharmaceuticals with a very long half-life due to the binding of their Fc portion to the neonatal receptor (FcRn), a pharmacokinetic property that can be further improved through engineering of the Fc portion, as demonstrated by the approval of several new drugs. Many Fc variants with increased binding to FcRn have been found using different methods, such as structure-guided design, random mutagenesis, or a combination of both, and are described in the literature as well as in patents. Our hypothesis is that this material could be subjected to a machine learning approach in order to generate new variants with similar properties. We therefore compiled 1323 Fc variants affecting the affinity for FcRn, which were disclosed in twenty patents. These data were used to train several algorithms, with two different models, in order to predict the affinity for FcRn of new randomly generated Fc variants 24. To determine which algorithm was the most robust, we first assessed the correlation between measured and predicted affinity in a 10-fold cross-validation test. We then generated variants by in silico random mutagenesis and compared the prediction made by the different algorithms. As a final validation, we produced variants, not described in any patent, and compared the predicted affinity with the experimental binding affinities measured by surface plasmon resonance (SPR). The best mean absolute error (MAE) between predicted and experimental values was obtained with a support vector regressor (SVR) using six features and trained on 1251 examples. With this setting, the error on the log(KD) was less than 0.17. The obtained results show that such an approach could be used to find new variants with better half-life properties that are different from those already extensively used in therapeutic antibody development.

In the framework of the international internship of Pamela Romero, we have used Machine Learning to predict BRET time series in the context of cell signaling. Bioluminescence Resonance Energy Transfer (BRET) is used in to measure dynamic events on the molecular scale, such as protein-protein interactions. We have selected Random Forest Regression models and tested different numerical experiments. The first case was based on a point-to-point prediction: for each time step in the series the next one is predicted, which requires knowing a lot of information, not available in practice. The second case introduced a feedback in the prediction: the result of the previous prediction is used as an input for the current prediction, which requires knowing only the first point of the time series, a much more realistic situation. The third case corresponds to a prediction spanning multiple time steps. The inputs of the different test cases are the BRET time series, and relative information on the cell signaling experiments, such as the nature and dose of the ligand (stimulus), the type of receptor, and possible pharmacological perturbations. In all three experiments, we obtained good results in the testing set with errors close to zero and accuracy between 80% and 98%.

Intracellular variable fragments from heavy-chain antibody from camelids (intra-VHH) have been successfully used as chaperones to solve the 3D structure of active G protein-coupled receptors bound to their transducers. However, their effect on signaling has been poorly explored, although they may provide a better understanding on the relationships between receptor conformation and activity. We have isolated and characterized iPRC1, the first intra-VHH recognizing a member of the large glycoprotein hormone receptors family, the follicle-stimulating hormone receptor (FSHR) 27. This intra-VHH recognizes the third intracellular loop of FSHR and decreases cAMP production in response to FSH, without altering G

Autism spectrum disorders (ASDs) are diagnosed in 1/100 children worldwide, based on two core symptoms: deficits in social interaction and communication, and stereotyped behaviors. G protein-coupled receptors (GPCRs) are the largest family of cell surface receptors that transduce extracellular signals to convergent intracellular signaling and downstream cellular responses that are commonly dysregulated in ASD. Despite hundreds of GPCRs being expressed in the brain, only 23 are genetically associated with ASD according to the Simons Foundation Autism Research Initiative (SFARI) gene database: oxytocin OTR; vasopressin V

Bio-Maths seminar, Institut Camille Jordan, Université Lyon 1–Claude Bernard

Explorations au coeur du système reproducteur, L'Edition de l'Université Paris-Saclay #20 Hiver 2022/2023

Des chercheurs italiens en immersion dans l’unité PRC, e-Confluence, Journal interne du Centre INRAE Val-de-Loire, n°12, Juillet 2023

Pharmacologie réverse à l'aide d'anticorps intracellulaires anti-RFSH actif, Echosciences