Date for Last Common Ancestor?
T&B Schmal
I am interested in bracketing the dates for the last common ancestor of
humankind. Would a good guess be somewhere between "Eve" and the
appearance of modern man - say, between 200 and 50 Kya? It seems to me
that dates outside this range would be impossible.
Is this on track and can this range be narrowed? Suggestions welcomed.
Tom Schmal
Whatever we see or feel or do is the effect
of whatever we have seen or felt or done.
Stephen Barnard
The attribute of being "the last common ancestor of humankind" has a
very peculiar property. That person can only be identified long after
the fact, and the identity and date of that person is liable to be
changed radically at any time. For example, if a new disease were to
decimate the world's human population, with only say 10% of the people
surviving because of an inherited resistance, then the date of the last
common ancestor would probably be moved forward in time considerably.
Steve Barnard
Susan S. Chin
Stephen Barnard (st...@megafauna.com) wrote:
: T&B Schmal wrote:
: >
: > I am interested in bracketing the dates for the last common ancestor of
: > humankind. Would a good guess be somewhere between "Eve" and the
: > appearance of modern man - say, between 200 and 50 Kya? It seems to me
: > that dates outside this range would be impossible.
: >
: > Is this on track and can this range be narrowed? Suggestions welcomed.
: >
: > Tom Schmal
: The attribute of being "the last common ancestor of humankind" has a
: the fact, and the identity and date of that person is liable to be
: changed radically at any time. For example, if a new disease were to
: decimate the world's human population, with only say 10% of the people
: surviving because of an inherited resistance, then the date of the last
: common ancestor would probably be moved forward in time considerably.
: Steve Barnard
Whether or not we will ever determine who and when this last common
ancestor occurred, I don't see how it can arbitrarily be moved forward in
time due to a scenario of catastrophic decimation of 90% of humankind. The
assumption is that by that time, the LCA had already split off to form
these various populations.
Even if you subscribe to the theory of multiple and separate evolutionary
roots of mankind per geographic regions, at one point, there was
a species of organism from which Homo sapiens sapiens arose.
The "Eve" that we hear about as the "mother of us all" is really a
population which contained the genes which ultimately resulted in all of
humakind as we see it today. It was not meant to imply (though of course
it does anyway) that we all arose from one female individual, but from a
population of individuals with those genes.
The question should be how reliable is the molecular data, and to what
extent should paleoanthropologists rely on it in developing their
theories of evolution and the LCA?
Susan
--
sus...@netcom.com
Stephen Barnard
It's really very simple. The mutation that gave rise to the immunity to the
disease could easily, and probably would have, occured much later than the
erstwhile LCA. Therefore, the new LCA would be that person who first exhibited
the mutation.
Meanwhile, all the other gross phenotypical characteristics that we commonly
attribute to "humanity" would be more-or-less unchanged.
This has nothing to do with the "theory of multiple and separate evolutionary
roots of mankind per geographic regions", which I consider to be highly
unlikely. I also don't quarrel with the concept that there *is* an LCA, which
is clearly the case from purely logical arguments. The point is that the
attribute of being the LCA has some very peculiar semantic properties.
Steve Barnard
Stephen Barnard
I also have to point out (reluctantly) that you are wrong about the claim that
"Eve" is actually a population of more than one. We all *do* descend from a
single female individual, which is trivial to prove. This is precisely why the
"Eve" hypothesis is so devastating to the "theory of multiple and separate
evolutionary roots of mankind per geographic regions".
Steve Barnard
T&B Schmal
I wrote:
I am interested in bracketing the dates for the last common ancestor of
humankind. Would a good guess be somewhere between "Eve" and the
appearance of modern man - say, between 200 and 50 Kya? It seems to me
that dates outside this range would be impossible.
Tom Schmal
CHESSONP asks:
>
> To establish the existence of a "Last Common Ancestor"by
> logical means only will require quite a lot more than you have supplied,
> starting with a clear definition of what is the "Last Common Ancestor" for
> a species.
What I ment by LCA is tough to define in genetic terms, but from a lineage
point of view, it is a person (well, really a pair of persons) from which
all of us living are directly descended, ie, each of us can trace our
ancestory to that one person. Now, we can also all trace our ancestory to
that person's mother but she would not be the last common ancestor, she
would be the second-to-last.
As an analogy - at the Schmal family reunion, all the great grandchildren
can trace their lineage to great grandma Schmal. They can also trace
their lineages to many other great grandmas (through my wife's parents,
for example). But for all the children at the reunion, great grandma
Schmal is the LCA.
Eve, like the missing link, the first primate, the first single-celled
organism, is one of our common ancestors. At this point she is the last
Known common ancestor. As more genetic analysis is done she will
(probably) be displaced by someone more recent. My question is what is
the most recent possible date that will be eventually determined to be the
date of the LCA.
I hope that does not muddy the waters too much.
Tom Schmal
Susan S. Chin
Stephen Barnard (st...@megafauna.com) wrote:
: Susan S. Chin wrote:
: >
: > Stephen Barnard (st...@megafauna.com) wrote:
: > : >
: > : > I am interested in bracketing the dates for the last common ancestor of
: > : > humankind. Would a good guess be somewhere between "Eve" and the
: > : > appearance of modern man - say, between 200 and 50 Kya? It seems to me
: > : > that dates outside this range would be impossible.
: > : > Is this on track and can this range be narrowed? Suggestions welcomed.
: > : >
: > : > Tom Schmal
: > :
: > : very peculiar property. That person can only be identified long after
: > : the fact, and the identity and date of that person is liable to be
: > : changed radically at any time. For example, if a new disease were to
: > : decimate the world's human population, with only say 10% of the people
: > : surviving because of an inherited resistance, then the date of the last
: > : common ancestor would probably be moved forward in time considerably.
: >
: > : Steve Barnard
: >
: > Whether or not we will ever determine who and when this last common
: > ancestor occurred, I don't see how it can arbitrarily be moved forward in
: > time due to a scenario of catastrophic decimation of 90% of humankind. The
: > assumption is that by that time, the LCA had already split off to form
: > these various populations.
: >
: > Even if you subscribe to the theory of multiple and separate evolutionary
: > roots of mankind per geographic regions, at one point, there was
: > a species of organism from which Homo sapiens sapiens arose.
: >
: > The "Eve" that we hear about as the "mother of us all" is really a
: > population which contained the genes which ultimately resulted in all of
: > humakind as we see it today. It was not meant to imply (though of course
: > it does anyway) that we all arose from one female individual, but from a
: > population of individuals with those genes.
: It's really very simple. The mutation that gave rise to the immunity to the
: disease could easily, and probably would have, occured much later than the
: erstwhile LCA. Therefore, the new LCA would be that person who first exhibited
: the mutation.
In this scenario though, the populations that exist, those with and those
without the immunity, are *already* commonly descended from the Last
Common Ancestor. The new LCA you are referring to, the one with the
beneficial mutation shares a common ancestry with those unfortunate ones
who didn't "survive." The emphasis should be on the Common Ancestor part
of LCA, not on Last, since that might create some confusion. In your
scenario, the Common Ancestor remains the same, regardless of the
mutation. That Common Ancestor is the LAST Common Ancestor. No more after
that. So, is that your point as well?
: This has nothing to do with the "theory of multiple and separate evolutionary
: roots of mankind per geographic regions", which I consider to be highly
: unlikely. I also don't quarrel with the concept that there *is* an LCA, which
: is clearly the case from purely logical arguments. The point is that the
: attribute of being the LCA has some very peculiar semantic properties.
: Steve Barnard
--
sus...@netcom.com
Susan S. Chin
: > > : >
: > > : > I am interested in bracketing the dates for the last common ancestor of
: > > : > humankind. Would a good guess be somewhere between "Eve" and the
: > > : > appearance of modern man - say, between 200 and 50 Kya? It seems to me
: > > : > that dates outside this range would be impossible.
: > > : >
(woops! I deleted my Eve claims)
: I also have to point out (reluctantly) that you are wrong about the claim that
: "Eve" is actually a population of more than one. We all *do* descend from a
: single female individual, which is trivial to prove. This is precisely why the
: "Eve" hypothesis is so devastating to the "theory of multiple and separate
: evolutionary roots of mankind per geographic regions".
: Steve Barnard
Well, I know that initially when the theory of an Eve came out, that was
the popular understanding, that we all descended from this one female.
But tell me if I'm wrong here, but aren't we getting awfully close to
Creationism here? If there was an "Eve," there must've been a male who
contributed his genes towards creating that first generation of humanity
(this sounds pretty ridiculous to me, but who am I to argue with
science?). We therefore theoretically are also descended from that one
male, but unfortunately his mitochondria is of no use to us. Well, it's
getting late, so maybe it'll make more sense tomorrow.
I did read about my Eve as a population and not an individual
ancestor recently, but don't recall the source.
What are you basing your one ancestral Eve idea?
Susan
--
sus...@netcom.com
Stephen Barnard
[a bunch of stuff clipped for brevity]
>
> : It's really very simple. The mutation that gave rise to the immunity to the
> : disease could easily, and probably would have, occured much later than the
> : erstwhile LCA. Therefore, the new LCA would be that person who first exhibited
> : the mutation.
>
> In this scenario though, the populations that exist, those with and those
> without the immunity, are *already* commonly descended from the Last
> Common Ancestor. The new LCA you are referring to, the one with the
> beneficial mutation shares a common ancestry with those unfortunate ones
> who didn't "survive." The emphasis should be on the Common Ancestor part
> of LCA, not on Last, since that might create some confusion. In your
> scenario, the Common Ancestor remains the same, regardless of the
> mutation. That Common Ancestor is the LAST Common Ancestor. No more after
> that. So, is that your point as well?
>
The "last" part is crucial to the concept of "last common ancestor". I you leave
out "last" then we have common ancestors going all the way back to the origin of
life. In my scenario the erstwhile LCA remains a CA, but no longer the LCA.
That's what is a little peculiar about the property of being an LCA.
Steve Barnard
Stephen Barnard
> : I also have to point out (reluctantly) that you are wrong about the claim that
> : "Eve" is actually a population of more than one. We all *do* descend from a
> : single female individual, which is trivial to prove. This is precisely why the
> : "Eve" hypothesis is so devastating to the "theory of multiple and separate
> : evolutionary roots of mankind per geographic regions".
>
> : Steve Barnard
>
> Well, I know that initially when the theory of an Eve came out, that was
> the popular understanding, that we all descended from this one female.
> But tell me if I'm wrong here, but aren't we getting awfully close to
> Creationism here? If there was an "Eve," there must've been a male who
> contributed his genes towards creating that first generation of humanity
> (this sounds pretty ridiculous to me, but who am I to argue with
> science?). We therefore theoretically are also descended from that one
> male, but unfortunately his mitochondria is of no use to us. Well, it's
> getting late, so maybe it'll make more sense tomorrow.
>
> I did read about my Eve as a population and not an individual
> ancestor recently, but don't recall the source.
>
> What are you basing your one ancestral Eve idea?
>
> Susan
> --
> sus...@netcom.com
I've been called a lot of nasty things, but never before have I been
called a creationist. :-) I'm just kidding. It's a good question.
Just because "Eve" was our last *female* common ancestor, that by no
means implies that her mate was our last *male* common ancestor. In
fact, the liklihood that he was is extremely small. Since one man can
produce many more offspring than one woman, it is not unlikely that our
last male common ancestor has a much more recent date than the
mitochondrial Eve.
Here's the logic that demonstrates that there is a single Last Common
(female) Ancestor:
Let S_1 be the set of all people whom are alive today. Let S_2 be the
set of mothers of the members of S_1. In general, let S_k be the set
of mothers of S_k-1. The size of these sets is nondecreasing (S_k <=
S_k-1), because everyone has only one mother, but some mothers have
more than one child. When the size of the set reaches 1 then we have
arrived at the mitochondrial Eve.
Steve Barnard
Stephen Barnard
Stephen Barnard wrote:
> Let S_1 be the set of all people whom are alive today. Let S_2 be the
> set of mothers of the members of S_1. In general, let S_k be the set
> of mothers of S_k-1. The size of these sets is nondecreasing (S_k <=
> S_k-1), because everyone has only one mother, but some mothers have
> more than one child.
Oops. I meant to say that the size of the sets in *nonincreasing*, or
better yet, they never get any bigger (and they are most likely to get
smaller).
Steve Barnard
Susan S. Chin
Stephen Barnard (st...@megafauna.com) wrote:
: [a bunch of stuff clipped for brevity]
: Steve Barnard
The "last" part of LCA is crucial in identifying when the split between
lineages occurred. At that point of splitting, branching, whatever you
want to call it, there is an ancestral species known as the LCA. If there
is further splitting or speciation which later occurs, that is totally
irrelevant to the original LCA which has already been identified wrt Homo
sapiens sapiens or any other species. Further splitting produces LCA's for
OTHER species which result from the split, but has nothing to do with the
original LCA. So if anything, LCA's are relative to what 2 organisms you
are comparing. The ancestral relationship to their LCA though is absolute.
In the original post, I believe the question was LCA for Homo sapiens
sapiens or mankind as we know it today. Therefore, when that crucial
event occurred which gave rise to modern man, it is assumed that it
occurred only once, on a species wide unit of evolution. But in any
case, we have our LCA. Further events which occur are irrelevant,
unless you are categorizing different races into various subspecies of
H.s.s. And even then, the LCA of H.s.s. as a species doesn't change.
It sounds to me like we have a disagreement over the unit of evolution
here. I'm referring to the LCA as a species, or a population of the
species which gave rise to modern man. Once this LCA appears, any later
purported LCA's are really artificially recognized units of evolution.
Susan
--
sus...@netcom.com
Stephen Barnard
You seem to be confusing two separate uses of the concept of Last Common
Ancestor. (Sorry, I don't mean to sound condescending, but I think you
really are mixing up two different things.)
1. The concept is used to define the relatedness of species by looking at
the LCA of two species. For example, chimps are thought to be more closely
related to humans than are gorillas because we share a more recent LCA with
chimps (probably).
2. The sense in which Mitochondrial Eve was our last female common ancestor
has nothing to do directly with the beginning of our species. Her
immediate ancestors were just as human as all of her descendants.
When used for the purpose of relating two species, it seems like the LCA is
absolute, as you say. It does have a another peculiar property, however.
It can only be identified well after the fact. For example, suppose a
lineage splits into two, leading eventually to two separate species. For a
long time after the split the members of the two lineages will remain in
the same species and will be capable of mixing reproductively in principle.
It will only be when the two species have finally split reproductively that
their LCA will be meaningful for relating them as species.
Steve Barnard
CHESSONP
"Let S_1 be the set of all people whom are alive today. Let S_2 be the
set of mothers of the members of S_1. In general, let S_k be the set
of mothers of S_k-1. The size of these sets is nondecreasing (S_k <=
S_k-1), because everyone has only one mother, but some mothers have
arrived at the mitochondrial Eve.
Steve Barnard"
This does not prove that the cardinality of the sequence of sets {S_k}
coverges to one. To establish the existence of a "Last Common Ancestor"by
John Hawks
Susan and Stephen (and whoever else),
Hi, I'm John Hawks, and I study with Dr. Wolpoff at the University of Michigan. I want to let you know that
you're both right about "last common ancestors." However, the topic is one that readily invites confusion,
so I'm glad to see this thread appear as it promises to be a very useful discussion.
The concept of a "last common ancestor" is of course perfectly clear to all of us. We would readily admit
that a person and his or her first cousin share at least one grandparent as their "last common ancestor." We
also may claim that A. africanus is the "last common ancestor" of A. boisei and A. robustus. Some of us may
even go so far as to suggest that all living humans share a single female "last common ancestor" that we
might call "Eve."
The point that many people miss (including the authors of far too many papers) is that the situations that I
just listed relate to each other only by analogy, not because of any feature of their sets that would cause
them to share a function called "last common ancestor." Indeed, our concept of "last common ancestor" as we
usually discuss it is an intuitive one. Rarely does anyone define it rigorously.
What you have been talking about up to this point is not a confusion over the concept of "last common
ancestor," but instead a confusion about how that concept maps upon two very different types of sets: the
set of species and the set of individuals.
> > Susan S. Chin wrote:
> >
> > > It sounds to me like we have a disagreement over the unit of evolution
> > > here. I'm referring to the LCA as a species, or a population of the
> > > species which gave rise to modern man. Once this LCA appears, any later
> > > purported LCA's are really artificially recognized units of evolution.
> > > . . .
> > > The emphasis should be on the Common Ancestor part
> > > of LCA, not on Last, since that might create some confusion.
Susan is referring to a speciation event here. The founder population of the new species is in a way a
"common ancestor population" for that species. Depending on the subsequent population history of the
species, it might even be the "last common ancestor population." As Susan notes, all members of a species,
living and dead, must be descended from *some* of the members of this population. However, it might be more
useful to refer to this population simply as the "founder population," since (1) not all of the members of
the founder population are necessarily ancestors of later individuals, and (2) the status of this population
as a "common ancestor" or even "last common ancestor" of later individuals depends on the subsequent
population history of the species. (Note: when Cann et al. and Vigilant et al. refer to the population in
which their "Eve" lived, they specifically mean it to be a hypothetical founder population of H. sapiens
sapiens.)
> Stephen Barnard wrote:
>
> > 2. The sense in which Mitochondrial Eve was our last female common ancestor
> > has nothing to do directly with the beginning of our species. Her
> > immediate ancestors were just as human as all of her descendants.
Stephen is, and has been, talking about individuals, not species. And he is perfectly right about the way in
which the "last common ancestor" can change as a consequence of demographic history (though the logic
solution in his earlier post is flawed). However, the sense in which "mitochondrial Eve (mtEve)" is our
"last female common ancestor" is not the way that he describes. In truth, we might even have a "last female
common ancestor" who is much more recent than mtEve. The reason for this is because of the types of sets
that we are talking about.
When we talk about "last common ancestors" in biology, we would like them to conform to Susan's concept of
them:
> > Susan S. Chin wrote:
> > >
> > > The "last" part of LCA is crucial in identifying when the split between
> > > lineages occurred. At that point of splitting, branching, whatever you
> > > want to call it, there is an ancestral species known as the LCA. If there
> > > is further splitting or speciation which later occurs, that is totally
> > > irrelevant to the original LCA which has already been identified wrt Homo
> > > sapiens sapiens or any other species. Further splitting produces LCA's for
> > > OTHER species which result from the split, but has nothing to do with the
> > > original LCA. So if anything, LCA's are relative to what 2 organisms you
> > > are comparing. The ancestral relationship to their LCA though is absolute.
This is true of species. Two species of animals *must* share a single "last common ancestor," which is
itself a species. This is because the set of animal species (and the set of all species of organisms if we
assume a single origin of life) exhibits a property called coalescence. Coalescence is a very confusing
concept, but simply put it is this: If we start at the present and look backward in time, then the diversity
of elements of our set will coalesce to a single element. Coalescence describes sets in which the only way
that new elements are created through time is by divergence. New species are created only by the splitting
of an old species. Each species has one and only one parent species. New species are not the product of a
mixture of two old species.
The set of animal species is a coalescent set. The set of sexual animal individuals is not. If you look back
at the history of individuals, they do not coalesce to a single individual at some time in the past.
Instead, they branch out into a family tree, increasing exponentially into the past. This is because the
process of sexual reproduction is not one of divergence. Instead, the genes of two individuals are mixed
with the creation of each new individual. Even if some of the branches of a person's family tree lead to the
same individual, even if most of the branches should happen to lead to some small number of individuals, the
"last common ancestor" of individuals in the geneological sense says nothing about the pattern of splitting
or branching of the individuals of a species. Indeed individuals cannot be said to split or branch in the
same way as species.
In a similar sense, the "last common ancestor" of populations cannot say anything about the branching
history of the set of populations of a species except in certain special cases (more discussion on this
point may continue at a later time), and so my preference of "founder population" for the idea Susan is
writing about.
Individuals make lousy subjects for study because they are the product of a mixture of their parents.
Specifically, they have alleles from each of their parents, and from each of their grandparents, and so on.
Alleles, on the other hand, are great for studying splitting and branching because they (outside of the
action of recombination mutations) *do* feature coalescence. The mitochondrial DNA is, as you all know, just
one big allele, not subject to recombination and *mostly* maternally inherited. When Cann et al. and
Vigilant et al. refer to mitochondrial Eve, what they are talking about is a female who contained the
coalescent "last common ancestor" of human mitochondrial DNA. Note that this is not necessarily the "last
female common ancestor" of living humans, and it is almost certainly not the "last female common ancestor"
of all H. sapiens sapiens--even if mtEve lived in the *hypothetical* founder population of H. sapiens
sapiens. Note also that this does not even require that mtEve herself be a member of H. sapiens sapiens. All
living humans *must* have a common mtDNA ancestor because the set of human mtDNA alleles is a coalescent
set. The date and species associated with this ancestor are thus matters for scientific investigation. (I
would add that anyone who wants to start a thread on mtEve would find an eager participant here.)
I will summarize:
"Last common ancestors" do not have the meaning we would like when talking about sets of biological entities
that are not coalescent sets. These include sexually reproducing individuals and populations (though certain
subsets of populations may be coalescent). In these sets, the "last common ancestor" of two elements takes
on a strictly genealogical meaning, one that is obtained only by analogy.
Coalecent sets include sets of species and sets of alleles (though in certain circumstances, new alleles may
be produced through recombination). In these sets, the "last common ancestor" of two elements refers
specifially to the shared element immediately preceding in time the split between the lineages leading to
the two elements.
Looking forward to further discussion, and
Apologetic for the length of the post,
John Hawks
Stephen Barnard
I suppose that in strict mathematical terms it doesn't. That would
require a strictly decreasing cardinality of sets.
From a practical point of view I'm satisfied with it. The group that
included the putative mitochondrial Eve would very probably have been
pretty small. But you never know. :-) I've seen speculation in this
newsgroup that relies on *much* flimsier assumptions.
Steve Barnard
Stephen Barnard
John Hawks wrote:
>
> Susan and Stephen (and whoever else),
>
> Hi, I'm John Hawks, and I study with Dr. Wolpoff at the University of Michigan. I want to let you know that
> you're both right about "last common ancestors." However, the topic is one that readily invites confusion,
> so I'm glad to see this thread appear as it promises to be a very useful discussion.
>
Thanks for clarification, John. It was very informative.
I don't *like* to sound like a know-it-all, but I know that I do at
times. You obviously know a lot more about this topic than I do.
BTW, the reason I'm somewhat up-to-speed on this topic is that I've been
reading Daniel Dennet's book, Darwins Dangerous Idea, which I like very
much so far.
Steve Barnard
Jane Andrews
On Mon, 12 Aug 1996, Stephen Barnard wrote:
> Susan S. Chin wrote:
> >
> > I did read about my Eve as a population and not an individual
> > ancestor recently, but don't recall the source.
> > > >
then stephen replied (amongst other things)
>
> Here's the logic that demonstrates that there is a single Last Common
> (female) Ancestor:
>
> Let S_1 be the set of all people whom are alive today. Let S_2 be the
> set of mothers of the members of S_1. In general, let S_k be the set
> of mothers of S_k-1. The size of these sets is nondecreasing (S_k <=
> S_k-1), because everyone has only one mother, but some mothers have
> more than one child. When the size of the set reaches 1 then we have
> arrived at the mitochondrial Eve.
>
> Steve Barnard
>
>
I with Susan on this one. Unfortunatly I can't remember the reference
either (I can go and look it up tomorrow if anyone's interested) but I
remember reading a paper which idicated that the proposed "eve" could have
come from an ancestoral population of up to 10 000 individuals. Her
mitochondrial sequence has been preserved by random genetic drift and all
the others lost by chance, stochastic processes. This means that there
is no need to invoke a huge bottle neck reducing, the effective
female population to one, in order to explain the apparent single female
ancestor.
Steve points out that mitochondrial DNA is not the whole story. Nuclear
DNA is inherited from both parents and so follows different patterns to
mtDNA. An individuals nuclear DNA has many more ancestors than his or her
mtDNA. Thus all those individuals in the ancesoral population whose mtDNA
was lost by drift may have contributed nuclear DNA to later individuals
carrying the remaining mtDNA - perhaps nuclear sequences that were not
present in any of the ancestoral population who did carry the surviving
mtDNA. Thus individuals who are not represented by an "Eve" character (ie
the didn't have the ancestoral mtDNA sequence) have contributed to the
genetic vartion of the desendent populations (us!).
I'm not sure that I've explained myself very clearly, but I hope that this
makes sence.
Jane Andrews.
John Hawks
Jane Andrews wrote:
>
> . . . I
> remember reading a paper which idicated that the proposed "eve" could have
> come from an ancestoral population of up to 10 000 individuals. Her
> mitochondrial sequence has been preserved by random genetic drift and all
> the others lost by chance, stochastic processes. This means that there
> is no need to invoke a huge bottle neck reducing, the effective
> female population to one, in order to explain the apparent single female
The idea of the population of 10,000 comes from the original mtEve paper:
Cann, Stoneking and Wilson, 1989. Mitochondrial DNA and human evolution.
Nature 325:31-6, Jan 1 '87.
The authors make it clear that there was no time in the past during which
the lineage leading to living humans consisted of only one couple. They
assert that the individual that carried the unique mtDNA source of all living
people was a member of a group of individuals, all of which are ancestors of
living people.
However, the authors viewed this population is a small, bottleneck population.
The reason for this bottleneck is the comparatively recent date that Cann et al.
associate with their mitochondrial ancestor. If the date had been much more
ancient (like 1Ma or more), they would perhaps have needed no special
explanation--a million years is a long time for random processes to knock out
all but one million-year-old lineage and its descendent lineages. It is
easily apparent that living humans must all share a single mitochondrial
ancestor at some point (take for instance, the original mitochondrial
organism). Cann et al. didn't have to prove that there was a single mitochondrial
ancestor. What they had to do was explain why their date (ca. 200ka) was so
young.
There are at least two potential explanations. One is that the mitochondrial
variant apparently shared by living humans conferred some sort of selective
advantage, leading to its replacement of other human mtDNA variants. Cann et al.
rejected this explanation, not without good reason. At the time, few
mitochondrial diseases had been described, and researchers were finding out
that the mtDNA molecule had had most of its functions abrogated by the nuclear
DNA. Cann et al. therefore assumed that the majority of mtDNA mutations were
neutral.
The other explanation is that there was a *large* bottleneck at about the
time that the mitochondrial ancestor lived. This is the explanation that
Cann et al. chose, because it made demographic sense, and because a group
of archaeologists and paleoanthropologists were suggesting a theory that
would fit well with just such a bottleneck. Out of Africa was born.
This is the explanation of the "population" that you had heard about. I'm
sure I needn't point out that many people have disagreed with the bottleneck
explanation, and many others have disputed the 200ka date. More discussion
can follow if anyone is interested.
John Hawks
Stephen Barnard
Stephen Barnard wrote:
>
> CHESSONP wrote:
> >
> > "Let S_1 be the set of all people whom are alive today. Let S_2 be the
> > set of mothers of the members of S_1. In general, let S_k be the set
> > of mothers of S_k-1. The size of these sets is nondecreasing (S_k <=
> > S_k-1), because everyone has only one mother, but some mothers have
> > more than one child. When the size of the set reaches 1 then we have
> > arrived at the mitochondrial Eve.
> >
> > Steve Barnard"
> >
> > coverges to one. To establish the existence of a "Last Common Ancestor"by
> > logical means only will require quite a lot more than you have supplied,
> > starting with a clear definition of what is the "Last Common Ancestor" for
> > a species.
>
> I suppose that in strict mathematical terms it doesn't. That would
> require a strictly decreasing cardinality of sets.
>
> From a practical point of view I'm satisfied with it. The group that
> included the putative mitochondrial Eve would very probably have been
> pretty small. But you never know. :-) I've seen speculation in this
> newsgroup that relies on *much* flimsier assumptions.
>
> Steve Barnard
I've been thinking about this some more, and I'm becoming convinced that the
argument for a single mitochondrial Eve, while not airtight, is about as close to
airtight as anything gets in paleoanthropology. Bear with me.
If the cardinality (i.e., the size) of the sets were strictly decreasing there
would be no problem. We would eventually arrive at the singleton set. That's
just simple mathematical induction.
There is no question that the sizes of the sets does not increase. That would
require at least one person to have more than one mother. Obviously, that's not
possible. I think I'm on pretty safe ground there.
The only problem arises when the sizes of the sets remains unchanged. Suppose
S_k = S_k-1 = N (where I mean the *sizes* of the sets). This would mean that
every member (i.e., each of the N mothers) in S_k had *exactly* N daughters who
were in S_k-1, who in turn must all bear female children (because they are
mothers of mother in subsequent sets).
Clearly, for large N this is extremely unlikely. But it's worse than that. Even
for small N (say N=2), for N never to reach one it would be required that both
mothers in set S_k-1 be the *only reproducing daughter* of the two mothers in set
S_k, and this state of affairs would have to be maintained backward in time *in
perpetuity*!
So it appears to me that the case for a single mitochondrial Eve is made. If
anyone can point out a flaw in this argument (assuming anyone has waded through
all this), I'd be very interested to hear it.
Steve Barnard
Stephen Barnard
Jane Andrews wrote:
>
> On Mon, 12 Aug 1996, Stephen Barnard wrote:
>
> > Susan S. Chin wrote:
> > >
> > > I did read about my Eve as a population and not an individual
> > > ancestor recently, but don't recall the source.
> > > > >
> then stephen replied (amongst other things)
> >
> > Here's the logic that demonstrates that there is a single Last Common
> > (female) Ancestor:
> >
> > set of mothers of the members of S_1. In general, let S_k be the set
> > of mothers of S_k-1. The size of these sets is nondecreasing (S_k <=
> > S_k-1), because everyone has only one mother, but some mothers have
> > more than one child. When the size of the set reaches 1 then we have
> > arrived at the mitochondrial Eve.
> >
> > Steve Barnard
> >
> >
> remember reading a paper which idicated that the proposed "eve" could have
> come from an ancestoral population of up to 10 000 individuals. Her
> mitochondrial sequence has been preserved by random genetic drift and all
> the others lost by chance, stochastic processes. This means that there
> is no need to invoke a huge bottle neck reducing, the effective
> female population to one, in order to explain the apparent single female
> ancestor.
[snip]
>
> Jane Andrews.
See my later post (and a slight correction) where I flesh out this argument.
I freely admit that there *may* be a flaw in it, but I don't see it, and I'd
really like someone to point it out if it exists.
The argument seems to demonstrate that, except for a chain of circumstances
that would be vanishingly unlikely, there really is a single last common
female ancestor.
Now, whether the date of the "Eve" that was deduced from mDNA is accurate, I
don't know.
Steve Barnard
Stephen Barnard
Stephen Barnard wrote:
>
[snipped for brevity]
>
> Clearly, for large N this is extremely unlikely. But it's worse than that. Even
> for small N (say N=2), for N never to reach one it would be required that both
> mothers in set S_k-1 be the *only reproducing daughter* of the two mothers in set
> S_k, and this state of affairs would have to be maintained backward in time *in
> perpetuity*!
>
Oops, there's a slight flaw here, but it's not fatal.
The mothers in S_k could have other reproducing daughters that weren't
in S_k-1 because their lineages died out before the present day.
Other than that, the argument still follows as before.
Steve Barnard
T&B Schmal
> Let S_1 be the set of all people whom are alive today. Let S_2 be the
> set of mothers of the members of S_1. In general, let S_k be the set
> of mothers of S_k-1. The size of these sets is nondecreasing (S_k <=
> S_k-1), because everyone has only one mother, but some mothers have
> more than one child. When the size of the set reaches 1 then we have
> arrived at the mitochondrial Eve.
I am with you until the set reaches 1. At that point we have the LCA, who
may or may not be "mitochondrial" Eve.
Suppose mitochondrial Eve living 200 Kya owned one of fifty possibly
mitrochondria at the time. Other than that she was nothing special. Her
descendants had no particular edge over anyone else and her mtDNA remained
a small percentge of the various kinds of mtDNA in existence. But say
one day 50 Kya a human who happened to have her mtDNA was blessed with the
gift of full speech and that person's descendants quickly populated the
earth. That human would be the "speech" Eve and she would be the LCA, the
set of 1 you calculate. Mitrochondrial Eve would be way out of the
running as the LCA.
I'm not sure if Last Common Ancestor can be defined genetically, but it
certainly can be defined through lineage. My thesis is that if you can
find the LCA you can find the point that true modern man emerged. So where
(when) would be a good place to look?
Tom Schmal
Susan S. Chin
Stephen Barnard (st...@megafauna.com) wrote:
: Stephen Barnard wrote:
: >
: > >
: > > "Let S_1 be the set of all people whom are alive today. Let S_2 be the
: > > set of mothers of the members of S_1. In general, let S_k be the set
: > > of mothers of S_k-1. The size of these sets is nondecreasing (S_k <=
: > > S_k-1), because everyone has only one mother, but some mothers have
: > > more than one child. When the size of the set reaches 1 then we have
: > > arrived at the mitochondrial Eve.
: > > Steve Barnard"
: Clearly, for large N this is extremely unlikely. But it's worse than that. Even
: for small N (say N=2), for N never to reach one it would be required that both
: mothers in set S_k-1 be the *only reproducing daughter* of the two mothers in set
: S_k, and this state of affairs would have to be maintained backward in time *in
: perpetuity*!
: So it appears to me that the case for a single mitochondrial Eve is made. If
: anyone can point out a flaw in this argument (assuming anyone has waded through
: all this), I'd be very interested to hear it.
: Steve Barnard
The problem I see with this argument is that evolution occurs in nature,
not in a mathematical equation (or whatever that was). As such,
hierarchies exist, meaning differing levels of organization produce
elements that can't possibly be accounted for by simple equations. It'd
be nice (no, it wouldn't actually) if things were as simple as you've
made it out. But I seriously doubt that they ever are in nature.
Susan
--
sus...@netcom.com
Susan S. Chin
F...@megafauna.com> <320F7F...@umich.edu> <320FEC...@megafauna.com>:
Organization: NETCOM On-line Communication Services (408 261-4700 guest)
Stephen Barnard (st...@megafauna.com) wrote:
: Steve Barnard
Gee Steve, now that you mention it, you do often come across like a know
it all. I don't know what your background in Paleoanthro is, but you will
find that, as a historical science, there is hardly ever any situations
where you will ever "know it all," "know it for certain," and least of
all, be in a position to tell someone flat out they are wrong (however
reluctantly of course).
I don't claim to know it all, but then again, I don't go around talking
condescendingly to others in the newsgroup either. Just something for you
to think about in future posts.
Susan
--
sus...@netcom.com
Stephen Barnard
Susan S. Chin wrote:
>
> I don't claim to know it all, but then again, I don't go around talking
> condescendingly to others in the newsgroup either.
Well, you just did, didn't you?
Steve Barnard
Stephen Barnard
Susan S. Chin wrote:
>
> The problem I see with this argument is that evolution occurs in nature,
> not in a mathematical equation (or whatever that was). As such,
> hierarchies exist, meaning differing levels of organization produce
> elements that can't possibly be accounted for by simple equations. It'd
> be nice (no, it wouldn't actually) if things were as simple as you've
> made it out. But I seriously doubt that they ever are in nature.
>
The only fact of nature that this argument relies on is that no one has
more than one mother. I suppose there are other hidden assumptions,
such as that all inheritance comes from one's parents, and not from (for
example) gene transfer via viruses. Otherwise, it's just an exercise in
logic. There may be an error in the logic, because I'm far from
infallible, but you haven't pointed it out. You merely questioned my
motives in another post.
Mathematics isn't a useless frippery. It allows us to draw conclusions
from assumptions. If the assumptions are true and the logic is correct
then the conclusions are correct.
Steve Barnard
Susan S. Chin
Stephen Barnard (st...@megafauna.com) wrote:
: Steve Barnard
As you've said, I'm not questioning your logic. What I do question though
is that the only "fact of nature used is that everyone has only one
mother," and whether such an exercise in logic proves anything at all.
What I see that it does "prove," if there are no logical flaws, is that
it is logistically possible that we all descended from a sole ancestor
popularly known as mitochondrial Eve. If that was your goal, and assuming
your argument is logically correct, then you would have accomplished it.
Susan
--
sus...@netcom.com
CHESSONP
In article <32105A...@megafauna.com>, Stephen Barnard
<st...@megafauna.com> writes:
An example. Let I_k be the interval of real numbers greater than -1/k and
less than 1+1/k, or in more familiar notation, (-1/k, 1+1/k). This
interval has length 1+2/k. If we consider the sequence of interval lengths
{1+2/k} obtained from the associated sequence of intervals {I_k}, we see
that the lengths are strictly decreasing. However, the sequence of lengths
does not go to zero, but rather to a non-zero number, namely one.
Geoffrey Norman Watson
On Sun, 11 Aug 1996, T&B Schmal wrote:
> I wrote:
>
> I am interested in bracketing the dates for the last common ancestor of
> humankind. Would a good guess be somewhere between "Eve" and the
> appearance of modern man - say, between 200 and 50 Kya? It seems to me
> that dates outside this range would be impossible.
>
> Tom Schmal
>
>
> CHESSONP asks:
> >
> > To establish the existence of a "Last Common Ancestor"by
> > logical means only will require quite a lot more than you have supplied,
> > starting with a clear definition of what is the "Last Common Ancestor" for
> > a species.
>
>
> What I ment by LCA is tough to define in genetic terms, but from a lineage
> point of view, it is a person (well, really a pair of persons) from which
> all of us living are directly descended, ie, each of us can trace our
> ancestory to that one person. Now, we can also all trace our ancestory to
> that person's mother but she would not be the last common ancestor, she
> would be the second-to-last.
>
The main problem with this definition of LCA is what possible interest or
significance can it have? Each part of the human DNA has a history which,
since lineages die out at random, may be traceable to a single origin - an
LCA in this sense. However there are possibly a large number of such LCAs
and their contibutions are only to a small fragment of our DNA - and in
most cases our copy has been modified by mutation anyway.
The only place for an LCA is on a Trivial Pursuit card :).
The significance of LCA analysis is in the process of deciphering the
lineages, and the tree that results. This may give us important information
about our history as a species.
--------------------------------------------------------------------------\
Geoffrey Watson gw...@cs.uq.edu.au
Stephen Barnard
CHESSONP wrote:
>
>
> An example. Let I_k be the interval of real numbers greater than -1/k and
> less than 1+1/k, or in more familiar notation, (-1/k, 1+1/k). This
> interval has length 1+2/k. If we consider the sequence of interval lengths
> {1+2/k} obtained from the associated sequence of intervals {I_k}, we see
> that the lengths are strictly decreasing. However, the sequence of lengths
> does not go to zero, but rather to a non-zero number, namely one.
I don't see how this applies to my argument. Your example operates in an
entirely different domain (infinite sets of real numbers in some bounded
interval), while my argument deals with finite sets. Furthermore, your
example doesn't make use of the critical inheritance property (everyone has
exactly one mother) which is the basis for my argument.
You can't disprove an argument by presenting an entirely different problem
and saying that it doesn't lead to the same answer.
Thanks for trying, though.
Steve Barnard
CHESSONP
In article <3213B0...@megafauna.com>, Stephen Barnard
<st...@megafauna.com> writes:
The point of the example is to illustrate the fault in your logic. A
decreasing sequence of numbers may not converge to anything. A decreasing
sequence of numbers that is bounded will converge to its greatest lower
bound which need not be zero. A decreasing sequence of positive whole
numbers will converge to zero, but any finite segment of this sequence
will not. A non-increasing sequence of positive whole numbers will
converge to its greatest lower bound, which may not be zero.
You cannot claim that the size of your set of great .... grand mothers
reaches one at some point simply because each set (going backward by
generation) is not larger than the previous one. On logical grounds alone
the number of females in this "lca founding population" could be any
number no larger than the smallest known human population size at some
time.
John Hawks
CHESSONP wrote:
> The point of the example is to illustrate the fault in your logic. A
> decreasing sequence of numbers may not converge to anything. A decreasing
> sequence of numbers that is bounded will converge to its greatest lower
> bound which need not be zero. A decreasing sequence of positive whole
> numbers will converge to zero, but any finite segment of this sequence
> will not. A non-increasing sequence of positive whole numbers will
> converge to its greatest lower bound, which may not be zero.
>
> You cannot claim that the size of your set of great .... grand mothers
> reaches one at some point simply because each set (going backward by
> generation) is not larger than the previous one. On logical grounds alone
> the number of females in this "lca founding population" could be any
> number no larger than the smallest known human population size at some
> time.
You are right about Steve's proof. In fact, to be accurate, Steve's proof must be
expressed in terms of probability. In each generation, each mother will, with a
certain probability, have one daughter, more than one daughter, or no daughters. The
use of probability is what introduces *stochasticity* (of which you will probably
have heard) into the proof. It cannot be demonstrated in this way that there *must*
be a single mitochondrial ancestor. However, it can be demonstrated with
*overwhelming* likelihood that such an ancestor existed.
In addition, we can evaluate the likelihood that such an ancestor existed during any
period of time, making whatever assumptions about population demography that we
want. In this way, hypotheses about past population history can be tested with
reference to the *actual* mtDNA distribution of living humans. Such a process of
inference can, with some alterations, be applied to alleles of other genes.
I will point out that a non-probabilistic demonstration of the existence of a single
mtDNA ancestor is a corollary of the assumption that the mitochondria of animal and
plant cells must have had a single origin as an independent mitochondrial organism.
John Hawks
Stephen Barnard
John Hawks wrote:
>
> CHESSONP wrote:
>
> > The point of the example is to illustrate the fault in your logic. A
> > decreasing sequence of numbers may not converge to anything. A decreasing
> > sequence of numbers that is bounded will converge to its greatest lower
> > bound which need not be zero. A decreasing sequence of positive whole
> > numbers will converge to zero, but any finite segment of this sequence
> > will not. A non-increasing sequence of positive whole numbers will
> > converge to its greatest lower bound, which may not be zero.
> >
> > You cannot claim that the size of your set of great .... grand mothers
> > reaches one at some point simply because each set (going backward by
> > generation) is not larger than the previous one. On logical grounds alone
> > the number of females in this "lca founding population" could be any
> > number no larger than the smallest known human population size at some
> > time.
>
> You are right about Steve's proof. In fact, to be accurate, Steve's proof must be
> expressed in terms of probability. In each generation, each mother will, with a
> certain probability, have one daughter, more than one daughter, or no daughters. The
> use of probability is what introduces *stochasticity* (of which you will probably
> have heard) into the proof. It cannot be demonstrated in this way that there *must*
> be a single mitochondrial ancestor. However, it can be demonstrated with
> *overwhelming* likelihood that such an ancestor existed.
>
Now just a darn minute, John. I *did* make a probabilistic argument.
Go back and read it again. I admit that the *original* argument
wasn't probabilistic and it was flawed, but I fixed it up. I think
we're saying exactly the same thing.
Steve Barnard
Stephen Barnard
> decreasing sequence of numbers may not converge to anything.
There are numbers (real numbers) and then there are NUMBERS (natural numbers).
It is very easy to define an iterated sequence of finite sets with a stochastic
transition rule that converges to a singleton set with probability one. The
sequence I defined is an example.
> A decreasing
> sequence of numbers that is bounded will converge to its greatest lower
> bound which need not be zero. A decreasing sequence of positive whole
> numbers will converge to zero,
The sequence is non-increasing. It converges to a singleton set -- not to a
null set. (Well, eventually it does, at the origin of life!)
> but any finite segment of this sequence
> will not. A non-increasing sequence of positive whole numbers will
> converge to its greatest lower bound, which may not be zero.
>
> You cannot claim that the size of your set of great .... grand mothers
> reaches one at some point simply because each set (going backward by
> generation) is not larger than the previous one. On logical grounds alone
> the number of females in this "lca founding population" could be any
> number no larger than the smallest known human population size at some
> time.
Are you claiming that the set of mothers will (each) have exactly one daughter
whose lineage continues to the present day, with probability one? And that
this amazing probability will extend back in time forever? That's what's
required for the size of the set to never decrease.
Steve Barnard
CHESSONP
In article <32153B...@megafauna.com>, Stephen Barnard
<st...@megafauna.com> writes:
Help me flush out the model.
Let H be the set of all human beings alive at time t_0. Let M_0 be
the set of mothers (living or dead) of the members of H. Represent the
functional relationship between H and M_0 by the symbol f, defined by
m=f(h) means that m "is the mother of" h.
The function so defined, f:H -->M_0, is onto, meaning that every element
in M_0 has a "preimage" in the set H. Since H is finite, this implies
logically that M_0 is finite, and, indeed, that the size of M_0 is no
greater than the size of H.
With this functional relationship we can define a sequence of sets
{M_k} for all positive numbers, k, by defining:
"m is an element of M_k, if, and only if, there exists an
element d
of the set M_k-1, such that m=f(d)."
About the sets M_k, we know only that the size of M_k is no greater
than the size of M_k-1. We will need further assumptions to get the
sequences of sizes of {M_k} to converge to one.
I apologize for the clumsiness of the notation, but without a decent
symbol set I think this is the best I can do.
T&B Schmal
After reading the following description Susan Chin suggested I use "most
recent common ancestor." LCA has a different meaning entirely in
paleoanthro.
A previous post read:
"What I ment by LCA is tough to define in genetic terms, but from a lineage
point of view, it is a person (well, really a pair of persons) from which
all of us living are directly descended, ie, each of us can trace our
ancestory to that one person. Now, we can also all trace our ancestory to
that person's mother but she would not be the last common ancestor, she
would be the second-to-last.
"As an analogy - at the Schmal family reunion, all the great grandchildren
can trace their lineage to great grandma Schmal. They can also trace
their lineages to many other great grandmas (through my wife's parents,
for example). But for all the children at the reunion, great grandma
Schmal is the LCA.
"Eve, like the missing link, the first primate, the first single-celled
organism, is one of our common ancestors. At this point she is the last
Known common ancestor. As more genetic analysis is done she will
(probably) be displaced by someone more recent. My question is what is
the most recent possible date that will be eventually determined to be the
date of the LCA."
So, with this definition, and LCA replaced by MRCA, will anyone venture a
guess and a possible reason for the date?
Tom Schmal
Stephen Barnard
CHESSONP wrote:
>
> Help me flush out the model.
>
> Let H be the set of all human beings alive at time t_0. Let M_0 be
> the set of mothers (living or dead) of the members of H. Represent the
> functional relationship between H and M_0 by the symbol f, defined by
>
> m=f(h) means that m "is the mother of" h.
>
> The function so defined, f:H -->M_0, is onto, meaning that every element
> in M_0 has a "preimage" in the set H. Since H is finite, this implies
> logically that M_0 is finite, and, indeed, that the size of M_0 is no
> greater than the size of H.
>
> With this functional relationship we can define a sequence of sets
> {M_k} for all positive numbers, k, by defining:
>
> "m is an element of M_k, if, and only if, there exists an
> element d
> of the set M_k-1, such that m=f(d)."
>
> About the sets M_k, we know only that the size of M_k is no greater
> than the size of M_k-1. We will need further assumptions to get the
> sequences of sizes of {M_k} to converge to one.
>
> I apologize for the clumsiness of the notation, but without a decent
> symbol set I think this is the best I can do.
Your definition is fine, as far as it goes. Like you say, you need further
assumptions to get the size to converge to one.
The assumption that I made, which seems perfectly reasonable to me, is that the
probability that every mother in M_k-1 produces *exactly* one mother in M_k is
less than one.
Steve Barnard
Geoffrey Norman Watson
On Wed, 21 Aug 1996, Stephen Barnard wrote:
>
> Your definition is fine, as far as it goes. Like you say, you need further
> assumptions to get the size to converge to one.
>
> The assumption that I made, which seems perfectly reasonable to me, is that the
> probability that every mother in M_k-1 produces *exactly* one mother in M_k is
> less than one.
>
> Steve Barnard
>
Your assumption is reasonable, but how to justify it? One defect of this analysis
is that it is difficult to see how it could be used for simulations. The
generation of the sets M_k is in the opposite direction to time, and a lot of
what is happening depends on the rest of the population, which is not modelled
here.
Forwards simulations from a small base population show that female-line-lineages
die out randomly. To eliminate them rapidly you need a small population, and
obviously the probability of losing one falls as the number of lineages left
reduces. This supports your requirement that the probability is less than one
but I can't see how you would derive it rigorously.
------------------------------------------------------------------------------
Geoffrey Watson gw...@cs.uq.edu.au
Stephen Barnard
Geoffrey Norman Watson wrote:
>
> On Wed, 21 Aug 1996, Stephen Barnard wrote:
>
> >
> > assumptions to get the size to converge to one.
> >
> > The assumption that I made, which seems perfectly reasonable to me, is that the
> > probability that every mother in M_k-1 produces *exactly* one mother in M_k is
> > less than one.
> >
> > Steve Barnard
> >
> Your assumption is reasonable, but how to justify it?
It can be justified on a priori statistical grounds. For large sets it can be
justified by the law of large numbers. For small sets it can be justified for
essentially the same reason, but by looking at averages over time. I haven't
actually *done* such a thing, but it's pretty clear to me that it can be done.
> One defect of this analysis
> is that it is difficult to see how it could be used for simulations. The
> generation of the sets M_k is in the opposite direction to time, and a lot of
> what is happening depends on the rest of the population, which is not modelled
> here.
>
This is true, but I question the value of simulations anyway. The date of the LCA
(female) is so contingent that I don't see how a simulation could give a meaningful
estimate. Maybe a large number of simulations could narrow it down. I don't know.
It seems like you would have to simulate forward in time, but maybe not.
One thing to realize is that all of the members of a set M_k are not necessarily
alive at the same time, so k cannot be exactly correlated with the date. As k
increases you would expect the dates of the members of M_k to spread out.
> Forwards simulations from a small base population show that female-line-lineages
> die out randomly. To eliminate them rapidly you need a small population, and
> obviously the probability of losing one falls as the number of lineages left
> reduces. This supports your requirement that the probability is less than one
> but I can't see how you would derive it rigorously.
>
> ------------------------------------------------------------------------------
> Geoffrey Watson gw...@cs.uq.edu.au
Steve Barnard