Algebraic Morphology of DNA–RNA Transcription and Regulation

Abstract

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.

Group Theory of Messenger RNA Metabolism and Disease

“…From our perspective, DNA/RNA sequences that lead to f p groups closely resembling a free group are considered healthy sequences. 19,21,23 The standard poly(A) sequences mentioned earlier play a regulatory role in producing mature mRNA during translation. Sequences that generate an f p group diverging from a free group F r may be indicative of a disease.…”

“…[15][16][17] To study the epigenetic code (hereinafter referred to as the ecode), we used infinite (finitely generated) groups denoted by f p , and their representations over the (2 × 2) matrix group SL 2 (C), where the entries are complex numbers. 18, 19 The significance of this group extends across all fields of physics, as it represents a DOI: 10.14218/GE.2023.00079 | Volume 00 Issue 00, Month Year 2 Planat M. et al: Group theory of messenger RNA metabolism Gene Expr space-time-spin group. In this study, we applied a mathematical field known as algebraic geometry to define the e-code, which has not been done before.…”

“…A surface within G containing isolated singularities indicates another potential disease that can be identified specifically, e.g., relating to an oncogene or a neurological disorder. 19 The e-code we define comprises such algebraic geometric characteristics.…”

“…An additional attribute of healthy sequences, which leads to a group f p approximating the free group F r and not mentioned in the study of Planat et al, 19 is their connection to aperiodicity. Schrodinger proposed the periodicity of living crystals.…”

“…Schrodinger proposed the periodicity of living crystals. 20 Planat et al 19 characterized aperiodic DNA sequences. 21 We advanced this concept by introducing the so-called profinite completion ˆr F of the free group F r .…”