# 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?

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.

• @WendyKrieger - your diagram is exactly correct - it's the same one I got back in 193-94. Regarding the missing vertices of the cube, simply recall that DNA must occur in a DOUBLE-helix in order to do its job and that the two strands of DNA MUST be ANTI-parallel for stereochemical reasons. Hence, the missing vertices are the four vertices on the OPPOSITE strand , with a below t, t above a, g below c, and above g in your diagram (repesenting what is termed "Watson-Crick' complementary base pairing. (RNA also forms functionally important anti-parallel double-helices.) see next comment also – David Halitsky Dec 29 '17 at 12:21
• @WendyKrieger - I never put forward this interpretation of "duplex" RNA or DNA because I couldn't motivate it fully, but if you look at the role of forward vs backward reading of a 9-tuple in my accumulation of the 112 and 128 in my recent post, you will see that there may in fact be sufficient motivation to fill in your/my diagram the way I just suggested. (One more comment - see next) – David Halitsky Dec 29 '17 at 12:23
• @DavidHalitsky You probably noticed i edited your table to put the - signs down the right hand side. If you invert the signs of any given row, you will see that a column becomes all - signs. +--- is a dead giveaway for a tetrahedron in this case. – wendy.krieger Dec 29 '17 at 12:25
• @WendyKrieger - your/my diagram here does NOT "contradict" the diagram in my paper on "codon:anticodon" recognition logic. Rather, as I see it, the situation is like that which obtains regarding Coxeter's various shadows of the 4-cube in 3-space - all equally valid - just different ones useful for different purposes (as he himself takes pains to point out), and some more symmetrical than others in different ways. – David Halitsky Dec 29 '17 at 12:27
• @WendyKrieger - regarding symmetries and swaps, here are two very important biomolecular terms: 1) a "transition" mutation is t<>c or a<>g; 2) a "transversion" mutation is t<>a, t<>g, c<>a, c<>g. Transitions are at least an order of magnitude more frequent than transversions because they preserve Y vs R identity while transversions don't, and if you look at the stereochemical difference between Y = t,c and R = a,g, you will see immediately why this frequency difference arises. – David Halitsky Dec 29 '17 at 17:22

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.

• -thank you so much for taking the time to respond, and to do so in such a thought-provoking way. I am going to be sure that one of my old mentors (Ray Dougherty - MSEE Dartmouth, PhD MIT with Noam as thesis supervisor) sees your response, because we've been disussing the signal-theoretic properties of biomolecular strings and linguistic strings for many years. Also, I want to apologize for the use of "a" to mean "+" or "-" - it's a habit from old linguistic training, where "a" means that a "distinctive (binary) feature" of a sound system element is "unspecified" – David Halitsky Nov 24 '17 at 18:13
• I'm sure that putting this in the context of linguistics makes more sense of the original notation. I did briefly consider trying to use regular languages (the computer science concept) in order to make sense of statements such as "$a = +$" (by treating $+$ and $-$ as symbols in the alphabet of the language--perfectly OK to do in that context). In any case, if you already have a well-developed system of notation in which those expressions make sense, I have no complaint about that. Perhaps the "parity bit" is really the key idea here. – David K Nov 24 '17 at 18:23
• Actually - no - I think I'm going to adopt your function idea to replace the a = +/- notation, which was just a temporary convenience. And I think the "parity" bit notion is, as you say, the key idea here. In this regard, see the following link for a use of the K/M opposition in what has been called "Halitsky symmetry" - I think the parity bit may be a check on the ENTIRE code - that's why I find it so very interesting . . . friedel-online.com/genetic_code_and_evolution/… – David Halitsky Nov 24 '17 at 18:33
• I don't know how busy or interested you are - but I think that we could very quickly knock out a short paper for Math Biosciences setting forth your "parity bit" interpretation of "Halitsky point symmetry" . . . my email is halitsky.d@att.net, and I assure you I won't be offended if I don't hear from you .... thanks so much again !!! – David Halitsky Nov 24 '17 at 18:52
• you may be interested in the role played by the Y vs R opposition in this recent question I posted to MO: mathoverflow.net/questions/288394/… – David Halitsky Dec 13 '17 at 22:52