Читать книгу General system theory of aging. Special role of the immune system - V. I. Dontsov - Страница 14

The first mathematical model of aging was created almost 200 years ago by B. Gompertz (1825) and still most accurately describes the age dynamics of human mortality and, apparently, of most other organisms. As a specialist in life insurance, Gompertz theoretically derived the practically necessary formula for increasing his mortality rate with age, which until now has been the most common quantitative description of aging itself.

Mortality, as “quantitative characterization of the inability to resist destruction,” can now be viewed as the reciprocal of vitality – the ability to withstand the totality of destructive processes.

A simple assumption about the stochasticity of the aging process is enough: the viability over time decreases in proportion to itself at each time point (formula 2) in order to obtain the basic law of aging: mortality increases with age by the exponent (formula 3). Such a nonspecific increase in the body’s vulnerability to all influences with age is called aging itself.

d X / d t = – k X, (2)

where k is a coefficient, X is viability, t is time.

Considering the mortality (μ) as an inverse viability value (μ = 1 / X), the basic formula (3) is obtained from formula (2) aging (B. Gompertz and W. Makekem) – with age, the overall mortality increases exponentially:

μ = Ro exp (α t) + A, (3)

where Ro is the initial mortality rate, α is the rate of increase in mortality, A is the coefficient characterizing the contribution of external influences to mortality, the effect of which weakly depends on age.

The approach to writing formula is now theoretically clear: it is an elementary differential equation that describes, for example, radioactive decay in physics and other simple probabilistic processes. The essence of the phenomenon lies in the fact that at each moment in time the state change does not depend on the prehistory, but only on the present state of the system.

The general mechanisms of such processes are also clear – these are principally probabilistic regularities associated with the ultimate stability of any elements delimited from the external environment; then a complex organism consisting of such elementary units can only lose them over time. The main issue is then the nature of such “elementary units of life.”

Gomperz himself noted the similarity of the curves of changes in mortality and entropy, and V. Perks (1932) directly wrote that “the inability to resist destruction has the same nature as energy dissipation” (that is, aging is equivalent to an increase in entropy, which serves as a measure of disorder any system); A. Comfort (1967) writes that viability can be reduced to a rather specific, though not material, substrate – information in cells, which is “just biological energy”.

Thus, the meaningful interpretation of the concept of “viability” was reduced from the very beginning, and is reduced now, not so much to the material content, but to the energy and information content – to the “entelechy” of the ancients.

For a population of animals or a human cohort, by definition:

μ = dN (t) / N (t),

where N (t) is the number of members of an endangered population at time t. By integrating the Gompertz-Makema equation, one can obtain a direct formula for calculating the number of survivors of a certain age (formula 4):

N (t) =N_oexp ((-A t – R_o/α (exp (α t) – 1)) (4)

The qualitative view of the survival, mortality and survival curves corresponding to the formulas presented above corresponds to the real survival curves of various human populations, as well as a number of other species. However, the Gompertz-Makema formula describes only the middle part of the mortality intensity curve, whereas the initial part of the curve (growth and development processes – up to 20—25 years) and the final part (older than 80—90 years, individuals with hereditary longevity) cannot be taken into account in this way.

The full mortality curve, which takes into account the period of growth and development and hereditary longevity, can be obtained from the systemic stochastic-regulatory theory of aging discussed below and proposed by us earlier (Dontsov, 1990, 2012, 2017).

The general reason for allowing entropy to work in any system is the principle delimitation of this system from the external one, which does not allow it to fully renew itself and puts a limit to its existence as a separate system.

Similarly, the global cause of aging is the discreteness of the existence of life in the form of individual forms – living organisms, their fundamental limitations (limits of adaptation of all homeostasis mechanisms) in comparison with the almost infinite variety of influences on each particular organism of the rest of the World. The quantitative and qualitative infinity of the effects of the World on a discrete organism can only partially be compensated by homeostasis, which leads to the accumulation of uncompensated damage – the most common mechanism of aging.

Self-renewal of an organism at all its levels is not a sufficient anti-aging factor since the self-renewal process itself is not absolute and has the same random mechanisms.

Some obvious and experimentally and demographically confirmed conclusions are interesting, however, sometimes paradoxically sounding. So from the above, it is obvious that the greatest absolute decrease in viability can be observed at an early age, which we can see from the curves of changes in the ontogenesis of the absolute value of many physiological functions. This means that prevention of aging should begin at the earliest ages. At the same time, in old age, even small absolute changes in viability lead to pronounced changes in mortality, so at older ages, it is convenient to study the effects of adaptogens and biostimulants, although a small vital resource may not lead to a significant increase in life expectancy.

The mathematical analysis of the theories of aging, based on the modeling of its essence – the age-related decline in overall viability, turned out to be surprisingly fruitful and suitable both for objectives of theoretical research and for practical research in population gerontology. At the same time, the common cause of aging is manifested by some general mechanisms that should be modeled and evaluated for their contribution to the overall aging of the system.

Another approach to the quantitative assessment of aging, based on the same definition – reducing overall viability with age, is to consider the overall viability of the system as an integral of the viability of its parts, which, as applied to the body, means that the overall viability of the body consists of maintaining vitality (functional resource) of its main organs and systems (formula 5).

Х = k₁ х₁ + k₂ х₂ +….+k_n х_n (5)

where k is the coefficient, x1 … n is the viability of organs and systems.

The definition of individual aging as a biological age is based on this.