arXiv subjects

Erwan Hingant

Publications and source records attributed to Erwan Hingant.

12 recordsLinked to original sources

arXiv2023

Long-time asymptotic of the Lifshitz-Slyozov equation with nucleation

We consider the Lifshitz-Slyozov model with inflow boundary conditions of nucleation type. We show 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 to an entropy functional. Moreover, we give 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.

math.AP↗

arXiv2023

Modeling compartmentalization within intracellular signaling pathway

We present a new modeling approach for G protein coupled receptors signaling systems, that take into account the compartmentalization of receptors and their effectors, both at plasma membrane and in dynamic intra-cellular vesicles called endosomes. 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 plasma membrane, degradation through transfer into lysosomes as well as endosomes fusion through coagulation dynamics. The second building block is biochemical reactions into each compartments and the transfer of molecules between the dynamical compartments. In this work, we prove sufficient conditions to obtain exponentially ergodicity for the size distribution of intracellular compartments. We futher design a finite volume scheme to simulate our model and show two application cases for receptor trafficking and spatially biased second effector signaling.

math.AP↗

arXiv2021

The Becker-Döring process: pathwise convergence and phase transition phenomena

In this note, we study an infinite reaction network called the stochastic Becker-Döring process, a sub-class of the general coagulation-fragmentation models. We prove pathwise convergence of the process towards the deterministic Becker-Döring equations which improves classical tightness-based results. Also, we show by studying the asymptotic behavior of the stationary distribution, that the phase transition property of the deterministic model is also present in the finite stochastic model. Such results might be interpreted closed to the so-called gelling phenomena in coagulation models. We end with few numerical illustrations that support our results.

math.PR↗

arXiv2020

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

We prove existence and uniqueness of solutions to the initial-boundary 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.

math.AP↗

arXiv2020

Quasi-stationary distribution and metastability for the stochastic Becker-Döring model

We study a stochastic version of the classical Becker-Döring model, a well-known kinetic model for cluster formation that predicts the existence of a long-lived metastable state before a thermodynamically unfavorable nucleation occurs, leading to a phase transition phenomena. This continuous-time Markov chain model has received little attention, compared to its deterministic differential equations counterpart. We show that the stochastic formulation leads to a precise and quantitative description of stochastic nucleation events thanks to an exponentially ergodic quasi-stationary distribution for the process conditionally on nucleation has not yet occurred.

math.PR↗

arXiv2016

Deterministic and Stochastic Becker-Döring equations: Past and Recent Mathematical Developments

We present a survey on the results on a particular coagulation-fragmentation model given by the Becker-Döring equations. For both the deterministic and stochastic versions, we include well-posedness, long-time behavior, convergence rate towards equilibrium, coarsening and relation to transport equations, time-dependent properties, metastability and classical nucleation theory. All along this survey, we highlight recent results and open questions.

math-ph↗

arXiv2016

Quasi steady state approximation of the small clusters in Becker-Döring equations leads to boundary conditions in the Lifshitz-Slyozov limit

This papers addresses the connection between two classical models of phase transition phenomena describing different stages of the growth of clusters. The Becker-Döring model (BD) describes discrete-sized clusters through an infinite set of ordinary differential equations. The Lifshitz-Slyozov equation (LS) is a transport partial differential equation on the continuous half-line $x\in (0,+\infty)$. We introduce a scaling parameter $\varepsilon>0$, which accounts for the grid size of the state space in the BD model, and recover the LS model in the limit $\varepsilon\to 0$. The connection has been already proven in the context of outgoing characteristic at the boundary $x=0$ for the LS model, when small clusters tend to shrink. The main novelty of this work resides in a new estimate on the growth of small clusters, which behave at a fast time scale. Through a rigorous quasi steady state approximation, we derive boundary conditions for the incoming characteristic case, when small clusters tend to grow.

math.AP↗

arXiv2015

First passage times in homogeneous nucleation: dependence on the total number of particles

Motivated by nucleation and molecular aggregation in physical, chemical and biological settings, we present an extension to a thorough analysis of the stochastic self-assembly of a fixed number of identical particles in a finite volume. We study the statistic of times it requires for maximal clusters to be completed, starting from a pure-monomeric particle configuration. For finite volume, we extend previous analytical approaches to the case of arbitrary size-dependent aggregation and fragmentation kinetic rates. For larger volume, we develop a scaling framework to study the behavior of the first assembly time as a function of the total quantity of particles. We find that the mean time to first completion of a maximum-sized cluster may have surprisingly a very weak dependency on the total number of particles. We highlight how the higher statistic (variance, distribution) of the first passage time may still help to infer key parameters (such as the size of the maximum cluster) from data. And last but not least, we present a framework to quantify the formation of cluster of macroscopic size, whose formation is (asymptotically) very unlikely and occurs as a large deviation phenomenon from the mean-field limit. We argue that this framework is suitable to describe phase transition phenomena, as inherent infrequent stochastic processes, in contrast to classical nucleation theory.

q-bio.BM↗

arXiv2015

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-Döring, 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-Döring 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 $\lim_{x\to 0} (a(x)u(t)-b(x))f(t,x) = αu(t)^2$ 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 $α$ 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.

math.PR↗

arXiv2013

Derivation and mathematical study of a sorption-coagulation equation

This work is devoted to the derivation and the matematical study of a new model for water-soluble polymers and metal ions interactions, which are used in chemistry for their wide range of applications. First, we motivate and derive a model that describes the evolution of the configurational distribution of polymers. One of the novelty resides in the configuration variables which consider both, the size of the polymers and the quantity of metal ions they captured through sorption. The model consists in a non-linear transport equation with a quadratic source term, the coagulation. Then, we prove the existence of solutions for all time to the problem thanks to classical fixed point theory. Next, we reformulate the coagulation operator under a conservative form which allows to write a finite volume scheme. The sequence of approximated solutions is proved to be convergent (toward a solution to the problem) thanks to a $L^1 - weak$ stability principle. Finally, we illustrate the behaviour of the solutions using this numerical scheme and we intend to discuss on the long-time behaviour.

math.NA↗

arXiv2013

Alzheimer's disease: analysis of a mathematical model incorporating the role of prions

We introduce a mathematical model of the in vivo progression of Alzheimer's disease with focus on the role of prions in memory impairment. Our model consists of differential equations that describe the dynamic formation of β-amyloid plaques based on the concentrations of Aβ oligomers, PrPC proteins, and the Aβ-x-PrPC complex, which are hypothesized to be responsible for synaptic toxicity. We prove the well-posedness of the model and provided stability results for its unique equilibrium, when the polymerization rate of β-amyloid is constant and also when it is described by a power law.

math.CA↗

arXiv2011

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.

math.AP↗