, Volume 20, Issue 6, pp 799–821 | Cite as

A 2D analysis of correlations between the parameters of the Gompertz–Makeham model (or law?) of relationships between aging, mortality, and longevity

  • A. GolubevEmail author
Research Article


When mortality (μ), aging rate (γ) and age (t) are treated according to the Gompertz model μ(t) = μ0eγt (GM), any mean age corresponds to a manifold of paired reciprocally changing μ0 and γ. Therefore, any noisiness of data used to derive GM parameters makes them negatively correlated. Besides this artifactual factor of the Strehler–Mildvan correlation (SMC), other factors emerge when the age-independent mortality C modifies survival according to the Gompertz–Makeham model μ(t) = C+μ0eγt (GMM), or body resources are partitioned between survival and protection from aging [the compensation effect of mortality (CEM)]. Theoretical curves in (γ, logμ0) coordinates show how μ0 decreases when γ increases upon a constant mean age. Within a species-specific range of γ, such “isoage” curves look as nearly parallel straight lines. The slopes of lines constructed by applying GM to survival curves modeled according to GMM upon changes in C are greater than the isoage slopes. When CEM is modeled, the slopes are still greater. Based on these observations, CEM is shown to contribute to SMC associated with sex differences in lifespan, with the effects of several life-extending drugs, and with recent trends in survival/mortality patterns in high-life-expectancy countries; whereas changes in C underlie differences between even high-life-expectancy countries, not only between high- and low-life-expectancy countries. Such interpretations make sense only if GM is not merely a statistical model, but rather reflects biological realities. Therefore, GM is discussed as derivable by applying certain constraints to a natural law termed the generalized Gompertz–Makeham law.


Aging Lifespan Mortality Survival Parametric analysis Gompertz–Makeham law Strehler–Mildvan correlation 



  1. Anisimov VN, Piskunova TS, Popovich IG, Zabezhinski MA, Tyndyk ML, Egormin PA, Yurova MN, Rosenfeld SV, Semenchenko AV, Kovalenko IG, Poroshina TE, Berstein LM (2010) Gender differences in metformin effect on aging, life span and spontaneous tumorigenesis in 129/Sv mice. Aging 2:945–958CrossRefGoogle Scholar
  2. Anisimov VN, Berstein LM, Popovich IG, Zabezhinski MA, Egormin PA, Piskunova TS, Semenchenko AV, Tyndyk ML, Yurova MN, Kovalenko IG, Poroshina TE (2011) If started early in life, metformin treatment increases life span and postpones tumors in female SHR mice. Aging 3:148–157CrossRefGoogle Scholar
  3. Austad SN, Fischer KE (2016) Sex differences in lifespan. Cell Metab 23:1022–1033. CrossRefPubMedPubMedCentralGoogle Scholar
  4. Austad SN, Hoffman JM (2019) Response to genes that improved fitness also cost modern humans: evidence for genes with antagonistic effects on longevity and disease. Evolut Med Public Health 2019:7–8. CrossRefGoogle Scholar
  5. Basellini U, Canudas-Romo V, Lenart A (2018) Location-scale models in demography: a useful re-parameterization of mortality models. Eur J Popul. CrossRefGoogle Scholar
  6. Börger M, Genz M, Ruß J (2018) Extension, compression, and beyond: a unique classification system for mortality evolution patterns. Demography 55:1343–1361. CrossRefPubMedGoogle Scholar
  7. Box GEP, Draper NR (1987) Empirical model building and response surfaces. Wiley, New YorkGoogle Scholar
  8. Brewer RA, Gibbs VK, Smith DL Jr (2016) Targeting glucose metabolism for healthy aging. Nutr Healthy Aging 4:31–46. CrossRefPubMedPubMedCentralGoogle Scholar
  9. Burger O, Missov TI (2016) Evolutionary theory of ageing and the problem of correlated Gompertz parameters. J Theor Biol 408:34–41. CrossRefPubMedGoogle Scholar
  10. Chapman JW, O’Callaghan CJ, Hu N, Ding K, Yothers GA, Catalano PJ, Shi Q, Gray RG, O’Connell MJ, Sargent DJ (2013) Innovative estimation of survival using log-normal survival modelling on ACCENT database. Br J Cancer 108:784–790. CrossRefPubMedPubMedCentralGoogle Scholar
  11. Cheng CJ, Gelfond JAL, Strong R, Nelson JF (2019) Genetically heterogeneous mice exhibit a female survival advantage that is age- and site-specific: results from a large multi-site study. Aging Cell. CrossRefPubMedPubMedCentralGoogle Scholar
  12. Cheung SL, Robine JM (2007) Increase in common longevity and the compression of mortality: the case of Japan. Popul Stud 61:85–97. CrossRefGoogle Scholar
  13. Crimmins EM, Shim H, Zhang YS, Kim JK (2019) Differences between men and women in mortality and the health dimensions of the morbidity process. Clin Chem 65:135–145. CrossRefPubMedGoogle Scholar
  14. de Beer J, Janssen F (2016) A new parametric model to assess delay and compression of mortality. Popul Health Metr 14:46. CrossRefPubMedPubMedCentralGoogle Scholar
  15. de Crécy-Lagard V, Haas D, Hanson AD (2018) Newly-discovered enzymes that function in metabolite damage-control. Curr Opin Chem Biol 47:101–108. CrossRefPubMedGoogle Scholar
  16. de Lorenzo V, Sekowska A, Danchin A (2014) Chemical reactivity drives spatiotemporal organisation of bacterial metabolism. FEMS Microbiol Rev 1:1. CrossRefGoogle Scholar
  17. de Magalhaes JP, Cabral JA, Magalhaes D (2005) The influence of genes on the aging process of mice: a statistical assessment of the genetics of aging. Genetics 169:265–274. CrossRefPubMedPubMedCentralGoogle Scholar
  18. De Paepe M, Taddei F (2006) Viruses’ life history: towards a mechanistic basis of a trade-off between survival and reproduction among phages. PLoS Biol 4:e193. CrossRefPubMedPubMedCentralGoogle Scholar
  19. Dukan S, Nyström T (1999) Oxidative stress defense and deterioration of growth-arrested Escherichia coli cells. J Biol Chem 274:26027–26032. CrossRefPubMedGoogle Scholar
  20. Ebeling M, Rau R, Baudisch A (2018) Rectangularization of the survival curve reconsidered: the maximum inner rectangle approach. Popul Stud 72:369–379. CrossRefGoogle Scholar
  21. Ediev DM, Sanderson WC, Scherbov S (2019) The inverse relationship between life expectancy-induced changes in the old-age dependency ratio and the prospective old-age dependency ratio. Theor Popul Biol 125:1–10. CrossRefPubMedGoogle Scholar
  22. Erkmen O (2009) Mathematical modeling of Salmonella typhimurium inactivation under high hydrostatic pressure at different temperatures. Food Bioprod Process 87:68–73. CrossRefGoogle Scholar
  23. Finkelstein M (2012) Discussing the Strehler–Mildvan model of mortality. Demogr Res 26:191–206CrossRefGoogle Scholar
  24. Gavrilov LA, Gavrilova NS (1991) The biology of life span: a quantitative approach. Harwood Academic Publisher, New YorkGoogle Scholar
  25. Gavrilov LA, Gavrilova NS (2001) The reliability theory of aging and longevity. J Theor Biol 213:527–545. CrossRefPubMedGoogle Scholar
  26. Goldsmith TC (2004) Aging as an evolved characteristic—Weismann’s theory reconsidered. Med Hypotheses 62:304–308. CrossRefPubMedGoogle Scholar
  27. Golubev AG (1996) The other side of metabolism: a review. Biochemistry 61:1443–1460Google Scholar
  28. Golubev A (2004) Does Makeham make sense? Biogerontology 5:159–167CrossRefGoogle Scholar
  29. Golubev A (2009) How could the Gompertz–Makeham law evolve. J Theor Biol 258:1–17. CrossRefPubMedGoogle Scholar
  30. Golubev AG (2012) The issue of the feasibility of a general theory of aging. III. Theory and practice of aging. Adv Gerontol 2:109–119. CrossRefGoogle Scholar
  31. Golubev AG (2019) Why and how do we age? A single answer to two questions. Adv Gerontol 9:1–14CrossRefGoogle Scholar
  32. Golubev A, Hanson AD, Gladyshev VN (2017a) Non-enzymatic molecular damage as a prototypic driver of aging. J Biol Chem 292:6029–6038. CrossRefPubMedPubMedCentralGoogle Scholar
  33. Golubev A, Hanson AD, Gladyshev VN (2017b) A tale of two concepts: harmonizing the free radical and antagonistic pleiotropy theories of aging. Antioxid Redox Signal. CrossRefPubMedGoogle Scholar
  34. Golubev A, Panchenko A, Anisimov V (2018) Applying parametric models to survival data: tradeoffs between statistical significance, biological plausibility, and common sense. Biogerontology 19:341–365. CrossRefPubMedGoogle Scholar
  35. Gompertz B (1825) 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. Phil Trans R Soc Lond 115:513–583. CrossRefGoogle Scholar
  36. Hamilton WD (1966) The moulding of senescence by natural selection. J Theor Biol 12:12–45. CrossRefPubMedGoogle Scholar
  37. Hamilton A (2007) Laws of biology, laws of nature: problems and (dis)solutions. Philos Compass 2:592–610. CrossRefGoogle Scholar
  38. Hanson AD, Henry CS, Fiehn O, Crécy-Lagard VD (2016) Metabolite damage and metabolite damage control in plants. Annu Rev Plant Biol 67:31–52. CrossRefGoogle Scholar
  39. Harrison DE, Strong R, Alavez S, Astle CM, DiGiovanni J, Fernandez E, Flurkey K, Garratt M, Gelfond JAL, Javors MA, Levi M, Lithgow GJ, Macchiarini F, Nelson JF, Sukoff Rizzo SJ, Slaga TJ, Stearns T, Wilkinson JE, Miller RA (2019) Acarbose improves health and lifespan in aging HET3 mice. Aging Cell. CrossRefPubMedPubMedCentralGoogle Scholar
  40. Hipkiss AR (2017) On the relationship between energy metabolism, proteostasis, aging and Parkinson’s disease: possible causative role of methylglyoxal and alleviative potential of carnosine. Aging Dis 8:334–345. CrossRefPubMedPubMedCentralGoogle Scholar
  41. Janssen F, de Beer J (2019) The timing of the transition from mortality compression to mortality delay in Europe, Japan and the United States. Genus 75:10. CrossRefGoogle Scholar
  42. Johnson ML (2000) Parameter correlations while curve fitting. Methods Enzymol 321:424–446CrossRefGoogle Scholar
  43. Jones OR, Scheuerlein A, Salguero-Gomez R, Camarda CG, Schaible R, Casper BB, Dahlgren JP, Ehrlen J, Garcia MB, Menges ES, Quintana-Ascencio PF, Caswell H, Baudisch A, Vaupel JW (2014) Diversity of ageing across the tree of life. Nature 505:169–173. CrossRefPubMedPubMedCentralGoogle Scholar
  44. Keller MA, Turchyn AV, Ralser M (2014) Non-enzymatic glycolysis and pentose phosphate pathway-like reactions in a plausible Archean ocean. Mol Syst Biol. CrossRefPubMedPubMedCentralGoogle Scholar
  45. Kirkwood TBL (2015) Deciphering death: a commentary on Gompertz (1825) ‘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’. Phil Trans R Soc B. CrossRefPubMedGoogle Scholar
  46. Kitadai N, Maruyama S (2018) Origins of building blocks of life: a review. Geosci Front 9:1117–1153. CrossRefGoogle Scholar
  47. Kosmachevskaya OV, Shumaev KB, Topunov AF (2015) Carbonyl stress in bacteria: causes and consequences. Biochemistry 80:1655–1671. CrossRefPubMedGoogle Scholar
  48. Kowald A, Kirkwood TBL (2015) Evolutionary significance of ageing in the wild. Exp Gerontol 71:89–94. CrossRefPubMedGoogle Scholar
  49. Kowald A, Kirkwood TB (2016) Can aging be programmed? A critical literature review. Aging Cell 15:986–998. CrossRefPubMedPubMedCentralGoogle Scholar
  50. Lemaître J-F, Berger V, Bonenfant C, Douhard M, Gamelon M, Plard F, Gaillard J-M (2015) Early-late life trade-offs and the evolution of ageing in the wild. Proc Roy Soc B. CrossRefGoogle Scholar
  51. Lenart P, Kuruczova D, Joshi PK, Bienertova Vasku J (2018) Rethinking mortality rates in men and women: do men age faster? bioRxiv. CrossRefGoogle Scholar
  52. Lerma-Ortiz C, Jeffryes James G, Cooper Arthur JL, Niehaus Thomas D, Thamm Antje MK, Frelin O, Aunins T, Fiehn O, de Crécy-Lagard V, Henry Christopher S, Hanson Andrew D (2016) ‘Nothing of chemistry disappears in biology’: the Top 30 damage-prone endogenous metabolites. Biochem Soc Trans 44:961–971. CrossRefPubMedGoogle Scholar
  53. Li T, Anderson JJ (2015) The Strehler–Mildvan correlation from the perspective of a two-process vitality model. Popul Stud 69:91–104. CrossRefGoogle Scholar
  54. Limpert E, Stahel WA (2017) The log-normal distribution. Significance 14:8–9. CrossRefGoogle Scholar
  55. Linster CL, Van Schaftingen E, Hanson AD (2013) Metabolite damage and its repair or pre-emption. Nat Chem Biol 9:72–80CrossRefGoogle Scholar
  56. Madeo F, Carmona-Gutierrez D, Hofer SJ, Kroemer G (2019) Caloric restriction mimetics against age-associated disease: targets, mechanisms, and therapeutic potential. Cell Metab 29:592–610. CrossRefPubMedGoogle Scholar
  57. Makeham WM (1860) On the law of mortality and the construction of annuity tables. J Inst Actuar 8:301–310CrossRefGoogle Scholar
  58. Markov AV, Naimark EB, Yakovleva EU (2016) Temporal scaling of age-dependent mortality: dynamics of aging in Caenorhabditis elegans is easy to speed up or slow down, but its overall trajectory is stable. Biochemistry 81:906–911. CrossRefPubMedGoogle Scholar
  59. Mitchell SJ, Madrigal-Matute J, Scheibye-Knudsen M, Fang E, Aon M, González-Reyes JA, Cortassa S, Kaushik S, Gonzalez-Freire M, Patel B, Wahl D, Ali A, Calvo-Rubio M, Burón MI, Guiterrez V, Ward TM, Palacios HH, Cai H, Frederick DW, Hine C, Broeskamp F, Habering L, Dawson J, Beasley TM, Wan J, Ikeno Y, Hubbard G, Becker KG, Zhang Y, Bohr VA, Longo DL, Navas P, Ferrucci L, Sinclair DA, Cohen P, Egan JM, Mitchell JR, Baur JA, Allison DB, Anson RM, Villalba JM, Madeo F, Cuervo AM, Pearson KJ, Ingram DK, Bernier M, de Cabo R (2016) Effects of sex, strain, and energy intake on hallmarks of aging in mice. Cell Metab 23:1093–1112. CrossRefPubMedPubMedCentralGoogle Scholar
  60. Muchowska KB, Varma SJ, Moran J (2019) Synthesis and breakdown of universal metabolic precursors promoted by iron. Nature 569:104–107. CrossRefPubMedGoogle Scholar
  61. Németh L, Missov TI (2018) Adequate life-expectancy reconstruction for adult human mortality data. PLoS ONE 13:e0198485. CrossRefPubMedPubMedCentralGoogle Scholar
  62. Pletcher SD, Khazaeli AA, Curtsinger JW (2000) Why do life spans differ? Partitioning mean longevity differences in terms of age-specific mortality parameters. J Gerontol Ser A 55:B381–B389CrossRefGoogle Scholar
  63. Rabbani N, Xue M, Thornalley Paul J (2016) Methylglyoxal-induced dicarbonyl stress in aging and disease: first steps towards glyoxalase 1-based treatments. Clin Sci 130:1677–1696. CrossRefPubMedGoogle Scholar
  64. Ralser M (2018) An appeal to magic? The discovery of a non-enzymatic metabolism and its role in the origins of life. Biochem J 475:2577–2592. CrossRefPubMedPubMedCentralGoogle Scholar
  65. Redman LM, Smith SR, Burton JH, Martin CK, Il’yasova D, Ravussin E (2018) Metabolic slowing and reduced oxidative damage with sustained caloric restriction support the rate of living and oxidative damage theories of aging. Cell Metab 27:805–815. CrossRefPubMedPubMedCentralGoogle Scholar
  66. Riggs JE, Hobbs GR (1998) Nonrandom sequence of slope-intercept estimates in longitudinal gompertzian analysis suggests biological relevance. Mech Ageing Dev 100:269–275CrossRefGoogle Scholar
  67. Royston P, Parmar MKB, Altman DG (2008) Visualizing length of survival in time-to-event studies: a complement to Kaplan–Meier plots. J Natl Cancer Inst 100:92–97. CrossRefPubMedGoogle Scholar
  68. Sasaki T, Kondo O (2016) An informative prior probability distribution of the gompertz parameters for bayesian approaches in paleodemography. Am J Phys Anthropol 159:523–533. CrossRefPubMedGoogle Scholar
  69. Shen J, Landis GN, Tower J (2017) Multiple metazoan life-span interventions exhibit a sex-specific Strehler–Mildvan inverse relationship between initial mortality rate and age-dependent mortality rate acceleration. J Gerontol Ser A 72:44–53. CrossRefGoogle Scholar
  70. Shilovsky GA, Putyatina TS, Lysenkov SN, Ashapkin VV, Luchkina OS, Markov AV, Skulachev VP (2016) Is it possible to prove the existence of an aging program by quantitative analysis of mortality dynamics? Biochemistry 81:1461–1476. CrossRefPubMedGoogle Scholar
  71. Shilovsky GA, Putyatina TS, Ashapkin VV, Luchkina OS, Markov AV (2017) Coefficient of variation of lifespan across the tree of life: is it a signature of programmed aging? Biochemistry 82:1480–1492. CrossRefPubMedGoogle Scholar
  72. Simons MJ, Koch W, Verhulst S (2013) Dietary restriction of rodents decreases aging rate without affecting initial mortality rate—a meta-analysis. Aging Cell 12:410–414. CrossRefPubMedGoogle Scholar
  73. Strehler BL (2000) Understanding aging. Methods Mol Med 38:1–19. CrossRefPubMedGoogle Scholar
  74. Strehler BL, Mildvan AS (1960) General theory of mortality and aging. Science 132:14–21CrossRefGoogle Scholar
  75. Strong R, Miller RA, Astle CM, Baur JA, de Cabo R, Fernandez E, Guo W, Javors M, Kirkland JL, Nelson JF, Sinclair DA, Teter B, Williams D, Zaveri N, Nadon NL, Harrison DE (2013) Evaluation of resveratrol, green tea extract, curcumin, oxaloacetic acid, and medium-chain triglyceride oil on life span of genetically heterogeneous mice. J Gerontol Ser A 68:6–16. CrossRefGoogle Scholar
  76. Stroustrup N (2018) Measuring and modeling interventions in aging. Curr Opin Cell Biol 55:129–138. CrossRefPubMedPubMedCentralGoogle Scholar
  77. Tarkhov AE, Menshikov LI, Fedichev PO (2017) Strehler–Mildvan correlation is a degenerate manifold of Gompertz fit. J Theor Biol 416:180–189. CrossRefPubMedGoogle Scholar
  78. Turturro A, Witt WW, Lewis S, Hass BS, Lipman RD, Hart RW (1999) Growth curves and survival characteristics of the animals used in the biomarkers of aging program. J Gerontol: Ser A 54:B492–B501. CrossRefGoogle Scholar
  79. Walker RF (2017) On the cause and mechanism of phenoptosis. Biochemistry 82:1462–1479. CrossRefPubMedGoogle Scholar
  80. Weiss KM, Buchanan AV (2012) Is life law-like? Genetics 188:761–771. CrossRefGoogle Scholar
  81. Wilmoth JR, Horiuchi S (1999) Rectangularization revisited: variability of age at death within human populations. Demography 36:475–495CrossRefGoogle Scholar
  82. Wrycza TF, Missov TI, Baudisch A (2015) Quantifying the shape of aging. PLoS ONE 10:e0119163. CrossRefPubMedPubMedCentralGoogle Scholar
  83. Yang Y, Santos AL, Xu L, Lotton C, Taddei F, Lindner AB (2018) Temporal scaling of ageing as an adaptive strategy of Escherichia coli. bioRxiv. CrossRefGoogle Scholar
  84. Yashin AI, Begun AS, Boiko SI, Ukraintseva SV, Oeppen J (2001) The new trends in survival improvement require a revision of traditional gerontological concepts. Exp Gerontol 37:157–167CrossRefGoogle Scholar
  85. Yen K, Steinsaltz D, Mobbs CV (2008) Validated analysis of mortality rates demonstrates distinct genetic mechanisms that influence lifespan. Exp Gerontol 43:1044–1051. CrossRefPubMedGoogle Scholar
  86. Zheng H, Yang Y, Land K (2012) Heterogeneity in the Strehler–Mildvan general theory of mortality and aging. Demography 48:267–290. CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Carcinogenesis and OncogerontologyN.N. Petrov National Medical Research Center of OncologySaint PetersburgRussia

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