Article

ISSN 0006-2979, Biochemistry (Moscow), 2024, Vol. 89, No. 2, pp. 371-376 © Pleiades Publishing, Ltd., 2024.

371

DISCUSSION

Calculating Aging: Analysis of Survival Curves

in the Norm and Pathology, Fluctuations in Mortality

Dynamics, Characteristics of Lifespan Distribution,

and Indicators of Lifespan Variation

Gregory A. Shilovsky

1,2,3

Belozersky Institute of Physico-Chemical Biology, Lomonosov Moscow State University, 119991 Moscow, Russia

Faculty of Biology, Lomonosov Moscow State University, 119234 Moscow, Russia

Institute for Information Transmission Problems, Russian Academy of Sciences, 127051 Moscow, Russia

e-mail: gregory_sh@list.ru; grgerontol@gmail.com

Received November 24, 2023

Revised November 24, 2023

Accepted December 29, 2023

Abstract— The article describes the history of studies of survival data carried out at the Research Institute of

Physico-Chemical Biology under the leadership of Academician V. P.  Skulachev from 1970s until present, with spe-

cial emphasis on the last decade. The use of accelerated failure time (AFT) model and analysis of coefficient of

variation of lifespan (CV

) in addition to the Gompertz methods of analysis, allows to assess survival curves for

the presence of temporal scaling (i.e., manifestation of accelerated aging), without changing the shape of survival

curve with the same coefficient of variation. A modification of the AFT model that uses temporal scaling as the null

hypothesis made it possible to distinguish between the quantitative and qualitative differences in the dynamics

of aging. It was also shown that it is possible to compare the data on the survival of species characterized by the

survival curves of the original shape (i.e., “flat” curves without a pronounced increase in the probability of death

with age typical of slowly aging species), when considering the distribution of lifespan as a statistical random

variable and comparing parameters of such distribution. Thus, it was demonstrated that the higher impact of mor-

tality caused by external factors (background mortality) in addition to the age-dependent mortality, the higher the

disorder of mortality values and the greater its difference from the calculated value characteristic of developed

countries (15-20%). For comparison, CV

for the Paraguayan Ache Indians is 100% (57% if we exclude prepuberty

individuals as suggested by Jones et al.). According to Skulachev, the next step is considering mortality fluctuations

as a measure for the disorder of survival data. Visual evaluation of survival curves can already provide important

data for subsequent analysis. Thus, Sokolov and Severin [1] found that mutations have different effects on the

shape of survival curves. TypeI survival curves generally retains their standard convex rectangular shape, while

typeII curves demonstrate a sharp increase in the mortality which makes them similar to a concave exponential

curve with a stably high mortality rate. It is noteworthy that despite these differences, mutations in groups I and II

are of a similar nature. They are associated (i) with “DNA metabolism” (DNA repair, transcription, and replication);

(ii) protection against oxidative stress, associated with the activity of the transcription factor Nrf2, and (iii) regula-

tion of proliferation, and (or these categories may overlap). However, these different mutations appear to produce

the same result at the organismal level, namely, accelerated aging according to the Gompertz’s law. This might

be explained by the fact that all these mutations, each in its own unique way, either reduce the lifespan of cells

or accelerate their transition to the senescent state, which supports the concept of Skulachev on the existence of

multiple pathways of aging (chronic phenoptosis).

DOI: 10.1134/S0006297924020159

Keywords: mortality curves, aging, lifespan inequality, phenoptosis, acute phenoptosis, biorhythms, chronobiology

Abbreviations: AFT, accelerated failure time model; CV, coefficient of variation; CV

, coefficient of variation of lifespan.

SHILOVSKY372

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

INTRODUCTION

This issue of Biochemistry (Moscow) is dedicated

to the memory of the outstanding gerontologist and

biochemist V. P. Skulachev. It includes the article [1]

that mentions the very dialogue, in which it was sug-

gested that not only the value of species-specific lifes-

pan [2], but also its trajectory in a cohort (survival

curve), as well as comparison of survival curves, are

important for solving the problems in the studies of

bio logy of lifespan, an area of gerontology developed

at the Belozersky Research Institute of Physico-Chem-

ical Biology.

The biology of lifespan has been the earliest re-

search direction at the Belozersky Institute. It was initi-

ated at the Department of Bioenergetics in the 1970s [3].

In 1991, a monograph was published in two languages,

which has since become a classic textbook on this area

of gerontology [4]. According to the Gompertz’s law, an

increase in the probability of death with age (at least,

within a certain age interval) is described by the ex-

ponential dependence. Survival curves obey this law,

at least in mammals and classical laboratory animals,

such as nematodes and fruit flies (for more details,

see[5, 6]).

As years passed by, it has become clear that not all

problems could be solved by analyzing life tables using

the Gompertz law (or Gompertz–Makeham equation)

as a model for the increase in the risk of death with

age, as it was revealed, for example, in the discussion

about the limits of application of the Makeham term

and derivatives of the Gompertz law (e.g., the Strehler–

Mildvan correlation) [4-6]. Skulachev has become in-

terested in the assumptions that are now discussed in

the article [1]. Ten years ago, he created and headed a

small group dedicated to studying these topics to which

he contributed much of his attention (see[7-10]).

Sokolov and Severin [1] used the simplest meth-

od of preliminary research, namely, visual comparison

of survival curves, in ten lines of mutant mice with

progeria [11-17] and observed two types of survival

curves. TypeI curves were similar to those for control

mice, while type II curves had (or resembled) an in-

verted exponential shape. Although it was only a pre-

liminary analysis of survival data, it allowed to assess

the severity of impact of the studied mutations on the

survival curve of mutation carriers. In fact, normal

survival curves are poorly approximated by an invert-

ed exponential. This function [exp(-ct)] lacks inflection

points, but has a long tail and is noticeably differ-

ent from a straight line. In survival curves present-

ed in[1], the inflection point is visible even on a step

graph. This fact cannot be ignored when analyzing

survival curves. After all, the presence of inflection

point means the presence of long-livers, which are

quite noticeable and cannot be explained by the error

of measurements. For further data analysis, the Sku-

lachev group used the methods discussed in the next

section.

DATA ANALYSIS MODELS

The studies of Skulachev group. Coefficient of

vari ation (CV). One of the main milestones in the re-

search of lifespan was the article [18] published by

Skulachev, Gavrilova, Gavrilov, and Severin (who is

one of the authors of [1]). Using national population

survey data from the United States and 14 most de-

veloped countries, the authors have moved from an-

alyzing survival curves to studying the lifespan as a

randomly distributed variable. They proposed that not

only the lifespan itself and its time-dependent dynam-

ics, but also its relative variation (e.g., CV) are import-

ant. It has been shown that the relative variabilities

of the parameters of human development and aging

are similar. Thus, the relative variabilities of age at

which such an ontogenically controlled event as fe-

male puberty (menarche) occurs, age of the onset of

aging- associated changes (menopause), and lifespan

(age at death) are approximately the same, and their

CVs fluctuate around 15-20% [18]. Later, analysis of data

on the survival of Japanese women from an open data-

base[19] produced similar results for the CV of lifes-

pan (CV

) [7]. Indeed, all studies of additional demo-

graphic indices, including those suggested by Baudish,

Vaupel, and Colchero [10, 20], began with the recogni-

tion of existence of methods capable of providing new

information on the trajectories of mortality. Another

significant step in the analysis of lifespan data was

the work of researchers from the Max Planck Institute

(Rostock, Germany/Odense, Denmark) published in

2014 in Nature [19]. Jones et al. [19] compared survival,

mortality, and fertility curves for a wide range of

systematic groups (23 vertebrate species, 10 inverte-

brates, 12 vascular plants, and one alga) from the age

of sexual maturation to the age at which only 5% of

individuals survived (terminal age).

Interestingly, Jones et al. in [19] failed to provide

in the summary table the survival curves (and cal-

culations of the studied parameters) for laboratory

mice, despite the abundance of such data. The authors

explain this by the fact that survival curves of mouse

strains are greatly distorted compared to other animals

(due to the load of mutations resulting from keeping

these animals under laboratory conditions). Skulachev

believed the work of Jones et al. to be a truly funda-

mental study that had summarized the most reliable

data on the survival curves of a wide variety of spe-

cies. He discussed the conclusions and ideas of this

work at gerontological seminars and in his articles

[10, 21] and even in his book “Life with No Aging” [22].

ANALYSIS OF SURVIVAL CURVES 373

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

However, Skulachev suggested that despite the repre-

sentativeness of the studied cohort, the conclusions

of the authors of [19] were not that unambiguous and

therefore, should be critically analyzed [7, 9]. Based on

the calculations, the survival, mortality, and fertility

curves published by the Institute of Demographic Re-

search [19] were divided into four large groups accord-

ing to the ratio between the mortality at the terminal

age and average mortality within the entire studied

period. GroupI included species with the smallest in-

crease in mortality with age (the ratio of the maximum

mortality to the average mortality within the studied

age interval), and groupIV– with the greatest one[7].

Thus, in our opinion, the signs of actuarial aging are

absent in plants and algae, some lower Metazoa (cni-

darians), and some vertebrates (e.g., amphibians and

reptiles). Beside mammals, species with a large number

of postmitotic cells, for example, insects, are most sus-

ceptible to actuarial aging (although the mortality rate

of other arthropods, such as crabs, may not increase

with age). The limitations of the method [19] are most

clearly seen in birds. Thus, contrary to the findings by

Jones et al. [19], large birds age, but their biological

aging might manifest itself at a late age to which less

than 5% of individuals survive in nature. The aging of

small birds, which have many enemies and high ex-

trinsic mortality, is supposed to be almost impossible

to detect in nature. Therefore, in the table, they would

be located next to slowly aging species. The literature

on biogerontology repeatedly mentions a situation

when mortality caused by external causes is so high

that it completely hides age-dependent component of

mortality (a case similar to typeII curves in[1]).

Semiparametric Cox proportional hazards

model and accelerated failure time model (AFT).

Thesurvival curves of mice with mutations that short-

en the lifespan differ (including the shape of the sur-

vival curve [pace and shape of aging]). They also look

different in mice with progeric mutations, which not

only reduce lifespan, but also lead to the development

of age-related pathologies characteristic of this species.

However, the influence of progeric mutations on the

shape of survival curves (shapes of aging) is different

in different cases. Since visual assessment of shapes

of survival curves “by eye” does not always lead to

accurate results (which is why at one time they aban-

doned eye tests, for example, the normality of distribu-

tion), special demographic methods began to be used

in biogerontology. The semiparametric Cox propor-

tional hazards model [23] is widely used in medicine

and epidemiology, but rarely in the studies of aging.

Itestimates the rate of age-dependent mortality under

different conditions by analyzing the log-cumulative

hazard plot, i.e., dependence of the hazard ratio (loga-

rithm of the hazard function) on the logarithm of time,

which is useful in medical research (for example, for

estimating mortality risks at various time points after

surgery), but not for the analysis of survival curves

[23, 24]. The AFT model compares entire survival

curves instead of immediate probabilities of death at

particular time points. In this case, survival curves

can be transformed into each other by changing the

variables: S

(λt)=S

(t), where λis a dimensionless co-

efficient that determines the magnitude of the effect,

which is the same for any quantile. The biological

meaning of this formula is that biological clock runs at

a different speed for the two compared groups of indi-

viduals. In this case, changes in the risk of death with

age remain qualitatively the same. Graphically, surviv-

al curves S

and S

look stretched/compressed relative

to each other along the time axis. At the same time, the

values of the mean, median, and maximum lifespan

also change proportionally, and the above-mentioned

lifespan remains [almost] constant [25]. In this case,

temporal scaling takes place, when various factors

that either increase or decrease the lifespan (oxidative

stress, changes in temperature or diet, mutations) do

not change the shape of the survival curve, but only

stretch or compress it along the time axis. According

to the authors of [26], this indicates an existence of

the aging program. In [25], raw data on the effects of

various genetic changes that increase the lifespan in

mice were analyzed using both models in order to se-

lect the model that most closely matched the experi-

mental data. According to the AFT model, homozygous

mutations Prop1 (λ=1.48) and Pit1 (λ=1.39) had the

greatest effect on the mouse lifespan. The homozygous

mutation PappA and heterozygous mutations Clk1

+/–

and Irs2

+/–

had a somewhat weaker effect in males

(1.20 <  λ < 1.40). The other genetic changes caused a sim-

ilar and rather weak effect (1.03 < λ  <  1.20). In the case

of Irs2

+/–

mutation, the effect was stronger in males

than in females, and in the case of Clk1

+/–

, it was dif-

ferent in two mouse strains.

In 2016, an automated system was developed that

allowed to accurately record the moment of death in

Caenorhabditis elegans nematodes [26]. It has been

shown that in nematodes, gene knockouts change the

time- scaling coefficient of lifespan (λ) 2 to 3-fold, perox-

ide– up to 17-fold, and temperature– up to 7-fold[26].

Markov et al. [27] analyzed the survival curves of wild-

type Drosophila flies, long-lived flies selected for slowed

down aging (and, consequently, increased longevity),

and short-lived flies that were cultivated on unfavor-

able food (selected for early reproduction). The authors

assumed that with a small difference in the lifespan

of the compared Drosophila groups, there would be a

temporal scaling of survival curves (the Markov’s rule).

Skulachev and his colleagues [8] compared the data on

the survival of Drosophila kept on normal and unfa-

vorable (starch and salt, respectively) media [27] using

the method proposed in [26]. Briefly, the lifespan data

SHILOVSKY374

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

were logarithmized and then normalized to a com-

mon time scale by dividing by the group mean lifes-

pan. Themean value for the data transformed in this

way was 1 in all groups. The deviations from the mean

in different groups were compared in pairs using the

Kolmogorov–Smirnov test with the Bonferroni correc-

tion [8] used for the multiple pairwise comparisons

to reduce the likelihood of false-positive results [28].

The Markov’s rule was shown to be valid for small dif-

ferences in CV

(~10%) and non-overlapping survival

curves and therefore, can be used for primary analysis

of survival data [8].

THEORIES AND MECHANISMS OF AGING.

KEY PROTEINS AND PATHWAYS

REGULATING AGING

Aging is associated with degeneration of tissues in

organs (e.g., atrophy of myofibrils in skeletal muscles

and their replacement by cells of the adipose and con-

nective tissues or an overall increase in the proportion

of senescent cells). Based on this fact, the authors of

[1] suggested that the primary cause of aging is aging

of individual cells, and not, for example, age-related

changes in intercellular structures. Sokolov and Sev-

erin [1] found that although mutations have different

effects on the shape of survival curves, these muta-

tions are of a similar nature. They are associated with

(i)  DNA metabolism (DNA repair, transcription, and

replication); (ii)  antioxidant protection related to the

activity of the transcription factor Nrf2, and (iii) reg-

ulation of cell proliferation (although these categories

can overlap as well). However, such diverse mutations

appear to have a similar effect at the organismal level,

namely, accelerated aging according to the Gompertz’s

law. This might be explained by the fact that such

mutations, including mutations in the two genes cod-

ing for proteins responsible for the “true progerias”

[Hutchinson–Gilford progeria (Lmna

G608G

) and XPD (xe-

roderma pigmentosum D], each in their own unique

way, either reduce cell lifespan or accelerate cell tran-

sition to the senescent state [1]. This supports the con-

cept on the existence of the multiple aging pathways

(chronic phenoptosis) proposed by Skulachev [10].

Considering the differences in the effect of mu-

tations, we can assume that the effect of type  II mu-

tations on the lifespan is more detrimental. Once the

probability of death ceases to depend on age (and

lifespan decreases), then the viability decreases so

much that animals die without having time to age

more. This explains the fact cited in [1] that type II

curves (with reduced lifespan) can be relatively easi-

ly changed. For example, deletion of the p21-encoding

gene (Cdkn

–/–

) increases the lifespan in mice deficient

by the Terc1 telomerase, making the shape of the sur-

vival curve similar to an inverted exponential [29].

Transformation of a typeI curve into a typeII curve

requires additional damaging effect and not a protec-

tive one.

Models of cell aging. According to [1], there are

two types of experimental cell models of aging: repli-

cative and chronological. Although both models have

their limitations, they have been successfully used for

testing geroprotectors [30]. However, A. N. Khokhlov

[30], who was mentioned by the authors of [1], stip-

ulated that the theory of replicative aging (and exis-

tence of the Hayflick limit) does not explain aging of

an organism [31]. The statement that cell aging is the

main cause of body aging is also a debatable issue.

Theauthors of [1] refer to the Khokhlov’s theory of ag-

ing. However, if we take his famous article in Bioger-

ontology [32], it says something different: “Apparently,

the impairment of regulatory processes, realized at

the neurohumoral level, still plays the main role in

the mechanisms of aging of multicellular organisms,

not just the accumulation of macromolecular defects

in individual cells. It seems that the quality of the cells

themselves does not worsen with age as much as reli-

ability of the organism control over cells, organs, and

tissues, which leads to an increase in the probability of

death”. Similarly, Skulachev has repeatedly noted the

importance of the control of aging at the body level,

namely the action of biological master clock located in

the cells of the suprachiasmatic nucleus of the hypo-

thalamus.

Indeed, when studying the effect of mutations, it

cannot be excluded that genes knockout will not af-

fect other processes at the same time, therefore, such

studies should be focused on individual genes or pro-

teins that play an important role in a specific mech-

anism investigated in each specific study. Thus, for

the Khokhlov’s group, which studies DNA damage as a

primary cause of aging, such protein is PARP1 (a uni-

versal sensor of DNA damage) [33]. Skulachev, in turn,

considered the Nrf2/Keap1/ARE system as one of the

most important representatives of the anti-aging pro-

grams (vitauct) [34]. This system controls almost entire

cell antioxidant defense [35], in particular, expression

of genes responsible for cell protection from the oxi-

dative stress and mentioned in[1] (expression of such

genes is under control of Nrf2 due to the presence of

ARE in their promoters; also, p62 protein protects Nrf2

from proteasomal degradation with the participation

of ubiquitin ligase adapter Keap1).

In conclusion, I would like to support Severin[1]

in his suggestion to continue gerontological research

in the directions outlined by the Skulachev’s group.

In particular, it would be very valuable to resume

our weekly gerontological seminars for discussion

of current research studies and news in the field of

biogerontology. These seminars had attracted not only

ANALYSIS OF SURVIVAL CURVES 375

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

employees of the Institute of Physico-Chemical Biology

and the Faculty of Biology of the Lomonosov Moscow

State University, but also many scientists from other

institutes, which once again confirms the relevance of

the topics discussed at these meetings.

Acknowledgments. The author thanks A. V. Seliver-

stov for valuable advice during writing of the article.

Contributions. G. A.  Shilovsky developed the con-

cept and wrote the article.

Funding. This work was supported by ongoing in-

stitutional funding. No additional grants to carry out

or direct this particular research were obtained.

Ethics declarations. This work does not contain

any studies involving human and animal subjects.

The author of this work declares that he has no con-

flicts of interest.

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