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"But if she's my grandmother, then who's my grandfather?"

"Isn't it obvious? ...YOU ARE!"

"Aaaaaahh! Aaaaaah! Aaaaaaaaahh!"

The main DVD commentary track for the Futurama episode Roswell That Ends Well briefly discusses the immensely complicated situation created in Philip J. Fry's DNA when he accidentally becomes his own grandfather, by travelling back in time to 1947 and sleeping with his paternal grandmother, thereby conceiving his father, Yancy Fry.

Here's Fry's family tree.

Fry = Mildred
    |
    |
  Yancy = Fry's mother
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        |
       Fry

Figuring out Fry's genetic makeup

Pick any one of Fry's genes and follow it forwards in time. Call this Gene A.

When Fry sleeps with Mildred in 1947, there is a 50% chance that Gene A gets carried on to Yancy, and a 50% chance that Gene A is discarded. That means Yancy's genetic makeup is 50% Fry's genes and 50% Mildred's genes.

Assume that Gene A did make it to Yancy. Now go forward another generation. We know that Fry is made up of 50% Yancy's genes and 50% Fry's mother's genes. But we also know for a fact that Gene A is present in Fry's genetic makeup.

That means, for Gene A, either:

  1. Fry inherits Gene A from Yancy and it becomes a causal loop: Yancy inherits it from Fry, Fry inherits it from Yancy, and so on forever. Let's say X% of Yancy's genes do this.
  2. Yancy does NOT pass Gene A on to Fry - but Fry receives an identical Gene A from his mother instead anyway. Then (50 - X)% of Yancy's genes do this.

Explain that again?

  • 50% of Fry's genes are regular genes, inherited from his mother
  • X% of Fry's genes are eternal genes with no origin, inherited from Yancy
  • (50 - X)% of Fry's genes are regular genes, inherited from Yancy, who originally inherited them from Mildred.

When Fry sleeps with Mildred, the eternal genes are among those Fry passes on:

  • 50% of Yancy's genes are regular genes, inherited from Mildred
  • X% of Yancy's genes are eternal genes with no origin, inherited from Fry
  • (50 - X)% of Yancy's genes are eternal genes with no origin, inherited from Fry, who originally inherited them from his mother.

When Yancy sleeps with his wife and conceives Fry, the eternal genes are among those Yancy passes on:

  • 50% of Fry's genes are regular genes, inherited from his mother
  • X% of Fry's genes are eternal genes with no origin, inherited from Yancy
  • (50 - X)% of Fry's genes are regular genes, inherited from Yancy, who originally inherited them from Mildred.

etc.

Consequences

  • The first major question this raises is: what is X? How many of these genes are eternal? This value can't be more than 50%, as only 50% of one's genome gets passed on to the next generation.

    However, we also know X can't be zero. Proof: Fry is male, but all his direct ancestors are ultimately female (apart from himself). That means he has nowhere to inherit the male Y-chromosome from except himself. So his Y-chromosome makes up part of the genes which are in the causal loop.

    Beyond that it is impossible to say what X is.

  • Fry and Mildred have (50 - X)% of their genetic code in common. Depending on what X is (see above), this is either disgustingly incestuous (for X ≅ 5) or entirely inoffensive and legal (for X = 50).

    But remember that the situation with Fry is exactly symmetrical with the situation with Yancy. Yancy is also his own grandfather. And Fry's mother and Yancy also have (50 - X)% of their genetic code in common, which makes their relationship just as bad!

  • It seems reasonable to assume that the eternal genes comprise more than just Fry's Y-chromosome. Fry is unique in the entire human species in that he lack the delta (intelligence) brainwave, instead cobbling together other brainwaves as a substitute, which makes him immune to the intelligence-draining powers of the evil brainspawn he fights in the episodes The Day The Earth Stood Stupid and The Why Of Fry. This could be either due to the eternal genes or due to the fact that he may be rather inbred. It seems unlikely that Fry would be the only inbred person in the future so we can probably put this phenomenon down to the eternal genes too. (Or perhaps a combination of the two.)

  • It is entirely possible that Fry's brother, Yancy Jr., has some of the eternal genes in his genetic makeup as well.

Why Huser's proof below doesn't quite work

Huser writes "Fry gets half his genes from his mother, a quarter from Grandma Mildred, and a quarter from himself", then substitutes the original equation in to figure out that the quarter-from-himself are ALSO "half from his mother, a quarter from Mildred, and a quarter from himself", and so on, and so on.

This doesn't work. But it took me quite a while to figure out why it doesn't work for myself. The reason is simple to give but difficult to comprehend.

In a nutshell: genes can't mix.

It is impossible for, say, gene #576 in Yancy's genome to be inherited from anywhere other than gene #576 in Fry's genome. And vice versa.

Pick gene #576 from Fry's genome and follow it backwards in time. Either it came from Fry's mother's gene #576 - in which case we stop. Or it came from Yancy gene #576, in which case we follow it back another generation. Either this gene came from Mildred - in which case we stop. Or it came from Fry gene #576, in which case we follow it back another generation.

Now watch carefully. We've gone back two generations and returned to Fry. Huser's calculations (though he may not realise) implictly require that we still have an equal chance of ending up in either Yancy or Fry's mother. But remember we are still looking at the gene #576. We KNOW where that came from. It came from Yancy, as we saw above. Then Fry again. Over and over. It is impossible for a gene to be more than two generations old. If it is more than two generations old, it is eternal, a factor not accounted for in Huser's calculations.

Philip J. Fry inherited two-thirds of his genes from his mother and one-third from his grandmother. How is this possible?

In the following equations, F represents Philip, M his mom, D his dad, and G his grandmother.

  1. F = 1/2*M + 1/2*D Fry gets half his genes from his mom and half from his dad.
  2. D = 1/2*G + 1/2*F Dad gets half his genes from his mom (Grandma Mildred) and half from his dad (Fry himself).
  3. 1/2*D = 1/4*G + 1/4*F Divide equation (2) in half.
  4. F = 1/2*M + 1/4*G + 1/4*F Substitute equation (3) into equation (1).
Therefore, Fry gets half his genes from his mother, a quarter from Grandma Mildred, and a quarter from himself. This is where it gets tricky:
  • F = 1/2*M + 1/4*G + 1/4*F Original equation.
  • F = 1/2*M + 1/4*G + 1/4*(1/2*M + 1/4*G + 1/4*F) Substitute the original equation into itself.
  • F = 1/2*M + 1/4*G + (1/8*M + 1/16*G + 1/16*F) Distribute the 1/4
  • F = (1/2+1/8)*M + (1/4+1/16)*G + 1/16*F Group like terms.
  • F = (1/2+1/8)*M + (1/4+1/16)*G + 1/16*F New equation.
  • F = (1/2+1/8)*M + (1/4+1/16)*G + 1/16*{(1/2+1/8)*M + (1/4+1/16)*G + 1/16*F} Substitute the new equation into itself.
  • F = (1/2+1/8)*M + (1/4+1/16)*G + {1/16*(1/2+1/8)*M + 1/16*(1/4+1/16)*G + 1/16*1/16*F} Distribute the 1/16.
  • F = (1/2+1/8)*M + (1/4+1/16)*G + {(1/32+1/128)*M + (1/64+1/256)*G + 1/256*F} Distribute the 1/16 again.
  • F = (1/2+1/8+1/32+1/128)*M + (1/4+1/16+1/64+1/256)*G + 1/256*F Group like terms.

Continue indefinitely to get:
F =
(1/2+1/8+1/32+1/128+1/512+1/2048...)*M +
(1/4+1/16+1/64+1/256+1/1024+1/4096...)*G +
(an infinitesimally small fraction)*F.

The sum (1/2+1/8+1/32+1/128+1/512+1/2048...) converges to 2/3. The sum (1/4+1/16+1/64+1/256+1/1024+1/4096...) converges to 1/3. Therefore, Fry's genetic makeup is 2/3 from his mother, 1/3 from his grandmother, and an infinitesimally small Y chromosome.

QED.

In collaboration with sam512, BaronWR, StrawberryFrog and Andrew Aguecheek

Huser's write-up above is completely wrong, for the reason that sam512 explains: the genes are discrete, and cannot be mixed. However, the analysis in the original write-up is not entirely correct either, since it uses an insufficiently accurate model for genetic inheritance.

Genetic model

The human DNA blueprint consists of a large number of genes. Each gene can occur in one or more versions called alleles. One can think of each gene as encoding some characteristic, e.g. eye colour, and of the alleles as the possible values of this characteristic, e.g. "blue eyes" or "brown eyes". (Actually, eye colour is not determined by a single gene, but that's beside the point.)

Each gene is located on a chromosome. Humans have 23 pairs of chromosomes, and both chromosomes in a pair contain alleles for the same genes. Any individual therefore has two alleles of each gene (except for genes on the sex chromosomes, as explained below). When a male and a female reproduce, their offspring inherits exactly one allele from each parent for each gene. The inherited allele is selected randomly from each parent's pair. The details of the process are irrelevant to the model.

The above holds for 22 of the 23 pairs, the so-called autosomes. The last pair are the sex chromosomes. Females have a pair of X chromosomes, while males have one X and one Y chromosome. Essentially (outside a small pseudoautosomal region) the X and Y chromosome are completely different, so there are X-linked and Y-linked genes. Offspring will essentially inherit either the X or Y chromosome of the father in its entirety, while the alleles in the X chromosome inherited from the mother are selected randomly from the mothers pair. A male will therefore have only a single allele of the X-linked genes, and this allele is always inherited from the mother. On the other hand, the Y-linked genes of a male are inherited directly from the great...grandfathers, modulo mutations. Most of the Y-linked genes are insignificant junk. Females have pairs of alleles also for the X-linked genes; one of them is the father's unique allele, while the other is selected randomly from the mother's pair.

Fry's inheritance

Now let us consider a some specific gene in Fry's genetic make-up. X-linked genes are uninteresting as they inherited from his mother. Y-linked genes are inherited from his grandfather, i.e. himself, giving rise to a causal loop. Most of these genes are insignificant though. The most fun case is therefore the autosomal genes.

Call Philip J. Fry's pair of alleles for the gene PF and PM, where PF is the allele inherited from Yancy and PM the one from Fry's mother. Similarly, let YF be Yancy's allele inherited from Fry, and YM the one he inherits from Mildred. There are two equally likely probably possibilities:

  1. PF := YM (the notation ":=" indicates a causal relationship: PF depends causally on YM). Fry inherits an allele from his paternal grandmother, just like any human would.
  2. PF := YF. Fry inherits a gene from his paternal grandfather, i.e. himself. There are two possibilities, each with probability 25%:
    1. YF := PM. Then PF = PM, i.e. Fry has inherited the same allele from his mother in two different ways. As will be explained below, this represents a degree of inbreeding.
    2. YF := PF. Then PF := PF, so there is a causal loop.

Inbreeding

As is commonly known, it is a bad thing for closely related individuals to produce offspring. The reason is that many genetic illnesses are caused by recessive alleles. This means that individuals with one "bad" allele B and one "good" allele G will be healthy, but B-homozygotes, i.e. individuals with two B alleles, are affected by the disease. Inbred offspring are more likely to be homozygotes for rare alleles, and thus more likely to be diseased.

To see why, consider for example the case of two siblings reproducing. Then each allele of the offspring is seleceted uniformly among the 4 alleles (total) of its two grandparents. 75% of the time the two alleles are inherited from different places, and may or may not be equal. But there is a 25% chance that the two alleles are inherited from the same place, and are therefore tautologically equal.

Hence, if a a certain allele B occurs in the general population with frequency 1 in 1,000, then a generic individual has a 1 in 1,000,000 probability of being a B-homozygote, while the offspring of a pair of siblings has a 1 in 4,000 probability of being a B-homozygote.

Conclusion

Fry's X chromosome is normal, while (essentially) all of his Y chromosome comes from a causal loop. The Y-linked part of Fry's DNA could be absolutely anything, but since there are few significant genes in the Y-linked part it is unclear how important this is.

Of the remaining genes, 50% are normal. 25% have a tautologically equal pair of alleles, so Fry is as inbred as the offspring of a pair of siblings. The remaining 25% of the genes include one allele that exists in a causal loop.

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