Mendel: The Last Genebender
This is what I'm thinking about as I watch The Legend of Korra, which takes place in the same fictional universe as Avatar: The Last Airbender.
Proposal 1: The Genetic Basis of Bending
There are five alleles: w for waterbending, e for earthbending, a for airbending, f for firebending, and N for no bending. The first four are recessive, i.e. ww is the genotype of a waterbender, aa of an airbender, but a wa displays no bending whatsoever.
The final allele, N, blocks all bending in the person who has it. If two people reproduce, one of whom is ff and the other of whom is NN, then despite one parent being a firebender, none of their children will ever be a bender, because all will exhibit the genotype Nf. N is a dominant allele.
In the original series, Sokka and Katara are both members of the water tribe, but Sokka is not a waterbender while Katara is. Their father Hakoda is not a bender, while their mother Kya is. We can therefore surmise that Hakoda's genotype is Nw, Kya's is ww, and, by Punnett square:
| N | w
-----------
w | Nw | ww
w | Nw | ww
Sokka's genotype is Nw, i.e. no expressed bending, while Katara's is ww, i.e. she is a waterbender.
This is consistent with Bolin and Mako in Korra. Bolin is an earthbender and Mako is a firebender, but the two are brothers. Consequently, Bolin must have ee and Mako ff, so both of their parents must be ef:
| e | f
-----------
e | ee | ef
f | fe | ff
This analysis seems to fit most of the show.
Proposal 2: Account for Aang's Descendents
Aang has three children: Kya, Bumi, and Tenzin. Kya is a waterbender, Tenzin is an airbender, and Bumi does not bend. Bumi is well-explained by the previous proposal, as his genotype would be aw, a non-bending type. But how to account for the other two?
Let us posit a sixth allele: C is the most recessive allele, and as such enables any other kind of bending. A person with the Ca genotype would be an airbender, Cw a waterbender, and so forth; CN would still be a non-bender. In effect, C enables the other allele to take effect.
Now, we revise our analysis of Sokka and Katara's lineage. Suppose that Hakoda, instead of Nw, were instead NC:
| N | C
-----------
w | Nw | Cw
w | Nw | Cw
Now, Sokka's genotype is Nw, while Katara's is Cw. Sokka is still a non-bender, and Katara is still a waterbender. Now Katara reproduces with Aang (Ca):
| C | w
-----------
C | CC | Cw
a | Ca | aw
Their children can have one of three possible phenotypes: either airbending (genotype Ca), waterbending (Cw), or neither (CC or aw), giving Tenzin's genotype as Ca. His wife Pema either has a recessive airbending trait, so that their children have aa genotypes, or a C allele, so that their children have Ca genotypes. The latter seems more likely.
Accounting for the Avatar
How do we make room for the Avatar, who can bend all four elements? Above I suggested that Aang's genotype was Ca, which enabled us to make the square for Aang and Katara's children work. But what enables the Avatar to bend all four elements?
Magic. Just magic. Fuck you.