Kinetic modelling of competition and depletion of shared miRNAs by competing endogenous RNAs

TL;DR

This paper reviews mathematical models of competing endogenous RNAs (ceRNAs) and their role in gene regulation, focusing on miRNA interactions, network effects, and implications for cellular proteome composition.

Contribution

It provides a comprehensive overview of kinetic models of miRNA-ceRNA interactions, highlighting how network topology influences ceRNA crosstalk and gene expression regulation.

Findings

Mathematical models elucidate conditions for significant ceRNA interactions.

Network topology affects the extent of miRNA-mediated crosstalk.

ceRNA interactions influence gene expression noise and cellular regulation.

Abstract

Non-conding RNAs play a key role in the post-transcriptional regulation of mRNA translation and turnover in eukaryotes. miRNAs, in particular, interact with their target RNAs through protein-mediated, sequence-specific binding, giving rise to extended and highly heterogeneous miRNA-RNA interaction networks. Within such networks, competition to bind miRNAs can generate an effective positive coupling between their targets. Competing endogenous RNAs (ceRNAs) can in turn regulate each other through miRNA-mediated crosstalk. Albeit potentially weak, ceRNA interactions can occur both dynamically, affecting e.g. the regulatory clock, and at stationarity, in which case ceRNA networks as a whole can be implicated in the composition of the cell's proteome. Many features of ceRNA interactions, including the conditions under which they become significant, can be unraveled by mathematical and in silico models. We review the understanding of the ceRNA effect obtained within such frameworks, focusing on the methods employed to quantify it, its role in the processing of gene expression noise, and how network topology can determine its reach.

Tables

Table 1. Table 1: Variables and parameters appearing in the basic model, Eq. ( 2 ). Note that the levels of molecular species can be specified by copy numbers (as indicated below) as well as by (continuous) concentrations, depending on whether the modeling framework is stochastic (see Section 2.3 ) or deterministic (as in Eq. ( 2 )), respectively.

Variable	Units	Description
\svhline $m_i$	molecules	Number of free copies of ceRNA species $i$
${\mu }_a$	molecules	Number of free copies of miRNA species $a$
$c_{ia}$	molecules	Number of copies of $i-a$ complex
Parameter	Units	Description
\svhline $b_i$	molecule min-1	Transcription rate of ceRNA species $i$
$d_i$	min-1	Degradation rate of ceRNA species $i$
${\beta }_a$	molecule min-1	Transcription rate of miRNA species $a$
${\delta }_a$	min-1	Degradation rate of miRNA species $i$
$k_{ia}^{+}$	molecule-1 min-1	$i-a$ complex association rate
$k_{ia}^{-}$	min-1	$i-a$ complex dissociation rate
${\kappa }_{ia}$	min-1	Catalytic decay rate (with miRNA re-cycling) of $i-a$ complex
${\sigma }_{ia}$	min-1	Stoichiometric decay rate (without miRNA re-cycling) of $i-a$ complex

Table 2. Table 2: Values of kinetic parameters used in the different figures.

Parameter	Fig. 2A–C	2D–F	3A	3B	3C	3D	5	6A,B	6C,D
\svhline $b_1$ [molec min-1]	10	–	10	20	2	1	1 (mean)	–	10
$b_2$ [molec min-1]	15	10	15	10	10	10	1 (mean)	0	0
${\beta }_1$ [molec min-1]	–	20	15	–	15	–	–	15	15
$d_1$ [min-1]	0.1	0.1	0.1	0.1	0.1	0.1	0.005	0.1	0.1
$d_2$ [min-1]	0.1	0.1	0.1	0.1	0.1	0.1	0.005	0	0.1
${\delta }_1$ [min-1]	0.1	0.1	0.1	0.1	0.1	0.1	0.01	0.1	0.1
$k_{11}^{+}$ [molec-1 min-1]	$e^{-3}$	$e^{-2}$	1	$e^{-2}$	$e^{15}$	$e^{15}$	$e^{-2}$	shown	caption
$k_{21}^{+}$ [molec-1 min-1]	$e^{-5}$	$e^{-4}$	$e^{-3}$	$e^{-3}$	$e^{-4}$	$e^{-4}$	$e^{-3}$	0	caption
$k_{11}^{-}$ [min-1]	0.001	0.001	0.001	0.001	0.001	0.001	0.1	0.001	0.001
$k_{21}^{-}$ [min-1]	0.001	0.001	0.001	0.001	0.001	0.001	0.1	0	0.001
${\kappa }_{11}$ [min-1]	0.001	0.001	0.001	0.001	0.1	0.1	0.05	0.001	0.001
${\kappa }_{21}$ [min-1]	0.001	0.001	0.001	0.001	0.001	0.001	0.05	0	0.001
${\sigma }_{11}$ [min-1]	1	1	1	1	1	1	0.001	1	1
${\sigma }_{21}$ [min-1]	1	1	1	1	1	1	0.001	0	1

Equations

$$\begin{array}[]{r@{}l}&{}\emptyset\xrightleftharpoons[d_{i}]{b_{i}}m_{i}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\emptyset\xrightleftharpoons[\delta_{a}]{\beta_{a}}\mu_{a}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mu_{a}+m_{i}\xrightleftharpoons[k_{ia}^{-}]{k_{ia}^{+}}c_{ia}\\ &{}c_{ia}\xrightharpoonup{\sigma_{ia}}\emptyset~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}c_{ia}\xrightharpoonup{\kappa_{ia}}\mu_{a}\end{array}~{}~{}~{}~{}~{}~{}~{}.$$

$$\begin{array}[]{r@{}l}\frac{dm_{i}}{dt}&{}=b_{i}-d_{i}m_{i}-\sum_{a}k_{ia}^{+}m_{i}\mu_{a}+\sum_{a}k_{ia}^{-}c_{ia}~{}~{},\\ \frac{d\mu_{a}}{dt}&{}=\beta_{a}-\delta_{a}\mu_{a}-\sum_{i}k_{ia}^{+}m_{i}\mu_{a}+\sum_{i}(k_{ia}^{-}+\kappa_{ia})c_{ia}~{}~{},\\ \frac{dc_{ia}}{dt}&{}=k_{ia}^{+}m_{i}\mu_{a}-(\sigma_{ia}+\kappa_{ia}+k_{ia}^{-})c_{ia}~{}~{},\end{array}$$

$${\alpha }_{ia}=\frac{{\sigma }_{ia}}{{\sigma }_{ia}+{\kappa }_{ia}}$$

$$\frac{d{n}_{\ell }}{dt}=k_{\text{on}}(1-n_{\ell })f_{\ell }^H-k_{\text{off}}{n}_{\ell },$$

$$⟨n_{\ell }⟩=\frac{f_{\ell }^H}{f_{\ell }^H+K^H},K={\left(\frac{k_{\text{off}}}{k_{\text{on}}}\right)}^{1/H}.$$

$$\begin{array}[]{r@{}l}\left\langle m_{i}\right\rangle&{}=\frac{b_{i}+\sum_{a}k_{ia}^{-}\left\langle c_{ia}\right\rangle}{d_{i}+\sum_{a}k_{ia}^{+}\left\langle\mu_{a}\right\rangle}{}{}\\ \left\langle\mu_{a}\right\rangle&{}=\frac{\beta_{a}+\sum_{i}(k_{ia}^{-}+\kappa_{ia})\left\langle c_{ia}\right\rangle}{\delta_{a}+\sum_{i}k^{+}_{ia}\left\langle m_{i}\right\rangle}{}{}\\ \left\langle c_{ia}\right\rangle&{}=\frac{k^{+}_{ia}\left\langle\mu_{a}\right\rangle\,\left\langle m_{i}\right\rangle}{\sigma_{ia}+\kappa_{ia}+k^{-}_{ia}}{}{}\end{array}~{}~{}.$$

$$\begin{array}[]{r@{}l}\left\langle m_{i}\right\rangle&{}=\frac{m_{i}^{\star}}{1+\sum_{a}\mu_{a}/\mu_{0,ia}}~{}~{},\\ \left\langle\mu_{a}\right\rangle&{}=\frac{\mu_{a}^{\star}}{1+\sum_{i}m_{i}/m_{0,ia}}~{}~{},\end{array}$$

$$\begin{array}[]{r@{}l}m_{0,ia}&{}=\frac{\delta_{a}}{k_{ia}^{+}}\left(1+\frac{k_{ia}^{-}+\kappa_{ia}}{\sigma_{ia}}\right)~{}~{},\\ \mu_{0,ia}&{}=\frac{d_{i}}{k_{ia}^{+}}\left(1+\frac{k_{ia}^{-}}{\sigma_{ia}+\kappa_{ia}}\right){}{}\end{array}$$

$$\frac{\partial P}{\partial t}=\sum _{a=1}^M{\beta }_a\left(P_{{\mu }_a-1}-P\right)$$

$$+\sum _{i=1}^N{b}_i\left(P_{m_i-1}-P\right)$$

$$+\sum _{a=1}^M{\delta }_a\left[({\mu }_a+1)P_{{\mu }_a+1}-{\mu }_{a}P\right]$$

$$+\sum _{i=1}^N{d}_i\left[(m_i+1)P_{m_i+1}-m_{i}P\right]$$

$$+\sum _{i=1}^N\sum _{a=1}^M{k}_{ia}^{+}\left[({\mu }_a+1)(m_i+1)P_{{\mu }_a+1,m_i+1,c_{ia}-1}-{\mu }_a{m}_{i}P\right]$$

$$+\sum _{i=1}^N\sum _{a=1}^M{k}_{ia}^{-}\left[(c_{ia}+1)P_{{\mu }_a-1,m_i-1,c_{ia}+1}-c_{ia}P\right]$$

$$+\sum _{i=1}^N\sum _{a=1}^M{\sigma }_{ia}\left[(c_{ia}+1)P_{c_{ia}+1}-c_{ia}P\right]$$

$$+\sum _{i=1}^N\sum _{a=1}^M\sum _{i=M+1}^{M+N}{\kappa }_{ia}\left[(c_{ia}+1)P_{{\mu }_a-1,c_{ia}+1}-c_{ia}P\right]$$

$$P(\{{\mu }_a\},\{m_i\},\{c_{ia}\},t)=\prod _{i=1}^N{P}_i(m_i)\prod _{a=1}^M{P}_a({\mu }_a)\prod _{\ell =1}^{N\cdot M}{P}_{\ell }(c_{\ell })$$

$$\vec{x}$$

$$P(\vec{x})\simeq G(\vec{x}|\vec{a},{\Sigma }^{-1})=\frac{\exp \left[-\frac{1}{2}{\left(\vec{x}-\vec{a}\right)}^T{\Sigma }^{-1}\left(\vec{x}-\vec{a}\right)\right]}{\sqrt{(2\pi {)}^{M+N+MN}\det (\Sigma )}}$$

$$F(\mathbf{z},t)=\sum _{\mathbf{x}}\prod _{i=1}^{N+M+N\cdot M}{z}_i^{x_i}P(\mathbf{x},t).$$

$${\partial }_{t}F(\mathbf{z},t)=\mathcal{H}(\mathbf{z})F(\mathbf{z},t),$$

$$\mathcal{H}(\mathbf{z})$$

$${\partial }_{z_{i_1}^{l_1},z_{i_2}^{l_2},\dots ,z_{i_k}^{l_k}}^{l_1+l_2+\cdots +l_k}F(\mathbf{z},t){|}_{\mathbf{z}=1}=⟨x_{i_1}^{l_1}{x}_{i_2}^{l_2}\cdots x_{i_k}^{l_k}{⟩}_{P(\mathbf{x},t)}.$$

$$\begin{array}[]{r@{}l}\frac{dm_{i}}{dt}&=b_{i}-d_{i}m_{i}\,\underline{\underline{+\xi_{i}}}\,-\sum_{a}k_{ia}^{+}m_{i}\mu_{a}\,\underline{\underline{+\sum_{a}\xi_{ia}^{+}}}\,+\sum_{a}k_{ia}^{-}c_{ia}\,\underline{\underline{+\sum_{a}\xi_{ia}^{-}}}~{}~{},\\ \frac{d\mu_{a}}{dt}&=\beta_{a}-\delta_{a}\mu_{a}\,\underline{\underline{+\xi_{a}}}\,-\sum_{i}k_{ia}^{+}m_{i}\mu_{a}\,\underline{\underline{+\sum_{i}\xi_{ia}^{+}}}\,+\sum_{i}(k_{ia}^{-}+\kappa_{ia})c_{ia}\,\underline{\underline{+\sum_{i}(\xi_{ia}^{-}+\xi_{ia}^{\rm cat})}}~{}~{},\\ \frac{dc_{ia}}{dt}&=k_{ia}^{+}m_{i}\mu_{a}\,\underline{\underline{+\xi_{ia}^{+}}}\,-(\sigma_{ia}+\kappa_{ia}+k_{ia}^{-})c_{ia}\,\underline{\underline{+(\xi_{ia}^{\rm st}+\xi_{ia}^{\rm cat}+\xi_{ia}^{-})}}~{}~{},\end{array}$$

$$\begin{array}[]{r@{}l}&{}\left\langle\xi_{i}(t)\xi_{i}(t^{\prime})\right\rangle=(b_{i}+d_{i}\left\langle m_{i}\right\rangle)~{}\delta(t-t^{\prime})~{}~{},\\ &{}\left\langle\xi_{a}(t)\xi_{a}(t^{\prime})\right\rangle=(\beta_{a}+\delta_{a}\left\langle\mu_{a}\right\rangle)~{}\delta(t-t^{\prime})~{}~{},\\ &{}\left\langle\xi_{ia}^{+}(t)\xi_{ia}^{+}(t^{\prime})\right\rangle=k_{ia}^{+}\left\langle m_{i}\right\rangle\left\langle\mu_{a}\right\rangle~{}\delta(t-t^{\prime})~{}~{},\\ &{}\left\langle\xi_{ia}^{-}(t)\xi_{ia}^{-}(t^{\prime})\right\rangle=k_{ia}^{-}\left\langle c_{ia}\right\rangle~{}\delta(t-t^{\prime})~{}~{},\\ &{}\left\langle\xi_{i}^{\rm cat}(t)\xi_{i}^{\rm cat}(t^{\prime})\right\rangle=\kappa_{ia}\left\langle c_{ia}\right\rangle~{}\delta(t-t^{\prime})~{}~{},\\ &{}\left\langle\xi_{i}^{\rm st}(t)\xi_{i}^{\rm st}(t^{\prime})\right\rangle=\sigma_{ia}\left\langle c_{ia}\right\rangle~{}\delta(t-t^{\prime})~{}~{},\end{array}$$

$$\frac{d\mathbf{x}}{dt}=\mathbf{f}(\mathbf{x})+\mathbf{\xi },$$

$$\frac{d}{dt}\delta\mathbf{x}=\mathbf{S}\delta\mathbf{x}+\bm{\xi}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}\mathbf{S}=\frac{d\mathbf{f}}{d\mathbf{x}}\bigg{|}_{\mathbf{x=\left\langle x\right\rangle}}~{}~{},$$

$$⟨\delta x_a\delta x_b⟩=-\sum _{i,l,s,r}{B}_{as}{B}_{br}\frac{{\Gamma }_{il}}{{\lambda }_s+{\lambda }_r}(B^{-1}{)}_{si}(B^{-1}{)}_{rl},$$

$$\begin{array}[]{r@{}l}P(k,\tau|\mathbf{x},t)d\tau=&\,\,\text{Prob}\{\text{no process between time $t$ and time $t+\tau$}\}\times\\ &\,\,\times\,\,\text{Prob}\{\text{process $k$ between time $t+\tau$ and time $t+\tau+d\tau$}\}\\ \equiv&\,\,P_{0}\times P_{k}~{}~{}.\end{array}$$

$$P_0={\left(1-\sum _{k^{\prime }}{f}_{k^{\prime }}\frac{\tau }{K}\right)}^K\simeq e^{-\tau {\sum }_{k^{\prime }}{f}_{k^{\prime }}}(K\gg 1).$$

$$P(k,\tau |\mathbf{x},t)\simeq f_k{e}^{-\tau {\sum }_{k^{\prime }}{f}_{k^{\prime }}},$$

$$P(k,\tau |\mathbf{x},t)\simeq \underset{\text{prob. of waiting time τ}}{\underbrace{\left(\sum _{k^{\prime }}{f}_{k^{\prime }}\right)e^{-\tau {\sum }_{k^{\prime }}{f}_{k^{\prime }}}}}\times \underset{\text{prob. of process k}}{\underbrace{\frac{f_k}{{\sum }_{k^{\prime }}{f}_{k^{\prime }}}}}.$$

$$\tau =-\frac{\ln (u)}{{\sum }_{k^{\prime }}{f}_{k^{\prime }}}$$

$${\rho }_{ij}=\frac{⟨m_i{m}_j⟩-⟨m_i⟩⟨m_j⟩}{\sqrt{⟨m_i^2⟩-{⟨m_i⟩}^2}\sqrt{⟨m_j^2⟩-{⟨m_j⟩}^2)}}\equiv \frac{\mathrm{cov}(m_i,m_j)}{\sqrt{⟨\delta m_i^2⟩}\sqrt{⟨\delta m_j^2⟩}},$$

Full text

Introduction

microRNAs (miRNAs) –short, endogenous, noncoding RNAs that operate post-transcriptionally via sequence-specific binding to target RNAs– are increasingly recognized as key actors in the regulation of eukaryotic gene expression bartel ; flynt ; cech ; gurt ; metaz . Following transcription (from either introns of protein-coding genes or from miRNA-specific genes) and maturation, miRNAs get incorporated into specialized, multiprotein complexes known as RISCs (short for RNA-induced silencing complexes) risc . Once within a RISC, the miRNA provides the pattern to bind specific sites called miRNA response elements (MREs) found on their target RNAs chan ; why . Effective base pairing typically requires 6- to 9-nucleotide complementarity, and leads to negative gene expression control through either mRNA destabilization or translational repression chek ; jona ; djur . The fact that miRNA expression is significantly tissue-specific places miRNAs at the center of the regulatory layer that controls the composition of the protein repertoire and cell type specificity bart ; liang ; fran ; eber . Still, many aspects of miRNA biology suggest that this role might be exerted through a broader and more complex, yet possibly more subtle, class of mechanisms.

In first place, miRNAs appear to be highly conserved in vertebrates and invertebrates, and their mRNA target structure displays a significant degree of conservation in higher organisms bere ; josh . For instance, more than half of human genes are conserved miRNA targets, including a large number of weak-interacting sites that appear to be under selective pressure to be maintained frie . Such a strong degree of conservation suggests that protein levels may need to be fine tuned within extremely precise ranges baek . Quantitative studies together with the statistical overrepresentation of noise-buffering motifs within the miRNA-RNA network indeed supports this idea shim ; tsang ; reda , and recent experiments have confirmed miRNA’s ability to stabilize output levels for lowly expressed proteins sici . Yet, the amount of noise reduction that can be achieved even in optimal conditions does not seem to justify a view of noise suppression as the key evolutionary driver for a significantly conserved miRNA targeting pattern wangs ; das ; ober ; schm .

Secondly, miRNA targets are known to include, together with messenger RNAs, a host of ncRNA species like lncRNAs as well as pseudogenes guil ; hans ; eber2 . On one hand, miRNA sponging by ncRNAs can clearly be critical in determining both miRNA levels and their potential for translational repression. On the other, it substantially increases the complexity of the network of miRNA-RNA interactions. It is now clear that each long RNA molecule can typically be targeted by multiple miRNAs, while every miRNA can interact with a very large number of distinct RNAs, generating an extended interaction network stretching across the entire transcriptome suma ; helw ; kimd ; zavo . Now the ability of miRNAs to regulate gene expression is ultimately linked to the overall target availability, and tends to get weaker as the number of targets (more precisely, of potential binding sites) increases, the so-called ‘dilution’ effect arve . This leaves room to search for alternative mechanisms through which miRNAs could exert a regulatory function, even at the non-local (up to system-scale) level.

The heterogeneity of the miRNA-RNA network and the fact that repression potential depends tightly on molecular levels suggest that competition to bind miRNAs might be a contributing factor in the establishment of robust protein profiles levine ; fzor . In rough terms, the essence of the so-called ceRNA hypothesis (whereby ‘ceRNA’ stands for ‘competing endogenous RNA’) is that, due to a cross-correlation of molecular levels, competition can induce an effective positive coupling between miRNA targets, such that a perturbation affecting the level of one target could be broadcast to its competitor via the subsequent shift in miRNA availability salm . In this respect, one might say that RNAs form a sort of ‘molecular ecosystem’, where mutual dependencies can be established post-transcriptionally via miRNA-mediated interactions driven by competition. The ceRNA scenario has received much attention since its formulation, both ex vivo and in synthetic systems (see e.g. tay ; vano ; karr ; tayy ; yuany ; sgro ). Effective interactions coupling RNAs targeted by the same miRNAs (which can be probed e.g. by over-expressing miRNAs or targets) are now known to be implicated in a variety of processes, from development and differentiation fati , to stress response stress and disease alva ; anas , and have been investigated in connection to their perspective therapeutic usefulness sanc .

Still, it has also become clear that the theoretical appeal of the ceRNA effect is not easily translated into quantitative understanding. A key issue is that of fine tuning. Several conditions clearly factor in the emergence of the ceRNA scenario. The possibility to turn competition between miRNA targets into an effective positive coupling between them presupposes for instance a cross-coordination of molecular levels, as a large excess (resp. scarcity) of miRNAs with respect to targets or binding sites will necessarily result into a completely repressed (resp. unrepressed) profile jens ; denzler . The ceRNA scenario would naturally become less realistic if kinetic parameters had to be tightly tuned in order to allow for ceRNA crosstalk conditions to arise. In addition, experiments suggest that a relatively small number of targets are usually sensitive to modulation in miRNA availability. Moreover, which targets are responsive depends on miRNA levels alau ; boss ; denz . The emergent selectivity and adaptability of ceRNA interactions should be reconciled with the heterogeneity observed in the miRNA-RNA interaction network in which each miRNA can regulate up to hundreds of targets.

Mathematical and in silico models developed in recent years have shed light on several of these issues and revealed many unexpected traits wang ; laix . This chapter aims at reviewing the methods employed and the key features of the ceRNA scenario that such studies suggest.

Our starting point is a generic, minimal deterministic mathematical model of post-transcriptional regulation whose steady states can be fully characterized analytically and numerically. Despite its roughness, it allows to precisely quantify the sensitivity of a ceRNA to alterations in the level of one of its competitors, sufficing to capture many of the central characteristics of miRNA-based regulation from basic assumptions about the underlying processes. In particular, miRNA-ceRNA interaction strengths and silencing/sequestration mechanisms emerge, together with the relative abundance of regulators and targets, as key factors for the onset and character of ceRNA crosstalk, including its selectivity. Moreover, heterogeneities in kinetic parameters as well as in miRNA-ceRNA interaction topology are major drivers of ceRNA crosstalk in a broad range of parameter values. The picture obtained at stationarity can be extended to out-of-equilibrium regimes. In particular, one can characterize a ‘dynamical’ ceRNA effect, which can be stronger than the equilibrium one, as well as the typical timescales required to reach stationary crosstalk.

Passing from a deterministic to a stochastic description, one can address the behaviour of fluctuations in molecular levels and evaluate the ability of miRNA-based regulatory elements to process noise. We will show in particular that the ceRNA mechanism can provide a generic pathway to the reduction of intrinsic noise both for individual proteins and for complexes formed by sub-units sharing a miRNA regulator (which might explain why interacting proteins are frequently regulated by miRNA clusters). The processing of extrinsic (transcriptional) noise is more involved. While ceRNA crosstalk is generically hampered by it, specific patterns of transcriptional correlations can actually result in enhanced noise buffering and in the emergence of complex (e.g. bistable) expression patterns. On the other hand, one can quantify the physical limits to crosstalk intensity by considering how different sources of noise affect it. It turns out that the size of target derepression upon the activation of its competitor is a crucial determinant of the strength of miRNA-mediated ceRNA regulation. When it is sufficiently large, post-transcriptional crosstalk can be as effective as direct transcriptional regulation in controlling expression levels. In specific cases, ceRNA crosstalk may even represent the most effective mechanism to tune gene expression.

An especially important question (and a difficult one, in view of the fact that the effect can be rather modest) concerns the quantification of ceRNA crosstalk intensity, and specifically the identification of unambiguous crosstalk markers that can be validated both experimentally and through the analysis of transcriptional data. We shall examine a few alternatives that have been employed, highlighting the different motivations underlying their use, their physical meaning and their respective limitations.

Models and methods

Deterministic model

The simplest mathematical representation of the dynamics of $N$ ceRNA species and $M$ miRNA species interacting in a miRNA-ceRNA network is based on deterministic mass-action kinetics. We shall denote by $m_{i}$ the level of ceRNA species $i$ (with $i$ ranging from 1 to $N$), by $\mu_{a}$ the level of miRNA species $a$ (ranging from 1 to $M$), and by $c_{ia}$ the levels of miRNA-ceRNA complexes. Based on experimental evidence, one can assume that all miRNA molecules are ‘active’, i.e. bound to an Argonaute protein and ready to attach to a target ceRNA. This allows to discard the kinetic steps leading to the formation of the RNA-induced silencing complex (RISC). In such conditions, concentration variables evolve in time due to

synthesis and degradation events, 2. 2.

complex binding and unbinding events, 3. 3.

the processing of complexes.

The latter in turn can follow two distinct pathways: a catalytic one, leading to the degradation of the ceRNA with the re-cycling of the miRNA; and a stoichiometric one, where both molecules are degraded, possibly after sequestration into P-bodies vale ; bacc . The relevant processes (see Fig. 1A and B for a sketch) are therefore

[TABLE]

Correspondingly, the mass action kinetic equations take the form (see e.g. figl ; bosi ; meht )

[TABLE]

where the physical meaning of parameters is summarized in Table 2.1 and where the indices $i$ and $a$ range from $1$ to $N$ and from $1$ to $M$, respectively.