Edited 1/21/2018 to add the following:
Here is a DropBox link
to a PDF showing how my team used biomolecular first principles to arrive at a set of 240 biomolecular objects (which we believe to be an instantiation of the roots of $E_8$), and more generally, how we arrived at related sets of biomolecular objects with the cardinalities of the Zumkeller numbers (176,240,336) and the correponding edge-magic injection label numbers (11,15,21).
The "first principles" which we used are those described in the original MSE question below.
The four DNA bases (t,c,a,g) and the four RNA bases (u,c,a,g) all share the following fundamental biochemical properties:
1) each base is a pyrimidine (Y) or purine (R)
2) each base is weak (W) or strong (S)
3) each base is keto (K) or amino (M)
For the sake of this discussion, let:
+Y = Y and -Y = R
+W = W and -W = S
+K = K and -K = M
a = + or -
Then the properties aY, aW, aK are distributed among the four bases (DNA or RNA) such that you only need to specify the value of a for two of these three properties in order to uniquely specify one of the four bases (DNA or RNA):
t c a g Y + + - - -R W + - + - -S K + - - + -M
What is the mathematical structure that best captures this redundancy in the specification of the four bases (DNA or RNA) by the three properties aY, aW, aK?