An In Silico Model to Simulate the Evolution of Biological Aging

An In Silico Model to Simulate the Evolution of Biological Aging

An In Silico Model to Simulate the Evolution of Biological AgingA. Sajina1, a) and D.R. Valenzano1, b)

Max Planck Institute for Biology of Ageing, Joseph-Stelzmann Strasse 9b, 50931 Cologne,Germany.

Biological aging is characterized by an age-dependent increase in the probability of death and by a decrease inthe reproductive capacity. Individual age-dependent rates of survival and reproduction have a strong impacton population dynamics, and the genetic elements determining survival and reproduction are under differentselective forces throughout an organism lifespan. Here we develop a highly versatile numerical model of genomeevolution — both asexual and sexual — for a population of virtual individuals with overlapping generations,where the genetic elements affecting survival and reproduction rate at different life stages are free to evolvedue to mutation and selection. Our model recapitulates several emerging properties of natural populations,developing longer reproductive lifespan under stable conditions and shorter survival and reproduction inunstable environments. Faster aging results as the consequence of the reduced strength of purifying selectionin more unstable populations, which have large portions of the genome that accumulate detrimental mutations.Unlike sexually reproducing populations under constant resources, asexually reproducing populations fail todevelop an age-dependent increase in death rates and decrease in reproduction rates, therefore escapingsenescence. Our model provides a powerful in silico framework to simulate how populations and genomeschange in the context of biological aging and opens a novel analytical opportunity to characterize how realpopulations evolve their specific aging dynamics.

Keywords: aging, longevity, fitness, mutation accumulation, selection, aging simulation.

I. INTRODUCTION

Natural populations evolve from the interaction be-tween external selective forces and the internal capac-ity to respond to them. External forces include preda-tors, parasites and available resources, and populationresponse to these forces importantly depends on ratesof mutation and recombination. Standard evolutionarytheory of aging predicts that deleterious mutations affect-ing early or late survival are differentially selected, withmutations negatively affecting survival in early life be-ing readily removed by purifying selection, as opposedto those affecting survival in late life, which insteadhave a tendency to accumulate due to weaker selec-tive forces [Charlesworth, 2000, Haldane, 1941, Hamil-ton, 1966, Medawar, 1952, Rose, 1991]. A consequenceof this phenomenon is that young individuals have lowerdeath rates (i.e. higher chances to survive) compared toold individuals. Furthermore, it is theoretically possiblethat selection could favor alleles that have a detrimentaleffect on survival in late life stages, as long as they havea beneficial effect early on in life – i.e. alleles that con-tribute to an aging phenotype can be fixed in the genepool if their overall effect on fitness is positive [Williams,1957].

Although these models greatly helped to frame thegenetics of biological aging on a solid evolutionary ba-sis, there are some important aspects that were left out,which are key for our understanding on how selectionshapes individual fitness in biological populations. In

a)Friedrich-Wilhelm Gymnasium, Severinstrasse 241, Cologne,Germany.b)Corresponding author: [email protected]

particular, exclusively focusing on how selection shapesage-specific survival, they did not analyze how genes thatregulate survival and reproduction might coevolve andhow their interaction could affect individual fitness andpopulation stability under different selective pressures,which are key aspects to understanding how natural pop-ulations evolve.

Unlike analytical methods, computer simulations – ornumerical models – can sustain high parameter complex-ity and are particularly suited to model the evolutionof dynamic systems, such as biological systems [Dzwineland Yuen, 2005, Penna, 1995].

To analyze the role of selection in shaping the geneticbasis of reproductive aging and age-dependent changesin death rate, we developed an in silico model that simu-lates the evolution of biological populations, where eachindividual agent is provided with a genome that definesthe probabilities to reproduce and survive. We let a ge-netically heterogeneous population evolve under differentenvironmental constraints, including limiting resources,sexual or asexual reproduction and different populationsizes. We find that, in a sexual model, the force of se-lection decreases with age after sexual maturation. How-ever, we demonstrate that, in stable environments, se-lection favors prolonged survival and reproduction aftersexual maturation more than in unstable conditions. Inunstable environments, characterized by continuous ex-pansions and compression of population size, we observethe emergence of increased early mortality. Importantly,we observe that the hallmarks of biological aging, i.e. thetime-dependent decrease in individual survival and re-production, are more dramatic in the sexual model anddo not evolve in the asexual model under constant re-source conditions. Our model offers the possibility toaddress key evolutionary questions regarding the evolu-

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An In Silico Model to Simulate the Evolution of Biological Aging 2

tion of biological aging at both individual and populationlevel, and its applications can give key insights on howgenomes contribute to the time-dependent deteriorationof biological functions.

II. THE MODEL

a. The genome. Our model simulates the evolutionof a population of agents, which evolves through a se-quence of discrete time intervals, called stages. For eachagent i, the probability to surviving and reproducing ateach stage is defined by a bit-string code that we namegenome (Gi). The probability to surviving from one

0 1

1 0

1 0

0 0

1 0

0 1

...

...

1 1

1 0

1 1

0 1

1 1

1 1

...

...

;Gi=

SURVIVAL REPRODUCTION NEUTRAL

S0

S70

R16

R70

...

0 0

1 1

0 1

1 0

0 0

1 0

...

...

1 0

0 0

0 1

1 1

1 1

1 1

...

...

;...

0 0

1 1

1 0

0 1

1 0

0 0

...

...

s0

s70

r16

r70

N

FIG. 1. Genome composition. The genome (Gi) containsa sequence of discrete bit arrays – Si and Ri – that definethe probability to survive from one stage to the next and toreproduce at each stage, respectively indicated as si and ri.Reproduction starts at age 16, therefore there are no R andr values with an index smaller than 16. A neutral portion ofthe genome, indicated by N , has no corresponding phenotype,and is used as a control region to measure neutral evolution.

stage to the next is indicated by sk, and we refer to itas transition probability. The transition probability forstage k is proportional to the numbers of 1′s present in acorrensponding 20-bits array, called Sk (FIG. 1). Anal-ogously, the probability to reproducing at stage k, indi-cated by rk, is proportional to the number of 1′s in the20-bits array Rk:

sk =

(maxsurv −minsurv

21

) 20∑b=0

Skb +minsurv

rk =

(maxrepr −minrepr

21

) 20∑b=0

Rkb +minrepr

with both Skb and Rkb that can be = 0 or 1 and maxsurvand maxrepr being the initially set maximum allowed

survival and reproduction probability, respectively.Gi also contains a bit array that is not translated in

any phenotype (indicated as N in FIG. 1), which is usedas a functionally-neutral portion of the genome.

For each agent i in the population, Gi is a continuousbit array, i.e. all Sk, Rk and N are contiguous. For repre-sentation purposes, in all the plots shown in results, bothsk and rk are represented in ascending order, i.e. withsk preceding sk+1. However, the actual order along thebit array is randomized, so that Sk does not necessarilypreceed Sk+1.

b. Simulation progression. The seed (or starting)population is composed of genetically heterogeneousagents, whose genomes are randomly generated bitarrays, which have on average an equal number of 1′sand 0′s. Each agent from the seed population has aspecified chronological age, and becomes one time-unitolder at every new stage. Resources are used as thepopulation growth-limiting factor. Resource amountis defined before the simulation starts and, at eachstage, the population can be in a state of resourceabundance or shortage, i.e. Nt ≤ Rt and Nt > Rt, withRt being resources available at stage t and Nt being thepopulations size (i.e. number of individuals) at stage t.In the former situation, for each individual, the deathrate will correspond to 1 − st, while in the latter it willcorrespond to µ (1 − st), with µ being a constant thatincreases actual death rate for all the agents in case ofresource shortage. Depending on the simulation type,resources are set to be either constant or variable. Inthe constant resource model, resources are the same atevery new stage, regardless from the leftover resourcesfrom the previous stage. However, population growthand contraction will affect the death rate depending onwhether population size exceeds resouce units. In thevariable resource model, the amount of available re-sources is the sum of two quantities. One is proportionalto the difference between resources and population sizefrom the previous stage (leftover resources), and theother is a fixed resource increment R:

Rt+1 =

{(Rt −Nt)k + R if Nt < Rt

(Rt −Nt) + R if Nt ≥ Rt

If the amount of resource units from the previ-ous stage exceeds population size, the leftover resourcesregenerate proportionally to a fixed proliferating valuek – set at the beginning of the simulation – and areadded to the fixed resource increment R. However, ifthe population size exceeds or is equal to the resourceunits, i.e. no resource is left from the previous stage,the difference between population size and resourcesavailable is subtracted from the fixed resource incrementR. If this difference is ≤ 0, resources are then set to 0.In this case, death rate is affected (see above).

Reproduction follows resource consumption (FIG.2). In the asexual model, each individual can repro-duce or not, based on the probability provided by the


individual age-specific rk value. In the sexual model,

STARTING POPULATION

(N0, V

G)

AGE += 1

(resources usage/renewal)

REPRODUCTION

(asex)

mutation

(sex)

pairing

recombination

mutation

o!spring

DEATH(EXTRINSIC DEATH CRISIS)

STA

GE

LO

OP

FIG. 2. Schematics of events sequence in the simulation. N0

and VG represent the initial number of individuals and thegenetic variance initially defined for any seed (or starting)population.

the individuals that, based on the stage-specific rkvalue, are selected to reproduce, undergo chromosomerecombination. Recombination is determined by a fixedrecombination frequency, and is followed by the randomselection of one of the two chromosomes (either theleft or the right in each S0:70, R16:70, and N from FIG.1) to match a corresponding recombining chromosomeselected from another individual. Chromosome matchingis followed by mutation, which is set by a fixed mutationrate before each simulation. Mutation transforms 0sto 1s and, vice versa, 1s to 0s. Since any 0 mutatinginto 1 in both Rk and Sk translates into an increase inthe probability of reproducing or surviving, respectively,corresponding to a phenotypically beneficial mutation,we let 0 to 1 mutations to be 10 times less likely than1 to 0 mutations, in order to let mutations to be morelikely detrimental than beneficial.

Once the mutation step takes place, a novel genomeis complete, and it represents the new survival andreproduction program for a newborn agent, which isadded to the current population with the starting age of0.

Following reproduction, each agent’s sk values con-tribute to determining whether the agent survives or iseliminated from the population. The actual probabilityof survival through any stage depends on i) each agent’ssk value for the actual age k at the given stage, ii) thedifference between resources and population, which canadd a weight on each agent’s death risk, and iii) the

extrinsic death crisis parameter, which can be initiallyset and determines whether – at specific stages – thepopulation undergoes massive death.

All individuals reach sexual maturation at stage 16,therefore Rk start at k = 16.

A. Fitness

We define individual genomic fitness (GFind) as thecumulative individual lifetime expected offspring contri-bution, given the genetically determined probabilities tosurvive and reproduce. For instance, the genomic fitnessfor an agent at age 50 corresponds to the product of twoterms, one that defines the probability to surviving toreproductive age:

SRA = s0s1s2...s15

and one that defines the probability to surviving and re-producing at all the stages that follow sexual maturation,up to the last stage (age 50 in this example):

SR16−50 = s16r16 + s16s17r17 + ...+ s16s17...s50r50

with s and r as the transition probability for survivaland reproduction, respectively, defined in the individualgenome by their corresponding S and R values. Moregenerally, this product can be written as:

GFind =

(15∏k=0

sk

)Age∑i=16

ri

i∏j=16

sj

with Age as the individual age at which fitness is mea-sured.

We also define, for any stage, an average populationgenome and its associated genomic fitness (GFavg), as:

GFavg =

(15∏k=0

sk

)Age∑i=16

ri

i∏j=16

sj

with r and s as the average reproduction and survivalprobabilities for each age.

Analogously, we can define relative individual genomicfitness as the ratio of individual fitness to the sum of allthe individual fitness:

RGFi =GFi∑N

ind=1GFind

Tracking population size throughout the simulationprogression, enables to measure rates of actual survivalat any stage, for each age class, defined as:

σak =

Na+1k+1

Nak


0 1000 2000 3000 4000 5000

010000

20000

Stage

N

0 1000 2000 3000 4000 5000

010000

20000

Stage

N

Population

Resources

Constant Resources Variable Resources

FIG. 3. Sample population growth and population size oscillations under constant (left) and variable (right) resource conditions.Only the first 5k stages of the simulations are displayed for simplicity. In the constant resource conditions resources are fixedand do not change as a function of population size, whereas, in the variable resource conditions, resources vary depending onan intrinsic growth rate, a fixed stagewise increment, and population size.

with Nak as the number of individuals of age k at stage

a. Replacing s with σ, allows a corrected GFavg, whichwe name population fitness or Fp:

Fp =

(15∏k=0

σk

)Age∑i=16

ri

i∏j=16

σj

III. RESULTS

During each simulation, an initial random gene pool isgenerated, and Sik, Rik and Ni (see FIG. 1) evolve freelydue to mutation and – in the sexual model – recombina-tion. However, this model also allows populations to bere-seeded from end-stages of previous simulations. Popu-lation size oscillates as a function of i) available resources(FIG. 3) and ii) evolving distributions of age-dependentreproduction and survival probabilities encoded in eachindividual genome, which affect the ratios of births overdeaths. The model does not bias the direction in whichpopulations will evolve higher or lower survival and re-production probabilities throughout age, and all the ob-served changes in age-specific reproduction and survivalprobabilities (rk and sk) are due to selection and drift.

A. Sexual model

Under constant resources, populations of agents thatreproduce sexually evolve to a state of higher Si beforesexual maturation and continuously declining Si and Ri

values when i > tm, with tm as the time of sexual matu-ration (FIG. 4).

Since si and ri are directly proportional to Si andRi, respectively, also the age-dependent probabilities ofsurviving and reproducing become accordingly higher inearly life and decline after sexual maturation. The phe-notypically neutral portion of the genome (indicated byN in FIG. 1, and by the dashed lines in FIG. 4 and inFIG. 5B), used as a control region to test the effects ofmutation without selection on the genome sequence evo-lution throughout the simulation, enables to show thatthe distribution of the Si and Ri values throughout ageis under strong selection before and after sexual matura-tion, for all the ages where Si and Ri values are abovethe curve corresponding to the expected values (horizon-tal dashed line in FIG. 4 and in FIG. 5B). For each age,the difference between Si, Ri and the expected valuescorresponding to the neutrally evolving portion of thegenome (Ni), is proportional to the strength of selection.

Population genetic variance for Si reaches low valuesfrom stage 0 until about sexual maturation, and has itsminimum around sexual maturation also for Ri, as a con-sequence of strong purifying selection. After the onset ofsexual maturation, genetic variance for both Si and Ri

increases rapidly, reflecting weakened purifying selectionat later ages (FIG. 4).

We tested the effects of resource availability on popu-lation size oscillation and genome evolution. Even withconstant resources, population size fluctuates due to theincreased overall mortality induced by population sizeexceeding available resources (see II.b). In the constantresource conditions, population oscillations amplitudeswere comparable when resources were set to 1k or 5k


Age

1 15 31 47 63 23 39 55

Age

0.8

1.0

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Age

Ge

ne

tic V

ari

an

ce

1 15 31 47 63 23 39 55

0.8

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Fre

qu

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f 1

’s

60k

5k10k0

1.6kStages

Si

Ri

Si

Ri


tm

tm

tm

tm

sd(Si) sd(R

i) sd(S

i) sd(R

i)

0.0

0.4

0.8

1 15 31 47 63 23 39 55

0.0

0.4

0.8

tm

tm

tm

tm

FIG. 4. Evolution of the genome in the sexual model. Above: The x-axis represents the position of Si (on the left) and Ri (onthe right) along the genome, sorted according to the age they correspond to. The y-axis represents the ratio of 1’s for eachtransition probability Si and Ri indicated on the x-axis. The genomes are randomly generated at stage 0 (grey line) and startwith an even distribution of 0’s and 1’s, for all the Si and Ri (average values only are shown). As the simulation proceeds(different color lines), mutation and selection change the frequencies of 1’s and 0’s for each age-specific Si and Ri. The geneticmodules relative to survival (Si) fix higher frequencies of 1’s before and about sexual maturation (vertical dashed line and tm),while reproduction (Ri, not defined before tm) fix higher frequencies of 1’s starting at tm. At later stages of the simulation,genetic modules for both survival and reproduction after sexual maturation accumulate more deleterious mutations than atearlier stages, as indicated by the lower values to the right of the cyan curve (60k stages). Below: population average of thegenetic variance for Si and Ri as a function of age after 60k stages of simulation. Genetic Variance before tm is low, andincreases for both survival and reproduction with Age > tm. Fixed resources are set to 5k units, mutation rate is set to 0.001per site and recombination rate is set to 0.01.

units (FIG. 5A). However, under variable resource condi-tions, population size is a function of the stage-dependentresources increment (R) parameter. Population size os-cillations had elevated amplitude in early stages, andstabilized to smaller amplitudes when R was high (5kunits) (FIG. 5A). However, when R was set to 1k units(or lower), constraining population size to lowever val-ues, population size oscillations did not stabilize, indi-cating that smaller populations are more unstable thanlarger populations. Importantly, in the variable resourcecondition, if R is large, resources do not oscillate with

large amplitude once the population stabilizes (FIG. 5A).Decreased oscillation amplitudes in both population sizeand resource units is due to Si and Ri values evolvingtowards a stable equilibrium (FIG. 5A). This was notobserved when resources were variable and R was small,in which case also the age-dependent distribution of Si inthe post-reproduction phase was lower than in larger pop-ulations or in the constant resource condition (FIG. 5B),highlighting that in our model genome evolutionary dy-namics not only depend on resource availability (constantor variable), but also on population size. Interestingly,


01

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Small Population

0.0

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0.0

0.4

0.8

Age

1 15 31 47 63 23 39 55

tm

tm

tm

tm

Si

Ri

Si

Ri

FIG. 5. Population, resources and genomic dynamics at different population sizes in the sexual model. A: Population andresource dynamics in larger and smaller populations, under constant (left) or variable (right) resource conditions. Populationand resource values are displayed on the y axis. Resources are shown in red and population in blue. Each plot is split in twoparts, with early stages on the left, and late stages on the right. Each run consists of 60,000 stages. Large and small populationsare determined by the resource value, set to 5k and 1k resource units, respectively. In the constant resource condition, theresource value is fixed, whereas in the variable resource condition the population size is directly affected by the fixed resourceincrement R. For sake of representation, R is not added to the resource values on the plot. B: Average Si and Ri values at thelast stage of the simulation (stage 60,000) in the constant (left) and variable (right) resource condition model. The horizontaldashed lines represent the expected Si and Ri values for for the portion of the genome that is not subject to selection, used asa control region, indicated by Ni in our model. Mutation rate is set to 0.001 per site and recombination rate is set to 0.01. tmcorresponds to the age of sexual maturity.


we observed that smaller populations, in the variable re-source condition, evolved to a status of increased earlymortality, even before sexual maturation. This result wasnot a rare event and emerged in all our simulations. Ad-ditionally, in our model the age-dependent reproductionevolves in a different fashion from survival as – in thevariable resource condition – Ri have the same behav-ior in small and large populations (FIG. 5B, right). Infact, while Si drop right after tm in small populations un-der variable resources, Ri dynamics are superimposablein large and small populations (FIG. 5B). This behaviormight be explained by the fact that in our model onlysurvival is penalised by a factor µ when population sizebecomes larger than available resources (see II.b).

B. Asexual model

In the asexual model, we asked what the effects areof population size and availability of constant or variableresources on population dynamics and genome evolution(FIG. 6), and how these dynamics are different from thesexual model. In the constant resource condition, popu-lation oscillations decreased amplitude during long sim-ulations (60k stages) both in larger and in smaller popu-lations (FIG. 6A). Interestingly, unlike what we observedin the sexual model, in the variable resource condition,small-scale population size differences did not result inbalanced equilibrium between resources and populationsize in the variable resource condition (FIG. 6A). How-ever, we did observe an increase in the oscillation periodbetween early and late stages of the simulation, as laterstages of the simulation have longer oscillation periodsthan earlier stages. Additionally, the genome of asexu-ally reproducing populations, both in the constant and,to a lesser degree, in the variable resource condition, didnot reach elevated values of Si and Ri both in large andsmall populations – at least in the tested population-size ranges – suggesting that, in populations reproducingasexually, selection for increased early survival and re-production was weaker or had a slower evolution than inpopulations reproducing sexually (FIG. 6B).

C. Survival and death rates

We calculated the observed death rate in both the sex-ual and asexual model, for the constant and variable re-source model, in small and large populations (FIG. 7).

Since in our model resources affect survival probabil-ities when the population size exceeds the available re-sources (see II.b), actual (i.e. observed) mortality at aspecific time point is expected to differ from the mortal-ity predicted solely by the distribution of the sk values.

In the sexual model, both in the constant resourcecondition and in the variable resource condition, oncereached equilibrium between population and resources,the age-dependent death rate remained low before sex-

ual maturation and increased after sexual maturation inlarge populations with slightly different dynamics, as inthe constant resource condition the increase in mortalityhappened later than in the variable resource condition(FIG. 7). Small populations in the variable resource con-dition evolved high early mortality – i.e. before sexualmaturation – but had an overall slower increase in theage-dependent death rate, as shown by the smaller slopein the death rate (FIG. 7).

In the asexual model, under constant resources, we findno age-dependent change in the rate of death both inlarge and small populations, except for an increase indeath rate after age 50 in small populations (FIG. 7).This might result from the evolution of Si in the asex-ual model under constant resources, which does not sig-nificantly deviate from the expected values reached inthe control genome region (FIG. 6B, left). However, theconstant death rate across all the ages in the constantresource condition is associated with extemely high pop-ulation oscillation stability, as indicated by the limitedpopulation size oscillation amplitude (FIG. 6A). Inter-estingly, we do observe an age-dependent increase in therate of death in both the small and large populationsfor the variable resource model (FIG 7). This behaviouris associated with high amplitude in population size os-cillation, which in our model leads to increased overallmortality (FIG. 6A).

IV. DISCUSSION

Studying the changes in individual probability of sur-vival and reproduction is the focus of gerontology, exper-imental aging research and evolutionary biology. Agingresearch is ultimately focused on identifying the molecu-lar mechanisms that affect the changes in the probabili-ties of surviving and reproducing throughout one individ-ual’s lifetime. On the other hand, evolutionary biology– and in particular population genetics – is interested inthe genetic changes that affect the probabilities of sur-vival and reproduction over several generations, as theydirectly determine biological fitness, i.e. how likely it isto transmit specific alleles from one generation to thenext.

In our model, we combine the “biology of aging” andthe “evolutionary biology” perspectives on survival andreproduction and develop a novel in silico system wheredistinct genomic elements separately determine the prob-abilities of surviving or reproducing at a specific lifestage. Using this approach, we can investigate how nat-ural selection molds the allele frequencies that determinesurvival and reproduction at different life stages withinthe lifetime of an individual – as well as throughout gen-erations – in the population’s gene pool.

We let populations evolve either sexually or asexu-ally, under the effect of mutation and recombination (inthe sexually reproducing populations). As populationsevolve under fixed or variable resource conditions, we


0 1000 2000 3000

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Small Population

Fre

qu

en

cy o

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Si

Ri

Si

Ri

A

B

tm

tm

tm

tm

FIG. 6. Population, resources and genomic dynamics at different population sizes in the asexual model. A: Population andresource dynamics in larger and smaller populations, under constant (left) or variable (right) resource conditions. Populationand resource values are displayed on the y axis. Resources are shown in red and population in blue. Each plot is split in twoparts, with early stages on the left, and late stages on the right. Each run consists of 60,000 stages. Large and small populationsare determined by the resource value, set to 5k and 1k resource units, respectively. In the constant resource condition, theresource value is fixed, whereas in the variable resource condition the population size is defined by the fixed resource incrementR. B: Average Si and Ri values at the last stage of the simulation (stage 60,000) values in the constant (left) and variable(right) resource condition model for small and large populations. In the constant resource model we added a third plot (dashedblack line) for very large populations, with 25k units of fixed resource. The horizontal shaded area represents the interval ofvalues for Si and Ri for the portion of the genome that is not subject to selection, used as a control region. Mutation rate isset to 0.001 per site and recombination rate is set to 0.01. tm corresponds to the age of sexual maturity.


0 10 20 30 40 50 60 70

0.0

0.4

0.8

Age

De

ath

Ra

te

0 10 20 30 40 50 60 70

0.0

0.4

0.8

Age

De

ath

Ra

te

tm

tm

Sexual Model

Constant Resources, Small Population

Variable Resources, Large Population

Asexual Model

Constant Resources, Large Population

Variable Resources, Small Population

FIG. 7. Observed death rate in the sexual and asexual models, with constant and variable resources, in large and small

populations. Death rate is expressed, for each age a at stage s, asN(a)s−N(a+1)s+1

N(a)s, with N(a)s as the number of individuals

of age a at stage s. The displayed values are the average death-rates for the last 100 stages of 60k stages-long simulations. tmcorresponds to the age of sexual maturity.

monitor population and genomic dynamics throughoutthe simulation. In particular, we monitor the evolutionof differential survival and reproduction probabilities atthe different individual ages allowed by our model. Wecompare the age-dependent values that survival and re-production probabilities reach throughout the simulationwith those achieved in a neutral portion of the genomethat is not subject to selection. This enables us to quan-tify the strength of selection, and to identify the por-tions of the genome that are under stronger selection,i.e. those that deviate from the neutrally evolving partsof the genome. We find that populations reproducingsexually evolve gene-pools that have high survival proba-bilities before sexual maturation and progressively lowersurvival probabilities after sexual maturation. This isin line with the expectation that early-acting deleteri-ous mutations are efficiently removed by selection fromthe gene pool [Medawar, 1952]. Importantly, genomicvariance for genetic elements affecting survival is lowerin the pre sexual-maturation phase – indicated by tmthroughout the paper – indicating strong purifying se-lection in early life stages. Additionally, the geneticelements affecting early survival accumulate high levelsof beneficial mutations. Importantly, in our simulationthe probabilities to survive evolve low values after sexualmaturation and the genetic variance for survival (as wellas for reproduction), increases dramatically after sexualmaturation, strongly indicating that the age-dependentdecreased probability of surviving and reproducing, i.e.aging, is directly correlated with increased genetic vari-ance, which results from the relaxed purifying selection

to maintain elevated survival and reproduction in late-lifestages. This result suggests that genes or gene variantsaffecting late-life survival and reproduction are expectedto be associated with higher population variance com-pared to genes affecting early-life phenotyes, such as de-velopment and early reproduction.

In the sexual model, populations evolving under vari-able resources undergo large oscillations in populationsize during the first few thousand simulation stages.However, at later stages, oscillations in population sizeand resources decrease in amplitude and reach an equi-librium (FIG. 5A). Additionally, we provided environ-mental perturbations to populations “in equilibrium” bystochastically imposing 90% mortality to the whole pop-ulation. This had no effect on population size oscillation(data not shown), i.e. the population-resource equilib-rium re-established rapidly after the perturbation. Oncereached this stable equilibrium between resources andpopulation size oscillations, the variable resource modelbehaved like the constant-resource model. In future stud-ies we will investigate how varying the fixed resourceincrement R affects population oscillations and genomecomposition. Intriguingly, we observed that populationsize affects the probability to reach a resource-populationequilibrium in the variable resource model, as smallerpopulations do not reach such balance. Larger popula-tions display therefore higher stability and genetic plas-ticity compared to smaller populations. Interestingly, wefind that sexually reproducing populations under vari-able resource conditions develop increased early-life in-fant mortality, i.e. the survival probabilities in the ages


preceding sexual maturation significantly drop under twoconditions: i) when population size is small and ii) inearly stages of large population simulations, before thepopulation-resource equilibrium is reached (FIG. 5B andFIG 7). Elevated initial infant mortality is associatedwith population instability and disappears when popu-lation and resource oscillations eventually balance eachother. We find the increase in early mortality paradox-ical, since the early ages preceeding sexual maturation,which in our model starts at age 16, should be underequally strong purifying selection, and any deleteriousmutation affecting survival probabilities in this life stageshould be rapidly removed from the gene pool. At thesame time, we are aware that many species – includ-ing our own – display elevated infant death rates, whichthen decline to a minimum around puberty. After pu-berty, death rate starts raising, as already observed byR.A. Fisher [Fisher, 1930]. A possible interpretation forthis surprising elevated infant mortality is that, in thevariable resource condition, large population oscillations,characterized by continuous population contractions andexpansions, weakens the action of selection to efficientlyremove deleterious mutations acting at early ages. Ad-ditionally, deleterious mutations become more likely toreach high frequency in the population due to continu-ous bottlenecks [Hughes, 2009]. Alternatively, it couldbe speculated that a potentially active mechanism couldfavour the fixation of early-acting deleterious mutationsto buffer large population oscillations by lowering thepopulation growth rate, ultimately benefiting populationstability. However, these hypotheses need to be rigor-ously tested in future studies.

Unlike populations reproducing sexually, asexually re-producing populations under constant resource condi-tions do not evolve in our model an age-dependent dif-ferentiation in the genomically-encoded probabilities tosurvive and reproduce. In particular, we did not ob-serve higher probability of surviving or reproducing inearly life stages, compared to late life stages. This phe-nomenon can be observed at the genomic level, wherethe genomically-encoded survival and reproduction prob-abilities do not dramatically differ before and after sex-ual maturation (FIG. 6B). Similarly to what seen in theSi, the measured death-rates show – particularly in theconstant resource condition – no variation (i.e. do notincrease) with age (FIG. 7). Since we define aging as anage-dependent increase in the probability of death andas a reduction in the probability of reproduction, asex-ually reproducing populations – particularly under con-stant resource conditions – seem to not undergo aging.In our model, asexually reproducing populations underfixed resource conditions are associated to the absenceof senescence. This phenomenon, which spontaneouslyemerged in our simulation, is surprisingly in line withthe evidences supporting extreme longevity and the lackof senescence in several organisms that reproduce clon-ally [Garcia-Cisneros et al., 2015]. However, some asex-ually reproducing species are also reported to undergo

senescence [Martınez and Levinton, 1992]. Interestingly,our model shows to some degree a detectable increase inthe age-dependent death-rate in asexually reproducingpopulations under the variable resources condition (FIG.7). In fact, both in large and small populations – us-ing the same resource and population ranges used for thesexually reproducing populations – population size andresource units oscillations did not evolve to a status ofequilibrium characterized by lower oscillation amplitude.This result can be interpreted as a lower stability of pop-ulations reproducing asexually, compared to populationsreproducing sexually, under varying resource conditions.In future studies it will be important to explore howsexually and asexually reproducing populations competeagainst each other under different resource conditions.

A. Conclusions

For aging to be considered an adaptation, we wouldexpect a decreased genetic variance for the genetic mod-ules that determine late-life decrease in survival and re-production. In contrast with this expectation, we ob-served an age-dependent increased genetic variance fol-lowing sexual maturation, which can be interpreted asdecreased purifying selection in late-life phenotypes. Thefitness effect of beneficial mutations acting in late-life isnegligible compared to mutations affecting survival andreproduction at earlier ages. Even in our model, whichdoes not constrain survival and reproduction to becomehigher or lower at different life-stages, individual fitnesswas not maximized, as survival and reproduction proba-bilities were elevated only in early life, before and aroundsexual maturation, and then decayed after sexual matu-ration, due to the effects of relaxed purifying selectionand consequent mutation load on late-life acting geneticmodules. This finding is in line with previous stud-ies that predicted a declining force of natural selectionthroughout lifespan [Charlesworth, 2000, Haldane, 1941,Medawar, 1952]. Since the genetic elements acting onsurvival and reproduction in late life-stages accumulatedmore deleterious mutations, compared to those respon-sible for survival and reproduction in earlier life-stages,we observed that a large portion of the genome that af-fected individual survival and reproduction at later ageswas strongly associated with high genetic variance.

In our model, the genetic modules that determine theprobabilities to survive and reproduce at different timestages are free to evolve due to mutation, recombination(in the sexual model) and selection. The simulations an-alyzed in this study were run at fixed recombination andmutation rates (indicated in the figure captions). Fu-ture studies will clarify how recombination and mutationrate optima evolve under different conditions and affectthe age-dependent probabilities of survival and reproduc-tion.

To note, our model does not allow for pleiotropic effectsof genetic modules acting at different ages, since each ge-


netic module is age-specific. We can hypothesize that inthe presence of antagonistic pleiotropism, increased late-life mortality and decreased late-life reproductive poten-tial could be potentially positively selected, granted thatthey provide a fitness advantage in early life stages.

Importantly, our model does not allow epistasis to takeplace in a direct genetic way, i.e. in our simulationsthere are no gene to gene interactions that lead to non-additive phenotypic effects, as each genetic module is in-dependently responsible for its own phenotypic output.Although epistasis plays a key role in real biology, wedeliberately did not implement it in our model for com-putational simplicity. The genetic modules for survivaland reproduction that are implemented in our model,rather than being an in silico version of individual genes,can be understood as approximations of gene networksor pathways that influence either survival or reproduc-tion at given ages. In this respect, epistasis might notbe applicable to our model. Importantly, despite the ab-sence of epistasis, our model reproduces several featuresof real population aging, such as increased early survivaland rapid increase of the death rate and decrease of fertil-ity after sexual maturation. Despite the absence of sensustrictu epistasis and pleiotropism, this model allows com-plex coevolution patterns between genetic modules thataffect survival and reproduction at different life-stages,and provides an indirect test for the necessity of geneticepistasis and pleiotropism for the evolution of genetically-encoded biological aging. In other words, since our modelevolves features typical of biological aging populationswithout implementing epistasis and pleiotropism, we canconclude that epistasis and pleiotropism – although im-portant in organism aging processes – might not be nec-essary for the development of biological aging. Addition-ally, adopting a model that associates to individual agesdiscrete probabilities to survive and reproduce, enables toaccurately calculate fitness at both individual and pop-ulation level. Fitness measures factor in both survivaland reproduction probabilities encoded in the genome,as well as observed survival and reproduction outputs.The study of how fitness varies throughout populationevolution under different resource conditions and popu-lation size will be further developed in future studies.This model provides on one hand a powerful in silicosystem to generate hypotheses to test on real-life data,and on the other hand provides a numerical platform totest parameters derived from real biology, such as mea-sured mutation and recombination rates, population size,

resource availability and overall extrinsic risk of mortal-ity. The use of this novel model to simulate the genomeevolution in the context of biological aging opens newpossibilities to understand how real populations evolvesuch a wide range of life history trait strategies.V. AUTHORS CONTRIBUTION ANDACKNOWLEDGEMENTS

D.R.V. developed the model, contributed to the codeand wrote the manuscript. A.S. wrote the python codeand critically contributed to the improvement of themodel.

We thank Fabio Iocco and Dmitri Petrov for the in-sightful suggestions and all the members of the Valen-zano laboratory at the Max Planck Institute for Biol-ogy of Ageing for their constructive comments on themanuscript.

This work has been entirely supported by the MaxPlanck Society and by the Max Planck Institute for Bi-ology of Ageing in Cologne, Germany.

VI. BIBLIOGRAPHY

B. Charlesworth. Fisher, Medawar, Hamilton and the Evolution ofAging. Genetics, 156:927–931, 2000.W. Dzwinel and D.A. Yuen. Aging in Hostile Environments Mod-eled by Cellular Automata with Genetic Dynamics. InternationalJournal of Modern Physics, 16:357–377, 2005.R.A. Fisher. The Genetical Theory of Natural Selection. OxfordUniversity Press, Oxford, 1930. ISBN 978-0-19-850440-5.A. Garcia-Cisneros, B. Perez-Portela, B.C. Almroth, S. Degerman,C. Palacın, and H. Nilsson Skold. Long telomeres are associatedwith clonality in wild populations of the fissiparous starfish Cosci-nasterias tenuispina. Heredity, 2015. doi:10.1038/hdy.2015.43.J.B.S. Haldane. New Paths in Genetics. George Allen and Unwin,Ltd., London, 1941.W.D. Hamilton. The Moulding of Senescence by Natural Selection.Journal of Theoretical Biology, 12(1):12–45, 1966.A.L. Hughes. Near-neutrality: the leading edge of the neutral the-ory of molecular evolution. Ann N Y Acad Sci, 1133:162–170, 2009.D.E Martınez and J.F. Levinton. Asexual metazoans undergosenescence. Proc. Natl. Acad. Sci. USA, 89:9920–9923, 1992.P.B. Medawar. An unsolved Problem of Biology. University Lec-ture, 1952.T.J. Penna. A Bit-String Model for Biological Aging. Journal ofStatistical Physics, 78(5/6):1629–1633, 1995.M.R. Rose. The Evolutionary Biology of Aging. Oxford UniversityPress, Oxford, 1991. ISBN 0-19-506133-0.G.C. Williams. Pleiotropy, Natural Selection, and the Evolution ofSenescence. Evolution, 11:398–411, 1957.

http://dx.doi.org/10.1038/hdy.2015.43

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An In Silico Model to Simulate the Evolution of Biological Aging