A Mathematical Approach on the Limits of ceRNA Hypothesis Through an Ordinary Differential Equations (ODE) Model of mRNA-microRNA Interactions

Background: MicroRNAs (miRNAs) are small, non-coding RNA molecules that regulate gene expression post-transcriptionally by binding to target messenger RNAs (mRNAs) and suppressing their expression. Competing endogenous RNAs (ceRNAs), including mRNAs and circular RNAs (circRNAs), modulate miRNA availability through competitive binding, forming regulatory networks that fine-tune gene expression. CircRNAs can act as miRNA sponges, reducing miRNA-mediated repression of other targets, a mechanism implicated in various pathophysiological processes, including oncogenesis. Methods: We propose a mathematical model describing the dynamics of miRNA–mRNA–protein interactions, extending existing frameworks for miRNA–mRNA regulation. A qualitative analysis of the associated nonlinear differential equations system is performed. Results: We prove the boundedness of all positive solutions, establish the existence of a unique positive attracting equilibrium, and provide a mathematical perspective on the crosstalk mechanism in protein production. Conclusions: The effectiveness of ceRNA interactions depends on the relative abundance of miRNAs and their targets. This highlights the ongoing debate regarding the biological impact of low-abundance RNA transcripts on miRNA-mediated regulation.