Computer Physics Communications 121–122 (1999) 108–112www.elsevier.nl/locate/cpc

Getting older with computers

T.J.P. Pennaa,b,1, A. Raccoa, M. Argollo de Menezesba Instituto de Física, Universidade Federal Fluminense, Av. Litorânea, s/n, Boa Viagem, Niterói 24210-340, RJ, Brazil

b Center for Polymer Studies, Boston University, Boston, MA 02215, USA

Abstract

Recently some of us, physicists, have turned our attention to the aging problem. Here, we review some results from a verysimple computational model of population dynamics that is able to reproduce many features displayed by real populations. 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

Everybody gets older. The reader of this work willbe older by the time he/she finishes the reading.However, this is not a reason for stopping the readingof this article right now. Aging is an inevitable process,and so is death. In spite of its importance and interest,many aspects of aging remain a mystery. We cannoteven predict the life span of species. As an exampleof the diversity of life spans, the golden splendorbeetle lives more than forty years. In California, the“King Clone”, a tree, has been living for more than12000 years. Some fishes, like the salmon, die soonafter reproduction. Many bacteria replicate every 20minutes. If food supply and a steady environment isprovided, some very primitive species will never die.

Aging is a complicated phenomenon because manyfactors seem to be important in the determination ofthe aging process: for example, an Asiatic elephantlives 50 years less if kept in captivity. Smoking candecrease longevity by many years. The huge num-ber of factors acting in the aging process lead to anequally huge number of theories about aging. Nev-ertheless, there are some phenomenological conclu-

1 E-mail: tjpp@bu.edu.

sions on aging that can give support to some theoriesand models. One of them is the so-called “Gompertzcurve”, that is represented by an exponential growth ofmortality with aging (the decrease of survival is alsoknow as “senescence”). The exponential behavior ofthe death rates holds for many species and even au-tomobiles [29]. This result is known since 1825 [9].There is another phenomenological law which is morerecent. Azbel studied many mortality curves for hu-mans (different countries and years) and flies [3]. Hefound some quantities that seem to be universal, i.e.,hold for human and flies whereas other ones seem tobe species-specific (we will give details on the defini-tions of these quantities further in this work).

In this paper, we will not focus on the many theoriesof aging; we will concentrate on some aspects ofthe evolutionary theories of aging [22,14]. Actually,we will be even more specific and we will presentresults from a model that is based on one of thosetheories, namely the mutation accumulation theory, byMedawar [11]. It is important to have a model based ononly one of the aspects of aging because, in this way,we can determine its role in the whole phenomenon.

Let us briefly describe the basis of the mutation ac-cumulation theory. A disease affecting an individual in

0010-4655/99/$ – see front matter 1999 Published by Elsevier Science B.V. All rights reserved.PII: S0010-4655(99)00291-X

T.J.P. Penna et al. / Computer Physics Communications 121–122 (1999) 108–112 109

the very first years of its lifetime is more dangerous tothe preservation of the species than a disease affectingthe same individual at ages beyond the one at whichthe individual becomes sexually mature. In this sense,the evolution pressure acts in the direction of push-ing the onset of diseases towards advanced age. Allgenetic information about individuals that suffer fataldiseases before sexual maturity are swept out of thepopulation in the next generation. On the other hand,there is no such strong pressure in order to extend life-time beyond the onset of sexual maturity. The ques-tion now is whether this weak pressure, if any, can beresponsible for the observed gradual senescence andconsequently by the exponential behavior of mortality.

2. Dynamics of the bit-string model

The model we are going to present in this sectionwas proposed in 1994 (published in 1995) [15] whenthe first results from computer simulations on agingstarted to be published [25]. We will emphasize heredetails for an efficient implementation on computers.

We start with a population ofN asexual individuals,each one characterized by a sequence ofL bits

S = {s1s2 . . . sL}, si = 0,1. (1)

Each bit corresponds to a given time interval in theindividual life history; typically one bit correspondsto one year if we consider human population, one daywhen referring toDrosophilaand so on. The lengthLis larger than that maximum age that any individualcould reach. UsuallyL is a multiple of 32 because thebit-string is stored as integer variables. An individualwill have a bit set to 1 in its bit-string at a positioncorresponding to the age when a disease decreases itssurvival probability. In the years that the individualdoes not suffer any effect of diseases, the bits in itsstring will be set to 0. A death rule is that the individualstays alive only if the number of deleterious mutationswhich became active up to its current age is lower thana thresholdT (a parameter of this model). Previoussimulations [15] have shown that the effect of, forinstance, increasingT changes only the populationsize by allowing that older individuals survive.

If the individual reaches the ageR (minimum ageat reproduction) it becomes sexually mature, and itcan generateb offspring at each further age (including

ageR). Each offspring inherits a copy of the parent’sbit-string except forM bits out of itsL bits chosenwith equal probabilities. TheseM bits can representpoint mutations. We usually consider an extreme casewhere only harmful mutations at birth take place. Thisis still valid because deleterious mutations are muchmore frequent than good ones. This corresponds, inthe present model, to setting to one those bits in theM selected positions. If the selected bit is previouslyset to one, it will stay set to one and consequently itsbit-string will be as good as its parent bit-string.

As usual in population dynamics studies, we keepthe population size within computer memory limitsby modeling the fight for territory and food. This isdone by introducing a Verhulst (logistic) term. Theindividual will survive and will age by one year withprobabilitypsurv= (1− N(t)/Nmax), whereN(t) isthe population size at timet andNmax is the maximumnumber of individuals allowed by food and spacerestrictions.

These simple rules are used in order to definethe dynamics of this model. During the dynamicevolution, the bit-strings with more bits set to 1 thanthe thresholdT before the minimum reproduction ageR will disappear. That is the way natural selectionpressure acts in this model. Individuals whose bit-strings have a small number of deleterious mutations,can survive beyond ageR and they will generateoffspring and eventually will dominate the population.

One time interval in our simulation is defined whenthe following steps are applied to all individuals of thepopulation:(1) We performed a test if the individual stays alive

by having accumulated less mutations – at itscurrent age – than the threshold. If it does not, it iseliminated from the population and we restart witha new individual. The population size is decreasedby one.

(2) The individual stays alive with probability givenby the Verhulst term. If it does not, we apply thesame rules for death as in the previous step. Thisis an age-independent condition for survival.

(3) If its current age is larger thanR, the individualbreeds, generatingb new individuals with their re-spective modified bit-strings. Here the populationsize increases accordingly.

(4) The age of the individual being tested is increasedby one.

110 T.J.P. Penna et al. / Computer Physics Communications 121–122 (1999) 108–112

For the sake of efficiency/speed, we use anotherinteger word, where we stored the current age and theage at which the individual is genetically determinedto die (it can die before because of the Verhulst factor).If we considerL= 32, we need only 10 bits for bothages.

At the end of each time interval, we determine theage distribution of our population,Nx(t). We simplycount the number of individuals with agex. It hasbeen shown that the dynamics described above leadsthe system to a stationary state, corresponding to astationary distribution of ages [1,27]. In this finalsituation, we can replaceNx(t) byNx .

A quantity of interest for aging studies is themortality rate [3]:

qx = ln

(Nx

Nx+1

), (2)

whereqx is the mortality at agex andNx is againthe number of individuals with agex in a stationarypopulation. As shown in Refs. [16,20], the ratioqx/q1suppresses the effect of death due to ecological causes(Verhulst factor), recovering the more realistic casewhere most deaths are due (or related) to geneticcauses.

In the next section, we review a few results of thispresent model. We have described here the asexualversion because it is the version to be used in thiswork. In the next section, we present also the sexualmodel and some results from this variant implementa-tion.

3. Results of the bit-string model

An old and well-known result from aging studies isthe Gompertz law. The exponential increase of mor-tality with age seems to be universal (many differentspecies present this feature). In this way, any modelproposed to study the dynamics of an age-structuredpopulation must reproduce this exponential behavior.The bit-string model, in particular, has succeeded wellin this task. Using as free parameters the mutation rateM and the threshold of mutationsT , Penna and Stauf-fer fit the mortality curve from an 1989 German pop-ulation table [17]. In that study, both versions (asexualand sexual) of this model are used. The exponentialbehavior is also confirmed by analytical calculationsby Almeida et al. [1].

Recently, Azbel proposed a new phenomenologicaltheory for mortality [3] based on the analysis ofmany demographic tables of human populations, fromdifferent countries, extended in a period of more thanone hundred years. Azbel started by fitting exponentialcurves such as:

qx =Abexp[b(x −X)], (3)

for all data available. He found that the free parametersA,b andX are related bya = lnA− bX. Moreover,he foundA = 11± 2 andX = 103± 1 years. Basedon this finding, Azbel proposed thatX and A aregenetic factors. Using some data forDrosophila,Azbel also showed thatX and A have differentvalues from the human parameters. Therefore it issuggested that these parameters should be species-dependent. Racco et al. [20] have shown that the bit-string model is compatible with these findings. Usingdifferent mutation rates, they have always found alinear correlation betweena and b. Therefore, theypropose that the mutation rateM should be usedto characterize different species. Another interestingpoint in the Racco et al. work is: the mutation rate usedin order to getX = 103, as found by Azbel for humanpopulations, is very close to the mutation rate usedby Penna and Stauffer to fit the German populationmortality curves (M = 0.001). The deviation fromthe exponential behavior for heterogeneous population(notably for flies) is suggested by Racco et al. as a non-equilibrium effect.

Another aging fact that can be reproduced by thepresent bit-string model is the phenomenon knownas “catastrophic senescence”. Catastrophic senescencemeans death by aging soon after reproduction. Thesalmon is maybe the most impressive example ofthis phenomenon. At the onset of the mature age(11 years), they are strong and healthy enough toswim for more than 3000 km and then to reproduce.Nevertheless, they die a few weeks later. The oldestsalmon fished was 13 years old. Using computersimulations of the model described here, but imposingthat the breeding attempt occurs only once in thelifetime (just like for semelparous species [14]), Pennaet al. [16] were able to reproduce the dramatic fallin the survival curves (see [19] for an alternativederivation of catastrophic senescence). These resultswere confirmed by exactly solving the model in thisvery special case.

T.J.P. Penna et al. / Computer Physics Communications 121–122 (1999) 108–112 111

The model has been used in many other situations.The longevity of trees and some fish seem to bethe converse of the salmon problem [2]. Also theheritable longevity of humans, as observed by Perlset al. [18] is presented in this model, as shown byOliveira et al. [13], in the first paper using this modelto be published in a medical journal. The suddendisappearance of the Northern Cod off Newfoundlanddue to overfishing was also reproduced by introducingthe probability of fishing. This system is shown tobe very unstable to a slight increase of fishing [12].Social disruption in the wolves populations due tointense hunting, first proposed by Feingold [8], wasintensively studied by other authors. Feingold hasalso applied this model to studies of populationsof caribous. The stability of the dynamics and thepossibility of chaos is pointed out by Bernardes etal. [5].

Aspects of sexual reproduction have been also stud-ied with a modified version of this model. There aresome ways of introducing sexual reproduction in thismodel. In all of these implementations the individual istreated through two bit-strings instead of only one. Theimportant concept here is the dominance. In the firstattempt of introducing sexual reproduction was pro-posed by Bernardes [4]. In this version, each positionin a bit-string has a probabilityh of being dominant.In the second way, proposed by Stauffer et al. [26],an extra string contains bits set to 1 in correspondingdominant positions. A third version was proposed bySchneider et al. [24]. Inspired by a previous work byCebrat [6], this more recent version explicitly distin-guishes the sexual chromosomesX andY . The authorsclaim to provide an explanation for why women havelower mortalities than men. This explanation is sim-ply based on the fact that for females a bad mutationin oneX chromosome can be masked by the otherX

chromosome. For males, in the other hand, a singlemutation in theX chromosome would always act as adominant disease.

Stauffer et al. [26] made use of the sexual versionof the bit-string model to show the advantages of sex-ual reproduction, in disagreement with a previous pa-per by Redfield [21]. Redfield found no advantagesat all for sexual compared to asexual reproduction inher simulations. She suggested that females should se-lect younger males for reproduction as an explana-tion for the predominance of sexual reproduction in

nature. In that work, however, Redfield neglected theeffects of dominance and even age-structured popula-tions. These points were taken into account by Stauf-fer et al. using the current model and they have shownthe drastic advantages of sexual reproduction. Moss deOliveira and Sá Martins have pointed out that sexualreproduction is advantageous also after catastrophicsituations [23]. Moreover they also explain why na-ture does not rely on meiotic partenogenesis, a ques-tion raised by Bernardes [4], a question ignored in therecent special issue of Science (October) about Evolu-tion and Sex.

4. Conclusions

In summary, the bit-string model has shown to bea good tool for studies of age-structured populationdynamics, in both asexual and sexual versions. Manyaspects of the evolutionary theory of aging have beentested in this framework. Besides being plausible froma biological point of view, the bit-string model alsoallows easy implementation on computers.

Acknowledgments

This work is partially supported by Brazilian agen-cies CNPq and CAPES. TJPP is grateful to H.E. Stan-ley for the hospitality at CPS/BU. The authors alsothank D. Stauffer, S. Moss de Oliveira, P.M.C. de Oli-veira, Naeen Jan and J.S. Sá Martins for suggestionsand criticism along these years.

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