Cases where ANY 2 of 3 +/- choices select one of four possible elements

Edited 1/21/2018 to add the following:

Here is a DropBox link

https://www.dropbox.com/s/7rtt0iqmgimsgzu/Zumkeller_edge-magic.pdf?dl=0

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.

Original Question

Reference:

https://www.ncbi.nlm.nih.gov/pubmed/8054766

Background:

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

Question:

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?

Solution 1:

It seems to me you can consider two of the properties to be the "encoding" of the base and treat the third property as a "parity check bit." You can make an arbitrary choice of which property to designate as the "parity check bit." I'm not sure if this qualifies as a "mathematical structure," however.

A possible difficulty in trying to discuss a mathematical structure for your observation about the bases of DNA or RNA is that the notation you have used is really not mathematical. For example, in normal usage an equation such as "$a = +$" is nonsense, and writing "$a = + \text{ or } -$" is no better. Also if $Y,$ $W,$ and $K$ are mathematical objects to which the operators $+$ or $-$ can be applied, how do we know that $+Y \neq +W$ or that $+Y \neq -K$ (among many other possibilities)?

What you could do instead is define functions named $Y,$ $W,$ and $K,$ each of which maps any base to either $1$ or $-1$ depending on whether the base has the property corresponding to that function's name or the opposite property. For example, $Y(t) = Y(c) = 1$ and $Y(a) = Y(g) = -1.$

Now that we have some actual mathematical definitions associated with the bases, we can say that for any arbitrary base $x$ in either DNA or RNA, $$ Y(x) \times W(x) \times K(x) = 1, $$ which is a constraint on the properties of the base. Using this constraint, it is clear that whenever you know two of the properties of the base, you can find the third property by solving this equation algebraically. Given that the three properties uniquely identify one of the bases of DNA (or one of the bases of RNA), this constraint tells us that any two of the properties are sufficient to identify the base.

This is not the only possible formulation, however. You could just as well use $1$ and $0$ as the indicators of "has the property" or "has the opposite property", so that $Y(t) = Y(c) = 1$ and $Y(a) = Y(g) = 0.$ Then the constraint would be $$ Y(x) + W(x) + K(x) \equiv 1 \pmod 2, $$ or in more ordinary language, the sum of the three function values must be an odd number. This is closer to the way "parity bits" are interpreted in telecommunication and computing, but the effect you are concerned with is the same, namely, by knowing any two of the three properties we can find the third.

Solution 2:

This particular group corresponds to a tetrahedron sitting in the vertices of a cube.

 t--------C        
 |\       |\           Y         pYrimidine    Caps are anti-forms
 | T------+-c        K |         puRine        or chiral opposites
 | |      | |         \|         Weak          as per DH comment below.
 | |      | |      W---*---S     Strong
 A-+------g |          |\        Keto          They were not part of 
  \|       \|          | M       aMino         the original answer.
   a--------G          R

The ruling symmetry here would be something like [3,3], supposing there are various interchanges like (ag)(tc); [diagonal swaps] and (acg) [cyclic rotation of t]. I really don't know. I'm just presenting the data in a geometric form.