DNA Sequence and Structure under the Prism of Group Theory and Algebraic Surfaces

Taking a DNA sequence, a word with letters/bases A, T, G and C, as the relation between the generators of an infinite group π, one can discriminate between two important families: (i) the cardinality structure for conjugacy classes of subgroups of π is that of a free group on one to four bases, and the DNA word, viewed as a substitution sequence, is aperiodic; (ii) the cardinality structure for conjugacy classes of subgroups of π is not that of a free group, the sequence is generally not aperiodic and topological properties of π have to be determined differently. The two cases rely on DNA conformations such as A-DNA, B-DNA, Z-DNA, G-quadruplexes, etc. We found a few salient results: Z-DNA, when involved in transcription, replication and regulation in a healthy situation, implies (i). The sequence of telomeric repeats comprising three distinct bases most of the time satisfies (i). For two-base sequences in the free case (i) or non-free case (ii), the topology of π may be found in terms of the SL(2,C) character variety of π and the attached algebraic surfaces. The linking of two unknotted curves—the Hopf link—may occur in the topology of π in cases of biological importance, in telomeres, G-quadruplexes, hairpins and junctions, a feature that we already found in the context of models of topological quantum computing. For three- and four-base sequences, other knotting configurations are noticed and a building block of the topology is the four-punctured sphere. Our methods have the potential to discriminate between potential diseases associated to the sequences.
About Klee Irwin

Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness.
As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics.
Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.
Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.
Group DNA Sequence character variety card seq transcription Factor
About Klee Irwin

Group DNA Sequence character variety card seq transcription Factor
Citation: Planat, M.; Amaral, M.M.;
Fang, F.; Chester, D.; Aschheim, R.;
Irwin, K. DNA Sequence and
Structure under the Prism of Group
Theory and Algebraic Surfaces. Int. J.
Mol. Sci. 2022, 23, 13290.
https://doi.org/
10.3390/ijms232113290
Academic Editors: Giuseppe Zanotti
and Zhongzhou Chen
Received: 16 September 2022
Accepted: 26 October 2022
Published: 31 October 2022
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Copyright: © 2022 by the authors.
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Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
 
International Journal of
 
Molecular Sciences
Article
DNA Sequence and Structure under the Prism of Group Theory
and Algebraic Surfaces
Michel Planat 1,*,†
, Marcelo M. Amaral 2,†
, Fang Fang 2
, David Chester 2
, Raymond Aschheim 2
and Klee Irwin 2
1
Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne-Franche-Comté, F-25044 Besançon, France
2 Quantum Gravity Research, Los Angeles, CA 90290, USA
* Correspondence: michel.planat@femto-st.fr
†
These authors contributed equally to this work.
Abstract: Taking a DNA sequence, a word with letters/bases A, T, G and C, as the relation between
the generators of an infinite group π, one can discriminate between two important families: (i)
the cardinality structure for conjugacy classes of subgroups of π is that of a free group on one to
four bases, and the DNA word, viewed as a substitution sequence, is aperiodic; (ii) the cardinality
structure for conjugacy classes of subgroups of π is not that of a free group, the sequence is generally
not aperiodic and topological properties of π have to be determined differently. The two cases rely on
DNA conformations such as A-DNA, B-DNA, Z-DNA, G-quadruplexes, etc. We found a few salient
results: Z-DNA, when involved in transcription, replication and regulation in a healthy situation,
implies (i). The sequence of telomeric repeats comprising three distinct bases most of the time satisfies
(i). For two-base sequences in the free case (i) or non-free case (ii), the topology of π may be found in
terms of the SL(2,C) character variety of π and the attached algebraic surfaces. The linking of two
unknotted curves—the Hopf link—may occur in the topology of π in cases of biological importance,
in telomeres, G-quadruplexes, hairpins and junctions, a feature that we already found in the context
of models of topological quantum computing. For three- and four-base sequences, other knotting
configurations are noticed and a building block of the topology is the four-punctured sphere. Our
methods have the potential to discriminate between potential diseases associated to the sequences.
Keywords: DNA conformations; transcription factors; telomeres; infinite groups; free groups;
algebraic surfaces; aperiodicity; character varieties
1. Introduction
Group theory and algebraic geometry serve the decipherment of ‘the book of life’ [1],
a book made of a language employing four letters/nucleotides: A (adenine), T (thymine),
G (guanine) and C (cytosine), as described in this work. There are finite groups, groups
made of a finite number of generators and a finite number of elements that may be used to
map the codons to amino acids, as carried out in our papers [2,3]. Such an approach toward
the genetic code is made possible by identifying the irreducible characters of the group to
the amino acids. The multiplets of codons attached to a selected amino acid correspond
to the irreducible characters having the corresponding dimension of the representation
(Table 3 in [2], Table 4 in [3]). A virtue of the approach is that the used irreducible characters
are also seen as quantum states carrying complete quantum information.
For modeling DNA in its various conformations taken in transcription factors, telom-
eres and other building blocks of molecular biology, we need infinite groups defined from
a motif. A sequence of the DNA nucleotides serves as the generator of the group [4]. In this
context, it has been found that a group that is not free is often the witness of a potential
disease. We coined the term ‘syntactical freedom’ for recognizing this property, with inspi-
ration from an earlier work [5]. We also showed that such free groups have the distinctive
Int. J. Mol. Sci. 2022, 23, 13290. https://doi.org/10.3390/ijms232113290
https://www.mdpi.com/journal/ijms
Int. J. Mol. Sci. 2022, 23, 13290
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property of generating an aperiodic substitution rule, providing a connection between
(group) syntactical freedom and irrational numbers (Section 4 in [4]). For an infinite group,
the representation cannot be based on characters but on the so-called character variety.
This topic leads to a relationship between DNA, algebraic topology and algebraic geometry.
Tools already proposed for topological quantum computing [6] are also used in the context
of DNA conformations.
In Section 2, we briefly account for the many types of topologies that DNA can show,
in terms of double strands or more strands. Then, we recall the mathematical concepts
employed in our paper with some redundancy with earlier work [4,6].
In Section 3, we explain the concept of an SL(2,C) character variety associated to
an infinite group with two or three generators. The former case corresponds to DNA
motifs having only two distinct nucleotides. In such a case, the variety often contains
the Cayley cubic associated to the Hopf link, the non disjoint union of two circles in the
three-dimensional space. In the later case, the variety contains the Fricke–Klein seventh
variable polynomial that is characteristic of the topology of the three-dimensional sphere
with four points removed.
In Section 4, we apply these mathematical methods to transcription factors, telomeric
sequences and a specific DNA decamer sequence, where almost all of its conformations
have been crystallized.
2. Materials and Methods
Mathematical calculations performed in this paper are on the software Magma [7] (for
groups) or on Sage software [8] (for character varieties).
2.1. DNA Conformations
DNA is a long polymer made from a chain of the nucleotides A, T, C or G. DNA exists
in many possible conformations, which include a double-stranded helix of A-DNA, B-DNA
and Z-DNA, although only B-DNA and Z-DNA have been directly observed in functional
organisms [9,10]. The B-DNA form is most common under the conditions found in cells,
but Z-DNA is often preferred when DNA binds to a protein. A view of a double helix
in the A-, B- and Z-DNA forms is given in Figure 1 Other DNA conformations also exist,
such as a single-stranded hairpin used mostly in macromolecular synthesis and repair,
a triple-stranded H-DNA found in peptides, a G-quadruplex structure found in telomeres
and a Holliday junction.
Figure 1. From left to right, the structures of A-, B- and Z-DNA. The view of the double helix from
above (or below) shows distinct symmetries, 11-fold for the A-DNA, 10-fold for the B-DNA and
6-fold for the Z-DNA [9,10].
Int. J. Mol. Sci. 2022, 23, 13290
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2.2. Finitely Generated Groups, Free Groups and Their Conjugacy Classes, and Aperiodicity
of Sequences
The free group Fr on r generators (of rank r) consists of all distinct words that can be
built from r letters where two words are different unless their equality follows from the
group axioms. The number of conjugacy classes of Fr of a given index d is known and is a
good signature of the isomorphism, or the closeness, of a group π to Fr. In the following,
the cardinality structure of conjugacy classes of index d in Fr is called the cardinality
sequence (card seq) of Fr, and we need the cases from r = 1 to 3 to correspond to the number
of distinct bases in a DNA sequence. The card seq of Fr is in Table 1 for the three sequences
of interest in the context of DNA [11].
Table 1. Number of conjugacy classes of subgroups of index d in free group of rank r = 1 to 3 [11].
The last column is the index of the sequence in the on-line encyclopedia of integer sequences [12].
r
Card Seq
Sequence Code
1
[1, 1, 1, 1, 1, 1, 1, 1, 1, · · · ]
A000012
2
[1, 3, 7, 26, 97, 624, 4163, 34470, 314493, · · · ]
A057005
3
[1, 7, 41, 604, 13753, 504243, 24824785, 1598346352, · · · ]
A057006
Next, given a finitely generated group f p with a relation (rel) given by the sequence
motif, we are interested in the card seq of its conjugacy classes. Often, the DNA motif in
the sequence under investigation is close to that of a free group Fr, with r + 1 being the
number of distinct bases involved in the motif. However, the finitely generated group fp =
〈x1, x2|rel(x1, x2)〉, or fp = 〈x1, x2, x3|rel(x1, x2, x3)〉 or fp = 〈x1, x2, x3, x4|rel(x1, x2, x3, x4)〉
(where the xi are taken in the four bases A, T, G and C, and rel is the motif), may not be the
free group F1 = 〈x1, x2|x1x2〉, or F2 = 〈x1, x2, x3|x1x2x3〉 or F3 = 〈x1, x2, x3, x4|x1x2x3x4〉.
The closeness of fp to Fr can be checked by its signature in the finite range of indices of the
card seq.
2.2.1. Groups fp Close to Free Groups and Aperiodicity of Sequences
According to reference [5], aperiodicity correlates to the syntactical freedom of ordering
rules. This statement was checked in the realm of transcription factors (Section 4 in [4]).
Let us introduce the concept of a general substitution rule in the context of free groups.
A general substitution rule ρ on a finite alphabet Ar on r letters is an endomorphism of the
corresponding free group Fr (Definition 4.1 in [13]). The endomorphism property means
the two relations ρ(uv) = ρ(u)ρ(v) and ρ(u−1) = ρ−1(u), for any u, v ∈ Fr.
A special role is played by the subgroup Aut(Fr) of automorphisms of Fr. We introduce
the map α : Fr → Zr from Fr to the Abelian group Zr in order to investigate the substitution
rule ρ with the tools of matrix algebra.
The map α induces a homomorphism M : End(Fr) → Mat(r,Z). Under M, Aut(Fr)
maps to the general linear group of matrices with integer entries GL(r,Z). Given ρ, there is
a unique mapping M(ρ) that makes the map diagram commutative [13] (p. 68). The substi-
tution matrix M(ρ) of ρ may be specified by its elements at row i and column j as follows:
(M(ρ))i,j = card(ρai (aj)).
This approach was applied to binding motifs of transcription factors [4]. The bind-
ing motif rel in the finitely presented group f p = 〈A, T, G, C|rel(A,T,G,C)〉 is split into
appropriate segments so that rel = relArelTrelGrelC with the substitution rules A→ relA,
T → relT , G → relG, C → relC.
We are interested in the sequence of finitely generated groups
f (l)
p = 〈A, T, G, C|rel(rel(rel · · · (A, T, G, C)))〉 (with rel applied l times)
Int. J. Mol. Sci. 2022, 23, 13290
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whose card seq is the same at each step l and equal to the card seq of the free group Fr (in
the finite range of indices that it is possible to check with the computer).
Under these conditions, (group) syntactical freedom correlates to the aperiodicity
of sequences.
2.2.2. Aperiodicity of Substitutions
There is no definitive classification of aperiodic order, the intermediate between
crystalline order and strong disorder, but in the context of substitution rules, some criteria
can be found. First, we need a few definitions.
A non-negative matrix M ∈ Mat(d,R) is one whose entries are non-negative numbers.
A positive matrix M (denoted M > 0) has at least one positive entry. A strictly positive
matrix (denoted M >> 0) has all positive entries. An irreducible matrix M = (Mij)1≤i,j≤d
is one for which there exists a non-negative integer k with (Mk)ij > 0 for each pair (i, j).
A primitive matrix M is one such that Mk is a strictly positive matrix for some k.
A Perron–Frobenius (PF for short) eigenvector v of an irreducible non-negative matrix
is the only one whose entries are positive: v > 0. The corresponding eigenvalue is called
the PF eigenvalue.
We will use the following criterion (Corollary 4.3 in [13]). A primitive substitution
rule ρ of substitution matrix M(ρ) with an irrational PF-eigenvalue is aperiodic.
A well-studied primitive substitution rule is the Fibonacci rule ρ = ρF : a→ ab, b→
a of substitution matrix MF =
(
1 1
1 0
)
and PF-eigenvalue equal to the golden ratio
λPF = φ = (
√
5 + 1)/2 (Example 4.6 in [13]). As expected, the irrationality of λ cor-
responds to the aperiodicity of the Fibonacci sequence.
The sequence of Fibonacci words is as follows:
a, b, ab, aba, abaab, abaababa, abaababaabaab, · · ·
The words have lengths equal to the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, · · ·
All finitely generated groups f (l)
p whose relations rel(a, b) = ab, aba, abaab, abaababa, · · ·
have a card seq whose elements are 1s, as for the card seq of the free group F1. The Fibonacci
sequence is our first example where group syntactical freedom correlates to aperiodicity.
2.2.3. A Four-Letter Sequence for the Transcription Factor of the Fos Gene
Let us now apply the method to a transcription factor of importance. The transcription
factor of gene Fos has selected motif rel = TGAGTCA [14]. For this case, the four-letter
generated group has a card seq similar to the free group F3 given in Table 1.
We split rel into four segments so that rel = relArelTrelGrelC with the substitution
maps A → relA = T, T → relT = G, G → relG = AGTC, C → relC = A to produce the
substitution sequence
A, T, G, C, ATGC, TGAGTCA, GAGTCTAGTCGAT · · ·
The substitution matrix for this sequence is M =

0 0 1 1
1 0 1 0
0 1 1 0
0 0 1 0
.
It is a primi-
tive matrix (M4 >> 0) whose eigenvalues follow from the vanishing of the polynomial
λ4 − λ3 − λ2 − λ− 1. There are two real eigenvalues λ1 ≈ 1.92756 and λ2 ≈ −0.77480, as
well as two complex conjugate eigenvalues λ3,4 ≈ −0.07637± 0.81470i. The PF-eigenvalue
is λPF = λ1, with an eigenvector of (positive) entries ≈ (1, 0.37298, 0.40211, 0.20861)T . It
follows that the selected sequence for the Fos gene is aperiodic.
All of the finitely generated groups f (l)
p whose relations are
rel(A, C, G, T) = ATGC, TGAGTCA, GAGTCTAGTCGAT, · · · ,
Int. J. Mol. Sci. 2022, 23, 13290
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have a card seq whose elements are
1, 7, 41, 604, 13753, 504243, · · · ,
which is the card seq of the free group F3. For the Fos transcription factor, group syntactical
freedom correlates to aperiodicity as expected.
Further examples are obtained in the context of DNA sequences for transcription
factors (Section 4 in [4]) and below, in relation to DNA conformations and telomeres.
3. Discussion
In the following, we make use of SL(2,C) representations of the infinite groups π
arising from specific DNA sequences. The character variety has many interpretations
in mathematics and physics. For instance, in mathematics, the variety is the space of
representations of hyperbolic structures of three-manifolds M with fundamental group
π(M), and the variety of the characters of SL(2,C) representations of π(M) is reflected
in the algebraic geometry of the character variety [15,16]. In physics, the group SL(2,C)
expresses the symmetries of fundamental physical laws. It is also known as the Lorentz
group; more precisely, the double cover of the restricted Lorenz group is SL(2,C), which is
the spin group.
Figure 2. (Left): the Hopf link. (Right): the link L = A ∪ B is attached to the plane R2 in the
half-space R3+ . It is not splittable. This can be proved by checking that the fundamental group
π = π2(L) is not free [17] and p. 90 in [18]. One gets π2 = 〈x, y, z|(x, (y, z)) = z〉, where (.,.)
means the group theoretical commutator. The cardinality sequence of cc of subgroups of π2 is
[1, 3, 10, 51, 164, 1365, 9422, 81594, 721305, · · · ] (Figure 3 in [4]).
3.1. SL(2,C) Character Varieties and Algebraic Surfaces
Recently, we found that the representation theory of finite groups with their character
table allows us to derive an approach of the genetic code [3].
For infinite groups π such as those defined by DNA sequences, it is useful to describe
the representations of π in the Lorentz group SL(2,C), the group of (2× 2) matrices with
complex entries and determinant 1. Such a group expresses the fundamental symmetry of
all known physical laws, apart from gravitation.
Representations of π in SL(2,C) are homomorphisms ρ : π → SL(2,C) with character
κρ(g) = tr(ρ(g)), g ∈ π. The set of characters allows us to define an algebraic set by taking
the quotient of the set of representations ρ by the group SL2(C), which acts by conjugation
on representations [15,19].
For two-generator groups, the character variety may be decomposed into the product
of surfaces, which reveals the topology of M. We recently found a connection between some
groups whose topology is based on the Hopf link and a model of topological quantum
computing [6]. The Hopf link underlies many DNA sequences whose group structure is
(or is not) that of the free group F1. The classification of the involved algebraic surfaces in
Int. J. Mol. Sci. 2022, 23, 13290
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the variety is performed using specific tools available in Magma [7]; see (Section 2.1 in [6])
for details.
For three-generator groups, we find that the Fricke–Klein quartic is part of the charac-
ter variety.
3.2. The Hopf link
Taking the linking of two unknotted curves as in Figure 2 (Left), the obtained link is
called the Hopf link H = L2a1, whose knot group is defined as the fundamental group of
the knot complement in the three-sphere S3
Π1(S3 \ L2a1) = 〈a, b|[a, b]〉 = Z2,
(1)
where [a, b] = abAB (with A = a−1, B = b−1) is the group theoretical commutator.
There are interesting properties of the knot group Π1 of the Hopf link.
Figure 3. Left: a three-dimensional picture of the SL2(C) character variety ΣH for the Hopf link
complement H. Right: a modified character variety of defining equation fH̃(x, y, z) with similar sin-
gularities.
First, the number of coverings of degree d of Π1 (which is also the number of conjugacy
classes of index d) is precisely the sum of divisor function σ(d) [20].
Second, an invariance of Π1 under a repetitive action of the golden ratio substitution
(the Fibonacci map) ρ : a → ab, b → a or under the silver ratio substitution ρ : a → aba,
b→ a exists. The terms golden and silver refer to the Perron–Frobenius eigenvalue of the
substitution matrix (Examples 4.5 and 4.6 in [13]). Such an observation links the Hopf link,
the group Π1 of the 2-torus and aperiodic substitutions.
Using Sage software [8] developed from Ref. [19], the SL2(C) character variety is the
polynomial corresponding to the so-called Cayley cubic
fH(x, y, z) = xyz− x2 − y2 − z2 + 4.
(2)
As expected, the three-dimensional surface Σ : fH(x, y, z) = 0 is the trace of the
commutator and is known to correspond to the reducible representations (Theorem 3.4.1
in [21]). A picture is given in Figure 3 (left).
In the perspective of algebraic geometry, we classify the homogenization of equation
fH as a rational surface of degree 3 del Pezzo type. It displays four simple singularities.
3.3. Beyond the Hopf Link
As shown in [6], the Hopf link is the irreducible component of many character va-
rieties relevant to a model of topological quantum computing. In the context of DNA
groups investigated in the next section, we also find another surface with similar simple
singularities as shown in Figure 3 (right). The defining polynomial is
fH̃(x, y, z) = z
4 − 2xyz (+z3) + 2x2 + 2y2 − 3z2(−4z)− 4.
(3)
Int. J. Mol. Sci. 2022, 23, 13290
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The homogenization of equation fH̃(x, y, z) allows us to classify it as a conic bundle in
the family of K3 surfaces.
For the DNA sequence, whose group πH̃ contains the component fH̃(x, y, z), we refer
to the third subsection of the Results section below and the first table in this subsection.
The relevant triplet nnn=CGG of the dodecameric sequence d(CCnnnN6N7N8GG) leads to
a DNA conformation with the label 1ZEY in the PDB bank.
The DNA dodecamer sequence d(CCCCCGCGGGGG) is also found in the PDB bank
with label 2D47, corresponding to a complete turn of A-DNA. The character variety for the
group defined by this sequence contains the polynomial fH and a polynomial similar to fH̃
without the third-order term z3 and the first-order term −4z, but in the same family.
3.4. The Fricke–Klein Seventh Variable Polynomial
The Cayley cubic is a subset of the character variety for the four-punctured three-
dimensional sphere S24 (the sphere minus four points). Its fundamental group Π1 is isomor-
phic to the free group F3 of rank 3, Π1(S24) = 〈α, β, γ, δ|αβγδ〉, where the four homotopy
classes α, β, γ, δ correspond to loops around the punctures.
The SL(2,C) character variety for Π1(S24) satisfies a quartic equation in terms of the
Fricke–Klein seventh variable polynomial [21] (p. 65) and [22]:
f (x, θ) = xyz+ x2 + y2 + z2 − θ1x− θ2x− θ3z+ θ4,
(4)
with θ1 = uv+ wk, θ2 = uw+ vk, θ3 = uk+ vw, θk = uvwk+ u2 + v2 + w2 − 4.
4. Results
In this section, we apply the SL(2,C) representation theory to specific non-canonical
DNA sequences having regulatory functions in gene expression (the transcription factors),
replication (the telomeres) and DNA conformations.
4.1. Group Structure and Topology of Transcription Factors
In a transcription factor, a motif-specific DNA binding factor controls the rate of the
transcription of a gene from DNA to messenger RNA by binding a protein to the DNA
motif. In reference [4], we found a correlation between motifs whose subgroup structure
is that of a free group and the lack of a potential disease while the gene is activated in
transcription, the property of ‘syntactical freedom’.
In Table 2, this idea is illustrated by restricting to a few transcription factors whose
motif comprises two bases. The card seq of the motif is either the free group F1, close to
F1 or away from a free group when the card seq is that of the modular group H3, of the
Baumslag–Solitar group BS(−1, 1) or that of groups π1 and π′1. Compared to the results
provided in [4], there is the additional fourth column that signals when the Groebner base
for the ideal ring of the SL(2,C) character variety contains the Cayley cubic, the unique
component in the case of the Hopf link [6], a degree 3 del Pezzo surface (denoted HL), or
not. An additional fifth column is filled to check the presence of a surface of type K3. Only
the last row of the table for the transcription factor of gene EHF does not show this property.
Int. J. Mol. Sci. 2022, 23, 13290
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Table 2. Group structure of motifs for a few two-letter transcription factors. The card seq for the mod-
ular group H3 is [1, 1, 2, 3, 2, 8, 7, 10, 18, 28, · · · ]. The Baumslag–Solitar group BS(−1, 1) is the funda-
mental group of the Klein bottle. The card seq for BS(−1, 1) is [1, 3, 2, 5, 2, 7, 2, 8, 3, 8, 2, 13, 2, 9, 4, · · · ].
The card seq
for π1
is
[1, 4, 1, 2, 4, 2, 1, 7, 2, 2, 4, 2, 2, 8, 1, 2, 7, 2, 3, · · · ];
for π′1,
it
is
[1, 1, 1, 2, 1, 3, 3, 1, 2, 2, 1, 1, 9, 2, 14, 2, 1, · · · ].
The symbol HL means that the Cayley cubic is
part of the Groebner base for the ideal ring of the corresponding SL(2,C) character variety.
For three-letter transcription factors, the ideal ring of the corresponding SL(2,C) character variety
contains the Fricke–Klein seventh variable polynomial 4, which is a feature of the four-punctured
sphere topology. The group structure of three-letter transcription factors not leading to free groups is
shown in (Table 5 in [4]).
Gene
Motif
Card Seq
Link
Type
Literature
DBX
TTTATTA
F1
HL
K3
[23], MA0174.1
SPT15
TATATATAT
.
.
.
., MA0386.1
PHOX2A
TAATTTAATTA
≈F1
.
.
., MA0713.1
FOXA
TGTTTGTTT
F1
.
.
[24,25]
FOXG
TTTGTTTTT
.
.
.
[24]
NKX6-2
TAATTAA
H3
no
K3
[23], [MA0675.1, MA0675.2]
FOXG
TGTTTG
BS(−1, 1)
no
K3
[23,26], MA1865.1
HoxA1, HoxA2
TAATTA
π1
no
K3
[23], [MA1495.1, MA0900.1]
POU6F1, Vax
., [MAO628.1, MA0722.1]
RUNX1
TGTGGT
.
no
.
., MA0511.1
RUNX1
TGTGGTT
π′1
no
K3
[23], MA0002.2
EHF
CCTTCCTC
.
HL
., MA0598.1
The Character Variety for the Transcription Factor of the DBX Gene
We explicitly show the SL(2,C) character variety for the transcription factor of the
DBX gene.
fDBX(x, y, z) = fH(x, y, z)(yz2 − y2 − xz− y+ 2)(xy2 − z3 − yz− x + 3z)
(5)
(y3 − z2 − 3y+ 2)(y2z− xy− yz+ x− z)(z4 − x2y+ xz− 4z2 + y+ 2)
The factors in (5) are three degree 3 del Pezzo surfaces (including the Cayley cubic fH),
two rational ruled surfaces and a K3 surface birationally equivalent to the projective plane,
respectively. The latter factor also belongs to the character variety of group Π1(S4 \ Ẽ6),
where S4 is the four-sphere and Ẽ6 is the singular fiber IV∗ in Kodaira’s classification of
minimal elliptic surfaces (Figure 4b in [6]).
It is important to mention that, for three-letter transcription factors, the ideal ring of
the corresponding SL(2,C) character variety contains the Fricke–Klein seventh variable
polynomial (4), which is a feature of the four-punctured sphere topology.
Table 3 provides a short account of the function or potential dysfunction of the genes
under consideration. As mentioned before, most of the time, such a dysfunction is corre-
lated to a card seq away from that of the free group F1.
In view of our results, it is interesting to correlate the presence of the Hopf link HL in
the character variety with the possible remodeling of B-DNA into Z-DNA or another DNA
conformation. To our knowledge, general information about this subject is still lacking.
From a biological point of view, it is known that some of the Z-DNA-forming conditions that
are relevant in vivo are the presence of DNA supercoiling, Z-DNA-binding proteins [27]
and base modifications. When transcription occurs, the movement of RNA polymerase II
along the DNA strand generates positive supercoiling in front of, and negative supercoiling
behind, the polymerase [28].
Int. J. Mol. Sci. 2022, 23, 13290
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Table 3. A short account of the function or dysfunction (through mutations or isoforms) of genes
associated with transcription factors and sections in Table 2.
Gene
Type
Function
Dysfunction
DBX
drosophila segmentation
SPT15
TATA-box
gene expression, regulation
binding protein
in Saccharomyces cerevisiae
PHOX2A
homeodomain
differentiation, maintenance
fibrosis
of noradrenergic phenotype
of extraocular muscles
FOX proteins
forkhead box
growth, differentiation,
FOXA2
.
insulin secretion
diabete
longevity
NKX6-2
homeobox
central nervous system, pancreas
spastic ataxia
FOXG
forkhead box
notochord (neural tube)
chordoma
HoxA1
homeobox
embryonic devt of face and hear
autism
HoxA2
.
.
cleft palate
Pou6F1
.
neuroendocrine system
clear cell adenocarcinoma
Vax
.
forebrain development
craniofacial malform.
RunX1
Runt-related
cell differentiation, pain neurons
myeloid leukemia
EHF
homeobox
epithelial expression
carcinogenesis, asthma
Perhaps the lack of HL in the character variety for transcription factors of genes in
Table 2 means that the Z-DNA-forming condition is not realized.
4.2. Group Structure and Topology of DNA Telomeric Sequences
Terminal structures of chromosomes are made of short highly repetitive G-rich se-
quences with proteins known as telomeres. They have a protective role against the shorten-
ing of chromosomes through successive divisions. Most organisms use a telomere-specific
DNA polymerase called telomerase that extends the 3’ end of the G-rich strand of the
telomere [29]. Telomere shortening is associated with aging, mortality and aging-related
diseases such as cancer.
A list of results obtained by using our group theoretical approach is in Table 4. For
two-letter telomere sequences, the SL(2,C) character variety contains the Cayley cubic,
the characteristic of the Hopf link HL, only in the first row. In addition to the Cayley
cubic, one finds surfaces of a general type. In the next two rows, the Cayley cubic is not
found. There are degree 3 del Pezzo surfaces in the factors of the character variety but not
general surfaces.
As for the Hopf link, the sequence is found to be aperiodic with the Perron–Frobenius
eigenvalue λPF equal to the golden ratio. For three-letter telomere sequences, the card seq
is that of the free group of F2, except for the last row, where the identified group is π2; see
Figure 2 (right) for the definition of such a group. In the former seven cases, the DNA
topology is known to be a G-quadruplex structure [30–36]. We could identify an aperiodic
structure of the telomere sequence with the Perron–Frobenius eigenvalue λPF as shown in
column 5. In the latter case, the topology is of the basket type [36] and no aperiodicity of
the telomere sequence could be found.
Figure 4, taken from the protein data bank (PDB 2HY9), illustrates the G-quadruplex
structure of the telomere sequence in vertebrates.
Int. J. Mol. Sci. 2022, 23, 13290
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Table 4. Group analysis of the telomere sequence found in some eukaryotes. The first column is
for the telomere repeat, the second column is the organism under investigation, the third column is
for the PDB code, the fourth column is for the card seq of the group π or that of the corresponding
group that is identified, the fifth column is for the Perron–Frobenius eigenvalue when the sequence
is found to be aperiodic, the sixth column identifies the presence of the Hopf link (in two-base
sequences) or the DNA conformation (in three-base sequences) and the seventh column is a relevant
reference. The notation G-quadr is for the G-quadruplex; see Figure 4. The card seq for π′′1 is
[1, 3, 2, 16, 16, 69, 118, 719, 1877, 8949 · · · ]. The Hecke group H4 is defined in (Table 2 in [4]).
Seq
Organism
PDB
Card Seq
λPF
Link/DNA Conf
Ref
G4T4G4
Oxytricha
1D59
π′′1
(
√
5 + 1)/2
HL
[37]
TG4T
universal
244D_1
H4
.
no
[38]
T2G4
Tetrahymena
230D
H4
.
no
[29]
T2AG3
Vertebrates
2HY9
F2
2.5468
G-quadr.
[30]
TAG3
Giardia
2KOW
F2
2.2055
G-quadr
[31]
T2AG2
Bombys mori
unknown
F2
no
G-quadr
[32]
T4AG3
Green algae
unknown
F2
3.07959
unknown
[33]
G2T2AG
Human
unknown
F2
2.5468
G-quadr
[34]
TAG3T2AG3
Human
2HRI
F2
3.3923
G-quadr
[35]
G3T2AG3T2AG3T
Human
unknown
F2
4.3186
G-quadr
[36]
(GGGTTA)3G3T
Human
unknown
π2
no
basket
[36]
Figure 4. Human telomere DNA quadruplex structure in K+ solution hybrid-1 form, PDB 2HY9 [30].
4.3. Group Structure and Topology of the DNA Decamer Sequence d(CCnnnN6N7N8GG) [10]
A challenging question of structural biology is to determine if and how a DNA (or
RNA) sequence defines the three-dimensional conformation, as well as the secondary and
tertiary structure of proteins. In the previous two subsections, we tackled the problem
with regard to transcription factors and telomeric sequences, respectively. In the former
case, we restricted to the DNA part of the transcription since the DNA motif is almost
exactly known from X-ray techniques while the secondary structure of the binding protein
strongly depends on the model employed and the choice made to recognize the sections
of the secondary structures (e.g., alpha helices, beta sheets and coils) [39]. In the latter
case, in many organisms, nature invented telomerase for taking care of the replication
without damaging the sequences at the 3-ends too much, while keeping the catalyzing
action of DNA polymerase. Again, there is a loop complex in telomerase comprising
telomere-binding proteins, with secondary structures not being analyzed so far with our
group theoretical approach.
Int. J. Mol. Sci. 2022, 23, 13290
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In this section, we also study DNA conformations and their relationship to algebraic
topology in a specific DNA decamer sequence investigated in reference [10] by a standard
crystallization technique followed by X-ray diffraction discrimination. In the sequence
d(CCnnnN6N7N8GG), the factors N6, N7 and N8 are taken in the two nucleotides G and
C, and nnn is specified in order to maintain the self-complementarity of the sequence.
This inverse repeated motif is the minimum motif used to distinguish between the double-
strand forms of B- and A-DNA, while excluding the Z-DNA forms. A third conformation
is allowed and called the four-stranded Holliday junction J. We refer to (Table 1 in [10]) for
the main results.
On our side, the card seq of each sequence was determined and the SL(2,C) character
variety was obtained. Our results are summarized in the four Tables 5–8.
Table 5. Group analysis of the sequence d(CCnnnN6N7N8GG), where N6, N7 and N8 are taken in
the two nucleotides G and C and nnn is specified in order to maintain the self-complementarity
of the sequence [10]. The first column is for the selected triplet N6N7N8, the second column is for
the code in the protein data bank, the third column is for the DNA conformation when known (see
Table 1 in [10]), the fourth column is for the cardinality structure of subgroups of π and the fifth
column checks the occurrence of a surface corresponding to the Hopf link in the factorization of the
SL(2,C) of π. The symbols A, B and J are for A-DNA, B-DNA and a four-stranded Holliday junction;
lowercase is used when the conformation is not confirmed in [10].
Triplet
PDB
Conformation
Card Seq (π)
Knot
CCC
1ZF1
A
[1, 1, 1, 1, 7, 1, 1, 2, 9, 6, · · · ]
HL
CCC
1ZF2
J
idem
HL
CCG
1ZEX
A
idem
HL
CGG
1ZEY
A
[1, 1, 1, 2, 6, 3, 1, 4, 2, 6, · · · ]
HL like
CGC
none
unknown
[1, 1, 2, 1, 6, 3, 2, 1, 3, 6, · · · ]
no
GGG
1ZF9
A
[1, 1, 1, 1, 10, 25, 25, 9, 2, 1798, · · · ]
no
GCC
none
b/J
[1, 1, 1, 1, 6, 1, 2, 1, 1, 6, · · · ]
HL
GCG
none
unknown
[1, 1, 2, 2, 7, 5, 1, 4, 5, 9, · · · ]
no
GGC
none
B/a
[1, 1, 1, 1, 6, 11, 9, 5, 2, 208, · · · ]
no
(card seq of Hecke group H5)
Table 6. Group analysis of the sequence d(CCnnnN6N7N8GG), where N6, N7 and N8 are taken
in the two nucleotides A,T [10]. Groups π3 and π′3 are as in (Table 5 in [4]). The card seq for
π′3 is [1, 7, 50, 867, 15906, 570528, · · · ]; for π′′3 , it is [1, 7, 50, 739, 15234, 548439, · · · ]; for π
(4)
3
, it is
[1, 7, 59, 1258, 24787, · · · ]. Groups π′′3 and π′3 may be simplified to a group whose card seq is that of
π2, the fundamental group of the link L = A ∪ B described in Figure 3 (right).
Triplet
PDB
Conformation
π
TTA
1ZFH
B
π′′3 → π2
TAA
none
B
π′′3 → π2
AAT
none
b
π′′3 → π2
ATT
none
unknown
π′′3 → π2
AAA
none
b
π′3 → π2
TTT
none
unknown
π′3 → π2
ATA
none
unknown
π
(4)
3
TAT
none
unknown
π
(4)
3
In Table 5, N6, N7 and N8 are taken in the two nucleotides G and C, forming eight
triplets and the associated two-letter decamer sequences. Note that the triplet CCC pro-
duces two distinct DNA conformations A and J. The character variety of the Hopf link HL
(the Cayley cubic) is present in the factors of the ideal ring of the SL(2,C) character variety
in five cases over the nine possibilities, where one case (with triplet CGG and code 1ZEY in
the protein data bank) shows an algebraic surface similar to the Cayley cubic (with four
Int. J. Mol. Sci. 2022, 23, 13290
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simple singularities) as defined in Equation (3) and as shown in Figure 3 (right). We do not
observe a clear correlation between the type of DNA conformation and the underlying HL
topology, but the presence of HL in the variety seems to exclude the B-DNA conformation.
In addition, the character variety always contains a surface of type K3 in its factors.
Table 7. Group analysis of the sequence d(CCnnnN6N7N8GG) [10], where N6, N7 and N8 are taken
in the two nucleotides A,G (left) and A,C (right). Groups π3 and π′3 are as in (Table 5 in [4]). The card
seq for π(3)
3
is [1, 7, 41, 668, 14969, · · · ] and, for π(5)
3
, it is [1, 7, 41, 604, 28153, · · · ].
Triplet
PDB
Conformation
π
Triplet
PDB
Conformation
π
AGA
1ZEW
B
F3
ACA
none
unknown
π
(3)
3 → π2
AGG
none
unknown
π
(5)
3
ACC
none
J
F3
GGA
1ZFA
A
F3
CCA
none
unknown
F3
AAG
none
unknown
F3
AAC
1ZF0
B
π3”→ π2
TGT
none
unknown
F3
TCT
none
b
π
(3)
3 → π2
TGG
1ZF6
A
F3
TCC
none
unknown
F3
GGT
1ZF8
A
F3
CCT
none
b
F3
TTG
none
unknown
π′3
TTC
none
B
π3”→ π2
Table 8. Group analysis of the sequence d(CCnnnN6N7N8GG) [10], where N6, N7 and N8 are
taken in the three nucleotides A ,G, C (left) and A, T, C (right). The card seq for π(6)
3
is
[1, 7, 59, 874, 20371, 748320 · · · ]
Triplet
PDB
Conformation
π
Triplet
PDB
Conformation
π
AGC
1ZFM
B
F3
ATC
1ZFC/1ZF3
B/J
π
(6)
3
ACG
none
unknown
F3
ACT
none
B
π
(3)
3 → π2
GCA
1ZFE
B
F3
TCA
none
unknown
π
(3)
3 → π2
GAC
1ZF7
B
F3
TAC
none
unknown
π
(6)
3
CAG
none
unknown
F3
CAT
none
unknown
F3
CGA
none
unknown
F3
CTA
none
unknown
F3
In Table 6, N6, N7 and N8 are taken in the two nucleotides A and T, forming eight
triplets. The DNA conformation (when known) is of type B. The card seq that we obtain
is groups π′3, π
′′
3 or π
(4)
3 as described in the caption of Table 5. In six cases over the eight
possibilities, the groups encapsulate the topology of the rank 2 group π2, whose associated
link is shown in Figure 2 (right). As already mentioned, the SL(2,C) character variety
contains the Fricke–Klein seventh variable polynomial 4.
In Table 7, N6, N7 and N8 are taken in the two nucleotides A, G (left part) and A,
C (right part). This time, either the card seq of the group π is that of the free group F3,
of rank 3 (10 cases over the 16 possibilities), or not. In the latter case, the group encapsulates
the topology of π2 only at the right side of the table. Similar conclusions hold in Table 8
when N6, N7 and N8 are taken in the two nucleotides A, G, C (left part) and A, T, C (right
part).
To summarize this section, no clear correlation is observed between the DNA con-
formations of the considered decamer and our group analysis. Longer sequences may be
needed to obtain such a correlation. For instance, the two-letter DNA dodecamer sequence
d(CCCCCGCGGGGG) (PDB 2D47) corresponding to a complete turn of A-DNA—see
Figure 5 (right)—features the polynomial fH (for HL) and a fourth-order polynomial simi-
lar to fH̃ with four simple singularities, as announced at the end of Section 3.
Int. J. Mol. Sci. 2022, 23, 13290
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Figure 5. (Left) The four-strand Holliday junction J: PDB 1ZF2, (Right) A complete turn of A-
DNA: PDB 2D47. It is associated to DNA dodecamer sequence d(CCCCCGCGGGGG) with SL(2,C)
containing the factor fH = xyz− x2 − y2 − z2 + 4 (the Cayley cubic) and the factor fH̃ = z4 − 2xyz+
2x2 + 2y2 − 3z2 − 4.
5. Conclusions
In the present paper, following earlier work about the genetic code [2,3] and about
the role of transcription factors in genetics [4], we made use of group theory applied to
appropriate DNA motifs and we computed the corresponding variety of SL(2,C) repre-
sentations. The DNA motifs under consideration may be canonical structures, such as
(double-stranded) B-DNA, or non-canonical DNA structures [40], such as (single-stranded)
telomeres, (double-stranded) A-DNA, Z-DNA or cruciforms, (triple-stranded) H-DNA,
(four-stranded) i-motifs or G-quadruplexes, etc. One objective of the approach is to establish
a correspondence between the algebraic geometry and topology of the SL(2,C) character
variety and the types of canonical or non-canonical DNA-forms. For example, for two-letter
transcription factors, one can correlate the presence of the Cayley cubic and/or a K3 surface
in the variety with DNA supercoiling in the remodeling of B-DNA to Z-DNA. For three- or
four-letter DNA structures, our work needs to be developed in order to put the features
of the variety and potential diseases in correspondence. In a separate work devoted to
topological quantum computing, the topology of the four-punctured sphere and the re-
lated Fricke surfaces (generalizing the Cayley cubic) are relevant [41]. In addition, Fricke
surfaces may be put in parallel with differential equations of the Painlevé VI type. It will
be important to compare these models with other qualitative models based on non-linear
differential equations [42,43]. In the near future, we intend to apply these mathematical
tools to the context of DNA structures.
Author Contributions: Conceptualization, M.P., F.F. and K.I.; methodology, M.P., D.C. and R.A.; soft-
ware, M.P.; validation, R.A., F.F., D.C. and M.M.A.; formal analysis, M.P. and M.M.A.; investigation,
M.P., D.C., 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, USA.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The datasets used and/or analyzed during the current study are
available from the corresponding author on reasonable request.
Conflicts of Interest: The authors declare no conflict of interest.
Int. J. Mol. Sci. 2022, 23, 13290
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