  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.
