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Proc Natl Acad Sci U S A. 1996 December 24; 93(26): 15276–15278. | PMCID: PMC26394 |
Copyright © 1996, The National Academy of Sciences of the USA Genetics The Malthusian parameter of ascents: What prevents the
exponential increase of one’s  ancestors? Susumu Ohno Beckman Research Institute of the City of Hope, 1450 East Duarte Road, Duarte, CA 91010-3000 Accepted October 23, 1996. The reason that the indefinite exponential increase in the number
of one’s ancestors does not take place is found in the law of sibling
interference, which can be expressed by the following simple
equation: where
Nn is the number of ancestors in the
nth generation, ASZ is the average sibling size
of these ancestors, and Nn+1 is the
number of ancestors in the next older generation ( n + 1).
Accordingly, the exponential increase in the number of one’s ancestors
is an initial anomaly that occurs while ASZ remains at 1.
Once ASZ begins to exceed 1, the rate of increase in the
number of ancestors is progressively curtailed, falling further and
further behind the exponential increase rate. Eventually,
ASZ reaches 2, and at that point, the number of ancestors
stops increasing for two generations. These two generations, named AN
SA and AN SA + 1, are the most critical in the ancestry, for one’s
ancestors at that point come to represent all the progeny-produced
adults of the entire ancestral population. Thereafter, the fate of
one’s ancestors becomes the fate of the entire population. If the
population to which one belongs is a successful, slowly expanding one,
the number of ancestors would slowly decline as you move toward the
remote past. This is because ABZ would exceed 2. Only when
ABZ is less than 2 would the number of ancestors increase
beyond the AN SA and AN SA + 1 generations. Since the above is an
indication of a failing population on the way to extinction, there had
to be the previous AN SA involving a far greater number of individuals
for such a population. Simulations indicated that for a member of a
continuously successful population, the AN SA ancestors might have
numbered as many as 5.2 million, the AN SA generation being the 28th
generation in the past. However, because of the law of increasingly
irrelevant remote ancestors, only a very small fraction of the AN SA
ancestors would have left genetic traces in the genome of each
descendant of today. Keywords: sibling interference, AN SA generation Because of the exponential nature of the increase in cell numbers
after successive cell divisions, a microscopic fertilized egg in
mammals grows into a fully formed newborn within a matter of weeks and
months; 30 cell divisions yield about 1 billion (1,073,421,824) cells,
and 40 cell divisions yield about 1 trillion (1,100,183,947,776) cells.
It is a curious fact that in spite of the early concern about potential
and actual population explosions, which was expressed by R. A. Fisher
as the Malthusian parameter of population genetics ( 1), little
attention has been paid to the potential for the equally exponential
increase in the number of one’s ancestors. Inasmuch as it takes two to
make one, this potential is present in all sexually reproducing
organisms. Fig. shows that as one looks into
the past, the number of ancestors should increase exponentially with
ascending generations in the exact same manner as the increase in cell
numbers after successive cell divisions. With the length of a
generation in our own species being 20 years, the number of our
ancestors should have passed the 1 billion mark at the end of 14th
century, which is 30 generations in the past. Although the worldwide
combined human population of today passed the 6 billion mark in 1996,
there is a little doubt that at 6 billion strong, human beings are
very, very close to the upper limit of the earth’s carrying capacity.
Yet, if we look back to 800 years ago, only 40 generations away, Fig.
shows that our ancestors should have numbered more than 1 trillion.
Clearly something is amiss in Fig. .
On the other hand, we also know that each modern species had its
predecessor and that predecessor had its own predecessor. Accordingly,
the ancestry should transcend a successive speciation process. Thus, we
are faced with a seemingly insurmountable dilemma. In the long term of
millions of years, the number of our ancestors should be enormous, yet
in the short term of hundreds of years, there should be far fewer
ancestors than the exponential expectation indicates. The Law of Sibling Interference The only reason that the actual number of ancestors starts to fall
behind the exponential expectation is the eventual inclusion of
siblings (brothers and sisters) among one’s ancestors. Needless to
say, in practice, half siblings sharing only one parent but not the
others should also be considered. The same individual appearing in
different positions (e.g., in the maternal and paternal halves) in the
genealogical tree should also be treated as siblings. This principle is
illustrated by the simplest example at the bottom of Fig. . In this
example, one is a product of the first-cousin mating. Accordingly, two
of the four grandparents are siblings of each other. As a consequence,
the number of great-grandparents is reduced from the exponential
expectation of eight to six. It was found that this principle of
sibling interference can be expressed in the following simple equation:
where Nn is the number of ancestors in the
nth generation in the past, ASZ is the average
sibling size of these ancestors, and
Nn+1 is the number of ancestors in the
next older generation ( n + 1). In the case of the
first-cousin mating, the average sibling size among the four grand
parents is 4/3, which is 1.3333. It follows that the number of
great-grandparents should be 4/1.3333 × 2, which is 6. It is apparent that how recently in one’s genealogy the sibling
interference begins is entirely dependent upon the degree of the
inbreeding in the past. At the very extreme of continuous
brother–sister matings, the sibling interference begins in the parents
and the number of ancestors remains at 2 for the duration in which
consecutive brother–sister matings have been practiced. This is
because 2/2 × 2 is still 2. Such an extreme of the inbreeding
need not be a laboratory oddity, for it has been shown that cheetahs of
South Africa are as inbred as any inbred strain of the laboratory mouse
( 2). The AN SA Generation and the Effective Population Number Once the sibling interference begins, it is expected that the
average sibling size among the ancestors would steadily increase as we
delve further back into ancestral generations of the more remote past.
This is because, as the absolute numbers of ancestors increase, more
and more siblings and half siblings would be included among the rank of
ancestors. Instances of the same individual appearing in different
branches of the ever-expanding genealogical tree should also increase.
Inevitably, there would eventually be a generation at which the average
sibling size reaches 2. The hallmark of this generation is that the
number of ancestors of this generation and that of the generation
immediately before it should remain exactly the same, for the simple
reason that Nn/2 × 2, which is
Nn+1, is still the same as
Nn. The earlier of the two generations is defined as
the AN SA generation and the older one as the AN SA + 1 generation. In
this nomenclature, AN stands for ancestry, and SA stands for saturated.
The above noted temporary halt in the number of ancestors is due to the
inclusion of all the progeny-produced adults of the ancestral
population as the ancestors. It follows that the number of the AN SA
ancestors becomes an apparent equivalent of the effective population
number in population genetics known as Ne ( 3). This
equivalence, however, is more apparent than real. At the extreme of the
inbreeding is a product of continuous brother–sister matings. For this
type of individual, the number of ancestors at the AN SA generation is
2 for the duration of inbreeding as already noted. Yet, there are still
several hundred progeny-producing cheetahs in the South Africa.
Accordingly, Ne for the cheetah of the South Africa
would be not 2 but several hundreds. Looking at this extreme, one
realizes that there can be more than one AN SA generation for a species
if not for a population. It has been stated that the cheetahs of South
Africa went through two successive bottlenecks, and only the first one,
occurring in the late Pleistocene some 10,000 years ago, affected the
cheetah as a whole ( 2). It follows that the pre-Pleistocene cheetah
must have had another AN SA generation, the AN SA ancestors of that
time numbering far greater then 2. Numbers of Ancestors in Generations Older than the AN SA + 1
Generation As all the progeny-produced adults of the entire population came
to be included in one’s ancestry at the AN SA generation, numbers of
ancestors of all generations older than the AN SA + 1 too became all
the progeny-produced adults of the entire population of each
generation. Consequently, the steady increase in the numbers of
ancestors in generations older than the AN SA also became a clear
revelation of a failing population in which the average sibling size
among the progeny-produced adults had fallen below 2. Such a population
had been shrinking for sometime. It follows that the present AN SA was
the secondary AN SA under this particular circumstance; therefore,
there must have been a previous AN SA involving a far larger number of
ancestors when that population enjoyed a greater success. This point
was already made in connection with a product of successive
brother–sister matings. From the above, it becomes quite clear that if it was the primary AN SA
of a reasonably successful population, the number of ancestors at the
AN SA had to be very large, since even with a very moderate average
progeny size of 2.033, the number of ancestors in the 12th generation
before the AN SA would have fallen to 83% of the AN SA number. Shown
in Fig. is the one simulation depicting
initial rise in the number of ancestors until the arrival of the AN SA
and subsequent decline, under the following conditions. (i)
The average sibling size among the ancestors became 1.01 at the 10th
generation from the present when the ancestors numbered 1,024.
(ii) Thereafter, there was an incremental increase of 0.01
per generation for 10 generations in the average sibling size.
(iii) After reaching the average sibling size of 1.10 at the
19th generation into the past, the incremental increase became 0.1 per
generation for the next 10 generations, thus, reaching the AN SA
generation at the 28th generation from the present; for human beings,
this generation falls in the middle of 15th century. (iv)
After reaching the AN SA, the average sibling size for the ancestors of
generations older than the AN SA stabilized at 2.033. As shown in Fig.
, it would be noted that the AN SA ancestors at the 28th generation
away from the present numbered roughly 5.2 million. This number is only
1% of the exponential expectation; nevertheless, it is large enough to
indicate that this is the primary AN SA of a rather successful
population that had steadily been expanding at a moderate rate of
1.0165 per generation until the AN SA.
The Validity of One Universal Genealogical Myth All the long-settled families in the Alsace region of France
invariably include heroic Charlemagne of eighth century in their
genealogical trees. Charlemagne, who was king of the Franks, later
became emperor of all the Romans. Similarly, all the Japanese families
of the Genji lineage include emperor Seiwa of ninth century as the most
illustrious ancestor in their genealogies, whereas another emperor,
Kanmu of eighth century, plays the same role in the genealogical trees
of those belonging to the Heike lineage. Such a genealogical claim has
traditionally been dismissed outright as an absurd fantasy borne of a
wishful delusion. Quite to the contrary, this study reveals that unlike
the Ne of population genetics, the number of
ancestors at the AN SA generation was very large, probably numbering in
the millions, and the ancestors of the AN SA and all generations
previous to the AN SA included all the progeny-produced adults of the
entire ancestral population. It follows that among them had to be all
the local kings of the times. Not to be forgotten, however, are other
ancestors of the times, for also included in the ancestry were
murderers, thieves, embezzlers, prostitutes, and all other social
misfits of the times. The Law of Increasingly Irrelevant Remote Ancestors The expression of “diluted blood” is frequently used to
lament the ineptness of a descendant in comparison with his or her
illustrious, but remote, ancestor. Indeed, as one’s ancestors fade
into the remote past, there is an ever-increasing chance that they have
become totally irrelevant in the genetic sense in that they left no
trace in the genome of one of their descendants of today. Individual genes of all eukaryotes can be placed as beads on a finite
number of strings; this finite number is 23 in the case of our own
species. Each chromosome (string), however, is not a heritable unit,
since each becomes a mixture of segments derived from paternal and
maternal grandparents due to crossovers after the chiasma formation
during first meiosis of germ cells in both parents. Accordingly, the
true unit of inheritance is a chromosomal segment of variable lengths
produced by meiotic crossovers. In the case of humans as well as the
mouse, the data suggest that a mean chiasma count at diakinesis is
about 50 ( 4). Although crossovers might be a little more frequent in
female germ cells than in male germ cells, such a small difference can
be ignored for the present. It follows that, on average, each human
individual inherits roughly 100 chromosomal segments from each of the
two parents, a total of 200 segments. Assuming that each crossover
occurs at random along the entire length of each chromosomal pair, the
total number of chromosomal segments to be inherited from one’s
grandparents is 400. It follows that at the maximum, only 4,000
ancestors of the 20th generation in the past left one tiny chromosomal
segment each in a given individual genome of today. The simulation
shown in Fig. suggests that the actual number of ancestors 20
generations ago likely numbered 616,315. It follows that 4,000
ancestors amounted to only 0.65% of the total ancestors of 20
generations ago. Consequently, at least 98.7% of those ancestors
became forever irrelevant, as they left no genetic trace in each of us
today. However, relevance and irrelevance being more or less random
affairs, of two overtly unrelated individuals of the same population
today, the relevant 4,000 ancestors as a set of one is bound to be very
different from those of the other. This might explain why overtly
unrelated individuals of a fairly homogeneous population still display
divergent phenotypes. 1. Fisher R A. The Genetical Theory of Natural Selection. Oxford: Oxford Univ. Press; 1939. 2. O’Brian A J, Wildt D E, Bush M, Caro T M, Fitzgibbon C. Proc Natl Acad Sci USA. 1987;84:508–511. [PubMed] 3. Kimura M, Ohta T. Theoretical Aspects of Population Genetics. Princeton: Princeton Univ. Press; 1981. 4. Polani E. In: Trisomy 21 (Down’s Syndrome) Research Perspectives. de la Crutz F F, Gerald P E, editors. Baltimore: Univ. Park Press; 1981. pp. 112–115.
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