# Algebraic Morphology of DNA–RNA Transcription and Regulation

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

Again since the one face, constant in symmetry, appears sometimes fair and sometimes not, can we doubt that beauty is something more than symmetry, that symmetry itself owes its beauty to a remoter principle?[1] (Ennead I, Sixt Tractate, p66).

## 2. Theory

#### 2.1. Finitely Generated Groups, Free Groups and Their Conjugacy Classes

#### 2.2. The $SL(2,\mathbb{C})$ Character Variety of a Finitely Generated Group and a Groebner Basis

#### 2.3. Algebraic Geometry and Topology of DNA/RNA Sequences

#### 2.3.1. Two-Base Sequences

#### 2.3.2. Three-Base Sequences

#### 2.3.3. Four-Base Sequences

## 3. Discussion

## 4. Results

#### 4.1. Algebraic Morphology of the Transcription Factor Prdm1

#### 4.1.1. The Character Variety

#### 4.1.2. The Groebner Basis

#### 4.2. Algebraic Morphology of Homeodomains for Nanog and Xvent

**Table 2.**A few (three-base) transcription factors whose group structure is away from a free group or whose Groebner basis of the $SL(2,\mathbb{C})$ character variety contains a (possibly almost) singular surface. The symbol gene is for the identification of the transcription factor in the Jaspar database [34], motif is for the consensus sequence of the transcription factor, card seq is for the cardinality sequence of conjugacy classes of subgroups of the group whose motif is the generator, simple sing is for the identification of a surface with simple singularities within the Groebner basis and the last column is for a reference paper and the corresponding disease. The group ${F}_{2}$ is the free group of rank two. The card seq for ${\pi}_{2}$ is $[1,3,10,51,164,1230,7829,59835,491145\cdots ]$, close to the card seq of the group $\left(\right)$. The latter group is found as governing the structure of many transcription factors and is associated to the link found in ([13], Figure 2). The card seq for ${\pi}_{3}$ is $[7,14,89,264,1987,11086,93086\cdots ]$. The surface ${f}_{b}^{\left({A}_{1}\right)}(x,y,z)={x}^{2}+{y}^{2}-6{z}^{2}+4xyz$ (not defined in the text) is part of the character variety for the genes Pitx1, OTX1, etc.

Gene | Motif | Card Seq | Simple Sing | Ref & Disease |
---|---|---|---|---|

Prdm1 | ACTTTC | ${F}_{2}$ | ${S}_{1},{S}_{2}(x,y,z)$ | [34], MA0508.2 lupus, rheumatoid arthritis MA1549.1 lung adenocarcinoma MA0076.2 gastric cancer [MA0712.2, MA0883.1] medulloblastomas [37] drug sensitivity |

POU6F1 | TAATGAG | ${\pi}_{2}$ | no | |

ELK4 | CTTCCGG | . | no, Fricke | |

OTX2 | GGATTA | ${\pi}_{3}$ | no | |

N-box | TTCCGG | . | no, Fricke | |

Pitx1,OTX1,⋯ | TAATCC | . | ${f}_{H}^{\left(4\right)},{f}_{b}^{\left({A}_{1}\right)}(x,y,z)$ | [34], [MA0682.1,MA0711.1] autism, epilepsy, ⋯ |

Nanog | TAATGG | . | ${f}_{H}^{\left(4\right)},{f}_{a}^{\left({A}_{1}\right)}(x,y,z)$ | [35] cancer cells |

Xvent | CTAATT | F2 | ${f}_{4,\left\{\right\}}^{\left(2{A}_{1}\right)},{f}^{\left({A}_{2}\right)}(x,y,z)$ | [36] |

#### 4.3. Algebraic Morphology of microRNAs

**Table 3.**A few human (prefix ‘hsa’) microRNAs whose group structure is away from a free group or whose Groebner basis of the $SL(2,\mathbb{C})$ character variety contains a singular surface. The symbol mir is for the identification in the Mir database [43], seed is for the seed of the miRNA, card seq is for the cardinality sequence of conjugacy classes of subgroups of the group whose seed is the generator, sing is the identification of a singular surface within the Groebner basis and the last column is for a reference paper and the corresponding disease [40]. The card seq for ${\pi}_{1}$ and ${\pi}_{1}^{\prime}$ are given in ([4], Table 5). The card seq for ${\pi}_{2}^{\prime}$ is $[1,3,7,34,139,931,5208,43867\cdots ]$. For hsa-mir-124-1-3p, one encounters the Fricke surface ${f}_{2,\left\{\right\}}^{\left({A}_{1}\right)}=xyz+{x}^{2}+{y}^{2}+{z}^{2}-2y$ in the character variety.

mir | Seed | Card Seq | Simple Sing | Ref & Disease |
---|---|---|---|---|

hsa-mir-193b-5p | GGGGUU | ${\pi}_{1}$ | no | [40,43] lung cancer |

GGGGUUU | ${\pi}_{1}^{\prime}$ | no | ||

hsa-mir-155-3p | UCCUAC | ${F}_{2}$ | ${f}_{b}^{\left({A}_{1}\right)}(x,y,z)$ | [40,41,43] multiple sclerosis |

UCCUACA | ${\pi}_{2}$ | no | ||

hsa-mir-193a-5p | GGGUCUU | ${F}_{2}$ | ${f}_{b}^{\left({A}_{1}\right)}(x,y,z)$ | [40,43] breast cancer |

hsa-mir-223-5p | GUGUAUU | . | . | . |

hsa-mir-133-3p | UUGGUC | ${F}_{2}$ | ${f}_{b}^{\left(3{A}_{1}\right)}(x,y,z)$ | [40,43] atrial fibrillation |

UUGGUCC | ${\pi}_{2}^{\prime}$ | no | ||

hsa-mir-124-3p | AAGGCA | ${F}_{2}$ | ${f}_{b}^{\left(3{A}_{1}\right)},{f}_{2,\left\{\right\}}^{\left({A}_{1}\right)}$ | [43,44] |

AAGGCAC | . | no sing | Alzheimer’s disease |

**Table 4.**The opposite strand of the microRNA considered in Table 3. The seed sequence is made of 4 distinct bases and the corresponding card seq is the free group ${F}_{3}$ of rank 3. The Groebner basis contains 4 copies of the generic collection of surfaces ${\kappa}_{4}(x,y,z)$, ${f}^{\left(3{A}_{1}\right)}(x,y,z)$, ${\kappa}_{3}(x,y,z)$, etc., as shown in Figure 5, except for the -5p strand of mir-133, where there are only 3 copies of the generic surfaces.

mir | Seed | Card Seq | Sing | Ref & Disease |
---|---|---|---|---|

hsa-mir-193b-3p | ACUGGCC | ${F}_{3}$ | $4\times $ generic | [40,43] |

hsa-mir-155-5p | UUAAUGCUA | . | . | [40,41,43] |

hsa-mir-193a-3p | ACUGGCC | . | . | [40,43] |

hsa-mir-223-3p | GUCAGUU | . | . | . |

hsa-mir-124-5p | GUGUUCA | . | . | . |

hsa-mir-133-5p | GCUGGUA | . | $3\times $ generic | [43,44] |

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Uždavinys, A. The Heart of Plotinus. The Essential Enneads; World Wisdom, Inc.: Bloomington, IN, USA, 2009. [Google Scholar]
- Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Graph coverings for investigating non local structures in protein, music and poems. Science
**2021**, 3, 39. [Google Scholar] [CrossRef] - Irwin, K. The code-theoretic axiom; the third ontology. Rep. Adv. Phys. Sci.
**2019**, 3, 39. [Google Scholar] [CrossRef] - Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K. Group theory of syntactical freedom in DNA transcription and genome decoding. Curr. Issues Mol. Biol.
**2022**, 44, 1417–1433. [Google Scholar] [CrossRef] [PubMed] - Dang, Y.; Gao, J.; Wang, J.; Heffernan, R.; Hanson, J.; Paliwal, K.; Zhou, Y. Sixty-five years of the long march in protein secondary structure prediction: The final strech? Brief. Bioinform.
**2018**, 19, 482–494. [Google Scholar] - Dyakin, V.V.; Wisniewski, T.M.; Lajtha, A. Racemization in Post-Translational Modifications Relevance to Protein Aging, Aggregation and Neurodegeneration: Tip of the Iceberg. Symmetry
**2021**, 13, 455. [Google Scholar] [CrossRef] - Abd El Nabi, M.L.R.; Jasim, M.W.; El-Bakry, H.M.; Taha, M.H.N.; Khalifa, N.E.M. Breast and colon cancer classification from gene expression profiles using data mining techniques. Symmetry
**2020**, 12, 408. [Google Scholar] [CrossRef] [Green Version] - Milhem, Z.; Chroi, P.; Nutu, A.; Ilea, M.; Lipse, M.; Zanoaga, O.; Berindan-Neagoe, I. Non-coding RNAs and reactive oxygen species—Symmetric players of the pathogenesis associated with bacterial and viral infections. Symmetry
**2021**, 13, 1307. [Google Scholar] [CrossRef] - Aldrich, P.R.; Horsley, R.K.; Turcic, S.M. Symmetry in the language of gene expression: A survey of gene promoter networks in multiple bacterial species and non-regulons. Symmetry
**2011**, 3, 750–766. [Google Scholar] [CrossRef] [Green Version] - Heinemann, U.; Roske, Y. Symmetry in nucleic-acid double helices. Symmetry
**2020**, 12, 737. [Google Scholar] [CrossRef] - Bansal, A.; Kaushik, S.; Kukreti, S. Non-canonical DNA structures: Diversity and disease association. Front. Genet.
**2022**, 13, 959258. [Google Scholar] [CrossRef] - Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Chester, D.; Irwin, K. Character varieties and algebraic surfaces for the topology of quantum computing. Symmetry
**2022**, 14, 915. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Gröbner Basis. Available online: https://en.wikipedia.org/wiki/Gröbner_basis (accessed on 1 August 2022).
- Martinez, N.J.; Walhout, A.J.M. The interplay between transcription factors and microRNAs in genome-scale regulatory networks. Bioessays
**2009**, 31, 435–445. [Google Scholar] [CrossRef] [Green Version] - miR-155. Available online: https://en.wikipedia.org/wiki/MiR-155 (accessed on 18 November 2022).
- Kwak, J.H.; Nedela, R. Graphs and their coverings. Lect. Notes Ser.
**2007**, 17, 118. [Google Scholar] - The On-Line Encyclopedia of Integer Sequences. Available online: https://oeis.org/book.html (accessed on 1 November 2022).
- Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. General relativity with spin and torsion: Foundations and prospects. Rev. Mod. Phys.
**1976**, 48, 393–416. [Google Scholar] [CrossRef] [Green Version] - Rovelli, C.; Vidotto, F. Covariant Loop Quantum Gravity, 1st ed.; Cambridge University Press: Cambridge, MA, USA, 2014. [Google Scholar]
- Goldman, W.M. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. Eur. Math. Soc.
**2009**, 13, 611–684. [Google Scholar] - Ashley, C.; Burelle, J.P.; Lawton, S. Rank 1 character varieties of finitely presented groups. Geom. Dedicata
**2018**, 192, 1–19. [Google Scholar] [CrossRef] [Green Version] - Python Code to Compute Character Varieties. Available online: http://math.gmu.edu/~slawton3/main.sagews (accessed on 1 May 2021).
- Bosma, W.; Cannon, J.J.; Fieker, C.; Steel, A. (Eds.) Handbook of Magma Functions, 2.23rd ed.; University of Sydney: Sydney, Australia, 2017; 5914p. [Google Scholar]
- Cantat, S. Bers and Hénon, Painlevé and Schrödinger. Duke Math. J.
**2009**, 149, 411–460. [Google Scholar] [CrossRef] [Green Version] - Farb, B.; Margalit, D. A Primer on Mapping Class Groups; Princeton University Press: Princeton, NJ, USA, 2012. [Google Scholar]
- Planat, M.; Chester, D.; Amaral, M.; Irwin, K. Fricke topological qubits. Quant. Rep.
**2022**, 4, 523–532. [Google Scholar] [CrossRef] - Benedetto, R.L.; Goldman, W.M. The topology of the relative character varieties of a quadruply-punctured sphere. Exp. Math.
**1999**, 8, 85–103. [Google Scholar] [CrossRef] - Iwasaki, K. An area-preserving action of the modular group on cubic surfaces and the Painlevé VI. Comm. Math. Phys.
**2003**, 242, 185–219. [Google Scholar] [CrossRef] - Inaba, M.; Iwasaki, K.; Saito, M.H. Dynamics of the sixth Painlevé equation. arXiv
**2005**, arXiv:math.AG/0501007. [Google Scholar] - ADE Classification. Available online: https://en.wikipedia.org/wiki/ADE_classification (accessed on 1 August 2022).
- Planat, M.; Amaral, M.M.; Chester, D.; Irwin, K. SL(2,ℂ) scheme processsing of singularities in quantum computing and genetics. Axioms
**2023**, 12, 233. [Google Scholar] [CrossRef] - Doody, G.M.; Care, M.A.; Burgoyne, N.J.; Bradford, J.R.; Bota, M.; Bonifer, C.; Westhead, D.R.; Tooze, R.M. An extended set of PRDM1/BLIMP1 target genes links binding motif type to dynamic repression. Nucl. Acids Res.
**2010**, 38, 5336–5350. [Google Scholar] [CrossRef] [PubMed] - Sandelin, A.; Alkema, W.; Engstrom, P.; Wasserman, W.W.; Lenhard, B. JASPAR: An open-access database for eukaryotic transcription factor binding profiles. Nucleic Acids Res.
**2004**, 32, D91–D94. Available online: https://jaspar.genereg.net/ (accessed on 1 November 2022). [CrossRef] [Green Version] - Jauch, R. Crystal tructure and DNA inding of the homeodomain of the stem cell transcription factor Nanog. J. Mol. Biol.
**2008**, 376, 758–770. [Google Scholar] [CrossRef] - Schuff, M.; Siegel, D.; Philipp, M.; Bunsschu, K.; Heymann, N.; Donow, C.; Knöchel, W. Characterization of Danio rerio Nanog and Functional Comparison to Xenopus Vents. Stem Cells Devt.
**2012**, 21, 1225–1238. [Google Scholar] [CrossRef] [Green Version] - Schaeffer, L.N.; Huchet-Dymanus, M.; Changeux, J.P. Implication of a multisubunit Ets-related transcription factor in synaptic expression of the nicotinic acetylcholine receptor. EMBO J.
**1998**, 17, 3078–3090. [Google Scholar] [CrossRef] - microRNA. Available online: https://en.wikipedia.org/wiki/MicroRNA (accessed on 1 September 2022).
- Fang, Y.; Pan, X.; Shen, H.B. Recent deep learning methodology development for RNA-RNA interaction prediction. Symmetry
**2022**, 14, 1302. [Google Scholar] [CrossRef] - Medley, C.M.; Panzade, G.; Zinovyeva, A.Y. MicroRNA stran selection: Unwinding the rules. WIREs RNA
**2021**, 12, e1627. [Google Scholar] [CrossRef] - Dawson, O.; Piccinini, A.M. miR-155-3p: Processing by-product or rising star in immunity and cancer? Open Biol.
**2022**, 12, 220070. [Google Scholar] [CrossRef] - Kozomara, A.; Birgaonu, M.; Griffiths-Jones, S. miRBase: From microRNA sequences to function. Nucl. Acids Res.
**2019**, 47, D155–D162. [Google Scholar] [CrossRef] - miRBase: The microRNA Database. Available online: https://www.mirbase.org/ (accessed on 1 November 2022).
- Kou, X.; Chen, D.; Chen, N. The regulation of microRNAs in Alzheimer’s disease. Front. Neurol.
**2020**, 11, 288. [Google Scholar] [CrossRef] - Dyakin, V.V. Fundamental Cause of Bio-Chirality: Space-Time Symmetry—Concept Review. Symmetry
**2023**, 15, 79. [Google Scholar] [CrossRef] - Sbitnev, V. Relativistic Fermion and Boson Fields: Bose-Einstein Condensate as a Time Crystal. Symmetry
**2023**, 15, 275. [Google Scholar] [CrossRef]

**Figure 1.**

**Left**: the Nanog transcription factor (PDB 9ANT).

**Right**: the pre-miR-155 secondary structure [16].

**Figure 2.**(

**Up**): Complementary base-pairing between miR-155-3p and the human Irak3 (interleukin-1 receptor-associated kinase 3) mRNA ([16], Figure 5). The requisite‘seed sequence’ base-pairing is denoted by the bold dashes. (

**Down**): the surface ${f}_{b}^{\left({A}_{1}\right)}(x,y,z)={x}^{2}+{y}^{2}-6{z}^{2}+4xyz$.

**Figure 3.**(

**Left**): the Cayley cubic ${\kappa}_{4}(x,y,z)$. (

**Right**): the surface ${f}_{a}^{\left({A}_{1}\right)}(x,y,z)$.

**Figure 4.**The Fricke surface ${V}_{1,1,1,1}(x,y,z)={f}_{a}^{\left(3{A}_{1}\right)}(x,y,z)$ (with three simple singularities of type ${A}_{1}$).

**Figure 5.**(

**Up**): Complementary base-pairing between miR-155-5p and the human Spi1 (spleen focus forming virus proviral integration oncogene) ([16], Figure 4). The requisite ‘seed sequence’ base-pairing is denoted by the bold dashes. (

**Down (from left to right)**): the surfaces ${f}_{H}^{\left(4\right)}={\kappa}_{4}(x,y,z)$, ${f}^{\left(3{A}_{1}\right)}(x,y,z)$ and ${\kappa}_{3}(x,y,z)$, four copies of them are contained within the Groebner basis for the character variety.

**Figure 7.**(

**Left**): the cubic surface ${f}_{4,\left\{\right\}}^{\left(2{A}_{1}\right)}(x,y,z)$. (

**Right**): the cubic surface ${f}_{b}^{\left(3{A}_{1}\right)}(x,y,z)$.

**Table 1.**The counting of conjugacy classes of subgroups of index d in the free group ${F}_{r}$ of rank r = 1 to 3. The last column is the index of the sequence in the on-line encyclopedia of integer sequences [18].

r | Card Seq | Sequence Code |
---|---|---|

1 | $[1,1,1,1,1,1,1,1,1,\cdots ]$ | A000012 |

2 | $[1,3,7,26,97,624,4163,34470,314493,\cdots ]$ | A057005 |

3 | $[1,7,41,604,13753,504243,24824785,1598346352,\cdots ]$ | A057006 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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/).

## Share and Cite

**MDPI and ACS Style**

Planat, M.; Amaral, M.M.; Irwin, K.
Algebraic Morphology of DNA–RNA Transcription and Regulation. *Symmetry* **2023**, *15*, 770.
https://doi.org/10.3390/sym15030770

**AMA Style**

Planat M, Amaral MM, Irwin K.
Algebraic Morphology of DNA–RNA Transcription and Regulation. *Symmetry*. 2023; 15(3):770.
https://doi.org/10.3390/sym15030770

**Chicago/Turabian Style**

Planat, Michel, Marcelo M. Amaral, and Klee Irwin.
2023. "Algebraic Morphology of DNA–RNA Transcription and Regulation" *Symmetry* 15, no. 3: 770.
https://doi.org/10.3390/sym15030770