Life, Love and Death: Models of Biological
Reproduction and Aging
Dietrich Stauffer
Institute for Theoretical Physics
Cologne University
D-50923 Köln, Euroland
April 14, 1999
The mutation accumulation theory of biological ageing is very suitable for Monte Carlo
simulations: Energy corresponds to Darwinian selection,. entropy to bad mutations. We
review the Penna bitstring model with particular emphasis on sex, and also the claimed
deceleration of mortality for very old humans.
1
Penna model
Why do we get old and die? Each age has its own answer to it, and here of course I talk about
computer simulations. There are many theories of biological aging [1], and I talk here about
the one most suited for computer simulation, the half-century old mutation-accumulation
hypothesis. This is somewhat similar to thermal physics with its balance between order
(energy minimization) and disorder (entropy). Darwinistic selection produces order: Only
the fittest will dominate in the population after a long time, provided their fitness is inherited
and given on to their offspring. Biological mutations due to copying errors during DNA
replication produce disorder, which also can be transmitted to the children. While over
several billion years these mutations have improved the first bacteria into better and better
life forms up to the creation of the German Herr Professor, over the shorter time scale of
one human life most mutations are hazardous to our health. The rare good mutations thus
are neglected in this aging theory, and all the mutations here are bad and inheritable.
If a mutation kills us at young age before we get any children, then it is not at all given
on to any offspring; if it kills us during the age we get children, selection pressure will soon
eliminate most of these mutations from the population. If, on the other hand, an inherited
mutation causes deadly cancer at age 60, most of us will not get new children anymore, no
selection pressure removes the mutation, and it will stay in the population; after all, who
would really be sorry I drop dead here before getting my travel expenses? So after many
1
generations the mutations active at old age will be much more numerous than those active at
young age. In this sense, we get old and finally die because of an accumulation of mutations
which become detrimental at late stages in our life.
Beauty, intelligence, or creativity are difficult to measure, while death rates are well
tabulated in many countries. Thus it is quantitatively best to define aging as the increase
of mortality q(x) for ages x after childhood. Mortality, normally defined as the fraction of
survivors dying within the next year, should better be defined as a derivative
q(x) = −d ln S(x)/dx
(1)
where S is the number of survivors at age x. (Sometimes this derivative is also called force
of mortality or hazard function.)
Figure 1a shows German mortalities for men (higher) and women (lower), extrapolated
by insurance mathematicians to the year 2000. This semilogarithmic plot shows that in
adulthood the mortality increases roughly exponentially with age, the famous Gompertz law
of the 19th century:
q = a exp(bx) .
(2)
And fig.1b gives similar results from Monte Carlo simulations [2] of the Penna model. Similar
exponential increase was found, less accurately, in many animals. We ignore for now the
special problems of childhood evident from fig.1a.
West Germans extrapolated by insurance scientists to the year 2000; and Gompertz slope 0.093
1
mortality
0.1
0.01
0.001
0.0001
0
20
40
60
80
100
age
Fig. 1a: West German data [37] for males (+) and females (diamonds); the straight line
indicates a Gompertz law in the semilogarithmic plot.
In this Penna model [3], today the most widespread simulation method for biological
aging and the first to reproduce the Gompertz law [2], the entire genome is represented by,
2
typically, the 32 bits of one computer word. Each bit position corresponds to a section of
the life of the individual, like 4 years for humans, or days for fruit flies. A zero bit means
health; a bit set to one signifies a life-threatening inherited disease which diminishes our
health in this and every following section of life. Three such active mutations kill us. When
the minimum reproductive age is reached, each individual gets one child per section, with
the same bit-string except that one randomly selected bit is toggled or set to one. Because
of lack of food and space, individuals not killed by the mutations die with the Verhulst
probability N/Nmax where N is the actual population size and the carrying capacity Nmax
is its theoretical limit. Together with 512 Cray-T3E processors this model produced fig.1b.
To find programs for the Penna model, for both sexual and asexual reproduction, as well as
many applications, you should buy our excellent book [2].
Mortality in asexual Penna model, and a fit of the Gompertz law. From Moss de Oliveira et al.
10
mortality
1
0.1
0.01
0.001
0
2
4
6
8
10
12
14
16
age
Fig. 1b: Asexual simulations with (diamonds) and without (+) deaths due to the Verhulst
factor; from [2]. The small deviations of the genetic deaths (+) from the Verhulst line are
statistically significant.
2
Sex and all that
Why did Nature invent something as complicated as sex; why do we not all proliferate like
bacteria and university professors do it most of the time: by cell division and by letting
students research what we are interested in. Is there an evolutionary advantage of sexual
compared with asexual reproduction? Again, I review some computer simulations, often
using the above Penna model. (This section is mainly taken from [4].)
A special section “Evolution of Sex” in Science magazine [5] addressed the traditional
arguments but failed to review the recent computer simulations of this problem, starting with
3
Redfield [6]. Any mutation which gives an advantage after thousands of generations, but kills
the population during the first few generations, is not evolutionary realistic: Nature does not
tunnel through a large energy barrier and did not learn multi-canonical Ising simulations; nor
has it invented cluster flip algorithms. Thus each small mutation should give an advantage,
or at least not too much of a disadvantage, to the individuals which have this mutation.
Redfield’s elegant computer model [6] assumed that every mutation reduces slightly the
survival probability, and on average a child gets half of the maternal mutations and half of
the paternal mutations. If the male and female mutation rate is the same, then the stationary
survival probability for the children is the same with sexual and with asexual reproduction.
But still the males give birth much more seldomly than they drink beer watching soccer;
so Nature would be much better off without them. (The female tradition of eating the no
longer used male [7] has not yet been adopted widely.) Things get even worse if the male
mutation rate is taken as much higher than the female one: Then sexual reproduction gives
much lower survival rates to the offspring than had the females followed the leadership of
the bdelloid rotifers and reproduced asexually since many millioons of years.
Obviously, mother Nature did not read Redford’s Nature article since sex is quite
widespread among life on Earth. A Brazilian group [8] found out why: Real mutations
can be divided into the common recessive and the rare dominant mutations. Thus the child
of a father with 8 and a mother with 4 bad mutations will not have on average 6 bad
mutations reducing its survival chances. Instead, if among the many genes of a species, one
of the father differs from the corresponding one of the mother, then it affects adversely the
child only if the mutation is dominant; recessive mutations affect the child only if both father
and mother had them. 80 or 90 percent of real mutations are recessive, perhaps because the
individuals with more dangerous dominant mutations were killed long ago. As soon as this
aspect was taken into account in the Redfield model [8], the advantages of sex became very
clear through improved survival rates, even if the male mutation rate was six times higher
than the female mutation rate.
This distinction between recessive and dominant mutations is not a DNA repair mechanism, nor does it remove the bad mutations by death. It is merely a cover-up: The hereditary
disease caused by the bad recessive mutation is stored in the genes of the child, but it does
not affect its health as long as one of the two sets of genes (from the mother and from the
father) is still unmutated. It seems we men play the role of back-up diskettes (for the main
disk represented by the female genome): Normally useless, but helpful in case of accidental
loss of information on the main disk. Similarly, Morris [8] compared sex with repeated
proofreading.
Gabriel [9] questioned these results since they came from separate computer simulations
for sexual and asexual reproductions, as if the inventors of sex simultaneously moved to
an island or other separate environment where they did not compete against the asexual
population. This is biologically not probable, and a simultaneous study of both sexual and
asexual individuals seems better. With males not giving birth, the asexual population has
a birth rate twice as high as that of the sexual one, when all other parameters are the
same. To answer Gabriel’s critique one needs therefore a study as in [10] where sexual and
4
asexual populations coevolved. This is simulated here for the Redfield model, not the more
complicated ones, based on the program Redfield kindly sent me years ago.
First, asexual reproduction of individuals with only one set of chromosomes was used
until the mean fitness, i.e. the average survival rate W , changed by less than 10−6 in one
time step; then the time was set to zero and sex was switched on for one percent of the
population, and the computer simulation stopped if the sexual W was nearly constant. We
monitored the ratio R(t) of sexual versus asexual individuals versus time t (= number of
generations), for various ratios α of male to female mutation rate.
Since the asexual population has already a time independent Wa , the sexual population
with changing fitness Ws (t) gives a ratio
R(t + 1) = R(t) · Ws (t)/(2Wa )
(2)
and changes exponentially in time if also the sexual fitness reached its equilibrium. (The
factor 2 comes from the fact that of all men only Arnold Schwarzenegger became pregnant.)
We thus only have to check if at the end of our simulations, the ratio R increases (sex
favoured) or decreases (sex dies out).
Without the distinction between dominant and recessive mutations, sex has little chance
to survive: We had to assume, without any biological justification, the male mutation rate
to be only 30 percent of the female one to find an equilibrium between sexual and asexual
population. With only 20 percent of the mutations dominant, Figure 2 gives an entirely
different picture: Even for α = 6, i.e. for a male mutation rate six times higher than the
female mutation rate (set equal to unity), sexual reproduction is still favoured over asexual
one; only for α = 8 is the situation reversed.
A more realistic simulation has to take into account the various stages in the life of an
individual, that means biological aging. Most of the simulations use Penna’s bit-string model
[3], and its asexual version was reviewed in [11]. It was also applied to sexual reproduction
[12,8]. With sex included the Fortran program becomes much longer (200 lines) but confirmed
[8] that sexual reproduction gives higher survival probabilities than asexual one; again rare
dominant mutations were distinguished from more common recessive ones. In line with the
tradition of aging theories, again all mutations were assumed detrimental.
Unfortunately, this justification of male existence was questioned [13] by the suggestion,
that Nature should have widely adopted the compromise of meiotic parthenogenesis. Here no
males are needed, females have two sets of genes in all their cells, and give on to their offspring
a random combination of them, just as sexual production produces a random combination
of male and female genes for the child: Genetic algorithm with only one sex. Now the
information is still stored twice, and an error in one of the two sets of genes can still be
covered up by the same trick of recessive versus dominant mutations. But now there are no
males taking away the food from the females without getting pregnant. Simulations showed
about the same survival chances with meiotic parthenogenesis as with sexual reproduction.
So why did Nature not follow this simple compromise?
The answer came quite late [14]: Sex gives more variety than meiotic parthenogenesis, and
this greater variety helps survival after catastrophic changes of the enviromnent. Imagine
5
20 % dominant, r=0.2; female mutation rate = 1, alpha = 6, 7, 8 from top to bottom
0.01
sexual fraction
0.001
0.0001
1e-05
0
5
10
15
20
25
30
35
time
Figure 2: Near-equilibrium of sexual and asexual reproduction in Redfield model with
distinction between many recessive and few dominant mutations; female mutation rate is
one; α as given in top line.
another big meteor like the one killing the dinosaurs would come not only to the movies
but to Earth, reducing drastically the surface temperature because of the big amount of
dust and smoke. If all individuals living before the crash would be adjusted to the same
temperature, the species might die out; if thanks to the greater variety produced by the
sexual combination of different genes different individuals of the same species are adjusted
to different temperature, some minority should survive. It was quite difficult for Sa Martins
and Moss de Oliveira [14] to confirm this plausible speculation by explicit simulation of
the Penna model and a sudden catastrophe, comparing meiotic parthenogenesis with sexual
reproduction. But even sex did not help the dinosaurs to survive the catastropic climate
change. And we still have to solve the problem of how to avoid the asexual version to win
with its twice as high birthrate over the sexual one, if both compete with each other in the
same space long before a catastrophe justifies sex (Sa Martins and Moss de Oliveira, priv.
comm.).
Why do women live longer than men? Do we have too much work, too much beer, too
many steaks, as some claim? Cebrat [15] explained also this effect through dominant versus
recessive mutations. Female mammals have two X chromosomes, and males combine one X
with one Y chromosome. So a bad mutation in the X chromosome usually will be recessive
for the females but is always dominant for the males. Computer simulations [15] confirmed
this idea and gave a male mortality twice as high as the female one, except for the highest
6
ages where equal rights prevail. Human reality is similar. Some question remains on whether
ref.15 was allowed to assume a higher mutation rate for the X chromosome, since [16] stated
that the X chomosome is mutated less than the other chromosomes.
A crucial biological test for this theory are birds, where the females have two different
and the males two identical chromosomes, reversed from the case of mammals. Here the
males should live longer than the females, provided we eat none of the chickens. Indeed,
Smith [15] (who anticipated the Cebrat theory qualitatively) cites several papers claiming
that male birds live longer than female birds.
Biologists debate since a long time why menopause exists; that means why women in
their later years no longer get children. Why did evolution not increase the population by
giving women the same possibility as men, who can father children in their 80’s? Or why
does nature not kill the women after menopause, similar to the rapid death of salmon after
reproduction? One reason is [8] that the above genetics cannot kill the women and leave men
alive at an age of about 50 years: In which chromosome should this genetically programmed
suicide be stored, if males and females share most of the genes? (This argument only explains
how menopause is possible, not why it is preferred by evolution. And Cebrat pointed out
that it works only if husbands betray their wifes often.) Human civilization is also hardly
a reason for menopause, since analogous effects are observed in most mammals, provided
they are protected against hunger and predators [17]. Presumably the best explanation
comes from a recent computer simulation [18] of a Penna-type model, with the need of the
offspring for some period of child care [19], together with a risk of reproduction increasing
with increasing age, that means a danger arising from a greater possibility that childbirth will
kill the mother as she gets older. This combination causes the maximum age of reproduction
to self-organize to some intermediate age.
Should you start a family soon or late in life [20]? Virginia opossums did both: When
some of them emigrated to an island without predators, those on the island married later and
lived longer than those remaining on the continent and threatened by predators, page 99 in
[1]. Computer simulations on the Penna model could reproduce this effect quite accurately
[21].
Regrettably, certain feminist propaganda is misusing the Penna model to claim that
husbands should stay faithful to their wifes [22]; these authors look at the California mouse
to find a rare example of monogamy. On the other hand, following some birds, [23] presents
simulations according to which females should have an affair with older males. This latter
paper may be a consolation for all men who are not satisfied to be biological teraflops and to
be regarded by these computer simulations as merely an insurance against catastophic loss
of information or change of climate. Perhaps the war between the sexes could end with the
peace proposal of Suzana Moss de Oliveira [2], that males are disgusting though needed.
3
Do old men die like flies?
In interdisciplinary research like this one, it is only natural if the experts who have worked in
this field since decades ignore the work of newcomers like us physicists who try to simulate
7
what the experts have been thinking about. So let us get closer to their style of work by
just looking at experimental data and drawing conclusions from them, instead of simulating
microscopic models of the genome. Thus this second part discusses the problem of the oldest
old for human beings: Does our mortality as a function of age have a maximum at very old
age, so that afterwards we get healthier and healthier every year?
For fruit flies this is well established [24,25], and for some other animals there are some
indications in the same direction [26]. But what about humans [27]? There are lot of exciting
claims in the literature: “ Humans who make it to 110 years of age appear to have truly
better survival rates than those who make it to 95 or 100” (page 14 of [1]); “beyond 85
years, the mortality rate stops increasing exponentially and becomes constant, or actually
decreases” (page 122 of [1]); “mortality decelerates at older ages. ... the rate of increase
slows down” (page 18 of [1]); “mortality continues to rise throughout adult life, but at a
decreasing rate after the age of 75 or 80” (page 47 of [1]); “death rates increase at a slowing
rate after age 80” [26]. Female life expectancy in West Germany is now above 80 years.
With more and more people reaching these old ages, these statements, if true, have great
importance for fixing the age of retirement (in countries where this is done), for pension
funds, and for unemployment. However, the facts are less rosy for us old people.
First, women life longer than men. Death tables of many rich countries [29] show that
over most of life, the female mortality is only half of the male mortality. At old age, however,
the ratio of the two mortalities gets closer to one. The men continue roughly to obey the
Gompertz law of a mortality increasing exponentially with age, while around the age of 80
the female mortality increases faster than the Gompertz law fitted to middle age predicts.
Would the female mortality follow the same slope d ln(q)/d(age) for centenarians as for
middle age, then in a plot which ignores ages below 80 [26,29], this normalization of female
mortality looks like a deceleration of mortality and like the key to nearly eternal youth;
but it is not. Thus it is better to ignore these lawless women and to concentrate on male
mortality which does not have the anomalous increase near the age of 80. Of course now the
statistics gets worse since there are much less old men around.
Second, a mortality defined as the fraction of people dying within the next year can by
definition not be larger than 100 % and thus necessarily reaches a plateau or maximum at
or below unity. This deviation from the Gompertz law is a mathematical triviality without
biological meaning. A proper definition uses a derivative q = −d ln(S)/dx (also called µ
[29]) where S(x) is the number of survivors up to the given age. Of course, we cannot take
infinitely small age increments on limited data; monthly instead of yearly mortalities would
already be much smaller than unity for old humans and would solve the problem, but seem
not to be available. The easy way out is to define [27-29] q(x + 1/2) = − ln[S(x + 1)/S(x)]
for age x in years; for old humans the higher-order terms in the Taylor expansion are not
important because of the smooth variation of S(x) and q(x). This definition was already
used in fig.1. Now, if we look at fig.1a, we still see a slight curvature in this logarithmic
plot of male mortalities, though weaker than had we plotted [S(x) − S(x + 1)]/S(x). These
mortalities of old German men are below those extrapolated from the Gompertz law. Is this
decrease of the slope reliable?
8
Third, one should not only look at some small age interval. For old age the best real data
I know, as opposed to the extrapolated ones in fig.1a, are published by Thatcher, Kannisto
and Vaupel [29] and use 13 rich countries, basically European Union plus Japan. They are
restricted to ages of 80 and more but fit reasonably onto the West German death tables of
1987 available for ages 0 to 100, as seen in fig.3. The fits of Thatcher et al using the Gompertz
law (a exp(bx)); straight line) or the Kannisto improvement (a exp(bx)/(1 + a(exp(bx) − 1));
curve) are quite bad in middle age since these fits ignore all ages below 80; but their raw data
again show some deviation from a straight line near age 100. Can we trust these points?
In their list of countries, the USA are absent; and an earlier book of one of the authors [30]
puts the USA into the group with unreliable statistics. But such statistics exist [31] and show
lots of Americans living beyond the age of 130 years, with mortality maxima just like for flies.
Such statistics have little credibility [1]. Thus the strongest deviations from the Gompertz
law, i.e. the US data, are very unreliable. This should make us suspicious: Are the slight
deviations of figure 2 due to similar but less strong inaccuracies? According to Thatcher et
al [29], the Scandinavian data seem best. If we look at their plots for these Scandinavian
countries and ignore particularly small Iceland, then we hardly see any deviations from the
Gompertz law. Because of their relatively small population, these countries contribute only
little to the average over 13 countries plotted in fig. 3. Thus for the best data we have no
significant deviations from the Gompertz law for old men; for slightly less reliable data, we
have slight deviations; and for very unreliable data we have strong deviations and curves
similar to flies. This comparison suggests that one should wait for better data before making
claims (ref.38, pages 18,19,24,25 in ref.1) about a deceleration of human mortality at old
age. For example, for a fraction of the population the age might have been recorded wrongly
a century ago, with an age error randomly positive or negative; this effect alone gives a
downward deviation from the Gompertz law at old ages. And if since 1910 births were
recorded more accurately, the effect would vanish soon.
If instead we trust the data from all the 13 countries together, as collected in [29]
and plotted in figure 3, then figure 4a suggests a roughly linear increase [32] of q(x) with
d2 q/dx2 = 0, i.e. neither an acceleration (positive d2 q/dx2 ) nor a deceleration (negative
d2 q/dx2 ); figure 4b shows a visual fit to a formula giving an exponential increase at middle
age and the linear one at old age. At age 120, these fits give a mortality ln[S(119.5)/S(120.5)]
near 1.2, above the extrapolation 0.7 to 1.0 of [29]. Of course, the scattering of the data does
not exclude many other possibilities, like a plateau or maximum, or in the opposite direction
a divergence to infinity at some maximum age [33].
Simulations alone cannot answer these problems; the usual Penna model gives a maximum
age, fig.1b, while some modifications [34] reduced the mortality at old age. Different
simulations, where survival probabilities as opposed to bit-string genomes are inherited,
also gave one or the other choice depending on parameters [35]. The phenomenological
Azbel theory does not yet answer the question how to avoid that the longer-lived families
(longevity is inheritable [36]) finally dominate in the population and let the less favored
families go extinct.
9
West Germany (diamonds) and data from Thatcher et al. (+) with their Gompertz and Kannisto fits
1
mortality
0.1
0.01
0.001
0.0001
0
20
40
60
age
80
100
Figure 3: Male mortalities in 13 rich countries 1980-1990 from Thatcher et al. [29] (above
80 years) and West German death tables of 1987. The two curves show their fits [29] based
on ages above 80 only and thus disagreeing with reality in middle age.
In short, scrutiny of existing data leaves little room for the exaggerations cited at
the beginning of this section, though it may suggest for the oldest old a slightly lower
mortality than an extrapolation of the Gompertz law would predict there. The often claimed
deceleration of human mortality is more than a downward deviation from its Gompertz-law
exponential acceleration and thus far seems unproven.
(The Kannisto ansatz for −d ln S/dx gives a good fit with very few parameters [29] but
converges to unity for old age, i.e. to a survival probability S ∝ exp(−x) if x is measured in
years. Why should the time unit “year” be so special for humans?)
Azbel [27] has analyzed numerous death tables of different countries and different
centuries and concluded that eq.(2) can be written as
q(x) = Ab exp(b(x − X))
(3)
where the characteristic age X ∼ 102 years and the amplitude A ∼ 10 is the same for all
humans, while the Gompertz slope b ∼ 10−1 year−1 increased with increased medical care
etc. This increase of b let the average life expectancy increase by a few decades over the
course of one century, up to about 80 years now, without changing X. Much smaller, only
by about five years, is the increase in the age of the oldest Swedish people, reviewed on page
46 of ref.1. This “maximum” age xmax can be calculated by setting S(x = xmax ) = 1 in a
10
Same data for men above 70 on linear scale for the (force of) mortality; and 0.03*(age-82)
1.6
1.4
1.2
mortality
1
0.8
0.6
0.4
0.2
0
70
75
80
85
90
95
100
105
110
115
age
Same data for men above 30 and exp-lin crossover 0.00004/(exp(-age/10)+0.003/age)
1
mortality
0.1
0.01
0.001
30
40
50
60
70
age
80
90
100
110
Figure 4: Same data as in Fig.3 plotted linearly together with a straight line (top) and
logarithmically together with a simple crossover function (bottom).
11
finite population with N babies of zero age. Inserting eq.(3) into eq.(1) we get (see also [39])
S(x) = N · exp[Ae−bX (1 − ebx )] ' N · exp(−Aeb(x−X) )
(4)
and setting this quantity equal to one for x = xmax we get
xmax − X ∝ 1/b .
(5)
Thus the Gompertz-Azbel law for homogeneous populations predicts the maximum age to
diminish towards the characterictic age X, instead of increasing, with improving wealth
of the nation. This prediction contradicts the observed slight increase in xmax and remains
even if we take into account a moderate growth of the population N ∝ exp(const · t) as well
as the decrease of b with time t. Thus eqs.(3-5) do not seem to describe the fine details of
xmax ; perhaps Azbel’s [27] more complicated “heterogeneous” description with a distribution
of Gompertz slopes b can.
References
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1999
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12
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