Bending Genetics! (yes really)
So, I think I have a workable system, modified from one that I saw somewhere else (either tumblr or CapsAtla, don't remember).
Bending is dependent on both genetics and on environment, with two gene loci and several alleles.
The two genes involved are the 'bending' gene, with two alleles (B,b) and the 'nation' gene, with five alleles (A,F,E,W,x). To be a bender, you must have both two copies of the recessive bending gene (b) and at least one dominant nation gene (A,F,E,W). The dominant bending gene will turn off bending, no matter what nation gene you have, and the recessive nation gene will cancel out bending as well. So, there are more ways to be a nonbender than a bender.
If you have two different, dominant nation genes, they 'compete,' and which one you bend is 'imprinted' early in life by some environmental factor. What exactly this is, I'm unsure of, but it may be exposure to the chi from another bender, practicing the forms early in life, or some other cultural variable. Regardless, if neither gene is imprinted during the time window in early development, no bending results.
This explains the twins from The Fortuneteller. They are probably double-dominants (EF, say), and one was 'imprinted' into earthbending while the other received no imprint. Why? Well, we can't know this until we know what the imprint is, but perhaps the nonbending twin was sickly and bedridden as a child. Or perhaps the nonbending twin is less spiritual.
The Kataang family
Both Aang and Katara are benders, so both have bb for their bending gene. Aang, an Air Nomad, may have Ax or AA, depending on his parentage.
Katara may have WW or Wx. Neither Kya I nor Hakoda are benders, but since their daughter is a bender, their lack of bending is likely due to the presence of the dominant ‘bending’ gene, rather than the recessive ‘nation’ gene. This means at least one of them must have had a W gene, but both must have carried a b gene making them both Bb.
When we look at Katara and Aang’s kids, Wx and Ax seem more likely for them. If she were WW, all her children would be AW, a double-dominant. And if they were all raised in similar ways, they would likely have all resulted with the same bending. Of course, it could be that there were disparities in the ways the kids were raised, in which case all bets are off. If Kya received the ‘waterbending’ imprint, Tenzin the ‘airbending’ imprint, and Bumi received no imprints or conflicting ones, then they could all be bb AW!
All the same, since Aang and Katara spawned one waterbender, one airbender, and one nonbender, it's more likely that they were both heterozygous for their ‘nation’ gene. Kya II is thus bb Wx (a deterministic waterbender), Tenzin is bb Ax (a deterministic airbender), and Bumi is bb xx. Any of them could also be bb AW, but received the imprint for their bending. AW, Wx, Ax, and xx all have equal ¼ chances to occur with an Ax x Wx mating.
Tenzin and Pema
Tenzin married Pema, a nonbender, who must carry, xx or B or conflicting nation genes, or a combination. But If Pema carries both xx and Bb, then there is only a 25% chance that any one child would be an airbender, and thus only a .3% chance that all four of them would be. This is thus a very unlikely possibility.
If Pema carries a B gene, she did not pass it on to any of her four children. Since Tenzin is a bb, there is a 50% chance that she will pass the nonbender B to any one child. For four kids, this is a 6.3 % chance. This is certainly possible, though unlikely.
If Pema is bb, however, she must be xx to be a nonbender. Otherwise, she’d have to carry conflicting dominant ‘nation’ genes. However, NONE of her offspring are any kind of bender other than air, which has a 6.3 % chance for all four children (that’s only if they all get an air ‘imprint’).
If Pema is double-homozygous, all of her children are bb Ax. However, again, there is a 6.3 % chance that all four children would not receive Tenzin’s x gene.
There is another possibility, however. Pema may carry two conflicting dominant nation genes, one of which is A. If we call Pema’s unknown other nation gene Z, then this results in four possibilities: AZ, AA, Ax, and Zx. If all children are exposed to the air ‘imprint,’ this raises the possibility of one child being an airbender to 75%, and of all four children being airbenders to 31.6%! Food for thought.
Because of the probabilities involved, it is thus unfortunately impossible to tell Pema’s genetics. This is a common problem for geneticists.
The results of a hypothetical Lin x Tenzin pairing cannot be determined without knowing Lin’s paternal line.
Mako and Bolin
This is a similar situation to the Kataang family. Mako and Bolin’s parents could have been one Ex and one Fx, resutling in an Fx for Mako and an Ex for Bolin. Another option is that one of them is an EF, and was ‘imprinted’ to one side or the other (fire imprint for Mako, earth imprint for Bolin).
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I may be overly complicating things, so if anyone more knowledgeable about genetics than I would like to comment, I would be most pleased!
Bending is dependent on both genetics and on environment, with two gene loci and several alleles.
The two genes involved are the 'bending' gene, with two alleles (B,b) and the 'nation' gene, with five alleles (A,F,E,W,x). To be a bender, you must have both two copies of the recessive bending gene (b) and at least one dominant nation gene (A,F,E,W). The dominant bending gene will turn off bending, no matter what nation gene you have, and the recessive nation gene will cancel out bending as well. So, there are more ways to be a nonbender than a bender.
If you have two different, dominant nation genes, they 'compete,' and which one you bend is 'imprinted' early in life by some environmental factor. What exactly this is, I'm unsure of, but it may be exposure to the chi from another bender, practicing the forms early in life, or some other cultural variable. Regardless, if neither gene is imprinted during the time window in early development, no bending results.
This explains the twins from The Fortuneteller. They are probably double-dominants (EF, say), and one was 'imprinted' into earthbending while the other received no imprint. Why? Well, we can't know this until we know what the imprint is, but perhaps the nonbending twin was sickly and bedridden as a child. Or perhaps the nonbending twin is less spiritual.
The Kataang family
Both Aang and Katara are benders, so both have bb for their bending gene. Aang, an Air Nomad, may have Ax or AA, depending on his parentage.
Katara may have WW or Wx. Neither Kya I nor Hakoda are benders, but since their daughter is a bender, their lack of bending is likely due to the presence of the dominant ‘bending’ gene, rather than the recessive ‘nation’ gene. This means at least one of them must have had a W gene, but both must have carried a b gene making them both Bb.
When we look at Katara and Aang’s kids, Wx and Ax seem more likely for them. If she were WW, all her children would be AW, a double-dominant. And if they were all raised in similar ways, they would likely have all resulted with the same bending. Of course, it could be that there were disparities in the ways the kids were raised, in which case all bets are off. If Kya received the ‘waterbending’ imprint, Tenzin the ‘airbending’ imprint, and Bumi received no imprints or conflicting ones, then they could all be bb AW!
All the same, since Aang and Katara spawned one waterbender, one airbender, and one nonbender, it's more likely that they were both heterozygous for their ‘nation’ gene. Kya II is thus bb Wx (a deterministic waterbender), Tenzin is bb Ax (a deterministic airbender), and Bumi is bb xx. Any of them could also be bb AW, but received the imprint for their bending. AW, Wx, Ax, and xx all have equal ¼ chances to occur with an Ax x Wx mating.
Tenzin and Pema
Tenzin married Pema, a nonbender, who must carry, xx or B or conflicting nation genes, or a combination. But If Pema carries both xx and Bb, then there is only a 25% chance that any one child would be an airbender, and thus only a .3% chance that all four of them would be. This is thus a very unlikely possibility.
If Pema carries a B gene, she did not pass it on to any of her four children. Since Tenzin is a bb, there is a 50% chance that she will pass the nonbender B to any one child. For four kids, this is a 6.3 % chance. This is certainly possible, though unlikely.
If Pema is bb, however, she must be xx to be a nonbender. Otherwise, she’d have to carry conflicting dominant ‘nation’ genes. However, NONE of her offspring are any kind of bender other than air, which has a 6.3 % chance for all four children (that’s only if they all get an air ‘imprint’).
If Pema is double-homozygous, all of her children are bb Ax. However, again, there is a 6.3 % chance that all four children would not receive Tenzin’s x gene.
There is another possibility, however. Pema may carry two conflicting dominant nation genes, one of which is A. If we call Pema’s unknown other nation gene Z, then this results in four possibilities: AZ, AA, Ax, and Zx. If all children are exposed to the air ‘imprint,’ this raises the possibility of one child being an airbender to 75%, and of all four children being airbenders to 31.6%! Food for thought.
Because of the probabilities involved, it is thus unfortunately impossible to tell Pema’s genetics. This is a common problem for geneticists.
The results of a hypothetical Lin x Tenzin pairing cannot be determined without knowing Lin’s paternal line.
Mako and Bolin
This is a similar situation to the Kataang family. Mako and Bolin’s parents could have been one Ex and one Fx, resutling in an Fx for Mako and an Ex for Bolin. Another option is that one of them is an EF, and was ‘imprinted’ to one side or the other (fire imprint for Mako, earth imprint for Bolin).
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I may be overly complicating things, so if anyone more knowledgeable about genetics than I would like to comment, I would be most pleased!