Mathematical modeling confirms the length-dependency of telomere shortening

Mechanisms of Ageing and Development 125 (2004) 437–444

Mathematical modeling confirms the length-dependency of telomere shortening Jorn op den Buijs a , Paul P.J. van den Bosch a,b , Mark W.J.M. Musters a , Natal A.W. van Riel a,b,∗ a

Department of Biomedical Engineering, Eindhoven University of Technology EH. 4.26, P.O. Box 513, 5600 MB Eindhoven, The Netherlands b Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands Received 13 November 2003; received in revised form 3 March 2004; accepted 12 March 2004 Available online 23 April 2004

Abstract 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. © 2004 Elsevier Ireland Ltd. All rights reserved. Keywords: Ageing; Computer simulation; Endothelial cells; Feedback; Fibroblasts; Telomere

1. Introduction Telomeres consist of tandem repeats of the nucleotide sequence TTAGGG at the ends of eukaryotic chromosomes. During each cell division, telomeres are shortened. As a consequence, the length of telomeric DNA inversely correlates with the number of population doublings (PDs) reached by cells in culture, such as human diploid fibroblasts (Allsopp et al., 1992; Lansdorp et al., 1996; Wright et al., 1997; Martens et al., 2000) and human umbilical vein endothelial cells (HUVEC) (Chang and Harley, 1995; Zhang et al., 2000). It is believed that one or more critically short telomeres fail to protect the cell from entering a state of growth arrest or senescence (Hemann et al., 2001; Karlseder et al., 2002). Replicative senescence induced by telomere erosion ∗ Corresponding author. Tel.: +31-40-247-5506; fax: +31-40-243-4582. E-mail address: [email protected] (N.A.W. van Riel).

might play a crucial role in life-threatening diseases such as atherosclerosis (Samani et al., 2001; Minamino et al., 2002) and heart failure (Oh et al., 2003). Several mechanisms have been proposed to explain telomere shortening. The first theory was based on the assumption that DNA polymerase is not able to fully replicate the genome at the telomeric endings: the so-called end-replication problem (Olovnikov, 1973). However, the contribution of the end-replication problem to telomere shortening could be much smaller than the generally observed telomeric attrition of 50–150 bp per PD. More recently, other causes of telomeric DNA loss have been revealed. von Zglinicki et al. (2000) showed that accumulation of single strand breaks induced by oxidative stress is probably the most important determinant of telomere shortening in cell cultures. Indeed, the rate of telomere shortening was elevated by exposure of cells to oxidative stress in many studies, while some investigations also showed a decline when anti-oxidants were applied (von Zglinicki, 2002). In

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addition, Huffman et al. (2000) demonstrated that the length of G-rich overhangs at the telomeres correlates with the rate of telomere shortening and ascribed this to a C-strand specific exonuclease. This exonuclease might be recruited by the telomere repeat binding factor 2 (TRF2) to form telomeric loops (t-loops) at the telomere ends (Karlseder, 2003). These t-loops are likely to prevent the initiation of a DNA damage response on one hand (Li et al., 2003), but might hamper elongation of the telomere by the enzyme telomerase on the other (Smogorzewska et al., 2000). A wide variety of mathematical models describing telomere shortening exists in the literature. Levy et al. (1992) and Arino et al. (1995) modeled the end-replication problem as a deterministic process, whereas Tan (1999) and Sozou and Kirkwood (2001) proposed stochastic models, in line with the wide variability of PDs reached by clonally-derived cell cultures. Besides the end-replication problem, Proctor and Kirkwood (2002) also incorporated telomere length reduction as a consequence of C-strand processing and single strand breaks in their mathematical description and they suggested a pivotal role for oxidative stress. Although above-mentioned mathematical models differ in type and complexity, they all describe the loss of telomeric repeats as a rather autonomous process, i.e. independent of the telomere length itself. A constant telomere loss seems to be compatible with measurements of the mean telomere length during cumulative PDs, however, the variance in these data does not exclude a mechanism with negative feedback where the telomere loss is dependent on the telomere length. In fact, there is evidence that telomere length regulation is a homeostatic process, involving telomerase and two related TTAGGG repeat binding factors, TRF1 and TRF2 (Smogorzewska et al., 2000). Telomerase adds back telomeric repeats at the telomere ends, while TRF1 and TRF2 bind along the telomere and are therefore assumed to measure telomere length. Previously published mathematical models of telomere shortening have assumed the absence of telomerase, since in most non-proliferating human somatic cells telomerase activity is undetectable. However, substantial telomerase activity has been detected in proliferating human epithelial cells (Belair et al., 1997), human endothelial cells (Vasa et al., 2000) and human fibroblasts (Masutomi et al., 2003) in vitro and fibroblasts and endothelial cells in human skin samples (Osanai et al., 2002). Furthermore, the positive skewness as observed in the telomere length distributions of cultured human fibroblasts (Lansdorp et al., 1996; Martens et al., 2000) is also in accordance with negative feedback regulation of telomere length. This suggests that a mathematical model of the erosion of telomeres needs a feedback loop to match experimental data more accurately than models using a constant loss, or a loss drawn from a statistical distribution that is independent of the telomere length. In this study, we modeled telomere length distributions using: (i) a normal distribution, which results from a constant telomere loss (Levy et al., 1992); (ii) a lognormal dis-

tribution, empirically used to describe fragmentation of particles (Brown and Wohletz, 1995); and (iii) a Weibull distribution, which assumes negative feedback regulation of telomere length (Oexle, 1998). These functions were compared to telomere length distributions measured by fluorescence in situ hybridization (FISH) in human fibroblasts (Martens et al., 2000) and by Southern blotting in human umbilical vein endothelial cells, reproduced from Zhang et al. (2000). The Weibull distribution resulted in the best fit, which suggests that telomeres are shortened proportional to their length. Therefore, we developed a relatively simple model of telomere shortening and simulated three cases. In the first case only a constant loss was integrated, the second case assumed constant relative shortening, and the third case was a combination of feedback and constant loss. In the models, the percentage of senescence was related to the percentage of telomeres below a certain critical telomere length. Published data of telomere length and senescence in endothelial cell cultures were mimicked and maximum likelihood estimation (MLE) was used to quantify the parameters of our models of telomere shortening. From these results it can be concluded that telomere erosion could occur in a length-dependent fashion.

2. Materials and methods 2.1. Normal distribution Levy et al. (1992) modeled a constant telomere loss, which resulted in a binomial distribution. The normal distribution approximates the binomial distribution with a large number of trials. Let T [kbp] be the telomere length, µ [kbp] the mean telomere length and s [kbp] the standard deviation. The normal distribution n(T) [kbp−1 ] becomes: 1 2 2 n(T) = √ e((−(T −µ) )/2σ ) σ 2π

(1)

2.2. Lognormal distribution According to Oexle (1998), distributions of telomere length in human diploid fibroblasts are lognormal, presumably as a result of breakage and recombination of telomeric DNA by oxidative stress damage. The lognormal telomere length distribution can be denoted as: 2 1 2 n(T) = √ e(−ln (T/b)/2a ) a 2πT

(2)

Here a and b [kbp] are positive constants. 2.3. Weibull distribution When assuming negative feedback regulation of telomere length, the length distribution can be derived as follows. Let N(T) be the fraction of telomeres with a length smaller than

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T; the fraction of remaining telomeres will be 1 − N(T). The change in 1−N(T) as a function of T depends on the fraction itself and, as a consequence of negative feedback, also on T:

Table 1 Summary of the simulated cases Case

Constraints

Model equation

Initial distribution

d(1 − N(T)) = (1 − N(T))aTb dT

1 2 3

fs = 0

T0 = 0 –

T = − T0

T = −(1 − fs )T

T = −{(1 − fs )T + T0 }

Normal Weibull Weibull

(3)

Again, a [kbp−(b +1) ] and b are positive constants. The Weibull distribution n(T) can be derived from above differential equation via d ln(1 − N(T)) = −aTb dT : n(T) =

dN(T) b+1 = aTb e−(a/(b+1))T dT

(4)

2.4. Comparison of length distributions with experimental data We used MLE to fit the normal, lognormal and the Weibull distributions to FISH measurements of telomeres in human fibroblasts (Martens et al., 2000) and Southern blots of telomeric restriction fragments (TRFs) in human endothelial cells (Zhang et al., 2000). In the case of Southern blotting, the distributions were obtained by normalizing the optical density by the molecular weight. Furthermore, TRFs contain a subtelomeric region of about 3 kbp (Hultdin et al., 1998). This was taken into account by introducing a subtelomeric region Ts [kbp]: T ∗ = T − Ts

(5)

We replaced T by T∗ in the three telomere length distributions in (1), (2) and (4) and estimated Ts using MLE in the case of Southern blotting. For the FISH measurements holds that there is no subtelomeric region measured, and we therefore set Ts to zero. 2.5. Comparison of models to experimental data We composed a relatively simple model of the regulation and shortening of telomere length. Simulations of three cases of this model were compared with the mean TRF length and the percentage of ␤-galactosidase (␤-gal) staining as measured in endothelial cells by Zhang et al. (2000). The level of ␤-gal staining was assumed to resemble the percentage of senescent cells. For the sake of parameter estimation, no separate mechanisms of telomere shortening were distinguished; instead, we proposed a single equation in which these processes were lumped:

T = −{(1 − fs )T + T0 }

(6)

Here T [bp PD−1 ] is the telomere loss per PD. The telomere length T was regulated using shortening factor fs [PD−1 ]. A constant loss T0 [bp PD−1 ] was introduced, to take autonomous processes into account, such as the end-replication problem. Because telomere ends contain single stranded overhangs, varying in length from a few to a few hundred nucleotides, it should be mentioned that the mean telomere length of a double stranded telomere end was modeled here.

When a fraction vs of the cells becomes senescent, the rate of telomere shortening per unit of time will depend on the fraction of remaining proliferating cells (1 − vs ). These (1 − vs ) cells will have to divide vs more times to reach the next PD. Hence, the rate of telomere shortening per PD will remain the same: 
vs

T(1 − vs ) 1 + = T (7) 1 − vs Three cases of the model were simulated (see Table 1). In the first case, the loss is completely autonomous: fs = 0. The second case assumed total negative feedback: T0 = 0. The third case simulated a combination of constant loss and negative feedback. Since data on telomere length and percentage senescence of Zhang et al. (2000) are available from PD 11, the simulations were started at this PD. For the first case with constant loss only, the normal distribution was taken as initial condition, as obtained after fitting it to the telomere length distribution at PD 11. For the other two cases, in which negative feedback was incorporated, we used the Weibull function as initial condition. Note that we corrected for the subtelomeric region estimated by MLE. During progressive PDs telomeres were shortened with T. The mean telomere length T [kbp] was determined as follows:  T = n(T)T dT (8) In the model, it was assumed that the telomere lengths were homogenously distributed over all cells in a culture. We modeled the presence of a small number (ns ) of so-called proliferation-restricting telomeres (PRTs; Tan, 1999). Senescence was triggered when these PRTs shortened below a critical telomere length Tc [kbp]. We modeled two hypotheses: 1. When all ns PRTs shorten below Tc , the cell enters the senescent state. 2. When at least one out of ns PRTs becomes too short, senescence is triggered. Let N(Tc ) be the fraction of telomeres below Tc :  Tc N(Tc ) = n(T) dT

(9)

0

When drawing one telomere out of the distribution, the probability that a telomere is shorter than Tc equals N(Tc ). For the first hypothesis, the fraction of senescent cells vs is given by: vs = {N(Tc )}ns

(10)

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For the second hypothesis holds: vs = 1 − {1 − N(Tc )}ns

(11)

Note that both hypotheses are equal in the case ns is one. If ns is much smaller than the total number of 92 telomeres in a cell, the telomere length distribution in non-cycling cells approximates that in cycling cells. In that case, the simulated distributions were not corrected for the fact that senescent cells do not shorten their telomeres any further. We did not constrain our model to estimate an exact number of PRTs, since the induction of senescence might be a stochastic process. It could result from the disruption of a single t-loop (Li et al., 2003), or from two or more short telomeres and subsequent chromosome fusions (Hemann et al., 2001). Using MLE, the percentage of senescence and the mean telomere length, as determined by the model, were fitted to experimental data. We used the estimated subtelomeric region Ts to subtract it from the experimental TRF length measurements and obtain the telomere length. This made the estimation of the most likely values for fs , T0 , Tc and ns feasible. The cost function fc to be minimized by MLE was a weighted nonlinear least squares error criterion. It was based on the difference between the model predictions of the mean telomere length (Tt ) and fraction of senescent cells (vs ) and the ‘true’ experimentally obtained Tt and vs,t . The errors between model and data were squared and summed over the number of data points for the telomere measurement (NT ) and percentage of senescence (Ns ). The squared errors were weighted by NT , Ns and the squared maximum of the data: N

fc =

T  1 (T − Tt )2 NT max2 (Tt ) 1

N

+

s  1 (vs − vs,t )2 Ns max2 (vs,t )

(12)

1

The variances of the most optimal model parameters were determined using the value of the cost function fˆ c and the ˆ of the cost function with respect inverse of the Hessian H to the parameters in the optimum. The variances of the estimated parameters are located on the diagonal of the matrix Vˆ : ˆ −1 fˆ c H Vˆ = (13) (NT + Ns ) 2.6. Technical information The simulations and parameter estimations were carried out in MATLAB 6.5 (The Mathworks Inc.), running under Microsoft Windows 2000 on a 1.0 GHz IBM compatible PC with 512 MB RAM. For simulating the models, a non-stiff differential equation solver, medium order method, was employed. We used MATLAB’s Optimization Toolbox 2.2 for MLE. The Levenberg-Marquardt algorithm was applied

for fitting the statistical distributions and the Nelder-Mead method for fitting the models. Parameters were estimated with non-negative constraints.

3. Results 3.1. Telomere length distributions We fitted the normal, lognormal and the Weibull distribution to measurements of telomere length in fibroblasts using FISH (Martens et al., 2000) and in endothelial cells using Southern blotting (Zhang et al., 2000). In fibroblasts, the Weibull and normal distributions are clearly superior to the lognormal distribution (Table 2). When comparing the normal and Weibull functions, the latter results in a relatively better fit upon increased number of PDs. Also in HUVECs, the Weibull distribution describes the experimental data most accurately. Fig. 1 shows the most optimal fit of the Weibull distribution in both cell types. In the Southern blot measurements, the subtelomeric region Ts was estimated to be 4.7±0.03 kbp for the normal distribution and 3.7±0.03 kbp for the Weibull distribution. Fitting the lognormal distribution failed in estimation of the subtelomeric region. Hultdin et al. (1998) estimated a mean value for the subtelomeric region of 3.2 kbp by comparing FISH measurements and Southern blotting in ten different cell strains. 3.2. Models of telomere shortening We simulated the three model cases for both hypotheses of senescence and fitted the model output to the data from Zhang et al. (2000). The obtained values for fs , T0 , Tc and ns , as well as the relative remaining values of the cost function fc in the optimum are summarized in Table 3. Model case 3 most accurately describes the experimental data, when assuming that all PRTs should be shorter than Tc to trigger senescence (see Fig. 2). For model case 1 with autonomous loss only, we found rates of telomere attrition of 110 and 112 bp PD−1 . Other values for the rate of telomere loss found in HUVECs are 101 bp PD−1 (Huffman et al., 2000), 110 bp PD−1 (Xu Table 2 Relation between remaining squared errors after model fits Cell type

PD

Normal

Lognormal

Weibull

Fibroblasts (FISH)

10 33 49

1.0 1.3 1.9

3.6 2.7 1.9

1.0 1.0 1.0

HUVECs (Southern blotting)

11 25 38

1.1 1.9 1.1

1.8 1.4 1.7

1.0 1.0 1.0

Errors were normalized to the remaining error in the case of the Weibull distribution.

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Fig. 1. Weibull distribution fitted on telomere length measurements in (A) human diploid fibroblasts (Martens et al., 2000) and (B) HUVECs (Zhang et al., 2000). Bars in (A) and solid line in (B) represent experimental data; dashed lines represent the fitted distribution.

et al., 2000), 160–170 bp PD−1 (Furumoto et al., 1998) and 190 bp PD−1 (Chang and Harley, 1995). Case 2 resulted in estimated values for the shortening factor of 0.975 and 0.978 PD−1 under the two different hypotheses. Martens et al. (2000) determined this parameter in human fibroblasts and found a value of 0.976 PD−1 . We computed optimal values for ns of 0.73–3.60. Thus, our models predicted that only a small number out of the 92 telomeres in a cell is responsible for triggering the senescent state. Hemann et al. (2001) showed that in mice, end-to-end fusions of chromosomes preferentially occur at a few specific chromosomes with the shortest telomeres. Computer simulations by Tan (1999) suggested that there are exactly 2 PRTs. Our most optimal model case estimated a mean value of 1.22 PRTs. This implies that in most cells only one critically short telomere suffices to trigger senescence, while in some cells more than one short telomere is required. Under the hypothesis that one out of ns telomeres suffices to trigger senescence, the model with a constant loss failed in

Fig. 2. Comparison of the third model case under the first hypothesis to telomere length and senescence measurements as reproduced from Zhang et al. (2000).

Table 3 Summary of estimated parameters Parameter

Estimated value Case 1

Unit

Description

Case 2

Case 3

All of ns PRTs are shorter than Tc fs – 110 ± 2

T0 2.45 ± 0.12 Tc ns 2.41 ± 0.10 – 1.11

0.975 ± 0.001 – 1.42 ± 0.05 0.73 ± 0.03 1.28

0.983 ± 0.000 21 ± 1 2.14 ± 0.05 1.22 ± 0.05 1.00

PD−1 bp PD−1 kbp – –

Shortening factor Autonomous telomere loss Critical telomere length Number of critical telomeres Relative remaining error

At least one of ns PRTs is shorter than Tc fs – 112 ± 3

T0 0.00 ± 8 Tc ns 0.93 ± 0.06 – 1.33

0.978 ± 0.000 – 1.22 ± 0.03 3.60 ± 0.20 1.02

0.978 ± 0.000 2±1 1.65 ± 0.04 1.58 ± 0.08 1.02

PD−1 bp PD−1 kbp – –

Shortening factor Autonomous telomere loss Critical telomere length Number of critical telomeres Relative remaining error

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Fig. 3. Telomere length distributions at different population doublings (PD) as produced by model case 3, under the first hypothesis of senescence.

determination of a most likely critical telomere length. For the other cases, critical telomere lengths were in the range of 1.22–2.45 kbp. Martens et al. (2000) suggested that this value is in the range of 1–2 kbp for human fibroblasts. Simulating combined feedback and autonomous loss resulted in a constant attrition of 21 bp PD−1 under hypothesis 1. Strikingly, von Zglinicki et al. (2000) were able to reduce the rate of telomere shortening in fibroblasts to about 30 bp PD−1 under antioxidative protection. The small constant loss of about 2 bp PD−1 found under hypothesis 2 might represent the length reduction of a few base pairs due to the end-replication problem. The telomere length distributions resulting from the most optimal model (case 3, hypothesis 1) are shown during three different PD levels in Fig. 3. Although we based our model on data of HUVECs, notice the resemblance of the forms of these distributions with those measured in fibroblasts using FISH (Fig. 1A; Martens et al., 2000).

4. Discussion In this paper, we have first shown that telomere length distributions as measured in human somatic cells could be accurately described by the Weibull distribution, which implies feedback regulation of telomere length. Thereafter, we proposed a relatively simple negative feedback model of telomere reduction. Three model cases were simulated and fit to measurements of mean telomere length and degree of senescence in human endothelial cells (Zhang et al., 2000). This resulted in the estimation of relevant parameters of telomere shortening, namely the autonomous loss, the shortening factor and the critical telomere length. Based on these findings, we suggest that telomere shortening in human somatic cells is dependent on telomere length itself. In the future, detailed mathematical models of telomeric loss should therefore consider the incorporation of one or more feedback loops.

Oexle (1998) compared several statistical distributions to telomere length distributions as measured by FISH (Lansdorp et al., 1996). Based on qualitative observations, it was concluded that telomere length distributions are lognormal. Here, we oppose this conclusion by fitting the normal, lognormal and Weibull distributions to more recent experimental data, measured in fibroblasts using FISH (Martens et al., 2000) and in endothelial cells using Southern blotting (Zhang et al., 2000). We quantitatively demonstrated that the Weibull distribution yields the most accurate fit. Our model assumed that senescence was induced by critical telomere shortening solely: telomere-independent activation of senescence mechanisms (Russo et al., 1998), and somatic mutations (Martin et al., 1996) were neglected. The percentage of senescence in a cell culture was calculated based on the fraction of telomeres below a certain critical telomere length. We tested two hypotheses for senescence: (i) cells stop dividing when a certain number of telomeres shorten below the critical length; and (ii) senescence is triggered by the critically shortening of at least one out of a few PRTs. The model with combined constant loss and regulation resulted in agreeable fits and estimated parameters values under both hypotheses. Hence, none of these mechanisms can be excluded. However, our model predicted for both hypotheses that only a small number of telomeres is involved in inducing senescence. Although we were able to satisfactorily fit all models to the data, we speculate that a model combining negative feedback and a small autonomous loss of telomeric DNA, most accurately describes the telomere loss in human somatic cells. It is likely that some of the processes involved in telomere attrition, such as incomplete replication and random primer positioning, are indeed autonomous. Feedback regulation of telomere length might have emerged from several mechanisms. One such mechanism is the action of the enzyme telomerase, which adds back telomeric repeats at the telomere. Telomerase has been shown to be active in proliferating human epithelial cells, human endothelial cells and human fibroblasts (Belair et al., 1997; Vasa et al., 2000; Masutomi et al., 2003). Its negative regulators TRF1 and TRF2 bind along the telomere (Smogorzewska et al., 2000). A plausible hypothesis is that longer telomeres bind more TRF1 and TRF2 and are therefore less well preserved by telomerase. Hemann et al. (2001) showed that telomerase preferentially elongates the shortest telomeres in mice, supporting this hypothesis. Additionally, TRF2 is suggested to recruit a C-strand specific exonuclease to trim the 5 -ended strand of the telomere, because overhangs are required for formation of t-loops, which are protective to senescence (Karlseder et al., 2002). Longer telomeres, binding more TRF2, might therefore suffer more loss from nuclease activity. Hence, regulated C-strand processing could contribute to controlled telomere attrition. In that case, the distribution of the overhang lengths should also follow a Weibull distribution, since a linear relationship exists between the overhang length of

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Fig. 4. Fitting the Weibull function to G-rich overhang length distributions adopted from Huffman et al. (2000). The hypothesis of no correlation between data and model was tested: P = 0.0019 for endothelial cells and P < 0.001 for the other cell types.

telomeres and the rate of shortening (Huffman et al., 2000). We tested this hypothesis by fitting the Weibull function to the overhang length in nucleotides [nt] in different cell types (Huffman et al., 2000). From the agreeable fits (see Fig. 4) it can be concluded that G-rich overhangs are likely involved in telomere regulation. Finally, the role of oxidative stress has been shown to be pivotal in the determination of the rate of telomere shortening (von Zglinicki, 2002). Erosion of telomeres by oxidative damage could be interpreted as a process of breakage and coalescence (Oexle, 1998). Although the lognormal shape has been empirically used to describe particle size distributions, Brown and Wohletz (1995) demonstrated that this was simply fortuitous. Instead, they suggested the Weibull distribution, which can be derived on a physical basis. Thus, also the oxidative stress theory of telomere shortening could well agree with a model containing negative feedback in a biophysical sense. There is a possibility that the increased skewness of telomere length distributions is an observational artifact. FISH requires metaphase chromosomes and staining intensities between different metaphases may differ. Graakjaer et al. (2003) normalized telomere values from each cell as measured by FISH in lymphocytes. These data are less positively skewed than the telomere distributions from Martens et al. (2000). Additionally, it may be questioned if telomere length is strictly proportional to the amount of telomeric DNA in Southern blotting, due to the large and variable fraction of the non-TTAGGG component of TRFs (Allsopp et al., 1992). Moreover, TRF analysis is biased against the detection of short telomeres. Single telomere length analysis (STELA) was recently developed by Baird et al. (2003) to overcome these limitations. They measured the

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cell-to-cell distributions of the XpYp telomere and these data do not show large positive skewness. By measuring telomere length distributions over a range of PDs, STELA could be used to support or undermine the existence of the length-dependency of telomere shortening. The model proposed here is relatively simple as compared to detailed descriptions of the mechanisms of telomere shortening (Tan, 1999; Sozou and Kirkwood, 2001; Proctor and Kirkwood, 2002, 2003). The reduced complexity served our goal of parameter estimation: to apply parameter estimation, the number of data points should be significantly larger than the number of parameters to be estimated. Furthermore, MLE techniques cannot distinguish between separate mechanisms contributing to the same type (regulated or autonomous) of telomere attrition. Thus, our modeling resulted in the quantification of lumped parameters. The standard deviations of the estimated parameters indicate whether the parameters are reliable, provided the model describes telomeric loss in a manner that represents the true biology. In summary, we have given strong evidence that telomere shortening is proportional to telomere length in human somatic cells. We have developed a mathematical model of telomere length regulation and applied this model to published data. This resulted in the quantification of physiologically relevant parameters.

Acknowledgements This research was financially supported by a grant from Senter (Dutch Ministry of Economic affairs, no. TSGE1028) and Unilever Research and Development Vlaardingen, The Netherlands. Two anonymous referees are acknowledged for their valuable comments.

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