turpin@cs.utexas.edu (Russell Turpin) (12/10/90)
----- The subject is the theory of an "Eve", one woman who is the matrilineal ancestress of all living humans. Since others may have some interest in this, I have cross-posted it to soc.history and sci.bio. The results of a simple statistical model are included below. ----- In a previous posting, I described how for any fixed population, an Eve is expected to emerge every N generations, where N is a number that is a statistical function of population and mating patterns. In article <1613@sun13.scri.fsu.edu> mayne@vsserv.scri.fsu.edu (William (Bill) Mayne) writes: > Yes, but for populations of significant size N will be very large. Mr Mayne guides his intuitions by one highly improbable case, and later tries incorrectly to extend it to the far more likely case. He writes: > The chance that in a population with n reproducing females n-1 > will have only male offspring is very remote for n of 1000 or > more. The exact probability will depend upon the average number > of offspring and the distribution. ... The probability is only > slightly less that there would arise a generation consisting > of only males, resulting in extinction. ... Actually every Eve > will represent a close brush with extinction, since there will > be only a few females in the generation after Eve. ... All of the above is based on the idea that Eve emerges as a result of all other contemporary women having only sons. This need never happen. Some of Eve's contemporaries might bear only sons, but most of them probably had descendants through matrilineal lines that lasted many generations. As Mr Mayne writes: > Granted, more complex scenarios are possible, such as other > females having female offspring but those female lines becoming > extinct as far as unbroken female lineages go some generations > later. ... This scenario, which is not the one Mr Mayne talks about above, is the *far* more probable one. > ...The calculation of your N is further complicated since human > population is not in a steady state. ... As a first approximation, this a fair assumption for most of human history. For hundreds of thousands of years prior to the neolithic revolution, the human population (like other animal populations) was limited by how much food could be gathered from the natural landscape. > ... Still, I would venture the conjecture that for a population > with 1000 females N would exceed the number of generations > since humans first appeared, probably by a huge factor. ... Mr Mayne guesses wrong. An analytic solution is beyond me, but writing a probabilistic model for this problem was a simple exercise, and provided me a good excuse to try out a new C compiler on my Macintosh. The number of generations required to eliminate all but one matrilineal line are shown below. F is the number of women in a steady-state population. (Except for the largest case, I made several runs for each value of F.) #generations (N) #years (15yr/gen) F least most least most ---- ----- ---- ----- ----- 8 5 37 75 1605 128 172 625 2580 9375 1024 2392 4163 35880 62445 Thus, for a population of a few hundred, an Eve appears every few millenia. For a population of two or three thousand, an Eve appears every several tens of thousands of years. One characteristic of matrilineal descent that was made clear from the model is that the presence of living matrilineal descendants from one woman is NOT a stable point, unless she is the sole matrilineal ancestress of the entire population. The greater the portion of the population that is matrilineally descended from her, the greater the likelihood of this portion increasing. But if a lesser portion of the population is matrilineally descended from her, then it tends to decrease. It is far from clear to me that a steady population is the most favorable assumption for the appearance of an Eve. My intuition is that a population that increases and decreases, and where relatives are more likely to survive or die out together, is more conducive to the appearance of an Eve. This is also a more likely description of population change in early human groups. It is not necessary to assume that this relatively small group was ever the only population of humans; only that all current humans are matrilineally descended from them. Other groups might have existed at the same time, but died out (or were killed off) with little matrilineal mixing with this one group. Finally, it should be kept in mind that social structures can impede or promote matrilineal mixing. For example, if women remain with their original group when they mate with men from another group (the man either remaining with his own group or changing to the woman's group), then there is NO matrilineal mixing between groups. This remains so even if people religiously mate outside their own group. This may have been an important factor in determining mitochondrial DNA inheritance. Russell
felsenst@milton.u.washington.edu (Joe Felsenstein) (12/10/90)
The theory of how far back a mitochondrial "Eve" should be is well-known. It involves the time until Nf female lineages have a common ancestor up the female line. It is a bit too complex to explain here but is based on old results in genetic drift theory. Basically with Nf females (N-sub-f) in a randomly reproducing population (one where each offspring comes from a randomly selected female independently of all others -- the classical Wright-Fisher model), the time until two randomly sampled females have a common ancestor up their female lines is on average Nf generations. For all females in the population the corresponding result is 2Nf generations. The result for two female lineages is easy to explain. Each generation going back there is a random chance with probability 1/Nf that they come to the same ancestor. Then it is just like tossing a coin with this probability of heads. The result for the time to first heads is a geometric distribution with mean time Nf generations. The result for all Nf females is more complex so I won't try to explain it unless there is some overwhelming demand, but it comes out as twice that time, on average. Keep in mind that this "Eve" will then be (1) not the common ancestor of other parts of the genome, and (2) by no means the only female in the population, and (3) by no means the only female in that generation who contributes genes to the population of the present. --- Joe Felsenstein, Dept. of Genetics, Univ. of Washington, Seattle WA 98195 DO NOT send to me at "milton" but instead please use: Internet/ARPANet: joe@genetics.washington.edu (IP No. 128.208.128.1) BITNET/EARN: FELSENST@UWALOCKE UUCP: ... uw-beaver!evolution.genetics!joe
elturner@phoenix.Princeton.EDU (Edwin L Turner) (12/11/90)
In article <15644@cs.utexas.edu> turpin@cs.utexas.edu (Russell Turpin) writes: >The subject is the theory of an "Eve", one woman who is the >matrilineal ancestress of all living humans. >In a previous posting, I described how for any fixed population, >an Eve is expected to emerge every N generations, where N is a >number that is a statistical function of population and mating >patterns. This problem is best thought of as one of asexual reproduction; basicly we are tracing the evolutionary tree of mitochondria, which could be thought of as asexual symbiotes living in human cells. It has a variety of analogies in other problems (e.g., the dwindling of the pool of family names in a culture which only passes on one gender's family name to offspring). >In article <1613@sun13.scri.fsu.edu> mayne@vsserv.scri.fsu.edu (William (Bill) Mayne) writes: >> ...The calculation of your N is further complicated since human >> population is not in a steady state. ... In a sufficiently rapidly growing population (see below), N goes to infinity; in other words, mulitple pure maternal lines survive indefinitely. >> ... Still, I would venture the conjecture that for a population >> with 1000 females N would exceed the number of generations >> since humans first appeared, probably by a huge factor. ... > >Mr Mayne guesses wrong. It sounds as though Mr. Mayne may be thinking vaguely about some sort of combinatorics problem, but this is actually a sort of random walk in log space problem, so N is not so sensitive to the population size. >An analytic solution is beyond me, but writing a probabilistic >model for this problem was a simple exercise, and provided me a >good excuse to try out a new C compiler on my Macintosh. I often wonder if computers will put an end to mathematics (as in that old Asimov story, title?). Anyway, a friend of mine and I worked out the problem when we first heard about the Eve stuff about 10 years ago. Unfortunately, I haven't easily been able to locate the calculation in my notes, but I remember the answer: A population with F females (or more accurately, F independent mitochondrial lines) will carry only a single maternal line (only a single mitochondrial) lines after just N=xF generations where x is a small numerical factor (2, 3/2, pi, ...) whose exact value I forget. Thus, a thousand females will produce an Eve in well less than 10^5 years. We did not solve the general problem in a growing population but did show that no Eve was ever likely to emerge if the population were growing geometricly with a doubling time short compared xF generations (which still allows very slow growth). I am sure all of this must be well known to biologists studying population dynamics. One interesting thing we do know about Eve is that she had two or more daughters both of whom had one or more daughters ... Ed Turner "Do you want to know a secret, phoenix!elturner Just between you and me? I don't know where I'm going; Yes, I don't know who I'm going to be." or elturner@phoenix.Princeton.EDU - The Other Side of This Life