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Conceived and designed the experiments: MO CB MC. Performed the experiments: MO CB. Analyzed the data: MO CB DC. Wrote the paper: MO CB MC.

The authors have declared that no competing interests exist.

MicroRNAs are endogenous non-coding RNAs which negatively regulate the expression of protein-coding genes in plants and animals. They are known to play an important role in several biological processes and, together with transcription factors, form a complex and highly interconnected regulatory network. Looking at the structure of this network, it is possible to recognize a few overrepresented motifs which are expected to perform important elementary regulatory functions. Among them, a special role is played by the microRNA-mediated feedforward loop in which a master transcription factor regulates a microRNA and, together with it, a set of target genes. In this paper we show analytically and through simulations that the incoherent version of this motif can couple the fine-tuning of a target protein level with an efficient noise control, thus conferring precision and stability to the overall gene expression program, especially in the presence of fluctuations in upstream regulators. Among the other results, a nontrivial prediction of our model is that the optimal attenuation of fluctuations coincides with a modest repression of the target expression. This feature is coherent with the expected fine-tuning function and in agreement with experimental observations of the actual impact of a wide class of microRNAs on the protein output of their targets. Finally, we describe the impact on noise-buffering efficiency of the cross-talk between microRNA targets that can naturally arise if the microRNA-mediated circuit is not considered as isolated, but embedded in a larger network of regulations.

The expression of protein-coding genes is controlled by a complex network of regulatory interactions. It is becoming increasingly appreciated that the post-transcriptional repression by microRNAs, a class of small non-coding RNAs, is a key layer of regulation in several biological processes. Since gene expression is a fundamentally stochastic process, the mixed network (comprising transcriptional and microRNA-mediated regulations) has to reliably perform its functions in the presence of noise. In this paper we investigate the function of one of the recurrent architectures of this network, the microRNA-mediated feedforward loops, using a detailed analytical model and simulations. With this approach we show that these regulatory circuits are appropriately designed so as to control noise, giving a rigorous mathematical proof of a previously proposed biological intuition. Moreover the theoretical framework introduced in this paper allows us to make nontrivial predictions that are presently in agreement with observed features of microRNA regulation and that could be more specifically tested experimentally in the future.

MicroRNAs (miRNAs) are endogenous small non-coding RNAs which negatively regulate the protein production of their targets in metazoans and plants. They are expected to target a substantial portion of the human genome

Recent work, reviewed in

(A) MiRNAs and corresponding targets can present different degrees of coexpression between the two extremes of concurrent expression (high correlation) and exclusive domains (high anticorrelation). These two opposite situations are expected to correspond to different functional roles (B) for the miRNA repression. A peculiar expression pattern, evidence of a functional aim, is a consequence of the network structure in which miRNAs are embedded. A high miRNA-target correlation can be achieved through the incoherent FFL (C), where the miRNA repression is opposite to the TF action. Whereas a failsafe control can be performed by a coherent FFL (D), in which the miRNA reinforces the TF action leading to mutually exclusive domains of miRNA-target expression.

It is by now well understood that gene espression is inherently a stochastic process

The main goal of our paper is to introduce and solve analytically a stochastic model describing these incoherent FFLs in order to give a proof to this conjecture. Our results show that with respect to the simple gene activation by a TF, the introduction of a miRNA-mediated repressing pathway can significantly dampen fluctuations in the target protein output for essentially all the choices of input parameters and initial conditions. As a test of our analysis we also performed extensive numerical simulations which nicely agree with our analytical results. It is important to stress (and we shall discuss this issue in detail in the following) that this noise buffering function is actually a precise consequence of the peculiar topolgy of the FFL. In fact, in order to fine-tune the level of a target protein one would not necessarily need a FFL topology. The same result could well be obtained with an independent miRNA (not under the control of the master TF which activates the target), but this choice would lead to strong fluctuations in the target expression. In the same theoretical framework we can show that the construction of an optimal noise filter does not necessarily imply a strong repression, in agreement with the observation that the miRNA down-regulation of a target is often modest

Here we focus on the incoherent FFL in

(A) A miRNA-mediated incoherent FFL that can be responsible for miRNA-target coexpression; (B) a gene activated by a TF; (C) an open circuit that leads to the same mean concentrations of the molecular species of the FFL in scheme A. (A′)(B′)(C′) Detailed representation of the modelization of the corresponding circuits. Rectangles represent DNA-genes, from which RNAs (

The miRNA action can direct translational repression or destabilization of target mRNAs

The parameter

The alternative choice of a degradation rate of mRNAs as a function of miRNA concentration does not yield significantly different results, as reported in

The probability of finding in our cell exactly

where

In order to solve the master equation for

We would like to emphasize that linearizing the Hill functions does not mean to linearize the model. In fact, even with a linearized dependence on the miRNA copy number, our model keeps a nonlinear contribution in the term encoding the target translation (due to the fact that it depends on both the number of miRNAs and mRNAs). As we will see later, this nonlinearity leads to non trivial consequences.

Despite this nonlinearity, the moment generating function approach

We now use the following properties of the moment generating function:

Noise in protein expression is originated by the combination of two types of fluctuations: intrinsic and extrinsic ones. Intrinsic fluctuations are essentially due to the stochasticity of the gene expression process. Extrinsic ones, instead, are due to the environment. In the latter case a prominent role is played by the noise transmitted by upstream genes

To show the noise buffering role of the miRNA-mediated incoherent FFL (

(A) An example of simulation results for the FFL (scheme on the right or more detailed in

The same fine-tuning of the mean target concentration achieved with a FFL could be equally obtained with an open circuit like the one in

(A) An example of simulation results for the FFL (scheme on the right or more detailed in

Stochastic equations are the natural formalism to describe a set of biochemical reactions when the number of molecules involved is small and thus fluctuations are important. As the number of molecules increases, the stochastic description smoothly converges, at least for linear systems, toward a deterministic one and stochastic equations can be substituited by ordinary differential equations (ODE). It is usually expected that even in the regime in which fluctuations cannot be neglected one could use ODE if interested only in the time evolution of the mean values. This approximation can be thought as a sort of “mean field” approach (by analogy with statistical mechanics where the mean field approximation is implemented by neglecting fluctuations). However, similarly to what happens in statistical mechanics in the proximity of a critical point, it may happen that, even at the level of mean values, the ODE description does not coincide with the (more rigorous) stochastic one. These breakdowns between the two descriptions are known as “deviant effects”

In a FFL, fluctuations of

In the previous sections we compared different regulatory circuits keeping the same amount of input noise, i.e. the same amount of fluctuations in the copy number of master TFs. Since the topology of a regulatory motif can have stronger effects on extrinsic rather than intrinsic noise

Setting the rates of degradation (

We report in

We maintain constant the number of TFs

The efficiency of the FFL in controlling the fluctuations of the target protein is a function of three main parameters: the number of master TFs (which in turn is a function of

The repression strength is defined as

(A) The coefficient of variation of the target protein

Another variable which can be tuned in order to achieve optimal noise reduction is the miRNA concentration. If we keep the number of TFs constant then the miRNA concentration

The last case that we consider in this section is the effect of different TF concentrations on the noise buffering properties of the FFL. It is expected that for high TF concentrations (i.e. high values of

To give a more comprehensive insight into the relation between noise control and target repression, we finally evaluate the buffering of fluctuations (

(A) The target noise

We consider the characterization of the optimal setting of miRNA-mediated incoherent FFLs for noise buffering, and the resulting U-shaped profile predicted for the noise reduction factor, as one of the major results of our analysis which, we expect, should be amenable of direct experimental validation. The fact that an optimal noise filtering is obtained with a rather modest reduction in the amount of the target protein also agrees with the recent experimental observation that miRNA down-regulation of targets is often modest

The capability of incoherent FFLs to reduce fluctuations was previously studied with simulations in the contest of purely transcriptional networks

(A) Detailed scheme of a purely transcriptional incoherent FFL. (B) The coefficient of variation of the target protein

Moreover some peculiarities (not currently included in our model) of the mixed-motif can further explain why it can be better suited for noise buffering. Firstly, fluctuations in RNA polymerase and ribosome abundance are possible sources of extrinsic noise in gene expression

Secondly, delays in the action of regulators (miRNA or proteins) in the indirect pathway from the master TF to the target can damage the noise buffering function (see

Finally, the presence of a nucleus makes the eukaryotic cell a two-compartment system with stochastic transport channels, with consequences on gene expression noise

A recent study pointed out that the action of a miRNA on a specific target gene expression is affected by the total number of miRNA targets and their mRNA abundance

(A) Scheme of a miRNA-mediated FFL with an additional independently transcribed target gene (second target). (B) The degree of protein downregulation

The model used so far for miRNA regulation was based on the hypothesis of perfectly catalytic action. The rate of translation of target mRNAs was assumed to be a nonlinear decreasing function of miRNA concentration, neglecting the details of mRNA-miRNA physical coupling with the implicit assumption that the downregulation process does not affect the available miRNA pool. A perfectly catalytic action does not predict any competition effect among multiple targets at equilibrium, since each target can only sense the available number of miRNAs without altering it. On the other hand, a stoichiometric model has been proposed in the context of sRNA regulation in bacteria

The corresponding stochastic model, of which equations 9 describe the mean-field limit, cannot be solved analytically starting from the master equation, therefore noise properties will be examined in the following with simulations only.

In the first place we evaluate the dependence of the target protein downregulation on the expression rate of the secondary target, starting from the model described by Equations 9. The dilution effect is shown in

As expected, a perfectly catalytic mode does not feel the effect of secondary mRNA targets (red line in

Since a high level of expression of secondary targets can determine a decrease of the average downregulation, it can potentially reduce the FFL ability in filtering out target fluctuations. In fact, also the noise reduction

While the corruption of the noise-buffering ability seems mainly due to the increase in the mean level of secondary messengers, there is another more subtle cause that gives a contribution: the uncorrelated fluctuations of secondary messengers. Since the secondary target is independently transcribed (not under the control of the master TF activating the miRNA gene) its fluctuations are expected to be completely uncorrelated with the miRNA ones, implying a random sequestration of miRNAs. To disentagle this contribution from the dilution effect, we studied the case of a secondary target transcribed at the same effective rate of the FFL target, but with different levels of fluctuations (see

Few cases of incoherent miRNA-mediated FFLs have been experimentally verified until now: a case involving c-Myc/E2F1 regulation

On the purely computational side, it is interesting to notice that in

Finally, a significant enrichment in oncogenes within the components of the FFLs was also observed

The type of regulatory action which a miRNA exerts on its targets can be rather well understood looking at the degree of coexpression with the targets

Our paper presents the first model explicitly built on the mixed version of the FFL. From a theoretical point of view, we addressed the detailed master equation describing the system (without neglecting the dynamics of mRNA), instead of the approximate Langevin description, and we were able to apply the moment generating function approach despite the presence of nonlinear terms that can give rise to deviant effects. This approach allowed us to take into account extrinsic fluctuations as the noise propagating from upstream genes, without an arbitrary definition of the extrinsic noise distribution. This strategy can be naturally extended to other circuits in the mixed network to test their potential role in the control of stochasticity.

Furthermore, we compared, in terms of noise buffering ability, miRNA-mediated FFLs with their purely transcriptional counterparts, where the miRNA is replaced by a protein that inhibits transcription rather than translation. This comparison shows that a miRNA regulator can be better suited for the noise buffering purpose.

Finally, we tryed to overcome the limitations in the analysis that can arise from considering a genetic circuit as operating in isolation. In this perspective, we evaluated the impact that the recently discovered dilution effect

The hypothesis of a role of miRNAs in noise buffering can shed new light on peculiar characteristics of miRNA regulation. As discussed in

Simulations were implemented by using Gillespie's first reaction algorithm

Details on the theoretical model, supplementary analysis, and simulations.

(0.75 MB PDF)

We would like to thank Mariama El Baroudi and Antonio Celani for useful discussions.