Modélisation et analyse stochastique multi-échelle en dynamique des populations : application à la population folliculaire ovarienne

This thesis aims to develop and analyze mathematically structured population models to explore the dynamics of the ovarian follicle population throughout the reproductive life of mammals.Firstly, we have built a birth, migration and death stochastic process structured in compartments: each follicle maturation stage is characterized by a compartment. In order to take into account interactions within the follicle population, some transition rates of the resulting continuous-time Markov chain depend on the whole or part of the population. Using a difference in abundances and time scales within the population, we have obtained a so-called slow/fast model. We have then proved the existence and uniqueness of a limit model in the framework of stochastic singular perturbations by using a Foster-Lyapunov criterion, a coupling and a well-chosen Poisson equation. The limit model consists of an ordinary-differential equation ruling the dynamics of the first (slow) compartment, coupled with a quasi-stationary distribution in the remaining (fast) compartments.In a second step, we have added to the model an intermediate timescale to refine the modeling of the continued acceleration of follicle growth. With the same mathematical tools, we have proven the existence and uniqueness of a more complex limit model than the previous one. The slowest and fastest timescales are always represented by an ordinary-differential equation and a quasi-stationary distribution respectively. The middle scale is represented by the fixed point of a system of ordinary differential equation. Each timescale depends on the previous one through an averaging, which makes the study of this model particularly difficult in a general framework. In the case of two and three timescales, we have then extended the study of the limit to the higher order by identifying the mean and variance of the error between the model and its limit.Finally, we have introduced a model (a system of ordinary-differential equation) adapted to the biological experimental data available in the mouse. This model captures more accurately the early step of follicle formation (by adding a compartment upstream the follicle dynamics) and takes into account the existence of two distinct subpopulations of follicles. Taking into consideration a study of theorical identifiability of the parameters, we have reparametrized the model and estimated the new parameters. We have illustrated the model potential in two ways: by suggesting the type of biological data needed to identify a larger number of parameters and by reproducing in silico experiments.Cette thèse vise à élaborer et analyser mathématiquement des modèles de populations structurées pour étudier l'évolution de la population de follicules ovariens tout au long de la vie reproductive des mammifères.Dans un premier temps, nous avons construit un modèle de naissance, mort et migration; structuré en compartiments : chaque stade de maturation des follicules est représenté par un compartiment. Afin de tenir compte des interactions existant au sein de la population folliculaire, les taux de transition de la chaîne de Markov à temps continu ainsi obtenue dépendent de l'ensemble ou d'une partie de la population. En utilisant une différence d’abondance et de vitesse de transit au sein des compartiments, nous obtenons un modèle dit lent/rapide. Nous démontrons alors l’existence et l’unicité d’un modèle limite dans le cadre de la théorie des perturbations stochastiques singulières en utilisant un critère de Foster-Lyapunov, un couplage ainsi qu’une équation de Poisson bien choisie. Ce modèle limite est composé d’une équation différentielle ordinaire décrivant la (lente) dynamique du premier compartiment, couplée avec une distribution quasi-stationnaire pour les compartiments (rapides) suivants.Dans un second temps, nous ajoutons au modèle une échelle de temps intermédiaire pour modéliser plus finement l’accélération continue de la croissance des follicules. À l’aide des mêmes outils mathématiques, nous prouvons l’existence et l’unicité d’un modèle limite plus complexe que le précédent. L’échelle la plus lente et la plus rapide sont toujours représentées respectivement par une équation différentielle ordinaire et par une distribution quasi-stationnaire. L’échelle intermédiaire est représenté par le point fixe d’un système d'équation différentielle ordinaire. Chaque échelle dépend de la précédente via une moyennisation ce qui rend l’étude de ce modèle dans un cadre général particulièrement difficile. Dans le cas de 2 ou de 3 échelles de temps, nous avons ensuite poussé l’étude de la limite à l’ordre supérieur en caractérisant l’espérance et la variance de l’erreur entre le modèle et sa limite.Enfin, nous avons introduit un modèle (un système d’équations différentielles ordinaires) adapté aux données biologiques expérimentales disponibles chez la souris. Ce modèle capture plus finement les premières étapes de formation des follicules (par l’ajout d’un compartiment en amont de la dynamique folliculaire) et tient compte de l’existence de deux sous-populations distinctes. En tenant compte d’une étude de l’identifiabilité théorique des paramètres, nous avons reparamétré le modèle et estimé les nouveaux paramètres du modèle. Nous avons finalement illustré le potentiel du modèle de deux façons : en suggérant le type de données biologiques à ajouter afin d’identifier un plus grand nombre de paramètres et en reproduisant des expériences in silico

Modeling and multiscale stochastic analysis in population dynamics : application to the ovarian follicle population

Cette thèse vise à élaborer et analyser mathématiquement des modèles de populations structurées pour étudier l'évolution de la population de follicules ovariens tout au long de la vie reproductive des mammifères.Dans un premier temps, nous avons construit un modèle de naissance, mort et migration; structuré en compartiments : chaque stade de maturation des follicules est représenté par un compartiment. Afin de tenir compte des interactions existant au sein de la population folliculaire, les taux de transition de la chaîne de Markov à temps continu ainsi obtenue dépendent de l'ensemble ou d'une partie de la population. En utilisant une différence d’abondance et de vitesse de transit au sein des compartiments, nous obtenons un modèle dit lent/rapide. Nous démontrons alors l’existence et l’unicité d’un modèle limite dans le cadre de la théorie des perturbations stochastiques singulières en utilisant un critère de Foster-Lyapunov, un couplage ainsi qu’une équation de Poisson bien choisie. Ce modèle limite est composé d’une équation différentielle ordinaire décrivant la (lente) dynamique du premier compartiment, couplée avec une distribution quasi-stationnaire pour les compartiments (rapides) suivants.Dans un second temps, nous ajoutons au modèle une échelle de temps intermédiaire pour modéliser plus finement l’accélération continue de la croissance des follicules. À l’aide des mêmes outils mathématiques, nous prouvons l’existence et l’unicité d’un modèle limite plus complexe que le précédent. L’échelle la plus lente et la plus rapide sont toujours représentées respectivement par une équation différentielle ordinaire et par une distribution quasi-stationnaire. L’échelle intermédiaire est représenté par le point fixe d’un système d'équation différentielle ordinaire. Chaque échelle dépend de la précédente via une moyennisation ce qui rend l’étude de ce modèle dans un cadre général particulièrement difficile. Dans le cas de 2 ou de 3 échelles de temps, nous avons ensuite poussé l’étude de la limite à l’ordre supérieur en caractérisant l’espérance et la variance de l’erreur entre le modèle et sa limite.Enfin, nous avons introduit un modèle (un système d’équations différentielles ordinaires) adapté aux données biologiques expérimentales disponibles chez la souris. Ce modèle capture plus finement les premières étapes de formation des follicules (par l’ajout d’un compartiment en amont de la dynamique folliculaire) et tient compte de l’existence de deux sous-populations distinctes. En tenant compte d’une étude de l’identifiabilité théorique des paramètres, nous avons reparamétré le modèle et estimé les nouveaux paramètres du modèle. Nous avons finalement illustré le potentiel du modèle de deux façons : en suggérant le type de données biologiques à ajouter afin d’identifier un plus grand nombre de paramètres et en reproduisant des expériences in silico.This thesis aims to develop and analyze mathematically structured population models to explore the dynamics of the ovarian follicle population throughout the reproductive life of mammals.Firstly, we have built a birth, migration and death stochastic process structured in compartments: each follicle maturation stage is characterized by a compartment. In order to take into account interactions within the follicle population, some transition rates of the resulting continuous-time Markov chain depend on the whole or part of the population. Using a difference in abundances and time scales within the population, we have obtained a so-called slow/fast model. We have then proved the existence and uniqueness of a limit model in the framework of stochastic singular perturbations by using a Foster-Lyapunov criterion, a coupling and a well-chosen Poisson equation. The limit model consists of an ordinary-differential equation ruling the dynamics of the first (slow) compartment, coupled with a quasi-stationary distribution in the remaining (fast) compartments.In a second step, we have added to the model an intermediate timescale to refine the modeling of the continued acceleration of follicle growth. With the same mathematical tools, we have proven the existence and uniqueness of a more complex limit model than the previous one. The slowest and fastest timescales are always represented by an ordinary-differential equation and a quasi-stationary distribution respectively. The middle scale is represented by the fixed point of a system of ordinary differential equation. Each timescale depends on the previous one through an averaging, which makes the study of this model particularly difficult in a general framework. In the case of two and three timescales, we have then extended the study of the limit to the higher order by identifying the mean and variance of the error between the model and its limit.Finally, we have introduced a model (a system of ordinary-differential equation) adapted to the biological experimental data available in the mouse. This model captures more accurately the early step of follicle formation (by adding a compartment upstream the follicle dynamics) and takes into account the existence of two distinct subpopulations of follicles. Taking into consideration a study of theorical identifiability of the parameters, we have reparametrized the model and estimated the new parameters. We have illustrated the model potential in two ways: by suggesting the type of biological data needed to identify a larger number of parameters and by reproducing in silico experiments

Modélisation et analyse stochastique multi-échelle en dynamique des populations : application à la population folliculaire ovarienne

This thesis aims to develop and analyze mathematically structured population models to explore the dynamics of the ovarian follicle population throughout the reproductive life of mammals.Firstly, we have built a birth, migration and death stochastic process structured in compartments: each follicle maturation stage is characterized by a compartment. In order to take into account interactions within the follicle population, some transition rates of the resulting continuous-time Markov chain depend on the whole or part of the population. Using a difference in abundances and time scales within the population, we have obtained a so-called slow/fast model. We have then proved the existence and uniqueness of a limit model in the framework of stochastic singular perturbations by using a Foster-Lyapunov criterion, a coupling and a well-chosen Poisson equation. The limit model consists of an ordinary-differential equation ruling the dynamics of the first (slow) compartment, coupled with a quasi-stationary distribution in the remaining (fast) compartments.In a second step, we have added to the model an intermediate timescale to refine the modeling of the continued acceleration of follicle growth. With the same mathematical tools, we have proven the existence and uniqueness of a more complex limit model than the previous one. The slowest and fastest timescales are always represented by an ordinary-differential equation and a quasi-stationary distribution respectively. The middle scale is represented by the fixed point of a system of ordinary differential equation. Each timescale depends on the previous one through an averaging, which makes the study of this model particularly difficult in a general framework. In the case of two and three timescales, we have then extended the study of the limit to the higher order by identifying the mean and variance of the error between the model and its limit.Finally, we have introduced a model (a system of ordinary-differential equation) adapted to the biological experimental data available in the mouse. This model captures more accurately the early step of follicle formation (by adding a compartment upstream the follicle dynamics) and takes into account the existence of two distinct subpopulations of follicles. Taking into consideration a study of theorical identifiability of the parameters, we have reparametrized the model and estimated the new parameters. We have illustrated the model potential in two ways: by suggesting the type of biological data needed to identify a larger number of parameters and by reproducing in silico experiments.Cette thèse vise à élaborer et analyser mathématiquement des modèles de populations structurées pour étudier l'évolution de la population de follicules ovariens tout au long de la vie reproductive des mammifères.Dans un premier temps, nous avons construit un modèle de naissance, mort et migration; structuré en compartiments : chaque stade de maturation des follicules est représenté par un compartiment. Afin de tenir compte des interactions existant au sein de la population folliculaire, les taux de transition de la chaîne de Markov à temps continu ainsi obtenue dépendent de l'ensemble ou d'une partie de la population. En utilisant une différence d’abondance et de vitesse de transit au sein des compartiments, nous obtenons un modèle dit lent/rapide. Nous démontrons alors l’existence et l’unicité d’un modèle limite dans le cadre de la théorie des perturbations stochastiques singulières en utilisant un critère de Foster-Lyapunov, un couplage ainsi qu’une équation de Poisson bien choisie. Ce modèle limite est composé d’une équation différentielle ordinaire décrivant la (lente) dynamique du premier compartiment, couplée avec une distribution quasi-stationnaire pour les compartiments (rapides) suivants.Dans un second temps, nous ajoutons au modèle une échelle de temps intermédiaire pour modéliser plus finement l’accélération continue de la croissance des follicules. À l’aide des mêmes outils mathématiques, nous prouvons l’existence et l’unicité d’un modèle limite plus complexe que le précédent. L’échelle la plus lente et la plus rapide sont toujours représentées respectivement par une équation différentielle ordinaire et par une distribution quasi-stationnaire. L’échelle intermédiaire est représenté par le point fixe d’un système d'équation différentielle ordinaire. Chaque échelle dépend de la précédente via une moyennisation ce qui rend l’étude de ce modèle dans un cadre général particulièrement difficile. Dans le cas de 2 ou de 3 échelles de temps, nous avons ensuite poussé l’étude de la limite à l’ordre supérieur en caractérisant l’espérance et la variance de l’erreur entre le modèle et sa limite.Enfin, nous avons introduit un modèle (un système d’équations différentielles ordinaires) adapté aux données biologiques expérimentales disponibles chez la souris. Ce modèle capture plus finement les premières étapes de formation des follicules (par l’ajout d’un compartiment en amont de la dynamique folliculaire) et tient compte de l’existence de deux sous-populations distinctes. En tenant compte d’une étude de l’identifiabilité théorique des paramètres, nous avons reparamétré le modèle et estimé les nouveaux paramètres du modèle. Nous avons finalement illustré le potentiel du modèle de deux façons : en suggérant le type de données biologiques à ajouter afin d’identifier un plus grand nombre de paramètres et en reproduisant des expériences in silico

Averaging of a stochastic slow-fast model for population dynamics: application to the development of ovarian follicles

International audienceWe analyze a birth, migration and death stochastic process modeling the dynamics of a finite population, in which individuals transit unidirectionally across successive compartments. The model is formulated as a continuous-time Markov chain, whose transition matrix involves multiscale effects; the whole (or part of the) population affects the rates of individual birth, migration and death events. Using the slow-fast property of the model, we prove the existence and uniqueness of the limit model in the framework of stochastic singular perturbations. The derivation of the limit model is based on compactness and coupling arguments. The uniqueness is handled by applying the ergodicity theory and studying a dedicated Poisson equation. The limit model consists of an ordinary-differential equation ruling the dynamics of the first (slow) compartment, coupled with a quasi-stationary distribution in the remaining (fast) compartments, which averages the contribution of the fast component of the Markov chain on the slow one. We illustrate numerically the convergence, and highlight the relevance of dealing with nonlinear event rates for our application in reproductive biology. The numerical simulations involve a simple integration scheme for the deterministic part, coupled with the nested algorithm to sample the quasi-stationary distribution

Nonlinear compartmental modeling to monitor ovarian follicle population dynamics on the whole lifespan

In this work, we introduce an ODE-based compartmental model of ovarian follicle development all along lifespan. 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 datadriven 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 coexistence of two follicle subpopulations of different embryonic origins. We then setup a complete estimation strategy, including the study of theoretical identifiability, 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 finally illustrate the model potential for experimental design (suggestion of targeted new data acquisition) and in silico experiments

Nonlinear compartmental modeling to monitor ovarian follicle population dynamics on the whole lifespan

In this work, we introduce an ODE-based compartmental model of ovarian follicle development all along lifespan. 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 datadriven 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 coexistence of two follicle subpopulations of different embryonic origins. We then setup a complete estimation strategy, including the study of theoretical identifiability, 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 finally illustrate the model potential for experimental design (suggestion of targeted new data acquisition) and in silico experiments

Nonlinear compartmental modeling to monitor ovarian follicle population dynamics on the whole lifespan

International audienceIn this work, we introduce a compartmental model of ovarian follicle development all along lifespan, based on ordinary differential equations. The model predicts 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 datadriven 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 coexistence of two follicle subpopulations of different embryonic origins. We then setup a complete estimation strategy, including the study of theoretical identifiability, 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 finally illustrate the model potential for experimental design (suggestion of targeted new data acquisition) and in silico experiments

Stochastic chemical kinetics of cell fate decision systems: from single cells to populations and back

International audienceStochastic chemical kinetics is a widely used formalism for studying stochasticity of chemical reactions inside single cells. Experimental studies of reaction networks are generally performed with cells that are part of a growing population, yet the population context is rarely taken into account when models are developed. Models that neglect the population context lose their validity whenever the studied system influences traits of cells that can be selected in the population, a property that naturally arises in the complex interplay between single-cell and population dynamics of cell fate decision systems. Here, we represent such systems as absorbing continuous-time Markov chains. We show that conditioning on non-absorption allows one to derive a modified master equation that tracks the time evolution of the expected population composition within a growing population. This allows us to derive consistent population dynamics models from a specification of the single-cell process. We use this approach to classify cell fate decision systems into two types that lead to different characteristic phases in emerging population dynamics. Subsequently, we deploy the gained insights to experimentally study a recurrent problem in biology: how to link plasmid copy number fluctuations and plasmid loss events inside single cells to growth of cell populations in dynamically changing environments