Do cells sense time by number of divisions?
aa r X i v : . [ q - b i o . S C ] N ov Do cells sense time by number of divisions?
Z. Schuss , K. Tor and D. Holcman Abstract
Do biological cells sense time by the number of their divisions, a process that ends atsenescence? We consider the question ”can the cell’s perception of time be expressedthrough the length of the shortest telomere?” The answer is that the absolute timebefore senescence cannot be expressed by the telomere’s length and that a cell cansurvive many more divisions than intuitively expected. This apparent paradox is dueto shortening and elongation of the telomere, which suggests a random walk modelof the telomere’s length. The model indicates two phases, first, a determinist driftof the length toward a quasi-equilibrium state, and second, persistence of the lengthnear an attracting state for the majority of divisions prior to senescence. The measureof stability of the latter phase is the expected number of divisions at the attractor(”lifetime”) prior to crossing a threshold to senescence. The telomerase regulates sta-bility by creating an effective potential barrier that separates statistically the shortestlifetime from the next shortest. The random walk has to overcome the barrier in orderto extend the range of the first regime. The model explains how random telomeredynamics underlies the extension of cell survival time.
Finding a measure of our sensation of time is an intriguing question in physical and lifesciences. In cell biology, the lifetime of a cell is reflected in the number of cell divisions priorto senescence, which is a measure of a cell’s lineage death and is a component of cellularaging [4]. The number of cell divisions is expressed by the length of telomeres, which protectthe ends of the chromosomes. Telomeres can lose between a few to hundreds of base pairsduring cell division, or increase their length through the action of the enzyme telomerase. Weare concerned here with the physical mechanism that regulates the number cell of divisionsprior to senescence.Several decades of research have revealed that telomeres are made of repetitive nu-cleotide sequences at each end of a chromatid, which protects the end of the chromosomefrom deterioration or from fusion with other chromosomes. Following each cell division, thetelomere ends become shorter on the average [1]. Telomeres are necessary for the mainte-nance of chromosomal integrity and overall genomic stability [11, 21] and in the absence of Tel-Aviv University, Tel-Aviv 69978, Israel. Ecole Normale Sup´erieure, 46 rue d’Ulm 75005 Paris, France. This research was supported by a FRMteam grant. Results
We adopt the random walk model of telomere dynamics [32, 12], in which the length x ofthe telomere can decrease or increase in each division. The model assumes that the lengthdecreases by a fixed length a with probability l ( x ) or, if recognized by a polymerase, itincreases by a fixed length b ≫ a with probability r ( x ) = 1 − l ( x ). The jump probability r ( x ) is assumed a decreasing function of x with r (0) = 1 [12]. Thus the length of the telomereat the n -th division is an asymmetric random walk x ( n ). In our simplified model, we assumethat the maximal length of a telomere is L ≫ b . We assume that senescence ensues whenthe length decreases below a critical value T , that is, the division process stops.The problem at hand is to determine the evolution of the telomere length, to study thedynamics of the shortest length, and to investigate the role of the probability r ( x ), whichcan be modulated by telomere diseases [3]. In particular, we study the statistics of thetrajectories x ( n ), their expected time to reach their quasi-stationary state, and the expectednumber of divisions before reaching the threshold T for the first time. The model of the telomere dynamics is x n +1 = x n − a w.p. l ( x n ) x n + b w.p. r ( x n ) , (1)where the right-probability r ( x ) can be approximated by r ( x ) = 11 + βx , (2)for some β >
0. The model (1) is simulated for all telomeres (with a maximum of 32), andthe statistics of the average, the longest, and the shortest trajectories are calculated. Therepresentative parameters are as follows. The shortening law is a = 3 and the elongationlaw is b = 30. The results are quite similar to an exponential law with mean b = 30 (see[12]). The results shown in Fig.1 reveal two phases: in the first one, the telomere’s lengthdecreases almost deterministically to a quasi-equilibrium length as described below. In thesecond phase, the length persists in the quasi-equilibrium state for the majority of divisions.In the absence of any stopping process, the telomere’s length stays near its minimum, soapparently the cell can live forever. 3 (A) T e l o m e r e l eng t h s ( ba s e pa i r s ) Number of divisions0 500 1000 1500 2000
Phase 1 Phase 2 (B) T e l o m e r e l eng t h s ( ba s e pa i r s ) Number of divisions0 500 1000 1500 2000MaximumMeanMininum 0 500 1000 1500 + + + + (C) Telomere length(base pair) C oun t Figure 1:
Telomere dynamics in yeast. (A)
Stochastic dynamics of the 32 telomeres (see(1) with elongation exponentially distributed with mean 40 bps.) (B)
Dynamics of the minimum,maximum, and the mean. Two phases are seen: phase 1 is characterized by constant decay andconvergence toward phase 2, which is a quasi steady-state due to an attractor, which prevents thecollapse of the telomere to a critical value. There is a large increase in the number of cell divisionsin phase 2. (C)
Distribution of telomere length at steady-state. The parameters are β = 0 .
045 and a = 3 ± / To characterize each phase, we use the scaling x n = y n L and setting ε = b/L , the dynamics(1) becomes y n +1 = ( y n − εab w.p. ˜ l ( y n ) y n + ε w.p. ˜ r ( y n ) , (3)where ˜ l ( y ) = l ( x ). In the limit ε ≪
1, the process y n moves in small steps. The dynamics(3) falls under the general scheme [16, 19, 18, 27] y n +1 = y n + εξ n , (4)where Pr { ξ n = ξ | y n = y, y n − = y , . . . } = w ( ξ | y, ε ) , (5) ε is a small parameter, and y is a random variable with a given pdf p ( y ). In the case athand the function w ( ξ | y ) defined in (5) is given by w ( ξ | y ) = (1 − ˜ r ( y )) δ (cid:16) ξ + ab (cid:17) + ˜ r ( y ) δ ( ξ − , (6)so the conditional jump moments are m n ( y ) = (cid:16) − ab (cid:17) n (1 − ˜ r ( y )) + ˜ r ( y ) . The probability density function (pdf) of y n satisfies the forward master equation p ε ( y, n + 1 | x, m ) = p ε (cid:16) y + ε ab , n | x, m (cid:17) ˜ l (cid:16) y + ε ab (cid:17) + p ε ( y − ε, n | x, m )˜ r ( y − ε ) (7)4nd the backward equation p ε ( y, n | x, m ) − p ε ( y, n | x, m + 1) (8)= p ε (cid:16) y, n | x − ε ab , m + 1 (cid:17) (1 − ˜ r ( x )) + p ε ( y, n | x + ε, m + 1)˜ r ( x ) − p ε ( y, n | x, m ) . The first conditional jump moment, m ( y ) = − a ˜ l ( y ) b + ˜ r ( y ) , (9)changes sign at z = bLβa . (10)If the process is terminated at the threshold T mentioned above, then εT /b < y <
1. Inthis case, the pdf p ε ( y, n ) converges to a quasi-stationary density p ε ( y ) for large n , prior tothe termination of the trajectory y n at y = T /L . The quasi-stationary master equation (7)becomes p ε ( y ) = p ε (cid:16) y + ε ab (cid:17) ˜ l (cid:16) y + ε ab (cid:17) + p ε ( y − ε ) ˜ r ( y − ε ) , (11)which for small ε is peaked near z , as shown in fig. 1C and 2A. Indeed, to find the asymptoticstructure of the quasi-stationary solution p ε ( y ) for ε ≪
1, we construct its approximation inthe WKB form [27] p ε ( y ) = K ε ( y ) exp {− ψ ( y ) /ε } , where ψ ( y ) is the solution of the eikonalequation ˜ l ( y ) exp n − ψ ′ ( y ) ab o + ˜ r ( y ) exp { ψ ′ ( y ) } = 1 . (12)The derivatives of ψ ( y ) at z shows a single peak near z . Indeed ψ ′ ( z ) = 0 and ψ ′′ ( z ) = 2 aLβa + b , (13) ψ ′′′ ( z ) = −
43 ( a + 2 b ) β L a b ( a + b ) . (14) The essence of the mentioned paradox is due to the behavior of the first conditional moment m ( y ) of the jump size (see (9)). Because m ( z ) = 0 and m ′ ( z ) < y , for example from y = 1, toward y = z , where it is trapped inquasi-equilibrium fluctuations about z for an expected number of jumps ¯ n , which may belarger than the expected number of jumps n y → z that is required to reach z from y for thefirst time. Specifically, the expected number of jumps (expected lifetime) n z → T to go from z over the threshold (or 0) may be larger than n → z .5o study this trapping phenomenon, we note first that the expected lifetime n ( y ) tocross the boundary y = z from an internal point z < y < L n n = (1 − y ) /ε Z ( z − y ) /ε n ( y + εξ ) w ( ξ | y ) dξ − n ( y ) = − z ≤ y ≤ , n ( y ) = 0 for y < z . (15)Setting τ ( y ) = εn ( y ), we write the Kramers-Moyal expansion of (15) as ∞ X k =1 ε k − m k ( y ) k ! d k τ ( y ) dy k = − z ≤ y ≤ , τ ( z ) = 0 , τ (1) = 0 , (16)and the diffusion approximation to (16) is L ε τ ( y ) = ε m ( y ) τ ′′ ( y ) + m ( y ) τ ′ ( y ) = − z ≤ y ≤ , τ ( z ) = 0 , τ ′ (1) = 0 . (17)To show that the solution of (17) is a valid approximation to that of (16) it suffices to showthat τ ( y ) has a convergent series representation τ ( y ) = ∞ X k =0 ε k τ k ( y ) , (18)where τ k ( y ) are bounded functions. The stochastic dynamics corresponding to (17) is˙ y = m ( y ) + p εm ( y ) ˙ w ( t ) , (19)where ˙ w ( t ) is δ -correlated Gaussian white noise and t is the smoothly interpolated εn . Inparticular, the effective potential well is V ( y ) = − ε Z y m ( z ) m ( z ) dz = (1 − y ) 2 bεa + 2( a + b ) bLa β log a βy + εba βz + εb , (20)for which the drift − V ′ ( y ) vanishes at z . Fig.2A shows the effective potential V ( y ) forvarious values of the parameter β . Changing β affects both the height and the location ofthe minimum of V ( y ).The solution of (17) is given by τ ( y ) = 2 ε y Z z exp (cid:26) bεa ( z − z ) − a + b ) ba β log a βz + εba βz + εb (cid:27) × Z z exp (cid:26) − bεa ( u − z ) + 2( a + b ) ba β log a βu + εba βz + εb (cid:27) a βbu + εb ( βbu + ε ) b du dz. (21)6 −
593 1185 2000 4000 4500 (A)
Phase 2 Phase 1Telomere lengths (base pairs) P o t en t i a l V b b b b
100 200 300 400 500 600 (B)
Threshold M ean F i r s t T i m e t o T h r e s ho l d ShortestSecond shortestThird shortest
100 200 300 400 500 600 (C)
Threshold M ean F i r s t T i m e t o T h r e s ho l d ShortestSecond shortest
Figure 2:
Statistical properties of telomere dynamics. (A)
Representation of the effectivepotential V ( y ) for different values of the parameter β . (B) The MFPTs of the first three shortesttelomeres. (C)
The MFPTs of the shortest and second shortest telomeres for different thresholds T . The MFPTs are statistically separated for small T (almost one standard deviation apart).Parameters are β = 0 . a = 3 ± / L = 1000. v = 2 b ( u − z ) aε , ζ = 2 b ( z − z ) aε and the expansion of the denominator in (16) give τ ( y ) = ε b ( y − z ) /aε Z e ζ (1 + Bζ ) A ∞ Z ζ e − v (1 + Bv ) A dv dζ [1 + O ( ε )] , (22)where A = 2( a + b ) ba β , B = a βa + b . The large ζ asymptotics e ζ (1 + Bζ ) A ∞ Z ζ e − v (1 + Bv ) A dv ∼ ∞ X j =0 Γ( A + 1)Γ( A − j + 1) (cid:18) B Bζ (cid:19) j and (22) give that for small ετ ( y ) ∼ ε b ( y − z ) /aε Z (cid:20) AB Bζ + · · · (cid:21) dζ = ba ( y − z ) + Aε B log (cid:18) B b ( y − z ) aε (cid:19) + · · · . To conclude, the explicit solution of (17) has the asymptotic representation for small ετ ( y ) = ba ( y − ζ ) + Aε B log (cid:18) B b ( y − z ) aε (cid:19) + · · · . (23)It is clear from (23) that the derivatives of τ ( y ) are uniformly bounded for y ≥ z and ε > n → z is given by n → z ∼ ¯ τ (1) ε , that is n ( y ) ∼ baε ( y − z ) + A B log (cid:18) B b ( y − z ) aε (cid:19) + · · · . (24)This analysis clarifies the first phase, which consists of noisy drifting to z . The second phasecorresponds to escape from z over the threshold T .8 (A) M ean F i r s t T i m e t o T h r e s ho l d Threshold240 400 600 800 10002 bbb Single telomereShortest telomere (B) M ean F i r s t T i m e t o T h r e s ho l d Threshold100 200 300 400 500L = 2000L = 1000L = 700L = 500 Single telomereShortest telomere
Figure 3:
Effect of changing the telomerase efficiency parameter β on the MFPTto threshold. (A) The MFPT of a telomere and of the shortest among 32 telomeres for β (green), β/ β (blue). (B) Effect of changing the initial length L . Parametersare β = 0 . a = 3 ± / L = 2000. Threshold T = 150, beta = 0.045
300 500 700 900 (A) C oun t ShortestSecond shortestThird shortest
Threshold T = 200, beta = 0.045
300 350 400 450 500 (B)
ShortestSecond shortestThird shortest
Threshold T = 300, beta = 0.045
250 300 350 400 (C)
ShortestSecond shortestThird shortest
Threshold T = 150, beta = 0.09
300 320 340 360 (D)
ShortestSecond shortestThird shortest
Threshold T = 200, beta = 0.09
270 280 290 300 310 320 330 (E)
ShortestSecond shortestThird shortest
Threshold T = 300, beta = 0.09
240 250 260 270 280 (F)
ShortestSecond shortestThird shortest
Figure 4:
Histogram of arrival time to a threshold T. (A,B,C)
T=300,200,150. Pa-rameters are β = 0 . a = 3 ± / L = 1000. Number of runs = 10000. (D,E,F) : β =0 .
09 9 .5 The role of the shortest telomere: time to senescence
To further investigate the influence of telomere length distribution on the time to senescence,we resort to numerical simulations of the model (1). Interestingly, as the threshold decreases,the difference between the time to threshold increases, leading to a clear gap between theshortest and the second shortest lifetimes, which is reduced between the second and thethird. This behavior can be interpreted by the two phases: as long as the threshold fallsinto the first phase, the decay is deterministic and the difference between the first, secondand third is insignificant. However, when the threshold is moved to the left of the criticalpoint z (e.g., z = 300 for β = 0 . N i.i.d. trajectories of (1), we use the diffusion approximation (19) toapproximate the pdf of the first passage time to threshold. The solutions of p t = L n p for 0 < y < , p ( y,
0) = p ( y ) , p (0 , t ) = p (1 , t ) = 0 (25) p ε,t = L ε p ε for 0 < y < , p ε ( y,
0) = p ( y ) , p ε (0 , t ) = p ε (1 , t ) = 0 , (26)where the operators L n and L ε are defined in (15) and (19), respectively, can be constructedby the method of separation of variable. The dependence on time is exponential with ex-ponents that are the eigenvalues of the boundary value problems for the two operators. Forsmall ε , these are singular perturbation problems and thus there is a big gap between thefirst and second eigenvalues in either case. The first eigenvalue is the reciprocal of the MFPT[27]. Thus the eigenfunction expansions of the pdfs are dominated by the first eigenfunction.Because the first eigenvalue is exponentially large in 1 /ε , the pdf of survival time in thepotential well can be approximated by a single exponential, which means that the survivaltime is Poissonian with mean τ ( z ). We use henceforward this approximation.We further investigate the gap between the first and the second MFPT by plotting inFig.3C the mean and the variance. We find that the standard deviations due to the shortestand to the second shortest overlap minimally, suggesting that the shortest telomere playsa key role in triggering senescence. This result is due to the randomness of the model anddepends on its parameters.For N Poissonian i.i.d. processes with escape time E [ τ ] = ¯ τ , the expected shortestescape time is ¯ τ N . The expected second shortest lifetime is 2 ¯ τ N . Thus the gap between thefirst and the second is ∆ = 2 ¯ τ N − ¯ τ N = ¯ τ N , (27)which is the standard deviation of the first time E [ τ first ] = ¯ τ /N . Fig.4 indicates a deviationfrom the Poissonian case. Thus to study the effect of the shortest telomere, we use the ratio R = | E [ τ second ] − E [ τ first ] | q(cid:0) E [ τ first ] − E [ τ first ] (cid:1) , (28)10here E [ τ first ] (resp. E [ τ second ]) is the MFPT for the first (resp. second) telomere length toreach the threshold T . Interestingly, for a threshold T = 150, we obtain that R = 1 .
92 (thevalue of β = 0 .
045 is given in table 1), while for the value β = 0 .
09, we get R = 1 .
16. Thisresult suggests that decreasing the efficiency of the telomerase reduces the isolation of theshortest telomere relative to the second.Parameters Symbol ValueThreshold T 150 150 200 200 300 300Beta β E [ τ T ,min ] 410 305 348 281 274 239MFPT 2ndshortest E [ τ T ,min ] 509 320 397 292 297 245MFPT 3rdshortest E [ τ T ,min ] 640 333 442 301 316 2511st separation R R To investigate the effect of telomerase on cell lifetime, we vary its efficiency by varying theparameter β and the initial telomere length L . The results are shown in Figure 3: decreasingthe parameter β changes the position of the two phases and thus the time to threshold. Wecompare the mean and shortest among 32 telomeres arrival times to T for three values of β (green), β (blue) and β/ β/ L . This effect of thetelomere length diminishes as the threshold decreases (Fig 3B). This paper analyzes telomere dynamics by an asymmetric random walk model of the length.The asymmetry of the length is due to the decrease by a small amount a , compared tothe large, but rare increase b after each step of the walk (with state-dependent transitionprobability). The main result of our model is that the time to senescence (measured by thelength of the shortest telomere) is not proportional to the initial telomere length. Moreover,the dynamics is divided into two phases: the first one is drift toward the quasi steady-state,where aging or the number of cell divisions is reflected in the telomere length. This couldaccount for 30% to 50% of the number of divisions. In the second phase, the telomere lengthno longer reflects the number of divisions, but rather stays near an equilibrium length main-tained close to the critical telomere size. Due to random breaks and repairs, the telomere’s11ength eventually reaches its critical size. The present stochastic model and computationsare generic and can be applied to any cells.Interestingly, the present analysis (Fig. 1B) reveals a clear difference between the meanand the shortest telomere length. Moreover, stochastic numerical simulations show a uni-versal gap between the MFPT of the shortest and the second one (Fig. 4), suggesting thatsenescence can be triggered by the shortest telomere that reaches a given threshold. Anydivision, past this threshold would affect irreversibly the protein synthesis due to DNA. Be-cause we do not know the exact threshold, we explore a continuum interval of values, butthe results seem identical, showing a pronounced effect of a low threshold.Although we previously studied the steady-state distribution of telomere length by theasymmetric random walk (model 1), we could not draw any conclusions about the dynamics,which requires a first passage time analysis of the shortest telomere. We found in [32, 12]that there was a statistical gap between the distribution of the shortest and the secondshortest telomere at steady state and that the gap depended on the parameters of the model.Computing analytically the gap between the MFPT of the shortest and the second shortestto a threshold T remains an open problem.To study the consequences of telomere syndromes [3], we varied the parameter β thatrepresents the telomerase efficiency: we found that increasing the value of β by a factor2, which is equivalent to decreasing the telomerase efficiency, has several consequences forsenescence (Figs 3 and 4): for example, with a threshold of T = 150, the time to senes-cence decreases from 410 to 305 (parameters in the figure legends). Another consequenceof increasing β is that the distribution of the shortest and the second shortest telomere getmixed, as shown above by the decrease of the ratio R defined by 28 from R = 1 .
92 (for β = 0 . R = 1 .
16 for β = 0 . T , which we varied, but in some knowncases the minimum telomere length needed to ensure human telomere protective stabilityin white blood cells is 3.81 kb [6]. We conclude with general consequences of telomereshortening, which could be seen as a deregulation of time sensing: it is known that leukocytetelomere length can be used as a bio-marker of cardiovascular diseases, confirming that thedistribution of telomere length and probably that of the shortest, plays a key role [6]. Stresshormones such as cortisol, is roughly inversely proportional to Leukocyte telomere length ina normal group of persons, but not in those suffering from Major Depressive Disorder [13].Similarly, adults suffering from major depression have shorter telomere length [6]. Finally,with aging, the average telomere length decreases and this is correlated with an increase inmortality. Thus the measure of telomere shortness is also a statistical indicator of humanmortality. To conclude, all these conditions can now be incorporated in modeling so thatthe distribution of telomere lengths and the shortest one can be predicted. References [1] Wikipedia-telomere, 2017. 122] T. Antal, K. Blagoev, S. Trugman, and S. Redner. Aging and immortality in a cellproliferation model.
Journal of theoretical biology , 248(3):411–417, 2007.[3] M. Armanios and E. H. Blackburn. The telomere syndromes.
Nature reviews. Genetics ,13(10):693, 2012.[4] G. Aubert and P. M. Lansdorp. Telomeres and aging.
Physiological reviews , 88(2):557–579, 2008.[5] B. J. Ballew and V. Lundblad. Multiple genetic pathways regulate replicative senescencein telomerase-deficient yeast.
Aging cell , 12(4):719–727, 2013.[6] E. H. Blackburn, E. S. Epel, and J. Lin. Human telomere biology: A contributory andinteractive factor in aging, disease risks, and protection.
Science , 350(6265):1193–1198,2015.[7] R. A. Blythe and C. E. MacPhee. The life and death of cells.
Physics , 6:129, 2013.[8] S. A. Booth and F. J. Charchar. Cardiac telomere length in heart development, function,and disease.
Physiological Genomics , pages physiolgenomics–00024, 2017.[9] A. Canela, E. Vera, P. Klatt, and M. A. Blasco. High-throughput telomere lengthquantification by fish and its application to human population studies.
Proceedings ofthe National Academy of Sciences , 104(13):5300–5305, 2007.[10] T. R. Cech. Beginning to understand the end of the chromosome.
Cell , 116(2):273–279,2004.[11] T. De Lange, L. Shiue, R. Myers, D. Cox, S. Naylor, A. Killery, and H. Varmus.Structure and variability of human chromosome ends.
Molecular and cellular biology ,10(2):518–527, 1990.[12] K. D. Duc and D. Holcman. Computing the length of the shortest telomere in thenucleus.
Physical review letters , 111(22):228104, 2013.[13] B. Fair, S. H. Mellon, E. S. Epel, J. Lin, D. R´ev´esz, J. E. Verhoeven, B. W. Penninx,V. I. Reus, R. Rosser, C. M. Hough, et al. Telomere length is inversely correlated withurinary stress hormone levels in healthy controls but not in un-medicated depressedindividuals-preliminary findings.
Journal of Psychosomatic Research , pages 177–180,2017.[14] M. T. Hemann, M. A. Strong, L.-Y. Hao, and C. W. Greider. The shortest telomere,not average telomere length, is critical for cell viability and chromosome stability.
Cell ,107(1):67–77, 2001.[15] S. S. Khan, B. D. Singer, and D. E. Vaughan. Molecular and physiological manifestationsand measurement of aging in humans.
Aging Cell , 2017.1316] C. Knessl, B. Matkowsky, Z. Schuss, and C. Tier. An asymptotic theory of large devi-ations for markov jump processes.
SIAM journal on applied mathematics , 45(6):1006–1028, 1985.[17] C. Knessl, B. Matkowsky, Z. Schuss, and C. Tier. Asymptotic analysis of a state-dependent m/g/1 queueing system.
SIAM Journal on Applied Mathematics , 46(3):483–505, 1986.[18] C. Knessl, B. Matkowsky, Z. Schuss, and C. Tier. Boundary behavior of diffusionapproximations to markov jump processes.
Journal of statistical physics , 45(1):245–266, 1986.[19] C. Knessl, B. Matkowsky, Z. Schuss, and C. Tier. A singular perturbation approach tofirst passage times for markov jump processes.
Journal of Statistical Physics , 42(1):169–184, 1986.[20] K. Lange.
Applied probability . Springer Science & Business Media, 2010.[21] S. Marcand, E. Gilson, and D. Shore. A protein-counting mechanism for telomere lengthregulation in yeast.
Science , 275(5302):986–990, 1997.[22] J. op den Buijs, P. P. van den Bosch, M. W. Musters, and N. A. van Riel. Mathematicalmodeling confirms the length-dependency of telomere shortening.
Mechanisms of ageingand development , 125(6):437–444, 2004.[23] C. J. Proctor and T. B. Kirkwood. Modelling telomere shortening and the role ofoxidative stress.
Mechanisms of ageing and development , 123(4):351–363, 2002.[24] C. J. Proctor and T. B. Kirkwood. Modelling cellular senescence as a result of telomerestate.
Aging cell , 2(3):151–157, 2003.[25] D. H. Rehkopf, B. L. Needham, J. Lin, E. H. Blackburn, A. R. Zota, J. M. Wojcicki, andE. S. Epel. Leukocyte telomere length in relation to 17 biomarkers of cardiovasculardisease risk: A cross-sectional study of us adults.
PLoS medicine , 13(11):e1002188,2016.[26] I. A. Rodriguez-Brenes and C. S. Peskin. Quantitative theory of telomere length reg-ulation and cellular senescence.
Proceedings of the National Academy of Sciences ,107(12):5387–5392, 2010.[27] Z. Schuss.
Theory and applications of stochastic processes: an analytical approach ,volume 170. Springer Science & Business Media, 2009.[28] Z. Tan. Intramitotic and intraclonal variation in proliferative potential of human diploidcells: explained by telomere shortening.
Journal of theoretical biology , 198(2):259–268,1999.[29] M. T. Teixeira, M. Arneric, P. Sperisen, and J. Lingner. Telomere length homeostasisis achieved via a switch between telomerase-extendible and-nonextendible states.
Cell ,117(3):323–335, 2004. 1430] J. D. Watson. Origin of concatemeric t7dna.
Nature , 239(94):197–201, 1972.[31] R. J. Wellinger and V. A. Zakian. Everything you ever wanted to know about saccha-romyces cerevisiae telomeres: beginning to end.
Genetics , 191(4):1073–1105, 2012.[32] Z. Xu, K. D. Duc, D. Holcman, and M. T. Teixeira. The length of the shortest telomereas the major determinant of the onset of replicative senescence.
Genetics , 194(4):847–857, 2013.[33] V. F. Zaitsev and A. D. Polyanin.
Handbook of exact solutions for ordinary differentialequations . CRC press, 2002. 15 hreshold T = 150, beta = 0.045
300 500 700 900 (A) C oun t Threshold T = 150, beta = 0.09
300 320 340 360 (D)
Exit time [a.u.]
Threshold T = 200, beta = 0.09
270 280 290 300 310 320 330 (E) C oun t ShortestSecond shortestThird shortest
Threshold T = 300,
240 250 (F)
Exit time [a.u.]Exit time [a.u.]
ShortestSecond shortestThird shortest
ShortestSecond shortestThird shortest
300 400 500 600 700 (B)
Exit time [a.u.] (C)
Exit time [a.u.]
ShortestSecond shortestThird shortest