 Research article
 Open Access
Dynamical modeling of microRNA action on the protein translation process
 Andrei Zinovyev†^{1, 2, 3}Email author,
 Nadya Morozova†^{4},
 Nora Nonne^{4},
 Emmanuel Barillot^{1, 2, 3},
 Annick HarelBellan^{4} and
 Alexander N Gorban^{5, 6}
https://doi.org/10.1186/17520509413
© Zinovyev et al; licensee BioMed Central Ltd. 2010
 Received: 27 September 2009
 Accepted: 24 February 2010
 Published: 24 February 2010
Abstract
Background
Protein translation is a multistep process which can be represented as a cascade of biochemical reactions (initiation, ribosome assembly, elongation, etc.), the rate of which can be regulated by small noncoding microRNAs through multiple mechanisms. It remains unclear what mechanisms of microRNA action are the most dominant: moreover, many experimental reports deliver controversial messages on what is the concrete mechanism actually observed in the experiment. Nissan and Parker have recently demonstrated that it might be impossible to distinguish alternative biological hypotheses using the steady state data on the rate of protein synthesis. For their analysis they used two simple kinetic models of protein translation.
Results
In contrary to the study by Nissan and Parker, we show that dynamical data allow discriminating some of the mechanisms of microRNA action. We demonstrate this using the same models as developed by Nissan and Parker for the sake of comparison but the methods developed (asymptotology of biochemical networks) can be used for other models. We formulate a hypothesis that the effect of microRNA action is measurable and observable only if it affects the dominant system (generalization of the limiting step notion for complex networks) of the protein translation machinery. The dominant system can vary in different experimental conditions that can partially explain the existing controversy of some of the experimental data.
Conclusions
Our analysis of the transient protein translation dynamics shows that it gives enough information to verify or reject a hypothesis about a particular molecular mechanism of microRNA action on protein translation. For multiscale systems only that action of microRNA is distinguishable which affects the parameters of dominant system (critical parameters), or changes the dominant system itself. Dominant systems generalize and further develop the old and very popular idea of limiting step. Algorithms for identifying dominant systems in multiscale kinetic models are straightforward but not trivial and depend only on the ordering of the model parameters but not on their concrete values. Asymptotic approach to kinetic models allows putting in order diverse experimental observations in complex situations when many alternative hypotheses coexist.
Keywords
 Reaction Network
 Protein Translation
 Ribosomal Subunit
 Elementary Reaction
 Steady State Rate
Background
MicroRNAs (miRNAs) are currently considered as key regulators of a wide variety of biological pathways, including development, differentiation and oncogenesis. Recently, remarkable progress was made in understanding of microRNA biogenesis, functions and mechanisms of action. Mature microRNAs are incorporated into the RISC effector complex, which includes as a key component an Argonaute protein. MicroRNAs affect gene expression by guiding the RISC complex toward specific target mRNAs. The exact mechanism of this inhibition is still a matter of debate. In the past few years, several mechanisms have been reported, and some of the reports contradict to each other (for review, see [1–3]). The inhibition mechanisms include, in particular, the inhibition of translation initiation (acting at the level of cap40S or 40SAUG60S association steps), the inhibition of translation elongation and the premature translation termination. MicroRNAmediated mRNA decay and sequestration of target mRNAs in Pbodies have been also proposed. Moreover, some microRNAs mediate target mRNA cleavage [4], chromatin reorganization followed by transcriptional repression or activation [5, 6], or translational activation [7, 8].
Concurrently, several reports have been published indicating an action of microRNAs at the level of initiation. An increasing number of papers reports that microRNAtargeted mRNAs shift towards the light fractions in polysomal profiles [18–20]. This shows a decrease of mature translating ribosomes, suggesting that microRNAs act at the initiation step. Moreover, several reports show that microRNAmediated inhibition is relieved when translation is driven by a capindependent mechanism such as IRESmRNA or AcappedmRNA [18, 20, 21]. This observation was confirmed in several invitro studies [22–25]. In particular, in one of those, an excess of eIF4F could relieve the inhibition, and the inhibition led to the decreased 80S in the polysomal gradient [22].
Most of the data indicating a shift towards the light polysomal fraction or the requirement for a capdependent translation are often interpreted in favour of involvement of microRNAs at early steps of translation, i.e., cap binding and 40S recruitment. However, some of them are also compatible with a block at the level of 60S subunit joining. This hypothesis is also supported by invitro experiments showing a lower amount of 60S relative to 40S on inhibited mRNAs. Moreover, toeprinting experiments show that 40S is positioned on the AUG [26]. Independently, it was shown that eIF6, an inhibitor of 60S joining, is required for microRNA action [27], but this was in contradiction with other studies [2].
Thus, the data on the exact step of translational inhibition are clearly contradictory. Taking also into account the data about mRNA degradation and Pbodies localization, it is difficult to draw a clear picture of the situation, and the exact mechanism by which microRNA represses mRNA expression is highly controversial, not mentioning the interrelations between the different mechanisms and their possible concomitant action. Several attempts to integrate the different hypotheses have been made [1–3, 28–30]. For example, it was proposed that one mechanism could act as a "primary" effect, and the other as a "secondary" mechanism, either used to reinforce the inhibition or as a backup mechanism. In others, the different mechanisms could all coexist, but occur differentially depending on some yet unidentified characteristics. For example, it has been observed than the same mRNA targeted by the same microRNA can be regulated either at the initiation or the elongation step depending on the mRNA promoter and thus on the mRNA nuclear history [31]. It was also proposed that technical (experimental) problems, including the variety of experimental systems used, may also account for these discrepancies [1–3]. However, this possibility does not seem to be sufficient to provide a simple and convincing explanation to the reported discrepancies.
A possible solution to exploit the experimental observations and to provide a rational and straightforward data interpretation is the use of mathematical models for microRNA action on protein translation. For many years, the process of protein synthesis is a subject of mathematical modeling with use of various approaches from chemical kinetics and theoretical physics. Many of the models created consider several stages of translation, however, most of them concentrate on the elongation and termination processes. In [32–34], a nonequilibrium statistical physics description of protein synthesis was proposed. Models taking into account gene sequence were developed in [33, 35–37]. These models can predict the probability of that a ribosome will completely terminate a transcript, spatiotemporal organization of ribosomes in polysome, dependence of the protein synthesis rate on various factors, such as presence of slow synonymous codons in the gene sequence [33, 37] and the frequency of nonsense errors [35]. Several models of the effect of microRNA on protein translation were proposed. Thus, in [38] the authors tried to determine which inhibition mechanism (via translation repression or transcript degradation) is the most abundant in mammalian cells using Bayesian modeling and microarray data. Quantitative features of sRNAmediated gene regulation were considered in [39]. A simple kinetic model of microRNAmediated mRNA degradation was proposed in [40] and compared to a temporal microarray dataset.
In this paper we will analyze two simple models of microRNA action on protein translation developed recently by Nissan and Parker [41]. They studied the microRNAdependent steady states rates of protein synthesis [41] and provided a critical analysis for the experiments with alternative mRNA cap structures and IRES elements [22, 23, 25]. This analysis led to a possible explanation of the conflicting results. The authors suggested that the relief of translational repression upon replacement of the cap structure can be explained if microRNA is acting on a step which is not ratelimiting in the modified system, in which case, the effect of microRNA can simply not be observed. It was claimed that it is impossible to discriminate between two alternative interpretations of the biological experiments with cap structure replacement, using sole monitoring of the steady state level of protein synthesis [41].
Two remarks can be made in this regard. Firstly, in practice not only the steady state level of protein can be observed but also other dynamical characteristics, such as the relaxation time, i.e. the time needed to achieve the steady state rate after a perturbation (such as restarting the translation process). We argue that having these measurements in hands, one can distinguish between two alternative interpretations. In this paper we provide such a method from the same models as constructed by Nissan and Parker, for comparison purposes. However, the method applied can be easily generalized for other models.
Secondly, even in the simple nonlinear model of protein translation, taking into account the recycling of ribosomal components, it is difficult to define what is the rate limiting step. It is known from the theory of asymptotology of biochemical networks [42] that even in complex linear systems the "rate limiting place" notion is not trivial and cannot be reduced to a single reaction step. Moreover, in nonlinear systems the "rate limiting place" can change with time and depend on the initial conditions. Hence, conclusions of [41] should be reconsidered for the nonlinear model, made more precise and general. The notion of rate limiting step should be replaced by the notion of dominant system.
In this paper we perform careful analysis of the Nissan and Parker's models and provide their approximate analytical solutions, which allows us to generalize the conclusions of [41] and make new checkable predictions on the identifiability of active mechanism of microRNAdependent protein translation inhibition.
The paper is organized in the following way. The Methods contain introduction, all necessary definitions and basic results of the asymptotology of biochemical reaction networks (quasiequilibrium, quasi steadystate, limiting step and dominant system asymptotics), used further in the Results. The Methods section is deliberately made rather detailed to make the reading selfsufficient. These details are necessary for reproducing the analytical calculations but not for understanding the interpretation of the modeling results. When the most important notions are introduced (such as dominant system, critical parameters), they are emphasized in bold. The Results section starts with listing model assumptions, followed by deriving semianalytical solutions of Nissan and Parker's model and interpretation of the analysis results and prediction formulations. For those readers who are interested only in the applied aspect of this work, it is possible to skip the details of deriving the analytical solutions and start reading from the "Model assumptions" section, look at the definition of the model parameters and variables and continue with 'the 'Effect of microRNA on the translation dynamics" section.
Results
Model assumptions
We consider two models of action of microRNA on protein translation process proposed in [41]: the simplest linear model, and the nonlinear model which explicitly takes into account recycling of ribosomal subunits and initiation factors.
Both models, of course, are significant simplifications of biological reality. Firstly, only a limited subset of all possible mechanisms of microRNA action on the translation process is considered (see Fig. 1). Secondly, all processes of synthesis and degradation of mRNA and microRNA are deliberately neglected. Thirdly, interaction of microRNA and mRNA is simplified: it is supposed that when microRNA is added to the experimental system then only mRNA with bound microRNAs are present (this also assumes that the concentration of microRNA is abundant with respect to mRNA). Concentrations of microRNA and mRNA are supposed to be constant. Interaction of only one type of microRNA and one type of mRNA is considered (not a mix of several microRNAs). The process of initiation is greatly simplified: all initiation factors are represented by only one molecule which is marked as eIF4F.
Finally, the classical chemical kinetics approach is applied, based on solutions of ordinary differential equations, which supposes sufficient and wellstirred amount of both microRNAs and mRNAs. Another assumption in the modeling is the mass action law assumed for the reaction kinetic rates.
It is important to underline the interpretation of certain chemical species considered in the system. The ribosomal subunits and the initiation factors in the model exist in free and bound forms, moreover, the ribosomal subunits can be bound to several regions of mRNA (the initiation site, the start codon, the coding part). Importantly, several copies of fully assembled ribosome can be bound to one mRNA. To model this situation, we have to introduce the following quantification rule for chemical species: amount of "ribosome bound to mRNA" means the total number of ribosomes translating proteins, which is not equal to the number of mRNAs with ribosome sitting on them, since one mRNA can hold several translating ribosomes (polyribosome). In this view, mRNAs act as places or catalyzers, where translation takes place, whereas mRNA itself formally is not consumed in the process of translation, but, of course, can be degraded or synthesized (which is, however, not considered in the models described further).
The simplest linear protein translation model
The list of chemical species in the model is the following:
1. 40S, free small ribosomal subunit.
2. mRNA:40S, small ribosomal subunit bound to the initiation site.
3. AUG, small ribosomal subunit bound to the start codon.
The catalytic cycle is formed by the following reactions:
1. 40S → mRNA:40S, Initiation complex assembly (rate k _{1}).
2. mRNA:40S → AUG, Some late and capindependent initiation steps, such as scanning the 5'UTR for the start AUG codon recognition (rate k _{2}) and 60S ribosomal unit joining.
3. AUG → 40S, combined processes of protein elongation and termination, which leads to production of the protein (rate k _{3}), and fall off of the ribosome from mRNA.
where Prsynth(t) is the rate of protein synthesis.
Following [41], let us assume that k_{3} >> k_{1}, k_{2}. This choice was justified by the following statement: "...The subunit joining and protein production rate (k_{3}) is faster than k_{1} and k_{2} since mRNA:40S complexes bound to the AUG without the 60S subunit are generally not observed in translation initiation unless this step is stalled by experimental methods, and elongation is generally thought to not be rate limiting in protein synthesis..." [41].
Now let us consider two experimental situations: 1) the rates of the two translation initiation steps are comparable k_{1} ≈ k_{2}; 2) the capdependent rate k_{1} is limiting: k_{1} << k_{2}. Accordingly to [41], the second situation can correspond to modified mRNA with an alternative capstructure, which is much less efficient for the assembly of the initiation factors, 40S ribosomal subunit and polyA binding proteins.
Interestingly, experiments with cap structure replacement were made and the effect of microRNA action on the translation was measured [22, 23]. No change in the protein rate synthesis after applying microRNA was observed. From this it was concluded that microRNA in this system should act through a capdependent mechanism (i.e., the normal "wildtype" cap is required for microRNA recruitment). It was argued that this could be a misinterpretation [41] since in the "modified" system, capdependent translation initiation is a rate limiting process (k_{1} << k_{2}), hence, even if microRNA acts in the capindependent manner (inhibiting k_{2}), it will have no effect on the final steady state protein synthesis rate. This was confirmed this by the graph similar to the Fig. 3a.
From the analytical solution (2) we can further develop this idea and claim that it is possible to detect the action of microRNA in the "modified" system if one measures the protein synthesis relaxation time: if it significantly increases then microRNA probably acts in the capindependent manner despite the fact that the steady state rate of the protein synthesis does not change (see the Fig. 3b). This is a simple consequence of the fact that the relaxation time in a cycle of biochemical reactions is limited by the second slowest reaction (see [42] or the "Dominant system for a simple irreversible catalytic cycle with limiting step" section in Methods). If the relaxation time is not changed in the presence of microRNA then we can conclude that none of the two alternative mechanisms of microRNAbased translation repression is activated in the system, hence, microRNA action is dependent on the structure of the "wildtype" transcript cap.
Modeling two mechanisms of microRNA action in the simplest linear model
Observable value  Initiation(k_{1})  Step after initiation, capindependent(k_{2})  Elongation (k_{3}) 

Wildtype cap  
Steadystate rate  decreases  decreases  no change 
Relaxation time  increases slightly  increases slightly  no change 
Acap  
Steadystate rate  decreases  no change  no change 
Relaxation time  no change  increases drastically  no change 
This conclusion suggests the notion of a kinetic signature of microRNA action mechanism which we define as the set of measurable characteristics of the translational machinery dynamics (features of time series for protein, mRNA, ribosomal subunits concentrations) and the predicted tendencies of their changes as a response to microRNA action through a particular biochemical mechanism.
The nonlinear protein translation model
To explain the effect of microRNA interference with translation initiation factors, a nonlinear version of the translation model was proposed [41] which explicitly takes into account recycling of initiation factors (eIF4F) and ribosomal subunits (40S and 60S).
1. 40S, free 40S ribosomal subunit.
2. 60S, free 60S ribosomal subunit.
3. eIF4F, free initiation factor.
4. mRNA:40S, formed initiation complex (containing 40S and the initiation factors), bound to the initiation site of mRNA.
5. AUG, initiation complex bound to the start codon of mRNA.
6. 80S, fully assembled ribosome translating protein.
There are four reactions in the model, all considered to be irreversible:
1. 40S + eIF4F → mRNA:40S, assembly of the initiation complex (rate k _{1}).
2. mRNA:40S → AUG, some late and capindependent initiation steps, such as scanning the 5'UTR by for the start codon AUG recognition (rate k _{2}).
3. AUG → 80S, assembly of ribosomes and protein translation (rate k _{3}).
4. 80S → 60S+40S, recycling of ribosomal subunits (rate k _{4}).
where [40S] and [60S] are the concentrations of free 40S and 60S ribosomal subunits, [eIF 4F] is a concentration of free translation initiation factors, [mRNA : 40S] is the concentration of 40S subunit bound to the initiation site of mRNA, [AUG] is the concentration of the initiation complex bound to the start codon, [80S] is the concentration of ribosomes translating protein, and Prsynth is the rate of protein synthesis.
with the following justification: "...The amount 40S ribosomal subunit was set arbitrarily high ... as it is thought to generally not be a limiting factor for translation initiation. In contrast, the level of eIF4F, as the canonical limiting factor, was set significantly lower so translation would be dependent on its concentration as observed experimentally... Finally, the amount of subunit joining factors for the 60S large ribosomal subunit were estimated to be more abundant than eIF4F but still substoichiometric when compared to 40S levels, consistent with in vivo levels... The k_{4} rate is relatively slower than the other rates in the model; nevertheless, the simulation's overall protein production was not altered by changes of several orders of magnitude around its value..." [41].
Notice that further in our paper we show that the last statement about the value of k_{4} is needed to be made more precise: in the model by Nissan and Parker, k_{4}is a critical parameter (see "Asymptotology and dynamical limitation theory for biochemical reaction networks" section in Methods). It does not affect the steady state protein synthesis rate only in one of the possible scenarios (inefficient initiation, deficit of the initiation factors).
Steady state solution
provided that α << 1  δ or α << 1  β. In the expression for x_{1} we cannot neglect the term proportional to a, to avoid zero values in (13).
The solution x_{2} is always negative, which means that one can have one positive solution x_{0} << 1 if and two positive solutions x_{0} and x_{1} if . However, from (12), (14) and (10) it is easy to check that if x_{1} > 0 then x_{0} does not correspond to a positive value of [eIF 4F]_{ s }. This means that for a given combination of parameters satisfying (10) we can have only one steady state (either x_{0} or x_{1}).
The two values x = x_{0} and x = x_{1} correspond to two different modes of translation. When, for example, the amount of the initiation factors [eIF 4F]_{0} is not enough to provide efficient initiation ( , x = x_{1}) then most of the 40S and 60S subunits remain in the free form, the initiation factor [eIF 4F] being always the limiting factor. If the initiation is efficient enough , then we have x = x_{0} << 1 when almost all 60S ribosomal subunits are engaged in the protein elongation, and [eIF 4F] being a limiting factor at the early stage, however, is liberated after and ribosomal subunits recycling becomes limiting in the initiation (see the next section for the analysis of the dynamics).
This explains the numerical results obtained in [41]: with low concentrations of [eIF 4F]_{0} microRNA action would be efficient only if it affects k_{2} or if it competes with eIF 4F for binding to the mRNA cap structure (thus, effectively further reducing the level [eIF 4F]_{0}) With higher concentrations of [eIF 4F]_{0}, other limiting factors become dominant: [60S]_{0} (availability of the heavy ribosomal subunit) and k_{4} (speed of ribosomal subunits recycling which is the slowest reaction rate in the system). Interestingly, in any situation the protein translation rate does not depend on the value of k_{1} directly (of course, unless it does not become "globally" rate limiting), but only through competing with eIF 4F (which makes the difference with the simplest linear protein translation model).
Equation (15) explains also some experimental results reported in [22]: increasing the concentration of [eIF 4F] translation initiation factor enhances protein synthesis but its effect is abruptly saturated above a certain level.
and N_{ polysome }changes in the same way as the protein synthesis steady state value.
Analysis of the dynamics
It was proposed to use the following model parameters in [41]: k_{1} = k_{2} = 2, k_{3} = 5, k_{4} = 1, [40S]_{0} = 100, [60S]_{0} = 25, [eIF 4F]_{0} = 6. As we have shown in the previous section, there are two scenarios of translation possible in the Nissan and Parker's model which we called "efficient" and "inefficient" initiation. The choice between these two scenarios is determined by the critical combination of parameters . For the original parameters from [41], β = 0.48 < 1 and this corresponds to the simpler onestage "inefficient" initiation scenario. To illustrate the alternative situation, we changed the value of k_{4} parameter, putting it to 0.1, which makes β = 4.8 > 1. The latter case corresponds to the "efficient" initiation scenario, the dynamics is more complex and goes in three stages (see below).
1) Stage 1: Relatively fast relaxation with conditions [40S] >> [eIF 4F], [60S] >> [AUG]. During this stage, the two nonlinear reactions 40S + eIF 4F → mRNA : 40S and AUG + 60S → 80S can be considered as pseudomonomolecular ones: eIF 4F → mRNA : 40S and AUG → 80S with rate constants dependent on [40S] and [60S] respectively. This stage is characterized by rapidly establishing quasiequilibrium of three first reactions (R1, R2 and R3 with k _{1}, k _{2} and k _{3} constants). Biologically, this stage corresponds to assembling of the translation initiation machinery, scanning for the start codon and assembly of the first full ribosome at the start codon position.
2) Transition between Stage 1 and Stage 2.
3) Stage 2: Relaxation with the conditions [40S] >> [eIF 4F], [60S] << [AUG]. During this stage, the reactions 40S + eIF 4F → mRNA : 40S and AUG + 60S → 80S can be considered as pseudomonomolecular eIF 4F → mRNA : 40S and 60S → 80S. This stage is characterized by two local quasisteady states established in the two network reaction cycles (formed from R1R2 and R3R4 reactions). Biologically, this stage corresponds to the first round of elongation, when first ribosomes moves along the coding region of mRNA. The small ribosomal subunit 40S is still in excess which keeps the initiation stage (reaction R1R2 fluxes) relatively fast.
4) Transition between Stage 2 and Stage 3.
5) Stage 3: Relaxation with the conditions [40S] << [eIF 4F], [60S] << [AUG]. During this stage, the reactions 40S + eIF 4F → mRNA : 40S and AUG + 60S → 80S can be considered as pseudomonomolecular 40S → mRNA : 40S and 60S → 80S. During this stage all reaction fluxes are balanced. Biologically, this stage corresponds to the stable production of the protein with constant recycling of the ribosomal subunits. Most of ribosomal subunits 40S are involved in protein elongation, so the initiation process should wait the end of elongation for that they would be recycled.
Stage 1: translation initiation and assembly of the first ribosome at the start codon
From these equations, one can determine the effective duration of the Stage 1: by definition, it will be finished when one of the two conditions ([40S] >> [eIF 4F], [60S] >> [AUG]) will be violated, which happens at times ~ and ~ correspondingly, hence, the second condition will be violated first (from [60S]_{0} < [40S]_{0}).
Stage 2: first stage of protein elongation, initiation is still rapid
At some point, the amount of free small ribosomal subunit 40S, which is abundant at the beginning of the Stage 2, will not be sufficient to support rapid translation initiation. Then the initiation factor eIF 4F will not be the limiting factor in the initiation and the condition [40S] >> [eIF 4F] will be violated. We can estimate this time as .
Stage 3: steady protein elongation, speed of initiation equals to speed of elongation
where t"' is the time when the Stage 3 of the relaxation starts. This relaxation goes relatively fast, since k_{3}[AUG]_{t=t'''}is relatively big. So, during the Stage 3, one can consider the cycle R 1  R 4 equilibrated, with [80S] = [80]_{ s }, [60S] = [60]_{ s }values.
The solution for the Stage 3 can be further simplified if k_{2} << k_{1}[eIF 4F]_{t = t'''}or k_{2} >> k_{1} [eIF 4F]_{t = t'''}.
Transitions between stages
Along the trajectory of the dynamical system (6) there are three dominant system each one transforming into another. At the transition between stages, two neighbor dominant systems are united and then split. Theoretically, there might be situations when the system can stay in these transition zones for long periods of time, even infinitely. However, in the model (6) this is not the case: the trajectory rapidly passes through the transition stages and jumps into the next dominant system approximation.
Three dominant approximations can be glued, using the concentration values at the times of the switching of dominant approximation as initial values for the next stage. Note that the Stage 2 has essentially one degree of freedom since it can be approximated by a single equation (23). Hence, one should only know one initial value [40S]_{t = t''}to glue the Stages 1 and 2. The same is applied to the gluing of Stages 2 and 3, since in the end of Stage 2 all variable values are determined by the value of [40S]_{t = t'''}.
Case of always limiting initiation
As it follows from our analysis, the most critical parameter of the nonlinear protein translation model is the ratio . Above we have considered the case β > 1 which is characterized by a switch of the limiting factor in the initiation (from eIF 4F at the Stages 1 and 2 to 40S at the Stage 3).
for which the solution derived above is not directly applicable. However, the analytical calculations in this case can be performed in the same fashion as above. The detailed derivation of the solution is given in Additional file 1. The effect of putting k_{1} very small on the steady state protein synthesis and the relaxation time is shown on Fig. 7.
In a similar way all possible solutions of the equations (6) with very strong inhibitory effect of microRNA on a particular translation step can be derived. These solutions will describe the situation when the effect of microRNA is so strong that it changes the dominant system (limiting place of the network) by violating the initial constraints (10) on the parameters (for example, by making k_{3} smaller than other k_{ i }s). Such possibility exists, however, it can require too strong (nonphysiological) effect of microRNAdependent translation inhibition.
Effect of microRNA on the translation dynamics
Our analysis of the nonlinear Nissan and Parker's model showed that the protein translation machinery can function in two qualitatively different modes, determined by the ratio . We call these two modes "efficient initiation" (β > 1) and "inefficient initiation" (β < 1) scenarios. Very roughly, this ratio determines the balance between the overall speeds of initiation and elongation processes. In the case of "efficient initiation" the rate of protein synthesis is limited by the speed of recycling of the ribosomal components (60S). In the case of "inefficient initiation" the rate of protein synthesis is limited by the speed of recycling of the initiation factors (eIF4F). Switching between two modes of translation can be achieved by changing the availability of the corresponding molecules ([60S]_{0} or [eIF 4F]_{0}) or by changing the critical kinetic parameters (k_{2} or k_{4}). For example, changing k_{4} from 1 (Fig. 7c) to 0.1 (Fig. 7a), performs such a switch for the original parameter values from [41].
As a result of the dynamical analysis, we can assemble an approximate solution of the nonlinear system under assumptions (10) about the parameters. An example of the approximate solution is given on Fig. 7. The advantage of such a semianalytical solution is that one can predict the effect of changing the system parameters. For example, on Fig. 7b the solution is compared to an exact numerical one, where the parameters have been changed but still obey the initial constraints (10).
One of the obvious predictions is that the dynamics of the system is not sensitive to variations of k_{3}, so if microRNA acts on the translation stage controlled by k_{3} then no microRNA effect could be observed looking at the system dynamics (being the fastest one, k_{3} is not a critical parameter in any scenario).
Analogously, decreasing the value of k_{2} can convert the "efficient" initiation scenario into the opposite after the threshold value . We can recapitulate the effect of decreasing k_{2} in the following way. 1) in the case of "efficient" initiation k_{2} does not affect the steady state protein synthesis rate up to the threshold value after which it affects it in a proportional way. The relaxation time drastically increases, because decreasing k_{2} leads to elongation of all dynamical stages duration (for example, we have estimated the time of the end of the dynamical Stage 2 as ). However, after the threshold value the relaxation time decreases together with k_{2}, quickly dropping to its unperturbed value (see Fig. 8CD). 2) in the case of "inefficient" initiation the steady state protein synthesis rate depends proportionally on the value of k_{2} (15), while the relaxation time is not affected (see Fig. 8AB).
MicroRNA action on k_{1} directly does not produce any strong effect neither on the relaxation time nor on the steady state protein synthesis rate. This is why in the original work [41] capdependent mechanism of microRNA action was taken into account through effective change of the [eIF 4F]_{0} value (total concentration of the translation initiation factors), which is a critical parameter of the model (see 15).
Modeling of four mechanisms of microRNA action in the nonlinear protein translation model
Observable value  Initiation(k_{1})  Step after initiation(k_{2})  Ribosome assembly (k_{3})  Elongation (k_{4}) 

Wildtype cap, inefficient initiation  
Steadystate rate  slightly decreases  decreases  no change  decreases after threshold 
Relaxation time  no change  no change  no change  goes up and down 
Wildtype cap, efficient initiation  
Steadystate rate  no change  slightly decreases after strong inhibition  no change  decreases 
Relaxation time  no change  goes up and down  no change  no change 
Acap, inefficient initiation  
Steadystate rate  decreases  decreases  no change  slightly decreases after strong inhibition 
Relaxation time  no change  no change  no change  goes up and down 
Acap, efficient initiation  
Steadystate rate  decreases after threshold  slightly decreases after strong inhibition  no change  decreases 
Relaxation time  goes up and down  goes up and down  no change  increases 
Available experimental data and possible experimental validation
It is important to underline that the Nissan and Parker's models analyzed in this paper are qualitative descriptions of the protein translation machinery. The parameter values used represent rough orderofmagnitude estimations or real kinetic rates. Moreover, these values should be considered as relative and unitless since they do not match any experimental time scale (see below). Nevertheless, such qualitative description already allows to make predictions on the relative changes of the steady states and relaxation times (see the Table 2), and in principle these predictions can be verified experimentally. Let us imagine an experiment in which it would be possible to verify such predictions. In this experiment, two time series should be compared: 1) one measured in a system in which microRNA acts on a normal "wildtype" protein translation machinery and 2) another system almost fully identical to the first one but in which one of the translation stages is modified (made slow and ratelimiting, or, opposite, very rapid). There are multiple possibilities to modify the rate of this or that translation stage. The initiation can be affected by changing the concentration of the initiation factors such as eIF4F as in [22]. The scanning stage can be affected by introducing various signals in the 5'UTR sequence of mRNA such as inframe AUG codons (see, for example, [43]). In principle, the elongation stage can be modified by introducing slow synonymous codons in the coding sequence (there even exist mathematical models of their effect [33, 35, 37] that can be used for the optimal experiment design). The stage of elongation termination can be influenced by varying the concentration of the corresponding release factors (ETF1 or ETF2), at least in vitro. The two time series measured after activation or introduction of microRNA should be characterized for the relative changes of steady state values and relaxation times of protein and mRNA concentrations, and, if possible, the number of ribosomes in the polysome. Also, ideally, it is desired to construct several experimental systems in which the amount of inhibition by microRNA can be gradually changed (for example, by changing the number of the corresponding seed sequences in the 3'UTR region).
To the best of our knowledge, there is no such a dataset published until so far, even partially. In several recent papers, one can find published time series of protein and mRNA concentrations or their relative changes measured after introducing microRNA. For example, the deadenylation time course is shown in [25]: translation decreases after 20 min and stops at 30 min, deadenylation begin at 30 min, goes around 1 h. In [44], the authors study the kinetics of degradation of mRNA. After adding microRNA to the system, the amount and the length of the targeted mRNA starts to decrease at around 35 hours, and decreases by 90% at 8 hours. In [45], the authors study the global change of protein after transfection of a microRNA. They described a small change at the mRNA level at 8 h after miRNA transfection, and the considerable decrease appeared only after 32 hours while the protein concentration change was apparent at the timecourse between 8 hours and 32 hours. In the in vitro system used in [22], at 15 min after incubation with microRNA there was already a 25% decrease of translation, indicating that the translational inhibition can be a relatively rapid mechanism.
These data on protein translation kinetics show that the relaxation time range could vary from several minutes to several hours and even tens of hours depending on the critical step affected, on various mRNA properties and on the whole biological system taken for the experiment (for example, the presence or absence of different effectors influencing different steps of the translation process). These data should be taken into account when constructing more realistic and quantitative models of microRNA action on protein translation.
Discussion
The role of microRNA in gene expression regulation is discovered and confirmed since ten years, however, there is still a lot of controversial results regarding the role of concrete mechanisms of microRNAmediated protein synthesis repression. Some authors argue that it is possible that the different modes of microRNA action reflect different interpretations and experimental approaches, but the possibility that microRNAs do indeed silence gene expression via multiple mechanisms also exists. Finally, microRNAs might silence gene expression by a common and unique mechanism; and the multiple modes of action represent secondary effects of this primary event [1–3].
The main reason for accepting a possible experimental bias could be the studies in vitro, where conditions are strongly different from situation in vivo. Indeed, inside the cell, mRNAs (microRNA targets) exist as ribonucleoprotein particles or mRNPs, and second, all proteins normally associated with mRNAs transcribed in vivo are absent or at least much different from that bound to the same mRNA in an in vitro system or following the microRNAs transfection into cultured cells. The fact that RNAbinding proteins strongly influence the final outcome of microRNA regulation is proved now by several studies [19, 46, 47]. The mathematical results provided in this paper suggests a complementary view on the coexistence of multiple microRNAmediated mechanisms of translation repression. Mathematical modeling suggests to us to ask a question: if multiple mechanisms act simultaneously, would all of them equally contribute to the final observable repression of protein synthesis or its dynamics? The dynamical limitation theory gives an answer: the effect of microRNA action will be observable and measurable in two cases: 1) if it affects the dominant system of the protein translationary machinery, or 2) if the effect of microRNA action is so strong that it changes the limiting place (the dominant system).
In a limited sense, this means, in particular, that the protein synthesis steady rate is determined by the limiting step in the translation process and any effect of microRNA will be measurable only if it affects the limiting step in translation, as it was demonstrated in [41]. Due to the variety of external conditions, cellular contexts and experimental systems the limiting step in principle can be any in the sequence of events in protein translation, hence, this or that microRNA mechanism can become dominant in a concrete environment. However, when put on the language of equations, the previous statement already becomes nontrivial in the case of nonlinear dynamical models of translation (and even linear reaction networks with nontrivial network structure). Our analysis demonstrates that the limiting step in translation can change with time, depends on the initial conditions and is not represented by a single reaction rate constant but rather by some combination of several model parameters. Methodology of dynamical limitation theory that we had developed [42, 48], allows to deal with these situations on a solid theoretical ground.
Furthermore, in the dynamical limitation theory, we generalize the notion of the limiting step to the notion of dominant system, and this gives us a possibility to consider not only the steady state rate but also some dynamical features of the system under study. One of the simplest measurable dynamical feature is the protein synthesis relaxation time, i.e. the time needed for protein synthesis to achieve its steady state rate. The general idea of "relaxation spectrometry" goes back to the works of Manfred Eigen, a Nobel laureate [49] and is still underestimated in systems biology. Calculation of the relaxation time (or times) requires careful analysis of time scales in the dynamical system, which is greatly facilitated by the recipes proposed in [42, 50]. As we have demonstrated in our semianalytical solutions, measuring the steady state rate and relaxation time at the same time allows to detect which step is possibly affected by the action of microRNA (resulting in effective slowing down of this step). To our knowledge, this idea was never considered before in the studies of microRNAdependent expression regulation. The Table 2 recapitulates predictions allowing to discriminate a particular mechanism of microRNA action.
Conclusions
The analysis of the transient dynamics gives enough information to verify or reject a hypothesis about a particular molecular mechanism of microRNA action on protein translation. For multiscale systems only that action of microRNA is distinguishable which affects the parameters of dominant system (critical parameters), or changes the dominant system itself. Dominant systems generalize and further develop the old and very popular idea of limiting step. Algorithms for identifying dominant systems in multiscale kinetic models are straightforward but not trivial and depend only on the ordering of the model parameters but not on their concrete values. Asymptotic approach to kinetic models of biological networks suggests new directions of thinking on a biological problem, making the mathematical model a useful tool accompanying biological reasoning and allowing to put in order diverse experimental observations.
However, to convert the methodological ideas presented in this paper into a working tool for experimental identification of the mechanisms of microRNAdependent protein translation inhibition, requires special efforts. Firstly, we need to construct a model which would include all known mechanisms of microRNA action. Secondly, realistic estimations on the parameter value intervals should be made. Thirdly, careful analysis of qualitatively different system behaviors should be performed and associated with the molecular mechanisms. Fourthly, a critical analysis of available quantitative information existing in the literature should be made. Lastly, the experimental protocols (sketched in the previous section) for measuring dynamical features such as the relaxation time should be developed. All these efforts makes a subject of a separate study which is an ongoing work.
Methods
Asymptotology and dynamical limitation theory for biochemical reaction networks
Most of mathematical models that really work are simplifications of the basic theoretical models and use in the backgrounds an assumption that some terms are big, and some other terms are small enough to neglect or almost neglect them. The closer consideration shows that such a simple separation on "small" and "big" terms should be used with precautions, and special culture was developed. The name "asymptotology" for this direction of science was proposed by [51] defined as "the art of handling applied mathematical systems in limiting cases".
In chemical kinetics three fundamental ideas were developed for model simplification: quasiequilibrium asymptotic (QE), quasi steadystate asymptotic (QSS) and the idea of limiting step.
In the IUPAC Compendium of Chemical Terminology (2007) one can find a definition of limiting step[52]: "A ratecontrolling (ratedetermining or ratelimiting) step in a reaction occurring by a composite reaction sequence is an elementary reaction the rate constant for which exerts a strong effect stronger than that of any other rate constant  on the overall rate."
Usually when people are talking about limiting step they expect significantly more: there exists a rate constant which exerts such a strong effect on the overall rate that the effect of all other rate constants together is significantly smaller. For the IUPAC Compendium definition a ratecontrolling step always exists, because among the control functions generically exists the biggest one. On the contrary, for the notion of limiting step that is used in practice, there exists a difference between systems with limiting step and systems without limiting step.
During XX century, the concept of the limiting step was revised several times. First simple idea of a "narrow place" (the least conductive step) could be applied without adaptation only to a simple cycle or a chain of irreversible steps that are of the first order (see Chap. 16 of the book [53] or [54] or the section "Dominant system for a simple irreversible catalytic cycle with limiting step" of this paper). When researchers try to apply this idea in more general situations they meet various difficulties.
Recently, we proposed a new theory of dynamic and static limitation in multiscale reaction networks [42, 48]. This approach allows to find the simplest network which can substitute a multiscale reaction network such that the dynamics of the complex network can be approximated by the simpler one. Following the asymptotology terminology [55], we call this simple network the dominant system (DS). In the simplest cases, the dominant system is a subsystem of the original model. However, in the general case, it also includes new reactions with kinetic rates expressed through the parameters of the original model, and rates of some reactions are renormalized: hence, in general, the dominant system is not a subsystem of the original model.
The dominant systems can be used for direct computation of steady states and relaxation dynamics, especially when kinetic information is incomplete, for design of experiments and mining of experimental data, and could serve as a robust first approximation in perturbation theory or for preconditioning.
Dominant systems serve as correct generalization of the limiting step notion in the case of complex multiscale networks. They can be used to answer an important question: given a network model, which are its critical parameters? Many of the parameters of the initial model are no longer present in the dominant system: these parameters are noncritical. Parameters of dominant systems (critical parameters) indicate putative targets to change the behavior of the large network.
Most of reaction networks are nonlinear, it is nevertheless useful to have an efficient algorithm for solving linear problems. First, nonlinear systems often include linear subsystems, containing reactions that are (pseudo)monomolecular with respect to species internal to the subsystem (at most one internal species is reactant and at most one is product). Second, for binary reactions A + B → ..., if concentrations of species A and B (c_{ A }, c_{ B }) are well separated, say c_{ A }>> c_{ B }then we can consider this reaction as B → ... with rate constant proportional to c_{ A }which is practically constant, because its relative changes are small in comparison to relative changes of c_{ B }. We can assume that this condition is satisfied for all but a small fraction of genuinely nonlinear reactions (the set of nonlinear reactions changes in time but remains small). Under such an assumption, nonlinear behavior can be approximated as a sequence of such systems, followed one each other in a sequence of "phase transitions". In these transitions, the order relation between some of species concentrations changes. Some applications of this approach to systems biology are presented by [50]. The idea of controllable linearization "by excess" of some reagents is in the background of the efficient experimental technique of Temporal Analysis of Products (TAP), which allows to decipher detailed mechanisms of catalytic reactions [56].
Below we give some details on the approaches used in this paper to analyze the models by Nissan and Parker [41].
Notations
To define a chemical reaction network, we have to introduce:

a list of components (species);

a list of elementary reactions;

a kinetic law of elementary reactions.
where s enumerates the elementary reactions, and the nonnegative integers α_{ si }, β_{ si }are the stoichiometric coefficients. A stoichiometric vector γ_{ s }with coordinates. γ_{ si }= β_{ si } α_{ si }is associated with each elementary reaction.
where T is the temperature, w_{ s }is the rate of the reaction s, v is the vector of external fluxes normalized to unite volume. It may be useful to represent external fluxes as elementary reactions by introduction of new component Ø together with income and outgoing reactions Ø → A_{ i }and A_{ i }→ Ø.
where k_{ s }is a "kinetic constant" of the reaction s.
Quasi steadystate and quasiequilibrium asymptotics
If they coincide, then the fast subsystem just dominates, and there is no fastslow separation for variables (all variables are either fast or constant). But if there exist additional linearly independent linear conservation laws for the fast system, then let us introduce new variables: linear functions b^{1}(c),...b^{ n }(c), where b^{1}(c),...b^{ m }(c) is the basis of the linear conservation laws for the initial system, and b^{1}(c),...b^{m+l}(c) is the basis of the linear conservation laws for the fast subsystem. Then b^{m+l+1}(c),...b^{ n }(c) are fast variables, b^{m+1}(c),...b^{m+l}(c) are slow variables, and b^{1}(c),...b^{ m }(c) are constant. The quasiequilibrium manifold is given by the equations and for small ε it serves as an approximation to a slow manifold.
The quasi steadystate (or pseudo steady state) assumption was invented in chemistry for description of systems with radicals or catalysts. In the most usual version the species are split in two groups with concentration vectors c^{s} ("slow" or basic components) and c^{f} ("fast intermediates"). For catalytic reactions there is additional balance for c^{f}, amount of catalyst, usually it is just a sum The amount of the fast intermediates is assumed much smaller than the amount of the basic components, but the reaction rates are of the same order, or even the same (both intermediates and slow components participate in the same reactions). This is the source of a small parameter in the system. Let us scale the concentrations c^{f} and c^{s} to the compatible amounts. After that, the fast and slow time appear and we could write c^{s} = W^{s}(c^{s}, c^{f}), , where ε is small parameter, and functions W^{s}, W^{f} are bounded and have bounded derivatives (are "of the same order"). We can apply the standard singular perturbation techniques. If dynamics of fast components under given values of slow concentrations is stable, then the slow attractive manifold exists, and its zero approximation is given by the system of equations W^{f}(c^{s}, c^{f}) = 0.
The QE approximation is also extremely popular and useful. It has simpler dynamical properties (respects thermodynamics, for example, and gives no critical effects in fast subsystems of closed systems).
Nevertheless, neither radicals in combustion, nor intermediates in catalytic kinetics are, in general, close to quasiequilibrium. They are just present in much smaller amount, and when this amount grows, then the QSS approximation fails.
The simplest demonstration of these two approximation gives the simple reaction: S + E ↔ SE → P + E with reaction rate constants and k_{2}. The only possible quasiequilibrium appears when the first equilibrium is fast: = κ^{±}/ε. The corresponding slow variable is C^{ s }= c_{ S }+ c_{ SE }, b_{ E }= c_{ E }+ c_{ SE }= const.
For the QE manifold we get a quadratic equation . This equation gives the explicit dependence C_{ SE }(C^{ s }), and the slow equation reads Ċ^{ s }= k_{2}c_{ SE }(C_{ s }), C^{ s }+ c_{ P }= b_{ S }= const.
For the QSS approximation of this reaction kinetics, under assumption b_{ E }<< b_{ S }, we have fast intermediates E and SE. For the QSS manifold there is a linear equation , which gives us the explicit expression for c_{ SE }(c_{ S }): (the standard MichaelisMenten formula). The slow kinetics reads . The difference between the QSS and the QE in this example is obvious.
The terminology is not rigorous, and often QSS is used for all singular perturbed systems, and QE is applied only for the thermodynamic exclusion of fast variables by the maximum entropy (or minimum of free energy, or extremum of another relevant thermodynamic function) principle (MaxEnt). This terminological convention may be convenient. Nevertheless, without any relation to terminology, the difference between these two types of introduction of a small parameter is huge. There exists plenty of generalizations of these approaches, which aim to construct a slow and (almost) invariant manifold, and to approximate fast motion as well. The following references can give a first impression about these methods: Method of Invariant Manifolds (MIM) ([57, 58], Method of Invariant Grids (MIG), a discrete analogue of invariant manifolds ([59]), Computational Singular Perturbations (CSP) ([60–62]) Intrinsic LowDimensional Manifolds (ILDM) by [63], developed further in series of works by [64]), methods based on the Lyapunov auxiliary theorem ([65]).
Multiscale monomolecular reaction networks
A monomolecular reaction is such that it has at most one reactant and at most one product. Let us consider a general network of monomolecular reactions. This network is represented as a directed graph (digraph) [66]: vertices correspond to components A_{ i }, edges correspond to reactions A_{ i }→ A_{ j }with kinetic constants k_{ ji }> 0. For each vertex, A_{ i }, a positive real variable c_{ i }(concentration) is defined.
where c^{ s }is the steady state of the system (36), i.e. when all = 0, and c(0) is the initial condition.
If all reaction constants k_{ ij }would be known with precision then the eigenvalues and the eigenvectors of the kinetic matrix can be easily calculated by standard numerical techniques. Furthermore, Singular Value Decomposition (SVD) can be used for model reduction. But in systems biology models often one has only approximate or relative values of the constants (information on which constant is bigger or smaller than another one). Let us consider the simplest case: when all kinetic constants are very different (separated), i.e. for any two different pairs of indices I = (i, j), J = (i', j') we have either k_{ I }>> k_{ J }or k_{ J }<< k_{ I }. In this case we say that the system is hierarchical with timescales (inverses of constants k_{ ij }, j ≠ 0) totally separated.
Linear network with totally separated constants can be represented as a digraph and a set of orders (integer numbers) associated to each arc (reaction). The lower the order, the more rapid is the reaction. It happens that in this case the special structure of the matrix K (originated from a reaction graph) allows us to exploit the strong relation between the dynamics (36) and the topological properties of the digraph. In this case, possible values of l_{ i }k are 0, 1 and the possible values of r_{ i }are 1, 0, 1 with high precision. In previous works, we provided an algorithm for finding nonzero components of l_{ i }, r_{ i }, based on the network topology and the constants ordering, which gives a good approximation to the problem solution [42, 48, 50].
Dominant system for a simple irreversible catalytic cycle with limiting step
A linear chain of reactions, A_{1} → A_{2} → ...A_{ n }, with reaction rate constants k_{ i }(for A_{ i }→ A_{i+1}), gives the first example of limiting steps. Let the reaction rate constant k_{ q }be the smallest one. Then we expect the following behavior of the reaction chain in time scale ≳1/k_{ q }: all the components A_{1},...A_{q1}transform fast into A_{ q }, and all the components A_{q+1},...A_{n1}transform fast into A_{ n }, only two components, A_{ q }and A_{ n }are present (concentrations of other components are small), and the whole dynamics in this time scale can be represented by a single reaction A_{ q }→ A_{ n }with reaction rate constant k_{ q }. This picture becomes more exact when k_{ q }becomes smaller with respect to other constants.
all coordinates of l^{0} are equal to 1, the only nonzero coordinate of r^{0} is and we represent vectorcolumn r^{0} in row.
Reduced form means that in reality some of these reaction are not monomolecular and include some other components (not from the list A_{1},... A_{ n }). But in the study of the isolated cycle dynamics, concentrations of these components are taken as constant and are included into kinetic constants of the cycle linear reactions.
This is trivial: all the concentration is collected at the starting point of the "narrow place", but may be useful as an origin point for various approximation procedures.
In that case we say that the cycle has a limiting step with constant k_{min}.
There is significant difference between the examples of limiting steps for the chain of reactions and for irreversible cycle. For the chain, the steady state does not depend on nonzero rate constants. It is just c_{ n }= b, c_{1} = c_{2} =... = c_{n1}= 0. The smallest rate constant k_{ q }gives the smallest positive eigenvalue, the relaxation time is τ = 1/k_{ q }. The corresponding approximation of eigenmode (right eigenvector) r^{1} has coordinates: . This exactly corresponds to the statement that the whole dynamics in the time scale ≳1/k_{ q }can be represented by a single reaction A_{ q }→ A_{ n }with reaction rate constant k_{ q }. The left eigenvector for eigenvalue k_{ q }has approximation l^{1} with coordinates . This vector provides the almost exact lumping on time scale ≳1/k_{ q }. Let us introduce a new variable c_{lump} = ∑_{ i }l_{ i }c_{ i }, i.e. c_{lump} = c_{1} + c_{2} +... + c_{ q }. For the time scale ≳1/k_{ q }we can write c_{lump} + c_{ n }≈ b, dc_{lump}/dt ≈ k_{ q }c_{lump}, dc_{ n }/dt ≈ k_{ q }c_{lump}.
In the example of a cycle, we approximate the steady state, that is, the right eigenvector r^{0} for zero eigenvalue (the left eigenvector is known and corresponds to the main linear balance b: ≡ 1). In the zeroorder approximation, this eigenvector has coordinates
If k_{ n }/k_{ i }is small for all i < n, then the kinetic behavior of the cycle is determined by a linear chain of n  1 reactions A_{1} → A_{2} → ...A_{ n }, which we obtain after cutting the limiting step. The characteristic equation for an irreversible cycle, , tends to the characteristic equation for the linear chain, , when k_{ n }→ 0.
where δ_{ i }→ 0 when .
A cycle with limiting step (41) has real eigenspectrum and demonstrates monotonic relaxation without damped oscillations. Of course, without limitation such oscillations could exist, for example, when all k_{ i }≡ k > 0, (i = 1,...n).
The relaxation time of a stable linear system (41) is, by definition, τ = 1/min{Re(λ_{ i })} (λ ≠ 0). For small k_{ n }, τ ≈ 1/k_{ τ }, k_{ τ }= min{k_{ i }}, (i = 1,...n  1). In other words, for a cycle with limiting step, k_{ τ }is the second slowest rate constant: k_{min} << k_{ τ }≤ ....
Notes
Declarations
Acknowledgements
We acknowledge support from Agence Nationale de la Recherche (project ANR08SYSC003 CALAMAR) and from the Projet Incitatif Collaboratif "Bioinformatics and Biostatistics of Cancer" at Institut Curie. AZ and EB are members of the team "Systems Biology of Cancer" Equipe labellis'ee par la Ligue Nationale Contre le Cancer. This work was supported by the European Commission Sixth Framework Programme Integrated Project SIROCCO contract number LSHGCT2006037900. We thank Vitaly Volpert and Laurence Calzone for inspiring and useful discussions.
Authors’ Affiliations
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