Inevitable Aging?: Contributions to Evolutionary-Demographic Theory (Demographic Research Monographs

Inevitable Aging?: Contributions to Evolutionary-Demographic Theory ( Demographic Research Monographs) edition by Baudisch, Annette () .
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The bounce-back is essential. Near the wall of death, hazards increase more rapidly than exponentially. Derivatives of all orders go to infinity. This behavior is particularly striking, because the linear approximate model would predict no wall of death at all. Mutation rates that drop sufficiently rapidly with ages of onset will prevent walls of death in the absence of an upper limit to ages of reproduction.

A natural way to avoid collapse, less contrived than tailoring the rate function, is to assume that action profiles do not entirely vanish at young ages even though their effects may be concentrated late in life. As long as the profiles are not themselves unbounded, such a condition implies the existence of late-age plateaus. A proof that early-age effects guarantee the existence of equilibria has already been given in ref. For completeness, we restate it here:.

Suppose the integral of q m is finite and sufficiently small and suppose for some set B with , and suppose is finite. Then there exists such that Lebesgue almost surely and is bounded by a constant times x. The results in this paper showing equilibrium collapse in their absence suggest that such effects are not merely allowed, but required. In the face of mutation accumulation, any equilibrium is a state in which inflow due to continuing mutation is balanced by outflow due to selection, allele by allele. In the linear approximate model, each mutant allele has a fixed cost, unaffected by other alleles.

Summing over alleles, the total inflow at equilibrium, that is, the total mutation rate, has to equal the sum of these fixed costs, which, under the linearity assumption, would be the total selective cost. In our nonlinear model, outflow of each allele is driven not by its fixed cost but by its marginal selective cost, which is reduced by the presence in the genome of other deleterious alleles.

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The greater the supply of mutant alleles, the less the marginal cost of each one. The total mutation rate at equilibrium still has to equal the sum of the costs. However, the sum of the marginal costs is not the total selective cost, but something less than it. Let on as from , with elsewhere and , with fertility rates that imply stationarity at equilibrium. Then the total mutation rate for mutations held in mutation—selection balance belonging to equals the sum of and the relative entropy of the equilibrium net maternity function relative to the normalized baseline net maternity function.

That is , with and. The proof in SI Text depends on being able to integrate the right-hand side of Eq. The right-hand side of Eq. The second term subtracts the portion resulting from zeroing out fitness contributions from ages beyond any wall of death.

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The third subtracts the contributions from interactions among alleles. The subtractions bring the total loss in fitness down to the total mutation rate for those mutations kept at finite intensity by natural selection. Closed-form expressions are available for the special family of cases characterized by Theorem 2.

The overall loss in fitness due to genetic load is. Their common value is. We can visualize the disappearance of equilibria in the absence of early-age effects in several ways: Consider the picture with respect to age. At sufficiently advanced ages, remaining net fertility necessarily drops below any mutation rate that is bounded below. If the force of selection depends only on remaining net fertility, selection cannot balance mutation, and such ages must lie beyond a wall of death. However, if we try to construct an equilibrium with some specific wall of death, we find too little remaining net fertility very close to the wall to balance mutation there.

Each wall of death implies an earlier one. The instability propagates down through the whole reproductive span, and our construction unravels. A complementary picture emerges with respect to time. A steady influx of mutations affects the whole reproductive span. Hazards begin to increase linearly with time at all ages. At older ages, where selective pressure is always low, this linear increase continues unabated, whereas at younger ages it is slowed for a while by outflow due to natural selection.

At a snapshot in time, hazards lie low for a stretch of ages, climb at ages where remaining net fertility is dropping off, and link up with the old-age segment. As time goes by, the climbing phase shifts down to younger and younger ages, until the hazard rate at every adult age comes to be marching toward infinity. Details of the dynamics depend on assumptions about fertility. We may hold fertility fixed over time, but we have to recognize that no fixed fertility level is sufficient for stationary population growth at equilibrium when there is no equilibrium.

Unbounded accumulation of mutations across the whole reproductive span drives any population to extinction. We may instead let fertility levels adjust over time to maintain stationarity with current values of the hazard, on the assumption that feedback between resources and population growth operates on a faster time scale than mutation and selection.

Under this scenario, the climbing phase in the snapshot of the hazard function steepens with age and time as it shifts to younger ages, and the fertility level required for stationarity heads toward infinity.

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A third picture relates to shape. The family of examples constructed in Theorem 2 have mutation rate functions that are nearly constant over age but that are pinched to zero at a point.


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The pinching has a characteristic steepness that turns out to produce an equilibrium on a subset of ages ending in a wall of death. The nearer the mutation rate function to constancy, the nearer is the wall of death to the age at maturity, the shorter the reproductive lifetime, and the higher the fertility level required for stationarity. To reach the case of a wholly constant mutation rate function the brief high burst of fertility before death would have to turn into a delta function or point mass at the age of maturity, followed by immediate death.

Nonlinear interactions make the Gompertz—Makeham form collapse. This outcome depends on recombination. Recombination spreads the deleterious alleles throughout the population, leaving no lineages untouched by a surfeit of late-acting mutant alleles. In the absence of recombination, as shown in ref. However, in the presence of recombination as specified in our model, even a tiny rate of production of mutant alleles with arbitrarily late ages of effects cannot be balanced by natural selection, and leads to collapse of the equilibrium.

The ratchet destroys any equilibrium distribution of genotypes in nonsexually reproducing populations of finite size subject to genetic drift. Our results pertain to sexually reproducing populations of infinite size, that is, of sufficient size that genetic drift can be ignored. As with the ratchet, the lesson from our results is to point to the evolutionary need for processes to counteract collapse. The generic antidote to collapse in our context comes in the form of small early-age manifestations of deleterious effects concentrated at later ages.

The fascinating concomitant is that such early-age trace effects leave their signature in late-age plateaus in hazard functions. Plateaus appear not as a convenient add-on but as something like an obligate feature of mutation accumulation models with equilibria. It seems logical that organisms should acquire mechanisms that push the age of onset of ill effects from numerous minor defects associated with genetic load toward later ages.

However, our analysis suggests that such mechanisms, if carried to extremes, have bad evolutionary repercussions. For species like humans, with substantial postreproductive survival, theories that emphasize mutation accumulation face the challenge of accounting for the absence of a wall of death at the end of reproduction. Early-age trace effects can remove walls of death, but the challenge remains in the persistence of exponentially increasing hazards well beyond the end of reproduction and the extreme late ages at which plateaus occur.


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The importance of postreproductive ages of nurturance, perhaps especially of grandmothers, is likely part of the explanation. In social species, transfers within sharing groups and across generations have a bearing on optimal life strategies. However, it would be a mistake to think that they can reshape the age-specific force of natural selection, linear or nonlinear, for the obvious reason that each mutation in the genome of an individual does not simultaneously appear in the genomes of all other members of a sharing group.

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Recombination erodes any potential for group selection in this setting. It seems to us likely that effects of accumulating mutations in humans are now seen at postreproductive ages bearing age-specific signatures that were imprinted at earlier epochs when their contributions were more rapidly lethal at younger ages. Sizes of effects under our models have little impact, and sometimes no impact, on predicted hazard functions, but they have major impact on rates of turnover. For each allele, the reciprocal of the nonlinear age-specific force of natural selection indicates a typical clearance time.

For the small effects posited by the theory, this can amount to thousands of generations. Future models may be able to make these ideas more concrete by supplementing mutation accumulation with explicit models of how genetic change affects vitality throughout the life course. Empirical studies among humans are beginning to characterize genetic diversity at lower and lower thresholds of allele frequency.

These are the kinds of alleles modeled by mutation accumulation theory. A long-term goal is to integrate the nonlinear models for mutation accumulation examined here with models for other contributors to senescent processes. Mutation accumulation does not act in isolation. It reshapes vital schedules that themselves reflect cellular and organismic processes and considerations of life-history optimization in interactions with environments.

Mathematical modeling of mutation accumulation is a point of departure for further enhancements of our evolutionary understanding of senescence. Our model is the general model in ref. For such cost functions theorems 2. Each individual in our infinite population carries some finite batch of deleterious mutant alleles specified by an element g in , the space of integer-valued Borel measures on.

Alleles in the batch are those with. Selective cost S , as defined below, is a function of g. The symbol denotes a unit mass at m , so the marginal selective cost of m is given by. Care is necessary with respect to the definition of fitness costs in the presence of heterogeneity. For nonlinear models, the choice does make a difference.

As Charlesworth points out ref. The alleles are not invading a population but are being held at equilibrium frequencies. Measuring selective cost by reductions in the NRR makes frequencies agree with classical formulas for single-locus models.

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The demographic meaning may be clarified through stable population theory. The population members who carry a particular collection of mutant alleles make a contribution to the next generation given by their mutation-dependent NRR. This is empirically observed especially for species that keep on growing during adult ages.

Perhaps the diversity of aging matches the diversity of life. My thesis, the central insight of this monograph, is: But the question is Springer Shop Bolero Ozon. Contributions to Evolutionary-Demographic Theory. An Optimization Model Based on Vitality. Your recently viewed items and featured recommendations.

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