[Paleopsych] Sigma Xi: Genealogy in the era of genomic
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Genealogy in the era of genomics: models of cultural and family traits reveal
human homogeneity and stand conventional beliefs about ancestry on their head.
Susanna C. Manrubia; Bernard Derrida; Damian H. Zanette. American Scientist,
March-April 2003 v91 i2 p158(8)
What plain proceeding is more plain than this?" asks the Earl of Warwick in the
Shakespearean play Henry VI, Part II. "Henry doth claim the crown from John of
Gaunt, the fourth son; York claims it from the third. Till Lionel's issue
fails, his [John of Gaunt's] should not reign."
In truth, nothing was plain for another two generations, as the ruling family,
the Plantagenets, nearly butchered themselves into extinction during England's
15th-century Wars of the Roses, precipitated by the competing claims of the
House of Clarence (descendants of Lionel, third son of Edward III), Lancaster
(founded by John of Gaunt, the fourth son) and York (the house of the fifth
son, Edmund). The smoke cleared only after Gaunt's descendant Henry VII of
Tudor defeated the last Plantagenet king, Richard Ill, in battle. He
consolidated his power with an intra-family marriage to Elizabeth of York.
Their son, Henry VIII, was descended from King Edward III (1312--1377) in four
different ways--each one marking a key alliance and a turning point in English
The story of the royal houses of England illustrates not only how the fate of
nations can turn on questions of genealogy but also how the phenomenon of
coalescence--the merging of the branches in a family tree--is staggeringly
common in any closed population. In fact the Plantagenets are in some ways
utterly typical. In a population of 1,000 people who choose their mates at
random, 10 generations are normally enough to guarantee that any two people
have some ancestor in common. Perhaps even more startlingly. 18 generations
normally guarantee that any two people in such a population have all their
ancestors in common. So it is not the least bit surprising, for example, that
every hereditary monarch in Europe at the beginning of the 20th century was a
descendant of Edward III.
In recent years, the field of genomics has revolutionized our perception of how
closely all human beings are related to each other. The study of mitochondrial
DNA, or mtDNA (passed on without change, except for mutations, from mother to
daughter), and certain genes on the Y chromosome (passed on from father to son)
has enabled geneticists to place the time of the "mitochondrial Eve" or the "Y
chromosome Adam" in the surprisingly recent past. The mitochondria carried in
all human cells are the global legacy of a single woman. In a pioneering study
carried out in 1987, the University of California, Berkeley, team of Rebecca
Cann, Mark Stoneking and Allan Wilson estimated that this woman lived between
140,000 and 290,000 years ago.
These analyses tell only part of the story, because they are based on mono
parental inheritance. The great majority of our genome is inherited from both
the mother and the father, and their genes are shuffled by the crossingover, or
recombination, of DNA. Our recombinant DNA tells a much richer story of our
past, if we could only learn to read it. Each one of us, if we could look far
enough back in the past, would find just as tangled a pedigree as Henry VIII's,
with many different coalescing branches.
Mitochondrial DNA is a powerful tool because it cuts through this thicket and
highlights a single vine--but for the very same reason, it misrepresents the
complexity of our past. To understand the full story of human ancestry, the way
that genes and lineages evolve over tens and hundreds of generations, we have
to use mathematical models and computer simulations, because we do not have
genealogical records that extend so far back into the past. These biparental
models show that mitochondrial DNA actually underestimates how quickly human
populations become homogeneous in ancestry.
The Extinction of Families
The first serious attempt to solve a genealogical problem mathematically
resulted from a controversy involving one of the most famous British scientists
of the Victorian era, Sir Francis Galton. Appropriately enough, Galton, a first
cousin of Charles Darwin, had written a book entitled Hereditary Genius, in
which he attempted to explain the oft-noted phenomenon of the decline of great
families. "The instances are very numerous in which surnames that were once
common have since become scarce or have wholly disappeared," Galton wrote
several years later. "The tendency is universal, and, in explanation of it, the
conclusion has been hastily drawn that a rise in physical comfort and
intellectual capacity is necessarily accompanied by diminution in 'fertility."'
Galton himself proposed an alternative explanation that (not surprisingly, for
that era) blamed the women. Men who had recently been elevated in status, he
wrote, would tend to consolidate their positions by marrying heiresses, who
were by definition women from families with no sons. Such women, he believed,
would themselves be less likely to produce sons.
However, a Swiss botanist named Alphonse de Candolle correctly pointed out that
there was another possible explanation for the failure of some family names to
perpetuate themselves: It could simply arise by chance. Until scientists knew
the likelihood of a surname dying out by random processes, they would not have
any way to tell whether the extinction of "famous" surnames was in any way
In 1874, Galton enlisted a mathematician, the Reverend Henry William Watson, to
resolve this question. The approach Watson took was ingenious, and he came
within a whisker of discovering a basic result of the 20th-century theory of
Because he wanted to assess the role of chance, Watson assumed that all males
had the same innate fertility, so that the differences in their numbers of
offspring were attributable purely to chance. Thus, each male had a certain
probability [p.sub.0] of having no sons; a probability [p.sub.1] of having one
son; a probability [p.sub.2] of having two sons; and so forth. Of course, if a
man had no sons, his lineage would die out immediately
So the probability of extinction after one generation--call it [q.sub.1]--would
be just the same as [p.sub.0].
But things get more complicated in the succeeding generations, and that is why
Galton had asked for help. For example, a man could have one son (probability
[p.sub.1]) who himself had no sons (with probability [p.sub.0]); the
probability of his line going extinct in this way would then be the product of
the probabilities, [p.sub.1][p.sub.0]. Or he could have two sons (with
probability [p.sub.2])' who both had no sons (with probability
[p.sup.2.sub.0]); the probability of this event would be [p.sub.2]
[p.sup.2.sub.0]. Adding up the probability of each of these events gives the
probability that the lineage is extinct after two generations, [q.sub.2]:
[q.sub.2] = [p.sub.0] + [p.sub.1] [p.sub.0] + [p.sub.2] [p.sup.2.sub.0] +
[p.sub.3] [p.sup.3.sub.0] ...
Watson's brilliant insight was that the expression on the right side of this
equation, called the generating function, contained all the information about
the probability of extinction in later generations as well. Computing the
probabilities of extinction was simply a matter of applying the generating
function over and over again. Mathematically, he defined the generating
function f(x) by replacing each [p.sub.0] (except the first) with a variable x:
f(x) = [p.sub.0] + [p.sub.1] x + [p.sub.2] [x.sup.2] + [p.sub.3] [x.sup.3] ...
Then he showed that the extinction probabilities for each generation are
obtained by feeding the previous generation's extinction probability back into
this function, a process called iteration:
[q.sub.1] =f(0), [q.sub.2] = f([q.sub.1]), [q.sub.3] = f([q.sub.2]), ...
And what would be the probability of extinction after an indefinite number of
generations, [q.sub.[infinity]]? It would simply be an iterate of itself! That
f([q.sub.[infinity]]) = [q.sub.[infinity]]
This is the equation that gives the probability that any lineage will
ultimately--whether after one generation, 10, or any number--become extinct.
Now, having come so close to a beautiful solution, Watson made his great
blunder. With no demographic data to tell him the probabilities of having zero
sons, one son, etc., he simply took a guess: f(x) [(3 + x).sup.5]/[4.sup.5].
The guess was not a bad one, but then he made a mathematical mistake by
overlooking a solution to his equation. He thought that the only solution was
f(l) = 1; in other words, [q.sub.[infinity]] = 1, meaning a 100 percent
probability that any lineage will eventually go extinct. How depressing! "All
the surnames, therefore, tend to extinction in an indefinite time," Watson
wrote. "This result might have been anticipated generally, for a surname once
lost can never be recovered, and there is an additional chance of loss in every
Watson's analysis was correct for shrinking or constant-size populations. But
in a growing population, a second solution for [q.sub.[infinity]] appears. For
the generating function Watson used, where the population was growing at a rate
of around 8 percent per generation, it turns out that f(0.55) = 0.55 as well,
meaning there is a 55 percent probability that any lineage will become extinct
and a 45 percent probability that it will survive forever. Very roughly, one
may say that a lineage (say, the Smiths or the Joneses) can reach a critical
mass where its survival is essentially assured. But because Watson seemed to
have resolved the debate, no one caught his mistake for another 50 years.
Surnames and Mitochondria
In the 1920s, a new generation of biologists and mathematicians laid the
foundations of population genetics, and soon discovered Watson's error. In a
growing population, any given lineage has a nonzero chance to survive
indefinitely. In 1939, Alfred Lotka used data from the 1920 U.S. census to
estimate [p.sub.0], [p.sub.1], etc., and then computed that [q.sub.[infinity]]
= 0.819. This meant that in the United States of that era, the probability for
indefinite survival of a surname, beginning with one progenitor, was about 18
percent. Or, if you prefer to look at it pessimistically, the probability of
eventual extinction was about 82 percent.
There is always an inherent danger in such pronouncements: They begin to sound
like absolute truths. It is important to remember that they are dependent on
particular mathematical assumptions, which may or may not conform to the real
world. In Watson's model, which has become known to population geneticists as a
Galton-Watson process, some of the assumptions are quite debatable. Do all
males really have the same innate fertility? Perhaps being a member of a
particular family confers some evolutionary advantage; in that case, the
process is no longer "neutral." (This becomes more likely when one applies
Galton-Watson processes to biological traits rather than surnames.) Is the
fertility of each male really independent of each other male, and unvarying
over time? And what happens if we allow "mutation" of surnames, either through
immigration or through fluidity in spelling?
Different cultures, in fact, show great differences in the mutability of
surnames. In China, surnames have been strongly conserved over thousands of
years. A survey by Emperor Tang Taizong in 627 A.D. found a total of 593
different surnames. In 960 A.D., the book Surnames of a Hundred Families
recorded 438 surnames. Today, about 40 percent of the population of China have
one of the 10 most common names, and 70 percent have one of the 45 most common
names. We believe that this lack of mutability is inherent to the Chinese
writing system, which represents each surname by a single character.
By contrast, the U.S. and Canada have the highest diversity of surnames in the
world, a legacy of their history as countries built by immigration. The extreme
mutability of English spellings has also increased the variety of surnames, as
the following excerpt from a World Wide Web page devoted to Hemingways (Figure
3 below) attests:
My most elusive Hemingway ancestor, Fisher Hemingway, born in 1819 or 1820 in
New York... is listed as: Hemensway, Fisher in the 1880 census; Hemingway,
Fisher when he married Catharine Chambers in 1858; Hemenway, Fisher in the
1845-46 Cleveland city directory, Henenway, Fisher in the 1840 census;
Hemmingway, Fisher when he married Elizabeth Elliott in 1839 ... My current
list of Hemingway variations runs to many pages, and I suspect that I have
overlooked many others.
We have studied the distribution of surnames using a simple model that allows
for a small probability of mutations at any time and that also includes a
flexible death rate that can be made equal to, greater than or less than the
birth rate. In this model, like the Galton-Watson model, we find dramatic
differences between growing populations and static ones, where the birth and
death rate are equal.
In a growing population, the diversity of names always increases over time.
Given enough time, the number of names that belong to exactly y people, or
n(y), becomes proportional to l/[y.sup.2] for large enough y, that is to say
for large family sizes. Thus, for example, there should be 100 times as many
names that belong to only 20 people as names that belong to 200 people.
In a static population, on the other hand, the mutation rate becomes very
important. If the mutation rate is too low, then the diversity is very likely
to decrease until there is only one dominant surname. On the other hand, if the
mutation rate is high, then the frequency function n(y) will approach a steady
state, but one that is much more biased toward small family sizes than is the
distribution for a growing population.
We emphasize that these steady-state distributions hold true only after many
generations. On the flip side of the coin, deviations from the expected steady
state can reflect recent historical events. For instance, modern Japanese
surnames began to appear only 120 years ago. Thus we would expect the
distribution of family sizes--particularly large families--to retain an imprint
of the "initial state" of a century ago.
A comparison with real data taken from three sources--the whole 1996 Argentine
phone book, the "A" entries of the 1996 Berlin phone book, and the whole list
of surnames from five Japanese cities circa 2000--seems to bear out these
conclusions. (In this study we defined a "family" as all people with the same
last name.) The Argentine data fit very nicely to the steady-state line n(y),
except for a slight deficit of very large families. This is consistent with
Argentina's demography, a generally pan-European population that has been
disturbed a bit by immigration in the late 1800s and after World War II. The
Berlin data have more scatter, because they come from a smaller data set, but
seem to follow the steady-state distribution. The Japanese data, however,
deviate from the steady-state distribution dramatically, with a significant
excess of large families. If we were to return a century or two from now, we
would probably find the distribution to be clustered more closely around the
straight line shown in Figur e 4. However, this prediction would not hold up if
Japan (or either of the other countries) went through a prolonged period of
zero population growth.
Modern scientists may not care as much as Victorian scientists did about
surnames or the death of "great families." They do, however, care about
mitochondrial DNA, which has the same mode of inheritance as a surname. A
mother's mtDNA is passed along intact, except for rare mutations, to all of her
children; only her daughters, though, can propagate that DNA to the next
generation. The simplest counting unit on a double strand of DNA is the base
pair, made up of a nucleotide on each strand. There is a special segment in
mtDNA, the so-called control region, about 500 base pairs in length, that
apparently evolves neutrally. This segment does not seem to have a specific
function, and the mutations do not offer any survival benefit. Thus the slow,
random genetic drift of mtDNA forms an excellent genetic clock that indicates
whether two people, or two groups of people, had a common ancestor, and how
long ago. The discovery of this clock has cleared up some important historical
debates, both from the recent and the distant past. For example, the mtDNA of a
woman whom many people believed to be the princess Anastasia, the daughter of
the last tsar of Russia, turned out to be unrelated to other living relatives
of the Romanov dynasty. The mitochondria of Pacific Islanders have mutations
common among Asiatic people, and thus prove that the Pacific Islanders came
from Asia, rather than from the Americas as some historians believed. And the
analysis of mtDNA from the upper arm of a 50,000-year-old Neandertal skeleton
established that Neandertals apparently split from the lineage leading to
modern human beings some 500,000 years ago and therefore do not contribute
mtDNA to modern humans. In general, mtDNA, just like surnames, can identify
demographic events in a population's past, such as migrations, population
bottlenecks or expansions.
One Parent or Two?
Mitochondrial DNA has provided groundbreaking insights into the history of
humans. However, mtDNA tells only part of the story: We know that we have,
potentially, as many contributors to our genes as ancestors in our genealogical
tree. "Mitochondrial Eve" and "Y-chromosome Adam" need not be contemporaries or
live in the same region, and they are not necessarily the most important
contributors to our genetic makeup. In fact, if we had one common ancestor at
some particular time, we almost certainly had many of them. Mitochondrial Eve
merely happens to be the one who is our mother's mother's mother's (repeat this
many thousand times) mother. Mitochondrial analysis cannot tell us who is our
mother's father's mother's father's (repeat this many thousand times) father.
Some of these undetectable ancestors may have lived a good deal more recently
than mitochondrial Eve.
It is also worth noting that common ancestors do not necessarily make equal
contributions to our genome. it is true that our parents each contribute 50
percent of our genetic material, but our grandparents do not necessarily each
contribute 25 percent. Going farther back, some ancestors may have their
genetic contribution enhanced by genealogical coalescence: More branches
leading to them translates to more opportunities to pass their DNA down to us.
Two recent studies, one by us and the other by Joseph Chang of Yale University,
have emphasized the difference between the genetic and genealogical approaches
to coalescence. The mathematical models of genealogy that we studied and that
Chang studied are very similar and can be extended, as we did, to populations
of varying sizes. The models work from present to past. We assume that each
individual randomly chooses two parents from the preceding generation.
"Of course [this model] is not meant to be particularly realistic," writes
Chang. "Still, one might worry that this simple model ignores considerations of
sex and allows impossible genealogies. If this seems bothersome, an alternative
interpretation of the same process is that each 'individual' is actually a
couple, and that the population consists of n monogamous couples. Then the
random choices cause no contradictions: the husband and wife each were born to
a couple from the previous generation." As further arguments for the validity
of this model, we might add that it gives a good match to census data on family
sizes, and that it can (if desired) be reformulated to move forward in time.
(The "forward" version is, however, slightly more complicated.)
With this model, one can study a variety of questions. For example, there is
the one Galton and Watson were interested in: What is the probability that your
line (now defined as all descendants, not just sons of sons of sons) will go
extinct? If you pick two people at random in the present, how many generations
back will you have to go to find a common ancestor? How far back do you have to
go until all the ancestors are the same? Figure 5 illustrates these questions.
In this constant population of 12 people, the first common ancestor of the two
individuals denoted by red and blue boxes is a grandmother, shown in red
bounded by blue. Going back to previous generations, we find the red+blue
common ancestors becoming more and more common, until after a mere 6
generations there is complete overlap between the two individuals' ancestry.
Notice that the mitochondrial lineages (shown in green) have not yet coalesced,
so that a geneticist studying mtDNA may or may not realize that the two
present-day individuals are so closely related.
This example is not unusual. The number of generations to the first common
ancestor, in a constant population of n people, is typically the logarithm of n
to the base 2. (The logarithm of 12 to the base 2 is 3.6, so we would expect a
common ancestor around three or four generations ago.) According to Chang, the
number of generations, G, until any two individuals have the same set of
ancestors, is 1.77 times the logarithm of n to the base 2. One might call this
the "coalescence time" of the population. (For a population of 12, it works out
to about 6.3 generations, which agrees with the example in Figure 5.) We
happened to choose a different approach from Chang's, comparing instead the
number of times that a given ancestor appears in two distinct genealogical
trees. We found that it takes on the order of log n generations for the number
of repetitions of each ancestor to become identical in any tree, with an abrupt
transition of about 14 generations (independent of the population size) where
the similarity jum ps from 1 to 99 percent. Finally, both Chang and our group
found that there is not only a universal common ancestor but a universal
ancestral population. At the coalescence time a complete dichotomy emerges, in
which every individual is either an ancestor of all people in the present
generation or none of them. (If the population is constant, about 80 percent of
the people in the Gth generation are universal ancestors, and the remaining 20
percent have had their lines go extinct. In a growing population, the
proportion of universal ancestors is higher and the proportion of extinct lines
Clearly, these model results stand the conventional wisdom about ancestry and
"mitochondrial Eves" on its head. It is therefore very important to scrutinize
the assumptions we made, to see what is reasonable and what is not. We heartily
concur with Chang when he writes, "What is the significance of these results?
An application to the world population of humans would be an obvious misuse.
In the real world, the selection of parents (or in the "forward" model, the
selection of mates) is, of course, not random. Geography, race, religion and
class have always played strong roles in biasing mate selection. Even so, the
models are telling us something important: In subpopulations where random
mating can take place, a common ancestor pool emerges with startling rapidity,
in hundreds rather than hundreds of thousands of years.
By contrast, genetic homogeneity in a population takes a great deal longer to
emerge. Although a genealogical tree has the property of doubling the ancestry
at each generation, this is not the case for individual genes, which
necessarily are inherited along single branches and thus conform to a
monoparental model. Thus one might define a genetic coalescence time to be the
number of generations required to reach a common ancestor for any particular
non-recombining allele. (This is essentially the same as the mitochondrial DNA
problem, in which each individual is linked to only one parent.) In his
pioneering contribution, Sir John Frank Charles Kingman, currently at the
University of Cambridge, has shown that this kind of coalescence takes a number
of generations equal to the population size itself. Thus, for example, a
randomly intermarrying population of 1,000 people will reach genealogical
coalescence in 18 generations, but will require a thousand generations to
achieve genetic coalescence. And even in this case, different genes may lead to
different common ancestors. Thus, once again, it is more appropriate to speak
of an entire ancestral population, rather than a single progenitor Eve.
The analysis of mitochondrial DNA has allowed scientists to obtain many
spectacular results regarding human evolution. MtDNA represents a small, though
essential, piece of our whole genome. Its relevance to the origin of and
relationships among human groups lies in its peculiar mode of transmission
through the maternal line, analogous to surnames. However, our genetic ancestry
is much broader, because we know that a large fraction of any population many
generations ago is included in our genealogical tree. Our surname, like mtDNA,
is only one small piece of information about our origins.
Mitochondrial genes contain information largely about energy production. But
most of the information that characterizes us as human beings resides in our
so-called nuclear genes, which constitute more than 99.99 percent of the human
genome. These genes mix every time a pair of humans reproduce, through the
process of recombination. If we could follow all the branches through which we
have inherited our genes, we would probably find that all those people included
in our genealogical tree have contributed--maybe in an extremely diluted
way--to our genetic inheritance. It is not only mitochondrial Eve, but probably
most of her contemporaries, who have left silent footprints in our extant
The next time you hear someone boasting of being descended from royalty, take
heart: There is a very good probability that you have noble ancestors too. The
rapid mixing of genealogical branches, within only a few tens of generations,
almost guarantees it. The real doubt is how much "royal blood" your friend (or
you) still carry in your genes. Genealogy does not mean genes. And how similar
we are genetically remains an issue of current research.
[FIGURE 4 OMITTED]
[FIGURE 6 OMITTED]
Cann, R, M. Stoneking and A. C. Wilson. 1987. Mitochondrial DNA and human
evolution. Nature 325:31-36.
Cavalli-Sforza, Luigi L., P. Menozzi and A. Piazza. 1994. History and Geography
of Huwan Genes. Princeton, N.J.: Princeton University Press.
Chang, Joseph T. 1999. Recent common ancestors of all present-day individuals.
Advances in Applied Probability 31:1002-1026.
Derrida, Bernard, Susanna C. Manrubia and Damian H. Zanette. 1999. Statistical
properties of genealogical trees. Physical Review Letters 82:1987-1990.
Derrida, Bernard, Susanna C. Manrubia and Damian H. Zanette. 2000a. On the
genealogy of a population of biparental individuals. Journal of Theoretical
Derrida, Bernard, Susanna C. Manrubia and Damian H. Zanette. 2000b.
Distribution of repetitions of ancestors in genealogical trees. Physica A
Harris, Theodore E. 1963. The Theory of Branching Processes. New York: Springer
Kingman, J. F. C. 1982. The coalescent. Stochastic Processes and Their
Manrubia, Susanna C., and Damian H. Zanette. 2002. At the boundary between
biological and cultural evolution: The origin of surname distributions. Journal
of Theoretical Biology 216:461-477.
Sykes, Brian. 2001. The Seven Daughters of Eve. New York W. W. Norton &
Watson, H. W., and Francis Galton. 1874. On the probability of the extinction
of families. The Journal of the Royal Anthropological Institute 4:138-144.
Zanette, Damian H., and Susanna C. Manrubia. 2001. Vertical transmission of
culture and the distribution of family names. Physica A 295:1-8.
Links to Internet resources for further exploration of "Genealogy in the Era of
Genomics" are available on the American Scientist Web site:
Susanna C. Manrubia received her Ph.D. in physics at the Universitat
Politecnica de Catalunya in 1996. She currently holds a tenure-track position
(Ramon y Cajal) at the Center for Astrobiology in Madrid. She is interested in
biological evolution and has applied statistical-mechanics methods and modeling
techniques to a variety of problems where regular collective patterns appear.
Bernard Derrida studied at Ecole normale superieure (ENS) in Paris and
completed his Ph.D. in 1979. Since 1993 he has been professor of physics at
Universite Pierre et Marie Curie and at ENS. He is an expert in statistical
mechanics who has adapted statistical-physics ideas to various problems in
biology. Damian H. Zanette is a researcher and professor at Centro Atomico
Bariloche and Instituto Balseiro in Argentina. He completed his Ph.D. in 1989
with a thesis on kinetic theory. He is interested in applying the methods of
statistical physics to revealing patterns of complexity in social and
biological systems. Address for Manrubia: Centro de Astrobiologia, INTA-CSIC,
Ctra. de Ajalvir Km. 4,28850 Torrejon de Ardoz, Madrid, Spain. Internet:
cuevasms at inta.es
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