Journal of Gerontology: BIOLOGICAL SCIENCES
2003, Vol. 58A, No. 6, 484–494
Copyright 2003 by The Gerontological Society of America
What Fecundity Patterns Indicate About Aging and
Longevity: Insights From Drosophila Studies
Vassily N. Novoseltsev,1,2 Janna A. Novoseltseva,1 Sergei I. Boyko,2 and Anatoli I. Yashin2
1
Institute of Control Sciences, Moscow, Russia.
Max-Planck Institute for Demographic Research, Rostock, Germany.
2
The age pattern of fecundity is represented as a result of a superposition of two processes: the
genetic fecundity program encoded in the organism’s reproductive machinery and senescence of
the reproductive system. Accumulation of oxidative damage produces the energy decline, which
could potentially be used in reproduction. As a result, the age-declining process arises in the
reproductive machinery at a critical age. We show that this mechanism is common for different
species. It establishes a connection between the decline of organism vitality and reproductive
senescence. We suggest a parametric description of a fecundity pattern that allows for prediction
of reproductive longevity. We apply the approach to Drosophila studies to analyze the relation
between fecundity and survival. We show that fecundity patterns may predict a mean life span in
Drosophila under specified environmental conditions.
R
EPRODUCTION is one of the most important processes in life history affecting the evolutionary success
of a genotype (1). That is why an analysis of fecundity
becomes an essential part of the studies focusing on evolution of life history traits in various species. In this article,
we address the question, Why is fecundity scheduling so
similar in disparate species? (Figure 1). We turn to Drosophila as a model organism to clarify the intrinsic mechanisms underlying such a similarity and to propose a unified
description of an age-related fecundity pattern.
Fecundity timing is studied in the most detail in Drosophila. It is well known that senescence results in a drop
in fecundity with advancing age (2–6). Maynard Smith (7)
suggested that a reduction in egg production increases
longevity in Drosophila. Lints and Lints (2) measured daily
fecundity patterns as well as life span and oxygen consumption in Drosophila females. Although they have found
no correlation between fecundity and life span, they hypothesized that ‘‘it is possible that more data or a more refined
technique would reveal the existence of such a correlation.’’
Since then, it has often been demonstrated that reproduction
is a costly activity having negative effects on longevity and
survival as well as on future fertility (8,9). It is also known
that a fecundity pattern measured in experimental studies
depends on environmental and experimental procedures.
Alteration in fecundity-related traits in Drosophila were found
in an artificial selection on postponed senescence (10–12), late
fecundity (13–15), reproduction at a ‘‘young’’ or ‘‘old’’ age
(6,16), longevity (17), high or low adult mortality (18,19), and
body size (20). Female fecundity patterns can be remarkably
modified by experimental manipulations of exposure to males
(9,21), nutrition (11,22), and ambient temperature (20).
Müller and coworkers (26) successfully modeled the
declining part of the age-related fecundity in medflies Ceratitis capitata by an exponential function, which starts at
a certain age with peak egg laying, but they did not describe
484
the early pattern of fecundity. Pretzlaff and Arking (15)
noted an exponential decrease of fecundity at advanced ages
in Drosophila. Various mathematical approaches were used
to simulate optimal resource allocation and reproductive
patterns (27–31). But to the best of our knowledge, there
were only a few attempts to confront the results of mathematical modeling on the experimentally measured fecundity
pattern (18,30,31).
A few models were developed for human fecundity and
fecundability with special attention to behavioral, social,
and physiological factors (32,33). Natural fertility, particularly in traditional populations, was also studied (34,35).
Nonetheless, relatively little is known about the mechanisms
that form specific features of fecundity patterns in disparate
species and cause their alteration in varying environments.
Thus, we argue that a more exact and refined description
of age-related fecundity patterns in animals is needed for
both a deeper understanding of the intrinsic processes involved in shaping age-related fecundity in evolution and
artificial selection and for an adequate processing of experimental fecundity-related data.
We had hypothesized earlier (36) that the reproductive
capacity of a female Drosophila may be characterized by
a genotype-specific maximal rate of egg production, which
is attainable in an optimal environment and is continuously
supported until the intrinsic deterioration causes its decrease. We have called this maximum a ‘‘fecundity plateau.’’
To enhance this hypothesis, we analyze in the following
paragraphs the genetic and physiological mechanisms that
underlie time scheduling of fecundity in an organism.
METHOD
Onset of Reproduction and Early Fecundity
It is well known that after the onset of reproduction,
fecundity in Drosophila populations achieves its peak. We
FECUNDITY PATTERNS AND AGING
485
productive system. Thus, the homeostatic approach predicts
that, at older ages, the maximum power attainable in this
system is limited by the exponential function of age:
ð2Þ
PðxÞ ¼ Pmax expðx=sL Þ;
where sL is a time constant of a process of power decrease in
the reproductive machinery, and Pmax is measured as joules/
second or llO2/day. To model fecundity processes in flies,
we convert Pmax and P(x) to units of eggs/day by use of
a corresponding scale.
Thus, we may assume that the sL value reflects reproductive senescence of an organism and characterizes ‘‘late’’
fecundity in an individual.
Figure 1. Age-related fecundity patterns in various species (dotted lines). To
underlie similarity of the patterns, a uniform two-stage approximation was
applied as described in the article (continuous lines). A, Medfly, mean population fecundity, eggs/day [modified from (23)]; B, Lion [modified from (24)];
C, Baboon [modified from (24)]; D, Human female fertility in traditional Ache
population [modified from (25)]. The values of parameters of approximation are
presented in Table 1.
have made sure that this peak of egg production can be
maintained in optimal environments for a rather long period
and have noted this period as a fecundity plateau. The level
of the plateau was named a reproductive capacity, RC (36).
Thus, we hypothesize that, in an individual female, a genetically endowed maximum progeny production rate exists
and is encoded in the female’s reproductive machinery. We
will describe the early fecundity as an exponential increase at
the onset of reproduction arriving at the steady-state level, RC:
ð1Þ
FG ðxÞ ¼ RC ½1 expfðx X0 Þ=s0 g;
where x is age, FG(x) is a number of eggs produced per day
(or other specified time in disparate species) prescribed by
the genetic program of an organism for a given environment, XO is the age of the onset of reproduction (x XO),
and sO is an onset time constant. As sO is usually small, the
process governed by Equation 1 rapidly increases, and at
ages (XO þ 3 sO) arrives very close to the RC value, thus
achieving a plateau level in fecundity.
We will use this approximation to describe the ‘‘early’’
fecundity in female Drosophila flies.
Reproductive Senescence and Late Fecundity
Our homeostatic model of aging (36,37) predicts
a roughly exponential deceleration of senescence at
advanced ages in an organism (see Appendix A for details).
Following Kirkwood’s conjecture that ‘‘The cells of the
reproductive system are subject to exactly the same kinds of
biochemical stresses (oxidative damage, mutation, etc.) that
cause damage in other cells,’’ (38) we assume that an exponential decrease of the energetic capacity exists in the re-
Critical Age: Onset of Reproductive Senescence
The last assumption in our hypothesizing is that a transition
from an early-fecundity mode to a late-fecundity one is a direct
result of senescence of the reproductive system. At early ages,
the power resources of the reproductive system exceed the
limits necessary for realizing the genotype-specific fecundity
program in a specified environment: P(x) . FG(x). The extra
energy, P(x)FG(x), is used to fill the ovaries with eggs and to
mature them (39–42). At these ages, the genotype-specified
reproductive program is successfully executed under given
environmental conditions. Equation 1 is held and the
fecundity is kept at the RC plateau level.
With the passing time, oxidative damage accumulates in
an organism’s tissues. As a result, P(x) decreases, and at
a certain age, T, P(x) becomes smaller than is required to
keep the endowed fecundity level, RC. This T point is a critical one, defining the fecundity switch from the early mode
to the late one. The energy resource now limits egg production and reproductive senescence becomes manifested.
Thus, the realized pattern F(x) at each age is a minimum of
two values, the genetic prescription and the energetic capacity attainable at this age:
ð3Þ
FðxÞ ¼ min½FG ðxÞ; PðxÞ;
as shown in Figure 2. Observations on Drosophila females
show that the length of the steady state plateau, before the
onset of reproductive senescence at age T, may be up to 30
days.
It is worth noting that the proposed approximation may be
applied both to individuals and to populations. It allows for
estimation of the mean population reproductive potential,
Z ‘
FðxÞ dx ¼ RC ½h þ sL s0 ;
ð4Þ
RP ¼
0
where h ¼ T X0 is a length of a plateau of a fecundity
pattern. This approximate formula describes the hypothetical reproductive success of a fly with unlimited life span,
and it is very exact for h 3s0.
RESULTS
Analysis of Fecundity Patterns
Fecundity patterns in Drosophila.—To shed light on the
characteristics of fecundity patterns common to various
strains of Drosophila, we discuss several special cases. We
NOVOSELTSEV ET AL.
486
deviations of the approximations are comparatively small,
thus demonstrating a satisfactory robustness of the technique,
which gives a high reliability in modeling and parameterization. Graphical presentation of the particular age trajectories and a sensitivity analysis (see Appendix B) supports this
thesis. We found a statistically significant correlation between the RC and T values in Figures 4 through 7, which
confirmed the general idea that the elevated oxygen
consumption linked to a higher progeny production correlates with an early onset of reproductive senescence (Figure
3). No significant correlation was found between other
fecundity-related parameters. In particular, the correlation
coefficient between the critical age T and sL is r ¼.024.
Figure 2. Mechanism underlying fecundity scheduling in a female fly. The
realized pattern F(x), shown with a bold line bordering the shadowed area, at
each age is a minimum of two time functions. One of them, FG(x), describes the
genetic program of egg production. It begins with a fast transient response
ending at a steady-state plateau, RC [reproductive capacity] ¼ 70 eggs/day. The
other time function, P(x), represents the decreasing maximum power, which can
be realized in the reproductive system. At critical age T ¼ 23 days, this power
drops below the level needed for full-strength work of the reproductive
machinery. Maintenance of the endowed rate of egg production becomes
impossible, thus manifesting the onset of reproductive senescence. The
parameters of the example correspond to Drosophila experiments.
assumed that in each particular case, the population heterogeneity did not mask an exponential-type fecundity decrease
observed at the individual level. The findings of Müller and
colleagues (26) on the similarity of quantile and mean population fecundity patterns in medflies confirmed this assumption.
We applied the approximation of Equations 1, 2, and 3
to the particular mean population fecundity patterns in
Drosophila melanogaster strains (17,18,20,22,43). To convert graphical data from figures in the cited papers into
numerical form, hard copy images were scanned and digitized. This procedure allowed us to obtain numerical values
of points constituting the curves on the flies’ fecundity. For
each particular data set, we then applied the standard least
square estimating procedure (44) to evaluate the reproductive capacity RC, critical age T, and time constant sL. Unfortunately, experimental data were not readily available for
a proper evaluation of the very initial part of the pattern.
Therefore, we chose one of the three values, 0.5, 1.0, or 2.0
days, for the value of the time constant sO. We found that
such a restriction did not noticeably affect the results of
estimation of other parameters.
The estimated parameters for the 17 fecundity curves related to different Drosophila melanogaster strains are presented in Table 2. One can see from Table 2 that the standard
Size-related variability of fecundity in Drosophila.—To
clarify how the proposed technique can be applied to particular studies, we analyze a number of case studies based
on published results. We want to stress that it is not our aim
to criticize these studies or improve the published findings.
The following analysis is intended to illustrate how the
published results might be presented in more strict terms.
A positive phenotypic correlation between the body size
of adult female flies and their fecundity is well known
(43,46). In particular, a strong positive correlation (r ¼ .603,
p , .001) between wing length and lifetime progeny production has been reported in artificially selected strains (17).
Nonetheless, a number of questions raised related to the
selection procedures employed and a possible genetic correlation remain unanswered (47), and ‘‘these effects await
further study’’ (17). We believe that the technique proposed
above makes it possible to clear up some of these effects
(Figure 4).
McCabe and Partridge observed that ‘‘the females from
large selection lines were relatively fitter at the colder
temperature.’’ Our analysis of the data is in agreement with
this result. To demonstrate that the evaluations are reliable,
we perform sensitivity analyses (Appendix B).
Applying the same approach to the data of Hillesheim and
Stearns (43) yields similar results. These authors studied the
effects of artificial selection on life history traits in populations of large and small Drosophila melanogaster using
regimens of rich and poor larval nutrition (Figure 5).
They found that ‘‘Larger flies laid more eggs early in life
and lived shorter lives than smaller flies, which not only
lived longer but also laid more eggs later in life.’’ Our
technique presents this fecundity-related finding in more
detail. Namely, the large flies in both cases have a higher
reproductive capacity relative to the small flies. The late
fecundity in the small flies is higher than in large flies.
Table 1. The Estimated Parameters for Different Species
Species
Medfly
Lion
Baboon
Human
Early Fecundity
Time Constant
sO
1.5
1.0
1.0
3.0
(d)
(y)
(y)
(y)
Genetically Programmed
Fecundity Rate
RC
32.15 (eggs/d)
0.861 (live offspring/y)
0.618 (live offspring/y)
0.286 (probability of live births)
Notes: RC ¼ reproductive capacity; SD ¼ standard deviation; y ¼ years; d ¼ days.
Reproductive
Plateau Length
h
9.0
8.6
14.8
29.0
(d)
(y)
(y)
(y)
Late Fecundity
Time Constant
sL
17.57
3.84
2.86
2.16
(d)
(y)
(y)
(y)
Overall
Approximation Error
SD
1.5465
0.1112
0.0829
0.0531
FECUNDITY PATTERNS AND AGING
487
Table 2. The Estimated Parameters for Different Drosophila Strains
Source
McCabe and Partridge
(20) 188C
McCabe and Partridge
(20) 258C
Hillesheim and
and Stearns (43)
Stearns et al. (18)
Partridge et al. (22)
Zwaan et al. (17)
Strain
sO
Early Fecundity
RC
Critical Age
T
Late Fecundity
sL
Standard Deviationa
SD
Large
Control
Small
Large
Control
Small
Large Ab
Large B
Small A
Small B
HAM
LAM
Group Ac
Group B
Slow
Control
Fast
1.0
1.0
1.0
0.5
0.5
0.5
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
2.0
2.0
32.91
32.87
29.76
52.77
50.37
44.43
119.8
108.5
69.88
62.01
34.43
36.40
77.82
81.62
11.96
10.03
10.36
14.6
10.7
9.6
4.8
4.7
4.1
3.1
3.0
9.6
9.5
19.0
22.1
15.9
12.3
17.5
19.5
19.5
13.26
12.72
15.39
7.62
6.31
9.12
14.50
15.23
22.67
24.63
23.85
9.55
13.67
19.59
5.50
10.93
8.38
2.73
2.50
2.29
2.31
1.53
2.76
5.11
2.97
3.42
1.61
2.94
3.74
6.68
7.79
2.38
2.27
1.78
Notes: RC ¼ reproductive capacity; HAM ¼ high adult mortality; LAM ¼ low adult mortality.
Standard deviations (SD) are calculated for the total curves.
b
Selection on rich (A) or on poor (B) food.
c
Females kept continuously exposed (A) or intermittently exposed (B) to males.
a
Effect of exposure to males.—Partridge and colleagues
(22) presented the results of how exposure to males affects
fecundity in Drosophila melanogaster females. They compared the costs of exposure to males in 2 cases, and measured age-related fecundity patterns for females that were
kept continuously exposed (Group A) or intermittently exposed (Group B) to males (Figure 6). They found that the
egg production rates of the 2 groups ‘‘show no significant
differences.’’
The results of our analysis are in wide agreement with this
finding. Indeed, the RC levels in the groups are very close to
each other. Nonetheless, the late fecundity in Group B females proved to be essentially higher relative to Group A,
thus providing Group B with a higher reproductive
potential.
Effect of artificial selection for developmental time.—
Zwaan and colleagues (17) measured the time course of
progeny production in female Drosophila melanogaster in
lines selected for both fast and slow larval development.
They noted ‘‘the increased early reproduction of the slow
Figure 3. Correlation between the observed critical ages T and reproductive
capacities RC for the age-related patterns depicted in Table 3 (r ¼ –.635, coefficient of determination D ¼ 0.403).
lines’’ but stressed that ‘‘the reproduction patterns can be
changed without concomitant changes in adult longevity.’’
In the slow line, indeed, the RC value is higher as related
to the control and fast lines (Figure 7). It is interesting that
both slow and fast selected lines have a lower late fecundity
in relation to the control. The absence of changes in adult
longevity may be related to the unchanged RP in the lines
(263, 285, and 268 eggs in the slow, control, and fast strains,
respectively). The differences between these figures are
smaller than the 6% observed for RP in the sensitivity
experiment (see Appendix B).
Linkage of Longevity and Fecundity Patterns
in Drosophila
‘‘Late survival sacrificed for reproduction’’ is one of most
commonplace of modern evolutionary theories of aging
(48). This assumes that a deeper understanding of
mechanisms that underlie fecundity scheduling gives us
a chance to use this knowledge for outlining the linkage
between reproduction and longevity. The central idea of
such a prediction may be formulated as follows. The
genotype-specific energetic resource of the reproductive
system reflects the vitality of the organism as a whole, at
least to some extent. Thus, we will evaluate the wholeorganism vitality level and the level of reproductive resources for a particular Drosophila strain from the available
experimental data, and then estimate the correlation between
the two values. We will describe the vitality of an organism
by its close correlate, the homeostatic capacity, thus representing the inherited vitality of a particular Drosophila
female by the value S0, its initial homeostatic capacity (see
Appendix A for details).
A strong correlation between the initial homeostatic
capacity and the level of reproductive resources may be used
for a prediction of a fly’s longevity based on specific
characteristics of its fecundity pattern.
488
NOVOSELTSEV ET AL.
Figure 4. Age-related fecundity patterns in female Drosophila melanogaster
selected for body size at 2 environmental temperatures, 18 and 258C [from
McCabe and Partridge (20), modified]. Left panel: large, control, and small
lines (from above) at 188C. Points are for experimental data and continuous lines
for approximation. McCabe and Partridge found that ‘‘at both experimental
temperatures, especially the lower one, the small-line females rescheduled they
progeny production to later ages.’’ Our analysis confirms this finding and refines
it. The strain-related early fecundity RC [reproductive capacity] decreases in the
small lines: in the control at 188C, 29.76 eggs/day (SD [standard deviation] ¼
2.29) versus 32.87 (SD ¼ 2.50); in the control at 258C, 44.43 (SD ¼ 2.76) versus
50.37 (SD ¼ 1.53). The late fecundity, sL, increases in the small-line females
related to the control (15.39 vs.12.72 days at 188C and 9.12 vs. 6.31 at 258C).
The result is the increased reproductive potential, RP ¼ 884 eggs in the large
line, 737 in control, and 714 in the small line at 188C. Sensitivity analysis
confirms the findings (see Appendix B).
Correlation between vitality and reproductive power.—
To find out the correlation between vitality and reproductive
power, we simulated the life histories for the 6 experiments
of McCabe and Partridge (20). In each case, we assumed
that the overall oxygen consumption rate was a sum of the
age-independent maintenance component Wm and the agedependent reproduction-related component Wr [simulations
are not presented here; for details see (36)]. Wr was taken
proportionally to age-related fecundity. In these simulations,
we used survival data on the control strain at 188C as a base
point (see Table 3). Given the experimental longevity for the
control strain at this temperature, LS50 ¼ 34.5, we found that
the oxidative vulnerability was b0 ¼ 3.51 104 [1/ll O2].
We kept this value for the other 5 strains. Then, for each of
the other patterns, we adjusted the maintenance component
Wm relative to the size using as a base the wing-area
proportions (20). For the high temperature (258C) we applied a temperature-related adjustment, using the coefficient
2.0 calculated from Miquel data [see, e.g., (49), Figure 6].
Given the longevity data specific for each simulated strain,
we adjusted the corresponding S0 values. Thus, we found
Figure 5. The effects of artificial selection for body weight in Drosophila
melanogaster on rich (left panel) and poor (right panel) larval food. Modified
from Hillesheim and Stearns (43). Points are for experimental data and
continuous lines for approximation. Our approximation allows for the refining of
the original conclusion that ‘‘Larger flies laid more eggs early in life and lived
shorter lives than smaller flies, which not only lived longer but also laid more
eggs later in life.’’ The large flies in the both cases have higher plateau level: RC
[reproductive capacity] ¼ 119.8 and 108.5 eggs/day (SD ¼ [standard deviation]
5.11 and 2.97, respectively) relative to the small flies: RC ¼ 69.88 and 62.01
eggs/day (SD ¼ 3.42 and 1.61). The time constants of the late fecundity are
14.50 and 15.23 days in the large flies versus 22.67 and 24.63 days in the small
ones. Notice that the scales in the figures are different in both cases.
the curve S(x), which was a quasi-exponent with a time coefficient b0 (Wm þ Wr). This curve is shown in Figure 8 for
a control population at 188C.
We then proceeded by evaluating the initial energetic
capacity of the reproductive system, Pmax. We assumed that
the exponential decrease of the reproductive power in
Drosophila at advanced ages could be uniformly expanded
to the early ages as well. This assumption was widely equivalent to taking an age-independent rate of senescence of
Figure 6. Age-related fecundity patterns for females Drosophila melanogaster
kept continuously exposed (left) or intermittently exposed (right) to males
[from (22), with modifications]. Our analysis allows for the refining of the
original finding that the patterns ‘‘show no significant differences.’’ Early
fecundity in both groups are very close to one another: RC [reproductive
capacity] levels are 77.87 eggs/day (SD [standard deviation] ¼ 6.68) and 81.62
(SD ¼ 7.79). The expected late fecundity in the females intermittently exposed to
males is a little higher (time constant of the tail is 15.9 vs. 12.3 days).
FECUNDITY PATTERNS AND AGING
Figure 7. Fecundity patterns in lines Drosophila melanogaster selected for
fast and slow larval development [modified from Zwaan and colleagues (17)].
Slow (left), control (middle), and fast line (right) are presented. Those authors
found ‘‘the increased early reproduction of the slow lines.’’ In the slow line, RC
[reproductive capacity] ¼ 11.96 eggs/day (SD [standard deviation] ¼ 2.38)
versus RC ¼ 10.03 and 10.36 (SD ¼ 2.27 and 1.78, respectively) in the control
and fast lines. In casees where SD values did not confine the result, Zwaan and
coworkers were correct, as sO in the slow strain (sO ¼ 1.0) was smaller than in
the control and fast strain (in both cases, sO ¼ 2.0).
the reproductive system. Under it, we could restore the reproductive power, Pmax, in a female fly at her emergence.
Namely, we used Equation 1 for a backward calculation of
Pmax from the experimentally observed fecundity pattern at
the critical age, T.
We have
Pmax ¼ PðTÞ expðT=sL Þ:
ð5Þ
Given the parameters of the fecundity patterns presented
in Table 2, we directly calculated the related Pmax values for
the 6 strains. The resulting values are also presented in
Table 3, and a particular example is given in Figure 8.
In each particular case, the observed age pattern of reproductive senescence drastically differed from the age pattern
of senescence for the entire organism. However, a strong
correlation existed between the initial homeostatic capacity
and the initial power capacity of the reproductive system.
In fact, the correlation coefficient r ¼ .973 for the 188C
experiments, and r ¼ .512 for the 258C experiments, was
calculated. For the whole data set, r ¼ .796 (see Figure 9).
Prediction of longevity using fecundity patterns.—We
have hypothesized elsewhere (36) that a deep intrinsic
linkage exists between the power resource invested in the
reproductive machinery and in the maintaining mechanisms.
Table 3. The Estimated Parameters S0 and Pmax for Different
Drosophila Strains
Straina
LS50
(days)
Wm
(lO2/day)
S0
(lO2/day mmHg)
Pmax
(eggs/day)
188C
L
C
S
42.0
34.5
31.0
75.86
70.00b
64.14
2.59
1.72c
1.38
98.96
76.22
55.53
258C
L
C
S
18.0
17.0
15.5
3.20
2.21
1.95
99.09
106.1
69.65
a
117.2
136.7
108.7
S, C, and L ¼ Small, Control, and Large strains.
b
Estimated from Ref. 20, Figure 3.
c
Basal value (36).
489
Figure 8. Vitality and reproductive power in a particular Drosophila strain.
Homeostatic capacity (solid line) is used as a measure of vitality. Data from
a control experiment by McCabe and Partridge at 188C (20) were used to
calculate the exponential decrease in the reproductive power (dotted line). To
enable a direct comparison of the two curves, the homeostatic capacity was
converted to units [eggs/day] by applying the coefficient 44.32.
It was hypothesized that a genetic constraint exists and
results in the direct proportionality of the two characteristics.
We applied this approach to predict longevity data based on
the experimentally observed fecundity patterns. We repeated
the prediction procedure twice, using in each case a ‘‘reference
line,’’ which described the hypothesized proportionality
between homeostatic capacity and reproductive power.
Next, we predicted the mean life spans of the large, control,
and small strains at 188C, given the measured fecundity patterns (and using the data-set from the 258C experiments to
calculate the needed correlation). Finally, we predicted
longevity at 258C using measured fecundity patterns at this
temperature and using the correlation coefficient evaluated
from the 188C data.
The prediction procedure of the 188C longevity was as
follows. For the 3 experimental fecundity patterns at 188C
shown in Figure 4, we calculated the corresponding Pmax
values as described above (Table 3). Based on the three 258C
points in Figure 9, we evaluated the reference line, S0predict ¼
0.0265 Pmax.
Then, we calculated the S0 values for the small, control,
and large strains at 188C, and solved the life history equations (Appendix A) to yield the ages at death, xD. The results
directly predicted the mean population life spans, LSpredict
(Figure 10). The prediction was quite accurate (correlation
coefficient, r ¼ .9732).
The ‘‘backward’’ prediction yielded analogous results.
Given the experimental points for 188C and the corresponding reference line S0predict ¼ 0.0250 Pmax, we calculated the
hypothetical S0 values for the 258C. We then modeled the
life histories, having the results LSpredict for the ages at
death, xD. It was seen at Figure 10 that the prediction in this
case was not as good as in the first one, but the overall accuracy of the prediction was still sufficient (r ¼ .9365).
DISCUSSION
The obvious reason for the increasing interest in reproduction scheduling is the role that the age pattern of
fecundity plays in modern evolutionary concepts and theo-
490
NOVOSELTSEV ET AL.
Figure 9. Correlation between the homeostatic capacity S0 and reproductive
power Pmax calculated for six Drosophila populations [McCabe and Partridge
(20)]. Squares represent 188C experiments; least square linear approximation
(r ¼ .973; k ¼ .0250, dotted line). Circles represent the 258C experiments (r ¼
.512; k ¼ .0265; solid line).
ries (1,8,10,49,50–52). This is why growing attention has
been paid to a detailed analysis of the reproductive machinery in flies during the last decade (39,40–42,53). The latest
analysis of how assay environment in life history experiments affects the parameters of life history (47) shows that
fecundity-related traits are very sensitive to the experimental
protocols. Thus, more reliable and robust techniques are
needed to move forward with experimental fecundity data.
The general purpose of this article was to create a uniform
technique for parameterization of fecundity patterns in animals, particularly in flies, and to bring the technique to the
attention of research teams, especially those working with
Drosophila. We assumed that a fecundity pattern was a result of the superposition of two processes, the genetic
fecundity program encoded in the organism’s reproductive
machinery and the time-dependent pattern of power
resources of the reproductive system susceptible to accumulation of oxidative damage. The accumulated damage
restricted the maximum energy flux so that, at some critical
age, the late fecundity phase arose. The proposed technique
was based on simple assumptions drawn from the physiological mechanisms underlying reproductive-related processes in flies. The main quantitative characteristic of the
reproductive machinery was its reproductive capacity, RC,
which was the result of the partition of the acquired energy
to reproduction and maintenance. The reproductive machinery designed and created at the developmental stage provided the power consumed to handle this machinery as the
adult aged. At the onset of reproduction, the machinery
consumed energy in accordance with genetic prescriptions,
and the rate of egg production in populations rapidly
achieved the ‘‘steady state’’ plateau defined by the genotype
programs and modified by the current environments. The
steady-state fecundity rate was maintained in an organism
until the accumulated damage caused reproductive senescence. After this critical age, fecundity decreases exponentially.
We hypothesize that the initial transient to the plateau is
close to a step-wise one, but deviations from this pattern can
probably be found in many particular cases (like a short
overshoot, which is seen in the patterns presented in Figure
Figure 10. Prediction of the longevity based on measurement of the fecundity
patterns of McCabe and Partridge (20). Squares represent the longevity
predicted for the 188C experiments based on the fecundity patterns observed
at 188C, given the longevity and fecundity data for 258C: LSpredicted ¼ 34.18,
42.25, and 42.21 days for S-[small], C-[control], and L-[large]strains,
respectively. Circles represent the mean life spans predicted for the 258C
experiments based on the fecundity patterns for 258C, given the longevity and
fecundity data for 188C: LSpredicted ¼ 11.55, 22.29, and 11.21 days, respectively.
The overall correlation coefficient, r ¼ .9365.
6). In some cases, a transient process is too short to exhibit
a plateau, such as in the fecundity patterns at a higher temperature in the size-related experiments (20), seen in Figure
4. Nonetheless, we will provide an extended approach to
individual fecundity patterns in a future article (70).
Phenotypic plasticity allows organisms to acquire different quantities of energy in various environments and then to
allocate it between reproduction and maintenance (54–57).
That is why the reproductive machinery displays coordinated
variations of the observed reproduction-related traits in
different environments. This means that the age pattern of
fecundity may provide us with important information about
individual rates of physiological decline and life span. We
studied several cases on Drosophila to analyze the mechanisms underlying the intrinsic relations between fecundity
scheduling and survival. The significant negative correlation
between the reproductive capacity and critical age, which is
found in this article, is in agreement with the general idea
that the elevated oxygen consumption correlates with a faster
reproductive senescence (60–63).
We demonstrate that fecundity patterns may be used to
predict mean life span in Drosophila under specified environmental conditions, basing on the linkage between fecundity and longevity found in this article. In evaluating the
initial energetic capacity of the reproductive system, Pmax,
we assumed that the exponential decline of the reproductive
power in Drosophila (a hypothesis confirmed by experimental data at advanced ages also takes place at early ages).
This assumption is a rather risky hypothesis. It can be justified by correlation between higher oxygen consumption rate
and lower oxidative vulnerability at these ages. These factors may compensate one another, thereby diminishing the
net effect [for medflies, see (37)].
We show that parameterization is an effective tool to
analyze reproductive patterns. It allows formal description
of properties, which were intensively discussed in the literature. For example, Stearns and coworkers (18) theoretically predict the optimal fecundity pattern in Drosophila
FECUNDITY PATTERNS AND AGING
population. The actual fecundity curves proved to be close
to the predictions; however, the answer did not present
a mathematical description of the fecundity pattern. Shanley
and Kirkwood (30) used a cosine approximation for a typical
fecundity pattern in mice. Since a fine approximation of the
pattern was not a primary goal of their study, the standard
errors proved to be rather high. Cichon (31) used dynamic
programming modeling to predict age-dependent patterns of
reproductive rates presumably formed by ‘‘optimal
lifetime strategies of resource partitioning.’’ In contrast with
Cichon’s opinion that the fecundity pattern from the experiments of Stearns and colleagues (45) ‘‘exactly matches the
patterns,’’ the calculated curves are too schematic to be
confronted with real experimental data [as reported in (18)].
The unified approach to specification of reproductive
scheduling opens up a new avenue in the analysis of agerelated fecundity, including the reproductive costs, acquisition costs, and evolutionary optimality in insects and other
species. We expect that a plateau representation, which was
given in this article, may produce a more thorough and
reliable way to analyze these problems. A wide experimental basis exists for such an analysis including numerous
findings related to fecundity patterns in individual female
fruit flies. In particular, the approach is valuable since it
allows for quantification of the very important relationship
between ‘‘early reproduction’’ and longevity in regard to
such main concepts of evolutionary theories as ‘‘trade-offs’’
or ‘‘antagonistic pleiotropy’’ (10,13,47,49). Probably one of
the most essential application areas is an analysis of the
evolutionary physiology of the cost of reproduction (58).
Well out of the scope of this study are several intriguing
issues related to artificial selection experiments for early
versus late reproduction or short versus long life (10,13,15,
17,59).
However, like Carlson and colleagues, ‘‘Ultimately, we
are interested in understanding the correlative, and possible
causative, relationship between female reproduction and
longevity’’ (41). We hope the presented study is a step in
this direction.
ACKNOWLEDGMENTS
The authors thank James W. Vaupel for the opportunity to complete this
work in the Max Planck Institute for Demographic Research, anonymous
referees for valuable comments and Jenae Tharaldson for reading the
manuscript. The work on this paper was partly supported by NIH/National
Institute on Aging grant PO1 AG08761-01.
Address correspondence to Anatoli I. Yashin, Max Planck Institute for
Demographic Research, 18057 Rostock, Germany. E-mail: yashin@
demogr.mpg.de
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Received February 10, 2003
Accepted February 27, 2003
Decision Editor: James R. Smith, PhD
APPENDIX A
Homeostatic Model of Reproductive Aging
The homeostatic model of aging describes senescence
as a decrease in the energetic capabilities of an organism
(36). This model can be applied to a particular system of an
organism as well as to a single organism as a whole. Being
applied to a system (e.g., a reproductive system), the model
describes ‘‘systemic senescence’’ (e.g., reproductive senescence), which ultimately yields stopping of the functioning
of the system. In an organism as a whole, the exhaustion of
energy resources results in death.
The basic notion of the model is that of homeostatic
capacity, which characterizes the overall ability of the
systemic mechanisms to convert substances delivered from
external sources (fuel and oxidizer) into energy to replenish
the energy expenditure in the system.
The oxygen level in the cells presents the oxygen resource of a system that can be converted into energy (ATP)
by the mitochondrial mechanisms. The overall ability of
a system to convert atmospheric oxygen into energy is
denoted as homeostatic capacity S(x). We assume that
oxidative stress (60) deteriorates the homeostatic mechanisms of a system from an initial value S0 at a rate that is
proportional to the rate of oxygen consumption, W(x).
It is known that a portion of the oxygen converted to free
radical particles in a fly is approximately 1% to 3% (61,62).
We denote this portion as a. Simultaneously, the antioxidant
defensive and reparative mechanisms decrease the actual
destruction of the system’s elements so that only a small part
of the total flux of oxidative particles, c ,,1, damages the
FECUNDITY PATTERNS AND AGING
cellular structures. Thus, at age x, the oxidative damagerelated decrease of the homeostatic capacity proceeds at
a rate a c(x) W(x), where a is assumed to be a constant
value, c(x) is the age-related antioxidant defense pattern,
and W(x) is the overall oxygen consumption rate. The agerelated trajectories of enzymatic activity, related to such
a defense, are well known in Drosophila (37,59,63,64).
Let us assume that at the subcellular level the energyproducing system, which determines the homeostatic
capacity of an organism, can be represented as a structure
consisting of N uniform elements. Free radical particles
injure intracellular elements so that only n(x) of them
continue normal functioning at age x. The portion N n (x)
represents the biological effect of accumulated damage.
Since the particles hit both normal and damaged elements,
the number of ‘‘newly damaged’’ elements per unit of time is
proportional to n(x)/N. Assuming that a single hit is enough
to stop the normal functioning of the element, one has
dnðxÞ=dx ¼ a cðxÞ WðxÞ nðxÞ=N:
ðAÞ
Denoting the functional (homeostatic) capacity of the
element as C yields the overall capacity of the organism S(x)
¼ C n(x). Then a quasi-exponential function,
Z x
RðtÞ dt ;
ðBÞ
SðxÞ ¼ S0 exp 0
describes the age-related senescence of the homeostatic
capacity S(x). Here R is a relative rate of aging, R(x) ¼
b(x) W(x), and b(x) is oxidative vulnerability, b(x) ¼
a c(x)/N.
In fact, the time course of this ‘‘exponential-looking’’
function can be very far from the real exponent since both
b(x) and W(x) may have substantial time-related alterations.
But it is true that in any case, S(x) decreases with age at
a decelerated rate.
A senescence-caused decrease of the whole-body S(x)
results in a progressive diminution of the ‘‘steady-state’’
oxygen resource Q(x), and at some age xD, Q(x) ¼ P W(xD)/S(xD) ¼ 0, where P is an external energy source, the
atmospheric oxygen pressure. When Q(x) ¼ 0, a natural
senescence-caused death occurs (36,37). The energy synthesis stops and the organism dies from a shortage in energy
that would have been needed to maintain the vital processes
in the cells. The age xD determined by equation Q(xD) ¼ 0,
and defines the life span of the organism. As hypothesized
by Ukraintseva and Yashin (65), the age-related deceleration
of senescence described by Equation B widely corresponds
to the universal basal decrease in the rate of living during an
individual life. Equation B is in agreement with the
experimental data of Shock (66), which motivated Strehler
and Mildwan (67) to develop a model of mortality and
aging. This model uses energy-related hypothetical vitality
index V(x), which declines with age and characterizes
homeostatic capacities of an organism. The new derivation
of the Strehler and Mildvan model and its recent applications to human mortality data are discussed by Yashin and
colleagues (68,69).
In accordance with the outlined model, we assume that, at
least by the end of reproductive activity, the product of b(x)
493
and W(x) becomes approximately constant in female flies so
that S(x) converges to a ‘‘pure’’ exponent,
SðxÞ ¼ S0 exp½x=sL ;
ðCÞ
with sL ¼ 1/(b W), b W being the ‘‘life span-averaged’’
value of the product b(x) W(x). Note that Strehler and
Mildwan (67) assumed linear decline of homeostatic
capacity, which is just linear approximation of S(x).
The power that can be generated in the reproductive
system for progeny production (given a time-constant
energy source) is proportional to S(x) and thus also
decreases exponentially at advanced ages. This process
characterizes senescence of the reproductive system.
APPENDIX B
Sensitivity Analysis
The direct processing of the experimental data yields the
estimates of values of the reproductive capacity RC, critical
age of transition early-to-late fecundity T, late-fecundity
time constant sL, and reproductive potential RP. The only
measure of the sparseness of the data calculated directly
from the experimental data is the overall standard deviation,
rexp, which characterizes the error between the experimental
data and the least squares approximation.
To evaluate how sensitive the estimates of the parameters
RC, T, sL, and RP are to measurement errors, we used
a Monte-Carlo procedure as follows. Given the age-related
fecundity curve F(x) as described by Equation C with the
estimated parameters RC, T, sO, and sL, we fixed n ages (x1,
x2, . . . , xn) in a way similar to that used in the animal
experiment. For example, if n measurements were made
within a 2-day interval starting at age xA, we take n points
(xA, xAþ2, xAþ4, . . .). In nonregular cases, one may locate the
points randomly. In each point the ‘‘measurement’’ of
fecundity is simulated, F(xi) ¼ F(xi) þ ni, where ni is
a Gaussian random value with mean zero and the standard
deviation rn equal to the experimental one. Each sequence
of n random values F(xi), i ¼ 1, . . ., n is considered as
a particular fecundity data set.
We created N such sets and estimated the N related
fecundity patterns. Figure A1 depicts random ‘fecundity
patterns’ simulated for the experiments of McCabe and
Partridge (20) (N ¼ 10, rn ¼ 2.5). The solid line is the
fecundity pattern evaluated directly from the experimental
data (control at 188C). The estimated parameters become
Table A1. Analysis of Sensitivity of Parameters Evaluation
Parameter
Early fecundity, RC
Critical age, T
Late fecundity, sL
Reproductive potential, RP
Monte-Carlo
Direct
Evaluationa Evaluationb Bias
32.87
10.70
12.72
736.9
32.79
11.99
12.89
782.9
reval reval/rn
0.08 1.030
þ1.29 0.885
þ0.17 1.219
þ46.0 29.07
0.410
0.354
0.487
—
Note: RC ¼ reproductive capacity.
rn ¼ 2.50.
b
N ¼ 100; n ¼ 24 (the points are 2, 4, . . . , 48); rn ¼ 2.368. In all the realizations sO ¼ 1.
a
494
NOVOSELTSEV ET AL.
Figure A2. Histograms of the evaluated parameters of the simulated fecundity
patterns of McCabe and Partridge (20) (control at 188C); n ¼ 100. Left to right,
reproductive capacity RC, critical age T, late fecundity time constant sL. In all
cases, the standard deviations were essentially smaller than the standard
deviations of the experimental data.
Figure A1. Randomly simulated fecundity patterns for the experiments of
McCabe and Partridge (20); n ¼ 10, rn ¼ 2.5. The thick line is for the fecundity
pattern evaluated directly from the experimental data (control at 188C). The
estimated fecundity demonstrates a bias, however, both the early and late
fecundities are very robust: RC ¼ 32.79 6 1.03; sL ¼ 12.89 6 1.219.
random variables also; they are presented in Figure A2 and
Table A1 (in all the estimations N ¼ 100, sO ¼ 1).
The results show that the errors in the evaluated
parameters are of the same order as the measurement errors.
It is also seen that the estimated mean values are biased, but
the expected bias values are not essential (,1% in early
fecundity indicator RC, and about 1.5% in late fecundity
indicator sL). Moreover, the standard deviations reval of the
evaluations are essentially smaller than the standard deviation of the noise rn ¼ 2.5. The evaluation of critical age T
demonstrates a higher susceptibility to the measurement
errors, more than 10% resulting in a rather high error in the
expected reproductive potential (about 6%).
Thus, the sensitivity analysis confirms the robustness of
the evaluation technique, especially as related to the two
main parameters RC and sL describing early and late
fecundity.