Stochastic models of telomere shortening

Stationary Distribution of Telomere Lengths in Cells with Telomere Length Maintenance and its Parametric Inference

Telomeres are nucleotide caps located at the ends of each eukaryotic chromosome. Under normal physiological conditions as well as in culture, they shorten during each DNA replication round. Short telomeres initiate a proliferative arrest of cells termed 'replicative senescence'. However, cancer cells possessing limitless replication potential can avoid senescence by the telomere maintenance mechanism, which offsets telomeric loss. Therefore, cancer cells have sufficiently long telomeres even though their lengths are significantly shorter than their normal counterparts. This implies that the attrition and elongation rates play crucial roles in deciding whether and when cells ultimately become carcinogenic. In this research, we propose a concise mathematical model that shows the shortest telomere length at each cell division and prove mathematical conditions related to the attrition and elongation rates, which are necessary and sufficient for the existence of stationary distribution of telomere lengths. Moreover, we estimate the parameters of the telomere length maintenance process based on frequentist and Bayesian approaches. This study expands our knowledge of the mathematical relationship between the telomere attrition and elongation rates in cancer cells, which is important because the telomere length dynamics is a useful biomarker of cancer diagnosis and prognosis.

Brain aging and garbage cleaning : Modelling the role of sleep, glymphatic system, and microglia senescence in the propagation of inflammaging

Brain aging is a complex process involving many functions of our body and described by the interplay of a sleep pattern and changes in the metabolic waste concentration regulated by the microglial function and the glymphatic system. We review the existing modelling approaches to this topic and derive a novel mathematical model to describe the crosstalk between these components within the conceptual framework of inflammaging. Analysis of the model gives insight into the dynamics of garbage concentration and linked microglial senescence process resulting from a normal or disrupted sleep pattern, hence, explaining an underlying mechanism behind healthy or unhealthy brain aging. The model incorporates accumulation and elimination of garbage, induction of glial activation by garbage, and glial senescence by over-activation, as well as the production of pro-inflammatory molecules by their senescence-associated secretory phenotype (SASP). Assuming that insufficient sleep leads to the increase of garbage concentration and promotes senescence, the model predicts that if the accumulation of senescent glia overcomes an inflammaging threshold, further progression of senescence becomes unstoppable even if a normal sleep pattern is restored. Inverting this process by "rejuvenating the brain" is only possible via a reset of concentration of senescent glia below this threshold. Our model approach enables analysis of space-time dynamics of senescence, and in this way, we show that heterogeneous patterns of inflammation will accelerate the propagation of senescence profile through a network, confirming a negative effect of heterogeneity.

Mathematical modelling of telomere length dynamics

Telomeres are repetitive DNA sequences located at the ends of chromosomes. During cell division, an incomplete copy of each chromosome’s DNA is made, causing telomeres to shorten on successive generations. When a threshold length is reached replication ceases and the cell becomes ‘senescent’. In this paper, we consider populations of telomeres and, from discrete models, we derive partial differential equations which describe how the distribution of telomere lengths evolves over many generations. We initially consider a population of cells each containing just a single telomere. We use continuum models to compare the effects of various mechanisms of telomere shortening and rates of cell division during normal ageing. For example, the rate (or probability) of cell replication may be fixed or it may decrease as the telomeres shorten. Furthermore, the length of telomere lost on each replication may be constant, or may decrease as the telomeres shorten. Where possible, explicit solutions for the evolution of the distribution of telomere lengths are presented. In other cases, expressions for the mean of the distribution are derived. We extend the models to describe cell populations in which each cell contains a distinct subpopulation of chromosomes. As for the simpler models, constant telomere shortening leads to a linear reduction in telomere length over time, whereas length-dependent shortening results in initially rapid telomere length reduction, slowing at later times. Our analysis also reveals that constant telomere loss leads to a Gaussian (normal) distribution of telomere lengths, whereas length-dependent loss leads to a log-normal distribution. We show that stochastic models, which include a replication probability, also lead to telomere length distributions which are skewed.

Mathematical model and computer simulations of telomere loss

The molecular mechanisms that control the limited number of human cell divisions has occupied researchers ever since its first description in 1961. There is evidence that this limited growth capacity, referred to as cellular or replicative senescence, is the basis for organismal ageing. Numerous studies point to the molecular mechanisms of telomere involvement in this phenomenon. A hallmark of cell senescence is high stochasticity where individual cells enter senescence in a completely random and stochastic fashion. Therefore, mathematical modelling and computational simulations of telomere dynamics are often used to explain this stochastic nature of cell ageing. Models published thus far were based on the molecular mechanisms of telomere biology and how they dictate the dynamics of cell culture proliferation. In the present work we propose an advanced model of telomere controlled cell senescence based on abrupt telomere shortening, thus explaining some important but thus far overlooked aspects of cell senescence. We test our theory by simulating the proliferative potential and two-sister experiment originally conducted by Smith and Whitney in 1980.

Modelling the molecular mechanisms of aging

The aging process is driven at the cellular level by random molecular damage that slowly accumulates with age. Although cells possess mechanisms to repair or remove damage, they are not 100% efficient and their efficiency declines with age. There are many molecular mechanisms involved and exogenous factors such as stress also contribute to the aging process. The complexity of the aging process has stimulated the use of computational modelling in order to increase our understanding of the system, test hypotheses and make testable predictions. As many different mechanisms are involved, a wide range of models have been developed. This paper gives an overview of the types of models that have been developed, the range of tools used, modelling standards and discusses many specific examples of models that have been grouped according to the main mechanisms that they address. We conclude by discussing the opportunities and challenges for future modelling in this field.

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.

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New stochastic model of telomere shortening and elongation by telomerase.

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Analysis of the distribution of telomere length in the presence of telomerase.

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Senescence onset expressed as a function of the initial telomere length distribution.

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The initial telomere length distribution variance defines two senescence regimes.

Evaluating biomarkers to model cancer risk post cosmic ray exposure

Robust predictive models are essential to manage the risk of radiation-induced carcinogenesis. Chronic exposure to cosmic rays in the context of the complex deep space environment may place astronauts at high cancer risk. To estimate this risk, it is critical to understand how radiation-induced cellular stress impacts cell fate decisions and how this in turn alters the risk of carcinogenesis. Exposure to the heavy ion component of cosmic rays triggers a multitude of cellular changes, depending on the rate of exposure, the type of damage incurred and individual susceptibility. Heterogeneity in dose, dose rate, radiation quality, energy and particle flux contribute to the complexity of risk assessment. To unravel the impact of each of these factors, it is critical to identify sensitive biomarkers that can serve as inputs for robust modeling of individual risk of cancer or other long-term health consequences of exposure. Limitations in sensitivity of biomarkers to dose and dose rate, and the complexity of longitudinal monitoring, are some of the factors that increase uncertainties in the output from risk prediction models. Here, we critically evaluate candidate early and late biomarkers of radiation exposure and discuss their usefulness in predicting cell fate decisions. Some of the biomarkers we have reviewed include complex clustered DNA damage, persistent DNA repair foci, reactive oxygen species, chromosome aberrations and inflammation. Other biomarkers discussed, often assayed for at longer points post exposure, include mutations, chromosome aberrations, reactive oxygen species and telomere length changes. We discuss the relationship of biomarkers to different potential cell fates, including proliferation, apoptosis, senescence, and loss of stemness, which can propagate genomic instability and alter tissue composition and the underlying mRNA signatures that contribute to cell fate decisions. Our goal is to highlight factors that are important in choosing biomarkers and to evaluate the potential for biomarkers to inform models of post exposure cancer risk. Because cellular stress response pathways to space radiation and environmental carcinogens share common nodes, biomarker-driven risk models may be broadly applicable for estimating risks for other carcinogens.

Quantifying replicative senescence as a tumor suppressor pathway and a target for cancer therapy

To study quantitatively replicative senescence as a tumor suppressor mechanism, we investigate the distribution of a growing clonal cell population restricted by Hayflick’s limit. We find that in the biologically relevant range of parameters, if the imbalance between cell division and death is moderate or low (high death-to-birth ratio), senescence offers significant protection against cancer by halting abnormal cell proliferation at early pre-diagnostic stages of tumor development. We also find that by the time tumors are typically detected, there is a high probability that telomerase is activated, even if the cell of origin was telomerase negative. Hence, the fact that most cancers are positive for telomerase is not necessarily an indication that cancer originated in a telomerase positive cell. Finally, we discuss how the population dynamics of cells can determine the outcomes of anti-telomerase cancer therapies, and provide guidelines on how the model could potentially be applied to develop clinically useful tools to predict the response to treatment by telomerase inhibitors in individual patients.

The asymmetry of telomere replication contributes to replicative senescence heterogeneity

In 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.

Reconstructing the in vivo dynamics of hematopoietic stem cells from telomere length distributions

We investigate the in vivo patterns of stem cell divisions in the human hematopoietic system throughout life. In particular, we analyze the shape of telomere length distributions underlying stem cell behavior within individuals. Our mathematical model shows that these distributions contain a fingerprint of the progressive telomere loss and the fraction of symmetric cell proliferations. Our predictions are tested against measured telomere length distributions in humans across all ages, collected from lymphocyte and granulocyte sorted telomere length data of 356 healthy individuals, including 47 cord blood and 28 bone marrow samples. We find an increasing stem cell pool during childhood and adolescence and an approximately maintained stem cell population in adults. Furthermore, our method is able to detect individual differences from a single tissue sample, i.e. a single snapshot. Prospectively, this allows us to compare cell proliferation between individuals and identify abnormal stem cell dynamics, which affects the risk of stem cell related diseases.

Minimizing the risk of cancer: tissue architecture and cellular replication limits

Normal somatic cells are capable of only a limited number of divisions, which prevents unlimited cell proliferation and the onset of tumours. Cancer cells find ways to circumvent this obstacle, typically by expressing the enzyme telomerase and less often by alternative recombination strategies. Given this fundamental link between cellular replication limits and cancer, it is important to understand how a tissue's architecture affects the replicative capacity of a cell population. We define this as the average number of remaining divisions at equilibrium. The lower the replication capacity, the lower the chances to escape the replication limit during abnormal growth when a tumour develops. In this paper, we examine how the replication capacity is influenced by defining characteristics of cell lineages, such as the number of intermediate cell compartments, self-renewal capability of cells and division rates. We describe an optimal tissue architecture that minimizes the replication capacity of dividing cells and thus the risk of cancer. Interestingly, some of the features that define an optimal tissue architecture have been documented in a variety of tissues, suggesting that they may have evolved as a cancer-protecting strategy in multicellular organisms.

Modelling the regulation of telomere length: the effects of telomerase and G-quadruplex stabilising drugs

Telomeres are guanine-rich sequences at the end of chromosomes which shorten during each replication event and trigger cell cycle arrest and/or controlled death (apoptosis) when reaching a threshold length. The enzyme telomerase replenishes the ends of telomeres and thus prolongs the life span of cells, but also causes cellular immortalisation in human cancer. G-quadruplex (G4) stabilising drugs are a potential anticancer treatment which work by changing the molecular structure of telomeres to inhibit the activity of telomerase. We investigate the dynamics of telomere length in different conformational states, namely t-loops, G-quadruplex structures and those being elongated by telomerase. By formulating deterministic differential equation models we study the effects of various levels of both telomerase and concentrations of a G4-stabilising drug on the distribution of telomere lengths, and analyse how these effects evolve over large numbers of cell generations. As well as calculating numerical solutions, we use quasicontinuum methods to approximate the behaviour of the system over time, and predict the shape of the telomere length distribution. We find those telomerase and G4-concentrations where telomere length maintenance is successfully regulated. Excessively high levels of telomerase lead to continuous telomere lengthening, whereas large concentrations of the drug lead to progressive telomere erosion. Furthermore, our models predict a positively skewed distribution of telomere lengths, that is, telomeres accumulate over lengths shorter than the mean telomere length at equilibrium. Our model results for telomere length distributions of telomerase-positive cells in drug-free assays are in good agreement with the limited amount of experimental data available.

Can telomere shortening explain sigmoidal growth curves?

A general branching process model is proposed to describe the shortening of telomeres in eukaryotic chromosomes. The model is flexible and incorporates many special cases to be found in the literature. In particular, we show how telomere shortening can give rise to sigmoidal growth curves, an idea first expressed by Portugal et al. [A computational model for telomere-dependent cell-replicative aging, BioSystems 91 (2008), pp. 262-267]. We also demonstrate how other types of growth curves arise if telomere shortening is mitigated by other cellular processes. We compare our results with published data sets from the biological literature.

Modeling growth and telomere dynamics in Saccharomyces cerevisiae

A general branching process is proposed to model a population of cells of the yeast Saccharomyces cerevisiae following loss of telomerase. Previously published experimental data indicate that a population of telomerase-deficient cells regain exponential growth after a period of slowing due to critical telomere shortening. The explanation for this phenomenon is that some cells engage telomerase-independent pathways to maintain telomeres that allow them to become "survivors." Our model takes into account random variation in individual cell cycle times, telomere length, finite replicative lifespan of mother cells, and survivorship. We identify and estimate crucial parameters such as the probability of an individual cell becoming a survivor, and compare our model predictions to experimental data.

Stabilization of telomeres in nonlinear models of proliferating cell lines

We analyse an age-structured model of telomere loss in a proliferating cell population. The cell population is divided into telomere classes, which shorten each round of division. The model consists of a nonlinear system of partial differential equations for the telomere classes. We prove that if the highest telomere class is exempted from mortality, then all the classes stabilize to a nontrivial equilibrium dependent on the initial state of cells in the highest telomere class.

The concept of telomeric non-reciprocal recombination (TENOR) applied to human fibroblasts grown in serial cultures: concordance with genealogical data

Since the discovery of the limited life span of human fibroblasts some 50 years ago, many genealogical studies have been undertaken to describe growth kinetics of fibroblasts in serial cultures by their individual division behavior. It is now accepted that proliferation capacities of human fibroblasts strongly depend on their telomere lengths and integrity. Telomeres shorten with each replication round, and there is a direct correlation between cell division capacity and telomere lengths; that is, the consumption of disposable telomeric DNA repeats during cell divisions progresses until critically short telomeres determining the replicative senescence of the cells are present. Recently, we have suggested that telomeres in fibroblasts can also become elongated during DNA replication by telomeric non-reciprocal recombination (TENOR). Here we discuss genealogical data collected over the last decades as well as more recent findings on the telomere-driven replicative senescence process, and we summarize both to give an integrated picture.

Modelling cellular senescence as a result of telomere state

Telomeres in mammalian cells end in large duplex T loops. These loops protect the single-strand overhangs from degradation and/or interactions with signalling proteins. This protection is sometimes referred to as capping. At each cell division, telomeres shorten and there is a general consensus that telomere shortening triggers cell cycle exit. However, the exact mechanism by which telomere shortening causes cell cycle arrest is not known. Mathematical models of telomere shortening have been developed to help us understand the processes involved. Until now most models have assumed that the trigger for cell cycle arrest is the first telomere or a group of telomeres reaching a critically short length. However, there is evidence that cells stop cycling over a wide range of telomere lengths. This suggests that telomere length per se may not in fact be the trigger for cellular senescence. In this paper we develop a model which examines the hypothesis that uncapping of a telomere is the main trigger. By letting the probability of uncapping depend upon telomere length, we show that the hypothesized model provides a good fit to experimental data.

Cancer cell dynamics in presence of telomerase inhibitors: analysis of in vitro data

The inhibition of telomerase activity in actively dividing cells leads to suppression of cell growth after a time delay (inhibitory delay) required to reach a threshold telomeric DNA size. We developed a mathematical model of the dynamics of telomere size distribution and cell growth in the presence of telomere inhibitors that allowed quantification of the inhibitory delay. The model based on the solution of a system of differential equations described quantitatively recent experimental data on dynamics of cultured cells in presence of telomerase inhibitors. The analysis of the data by this model suggested the existence of at least two distinct subpopulations of cells with different proliferative activity. Size distribution of telomeres, fraction of proliferating cells, and tumor doubling times are of critical importance for the dynamics of cancer cells growth in presence of telomerase inhibitors. Rapidly growing cells with large telomeric DNA heterogeneity and small proliferating fractions as well as those with very short homogeneous telomeres would be the most sensitive to telomerase inhibitors.

Modelling perspectives on aging: can mathematics help us stay young?

We survey several types of mathematical models that keep track of age distributions in a population, or follow some aspects of aging, such as loss of replicative potential of stem cells. The properties of a class of linear models of this type are discussed and compared. We illustrate the applicability of such models with a simple example based on hypothetical stem cell dynamics developed to address age-related telomere loss in the human granulocyte pool. We then describe the contrasting behaviour of nonlinear systems. Examples are drawn from the class of "dynamical diseases" to illustrate some of the aspects of nonlinear systems. Applications of these, and other models to the problems of aging and replicative aging are discussed.

Estimation of replicative senescence via a population dynamics model of cells in culture

A simple mathematical model is developed for determining the time-varying fraction of senescent cells in culture in terms of the underlying probability distribution of the number of population doublings until senescence. This functional relationship is inverted, which allows for the estimation of the probability distribution of the number of population doublings until senescence given experimental data on the time-varying fraction of senescent cells. The relationship - in particular, the lag - between these two quantities is analyzed under the assumption that the number of population doublings until senescence follows the Weibull distribution. If the number of population doublings until senescence is geometrically distributed (i.e. the Weibull with shape parameter equal to one) then the cell culture appears immortal.

Accumulation of short telomeres in human fibroblasts prior to replicative senescence

The loss of telomere repeats has been causally linked to in vitro replicative senescence of human diploid fibroblasts (HDFs). In order to study the mechanism(s) by which telomere shortening signals cell senescence, we analyzed the telomere length at specific chromosome ends at cumulative population doublings in polyclonal and clonal HDFs by quantitative fluorescence in situ hybridization. The rate of telomere shortening at individual telomeres varied between 50 and 150 bp per population doubling and short telomeres with an estimated 1-2 kb of telomere repeats accumulated prior to senescence. The average telomere length in specific chromosome ends was remarkably similar between clones. However, some exceptions with individual telomeres measuring 0.5-1 kb were observed. In the fibroblast clones, the onset of replicative senescence was significantly correlated with the mean telomere fluorescence but, strikingly, not with chromosomes with the shortest telomere length. The accumulation of short telomeres in late passages of cultured HDFs is compatible with selection of cells on the basis of telomere length and limited recombination between telomeres prior to senescence.