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The authors have declared that no competing interests exist.

Conceived and designed the experiments: YLC AB KP. Performed the experiments: YLC KP. Analyzed the data: YLC KP. Contributed reagents/materials/analysis tools: YLC KP. Wrote the paper: YLC AB KP.

It is well established that individuals age differently. Yet the nature of these inter-individual differences is still largely unknown. For humans, two main hypotheses have been recently formulated: individuals may experience differences in aging rate or aging timing. This issue is central because it directly influences predictions for human lifespan and provides strong insights into the biological determinants of aging. In this article, we propose a model which lets population heterogeneity emerge from an evolutionary algorithm. We find that whether individuals differ in (i) aging rate or (ii) timing leads to different emerging population heterogeneity. Yet, in both cases, the same mortality patterns are observed at the population level. These patterns qualitatively reproduce those of yeasts, flies, worms and humans. Such findings, supported by an extensive parameter exploration, suggest that mortality patterns across species and their potential shapes belong to a limited and robust set of possible curves. In addition, we use our model to shed light on the notion of subpopulations, link population heterogeneity with the experimental results of stress induction experiments and provide predictions about the expected mortality patterns. As biology is moving towards the study of the distribution of individual-based measures, the model and framework we propose here paves the way for evolutionary interpretations of empirical and experimental data linking the individual level to the population level.

Aging is a widespread phenomenon across the tree of life. From yeast to humans, mortality changes over age have been widely documented. Interestingly, all individuals are not equal with respect to the aging process: large variability in individual life span has been reported, even in clonal populations. Understanding the nature of these differences is of great interest for medical research. So far, two hypotheses have been proposed: individuals may differ in their aging rate or timing. Here, we show that these two hypotheses can reproduce experimental and empirical mortality patterns as a result of natural selection. We also predict the corresponding population heterogeneity in aging. Many studies define subpopulations ad hoc, the work we present provides insight into a more accurate description of inter-individual differences in aging. Finally, our analysis also predicts the modifications of these mortality patterns under stressful conditions. This exploration reproduces experimental data obtained with heat shocks and permits to foresee new mortality patterns that could be observed with other perturbations.

Aging can be generally defined as age-related changes in a set of variables, from growth rate to reproductive effort, which influence the fitness of an organism. Aging is a multiscale process which can be measured at almost every level of the individual organism. Individual metrics of aging include a broad range of processes from damage to DNA and proteins

Age-specific mortality is arguably one of the most documented measure of aging. Since Gompertz' seminal work on human data

The nature of inter-individual differences in aging is crucial to study and forecast a population's life expectancy, but is as yet still largely unknown. For humans, two main hypotheses have been proposed recently

Because of these exponential patterns, differences in aging rates and timing can be visualized respectively as changes in slope and shift in mortality patterns

These mortality curves are population measures: they result from the aggregation of individual's aging. Biodemographic studies of aging have shown that specific population heterogeneity can reproduce the main features of qualitatively different mortality patterns, such as late-age mortality plateaus

In our model, we include Gompertz aging, that is exponential increase in mortality hazard with age, and implement heterogeneity in aging rates and timing. Population heterogeneity is allowed to evolve over generations under the influence of life-history trade-offs. Following the disposable soma theory

In section ‘Transitions in mortality curves’, we discuss predictions concerning population heterogeneity and the corresponding mortality patterns under different assumptions about inter-individual differences. We compare our results to mortality patterns of yeast, flies, worms and humans respectively. In section ‘Influence of mutation rate’, we test the robustness of our results with respect to mutation rates and highlight new features for the distribution of heterogeneity. Finally, in section ‘The notion of subpopulation’, we exploit the predictions of the model to shed light on the notion of subpopulations which is invoked in numerous experimental studies

The model we propose describes evolving populations in which individuals age, die and compete for reproduction while they are alive. The offspring they produce fill the next non-overlapping generation until the desired population size is reached. Survival and reproductive success of each individual are connected following the disposable soma theory. The more an individual invests in reproduction, the shorter its lifespan. For each individual, a single parameter

For each generation, all the individuals are synchronized, starting with age zero. While an individual

Competition for reproduction has been documented in a broad range of species and can take numerous forms, from limited access to habitat

Once an individual is chosen for reproduction, it produces one offspring which inherits its

The model we propose simulates evolving populations with a fixed size and non-overlapping generations. We start studying individuals at maturity, so that they are able to reproduce since time

Building the generation

Following the disposable soma theory, we implement a trade-off at the individual level between survival and reproduction. For each individual, the hazard of death between age

First, we implement

Second, to derive a mortality function for the Heterogeneity in aging Timing Model (HTM), let us now consider linear dependence on

The two hypotheses concerning heterogeneity in aging postulate different underlying trade-off mechanisms. In the case of heterogeneity in aging rates, investing in maintenance mechanisms slows down the aging process whereas in the case of heterogeneity in aging timing, the same investment delays the aging process. In

Over generations, the evolutionary process reshapes the distribution of

After several hundreds of generations (400 in the simulations below), the distribution stabilizes and mortality curves can be estimated. The probability for a whole population to disappear in the course of evolution is strictly positive. Indeed, each individual has a non-zero probability to die at each time step and this could happen before it reproduces at all. Population size can theoretically decrease over generations until extinction. Yet, for a broad range of parameters, this does not occur and a quasi-stationary distribution of strategies is reached. In

In this paper, we focus on adult mortality patterns and, as such, we do not explicitly address the heterogeneity arising from developmental processes. We assume that development leads to the expression of the inherited phenotype

In this section, we present the results of evolution in different environments under two distinct assumptions, heterogeneity in aging rates (HRM) and heterogeneity in aging timing (HTM), and compare the outcome with empirical and experimental data. In

The qualitative pattern goes from an exponential-exponential pattern to an exponential-U-shape-exponential pattern. Experimental data adapted from Vaupel et al.

Simulation results with both the HRM and HTM, presented in

In

The transitions in mortality patterns presented in

The scales of the y-axis differ in both graphs in order to zoom around the region of interest. (Experimental data adapted from Vaupel et al.

In both models, population heterogeneity is shaped by natural selection, via a life-history trade-off between survival and reproduction. The transitions observed in the two versions of the model rely on a change of the underlying heterogeneity in

These interactions are captured by the parameter

In the case of heterogeneity in aging timing, population heterogeneity evolves with the parameter

With heterogeneity in aging timing, bimodal distributions emerge from the evolutionary algorithm. Only the ratio between the two peaks changes with the environment.

In our models, individuals are competing with each other to access reproduction. Different

Mutations occurring on

First, we find that the properties of asymmetry in heterogeneity described in

The left part of the distribution remains unchanged, while the right tail gets closer to a uniform distribution as the mutation rate (

The left part of the distribution remains unchanged, while the right tail gets closer to a uniform distribution as the mutation rate (

Results in sections ‘Transitions in mortality curves’ and ‘Evolution of heterogeneity’ suggest that changes in population heterogeneity have a considerable influence on mortality patterns, providing the transitions in shape previously described. Yet, we find here that if the left-side of the

(HRM, 400 generations,

(HTM, 400 generations,

Because it explicitly links the individual level and population measures of aging, the model we propose also allows for study of what is called ‘subpopulations’ in several experimental studies. In the case of heterogeneity in aging rates, the distribution of individual lifespan in evolved populations is bimodal for

If the link between this aging marker and mortality is non-linear, then a unimodal and long-tailed distribution can lead to a bimodal distribution of individual lifespans. In the HRM, all the individuals in the right tail of the distribution contribute to the same peak in the distribution of lifespan (HRM, 400 generations,

When the right tail of the

A subpopulation also arises in studies focused on stress induction experiments

As a case study, we focus here on stress induction experiments in C. elegans.

The two curves reproduce the main features of survival curves corresponding to heat-shock experiments in C. elegans (filled diamonds, adapted from

Whether stress induction experiments induce adaptation which modifies population heterogeneity is a burning issue

Modifying

Modifying

In both figures, the distribution of

The results presented in the other columns provide predictions concerning the expected mortality patterns for other population heterogeneity. The interests of this exploration are twofold. First, it allows one to determine whether a given stress induction experiment will modify population heterogeneity. If the same distribution of

Studying the evolution of heterogeneity in resource allocation strategies is at the heart of a broad range of experimental studies and today's aging research

Moreover, in terms of medical treatments, focusing on aging rate or timing has distinct implications. On the one hand, modifying the rate of aging results in a high potential increase in life expectancy as aging is slowed during the whole life time. On the other hand, such changes in rate would not only extend the period of ‘healthy aging’ but also morbidity. These pros and cons have arguably significant implications in terms of public policy and need to be considered with regards to population heterogeneity. If the rate of aging is a fundamental constant in humans while timing is highly variable, being able to modify aging rate via medical progress will require substantial investment while timing would be presumably more flexible and thus cheaper to modify.

Numerous studies in current aging research have found mutations or treatments which dramatically influence aging dynamics

In this model, we analyze different possible heterogeneity in aging, only assuming Gompertz dependence on time. Our results show that assumptions about internal life-history trade-offs modulate the shape of population heterogeneity. Yet, in a wide array of scenarios, the same set of mortality and survival curves emerge. Population heterogeneity adapts to the environment according to the individual mortality functions but, at the population level, no qualitative changes can be observed. The transitions observed always go from a ‘kink’ shaped mortality curve to an exponential-U shape-exponential pattern. This robustness of the mortality patterns also echoes experimental observations of mortality and survival in different environments. In many organisms, the modifications observed in mortality patterns as a result of genetic manipulations

The systematic exploration of different forms of inter-individual differences in aging also provides predictions about the expected distributions of heterogeneity. Even though the mortality patterns look alike in the different versions of the model, the evolved heterogeneity distributions do not. Current technical advances, such as microfluidics set-ups

Previous works have used the Price equation to link population heterogeneity and natural selection

Moreover, the shape of these distributions also provides information about the link between the marker observed at the individual level and the measure of interest, such as individual lifespan. Comparing the distributions emerging from our evolutionary algorithm shows that to obtain the exponential-plateau-exponential pattern (which corresponds to a bimodal distribution of lifespans) with a unimodal distribution of the aging marker, the link between the marker and individual lifespan needs to be strongly non-linear. The exploration we propose here also provides milestones to interpret the individual-based data and their corresponding distributions.

Finally, we believe that the model is also flexible enough to allow the exploration of other types of life-history trade-offs, based on the key idea of evolving heterogeneity. As such, it paves the way for future interpretations of coming individual-based observations in evolving biological systems.

The simplicity of the framework we propose also enables the formalization of intuitive notions, such as a ‘subpopulation’ and the extensive exploration of mortality functions provides predictions of possible curves for organisms yet to be studied, along with expected distributions of heterogeneity. The robustness of mortality patterns observed suggests that aging is itself a robust process, relying on similar processes across species. We hope that this work paves the way for (i) faster understanding and classification of heterogeneity distributions across species which are not model organisms and (ii) opens up new prospects in terms of understanding the evolution of aging and its robustness. The set-up allows easy changes and explorations, as well as creating space to make the interactions between aging and reproduction more complex. In that sense, it complements previous approaches combining evolutionary theories and heterogeneity, as it provides a framework to explore yet to be explained aging dynamics.

We have simulated our model with both a continuous time and a discrete time framework, in both a stochastic and a deterministic manner, in order to ensure the robustness of our results. All models have been implemented in C and this section provides numerical methods for the algorithm used in our simulations.

In the stochastic models, the population has a finite size which is usually 500 individuals, unless otherwise mentioned. Modifying this parameter does not alter the conclusion of the models and the same results can be obtained for different population sizes. The mortality curve we observe as a result of the evolutionary algorithm depends on the parameter

In the discrete time version of the stochastic model, the age of each individual is incremented by one at each time step and followed by a survival test. This test consists of drawing a random number

The continuous time version of the stochastic model follows the same principles. The time of death and reproduction events are drawn following continuous distributions. First, the time of death for each individual is randomly drawn following an exponential distribution which corresponds to its

We have also implemented a deterministic model which corresponds to the stochastic models presented in the main text in the case of infinite populations. The purpose is twofold: (i) it ensures the robustness of our results and (ii) it paves the way for an analytical analysis of the evolutionary algorithm we present in this paper. For instance, the formalisation presented in this section allows one to study from a formal standpoint the convergence to a stationary distribution with infinite populations. The deterministic model is defined at the population level and describes the changes in

For computational reasons, the possible values of

In supplementary material, we also describe the explorations mentioned in the main article (

Exploration of the parameter space A - Mortality pattern for a mortality function

(EPS)

HRM where mutations consist in perturbing the previous value of

(EPS)

In the HRM, adding mild extrinsic mortality (

(EPS)

Transitions with the

(EPS)

Mortality pattern and stationary distribution obtained with sexual reproduction. The horizontal line shows 0.01 to highlight the decrease of the tail close to 1. (

(EPS)

Mortality pattern and stationary distribution obtained with a maturation time of 10. The horizontal line shows 0.01 to highlight the decrease of the tail close to 1. (

(EPS)

In the absence of mutations, the distribution of

(EPS)

For small mutation rates, population heterogeneity remains bimodal in the case of heterogeneity in timing (HTM). Heterogeneity is maintained because of the time-dependent competition, as described in the mathematical model in

(EPS)

The grey area represents the set of valid couple

(EPS)

Simulation Code. Program S1 contains a C code allowing to reproduce the mortality curves of the paper.

(ZIP)

Exploration of the parameter space and mathematical models.

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