Article
Version 1
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Topology and Dynamics of Transcriptome (Dys)regulation
Version 1
: Received: 27 March 2024 / Approved: 28 March 2024 / Online: 28 March 2024 (10:18:13 CET)
How to cite: Planat, M.; Chester, D. Topology and Dynamics of Transcriptome (Dys)regulation. Preprints 2024, 2024031741. https://doi.org/10.20944/preprints202403.1741.v1 Planat, M.; Chester, D. Topology and Dynamics of Transcriptome (Dys)regulation. Preprints 2024, 2024031741. https://doi.org/10.20944/preprints202403.1741.v1
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’s delve into the mathematical intricacies of these sequences. We’ve previously devised a mathematical approach for defining a “healthy” sequence (10.14218/GE.2023.00079). 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, fp should not have isolated singularities within its SL2(C) character variety (specifically within the corresponding Groebner basis). Now, let’s 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 (10.3390/dynamics4010001). 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 applications in cancer research.
Keywords
group theory; character variety; Painlevé equations; transcriptome; microRNAs; diseases; cancer research
Subject
Biology and Life Sciences, Biochemistry and Molecular Biology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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