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. In this paper, we describe the algebraic geometry of both TFs and miRNAs thanks to group theory. 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 $\pi$, we compute the $SL(2,\mathbb{C})$ character variety, where $SL(2,\mathbb{C})$ is simultaneously a \lq space-time' (a Lorentz group) and a \lq quantum' (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when $\pi$ looks like a free group $F_r$ ($r =1$ 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 $\mathcal{G}$ of the variety. A surface with simple singularities (like the well known Cayley cubic) within $\mathcal{G}$ is a signature of a potential disease even when $G$ looks like a free group $F_r$ in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in $\{A,T/U,G,C\}$. Several human TFs and miRNAs are investigated in detail thanks to our approach.

SL(2, C) Scheme Processing of Singularities in Quantum Computing and Genetics

“…In this section, we focus on the SL(2, C) character varieties attached to microRNAs (miRNAs for short). This case was already tackled in our recent paper [6].…”

“…More precisely, group theory and representations of symmetries with characters of finite groups have been used for topological quantum computing (TQC) [19] or elementary particles [29], and the the encoding of proteins [30]. Methods for dealing with infinite groups and SL(2, C) representations of such groups in TQC papers [5,31] were similarly employed for transcription factors [6] and miRNAs [6].…”

“…The Cayley cubic encountered in our previous paper is κ 16]. Many other examples can be found in [6].…”

“…Disregulation of miRNAs may lead to a disease like cancer. A key microRNA known as an oncomir (involved in immunity and cancer) is mir-155 [6].…”

“…Section 3 refers to the symmetries and the related representations of an hyperbolic 3-manifold found in the context of magic states in quantum computing. Section 4 investigates a third class of objects, which are a family of biological molecules called microRNAs that regulate the types and amounts of proteins [6]. In Section 5, we provide commentary on our results.…”