Thursday, November 13, 2014

Genes, music, and icosahedron (and tetrahedron too!)

For some time ago I introduced the notion of Hamiltonian cycle as a mathematical model for musical harmony and also proposed connection with biology: motivations came from two observations. The number of icosahedral vertices is 12 and corresponds to the number of notes in 12-note system and the number of triangular faces of icosahedron is 20, the number of amino-acids and the number of basic chords for the proposed notion of harmony. This led to a group theoretical model of genetic code and replacement of icosahedron with tetra-icosahedron to explain also the 21st and 22nd amino-acid and solve the problem of simplest model due to the fact that the required Hamilton's cycle does not exist.

This article was meant to be a continuation to the eralier article providing a proposal for a theory of harmony and detailed calculations. It however turned out that the proposed notion of bio-harmony was too restricted: all isosahedral Hamilton cycles with symmetries turned out to be possible rather than only the 3 cycles forced by the assumption that the polarity characteristics of the amino-acids correlate with the properties of the Hamiltonian cycle. This working hypothesis had to be given up. The fuel of the minirevolution was the observation the symmetries of the Hamiltonian cycles (Z6, Z4, Z2) are nothing but the icosahedral symmetries needed to predict the basic numbers of the genetic code and its extension to include also 12st and 22nd amino-acids! Thus icosahedral Hamiltonian cycles predict genetic code without further assumptions. Mathematician cannot simply neglect this kind of connection!

One also ends up with a proposal for what harmony is leading to non-trivial predictions both at DNA and amino-acid level.

1. 3-adicity and also 2-adicity are essential concepts allowing to understand the basic facts about harmony. The notion of harmony at the level of chords is suggested to reduce to the notion of closeness in the 3-adic metric using as distance the distance between notes measures as the minimal number of quints allowing to connect them along the Hamilton's cycle. In ideal case, harmonic progressions correspond to paths connecting vertex or edge neighbors of the triangular faces of icosahedron.

2. An extension of icosahedral harmony to tetra-icosahedral harmony was proposed as an extension of harmony allowing to solve some issues of icosahedral harmony relying on quint identified as rational frequency scaling by factor 3/2.

This extension is kept also now. One must however give up the idea about correlation between polarity characteristics of proteins and properties of Hamilton cycles. One must allow all 11 icosahedral harmonies with symmetries as bio-harmonies: their symmetry groups Z6, Z4, Z2 can be identified as the symmetry groups defined the decomposition of 60 DNA codons to 20+20+20 codons in the model of the genetic code. The 4 remaining DNAs and amino-acids can be assigned to both tetra-icosahedron and tetrahedron and icosahedron regarded as defining separate genetic codes. This explains why stopping codons can code for the 21st and 22nd amino-acid under some circumstances.

Tetrahedral code is second member in the hierarchy of genetic codes inspired by the notion of Combinatorial Hierarchy M(n+1)= MM(n)= 2M(n)-1 giving the numbers 2, 4,7, 64, 2126,... as numbers of DNA codons. The fourth member would correspond to what I called "memetic code" allowing representation of codons as sequences of 21 DNAs. It is not known whether the Combinatorial Hierarchy of Mersenne primes continues as Hilbert conjectured.

3. The notion of bio-harmony is partially characterized by the triplet n= (n0,n1, n2), characterizing the numbers of 0-, 1-, and 2-quint chords which in turn correspond to DNA codons in consistency with the observation that codons indeed correspond to triplets of nucleotides. n-quint chord corresponds to a triangle (face of icosahedron) containing n edges of the Hamiltonian. Particular bio-harmony requires a selection of a specific Hamiltonian cycle from each class of cycles (1 Z6 symmetric cycle having n= (2,12,6), 2 Z4 symmetric cycles n ∈{(0,16,4), (4,8,8)}, 3 Z2=Z2rot with n∈{(0,16,4),(2,12,6),(4,8,8)} and 5 Z2=Z2refl symmetric cycles with (n∈ {(2,12,6), (4,8,8)}. Note that the are only three different triplets n.

4. The model gives for the fusion of icosahedral and tetrahedral cycles just the ordinary genetic code so that it is consistent with the proposal that genetic code is realized also by dark proton sequences. For de-fused icosahedral and tetrahedral codes the common face would code for Pyl and Sec, the well-known 21st and 22nd amino-acid. An amusing "co-incidence" is that met to which genes realized as mRNA code is the first codon of gene. At the level of music met would correspond to the basic chord, "home" from which the simple music pieces often begin!

5. The original idea was that the rules of bio-harmony could be applied to amino-acid sequences interpreted as sequences of basic 3-chords. DNA would have represented the notes of the music. For given choice of harmony as Hamiltonian cycle meaning selection of of 4, 5 or 10 amino-acids coded by the 20 DNAs in question, the hypothesis had to be modified by replacing amino-acid sequences with DNA sequences.

These DNA sequences however define also amino-acid sequences identifiable as specific triangle at the orbit of Zn defining the DNA codons assigned to that amino-acid (there is a singular fiber space structure). Together the three 20-plets of DNAs define an amino-acid harmony with (4+5+10 =19 chords with tetrahedral extension defining a harmony with 22 chords/amino-acids). Hence both DNA sequences and amino-acid sequences define "bio-music".

6. The assumption that harmonic transitions between chords (DNA codons) minimize the distance between chords defined by quint-metric leads to highly non-trivial and testable predictions about both DNA sequences and amino-acid sequences. Negentropy Maximization Principle (NMP) suggests that evolution favors the generation of harmony which should thus increase in the proposed sense for DNA sequences defining particular genes or other functional units of DNA during evolution. Large quint-distances between subsequent codons/chords would tend to polished out under evolutionary pressures.

7. Could icosahedron, tetrahedron, and tetra-icosahedron have direct physical counterparts in living matter? For instance, water molecules form icosahedral clusters and the chlathrates associated with synaptic contacts have icosahedral symmetries. Tetra-icosahedron has 13 vertices with the added vertex representing one note- say E- in C-key as note with slightly different frequency to resolve the basic problem of rational number based 12-note scale (12 quints give slightly more that 7 octaves). Intriguingly, microtubules consist of basic structures consisting of 13 tubulins with 2 states defining bit: could these bit sequences define representation for the 3-chords and thus representation of sequence of DNA codons and realization of genetic code.

8. Music is language of emotions and peptides are molecules of emotion as Candace Pert expressed it. Could bio-harmonies serve as direct correlates for emotions? What is bio-music? A natural TGD inspired guess is that sounds can be replaced with heff=n× h dark photons with low frequencies and having energies in the range of bio-photons (visible and UV range maximally effective biologically) as proposed on basis of some physical facts and theoretical ideas kenociteallb/hearing. The frequency spectrum of dark cyclotron photons along magnetic flux tubes would define bio-music as "music of dark light" and bio-harmonies would correlate with emotions and moods.

If one can find various icosahedral Hamilton's cycles one can immediately deduce corresponding harmonies. This would require computer program and a considerable amount of analysis. My luck was that the all this has been done. One can find material about icosahedral Hamilton's cycles in web, in particular the list of all 1024 Hamilton's cycles with one edge fixed (this has no relevance since only shape matters). If one identifies cycles with opposite internal orientations, there are only 512 cycles. If the cycle is identified as a representation of quint cycle giving representation of 12 note scale, one cannot make this identification since quint is mapped to fourth when orientation is reversed. The earlier article about icosahedral Hamiltonian cycles as representations of different notions of harmony is helpful.

The tables listing the 20 3-chords of associated with a given Hamilton's cycle make it possible for anyone with needed computer facilities and music generator to test whether the proposed rules produce aesthetically appealing harmonies for the icosahedral Hamiltonian cycles. Biologist with access to DNA sequences could experiment with DNA codons to see whether their are harmonious in the sense that the distance between subsequent chords assignable to DNA codons tend to be small in quint metric. Note that DNA decomposes to pieces corresponding to different Hamiltonian cycles (harmonies) so that the comparison is not quite straightforward.

For details see the article Theory of Harmony or the chapter "Quantum model of hearing" of "TGD and EEG".

At 9:02 AM,  Ulla said...

http://arxiv-web3.library.cornell.edu/pdf/1403.5342v1.pdf
Orthodox type II endonucleases recognize four-to-eight base
pair sequences of palindromic DNA, cut both strands symmetrically, and act without an external
metabolite such as ATP.
+ the enzyme serving as a decoherence shield.

Interesting.

At 5:38 PM,  L. Edgar Otto said...

Matti,

Strange things happen when we try to divine hidden codes like Newton did for Bible prophesy. See 1290 days.

So from the numerology of it all we strangely find the same sort of numbers to interpret. This is true of secular codes such as that of DNA.

The Quasic Principle said much about this independently years ago and explains the DNA mechanisms, some before hand. There are 24 such basic codons to deduce- 20 if you look at structures that way. The informational approach corresponds to the binary code of the I Ching as Leibniz noted at the beginning of the digital age.

In those cults who try to image a day equal a year 61 is the number of Christ (61 89 107 not just materialism?)

History seems to fit very well in this tension between Judaeo-Christian and Greek philosophy and its analogs in the East. The can make a unified theory of sorts.

If you ever want to see my code of 74 of which I only found the Glass Bead Game as a cultural precedence at the time and the higher dimensional implications - well, it has been posted on my blog a long time.

At 8:51 PM,  matpitka@luukku.com said...

Amusingly, palindromic DNA would have as analog a sequence of 3-chords defining a piece which sounds the same if played backwards. No breaking of time reversal invariance, no dissipation, no loss of coherence;-). Bach was a mathematician as composer: he also composed piece - maybe fugue- which is musical palindrome.

At 5:33 PM,  L. Edgar Otto said...

Hi Matti,

I cannot see here your posting on this for 11-17-14 as I can on the smart phone where I cannot post.

There, Ulla has a most excellent point of reading the code as four codons.

Now it is beautiful to connect three things this way and see it corresponds to 20 faces of the icosahedron.

Yes there are 60 things, I in the binary code of the I Ching. 4 for the four seasons with 384 lines for the rotations and inversions of the hypercube. But the binary code of the I Ching does not use a 256 symbol set. Why?

An alternative topology sees the 24 cell in 4D matching the colors of 24 four colored squares...the problem then of realized physical or biological solutions has to consider at least 4D analogs to the paths if not more. Then where these are realized at not necessarily expressed symmetrically.

But your intuitions are sound and suggest (say with n-codon bases) what you mean by say dark photons etc does rationally relate to hyperbolic spaces and Mersenne arithmetic.

Why 21 or 22? You can ask this of why 24 or 25 so that by an extension of your tetrahedron plus icosahedron projected down we incorporate the particle structure and question of the standard model to more general spaces.

The relation of 2 and 3 here is a profound realization that the usual group treatment does not address. Yes, 240 8D reduced lattices are involved as well for a sort of quintessence issue- perhaps a matter of emotions- hard to say.

Some things played in reverse we may feel but not hear at all... so too dark particles. Some sounds can seem like they are endlessly rising.

I use the same color notation for a triad of bicolors as a set of 30 as for the twenty but from different corners of an orthogon.

But what all this amounts to for now is how we see discrete vs continuous group mathematics... some field or some kind of particle so need both.
The combinational and the reduced field approaches seem to have a conflict of believers. But Gammow and Crick-Watson both apply yet it is very hard to see at first or find in experiment how Gammow was also right about the 20 derived from combinations.

Lubos seems very brilliant in the way he can pick only one, and likely the wrong side of things. The galactic core is or was once a matter factory. The symmetry breaking is after all part of the Fabonacci ratios... and duality aside (2) even planets can be formed in pairs and that reduced to 3 to 5 symmetry.

But as you said once in a post there is so much to learn in math (even of what we know)... so enjoy the moment of seeing something beautiful. I drew a lot of this out on my blog with my informal math recreations - but the beauty wares thin then reaches higher- how far I cannot now say.

At 7:48 PM,  Matpitka@luukku.com said...

To Edgar:

The 20+20+20 division is only for the 60 DNA codons if icosahedral part of the code. The tetrahedral part gives the remaining 4 codons.
Both the fusion of isosahedral and tetrahedral code and the two separately are needed to undersand also the 21st and 22nd amino-acid Pyl and Sec. Ordinary genetic code corresponds to their fusion.Pyl and Sec coded by stopping codons correspond to them separately.

At 7:03 AM,  L. Edgar Otto said...

Matti,

Again this is only one side of the wider space picture wherein if that is all there is we do have to ask the question how do we account for more than the 20 combinational codons.

Alternatively we could ask how we must account for the 25 particle we image as the Higgs in the standard model.

Now, from evidence- to explain you question of why "inorganic" mechanisms need chirality to define life (the same structure chiral image does not have the same energy and it is an open question what directs it ultimately in one right or left handed form.) We find that 60 bucky like structures with an attachment of the tetrahedral 4 structures do indeed show relationships as you describle... but here we are talking about such asymmetry intrinsic to space of inorganic nanoparticles as if a property of space that does not have necessary direct relation to the apparent emergence of life... in fact inorgainc things have an analogy to 4 base DNA code reading. So your intuition concludes correctly but not from a more foundational theory.

Lubos btw compares feminism with nazism in a recent post... I wonder if he will eventually coin Rush Limbaugh's word feminazi? Again even his politics seems unoriginal.

Cold fusion is theoretically possible as well known via quantum theory already. Beyond rotation and reflection there may be new operations but in any case 11 dimensional SUSY is very far from the foundations to explain anything said as complete as a theory of everything. It is but vague intuition.

The evidently vague NP-hard problem of Protein folding and defects and overcompensation as destructive inflammation (overkill) is precisely an analog to the DNA structure and reading on many such levels. Your general approach is very bold and instructive (even your Newton like archaeology of Bible codes which put off reductionist physicists). But it is not clear to me the direction of scale is only as you say it is but maybe the other way around- I mean 61 can be seen as applying to organic scales or some have said 61 is the number of Christ in an 11 by 11 magic square matrix which is based on the old luna-solar calenders.

But I realized lately I was asking you for the math details and not the physical interpretaions, perhaps unfairly... then again such number theory should be common knowledge that for some reason our best physicists seem to be unaware.

At 7:42 AM,  Matpitka@luukku.com said...

I want to make clear that this is just a mathematical model, not random numerology. Symmetries of icosahedron, which characterise the Hamiltonian cycles turn out to be symmetries of genetic code. This is the main result. The manner how I ended up with these symmetries was not direct. I identified the symmetries by looking the code table, and later rediscovered them as symmetries of Hamiltonian cycles.

One usually sees DNA as 64 codons: 2^6. Nothing wrong with this. This is however not enough. To see DNA as as separate codes containing 60 and 4 codons and being fused or non-fused (explaining Sec and Pyl and why stop codons code for them) was difficult.

The fascinating analogues with religious myths is for me evidence that these myths reflect the basic structures of consciousness. My sincere hope that even the worst skeptic would realize this. Religion as a phenomenon of (collective) consciousness is fascinating irrespective of whether one is religious person or not.

I also dare to hope that I would not be labelled as a numerologist. TGD predicts a lot of numbers, typically integers, not only primes. They emerge from very general predictions rather than being random numerology. This kind of predictions are of course dangerous since standard physicist does have number theory in his mathematical repertoire. In the era of superstrings, it is sensational that theory predicts anything at all, and I understand the temptation of mainstream theorists to label TGD as a numerology;-).