Background Telomerase-negative cells have limited proliferation potential. In these cells, telomeres shorten until they reach a critical length and induce a permanently arrested state. This process called replicative senescence is associated with genomic instability and participates in tissue and organismal ageing. Experimental data using single-cell approaches in the budding yeast model organism show that telomerase-negative cells often experience abnormally long cell cycles, which can be followed by cell cycles of normal duration, before reaching the terminal senescent state. These series of non-terminal cell cycle arrests contribute to the heterogeneity of senescence and likely magnify its genomic instability. Due to their apparent stochastic nature, investigating the dynamics and the molecular origins of these arrests has been difficult. In particular, whether the non-terminal arrests series stem from a mechanism similar to the one that triggers terminal senescence is not known. Results Here, we provide a mathematical description of sequences of non-terminal arrests to understand how they appear. We take advantage of an experimental data set of cell cycle duration measurements performed in individual telomerase-negative yeast cells that keep track of the number of generations since telomerase inactivation. Using numerical simulations, we show that the occurrence of non-terminal arrests is a generation-dependent process that can be explained by the shortest telomere reaching a probabilistic threshold length. While the onset of senescence is also triggered by telomere shortening, we highlight differences in the laws that describe the number of consecutive arrests in non-terminal arrests compared to senescence arrests, suggesting distinct underlying mechanisms and cellular states. Conclusions Replicative senescence is a complex process that affects cell divisions earlier than anticipated, as exemplified by the frequent occurrence of non-terminal arrests early after telomerase inactivation. The present work unravels two kinetically and mechanistically distinct generation-dependent processes underlying non-terminal and terminal senescence arrests. We suggest that these two processes are responsible for two consequences of senescence at the population level, the increase of genome instability on the one hand, and the limitation of proliferation capacity on the other hand.

We study the mathematical properties of a model of cell division structured by two variables -the size and the size increment -in the case of a linear growth rate and a self-similar fragmentation kernel. We first show that one can construct a solution to the related two dimensional eigenproblem associated to the eigenvalue 1 from a solution of a certain one dimensional fixed point problem. Then we prove the existence and uniqueness of this fixed point in the appropriate L 1 weighted space under general hypotheses on the division rate. Knowing such an eigenfunction proves useful as a first step in studying the long time asymptotic behaviour of the Cauchy problem.

We are interested in a non-local partial differential equation modeling equal mitosis. We prove that the solutions present persistent asymptotic oscillations and that the convergence to this periodic behavior, in suitable spaces of weighted signed measures, occurs exponentially fast. It can be seen as a spectral gap result between the countable set of dominant eigenvalues and the rest of the spectrum, which is to our knowledge completely new. The two main difficulties in the proof are to define the projection onto the subspace of periodic (rescaled) solutions and to estimate the speed of convergence to this projection. The first one is addressed by using the generalized relative entropy structure of the dual equation, and the second is tackled by applying Harris's ergodic theorem on sub-problems.

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<p>Numerical simulations of granular flows have been widely developed and used during the last two decades. Depending on the situation and scale of the simulations, different methods are used, each having specific pros and cons. Among them, three main methods can be distinguished such as; discrete, continuous or depth-averaged approach. At the laboratory scale, discrete approach consists of representing all the grains and contacts. When the amount of grains are important enough to consider the granular medium as an effective fluid, Navier-Stokes simulations can be performed using an appropriate rheology for the fluid, like the -rheology. However, when simulations are performed on geophysical scales none of these two methods can be used because of the enormous computation time required to solve them. To cope up with this issue, the depth-averaged approachs wherein the normal velocities are neglected, considerably reduce the computation time.</p><p>Even though all these models have been widely used, it is not clear exactly what information can be extracted about the forces exerted to the ground. These forces represent a new way of visualising a geophysical granular flow. Indeed, very recently, the recorded seismic signals from geophysical granular flows were used to interpret these forces. As a result, seismic data can be used to extract information on the flow dynamics which was missing due to the difficulties of direct observation (ashes, dust, etc…). Being able to compute and interpret the forces generated by a granular flow on<br>the ground represents a new way for calibrations of numerical methods and is a key point in analysing seismic data generated by granular flows and subsequently in understanding the landslide dynamics at the geophysical scale.</p><p>After a quick presentation of the numerical differences between the three models, we present comparisons between discrete, continuous [1] and depth-averaged [2] models. Besides, we put forward this study on the values taken by the forces generated on the ground during the evolution of granular dam breaks. Although, these three methods give relatively the same final deposits, in good agreement with the experiments, we observe they lead to very different dynamics in terms of flow acceleration, forces and histories.</p><p>1. http:basilisk.fr.</p><p>2. A. Mangeney et al., JGR 112 F02017 (2007)</p>