Practical identifiability in the frame of nonlinear mixed effects models: the example of the in vitro erythropoiesis.

BACKGROUND: Nonlinear mixed effects models provide a way to mathematically describe experimental data involving a lot of inter-individual heterogeneity. In order to assess their practical identifiability and estimate confidence intervals for their parameters, most mixed effects modelling programs use the Fisher Information Matrix. However, in complex nonlinear models, this approach can mask practical unidentifiabilities. RESULTS: Herein we rather propose a multistart approach, and use it to simplify our model by reducing the number of its parameters, in order to make it identifiable. Our model describes several cell populations involved in the in vitro differentiation of chicken erythroid progenitors grown in the same environment. Inter-individual variability observed in cell population counts is explained by variations of the differentiation and proliferation rates between replicates of the experiment. Alternatively, we test a model with varying initial condition. CONCLUSIONS: We conclude by relating experimental variability to precise and identifiable variations between the replicates of the experiment of some model parameters.

Modeling and characterization of inter-individual variability in CD8 T cell responses in mice.

To develop vaccines it is mandatory yet challenging to account for inter-individual variability during immune responses. Even in laboratory mice, T cell responses of single individuals exhibit a high heterogeneity that may come from genetic backgrounds, intra-specific processes (e.g. antigen-processing and presentation) and immunization protocols.To account for inter-individual variability in CD8 T cell responses in mice, we propose a dynamical model coupled to a statistical, nonlinear mixed effects model. Average and individual dynamics during a CD8 T cell response are characterized in different immunization contexts (vaccinia virus and tumor). On one hand, we identify biological processes that generate inter-individual variability (activation rate of naive cells, the mortality rate of effector cells, and dynamics of the immunogen). On the other hand, introducing categorical covariates to analyze two different immunization regimens, we highlight the steps of the response impacted by immunogens (priming, differentiation of naive cells, expansion of effector cells and generation of memory cells). The robustness of the model is assessed by confrontation to new experimental data.Our approach allows to investigate immune responses in various immunization contexts, when measurements are scarce or missing, and contributes to a better understanding of inter-individual variability in CD8 T cell immune responses.

A model for establishment, maintenance and reactivation of the immune response after vaccination against Ebola virus.

The 2014-2016 Ebola outbreak in West Africa has triggered accelerated development of several preventive vaccines against Ebola virus. Under the EBOVAC1 consortium, three phase I studies were carried out to assess safety and immunogenicity of a two-dose heterologous vaccination regimen developed by Janssen Vaccines and Prevention in collaboration with Bavarian Nordic. To describe the immune response induced by the two-dose heterologous vaccine regimen, we propose a mechanistic ODE based model, which takes into account the role of immunological memory. We perform identifiability and sensitivity analysis of the proposed model to establish which kind of biological data are ideally needed in order to accurately estimate parameters, and additionally, which of those are non-identifiable based on the available data. Antibody concentrations data from phase I studies have been used to calibrate the model and show its ability in reproducing the observed antibody dynamics. Together with other factors, the establishment of an effective and reactive immunological memory is of pivotal importance for several prophylactic vaccines. We show that introducing a memory compartment in our calibrated model allows to evaluate the magnitude of the immune response induced by a booster dose and its long-term persistence afterwards.

Calibration, Selection and Identifiability Analysis of a Mathematical Model of the in vitro Erythropoiesis in Normal and Perturbed Contexts.

The in vivo erythropoiesis, which is the generation of mature red blood cells in the bone marrow of whole organisms, has been described by a variety of mathematical models in the past decades. However, the in vitro erythropoiesis, which produces red blood cells in cultures, has received much less attention from the modelling community. In this paper, we propose the first mathematical model of in vitro erythropoiesis. We start by formulating different models and select the best one at fitting experimental data of in vitro erythropoietic differentiation obtained from chicken erythroid progenitor cells. It is based on a set of linear ODE, describing 3 hypothetical populations of cells at different stages of differentiation. We then compute confidence intervals for all of its parameters estimates, and conclude that our model is fully identifiable. Finally, we use this model to compute the effect of a chemical drug called Rapamycin, which affects all states of differentiation in the culture, and relate these effects to specific parameter variations. We provide the first model for the kinetics of in vitro cellular differentiation which is proven to be identifiable. It will serve as a basis for a model which will better account for the variability which is inherent to the experimental protocol used for the model calibration.

Existence and stability of periodic solutions of an impulsive differential equation and application to CD8 T-cell differentiation.

Unequal partitioning of the molecular content at cell division has been shown to be a source of heterogeneity in a cell population. We propose to model this phenomenon with the help of a scalar, nonlinear impulsive differential equation (IDE). To study the effect of molecular partitioning at cell division on the effector/memory cell-fate decision in a CD8 T-cell lineage, we study an IDE describing the concentration of the protein Tbet in a CD8 T-cell, where impulses are associated to cell division. We discuss how the degree of asymmetry of molecular partitioning can affect the process of cell differentiation and the phenotypical heterogeneity of a cell population. We show that a moderate degree of asymmetry is necessary and sufficient to observe irreversible differentiation. We consider, in a second part, a general autonomous IDE with fixed times of impulse and a specific form of impulse function. We establish properties of the solutions of that equation, most of them obtained under the hypothesis that impulses occur periodically. In particular, we show how to investigate the existence of periodic solutions and their stability by studying the flow of an autonomous differential equation. Then we apply those properties to prove the results presented in the first part.

Stochastic modeling of a gene regulatory network driving B cell development in germinal centers.

Germinal centers (GCs) are the key histological structures of the adaptive immune system, responsible for the development and selection of B cells producing high-affinity antibodies against antigens. Due to their level of complexity, unexpected malfunctioning may lead to a range of pathologies, including various malignant formations. One promising way to improve the understanding of malignant transformation is to study the underlying gene regulatory networks (GRNs) associated with cell development and differentiation. Evaluation and inference of the GRN structure from gene expression data is a challenging task in systems biology: recent achievements in single-cell (SC) transcriptomics allow the generation of SC gene expression data, which can be used to sharpen the knowledge on GRN structure. In order to understand whether a particular network of three key gene regulators (BCL6, IRF4, BLIMP1), influenced by two external stimuli signals (surface receptors BCR and CD40), is able to describe GC B cell differentiation, we used a stochastic model to fit SC transcriptomic data from a human lymphoid organ dataset. The model is defined mathematically as a piecewise-deterministic Markov process. We showed that after parameter tuning, the model qualitatively recapitulates mRNA distributions corresponding to GC and plasmablast stages of B cell differentiation. Thus, the model can assist in validating the GRN structure and, in the future, could lead to better understanding of the different types of dysfunction of the regulatory mechanisms.

Exogenous IL-2 delays memory precursors generation and is essential for enhancing memory cells effector functions.

To investigate the impact of paracrine IL-2 signals on memory precursor (MP) cell differentiation, we activated CD8 T cell in vitro in the presence or absence of exogenous IL-2 (ex-IL-2). We assessed memory differentiation by transferring these cells into virus-infected mice. Both conditions generated CD8 T cells that participate in the ongoing response and gave rise to similar memory cells. Nevertheless, when transferred into a naive host, T cells activated with ex-IL-2 generated a higher frequency of memory cells displaying increased functional memory traits. Single-cell RNA-seq analysis indicated that without ex-IL-2, cells rapidly acquire an MP signature, while in its presence they adopted an effector signature. This was confirmed at the protein level and in a functional assay. Overall, ex-IL-2 delays the transition into MP cells, allowing the acquisition of effector functions that become imprinted in their progeny. These findings may help to optimize the generation of therapeutic T cells.

Mathematical model of the primary CD8 T cell immune response: stability analysis of a nonlinear age-structured system.

The primary CD8 T cell immune response, due to a first encounter with a pathogen, happens in two phases: an expansion phase, with a fast increase of T cell count, followed by a contraction phase. This contraction phase is followed by the generation of memory cells. These latter are specific of the antigen and will allow a faster and stronger response when encountering the antigen for the second time. We propose a nonlinear mathematical model describing the T CD8 immune response to a primary infection, based on three nonlinear ordinary differential equations and one nonlinear age-structured partial differential equation, describing the evolution of CD8 T cell count and pathogen amount. We discuss in particular the roles and relevance of feedback controls that regulate the response. First we reduce our system to a system with a nonlinear differential equation with a distributed delay. We study the existence of two steady states, and we analyze the asymptotic stability of these steady states. Second we study the system with a discrete delay, and analyze global asymptotic stability of steady states. Finally, we show some simulations that we can obtain from the model and confront them to experimental data.

CD8 memory precursor cell generation is a continuous process.

In this work, we studied the generation of memory precursor cells following an acute infection by analyzing single-cell RNA-seq data that contained CD8 T cells collected during the postinfection expansion phase. We used different tools to reconstruct the developmental trajectory that CD8 T cells followed after activation. Cells that exhibited a memory precursor signature were identified and positioned on this trajectory. We found that these memory precursors are generated continuously with increasing numbers arising over time. Similarly, expression of genes associated with effector functions was also found to be raised in memory precursors at later time points. The ability of cells to enter quiescence and differentiate into memory cells was confirmed by BrdU pulse-chase experiment in vivo. Analysis of cell counts indicates that the vast majority of memory cells are generated at later time points from cells that have extensively divided.

Mathematical study of feedback control roles and relevance in stress erythropoiesis.

This work is devoted to mathematical modelling of erythropoiesis. We propose a new multi-scale model, in which we bring together erythroid progenitor dynamics and intracellular regulatory network that determines erythroid cell fate. All erythroid progenitors are divided into several sub-populations according to their maturity. Two intracellular proteins, Erk and Fas, are supposed to be determinant for regulation of self-renewal, differentiation and apoptosis. We consider two growth factors, erythropoietin and glucocorticoids, and describe their dynamics. Several feedback controls are introduced in the model. We carry out computer simulations of anaemia and compare the obtained results with available experimental data on induced anaemia in mice. The main objective of this work is to evaluate the roles of the feedback controls in order to provide more insights into the regulation of erythropoiesis. Feedback by Epo on apoptosis is shown to be determinant in the early stages of the response to anaemia, whereas regulation through intracellular regulatory network, based on Erk and Fas, appears to operate on a long-term scale.

Multiscale Modeling of Germinal Center Recapitulates the Temporal Transition From Memory B Cells to Plasma Cells Differentiation as Regulated by Antigen Affinity-Based Tfh Cell Help.

Germinal centers play a key role in the adaptive immune system since they are able to produce memory B cells and plasma cells that produce high affinity antibodies for an effective immune protection. The mechanisms underlying cell-fate decisions are not well understood but asymmetric division of antigen, B-cell receptor affinity, interactions between B-cells and T follicular helper cells (triggering CD40 signaling), and regulatory interactions of transcription factors have all been proposed to play a role. In addition, a temporal switch from memory B-cell to plasma cell differentiation during the germinal center reaction has been shown. To investigate if antigen affinity-based Tfh cell help recapitulates the temporal switch we implemented a multiscale model that integrates cellular interactions with a core gene regulatory network comprising BCL6, IRF4, and BLIMP1. Using this model we show that affinity-based CD40 signaling in combination with asymmetric division of B-cells result in switch from memory B-cell to plasma cell generation during the course of the germinal center reaction. We also show that cell fate division is unlikely to be (solely) based on asymmetric division of Ag but that BLIMP1 is a more important factor. Altogether, our model enables to test the influence of molecular modulations of the CD40 signaling pathway on the production of germinal center output cells.

Model-Based Assessment of the Role of Uneven Partitioning of Molecular Content on Heterogeneity and Regulation of Differentiation in CD8 T-Cell Immune Responses.

Activation of naive CD8 T-cells can lead to the generation of multiple effector and memory subsets. Multiple parameters associated with activation conditions are involved in generating this diversity that is associated with heterogeneous molecular contents of activated cells. Although naive cell polarisation upon antigenic stimulation and the resulting asymmetric division are known to be a major source of heterogeneity and cell fate regulation, the consequences of stochastic uneven partitioning of molecular content upon subsequent divisions remain unclear yet. Here we aim at studying the impact of uneven partitioning on molecular-content heterogeneity and then on the immune response dynamics at the cellular level. To do so, we introduce a multiscale mathematical model of the CD8 T-cell immune response in the lymph node. In the model, cells are described as agents evolving and interacting in a 2D environment while a set of differential equations, embedded in each cell, models the regulation of intra and extracellular proteins involved in cell differentiation. Based on the analysis of in silico data at the single cell level, we show that immune response dynamics can be explained by the molecular-content heterogeneity generated by uneven partitioning at cell division. In particular, uneven partitioning acts as a regulator of cell differentiation and induces the emergence of two coexisting sub-populations of cells exhibiting antagonistic fates. We show that the degree of unevenness of molecular partitioning, along all cell divisions, affects the outcome of the immune response and can promote the generation of memory cells.

In Silico Methods for Studying T Cell Biology.

T cell function is dictated by a delicate balance of stimuli that shapes T-cell phenotype, the latter characterizable using a number of immunogenic assays. Thanks to advancements in next-generation sequencing technology, and the intersection between genetics, engineering, computer science, biostatistics, and immunology, it is now possible to profile immune cells residing in any organ or disease site at single cell resolution. Herein we review the most common approaches available to describe T cell activation, T cell molecular heterogeneity, and T cell function. We present informatic tools useful to both seasoned bioinformaticians and novices alike, providing a comprehensive overview of the in silico methods used to study T cell biology. With the goal of making this manuscript useful to a broad range of readers, we focus only on freely available tools and algorithms, and we describe the concepts using simple and direct language. We hope this work will serve to assist and inspire researchers interested in T cell biology.

Drugs modulating stochastic gene expression affect the erythroid differentiation process.

To better understand the mechanisms behind cells decision-making to differentiate, we assessed the influence of stochastic gene expression (SGE) modulation on the erythroid differentiation process. It has been suggested that stochastic gene expression has a role in cell fate decision-making which is revealed by single-cell analyses but studies dedicated to demonstrate the consistency of this link are still lacking. Recent observations showed that SGE significantly increased during differentiation and a few showed that an increase of the level of SGE is accompanied by an increase in the differentiation process. However, a consistent relation in both increasing and decreasing directions has never been shown in the same cellular system. Such demonstration would require to be able to experimentally manipulate simultaneously the level of SGE and cell differentiation in order to observe if cell behavior matches with the current theory. We identified three drugs that modulate SGE in primary erythroid progenitor cells. Both Artemisinin and Indomethacin decreased SGE and reduced the amount of differentiated cells. On the contrary, a third component called MB-3 simultaneously increased the level of SGE and the amount of differentiated cells. We then used a dynamical modelling approach which confirmed that differentiation rates were indeed affected by the drug treatment. Using single-cell analysis and modeling tools, we provide experimental evidence that, in a physiologically relevant cellular system, SGE is linked to differentiation.

Adding self-renewal in committed erythroid progenitors improves the biological relevance of a mathematical model of erythropoiesis.

We propose a new mathematical model of erythropoiesis that takes a positive feedback of erythrocytes on progenitor apoptosis into account, and incorporates a negative feedback of erythrocytes on progenitor self-renewal. The resulting model is a system of age-structured equations that reduces to a system of delay differential equations where the delays account for progenitor compartment duration and cell cycle length. We compare this model with experimental data on an induced-anemia in mice that exhibit damped oscillations of the hematocrit before it returns to equilibrium. When we assume no self-renewal of progenitors, we obtain an inaccurate fitting of the model with experimental data. Adding self-renewal in the progenitor compartment gives better approximations, with the main features of experimental data correctly fitted. Our results indicate the importance of progenitor self-renewal in the modelling of erythropoiesis. Moreover, the model makes testable predictions on the lifespan of erythrocytes confronted to a severe anemia, and on the progenitors behavior.

A delay reaction-diffusion model of the dynamics of botulinum in fish.

A model of interaction between fish and a bacterium (Clostridium botulinum) responsible for avian botulism is introduced, considering diffusion of both fish and bacterium in water. The fish population moves randomly in water. Death fish disintegrate in water, at different locations, causing bacteria to diffuse through water and infect other fish. Existence of uniform steady states is investigated and the linearized stability of the positive uniform steady state is analyzed. A Hopf bifurcation is proved to occur from the uniform steady state when the bifurcation parameter, here the time delay, passes through a critical value and diffusion coefficients satisfy some conditions, that induces time oscillations of the populations. Comments on diffusion-driven instability are provided, and numerical simulations are carried out to illustrate the results.

Identification of Nascent Memory CD8 T Cells and Modeling of Their Ontogeny.

Primary immune responses generate short-term effectors and long-term protective memory cells. The delineation of the genealogy linking naive, effector, and memory cells has been complicated by the lack of phenotypes discriminating effector from memory differentiation stages. Using transcriptomics and phenotypic analyses, we identify Bcl2 and Mki67 as a marker combination that enables the tracking of nascent memory cells within the effector phase. We then use a formal approach based on mathematical models describing the dynamics of population size evolution to test potential progeny links and demonstrate that most cells follow a linear naive→early effector→late effector→memory pathway. Moreover, our mathematical model allows long-term prediction of memory cell numbers from a few early experimental measurements. Our work thus provides a phenotypic means to identify effector and memory cells, as well as a mathematical framework to investigate their genealogy and to predict the outcome of immunization regimens in terms of memory cell numbers generated.

IL-2 sensitivity and exogenous IL-2 concentration gradient tune the productive contact duration of CD8(+) T cell-APC: a multiscale modeling study.

BACKGROUND: The CD8(+) T cell immune response fights acute infections by intracellular pathogens and, by generating an immune memory, enables immune responses against secondary infections. Activation of the CD8(+) T cell immune response involves a succession of molecular events leading to modifications of CD8(+) T cell population. To understand the endogenous and exogenous mechanisms controlling the activation of CD8(+) T cells and to investigate the influence of early molecular events on the long-term cell population behavior, we developed a multiscale computational model. It integrates three levels of description: a Cellular Potts model describing the individual behavior of CD8(+) T cells, a system of ordinary differential equations describing a decision-making molecular regulatory network at the intracellular level, and a partial differential equation describing the diffusion of IL-2 in the extracellular environment. RESULTS: We first calibrated the model parameters based on in vivo data and showed the model's ability to reproduce early dynamics of CD8(+) T cells in murine lymph nodes after influenza infection, both at the cell population and intracellular levels. We then showed the model's ability to reproduce the proliferative responses of CD5(hi) and CD5(lo) CD8(+) T cells to exogenous IL-2 under a weak TCR stimulation. This stressed the role of short-lasting molecular events and the relevance of explicitly describing both intracellular and cellular scale dynamics. Our results suggest that the productive contact duration of CD8(+) T cell-APC is influenced by the sensitivity of individual CD8(+) T cells to the activation signal and by the IL-2 concentration in the extracellular environment. CONCLUSIONS: The multiscale nature of our model allows the reproduction and explanation of some acquired characteristics and functions of CD8(+) T cells, and of their responses to multiple stimulation conditions, that would not be accessible in a classical description of cell population dynamics that would not consider intracellular dynamics.

A mathematical model of leptin resistance.

Obesity is often associated with leptin resistance, which leads to a physiological system with high leptin concentration but unable to respond to leptin signals and to regulate food intake. We propose a mathematical model of the leptin-leptin receptors system, based on the assumption that leptin is a regulator of its own receptor activity, and investigate its qualitative behavior. Based on current knowledge and previous models developed for body weight dynamics in rodents, the model includes the dynamics of leptin, leptin receptors and the regulation of food intake and body weight. It displays two stable equilibria, one representing a healthy state and the other one an obese and leptin resistant state. We show that a constant leptin injection can lead to leptin resistance and that a temporal variation in some parameter values influencing food intake can induce a change of equilibrium and a pathway to leptin resistance and obesity.

A predictive model of the dynamics of body weight and food intake in rats submitted to caloric restrictions.

Dynamics of body weight and food intake can be studied by temporally perturbing food availability. This perturbation can be obtained by modifying the amount of available food over time while keeping the overall food quantity constant. To describe food intake dynamics, we developed a mathematical model that describes body weight, fat mass, fat-free mass, energy expenditure and food intake dynamics in rats. In addition, the model considers regulation of food intake by leptin, ghrelin and glucose. We tested our model on rats experiencing temporally variable food availability. Our model is able to predict body weight and food intake variations by taking into account energy expenditure dynamics based on a memory of the previous food intake. This model allowed us to estimate this memory lag to approximately 8 days. It also explains how important variations in food availability during periods longer than these 8 days can induce body weight gains.