The optimal strategy balancing risk and speed predicts DNA damage checkpoint override times

Abstract

Checkpoints arrest biological processes allowing time for error correction. The phenomenon of checkpoint override (also known as checkpoint adaptation, slippage, or leakage), during cellular self-replication is biologically critical but currently lacks a quantitative, functional, or system-level understanding. To uncover fundamental laws governing error-correction systems, we derived a general theory of optimal checkpoint strategies, balancing the trade-off between risk and self-replication speed. Mathematically, the problem maps onto the optimization of an absorbing boundary for a random walk. We applied the theory to the DNA damage checkpoint (DDC) in budding yeast, an intensively researched model checkpoint. Using novel reporters for double-strand DNA breaks (DSBs), we first quantified the probability distribution of DSB repair in time including rare events and, secondly, the survival probability after override. With these inputs, the optimal theory predicted remarkably accurately override times as a function of DSB numbers, which we measured precisely for the first time. Thus, a first-principles calculation revealed undiscovered patterns underlying highly noisy override processes. Our multi-DSB measurements revise well-known past results and show that override is more general than previously thought.

Introduction

Checkpoints control a fundamental trade-off between risk and speed: Whenever self-replication arrests because of errors, opportunities for producing offspring decrease because of the delay. On the other hand, delays may be necessary for correcting errors, e.g., for DNA damage repair.¹ We wondered whether this trade-off could explain checkpoint override (also known as ‘adaptation’, ‘slippage’, or ‘leakage’²), which occurs in different biological systems: Cell cycle checkpoints^3–5 are overridden in the continued presence of errors, which is associated with genomic instability and aneuploidy.^2,6–10 Superficially similar phenomena also occur in animal development.^11,12

While trade-offs in checkpoint override have been described qualitatively^6,10,13,14, quantitative theories have not been developed so far. Molecular models^15–17 do not readily point to trade-offs. Furthermore, trade-offs in sensing or kinetic proofreading^18–34, which address processing or decision making in noisy environments, do not apply since checkpoints sense DNA damage or poor chromosome-spindle attachment reliably;³ uncertainty arises from the stochastic nature of the outcomes (survival, sickness, or death).

Override of the DDC in budding yeast^3,5 is particularly suitable for quantitative, system-level analyses. In haploid cells, which cannot repair generic DSBs efficiently using homologous recombination in the absence of a template on another chromosome, DSBs remain unrepaired in a substantial fraction of cells.³⁵ An unrepaired DSB causes an ≈8 hr arrest at the G2/M checkpoint, after which the cell cycle proceeds.^13,36,37 Arrests due to DNA damage caused by X-rays⁶ or telomere dysfunction^9,37 are also overridden.

A number of mutations such as cdc5-ad³⁷ are known to prevent or reduce checkpoint over-ride.^37–39 While this highlights the possibility of a biological function, the benefit of DDC override for wild-type yeast cells has not been established: DDC override did not increase viable offspring of haploid yeast⁶ unless DNA repair (rad52Δ) or telomere maintenance (tlc1Δ) were blocked^6,10,14. Furthermore, DDC override was observed only with exactly one endonuclease-generated DSB but not with two¹³, indicating that override is not a universal outcome after DNA damage in yeast. Thus, the functional relevance of override has remained unclear.

Additionally, the statistics of repair events after long arrests are unknown. Generally, the kinetics and efficiency of non-homologous end joining (NHEJ)⁴⁰, which is the dominant repair pathway for generic DSBs in haploid yeast, have been measured in bulk culture^41–45, which is too coarse to capture late, rare repairs.

Results

Theoretical results

Arrest implies an exponential fitness penalty

To compute the effects of different checkpoint strategies, we compare an arrested cell to other cells in the population. The two canonical scenarios of population genetics are exponential growth (Fig. 1 A) and stochastic birth-death processes in a fixed-size population (Fig. 1 B).⁴⁶ In exponential growth, the expected number of progeny of a checkpoint-arrested cell is reduced by 2^−t/T (P₂+P_1/2), where t is the arrest time and P_i the probability that i = 1 or 2 progeny proliferate after the arrest (Fig. 1 A). (We only consider two outcomes, survival or death, see Supplementary Note 1.) In two classical fixed-size population models, the Wright-Fisher and the Moran models⁴⁶, the expected number of progeny after checkpoint arrest also decays exponentially in t (Supplementary note 2); thus, the e^−t/τ penalty emerges generically and τ absorbs model-specific parameters, e.g., τ = T/log 2 for exponential growth.

Fig. 1. Scenarios for the self-replication dynamics of a checkpoint-arrested cell (magenta circle) in a population of cells (white) with generation time T.

A: The exponential growth with doubling time T is delayed by a checkpoint arrest of duration t (= T in this illustration) after which both, one, or no progeny (yellow circles) remains viable. Only the case where both progeny are viable is illustrated. B: In a population of fixed size N, random birth-death processes change the proportions of different clones, as in the Wright-Fisher or Moran models.

Winning strategy maximizes the arithmetic mean of the future progeny

To evaluate how different strategies influence an organism’s reproductive success, one of two quantities is commonly taken as a fitness function, i) the number of offspring averaged over all possible outcomes or ii) the expected number of offspring in a typical scenario.^25,46,47 For example, in fluctuating environments, (ii) is thought to be more relevant for evolution^46,47,49, favoring bet hedging⁴⁸. However, when errors occur randomly in a population, as we assume for DNA damage, (i) is the appropriate fitness function, see Supplementary note 3 and Fig. S1.

Checkpoint strategy parameters

In addition to the arrest penalty e^−t/τ, the following parameters are required:

• The current number of errors as a function of the arrest time t is denoted by E(t), representing, for example, the number of DSBs. E₀ = E(0) denotes the initial number of errors, and E(t) decreases with the arrest time t as errors are corrected.

• The probability that the errors are repaired at times {t_i} = {t₁, … ,t_E0 is denoted by r({t_i}). To simplify, we will assume that an error is repaired with a cumulative probability density function ρ(E, t) that depends on the number of remaining errors E(t) and the arrest time t. Thus, different repair events can be correlated but their likelihood and kinetics are identical. However, to compare with data, it will suffice to assume independent repair probabilities for each error, ρ(E, t) = ${\rho }^E(t)$.

• The probability of survival if a checkpoint is overridden after an arrest of duration t given that repairs take place at times {t_i} is denoted by s (t∣ {t_i}). A repair time t_i after t (t_i > t) indicates that error i was not fixed before checkpoint override. To simplify, we will write s(E) when the survival probability only depends on the number of remaining errors E(t). If each error reduces the survival probability equally, we will write s(E) = σ^E.

• The probability of advancement, that is, of override or continuation of the self-replication cycle, is a (t∣ {t_i}), which represents the checkpoint strategy which we seek to compute.

E(t), r(t∣ {t_i}), and s(t∣ {t_i}) are determined by biochemical processes, which we consider fixed. a (t∣ {t_i}) represents the behavior of the checkpoint which could maximize fitness.

Calculating the optimal checkpoint strategy

Combining the above probabilities, the fitness coefficient f represents the change in the expected number of progeny due to damage and checkpoint arrest (Supplementary note 4):

$$f[a]={{{\displaystyle \int }}^{\text{}}}_0^{\infty }dt\underset{E_0}{\overset{E=1}{{{\displaystyle \prod }}^{\text{}}}}d{t}_{E}r\left(\left\{t_i\right\}\right)s\left(t\mid \left\{t_i\right\}\right)a\left(t\mid \left\{t_i\right\}\right)e^{-t/\tau }.$$ (1)

If we make two simplifications, that each error is identical with respect to repair and post-override survival, the calculation of f as a functional of a (t∣ {t_i}) can be represented graphically (Fig. 2 A). In the error-time plane, two biologically plausible checkpoint strategies, the ‘timer’ override strategy and the ‘error-threshold’ override strategy (Figs. 2 B, C and Supplementary notes 5, 6), can be represented as vertical and horizontal lines, respectively. If the checkpoint follows either of these strategies, the cell cycle advances when these lines are reached.

Fig. 2. Pictorial representation of checkpoint strategies.

A: A checkpoint-arrested cell (red) moves to the right as time increases and moves up as repairs occur probabilistically. As prescribed by the checkpoint strategy, the cell divides into two cells (yellow). The progeny may or may not be viable. B: In the timer strategy, once a checkpoint is activated, the cell arrests until time t′ and then divides. The advancement boundary is a vertical line. C: In the error threshold strategy, cells arrest until E′ errors remain and then divide. The advancement boundary is a horizontal line. D, E: The optimal strategy could, in principle, be probabilistic (panel D) or deterministic (panel E).

However, in principle, the optimal checkpoint strategy could be stochastic; that is, at any point, the optimal decision could be to advance the cell cycle with a probability between zero and one. Nevertheless, in Supplementary note 7 and Fig. S2, we show that the optimal strategy is not stochastic (gray region in Fig. 2 D) but must be deterministic (dashed lines in Fig. 2 E). Thus, the optimal a (t∣ {t_i}) is a Dirac δ-function in t; the optimal strategy is represented by a sharp boundary (Fig. 2 E), which we refer to as the ‘optimal advancement boundary’.

The problem of solving for this boundary is, in principle, complicated because of the branching nature of the decisions; a decision at time t and with E(t) errors needs to be compared to decisions at an infinite number of future points. However, the optimal strategy can be found recursively starting from the future and working backwards by analyzing points from right to left (future-to-present) and then from top to bottom (all-to-no errors corrected) (Supplementary note 8, Figs. S3, S4). In this way, we arrive at the key result, which determines the optimal override time t* implicitly:

$$\frac{{\partial }_t\rho (t^{\ast })}{1-\rho (t^{\ast })}\approx \frac{\sigma }{E\tau }.$$ (2)

Experimental results

Experimental system

To apply the theory, we needed to measure the parameters in Table 1. For this, we created budding yeast strains, which could be blocked in pre-Start (G1) phase prior to DNA replication by adding methionine to the growth medium (+Met) or which could be released to proliferate by removing methionine (-Met), see Methods.⁵⁰ A GAL1pr-HO construct was used to express the endonuclease Ho, which creates DSBs at locations in the genome where we inserted the Ho recognition sequence,^35,45,51 by switching to galactose (Gal) medium. (The endogenous HO cut site at the mating type locus (MAT) was abolished by synonymous mutations (MATα-syn).)

Table 1. Correspondence between theoretical and experimental parameters.

Variable	Description	Experimental representation	Way of measurement or control
t	arrest time	arrest time	clock
E(t)	number of errors	double-strand breaks (DSBs)	generated by HO induction, which is either continuously induced in Gal medium or briefly induced by a short Gal pulse and monitored with the DSB sensor
$\frac{{\partial }_t\rho }{1-\rho }$	conditional repair probability density function	DSB repair probability per time, normalized with respect to future/unrealized repairs	measured by microscopy or combined flow cytometry and plating assay
σ	survival probability with one error	survival probability with one DSB	measured by combined flow cytometry and plating assay

To detect DSB induction and repair, we created a DSB sensor by inserting an HO cut site (HOcs) and a destabilized version of the yellow fluorescent protein gene yEVenus between the promoter and the ORF of the strongly expressed, non-essential ADH1 gene, creating an ADH1pr-HOcs-yEVenus-ADH1 fusion (Fig. 3 A). (We refer to the yEVenus protein as YFP for brevity.) When the sensor is cut, YFP fluorescence should be low; when the DSB is repaired, fluorescence should turn back on. The DSB sensor allowed us to induce HO briefly in galactose medium, switch to the preferred carbon source glucose, and focus on cells with DSBs based on fluorescence. Continuous HO induction was not needed to study override, unlike in past studies.

Fig. 3. A DSB-sensor (ADH1pr-HOcs-yEVenus-ADH1) reports the presence of a DSB.

A: Design of the sensor. B: Basic experimental protocol illustrated with clnΔ MET3pr-CLN2 GAL1pr-HO ADH1pr-HOcs-yEVenus-ADH1 strain. Raff = raffinose, Gal = galactose, Glu = glucose. C: YFP time courses in cells which additionally carry the cdc5-ad mutation. Green/black: cells that did/did not complete nuclear division. Four dead cells indicated by dotted lines. Switch to Glu-Met occurred at time 0 h. (n = 75) D: Cells 5 hrs after media switch to Glu-Met. Nuclear marker Htb2-mCherry in red. Scale bar: 5 μm. E: Examples of YFP time courses for cells that divided nuclei late but seemed to repair the DSB early. Time point of rise in fluorescence indicated by a blue circle; nuclear division indicated by a red circle.

In summary, the basic genotype for the experiments in this study is cln1-3Δ MET3pr-CLN2 GAL1pr-HO ADH1pr-HOcs-yEVenus-ADH1 HTB2-mCherry MATα-syn, and only modifications of this strain are highlighted explicitly.

DSB repair distributions

To apply the checkpoint theory, we needed to measure the timing of DSB repairs, i.e., the repair probability distribution ρ(t in Eq. (2). We used the protocol depicted in Fig. 3 B: i) cells were arrested in G1, ii) Ho was induced to cause a break in ADH1pr-HOcs-yEVenus-ADH1, iii) the cell cycle was restarted, and iv) a new cell cycle was prevented. Step (iv) allowed us to monitor cells under the microscope or in bulk culture for many hours without intact cells overcrowding the population. This was necessary to detect late, rare repair events. To measure DSB repair kinetics, we used cells which additionally had the override-suppressing cdc5-ad³⁷ mutation.

To characterize the system, we performed single-cell fluorescence microscopy and analyzed the images with YeaZ⁵² (Fig. 3 C-E, Supplementary note 9, Fig. S5). The long tails of the over-ride distributions (Fig. S6) were potentially biologically critical but analyzing the rare events that the tail represented was challenging by microscopy. Thus, we took advantage of the strong signal from the ADH1pr-HOcs-yEVenus-ADH1 reporter for fluorescence-activated cell sorting (FACS) (Figs. 4 A, S7). At +4 hrs after the switch to Glu-Met (Fig. 3 B), >10⁶ YFP- cells were isolated.

Fig. 4. Measurements of late DSB repair statistics and survival rates after checkpoint override.

A: Schematic of FACS experiments to measure the tail of the DSB repair time distribution. Horizontal and vertical axes on FACS plots are Venus-A and FSC-A, respectively. See Fig. S7 for more details. B: Measurements of the fraction of repaired cells (YFP+ colonies) compared to the number of cells plated (5 ⋅ 10⁴). Each circle represents the fraction of YFP+ colonies among 50k cells on one plate. Circles are stacked horizontally when experimental replicas had the same numbers of colonies. The thick black line represents a spline fit to the mean values for cdc5-ad cells at each time point. C: The probability distribution function of repair (black dashed line) is the negative of the derivative of the fit in panel B. Horizontal lines represent σ/(Eτ) for E = 1 (red), 2 (magenta), 3 (orange), 4 (yellow).

To avoid contamination by YFP+ cells, we chose the gates for FACS conservatively, based on the distribution of fluorescence in a no-cut-site control strain (Fig. S7 C). Then, from this population, batches of 50 000 YFP- cells were sorted and plated on Glu-Met plates every 2 hours. On Glu-Met plates, cells could generate colonies if they repaired the DSB. The double sorting, once at 4 hrs and once before plating, served to minimize the possibility that sorting errors let bright (YFP+) cells slip through as YFP- cells. After 3 days, we counted the number of all colonies and the subset of YFP+ colonies on the plates. With cdc5-ad yku70Δ cells, substantially fewer colonies emerged, showing that nearly all colonies represented YKU70-dependent repairs by non-homologous end joining (Figs. 4 B, S7 B), and not technical artefacts; thus, stray low-fluorescence cells, which did not have a DSB and which would contaminate the results, were dead or negligible. The number of YFP+ colonies decreased rapidly between the 6 h to 12 h time points (Fig. 4 B), showing that some YFP- cells had repaired the DSB between sorting events, had turned YFP back on, and had left the population of YFP- cells between time points. We estimate that for a typical YFP- cell, it took approximately 30 min to leave the YFP- population after repairing the DSB (see Methods). The total number of colonies, that is, YFP+ and YFP- combined, did not drop appreciably between the 6 h to 16 h time points (Fig. S7 D); this shows that the precipitous drop in YFP+ cells with time was not because cells were generally dying during the long arrests.

To extract the repair probability, we fit a spline through the mean fractions of repaired cells (Fig. 4 B). The negative of the derivative of the fit represents the conditional probability density function for DSB repair, ∂_tρ/(1 − ρ), indicated by a black dashed line in Fig. 4 C.

Statistics of survival after checkpoint override

Another important quantity for the risk-speed trade-off is the lethality of overriding the checkpoint with unrepaired damage (Eq. (2)). We measured the survival probability s(E) for one DSB (s(1) = σ) by repeating the experiment in Fig. 3 with wt-CDC5 cells. We plated CDC5 cells at the 6 h time point, just before checkpoint override (Figs. S6 B, S7 E). Subtracting the survival probability of cdc5-ad cells, also sorted at +6 h, we arrived at the mean survival probability σ. The best estimate of this quantity is represented, after dividing by the cell cycle time τ = 90 min log 2, by the red horizontal line in Fig. 4 C.

Quantitative experimental validation of theoretical predictions

We begin with a comparison of theory and experiment for one DSB (E = 1). The red (σ/τ) and black dashed lines (∂_tρ/(1 − ρ)) overlap well in the time window 6-9.5 h (95% bootstrap confidence intervals: ∂_tρ/(1 − ρ) is within 33% of σ/τ in the 6-10 h window, see Methods). Remarkably, the measured override time is 7.4 ± 1.4 h (mean ± STD) (n = 188, Figs. 5 A, S6 B). (Note that microscopy results reflect bud-to-nuclear-division times, to which the time from +0 until budding has to be added (≈50 min) to compare to FACS results.) Our measured override times for 1 DSB agree with reported override times of ≈8 h.^13,36,37 Thus, the optimal checkpoint theory explains the mean as well as the width of the override distribution for 1 DSB.

Fig. 5. The optimal checkpoint theory predicts DNA damage checkpoint override times.

A-F: Histograms of budding-to-nuclear-division probabilities scored by timelapse microscopy for single cells with the indicated cut sites (n = 188, 75, 88, 38, 85, 92). Panel A contains the same data as in Fig. S6 B but with bin sizes adjusted for easier comparison with the other panels. G: Comparison of the optimal checkpoint theory with the experimental data. Gray circles indicate observed checkpoint override times for different numbers of induced DSBs. Same experimental data as in panels A-F. Red bars represent means. Error bars indicate +/- SEM. The optimal override time is represented by dashed horizontal lines. The optimal override times were calculated by solving Eq. (2) using the measured parameters of the theory, which are shown in Fig. 4 C. The values are quoted in the main text. H: Representation of the results in panel G as the optimal advancement boundary and the observed advancement boundary (mean checkpoint override times for different numbers of DSBs). Horizontal red error bars indicate 95% confidence intervals. G, H: 50 min have been added to the means from the bud-to-nuclear-division histograms because the FACS time points are with respect to time 0 in the experimental protocol.

Next, we predicted the optimal override times for multiple DSBs. Assuming that each DSB reduces the survival probability independently, s(E) = σ^E, we predicted the optimal override time for multiple DSBs using Eq. (2). ∂_tρ/(1 − ρ) intersects σ/(Eτ) at approximately 13.8 h for E = 2 DSBs and 16.2 h for E = 3 DSBs. A linear extrapolation of the last half hour of ∂_tρ)/(1 − ρ) intersects σ/Eτ at 19.1 h for E = 4 DSBs.

To create multiple DSBs, we inserted additional cut sites at the URA3 locus or in the promoters of DLD2 or MIC60. URA3 was chosen for comparison with past work¹³, and DLD2 and MIC60 are in long genomic regions that are not essential.⁵³ Because in these strains, not all cut sites had a fluorescent reporter, we had to modify the protocol in Fig. 3 B and kept cells in galactose medium after break induction, see Methods.

Strikingly, our results agree with the predictions of the optimal checkpoint theory (Fig. 5 B-E): For two DSBs at ADH1 and URA3 or, alternatively, at DLD2 and MIC60, we observed that 80% of cells overrode with bud-to-nuclear-division time 13.4 ± 0.4 h (mean ± SEM, n = 75) (Fig. 5 B) for the former pair of cut sites and 13.6 ± 0.4 h (mean ± SEM, n = 88, bud-to-nuclear division) for the latter (Fig. 5 C). For three and four DSBs, the fraction of overriding cells decreased (34% and 42%, respectively) while the proportion of clearly dead cells, e.g., cell wall ruptured, increased. Nevertheless, for cells which performed nuclear divisions, we measured override times of 16.6 ± 0.7 h (mean ± SEM, n = 38, bud-to-nuclear division) for 3 DSBs and 20.2 ± 0.6 h (mean ± SEM, n = 85, bud-to-nuclear division) for 4 DSBs (Fig. 5 D, E). Note that these shifts in override timing as a function of DSB numbers cannot be explained by the greater lethality of increasing numbers of DSBs since non-overriding or dead cells are excluded from the averages. Thus, the match between theory and experiment is remarkably close (Fig. 5 G, H), especially considering the disparate nature of the experiments involved and the substantial cell-to-cell variability. (Also, compare to the alternative timer (Fig. 2 B) and the error-threshold (Fig. 2 C) strategies (Fig. S8).)

We have thus far compared theory and experiments in cells with a fixed number of DSBs. We next tested whether the override decision was adjusted dynamically. We created a strain in which one DSB would be repaired efficiently while the DSB at ADH1pr-HOcs-yEVenus-ADH1 would not, as before. For the repairable cut site, we switched the Ho-resistant MATα-syn out for wild-type MATa. When MATa is cut, the DSB is repaired efficiently by homologous recombination utilizing the HML and HMR cassettes.⁵⁴ We returned to our original experimental protocol in which Ho is shut back off (Fig. 3 B) but extended the GAL1pr-HO induction to 3 hrs to ensure high cut rates. Control cells without ADH1pr-HOcs-yEVenus-ADH1 divided their nuclei quickly after switching to Glu-Met (median: 75 min), showing that, indeed, MATa was repaired efficiently. However, YFP- cells with both MATa and ADH1pr-HOcs-yEVenus-ADH1 overrode after 8.1 ± 0.2 h (mean ± SEM, n = 92, bud-to-nuclear division, Fig. 5 F), as expected for 1 DSB and unlike for 2 DSBs (Fig. 5 A-C). These results indicate that cells compute the override time continuously as they move in the error-time plane and dynamically adjust their override time based on E(t), the number of DSBs at any given time (Fig. 2 E).

Override is an advantageous strategy

The close match between optimal and measured override times suggests that checkpoint override may be advantageous. To demonstrate this directly, we used FACS to compare override-competent CDC5 and -incompetent cdc5-ad cells with an unrepaired DSB at ADH1 at the 6 h time point, just before checkpoint override. We found a significantly higher probability of survival with 1 DSB for CDC5 cells versus cdc5-ad cells (Fig. S9) showing that override is beneficial for cells with wild-type repair genes when they continue to have an unrepaired DSB late into the checkpoint arrest. Because we expect the survival likelihood to be multiplicative, we could not verify the survival advantage of override with two or more DSBs because current FACS instruments cannot sort the needed number of cells (>10⁸) quickly enough.

DNA resection and checkpoint override

How the amount of DNA damage sets the override time is currently unclear.³⁹ Lee et al. proposed that the amount or rate of DNA resection determines the override time.¹³ However, further research into this question is needed.³⁹ To reexamine this in our system, we introduced the sgs1Δ exo1Δ double mutations, which nearly eliminates resection,⁵⁵ into our strains. In these cells, we found a shortening of the override time, 7.0 ± 0.3 h (mean ± SEM, n = 114) for 1 DSB and 8.6 ± 0.6 h, n = 36 for 2 DSBs (Fig. S10). The shorter – but robust – checkpoint arrests suggest that both DNA resection and the number of DSBs set the override times.

Discussion

We present a theory of checkpoint strategies, which reflects the fundamental balance between risk and speed and which led us to discover timing hierarchies in DNA damage checkpoint override. The theory is based on the simple principle of offspring maximization, which during the course of evolution would presumably have been performed by natural selection.

We showed mathematically that a deterministic strategy is optimal, which is based on the result that the average fitness has to be maximized. Such deterministic strategies, which maximize the average payoff, also arise in game theory and are referred to as ‘pure strategies’, where they may represent Nash equilibria, e.g., in Prisoner’s dilemma or Stag Hunt.⁵⁶

A deterministic strategy being optimal can be reconciled with the observed broad distribution of override times (Figs. 5 A-F, S6 B) by noting that the DSB repair probability is roughly flat for 6 h-10 h and then decays slowly (Fig. 4 C), suggesting that the differences in fitness between different override times may not be large. Indeed, the peaks in the fitness functions are wide (Fig. S11).

The optimal strategy did not need to be computed by simulations. Instead, we found a recursive solution, which shares similarities with dynamic programming. In this approach, the overall problem is broken down into sub-problems, which are solved, the solutions are stored and then used to solve the dependent problem.⁵⁷ Analogously, to solve for the optimal strategy, one needs to go from the future to the present and solve from 0 to E₀ errors. The algorithm requires that the damage is discrete, that there is a finite number of errors, that errors are identical, that each error reduces the survival probability independently, and that there is a time point when waiting ceases to be worthwhile.

It is also interesting that our theoretical predictions compelled us to reexamine the current view^9,10,39 that override only occurs for 1 DSB and not 2 DSBs¹³, which implied that checkpoint override is not a universal response to DNA damage in wild-type cells. Instead, we found override to occur with up to 4 DSBs, demonstrating that DNA damage checkpoint override is more general than previously thought. In Supplementary note 10, we discuss the differences with previous work.

The quantitative, system-level study we present here complements efforts to elucidate checkpoint override at the molecular level.^5,39 We expect traditional bottom-up and our top-down approaches to drive each other forward.

Methods

Strains

All experiments were performed with budding yeast strains with the W303 background. Genetic manipulations and crosses were performed by standard methods. To avoid inserting extraneous DNA which could affect DNA repair processes or the checkpoint, we repeatedly used the URA3-insertion/5-FOA-pop-out method to leave no plasmid or marker sequences behind in the genome. We deleted the Start cyclin CLN1,3 ORFs and replaced the CLN2 promoter by the MET3 promoter. We abolished the HO cut site in cells of mating type α by 10 synonymous mutations in the alpha1 gene (MATα-syn). We added a PEST degron to yEVenus to increase its dynamic responsiveness. The cdc5-ad mutation was cloned into our strain background from a strain given to us by Achille Pellicioli. A GAL1pr-HO plasmid was given to us by Eric Alani.

The basic genotype for all strains was cln1Δ0 cln2Δ0::MET3pr-CLN2 (promoter replacement) cln3Δ0::GAL1pr-HO ADH1pr-HOcs-yEVenus-ADH1 HTB2-mCherry::HIS5 MATα-syn. The modifications to this strain are indicated in list below. After constructing the basic strain, we performed three backcrosses with our wild-type W303 strains to reduce the likelihood of mutations.

Strain list (genotypes indicate modifications to the basic genotype):

• Name: AS18, Genotype: rad5-G535R

• Name: ET47, Genotype: RAD5

• Name: AS20.1, Genotype: cdc5-ad rad5-G535R

• Name: ET45, Genotype: cdc5-ad RAD5

• Name: AS30, Genotype: cdc5-ad yku70Δ0 rad5-G535R

• Name: ET46, Genotype: cdc5-ad yku70Δ0 RAD5

• Name: ET44, Genotype: ADH1pr-yEVenus-ADH1 (no ADH1pr-HOcs-yEVenus-ADH1) RAD5

• Name: AS31.9, Genotype: ura3Δ0::HOcs::KanMX rad5-G535R

• Name: AS32.1, Genotype: ura3Δ0::HOcs::KanMX RAD5

• Name: ET32, Genotype: MIC60pr::HOcs DLD2pr::HOcs (no ADH1pr-HOcs-yEVenus-ADH1) rad5-G535R

• Name: RD58, Genotype: MIC60pr::HOcs DLD2pr::HOcs (no ADH1pr-HOcs-yEVenus-ADH1) RAD5

• Name: RD53, Genotype: ura3Δ0::HOcs::KanMX MIC60pr::HOcs RAD5

• Name: RD54, Genotype: ura3Δ0::HOcs::KanMX MIC60pr::HOcs DLD2pr::HOcs RAD5

• Name: AS14.2, Genotype: MATa (no ADH1pr-HOcs-yEVenus-ADH1) RAD5

• Name: AS35.2, Genotype: MATa RAD5

• Name: AS106.1, Genotype: MIC60pr::HOcs exo1Δ sgs1Δ

• Name: AS104.1, Genotype: ura3Δ::HOcs::KanMX DLD2pr::HOcs exo1Δ sgs1Δ HOcs stands for the 30 bp sequence: TTCAGCTTTCCGCAACAGTATAATTTTATA.

• We performed experiments with cells carrying the W303-specific rad5-G535R mutation [1] as well as with cells with a corrected RAD5 gene wherever we indicate both versions of the strain in the list above. Since we observed no systematic or noticeable differences between results with rad5-G535R and RAD5, we pooled the measurements for higher statistical certainty. All other experiments were performed with cells with a corrected RAD5 gene only.

Media switch protocol

For almost all experiments, we followed the protocol illustrated in Fig. 3 B. To measure checkpoint override times in cells with two or more cut sites, where we could not monitor all breaks with a fluorescent DSB sensor, we modified the protocol as follows. We kept GAL1pr-HO on by switching to Gal-Met medium instead of Glu-Met after time 0, ensuring that cut sites were recut and not repaired.³⁵ With successful repairs suppressed, practically all nuclear divisions now occurred after checkpoint override. The alternative possibility of error-prone repair preventing recutting is too rare (≈ 10⁻³) [1] to affect our statistics. We also assessed the effect of continuous galactose medium on override times for one DSB and found only a small change, 8.9 ± 0.3 h (mean ± SEM, n = 53, not shown).

Microscopy

Microscopy experiments were performed with commercial microfluidic chips. Timelapse recordings were carried out with a 60x objective and a Hamamatsu Orca-Flash4.0 camera. The interval between images was 10 min except for experiments with ≥2 cut site where the time between images was increased to 15 min to decrease potential phototoxicity during the long arrests. The override time was recorded as the first time point at which the Htb2-marked nuclei separated. In microscopy experiments where we released cells from Gal+Met into Gal-Met (Fig. 5 A-C), there were sometimes 1-2 cells that divided immediately and then arrested in the next cycle. We excluded these if this first division occurred within the first 100 min.

Image processing

For segmenting the microscopy images, we used the YeaZ convolutional neural network and GUI⁵².

FACS

Fluorescence-activated cell sorting was performed with a Sony SH800S instrument. The protocol and all the gates are illustrated in Figs. 4 and S7.

The instrument was set to room temperature for sorting events. A 100 μm microfluidic sorting chip was used. The purity level was set to ‘ultra purity’. We used the excitation/emission filters for ‘Venus’.

Before each sort, cells were centrifuges at 900 g for 1.5 min, medium was removed to concentrate cells, and the cell culture was sonicated for 8 sec. This was done to speed up the sorting. At the 4 h time point, 1.0 − 1.8 · 10⁶ YFP- cells were isolated per biological replica, which took about 20 − 40 min. After sorting, cells were centrifuged, the sheath medium was discarded, and ≈5 ml SCD+Met medium was added. Cells were always kept on a nutator at 30° C between sorting events. The second sorting events of 50k YFP- cells at 6 h, 8 h, 10 h, 12 h, 14 h, or 16 h took 3 − 8 min. After the second sorting, 100 μl of SCD-Met medium was immediately added to the isolated cells and each batch of 50k YFP- cells was spread on a different Petri dish with SCD-Met agar medium.

To estimate the time it took cells to escape the YFP- gate after repairing the cut site at ADH1pr-HOcs-yEVenus-ADH1, we noted that at the 4 h sorting event, YFP- and YFP+ cells showed fluorescence levels of 1700 ± 700 [au] (median ± SD) and 7500 [au] (median), respectively. The speed of increase of fluorescence per hour therefore was about 1500 [au]/h, which is more than twice the standard deviation of fluorescence values in the YFP- gate. Hence, we estimate the escape time for repaired cells from the YFP- gate to be about 30 min.

Fit to data

In Fig. 4 B, we fit a ‘SmoothingSpline’ through the means at each time point using the Matlab ‘fit’ function with parameter ‘SmoothingParam’ set to 0.1. A spline was used to avoid any biases regarding the functional form, e.g., type of decay, we might have expected. The smoothing parameter was chosen without fine-tuning; thus, it only has one significant digit. It was chosen among such one-significant digit values because it was the highest such number (generating the least amount of smoothing) that removed the bumps from the fit.

Statistical tests

All p value tests were one-tailed. For the confidence interval analysis of the repair probability versus σ/τ in Fig. 4 C, we picked 10³ random alternative means for each time point (6 h, 8 h, 10 h, 12 h, 14 h, and 16 h) by bootstrapping. We fit the same spline as described above through 10³ combinations of these means and computed the derivative. We calculated how close the closest 90% and 95% of these conditional repair probability densities were to the σ/τ line between the 6 h and 10 h interval, which gave us the values quoted in the text (25% and 33%, respectively).

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.