COMPLETE QUANTUM INFORMATION IN THE DNA
GENETIC CODE
MICHEL PLANAT†, RAYMOND ASCHHEIM‡,
MARCELO M. AMARAL‡, FANG FANG‡ AND KLEE IRWIN‡
Abstract. We find that the degeneracies and many peculiarities of
the DNA genetic code may be described thanks to two closely related
(fivefold symmetric) finite groups. The first group has signature G =
Z5 oH where H = Z2.S4 ∼= 2O is isomorphic to the binary octahedral
group 2O and S4 is the symmetric group on four letters/bases. The
second group has signature G = Z5oGL(2, 3) and points out a threefold
symmetry of base pairings. For those groups, the representations for
the 22 conjugacy classes of G are in one-to-one correspondence with
the multiplets encoding the proteinogenic amino acids. Additionally,
most of the 22 characters of G attached to those representations are
informationally complete. The biological meaning of these coincidences
is discussed.
Arxiv: q-bio.OT, quant-ph, math.GR, math.AG
PACS: 02. 20.-a, 82.39.Pj, 87.10.Vg, 02.10.De
MSC codes: 20C15, 92D20, 20E45, 14H52, 14G05
Keywords: DNA genetic code, informationally complete characters, finite groups,
hyperelliptic curve
1. Introduction
La filosofiaè scritta in questo grandissimo libro che continuamente ci sta
aperto innanzi a gli occhi (io dico l’universo), ma non si può intendere se
prima non s’impara a intender la lingua, e conoscer i caratteri, ne’ quali
scritto. Egliè scritto in lingua matematica, e i caratteri son triangoli, cer-
chi, ed altre figure geometriche, senza i qualimezi impossibile a intenderne
umanamente parola; senza questiè un aggirarsi vanamente per un oscuro
laberinto. (Galileo Galilei (1564-1642), Il Saggiatore, cap. 6).
Until now, deciphering the code of life [1]-[8] –the genetic code– was still
unsuccessful although mathematical theories have been proposed before [9]-
[13]. How is our new attempt different from earlier trials? Our mathematical
approach lies at the crossroads of finite group theory and quantum informa-
tion in the line of other papers mainly devoted to quantum computing [14]
but also focused on elementary particles [15].
Life cells need a macromolecule called deoxyribonucleic (or DNA) packed
in a chromosome during the cell mitosis. But DNA unwinds to be copied
during DNA replication or when its code is used to make proteins. DNA
is an helix consisting of two parallel polynucleotide chains carrying genetic
instructions in 4 nitrogeneous bases for the growth and reproduction of all
living organisms. The genetic code is organized in triples of bases called
1
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© 2020 by the author(s). Distributed under a Creative Commons CC BY license.
2MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡, FANG FANG‡ANDKLEE IRWIN‡
codons. There are 43 = 64 codons but only 20 standard amino acids, mean-
ing a high redundancy of the code. In our approach, we use the characters
and the corresponding representations of a well defined finite group G to ex-
plain the mapping of the nitrogeneous bases to amino acids. The group char-
acters may be used to build minimal and informationally complete quantum
measurements, one for each character and the corresponding amino acid.
In Section 2, one details the discovery and main properties of DNA and
the genetic code. In Section 3, one recalls how minimal informationally com-
plete quantum measurements are performed and the necessary elements of
character theory of finite groups. In Section 4, the assignment of characters
of the small group G = (240, 105) to amino acids is detailed. In Section
5, we provide a justification of the fact that minor and major grooves in
the DNA double helix have periods whose ratio is the Golden ratio. This
is based on the study of points over a hyperelliptic curve occuring in the
character table.
2. DNA and the genetic code
The discovery of DNA is attributed to Watson and Crick in 1953 [1]. But
the phosphorous-containing substance now called DNA was first isolated by
F. Miescher in 1869 in the nuclei of white blood cells under the name of
‘nuclein’ (a nucleic acid) paving the way of its recognition as the carrier of
inheritance [2]. Only in 1909, P. Levene found that DNA contains pentoses
such adenine (A), guanine (G), thymine (T), cytosine (C), a deoxyribose
and a phosphate group. At that time, it was still believed that the protein
component of chromosomes was the true basis of heredity. The recognition
that DNA rather than proteins could be the genetic material was suggested
by the E. Chargaff’s rules, proposed in 1940 that the four bases are present
in the percentages: A ≈ T ≈ 30% and G ≈ C ≈ 20% for all species [3].
Subsequent X-ray crystallography work by English researchers R. Franklin
and M. Wilkins contributed to Watson and Crick’s derivation of the three-
dimensional, double-helical model for the structure of DNA [4].
As shown in Fig. 1, DNA is a double helix, with the two strands connected
by hydrogen bonds. In the most current form of DNA (called B-DNA), the
ratio of the diameter D ≈ 21Å to the period l+L and the ratio between the
minor groove l ≈ 13Å and the major groove L ≈ D ≈ 21Å are both close to
the golden ratio φ = (
√
5− 1)/2 ≈ 0.618 [17].
G. Gamow (also a cosmologist) observed that the 43 = 64 possible per-
mutations of the four DNA bases A, T, G and C, taken three at a time (as
codons), could be reduced to 20 distinct combinations and might code for the
twenty amino acids which, he suggested, might well be the sole constituents
of all proteins [5]. But the lack of overlapping of codons (not assumed by
Gamow) and the demonstration that the genetic code is made up of a series
of three base pair codons, which code for individual amino acids, dates back
to 1961 with the Crick, Brenner et al. experiment [6]. Now we know that
the peculiar non ambiguous and non overlapping assignment of all 64 codons
to the 20 amino acids is nearly universal [7].
The ‘genetic code’ is the set of rules used by living cells to translate infor-
mation encoded within genetic material (DNA or messenger RNA sequences
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COMPLETE QUANTUM INFORMATION IN THE DNA GENETIC CODE
3
Figure 1. The double-helical structure of DNA and base
pairing. Complementary bases are held together as a pair by
hydrogen bonds. Two hydrogen bonds connect T to A; three
hydrogen bonds connect G to C. The image is borrowed to
Wikipedia [16].
of nucleotide triplets, or codons) into proteins. Translation is accomplished
by the ribosome, which links amino acids in an order specified by messenger
RNA (mRNA), using transfer RNA (tRNA) molecules to carry amino acids
and to read the mRNA, three nucleotides at a time.
The codons which code for the same amino acids form multiplets and are
organized as
* Met, Trp: 2 singlets
* Asn, Asp, Cys, Gln, Glu, His, Lys, Phe, Tyr: 9 doublets
* Ile, Term: 2 triplets
* Ala, Gly, Pro, Thr, Val: 5 quadruplets
* Arg, Leu, Ser: 3 sextets.
A 21st proteinogenic amino acid Selenocysteine (symbol Sec) was found
present in some enzymes and a 22st amino acid Pyrrolysine (Symbol Pyl)
in some archaea and bacteria.
The assignment of three-letter codons to the standard 20 amino acids is
shown in Fig. 2a. It is clearly seen in this table that the assignment of codons
to amino acids is not random but follows underlying rules to be discovered.
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4MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡, FANG FANG‡ANDKLEE IRWIN‡
Woese and collaborators introduced the concept of polar requirement to
express stereochemical associations between amino acids and nucleobases
in solution [8]. Taking dimethylpyridine (or DMP) as a solvant, the polar
requirement is found as in Fig. 2b [18, 19]. For more recent predictors of
the stereochemical associators the reader may read reference [20].
Figure 2. (a) the generic codon table for RNA, (b) the
codon table colored by amino acid polar requirement values
[19].
3. Symmetries and quantum information from the characters
of a finite group
There are two important ingredients in our approach. One topic follows
from our earlier work on ‘magic state’ quantum computing [14] and consists
of performing a generalized quantum measurement while keeping a complete
information in the process, as explained in Sec. 3.1. The second (more
recent) topic consists of finding the ‘magic states’ in the bank of characters
of a selected finite group. The latter approach recalled in Section 3.2 was
found successful in the context of symmetries of elementary matter particles
[15].
3.1. Minimal informationally complete quantum measurements. Let
Hd be a d-dimensional complex Hilbert space and {E1, . . . , Em} be a collec-
tion of positive semi-definite operators (POVM) that sum to the identity.
Taking the unknown quantum state as a rank 1 projector ρ = |ψ〉 〈ψ| (with
ρ2 = ρ and tr(ρ) = 1), the i-th outcome is obtained with a probability given
by the Born rule p(i) = tr(ρEi). A minimal and informationally complete
POVM (or MIC) requires d2 one-dimensional projectors Πi = |ψi〉 〈ψi|, with
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COMPLETE QUANTUM INFORMATION IN THE DNA GENETIC CODE
5
Πi = dEi, such that the rank of the Gram matrix with elements tr(ΠiΠj),
is precisely d2.
With a MIC, the complete recovery of a state ρ is possible at a minimal
cost from the probabilities p(i). In the best case, the MIC is symmetric and
called a SIC with a further relation |〈ψi|ψj〉|2 = tr(ΠiΠj) = dδij+1
d+1
so that
the density matrix ρ can be made explicit [21].
In our earlier references, starting with [14], a large collection of MICs
are derived. They correspond to Hermitian angles |〈ψi|ψj〉|i
6=j ∈ A =
{a1, . . . , al} belonging to a discrete set of values of small cardinality l. They
arise from the action of a Pauli group Pd [22] on an appropriate magic state
pertaining to the coset structure of subgroups of index d of a free group with
relations.
In reference [15], an entirely new class of MICs in the Hilbert space Hd,
relevant for the lepton and quark mixing patterns, is obtained by taking
fiducial/magic states as characters of a finite group G possessing d conjugacy
classes and using the action of a Pauli group Pd on them. The same approach
is used in this paper.
3.2. The character table of a finite group and quantum informa-
tion. Let G be a finite group, V a finite-dimensional vector space over a
field F and let r : G→ GL(V ) be a representation of G on V . The character
κr(g) associates to an element g ∈ G the trace of the corresponding matrix
r(g). A character κr is called irreducible if r is an irreducible representation.
The degree of the character κ is called the dimension of the representation.
Assume that G is made of d conjugacy classes of elements. One introduces
class functions on G from the ring of complex-valued functions on G that
are constant on conjugacy classes. In fact we restrict ourselves to functions
with values in cyclotomic fields [23].
The table of irreducible characters of G, or character table for short, is a
way to summarize the properties of irreducible representations of elements
g of G.
Let us take the case of the symmetric group S4 also called the small group
(24, 12) (number 12 in the range of 15 groups of order 24). The character
table is in Table 1. It makes explicit the five irreducible characters of G, the
corresponding dimension dim of the representation, as well as the size and
order of an element in each class. For each class, we added the calculation
of the rank of the Gram matrix that corresponds to the POVM obtained by
the action of the 5-dit Pauli group over the irreducible character κi of the
class. For the classes 3 to 5, the rank of the Gram matrix is d2 = 25 so that
the POVM is a MIC.
As a second example, one takes the binary octahedral group 2O also called
the small group (48, 28) (number 28 in the range of 52 groups of order 48).
Group 2O is an important ingredient of our model of the genetic code, as
shown in section 4. The character table is shown in Table 2.
It makes
explicit the eight irreducible characters of G, the corresponding dimension
of the representation, as well as the size and order of an element in each
class. At the right hand side column, one finds the rank of the Gram matrix
that corresponds to the POVM obtained by the action of the 8-dit Pauli
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6MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡, FANG FANG‡ANDKLEE IRWIN‡
Class
1
2 3
4 5
Class
Size
1
3 6
8 6
Size
Order
1
2 2
3 4
Order
Char
dim
Gram
κ1
1
1 1
1 1
5
κ2
1
1
-1 1
- 1
21
κ3
2
2 0
-1 0
d2
κ4
3
-1
-1 0 1
d2
κ5
3
-1 1
0
-1
d2
Table 1. The character table for the symmetric group G =
S4. There are five conjugacy classes of elements of dimensions
1, 1, 2, 3 and 3, respectively. The size and the order of an
element in the class is as shown. The last five rows of the
table are the irreducible characters κi, i = 1..5, of G. To each
of them is associated the rank of the Gram matrix defined in
subsection 3.1.
group over the irreducible character κi of the class. None of the characters
of the group 2O is informationally complete.
Class
1
2 3
4 5
6 7
8
Class
Size
1
1 8
6 12
8 6
6
Size
Order
1
2 3
4 4
6 8
8
Order
Char
dim
Gram
κ1
1
1 1
1 1
1 1
1
8
κ2
1
1 1
1
- 1 1
-1
-1
45
κ3
2
2
-1 2 0
-1 0
0
56
κ4
2
-2
-1 0 0
1
√
2 −
√
2 59
κ5
2
-2
-1 0 0
1 −
√
2
√
2
59
κ6
3
3 0
-1 1
0
-1
-1
59
κ7
3
3 0
-1
-1
0 1
1
59
κ8
4
-4 1
0 0
-1 0
0
59
Table 2. The character table for the symmetric group G =
2O = (48, 28). There are eight conjugacy classes of elements
of dimensions 1(×2), 2(×3), 3(×2) and 4. The size and the
order of an element in the class is as shown. The last five
rows of the table are the irreducible characters κi, i = 1..8,
of G. To each of them is associated the rank of the Gram
matrix defined in subsection 3.1.
4. Symmetries and quantum information in the genetic code
As shown in the previous section, there may exist several representations
of a given dimension in a finite group. Does it exist a group G whose
multiplets map to the multiplets of the genetic code in a satisfactorily way?
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COMPLETE QUANTUM INFORMATION IN THE DNA GENETIC CODE
7
We shall impose the constraint that there are enough n-plets to embed the
mappings to the proteigenomic amino acids for dimensions n = 1, 2, 3 and
4 and that there are at least 2 sextuplets. Small groups (240, 105) and
(240, 106) (with 22 conjugacy classes) and (240, 107) and (240, 108) (with
28 conjugacy classes) are the group candidates with the smallest cardinality.
All of them have two 6-dimensional representations. We could not find other
candidate group of higher cardinality. The most economical groups are thus
the former two groups and the latter two are different only in the sense that
there are 6 one-dimensional representations instead of two.
The first group (240, 105) has signature G = Z5 oH where H = Z2.S4 ∼=
2O is isomorphic to the binary octahedral group 2O and S4 is the symmetric
group on four letters/bases. The second group (240, 106) has signature G =
Z5 o GL(2, 3) and points out a threefold symmetry of base pairings. For
those groups, the representations for the 22 conjugacy classes of G are almost
in one-to-one correspondence with the multiplets encoding the proteinogenic
amino acids. The assignment is not perfect in the sense that there are only
two representations of dimension 6 while there are three sextuplets in the
genetic code.
(240,105)
dimension
1
1
2
2
2
2
2
2
2
2
2
Z5 o (Z2.S4)
d-dit, d=22
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
d2
∼= Z5 o 2O
amino acid
Met
Trp Cys
Phe Tyr His Gln Asn Lys
Glu Asp
order
1
2
3
4
4
5
5
6
8
8
10
polar req.
5.3
5.2
4.8
5.0
5.4
8.4
8.6
10.0
10.1
12.5
13.0
(240,105)
dimension
3
3
4
4
4
4
4
4
4
6
6
d-dit, d=22
d2
475
483
480
d2
d2
d2
d2
d2
d2
d2
amino acid
Ile
Stop Leu,Pyl,Sec Leu Val
Pro Thr Ala Gly
Ser
Arg
order
10
15
15
15
15
20
20
30
30
30
30
polar req.
4.9
4.9
5.6
6.6
6.6
7.0
7.9
7.5
9.1
Table 3. For the group G = (240, 105) ∼= Z5 o 2O, the
table provides the dimension of the representation, the rank
of the Gram matrix obtained under the action of the 22 -
dimensional Pauli group, the order of a group element in the
class and a good assignment to an amino acid according to
its polar requirement value. Bold characters are for faithful
representations. There is an ‘exception’ for the assignment of
the sextet ‘Leu’ that is assumed to occupy two 4-dimensional
slots. All characters are informationally complete except for
the ones assigned to ‘Stop’, ‘Leu’, ‘Pyl’ and ‘Sec’.
The 22 characters of the group G = (240, 105) are investigated in Section
5. A sketch of the character properties of G is in Table 3. It includes the
dimension of the representation associated to each character, the order of an
element in each class, and additionally the rank of the Gram matrix when
one builds a POVM based on the character as a magic/fiducial state as in
section 3.1. Table 3 proposes a one-to-one assignment of the representations
of G to amino acids. A given degeneracy of an amino acid in the genetic
code corresponds to the dimension of the selected representations of G. The
amino acids of a given degeneracy/dimension in the table are ordered ac-
cording to their increasing polar requirement value (given in Fig. 2b) and
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8MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡, FANG FANG‡ANDKLEE IRWIN‡
(240,105)
dimension
1
1
2
2
2
2
2
2
2
2
2
∼= Z5 o 2O
amino acid Met
Trp
Cys
Phe
Tyr
His
Gln
Asn
Lys
Glue Asp
bases
AUG UGG UGU
UUU UAU CAU CAA AAU AAA GAA GAU
UGC
UUC UAC CAC CAG AAC AAG GAG GAC
(240,105)
dimension
3
3
4
4
4
4
4
4
4
6
6
amino acid
Ile
Stop
Leu
Leu
Val
Pro
Thr
Ala
Gly
Ser
Arg
Pyl,Sec
bases
AUU UAA UUA
CUU GUU CCU ACU CGU GGU UCU CGU
AUC UAG UUG
CUC GUC CCC ACC GCC GGC UCC CGC
AUA UGA UAG
CUA GUA CCA ACA GCA GGA UCA GGA
UGA
CUG GUG CCG ACG GCG GGG UCG CGG
AGU AGA
AGC AGG
Table 4. Excerpt of Table 3 with the assignment of codons
to amino acids according to the standard genetic code. There
is no ambiguity in such assignments except for codons, UAG
and UGA that code for ‘Stop’, as well as for ‘Pyl’ and ‘Sec’,
respectively. According to our approach, this ambiguity is re-
flected in the characters being non informationally complete
for the corresponding two slots.
(240,106)
dimension
1
1
2
2
2
2
2
2
2
2
2
∼= Z5 oGL(2, 3)
order
1
2
2
3
4
5
5
6
8
8
10
d-dit, d=22
d2
d2
483 d2
d2
d2
d2
d2
d2
d2
d2
(240,106)
dimension
3
3
4
4
4
4
4
4
4
6
6
order
10
15
15
15
15
20
20
30
30
30
30
d-dit, d=22
482 d2
d2
d2
d2
d2
d2
d2
d2
440
440
Table 5. The group (240, 106) ∼= Z5 o GL(2, 3) as another
candidate for the assignments of amino acids.
It has the
same dimensions of representations and closely related group
orders. One may postulate that the group (240, 106) encodes
D-amino acids while the group (240, 105) encodes L-amino
acids. For most naturally-occurring amino acids, the carbon
alpha to the amino group has the L-configuration.
thus follows the order of a group element in the corresponding class. Most
representations are informationally complete with the rank of the Gram ma-
trix equal to d2. The exceptions are for the 3-dimensional slot for the Stop
codon and the 4-dimensional slot that we associate to the amino-acid Leu
and the 2 extra amino acids Pyl and Sec. There is an ambiguity in the as-
signment of codons UAG and UGA that code for Stop, and for Pyl (with the
codon UAG) and Sec (with the codon UGA). Thus the slight ambiguity of
nature in coding the amino acids is recovered in our approach. The assign-
ments to codons is displayed in detail in Table 4. According to our approach,
this ambiguity is reflected into three characters being non informationally
complete.
Another possible assignment of amino acids to the representations of
group (240, 106) is possible. This group has similar dimensions of the rep-
resentations than group (240, 105) with slight differences in the order of an
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COMPLETE QUANTUM INFORMATION IN THE DNA GENETIC CODE
9
element in each class and the rank of the Gram matrix for the correspond-
ing POVM, as shown in Table 5. One may postulate that it encodes the
D-amino acids while the group (240, 105) encodes proteinogenic L-amino
acids. For most naturally-occurring amino acids, the carbon alpha to the
amino group has the L-configuration. D-amino acids are most occasionally
found in nature as residues in proteins.
Normal subgroups (48, 28) ∼= 2O of the group (240, 105) and GL(2, 3) ∼=
(48, 29) of the group (240, 106) may be used to encode most of the so-called
essential amino-acids as shown in Table 6. One observes that apart from
two exceptions for the 4-dimensional slots of GL(2, 3) the characters of these
groups are not informationally complete.
dimension
1
1
2
2
2
3
3
4
order
1
2
3
4
4
6
8
8
(48,28)=2O
d-dit, d=8
8
45
56
59
59
59 59
59
(48,29)=GL(2,3) d-dit, d=8
56
58
59
d2
d2
63 63
61
amino acid Met Trp Phe His Lys
Ile
Stop Thr
polar req.
5.3
5.2
5.0
8.4 10.1 4.9
6.6
Table 6. The assignment of representations of groups 2O
or GL(2, 3) to essential amino acids. There are two extra
essential amino acids Val and Leu but they are not strictly
essential from a metabolic perspective. Strictly indispensable
amino acids are Trp, Lys and Thr which cannot be submitted
to transamination.
5. Symmetries of the genetic code, the Golden ratio and a
hyperelliptic curve
If one accepts the validity of the aforementioned group theoretical model
of the genetic code, one can provide clues why nature selected these par-
ticular symmetries. One clue is contained in the detailed structure of the
characters of the group (240, 105) and its cousin group (240, 106). The
golden ratio is playing a role in accordance with the DNA structure.
The list of 22 characters for the group (240, 105) is in Table 7. A closely
related table (not shown) is found for the group (240, 106). The entries in
the character table are shown to be either rational numbers, quadratic ir-
rationalities such as z1 = −(
√
5 + 1)/2, z2 =
√
5 − 1, z3 = 3(1 +
√
5)/2,
z4 =
√
2 and their algebraic conjugates, or ±ri where ri is one of the four
roots ri of the quartic polynomial x
4 − x3 − 4x2 + 4x + 1 = 0, where
r1 = −2 cos(π/15) ≈ −1.956295, r2 = −2 cos(11π/15) ≈ 1.338261, r3 =
−2 cos(7π/15) ≈ −0.209056 and r4 = −2 cos(13π/15) ≈ 1.827091.
It is known that such a quartic contains the golden ratio φ = z2/2 and
the irrational
√
2 in its structure. Following [24, Section 3], the inflection
secant S has segments whose length ratio is φ. Also the double tangent,
the inflection secant and the straight line passing through the third tangent
point and parallel to them separate areas whose ratio is
√
2.
It may be that some other clues in the mystery of the genetic code (and
a possible connection to the present approach) are in a concept introduced
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10MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡, FANG FANG‡ANDKLEE IRWIN‡
under the name of BoerdijkCoxeter (BC) helix –an helix made of contiguous
tetrahedra– more or less at the same time than the discovery of DNA [25].
The concept was revisited to arrive at a BC helix exhibiting a connection
to aperiodicity and the golden ratio [26, 27].
From now, we consider the (genus one) hyperelliptic curve C
C : y2 = x4 − x3 − 4x2 + 4x+ 1
as a potential model of many features of DNA double helix.
The characters responsible of the quadratic irrationalities and the roots
of the quartic are defined over the cyclotomic fields Q(ζ5) and Q(ζ15), re-
spectively, where ζn is a primitive n-th complex root of unity. Thus we focus
on C with points over such cyclotomic fields.
Points of C over cyclotomic fields. As usual for elliptic and hyperelliptic
curves of genus g, C 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 C [23]. For the points of C,
there is a parameter called ‘bound’ that loosely follows the heights of the
x-coordinates found by the search algorithm.
Let us start with points on C with entries over the cyclotomic field Q(ζ5).
For points whose entries are bounded by modulus 1, one gets 8 cases as
follows (1 : ±1 : 0) (the points at infinity), (0 : ±1 : 1), (1 : ±1 : 1) and
(2 + φ : ±(3 + φ) : 1). For points whose entries are bounded by modulus
2, one gets 10 extra cases (±2 : −1 : 0), (±2 : 1 : 1), (2φ : ±(3 − 4φ) : 1),
(−2(1 + φ) : ±(7 + 4φ) : 1) and (1 − φ : ±(4 + φ) : 1). For points whose
entries are bounded by modulus 10, one gets 8 extra cases (4/3 : ±1/9 : 1),
(6 : ±31 : 1), (4/5 : ±31/25 : 1), (−1/7 : ±29/49 : 1).
Let us continue with points on C with entries over Q(ζ15). Points with
integer entries are as those given over Q(ζ5). In addition, the points whose
entries are bounded by modulus 1 are (r1 : 0 : 1), (r2 : 0 : 1), (r3 : 0 : 1),
(r4 : 0 : 1) (where ri is one of the four roots of the quartic polynomial
above), (2 + φ : ±(3 + φ) : 1) [the latter 2 points are on Q(ζ5)] The non
rational points whose entries are bounded by modulus 2 are (2φ : ±(3−4φ) :
1), (−2(1 + φ) : ±(7 + 4φ) : 1) [the latter 2 points are on Q(ζ5)] and
(φ2 : ±(1 + φ2) : 1. For non rational points whose entries are bounded by
modulus 7, one gets 8 extra cases and no more up to bound 10. The entries
could not be found related to simple irrationalities and are approximated as
(3.02234 : ±5.69063 : 1), (0.90536 : ±1.12804 : 1), (0.50552 : ±1.39140 : 1)
and (11.56677 : ±125.95399).
Over both cyclotomic fields, there are plenty extra points where the entries
have higher modulus but we do not list them here.
The group law on the Jacobian J of the hyperelliptic curve C.
There exists a group law on the Jacobian J of a hyperelliptic curve [28].
Using Magma we provide some results for the operations in J(C) defined
over Q(ζ5).
Let focus on the 8 elements ai of bound 1 in the Jacobian. In the Mumford
representation of Jacobian elements, a1 = (1, 0, 0), a2 = (x
2 − 15/8,−x2 +
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COMPLETE QUANTUM INFORMATION IN THE DNA GENETIC CODE
11
49/16, 2), a3 = (x−4/5,−x2−3/5, 2), a4 = (x−4/3,−x2+5/3, 2), a5 = (x+
1/7,−x2−4/7, 2), a6 = (x−1,−x2+2, 2), a7 = (x−(2+φ),−x2+2(1+φ), 2),
a8 = (x+ 2.25123,−x2 + 8.03750, 2). In Table 8, the inverses or elements ai
are shown. In Table 9, one explicits the sums ai + aj between two elements
of bound 1 in the Jacobian.
A similar group law exists for C defined over Q(ζ15) but it is more difficult
to explicit the points and we do not give details here.
6. Conclusion
We unveiled part of the machinery of DNA in building the genetic code
thanks to the representations of a distinguished finite group of signature
G = Z5o2O, where 2O is the binary octahedral group. The group characters
κ of G are found to map to the proteigenomic amino acids and this is
performed by almost keeping a complete quantum information when the
κ’s are seen as magic states, as in quantum computing. Moreover, there
exists a unique quartic and a related hyperelliptic curve C behind the DNA
scene which gives sense to the irrationalities found in the character table
(and in the real biological world). It is still too early to conclude that DNA
calculates on points of C or on points of the Jacobian J(C) but it is tempting
to work in this direction in future papers. It may be that other finite groups
and their attached hyperelliptic curve are governing the secondary structure
of proteins such that the alpha helix or the DNA packing into chromatin
and chromosomes.
References
[1] J. D. Watson and F. H.C. Crick, A structure for deoxyribose nucleic acid, Nature 171
737-738 (1953).
[2] L. A. Pray, Discovery of DNA Structure and Function:
Watson and
Crick, Nature Education 1 100 (2008),
paper available at
the address
https://www.nature.com/scitable/topicpage/discovery-of-dna-structure-and-
function-watson-397/.
[3] E. Chargaff, Chemical specificity of nucleic acids and mechanism of their enzymatic
degradation, Experientia 6 201209 (1950).
[4] B. Maddox, The double helix and the ‘wronged heroine’, Nature 421 407-408 (2003).
[5] G. Gamov, Possible Relation between deoxyribonucleic acid and protein structures,
Nature 173 318 (1954).
[6] F. H. Crick, L. Barnett, S. Brenner, R. J. Watts-Tobin, General nature of the genetic
code for proteins, Nature, 192 12276-32 (1961).
[7] E. V. Koonin ans A. S. Novozhikov, Origin and evolution of the genetic code: the
universal enigma, IUBMB Life 61 99-111 (2009).
[8] C. R. Woese, D. H. Dugre, W. C. Saxinger and S. A. Dugre, The molecular basis of
the genetic code, Proc. Natl Acad Sci 55 966-974 (1966).
[9] S. V. Petoukhov, Matrix genetics, part 2:
the degeneracy of the genetic code
and the octave algebra with two quasi-real units (the Yin-Yang octave algebra),
Preprint0803.3330 [q-bio].
[10] J. E.M. Hornos, Y. M.M. Hornos, Algebraic model for the evolution of the genetic
code, Phys. Rev. Lett. 71 44014404 (1993).
[11] M. Di Giulio, The origin of the genetic code: theories and their relationships, Biosys-
tems 80 175-184 (2005).
[12] C. M. Carlevaro, R. M. Irastorza and F. Vericat, Quaternionic representation of the
genetic code, Biosystems 141 10-19 (2016).
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12MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡, FANG FANG‡ANDKLEE IRWIN‡
[13] F. Antonelli and M. Forger, Symmetry breaking in the genetic code: Finite groups,
Math. Comp. Mod. 53 1469-1488 (2011).
[14] M. Planat and Z. Gedik, Magic informationally complete POVMs with permutations,
R. Soc. open sci. 4 170387 (2017).
[15] M. Planat, R. Aschheim, M. M. Amaral and K. Irwin, Informationally complete
characters for quark and lepton mixings, Symmetry 12 1000 (2020).
[16] https://en.wikipedia.org/wiki/DNA (accessed on 1 May 2020).
[17] D. Harel, R. Unger and J. L. Sussman, Beauty is in the genes of the beholder, Trends
in Bioch. Sc. 11 155-156 (1986).
[18] J. R. Jungck, The Genetic Code as a Periodic Table, J. Mol. Evol. 11 211–224 (1978).
[19] D. C. Mathew and Z. Luthey-Schulten, On the physical basis of the amino acid polar
requirement, J. Mol. Evol. 66 519-528 (2008).
[20] C. W. Carter Jr, Coding of Class I and II aminoacyl-tRNA synthetases, Adv Exp
Med Biol. 966 103148 (2017).
[21] C. A. Fuchs, On the quantumness of a Hibert space, Quant. Inf. Comp. 4 467-478
(2004).
[22] Planat, M. Pauli graphs when the Hilbert space dimension contains a square: Why
the Dedekind psi function? J. Phys. A Math. Theor. 2011, 44, 045301.
[23] Bosma, W.; Cannon, J. J.; Fieker, C. ; Steel, A. (eds). Handbook of Magma functions,
Edition 2.23 (2017), 5914pp (accessed on 1 January 2019).
[24] H. TR Aude, Notes on quartic curves, Am. Math. Month. 56 165-170 (1949).
[25] A. H. Boerdijk, Some remarks concerning close-packing of equal spheres, Philips Res.
Rep. 7 30 (1952).
[26] G. Sadler, F. Fang, R. Clawson and K. Irwin, Periodic modification of the Boerdijk-
Coxeter helix (tetrahelix), Mathematics 7 1001 (2019).
[27] Fang Fang, K. Irwin, J. Kovacs and G. Sadler, Cabinet of curiosities: the interesting
geometry of the angle β = arcos(3φ− 1)/4), Fractal Fract. 3 48 (2019).
[28] C. Costello and K. Lauter, Group law computations on Jacobians of hyperelliptic
curves, in T. Johansson and P. Q.Nguyen, editors, EUROCRYPT, volume 7881 of
Lecture Notes in Computer Science(Springer, 2013),pp. 194210.
† Université de Bourgogne/Franche-Comté, Institut FEMTO-ST CNRS UMR
6174, 15 B Avenue des Montboucons, F-25044 Besançon, France.
E-mail address: michel.planat@femto-st.fr
‡ Quantum Gravity Research, Los Angeles, CA 90290, USA
E-mail address: raymond@QuantumGravityResearch.org
E-mail address: Klee@quantumgravityresearch.org
E-mail address: Marcelo@quantumgravityresearch.org
E-mail address: Fang@QuantumGravityResearch.org
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COMPLETE QUANTUM INFORMATION IN THE DNA GENETIC CODE
13
κ1
1
1
1
1
1
1
1
1
1
1
1
1
κ2
1
1
1
1
-1
1
1
1
-1
-1
1
1
κ3
2
2
-1
2
0
2
2
-1
0
0
2
2
κ4
2
2
2
2
0
z1
z∗1
2
0
0
z1
z∗1
κ5
2
2
2
2
0
z∗1
z1
2
0
0
z∗1
z1
κ6
2
-2
-1
0
0
2
2
1
z4
−z4
-2
-2
κ7
2
-2
-1
0
0
2
2
1 −z4
z4
-2
-2
κ8
2
2
-1
2
0
z1
z∗1
-1
0
0
z1
z∗1
κ9
2
2
-1
2
0
z∗1
z1
-1
0
0
z∗1
z1
κ10
2
2
-1
2
0
z1
z∗1
-1
0
0
z1
z∗1
κ11
2
2
-1
2
0
z∗1
z1
-1
0
0
z∗1
z1
κ12
3
3
0
-1
-1
3
3
0
1
1
3
3
κ13
3
3
0
-1
1
3
3
0
-1
-1
3
3
κ14
4
-4
1
0
0
4
4
-1
0
0
-4
-4
κ15
4
-4
-2
0
0
z2
z∗2
2
0
0 −z2
−z∗2
κ16
4
-4
-2
0
0
z∗2
z2
2
0
0 −z∗2
−z2
κ17
4
-4
1
0
0
z2
z∗2
-1
0
0 −z2
−z∗2
κ18
4
-4
1
0
0
z2
z∗2
-1
0
0 −z2
−z∗2
κ19
4
-4
1
0
0
z∗2
z2
-1
0
0 −z∗2
−z2
κ20
4
-4
1
0
0
z∗2
z2
-1
0
0 −z∗2
−z2
κ21
6
6
0
-2
0
z3
z∗3
0
0
0
z3
z∗3
κ22
6
6
0
-2
0
z∗3
z3
0
0
0
z∗3
z3
→
1
1
1
1
1
1
1
1
1
1
d2
Met
→
1
1
1
1
1
1
1
1
1
1
d2
Trp
→
-1
-1
-1
-1
2
2
-1
-1
-1
- 1
d2
Cys
→
z∗1
z∗1
z1
z1
z1
z∗1
z1
z1
z∗1
z∗1
d2
Phe
→
z1
z1
z∗1
z∗1
z∗1
z1
z∗1
z∗1
z1
z1
d2
Tyr
→
-1
-1
-1
-1
0
0
1
1
1
1
d2
His
→
-1
-1
-1
-1
0
0
1
1
1
1
d2
Gln
→
r1
r2
r3
r4
z1
z∗1
r3
r4
r2
r1
d2
Asn
→
r4
r3
r1
r2
z∗1
z1
r1
r2
r3
r4
d2
Lys
→
r2
r1
r4
r3
z1
z∗1
r4
r3
r1
r2
d2
Glu
→
r3
r4
r2
r1
z∗1
z1
r2
r1
r4
r3
d2
Asp
→
0
0
0
0
-1
-1
0
0
0
0
d2
Ile
→
0
0
0
0
-1
-1
0
0
0
0
475
Stop
→
1
1
1
1
0
0
-1
-1
-1
-1
483
Leu, Pyl, Sec
→ −z1 −z1 −z∗1 −z∗1
0
0
z∗1
z∗1
z1
z1
480
Leu
→ −z∗1 −z∗1 −z1 −z1
0
0
z1
z1
z∗1
z∗1
d2
Val
→ −r3 −r4 −r2 −r1
0
0
r2
r1
r4
r3
d2
Pro
→ −r4 −r3 −r1 −r2
0
0
r1
r2
r3
r4
d2
Thr
→ −r1 −r2 −r3 −r4
0
0
r3
r4
r2
r1
d2
Ala
→ −r2 −r1 −r4 −r3
0
0
r4
r3
r1
r2
d2
Gly
→
0
0
0
0 −z1 −z∗1
0
0
0
0
d2
Ser
→
0
0
0
0 −z∗1 −z1
0
0
0
0
d2
Arg
Table 7. The character table
for the group G =
(240, 105) ∼= Z5 o 2O. The two last hand side columns are
for the rank of the Gram matrix (with d = 22) for the cor-
responding character and the assignment of an amino acid
of the genetic code, respectively (see also Table 3 and 4
for further details). The notation in the entries is as fol-
lows: z1 = −(
√
5 + 1)/2, z2 =
√
5 − 1, z3 = 3(1 +
√
5)/2,
z4 =
√
2, the exponent ∗ means the algebraic conjugation,
e.g. z∗1 = (
√
5 − 1)/2, the ri’s are the four real roots of the
quartic curve x4−x3−4x2 +4x+1 = 0, i.e. r1 ≈ −1.956295,
r2 ≈ 1.338261, r3 ≈ −0.209056 and r4 ≈ 1.827091.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 July 2020 doi:10.20944/preprints202007.0403.v1
14MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡, FANG FANG‡ANDKLEE IRWIN‡
points of J a1 a2 a3 a4 a5 a6 a7 a8
inverse
a1
4
5
15 7
a6 a7
?
Table 8. The inverse of the 8 elements ai of bound 1 in
the Jacobian J of the hyperelliptic curve C over Q(ζ5). One
gets −a2 = (1,−x2 + x/2, 2), −a3 = (x,−x2 − 1, 2), −a4 =
(x,−x2 + 1, 2) and −a5 = (x − 1,−x2, 2) of bounds 4, 5, 15
and 7, respectively.
sum a1 a2 a3 a4 a5 a6 a7
a8
a1
a1 a2 a3 a4 a5 a6 a7
a8
a2
.
15 5
4
7
a5 a8
?
a3
.
.
b6 a2 6
b8
b10 54
a4
.
.
.
b8
b6
b1
b2
14
a5
.
.
.
.
15 a2 b9
?
a6
.
.
.
.
.
a1 b4
b9
a7
.
.
.
.
.
.
a1
a2
a8
.
.
.
.
.
.
.
15
Table 9. The addition table between the 8 elements ai of
bound 1 in the Jacobian J of the hyperelliptic curve C over
Q(ζ5). The 10 elements with bound 2 in the Jacobian are
denoted bi. For entries with bound not 1 or 2, only the
bound is given. The points of the Jacobian with bounds 2
used in the table are b1 = (x + 2,−x2 + 3, 2), b6 = (x −
70/31,−x2 + 221/31, 2), b8 = (x − 6,−x2 + 5, 2), b2 ≈ (x +
0.19821,−x2+0.28285, 2), b9 ≈ (x−1.30757,−x2−2.47363, 2)
and b10 ≈ (x− 2.00889,−x2 − 1.01292, 2).
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