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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 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons 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 2 of 16 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 3 of 16 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 4 of 16 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 5 of 16 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 6 of 16 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 7 of 16 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 8 of 16 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 9 of 16 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 10 of 16 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 11 of 16 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 12 of 16 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 13 of 16 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 14 of 16 References 1. 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