Michel Planat, Marcelo M. Amaral, Klee Irwin (2022)

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.