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

Abstract

We explore the structural similarities in three different languages, first in the protein language whose primary letters are the amino acids, second in the musical language whose primary letters are the notes, and third in the poetry language whose primary letters are the alphabet. For proteins, the non local (secondary) letters are the types of foldings in space (α-helices, β-sheets, etc.); for music, one is dealing with clear-cut repetition units called musical forms and for poems the structure consists of grammatical forms (names, verbs, etc.). We show in this paper that the mathematics of such secondary structures relies on finitely presented groups fp on r letters, where r counts the number of types of such secondary non local segments. The number of conjugacy classes of a given index (also the number of graph coverings over a base graph) of a group fp is found to be close to the number of conjugacy classes of the same index in the free group Fr−1 on r−1 generators. In a concrete way, we explore the group structure of a variant of the SARS-Cov-2 spike protein and the group structure of apolipoprotein-H, passing from the primary code with amino acids to the secondary structure organizing the foldings. Then, we look at the musical forms employed in the classical and contemporary periods. Finally, we investigate in much detail the group structure of a small poem in prose by Charles Baudelaire and that of the Bateau Ivre by Arthur Rimbaud.

Group Theory of Syntactical Freedom in DNA Transcription and Genome Decoding

“…[22] and ( [23] p. 90) and follows from the fact that π 2 is not a free group. The group f p is close to π 2 in the sense that the cardinality sequence [1,3,10,51,164,1365,9422,81594, 721305, • • • ] of the cc of subgroups is that of π 2 , up to the higher index 9 that we could reach in our calculations.…”

“…The central portion of the protein contains 86 = 30 + 28 + 28 aa decomposed into 3 zinc fingers with the following secondary structure (letter H is for the α-helix segment, letter E is for the β-sheet segment and letter C is for the random coil segment) CCCEECCCCCCCCEECHHHHHHHHHHHHHH CCCEECCCCCCEECHHHHH-HHHHHHHHH CCCEECCCCCCEECHHHHHHHHHHHHHC Taking the former 3-letter chain as the relation of a finitely generated group on 3 letters (and rank 2), we get the cardinality sequence for the cc of its subgroups as [1,3,7,26,112, 717, • • • ], which fits the cardinality sequence of cc of subgroups of the free group F 2 only up to the index 4.…”

“…Our last papers focused on the relevance of free groups in the encoding of the secondary structure of proteins [2] and in the encoding of tonal music and poetry [3].…”

“…AAGGAAATTT[1,3,7,26,195, 1692, 11803, 73192 • • • ] ., A AR GTGGAAGATT [1, 3, 7,34, 139, 931, 5208, 43867 • • • ] Table 7 in [15], A . CCACGACCCG [1,7, 20, 167, 754, 7232, 60860, 683597 • • • ]…”

Algebraic Morphology of DNA–RNA Transcription and Regulation

“…The remote principle envisaged by Plotinus is still a symmetry principle but in a modern definition involving group theory and algebraic geometry. Recently, we wrote a paper about a common algebra possibly ruling the beauty and structure in poems, music and proteins [2]. We found that free groups govern the structure of such disparate topics where a language emerges from pure randomness.…”