# Topology and Dynamics of Transcriptome (Dys)Regulation

## TL;DR

This paper explores the mathematical patterns in RNA sequences to understand how disruptions in gene regulation might be linked to diseases like cancer.

## Contribution

The paper introduces a novel mathematical framework linking RNA sequence properties to algebraic structures like Painlevé equations and character varieties.

## Key findings

- RNA sequences with at most four distinct nucleotides generate groups with specific algebraic properties.
- Singularities in character varieties are connected to solutions of the Painlevé VI Equation.
- Decorated character varieties of Painlevé equations relate to the character variety of oncomir seeds.

## Abstract

RNA transcripts play a crucial role as witnesses of gene expression health. Identifying disruptive short sequences in RNA transcription and regulation is essential for potentially treating diseases. Let us delve into the mathematical intricacies of these sequences. We have previously devised a mathematical approach for defining a “healthy” sequence. This sequence is characterized by having at most four distinct nucleotides (denoted as nt≤4). It serves as the generator of a group denoted as fp. The desired properties of this sequence are as follows: fp should be close to a free group of rank nt−1, it must be aperiodic, and fp should not have isolated singularities within its SL2(C) character variety (specifically within the corresponding Groebner basis). Now, let us explore the concept of singularities. There are cubic surfaces associated with the character variety of a four-punctured sphere denoted as S24. When we encounter these singularities, we find ourselves dealing with some algebraic solutions of a dynamical second-order differential (and transcendental) equation known as the Painlevé VI Equation. In certain cases, S24 degenerates, in the sense that two punctures collapse, resulting in a “wild” dynamics governed by the Painlevé equations of an index lower than VI. In our paper, we provide examples of these fascinating mathematical structures within the context of miRNAs. Specifically, we find a clear relationship between decorated character varieties of Painlevé equations and the character variety calculated from the seed of oncomirs. These findings should find many applications including cancer research and the investigation of neurodegenative diseases.