Group Theory of Syntactical Freedom in DNA Transcription and Genome Decoding

Abstract

Transcription factors (TFs) are proteins that recognize specific DNA fragments in order to decode the genome and ensure its optimal functioning. TFs work at the local and global scales by specifying cell type, cell growth and death, cell migration, organization and timely tasks. We investigate the structure of DNA-binding motifs with the theory of finitely generated groups. The DNA ‘word’ in the binding domain -the motif- may be seen as the generator of a finitely generated group Fdna on four letters, the bases A, T, G and C. It is shown that, most of the time, the DNA-binding motifs have subgroup structure close to free groups of rank three or less, a property that we call ‘syntactical freedom’. Such a property is associated to the aperiodicity of the motif when it is seen as a substitution sequence. Examples are provided for the major families of TFs such as leucine zipper factors, zinc finger factors, homeo-domain factors, etc. We also discuss the exceptions to the existence of such a DNA syntactical rule and their functional role. This includes the TATA box in the promoter region of some genes, the single nucleotide markers (SNP) and the motifs of some genes of ubiquitous role in transcription and regulation.

DNA sequence and structure under the prism of group theory and algebraic surfaces

“…Groups f p 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 [4,Section 4]. Let us introduce the concept of a general substitution rule in the context of free groups.…”

“…This approach was applied to binding motifs of transcription factors [4]. The binding 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 = rel A rel T rel G rel C with the substitution rules We are interested in the sequence of finitely generated groups…”

“…For modeling DNA in its various conformations taken in transcription factors, telomeres, 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 which is not free is often the witness of a potential disease.…”

“…We coined the term 'syntactical freedom' for recognizing this property with inspiration from an earlier work [5]. We also showed that such free groups have the distinctive property of generating an aperiodic substitution rule providing a connection between (group) syntactical freedom and irrational numbers [4,Section 4]. For an infinite group, the representation cannot be based on characters but on the so-called character variety.…”

“…In Section 1, we briefly account for the many types of topology 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].…”