# Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems

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## Abstract

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## 1. Introduction

#### A Brief Review of the Literature

## 2. Graph Coverings and Conjugacy Classes of a Finitely Generated Group

## 3. Graph Coverings for Proteins

#### 3.1. The D614G Variant (Minus RBD) of the SARS-CoV-2 Spike Protein

AYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHDNPVLPF…

AYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVV.

NTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKV…

FVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTV

rel(H,E,C,G,I,T,4) =

CCCCCCCCEEEEEECCCCCCCEEEEECCCCCCCCCCEEEEEECCCCCCCC…

HHHHHHHHCC444444CHHHHHHHHHHHHHHHHHHHHCCCGGGGGHHHHH

HHIIIIICCCCCCCCCCCCCCCCCCTTTTTCCCCCCCCCHHHHHHHHHHH…

CCCTTTTTCCCCCTTTTTCCCC44444EEEEEECC,

#### 3.2. The $\beta $-2-Glycoprotein 1 or Apolipoprotein-H

## 4. Graph Coverings for Musical Forms

#### 4.1. The Sequence Isoc$(X;1)$, the Golden Ratio and More

#### 4.1.1. The Fibonacci Sequence

#### 4.1.2. The Period Doubling Cascade

#### 4.1.3. Musical Forms of the Classical Age

#### 4.2. The Sequence Isoc$(X;2)$ in Twentieth Century Music and Jazz

## 5. Graph Coverings for Prose and Poems

#### 5.1. Graph Coverings for Prose

- Le gamin du céleste Empire hésita d’abord; puis, se ravisant, il répondit: “Je vais vous le dire ”. Peu d’instants après, il reparut, tenant dans ses bras un fort gros chat, et le regardant, comme on dit, dans le blanc des yeux, il affirma sans hésiter: “Il n’est pas encore tout à fait midi.” Ce qui était vrai.

#### 5.2. Graph Coverings for Poems

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- 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] - Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Complete quantum information in the DNA genetic code. Symmetry
**2020**, 12, 1993. [Google Scholar] [CrossRef] - Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Quantum information in the protein codes, 3-manifolds and the Kummer surface. Symmetry
**2020**, 13, 1146. [Google Scholar] [CrossRef] - The Protein Data Bank. Available online: Https://pdb101.rcsb.org/ (accessed on 1 January 2021).
- ChromaticScale. Available online: Https://en.wikipedia.org/wiki/Chromatic_scale (accessed on 1 April 2021).
- Musical Form. Available online: Https://en.wikipedia.org/wiki/Musical_form (accessed on 1 April 2021).
- Growney, J. Mathematics in poetry. J. Online Math. Appl.
**2006**, 6, 1262. [Google Scholar] [CrossRef] - Glaz, S. Poetry inspired by mathematics: A brief journey through history. J. Math. Arts
**2011**, 5, 171–183. [Google Scholar] [CrossRef] - Henderson, C. For the love of poetry and mathematics: January 6, 2012. J. Humanist. Math.
**2012**, 2, 27–46. [Google Scholar] - Nichita, F.F. Mathematics and poetry. Unification, unity, union. Sci
**2020**, 2, 84. [Google Scholar] [CrossRef] - Aharony, R. Mathematics, poetry and beauty. J. Math. Arts
**2014**, 8, 5–12. [Google Scholar] [CrossRef] - Kornai, A. Mathematical Linguistics; Springer: London, UK, 2008. [Google Scholar]
- Monte-Serrat, D.M.; Ruiz, E.E.S.; Cattani, C. Linguistic Theoretical Frameworks for Dealing with AI. Available online: https://www.researchgate.net/project/Linguistic-theoretical-frameworks-for-dealing-with-AI (accessed on 28 September 2021).
- Andreatta, M. From Music to Mathematics and Backwards: Introducing Algebra, Topology and Category Theory into Computational Musicology. In Imagine Math 6; Emmer, M., Abate, M., Eds.; Springer: Cham, Switzerland, 2018. [Google Scholar] [CrossRef][Green Version]
- Shakhovska, N.; Fedushko, S. Data analysis of music preferences of web users based on social and demographic factors. Procedia Comput. Sci.
**2021**, in press. [Google Scholar] - Mednykh, A. Counting conjugacy classes of subgroups in a finitely generated group. J. Algebra
**2008**, 320, 2209–2217. [Google Scholar] [CrossRef][Green Version] - Kwak, J.H.; Nedela, R. Graphs and their coverings. Lect. Notes Ser.
**2007**, 17, 118. [Google Scholar] - Hardy, G.H.; Wright, E.M. An Introduction to the Theory of Numbers, 6th ed.; Oxford University Press: Oxford, UK, 2008; pp. 361–392. [Google Scholar]
- Planat, M. Quantum 1/f noise in equilibrium: From Planck to Ramanujan. Phys. A
**2003**, 318, 371. [Google Scholar] [CrossRef][Green Version] - Vrna, P. On the algebra of local unitary invariants of pure and mixed quantum states. J. Phys. A Math. Theor.
**2011**, 44, 225304. [Google Scholar] - Hall, M., Jr. Subgroups of finite index in free groups. Can. J. Math.
**1949**, 1, 187–190. [Google Scholar] - SARS-CoV-2 Spike D614G Variant, Minus RBD, in Protein Data Bank in Europe, Bringing Structure to Biology. Available online: https://www.ebi.ac.uk/pdbe/entry/pdb/6xs6 (accessed on 1 May 2021).
- Protein Secondary Structure. Available online: https://en.wikipedia.org/wiki/Protein_secondary_structure (accessed on 1 May 2021).
- Konagurthu, A.S.; Lesk, A.M.; Allison, L. Minimum message length inference of secondary structure fromprotein coordinate data. Bioinformatics
**2012**, 28, i97–i105. Available online: https://lcb.infotech.monash.edu/sstweb2/Submission_page.html (accessed on 1 May 2021). [CrossRef] - McDonnell, T.; Wincup, C.; Bucholz, I.; Pericleous, C.; Giles, I.; Ripoll, V.; Cohen, H.; Delcea, M.; Rahman, A. The role of beta-2-glycoprotein I in health and disease associating structure with function: More than just APS. Blood Rev.
**2020**, 39, 100610. [Google Scholar] [CrossRef] - Crystal Structure of Beta-2 Glycoprotein I Purified from Plasma (pB2GPI). Available online: https://www.rcsb.org/structure/6V06 (accessed on 1 May 2021).
- Mirabello, C.; Pollastri, G. Porter, PaleAle 4.0: High-accuracy prediction of protein secondary structure and relative solvent accessibility. Bioinformatics
**2013**, 29, 2056–2058. [Google Scholar] [CrossRef][Green Version] - Flicker, F. Time quasilattices in dissipative dynamical systems. SciPost Phys.
**2018**, 5, 001. [Google Scholar] - Putz, J.F. The Golden section and the piano sonatas of Mozart. Math. Mag.
**1995**, 68, 275–282. [Google Scholar] - Music for Strings, Percussion and Celesta. Available online: https://en.wikipedia.org/wiki/Music_for_Strings,_Percussion_and_Celesta (accessed on 1 May 2021).
- Twelve-Bar Blues. Available online: https://en.wikipedia.org/wiki/Twelve-bar_blues (accessed on 1 May 2021).
- Haydn-String Quartet, Op. 76, No. 3. Available online: https://www.youtube.com/watch?v=qoWdtGUe5fc (accessed on 1 May 2021).
- Barwick, L. Musical Form and Style in Murriny Patha Djanba Songs at Wadeye (Northern Territory, Australia). Available online: https://core.ac.uk/download/pdf/41240492.pdf (accessed on 1 May 2021).
- Baudelaire, C. Le vieux saltimbanque. In Petits Poèmes en Prose; Michel Lévy fréres: Paris, France, 1869. [Google Scholar]
- Irwin, K.; Amaral, M.; Chester, D. The Self-Simulation hypothesis interpretation of quantum mechanics. Entropy
**2020**, 22, 247. [Google Scholar] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Informationally complete characters for quark and lepton mixings. Symmetry
**2020**, 12, 1000. [Google Scholar] [CrossRef] - Planat, M.; Chester, D.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Finite groups for the Kummer surface: The genetic code and quantum gravity. Quantum Rep.
**2021**, 3, 68–79. [Google Scholar] [CrossRef] - Planat, M.; Chester, D.; Aschheim, R.; Amaral, M.M.; Fang, F. Irwin, Character varieties and algebraic surfaces for the topology of quantum computing. in preparation.

**Figure 1.**A picture of the secondary structure of D614G variant (minus RBD) of the SARS-CoV-2 spike protein found in the protein data bank in Europe [22].

**Figure 2.**A picture of the secondary structure of the apolipoprotein-H obtained with the software [24].

**Table 1.**The number Isoc$(X;d)$ for small values of first Betti number r (alias the number of generators of the free group ${F}_{r}$) and index d. Thus, the columns correspond to the number of conjugacy classes of subgroups of index d in the free group of rank r.

r | d = 1 | d = 2 | d = 3 | d = 4 | d = 5 | d = 6 | d = 7 |
---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 3 | 7 | 26 | 97 | 624 | 4163 |

3 | 1 | 7 | 41 | 604 | 13,753 | 504,243 | 24,824,785 |

4 | 1 | 15 | 235 | 14,120 | 1,712,845 | 371,515,454 | 127,635,996,839 |

5 | 1 | 31 | 1361 | 334,576 | 207,009,649 | 268,530,771,271 | 644,969,015,852,641 |

**Table 2.**Group analysis of the D614G variant (minus RBD) of the SARS-CoV-2 spike protein. The bold numbers mean that the cardinality structure of cc of subgroups of G fits that of the free group ${F}_{r-1}$ when the encoding makes use of r letters. In the last column, r is the first Betti number of the generating group ${f}_{p}$.

PDB 6XS6: AYTNSFTRGVYYPDKVFRSSVLHSTQDL … | Cardinality Structure of cc of Subgroups | r |
---|---|---|

6 letters H, E, C, G, I, T | [1,31,1361,334576] | 5 |

5 letters H, E, C, G, I | [1,15,235,14120] | 4 |

4 letters H, E, C, G | [1,7,41 604,14720] | 3 |

3 letters H, E, C | [1,3,7,30,127, 926] | 2 |

**Table 3.**Group analysis of apolipoprotein-H (PDB 6V06). The bold numbers means that the cardinality structure of cc of subgroups of ${f}_{p}$ fits that of the free group ${F}_{3}$ when the encoding makes use of 2 letters. The first model is the one used in the previous Section [24] where we took $4=H$ and $T=C$. The other models of secondary structures with segments E, H and C are from softwares PORTER, PHYRE2 and RAPTORX. The references to these softwares may be found in our recent paper [3]. The notation r in column 3 means the first Betti number of ${f}_{p}$.

PDB 6V06: GRTCPKPDDLPFSTVVPLKTFYEPG… | Cardinality Structure of cc of Subgroups | r |
---|---|---|

Konagurthu | [1,3,7,26,218,2241] | 2 |

PORTER | [1,3,7,26,97,624] | . |

PHYRE2 | [1,3,7,26,157,1046] | . |

RAPTORX | [1,7,17,134,923,13317] | 3 |

**Table 4.**Group analysis of a few musical forms whose structure of subgroups, apart from exceptions, is close to Isoc$(X;d)$ with $d=2$ (at the upper part of the table) or $d=3$ (at the lower part of the table). Of course, the forms A-B-C and A-B-C-D have the cardinality sequence of cc of subgroups exactly equal to Isoc$(X;2)$ and Isoc$(X;3)$, respectively.

Musical Form | Ref | Card. Struct. of cc of Subgr. | r |
---|---|---|---|

A-B-C-B-A | arch, Belá Bartók | [1,3,7,26,97,624, | 2 |

. | . | 4163,34470,314493] | . |

A-B-A-C-A-B-A | . | . | . |

A-B-A-C-A, A-B-A-C-A-B-A | rondo | . | . |

A-B-A-C | . | . | |

A-A-B-C-C | Haydn [32], | . | . |

. | djanba ([33], Figure 9.8) | . | . |

A-A-A-A-B-B-A-A-C-C-A-A | twelve-bar blues, | [1,7,14,109,396,3347, | 3 |

. | standard | 19758,287340] | . |

A-A-A-A-B-B-A-A-C-B-A-A | twelve-bar blues, | [1,3,7,26,97,624, | 2 |

. | variation 1 | 4163,34470,314493] | . |

A-A-A-A-B-B-A-A-B-C-A-C | twelve-bar blues, | [1,3,7,26,127, 799, | . |

. | variation 2 | 5168, 42879] | . |

A-B-C | Isoc$(X;2)$ | [1,3,7,26,97,624, | 2 |

. | . | 4163,34470,314493] | . |

A-A-B-B-C-C-D-D | pot pourri | [1,15,82,1583,30242] | 4 |

A-B-A-C-A-D-A | rondo | [1,7,41,604,13753,504243] | 3 |

A-B-C-D | Isoc$(X;3)$ | [1,7,41,604,13753, | 3 |

. | . | 504243,24824785] | . |

**Table 5.**Group analysis of an excerpt of a small poem in prose Le vieux saltimbanque by Charles Baudelaire. The text is split into segments encoded by the symbol H (for names and adjectives), E (for verbs), A for prepositions, B for adverbs, or C (for the other types: conjunctions, punctuation marks and so on). The cardinality structure of the cc of subgroups of a small index is compared to the one obtained with 10 runs of a sequence of words of a similar length (i.e., the length 250) with the corresponding number of letters.

Le Gamin du Céleste Empire …Ce Qui était Vrai. | Card. Seq. of cc of Subgroups | r |
---|---|---|

3 letters: rel=${C}^{2}{H}^{5}{C}^{2}{H}^{7}{H}^{6}{E}^{6}{C}^{7}C{C}^{4}C{C}^{2}{E}^{8}C\cdots $ | [1,3,7,34,131] | 2 |

4 letters: rel=${C}^{2}{H}^{5}{A}^{2}{H}^{7}{H}^{6}{E}^{6}{C}^{7}C{C}^{4}C{C}^{2}{E}^{8}C\cdots $ | [1,7,41,636,14364] | 3 |

5 letters: rel=${C}^{2}{H}^{5}{A}^{2}{H}^{7}{H}^{6}{E}^{6}{B}^{7}C{B}^{4}C{C}^{2}{E}^{8}C\cdots $ | [1,15,235,14376,.] | 4 |

[Random[1,3]: i in [1..250]] | [1,1,1,2,4,4] | 1 |

(10 runs) | [1,3,2,9,5,20] | 2 |

[1,3,1,6,6,15] | . | |

[1,3,7,30,124,987] | . | |

[1,7,17,126,323,2445] | 3 | |

etc. | ||

Isoc(X;2) | [1,3,7,26,97,624] | 2 |

[Random[1,4]: i in [1..250]] | [1,3,7,30,.] ($\times 3)$ | 2 |

(10 runs) | [1,3,10,51,.] ($\times 3)$ | . |

[1,3,7,26,457] | . | |

[1,3,10,39,.] | . | |

[1,3,13,52,.] | . | |

[1,7,20,143,.] | 3 | |

Isoc(X;3) | [1,7,41,604,13573] | 3 |

[Random[1,5]: i in [1..250]] | [1,7,41,620,.] ($\times 3)$ | 3 |

(10 runs) | [1,7,41,636,.] ($\times 3)$ | . |

[1,7,41,604,.] ($\times 2)$ | . | |

[1,7,41,668,.] | . | |

[1,7,50,819,.] | . | |

Isoc(X;4) | [1,15,235,14120,1712845] | 4 |

**Table 6.**Group structure of the poem Le Bateau Ivre’ (The Drunken Boat) by Arthur Rimbaud. Only the first strophe (that has four lines) is analyzed, firstly in its original form, then in an English translation. Each line is split into segments encoded by the symbol H (for names and adjectives), E (for verbs) or C (for the other types: conjunctions, adverbs, prepositions, punctuation marks and so on). The group relation is displayed for the first line only.) The cardinality structure of cc of subgroups of a small index is compared to the one obtained with 10 runs of a sequence of random 3-letter words of similar length (i.e., the length 35).

Comme je descendais des fleuves impassibles, | [1,1,7,17,114,1395,36973] | 1 |

rel=${C}^{4}{C}^{2}{E}^{10}{C}^{3}{H}^{7}{H}^{11}C$ | ||

Je ne me sentis plus guidé par les haleurs: | [1,3,7,26,97, 624,4171] | 2 |

Des Peaux-Rouges criards les avaient pris pour cibles | [1,3,7,26,97, 624,4163] | . |

Les ayant cloués nus aux poteaux de couleurs. | [1,3,7,26,97,624,4163] | . |

As I was floating down unconcerned rivers | [1,3,7,26,97, 624,4163,34470] | 2 |

rel=${C}^{2}\ast C\ast {E}^{3}\ast {E}^{8}\ast {C}^{4}\ast {E}^{1}1\ast {H}^{6}$ | ||

I now longer felt myself steered by the haulers: | [1,3,7,26,101,656,4227] | 2 |

Gaudy Redskins had taken them for targets | [1,3,7,26,97,624,4163,324935] | . |

Nailing them naked to coloured states. | [1,3,7,42,202,1682,9204] | . |

[Random[1,3]: i in [1..35]] | [1,3,7,30,.] ($\times 3)$ | 2 |

(10 runs) | [1,3,7,26,.]( $\times 3)$ | . |

[1,3,7,.,.,] | . | |

[1,3,10,.,.]($\times 2)$ | . | |

[1,3,13,.,.] | . | |

Isoc(X;2) | [1,3,7,26,97,624,4163,34470] | 2 |

**Table 7.**The same as in Table 6, but each line is split into segments encoded by the symbol H (for names and adjectives), E (for verbs), A for prepositions, or C (for the other types: conjunctions, adverbs, punctuation marks and so on). The cardinality structure of cc of subgroups of a small index is compared to the one obtained with 10 runs of a sequence of random 4-letter words of similar length (i.e., the length 35).

Comme je descendais des fleuves impassibles, | [1,7,41,604,13753] | 3 |

rel=${C}^{4}{C}^{2}{E}^{10}{A}^{3}{H}^{7}{H}^{11}C$ | ||

Je ne me sentis plus guidé par les haleurs: | [1,7,41,604,13753] | . |

Des Peaux-Rouges criards les avaient pris pour cibles | [1,7,41,604,13753] | . |

Les ayant cloués nus aux poteaux de couleurs. | [1,7,41,604,13753] | . |

As I was floating down unconcerned rivers | [1,7,59,1386,27011] | 3 |

rel=${C}^{2}C{E}^{3}{E}^{8}{A}^{4}{E}^{11}{H}^{6}$ | ||

I no longer felt myself steered by the haulers: | [1,7,41,604,13753] | . |

Gaudy Redskins had taken them for targets | [1,7,50,1763,51582] | . |

Nailing them naked to coloured states. | [1,7,59,1002,18671] | . |

[Random[1,4]: i in [1..35]] | [1,7,50,755,.] (×2) | 3 |

(10 runs) | [1,7,41,604,.] $(\times 3)$ | . |

[ 1,7,41,.,.]($\times 2)$ | . | |

[1,7,50,739,.]($\times 2)$ | . | |

[1,7,59,.,.] | . | |

Isoc(X;3) | [1,7,41,604,13753] | 3 |

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**MDPI and ACS Style**

Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems. *Sci* **2021**, *3*, 39.
https://doi.org/10.3390/sci3040039

**AMA Style**

Planat M, Aschheim R, Amaral MM, Fang F, Irwin K. Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems. *Sci*. 2021; 3(4):39.
https://doi.org/10.3390/sci3040039

**Chicago/Turabian Style**

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, Fang Fang, and Klee Irwin. 2021. "Graph Coverings for Investigating Non Local Structures in Proteins, Music and Poems" *Sci* 3, no. 4: 39.
https://doi.org/10.3390/sci3040039