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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.

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.

We study a stochastic version of the classical Becker-D��ring 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., 14 pages

In mammals, female germ cells are sheltered within somatic structures called ovarian follicles, which remain in a quiescent state until they get activated, all along reproductive life. We investigate the sequence of somatic cell events occurring just after follicle activation, starting by the awakening of precursor somatic cells, and their transformation into proliferative cells. We introduce a nonlinear stochastic model accounting for the joint dynamics of the two cell types, and allowing us to investigate the potential impact of a feedback from proliferative cells onto precursor cells. To tackle the key issue of whether cell proliferation is concomitant or posterior to cell awakening, we assess both the time needed for all precursor cells to awake, and the corresponding increase in the total cell number with respect to the initial cell number. Using the probabilistic theory of first passage times, we design a numerical scheme based on a rigorous Finite State Projection and coupling techniques to compute the mean extinction time and the cell number at extinction time. We find that the feedback term clearly lowers the number of proliferative cells at the extinction time. We calibrate the model parameters using an exact likelihood approach. We carry out a comprehensive comparison between the initial model and a series of submodels, which helps to select the critical cell events taking place during activation, and suggests that awakening is prominent over proliferation., Accepted in Journal of Mathematical Biology

In this note, we study an infinite reaction network called the stochastic Becker-D��ring process, a sub-class of the general coagulation-fragmentation models. We prove pathwise convergence of the process towards the deterministic Becker-D��ring 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.

We analyze a multi-type age dependent model for cell populations subject to unidirectional motion, in both a stochastic and deterministic framework. Cells are distributed into successive layers; they may divide and move irreversibly from one layer to the next. We adapt results on the large-time convergence of PDE systems and branching processes to our context, where the Perron-Frobenius or Krein-Rutman theorem can not be applied. We derive explicit analytical formulas for the asymptotic cell number moments, and the stable age distribution. We illustrate these results numerically and we apply them to the study of the morphodynamics of ovarian follicles. We prove the structural parameter identifiability of our model in the case of age independent division rates. Using a set of experimental biological data, we estimate the model parameters to fit the changes in the cell numbers in each layer during the early stages of follicle development.

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.html

This paper considers adiabatic reduction in both discrete and continuous models of stochastic gene expression. In gene expression models, the concept of bursting is a production of several molecules simultaneously and is generally represented as a compound Poisson process of random size. In a general two-dimensional birth and death discrete model, we prove that under specific assumptions and scaling (that are characteristics of the mRNA-protein system) an adiabatic reduction leads to a one-dimensional discrete-state space model with bursting production. The burst term appears through the reduction of the first variable. In a two-dimensional continuous model, we also prove that an adiabatic reduction can be performed in a stochastic slow/fast system. In this gene expression model, the production of mRNA (the fast variable) is assumed to be bursty and the production of protein (the slow variable) is linear as a function of mRNA. When the dynamics of mRNA is assumed to be faster than the protein dynamics (due to a mRNA degradation rate larger than for the protein) we prove that, with the appropriate scaling, the bursting phenomena can be transmitted to the slow variable. We show that the reduced equation is either a stochastic differential equation with a jump Markov process or a deterministic ordinary differential equation depending on the scaling that is appropriate. These results are significant because adiabatic reduction techniques seem to have not been applied to a stochastic differential system containing a jump Markov process. Last but not least, for our particular system, the adiabatic reduction allows us to understand what are the necessary conditions for the bursting production-like of protein to occur., Comment: 24 pages

In this article we demonstrate that the so-called bursting production of molecular species during gene expression may be an artifact caused by low time resolution in experimental data collection and not an actual burst in production. We reach this conclusion through an analysis of a two-stage and binary model for gene expression, and demonstrate that in the limit when mRNA degradation is much faster than protein degradation they are equivalent. The negative binomial distribution is shown to be a limiting case of the binary model for fast "on to off" state transitions and high values of the ratio between protein synthesis and degradation rates. The gene products population increases by unity but multiple times in a time interval orders of magnitude smaller than protein half-life or the precision of the experimental apparatus employed in its detection. This rare-and-fast one-by-one protein synthesis has been interpreted as bursting., 13 pages