Modeling competing endogenous RNAs networks

MicroRNAs (miRNAs) are small RNA molecules, about 22 nucleotide long, which post-transcriptionally regulate their target messenger RNAs (mRNAs). They accomplish key roles in gene regulatory networks, ranging from signaling pathways to tissue morphogenesis, and their aberrant behavior is often associated with the development of various diseases. Recently it has been shown that, in analogy with the better understood case of small RNAs in bacteria, the way miRNAs interact with their targets can be described in terms of a titration mechanism characterized by threshold effects, hypersensitivity of the system near the threshold, and prioritized cross-talk among targets. The latter characteristic has been lately identified as competing endogenous RNA (ceRNA) effect to mark those indirect interactions among targets of a common pool of miRNAs they are in competition for. Here we analyze the equilibrium and out-of-equilibrium properties of a general stochastic model of $M$ miRNAs interacting with $N$ mRNA targets. In particular we are able to describe in details the peculiar equilibrium and non-equilibrium phenomena that the system displays around the threshold: (i) maximal cross-talk and correlation between targets, (ii) robustness of ceRNA effect with respect to the model's parameters and in particular to the catalyticity of the miRNA-mRNA interaction, and (iii) anomalous response-time to external perturbations.