To model the dynamics of polymers formed through nucleation, elongated by polymerisation, shortened by depolymerisation and subject to aggregation reactions, we study a nonlinear integro-differential equation. Growth and shrinkage are described by transport terms, nucleation by a positive boundary condition, and aggregation by a Smoluchowski coagulation kernel. Our main result is the existence of steady states for the multiplicative coagulation kernel despite this kernel producing gelation in finite time for the pure coagulation equation. This is made possible by a sufficiently strong decay rate for large polymers. Beyond the existence result, the qualitative properties of the steady states are illustrated through explicit examples and numerical experiments. The analytical results connect the growth behaviour of the transport velocity and of the coagulation kernel to the decay properties of steady states.

Depolymerisation reactions constitute frequent experiments, for instance in biochemistry for the study of amyloid fibrils. The quantities experimentally observed are related to the time dynamics of a quantity averaged over all polymer sizes, such as the total polymerised mass or the mean size of particles. The question analysed here is to link this measurement to the initial size distribution. To do so, we first derive from the initial reaction system, two asymptotic models: At first order, a backward transport equation, and at second order, an advection-diffusion/Fokker-Planck equation complemented with a mixed boundary condition at the origin. We estimate their distance to the original system solution. We then turn to the inverse problem, i.e., how to estimate the initial size distribution from the time measurement of an average quantity, given by a moment of the solution. This question has been studied for the first order asymptotic model, and we analyse here the second order asymptotic. Thanks to Carleman inequalities and log-convexity estimates, we prove observability results and error estimates for a Tikhonov regularization. We then develop a Kalman-based observer approach, and implement it on simulated observations. Despite its severely ill-posed character, the second-order approach appears numerically more accurate than the first-order one.

AbstractIs there an advantage to heterogeneity in a population where individuals grow and divide by fission? This is a broad question, to which there is no easy universal answer. This article aims to provide a quantitative answer in the specific context of growth rate heterogeneity, which can represent genetic polymorphism, by comparing the fitness of homogeneous versus heterogeneous populations. We focus on size-structured populations, where the growth rate of each individual is set at birth by heredity and/or mutations. The fitness (or Malthus parameter, or effective fitness) of such heterogeneous populations is defined by its long-term behaviour, and we introduce the effective growth rate as the individual growth rate in the homogeneous population with the same fitness. We derive analytical formulae linking effective and individual growth rates in two paradigmatic cases: first, constant growth and division rates; second, linear growth rates and uniform fragmentation. Surprisingly, both cases yield similar expressions. By comparing the fitness and effective growth rates of populations with different degrees of heterogeneity or different heredity/mutation laws to those of average homogeneous populations, we quantitatively investigate the combined influence of heredity and heterogeneity and revisit previous results stating that heterogeneity is beneficial in the case of strong heredity.

In the absence of telomerase, telomere shortening triggers replicative senescence, a tumor suppressor mechanism that is also associated with oncogenic genomic instability. Yet, the precise mechanism that connects these seemingly opposing forces remains poorly understood. To directly study the complex interplay between senescence, telomere dynamics, and genomic instability, we develop a system in Saccharomyces cerevisiae to generate and track telomeres of precise length in the absence of telomerase. Using single-telomere and single-cell analyses combined with mathematical modeling, we identify a threshold length at which telomeres switch into dysfunction. A single shortest telomere below the threshold length is necessary and sufficient to trigger the onset of replicative senescence in a majority of cells. At population level, fluctuation assays establish that rare genomic instability arises predominantly in cis to the shortest telomere as Pol32-dependent non-reciprocal translocations that result in re-elongation of the shortest telomere and likely transient escape from senescence. The switch of the shortest telomere into dysfunction and subsequent processing in telomerase-negative cells thus serves as the mechanistic link between replicative senescence onset, genomic instability and the initiation of post-senescence survival.

Telomere shortening ultimately causes replicative senescence. However, identifying the mechanisms driving replicative senescence in cell populations is challenging due to the heterogeneity of telomere lengths and the asynchrony of senescence onset. Here, we present a mathematical model of telomere shortening and replicative senescence in Saccharomyces cerevisiae which is quantitatively calibrated and validated using data of telomerase-deficient single cells. Simulations of yeast populations, where cells with varying proliferation capacities compete against each other, show that the distribution of telomere lengths of the initial population shapes population growth, especially through the distribution of cells' shortest telomere lengths. We also quantified how factors influencing cell viability independently of telomeres can impact senescence rates. Overall, we demonstrate a temporal evolution in the composition of senescent cell populations-from a state directly linked to critically short telomeres to a state where senescence onset becomes stochastic. This population structure may promote genome instability and facilitate senescence escape.

To provide a mechanistic explanation of sustained then damped oscillations observed in a depolymerisation experiment, a bi-monomeric variant of the seminal Becker–Döring system has been proposed in Doumic et al (2019 J. Theor. Biol. 480 241–61). When all reaction rates are constant, the equations are the following: d v d t = − v w + v ∑ j = 2 ∞ c j , d w d t = v w − w ∑ j = 1 ∞ c j , d c j d t = J j − 1 − J j , j ⩾ 1 , J j = w c j − v c j + 1 , j ⩾ 1 , J 0 = 0 , where v and w are two distinct unit species, and c i represents the concentration of clusters containing i units. We study in detail the mechanisms leading to such oscillations and characterise the different phases of the dynamics, from the initial high-amplitude oscillations to the progressive damping leading to the convergence towards the unique positive stationary solution. We give quantitative approximations for the main quantities of interest: period of the oscillations, size of the damping (corresponding to a loss of energy), number of oscillations characterising each phase. We illustrate these results by numerical simulation, in line with the theoretical results, and provide numerical methods to solve the system.

In the absence of telomerase, telomere shortening triggers the DNA damage checkpoint and replicative senescence, a potent tumor suppressor mechanism. Paradoxically, this same process is also associated with oncogenic genomic instability. Yet, the precise mechanism that connects these seemingly opposing forces remains poorly understood. To directly study the complex interplay between senescence, telomere dynamics and genomic instability, we developed a system in Saccharomyces cerevisiae to generate and track the dynamics of telomeres of precise length in the absence of telomerase. Using single-telomere and single-cell analyses combined with mathematical modeling, we identify a threshold length at which telomeres switch into dysfunction. A single shortest telomere below the threshold length is necessary and sufficient to trigger the onset of replicative senescence in a majority of cells. At population level, fluctuation assays establish that rare genomic instability arises predominantly in cis to the shortest telomere as non-reciprocal translocations that result in re-elongation of the shortest telomere and likely escape from senescence. The switch of the shortest telomere into dysfunction and subsequent processing in telomerase-negative cells thus serves as the mechanistic link between replicative senescence onset, genomic instability and the initiation of post-senescence survival, explaining the contradictory roles of replicative senescence in oncogenesis. ### Competing Interest Statement The authors have declared no competing interest.

Originally motivated by the morphogenesis of bacterial microcolonies, the aim of this paper is to explore models through different scales for a spatial population of interacting, growing and dividing particles. We start from a microscopic stochastic model, write the corresponding stochastic differential equation satisfied by the empirical measure, and rigorously derive its mesoscopic (mean-field) limit. Under smoothness and symmetry assumptions for the interaction kernel, we then obtain entropy estimates, which provide us with a localization limit at the macroscopic level. Finally, we perform a thorough numerical study in order to compare the three modeling scales.

While significant efforts have been made to model bacterial cell division, few models have incorporated DNA replication into the control of this process. To date, models that attempt to capture the coordination between replication and division cycles are based on fundamentally different assumptions, and yet conflicting results have emerged. As a result, key questions regarding how replication affects cell size at division remain unclear. To address this, we develop in a first part, a robust mathematical framework to study models of coordination of replication and division cycles proposed in the literature. Through theoretical analysis, we highlight necessary and sufficient conditions to apply to replication-agnostic and division-agnostic models to ensure physiologically-coherent behaviors. Then, in a second part, we lead a comprehensive statistical analysis to assess the ability of the models to reproduce the division volume distribution conditioned on cell covariates. This in-depth analysis highlighted remarkable performances of a novel model, yielding promising results for future refinements toward a universal model of replication-division coordination in E. coli . ### Competing Interest Statement The authors have declared no competing interest. European Research Council, SINGER, 101054787

Shrinkage of large particles, either through depolymerisation (i.e. progressive shortening) or through fragmentation (breakage into smaller pieces) may be modelled by discrete equations, of Becker-D'́oring type, or by continuous ones. In this note, we review two kinds of inverse problems: the first is the estimation of the initial size-distribution from moments measurements in a depolymerising system, in collaboration with Philippe Moireau and inspired by experiments carried out by Human Rezaei's team; the second is the inference of fragmentation characteristics from size distribution samples, in collaboration with Miguel Escobedo and Magali Tournus, based on biological questions and experiments of Wei-Feng Xue's team.

The present paper provides a new representation of the solution to the fragmentation equation as a power series in the Banach space of Radon measures endowed with the total variation norm. This representation is used to justify how the fragmentation kernel, which is one of the two key parameters of the fragmentation equation, can be recovered from short-time experimental measurements of the particle size distributions when the initial condition is a delta function. A new stability result for this equation is also provided using a Wasserstein-type norm. We exploit this stability to prove the robustness of our reconstruction formula with respect to noise and initial data.

Systems describing the long-range interaction between individuals have attracted a lot of attention in the last years, in particular in relation with living systems. These systems are quadratic, written under the form of transport equations with a nonlocal self-generated drift. We establish the localisation limit, that is the convergence of nonlocal to local systems, when the range of interaction tends to 0. These theoretical results are sustained by numerical simulations. The major new feature in our analysis is that we do not need diffusion to gain compactness, at odd with the existing literature. The central compactness result is provided by a full rank assumption on the interaction kernels. In turn, we prove existence of weak solutions for the resulting system, a cross-diffusion system of quadratic type.

Progressive shortening of telomeres ultimately causes replicative senescence and is linked with aging and tumor suppression. Studying the intricate link between telomere shortening and senescence at the molecular level and its population-scale effects over time is challenging with current approaches but crucial for understanding behavior at the organ or tissue level. In this study, we developed a mathematical model for telomere shortening and the onset of replicative senescence using data from Saccharomyces cerevisiae without telomerase. Our model tracks individual cell states, their telomere length dynamics, and lifespan over time, revealing selection forces within a population. We discovered that both cell genealogy and global telomere length distribution are key to determine the population proliferation capacity. We also discovered that cell growth defects unrelated to telomeres also affect subsequent proliferation and may act as confounding variables in replicative senescence assays. Overall, while there is a deterministic limit for the shortest telomere length, the stochastic occurrence of non-terminal arrests drive cells into a totally different regime, which may promote genome instability and senescence escape. Our results offer a comprehensive framework for investigating the implications of telomere length on human diseases. One-sentence Summary Key determinants of population proliferation capacity in the context of telomere shortening.

Modelling, analysing and inferring triggering mechanisms in population reproduction is fundamental in many biological applications. It is also an active and growing research domain in mathematical biology. In this chapter, we review the main results developed over the last decade for the estimation of the division rate in growing and dividing populations in a steady environment. These methods combine tools borrowed from PDE’s and stochastic processes, with a certain view that emerges from mathematical statistics. A focus on the application to the bacterial cell division cycle provides a concrete presentation, and may help the reader to identify major new challenges in the field.

Aggregation of ubiquitinated cargo by oligomers of the protein p62 is an important preparatory step in cellular autophagy. In this work a mathematical model for the dynamics of these heterogeneous aggregates in the form of a system of ordinary differential equations is derived and analyzed. Three different parameter regimes are identified, where either aggregates are unstable, or their size saturates at a finite value, or their size grows indefinitely as long as free particles are abundant. The boundaries of these regimes as well as the finite size in the second case can be computed explicitly. The growth in the third case (quadratic in time) can also be made explicit by formal asymptotic methods. In the absence of rigorous results the dynamic stability of these structures has been investigated by numerical simulations. A comparison with recent experimental results permits a partial parametrization of the model.

The dynamics by which polymeric protein filaments divide in the presence of negligible growth, for example due to the depletion of free monomeric precursors, can be described by the universal mathematical equations of 'pure fragmentation'. The rates of fragmentation reactions reflect the stability of the protein filaments towards breakage, which is of importance in biology and biomedicine for instance in governing the creation of amyloid seeds and the propagation of prions. Here, we devised from mathematical theory inversion formulae to recover the division rates and division kernel information from time-dependent experimental measurements of filament size distribution. The numerical approach to systematically analyze the behaviour of pure fragmentation trajectories was also developed. We illustrate how these formulae can be used, provide some insights on their robustness, and show how they inform the design of experiments to measure fibril fragmentation dynamics. These advances are made possible by our central theoretical result on how the length distribution profile of the solution to the pure fragmentation equation aligns with a steady distribution profile for large times.

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.