Sociality, Hierarchy, Health: Comparative Biodemography: A Collection of Papers (2014)

2


Alleles, Mortality Schedules, and the Evolutionary Theory of Senescence

Kenneth W. Wachter

EVOLUTIONARY SENESCENCE IN THE DAYS OF ZEUS AND THE SALMON

The publication of the collection Between Zeus and the Salmon edited by Wachter and Finch (National Research Council, 1997) marked a moment of consolidation for the field of biodemography. Regularities in the shapes of demographic schedules as functions of age across a small set of highly diverse species had been established and confirmed. Demographers were learning about classical evolutionary theory of senescence. They were becoming familiar with Medawar’s ideas about the process called mutation accumulation, with Williams’ ideas about the phenomenon called antagonistic pleiotropy, and with the need to understand commonalities in demographic outcomes across species within evolutionary frameworks.

In 1997, on the other hand, quantitative models for evolutionary processes affecting senescence were short on demographic structure. Overall scientific knowledge of the character of genetic variation across the genome was limited. Demographic thinking stood in need of greater clarity about the distinct roles of genetic variation maintained in equilibrium and erstwhile genetic variants, now omnipresent or gone to “fixation.”

“Allele” in the title is a word for a genetic variant. In this paper, I generally refer to the most common kind of alleles, found in Single Nucleotide Polymorphisms (SNPs) and Single Nucleotide Variants (SNVs). An individual’s genetic code can be pictured as a string of letters; at most positions or “sites,” all members of a species have the same letter, but at some mil-

lions of sites, a majority has one letter and a minority a different letter. Any minority, no matter how small, counts as an SNV. Substantial minorities, say 1 percent or more, count as SNPs. Alleles with multiple effects, for instance, effects on both fertility and mortality, are called pleiotropic; when some are beneficial and some detrimental, they are called antagonistic. These words have become important to demography, because biodemographic theory is generating hypotheses about the statistical properties of alleles and their relationship to demographic schedules while genetic sequencing technology is making it possible to confront hypotheses with observations.

Demographers turn to evolutionary theory for two purposes: to understand commonalities and to understand differences. In this paper, I concentrate on commonalities. I consider research aimed at understanding general patterns shared across very different kinds of organisms in very different environments. The genetic process of mutation accumulation introduced by Medawar offers some of the best material for this purpose, for finding what flies, worms, elephants, and humans have in common. Other papers in this volume concentrate on differences. They scrutinize research aimed at making sense of specific contrasts, the differences in adaptive strategies and body plans that separate flies, worms, elephants, and humans. A rich literature drawing on a long tradition of fieldwork in biology employs the tools of optimal life history theory. Demographers have been extending this direction of inquiry with studies of the implications of sociality and intergroup and intergenerational transfers in social species. This part of evolutionary demography is described in this volume in the papers by Ronald Lee and by Paul Hooper, Michael Gurven, and Hillard Kaplan. Broader implications of biodemography are described by James W. Vaupel (2010).

A word is in order about fixation versus persisting genetic variation. Consider the treatment in Kaplan and Robson (2002) of symbiotic relationships between investment in brain size, intelligence, and longevity, described further in papers in this volume. The legacy of evolutionary tradeoffs is presumably encoded into alleles that have long since progressed to fixation. An association between brain size and intelligence driven by genetic variation is not to be expected across a present-day class of undergraduates. By the same token, age-specific demographic outcomes may easily reflect optimal tradeoffs across the lifespan without being driven by genes that differ from person to person. Within social species like humans, transfers within groups and across generations may be seen as shifting the optimal balance of investments in growth, reproduction, survival, and repair under hypothetical physiological constraints (Chu and Lee, 2006; Baudisch, 2008) without a mandate for genetic specification. Tradeoffs may be realized in strategies pursued by individuals within environments during their lives in accordance with evolved norms of reaction. Allelic variation is not required for optimal life history theory to make sense.

Successor volumes to Between Zeus and the Salmon, entitled Cells and Surveys, Offspring, and Biosocial Surveys, edited respectively by Finch, Vaupel, and Kinsella (National Research Council, 2001), Wachter and Bulatao (National Research Council, 2003), and Weinstein, Vaupel, and Wachter (National Research Council, 2007), emphasized the need for collection of genetic markers in conjunction with longitudinal social science surveys rich in demographic and behavioral information. The hurdles standing in the way of these enterprises were surmounted, and we are starting to reap the benefits of the new data. Analysis of genome-wide data is a huge enterprise pursued by giant research teams all over the world. The scientific returns from biodemographers doing the same kinds of studies that biomedical researchers are already doing seem modest. Many groups are looking for “silver bullets” or immediately valuable needles in haystacks, small numbers of candidate genes where different alleles entail big effects. They are, in other words, looking for cases where allelic variation in a population is strongly associated with medical and behavioral outcomes. Big effects mean “bingo.”

Biodemographers, on the other hand, while sometimes sharing such ambitions, also bring a different perspective. We are interested in small effects, in haystacks rather than needles. It is the accumulation of large numbers of somewhat independent small age-specific effects, whose distribution has been shaped over long stretches of time by natural selection, that seems most plausibly responsible for shared regularities in age-specific demographic schedules. The obvious comparison is with the heights of members of a population, going back to Francis Galton, where the observation of Gaussian distributions among adult members of populations is not ascribed to the detailed action of a few alleles but to the Central Limit Theorem applying to a large collection of small statistically independent contributions. It is generally a disappointment to biomedical and behavioral researchers when few, if any, effects on some trait from a large suite of genetic polymorphisms prove statistically significant. But for the program of evolutionary demography, when analysis pertains to traits that are known from twin studies to be heritable, bounds that show that genetic effects are individually small, but numerous and various, have payoff. They underwrite the relevance of statistical sources for regularity. Small effects mean “bingo.”

Ideas presented here are developed with more formal structure in Wachter et al. (2014), based on the January 2014 Sackler Colloquium of the National Academy of Sciences. Our discussion is restricted to adult, senescent mortality, where most progress has been made so far. Extensions to the analysis of regularities in infant and child mortality and in age-specific fertility should be possible, but the requirements and concomitants of growth and development and of mating practices make for complexities that are prudent to postpone.

Different kinds of heterogeneity are at stake in different segments of biodemographic modeling. The next section begins with a context familiar to many demographers, proportional hazards, and shows how new models can be seen as attempts to overcome limitations of fixed-frailty specifications in hazard analysis. Succeeding sections take up in turn three parts of the modeling repertoire: stochastic vitality, Genome-Wide Association Studies, and mutation accumulation. The final section draws back from details to survey some big questions and answers that evolutionary demography is offering. It concludes with a glance at early results from genome sequencing and the promise they hold for connecting some biodemographic models with empirical genetic data.

HETEROGENEITY: LIFELUCK, RISK, AND FRAILTY

Natural selection operates on phenotypic variability correlated with genetic variation. The underpinnings of evolutionary demography are therefore found in demographic treatments of population heterogeneity. For demographers the familiar treatment of heterogeneity is the fixed frailty formulation of Vaupel, Manton, and Stallard within the framework of proportional hazard models (Vaupel et al., 1979). The advantages and limitations of fixed frailty models serve as a convenient point of departure.

The appeal of proportional hazard modeling, on top of its empirical success, is the separation it allows between the characterization of differentials and the characterization of age patterns. Sir David Cox launched the subject with partial likelihood methods by separating estimation of age-independent effects of covariates from estimation of an age-dependent baseline hazard function. More generally, who dies early and who dies late can be studied on a percentile basis separately from studying the ages to which percentiles correspond. In one modern formulation (Wachter, 2014b), distinctions are drawn among three sources of heterogeneity called “lifeluck,” “risk,” and “frailty.”

Lifeluck is represented by a unit exponential random variable denoting on a logarithmic scale the relative position of an individual’s duration till death among people entirely alike in observed and unobserved covariates. It is the quantity in use for simulating an age at death, once the survival curve is completely specified. Risk encapsulates effects of an individual’s observed covariates, familiar from the hazard multiplier in any Cox model. Frailty is the lifelong fixed multiplier of Vaupel et al. (1979), a random statistical effect meant to capture unobserved heterogeneity, generally assumed to be drawn from a Gamma probability distribution. These three components of heterogeneity within the context of proportional hazards adumbrate the processes described in the following sections on stochastic vitality, effects in Genome-Wide Association Studies, and mutation accumulation.

The mechanics of hazard modeling via lifeluck, risk, and frailty may be followed in the textbook treatment in Chapter 8 of Essential Demographic Methods (Wachter, 2014a). For this paper’s purposes, what matters is the metric provided by the quotient of lifeluck divided by the product of risk and frailty. In this metric, for studying frailty distributions, populations with different baseline hazard functions can be pooled.

Variance in frailty manifests itself in a tendency for aggregate population hazard functions to taper off into plateaus at extreme ages. As population members age and die, demographic selection is culling high-frailty individuals early, leaving a set of survivors with lower frailties. The observation of plateaus in survival data for a number of model organisms and for humans, findings that did much to launch the field of biodemography, is a chief empirical phenomenon whose explanation is at stake in evolutionary models. It is well understood, however, that most kinds of heterogeneity, and not solely fixed frailty, tend to promote plateaus.

The fixed frailty framework has advantages and limitations, advantages notable in early studies of the heritability of longevity and limitations coming to the fore as data and modeling progress. In Yashin et al. (1995), frailties for a pair of related individuals are treated as the sum of a gamma variate shared between them and gamma variates unique to each of them, with the same rate parameter and differing shape parameters estimable from twin data. The sum of shape parameters, estimated at 4.1 in that study, is often rounded up to 5 to fit observations of plateaus among cohorts now reaching old age. Twin-study estimates of the heritability of adult lifespan at around 25 percent are described on page 537 of Vaupel (2010). Parameters can be chosen so that demographic selection with fixed Gamma frailty and Gompertz baseline hazards can fit observed survival curves and so in a formal sense could fully account for observed plateaus, but in a substantive sense such an explanation for plateaus is unsatisfactory.

In particular, if frailty is taken as a quantity fixed across the lifespan, the extent of heterogeneity required to promote plateaus at old ages implies some individuals with such low frailties as to have implausibly low death rates at early adult ages. Thanks to the pooling property already mentioned, it is reasonable to think about samples of a million or even a billion individuals. The least-frail member in a sample of a million with Gamma frailty with shape parameter 5 has average frailty 1/33. Such factors are many times smaller than those for the most potent observed risk factors and they entail the presence of a few individuals with superhuman low mortality at early adult ages.

A second substantive limitation of the fixed frailty formulation is the implication that heterogeneity among survivors always decreases with age. It is hard to maintain that the physiological condition of 90-year-olds is more homogeneous than the condition of 25-year-olds, even bearing in

mind that the proportional effect of higher old-age hazards amplifies whatever heterogeneity is there.

These two limitations of the fixed frailty perspective point the way toward two main refinements of the treatment of heterogeneity in mortality. The first limitation can be avoided by viewing a baseline hazard as a sum of cause-specific components, each with its own profile of age-specific effects within a competing risk framework. If separate random components of frailty multiply each age-specific component of baseline, heterogeneity driving plateaus at older ages is partially decoupled from heterogeneity at young adult ages, relieving the sense of implausibility. For components associated with genetic influences, this generalization of frailty points forward toward models of mutation accumulation.

The second limitation of the fixed frailty perspective is the enforced decline of heterogeneity among survivors. That can be avoided by replacing fixed frailty with a form of frailty changing over time. Initial randomness in physiological condition can be supplemented by incremental randomness across the lifecourse from the “thousand natural shocks that flesh is heir to.” This generalization of frailty points forward toward models of stochastic vitality.

STOCHASTIC VITALITY

In stochastic vitality models of survival through the lifecourse, heterogeneity present at birth is supplemented age by age by systematic trends and random shocks, usually modeled with a Markov process. Vitality may be modeled as a high-dimensional vector including physiological indices whose transitions can be estimated from longitudinal surveys, as in the sophisticated stochastic risk factors models of Kenneth Manton, Anatoli Yashin, and their collaborators. Generic properties are readily examined with stylized unidimensional models building on the pioneering work of Le Bras (1976). For purposes of visualization, variations on Brownian motion processes along the lines of Weitz and Fraser (2001) supply easy examples. A Brownian motion is a continuous counterpart of a random walk. Picture a variable called “vitality” summarizing a person’s state of physical robustness and vulnerability to death. At birth, for each person, it has some initial value, whose unobserved distribution across people is somewhat akin to the frailty in Vaupel et al. (1979). Vitality, however, does not stay fixed across life but heads up and down in haphazard little random steps superimposed on a long-term trend. The accumulation of many kinds of illnesses, physiological insults, treatments, nutrition, stress, and the like can be imagined as driving a person’s trajectory of vitality.

In such models, unobserved heterogeneity among individuals is being augmented across the lifecourse by new variation even as it is being

diminished by demographic selection. The observable outcome, death, is determined to occur when the trajectory or “sample path” of vitality hits some lower boundary or hovers below it for a lethal amount of time. The boundary may be some fixed minimal level or some curve varying with age. In sophisticated versions, probabilities of death may vary smoothly with distance from the boundary rather than impinging only when the boundary is crossed.

For demographers, the most noteworthy property shared by models of this kind is a generic prediction of plateaus in hazard rates at extreme ages. Mathematical proofs for a wide class of models have been offered by Aalen and Gjessing (2001), Gjessing et al. (2003), and Steinsaltz and Evans (2004). As heterogeneity in vitality accumulates, deaths progressively cull individuals with lower vitality, and under a broad range of specifications the distribution of vitality converges to what is called a “quasi-stationary” state. In the true stationary state, everyone is dead. At any particular advanced age, the vitalities of those still surviving come to have a distribution that persists. Vitalities at the low end of the distribution are removed by death but replenished by the random arrival of individuals whose state of health is going downhill.

The move from fixed frailty to stochastic vitality avoids the drawback I have discussed from implausibly low mortality rates at early ages from individuals at the extremes of low frailty. In vitality models, heterogeneity accumulates. Individuals with modest survival advantages in middle age may find themselves with palpable advantages in extreme old age, but only a rare few of them for whom the luck of the draw has happened to come out in their favor over and over. This picture accords with common sense. Some inborn advantages are erased over the lifecourse, others bolstered. It is only reasonable to think that a lifetime of ups and downs increases the power of demographic selection at extreme ages and reinforces the formation of plateaus. The demographic relevance of stochastic vitality models is enhanced by another property described in the following section. They are particularly amenable for combination with models of mutation accumulation, potentially allowing informative connections with genomic observations.

EFFECTS IN GENOME-WIDE ASSOCIATION STUDIES

In public consciousness, the subject of alleles and mortality calls up a picture of the gene for X, the gene for Y, of a handful of “genes for longevity.” Some individual alleles with palpable effects on longevity have been discovered in model organisms and a few have been detected for humans. A careful account is given by Vaupel (2010). Substantial effects from specific genetic variants can be treated as statistical fixed effects, that

is to say, observable covariates in stochastic risk factor models. They are the counterpart of risk in the trio of lifeluck, risk, and frailty. The techniques of Genome-Wide Association Studies (GWAS) are devoted to ferreting out such effects from data on millions of SNPs for thousands of survey respondents.

The APO-E polymorphism is the most famous example for demographers. From the point of view of evolutionary demography, however, such alleles are not very informative about the origin of common regularities in demographic schedules across species with widely different body plans and environmental niches. Each specific causal effect presumably has its own specific age pattern and presumably depends heavily on environmental context, affecting the hazard rates of nematode worms in different ways than, say, Icelanders. Understanding such effects is important for health and welfare, but it is not the big part of evolutionary demography. First, alleles with large detectable effects on outcomes studied by demographers and social scientists appear to be very rare. Second, for explaining statistical regularities, it seems natural to examine statistical properties, in this case properties characterizing the accumulation of small, somewhat independent effects in large numbers, in the tradition of explaining Gaussian distributions for human heights via the Central Limit Theorem.

In most social science GWAS studies, few or no alleles have effects whose statistical significance exceeds the stringent cutoffs imposed by statistical adjustments for multiple comparisons. However, from the demographic point of view, these are not “null” results per se. They allow detectability bounds to be calculated, bounding the number of alleles that can be having effects exceeding some given size. A case is made in the final section of this paper that large numbers of small effects, rather than small numbers of large effects, are more promising sources for regularities in demographic schedules, so bounds on numbers of large effects offer encouragement to the biodemographic agenda.

These approaches are being tested on outcome variables other than longevity. Some early results pertaining to longevity have been published from applications of GWAS studies to genetic markers from the first wave of the Framingham Heart Study, including Yashin et al. (2010). However, these remain controversial for technical reasons and are awaiting confirmation.

The most clear-cut bounds on effect sizes to date come from a GWAS by Rietveld et al. (2013) combining samples from many previous studies to obtain a total of 126,559 respondents. The trait under study is educational attainment, one of the few traits measured in somewhat consistent ways across many surveys. Only three SNPs were found to achieve genome-wide statistical significance with this sample size, and estimated effects of those polymorphisms were small. Most of the alleles accounting for the moderate heritability of educational attainment as assessed from twin studies must be below thresholds of detectability and thus very small.

Mutation accumulation theory makes assumptions not only about the dominant role of small effect sizes, but also about the prominence of effects with age-specific signatures. Small physiological differences originating from small differences in protein coding or regulation play out through different causal pathways. It is reasonable to expect that different pathways are important for viability at different stages and ages, making ultimate effects on demographic schedule age-specific. As far as it goes, there is also a little direct empirical evidence for the age-specificity of genetic effects, in this case from one of the few alleles with sufficiently large effects on survival to be individually detectable. In unpublished work, I have examined tabulations published by Flachsbart et al. (2009) of the frequency of a particular SNP in the Human Forkhead Box 03A Gene “FOX03A” among respondents in four age groups from 60- to 75-year-olds up to 105- to 110-year-olds. The gene is involved in insulin-signaling pathways. The increase with age in the frequency of the minor allele for this polymorphism is consistent with small early age effects on survival and rapidly increasing effects at later ages.

MUTATION ACCUMULATION WITH DEMOGRAPHIC COSTS

The route from the days of Between Zeus and the Salmon to the second decade of the 21st century goes from an impressionistic application of evolutionary theories in demography to quantitative theories with predictive content capable of calibration with emerging data. A critical early step was taken by Brian Charlesworth (2001), who put forward a stylized model connecting mutation accumulation to Gompertzian exponential increases in age-specific hazards and to possible plateaus. The basic idea is to consider a setting dominated by heavy exogenous mortality from predation and mishaps affecting mature individuals largely without regard to age. A hazard function constant over age implies an exponentially decreasing proportion of survivors by age. If fertility rates are more or less constant over ages with significant numbers of survivors, then a burst of mortality at a single age implies a loss in net reproduction that is an exponentially decreasing function of age.

In the tradition of the Oxford geneticist W.D. Hamilton (see Baudisch, 2008; Charlesworth, 2001), a linear approximation to a quantity called the “age-specific force of natural selection” can be computed. Its reciprocal is meant to be proportional to the number of mildly deleterious mutant alleles that will be found in “mutation-selection equilibrium,” among alleles whose effect is to impose a burst in probabilities of dying at the given age. The equilibrium is established when new mutations are appearing in the genomes of population members generation by generation at a rate that balances the disappearance of mutant alleles as they are weeded out of the population by natural selection. The assumption that alleles act by impos-

ing bursts of mortality can be relaxed. What is critical to Charlesworth’s idea is that age-specific effects of each of the relevant alleles are restricted to ages beyond some cutoff depending on the allele, and that every cutoff has at least some minimal rate of corresponding mutations.

The fascination of Charlesworth’s model is that it predicts exponentially rising Gompertz hazards at adult ages. Of course, there are numberless other ways proposed by biologists and demographers to explain Gompertz hazards. However, Charlesworth’s idea has two special advantages. First, it is highly generic. It is the kind of process that could pertain across a wide range of body plans and evolutionary environments. Second, it does not require putting some exponential function in somewhere in the model in order to get an exponential function out.

Charlesworth went on to observe that one could optionally posit a small fixed selective cost to every allele on top of the age-specific costs, and the result would be hazard functions whose exponential increases tapered into plateaus at extreme ages as observed. Later parts of this section describe results that put these ideas in a new light. A huge literature on mutation-selection balance is surveyed by Buerger (2000); most of it is rich in genetic detail but limited in demographic detail. Stimulated by Charlesworth’s idea, David Steinsaltz, Steve Evans, and I have worked to develop and analyze a model originally formulated by David Steinsaltz that incorporates age-specific demographic structure, population heterogeneity, and non-linear interactions into a general treatment of mutation, selection, and recombination.

The important word here is “non-linear.” The approaches of Hamilton and Charlesworth employ a linear approximation. The loss in net reproduction from the effects of two mutant alleles acting together is set equal to the sum of their effects acting alone. But mutation accumulation is inherently non-linear. Reduction in survival at one reproductive age reduces the remaining reproductive potential that can be lost by a reduction in survival at later reproductive ages. In other words, effects of alleles interact with each other. In this setting, linear approximations cannot be justified in the usual way by claims that total effects are small, because the predictions of the formulas are for large total effects cumulating out of small individual effects, whenever rates of mutation are not negligible. The linear theory is inconsistent and a non-linear theory is required.

Non-linear theory for mutation accumulation raises many mathematical challenges. Over the decade from 2000 to 2010, the challenges were eventually overcome. The theory developed most fully in Evans et al. (2013) now allows predictions of age-specific mortality schedules generated from assumptions about mutation rates and the profiles of mutational effects in the presence of recombination. Specifically, a demographer can specify a family of curves to serve as profiles of age-specific action. Many sites in the genome are assumed to share the same action profile, so that the

alleles found in SNVs come, as it were, in teams, each team with its own characteristic pattern of deleterious demographic impacts. The relationship between assumptions about profiles and empirical evidence is discussed in the next section. Each individual in a population is assumed to carry some random batch of mutant alleles drawn from each of the teams of alleles. The cumulative hazard function applying to the individual is formed by adding up the profiles for the alleles that the individual carries along with a baseline exogenous hazard. Decrements from alleles that affect age-specific fertility can also be included, but the present exposition focuses on hazard functions. Once mutation rates and baseline fertility and mortality are specified, the mathematical formulas predict the aggregate population hazard function to be found over time as mutations accumulate. They also predict the aggregate population survivorship at equilibrium, if a mutation-selection equilibrium occurs.

This framework describes a heterogeneous population. Different individuals carry different random batches of alleles. These batches turn out to be realizations of a Poisson process, a consequence of an assumption that genetic recombination operates more rapidly (in a specified sense) than mutation and selection and that the pool of SNVs corresponding to each action profile is large. The description is reasonable, inasmuch as mildly deleterious alleles with age-specific effects, however numerous in total, are expected to be fairly sparse and well-separated on chromosomes, avoiding any strong role for the correlations called linkage disequilibrium. The linear approximate theory deployed by Charlesworth assumed a similar kind of Poisson variability and seemed likely to provide a reasonable rough guide to the non-linear case.

Results from new modeling, as mentioned above, turn out to transform this picture. Predictions from the full non-linear theory overturn these predictions from linear theory. Mathematical proofs are found in Wachter et al. (2013) and Wachter et al. (2014). When it is the case that for any adult age there are mutant alleles whose effects are entirely restricted to later ages, as envisioned by Charlesworth, then the non-linear interactions among effects erode the selective pressure that holds in check the representation of late-acting alleles. In turn, interactions erode the selective pressure for earlier-acting alleles, and mutation-selection equilibrium is destroyed. The destruction of the equilibrium is predicted no matter how low are the rates of new mutations, so long as they are bounded below.

This finding explains that a far-reaching demographic difference emerges from what has seemed to be a minor difference for phenotypes, namely whether there are non-negligible deleterious effects at young ages for most alleles whose primary action is to raise hazard rates at old ages. Small early-age consequences of deleterious alleles can, in principle, exert small but sufficient selective pressure to keep the representation of these alleles in check

and rescue mutation-selection equilibrium. Effects at young ages may be undetectable, but through natural selection their role is expressed in the shape of hazards at late ages: They produce plateaus in hazard rates.

It follows that the logic of mutation accumulation argues for the existence of plateaus at extreme ages as a typical concomitant of mortality acceleration at middle ages. What was an optional feature in Charlesworth’s linear account comes out appearing to be a feature plausibly promoted by natural selection. It may have been a reasonable expectation that anything that pushes deleterious genetic effects back to later and later ages would be advantageous. But the results on mutation accumulation indicate that such processes, carried too far, have undesirable consequences. They also indicate that some part of an observable constant term in hazard functions might be attributable to genetic load, reflecting the accumulation of minimal effects at all ages.

In a paper in PLOS One, Danko et al. (2012) take issue with the approach described here, claiming that “mutation accumulation may be a minor force in shaping life history traits.” However, their model supports the opposite conclusion. For their simulations they impose an arbitrary bound of 10 on the maximal number of alleles and thus severely restrict the maximal allelic contribution, moving from a model for mutation accumulation—allowing large numbers of small effects—to a model for mutation nonaccumulation—allowing only small numbers of small effects. In their model with the bound removed, mutations do accumulate and reshape the survival schedule. Life history optimization has modest effects. However, in other models, it could have more importance.

In no sense does mutation accumulation operate alone. It has to operate within a framework determined by the portfolio of physiologically feasible adaptations. A better understanding of how genetic load interacts with optimal life history allocations and strategies in the presence of environmental variation across time and space is a high priority. By the same token, combination of mutation accumulation with stochastic vitality models likely holds a key to understanding the present-day post-reproductive signatures of deleterious alleles whose equilibrium frequencies were established over evolutionary time when their lethal consequences were often felt at prime reproductive and nurturing ages.

The selective pressure that holds deleterious alleles in check has to operate over evolutionary time at ages that matter for procreation, parenting, and grandparenting, that is, that matter to the bequeathing of genes to descendant generations. Ages of nurturing extend beyond ages of reproduction. However, it is clear that the extreme ages at which regularities in mortality schedules continue to be observed in humans are too late to have been directly subject to selective pressure over evolutionary time. Some account is therefore required for processes that have transferred the imprint of

natural selection operating at younger ages in bygone epochs into regularities now observed in schedules at old ages after enormous environmental changes and reductions in the level of mortality.

For natural selection to do its work, deleterious effects of mutant alleles have to be ultimately expressed in fertility and survival. But it makes sense to think of the effects of alleles operating through a number of intermediate causal pathways that might be reflected in the transition rates for a stochastic vitality model. What is gained from this point of view is the possibility of modeling changes in mortality concomitant with the advantages of civilization through changes in the boundary curve representing lethality. In such models, one would expect the imprint of natural selection over evolutionary time to be expressed at later ages in recent times, as the trajectory of vitality takes more time to reach a more distant boundary. The enterprise of constructing and analyzing such combined models is just beginning and may lead to new insights in the coming years.

EVOLUTIONARY DEMOGRAPHY: TENETS AND HYPOTHESES

The demographic relevance of evolutionary demography does not go without saying. It is easy to imagine that humans might have so drastically reshaped the environment and their physiological capacities as to make the general genetic legacy from the distant past irrelevant. On the other hand, it is also easy to imagine the contrary, that regularities in age-specific patterns shared to some extent with utterly different species have deep origins that partly override the vicissitudes of modern human life.

Biodemographic models are appealing, but they are technical, continually developing, and full of detail. In this final section, before surveying empirical evidence from genome sequencing, it may be helpful to pull back and venture a brash sketch of the big picture. Why do demographers want to know the things that genome sequencing may help explain? My sketch takes the form of a list of questions and the answers to these questions that evolutionary demography proposes:

Making this picture concrete requires confronting theory with data. Each main tenet of mutation accumulation theory constitutes a hypothesis about what is to be found in genome sequence data from samples of living individuals. This area of science is in rapid development and definitive results are still in the future. What can be done now is to sketch out how different pieces of a jigsaw puzzle look as if they might be able to be joined together.

Which hypotheses, suggested by demographic theory, are on the table?

1. Mutation accumulation theory proposes that each individual carries a personal sample from a large pool of mildly deleterious alleles, leading to the hypothesis that such a pool exists.
2. The theory proposes small effects allele by allele for alleles whose cumulative impact shapes age-specific adult human mortality, leading to the hypothesis that fitness costs for many deleterious alleles are individually small.
3. The theory proposes that most relevant alleles arose from mutations fairly far in the past, well before dramatic reductions in human mortality and likely before many of the transformations accompanying sustained population growth, leading to the hypothesis that age estimates for relevant alleles should often come out at more than several hundreds of generations.

What are genome studies published so far telling us with respect to these hypotheses? For the first hypothesis, concerning the existence of a large pool of mildly deleterious alleles, there is some striking confirmation. It comes from genetic sequencing of the roughly 2 percent of the genome that directly codes for proteins. A quest for low-frequency alleles by Tennessen et al. (2012) in the coding regions in the genomes of 2,044 individuals of European and African descent turned up a pool of 503,481 Single Nucleotide Variants (SNVs, as described at the outset). Individuals in the study were found to carry 13,959 SNVs on average. The overwhelming majority of these SNVs were deemed to be neutral or nearly neutral, with no selective cost or with negligible selective cost in comparison to the force of genetic drift. But an important minority of SNVs were classified as functional and deleterious. The classification depends on whether an SNV implies no change or some change in a protein product, whether it occurs in certain kinds of positions, and whether it meets various other criteria. The average number of SNVs per individual classified as functional and deleterious in protein-coding regions of the genome was estimated conservatively at around 318 and less conservatively at around 580. The proportion of rare SNVs among these functional SNVs was found to be higher than the proportion of rare SNVs among all SNVs, implying that the pool from which the functional SNVs are drawn is substantial. Thus, the first hypothesis is borne out by these early data.

The study by Tennessen et al. (2012) does not provide estimates of selective costs for putative deleterious alleles. What makes alleles deleterious are effects on the net reproduction of those who carry them, either on adult survival, on infant and child survival, on fertility and mating success, or on some combination. The term “MA-alleles” denotes the subset of deleterious alleles that affect adult hazard functions and, under equilibrium conditions, are held in equilibrium by mutation-selection balance. From the estimated numbers in the Tennessen paper, it seems reasonable to expect that numbers of MA-alleles should be in the hundreds or more. Only some fraction of deleterious alleles in coding regions belong to the subset of MA-alleles, but it is also likely that MA-alleles are found outside coding regions, interfering in small ways with the efficiency of transcription and the regulation of gene expression. The model of mutation accumulation motivates some informed guesses about a relationship between numbers of MA-alleles and average sizes of effects, as will shortly be described. These guesses come out consistent with the second hypothesis, that effects are small.

The third hypothesis concerns typical numbers of generations elapsing from the events of mutation which introduced each allele into the population. Alleles that descend from a given mutant are “derived alleles” and the number of generations is the “age” of the derived allele. Relationships between age, present-day frequency, past population size, and selective

cost are complicated by many factors, but arguments are described here that suggest that MA-alleles with average selective costs have average ages of more than several hundred generations, consistent with the third hypothesis.

A review of methods of age estimation for alleles is found in Slatkin and Rannala (2000). The complications of the subject emphasized in recent literature include expansion in effective population size since the beginnings of agriculture about 400 generations ago. These relationships are current subjects for genetic research. For demographers, the interpretation of results is further complicated by the uncertain translation between “effective population size,” a quantity entering genetic models that assume random mating among all population members, and “census population size,” half the number of feet on the ground that a prehistoric demographer would have found if he or she had completed an actual enumeration. The last 20 generations or so of very rapid world population growth are too few to have much bearing on the genetic results, but growth over the last 400 generations or so has implications that will shortly be discussed.

Mediation between results from the genetic literature and interpretations of demographic processes is aided by the model of Evans et al. (2013). One new tool for calibration is supplied by a generalization of Haldane’s Principle. In its older, classical form, Haldane’s Principle for deleterious alleles held mutation-selection balance equates the total mutation rate to the total loss in fitness. This equality fails in the presence of non-linear interactions. However, a generalized form for the non-linear setting has been proved in Wachter et al. (2013) equating the relevant total mutation rate to a quantity related to lifetable entropy.

This generalized Haldane’s Principle allows for inference of a total mutation rate for relevant deleterious alleles from a combination of theory and data. The rate is independent of the size and age-specific profile of allelic effects. The application of the principle proceeds from a comparison of a hypothetical baseline survival schedule dominated by exogenous mortality in the absence of MA-alleles with a hypothetical observable mortality schedule for populations over evolutionary time when mutation-selection equilibria could have prevailed. Guesses at the hypothetical observable schedule can be informed by anthropological lifetable estimates for present-day hunter-gatherers summarized, for example, by Gurven and Kaplan (2007).

For application of the generalized Haldane’s Principle, assumptions about the effective age-specific shape of contributions to net reproduction also have to be made. Such contributions reasonably include allowances for the protective effects of care from surviving parents and grandparents. Conceptually, an “effective Net Reproduction Ratio” might be defined to equal the cube root of the average number of daughters of daughters of

daughters born per newborn prospective great-grandmother, or in some other comparable way. It is the age-specific shape of contributions, not the level of fertility, that is important, because over long time periods, homeostatic mechanisms may be assumed to adjust population growth to near stationarity.

This paper is too brief to develop detailed estimates from Haldane’s Principle. However, a simple version of the calculation, coming out with an estimated rate of 0.20, may serve as a rough guide to an order of magnitude. The rate in question is the rate of new mutations per generation per individual for MA-alleles held in mutation-selection balance. The simple version takes as its starting point the average mortality schedule for hunter-gatherers in Gurven and Kaplan (2007). The version attributes half of the Makeham constant term to exogenous mortality and half to MA-alleles and adds moderate allowances for nurturing on top of a Coale-Trussell schedule for natural fertility. Implied reductions in effective net reproduction around 0.13 from genetic load are multiplied by an adjustment for non-linearity of around 1.5 for a mutation rate of 0.13 * 1.5 ≈ 0.20. Inasmuch as optimal life history models positing substantial costs to repair often imply increasing hazards due to factors other than genetic load, it could be argued that this estimate should be even lower. None of the more extreme population lifetables in Gurven and Kaplan (2007) would justify a much higher estimate, beyond, say 0.50.

An estimate of MA-alleles carried per individual is also needed. The estimate around 300 given conservatively by Tennessen et al. (2012) has to be adjusted upward for MA-alleles outside of coding regions and adjusted downward for deleterious alleles that are not MA-alleles. A simple version proposes that the two adjustments may roughly offset each other. Then the dynamic equation in Evans et al. (2013) gives an average selective cost equal to the quotient 0.20/300 ≈ 1/1500. More formally, this quantity is an estimate for the average over all MA-alleles of the generalized age-specific force of natural selection at equilibrium in the model.

The model of Evans et al. (2013) is a model for mean values, not for the random fluctuations in allele frequencies arising from genetic drift in populations with finite, moderate effective sizes. The age distribution of alleles is shaped by drift and by effective population size over time as well as by selective cost. However, the quantities of demographic interest are not frequencies of alleles at single genetic sites but aggregated counts of sets or “teams” of alleles sharing a single age-specific effect profile. Furthermore, the model assumes that recombination operates on a faster timescale than selection and mutation, a reasonable assumption when hundreds of generations are at issue. An effect of recombination is to make the distributions (across the population) of alleles at single sites appear statistically independent. The upshot is that a sum over all the alleles in a team of substantial

size is plausibly described by the model for mean values. No theorem is yet in hand, but simulations support this expectation.

An implication of this picture is an estimate for average age measured in generations for a set of deleterious alleles sharing an age-specific profile that comes out to be on the order of the reciprocal of the age-specific force of natural selection. This calculation presumes a stationary population and the establishment of mutation-selection equilibrium. Population growth since the beginnings of agriculture some 400 generations ago alters the calculation as it disturbs any equilibrium. Impacts of such growth have been assessed with simulation studies by Gazave et al. (2013) in the context of European populations. As compared to an imaginary population that remained stationary, they find population growth over 400 generations to entail much larger total numbers of mutant alleles present in the much larger total population at the end. However, they find only modestly larger numbers of alleles carried per person resulting from the growth. Most alleles are neutral. For functional deleterious alleles, they find only small changes in distributions. Their results are tailored to the case of European population growth as reconstructed by Gazave et al. (2014).

Taken together, all these results combine to suggest an average age for MA-alleles with average selective costs on the order of 1,500 generations or about 37,500 years. Variations around this average are bound to be large, both because of the variations in selective costs between alleles whose actions on hazard rates are concentrated at different ages and because of the intrinsic randomness of natural selection. However, it does appear that a substantial portion of genetic load affecting adult hazard rates is a legacy from long before the transformations of modern times.

In summary, then, data from studies of variation in human genome sequences are beginning to permit confrontation between predictions from evolutionary demographic models and empirical findings. Results from the first large sample study with high resolution on variants in coding regions of the genome, Tennessen et al. (2012), appear consistent with the main tenets of the existing demographic model for mutation accumulation. A large pool of variants of the kind posited by the model appears to exist. The number of functional deleterious alleles estimated to be carried per individual appears consistent with the smallness, on average, of the effects posited by the model. Rough early estimates of ages for such alleles are consistent with the idea of a legacy of genetic variation structured by natural selection long before the advent of modern mortality decline.

Opportunities are coming on the scene for more detailed confrontations between theory and data. Specific implementations of the mutation accumulation model make predictions about plausible distributions of selective cost across alleles from sets with similar age-specific profiles. Genetic sequencing studies can provide data on frequency distributions for alleles deemed to be

functional and deleterious, which should in due course allow more specific estimates of ages of alleles.

The 17 years since the volume Between Zeus and the Salmon have brought advances in theory and advances in genetic information that put the concept of an evolutionary origin for demographic regularities on a concrete footing. There is now a refined predictive model for mutation accumulation. Biodemographers have an agenda for combining mutation accumulation theory with stochastic vitality models to relate present-day patterns of post-reproductive survival to earlier patterns of reproductive-age survival over evolutionary time. We have an account that links the ubiquity of Gompertzian increases in hazard rates at medium-old ages to the plateaus at older ages. We have empirical evidence from genomic analyses that the kinds of genetic variants posited by mutation accumulation are indeed a predominant kind observed. Along with progress in understanding fundamental relationships between alleles and regularities in demographic schedules, surveyed in this paper, biodemographers are breaking new ground in understanding the fine-grained implications of sociality and adaptive strategies for the biodemography of health and lifecourse. This new ground is explored in this volume.

ACKNOWLEDGMENTS

I appreciate support from the U.C. Berkeley Center for the Economics and Demography of Aging (CEDA) funded by Grant P30 AG12839 from the U.S. National Institute on Aging. Ideas in this chapter have been worked out in collaboration with David Steinsaltz and Steve Evans. Robert Pickett helped clarify the discussion of frailty models and Ronald Lee pointed out the bearing of optimal history models on the Haldane calculation.

REFERENCES

Aalen, O.O., and Gjessing, H.K. (2001). Understanding the shape of the hazard rate: A process point of view. Statistical Science, 16(1), 1-22.

Baudisch, A. (2008). Inevitable Senescence? Contributions to Evolutionary-Demographic Theory. Berlin, Germany: Springer-Verlag.

Buerger, R. (2000). The Mathematical Theory of Selection, Recombination, and Mutation. Chichester, NY: John Wiley & Sons.

Charlesworth, B. (2001). Patterns of age-specific means and genetic variances of mortality rates predicted by the mutation-accumulation theory of ageing. Journal of Theoretical Biology, 210(1), 47-65.

Chu, C.Y., and Lee, R.D. (2006). The co-evolution of intergenerational transfers and longevity: An optimal life history approach. Theoretical Population Biology, 69(2), 193-201.

Danko, M.J., Kozlowski, J., Vaupel, J.W., and Baudisch, A. (2012). Mutation accumulation may be a minor force in shaping life history traits. PLOS ONE, 7(4):e34146.

Evans, S.N., Steinsaltz, D.R., and Wachter, K.W. (2013). A Mutation-Selection Model for General Genotypes with Recombination. Memoir 222. Providence, RI: American Mathematical Society.

Flachsbart, F., Caliebe, A., Kleindorpa, R., Blanché, H., von Eller-Eberstein, H., Nikolaus, S., Schreiber, S., and Nebel, A. (2009). Association of FOXO3a variation with human longevity confirmed in German centenarians. Proceedings of the National Academy of Sciences of the United States of America, 106(8), 655-691.

Gazave, E., Chang, D., Clark, A., and Keinan, A. (2013). Population growth inflates the per-individual number of deleterious mutations and reduces their mean effect. Genetics, 195, 969-978.

Gazave, E., Ma, L., Chang, D., Coventry, A., Gao, F., Muzny, D., Boerwinkle, E., Gibbs, R., Sing, C., Clark, A., and Keinan, A. (2014). Neutral genomic regions refine models of recent rapid human population growth. Proceedings of the National Academy of Sciences of the United States of America, 111(2), 757-762.

Gjessing, H.K., Aalen, O.O., and Hjort, N.L. (2003). Frailty models based on Lévy processes. Advances in Applied Probability, 35, 532-550.

Gurven, M., and Kaplan, H. (2007). Longevity among hunter-gatherers: A cross-cultural examination. Population and Development Review, 33(2), 321-365.

Kaplan, H.S., and Robson, A.J. (2002). The emergence of humans: The coevolution of intelligence and longevity with intergenerational transfers. Proceedings of the National Academy of Sciences of the United States of America, 99(15), 10221-10226.

Le Bras, H. (1976). Lois de mortalité et âge limite. Population, 31(3), 655-691.

National Research Council. (1997). Between Zeus and the Salmon: The Biodemography of Longevity. Committee on Population, K.W. Wachter and C.E. Finch (Eds.). Commission on Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

National Research Council. (2001). Cells and Surveys: Should Biological Measures Be Included in Social Science Research? Committee on Population, C. Finch, J.W. Vaupel, and K. Kinsella (Eds.). Commission on Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

National Research Council. (2003). Offspring: Human Fertility Behavior in Biodemographic Perspective. Panel for the Workshop on the Biodemography of Fertility and Family Behavior, K.W. Wachter and R.A. Bulatao (Eds.). Committee on Population, Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press.

National Research Council. (2007). Biosocial Surveys. Committee on Advances in Collecting and Utilizing Biological Indicators and Genetic Information in Social Science Surveys and Committee on Population, M. Weinstein, J.W. Vaupel, and K.W. Wachter (Eds.). Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press.

Rietveld, C., Medland, S., Derringer, J., Yang, J., Esko, T., Martin, N.W., Westra, H.J., Shakhbazov, K., et al. (2013). GWAS of 126,559 individuals identifies genetic variants associated with educational attainment. Science, 340(6139), 1467-1471.

Slatkin, M., and Rannala, B. (2000). Estimating allele age. Annual Review of Genomics, Human Genetics, 1, 225-249.

Steinsaltz, D., and Evans, S.N. (2004). Markov mortality models: Implications of quasi-stationarity and varying initial conditions. Theoretical Population Biology, 65(4), 319-337.

Tennessen, J., Bigham, A., O’Connor, T., Fu, W., Kenny, E.E., Gravel, S., McGee, S., Do, R., et al. (2012). Evolution and functional impact of rare coding variation from deep sequencing of human exomes. Science, 337(6090), 64-69.

Vaupel, J.W. (2010). Biodemography of human ageing. Nature, 464, 536-542.

Vaupel, J.W., Manton, K.G., and Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16(3), 439-454.

Wachter, K.W. (2014a). Essential Demographic Methods. Cambridge, MA: Harvard University Press.

Wachter, K.W. (2014b). Population heterogeneity in the spotlight of biodemography. In P. Kreager, B. Winney, S. Ulijaszek, and C. Capelli. (Eds.), Population in the Human Sciences: Concepts, Models, Evidence (pp. [TK]). Oxford, England: Oxford University Press.

Wachter, K.W., Evans, S.N., and Steinsaltz, D.R. (2013). The age-specific force of natural selection and biodemographic walls of death. Proceedings of the National Academy of Sciences of the United States of America, 110(25), 10141-10146.

Wachter, K.W., Steinsaltz, D.R., and Evans, S.N. (2014). Evolutionary shaping of demographic schedules. Proceedings of the National Academy of Sciences of the United States of America, 111(Suppl. 3), 10846-10853.

Weitz, J., and Fraser, H. (2001). Explaining mortality rate plateaus. Proceedings of the National Academy of Sciences of the United States of America, 98(26), 15383-15386.

Yashin, A.I., Vaupel, J.W., and Iachine, I.A. (1995). Correlated individual frailty: An advantageous approach to survival analysis of bivariate data. Mathematical Population Studies, 5(2), 145-159.

Yashin, A.I., Wu, D., Arbeev, K.G., and Ukraintseva, S.V. (2010). Joint influence of small-effect genetic variants on human longevity. Aging, 2(9), 612-620.

This page intentionally left blank.