## The Simplest Reliability Model Of Aging

In this section we show that a system built of nonaging components demonstrates an aging behavior (mortality growth with age) and subsequent mortality leveling-off.

Consider a parallel system built of n nonaging elements with a constant failure rate ^ and reliability (survival) function e-^x (see also Figure 5.3b). We already showed (see the system's failure and reliability structure section) that in this case the reliability function of the entire parallel system is s(x) = 1 - (1 - p)n = 1 - (1 - e-^y

This formula corresponds to the simplest case when the failure of elements is statistically independent. More complex models would require specific assumptions or prior knowledge on the exact type of interdependence in elements failure. One such model known as ''the model of the avalanche-like destruction'' is described elsewhere (see pp. 246-251 in Gavrilov, Gavrilova, 1991) and is briefly summarized in the theoretical models of systems failure in aging section.

Consequently, the failure rate of the entire system ^s(x), can be written as follows:

when x ^ 1/^ (early-life period approximation, when 1 - e-^x ^ ^x);

when x ^ 1/^ (late-life period approximation, when 1 - e-^x ^ 1).

Thus, the failure rate of a system initially grows as a power function n of age (the Weibull law). Then the tempo at which the failure rate grows declines, and the failure rate approaches asymptotically an upper limit equal to

Here we should pay attention to three significant points. First, a system constructed of nonaging elements is now behaving like an aging object: aging is a direct consequence of the redundancy of the system (redundancy in the number of elements). Second, at very high ages the phenomenon of aging apparently disappears (failure rate levels off), as redundancy in the number of elements vanishes. The failure rate approaches an upper limit, which is totally independent of the initial number of elements, but coincides with the rate of their loss (parameter Third, the systems with different initial levels of redundancy (parameter n) will have very different failure rates in early life, but these differences will eventually vanish as failure rates approach the upper limit determined by the rate of elements' loss (parameter Thus, the compensation law of mortality (in its weak form) is an expected outcome of this illustrative model.

Note also that the identical parallel systems in this example do not die simultaneously when their elements fail by chance. A common view in biology is the idea that all the members of homogeneous population in a hypothetical constant environment should die simultaneously so that the survival curve of such a population would look like a rectangle. This idea stems from the basic principles of quantitative genetics, which assume implicitly that every animal of a given genotype has the same genetically determined lifespan so that all variation of survival time around a genotype mean results from the environmental variance. George Sacher (1977) pointed out that this concept is not applicable to longevity and used an analogy with radioactive decay in his arguments.

Even the simplest parallel system has a specific lifespan distribution determined entirely by a stochastic nature of the aging process. In order to account for this stochasticity it was proposed to use a stochastic variance component of lifespan in addition to genetic and environmental components of phenotypic lifespan variance (Gavrilov and Gavrilova, 1991). The stochastic nature of the system's destruction also produces heterogeneity in an initially homogeneous population. This kind of induced heterogeneity was observed in isogenic strains of nematodes, in which aging resulted in substantial heterogeneity in behavioral capacity among initially homogeneous worms kept in controlled environmental conditions (Herndon et al., 2002).

The graph shown in Figure 5.8 depicts mortality trajectories for five systems with different degrees of redundancy.

System 1 has only one unique element (no redundancy), and it has the highest failure rate, which does not depend on age (no aging). System 2 has two elements connected in parallel (one extra element is redundant), and the failure rate initially increases with age (aging

Age, in dimensionless units, log scale

Age, in dimensionless units, log scale

Figure 5.8. Failure kinetics of systems with different levels of redundancy. The dependence of the logarithm of mortality force (failure rate) on the logarithm of age in five systems with different levels of redundancy (computer simulation experiment). The scales for mortality rates (vertical axis) and for age (horizontal axis) are presented in dimensionless units for mortality rates, and jU.x for age), to ensure the generalizability of the results (invariance of graphs on failure rate of the elements in the system, parameter Also, the log scale is used to explore the system behavior in a wide range of ages (0.01 —10 units) and failure rates ( 0.00000001—1.0 units). Dependence 1 is for the system containing only one unique element (no redundancy). Dependence 2 is for the system containing two elements connected in parallel (degree of redundancy = 1). Dependencies 3, 4 and 5 are for systems containing, respectively, 3, 4 and 5 elements connected in parallel (with increasing levels of redundancy). Source: Gavrilovand Gavrilova, 2005.

appears). The apparent rate of aging can be characterized by a slope coefficient that is equal to 1. Finally, the failure rate levels off at advanced ages. Systems 3, 4, and 5 have, respectively, three, four, and five elements connected in parallel (two, three, and four extra elements are redundant), and the failure rate initially increases with age at an apparent aging rate (slope coefficient) of 2, 3, and 4, respectively. Finally, the mortality trajectories of each system level off at advanced ages at exactly the same upper limit to the mortality rate.

This computational example illustrates the following statements: (1) Aging is a direct consequence of a system's redundancy, and the expression of aging is directly related to the degree of a system's redundancy. Specifically, an apparent relative aging rate is equal to the degree of redundancy in parallel systems. (2) All mortality trajectories tend to converge with age, so that the compensation law of mortality is observed. (3) All mortality trajectories level off at advanced ages, and a mortality plateau is observed. Thus, the major aging phenomena (aging itself, the compensation law of mortality, late-life mortality deceleration, and late-life mortality plateaus) are already observed in the simplest redundant systems. However, to explain the Gompertz law of mortality, an additional idea of initial damage load should be taken into account (see next section).

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