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Suppose we took all living humans and found the set of their mothers, mothers' mothers, etc. and then traced down as far as possible. Is there a logical reason that this tree has to converge to one 'mitochondrial eve'?

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Yes; she wouldn't necessarily have to be of the same species, but "mitochondrial Eve" must exist. The proof is pretty simple if you assume no one has more than one mother and that some mothers share a mother (and some other reasonable biological assumptions like the finite number of offspring).

Consider all living maternal lineages at any slice of time. This would effectively be a slice of all mothers, let's call it Generation N. In turn, these mothers all have mothers. We can call this Generation N-1. Not all women who have children in this generation are included: N-1 only contains mothers of the Generation N mothers. Generation N-1 must then be smaller than Generation N, because no mother has more than one mother. In practice, it's likely much smaller, since any mother who has more than one daughter who herself becomes a mother reduces the total possible by 1 more. You can recursively go through maternal generations N-2, N-3, etc this way and will find that every successive generation gets smaller. Eventually it will converge to 1 person, and that's your "Eve".

You can think about it similarly in the opposite direction, and find that as you go forward, extant maternal lineages can disappear but once a lineage disappears it cannot come back.

See also the Galton-Watson process.

A lot of "mothers" the same age as mitochondrial Eve have descendants today (in fact, they are likely to be either ancestors of everyone or ancestors of no one), but those lineages did not pass maternally. See also https://en.wikipedia.org/wiki/Pedigree_collapse

Bryan Krause
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  • well what if at some stage none of the mothers are sisters, for every generation until the beginning of life –  Apr 29 '21 at 10:07
  • This isn't so far fetched. What you're asking for N is that for all living people, in the set of all mothers there is at least some who are sisters. Then for N-1 you are asking for the set of all matrilineal grandmas there is at least some who are sisters, then the same for great great grandmas etc. Remember we are starting with 7.5 B mothers though. –  Apr 29 '21 at 10:31
  • what's to say that if we go all the way down, we don't find multiple sources for life –  Apr 29 '21 at 10:32
  • Brian. I’m sorry but you need to improve this answer, because I happen not to know about this and cannot understand your answer, despite a background education in molecular biology. If it is better answered by a text, close it as homework. Otherwise I need some sort of diagram and a comparison of alternatives (I presume the poster has an alternative in mind, but he does not state it clearly.) – David Apr 29 '21 at 18:51
  • @David I don't think any molecular biology is necessary here, just a recognition that each prior maternal generation must be smaller (or equal) in size. Of course molecular biology is useful in estimating just how long ago that common maternal ancestor would be, but that's not really what the question asked about. – Bryan Krause Apr 29 '21 at 18:57
  • @ggmate "what if at some stage none of the mothers are sisters, for every generation until the beginning of life" - there is no biological basis for this organization. It would require every mother to have at least one daughter, otherwise the population as a whole is still decreasing and so the number of unique maternal generations is decreasing. This can happen rarely, but there is no process to enforce it in every pairing across a population over history. – Bryan Krause Apr 29 '21 at 19:00
  • Hmm not sure I quite understand. If we take the set of all matrilineal great^n grandmas of living offspring. Why is it likely for a large number of them to be sisters? –  Apr 29 '21 at 22:32
  • @ggmate It's not just that a large number are likely to be sisters (though in practice they are, because not everyone lives long enough to have children and many live long enough to have multiple children), it's that the number of unique matrilineal lines with continuity to the future can only decrease as you go back in history, never increase. – Bryan Krause Apr 29 '21 at 22:35
  • But the only way they can decrease is by being sisters right? –  Apr 29 '21 at 22:36
  • @ggmate Yes, which you must see is quite common, right? – Bryan Krause Apr 29 '21 at 22:39
  • I can see it for early generations but it’s not so clear why for going back many many generations –  Apr 29 '21 at 22:41
  • @ggmate Does it help if you realize that the further back you go, the more childbearing women of that generation have no maternal lineage offspring in the future/present generation? There is nothing conceptually different about going back 1 generation and going back 100, except that you start with fewer and fewer "candidates" to be the mitochondrial Eve. – Bryan Krause Apr 29 '21 at 22:42
  • it's still if you think of it from the perspective that you're telling me that out of 7.5 billion mothers, and maternal grandmothers, great grandmothers etc of living people. they all end up being sisters. Even if on average every mother has 2 sisters who is a mother, you have to divide by 3 = 2+1. but the average number of sisters who are grandmothers is lower, maybe 1.5. At every generation this probability gets smaller and smaller. until it gets close to 0, at which point there might be 10 000 offspring and very few being related no matter how far back you go –  May 06 '21 at 07:42
  • it's not trivial that the reducing power is non negligible, from a mathematical perspective. Unless you specify details about the type of growth that is occurring with humans? –  May 06 '21 at 07:45