Boundary value for a nonlinear transport equation emerging from a stochastic coagulation-fragmentation type model

We investigate the connection between two classical models of phase transition phenomena, the (discrete size) stochastic Becker-D\"oring, a continous time Markov chain model, and the (continuous size) deterministic Lifshitz-Slyozov model, a nonlinear transport partial differential equation. For general coefficients and initial data, we introduce a scaling parameter and prove that the empirical measure associated to the stochastic Becker-D\"oring system converges in law to the weak solution of the Lifshitz-Slyozov equation when the parameter goes to 0. Contrary to previous studies, we use a weak topology that includes the boundary of the state space (\ie\ the size $x=0$) allowing us to rigorously derive a boundary value for the Lifshitz-Slyozov model in the case of incoming characteristics. The condition reads $\lim_{x\to 0} (a(x)u(t)-b(x))f(t,x) = \alpha u(t)^2$ where $f$ is the volume distribution function, solution of the Lifshitz-Slyozov equation, $a$ and $b$ the aggregation and fragmentation rates, $u$ the concentration of free particles and $\alpha$ a nucleation constant emerging from the microscopic model. It is the main novelty of this work and it answers to a question that has been conjectured or suggested by both mathematicians and physicists. We emphasize that this boundary value depends on a particular scaling (as opposed to a modeling choice) and is the result of a separation of time scale and an averaging of fast (fluctuating) variables.Comment: 42 pages, 3 figures, video on supplementary materials at http://yvinec.perso.math.cnrs.fr/video.htm

The Becker-Döring process: law of large numbers and non-equilibrium potential

In this note, we prove alaw of large numbersfor an infinite chemical reactionnetwork for phase transition problems called the stochastic Becker-Döring process.Under a general condition on the rate constants we show the convergence in lawand pathwise convergence of the process towards the deterministic Becker-Döringequations. Moreover, we prove that the non-equilibrium potential, associated to thestationary distribution of the stochastic Becker-Döring process, approaches the rela-tive entropy of the deterministic limit model. Thus, the phase transition phenomenathat occurs in the infinite dimensional deterministic modelis also present in the finitestochastic model.In this note, we prove alaw of large numbersfor an infinite chemical reactionnetwork for phase transition problems called the stochastic Becker-Döring process.Under a general condition on the rate constants we show the convergence in lawand pathwise convergence of the process towards the deterministic Becker-Döringequations. Moreover, we prove that the non-equilibrium potential, associated to thestationary distribution of the stochastic Becker-Döring process, approaches the rela-tive entropy of the deterministic limit model. Thus, the phase transition phenomenathat occurs in the infinite dimensional deterministic modelis also present in the finitestochastic model

Contributions à la modélisation mathématique et numérique de problèmes issus de la biologie : applications aux Prions et à la maladie d’Alzheimer

The aim of this thesis is to study, under several aspects, the formation of amyloids from proteins polymerization. The mathematical modelling of these phenomena in the case of in vitro or in vivo polymerisation remains questioned. We then propose here several models, which are also investigated from theoritical and numerical point of view. In the first part we present works done in collaboration with biologists. We propose two models based on the current theory on Prion phenomena that are designed for specific experimental conditions. These models allow us to analyse the experimental data obtained in laboratory and raise phenomena that remain unexplained by the theory. Then, from these results and biophysical considerations, we introduce a model which corroborates with data and provides a new approach on the amyloid formation in the particular case of Prion. This part is ended by the mathematical analysis of the model consisting of an infinite set of differentials equations. The system analysed is a Becker-Doring system coupled to a discrete growth-fragmentation system. The second part is dedicated to the analysis of a new model for polymerization of proteins with fragmentation subject to the surrounding variations of the fluid. Thus, we propose a model which is close to the experimental conditions and introduce new measurable macroscopic quantities to study the polymerization. The first introductory chapter states the stochastic description of the problem. We give the equations of motion for each polymers and monomers as well as a general formalism to study the limit in large number. Next, we give the mathematical framework and prove the existence of solutions to the Fokker-Planck-Smoluchowski equation for the configurational density of polymers coupled to the diffusion equation for monomers. The last chapter provides a numerical method adapted to this problem with numerical simulations In the last part, we are interested in modelling Alzheimer’s disease. We introduce a model that describes the formation of amyloids plaques in the brain and the interactions between Aβ-oligomers and Prion proteins which might be responsible of the memory impairment. We carry out the mathematical analysis of the model. Namely, for a constant polymerization rate, we provide existence and uniqueness together with stability of the equilibrium. Finally we study the existence in a more general and biological relevant case, that is when the polymerization depends on the size of the amyloidL’objectif de cette thèse est d’étudier, sous divers aspects, le processus de formation d’amyloïde à partir de la polymérisation de protéines. Ces phénomènes, aussi bien in vitro que in vivo, posent des questions de modélisation mathématique. Il s’agit ensuite de conduire une analyse des modèles obtenus. Dans la première partie nous présentons des travaux effectués en collaboration avec une équipe de biologistes. Deux modèles sont introduits, basés sur la théorie en vigueur du phénomène Prions, que nous ajustons aux conditions expérimentales. Ces modèles nous permettent d’analyser les données obtenues à partir d’expériences conduites en laboratoire. Cependant celles-ci soulèvent certains phénomènes encore inexpliqués par la théorie actuelle. Nous proposons donc un autre modèle qui corrobore les données et donne une nouvelle approche de la formation d’amyloïde dans le cas du Prion. Nous terminons cette partie par l’analyse mathématique de ce système compose d’une infinité d’équations différentielles. Ce dernier consiste en un couplage entre un système de type Becker-Doring et un système de polymérisation-fragmentation discrète. La seconde partie s’attache à l’analyse d’un nouveau modèle pour la polymérisation de protéines dont la fragmentation est sujette aux variations du fluide environnant. L’idée est de décrire au plus près les conditions expérimentales mais aussi d’introduire de nouvelles quantités macroscopiques mesurables pour l’étude de la polymérisation. Le premier chapitre de cette partie présente une description stochastique du problème. On y établit les équations du mouvement des polymères et des monomères (de type Langevin) ainsi que le formalisme pour l’étude du problème limite en grand nombre. Le deuxième chapitre pose le cadre fonctionnel et l’existence de solutions pour l équation de Fokker-Planck- Smoluchowski décrivant la densité de configuration des polymères, elle-même couplée a une équation de diffusion pour les monomères. Le dernier chapitre propose une méthode numérique pour traiter ce problème. On s’intéresse dans la dernière partie à la modélisation de la maladie d’Alzheimer. On construit un modèle qui décrit d’une part la formation de plaque amyloïde in vivo, et d’autre part les interactions entre les oligomères d’Aβet la protéine prion qui induiraient la perte de mémoire. On mène l’analyse mathématique de ce modèle dans un cas particulier puis dans un cas plus général ou le taux de polymérisation est une loi de puissanc

Fragmentation and monomer lengthening of rod-like polymers, a relevant model for prion proliferation

The Greer, Pujo-Menjouet andWebb model [Greer et al., J. Theoret. Biol., 242 (2006), 598-606] for prion dynamics was found to be in good agreement with experimental observations under no-flow conditions. The objective of this work is to generalize the problem to the framework of general polymerization-fragmentation under flow motion, motivated by the fact that laboratory work often involves prion dynamics under flow conditions in order to observe faster processes. Moreover, understanding and modelling the microstructure influence of macroscopically monitored non-Newtonian behaviour is crucial for sensor design, with the goal to provide practical information about ongoing molecular evolution. This paper's results can then be considered as one step in the mathematical understanding of such models, namely the proof of positivity and existence of solutions in suitable functional spaces. To that purpose, we introduce a new model based on the rigid-rod polymer theory to account for the polymer dynamics under flow conditions. As expected, when applied to the prion problem, in the absence of motion it reduces to that in Greer et al. (2006). At the heart of any polymer kinetical theory there is a configurational probability diffusion partial differential equation (PDE) of Fokker-Planck-Smoluchowski type. The main mathematical result of this paper is the proof of existence of positive solutions to the aforementioned PDE for a class of flows of practical interest, taking into account the flow induced splitting/lengthening of polymers in general, and prions in particular.Comment: Discrete and Continuous Dynamical Systems - Series B (2012) XX-X

A numerical scheme for rod-like polymers with fragmentation and monomers lengthening

In this paper, we introduce a new numerical method to solve a Fokker-Planck-Smolukowsky equation arising in the simulation of dilute polymeric liquids. The very recent model, considered here and introduced in a former paper, deals with rod-like polymers with fragmentation and monomers lengthening arising in biology. We formulate a spectral decomposition leading to a set of lengthening-fragmentation equations. The resulting system is solved by integration along the characteristic curves

Contributions to the mathematical and numerical modelling of biological problems : applications to Prions and Alzheimer's disease

L’objectif de cette thèse est d’étudier, sous divers aspects, le processus de formation d’amyloïde à partir de la polymérisation de protéines. Ces phénomènes, aussi bien in vitro que in vivo, posent des questions de modélisation mathématique. Il s’agit ensuite de conduire une analyse des modèles obtenus. Dans la première partie nous présentons des travaux effectués en collaboration avec une équipe de biologistes. Deux modèles sont introduits, basés sur la théorie en vigueur du phénomène Prions, que nous ajustons aux conditions expérimentales. Ces modèles nous permettent d’analyser les données obtenues à partir d’expériences conduites en laboratoire. Cependant celles-ci soulèvent certains phénomènes encore inexpliqués par la théorie actuelle. Nous proposons donc un autre modèle qui corrobore les données et donne une nouvelle approche de la formation d’amyloïde dans le cas du Prion. Nous terminons cette partie par l’analyse mathématique de ce système compose d’une infinité d’équations différentielles. Ce dernier consiste en un couplage entre un système de type Becker-Doring et un système de polymérisation-fragmentation discrète. La seconde partie s’attache à l’analyse d’un nouveau modèle pour la polymérisation de protéines dont la fragmentation est sujette aux variations du fluide environnant. L’idée est de décrire au plus près les conditions expérimentales mais aussi d’introduire de nouvelles quantités macroscopiques mesurables pour l’étude de la polymérisation. Le premier chapitre de cette partie présente une description stochastique du problème. On y établit les équations du mouvement des polymères et des monomères (de type Langevin) ainsi que le formalisme pour l’étude du problème limite en grand nombre. Le deuxième chapitre pose le cadre fonctionnel et l’existence de solutions pour l équation de Fokker-Planck- Smoluchowski décrivant la densité de configuration des polymères, elle-même couplée a une équation de diffusion pour les monomères. Le dernier chapitre propose une méthode numérique pour traiter ce problème. On s’intéresse dans la dernière partie à la modélisation de la maladie d’Alzheimer. On construit un modèle qui décrit d’une part la formation de plaque amyloïde in vivo, et d’autre part les interactions entre les oligomères d’Aβet la protéine prion qui induiraient la perte de mémoire. On mène l’analyse mathématique de ce modèle dans un cas particulier puis dans un cas plus général ou le taux de polymérisation est une loi de puissanceThe aim of this thesis is to study, under several aspects, the formation of amyloids from proteins polymerization. The mathematical modelling of these phenomena in the case of in vitro or in vivo polymerisation remains questioned. We then propose here several models, which are also investigated from theoritical and numerical point of view. In the first part we present works done in collaboration with biologists. We propose two models based on the current theory on Prion phenomena that are designed for specific experimental conditions. These models allow us to analyse the experimental data obtained in laboratory and raise phenomena that remain unexplained by the theory. Then, from these results and biophysical considerations, we introduce a model which corroborates with data and provides a new approach on the amyloid formation in the particular case of Prion. This part is ended by the mathematical analysis of the model consisting of an infinite set of differentials equations. The system analysed is a Becker-Doring system coupled to a discrete growth-fragmentation system. The second part is dedicated to the analysis of a new model for polymerization of proteins with fragmentation subject to the surrounding variations of the fluid. Thus, we propose a model which is close to the experimental conditions and introduce new measurable macroscopic quantities to study the polymerization. The first introductory chapter states the stochastic description of the problem. We give the equations of motion for each polymers and monomers as well as a general formalism to study the limit in large number. Next, we give the mathematical framework and prove the existence of solutions to the Fokker-Planck-Smoluchowski equation for the configurational density of polymers coupled to the diffusion equation for monomers. The last chapter provides a numerical method adapted to this problem with numerical simulations In the last part, we are interested in modelling Alzheimer’s disease. We introduce a model that describes the formation of amyloids plaques in the brain and the interactions between Aβ-oligomers and Prion proteins which might be responsible of the memory impairment. We carry out the mathematical analysis of the model. Namely, for a constant polymerization rate, we provide existence and uniqueness together with stability of the equilibrium. Finally we study the existence in a more general and biological relevant case, that is when the polymerization depends on the size of the amyloi

The initial-boundary value problem for the Lifshitz–Slyozov equation with non-smooth rates at the boundary

The authors would like to thank Boris Andreianov and Guy Barles (Institut Denis Poisson, Universit´e de Tours) for many interesting and helpful discussions on the subject. We also warmly thank the reviewers for their careful reading of the manuscript, and their valuable remarks that help us improve its quality. J. C. acknowledges support from MICINN, projects MTM2017-91054-EXP and RTI2018-098850-B-IOO; he also acknowledges support from Plan Propio de Investigaci ´on, Universidad de Granada, Programa 9 -partially through FEDER (ERDF) funds-. E. H. acknowledges support from FONDECYT Iniciaci´on n◦ 11170655. R. Y. does not have to thank the French National Research Agency for its financial support but he kindly thanks it for the excellent reviews embellished with arguments based on scientific and cultural novelties in the expertise of his yearly application file during the last four years. Part of this work was done while J. C. and R. Y. were visiting the Departamento de Matem´atica at Universidad del B´ıo-B´ıo and while E. H. and J. C. were visiting Institut Denis Poisson at Universit´e de Tours and INRAE Nouzilly. J.C. thanks Universit´e de Tours for a visiting position during last winter.We prove existence and uniqueness of solutions to the initialboundary value problem for the Lifshitz–Slyozov equation (a nonlinear transport equation on the half-line), focusing on the case of kinetic rates with unbounded derivative at the origin. Our theory covers in particular those cases with rates behaving as power laws at the origin, for which an inflow behavior is expected and a boundary condition describing nucleation phenomena needs to be imposed. The method we introduce here to prove existence is based on a formulation in terms of characteristics, with a careful analysis on the behavior near the singular boundary. As a byproduct we provide a general theory for linear continuity equations on a half-line with transport fields that degenerate at the boundary. We also address both the maximality and the uniqueness of inflow solutions to the Lifshitz–Slyozov model, exploiting monotonicity properties of the associated transport equation.Spanish Government European Commission MTM2017-91054-EXP RTI2018-098850-B-IOOPlan Propio de Investigacion, Universidad de Granada, Programa 9 - partially through FEDER (ERDF) fundsJunta de Andalucia European Commission P18-RT-2422 A-FQM-311-UGR18Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 1117065