Abstract—
Describing mortality dynamics using average indicators without considering variability can yield average results, impeding the analysis of survival-curve patterns during periods of significant mortality spikes, especially at the oldest or youngest ages. Therefore, instead of the generally accepted Gompertz method, other methods are increasingly used, which rely on comparisons of various demographic indicators. In humans, chronic phenoptosis, in contrast to age-independent acute phenoptosis, manifests as a rectangularization of the survival curve with a simultaneous increase in the life expectancy at birth due to the advancement of social, scientific, and technological progress. Rectangularization is difficult to notice solely by examining the optimal coefficients in the Gompertz—Makeham equation, primarily because of the inevitable calculation errors. This can be avoided by calculating demographic indicators based on the spread of the life expectancy: Keyfitz entropy, Gini coefficient, and coefficient of variation of lifespan. We examine several sub-Gompertzian models of mortality growth with age, which describe the aging of nematodes and insects. Within the sub-Gompertzian model of aging, the increase in mortality with age in invertebrates is quantified as a rectangularization of the survival function estimated by these demographic indicators. On the other hand, the increasing rectangularization of the survival function with the development of scientific and technological progress, demonstrated by a decrease in the Keyfitz entropy, along with a simultaneous increase in the life expectancy in humans, also aligns well with the hypothesis of an age-dependent increase in mortality in mammals overall. Calculations on aging models demonstrate the effectiveness of using Keyfitz entropy and the Gini coefficient as important demographic indicators. The use of these indicators seems preferable, especially for nematodes, where the sub-Gompertzian model of aging is applicable, and for vertebrates, primarily mammals, with certain restrictions, the Gompertz–Makeham law is applicable. Approaches that consider dynamic age-related shifts in improved survival, such as studying imbalances in lifespan, enhance our understanding of the mechanisms of aging. This, in turn, will contribute to the development of more accurate methods for assessing the effects of biologically active substances used in gerontology, such as anti-aging drugs and geroprotectors.
REFERENCES
Gompertz, B., On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Philos. Trans. R. Soc. L. A, 1825, vol. 115, no. 1, pp. 513–585. https://doi.org/10.1098/rstl.1825.0026
Deevey, E.S., Life tables for natural populations of animals, Q. Rev. Biol., 1947, vol. 22, no. 4, pp. 283–314. https://doi.org/10.1086/395888
Gavrilov, L.A. and Gavrilova, N.S., The Biology of Life Span: A Quantitative Approach, N. Y.: Harwood Academic Publisher, 1991.
Vaupel, J.W., Carey, J.R., Christensen, K., Johnson, T.E., Yashin, A.I., Holm, N.V., Iachine, I.A., Kannisto, V., Khazaeli, A.A., Liedo, P., Longo, V.D., Zeng, Y., Manton, K.G., and Curtsinger, J.W., Biodemographic trajectories of longevity, Science, 1998, vol. 280, no. 5365, pp. 855–860. https://doi.org/10.1126/science.280.5365.855
Khalyavkin, A.V., Influence of environment on the mortality pattern of potentially non-senescent organisms. General approach and comparison with real populations, Adv. Gerontol., 2001, vol. 7, pp. 46–49.
Jones, O.R., Gaillard, J.M., Tuljapurkar, S., et al., Senescence rates are determined by ranking on the fast-slow life-history continuum, Ecol. Lett., 2008, vol. 11, no. 7, pp. 664–673. https://doi.org/10.1111/j.1461-0248.2008.01187.x
Jones, O.R., Scheuerlein, A., Salguero-Gómez, R., Camarda, C.G., Schaible, R., Casper, B.B., Dahlgren, J.P., Ehrlén, J., García, M.B., Menges, E., Quintana-Ascencio, P.F., Caswell, H., Baudisch, A., and Vaupel, J.W., Diversity of ageing across the tree of life, Nature, 2014, vol. 505, no. 7482, pp. 169–173. https://doi.org/10.1038/nature12789
Ricklefs, R.E., Life-history connections to rates of aging in terrestrial vertebrates, Proc. Natl. Acad. Sci. U.S.A., 2010, vol. 107, no. 22, pp. 10 314–10 319. https://doi.org/10.1073/pnas.1005862107
Myl’nikov, S.V., Towards the estimation of survival curves parameters and geroprotectors classification, Adv. Gerontol., 2011, vol. 24, no. 4, pp. 563–569.
Akif’ev, A.P. and Potapenko, A.I., Nuclear genetic material as an initial substrate for animal aging, Genetika, 2001, vol. 37, no. 11, pp. 1445–1458.
Markov, A.V., Can kin selection facilitate the evolution of the genetic program of senescence?, Biochemistry (Moscow), 2012, vol. 77, no. 7, pp. 733–741. https://doi.org/10.1134/S0006297912070061
Strehler, B.L. and Mildvan, A.S., General theory of mortality and aging, Science, 1960, vol. 132, no. 3418, pp. 14–21. https://doi.org/10.1126/science.132.3418.14
Seliverstov, A.V., Heuristic algorithms for recognition of some cubic hypersurfaces, Program. Comput. Softw., 2021, vol. 47, no. 1, pp. 50–55. https://doi.org/10.1134/S0361768821010096
Makeham, W.M., On the law of mortality and the construction of annuity tables, The Assurance Magazine, and Journal of the Institute of Actuaries, 1860, vol. 8, no. 6, pp. 301–310. https://doi.org/10.1017/S204616580000126X
Gavrilov, L.A. and Gavrilova, N.S., Mortality measurement at advanced ages: A study of the social security administration death master file, N. Am. Actuar. J., 2011, vol. 15, no. 3, pp. 432–447. https://doi.org/10.1080/10920277.2011.10597629
Gavrilova, N.S. and Gavrilov, L.A., Are we approaching a biological limit to human longevity?, J. Gerontol. Series A, 2020, vol. 75, no. 1, pp. 1061–1067. https://doi.org/10.1093/gerona/glz164
Oeppen, J. and Vaupel, J.W., Demography. Broken limits to life expectancy, Science, 2002, vol. 296, no. 1, pp. 1029–1031. https://doi.org/10.1126/science.1069675
Shilovsky, G.A., Putyatina, T.S., Markov, A.V., and Skulachev, V.P., Contribution of quantitative methods of estimating mortality dynamics to explaining mechanisms of aging, Biochemistry (Moscow), 2015, vol. 80, no. 12, pp. 1547–1559. https://doi.org/10.1134/S0006297915120020
Golubev, A., A 2D analysis of correlations between the parameters of the Gompertz–Makeham model (or law?) of relationships between aging, mortality, and longevity, Biogerontology, 2019, vol. 20, no. 6, pp. 799–821. https://doi.org/10.1007/s10522-019-09828-z
Bohk-Ewald, C., Ebeling, M., and Rau, R., Lifespan disparity as an additional indicator for evaluating mortality forecasts, Demography, 2017, vol. 54, no. 4, pp. 1559–1577. https://doi.org/10.1007/s13524-017-0584-0
Frolkis, V.V., Aging and Life-Prolonging Processes, Wien, New York: Springer Verlag, 1982. https://doi.org/10.1007/978-3-7091-8649-7
Wrycza, T.F., Missov, T.I., and Baudisch, A., Quantifying the shape of aging, PLoS One, 2015, vol. 10, no. 3, p. e0119163. https://doi.org/10.1371/journal.pone.0119163
Burger, O., Baudisch, A., and Vaupel, J.W., Human mortality improvement in evolutionary context, Proc. Natl. Acad. Sci. U.S.A., 2012, vol. 109, no. 44, pp. 18210–18214. https://doi.org/10.1073/pnas.1215627109
Burger O., Evolutionary demography of the human mortality profile, in The Evolution of Senescence in the Tree of Life, Shefferson, R.P., Jones, O.R., and Salgnero-Gomez, R., Eds., Cambridge: Cambridge Univ. Press, 2017. https://doi.org/10.1017/9781139939867.006
Skulachev, M.V. and Skulachev, V.P., New data on programmed aging—slow phenoptosis, Biochemistry (Moscow), 2014, vol. 79, no. 1, pp. 977–993. https://doi.org/10.1134/S0006297914100010
Galimov, E.R., Lohr, J.N., and Gems, D., When and how can death be an adaptation?, Biochemistry (Moscow), 2019, vol. 84, no. 12, pp. 1433–1437. https://doi.org/10.1134/S0006297919120010
Skulachev, V.P., Shilovsky, G.A., Putyatina, T.S., Popov, N.A., Markov, A.V., Skulachev, M.V., and Sadovnichii, V.A., Perspectives of Homo sapiens lifespan extension: Focus on external or internal resources?, Aging (Albany, New York), 2020, vol. 12, no. 6, pp. 5566–5584. https://doi.org/10.18632/aging.102981
Keyfitz, N., What difference would it make if cancer were eradicated? An examination of the Taeuber paradox, Demography, 1977, vol. 14, no. 4, pp. 411–418.
Aburto, J.M., Alvarez, J.-A., Villavicencio, F., and Vaupel, J.W., The threshold age of lifetable entropy, Demogr. Res., 2019, vol. 41, no. 4, pp. 83–102. https://doi.org/10.4054/DemRes.2019.41.4
Demetrius, L., Adaptive value, entropy and survivorship curves, Nature, 1978, vol. 275, no. 2677, pp. 213–214. https://doi.org/10.1038/275213a0
Zhang, Z. and Vaupel, J.W., The age separating early deaths from late deaths, Demogr. Res., 2009, vol. 20, no. 29, pp. 721–730. https://doi.org/10.4054/DemRes.2009.20.29
Boldrini, M., Corrado Gini, J. R. Stat. Soc. Ser. A Stat. Soc., 1966, vol. 129, no. 1, pp. 148–150. https://doi.org/10.1111/j.2397-2327.1966.tb02144.x
Shkolnikov, V.M., Andreev, E.M., and Begun, A.Z., Gini coefficient as a life table function: Computation from discrete data, decomposition of differences and empirical examples, Demogr. Res., 2003, vol. 8, no. 11, pp. 305–358. https://doi.org/10.4054/DemRes.2003.8.11
Smits, J. and Monden, C., Length of life inequality around the globe, Soc. Sci. Med., 2009, vol. 68, no. 6, рр. 1114–1123.
Gavrilova, N.S., Gavrilov, L.A., Severin, F.F., and Skulachev, V.P., Testing predictions of the programmed and stochastic theories of aging: Comparison of variation in age at death, menopause, and sexual maturation, Biochemistry (Moscow), 2012, vol. 77, no. 7, pp. 754–760. https://doi.org/10.1134/S0006297912070085
Shilovsky, G.A., Putyatina, T.S., Lysenkov, S.N., Ashapkin, V.V., Luchkina, O.S., Markov, A.V., and Skulachev, V.P., Is it possible to prove the existence of an aging program by quantitative analysis of mortality dynamics?, Biochemistry (Moscow), 2016, vol. 81, no. 12, pp. 1461–1476. https://doi.org/10.1134/S0006297916120075
Shilovsky, G.A., Putyatina, T.S., Ashapkin, V.V., Luchkina, O.S., and Markov, A.V., Coefficient of variation of lifespan across the tree of life: Is it a signature of programmed aging?, Biochemistry (Moscow), 2017, vol. 82, no. 1, pp. 1480–1492. https://doi.org/10.1134/S0006297917120070
Rubanov, L.I. and Seliverstov, A.V., Projective-invariant description of a meandering river, J. Commun. Technol. Electron., 2017, vol. 62, no. 6, pp. 663–668. https://doi.org/10.1134/S1064226917060201
Chen, J., Senturk, D., Wang, J.L., Müller, H.G., Carey, J.R., Caswell, H., and Caswell-Chen, E.P., A demographic analysis of the fitness cost of extended longevity in Caenorhabditis elegans, J. Gerontol. A Biol. Sci. Med. Sci., 2007, vol. 62, no. 2, pp. 126–135. https://doi.org/10.1093/gerona/62.2.126
Evans, F.C. and Smith, F.E., The intrinsic rate of natural increase for the human louse, Pediculus humanus L., Amer. Naturalist, 1952, vol. 86, no. 830, pp. 299–310. https://doi.org/10.1086/281737
Comfort, A., The Biology of Senescence, New York: Elsevier, 1979.
Lewis, K.N., Mele, J., Hayes, J.D., and Buffenstein, R., Nrf2, a guardian of health span and gatekeeper of species longevity, Integr. Comp. Biol., 2010, vol. 50, no. 5, pp. 829–843. https://doi.org/10.1093/icb/icq034
Lewis, K.N., Wason, E., Edrey, Y.H., Kristan, D.M., Nevo, E., and Buffenstein, R., Regulation of Nrf2 signaling and longevity in naturally long-lived rodents, Proc. Natl. Acad. Sci. U.S.A., 2015, vol. 112, no. 12, pp. 3722–3727. https://doi.org/10.1073/pnas.1417566112
Ruby, J.G., Smith, M., and Buffenstein, R., Naked mole-rat mortality rates defy gompertzian laws by not increasing with age, Elife, 2018, vol. 7, p. e31157. https://doi.org/10.7554/eLife.31157
Shilovsky, G.A., Lability of the Nrf2/Keap/ARE cell defense system in different models of cell aging and age-related pathologies, Biochemistry (Moscow), 2022, vol. 87, no. 1, pp. 70–85. https://doi.org/10.1134/S0006297922010060
Zinovkin, R.A., Kondratenko, N.D., and Zinovkina, L.A., Does Nrf2 play a role of a master regulator of mammalian aging?, Biochemistry (Moscow), 2022, vol. 87, no. 12, pp.1465–1476. https://doi.org/10.1134/S0006297922120045
Ulasov, A.V., Rosenkranz, A.A., Georgiev, G.P., and Sobolev, A.S., Keap1/ARE signaling: Towards specific regulation, Life Sci., 2021, vol. 291, p. 120111. https://doi.org/10.1016/j.lfs.2021.120111
Hushpulian, D.M., Ammal Kaidery, N., Ahuja, M., Poloznikov, A.A., Sharma, S.M., et al., Challenges and limitations of targeting the Keap1-Nrf2 pathway for neurotherapeutics: Bach1 derepression to the rescue, Front. Aging Neurosci., 2021, vol. 13, p. 673205.https://doi.org/10.3389/fnagi.2021.673205
Dilman, V.M., Ontogenetic model of ageing and disease formation and mechanisms of natural selection, J. Theor. Biol., 1986, vol. 118, no. 1, pp. 73–81. https://doi.org/10.1016/S0022-5193(86)80009-1
Skulachev, V.P., Holtze, S., Vyssokikh, M.Y., Bakeeva, L.E., Skulachev, M.V., Markov, A.V., Hildebrandt, T.B., and Sadovnichii, V.A., Neoteny, prolongation of youth: From naked mole rats to “naked apes” (humans), Physiol. Rev., 2017, vol. 97, no. 1, pp. 699–720. https://doi.org/10.1152/physrev.00040.2015
Vyssokikh, M.Y., Holtze, S., Averina, O.A., Lyamzaev, K.G., Panteleeva, A.A., Marey, M.V., Zinovkin, R.A., Severin, F.F., Skulachev, M.V., Fasel, N., Hildebrandt, T.B., and Skulachev, V.P., Mild depolarization of the inner mitochondrial membrane is a crucial component of an anti-aging program, Proc. Natl. Acad. Sci. U.S.A., 2020, vol. 117, no. 1, pp. 6491–6501. https://doi.org/10.1073/pnas.1916414117
Colchero, F., Rau, R., Jones, O.R., et al., The emergence of longevous populations, Proc. Natl. Acad. Sci. U.S.A., 2016, vol. 113, no. 48, pp. 7681–7690. https://doi.org/10.1073/pnas.1612191113
Skulachev, V.P., Aging is a specific biological function rather than the result of a disorder in complex living systems: Biochemical evidence in support of Weismann’s hypothesis, Biochemistry (Moscow), 1997, vol. 62, no. 11, pp. 1191–1195.
Németh, L., Life expectancy versus lifespan inequality: A smudge or a clear relationship?, PLoS One, 2017, vol. 12, no. 1, p. e0185702. https://doi.org/10.1371/journal.pone.0185702
Shilovsky, G.A., Seliverstov, A.V., and Zverkov, O.A., Demographic indicators, models, and testing, Discrete Contin. Models Appl. Comput. Sci., 2023, vol. 31, no. 4, pp. 359–374. https://doi.org/10.22363/2658-4670-2023-31-4-359-374
Skulachev, M.V., Severin, F.F., and Skulachev, V.P., Aging as an evolvability-increasing program which can be switched off by organism to mobilize additional resources for survival, Curr. Aging Sci., 2015, vol. 8, no. 1, p. 95109. https://doi.org/10.2174/1874609808666150422122401
Neumann, J.T., Thao, L.T.P., Murray, A.M., Callander, E., Carr, P.R., Nelson, M.R., Wolfe, R., Woods, R.L., Reid, C.M., Shah, R.C., Newman, A.B., Williamson, J.D., Tonkin, A.M., and McNeil, J.J., ASPREE investigators. Prediction of disability-free survival in healthy older people, Geroscience, 2022, vol. 44, no. 3, pp. 1641–1655. https://doi.org/10.1007/s11357-022-00547-x
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Shilovsky, G.A., Seliverstov, A.V. Demographic Indicators of Probability Models. Adv Gerontol 13, 164–177 (2023). https://doi.org/10.1134/S2079057024600307
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DOI: https://doi.org/10.1134/S2079057024600307