symmetry
S S
Article
Quantum Information in the Protein Codes, 3-Manifolds and
the Kummer Surface
Michel Planat 1,*
, Raymond Aschheim 2
, Marcelo M. Amaral 2
, Fang Fang 2
and Klee Irwin 2






Citation: Planat, M.; Aschheim, R.;
Amaral, M.M.; Fang, F.; Irwin, K.
Quantum Information in the Protein
Codes, 3-Manifolds and the Kummer
Surface. Symmetry 2021, 13, 1146.
https://doi.org/10.3390/sym13071146
Academic Editor: Sergei D. Odintsov
Received: 22 April 2021
Accepted: 9 June 2021
Published: 26 June 2021
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1
Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne/Franche-Comté, 15 B Avenue
des Montboucons, F-25044 Besançon, France
2 Quantum Gravity Research, Los Angeles, CA 90290, USA; raymond@QuantumGravityResearch.org (R.A.);
Marcelo@quantumgravityresearch.org (M.M.A.); Fang@QuantumGravityResearch.org (F.F.);
Klee@quantumgravityresearch.org (K.I.)
* Correspondence: michel.planat@femto-st.fr
Abstract: Every protein consists of a linear sequence over an alphabet of 20 letters/amino acids. The
sequence unfolds in the 3-dimensional space through secondary (local foldings), tertiary (bonds) and
quaternary (disjoint multiple) structures. The mere existence of the genetic code for the 20 letters
of the linear chain could be predicted with the (informationally complete) irreducible characters
of the finite group Gn := Zn o 2O (with n = 5 or 7 and 2O the binary octahedral group) in our
previous two papers. It turns out that some quaternary structures of protein complexes display n-fold
symmetries. We propose an approach of secondary structures based on free group theory. Our results
are compared to other approaches of predicting secondary structures of proteins in terms of α helices,
β sheets and coils, or more refined techniques. It is shown that the secondary structure of proteins
shows similarities to the structure of some hyperbolic 3-manifolds. The hyperbolic 3-manifold of
smallest volume—Gieseking manifold—some other 3 manifolds and the oriented hypercartographic
group are singled out as tentative models of such secondary structures. For the quaternary structure,
there are links to the Kummer surface.
Keywords: protein structure; DNA genetic code; informationally complete characters; finite groups;
3-manifolds; Kummer surface; cartographic group
1. Introduction
We found in a previous work that the approach of quantum computation based on
magic states [1–3] may also be used to explore the symmetries and the structure of the
genetic code [4–6]. Given an appropriate finite group G with d conjugacy classes, one
takes an irreducible character κ = κr and a corresponding r-dimensional representation
in the conjugacy class. For the application to the genetic code, one takes the finite group
Gn := Zn o 2O (with n = 5 or 7 and 2O the binary octahedral group) [4,5]. For such a
group, the dimension r may be 1, 2, 3, 4, or 6 and the relevant conjugacy classes may be
mapped to the amino acids of degeneracy r in their relation to codons. Then one defines d2
one-dimensional projectors Πi = |ψi〉〈ψi|, where the |ψi〉 are the d2 states obtained from
the action of a d-dimensional Pauli group Pd on the character κ. When the rank of the
Gram matrix G with elements tr(ΠiΠj) is d2, the character κ corresponds to a minimal
informationally complete quantum measurement (or MIC), see, e.g., ([4], Section 3).
The second step of our work deals about the (secondary) genetic code found in the
protein structure.
Proteins are long polymeric linear chains encoded with the 20 amino acid residues
arranged in a biologically functional way. Today the protein database (or PDB) contain
about 1.8× 105 entries [7]. Proteins may perform a large variety of functions in living
cells and organisms including molecular recognition, catalyzing metabolic reactions, DNA
replication and structural support for molecules. The sequence of amino acids leads to
Symmetry 2021, 13, 1146. https://doi.org/10.3390/sym13071146
https://www.mdpi.com/journal/symmetry
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many different three-dimensional foldings that happen to be more conserved during evo-
lution than the sequences themselves. The structure of proteins determines their biological
function [8].
A coarse-grained representation of the backbone structure of the linear chain in a
protein—a secondary code—contains three main elements that are α helices and β pleated
sheets, due to the interactions between atoms and backbones, and random coils that
indicate an absence of a regular structure. The ordered structures are held in shape by
hydrogen bonds, which form between the carbonyl of one amino acid and the amino of
another. In an α helix, there is a pattern of bonds that puts the polypeptide chain into a
helical structure with each turn of the helix containing 3.6 amino acids [9]. In a β pleated
sheet, two or more segments of a polypeptide chain line up next to each other, forming
a sheet-like structure held together by hydrogen bonds [10]. The three main elements of
a protein linear chain are usually denoted H (if the segments form an α helix), E (if the
segments form a β pleated sheet) and C (if the segments form a coil) and constitute what is
called the secondary structure of the protein.
The protein secondary structure is an algebraic notation that is useful when working
with x-ray diffraction and NMR structures from PDB. However in vivo proteins encounter
a wide variety of effects (solvent effects, anionic and cationic concentration effects, van
der Waals forces, binding to other proteins and nucleic acids) to name a few. The scheme
below does lend itself to defining algebraic operations of transformations or projections
that could be performed to account for some of these effects.
In this paper, we are interested in the universality of the two- or three-letter secondary
code found in proteins. The letters are segments of the protein that correspond to an α helix
H, a β pleated sheet E or a random coil C. Our view of the connection of proteins as words
with two letters (or three letters) and free group theory is as follows. One defines the two-
letter group G := 〈H, C|rel(H, C)〉 or the three-letter group G := 〈H, E, C|rel(H, E, C)〉,
where rel(H, C) or rel(H,E,C) is the model of the protein secondary structure. For ex-
ample, a hypothetical secondary code, such as HHCCC, would correspond to the group
G :=
〈
H, C|H2C3
〉
which is called the modular group. Sometimes the group G corresponds
(or is close in its structure) to the fundamental group of a three-dimensional manifoldM
so that we takeM as a candidate manifold of the protein foldings. For the aforementioned
example, the candidate manifold would be the trefoil knot complement.
We find, from several protein examples belonging to highly symmetric complexes,
that the secondary code has to obey some structural algebraic constraints relying to free
group theory. Our first investigation points out the possible role of two algebraic building
blocks. The first one is the hyperbolic (unoriented) 3-manifold of smallest volume known
as the Gieseking manifold [11], when the secondary code only consists of two letters H
and C. The second one is the oriented hypercartographic group H+2 [12–14] (alias the
two-generator free group), when the secondary code needs the three letters H, E, and C.
The consistency of the (primary) genetic code and the secondary code is studied under
the light of the Kummer surface that we already assumed to play a role in the quaternary
structure of protein complexes [5].
In Section 2, we provide a few elements about free group theory, finitely generated
subgroups of a free group and the fundamental group of a 3-manifold. We single out the
mathematical objects that will be useful for our approach of the secondary structures of proteins.
In Section 3, we feature a protein example—the histone H3 of drosophila melanogaster—
with a short sequence of 136 amino acids (136 aa) only comprising H and C segments in
the secondary pattern. We compare the results obtained from four different models and
softwares and how well they fit the cardinality sequence of subgroups of a few candidate
3-manifolds. The Gieseking manifold m000 is a good candidate (obtained from one model)
not only in terms of the cardinality sequence but also in terms of the structure of the
corresponding subgroups.
In Section 4, we pass to more examples of proteins comprising H, E, and C patterns.
In Section 4.1, we look at the secondary pattern of myelin P2 in homo sapiens with 133 aa.
Symmetry 2021, 13, 1146
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In Section 4.2, we look at the case of the gamma-carbonic anhydrase (247 aa long) within
its 3-fold symmetric complex. Then, in Section 4.3, we study the Hfq protein with 74 aa
in each arm of the Hfq 6-fold symmetric complex. In both cases, a theory close to the
observed patterns is based on the oriented hypercartographic groupH+2 , a straightforward
generalization of the cartographic group C2 introduced by A. Grothendieck in his essay [12].
In the latter case, the subgroup sequence ofH+2 perfectly fits the secondary pattern of Hfq
protein predicted by one particular model. In Section 4.4, we study the secondary patterns
obtained for proteins belonging to 5-fold and 7-fold symmetric complexes. In particular,
we provide the comparison of models for the H2A-H2B complex in nucleoplasmin and
the acetylcholine receptor (with n = 5) and the Lsm 1-7 complex (with n = 7). In addition,
one proposes a local mapping of the amino acids to a protein secondary structure with
pseudo-helices, sheets and coils based on the characters of the group G7.
In Section 5, we investigate the nucleosome complex which is 8-fold symmetric.
Following our previous work in [4,5], we find that the nucleosome complex allows to
define another group theoretical model of the genetic code based on the characters of the
group G8. In addition, one can map the DNA double helix scaffold of the nucleosome
complex to the 16 singular points of a Kummer surface.
In Section 6, we briefly comment about the absolute Galois group over the rationals
G = Gal(Q̄/Q) as an object worthwhile to be used in the context of protein sequences.
2. Algebraic Geometrical Models of Secondary Structures
Let G = 〈x1, x2, · · · xl〉 be the free group on l generators.
It is known that every group is a quotient of some free group. One constructs a finitely
presented group f p as the quotient of a free group G by the normal subgroup defined by a
set of relations rels between the generators xl
f p := 〈x1, x2, · · · xl |rels(x1, x2, · · · xl)〉.
One also needs to define subgroups of finite index in a f p group. A subgroup Gs
of the finitely presented group f p is generated by the words specified by a generator list
Lr = L1 · · · Lr that may contain words or subgroups. In the following, we are interested by
the cardinality sequence ηd( f p) that counts the number of subgroups of a finite index d up
to some maximal index. This sequence allows us to identify a group f p (potentially as the
fundamental group of a 3-manifold).
Then, to a pair ( f p, Gs) corresponds the permutation group P that organizes the cosets.
With the Todd-Coxeter procedure, one can obtain a permutation representation P of the
pair from the action of f p on the coset space. In many cases, the finite group P has a
geometrical meaning in the sense that it corresponds to a finite geometry [15].
Finally, the group theoretical approach may be related to the theory of 3-manifolds.
According to the Poincaré conjecture (now a theorem) every simply connected closed
3-manifold is homeomorphic to the 3-sphere S3, alias the house of qubits [16]. However,
one can dress S3 as a 3-manifoldM that looses the homeomorphism to S3 following the
work of W. Thurston [17]. For instance, the three-dimensional space surrounding the
tubular neighborhood of a knot—the knot complement S3 \ K—is a 3-manifold. Among
the invariants characterizing a 3-manifold, there is the fundamental group π1(M) which
accounts for the first homotopy ofM. Finding a 3-manifoldM whose π1 is the current f p
is a way to identify the nature of the object under study.
Below we introduce two algebraic geometric objects playing a role in our description
of protein secondary structures. The first object is the hyperbolic 3-manifold of the smallest
volume [11,18]. The second one is the group of oriented hypermaps, a generalization of
Grothendieck’s cartographic group [12,14].
2.1. The Gieseking Manifold m000
This 3-manifold was described by Gieseking in his 1912 thesis. One takes an ideal
regular tetrahedron in the 3-dimensional hyperbolic space, that is a tetrahedron with
Symmetry 2021, 13, 1146
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all four vertices on the sphere at infinity and all dihedral angles equal to π/3. Then,
one identifies adjacent faces so that the orientation on the edges match ([11], Figure 1).
The resulting hyperbolic manifold has minimal volume among non-compact hyperbolic
manifolds. This volume is Gieseking’s constant
∫ 2π/3
0
ln(2 cos(x/2))dx = 1.01494160 · · · .
Remarkably, this constant also equals ζQ(i
√
3)(2), which is the Dedekind zeta function at 2
for the field Q(i
√
3) [18,19].
The fundamental group for the Gieseking manifold is denoted m000 in SnapPy soft-
ware [20]. The fundamental group is
π1(m000) :=
〈
x, y|x2y2 = yx
〉
.
The cardinality sequence ηd(π1(m000)) of subgroups of index d < 15 of π1(m000) is
given in Table 2. The permutation groups organizing the cosets of subgroups of π1(m000)
up to index 10 are in Table 1. The identification of sub-manifolds follows from SnapPy.
Table 1. The d-coverings (d = 1 . . . 10) of the Gieseking manifold m000. The corresponding 3-manifolds (3-man) are
identified thanks to SnapPy. The finite group P organizing the cosets of the index d fundamental group is given. It is shared
by almost all subgroups (see lacking P) of the free group associated to the PORTER model of secondary structures of histone
H3 (PDB; 6PWE_1). Some extra groups appear in the PORTER model (see extra P).
Index
1
2
3
4
5
3-man
m000
K4a1, ooct02_00001
ntet03_00000
m206, otet04_00002 m407, ntet05_00007
m204, ntet04_00000 m405, ncube01_00001
P
(1,1)
(2,1)
(3,1)
(4,1)
(5,1)
(12,3)
(20,3)
Index
6
7
8
9
10
3-man
s961, otet06_00003 y886, ntet07_00000 t12839, otet06_00007
x252, ntet06_00004
t12840, otet08_00002
ntet06_00005
ntet08_00002
P
(6,2)
(7,1)
(8,1)
(9,1)
(10,2)
(12,3)
(24,3) ×2
(24,13)
(24,13)
(96,70), (192,201)
(9,1), (648,705)
(10,2), (20,3), G14400
lacking P
(72,39)
(320,1635)
extra P
A8, S8
(216,53), A9, S9
S10, G7200
In the next section, we find that a model of the secondary structure in histone H3 (PDB
6PWE_1) (obtained with the software PORTER) is the group
G :=
〈
C, H|C44H12C4H3C3H12C8H28C7H10C5
〉
.
It is shown in Tables 1 and 2 that this model fits perfectly the Gieseking fundamental
group at the first 7 places and approximately at the subsequent 3 places. Up to index 7 the
permutation groups P are the same. At index 8, all P’s related to subgroups of π1(m000)
are also those related to subgroups of G, but A8 and S8 which are related to subgroups
of G are not in subgroups related to π1(m000). There are also a few differences between
subgroups of π1(m000) and G at index 9 and 10.
2.2. The Hypercartographic GroupH+2
The cartographic group is defined as
C2 :=
〈
x, y, z|x2 = y2 = z2 = (xz)2
〉
.
Symmetry 2021, 13, 1146
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The terminology comes from Grothendieck’s Esquisse d’un programme [12,13]. It
was motivated by the fact that conjugacy classes of transitive subgroups of the oriented
subgroup C+
2 of index 2 of the unoriented group C2 can be identified to topological maps on
connected, oriented surfaces without boundary, while more generally, conjugacy classes of
C2 can be identified with maps on connected surfaces which may or may not be orientable
or have a boundary. The group C+
2 was investigated by the first author in relation to
quantum contextuality in quantum information [15].
Here, we are concerned with a slight generalization of the cartographic group C2. To
interpret our results we need the oriented hypercartographic groupH+2 whose definition is
H+2
:= 〈x, y, z|xyz〉.
This group is intimately related to the so-called Belyi’s theorem. The latter theorem
states that a complex algebraic curve is defined over the fieldQ̄ of algebraic numbers if
and only if it may be uniformized by a subgroup of finite index in a triangle group. See [14]
and the conclusion of the present paper for additional details.
In the section below, the group defined from the PORTER model of the secondary
structure in protein Hfq (PDB 1HK9) is as follows
G :=
〈
C, H, E|C8H11C4E6C2E10CE7C3E13C9
〉
.
It is shown in Table 3 that this group perfectly fits the hypercartographic groupH+2
in terms of the cardinality of subgroups up to the higher index 7 that could be calculated.
In addition, the corresponding permutation groups organizing the cosets of subgroups in
both the cases ofH+2 and G fit as well.
Table 2. The models of the secondary structure for protein H3 of drosophila melanogaster and the
cardinality list of d-coverings (alias conjugacy classes of subgroups) of the associated fundamental
group. T1 is the trefoil knot, K0 is the figure-of-eight knot, the 0-surgery on K0 is the Akbulut
manifold ΣY ,Ẽ8 is the singular fiber of type II* and m000 is the Gieseking manifold. One restricts to
two-generator groups since histone H3 only consists of sections with α helices and coils. Observe that
the series of cardinalities for the secondary structure of H3 fits the series of the Gieseking manifolds
up to the first 7 indices. Bold characters are for partial sequences matching the cardinality sequence
for subgroups of the fundamental group of Gieseking manifold m000.
Protein
Model
ηd(T)
H3 (6PWE_1)
PSIPRED
[1,1,1,1,2, 2,1,3,5,5 .,.,.,.,.]
H3
PHYRE2
[1,1,1,1,3, 4,1,5,10,10 .,.,.,.,.]
H3
PORTER
[ 1,1,1,2,2, 3,1,12,6,5 .,.,.,.,.]
H3
RAPTORX
[1,1,1,1,2, 1,1,2,3,3 .,.,.,.,.]
m000
Gieseking
[1,1,1,2,2, 3,1,4,3,5, 4,14,1,5,10]
T1
trefoil
[1,1,2,3,2, 8,7,10,18,28, 27,88,134,171,354]
K0
figure-of-eight
[1,1,1,2,4, 11,9,10,11,38, 26,62,39,89,228]
K0(0,1)
ΣY
[1,1,1,2,2, 5,1,2,2,4, 3,17,1,1,2]
Ẽ8
singular fiber II*
[1,1,2,2,1, 5,3,2,4,1, 1,12,3,3,4]
2.3. Fundamental Groups of 3-Manifolds
Hyperbolic 3-manifolds that can be decomposed into regular ideal tetrahedra (up to
25 for the orientable case and up to 21 for the non-orientable case) have been investigated
in [21]. Details can be found in SnapPy [20]. In Tables 2 and 3, we collected a few 3-
manifolds whose number of subgroups ηd(π(M)) of index d of their fundamental group
π1(M) is close to that of the group arising from the secondary structure of the protein in
question. For example, the figure-of-eight knot K0 = K4a1 = 41, which is the subgroup of
Symmetry 2021, 13, 1146
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index 2 in π1(m000), corresponds to the manifold ooct_00001 in SnapPy (see Tables 1 and 2)
and ΣY = K0(0, 1) is the 0-surgery on K0 [22].
Table 3. A few proteins, the software used for determining their secondary structure and the
cardinality list of d-coverings (alias conjugacy classes of subgroups of index d) of the associated
group. One takes proteins that contain sections with α helices, β sheets and coils. The groups
obtained by mapping the appropriate characters of G7 = (336, 118) and G8 = (384, 5589) to amino
acids are also considered. Bold characters are for partial sequences matching the sequence of the
hypercartographic groupH+2 .
Protein
aa
Model
ηd(T)
myelin P2 (2WUT)
133
PSIPRED
[1, 3, 13, 84, 336, 4216]
2WUT
PHYRE2
[1, 3, 7, 26, 164, 10,669]
2WUT
PORTER
[1, 3, 7, 26, 135, 871]
2WUT
RAPTORX
[1, 3, 10, 59, 348, 2899]
.
(336,118)
[1, 3, 7, 30, 122, 991]
.
(384,5589)
[1, 3, 7, 34, 130, 999]
carbonic anhydrase (1QRE_1) 247
PSIPRED
[1, 3, 10, 43, 135, 1071]
1QRE_1
PHYRE2
[1, 3, 7, 26, 149, 1085]
1QRE_1
PORTER
[1, 3, 7, 26, 415, 4382]
1QRE_1
RAPTORX
[1, 3, 10, 35, 106, 804]
.
(336,118)
[1,3,7,30,150, 883]
.
(384,5589)
[1,3,10,47,148, 1015]
protein Hfq (1HK9_1)
74
PSIPRED
[1, 7, 17, 114, 1145, 14,275]
1HK9_1
PHYRE2
[1, 7, 14, 149, 1458, 21,756]
1HK9_1
PORTER
[1, 3, 7, 26, 97, 624, 4163, 34,470]
1HK9_1
RAPTORX
[1, 3, 10, 51, 162, 1434]
.
(336,118)
[1, 3, 7, 26, 134, 912]
.
(384,5589)
[1, 3, 7, 34, 146, 894]
H2A-H2B (2XQL_1)
91
PHYRE2
[1, 3, 7, 26, 103, 688]
2XQL_1
RAPTORX
[1, 3, 7, 26, 165, 2272]
.
(336,118)
[1, 3, 7, 26, 130, 943]
.
(384,5589)
[1, 3, 7, 26, 136, 967]
acetylcholin receptor (2BG9_1) 370
PSIPRED
[1, 3, 10, 35, 151, 1023]
2BG9_1
PHYRE2
[1, 7, 11, 92, 288, 2087]
2BG9_1
PORTER
[1, 7, 11, 92, 239, 2058]
2BG9_1
RAPTORX
[1, 3, 7, 34, 169, 1432]
.
(336, 118)
[1, 3, 10, 47, 124, 1026]
.
(384, 5589)
[1, 3, 7, 30, 140, 931]
Lsm 1-7 complex (4M75_1)
144
PSIPRED
[1, 3, 16, 81, 184, 1800]
4M75_1
PHYRE2
[1, 7, 14, 201, 705, 8850]
4M75_1
PORTER
[1, 3, 7, 26, 139, 1118]
4M75_1
RAPTORX
[1, 3, 7, 26, 125, 747]
.
(336, 118)
[1,3,7,34,145, 948]
.
(384, 5589)
[1,3,10,35,135, 975]
H+2
na oriented hypermaps [1, 3, 7, 26, 97, 624, 4163, 34,470]
ooct02_00017
3-manifold
[1, 3, 7, 26, 40, 231]
ooct02_00006
3-manifold
[1, 3, 10, 43, 112, 802]
noct02_00024
3-manifold
[1, 3, 10, 43, 117, 804]
ooct02_00009
3-manifold
[1, 3, 7, 30, 105, 649]
ooct04_00001
3-manifold
[1, 3, 7, 34, 43, 240, 254]
L7a1
3-manifold link
[1, 3, 7, 34, 75, 377, 807]
ooct03_00019
3-manifold
[1, 7, 11, 85, 95, 240, 492]
Symmetry 2021, 13, 1146
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3. Secondary Structure with α Helices: Drosophila Melanogaster Histone H3
(PDB 6PWE_1)
Now we show how the theory of the former section may be applied to concrete
secondary structures of proteins. One starts with a simple example with two generators
(α helices H and coils C). At the next section, we will study a simple example with three
generators (α helices H, β sheets E and coils C). Both examples are generic and provide
a good credit to our models based on the unoriented hyperbolic manifold m000 and the
oriented hypercartographic groupH+2 .
A review of the state of the art in the modeling of secondary structure is given in [8].
It is admitted that there is a limit imposed on the secondary structure prediction due to the
somewhat arbitrary definition of three states H, E, and C. It is true that there exist other
fine structures in the secondary protein pattern such as a 310 helix, a π helix and other
structures belonging to DSSP (the Dictionary of Protein Secondary Structures). As a result,
the assignment inconsistency would limit the highest accuracy based on three states to
about 90%. In practice, the best softwares achieve a precision about 80%.
We used the softwares PSIPRED 4.0 [23], PORTER 4.0 [24], PHYRE2 [25], and RAP-
TORX [26]. We do not enter into the details about the theory of these softwares. Below, we
we find that PORTER 4.0 is often well adapted to our goal of identifying an algebraic sec-
ondary structure. PORTER 4.0 uses two cascaded bidirectional recurrent neural networks:
one for prediction and one for filtering. The method has been trained and benchmarked by
cross-validation on a set of many non redundant proteins.
3.1. The Primary (Linear) Structure
The mRNA sequence for histone H3 of drosophila melanogaster may be found in [27]
with the reference NM_001032216.2. It contains 529 base pairings (529 bp). A convenient
way to pass from the NCBI format (with line feeds, numbers and blank spaces) to the bare
linear sequence is to make use of a software such as Massager [28]. Then, a reading frame
such as Expasy [29] allows to extract the candidate proteins.
The 5′3′ Frame 1 for sequence NM_001032216.2 is as follows:
IVFSNVK–T-TLVKPKSE
MARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRP
GTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVM
ALQEASEAYLVGLFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA
-ADTALTCR-SASVLYNRSFS
The partial sequence (in bold) beginning at the start codon M and ending at the stop
codon ‘-’ is the histone protein H3 with the NCBI reference NP_001027387.1. It can also be
found at the protein data base PDB [7] with reference 6PWE_1. The sequence consists of
136 amino acids (136 aa).
3.2. The Secondary Structure
According to most models, the secondary structure of histone protein H3 only consists
of subsections with an α helix H or a coil C.
The predicted secondary structures obtained from the three softwares for the histone
H3 protein are as follows:
CCCCCCCCCCCCCCCCCHHHHCHHHHCCCCCCCCCCCCCCCCCCCCHHHHHHHCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCHHHHHHHHHHHHCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCHHHHHHHHHHHHCCC
CCCCCCCCCCCCCCCCCCCCHHHHHCCCCCCCCCCCCCCCCCCHHHHHHHHHHHHHCC
HHHHHCCCCHHHHHHHHHHHCCCCCCCCHHHHHHHHHHHHHHHHHHHHHHHHCHHHH
CCHHHCCCHHHHHHHHHHHHCCCCCCCCHHHHHHHHHHHHHHHHHHHHHHHHHHHHC
HHHHHHHHHHHHHHHHHHHHHCCCCCCCHHHHHHHHHHHHHHHHHHHHHHHHHHHHC
HHHHHHHHHHHHHHHHHHCCCCCCCCCCHHHHHHHHHHHHHHHHHHHHHHHHHHHHC
CCCCCCHHHHHHHHHHCCCCC
CCCCCCHHHHHHHHHHCCCCC
CCCCCCHHHHHHHHHHCCCCC
CCCCCCHHHHHHHHHHHCCCC
Symmetry 2021, 13, 1146
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The first line is from PSIPRED, the second one is from PORTER, the third one is
from PHYRE2, and the last one is from RAPTORX. One can visually check how close
are the predictions.
Figure 1 is a sketch of the secondary structure of histone H3. In Table 2, it is found that
the best model happens to come from the fundamental group π1(m000) of the Gieseking
manifold m000 described in Section 2.1.
Figure 1. A picture of the secondary structure of histone H3 as predicted from PHYRE2.
4. Secondary Structures with α Helices and β Sheets: Myelin P2, Carbonic Anhydrase
and the Lsm 1-7 Complex
4.1. Myelin P2 for Homo Sapiens (PDB 2WUT)
The sequence of myelin P2 in homo sapiens comprises 133 amino acids as follows. As
before, the corresponding four rows for the secondary structures are from PSIPRED, PORTER,
PHYRE2, and RAPTORX, respectively. One can visually check how close are the predictions.
GMSNKFLGTWKLVSSENFDDYMKALGVGLATRKLGNLAKPTVIISKKGDIITIRTESTFKN
CCCHHCCEEEEEEEECCHHHHHHHCCCCHHHHHHHHHCCCEEEEEEECCEEEEEEECCCC
CCCHHCCEEEEEECCCCHHHHHHHCCCCHHHHHHHHHCCCEEEEEEECCEEEEEEECCCC
CCCCCCEEEEEEEEECCHHHHHHHHCCCHHHHHHHHCCCCEEEEEEECCEEEEEEECCCC
CCCCCCEEEEEEEEECCHHHHHHHCCCCHHHHHHHHCCCCEEEEEEECCEEEEEEECCCC
TEISFKLGQEFEETTADNRKTKSIVTLQRGSLNQVQRWDGKETTIKRKLVNGKMVAECKM
CCCHHCCEEEEEEEECCHHHHHHHCCCCHHHHHHHHHCCCEEEEEEECCEEEEEEECCCC
EEEEEEEECCEEEEECCCCCEEEEEEEEECCEEEEEEECCCCEEEEEEEEECCEEEEEEEE
EEEEEEECCCEEEEECCCCCEEEEEEEEECCEEEEEEECCCCCEEEEEEEECCEEEEEEEE
EEEEEEECCCEEEEECCCCCEEEEEEEEECCEEEEEEECCCCCEEEEEEEECCEEEEEEEE
KGVVCTRIYEKV
CCEEEEEEEEEC
CCEEEEEEEEEC
CCEEEEEEEEEC
CCEEEEEEEEEC
Figure 2 is a sketch of the secondary structure of myelin P2. Using Table 3, one
observes that the cardinality sequence of subgroups in the PHYRE2 and PORTER models
of the secondary structure of myelin P2 corresponds to that of the hypercartographic group
H+2 up to index 4. Up to this index, one can also show that the permutation groups P for
the structure of cosets in PHYRE2 and PORTER models correspond to that ofH+2 .
4.2. The 3-Fold Symmetric Complex for Gamma-Carbonic Anhydrase (PDB 1QRE)
In the protein data bank, the gamma-carbonic anhydrase for methanosarcina ther-
mophila (PDB 1QRE_1) is a sequence with 247 aa. As for myelin P2, using Table 3, one
observes that the cardinality sequence of subgroups in the PHYRE2 and PORTER models
of the secondary structure of 1QRE_1 corresponds to that of the hypercartographic group
H+2 up to index 4. The complex is 3-fold symmetric as shown in Figure 3a.
Symmetry 2021, 13, 1146
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Figure 2. A picture of the secondary structure of myelin P2 in homo sapiens (PDB 2WUT) as predicted
from PHYRE2.
4.3. The Hfq Protein Complex of Escherichia coli (PDB 1HK9)
The sequence of Hfq protein of Escherichia coli (PDB 1HK9_1) comprises 74 amino
acids. As before, the corresponding four rows for the secondary structures are from
PSIPRED, PORTER, PHYRE2, and RAPTORX, respectively. One can visually check how
close are the predictions.
GAMAKGQSLQDPFLNALRRERVPVSIYLVNGIKLQGQIESFDQFVILLKNTVSQMVYKHAISTVVPSRPVSHHS
CCCCCCCCCHHHHHHHHHHCCCCEEEEEECCCEEEEEEEECCCEEEEEECCCEEEEEEEEEEEEEECCCCCCCC
CCCCCCCCHHHHHHHHHHHCCCCEEEEEECCEEEEEEEEEECEEEEEEECCCEEEEEEEEEEEEECCCCCCCCC
CCCCCCCCCHHHHHHHHHHCCCEEEEEEECCEEEEEEEEEECCEEEEEECCCCEEEEEEEEEEEEECCEEEECC
CCCCCCCCCCHHHHHHHHHCCCCEEEEECCCCEEEEEEEEECCCEEEEEECCCEEEEEEEEEEEEECCCCCCCC
The PORTER model for this protein happens to coincide with that of the hypercarto-
graphic groupH+2 described in the Section 2.2.
As shown in Figure 3b, the Hfq complex consists of a quaternary structure with 6-fold
symmetry where each arm contains the protein Hfq. This object was studied in our recent
paper ([5], Section 2.2) as leading to a Kummer surface related to the character table of the
finite group G6 = (288, 69) ≡ Z6 o 2O.
Figure 3. (a) A picture of the structure of carbonic anhydrase (PDB 1QRE), (b) A picture of the
structure of Hfq protein complex of Escherichia coli (PDB 1HK9).
4.4. Other n-Fold Symmetric Complexes
4.4.1. The 5-Fold Symmetric H2A-H2B Complex in Nucleoplasmin (PDB 2XQL)
Molecular chaperones are proteins that help the folding or unfolding and the disassem-
bly of other molecular structures. Nucleoplasmin, the first identified molecular chaperone,
promotes the in vitro assembly of nucleosomes. The latter are the topic of our next section.
There is a histone octamer comprising two H2A-H2B dimers and an H3-H4 tetramer. The
Symmetry 2021, 13, 1146
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H2A-H2B histone complex is investigated in [30]. It has a pentameric structure as shown
in Figure 4a and is referred as 2XQL in the protein databank.
Figure 4. (a) the nucleoplasmin H2A-H2B: 2XQL in the protein databank, (b) the acetylcholine
receptor: 2BG9 in the protein databank, (c) the Lsm 1-7 complex in the spliceosome: 4M75 in the
protein databank.
We performed an investigation of the secondary structure of the 2XQL_1 protein that
one finds in each of the 5 arms of the complex. PSIPRED and PORTER models predict a
secondary structure with α helices and coils only that we could not compare to a known
group theoretical sequence. The PHYRE2 and RAPTORX models, as well as our approach
based on the mapping of amino acids to the characters of group G7 and G8 (explained
below), predict a cardinality sequence which fits that of the hypercartographic groupH+2 ,
as shown in Table 3.
4.4.2. The 5-Fold Symmetric Acetylcholine Receptor (PDB 2BG9)
The acetylcholine receptor is an integral membrane protein that responds to the
binding of the acetylcholine neurotransmitter. This receptor is also sensitive to nicotine
and muscarine. It has a pentameric structure shown in Figure 4b and is refereed as 2BG9 in
the protein databank.
We performed an investigation of the secondary structure of the 2BG9_1 protein that
one finds in the 5 arms of the complex. As shown in Table 3, all models predict a secondary
structure with α helices, β sheets and coils. One does not observe a good fit to a group
theoretical structure shared by all models. The best fit is between the RAPTORX model
and the fundamental group of the 3-manifold ooct_00001 where the cardinality (and the
structure) of subgroups coincide up to 4 places.
4.4.3. The 7-Fold Symmetric Lsm 1-7 Complex in the Spliceosome (PDB 4M75)
In molecular biology, there exists an ubiquitous family of RNA-binding proteins called
LSM proteins whose function is to serve as scaffolds for RNA oligonucleotides, assisting
the RNA to maintain the proper three-dimensional structure. Such proteins organize as
Symmetry 2021, 13, 1146
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rings of six or seven subunits. The Hfq protein complex was discovered in 1968 as an
Escherichia coli host factor that was essential for replication of the bacteriophage Qβ [31],
it displays an hexameric ring shape shown in Figure 3b of the previous subsection. As
already mentioned it is remarkable that the secondary structure of Hfq protein is so close
to the hypercartographic group model.
It is known that, in the process of transcription of DNA to proteins through messenger
RNA sequences (mRNA), there is an important step performed in the spliceosome [32].
It includes removing the non-coding intron sequences for obtaining the exons that code
for the proteinogenic amino acids. A ribonucleoprotein (RNP)—a complex of ribonucleic
acid and RNA-binding protein—plays a vital role in a number of biological functions that
include transcription, translation, the regulation of gene expression, and the metabolism of
RNA. Individual LSm proteins assemble into a six or seven member doughnut ring which
usually binds to a small RNA molecule to form a ribonucleoprotein complex.
In our previous paper [5], it was shown that 7-fold symmetry may be mirrored in the
finite group G7 = Z7 × 2O (with 2O the binary octahedral group) whose characters may be
mapped to the amino acids of the genetic code. Such a mapping is reproduced in Table 4.
It is important to mention that the characters of G7 are informationally complete except
for the trivial character that is not used in the mapping to amino acids and the character
mapped to the starting amino acid M.
It was also determined an algebraic object called a Kummer surface playing a role in
the mapping of characters to amino acids.
Table 4. For the group G7 := (336, 118) ∼= Z7 o 2O, the table provides the dimension of the representation, the rank of the
Gram matrix obtained under the action of the 29-dimensional Pauli group, the order of a group element in the class, the
angles involved in the character and a good assignment to an amino acid according to its polar requirement value. All
characters are informationally complete except for the trivial character and the one assigned to M. The entries involved
in the characters are z1 = 2 cos(2π/7), z2 = 2z1, z3 = −6 cos(π/7), z4 =
√
2, and z5 = 2 cos(2π/21) featuring the angles
2π/8 (in z4), 2π/7 and 2π/21.
(336,118)
dimension
1
1
1
2
2
2
2
2
2
2
Z7 o (Z2.S4)
d-dit, d = 29
29
785
d2
d2
d2
d2
d2
d2
d2
d2
∼= Z7 o 2O
amino acid
.
M
W
C
F
Y
.
.
H
Q
order
1
2
3
4
4
6
7
7
7
8
char
Cte
Cte
Cte
z1
z1
z1
z4
z4
z1,5
z1,5
polar req.
.
5.3
5.2
4.8
5.0
5.4
.
.
8.4
8.6
(336,118)
dimension
2
2
2
2
3
3
4
4
4
4
d-dit, d = 29
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
amino acid
N
K
E
D
I
Stop
.
.
.
.
order
14
14
14
21
21
21
21
21
21
21
char
z1,5
z1,5
z1,5
z1,5
Cte
Cte
Cte
z1,2
z1,2
z1,2
polar req.
10.0
10.1
12.5
13.0
10
15
.
.
.
.
(336,118)
dimension
4
4
4
f 4
4
4
6
6
6
d-dit, d = 29
d2
d2
d2
d2
d2
d2
d2
d2
d2
amino acid
V
P
T
A
G
.
L
S
R
order
28
28
28
42
42
42
42
42
42
char
z2,5
z2,5
z2,5
z2,5
z2,5
z2,5
z1,3
z1,3
z1,3
polar req.
5.6
6.6
6.6
7.0
7.9
.
4.9
7.5
9.1
4.4.4. Encoding a Protein with the Characters of the Finite Group G7
Since the group G7 is successful for encoding the genetic code and that, at the same
time, it provides an assignment to the 20 amino acids through the corresponding characters,
Symmetry 2021, 13, 1146
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one can ask ourselves if G7 may also be used to define a secondary structure in a protein.
Indeed we can get a secondary structure from the character table in the following way.
Observe that, to a character in Table 4, corresponds an entry denoted z1, z4, z1,2, z1,3,
z1,5, or z2,5 which expresses which zi appears in the slot or character. This entry mainly
reflects the character field associated to the character. For example, there are 11 slots
(and 11 amino acids) containing z5 and from these characters one can also define the
aforementioned Kummer surface. Let us choose to assign to these slots a secondary
structure H0 and to assign a secondary structure C0 to the remaining slots encoding an
amino acid. This method allows to encode the protein under examination with pseudo-
helices H0 and pseudo-coils C0.
We can refine the technique by introducing more structure in the pseudo coil seg-
ments. Some of the slots/amino acids correspond to a character with constant entries and
we choose to encode them as C0 as before and the remaining slots/amino acids which
correspond to a non constant entry (z1 or z1,3) are encoded with E0, that we consider as
a pseudo-sheet.
Then we can define the group
G0 := 〈H0, E0, C0|rel(H0, E0, C0)〉, where rel(H0, E0, C0) is the new model of the pro-
tein secondary structure obtained by our definition of pseudo-helices H0, pseudo-sheets
E0, and pseudo-coils C0. In Table 3, the cardinality structure of group G0 is compared to
that of the other models PSIPRED, PHYRE2, PORTER, and RAPTORX. One finds that the
cardinality sequence either fits, at the first few places, the hypercartographic groupH+2 or
that of a 3-manifold. It leaves open the question whether one of the standard models or
our own model is the most efficient.
5. The 8-Fold Symmetric Histone Complex of the Nucleosome: 3WKJ in the Protein
Data Bank
Strong DNA packaging is found in the nucleosome of eukaryotes. The nucleosome
complex consists of a double helix wrapped around a set of eight histone proteins com-
prising two copies of H2A, H2B, H3, and H4. The nucleosome is the fundamental sub-unit
of chromatin. Eukaryotic chromatin is further compacted by being folded into more com-
plex structures eventually forming a chromosome. Nucleosomes are considered to be the
support of epigenetic information. The nucleosome core particle contains approximately
146 base pairs (bp) of DNA wrapped in 1.67 left-handed superhelical turns around the
histone octamer as shown in Figure 5a.
We already met histone H3 of a different specie (drosophila melanogaster) in Section 3
as the preliminary example of a protein only containing α helices and random coils. In the
histone complex 3WKJ of the nucleosome, the secondary structure of histone H3 is also
found to be made of segments with α helices and coils but with a different organization
according to our group theoretical approach. This is also true for the other histones H4,
H2A, and H2B of the histone octamer.
In this section, we do not enter into the secondary structure of histones. We rather
focus on the 8-fold symmetry of the core particle in the histone complex. What interests
us about the double helix is the fact that their projection is a set of 16 double points
as shown by the arrows in Figure 5a. The reader may be familiar with our previous
paper [5] in which 16 double points occur in a beautiful algebraic object called a Kummer
surface. Such a Kummer surface was constructed from the character table of the group
G7 = (336, 118) ∼= Z7 o 2O in the context of the spliceosome complex that we investigated
in Section 4.4. Below, we pursue in the same line of ideas and build another model of
the genetic code based on the group G8 = (384, 5589) ∼= Z8 o 2O and a corresponding
Kummer surface.
Symmetry 2021, 13, 1146
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Figure 5. (a) The structure of a nucleosome consists of a DNA double helix wound around eight
histone proteins. There are eight periods (as shown in the picture) so that the two helices meet at
16 points . They map to the 16 double points of the Kummer surface. (b) A section at constant x4 of
the Kummer surface for the group G8.
The character table for the group G8 is in Table 5. As before for the group G7, Table 5
contains a good assignment to the 20 amino acids and some details about the character fields
through the entries zi. For dimensions 2 and 4, the assignments correspond to characters
that are informationally complete. However, it is not the case for the assignments of amino
acids in dimensions 1, 3, and 6.
Table 5. For the group G8 = (384, 5589) ∼= Z8 o 2O, the table provides the dimension of the representation, the rank of the
Gram matrix obtained under the action of the 37-dimensional Pauli group and the entries involved in the characters. The
notation is z1 = −
√
2, z2 = 2
√
2, z3 = 3
√
2, z4 = −
√
3 and z5 = −2 cos(5π/12). All characters having z4 and z5 in their
entries are informationally complete and are at the origin of the Kummer surface. All characters having entries with z2 or z4
are also informationally complete. A good matching to the amino acids (ordered according to their polar requirement and
simultaneously to the order of a group element) is given.
(384,5589)
dimension
1
1
1
1
2
2
2
2
2
2
Z8 o (48, 28)
d-dit, d = 37
37
1333
1333
1333
1361
d2
d2
1367
d2
d2.
∼= Z8 o 2O
amino acid
.
.
M
W
.
.
.
.
.
.
char
Cte
Cte
Cte
Cte
Cte
Cte
Cte
z1
z1
z1
(384,5589)
dimension
2
2
2
2
2
2
2
2
2
3
d-dit, d = 37
d2
d2
d2
d2
d2
d2
d2
d2
d2
1367
amino acid
C
F
Y
H
Q
N
K
E
D
.
char
z1
z1
z1
z4
z4
z1,4,5
z1,4,5
z1,4,5
z1,4,5
Cte
(384,5589)
dimension
3
3
3
4
4
4
4
4
4
4
d-dit, d = 37
d2
1367
1367
d2
1367
1367
d2
d2
d2
d2
amino acid I
Stop
.
.
.
.
.
.
.
V
char
Cte
Cte
Cte
Cte
Cte
Cte
z1,2
z1,2
z4
z4
(384,5589)
dimension
4
4
4
4
6
6
6
d-dit, d = 37
d2
d2
d2
d2
701
1365
1365
amino acid
P
T
A
G
L
S
R
char
z2,4,5
z2,4,5
z2,4,5
z2,4,5
Cte
z1,3
z1,3
Symmetry 2021, 13, 1146
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All 8 characters having z4 =
√
3 and z5 = −2 cos(5π/12) in their entries are informa-
tionally complete and are at the origin of the Kummer surface. We now show an important
characteristics of such characters. As an example, let us write the character number 16 as
obtained from Magma [33]
κ16 = [2,−2,−2, 2,−1, 0, 0, 2,−2, 0, 0, 0, 1,−1, 1, z1,−z1, z1,−z1, z1,−z1
0, 0, 0, 0, z4,−z4,−z4, z4, z5, z5, z5#5, z5#5,−z5,−z5#5,−z5 − z5#5]
where # denotes the algebraic conjugation, that is #k indicates replacing the root of unity w
by wk.
One defines a genus 2 hyper-elliptic curve C8 : y2 = f (x) defined over the group G8
from the equation
y2 = f (x) = (x + k)(x− k)(x + l)(x− l)(x +m)(x−m),
with k =
√
3, l = 2 cos(5π/12) and m = 2 cos(π/12). Explicitly,
C8 : y2 = x6 − 7x4 + 13x2 − 3,
leading to the polynomial definition of the Kummer surface S(x1, x2, x3, x4) as
S(x1, x2, x3, x4) = 156x41 + 12x
3
1x4 − 84x21x22 + 376x21x23 − 52x21x3x4
24x1x22x3 + 28x1x
2
3x4 − 4x1x3x24 + 12x42 − 52x22x23 + x22x24 + 28x43 − 4x33x4.
The de-singularization of the Kummer surface is obtained in a simple way by restrict-
ing the product f (x) to the five first factors.
As usual for elliptic and hyper-elliptic curves of genus g, C8 is embedded in a weighted
projective plane, with weights 1, g + 1, and 1, respectively, on coordinates x, y, and z.
Therefore, point triples are such that (x : y : z) = (µx : µy : µz), µ in the field of definition,
and the points at infinity take the form (1 : y : 0). Below, the software Magma is used for
the calculation of points of C8 [33]. For the points of C8, there is a parameter called ‘bound’
that loosely follows the heights of the x-coordinates found by the search algorithm.
It is found that the corresponding Jacobian of C8 has 16 = 6 + 10 points as follows:
* the 6 points bounded by the modulus 1:
Id := (1, 0, 0), K±1 := (x± k, 0, 1), L±1 := (x± l, 0, 1), and M = (x−m, 0, 1).
* the 10 points of modulus > 1:
a1 := K1 + K−1, a2 := K1 + M, a3 := K1 + K−1 + L1, a4 := K1 + L1, a5 := K−1 + M,
a6
:= K1 + K−1 + L−1, a7
:= K−1 + L1, a8
:= K−1 + L−1, a9
:= K1 + K−1 + M and
a10 := K1 + L−1.
The 16 points organize as a commutative group isomorphic to the maximally abelian
group Z42 as shown in the following Jacobian addition Table 6.
Table 6. The structure of the addition table for the 16 singular Jacobian points of the hyper-elliptic
curves C8.
A
B
C
D
B
A
D
C
C
D
A
B
D
C
B
A
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Where the blocks are given explicitly as
A :

Id
K1 K−1
a1
K1
Id
a1 K−1
K−1
a1
Id
K1
a1 K−1 K1
Id
, B :

M a2
a5
a9
a2 M a9
a5
a5
a9 M a2
a9
a5
a2 M
,
C :

L1
a4
a7
a3
a4 L1
a3
a7
a7
a3 L1
a4
a3
a7
a4 L1
, D :

a6
a8
a10
L−1
a8
a6
L−1
a10
a10
L−1
a6
a8
L−1
a10
a8
a6
.
To conclude this section, we can define a model of the secondary structure of nucleo-
some complex based on the character table of G8 as we did for the spliceosome complex
with the character table of G7. The amino acids that are mapped to characters containing
z5 should belong to a pseudo-helix H0 of the secondary structure. The other amino acids
either correspond to a constant entry in the character table and belong to a pseudo-coil
C0 or to a non-constant entry (which is either z1, z4, or z1,3) and belong to a pseudo-sheet
E0. In Table 3, the cardinality structure of subgroups of finite index of G8 obtained with
this model is compared to that of the other models PSIPRED, PHYRE2, PORTER, and
RAPTORX. One, again, observes that the cardinality sequence either fits, at the first few
places, the hypercartographic groupH+2 or that of a 3-manifold.
6. Discussion
The (primary) genetic code maps the 4-base words of DNA to the 20 proteinogenic
amino acids, a feature that we could model by using concepts of quantum information
theory associated to finite group representations. The (mostly informationally complete)
characters of finite groups Gn of signature Zn o 2O (2O the binary octahedral group) are
able to account for the degeneracies and many properties of the code (see [4] when n = 5,
see [5] when n = 6 and Section 5 of this paper when n = 7).
The secondary ‘genetic code’ lacks the universality of the primary code. In the
standard models of the secondary structure of proteins, the mapping from the 20 amino
acids to segments of α helices H, β sheet strands E, and coils C is not pointwise. The present
generation of softwares is defined by the evolutionary information derived from alignment
of multiple homologous sequences and the highest reported accuracy uses neural networks
for the optimal comparison of the sequences [8].
We could identify algebraic structures in the secondary code of proteins by employ-
ing the theory of infinite groups with generators H, E, and C and the protein relation
induced by the chosen model. Some hyperbolic 3-manifolds have been found as possi-
ble models of such a secondary structure. There exists a correspondence between the
3-sphere and the Bloch sphere of qubits so that a 3-manifold may be seen as a ‘dressing’
of qubits ([16], Section 1.1). In this view, quantum information controls the secondary
structure. Notice that topological dynamics and negative-curvature manifolds have been
proposed for modeling the brain in Reference [34].
It was unexpected that the oriented hypercartographic group H+2 seems to play a
major role in the secondary structure. Why are we interested by this feature?
We are interested in geometric physical codes or languages in action [35] and their
connection to the concept of emergence. Group representations arise here as a formal way
to describe those geometrical codes. Back to the secondary structure of proteins, we already
mentioned in the introduction that oriented hypermaps on surfaces are organized as the
oriented hypercartographic groupH+2 . Another important aspect is thatH
+
2 is related to
the so called absolute Galois group G = Gal(Q̄/Q), the group of field-automorphisms
of the field extensionQ̄ of the rational field Q. In the Esquisse d’un programme [12,13,36],
Grothendieck emphasizes the interest of looking at the action of G on topological, geometric
and even combinatorial structures. The highest level is the so-called ‘Teichmüller tower’.
Symmetry 2021, 13, 1146
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The simplest level concerns bipartite (hyper)maps called ‘dessins d’enfants’. To any dessin
D corresponds a (so-called) Belyi function f (x), where f (x) is a rational function of the
complex variable x whose structure reflects the critical points and the topology of D. The
remarkable result is that G acts faithfully on D, that is, each non-identity element of G
sends two non-isomorphic dessins to two inequivalent Belyi functions f (x), so that none
of the structure of G is lost by proceeding in this way. In passing, it is good to mention that
the theory of ‘dessins d’enfants’ can be used to account for geometric contextuality, the
counterpart of quantum contextuality [15,37].
Let us go back to the secondary structure of protein Hfq in Section 4.3 that builds one
of the 7 arms of the Lsm 1-7 complex in Figure 3b. According to our theory, there is a group
structure of the protein that intimately reflects that ofH+2 . Every subgroup of index d of
H+2 can be seen as permutation group on d elements, it can be drawn as a dessin D and
there is a faithful action of G on all dessins and permutation groups. In other words, the
protein Hfq contains in its structure the topology and algebra of G. The biological meaning
of this algebraic geometric structure needs further work. We leave it open at this stage.
It may be that the constraint of approximating the secondary structure with three letter
segments H, E, and C implies that every protein has to obey the G rules. We believe that
this rule may be seen as a support of the connection of biology to quantum gravity. In [38],
it is shown how a theory of quantum gravity may connect to G. We already proposed
a connection of our approach of the genetic code (see [5] and Section 5 of this paper) to
the Kummer surfaces that are K3 surfaces and play a role in some models of quantum
gravity [39].
Author Contributions: Conceptualization, M.P., F.F. and K.I.; methodology, M.P. and R.A.; software,
M.P.; validation, R.A., F.F. and M.M.A.; formal analysis, M.P. and M.M.A.; investigation, M.P., F.F. and
M.M.A.; writing—original draft preparation, M.P.; writing—review and editing, M.P.; visualization,
F.F. and R.A.; supervision, M.P. and K.I.; project administration, K.I.; funding acquisition, K.I. All
authors have read and agreed to the published version of the manuscript.
Funding: Funding was obtained from Quantum Gravity Research in Los Angeles, CA.
Conflicts of Interest: The authors declare no conflict of interest.
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