Modelling cellular senescence as a result of telomere state

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.

Telomere-dependent and telomere-independent roles of RAP1 in regulating human stem cell homeostasis

RAP1 is a well-known telomere-binding protein, but its functions in human stem cells have remained unclear. Here we generated RAP1-deficient human embryonic stem cells (hESCs) by using CRISPR/Cas9 technique and obtained RAP1-deficient human mesenchymal stem cells (hMSCs) and neural stem cells (hNSCs) via directed differentiation. In both hMSCs and hNSCs, RAP1 not only negatively regulated telomere length but also acted as a transcriptional regulator of RELN by tuning the methylation status of its gene promoter. RAP1 deficiency enhanced self-renewal and delayed senescence in hMSCs, but not in hNSCs, suggesting complicated lineage-specific effects of RAP1 in adult stem cells. Altogether, these results demonstrate for the first time that RAP1 plays both telomeric and nontelomeric roles in regulating human stem cell homeostasis.

Redundancy principle and the role of extreme statistics in molecular and cellular biology

The paradigm of chemical activation rates in cellular biology has been shifted from the mean arrival time of a single particle to the mean of the first among many particles to arrive at a small activation site. The activation rate is set by extremely rare events, which have drastically different time scales from the mean times between activations, and depends on different structural parameters. This shift calls for reconsideration of physical processes used in deterministic and stochastic modeling of chemical reactions that are based on the traditional forward rate, especially for fast activation processes in living cells. Consequently, the biological activation time is not necessarily exponentially distributed. We review here the physical models, the mathematical analysis and the new paradigm of setting the scale to be the shortest time for activation that clarifies the role of population redundancy in selecting and accelerating transient cellular search processes. We provide examples in cellular transduction, gene activation, cell senescence activation or spermatozoa selection during fertilization, where the rate depends on numbers. We conclude that the statistics of the minimal time to activation set kinetic laws in biology, which can be very different from the ones associated to average times.

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.

Aging and computational systems biology

Aging research is undergoing a paradigm shift, which has led to new and innovative methods of exploring this complex phenomenon. The systems biology approach endeavors to understand biological systems in a holistic manner, by taking account of intrinsic interactions, while also attempting to account for the impact of external inputs, such as diet. A key technique employed in systems biology is computational modeling, which involves mathematically describing and simulating the dynamics of biological systems. Although a large number of computational models have been developed in recent years, these models have focused on various discrete components of the aging process, and to date no model has succeeded in completely representing the full scope of aging. Combining existing models or developing new models may help to address this need and in so doing could help achieve an improved understanding of the intrinsic mechanisms which underpin aging.

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.

Computing the length of the shortest telomere in the nucleus

The telomere length can either be shortened or elongated by an enzyme called telomerase after each cell division. Interestingly, the shortest telomere is involved in controlling the ability of a cell to divide. Yet, its dynamics remains elusive. We present here a stochastic approach where we model this dynamics using a Markov jump process. We solve the forward Fokker-Planck equation to obtain the steady state distribution and the statistical moments of telomere lengths. We focus specifically on the shortest one and we estimate its length difference with the second shortest telomere. After extracting key parameters such as elongation and shortening dynamics from experimental data, we compute the length of telomeres in yeast and obtain as a possible prediction the minimum concentration of telomerase required to ensure a proper cell division.

The length of the shortest telomere as the major determinant of the onset of replicative senescence

The absence of telomerase in many eukaryotes leads to the gradual shortening of telomeres, causing replicative senescence. In humans, this proliferation barrier constitutes a tumor suppressor mechanism and may be involved in cellular aging. Yet the heterogeneity of the senescence phenotype has hindered the understanding of its onset. Here we investigated the regulation of telomere length and its control of senescence heterogeneity. Because the length of the shortest telomeres can potentially regulate cell fate, we focus on their dynamics in Saccharomyces cerevisiae. We developed a stochastic model of telomere dynamics built on the protein-counting model, where an increasing number of protein-bound telomeric repeats shift telomeres into a nonextendable state by telomerase. Using numerical simulations, we found that the length of the shortest telomere is well separated from the length of the others, suggesting a prominent role in triggering senescence. We evaluated this possibility using classical genetic analyses of tetrads, combined with a quantitative and sensitive assay for senescence. In contrast to mitosis of telomerase-negative cells, which produces two cells with identical senescence onset, meiosis is able to segregate a determinant of senescence onset among the telomerase-negative spores. The frequency of such segregation is in accordance with this determinant being the length of the shortest telomere. Taken together, our results substantiate the length of the shortest telomere as being the key genetic marker determining senescence onset in S. cerevisiae.

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.

Systems biology of ageing and longevity

Ageing is intrinsically complex, being driven by multiple causal mechanisms. Each mechanism tends to be partially supported by data indicating that it has a role in the overall cellular and molecular pathways underlying the ageing process. However, the magnitude of this role is usually modest. The systems biology approach combines (i) data-driven modelling, often using the large volumes of data generated by functional genomics technologies, and (ii) hypothesis-driven experimental studies to investigate causal pathways and identify their parameter values in an unusually quantitative manner, which enables the contributions of individual mechanisms and their interactions to be better understood, and allows for the design of experiments explicitly to test the complex predictions arising from such models. A clear example of the success of the systems biology approach in unravelling the complexity of ageing can be seen in recent studies on cell replicative senescence, revealing interactions between mitochondrial dysfunction, telomere erosion and DNA damage. An important challenge also exists in connecting the network of (random) damage-driven proximate mechanisms of ageing with the higher level (genetically specified) signalling pathways that influence longevity. This connection is informed by actions of natural selection on the determinants of ageing and longevity.

gamma-Tocotrienol prevents oxidative stress-induced telomere shortening in human fibroblasts derived from different aged individuals

The effects of palm gamma-tocotrienol (GGT) on oxidative stress-induced cellular ageing was investigated in normal human skin fibroblast cell lines derived from different age groups; young (21-year-old, YF), middle (40-year-old, MF) and old (68-year-old, OF). Fibroblast cells were treated with gamma-tocotrienol for 24 hours before or after incubation with IC50 dose of H2O2 for 2 hours. Changes in cell viability, telomere length and telomerase activity were assessed using the MTS assay (Promega, USA), Southern blot analysis and telomere repeat amplification protocol respectively. Results showed that treatment with different concentrations of gamma-tocotrienol increased fibroblasts viability with optimum dose of 80 microM for YF and 40 microM for both MF and OF. At higher concentrations, gamma-tocotrienol treatment caused marked decrease in cell viability with IC50 value of 200 microM (YF), 300 microM (MF) and 100 microM (OF). Exposure to H2O2 decreased cell viability in dose dependent manner, shortened telomere length and reduced telomerase activity in all age groups. The IC50 of H2O2 was found to be; YF (700 microM), MF (400 microM) and OF (100 microM). Results showed that viability increased significantly (p < 0.05) when cells were treated with 80 microM and 40 microM gamma-tocotrienol prior or after H2O2-induced oxidative stress in all age groups. In YF and OF, pretreatment with gamma-tocotrienol prevented shortening of telomere length and reduction in telomerase activity. In MF, telomerase activity increased while no changes in telomere length was observed. However, post-treatment of gamma-tocotrienol did not exert any significant effects on telomere length and telomerase activity. Thus, these data suggest that gamma-tocotrienol protects against oxidative stress-induced cellular ageing by modulating the telomere length possibly via telomerase.

Quantitative theory of telomere length regulation and cellular senescence

In normal somatic cells, telomere length shortens with each cell replication. This progressive shortening is associated with cellular senescence and apoptosis. Germ cells, stem cells, and the majority of cancer cells express telomerase, an enzyme that extends telomere length and, when expressed at sufficient levels, can immortalize or extend the life span of a cell line. It is believed that telomeres switch between two states: capped and uncapped. The telomere state determines its accessibility to telomerase and also the onset of senescence. One hypothesis is that the t loop, a large lariat-like structure, represents the capped state. In this paper we model a telomere state on the basis of the biophysics of t-loop formation, allowing us to develop a single mathematical model that accounts for two processes: telomere length regulation for telomerase positive cells and cellular senescence in somatic cells. The model predicts the steady-state length distribution for telomerase positive cells, describes the time evolution of telomere length, and computes the life span of a cell line on the basis of the levels of TRF2 and telomerase expression. The model reproduces a wide range of experimental behavior and fits data from immortal cell lines (HeLa S3 and 293T) and somatic cells (human diploid fibroblasts) well. We conclude that the t loop as the capped state is a quantitatively reasonable model of telomere length regulation and cellular senescence.

Immortalization of human precartilaginous stem cells by transfecting SV40Tag

Immortalized human precartilaginous stem cells (IPSCs) were established to provide stable cell resource for the study of the molecular mechanism of gene targeting on the differentiation of PSCs. Plasmid pCMVSV40T/PUR containing simian virus 40 large T antigen gene (SV40Tag) was transfected into human PSCs by using lipofectin transfection. Colonies were isolated by puromycin selection and expanded by multiple passages. Immunohistochemistry, RT-PCR and Southern blotting were used to identify the transfected cells and to detect the expression and integration of SV40Tag in expanded cell lines. The positive colonies were isolated and subcultured, designated immortalized precartilaginous stem cells (IPSCs), which were confirmed as fibroblast growth factor receptor-3 (FGFR-3) positive cells by immunohistochemistry and RT-PCR. SV40Tag cDNA was found in cultured IPSCs of passage 8 by Southern blotting, and the expressions of SV40Tag mRNA and protein were confirmed by RT-PCR. These findings suggested that IPSCs strain with SV40Tag was constructed successfully.

Mitochondrial dysfunction accounts for the stochastic heterogeneity in telomere-dependent senescence

Aging is an inherently stochastic process, and its hallmark is heterogeneity between organisms, cell types, and clonal populations, even in identical environments. The replicative lifespan of primary human cells is telomere dependent; however, its heterogeneity is not understood. We show that mitochondrial superoxide production increases with replicative age in human fibroblasts despite an adaptive UCP-2–dependent mitochondrial uncoupling. This mitochondrial dysfunction is accompanied by compromised [Ca²⁺]_i homeostasis and other indicators of a retrograde response in senescent cells. Replicative senescence of human fibroblasts is delayed by mild mitochondrial uncoupling. Uncoupling reduces mitochondrial superoxide generation, slows down telomere shortening, and delays formation of telomeric γ-H2A.X foci. This indicates mitochondrial production of reactive oxygen species (ROS) as one of the causes of replicative senescence. By sorting early senescent (SES) cells from young proliferating fibroblast cultures, we show that SES cells have higher ROS levels, dysfunctional mitochondria, shorter telomeres, and telomeric γ-H2A.X foci. We propose that mitochondrial ROS is a major determinant of telomere-dependent senescence at the single-cell level that is responsible for cell-to-cell variation in replicative lifespan.

After a limited number of cell divisions, somatic cells lose the capacity for proliferation, called cellular replicative senescence. Senescence, which is triggered by the loss of DNA sequences at the ends of chromosomes (telomeres), is often seen as an example of a regular “biological clock.” However, cell senescence is heterogeneous, with large differences in lifespan between individual cell lineages. This heterogeneity is clearly related to stress, specifically oxidative stress. It was not known, however, whether stress-induced “premature” senescence involves telomeres or is caused by telomere-independent DNA damage responses. Mitochondria are the most important source of reactive oxygen species (ROS) in cells under physiological conditions. We found that mitochondrial function deteriorated while cells approached senescence, leading to increased ROS production. Delaying mitochondrial dysfunction led to postponed replicative senescence and slowing of telomere shortening. Prematurely senescing cells sorted out of young cultures displayed mitochondrial dysfunction, increased oxidative stress, and short telomeres. We propose that replicative telomere-dependent senescence is not “clocked,” but rather is a stochastic process triggered largely by random mitochondrial dysfunction.

Mitochondrial uncoupling is used to study the relationship between telomere length, the production of reactive oxygen species, and replicative senescence.

The role of intracellular peroxide levels on the development and maintenance of telomere-dependent senescence

The fact that reactive oxygen species (ROS) influences telomere-shortening and with it the onset of senescence in cells is well known, but a detailed model describing this correlation has not been proposed so far. Based on experimental data, that span a wide range of intracellular peroxide levels we formulate here a deterministic equation and a stochastic model that describe this connection, taking into account biological functions such as DNA damage and repair. Through simulations of population development under oxidative stress and dynamics of telomere length distributions, we show that a subset of uncapped telomeres is required for cell cycle arrest. Our model also supports a possible mechanism by which the generation of ROS as a consequence of telomere dysfunction leads to a positive feedback that accelerates telomere erosion. In this model, telomere-state and ROS would mutually influence each other.

Immortalization of human umbilical vein endothelial cells with telomerase reverse transcriptase and simian virus 40 large T antigen

OBJECTIVE

To establish normally conditionally-immortalized human umbilical vein endothelial cells (HUVECs) by ectopic expression of the human telomerase catalytic enzyme (hTERT) and simian virus 40 large T (SV40 LT) antigen.

METHODS

Primary HUVECs were transfected with recombinant retrovirus containing hTERT or SV40 LT respectively. Subsequently drug resistant cell clones were screened and expanded for further studies. Endothelial cell biomarkers were confirmed by examination.

RESULTS

The morphological phenotype of the transfected cells was similar to the non-transfected cells. Von Willebrand factor, hTERT and SV40 LT could be detected in transfected HUVECs. Moreover, higher telomerase activity in transfected cells was maintained for over 50 population doublings compared with only low level of endogenous telomerase transiently at early population doublings in primary HUVECs. When exposed to TNF-alpha (tumor necrosis factor-alpha), the expression of E-selectin in transfected cells was significantly up-regulated, but no alteration of endothelial lipase was found.

CONCLUSION

Ectopic coexpression of hTERT and SV40 LT can effectively immortalize HUVECs without tumorigenicity in vitro. Immortalized HUVECs may be an ideal target of further molecular function studies.

Telomerase expression is sufficient for chromosomal integrity in cells lacking p53 dependent G1 checkpoint function

Background

Secondary cultures of human fibroblasts display a finite lifespan ending at senescence. Loss of p53 function by mutation or viral oncogene expression bypasses senescence, allowing cell division to continue for an additional 10 – 20 doublings. During this time chromosomal aberrations seen in mitotic cells increase while DNA damage and decatenation checkpoint functions in G₂ cells decrease.

Methods

To explore this complex interplay between chromosomal instability and checkpoint dysfunction, human fibroblast lines were derived that expressed HPV16E6 oncoprotein or dominant-negative alleles of p53 (A143V and H179Q) with or without the catalytic subunit of telomerase.

Results

Cells with normal p53 function displayed 86 – 93% G₁ arrest after exposure to 1.5 Gy ionizing radiation (IR). Expression of HPV16E6 or p53-H179Q severely attenuated G₁ checkpoint function (3 – 20% arrest) while p53-A143V expression induced intermediate attenuation (55 – 57% arrest) irrespective of telomerase expression. All cell lines, regardless of telomerase expression or p53 status, exhibited a normal DNA damage G₂ checkpoint response following exposure to 1.5 Gy IR prior to the senescence checkpoint. As telomerase-negative cells bypassed senescence, the frequencies of chromosomal aberrations increased generally congruent with attenuation of G₂ checkpoint function. Telomerase expression allowed cells with defective p53 function to grow >175 doublings without chromosomal aberrations or attenuation of G₂ checkpoint function.

Conclusion

Thus, chromosomal instability in cells with defective p53 function appears to depend upon telomere erosion not loss of the DNA damage induced G₁ checkpoint.

Replicative senescence in sheep fibroblasts is a p53 dependent process

Studies on telomere and telomerase biology are fundamental to the understanding of human ageing, and age-related diseases such as cancer. However, human studies are hampered by the lack of fully reflective animal model systems. Here we describe basic studies of telomere length and telomerase activity in sheep tissues and cells. Terminal restriction fragment lengths from sheep tissues ranged from 9 to 23 kb, with telomerase activity present in testis but suppressed in somatic tissues. Sheep fibroblasts had a finite lifespan in culture, after which the cells entered senescence. During in vitro growth the mean terminal restriction fragment lengths decreased in size at a rate of 210 and 350 bp per population doubling (PD). Senescent skin fibroblasts had increased levels of p53 and p21WAF1 compared to young cells. Incubation of senescent cells with siRNA duplexes specific for p53 suppressed p53 expression and allowed the cells to re-enter the cell cycle. Five PDs beyond senescence the siRNA-treated cells reached a second proliferative barrier. This study shows that telomere biology in sheep is similar to that in humans, with senescence in sheep GM03550 fibroblasts being a telomere-driven, p53-(p21WAF1)-dependent process. Therefore sheep may represent an alternative model system for studying telomere biology, replicative senescence, and by implication human ageing.

Lymphocyte telomere dynamics and telomerase activity in inflammatory bowel disease: effect of drugs and smoking

BACKGROUND

The chromosome instability observed in peripheral blood lymphocytes in ulcerative colitis could be a biomarker of cancer susceptibility.

AIM

To determine whether accelerated telomere shortening could explain chromosome instability and assess the effect of drugs and smoking on telomere dynamics in these cells.

METHODS

Peripheral blood lymphocytes were isolated from ulcerative colitis, Crohn's disease and non-inflammatory bowel disease control patients. Telomere lengths were measured by quantitative real-time polymerase chain reaction. After activation and cell separation, telomerase activity and human telomerase reverse transcriptase messenger ribonucleic acid were measured by telomerase repeat amplification protocol enzyme-linked immunosorbent serological assay and quantitative real-time polymerase chain reaction, respectively.

RESULTS

Age-related telomere loss in peripheral blood lymphocytes was similar in ulcerative colitis, Crohn's disease and control patients. Telomerase activity decreased with age in all groups and correlated positively with telomere length (r = 0.489, P = 0.006). Among Crohn's disease patients, azathioprine was associated with decreased telomerase activity (0.66 vs. 1.54, P = 0.026, P < 0.05) and smoking was associated with decreased human telomerase reverse transcriptase mRNA expression (10.5 vs. 33.3, P = 0.036, P < 0.05).

CONCLUSIONS

Telomere shortening is not accelerated and therefore cannot be the cause of the chromosome instability observed in ulcerative colitis peripheral blood lymphocytes. Azathioprine and cigarette smoking modify telomerase expression in these cells.

Mathematical modeling confirms the length-dependency of telomere shortening

Telomeres, the ends of chromosomes, shorten with each cell division in human somatic cells, because of the end-replication problem, C-strand processing and oxidative damage. On the other hand, the reverse transcriptase telomerase can add back telomeric repeats at the telomere ends. It has been suggested that once telomeres have reached a critical length, cells cease proliferation, also known as senescence. Evidence is accumulating that telomere shortening and subsequent senescence might play a crucial role in life-threatening diseases. So far, mathematical models described telomere shortening as an autonomous process, where the loss per cell division does not depend on the telomere length itself. In this study, published measurements of telomere distributions in human fibroblasts and human endothelial cells were used to show that telomeres shorten in a length-dependent fashion. Thereafter, a mathematical model of telomere attrition was composed, in which a shortening factor and an autonomous loss were incorporated. It was assumed that the percentage of senescence was related to the percentage of telomeres below a critical length. The model was compared with published data of telomere length and senescence of human endothelial cells using the maximum likelihood method. This enabled the estimation of physiologically important parameters and confirmed the length-dependency of telomere shortening.