Our findings show that the simple Hill function described by Model 4 is sufficient to quantitatively describe the effect of RNA interference, at the mRNA and protein level, in mammalian cells in vitro, for varying concentration of siRNA oligomers.

One significant feature of Model 4 is that it can predict the saturation effect of the RNAi process that we observed experimentally. We considered the possibility that this saturation could be in fact due to the inability of the cell to uptake high concentration of siRNA oligomers, however recent experiments [27], prove that uptake of siRNA oligomers in cells is linear with the concentration of siRNA oligomers transfected, at least in the concentration range we used. Additionally, Khan *et al* in [31], observed upregulation of mRNA targets of endogenous micro-RNA when transfecting siRNA oligomers in mammalian cells. In order to explain this effect, they suggested a saturation of the RISC complex (or other necessary small RNA processing or transport machinery).

It has been demonstrated in [23, 26, 27] that the enzymatic activity of RISC can be efficiently modeled in-vitro as a classic Michaelis-Menten reaction, where the target mRNA is the substrate, the siRNA-loaded RISC is the active enzyme (at a constant concentration), and the product is the degraded mRNA. This is one feature that Model 4 does not capture; namely for a fixed amount of siRNA-loaded RISC (i.e. *X*
_{
s
}), Eq. (11) should approximate the Michaelis-Menten in Eq. (10), instead Model 4 becomes simply proportional to *X*
_{
m
}, as Model 1 and 2. Nevertheless, Model 4 approximates very well the experimental data. We believe this happens because enzymatic reactions have a typical *K*
_{
M
} much greater that the physiological concentration of their substrate (*X*
_{
m
}) [23], and the same happens for the RISC complex [23, 26]. In this condition, the Michaelis-Menten equation becomes
and therefore it is linear in *X*
_{
m
}, as predicted by Model 4. When the siRNA concentration varies, Model 4 predicts that the parameters
will change as a function of *X*
_{
s
} as described by Eq. (11).

The three parameters of Model 4 have a straightforward biological interpretation, and their values can be easily tuned to accommodate for different efficiencies of RNAi. For example, the parameter *d*
_{4} can be used to weigh the degradation due to the RNAi compared to the endogeneous mRNA degradation, and its strength, i.e. what is the maximal degradation rate that can be achieved. *θ*
_{4} quantifies the siRNA oligomers concentration needed to achieve half of the maximal degradation of the targeted mRNA. The *h*
_{4} coefficient can accommodate for multiple target sites on the same mRNA, or for the cooperativity of the RISC complex.

Clearly, the RNAi process is very complex and no one-to-one relationship can be found between parameters of Model 4 and RNAi biological components. Nevertheless, it has been shown in [27] that between 10^{4} and 10^{5} siRNA oligomers per cell (corresponding to a concentration in the range 10 pM-100 pM) are sufficient to reach half-maximal mRNA target degradation. Model 4 predicts that half-maximal degradation is achieved for an amount of siRNA oligomers equal to *θ*
_{4}. The value of this parameter when fitting mRNA levels (Table 2 and Additional file 1 Table A1) is *θ*
_{4} ≈ 0.1 pmol despite of the different cell-lines and mRNA-siRNA pairs tested (EGFP and tTA). This value corresponds to a concentration of 50 pM in our experimental setting, hence in good agreement with the previously reported range. Altogether these observations suggest that the quantity *θ*
_{4} could be cell-type, mRNA, and siRNA indipendent.

It is estimated that the concentration of active RISC in a cell is about 3 - 5 nM [23, 26, 32]. Taking into account that the volume of a mammalian cell is in the range 10^{-13}
*L* - 10^{-12}
*L*, then we can estimate that the number of active RISC in a cell is in the range 10^{3} - 10^{4}. The above observations suggest that saturation begins when the number of siRNA oligomers in a cell becomes comparable to the number of RISC molecules.

We observed that the parameters of Model 4 estimated when fitting protein levels (set II experiments) are very close to the ones estimated when fitting mRNA levels (set I experiments). Namely, the optimized values of *d*
_{4} and *h*
_{4} are very similar for both experimental data. This is important since these are two independent biological experiments. This proves the mathematical robustness of Model 4. The only parameter changing between the two sets of experiment is *θ*
_{4}, which represents the concentration of siRNA oligomers needed to achieve half of the maximal degradation rate (*d*
_{4}). This is reflected in Figure 2and Figure 3, where it is clear that saturation is achieved at about 1 pmol for the mRNA data (Figure 2) and at about 20 pmol for the protein data (Figure 3). This difference may be due to biological variability, or to the simplified model of protein translation dynamics we used (steady-state approximation).

We also conformed that Model 4 is cell-line-independent, mRNA-independent, and siRNA-independent, since it can accurately describe the RNA interference process on a different cell-line (CHO) expressing a different mRNA (tTA), silenced by a different siRNA oligomer.

Interestingly, the difference in Model 4 parameters, when testing a different mRNA-siRNA pair (i.e. tTA versus EGFP), shows that only *d*
_{4} (the maximal degradation rate) and *h*
_{4} (the cooperativity) change significantly, suggesting that these two parameters can be used to describe changes in siRNA-mRNA silencing specific strength, whereas *θ*
_{4} may be kept constant.

Recently it has been proposed that siRNA and microRNA efficacy, defined as the percentage decrease in the target mRNA level due to the silencing reaction, could be limited due to mRNA abundance [33]or to mRNA degradation rate [34].

Model 4 predicts that the percentage decrease in target mRNA level (obtained from Eq. (13) simply dividing by *k*
_{
m
}/*d*
_{
m
}) is indeed sensitive to *d*
_{
m
} (the target mRNA degradation rate), with a higher degradation rate corresponding to a weaker effect of the silencing reaction, and vice-versa. This result is in line with the experimental observation described in Larsson et al [34]. In addition, according to Model 4, the transcription rate *k*
_{
m
} of the target mRNA does not have any influence on the silencing reaction efficacy. The target mRNA abundance, in absence of the silencing reaction, is simply obtained from Eq. 1 as *k*
_{
m
}/*d*
_{
m
}. Smaller *d*
_{
m
} will correspond to a higher mRNA abundance (for a constant *k*
_{
m
}) therefore a correlation between mRNA abundance and sensitivity to mRNA can be found [33], but this is only an indirect effect mediated by the degradation rate, at least according to our model. Our conclusion is that siRNA-mediated degradation in mammalian cells can always be best represented as an enzymatic reaction described by an Hill function, whose parameters have to be tuned to the specific siRNA-mRNA pair.

The models discussed so far consider the average behavior of a population of cells. In the case of singe-cell experiments, these models might not be efficient enough due to their deterministic nature and will not be able to capture any stochastic effects.

Since RNA has a plethora of functional properties and plays many of roles in regulating gene expression, it has been used in a number of different studies as a tool for elucidating gene functions. In fact with RNAi it is possible to selectively knock-down any gene and even modulate its dosage [35]. RNA has also been used in the design of therapeutic molecules as well as metabolic reprogramming [36]. The potential uses of this versatile molecule are still very much under study, but their effectiveness depend on many variables such as, the concentration of the silencing reagent, the transfection techniques, the cell type used and the target type selection. In the present study we biologically validated for the first time a mathematical model (Model 4) that has a simple mathematical form, amenable to analytical investigations and a small set of parameters with an intuitive physical meaning that can be used both by the computational and the experimental community interested in the analysis and application of RNA interference.