Table 1.

Theoretical background of the evolution-based model of causation; summary of fundamental concepts and arguments of the evolutionary theory of aging and the statistical theory of extreme values.

The evolutionary theory of aging (4–8)

1. Among ultimate theories of aging, which seek to answer why aging occurs as opposed to theories addressing the proximate mechanisms of aging, the evolutionary theory of aging had its basic argument initially articulated by Medawar (6, 7). At first sight, aging seems paradoxical from an evolutionary perspective, because natural selection acting on individuals supposedly causes the evolution of increased, not decreased, fitness. Medawar's reasoning reconciled evolution with the fact that aging is non-adaptive.

2. In Medawar's thought experiment, if one starts by considering a theoretical potentially immortal and ever-reproducing population, one can envision that the older the members of this population are, the fewer there will be of them simply because they are exposed for a longer time to the hazard of death due to extrinsic causes such as accident, predation, starvation, and infectious disease, which prevailed throughout human evolutionary history. Thus, older individuals make progressively less contribution in terms of reproduction to the next generation, implying that “the force of natural selection” weakens with increasing age.

3. The force of natural selection is a measure of the intensity of selection on genes. If we now consider a population with a window of reproductive ages, a lethal mutation whose effect occurs before the earliest age of reproduction in the population is not passed to the next generation (meaning that the force of natural selection is maximum), while a lethal mutation whose effect occurs after the end of reproduction in the population would freely pass to the next generation (the force of natural selection is zero).

4. Between those two ages, the force of natural selection declines with increasing age because of a decreasing contribution to reproductive output. In humans, due to the extreme dependence of human offspring during infancy and early childhood, the force of natural selection is dependent not only on reproductive output but also on transfers of food and care (e.g., parental care and help from others such as older siblings or grandparents) (5).

5. From the preceding points, the central idea of the evolutionary theory of aging can be stated as follows: the force of natural selection acting on genes whose effects occur at a given age (i.e., age-specific genetic effects) declines with increasing adult age, such that aging results from the accumulation of deleterious mutations with late age-specific effects.

6. This evolutionary process accounts for the progressive deterioration of physiological function characteristic of aging (9, 10). Aging-related diseases result from the same general process, but without necessarily sharing proximate mechanisms with aging (11–13), as initially put forward by Medawar (7) with respect to cardiovascular diseases and cancer.

7. Starting with Fisher (14), and mostly through the works of Hamilton (4) and Charlesworth (15), population genetics has provided the evolutionary theory of aging with a mathematically explicit basis. Hamilton (4) formally showed that the force of natural selection acting on a mutation that reduces survival decreases with age starting at the earliest age of reproduction in the population. This was illustrated using data of the population of the United States (16).

8. Two population-genetic mechanisms have been proposed for the evolutionary outcome of aging. Mutation accumulation is the passive accumulation of mutations with late-onset deleterious effects (when selection is weak or absent) (7). Antagonistic pleiotropy is the active fixation of mutations with early beneficial effects (when selection is intense) and late deleterious effects (17). Experimental and comparative biology studies have provided empirical evidence for the operation of both mechanisms in the evolution of aging (8). Studies have also explored their operation in diseases including Alzheimer's disease (18–20).

The statistical theory of extreme values (21–24)

1. The statistical theory of extreme values is concerned with the distribution of the maximum, the minimum, or other extreme order statistics derived from a collection of random variables following an initial distribution.

2. The cumulative distribution function (c.d.f.) of a random variable X, denoted by F(x), is the probability of observing a value of X no greater than x. In symbols, F(x) = P[X ≤ x]. Most of the common c.d.f.'s used in biological modeling are differentiable and for those, the derivative f(x) = dF(x) / dx is called the probability density function (p.d.f.) of x. If X1,..., Xn are n independent observations from an initial distribution with c.d.f. F(x), the probability that each is no greater than x is {F(x)}n; this is the same as the probability that the maximum Mn= max{X1,..., Xn} is no greater than x. Therefore, {F(x)}n is the distribution function of Mn.

3. By the same reasoning, the probability that an observation with c.d.f. F(x) exceeds x is given by 1 – F(x); this is called the survival function and is denoted by S(x) = 1 – F(x) = P[X > x]. The probability that n independent observations X1,..., Xn from the initial distribution all exceed x is {1 – F(x)}n = {S(x)}n; this is the same as the probability that the minimum mn= min{X1,..., Xn} exceeds x. Therefore, {S(x)}n is the survival function of the minimum mn.

4. If the independent observations come from n not necessarily identical initial distributions, say F1(x),..., Fn(x), the c.d.f.'s of the maximum and minimum are given, respectively, by F1(x)···Fn(x) and 1 – {S1(x)}···{Sn(x)}.

5. The above expressions pertain to the exact statistical theory of extreme values and show that the distributions of the maximum or minimum depend both on the initial distribution and on n. The asymptotic statistical theory of extreme values covers results about the limiting distribution of an extreme order statistic as n tends to infinity under some centering and scaling of the extreme order statistic.

6. Centering and scaling is required because for any fixed value of x, lim n → ∞ F n(x) = 0 or 1 if, respectively, F(x) <1 or F(x) = 1, i.e., the limiting distribution of the maximum is degenerate (similarly for the minimum). Thus, a non-degenerate limiting distribution must be found as the distribution of some sequence of transformed values of the maximum (or minimum) that depend on n but not on x.

7. For a given F(x), if such a limiting distribution of the maximum (or minimum) exists and we denote it G(x), we say that F is in the maximum (or minimum) domain of attraction of G. For example, if X1,..., Xn are independent observations with an exponential distribution with c.d.f. F(x) = 1 – exp(–x), then n times the minimum mn has a limiting distribution which is also exponential, so F is in the minimum domain of attraction of itself (though this is usually not the case for other distributions).