Fricke Topological Qubits

Planat, Michel; Chester, David; Amaral, Marcelo M.; Irwin, Klee

doi:10.3390/quantum4040037

Cited by 8 publications

(14 citation statements)

References 34 publications

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“…The surface κ 3 (x, y, z) lies within the character variety for the fundamental group of the link L6a1 [27]. We show below that this surface also lies in the generic Groebner basis obtained for 4-base sequences; see Section 4 below.…”

Section: Algebraic Geometry and Topology Of Dna/rna Sequences 231 Two...mentioning

confidence: 68%

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Algebraic Morphology of DNA–RNA Transcription and Regulation

2023

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Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. To discover the symmetries and invariants of the transcription and regulation at the scale of the genome, in this paper, we introduce tools of infinite group theory and of algebraic geometry to describe both TFs and miRNAs. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group π, we compute the SL(2,C) character variety, where SL(2,C) is simultaneously a ‘space-time’ (a Lorentz group) and a ‘quantum’ (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when π looks similar to a free group Fr (r=1 to 3) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non-free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis G of the variety. A surface with simple singularities (such as the well known Cayley cubic) within G is a signature of a potential disease even when π looks similar to a free group Fr in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in {A,T/U,G,C}. We produce a few tables of human TFs and miRNAs showing that a disease may occur when either π is away from a free group or G contains surfaces with isolated singularities.

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Section: Algebraic Geometry and Topology Of Dna/rna Sequences 231 Two...mentioning

confidence: 68%

“…where κ −2 (x, y, z), κ −3 (x, y, z) are Fricke surfaces [27] and f 4. The subscript 3A 1 is for featuring the three singularities of type A 1 .…”

Section: The Character Varietymentioning

confidence: 99%

Algebraic Morphology of DNA–RNA Transcription and Regulation

2023

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“…For this group, the Groebner basis with parameters (a,b,c,d) = (0,0,0,0) is quite simple: G mir−503−5p (0,0,0,0) = S (4A1) (x,y,z), which is the already mentioned Cayley cubic. For (a,b,c,d) = (1,1,0,0), G mir−503−5p (1,1,0,0) = −3xyzκ 3 (x,y,z), where κ 3 (x,y,z) is the Fricke surface described by Planat et al 38 For (a,b,c,d) = (1,1,1,1), there are several more polynomials. One of which defines the Fricke surface xyz + x 2 + y 2 +z 2 − 2x − y -2 = 0.…”

Section: Mirna Hsa-mir-503mentioning

confidence: 99%

Group Theory of Messenger RNA Metabolism and Disease

Planat,

Amaral,

Chester

et al. 2024

Gene Expr

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Our recent work has focused on the application of infinite group theory and related algebraic geometric tools in the context of transcription factors and microRNAs. We were able to differentiate between "healthy" nucleotide sequences and disrupted sequences that may be associated with various diseases. In this paper, we extend our efforts to the study of messenger RNA (mRNA) metabolism, showcasing the power of our approach. We investigate (1) mRNA translation in prokaryotes and eukaryotes, (2) polyadenylation in eukaryotes, which is crucial for nuclear export, translation, stability, and splicing of mRNA, (3) microRNAs involved in RNA silencing and post-transcriptional regulation of gene expression, and (4) identification of disrupted sequences that could lead to potential illnesses. To achieve this, we used: (a) infinite (finitely generated) groups f p , with generators representing the r + 1 distinct nucleotides and a relation between them [e.g., the consensus sequence in the mRNA translation (i), the poly (A) tail in item (ii), and the microRNA seed in item (iii)]; (b) aperiodicity theory, which connects healthy groups f p to free groups F r of rank r and their profinite completion ˆr F , and (c) the representation theory of groups f p over the space-time-spin group SL 2 (C), highlighting the role of surfaces with isolated singularities in the character variety. Our approach could potentially contribute to the understanding of the molecular mechanisms underlying various diseases and help develop new diagnostic or therapeutic strategies.

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“…Free groups F r of rank r = 2 and 3 have been found to be important in our earlier work about topological quantum computing (TQC) [1] and biology at the DNA/RNA genomic scale [2]. In the first context, one motivation is that an elementary link, the Hopf link L = L2a1 made of two unknotted curves may serve as naive approach of TQC, corresponding to one qubit on either curves, as in [3].…”

Section: Introductionmentioning

confidence: 99%

“…In the first context, one motivation is that an elementary link, the Hopf link L = L2a1 made of two unknotted curves may serve as naive approach of TQC, corresponding to one qubit on either curves, as in [3]. Representation theory of the fundamental group π 1 (L) over the group SL 2 (C) puts the punctured torus T 1 1 whose group is π 1 (T 1 1 ) ∼ = F 2 into focus. In the second context, at least in a first approximation, a finitely generated group f p defined from an appropriate DNA/RNA sequence turns out to be close to F 2 (for a sequence built from two distinct nucleotides) or to F 3 (for a sequence built from three distinct nucleotides).…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Fricke-Painlevé VI surfaces

Planat,

Chester,

Irwin

2023

Preprint

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The symmetries of a Riemann surface $\Sigma \setminus \{a_i\}$ with $n$ punctures $a_i$ are encoded in its fundamental group $\pi_1(\Sigma)$. Further structure may be described through representations (homomorphisms) of $\pi_1$ over a Lie group $G$ as globalized by the character variety $\mathcal{C}=\mbox{Hom} (\pi_1,G)/G$. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the $4$-punctured Riemann sphere $\Sigma=S_2^{(4)}$ and the \lq space-time-spin' group $G=SL_2(\mathbb{C})$. In such a situation, $\mathcal{C}$ possesses remarkable properties (i) a representation is described by a $3$-dimensional cubic surface $V_{a,b,c,d}(x,y,z)$ with $3$ variables and $4$ parameters, (ii) the automorphisms of the surface satisfy the dynamical (non linear and transcendental) Painlev\'e VI equation (or $P_{VI}$), (iii) there exists a finite set of $1$ (\mbox{Cayley-Picard})+$3$ (\mbox{continuous platonic})+$45$ (\mbox{icosahedral}) solutions of $P_{VI}$. In this paper we feature on the parametric representation of some solutions of $P_{VI}$, (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups $f_p$ encountered in TQC or DNA/RNA sequences are proposed.

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Fricke Topological Qubits

Cited by 8 publications

References 34 publications

Algebraic Morphology of DNA–RNA Transcription and Regulation

Algebraic Morphology of DNA–RNA Transcription and Regulation

Group Theory of Messenger RNA Metabolism and Disease

Dynamics of Fricke-Painlevé VI surfaces

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