Effects of initial telomere length distribution on senescence onset and heterogeneity

Replicative senescence, induced by telomere shortening, exhibits considerable asynchrony and heterogeneity, the origins of which remain unclear. Here, we formally study how telomere shortening mechanisms impact on senescence kinetics and define two regimes of senescence, depending on the initial telomere length variance. We provide analytical solutions to the model, highlighting a non-linear relationship between senescence onset and initial telomere length distribution. This study reveals the complexity of the collective behavior of telomeres as they shorten, leading to senescence heterogeneity

Estimating the Division Rate of the Self-Similar Growth-Fragmentation Equation

International audienceWe consider the growth-fragmentation equation and we address the problem of finding the division rate from the stable size distribution of the population, which is easily measured, but non-smooth. We propose a method based on the Mellin transform for growth-fragmentation equations with self-similar kernels. We build a sequence of functions which converges to the density of the population in division, simultaneously in several weighted L2 spaces, as the measurement error goes to 0. This improves previous results for self-similar kernels and allows us to understand the partial results for general fragmentation kernels. Numerical simulations confirm the theoretical results. Moreover, our numerical method is tested on real biological data, arising from a bacteria growth and fission experiment

Direct and inverse problems for some integro-differential equations arising in biology

Les phénomènes de croissance et de fragmentation jouent un rôle central dans de nombreux phénomènes biologiques. La première partie de ce manuscript concerne un modèle de l'activité électrique d'un réseau de neurones. Il s'agit d'une équation de croissance-fragmentation non linéaire. Grâce à une technique introduite par B. Perthame et L. Ryzhik, dans un cas particulier, nous quantifions des régimes dans lequels cette équation se relaxe avec un taux exponentiel. La deuxième et la troisième parties traitent de problèmes inverses. Le premier concerne l'équation de croissance-fragmentation ayant un noyau auto-similaire et le second un modèle de transduction olfactive. Après régularisations adéquates, ces deux problèmes reviennent à inverser des opérateurs intégraux dont les noyaux ont une structure auto-similaire. Grâce à la transformée de Mellin des inégalités de continuité et controlabilité de l'opérateur intégral sont établies. à partir de données expérimentales, ces études permettent d'estimer des paramètres importants des équations pour lesquels aucune mesure expérimentale directe n'est possible. La quatrième partie traite d'un modèle probabiliste de sénescence réplicative d'une lignée aléatoire de levures. En se basant sur des données expérimentales et des simulations numériques, le signal de sénescence est identifié et quantifié, et les sources de l'hétérogénéité des tailles des télomères sont analysées. Le modèle permet de mener une analyse complète de l'évolution des tailles des télomères.Growth and fragmentation phenomena play a central role in several biological phenomena. The first part of this thesis introduces a model of the electrical activity of a neural network. The equation involved is a non-linear growth-fragmentation equation. Thanks to a technique introduced by B. Perthame and L. Ryzhik, in a particular case, we are able to quantify regimes in which the equation has an exponential relaxation. The second and third part of this thesis both deal with inverse problems. The first one involves a growth-fragmentation equation with a self-similar kernel and the second one is a model of olfactive transduction. After regularization steps, these two problems come down to invert some integral operators whose kernels have a self-similar structure. Thanks to the use of the Mellin transform, some continuity and controllability inequalities are established. Using experimental data, these studies make it possible to estimate important parameters of the equations for which no direct experimental measurements can be obtained. The fourth part deals with a probabilistic model of replicative senescence of a random yeast lineage. Based on experimental data and numerical simulations, the senescence signal is identified and quantified, and the different sources of heterogeneity in the lengths of the telomeres are analyzed. This model allows us to completely analyze the evolution of the lengths of the telomeres

Determining the distribution of ion channels from experimental data

The authors study an integral inverse problem arising in the biology of the olfactory system. The transduction of an odor into an electrical signal is accomplished by a depolarising influx of ions through cyclic-nucleotide-gated (CNG for short) channels on the cilium membrane. The inverse problem studied in this paper consists in finding the spatial distribution of the CNG channels from the measured transduce electrical signals. The Mellin transform allows us to write an explicit formula for its solution. Proving observability and continuity inequalities is then a question of estimating the Mellin transform of the kernel of this integral equation on vertical lines. New estimates using arguments in the spirit of the stationary phase method are proven and a numerical scheme is proposed to reconstruct the density of CNG channels from modeled current representing experimental data, for an approximated model. For the original model an identifiability and a non observability (in some weighted L2 spaces) results are proven

Determining the distribution of calcium channels in the olfactory system

Cilia are long thin cylindrical structures that extend from an olfactory receptor into the nasal mucus. The transduction of an odor into an electrical signal occurs in the membranes of the cilia. The cyclic-nucleotide-gated (CNG) channels, activated by cyclic adenosine monophosphate (cAMP), allow a depolarizing influx of sodium ions, which initiate the electrical signal. A model for experiments involving the diffusion of cAMP into cilia and the resulting current activity has been developed in [8]. In this paper we study the inverse problem which consists in finding the spatial distribution of CNG ion channels from the experimental current data. This problem comes down to solving a Fredholm integral equation of the first kind. The Mellin transform allows us to write an explicit formula for its solution. Proving observability and continuity inequalities is then a question of estimating the Mellin transform of the kernel of this integral equation on vertical lines. We prove new estimates using arguments in the spirit of the stationary phase method. We complete the continuity and observability results obtained in [4] for an approximated kernel. We also introduce a better approximated kernel for which we prove continuity and observability results, and for which we perform numerical computations using experimental data. Lastly, we prove that no observability inequality, in certain weighted L 2 spaces, can be found for the original model. Throughout this paper we present numerical simulations illustrating the theoretical results

Effects of initial telomere length distribution on senescence onset and heterogeneity

International audienceReplicative senescence, induced by telomere shortening, exhibits considerable asynchrony and heterogeneity, the origins of which remain unclear. Here, we formally study how telomere shortening mechanisms impact on senescence kinetics and define two regimes of senescence, depending on the initial telomere length variance. We provide analytical solutions to the model, highlighting a non-linear relationship between senescence onset and initial telomere length distribution. This study reveals the complexity of the collective behavior of telomeres as they shorten, leading to senescence heterogeneity

Existence of recombination-selection equilibria for sexual populations

We study a birth and death model for the adapatation of a sexual population to an environment. The population is structured by a phenotypical trait, and, possibly, an age variable. Recombination is modeled by Fisher's infinitesimal operator. We prove the existence of principal eigenelements for the corresponding eigenproblem. As the infinitesimal operator is 1-homogeneous but nor linear nor monotone, the general Kre˘ ın-Rutman theory cannot be applied to this problem

The asymmetry of telomere replication contributes to replicative senescence heterogeneity

International audienceIn eukaryotes, the absence of telomerase results in telomere shortening, eventually leading to replicative senescence, an arrested state that prevents further cell divisions. While replicative senescence is mainly controlled by telomere length, the heterogeneity of its onset is not well understood. This study proposes a mathematical model based on the molecular mechanisms of telomere replication and shortening to decipher the causes of this heterogeneity. Using simulations fitted on experimental data obtained from individual lineages of senescent Saccharomyces cerevisiae cells, we decompose the sources of senescence heterogeneity into interclonal and intraclonal components, and show that the latter is based on the asymmetry of the telomere replication mechanism. We also evidence telomere rank-switching events with distinct frequencies in short-lived versus long-lived lineages, revealing that telomere shortening dynamics display important variations. Thus, the intrinsic heterogeneity of replicative senescence and its consequences find their roots in the asymmetric structure of telomeres