The synergy of damage repair and retention promotes rejuvenation and prolongs healthy lifespans in cell lineages

Damaged proteins are inherited asymmetrically during cell division in the yeast Saccharomyces cerevisiae, such that most damage is retained within the mother cell. The consequence is an ageing mother and a rejuvenated daughter cell with full replicative potential. Daughters of old and damaged mothers are however born with increasing levels of damage resulting in lowered replicative lifespans. Remarkably, these prematurely old daughters can give rise to rejuvenated cells with low damage levels and recovered lifespans, called second-degree rejuvenation. We aimed to investigate how damage repair and retention together can promote rejuvenation and at the same time ensure low damage levels in mother cells, reflected in longer health spans. We developed a dynamic model for damage accumulation over successive divisions in individual cells as part of a dynamically growing cell lineage. With detailed knowledge about single-cell dynamics and relationships between all cells in the lineage, we can infer how individual damage repair and retention strategies affect the propagation of damage in the population. We show that damage retention lowers damage levels in the population by reducing the variability across the lineage, and results in larger population sizes. Repairing damage efficiently in early life, as opposed to investing in repair when damage has already accumulated, counteracts accelerated ageing caused by damage retention. It prolongs the health span of individual cells which are moreover less prone to stress. In combination, damage retention and early investment in repair are beneficial for healthy ageing in yeast cell populations.

Note that the value of R for unlimited repair is an approximation for R → ∞. P and D are bounded by 1 and within that regime the deviations are neglectable.

Create lineage
In most simulations, we generate lineages up to 3 generations. The ODEs of a cell are not coupled to any other cell. Therefore, each cell can be solved individ-ually. In case there is some coupling between the cells, e.g. time or shared food resources, all ODEs of alive cells have to be solved at the same time instead and dead cells have to be removed from the system.

Finding wildtype cells in the parameter space
In order to find parameter combinations in the k 1 , k 2 and re space that lead to 24 divisions we use an iterative process with adaptive step size. Typically, we fix two of the three dimensions to a value and find the third one. Algorithm 2 shows an example for adapting k 2 if all other parameters are set.
Run single-cell model 20: if (rls == rls * or ∆k 2 < δ) then 29: wildtype is found if rls == rls * and the corresponding parameter is k 2 In the same manner, one can start with many cells and allow for parameter variations in k 1 and k 2 according to non-linear mixed effects (equation (2) in the paper) and check if the average value of the respective parameter is around the wanted replicative lifespan with some tolerance. The algorithm does not necessarily have to be 24 as it is for the wildtype yeast cells, but works for any preset replicative lifespan. Note that when varying the retention factor, it is computationally more efficient to start from a high value and reduce it stepwise. Corresponding signs have to be adapted.
In the paper, we often compare four cases, which correspond to following ii parameter combinations. We chose to fix the damage formation rate k 1 = 0.4 and adapt the repair rate correspondingly (see also S1 Fig).
repair mechanism retention (re, k 1 , k 2 , R)  The range of values of k 1 and k 2 (Fig 2) generating wild-type cells is in agreement with previous computational and experimental work [2][3][4] underlining the validity of the conclusions. However, often the estimation relies on various assumptions. Most importantly, experiments usually contain one specific type of damage such that we have to assume that the rate is similar even for other types. Further parameters are sometimes estimated indirectly and we have to assume that the underlying dependency is linear, which is not necessarily true. Clegg and colleagues implemented a similar damage formation term using values in the same range as we do [2]. The response of chaperones to damage formation measured by Saarikagas and colleagues (≈ 0.1 1 h ), can be interpreted as the increase of the damage while the clearance rate of protein deposits (≈ 0.07 − 0.15 1 h ) as the repair rate [4]. Further, the estimated aggregation formation rate by Paoletti et al (≈ 0.21 1 h ) can be interpreted as the damage formation rate [3]. In E.Coli, another organism often used in ageing studies, a value of ≈ 0.72 1 h for the rate of protein misfolding has been obtained [5].
Note that k 1 and k 2 in our model are non-dimensionalised, such that these values have to be multiplied by the growth rate µ (≈ 0.5 1 h , estimated in [6] assuming full availability of resources) to go back to full dimension and to be compared with experimental values. Consequently, we can justify that the order of the values in our model is realistic.

Average initial conditions in a cell lineage
For all population studies the founder cells start with initial conditions P (0) and D(0) which are specific to a set of parameters. Starting with an average cell gives rise to more realistic populations and facilitates the analysis since it is not necessary to simulate a large number of generations to get a representative view of the populations. The initial conditions are found according to algorithm 3. The algorithm converges independent of the age of the founder cell up to a iii certain precision. With = 10 −3 no numerical issues were faced and the values are sufficiently precise for our purpose.