### Generation-dependent accumulation of cell cycle arrests is specific to the absence of telomerase

We took advantage of a data set we recently published [6], where the successive cell divisions of individual lineages of yeast cells were tracked using a microfluidics-based live imaging approach. The analysis of the resulting movies allowed the measurement of the duration of each consecutive cell cycle, with a 10 min time resolution. We previously highlighted the presence of abnormally long cell cycles in virtually all telomerase-negative cell lineages, which were interchangeably called cell cycle arrests because they relied on the DNA damage checkpoint [5, 6] (Fig. 1A, B). Long cell cycles were extremely rare in cells expressing telomerase and their frequency did not increase with the number of generations.

We set a duration threshold *D* to define what was considered to be an abnormally long cell cycle. We observed that some of the abnormally long cell cycles, when appearing for the first time, were located at the end of the lineages and were never followed by a cell cycle of normal duration (Fig. 1C), consistent with first terminal senescence arrests being caused by the first telomere reaching a critical short length [16, 18, 19]. These long cell cycles were thus called terminal or senescence arrests (see "Methods" section"). Others, termed non-senescent, non-terminal or early arrests, were followed immediately or later on by at least one cell cycle of normal duration below *D* (Fig. 1C). In [5, 6], *D* was arbitrarily chosen to be the mean cell cycle duration of telomerase-positive cells plus three standard deviations. In the present work, unless otherwise stated, we also refer to long cell cycles or arrests using the same threshold *D*, which was computed from the telomerase-positive data set and was estimated to be 180 min. In addition, whenever possible, we challenged the robustness of the results with respect to this threshold. The fact that the first non-terminal arrests started generally earlier than the terminal arrests (Fig. 1D) suggested that they might be caused by a different mechanism and we therefore set out to characterize the law governing their appearance.

### The probability of appearance of the first non-terminal arrest increases over generations

We asked whether the first non-terminal arrest could be caused by a stochastic DNA damage event. Following this hypothesis, at each generation *j*, there would be a probability *p(j)* (0 ≤ *p(j)* ≤ 1) for an arrest to occur. Let *X* be the random variable describing the generation at which the first non-terminal arrest occurs. Then, for every positive integer k, we have:

$$\mathbb{P}\left(X=k\right)=p\left(k\right)\prod _{j=1}^{k-1}(1-p\left(j\right))$$

(1)

We first tested whether the event causing the arrest was independent of the number of generations, that is *p(j)* = *p* constant and *X* corresponds to Bernoulli trials. The generation of the first arrest would then follow a geometric distribution, of parameter *p* = 0.0465. However, a χ² goodness-of-fit test rejected this hypothesis (p-value < 0.05), robustly with regards to the threshold *D* (Fig. 2A and Additional file 1: Fig. S1).

Plotting the fraction of lineages in which the first non-terminal arrest occurred at generation *j* out of all lineages that have not yet experienced an arrest at that point confirmed that a constant *p(j)* did not describe the experimental data accurately (Fig. 2B). Instead, *p(j)* increased in a non-linear, possibly exponential, manner. We tested this hypothesis and to ensure that 0 ≤ *p(j)* ≤ 1 for every *j*, we chose to fit the data with a logistic function that approximates an exponential growth for small values of *j* (Fig. 2B):

$$p\left(j\right)=\frac{1}{1+exp(-\frac{j-a}{b})}$$

(2)

This approximation of *p(j)* allowed us to compute the generation of the first non-terminal arrest in simulated lineages. We found a good agreement between the simulation and the experimental data for the probability mass distribution function (Fig. 2A), in contrast to a geometric law, as well as for the cumulative distribution function (Fig. 2C), for a wide and biologically relevant range of *D* between 180 and 270 min (χ² goodness-of-fit tests, Fig. 2D). Therefore, while the first non-terminal arrest seemed to occur stochastically, the probability of its appearance increased exponentially over generations, which might be consistent with a time-dependent or telomere-length-dependent damage as a plausible cause of the first arrest.

### A telomere-shortening-dependent model fits the timing of the first non-terminal arrest

We previously modelled the telomere shortening mechanism at the molecular level and found that a model with the shortest telomere in the cell reaching a signalling threshold explains the onset of the first terminal senescence arrests and their heterogeneity [16]. Here, we asked whether a similar model of telomere shortening could also explain the first non-terminal arrest. Such a model would provide a mechanistic explanation for these arrests and would be consistent with the phenomenological description of the increase in their probability of occurrence over generations (Fig. 2B).

We therefore applied the telomere shortening model to look for a potential length threshold at which the first arrest would be induced. Each lineage started with an initial steady-state telomere length distribution simulated based on the length-dependent regulation of telomerase action. Telomeres were then allowed to shorten at each cell division, taking the asymmetry of telomere replication into account. We used the same approach as in [16, 17] and tested how the shortening dynamics of a linear combination of the lengths of the two shortest telomeres, that is *L*_{1} *+ α L*_{2} (where *L*_{1} and *L*_{2} represent the length of the shortest and second shortest telomeres, respectively, and *α* is a positive scalar) reaching a threshold *L*_{min} could fit the data. To do so, we simulated the senescence onsets of individual lineages according to [16, 17] and compared them to the experiments by minimizing the following error value, describing the distance between the simulation and the experimental data of the onset of the first arrest:

$$e\left(\alpha,{L}_{min}\right)={\sum }_{j=1}^{N}{\sum }_{i=1}^{n}{\left(G\left(i,j; \alpha,{L}_{min}\right)-\overline{G}\left(i\right)\right)}^{2},$$

(3)

where *N* is the number of simulations, *n* is the number of experimental lineages (here *n* = 115) and for the *j*^{th} simulation, *G(i,j;a)* represents the *i*^{th} lineage out of *n* lineages, ordered from shortest to longest and simulated with the law of telomeric signalling *L*_{1} *+ α L*_{2} *= L*_{min}, and \(\overline{G}\left(i\right)\) is the *i*^{th} shortest lineage in the ordered set of the *n* experimental lineages.

Interestingly, as in [16] for the onset of senescence, the parameter *α* that minimized *e* was always 0, indicating that taking only the length of the shortest telomere into account, and not the second, led to the best fit. We therefore chose *α = 0* for the rest of the present work. The *L*_{min} that minimized *e* was 61 bp, larger than the value 19 bp found in [16], which was consistent with the fact that the first arrest appeared earlier than senescence onset (Fig. 1C). The simulated lineage data were ordered based on their length and the area representing the 95% quantile of *N =* 1000 simulations was represented (Additional file 1: Fig. S2). However, the simulations failed to capture the variation of the generation of the first non-terminal arrest (Additional file 1: Fig. S2). We concluded that a deterministic *L*_{min} was not a reasonable assumption given the variable nature of the first arrest.

At a molecular level, it would not be surprising that the signalling threshold for the shortest telomere might be probabilistic rather than deterministic (see discussion). To take into account this possibility, we chose *L*_{min} from a distribution defined by:

$$p\left(L\right)=b.exp(-aL),$$

(4)

where *a > 0* and 0 < *b < 1* are parameters of the distribution. This distribution was chosen as a simple way to model a decreasing probability with only two parameters. At each division, one computes the shortest telomere length *L* and draws randomly, according to the probability *p(L)*, whether it signals the first arrest or not. Equation (4) shows that we assume that the probability of signalling senescence increases when *L* decreases. If a telomere happens to reach a length of 0 bp, which is a physiologically relevant situation [18], the arrest is immediately triggered regardless of this probability distribution.

Using numerical simulations (*N* = 1000), we fitted the parameters *a* and *b* on the experimental data and found the best fit to be *a* = 0.023 and *b* = 0.276, giving a signalling probability that is small if the shortest telomere is greater than 150 bp (*p*(150) = 0.009), and takes increasingly larger values as the shortest telomere decreases in length (Fig. 3A). This fit was in excellent agreement with the heterogeneity of the experimental data (Fig. 3B).

We conclude that a stochastic process or event depending on the length of the telomeres getting progressively shorter might explain the timing of occurrence of the first non-terminal arrest. Therefore, while the non-terminal arrests occurred slightly earlier than the terminal senescence arrests, both terminal and non-terminal arrests might be triggered by telomere-length-dependent processes (Fig. 1C and [16]). However, we do not exclude other sources of heterogeneity that might contribute to the observed timings. We speculate in the discussion about possible molecular mechanisms that would be consistent with the probability distribution we tested.

### The first non-terminal arrest is often followed by consecutive arrests

In the experimental data and our previous analyses, we observed the presence of consecutive long cell cycles corresponding to cells that managed to divide without repair, through adaptation [6]. We also proposed that more generally, a first arrest increased the probability of occurrence of a second one [5]. We thus closely examined the relationship between the first two non-terminal arrests. We first looked at the number of generations separating them and observed that the second arrest often immediately followed the first one (Fig. 4A). The overrepresentation of pairs of juxtaposed or closely positioned arrests (x = 1 or 2 in Fig. 4A) could not be explained by a random timing of occurrence of the second one as evidenced by the failure to fit a geometric distribution for a wide range of *D* (p-values of χ² goodness-of-fit tests < 0.05 most of the time; Additional file 1: Fig. S3).

We then asked whether the generation of the first arrest affected the number of divisions before the next arrest. We plotted the number of divisions between the two first arrests against the generation of the first arrest and observed no significant correlation (Pearson correlation coefficient *R* = -0.029, *p*-value = 0.77, Fig. 4B). The overrepresentation of juxtaposed first two arrests were not just coincidentally due to setting the threshold *D* at 180 min since the fraction of lineages displaying this situation varied little as a function of *D* (Fig. 4C). These results are consistent with the cells dividing while transmitting an arrest-inducing defect, such as a signalling telomere.

### The non-terminal and senescent consecutive arrests correspond to two distinct regimes

Consecutive arrests are often associated with adaptation events and, after non-terminal arrests, the eventual return to cell cycles of normal duration suggests the repair of the initial damage [6]. We therefore wondered if, for a sequence of non-terminal arrests, we could model the repair probability after a first arrest as constant, that is, we asked if the number of consecutive non-terminal arrests followed a geometric distribution (Fig. 5A). This hypothesis was not rejected (p-value > 0.05 for χ² goodness-of-fit tests) for *D* between 140 and 330 min (Fig. 5A, B). For the standard *D* = 180 min, the estimated probability of repair was 0.65 per generation. We concluded that after the initial arrest, each further division is associated with a constant probability of repair and recovery.

All lineages, whether they experienced non-terminal arrests or not, eventually enter senescence and undergo consecutive arrests before cell death. We asked whether each cell division in senescence would also have a constant probability of terminating the sequence of consecutive arrests. We found that this hypothesis of a geometric law was rejected for the standard *D* = 180 min (Fig. 5C, D). Therefore, the sequences of terminal and non-terminal arrests do not follow the same law, suggesting different underlying processes.

However, we observed that the hypothesis of geometric law for senescent arrests was not rejected for values of *D* between 200 and 350 min (Fig. 5D), values which were larger than for non-terminal arrests and more consistent with the timescale of persistent damage. Even at values of *D* where both non-terminal and terminal senescence arrests followed a geometric distribution such as 240 min, the parameter (between 0 and 1) defining the geometric laws and representing the probability of ending the consecutive arrests were different for the two processes: for example, 0.71 and 0.58 for the non-terminal and terminal senescence arrests, respectively, for *D* = 240 min. We then measured the difference in duration between non-terminal and senescence arrests and found that for a sequence of non-terminal consecutive arrests, the last long cell cycle had a duration on average similar to the preceding long cell cycles (medians of 250 min vs. 240 min, respectively; Fig. 5E). In contrast, in senescence, the last cell cycle leading to cell death was much longer than the preceding ones (medians of 730 min vs. 290 min, respectively; Fig. 5E). More generally, the cell cycle durations of non-terminal arrests were significantly shorter than for senescence arrests, even without taking the last arrest of the sequence into account (Fig. 5E).

Taken together, these results show that the dynamics of consecutive cell cycle arrests are kinetically different between non-terminal arrests and terminal senescence arrests.