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Mortality plateaus and the evolution of senescence: why are old-age mortality rates so low?

For the last several decades evolutionary thinking about senescence has been completely dominated by two models: antagonistic pleiotropy (AP) and mutation accumulation (MA). The AP model proposes that senescence arises as a result of mutations that increase the fitness of the young and, through their pleiotropic effects, also decrease the fitness of the old (Williams 1957). The MA model proposes that senescence arises from mutations that are selectively neutral in the early stages of the life cycle, but deleterious at older ages (Medawar 1952). Both models propose that the force of natural selection declines at older ages, allowing the buildup of deleterious genetic effects confined to the later stages of the life cycle (Hamilton 1966; Charlesworth 1993, 1994; Partridge and Barton 1993).

Experimental support for the MA and AP models is mixed (Hughes and Charlesworth 1994; Curtsinger et al. 1995; Promislow et al. 1996). In this paper we discuss recent experimental studies that raise a new challenge for the standard models. Large-scale survival studies in medflies, fruit flies, and nematodes demonstrate that mortality rates increase rapidly with increasing age in the early stages of the life cycle, and then level off at advanced ages. Further, the leveling off, or "mortality plateau," occurs at roughly 20% daily mortality, far below 100%. Here we summarize the relevant data, concentrating on Drosophila, and investigate, by analytical models and computer simulations that incorporate various assumptions about mutational effects, the extent to which MA and AP generate late-life mortality plateaus. We rebut the claim by Mueller and Rose (1996) that both MA and AP models robustly explain the demographic data, and we suggest ways that the standard theory can be modified to take into account the new observations.


In Drosophila, age-specific mortality rates are estimated by establishing experimental cohorts of age-synchronized flies and recording the numbers of deaths at daily intervals as the cohort ages, continuing until the last animal dies (e.g., Pearl and Parker 1924). From the observations of complete survival times, one can estimate probabilities of death for each sample period. Although there are exceptions (Finch 1990), it is often observed that estimated death rates fit reasonably well to the predictions of the Gompertz or Gompertz-Makeham model (Finch 1990), which predict exponentially increasing mortality rates with increasing age.

In contrast to demographers, who typically work with hundreds of thousands of observations, experimental gerontologists and students of life history evolution typically employ relatively small numbers of animals for survival experiments. Initial cohort sizes of 50-100 animals are common (e.g., Graves and Mueller 1993). Small experiments are sufficient for estimating mean life spans, but are not adequate to provide accurate estimates of age-specific mortality rates, particularly at advanced ages (Pletcher, unpubl. manuscript). For instance, if one sets up a survival experiment with 100 flies, typical mortality rates will result in only a handful of live flies at 40 days postemergence. The details of mortality rates at advanced ages are not detectable unless larger experiments are undertaken.

In the early 1990s two research projects were initiated with the specific goal of obtaining accurate estimates of mortality rates among the oldest old in experimental systems. Carey et al. (1992) studied genetically heterogeneous populations of the medfly Ceratitus capitata. Curtsinger et al. (1992) studied genetically homogeneous populations of D. melanogaster. Both experiments reported that the daily mortality rate increases rapidly at early ages, and then levels off at [approximately equal to]20% at advanced ages. Subsequent observations on the bean beetle Callosobruchus maculatus (Tatar et al. 1993) and the nematode Caenorhabditis elegans (Vaupel et al. 1994) reported qualitatively similar results.

Some recent Drosophila data demonstrating mortality plateaus [TABULAR DATA FOR TABLE 1 OMITTED] are listed in Table 1. Studies have employed numerous types of genetically defined stocks: inbred lines, [F.sub.1] hybrid crosses of inbred lines, control and experimental stocks selected by Luckinbill et al. (1984) and Rose (1984) for long life span via late fertility, chromosomal heterozygotes carrying balancer chromosomes, stocks allowed to accumulate spontaneous mutations (Pletcher et al. 1998), and stocks generated through P-element mutagenesis (Clark and Guadalupe 1995). In all cases, flies were reared under controlled larval densities. Survival was studied in either vials or population cages under controlled initial density, temperature, illumination, and humidity.

The statistical criterion for leveling off involves maximum likelihood tests of the fit of Gompertz and alternative mortality models (Fukui et al. 1993; Pletcher, unpubl. manuscript). The alternative models that we have used are the two-stage Gompertz model (Curtsinger et al. 1992) and the logistic model (Fukui et al. 1993). The former model fits a Gompertz curve to the early data, finds a break-point age, and then fits a second Gompertz curve with a different slope to the older age mortality. The latter model is related to the Gompertz, but includes an additional parameter that causes the curve to bend over at advanced ages. For every study listed in Table 1, either the two-stage Gompertz or the logistic model provides a better fit to the data than the Gompertz model, by log-likelihood ratio test. Whenever the two-stage Gompertz model provides the best fit, it always involves a shallower slope at older ages, that is, mortality rates level off at older ages.

Figure 1 summarizes all data from the Curtsinger lab in which males and females were reared together in population cages, pooling over experimental blocks and genotypes; the curves should therefore be regarded as generic rather than specific to a genotype. The figures do not include data collected from vials or sexes reared separately, though those data are qualitatively similar. For both sexes, mortality rates increase approximately exponentially in the early stages of adult life, up to about 50 days postemergence, and then level off. At the age when leveling off occurred, there were approximately 6000 males and 8000 females still alive, which is a sufficient sample size to give considerable confidence in the reality of the phenomenon.


Do mortality plateaus exist because of a slowdown in the rate of aging in individual flies? If so, the almost universal belief that mortality inexorably climbs with age is flawed. Because individuals only die once, mortality rates are necessarily a population measure, and as a result, several population-level phenomena have been invoked to explain the demographic observations. Brooks et al. (1994) suggested that leveling off was due to contamination of experiments by young flies, but that objection overlooks that fact that the leveling-off phenomenon is based on thousands of very old flies, not just a few potential contaminants; further, in our earliest experiments both dead and live flies were counted on a daily basis, specifically to rule out contamination (Curtsinger et al. 1994). Hughes and Charlesworth (1994) suggested that leveling off occurs only in inbred lines, because of the increased sensitivity to environmental variance, but recent observations on hybrid genotypes suggest this is not the case (Table 1).

The most common objection has been that leveling off is an artifact of density. That is, if adult flies interact and affect each other's viability in a way that depends on population density, then that source of physical stress would be expected to decline as the flies age, because the cohort density decreases over time. The result, it has been proposed, would be lower mortality rates at older ages. Two experiments rule out density as the explanatory variable for leveling off in Drosophila. Khazaeli et al. (1995a) set up experimental cohorts at three different initial densities and found that mortality rates leveled off at older ages regardless of initial density. Khazaeli et al. (1996) studied mortality rates at constant adult densities, which was accomplished by replacing dead flies with marked mutants on a daily basis. Mortality rates leveled off in both experimental (supplemented) and control (nonsupplemented) populations.

Finally, if all individuals in a population age according to the Gompertz model, but there are different Gompertz curves for different subpopulations, then observed mortality rates will decelerate at late ages for the mixed population (Vaupel and Yashin 1985; Vaupel 1990). A wide range of Gompertz curves in the population is required to generate the degree of leveling off seen in the Drosophila and medfly data: Vaupel and Carey (1993) assumed a range of nearly 10 orders of magnitude in the constant term of the Gompertz model - a value greatly exceeding the normal range of quantitative characters (Falconer 1989). Although the degree to which heterogeneity contributes to mortality plateaus is still an open question, predictions of the heterogeneity models are experimentally testable (Khazaeli et al. 1995b; Curtsinger and Khazaeli 1998). Recent results in Drosophila suggest that environmental heterogeneity acquired in the larval stage contributes little to leveling off at older ages in this species (Khazaeli et al., unpubl. manuscript).


The existence of mortality plateaus far below 100% poses a problem for standard models of the evolution of senescence. The fundamental difficulty is that, under a wide variety of assumptions about mutational effects, evolutionary models predict post-reproductive mortality rates near 100% in equilibrium populations (Curtsinger 1995).

Mutation Accumulation

The first genetic model explicitly incorporating mutation pressure on the evolution of age-specific mortality patterns was developed by Charlesworth (1990). For mathematical reasons, Charlesworth assumed that spontaneous mutations were completely age specific in their effects. This implies that individual mutations have effects at only one age, and mutation-induced changes in mortality across age classes are uncorrelated. Two relevant predictions about mortality patterns came from this work. First, under mutation accumulation, additive genetic variance for mortality rates should increase with age (see also Promislow et al. 1996). More important for our discussion, the model predicts that if deleterious mutations are completely age specific in their effects, then a total collapse of survival is predicted at later ages - mortality rates would inevitably reach unity (Charlesworth 1990). Charlesworth concluded that this feature of the model likely reflects the arbitrary assumption about the age range of individual mutational effects.

Partridge and Barton (1993) presented a model of mutation accumulation in an organism with two age classes (juvenile and adult). Trade-offs were introduced by assuming that both high juvenile survival and high adult survival could not be maintained simultaneously. Two key points emerge from this analysis (Partridge and Barton 1993). If deleterious mutations have pleiotropic effects on both juvenile and adult age classes, mutation pressure will push adult mortality above its optimal value. Mortality rates would be expected to be quite high, but less than 100%. However, if mutations affect only the adult age class (i.e., they are completely age specific) an extremely low mutation rate is sufficient to generate what they term "catastrophic senescence." Adult mortality rates reach 100%. It is difficult to extend these results to a larger number of age classes, but it is reasonable to expect that, if mutations with effects confined to older ages are frequent enough, mutation accumulation will result in mortality rates at or near 100% after reproduction has ceased.

From these theoretical arguments, it appears that a theory involving deleterious mutation pressure alone as the cause for senescence is not consistent with mortality plateaus far below 100%. To further illustrate this, we employ a simple computer program that simulates the evolution of mortality patterns under the standard assumptions of the MA theory.

Consider a hypothetical, haploid organism with 10 age classes. The organism is assumed to be semelparous and to reproduce asexually while in age class 5. Assumptions about the number of age classes and the mode and timing of reproduction are arbitrary, but the qualitative behavior of the model is not sensitive to those parameters. In Monte Carlo simulations, individual organisms are represented in the computer by a vector containing the probabilities of death, q(x), in each age class x. A population of 200 organisms is founded with q(x) = 0:50 for all x, that is, there is initially no senescence with respect to mortality. Every generation, each organism is tested for survival to the reproductive age by comparing a random number drawn from a uniform distribution on the interval (0,1) with evolutionary fitness, w, computed as:

[Mathematical Expression Omitted]. (1)

If the organism survives to reproduce, it is tested for mutations in the progeny. Mutations affect mortality rate in one randomly chosen age class, either prereproductive or postreproductive.

Simulations were executed in Turbo Pascal. Pseudorandom number generation was done with the "ran0" generator of Press et al. (1986). We assumed that the probability of occurrence of a mutation is 10% per individual per generation (one mutation per genome per generation). Given that a mutation occurs, the probability of a deleterious effect is assumed to be 0.99, while the probability of a beneficial effect is 0.01. An age class x is randomly chosen by sampling the uniform distribution, and then the mortality rate for that age class is either increased (deleterious) or decreased (beneficial) by 0.05, subject to the constraint that all q(x) lie in the interval (0,1). Organisms that survive until age class 5 are randomly chosen for reproduction and mutation until 200 progeny are produced, each carrying the parental q(x) vector with possible mutational modifications. The generational cycle then starts over.

The evolution of age-specific mortality curves for the MA scenario is shown in Figure 2. As mutations occur and the population evolves in the first [approximately equal to]750 generations, prereproductive q(x) decline slightly and postreproductive mortality increases dramatically. This occurs because rare, favorable mutations that improve prereproductive survival are selectively favored, while postreproductive survival, which is irrelevant to evolutionary fitness, is inflated by random mutations that are predominantly deleterious. In the long run, by generation 30,000, the mutation accumulation process results in prereproductive mortality rates near 10%, while postreproductive mortality rates approach 100%.

Using computer simulations, Mueller and Rose (1996) briefly consider a mutation accumulation model, and suggest the "process . . . produces a late-life mortality plateau in mortality rates." Considering that the plateau occurs at the greatest mortality rate allowable in the simulation, they also conclude that "if this [MA] process were the only important one determining mortality rates then the mortality rates in the plateau would eventually rise to 100%" (p. 15252-15253). Thus, there is general agreement from a variety of authors that the MA model predicts a mortality "wall" at older ages.

Antagonistic Pleiotropy

Theoretical developments explicitly investigating the results of antagonistically pleiotropic mutations are lacking. Current arguments for AP are based on optimality models involving physiological constraints (see below; Abrams and Ludwig 1995). Recently, Mueller and Rose (1996) (hereafter MR) used the results of computer simulations to conclude that pleiotropic mutations with beneficial effects on fitness at one age and deleterious effect at another are sufficient to generate mortality plateaus at levels below 100%. Referring to leveling off, the claim is made (Mueller and Rose 1996, p. 15253) that standard "population genetics theory for evolution in age-structured populations (i.e., Charlesworth 1994) ... can readily account for the findings now being obtained by experimental gerontologists and others." In this section, we examine the results of MR, and show by a combination of computer simulation and analytical methods that the results are not consistent with experimental data and are quite sensitive to assumptions about mutational effects. We will show that, even when AP mutations are common, the vast majority of mutational effect scenarios result in mortality rates evolving to nearly 100% at older ages.

The antagonistic pleiotropy simulations in MR take an infinite sites approach by assuming that each mutation occurs at a distinct genetic locus. In the simplest case, each locus is assumed to influence survival probabilities at exactly two randomly chosen ages. New mutations at those loci are assumed to increase survival at one age and decrease it at another. We emphasize that, with one exception, the MR models assume every new mutation has antagonistic and pleiotropic effects; that is, every new mutation improves survival in one age class, while reducing it another.

Once a mutation has occurred, the MR model assumes that survival from age x to age x + 1 is modified according to two equations, one describing the beneficial effect and one the deleterious effect for the two randomly chosen age classes. For beneficial effects

[Mathematical Expression Omitted], (2)

and for deleterious effects

[Mathematical Expression Omitted], (3)

(see MR eqs. 4 and 5). [P.sub.x] is survival in age class x before fixation of the new mutation, [Mathematical Expression Omitted] is survival after fixation of the new mutant, and [Lambda] is an arbitrary constant. These equations imply that for an age with low survival (high mortality rate) deleterious effects have a smaller magnitude than beneficial effects, while the opposite is true for ages with high survival (low mortality). For instance, if [P.sub.x] = 0.20 and [Lambda], = 0.1, mutations increase survival by 0.08 or decrease survival by 0.02. If [P.sub.x] = 0.80 and [Lambda] = 0.1, then mutations increase survival by 0.02 or decrease survival by 0.08. Thus, in the MR model, the effect of a mutation is dependent upon the magnitude of the survival parameter for the affected age class. We show below that it is this assumption about mutations that forces the model to equilibrate at intermediate mortality levels.

Mathematically, equations (2) and (3) imply that there is an equilibrium mortality rate at all ages imposed by the proportional effects of mutation, the level of which depends on the ratio of the probabilities of fixing a mutation that decreases mortality versus fixing one that increases mortality for a particular age class. To illustrate, consider a simple but relevant case: the equilibrium level of mortality at an age that is unaffected by natural selection. This case is relevant because mutations affecting old ages, after reproduction has ceased, are expected to be essentially neutral, having an effect on mortality rates but no effect on evolutionary fitness (Charlesworth 1994). Under neutrality, the probability of fixing a mutation that increases mortality for a particular age class is the same as the probability of fixing a mutation that decreases mortality for that age class. The relative proportion of "beneficial" mutations that go to fixation is simply equal to the proportion of total mutations that decrease mortality at that age. Letting the proportion of "beneficial" mutations affecting age x be [Pi], on average

[Mathematical Expression Omitted]. (4)

After simplification, equation (4) becomes

[Mathematical Expression Omitted]. (5)

At equilibrium, [Mathematical Expression Omitted] and equation (5) becomes

[Mathematical Expression Omitted], (6)


[Mathematical Expression Omitted], (7)

where [Mathematical Expression Omitted] is the equilibrium mortality rate at age x. Thus, for any age influenced only by the fixation of neutral mutations, mortality rates will converge to an equilibrium value equal to the probability that any single mutation increases mortality. If beneficial and deleterious mutations are equally likely, late-life mortality rates will evolve to [Mathematical Expression Omitted]. This equilibrium exists because of the assumption of proportional mutational effects embodied in equations (2) and (3).

To investigate the sensitivity of the MR results to slight deviations from the conditions set forth in their simulations, we modified our mutation accumulation simulation to allow for pleiotropic mutations. Figure 3 shows the results of a simulation in which we assume that two different classes of mutations are possible: (1) deleterious mutations that increase mortality rate in one randomly chosen age class, and (2) pleiotropic mutations that improve survival in one randomly chosen age class and decrease survival in another randomly chosen age class. Although in this case the frequency of pleiotropic mutations was assumed to be low, the results are qualitatively similar when age-specific deleterious mutations are rare (Pletcher, unpubl. data). Mortality rates evolve to near unity at postreproductive ages [ILLUSTRATION FOR FIGURE 3 OMITTED].

Using the ideas presented in equations (2-6) we have developed analytical methods to determine equilibrium mortality rates under different pleiotropic mutation regimes. Details of the analytical methods are presented in the Appendix. From these analyses we are able to demonstrate the following for the antagonistic pleiotropy model. (1) If the MR assumption of proportional mutational effect is changed to constant mutational effect then the mortality rates at postreproductive ages evolve to near 100%. (2) The MR result is dependent upon pleiotropic mutations that affect only late ages. That is, pleiotropic mutations that increase mortality rate at one postreproductive age and decrease mortality rate at another postreproductive age tend to drive the equilibrium mortality to 50%, because of the proportionality assumption. If such late-late mutations are excluded from the model, postreproductive mortality rates evolve to near 100%. (3) The MR model assumes that all mutations are pleiotropic. If this is changed to the assumption that a fraction of mutations are unconditionally deleterious and the rest pleiotropic, then postreproductive mortality rates evolve to near 100%. (4) The MR model predicts that populations evolving under different effective population sizes (or age-specific selection regimes) should exhibit late-life mortality plateaus at different levels (see Appendix). The experimental data do not support this prediction. Mortality plateaus are observed to occur at about the same level in stocks maintained under vastly different demographic regimes (see Curtsinger et al. 1995; Pletcher et al. 1998). In short, the results of the MR model for the antagonistic pleiotropy case are quite sensitive to assumptions about mutational effects and do not robustly explain experimental observations.

What can we conclude from these studies? Without making a priori assumptions about the number of loci affecting mortality rates at different ages, the majority of mutation scenarios incorporated into standard population genetic models of evolution in age-structured populations do not predict low level mortality plateaus late in life. Unfortunately, at the present time, we can not say which scenario(s) are more likely. Very little is known about the age-specific effects of spontaneous mutations. To date, only three studies have addressed the effects of mutation on age-specific mortality characters (Clark and Guadalupe 1995; Houle et al. 1994; Pletcher et al. 1998). Further, classic ideas about mutation pressure (Mukai et al. 1972) are being challenged by recent experimental results (Fernandez and Lopez-Fanjul 1996; Keightley and Caballero 1997) and statistical analyses (Keightley 1996). In the next section we discuss how current ideas about the properties of spontaneous mutations relate to the evolution of mortality plateaus.


Classic experiments on the effects of mutation, including the work of Mukai et al. (1972) on Drosophila, have led most evolutionary biologists to believe that the majority of spontaneous mutations are deleterious with respect to fitness. These deleterious mutations are thought to occur at a rate of about one per offspring and to cause a decline in fitness (as measured by larval viability) of about 1-2% per generation (Mukai et al. 1972; Simmons and Crow 1977). More recently, a reanalysis of the original Mukai et al. (1972) data by Keightley (1996), coupled with experimental results from Fernandez and Lopez-Fanjul (1996) and Keightley and Caballero (1997), have put the classic results under scrutiny. Keightley and Caballero (1997) suggest that the deleterious effects of spontaneous mutations in Caenorhabditis elegans are about 100 times less than calculated by Mukai et al. for Drosophila. Thus, it seems that even for classically studied characters, the matter of the distribution of spontaneous mutational effects is far from settled (Peck and Eyre-Walker 1997).

Unfortunately, even less is know about the effects of mutation on age-specific characters such as mortality. Houle et al. (1994) analyzed 48 mutation accumulation lines for average life span and a number of age-related life-history traits including early and late fecundity, and early and late male mating ability. They observed positively correlated mutational effects among early and late age classes and suggested that mutation accumulation as a mechanism for the persistence of senescence is not compatible with these results. In a Drosophila study aimed directly at examining the effects of novel mutations on age-specific mortality, Pletcher et al. (1998) found that mutational effects were strongly positively correlated between ages separated by 7-14 days, but for greater time intervals the correlation declined. Ages separated by four weeks or more showed no significant mutational correlation. More interestingly, they found little evidence for any mutational effects on mortality rates at later ages. Early age mortality in one sex (females) was shown to experience a significant increase as a result of the accumulation of deleterious mutations.

In a recent review of the evidence for age-specific effects of mutations and their influence on the evolution of senescence, Promislow and Tatar (1998) suggest that mutations act additively on the ln([[Mu].sub.x]) where [[Mu].sub.x] is the age-specific hazard or risk of death and is estimated by -ln([P.sub.x]) (Lee 1992). They note that phenotypic manipulations, such as dietary restriction (Weindruch and Walford 1982), display nearly equal effects on ln([[Mu].sub.x]) at all ages. Further, if mutations act additively on ln([[Mu].sub.x]), then genetic variance generated by new mutations should be normally distributed on this scale. Data from two experiments (Clark and Guadalupe 1995; Pletcher et al. 1998) support this prediction (Promislow and Tatar 1997).

In summary, much more information on the distribution of mutational effects on fitness and on age-specific mortality is needed before we can begin to discern the types of mutations that might influence the evolution of age-specific mortality patterns. For now, we suggest that models investigating mortality plateaus use a variety of different mutation scenarios, as it is likely that many different classes of mutations are occurring simultaneously. Conclusions should focus on how robust the results of specific models are to variation in classes of mutational effects.


How can the evolutionary models be modified to account for mortality plateaus around 20% per day? There are several reasonable avenues of research that merit investigation. First, populations might not be at equilibrium; perhaps old-age mortality rates are far below 100% because the mutation accumulation process is particularly slow or because mutation has little effect on mortality at these ages. Although this explanation is not likely as mortality rates have been evolving for hundreds of millions of years, it is testable. Pletcher et al. (1998) have documented a significant input of mutational variation per generation for age-specific mortality rates early in life. Further, female early-age mortality increased nearly 2% per generation under mutation pressure (Pletcher et al. 1998). Thus, it appears mutation is a significant force for senescence with the potential for rapidly increasing mortality rates in the absence of selection. However, late-life mortality appears to be relatively unaffected by the input of new mutations (Pletcher et al. 1998). With further observations, which are ongoing, we will be able to make quantitative predictions about the rate at which mortality rates should evolve under mutation pressure.

A second possibility is that mortality rates in adjacent age classes may be constrained to be similar (positive pleiotropy). In any single organism, the probability of death at one age is not likely to be independent of the probability of death at another. Susceptibility to sickness, disease, reproductive costs, etc. is dependent on the physiological state of the organism; we do not expect large, instantaneous changes in state. An individual who is healthy today is likely to be healthy tomorrow. Support for a flexibility constraint between mortality rates at adjacent ages is provided by Pletcher et al. (1998) using Drosophila, who suggest that the average mutation may effect mortality rates over a range of age classes spanning three to four weeks.

The results of computer simulations that incorporate one form of positive pleiotropy are presented in Figure 4. For these simulations, mortality rates are allowed to evolve under the same assumptions as the previous mutation accumulation simulation with one exception: mortality rates in adjacent age classes are constrained to be within [+ or -] 10% of each other. After 300,000 generations of selection, drift, and mutation, the evolved mortality curve has two qualitative features that we seek to explain: low prereproductive mortality and moderate postreproductive mortality [ILLUSTRATION FOR FIGURE 4 OMITTED]. However, the curve does not level off. In general, even these sorts of constraints are not expected to stop the rise in mortality rates at ages so advanced that they are not genetically correlated with ages experiencing significant selection pressure.

More realistic, functional constraints on the aging process have been investigated using optimality arguments. Abrams and Ludwig (1995) explored the relationship between mortality and age that would be optimal if senescence results from a diversion of energy and resources from repair (or maintenance) to reproduction. They concluded that a wide variety of mortality patterns could be expected if the repair-reproduction trade-off were a major determinant of senescence. In several specific cases a marked deceleration of mortality rates late in life is predicted. It would be interesting to extend these results to include how the influence of different types of mutations would change optimal mortality patterns, and what variation in patterns of mortality would be expected if parameters of the model were subject to small changes (mutation) in the absence of selection.

Third, redundancy models, which are closely related to the heterogeneity models mentioned above, predict a deceleration of mortality rates at older ages. These models assume that an organism is composed of multiple subsystems, and the organism will survive as long as a subset of systems is in working order (Gavrilov and Gavrilova 1991). A simple example is an organism with n systems connected in parallel such that the organism will survive as long as at least one system is functioning. In this case, mortality rates tend to increase exponentially early in life and level off at older ages (Gavrilov and Gavrilova 1991, p. 253). These models originate from reliability theory and view senescence as a purely physical process, unrelated to the fact that the intensity of natural selection declines with advancing age. The large body of evidence in favor of the role of natural selection in the evolution of senescence makes this view rather questionable (Luckinbill et al. 1984; Rose 1984; Promislow 1991). Nevertheless, redundancy models may be testable by "stress" experiments (Curtsinger and Khazaeli 1997).


The data and simulations presented here demonstrate that standard models of the evolution of senescence are not sufficient to robustly account for the observation that mortality rates level off at values far below 100% for advanced ages. All of the models presented here agree on one point: MA unmodified and in its simplest form predicts a wall of mortality at advanced ages; we believe AP does also, but that is a point of some contention. As with most successful evolutionary theories, the forging of a modified theory of senescence that is consistent with the new demographic data will require a cooperative effort of theoreticians and experimentalists from several disciplines. On the theoretical side, models incorporating positive pleiotropy and/or mechanistic constraints on mortality rates at different ages appear to be the most promising. Combining classic demographic (nonevolutionary) methods, including heterogeneity and redundancy models, with optimality and quantitative genetic (evolutionary) arguments is likely to provide interesting insights. Regarding future experimental work, we need to know how specific genes and mutations in those genes affect survival at different ages. The power of molecular genetic techniques make it possible to investigate specific genetic and biochemical mechanisms that influence mortality rates throughout the life span. Quantitative genetic experiments that examine the influence of individual, induced mutations and of the accumulation of mutations that occur naturally should provide evidence for different classes of mutational effects and the rates of mutation for each of these classes. Of course, all these experimental techniques must be combined with demographic experiments that are large enough to estimate mortality rates, especially at advanced ages.


We thank J. Vaupel for introducing us to problems in aging and demographic techniques. Comments that greatly improved the manuscript were provided by P. Abrams and two anonymous reviewers. Mountains of survival data were collected by A. Khazaeli, H. Fukui, S. Pletcher, D. Promislow, M. Tatar, D. Townsend, A. Resler, C. Misiak, K. Kelly, K. Broz, C. Gendron, G. Kantor, L. Ackert, and A. Kirscher. Research is supported by grants from the National Institutes of Health (AG 08761 and AG 11722).

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This appendix will illustrate how we calculate expected equilibrium levels of age-specific mortality for a variety of mutational effects, with special attention given the assumptions set forth in Mueller and Rose (1996).

General Considerations

Mueller and Rose (1996) (hereafter MR) simulate mutational events, and for a discrete age model, we can express the age-specific survivorship after the introduction of a novel mutation as changing according to the following rules

[Mathematical Expression Omitted]. (A1)

where x is the age in question and t represents time according to the number of mutational events (see p. 15250 of Mueller and Rose 1997).

Assuming the fitness of the current life-history is one, the fitness of the mutant homozygote life-history is 1 - s, and that of the mutant heterozygote is 1 - hs, the probability a novel mutation goes to fixation in a diploid population of size N is given by

[Pi](q) = [integral of] [Phi](y)dy between limits q and 0 / [integral of] [Phi](y) dy between limits [infinity] and 0 (A2)

where q is the initial frequency of the mutant (assumed to be 0.1 in MR) and [Phi](y) = exp{-2Nsy[2h + y(1 - 2h)]} (Ewens 1979). MR assume each new mutation is either completely dominant or completely recessive. In either case, the model reduces to an equivalent haploid model where

[Mathematical Expression Omitted]. (A3)

The fitness of each life-history, r, is obtained using standard theory describing evolution in age-structured populations (Charlesworth 1994) and is given by the solution to

1 = [summation of] [e.sup.-rx][l.sub.x][m.sub.x] where x = 0 to [infinity], (A4)

Where [l.sub.x] is the probability of survival from age 0 to age x

[Mathematical Expression Omitted]

and [m.sub.x] is the expected number of offspring produced by an individual of age x. Thus, for a mutant life history with fitness [r.sub.m] invading a population of individuals with fitness [r.sub.p], s = [r.sub.m] - [r.sub.p].

Using equations (A1), (A3) and (A4) to express the expected survivorship at time t + 1

[Mathematical Expression Omitted], (A5)

Where [x.sub.max] represents the number of age classes, [[Pi].sub.b] is the probability of fixing a mutation that increases survivorship at age x, [[Pi].sub.d] is the probability of fixing a new mutant that decreases survivorship at age x, [Lambda] is a constant, and the ratio of deleterious to beneficial mutations is v: 1

At the equilibrium, [Mathematical Expression Omitted]. Substituting this relation into (A5) and solving for [Mathematical Expression Omitted]

[Mathematical Expression Omitted] (A6)

Solving (A6) simultaneously for all x will provide the equilibrium age-specific survival values [Mathematical Expression Omitted] expected to evolve under mutation-selection-drift equilibrium. Simultaneous expected equilibria were solved using Mathematica version 3.0 (Wolfram Research 1997). These equilibria are the values below which the expected age-specific survivorship increases and above which the expected survivorship decreases. They do not imply that the mortality curve will no longer change under mutation pressure, but when mutational effects are small it will always tend to converge on these values.

Mutation Accumulation Case

Under a simple model of mutation accumulation - each mutation affects a single-age - late-age mortality rates are essentially neutral and [[Pi].sub.g](q) = [[Pi].sub.b](q). Thus, the equilibrium level of late-age survivorship is

[Mathematical Expression Omitted]. (A7)

If we assume deleterious mutations and beneficial mutations are equally likely, v = 1 and [Mathematical Expression Omitted] [ILLUSTRATION FOR FIGURE A1 OMITTED]. If deleterious mutations are 10 times more likey than beneficial mutations, [Mathematical Expression Omitted]. Thus, even in the face of proportional effects forcing survivorship to 0.5, if deleterious mutations are much more prevalent than beneficial mutations - a common assumption applied throughout evolutionary theory - age-specific mortality rates would be expected to reach very high levels late in life.

If we assume mutational effects are not proportional such that the age-specific survivorship after the introduction of a novel mutation changes in the following manner

[Mathematical Expression Omitted], (A8)

then there is no equilibrium survivorship/mortality at older ages If beneficial mutations and deleterious mutations are equally likely, mortality rates would drift around 05. If deleterious mutations are even slightly more common, late-life mortality rates would evolve to a probability of one.

The mutation accumulation simulations in MR assume that each age-specific mortality probability is determined by a fixed number of diallelic loci. At each locus there is a "good" allele and a "bad" allele with respect to mortality, and thus there is a predefined maximum and minimum mortality rate for each age. All mutations are deleterious, and the simulation results are exactly what we might expect Late-life mortality increases until individuals are fixed for the "bad" allele at all relevant loci. The representation of constant and relatively low levels of mortality at older ages in MR (MR fig. 6) is a result of defining only two alleles for a fixed number of loci - mortality rates continue to increase until they can no longer do so.

Antagonistic Pleiotropy Case

MR assume all mutations have effects on exactly two age classes. One age experiences a proportional increase in survivorship while the second age experiences a proportional decrease. From the perspective of a single, late age class, a mutation that increases survivorship will be deleterious on average due to decreases in survivorship early in life, while a mutation that decreases survivorship at older ages will have beneficial effects due to the pleiotropic increases in survivorship at early ages.

The calculation of the average s for a mutation at a specific age under the assumptions of antagonistic pleiotropy is complicated and requires averaging over the possible pleiotropic effects at all other ages. Nevertheless, this can be done numerically, and the probability of fixing a beneficial mutation or a deleterious mutation can be calculated for each age using equation (A3), and equation (A6) can be solved to hold simultaneously for each age.

Although it is difficult to intuitively deduce late-life equilibrium mortality under this scenario, there are a number of points worth mentioning. Mutations that increase mortality at old ages will tend to be beneficial when early life mortality is greater than zero due to their tendency to be associated with decreases in mortality at early ages. When early age mortality evolves to zero under the force of natural selection, these mutations are nearly neutral; and although they may drift to fixation, this fate is not assisted by natural selection. Conversely, mutations that decrease mortality late in life will continue to be selected against due to their association with increases in mortality at early ages. Thus, the decline of early age mortality results in a net mutation pressure in favor of increasing mortality rates at older ages. This mutation pressure is opposed by those mutations with beneficial and deleterious effects at old ages. Because the mutation pressure described above pushes mortality to values above 0.5, these double late-age mutations will cause an overall decrease in mortality and keep equilibrium values below unity. Thus, the greater the proportion of age classes in which selection has reduced mortality to near zero, the larger the influence of the neutral pleiotropic mutations and the greater the late-life mortality equilibrium.

Equilibrium mortality curves for two population sizes (2000 and 200) of an organism with 10 adult age classes are presented in Figure A2. These population sizes were chosen to visually correspond with the conditions put forth by MR for a hypothetical organism with 101 age classes. The strength of natural selection is expected to decline more slowly with age in the larger population (Charlesworth 1994). Since the number of age classes is the same for both population sizes, the fraction of ages under strong selection is greater for the larger population. As a result, mutations with deleterious effects at older ages have a greater marginal benefit, and we expect old-age mortality rates to equilibrate at higher values. This prediction is borne out as [Mathematical Expression Omitted] for the population of size 2000, while [Mathematical Expression Omitted] for the population of size 200. These results are in close agreement with the results of the simulations presented in MR (Mueller and Rose 1996, fig. 2). Further, if we exclude pleiotropic mutations with double effects late in life and allow only those with one effect at young ages and the opposite effect at older ages, equilibrium mortality rates are 100% due to the mutation pressure generated by the creation of the essentially neutral class of mutations mentioned above [ILLUSTRATION FOR FIGURE A3 OMITTED].
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Date:Apr 1, 1998
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