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Maximal Genetic Diversity from a Very Small Founding Generation - Gtx0 ?>
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Maximal Genetic Diversity from a Very Small Founding Generation
Posted: Posted June 7th by chiarizio
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Suppose an inhabitable area, far removed from any other inhabited area, is settled by a parental generation consisting of four fertile people; two men (A and B) and two women (Z and Y).

Suppose in the first filial generation each of the women has both a son and a daughter by each of the men.
A and Z have a son C(AZ) and a daughter X(AZ)
A and Y have a son D(AY) and a daughter W(AY)
B and Z have a son E(BZ) and a daughter V(BZ)
B and Y have a son F(BY) and a daughter U(BY).

Suppose in the second filial generation each woman has both a son and a daughter by each man who is not her father nor her son nor her full brother; and each man has a son and a daughter by each woman who is not his mother nor his daughter nor his full sister.
A and V have a son G(AV) and a daughter T(AV).
A and U have a son H(AU) and a daughter S(AU).
B and X have a son I(BX) and a daughter R(BX).
B and W have a son J(BW) and a daughter Q(BW).
C and Y have a son K(CY) and a daughter P(CY).
C and W have a son L(CW) and a daughter O(CW).
C and V have a son M(CV) and a daughter N(CV).
C and U have a son and a daughter. But now I’m running out of letters. I’ll have to try this again later.

Anyway, in the third filial generation, everyone has both a son and a daughter by each person of the opposite sex who is not their parent nor child nor full-sibling, nor grandparent nor grandchild nor half-sibling, nor full nibling (child of a full sibling) nor full auncle (full sibling of a parent); nor someone by whom they have previously had a child.

I haven’t got non-nebulous non-vague ideas about the fourth filial generation yet.

....

I have to come up with a better naming convention and a better way to track who has which common ancestors with whom.

....

Oh well.
If anyone has any suggestions, let me know.
I might use them. Even if I don’t use them as suggested, they might inspire me to think of whatever I do eventually use!

....

Stay safe and healthy, and wash your hands!


There are 6 Replies

I have to come up with a better naming convention and a better way to track who has which common ancestors with whom.


I suggest coding it out.

Here's some of the stuff I came up with in my notes:

Setup

Instead of assigning people new names, assign them percentages of the original four people, as well as their genders. So for example,

in generation one:

A M-A100
B M-B100
Z F-Z100
Y F-Y100


In generation two...

son C(AZ) M-A50Z50
daughter X(AZ) F-A50Z50
son D(AY) M-A50Y50
daughter W(AY) F-A50Y50
son E(BZ) M-B50Z50
daughter V(BZ) F-B50Z50
son F(BY) M-B50Y50
daughter U(BY) F-B50Y50


In this way you never run out of letters, and you can also keep up with who's distinct from who.

Now onto the rules...

Generation 1 Rules

Suppose in the first filial generation each of the women has both a son and a daughter by each of the men.


Nothing weird here, just pair all of the men with all of the women.

Generation 2 Rules

each woman has both a son and a daughter by each man who is not her father


M-B100 F-B50Y50


Can't share your genes with someone who only has those genes

nor her son


M-A100Z50 F-Z100


Can't share your genes with someone who only has those genes

nor her full brother


M-A100Y100 F-A100Y100


Can't have all of your genes in common

Based on that (and the fact that male-female and female-male are basically the same thing), there's really only two rules for generation two:

  • 1. Cant share your genes with someone who only has those genes
  • 2. Can't have all of your genes in common with someone else.

    Generation 3 Rules

    everyone has both a son and a daughter by each person of the opposite sex who is not their parent nor child nor full-sibling,


    These are covered by previous rules.

    nor grandparent nor grandchild


    Okay so with this one, there are two possibilities:

    Grandparent parent child
    A100 A50Z50 B50A25Z25
    A100 A50Z50 A25Z25B25Y25


    I'm going to omit genders for the time being because they don't matter.

    You can't share your genes with someone who only has those genes

    Oddly enough, this rule follows under the previous rule, so you don't actually need a new rule.

    nor half-sibling,


    This one was interesting to figure out the different possibilities for. I arrived at three combinations:

    A50Z50 B50Z50
    B50A25Z25 B50A25Y25
    A25Z25B25Y25 A25Z25B25Z25


    Weirdly, the last two combinations are indeed half-siblings despite both sharing 75% of their genes. The first of the last two is valid because AZ and AY are half-siblings, and B breeding with both of them separately is allowed. the last one is valid because AZ and BZ are half-siblings and half-siblings were allowed under generation two.

    You can't share your genes with someone who has 50% or more of them

    nor full nibling (child of a full sibling)


    You Your sibling Your child
    A50Z50 A50Z50 A25Z25B50


    Without doing much here, it's easy to see that you can omit the second step entirely, and this just falls under the "you can't breed with your children" rule. Your nibling's genes are going to be equivalent to your own child's genes.

    nor full auncle (full sibling of a parent);


    This is just the reverse of the above, so it's covered by the same rule.

    nor someone by whom they have previously had a child.


    This can be retranslated as:

    You can't create anyone new whose genes existed in generation 2 or before.

    However, if it's genetic diversity you're after, you can omit this rule completely because genetically, they're going to function as full siblings to whoever you had made previously.

    So your full ruleset for generation 3 looks like this:

  • 1. Cant share your genes with someone who only has those genes
  • 2. Can't have all of your genes in common with someone else.
  • 3. Can't share your genes with someone who has 50% or more of them
  • 4. Can't create anyone new whose genes existed in the previous generation or before

    Generalizing for future generations

    So you have two rulesets essentially:

    Generation 2

  • 1. Cant share your genes with someone who only has those genes
  • 2. Can't have all of your genes in common with someone else.

    Generation 3

  • 1. Cant share your genes with someone who only has those genes
  • 2. Can't have all of your genes in common with someone else.
  • 3. Can't share your genes with someone who has 50% or more of them
  • 4. Can't create anyone new whose genes existed in the previous generation or before

    Rule #4 there doesn't change anything if you add it to generation #2.

    So really the only difference between the two is rule #3. However, what's interesting is that rule #3 is just rule #2 with a different percentage.

    So the rulesets actually look like this:

    Generation 2

  • 1. Cant share your genes with someone who only has those genes
  • 2. Can't have 100% of your genes in common with someone else.
  • 3. Can't create anyone new whose genes existed in the previous generation or before

    Generation 3

  • 1. Cant share your genes with someone who only has those genes
  • 2. Can't share your genes with someone who has 50% or more of them
  • 3. Can't create anyone new whose genes existed in the previous generation or before

    Future Predictions

    Given this, it's easy to predict where other generations would go:

    Generation 4

  • 1. Cant share your genes with someone who only has those genes
  • 2. Can't share your genes with someone who has 25% or more of them
  • 3. Can't create anyone new whose genes existed in the previous generation or before

    Generation 5

  • 1. Cant share your genes with someone who only has those genes
  • 2. Can't share your genes with someone who has 12.5% or more of them
  • 3. Can't create anyone new whose genes existed in the previous generation or before

  • Edited June 7th by Xhin
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    Xhin
     

    @Xhin:
    I was indeed hoping for something like that!
    You understand what I was going for.
    I’m not sure, without actually working it out, that it’s going to work out — if you understand what I mean.
    I’m hoping to avoid too much avoidable genetic drift.
    I’m aware some may be in fact unavoidable.

    ____________________

    Thanks!


    Edited June 7th by chiarizio
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    @Xhin:
    I don’t think that rule for Generation 5 can work at all.
    And I think the rule for Generation 4 is just barely too strict.
    I think the Generation 4 rule would be, don’t mate with anyone if you and they have >25% of each other’s genes.
    The most-mixed individuals will have A25B25Y25Z25.
    We would have to allow two such persons to mate.
    ....
    But suppose in Generation 5 we had one of those people who wanted to mate with a A37.5B12.5Y37.5Z12.5.
    Should they be allowed to mate?


    Edited June 17th by chiarizio
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    @Xhin:


    Posted June 15th by chiarizio
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    I’m going to try this with three males and three females in the founding generation.
    ....
    Generation zero:
    Males include
    A100
    B100
    C100

    Females include
    X100
    Y100
    Z100

    .....

    Generation 1
    Nine new kinds of Male;
    A50X50
    A50Y50
    A50Z50
    B50X50
    B50Y50
    B50Z50
    C50X50
    C50Y50
    C50Z50

    Nine new kinds of female too;
    basically the full-sisters of those same kinds of Male.

    ......

    Generation 2
    Some A50X50 Male should mate with a female of each of the following kinds:
    Y100
    Z100
    A50Y50
    A50Z50
    B50X50
    C50X50
    and of course
    B50Y50
    B50Z50
    C50Y50
    C50Z50

    For each of the following kinds of Male, some A50X50 female should mate with one.
    B100
    C100
    A50Y50
    A50Z50
    B50X50
    C50X50
    and of course
    B50Y50
    B50Z50
    C50Y50
    C50Z50

    Similar things should be true of males and females of each of the other eight kinds that were new in Generation 1.

    This will give rise to 27 new kinds of Male and 27 new kinds of female.

    We’ll have
    A25B25X25Y25
    A25B25X25Z25
    A25B25Y25Z25
    A25C25X25Y25
    ...
    ...
    B25C25X25Y25
    ...
    ...

    And
    A50X25Y25
    A50X25Z25
    A50Y25Z25
    B50X25Y25
    ...
    ...
    C50X25Y25
    ....
    ...

    And we’ll have
    A25B25X50
    A25C25X50
    B25C25X50
    A25B25Y50
    ...
    ...
    A25B25Z50
    ...
    ...

    ........

    Generation 3

    No A25B25X25Y25 male should mate with any female of any of the following types:
    X100
    Y100
    A50X50
    A50Y50
    B50X50
    B50Y50
    A50X25Y25
    B50X25Y25
    A25B25X50
    A25B25Y50
    A25B25X25Y25
    But for every other type of female I think at least one A25B25X25Y25 Male should mate with a female of that type.

    Practically the same should be said of A25B25X25Y25 females.
    They shouldn’t mate with males of types
    A100
    B100
    A50X50
    A50Y50
    B50X50
    B50Y50
    A50X25Y25
    B50X25Y25
    A25B25X50
    A25B25Y50
    A25B25X25Y25
    But I think for any other type of Male at least one A25B25X25Y25 female should mate with a male of that type.

    So that means some A25B25X25Y25 male will mate with some female of each of the following types:
    Z100
    C50Z50
    A50Z50
    B50Z50
    C50X50
    C50Y50
    A25B25Z50
    A25C25Z50
    B25C25Z50
    C50X25Y25
    C50X25Z25
    C50Y25Z25
    A25C25X25Z25
    A25C25Y25Z25
    B25C25X25Z25
    B25C25Y25Z25
    and I guess
    A25C25X50
    A25C25Y50
    B25C25X50
    B25C25Y50
    A50X25Z25
    A50Y25Z25
    B50X25Z25
    B50Y25Z25
    A25B25X25Z25
    A25B25Y25Z25
    A25C25X25Y25
    B25C25X25Y25

    It should be easy to figure out which types of females each male of a type with 25% inheritance from each of two founders and two foundresses, should not mate with, and therefore which types one should mate with.

    ......

    A person of type A50X25Y25 should not mate with anyone who doesn’t have B100 nor C100 nor Z100 in their ancestry.
    But maybe someone of that type should mate with someone of each type who is descended from B100 and/or C100 and/or Z100.
    ...
    Likewise a person of type A25B25X50 shouldn’t mate with anyone who doesn’t have C100 nor Y100 nor Z100 as an ancestor or ancestress. But maybe someone of that type should mate with someone of each type that has that ancestor and/or one or both of those ancestresses.

    ............
    ............

    Generation 4

    The newly of-age people ready to become the parents of generation 4, are going to include several well-mixed types of individuals.
    For instance these nine:
    A25B12.5C12.5X25Y12.5Z12.5
    A25B12.5C12.5X12.5Y25Z12.5
    A25B12.5C12.5X12.5Y12.5Z25
    A12.5B25C12.5X25Y12.5Z12.5
    ...
    ..l
    A12.5B12.5C25X25Y12.5Z12.5
    ...
    ...

    There’ll be lots of other variety too.
    Any of 15 distributions of inheritance from the male founders such as
    A50
    A37.5B12.5
    A25B25
    A25B12.5C12.5

    could be paired with any of 15 distributions of inheritance from the female ancestresses such as
    X50
    X37.5Y12.5
    X25Y25
    X25Y12.5Z12.5

    6*6=36 of these 15*15=225 pairings were already members of previous generations; so that’s a maximum of 189 new types.

    We might recommend that, of each type who don’t have a particular founder as their ancestor, at least one male (if possible) and at least one female (if possible) mate with that founder, and at least one of each sex mate with some opposite-sex representative of each type who do have that founder as an ancestor.

    We should also recommend that no-one mate with anyone of exactly the same type.

    So far we’ve not recommended any parent-child nor full-sibling matings; and we’ve ruled out any full-sibling matings. But I don’t think we’ve ruled out all parent-child matings. I’ll have to think about it.

    ......

    =====

    Any way if an
    A25B12.5C12.5X25Y12.5Z12.5
    mates with an
    A12.5B25C12.5X12.5Y25Z12.5

    their offspring will be of type
    A18.75B18.75C12.5X18.75Y18.75Z12.5
    .....

    And in generation 5, if an
    A18.75B18.75C12.5X18.75Y18.75Z12.5
    mates with an
    A18.75B12.5C18.75X18.75Y12.5Z18.75

    their offspring will be of type
    A18.75B15.625C15.625X18.75Y15.625Z15.625

    ...

    And in generation 6 we might have a mating between an
    A18.75B15.625C15.625X18.75Y15.625Z15.625
    and an
    A15.625B18.75C15.625X15.625Y18.75Z15.625
    whose offspring would be of type
    A17.1875B17.1875C15.625X17.1875Y17.1875Z15.625

    And so on.

    The most well-mixed types would get closer and closer to having one-sixth (about 16.6667%) of their inheritance from each of the founders.

    Once the population is numerous enough we might just say that no-one can mate with an ancestor or descendant or child of an ancestor or descendant of a parent or grandchild of an ancestor or descendant of a grandparent or anyone with whom they share more than one great-grandparent.
    That would prohibit, among others, full- and half- first-cousins no matter how many generations removed, and full second-cousins, and double-half-second-cousins.
    But it would allow half-second-cousins, and full-second-cousins-once-removed, etc.

    To get to that point, though, the founding generation — generation 0 — would all have to have at least six children apiece; a son and a daughter by each of the three opposite-sex founders. In the first generation.
    And then in the second generation each of the founders would have to have 12 children apiece; a son and a daughter by each of six opposite-sex people in the first generation.
    For instance A100 would have to have a son and a daughter with a girl of each of the types B50X50, B50Y50, B50Z50, C50X50, C50Y50, C50Z50.
    This might be possible for the males A100, B100, C100; but it’s bound to be hard on the female foundresses X100, Y100, and Z100. Having already given birth to and raised six children, now they have to have a dozen more? I think at best they could go halfway.
    And then in generation 2, for each founder, there’ll be some newly mature people who are not descendants of that founder; probably three types of the opposite sex (at least). So the male founders will Each be called on to sire another six children, a son and a daughter by each of three newly adult women; and the female foundresses would also theoretically need to conceive another six children, which I just don’t think they could do.
    By Generation 3 some of the newly mature adults will have all six founders as ancestors, so their may be no need for the original founders to continue to reproduce. We might want a rule that if two people both had 37.5% or more of their genes from the same founder, they shouldn’t mate; or if they both had 25% or more of their genes from each of the same two founders, they shouldn’t mate; or something like that.

    Eventually we might be able to consider a rule that if two people both have more than one-sixth of their genes from the same founder they shouldn’t mate. But I’m not sure that would actually rule out parent-child incest. (It would rule out full-sibling incest.)

    ____________________

    [edit]looking back on this I may have made some errors in details. The general drift though is sorta close to the truth.[/edit]



    Edited August 28th by chiarizio
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    If we used the mating-circle idea, which we might if we can’t let the population grow exponentially with a high common ratio (counting “3” as high) even for the first several generations, the circle at generation 0 — the founders’ generation — might look like this:
    A100 — Z100 — B100 — X100 — C100 — Y100 — A100

    So the first generation of offspring would contain the following types:
    A50Y50 — A50Z50 — B50Z50 — B50X50 — C50X50 — C50Y50 — A50Y50

    Once they reach adulthood,
    A50Y50s will mate both with A50Z50s and with C50Y50s; producing A50Y25Z25s and A25C25Y50s:
    A50Z50s will mate both with A50Y50s and with B50Z50s; producing A50Y25Z25s and A25B25Z50s:
    B50X50s will mate both with B50Z50s and with C50X50s; producing B50X25Z25s and B25C25X50s:
    B50Z50s will mate both with A50Z50s and with B50X50s; producing A25B25Z50s and B50X25Z25s:
    C50X50s will mate both with B50X50s and with C50Y50s; producing B25C25X50s and C50X25Y25s: and,
    C50Y50s will mate both with A50Y50s and with C50X50s; producing A25C25Y50s and C50X25Y25s.

    So the second generation of offspring will form the following circle:
    A50Y25Z25 — A25B25Z50 — B50X25Z25 — B25C25X50 — C50X25Y25 — A25C25Y50 — A50Y25Z25.

    When they reach adulthood they’ll produce the third generation of offspring, who will form the following circle:
    A37.5C12.5Y37.5Z12.5 — A37.5B12.5Y12.5Z37.5 — A12.5B37.5X12.5Z37.5 — B37.5C12.5X37.5Z12.5 —
    — B12.5C27.5X37.5Y12.5 — A12.5C37.5X12.5Y37.5 — A37.5C12.5Y37.5Z12.5

    And so on.

    Note that IRL mating circles, in each generation, breeders keep only one male and no female from each of half of the types, and only one female and no male from each of the other half of the types. So the population doesn’t grow.

    In my experiment we could assume instead that in each generation two males of each type and two females of each type are kept. The population would quadruple in the first generation but then the breeding population would remain constant at 12 adult males and 12 adult females of each generation.

    ________________________________________


    However, if we’re really pursuing maximum diversity, we need some individual of each new type in each generation, to reproduce with some individual of each not-too-closely-related other new type of their own generation, AND OF THE PREVIOUS GENERATION!

    So except at start-up, an individual of every type will reproduce with partners of three generations.
  • their own generation
  • their younger parent’s generation
  • their children’s generation, unless they are a generation older than their child’s other parent.

    When we get to the third generation of offspring, some of them will be descended from every founder.
    That is, will have every on of the six founders as one of their eight great-grandparents.
    Maybe
    A37.5B12.5C12.5X12.5Y12.5Z12.5
    or
    A25B25C12.5X12.5Y12.5Z12.5
    or something.

    When we get to the fifth generation of offspring, some of them will have at least one of their sixteen greatgreatgrandparents from each of the nine types of the first generation of offspring.
    By the sixth generation, they can have both a greatgreatgreatgrandmother of each F1 type, and a GGGreatGrandfather of each F1 type.

    — — — — — — — — — —

    Past that point it may be pointless to be tracking the types so closely.

    I think if we can make sure that every newborn child has no maternal ancestor in the last five generations who is also a paternal ancestor in the last six generations, and vice-versa, we’ve done as much as will actually make a difference.

    The question is how to reach that point as soon as possible.


  • Edited June 19th by chiarizio
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