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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

Article

Molecular Signaling from the TGD Viewpoint

Matti Pitkänen 1

Abstract

The findings of Elowitz et al lead to a formal model suggesting that ligands of type BMP (bone
morphogenetic protein) have interactions. The interactions would be non-local so that it is difficult
to imagine that they could have chemical origin. The TGD based model for these long range inter-
actions is based on dark photon resonance. For the simplest, receptors would correspond to fixed
bio-harmonies. In a single ligand system the ligand would have the bio-harmony of its preferred
receptor. The interaction between ligand magnetic bodies would be re-tuning and could replace the
preferred bio-harmonies assignable to the participating ligands with distributions of bio-harmonies.
Therefore the ligands of the multi-ligand system would couple by bio-resonance also to other than
preferred receptors. The model stimulates questions, which lead to a rather detailed model for the
re-tuning and tuning processes at the level of codons and amino acids. The model suggests that the
tuning to a given bioharmony for the dark counterparts of basic biomolecules and its stabilization
involves epigenetic control based on the methylation of some special DNA and RNA nucleotides and
amino-acids acting as analogs of tuning forks. The proposal that bioharmonies are molecular corre-
lates for emotions suggests that this process involves minimal number of methylations, which define
the seed of phase transition to a bioharmony in the scale of the basic unit of genome (such as gene),
mRNA sub-unit (splicing) and protein sub-unit.

1 Introduction

I learned recently about interesting findings about communications of information molecules. The Quanta
Magazine article ”Biologists rethink the logic behind cells molecular signals” (https://cutt.ly/iA28lqn)
summarizes the findings of Elowitz et al described in the article ”The context-dependent, combinatorial
logic of BMP signaling” [4] (https://cutt.ly/yA8r07b).

1.1 Observations

Messenger molecules attaching to receptors are thought to be responsible for chemical communications.
Intercellular communications would involve first messengers (hormones, neurotransmitters,...) and intra-
cellular communications second transmitters, which are not proteins but rather light molecules.

The standard interpretation has been that messenger molecules themselves define the message. Lock-
key hypothesis states that the ligand has a special region (key), which attaches to the receptor in a context
independent way determined by the geometries of these regions. Induced-fit hypothesis states that the
regions in question can modify their surfaces to achieve a perfect fit. For bacteria only intracellular
communications are possible and for them there is evidence that in some special cases lock-and-key
principle works as was demonstrated by Michael Elowitz, the leader of the research group behind the
recent work.

The findings of the Elowitz and his coworkers force them to conclude that this model fails for the
multi-cellulars.

1. The group studied so-called bone morphogenetic proteins (BMP) (https://cutt.ly/oA7kZna),
which regulate how cells proliferate and differentiate in various tissues by directing them to turn
sets of genes on and off. These proteins have many other functions than bone growth.

1Correspondence: Matti Pitkänen http://tgdtheory.com/. Address: Rinnekatu 2-4 A8, 03620, Karkkila, Finland. Email:
matpitka6@gamail.com.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

2. BMPs are so-called 1st messengers and mediate communications between cells. BMPs attach to the
receptors of various types at the surface of the cell. This step is followed by signal transduction acti-
nating the corresponding signalling pathway. Eventually this leads to a generation of transcription
factors in the cell nucleus controlling the genetic response. The work concentrated on the study of
the binding of BMPs to the receptors at the cell membrane.

3. Mammals have genes that encode 11 or more distinct BMP proteins. BMPs occurs dimeres of the
same or different proteins and also these pairs can pair up. The family of BMP proteins sticks
to the associated family of receptor proteins, which also appear as dimers of pairs of them. BMP
molecules are not very selective but given BMP sticks to several dimer pairs of receptors.

Several interesting findings were made.

1. The response of a cell to several ligands is not simply the sum of responses of individual ligands.
The lock-key mechanism assuming 1-1 correspondence between ligands and receptors fails in the
presence of several ligands whereas for a single ligand there is strongly preferred receptor for a
giveng ligand.

The simplest chemical (and thus local) explanation that in presence of several ligands the affinities
Kijk of ligands Li to the receptors Rjk formed by dimers (j,k) of homologous or nearly homologous
molecules change so that approximate 1-to-1 correspondence becomes 1-to-many. This is called
promiscuity.

The responses as concentrations at the cell membrane for the activated signalling pathways Pijk

associated with receptor i+jk would be still linear in concentrations of Li but the matrix character-
izing the rate for the creation of Pijk would not be diagonal matrix anymore with respect to pairs
(i, jk).

2. This situation is easy to model mathematically but it is difficult to understand the physical mech-
anism behind the promiscuity.

3. The affinities are context dependent in the sense that they depend on the target cell and the
developmental stage of the cell.

4. One can classify the ligands in terms of whether they tend to increase or reduce receptor expression.
Ligands can be also classified in terms of their positive, or negative synergies with other ligands.
In the simplest situation one studies all possible pairs of ligands and finds their responses. Two
ligands Li and Li are regarded as equivalent if the responses for the pairs (Li, Lk) and (Lj , Lk) are
identical for all k. This defines functional equivalence. Sequence similarity (biological homology)
does not in general reflect the functional equivalence.

The effects of the ligands in equivalence classes depend on the context (cell type and cell age corre-
lating with receptor concentrations). However, it is found that the equivalence classes are context
independent. The proposal is that a single linear parameter could characterize the equivalence
classes for BMPs considered.

5. This inspires a chemical model for the situation. The basic parameters would be affinitiesKijk telling
the tendency of ligand Li to attach to dimer (j, k), signal complex activities εijk characterizing the
rate for the formation of signal complex Pijk. Also the receptor concentrations Ai and Bi for the
receptors of type I and II appear as parameters. The empirical data makes it possible to find the
best fit for the parameters. Promiscuity is the basic predictions. The model could be understood
in terms of the competition of ligands for receptors (j,k).

An inverse relationship between affinities and activities is predicted. Small number of affine ligands
with weak activity or vice versa.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

1.2 TGD view about the findings

What can one say about the situation in the TGD framework? Here only the key ideas of the TGD
inspired quantum biology are described. More detailed summaries can be found in various articles related
to dark variant of biochemistry [40, 31, 37], to dark realizations of genetic code [29, 34, 33, 36, 39], and
to the models for cell membrane as generalized Josephson junction [17, 18, 19, 28], of Pollack effect [30],
and of water memory and morphogenesis [23, 38].

1. The basic notion is magnetic body (MB) carrying dark matter as heff = nh0 phases of the ordinary
matter and behaving quantum coherently in length scale proportional to heff . MB would control
biomatter.

Communications to MB, sensory input, would be in terms of generalized dark Josephson radiation
from the cell membrane and the control by MB in terms of dark cyclotron radiation. The sensory
input would be from the entire cell membrane and induced by the attachment of the ligands to the
receptors.

The earlier proposal was that the control signals from MB affect directly the genome. The existing
chemical picture based on signal pathways activated at the cell membrane however suggests that
the situation is not so simple. The control signal arrives from MB to the receptors and activates
signal pathways. At the nuclear membrane similar processes would occur and lead to the activation
of transcription factors by similar signal pathways.

2. If the MB determines the response of receptors in a non-local way, promiscuity could be only
effective. Another option is that MB can control the affinities of receptors (by modifying their
surface geometries as in the induced fit model) so that the diagonal L-R matrix becomes non-
diagonal.

3. Context dependence would conform with the idea that MB determines the response and changes
during aging. The aging can be understood in the TGD framework as slow thermalization of MB so
that its temperature approaches the Hagedorn temperature of magnetic flux tubes. Physiological
temperature would be related very closely to Hagedorn temperature of MB.

The almost computer program-like determinism of biochemistry is in a sharp conflict with the stochas-
ticity expected to result from the locality and statistical nature of chemistry.

1. In the TGD framework and at a given level of scale hierarchy the dynamics of the space-time
surface as a preferred extremal is deterministic apart from small violations of determinism. Space-
time region as a preferred extremal is a minimal surface with singularities, which would bring in
the failure of determinism. Soap film with frames serves as a good analogy.

2. The notion of a magnetic body having flux tubes as body parts leads to a model of bio-catalysis
in which molecules are replaced with nodes of flux tube network. Molecules can find each other as
part of this dynamical network involving reconnection of U-shaped flux tubes and their shortening
in a reduction of heff liberating energy making to overcome the potential wall making the reaction
low.

3. One can argue that the second law implies stochasticity in molecular scales. Zero energy ontology
(ZEO) is another possible source of determinism. In so called ”big” state function reductions
(BSFRs) the arrow of time changes and the time evolution leads to the direction of geometric past
so that for the observer with the standard arrow of time the time evolution obeys second law in
wrong time direction and looks like self-organization which is basic characteristic of living matter
and usually thought to involve metabolic energy feed in an essential manner. In fact the time
reversed time evolution would change dissipation as loss of energy with extraction of energy from
the environment.

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DNA Decipher Journal | July 2022 | Volume 12 | Issue 2 | pp. 101-123 104

Pitkänen, M., Molecular Signaling from the TGD Viewpoint

The findings of Elowitz et al [4] lead to a formal model suggesting that ligands of type BMP have
interactions. The interactions would be non-local so that that they could have chemical origin. The
TGD based model for these long range interactions is based on dark photon resonance. For the simplest,
receptors would correspond to fixed bio-harmonies. In a single ligand system the ligand would have the
bio-harmony of its preferred receptor. The nteraction between ligand magnetic bodies would be re-tuning
and could replace the preferred bio-harmonies assignable to the participating ligands with distributions
of bio-harmonies. Therefore the ligands of the multi-ligand system would couple by bio-resonance also to
other than preferred receptors.

The model stimulates questions, which lead to a rather detailed model for the re-tuning and tuning
processes at the level of codons and amino acids. The model suggests that the tuning to a given bio-
harmony for the dark counterparts of basic biomolecules and its stabilization involves epigenetic control
based on the methylation of some special DNA and RNA nucleotides and amino-acids acting as analogs
of tuning forks.

The proposal that bioharmonies are molecular correlates for emotions suggests that this process in-
volves minimal number of methylations, which define the seed of phase transition to a bio-harmony in
the scale of the basic unit of genome (such as gene), mRNA sub-unit (splicing) and protein sub-unit.

2 Bio-harmony and context dependence

Also bio-harmony might relate to context dependence if fundamental communication and control signals
take place at the dark level that is between DAAs instead of AAs (amino acids) as parts of proteins by
energy and frequency resonances. DAAs would pair with AAs and communicate with energy resonance.

2.1 Bio-harmony

Consider first the bio-harmony [29, 33, 34, 39] in more detail.

1. Z6, Z4, and Z2,rot or Z2,refl act as symmetry groups for the 3 icosahedral Hamiltonian cycles. Each
cycle, one of type XZ6, one of type Z4, and one of type Z2, defines 12-note scale and 20 3-chords
identified as icosahedral faces and DNA codons. The notes along the cycle are obtained as a quint
cycle, that is by scaling the frequency of the note by factor 3/2 at each edge of the oriented cycle.

The orbits of faces under Zn are assigned with amino-acids (AAs). This assumption has a concrete
interpretation in terms of resonance mechanism for bio-communications [39].

This gives 60 3-chords and the numbers of triangles at the orbits of triangles and the numbers of
triangles at orbits correspond nicely to the numbers of DNA codons coding for AAs. 4 codons are
however missing. The fusion with a unique tetrahedral code gives 64 chords and a dark 3-photon
realization of the genetic code.

Also the identification of DtRNA, dtRNA, and DAA in terms of icosahedral code is possible as found
quite recently [39]. Also the dark realization of genetic codons in terms of dark proton triplets allows
this.

2. Z6 allows unique icosahedral harmony defined by 12-note scale realized as an icosahedral Hamil-
tonian cycle. The corresponding AAs correspond to 3 DNA 6-plets and one DNA 2-plet. Z4

corresponds to 2 bioharmomies with 5 amino-acids which correspond to DNA 4-plets. Z2,rot and
Z2,refl correspond to 10 2-plets both. Z2,rot corresponds to 3 icosahedral harmonies and Z2,refl to
5 icosahedral harmonies. This makes 1× 2× (3 + 5) = 16 bio-harmonies if the common key of the
12-note scale for the 3 icosahedral harmonies does not matter and the orientation of the Hamiltonian
cycle does not matter. One can also consider the possibility that the key and the orientation of the
cycle for the 3 icosahedral harmonies matter. The change of the orientation replaces quint cycle
with quart cycle (CG corresponds to quint and CF to quart).

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

3. The interpretation of bioharmonies is as correlates for moods, emotional states. There is evidence
for this interpretation from the strange finding that RNA is able to transmit conditioning based
on negative or positive emotions generated by stimulus (https://cutt.ly/6SuLNqk) discussed in
the TGD framework in [32, 35]. The interpretation would be that DRNA represents the effect of
stimulus by its bio-harmony characterizing emotional state, and can induce molecular emotional
expression in DDNA-DNA pairing and also in DAA-AA pairing and DX-X pairing in general.

DX-X pairing by energy resonance mechanism would correspond to emotional expression. Some-
thing in X would depend on bio-harmony. In the case of DNA and RNA this something could be
the methylation state and its analogs so that there would be a direct connection with epigenesis.
Epigenesis would realize the dynamics of emotional expression.

One can raise several questions about bio-harmony.

1. How fast is the dynamics of the molecular and higher level emotions and moods? If epigenetics
controls the dynamics of emotions, it could be rather fast at the molecular level. Note that the
hierarchy of Planck constants predicts a hierarchy of time scales.

2. How large parts of a given organism a given bio-harmony could characterize? Biomolecules, cell
nucleus, cell, organelle, ...? Is there a hierarchy of harmonies so that the harmonies in different
scales need not be identical?

Concerning molecular bio-harmonies, epigenetics could help to answer the question. For instance,
one can ask whether bio-harmony characterizes individual bio-molecules such as enzymes and re-
ceptors.

3. Could bio-harmony explain at least part of the context dependence found in the ligand-receptor
dynamics by the group of? It would seem that bio-harmony appears as an additional aspect of the
ligand-receptor pairing involving geometric constraints modelled in terms of lock-key or induced-fit
mechanisms. The enzyme and substrate would be like daters. The resonance mechanism would
allow E and S to meet and geometric constraints would determine whether this can lead anywhere.

4. This inspires several questions. Could the affinities and signal complex activities be determined
by the molecular emotional state of the L-R composite coded by the bio-harmony of the DX-X
complex? Could the ”emotional” state DX-X control affect the state of X complex? How? Could
this coupling have interpretation as emotional expression in a generalized sense?

5. For the simplest model this would predict that for Z4 the 5 AAs coded by 4 DNA codons would have
two emotional states and for Z2 10 DAA-AA pairs could have 3 resp. 4 emotional states depending
on whether one has Z2,rot resp. Z2,refl.

2.2 Could ligand interactions reduce to a re-tuning of ligand harmonies?

The notion of ligand interaction has been introduced as a purely formal notion in the article and it is
difficult to imagine a local chemical realization for it. However, the fact is that ligands change their
behavior in the presence of other ligands. Could the ligand interactions be realized at the level of their
MBs?

2.2.1 Ligand interactions as re-tuning

Could ligand interactions reduce to the re-tuning of ligand harmonies by the resonant dark photon inter-
actions between DAA flux tubes?

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

1. Assume that ligands and receptors can have several bio-harmonies but that free ligands (single lig-
and situation) and in the absence of other ligands they correspond to single preferred bio-harmony.
Assume that each receptors corresponds to a single bio-harmony (also this assumption could be
relaxed). Free energy minimization could imply preferred bio-harmonies for both receptors and lig-
ands. Assume that ligands can only pair with receptors with the same bio-harmony. The immediate
question is whether the 3+4 receptors assigned with BMPs could relate to 3+5 Z2 type harmonies.
The problem is that one Z2,refl harmony would not correspond to a receptor.

2. Interactions between two ligands L1 and L2 with different bioharmonies could induce a re-tuning
of L1 to the bio-harmony of L2 or vice versa. This tuning must respect the symmetry group Zn,
n = 6, 4, 2 in question. The Zn orbits would be preserved but the corresponding 3-chords would be
modified.

Some findings about water memory [38] support re-tuning as a basic mechanism of communications
between dark biomolecules and it is very natural in the resonance picture. Note that re-tuning is a
basic mechanism in radio communications.

Re-tuning would replace the ensemble of ligands with an ensemble in which also non-preferred L-R
pairings are possible. It would make the affinity matrixKi(jk) and activity matrix εi(jk) non-diagonal
and induce promiscuity. Probability distribution for bio-harmonies of ligands would emerge in this
way.

3. The large-scale quantum coherence at the level of MBs inspires the question whether the quantum
superposition of bioharmonies could occur for DAAs.

Could quantum superposition allow to understand the observation that the increase of the param-
eters Ki(jk) is accompanied by the decrease of εi(jk) and vice versa. Could one think that with a
suitable normalization one has

∑
(jk)Ki(jk)εi(jk) = constant. In ZEO one could regard the entire

signal complex, which involves both ligand, receptor and what it induces, as a single zero energy
state as a superposition of deterministic time evolutions.

If the formation of signal complexes involves a quantum transition from a single ligand-receptor
pair to a their quantum superposition involving delocalization at the cell membrane followed by
state function reduction involving localiation that is selection of the complexes, the condition∑

(jk)Ki(jk)εi(jk) = constant could reflect probability conservation.

4. Re-tuning of the icosahedral harmony for Z4 and Z2 should have a counterpart affecting the physics
of AAs. Could the re-tuning be generated at the level of DAA and result from the variation of flux
tube thickness as a motor action of MB? Or could it be induced by re-tuning at the level of DDNA?
Tuning must be visible at the level of AAs since DX-X resonance energies must be modified.

2.2.2 Tuning and re-tuning at the level of DX-X pairing

What could the re-tuning mean for the DX-X pairing?

1. For DDNA-DNA pairing dark cyclotron photons must couple to some degrees of freedom of DNA.
In the TGD framework, DNA can be magnetized [38]. The pairing with DDNA flux tubes carrying
a monopole flux with DNA strands is expected to induce magnetization along DNA due to the ring
currents of electron pairs of the aromatic rings analogous to Cooper pairs.

The simplest candidates for re-tuned frequencies are cyclotron frequencies for magnetized nucle-
obases. In re-tuning the cyclotron frequencies for electron pairs) assignable to aromatic rings of
nucleotides would be modified in re-tuning. The change of the thickness of the monopole flux tubes
defining the 12-note scale would automatically induce the re-tuning at DNA level. The re-tuning
could be induced by DDNA, DRNA, and DtRNA and would not require chemistry.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

2. What about the DAA-AA tuning? The only AAs with aromatic rings are Phe, tyr and trp. Could
DAA-AA resonance coupling between cyclotron radiation of DAA and vibrational modes of AA
with energies in the range .45-.045 eV spanning slightly more than 3 octaves?

The general forms of the vibrational and cyclotron energy spectra are the same and for a proper
value of heff the scale of the DAA spectrum is the same and resonance is possible.

Re-tuning would require change of the conformations of the AAs so that the elastic constants would
be modified. MB could induce this re-tuning as a kind of entrainment. As already proposed, this
could be achieved at the level of DNA by methylation of the start codon fixing the bio-harmony.

2.2.3 Could special 3-chords act as tuning forks

Physical model for the tuning and re-tuning should be based on resonance model.

1. Tuning to a particular 3-chord or 3-chords should force the entrainment to the bio-harmony. These
3-chords would serve as an analog of a tuning fork.

The simplest, and perhaps unrealistic, option is that the met 3-chord associated with the start codon
alone fixes the bio-harmony uniquely. The met 3-chord should be different for all Z2 harmonies.

2. The chords fixing the bio-harmony (the tables for the 3-chords of bio-harmonies are given in [29])
should be very special and thereindeed are very special chords in the icosahedral harmonies. The
epigenetic modification of the amino-acids corresponding to these 3-chords could force the re-tuning
of the bio-harmony.

The triangles, whose edges do not belong to the Hamiltonian cycle, define 0-quint 3-chords containing
no quint. These chords include dissonant chords possibly having semitones or tones intervals between the
notes (octave equivalence and quit cycle along the Hamiltonian cycle is assumed). There are 8 different
types of 0-quint chords with basic note X in 12-note scale labeled as Xexk, k = 1, .., 8, if the key does
not matter.

1. From the Appendix one learns that there the first possess no 0-quint chords Z4 harmony. In this
case, one could argue that the bit defined by the presence or absence of the 0-quint chord defines
the tuning fork, which could correspond to a methylation of some codon coding for one of the 5
AAs coded by DNA 4-plet. It is not clear, whether the choise of the codon matters.

2. By looking at the tables of 3-chords in the chord tables of the Appendix, one finds that if key
matters, it is easy to distinguish between harmonies using a single 0-quint chord. If the key does
not matter, it is in principle almost possible to assign different 0-quint chords to, say, met. There
are 2 Z2,refl harmonies with 2 0-quint chords, which cannot be distinguished in this manner. If
one introduces a fixed key or uses a second special 0-quint chord as a turning fork, also Z2,refl

harmonies can be distinguished from each other.

Interestingly, the number of BMP receptors possibly assignable to Z2,refl is 4 rather than 5.

3. Also tetrahedral codons define special chords in the sense that the intervals are separated by minor
third. These 3-chords are identical under the octave equivalence. In the model considered in [39],
3 of them correspond to stop codons whereas the remaining codon corresponds to trp.

2.2.4 Re-tuning as an epigenetic process

Re-tuning is an epigenetic process and can be seen as a control of MB. Methylation and its analogs are
basic tools of epigenesis.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

1. mRNA methylation (https://cutt.ly/1Srm06F) occurs after transcription and is controlled by
genes coding the needed enzymes. The methylated RNA nucleobase is often called the ”fifth RNA”
base. Start codon AUG coding for met is methylated as also the 3-prime untranslated regions
(3’-UTRs) immediately after the stop codon. This region post-transcriptionally influences gene
expression.

2. The findings that the RNA of a conditioned sea snail scattered over neurons of second sea snail
in Petri dish generate neuronal correlates of conditioning (https://cutt.ly/6SuLNqk), discussed
from the TGD point of view in [32, 35], support the view that the magnetic body of the RNA of sea
snail infects the emotion/mood related to the conditioning. The emotional state, mood, of DNA
and RNA would affect gene expression. Epigenesis could be based on emotional states lasting for
several generations. This is natural in ZEO [29, 35].

Hints about how the methylation could be involved with the tuning to a particular bio-harmony comes
from the research of the group led by Matthias Soller [3] (https://cutt.ly/OSeGnJu).

1. Post transcriptional methylation is known to occur for the few nucleotides of mRNA following the
cap of mRNA, whose function has remained poorly understood. Soller and collaborators demon-
strated that the two enzymes coding for the methylation of these nucleotides played an important
role in the animals’ reward learning process. The flies without the genes coding for the methylation
showed a defect in their ability to learn the association of a specific odour with a sugar reward.

2. Earlier work by one of paper’s co-authors, Prof. Rupert Fray, has demonstrated that that cap
modifications are highly dynamic in mice and that these modifications played a role in transporting
the mRNAs to synapses.

3. The lack of methylation implies a lack of the desired conditioning. Conditioning involves emotions,
perhaps also at the molecular level: could the bio-harmony of proteins involved with the process
differ from that associated with the protein activated by the odour molecules? The proteins would
be out-of-tune and conditioning would not happen.

The role of cap modifications in the mRNA transport would conform with the assumption that dark
photon resonance allows the mRNA to find synapses. If the bioharmony for them is wrong there is
no resonance and the transport fails.

These findings suggest the following interpretation in the TGD framework.

1. The resonance mechanism would force DDNA and DmRNA to have the same bio-harmony. The
post-transcriptional methylation of the first RNA codon could re-tune and stabilize mRNA bio-
harmony.

Stabilization could involve a methylation of a large enough number of special RNA codons so that
it would serve as a seed of a phase transition forcing the same bio-harmony for all codons. If bio-
harmonies correspond to molecular moods, this would be analogous to the spread of an emotional
mood in crowd. The special codons as signatures of the mood could be especially effective inducers
of this phase transition.

2. Could a 0-quint 3-chord assigned to met in the beginning of mRNA fix the Z2 harmony almost
uniquely by acting as a tuning fork. Z4 harmony could be fixed by the absence of methylation
in some mRNA nucleotide in codon coding for one of the 5 AAs [(val,pro,thr,ala,gly)] coded by 4
codons.

Note that the methylation of 2 AUG nucleotides of met affecting the cyclotron frequencies of AUG
could in principle select between the 16 bio-harmonies predicted by the simplest model. This

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

estimate is however based on counting of bits and bio-molecules need not see each other as bit
sequences as we do.

The methylation of the mRNAs associated with several 0-quint chords could help to stabilize the Z2

harmony at the level of DmRNA. Could the proteins obtained by splicing and involving methylation
in the beginning of mRNA portions coding them consist of functional sub-units with different bio-
harmony?

3. What about DNA? Could the methylation of the start codon also now help to stabilize the Z2

bio-harmony. Only A and C DNA nucleotides of DNA strand can be methylated (as also T and G
nucleotides of the conjugate strand). Note that A and G appear often in DNA repeats defining part
of what was called junk DNA. One can ask whether the methylation of A and C could stabilize the
bio-harmony and DNA level.

The corresponding RNA codon contains at most one U or G nucleotide. Note that met corresponds
to AUG whereas AGU corresponds to cys which together with trp (coded by tetrahedral codon) are
the only sulphur containing amino-acids. Met is special in the sense that it belongs to a symmetry
broken codon doublet for which ile has replaced met.

4. The first mRNA codon AUG codes for met so that the D(AUG)-Dmet pairing could induce the
DAA bio-harmony and affect the vibrational frequencies of AA. This is perhaps enough for the
stability of the bio-harmony. Could pretein methylation help to stabilize the bio-harmony of pro-
teins? According to Wikipedia (https://cutt.ly/uSiVACT), protein methylation is a type of post-
translational modification featuring the addition of methyl groups to proteins. It can occur on the
nitrogen-containing side-chains of arginine and lysine but also at the amino- and carboxy-termini
of a number of different proteins.

One can imagine 2 options for changing the bio-harmony at DAA-AA level. For the bureaucratic
option, the re-tuning would occur at the DNA level. This would require enzymes coded by appropriate
genes to re-tune the first codon of mRNA coding for AA.

For the non-bureaucratic option, DAA would re-tune AA directly by entrainment and this could
involve re-methylation.

2.3 An attempt to concretize the model of ligand interactions

The following is a very naive first attempt to concretize the idea about ligand interactions as a re-tuning,
whhich affects the matrices Ki(jk) and εi(jk). Reader should take the following considerations as as free
associations.

1. BMPs couple to 4+3 receptors. There are 3 Hamiltonian cycles with Z2,rot symmetry and 5 cycles
with Z2,refl symmetry assignable to 10 amino-acids coded by 2 or single DNA (met) have 3. There
are 4+3 receptors and 5+3 bioharmonies: could it be that the considered 4 receptors correspond
to 4+3 Z1 harmonies with the same Z4 harmony and that there is also a fifth receptor of this kind
but not considered?

A priori, any protein could correspond to any bio-harmony but the correlation of DAA and bio-
harmony could be forced by dynamics since the DAA-AA resonances might be possible only for
certain Z2 harmonies (and only for one of the 2 Z4 harmonies). Suppose that the receptors indeed
correspond to one particular Z2 harmony each.

2. If the binding sites for BMP-receptor pairs correspond to single AA (in analogy with tRNA-mRNA
binding), the binding site for Z2 harmonies should correspond to a AA which is one of the 10 AAs
coded by DNA doublet or singlet. The reduction of correspondence to the level of binding site
AA would conform with the finding that the functional similarity of BMPs does not very closely
correspond to the sequence similarity.

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In the code table there are 9 doublet AAs and 1 singlet. Symmetry breaking is present [39]. It is
not quite clear which doublets correspond to Z2. For instance, phe could correspond to the doublet
for Z6 leaving 8 doublet AAs plus (ile,met) as a doublet with a broken Z2 symmetry. UGG coding
for trp and 3 stop codons would correspond to the tetrahedral cycle.

By resonance condition, at most 3 receptors should correspond to more than 1 BMP as their
preferred receptor.

There are also chemical constraints on the AAs acting as a binding site. Resonance condition for
DAAs implies that pairing AAs are identical. The pairing AAs must be neutral and must be coded by
DNA doublets or singlets. This leaves the following cases under consideration.

1. Two amino acids have amide side-chains.

• Asparagine (Asn): NH22COCH2-

• Glutamine (Gln): NH22COCH22CH2

These side-chains do not ionize in the normal range of pH.

2. Two side-chains contain sulfur atoms, of which one ionizes in the normal range.

• Cysteine (Cys): HSCH2-

• Glutamine (Gln): NH2-COCH2-CH2-

3. Three amino acids have aromatic ring structures as side-chains. Of these, tyrosine ionizes in the
normal range; the other two do not.

• Phenylalanine (Phe)

• Tyrosine (Tyr)

• Tryptophan (Trp)

This would give 2+2+3 =4+3 AAs. In the above mentioned option Phe is however assigned with Z6

harmony but any other doublet AA could correspond to Z6 harmony. This would suggest that AAs with
amide side chains and containing sulphur correspond to 4 Z2,rot harmonies.

There are 10 BMPs with the decomposition 10=3+3+2+1+1. Using the standard biological notation,
this correspondings to the decomposition [GDF5,GDF6,GDF7], [BMP5,BMP6,BMP7], —BMP2,BMP4],[BMP9],
[BMP10]) to functional equivalence classes [4]. Could the two 3:s correspond to the 3 Z2,rot harmonies
and 2+1+1 to 4 of the 5 Z2,refl harmonies?

The two triplets [GDF5,GDF6,GDF7] resp. [BMP5,BMP6,BMP7] are weak resp. strong activators.
Both GDFs (growth differentiation factors) and BMPs (bone morphogenetic proteins) belong to the
transforming growth factor beta superfamily (TGF). If GDFs are excluded the correspondence between
BMPs and receptor proteins is 1-to-1.

2.4 Could the dark matter hierarchy relate to the bio-harmony?

One can wonder how the hierarchy of algebraic extensions and algebraic evolution defining the evolution-
ary state for a given layer of MB affects the L-R pairings.

1. 3N-photons and 3N-nucleons as dark variants of basic information molecules would correspond to
Galois confined states for which the 4-momenta for components are algebraic integers summing up
to ordinary integer when the momentum unit is defined by the p-adic length scale associated with
the extension. Also frequencies would correspond to rational integers for Galois confined states.

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Table 1: Notation of chords inspired by popular music notations.

CEG ≡ C , CD]G ≡ Cm , CD]F] ≡ Co , CEG] ≡ Caug ,
CFG ≡ C4 , CF]G ≡ C4+ , CGG] ≡ C6− , CGA ≡ C6 ,
CGB[ ≡ C7 , CGB ≡ Cmaj7 , CGC] ≡ C9− , CGD ≡ C9 .

(3.1)

2. These states depend on the algebraic extension of rationals defining n = heff/h0 as its dimension
although mass squared values and momenta are integer valued as also frequencies. This would give
an additional context dependence. For instance, organisms at higher levels of evolution could have
larger values of heff associated with the dark variants of the basic biomolecules.

3 Appendix: Tables of basic types of 3-chords for icosahedral
harmonies

3.1 Icosahedral harmonies as Hamiltonian cycles

One can find the list of Hamiltonian cycles at http://tinyurl.com/yacgzm9x. The edge {1, 2} is fixed
and cycles are oriented so that there are 1024 of them. All of them are relevant from the point of music
interpretation and the change of orientation corresponds to major-minor duality, albeit not in the simplest
sense. Note that this duality does not affect the characteristics listed above.

The general following general results hold true as one can learn at http://tinyurl.com/pmghcwd.
One can classify the cycles using their symmetries which can correspond to isometries of icosahedron
leaving them fixed or to a reflection taking the vertex n at the cycle to vertex 12− n. This symmetry is
not same as change of orientation which is purely internal operation and cannot change the cycle.

One can even find images of the cycles possessing symmetries at http://tinyurl.com/y8ek7ak8 and
deduce the triplets n and p characterizing them by visual inspection. Also one can write explicitly the
3-chords defined by the three kinds of faces. I have deduced the triplets n and the 3-chords defining
the harmony by the inspection of the images. ”Bio-harmony” (4, 8, 8) forced by the model of extended
genetic code involving also the 21st and 22nd amino-acids is of special interest.The classes of cycles with
symmetries 6-fold rotational symmetry and two distinct reflection symmetries realize it.

Before continuing some terminology and notation is in order. Take C as the major key. Submediant
or relative minor corresponds to Am, subdominant (sharp or flat) to F major (F ) or Fminor (Fm),
dominant to G. The notation for chords is such that quints correspond to subsequent notes in the chord.
For 1-quint chords this means that first two notes define the quint.

Table 1 summarizes the notation inspired by the popular music notation. The basic difference is that
the third is in most cases excluded so that the emotional character of the chord is not fixed.
Besides these notions it is convenient to introduce additional notations for various dissonant chords
appearing as 0-quint chords.

CC]D ≡ Cex1 , CC]D] ≡ Cex2 , CDD] ≡ Cex3 , CDE ≡ Cex4 ,
CD]E ≡ Cex5 , CC]E ≡ Cex6 , CDF] ≡ Cex7 , CDG] ≡ Cex8 .

(3.2)

Clearly, the sets {ex1}, {ex2, ex3}, {ex4, ex5, ex6}, {ex7}, {ex8}, corresponds to the span of 2, 3, 4, 6, 8
half notes for the chord.

The following summarizes the results. Note that Cex7 can be seen as part of D7 chord.

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1. There are 6 collections of cycles without any symmetries containing 48 cycles each: these 48 cycle
are mutually isometric so that one can say that there 6 different harmonies.

2. There is a collection with 6-fold rotational symmetry, 48/6=8 examples. n = (2, 12, 6). The chords of
this scale define 6-note scale involving only total steps. CDF and its 6 translates by integer number
of steps define 6 1-quint chords. CE[G (Cm) and its 6 translates (they obviously correspond to the
6-fold rotational symmetry) define also 6 1-quint chords. The reflection transforms these series to
those defined by GB[G and its translate and byFAC (F major) and its translates. Impressionists
like Debussy used 6-note scale of this kind. Half-octave shift is an exact symmetry. 1-chords lack
the third so that one cannot assign to 3-chords any emotional quality. The extension to 4-chord can
however bring either ”happy” or ”sad” quality. Clearly, these harmonies have ”jazzy” character.

0-quint chords are Faug ≡ FAC] and Gaug ≡ GHD] are transformed to each other by both
half-octave shift and inversion.

3. There are 2 collections with 2 distinct reflectional symmetries with 12=48/4 representatives in each.
Half-octave scaling is a symmetry of both these scales as one might guess.

The first cycle (see Fig. 3.3 ) has n = (0, 16, 4) so that there are no 0-quint chords which in general
are dissonant. Second cycle (see Fig. 3.3 ) realizes n = (4, 8, 8) bio-harmony and deserves some
comments. It will be discussed in detail later.

(a) The 8 2-quint chords consist of B[FG ≡ B[9, C9, F9, G9 and their half-octave scalings.
Clearly, the simple four-note scale appears here.

(b) Using the popular notion introduced earlier, 1-quint chords consist of two 4-plets Dmaj7, E9−,
A7, A6 and G]maj7, B[9−, D]7, D]6 related by half-octave shift. The harmony contains no
”simple” major or minor chord and only the extension to tetrahedral harmony can provide
them. The same is true for the second bio-harmony.

(c) The 4 0-quint chords are Cex3 ≡ CDD] and Eex2 ≡ EFG and their half-octave scalings
F]ex3 ≡ F]G]A and B[ex2 ≡ B[BC]G.

4. There are 3 collections with Z2 rotational symmetry with 48/2 = 24 representatives in each. The
triplets n are (0, 16, 4) (see Fig. 3.3 ), (2, 12, 6) (see Fig. 3.3 ), and (4, 8, 8) (see Fig. 3.3 ).

All these harmonies are symmetric with respect to half-octave shift (tritonus), which obviously
corresponds to the Z2 rotation. Tritonus would not have been tolerated by catholic church! This
symmetry characterizes all 3 harmonies. Basic 3-chords do not contain pure minor and major
chords. The reflection of the scale does not leave the collection of chords invariant but it is not clear
whether this corresponds only to a change of scale, probably not.

Consider the (4, 8, 8) case (see Fig. 3.3 ).

(a) The 8 2-quint chords appear as four-plet H9, C]9, D]9, F9 and its half octave shift (tritonus
interval) acting as a symmetry of the harmony. 2-quint chords are always of type X9 (note
that the third is missing) but also 1-quint chord can be of form X9 as explicit construction of
chords demonstrates: I have denoted these 1-quint chords by symbol X4 (CDG is obviously
equivalent with CDG).

(b) Using the popular music notation introduced earlier, the 8 1-quint chords areD7, Amaj7, A4+, E7
and their half-octave shifts G]7, D]7, D]4+, B[7.

No major and minor chords are included and only the extension to tetra-icosahedral harmony can
provide them and also break the symmetry giving rise to well-defined key.

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Table 2: Table gives various types of 3-chords for harmonies with Z6 rotational symmetry. Note that
half-octave shift is an exat symmetry. Note that Gaug = CEG], F aug act as bridges between the groups
related by half octave shift. The chords have been arranged so that they form orbits of Z6. ”Amino-acid
chords” correspond to preferred chords at the orbits.

(n0,n1,n2) 0-chords 1-chords 2-chords

(2, 12, 6) (Faug,Gaug) (Cm,Dm,Em,F]m,G]m,B[m), (C9, D9, E9, F ]9, G]9, B[9).
(F6, G6, A6, B6, C]6, D]6).

5. The four 0-quint chords appear in two types. D]ex2 ≡ D]EF] and its half-octave shift Aex2 ≡
AB[C plus Hex3 ≡ HC]G and its half-octave shift Fex3 ≡ FGC]. According to usual thinking
these chords involve dissonances. This dissonance character is a rather general phenomenon for the
harmonic loners and classical views about harmony would exclude them as asocial cases! In the
case of maximally symmetric harmony the loners are diminished chords and thus not so dissonant.
In some cases there are no 0-quint chords.

There are 5 collections of 20 chords with Z2 reflection symmetry (see Figs. 3.3, 3.3, 3.3, 3.3,
3.3 ). The integer triplets n are (2, 12, 6), (2, 12, 6), (4, 10, 6), (2, 12, 6), (2, 12, 6). Bio-harmony has a
representative also in this class (see Fig. 3.3 ). The half-octave scaling symmetry is broken for these
harmonies.

Some comments (4, 8, 8) case are in order (see Fig. 3.3 ).

1. 2-quint chords appear as reflection related multiplets C9, D9, H]9, D]9 and C]9, H9, F9, B[9.

2. 1-quint chords appear as symmetry related mutiplets G,D7, Amaj7, E7 and C]m,F]6, H6−, E6.
Key G major and C] minor would be natural looking keys even without tetrahedral extension. For
the mirror image B[ minor and E major would be the natural looking keys. For extension E major
would be the key.

To sum up, half octave shift is a symmetry of all harmonies expected those having only Z2 reflection
symmetry, and fails thus also for the corresponding bio-harmonies. The tables below give list for the three
types of 3-chords for the 11 harmonies possessing symmetries. A 3-chord with n quints is called n-quint
chord. The harmonies are labelled by integer triplets (n0, n1, n2), ni gives the number of n-quint chords.

The reversal of the orientation for the cycle induces the transformation C ↔ C, F] ↔ F], H ↔ C],
F ↔ G, D ↔ B[, E ↔ G], A↔ D] and produces a new scale with minor type chords mapped to major
type chords and vice versa.

The standard notation of chords used in popular music is used. One must however remember that
all 3-chords except those which are simple majors or minors lack the third so that their emotional tone
remains uncharacterized. For instance, C6 does could be replaced with Cm6 and G7 with Gm7. The
reader can check the chords by direct inspection of the figures. The convention used is that vertex number
1 in Hamiltonian cycle corresponds to C note.

3.2 Tables for the 3-chords of icosahedral harmonies

The following tables give the 3-chords of the icosahedral harmonies.

3.3 Illustrations of icosahedral Hamiltonian cycles with symmetries

The figures below illustrate the Hamiltonian cycles involved. Quite generally, the Zn symmetry acts by
a shift by 12/n quints along the cycle and the orbits of chords consist of at most n chords of same type
as the reader is encouraged to verify.

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Table 3: Table gives various types of 3-chords for the two harmonies with Z4 = Zrot
2 × Zrefl

2 symmetry.
4-plets represent the orbits. First cycle has no harmonic loners. Second cycle gives rise to bio-harmony
(4, 8, 8) for which 0-quint chords are dissonant. Both cycles have Z2 rotation symmetry acting as a
vertical reflection symmetry in figures and realized also as half-octave shift so that 4-plets contains chords
and their half-octave shifts. The genuine reflection symmetry acts as a horizontal reflection symmetry in
figures. The cycles correspond to figures 3.3, 3.3

.
(n0,n1,n2) 0-chords 1-chords 2-chords

(0, 16, 4) (D7, D6, G]7, G]6), (B[9, B9, E9, F9).
(G4+, A9−, C]4+, D]9−),
(Emaj7, Gmaj7, B[maj7, C]maj7),
(C9−, A9−, F ]9−, D]9−).

(4, 8, 8) (Cex3, Eex2, F ]ex3, B[ex2). (Dmaj7, E9−, A7, A6), (B[9, F9, C9, G9).
(G]maj7, B[9−, D]7, D]6). (E9, B9, F ]9, C]9).

Table 4: Table gives various types of 3-chords for harmonies with Z2 rotation symmetry acting as half-
octave shift. The doublets represent 2-chord orbits. The cycles correspond to figures 3.3, 3.3, and 3.3.

(n0,n1,n2) 0-chords 1-chords 2-chords

(0, 16, 4) (Em,B[m), (Cm,F]m), (D9, G]9),
(G6, C]6), (A6, D]6), (E9, B[9).
(D4+, G]4+), (B4+, F4+),
(Cmaj7, F ]maj7), (G6−, C]6−).

(2, 12, 6) (Aex4, D]ex2). (Am,D]m), (G9−, C]9−), (C9, F ]9),
(C4, F ]4), (E4+, B[4+), (A9, D]9),
(Dmaj7, G]maj7), (D9, G]9).
(Bmaj7, Fmaj7).

(4, 8, 8) (Aex2, Hex8, D]ex2, F ex8). (D7, G]7), (Amaj7, D]maj7), (G9, C]9), (A9, D]9),
(A4+, D]4+), (E7, B[7). (B9, F9), (E9, B[9).

Table 5: Table gives various types of 3-chords for harmonies with single reflection symmetry. The cycles
correspond to figures 3.3, 3.3, 3.3, 3.3, 3.3.

(n0,n1,n2) 0-chords 1-chords 2-chords

(2, 12, 6) (F]ex3, Hex4), (Am,D]), (A6, D]7), (C9, F9), (B9, F ]9),
(D7, B[6), (G6−, Fmaj7), (E9, C]9).
(D4+, B[9−), (E9−, G]4+),

(2, 12, 6) (Dex4, Hex4). (F, Fm), (C6−, B[maj7), (C9, D]9),
(D7, G]6), (Gmaj7, D]6−). (D]9, C]9),
(C]4−, A4+), (E4+, F ]6). (E9, B9).

(4, 8, 8) (Fex1, D]ex3, G]ex1, Aex2). (E7, E6), (Amaj7, B9−), (D9, B9), (C9, C]9),
(G,C]m), (D7, F ]6). (F9, G]9), (D]9, B[9).

(2, 12, 6) (Hex3, Eex7). (D7, G]6), (G,D]m), (C9, D]9),
(F, Fm), (C6−, B[maj7), (D9, C]9),
(A9−, C]4+), (E7, F ]6). (E9, B9).

(2, 12, 6) (F]ex2, F ex3). (F,B[m), (C7, G]6), (B[9, D]9),
(Amaj7, B9−), (E6, E7), (C9, C]9),
(G,C]m), (D7, B6). (D9, H9).

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

Figure 1: (n0, n1, n2) = (2, 12, 6) Hamiltonian cycle with 6-fold rotation symmetry acting shifts generated
by a shift of 2 quints.

Figure 2: (n0, n1, n2) = (0, 16, 4) Hamiltonian cycle with 4 reflection symmetries generated by reflections
in vertical and horizontal directions.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

Figure 3: (n0, n1, n2) = (4, 8, 8) Hamiltonian cycle with 4 reflection symmetries.

Figure 4: (n0, n1, n2) = (0, 16, 4) Hamiltonian cycle with 2-fold rotational symmetry realized as 6-quint
shift along the cycle.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

Figure 5: (n0, n1, n2) = (2, 12, 6) Hamiltonian cycle with 2-fold rotation symmetry.

Figure 6: (n0, n1, n2) = (4, 8, 8) Hamiltonian cycle with 2-fold rotation symmetry.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

Figure 7: (n0, n1, n2) = (2, 12, 6) Hamiltonian cycle with 2-fold reflection symmetry realized as horizontal
reflection

Figure 8: (n0, n1, n2) = (2, 12, 6) Hamiltonian cycle with 2-fold reflection symmetry.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

Figure 9: (n0, n1, n2) = (4, 8, 8) Hamiltonian cycle with 2-fold reflection symmetry.

Figure 10: (n0, n1, n2) = (2, 12, 6) Hamiltonian cycle with 2-fold reflection symmetry.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

Figure 11: (n0, n1, n2) = (2, 12, 6) Hamiltonian cycle with 2-fold reflection symmetry.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

Received April 4, 2022; Accepted July 23, 2022

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ISSN: 2159-046X DNA Decipher Journal www.www.dnadecipher.com

Published by QuantumDream, Inc.

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Pitkänen, M., Molecular Signaling from the TGD Viewpoint

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ISSN: 2159-046X DNA Decipher Journal www.www.dnadecipher.com

Published by QuantumDream, Inc.

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	Introduction
	Observations
	TGD view about the findings

	Bio-harmony and context dependence
	Bio-harmony
	Could ligand interactions reduce to a re-tuning of ligand harmonies?
	Ligand interactions as re-tuning
	Tuning and re-tuning at the level of DX-X pairing
	Could special 3-chords act as tuning forks
	Re-tuning as an epigenetic process

	An attempt to concretize the model of ligand interactions
	Could the dark matter hierarchy relate to the bio-harmony?

	Appendix: Tables of basic types of 3-chords for icosahedral harmonies
	Icosahedral harmonies as Hamiltonian cycles
	Tables for the 3-chords of icosahedral harmonies
	Illustrations of icosahedral Hamiltonian cycles with symmetries


