Institut Curie, Bioinformatics and Computational Systems Biology Of Cancer, Paris, France

INSERM, U900, Paris, F-75248 France

Mines ParisTech, Centre for Computational Biology, Fontainebleau, F-77300 France

CNRS FRE 2944, Institut André Lwoff, Villejuif, France

University of Leicester, Center for Mathematical Modeling, Leicester, UK

Institute of Computational Modeling SB RAS, Department of nonequilibrium systems, Krasnoyarsk, Russia

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 non-coding 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 co-exist.

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

The most frequently reported, but also much debated, is the mechanism of gene repression by microRNAs which occurs at the level of mRNA translation. At this level, several mode of actions have been suggested (see Fig.

Interaction of microRNA with protein translation process

**Interaction of microRNA with protein translation process**. Four mechanisms of translation repression which are considered in the mathematical modeling are indicated: 1) on the initiation process, preventing assembling of the initiation complex; 2) on a late initiation step, such as searching for the start codon; 3) on the ribosome assembly; 4) on the translation process. There exist other mechanisms of microRNA action on protein translation (transcriptional, transport to P-bodies, ribosome drop-off, co-translational protein degradation and others) that are not considered in this paper. Here 40S and 60S are light and heavy components of the ribosome, 80S is the assembled ribosome bound to mRNA, eIF4F is an translation initiation factor, PABC1 is the Poly-A binding protein, "cap" is the mRNA cap structure needed for mRNA circularization, RISC is the RNA-induced silencing complex.

Concurrently, several reports have been published indicating an action of microRNAs at the level of initiation. An increasing number of papers reports that microRNA-targeted mRNAs shift towards the light fractions in polysomal profiles

Most of the data indicating a shift towards the light polysomal fraction or the requirement for a cap-dependent 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 in-vitro experiments showing a lower amount of 60S relative to 40S on inhibited mRNAs. Moreover, toe-printing experiments show that 40S is positioned on the AUG

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

In this paper we will analyze two simple models of microRNA action on protein translation developed recently by Nissan and Parker

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

Secondly, even in the simple non-linear 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

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

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 (

Results

Model assumptions

We consider two models of action of microRNA on protein translation process proposed in

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.

Finally, the classical chemical kinetics approach is applied, based on solutions of ordinary differential equations, which supposes sufficient and well-stirred 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

The simplest linear protein translation model

The simplest representation of the translation process has the form of a circular cascade of reactions

The simplest model of microRNA action on the protein translation

**The simplest model of microRNA action on the protein translation**. The simplest model of microRNA action on the protein translation, represented with use of Systems Biology Graphical Notation (a) and schematically with the condition on the constants (b). The two mechanisms of microRNA action (cap-dependent and cap-independent) are depicted.

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 _{1}).

2. mRNA:40S → AUG, Some late and cap-independent initiation steps, such as scanning the 5'UTR for the start AUG codon recognition (rate _{2}) and 60S ribosomal unit joining.

3. AUG → 40S, combined processes of protein elongation and termination, which leads to production of the protein (rate _{3}), and fall off of the ribosome from mRNA.

The model is described by the following system of equations

where

Following _{3 }>> _{1}, _{2}. This choice was justified by the following statement: "...The subunit joining and protein production rate (_{3}) is faster than _{1 }and _{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..."

Under this condition, the equations (1) have the following approximate solution (which becomes the more exact the smaller the

for the initial condition

From the solution (2) it follows that the dynamics of the system evolves on two time scales: 1) fast elongation dynamics on the time scale ≈ 1/_{3}; and 2) relatively slow translation initiation dynamics with the relaxation time _{3 }rate, since it is neglected with respect to _{1}, _{2 }values. From (3) we can extract the formula for the protein synthesis steady-state rate _{
rel
}for it (inverse of the exponent power):

Now let us consider two experimental situations: 1) the rates of the two translation initiation steps are comparable _{1 }≈ _{2}; 2) the cap-dependent rate _{1 }is limiting: _{1 }<< _{2}. Accordingly to

For these two experimental systems (let us call them "wild-type" and "modified" correspondingly), let us study the effect of microRNA action. We will model the microRNA action by diminishing the value of a kinetic rate coefficient for the reaction representing the step on which the microRNA is acting. Let us assume that there are two alternative mechanisms: 1) microRNA acts in a cap-dependent manner (thus, reducing the _{1 }constant) and 2) microRNA acts in a cap-independent manner, for example, through interfering with 60S subunit joining (thus, reducing the _{2 }constant). The dependence of the steady rate of protein synthesis

Predicted change in the steady-state rate of protein synthesis and its relaxation time

**Predicted change in the steady-state rate of protein synthesis and its relaxation time**. Graphs illustrating the predicted change in the steady-state rate of protein synthesis (left), and its relaxation time, i.e., the time needed to recover from a perturbation to the steady state value (right). Four curves are presented. The black ones are for the wild-type cap structure, which is modeled by _{1 }= _{2}. The red ones are for the modified structure, when _{1 }<<_{2}. The main conclusion from the left graph is that if microRNA acts on a late initiation step, diminishing _{2}, then its effect is not measurable unless _{2 }is very strongly suppressed (as reported in

Interestingly, experiments with cap structure replacement were made and the effect of microRNA action on the translation was measured _{1 }<< _{2}), hence, even if microRNA acts in the cap-independent manner (inhibiting _{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.

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 cap-independent manner despite the fact that the steady state rate of the protein synthesis does not change (see the Fig.

The observations from the Fig.

Modeling two mechanisms of microRNA action in the simplest linear model

**Observable value**

**Initiation( k _{1})**

**Step after initiation, cap-independent( k _{2})**

**Elongation ( k _{3})**

**Wild-type cap**

decreases

decreases

no change

increases slightly

increases slightly

no change

**A-cap**

decreases

no change

no change

no change

increases drastically

no change

MicroRNA action effect is described for the protein synthesis steady rate and the relaxation time. It is assumed that the ribosome assembly+elongation step in protein translation, described by the _{3 }rate constant, is not rate limiting.

This conclusion suggests the notion of a **kinetic signature of microRNA action mechanism **which we define as

The non-linear protein translation model

To explain the effect of microRNA interference with translation initiation factors, a non-linear version of the translation model was proposed

The model contains the following list of chemical species (see also Fig.

Non-linear model of microRNA action on the protein translation

**Non-linear model of microRNA action on the protein translation**. Non-linear model of microRNA action on the protein translation, represented with use of Systems Biology Graphical Notation (a) and schematically with the condition on the constants (b). The difference from the simplest model (Fig. 2) is in the explicit description of initiation factors eIF4F, and ribosomal subunits 40S and 60S recycling.

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 _{1}).

2. mRNA:40S → AUG, some late and cap-independent initiation steps, such as scanning the 5'UTR by for the start codon AUG recognition (rate _{2}).

3. AUG → 80S, assembly of ribosomes and protein translation (rate _{3}).

4. 80S → 60S+40S, recycling of ribosomal subunits (rate _{4}).

The model is described by the following system of equations

where [40

The model (6) contains three independent conservations laws:

The following assumptions on the model parameters were suggested in

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 _{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..."

Notice that further in our paper we show that the last statement about the value of _{4 }is needed to be made more precise: in the model by Nissan and Parker, _{4 }

Steady state solution

The final steady state of the system can be calculated from the conservation laws and the balance equations among all the reaction fluxes:

where "s" index stands for the steady state value. Let us designate a fraction of the free [60S] ribosomal subunit in the steady state as

and the equation to determine

From the inequalities on the parameters of the model, we have _{1 }>> _{4}/[_{0 }then

provided that _{1 }we cannot neglect the term proportional to

The solution _{2 }is always negative, which means that one can have one positive solution _{0 }<< 1 if _{0 }and _{1 }if _{1 }> 0 then _{0 }does not correspond to a positive value of [_{
s
}. This means that for a given combination of parameters satisfying (10) we can have only one steady state (either _{0 }or _{1}).

The two values _{0 }and _{1 }correspond to **two different modes of translation**. When, for example, the amount of the initiation factors [_{0 }is **not enough to provide efficient initiation **(_{1}) then most of the 40**initiation is efficient **enough _{0 }<< 1 when almost all 60

Let us notice that the steady state protein synthesis rate under these assumptions is

This explains the numerical results obtained in _{0 }microRNA action would be efficient only if it affects _{2 }or if it competes with _{0}) With higher concentrations of [_{0}, other limiting factors become dominant: [60_{0 }(availability of the heavy ribosomal subunit) and _{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 _{1 }directly (of course, unless it does not become "globally" rate limiting), but only through competing with

Equation (15) explains also some experimental results reported in

It would be interesting to make some conclusions on the shift of the polysomal profile from the steady state solutions (14). In this model, the number of ribosomes sitting on mRNA _{
polysome
}is defined by _{1}, together with the effective volume of cytoplasmic space considered in the model. Nevertheless, the model can predict the relative shift of the polysome profile. In the steady state

and _{
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 _{1 }= _{2 }= 2, _{3 }= 5, _{4 }= 1, [40_{0 }= 100, [60_{0 }= 25, [_{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 _{4 }parameter, putting it to 0.1, which makes

Simulations of the protein translation model with these parameters and the initial conditions

are shown on the Fig.

Simulation of the non-linear protein translation model

**Simulation of the non-linear protein translation model**. Simulation of the non-linear protein translation model with parameters _{1 }= 2, _{2 }= 2, _{3 }= 5, _{4 }= 0.1, [40_{0 }= 100, [60_{0 }= 25, [_{0 }= 6. a) and b) chemical species concentrations at logarithmic and linear scales; c) and d) reaction fluxes at logarithmic and linear scales. By the dashed line several stages are delimited during which the dynamics can be considered as (pseudo-)linear. To determine where ">>" and "<<" conditions are violated, we arbitrarily consider "much bigger" or "much smaller" as difference in one order of magnitude (by factor 10).

1) Stage 1: Relatively fast relaxation with conditions [40_{1}, _{2 }and _{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 [40

4) Transition between Stage 2 and Stage 3.

5) Stage 3: Relaxation with the conditions [40

Stages 1-3 can be associated with the corresponding dominant systems

Dominant systems for three stages of relaxation

**Dominant systems for three stages of relaxation**. Dominant systems for three stages of relaxation of the model (6). Stage 1) The dominant system is a pseudo-linear network of reactions. Stage 2) The dominant system is a quasi-steady state approximation, where one supposes that the fluxes in two network cycles are balanced. Stage 3) The dominant system is a pseudo-linear network of reactions.

Stage 1: translation initiation and assembly of the first ribosome at the start codon

The dominant system of the Stage 1 (Fig.

where _{1 }· [40_{3 }· [60_{2 }<< _{4 }<< _{2}, also assuming _{2 }<< |

we have

From this solution, one can conclude that the relaxation of this model goes at several time scales (very rapid _{4}) and that when eIF4F, mRNA:40S and AUG already reached their quasiequilibrium values, [80S] continues to grow. This corresponds to the quasiequilibrium approximation asymptotics (see the "Quasi steady-state and quasiequilibrium asymptotics" section of the Methods). At some point 80S will reach such a value that it would be not possible to consider 60S constant: otherwise the conservation law (9) will be violated. This will happen when [80_{0 }>> [_{0 }and [40_{0 }> [60_{0 }we have

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 ([40_{0 }< [40_{0}).

Stage 2: first stage of protein elongation, initiation is still rapid

The Stage 2 is characterized by conditions [

Then (6) is simplified and, using the conservation laws, we have a single equation on [40

where _{4 }<< _{3 }and assuming that at the beginning of the Stage 2 [

and, further, assuming that at the beginning of the Stage 2 we have [40

where [40_{
t=t"
}is the amount of 40

where _{1}, _{2 }are linear and exponential slopes and [40_{
s2 }is the quasi-steady state value of [40

Other dynamic variables are expressed through [40

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

Stage 3: steady protein elongation, speed of initiation equals to speed of elongation

During the Stage 3 all fluxes in the network become balanced and the translation arrives at the steady state. From Fig.

where _{3}[_{
t=t'''
}is relatively big. So, during the Stage 3, one can consider the cycle _{
s
}, [60_{
s
}values.

Hence, the relaxation during the Stage 3 consists in redistributing concentrations of 40S and mRNA:40S to their steady states in a linear chain of reactions

where _{
s
}- [40_{
t = t'''
}). [40]_{
s
}and [_{
s
}are the steady-state values of the corresponding variables, see (12). The values [60_{
t = t''', }[_{
t = t''', }[_{
t = t'''
}and [_{
t = t″' }can be estimated from (28), using the [40_{
t = t'''
}value. The relaxation time at this stage equals

The solution for the Stage 3 can be further simplified if _{2 }<< _{1}[_{
t = t'''
}or _{2 }>> _{1 }[_{
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 [40_{
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 [40_{
t = t'''
}.

Case of always limiting initiation

As it follows from our analysis, the most critical parameter of the non-linear protein translation model is the ratio

In the case

Comparison of the numerical and approximate analytical solutions of the non-linear protein translation model

**Comparison of the numerical and approximate analytical solutions of the non-linear protein translation model**. Examples of the exact numerical (circles) and approximate analytical (solid lines) solutions of the non-linear protein translation model. a) For the set of parameters _{1 }= 2, _{2 }= 2, _{3 }= 5, _{4 }= 0.1; b) For parameters _{1 }= 1, _{2 }= 5, _{3 }= 50, _{4 }= 0.01; c) For the set of parameters from _{1 }= 2, _{2 }= 2, _{3 }= 5, _{4 }= 1; d) Reaction fluxes for the set of parameters c). Dashed black vertical lines denote evaluated transition points between the dynamics stages. Dashed red vertical points denote the time points where [40

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 }very small on the steady state protein synthesis and the relaxation time is shown on Fig.

**Analytical analysis of the case of very inefficient cap structure**. In this text we derive an asymptotical solution for the case when _{1 }is very small corresponding to the case of very inefficient translation initiation (for example, in the case of A-cap structure replacement experiment)

Click here for file

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 _{3 }smaller than other _{
i
}s). Such possibility exists, however, it can require too strong (non-physiological) effect of microRNA-dependent translation inhibition.

Effect of microRNA on the translation dynamics

Our analysis of the non-linear Nissan and Parker's model showed that the protein translation machinery can function in two qualitatively different modes, determined by the ratio _{0 }or [_{0}) or by changing the critical kinetic parameters (_{2 }or _{4}). For example, changing _{4 }from 1 (Fig.

As a result of the dynamical analysis, we can assemble an approximate solution of the non-linear system under assumptions (10) about the parameters. An example of the approximate solution is given on Fig.

One of the obvious predictions is that the dynamics of the system is not sensitive to variations of _{3}, so if microRNA acts on the translation stage controlled by _{3 }then no microRNA effect could be observed looking at the system dynamics (being the fastest one, _{3 }is not a critical parameter in any scenario).

If microRNA acts on the translation stage controlled by _{4 }(for example, by ribosome stalling mechanism) then we should consider two cases of efficient (_{4 }(as the slowest, limiting step) and any effect on _{4 }would lead to the proportional change in the steady state of protein production. By contrast, in the case of inefficient initiation, the steady state protein synthesis is not affected by _{4}. Instead, the relaxation time is affected, being ~ _{4 }increases the _{4 }critical for the steady state protein synthesis anyway when _{4 }becomes smaller than _{4 }value firstly leads to no change in the steady state rate of protein synthesis, whereas the relaxation time increases and, secondly, after the threshold value _{4 }by several orders of magnitude does not change the steady state rate of protein synthesis.

Effect of mimicking different mechanisms of miRNA action on translation

**Effect of mimicking different mechanisms of miRNA action on translation**. Effect of decreasing some model parameters mimicking different mechanisms of miRNA action on translation. Relaxation time here is defined as the latest time at which any chemical species in the model differs from its final steady state by 10% A) and B) correspond to the scenario with "inefficient" initiation, with use of the model parameters proposed in _{1 }= _{2 }= 2, _{3 }= 5, _{4 }= 1, [_{0 }= 6, [60_{0 }= 25, [40_{0 }= 100), which gives _{1 }= 2, _{2 }= 3, _{3 }= 50, _{4 }= 0.1, [_{0 }= 6, [60_{0 }= 25, [40_{0 }= 100), which gives _{1 }parameter: _{1 }= 0.01 for these curves, the other parameters are the same as on A-D) correspondingly.

Analogously, decreasing the value of _{2 }can convert the "efficient" initiation scenario into the opposite after the threshold value _{2 }in the following way. 1) in the case of "efficient" initiation _{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 _{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 _{2}, quickly dropping to its unperturbed value (see Fig. _{2 }(15), while the relaxation time is not affected (see Fig.

MicroRNA action on _{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 _{0 }value (total concentration of the translation initiation factors), which is a critical parameter of the model (see 15).

The effect of microRNA on various mechanism and in various experimental settings (excess or deficit of eIF4F, normal cap or A-cap) is recapitulated in Table _{3 }in our case). Nevertheless, if any change in the steady-state protein synthesis or the relaxation time is observed, theoretically, it will be possible to specify the mechanism responsible for it.

Modeling of four mechanisms of microRNA action in the non-linear protein translation model

**Observable value**

**Initiation( k _{1})**

**Step after initiation( k _{2})**

**Ribosome assembly ( k _{3})**

**Elongation ( k _{4})**

**Wild-type cap, inefficient initiation**

slightly decreases

decreases

no change

decreases after threshold

no change

no change

no change

goes up and down

**Wild-type cap, efficient initiation**

no change

slightly decreases after strong inhibition

no change

decreases

no change

goes up and down

no change

no change

**A-cap, inefficient initiation**

decreases

decreases

no change

slightly decreases after strong inhibition

no change

no change

no change

goes up and down

**A-cap, efficient initiation**

decreases after threshold

slightly decreases after strong inhibition

no change

decreases

goes up and down

goes up and down

no change

increases

MicroRNA action effect is described for the protein synthesis steady rate and the relaxation time (see also Fig. 8). "Efficient initiation" and "inefficient initiation" correspond to two qualitatively different solution types of Nissan and Parker's model (see the beginning of "Effect of microRNA on the translation dynamics" section). The effect of A-cap structure is modeled by making the cap-dependent initiation step very slow (by making the _{1 }parameter very small). It is assumed that the ribosome assembly step in protein translation, described by the _{3 }rate constant, is not rate limiting.

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 order-of-magnitude 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

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

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 microRNA-mediated 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

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 RNA-binding proteins strongly influence the final outcome of microRNA regulation is proved now by several studies

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

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

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 microRNA-dependent 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

In chemical kinetics three fundamental ideas were developed for model simplification:

In the IUPAC Compendium of Chemical Terminology (2007) one can find **a definition of limiting step **

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 rate-controlling 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

Recently, we proposed a new theory of dynamic and static limitation in multiscale reaction networks **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,

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.

**non-critical**. 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
}) are well separated, say _{
A
}>> _{
B
}then we can consider this reaction as _{
A
}which is practically constant, because its relative changes are small in comparison to relative changes of _{
B
}. We can assume that this condition is satisfied for all but a small fraction of genuinely non-linear reactions (the set of non-linear reactions changes in time but remains small). Under such an assumption, non-linear 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

Below we give some details on the approaches used in this paper to analyze the models by Nissan and Parker

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.

The list of components is just a list of symbols (labels) _{1},..._{
n
}. Each elementary reaction is represented by its

where _{
si
}, _{
si
}are the _{
s
}with coordinates. _{
si
}= _{
si
}- _{
si
}is associated with each elementary reaction.

A non-negative real _{
i
}≥ 0, amount of _{
i
}, is associated with each component _{
i
}. It measures "the number of particles of that species" (in particles, or in moles). The concentration of _{
i
}is an _{
i
}= _{
i
}/

where _{
s
}is the rate of the reaction _{
i
}and _{
i
}→ Ø.

The most popular

where _{
s
}is a "kinetic constant" of the reaction

Quasi steady-state and quasiequilibrium asymptotics

^{f}' and '^{s}' to distinguish fast and slow reactions. A small parameter appears in the following form

To separate variables, we have to study the spaces of linear conservation law of the initial system (35) and of the fast subsystem

If they coincide, then the fast subsystem just dominates, and there is no fast-slow 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 ^{1}(^{
n
}(^{1}(^{
m
}(^{1}(^{
m+l
}(^{
m+l+1}(^{
n
}(^{
m+1}(^{
m+l
}(^{1}(^{
m
}(

The **quasi steady-state **(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 ^{s }("slow" or basic components) and ^{f }("fast intermediates"). For catalytic reactions there is additional balance for ^{f}, amount of catalyst, usually it is just a sum ^{f }and ^{s }to the compatible amounts. After that, the fast and slow time appear and we could write ^{s }= ^{s}(^{s}, ^{f}), ^{s}, ^{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 ^{f}(^{s}, ^{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: _{2}. The only possible quasiequilibrium appears when the first equilibrium is fast: ^{±}/^{
s
}= _{
S
}+ _{
SE
}, _{
E
}= _{
E
}+ _{
SE
}=

For the QE manifold we get a quadratic equation _{
SE
}(^{
s
}), and the slow equation reads ^{
s
}= -_{2}
_{
SE
}(_{
s
}), ^{
s
}+ _{
P
}= _{
S
}=

For the QSS approximation of this reaction kinetics, under assumption _{
E
}<< _{
S
}, we have fast intermediates _{
SE
}(_{
S
}):

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) (

Multiscale monomolecular reaction networks

_{
i
}, edges correspond to reactions _{
i
}→ _{
j
}with kinetic constants _{
ji
}> 0. For each vertex, _{
i
}, a positive real variable _{
i
}(concentration) is defined.

"Pseudo-species" (labeled Ø) can be defined to collect all degraded products, and degradation reactions can be written as _{
i
}→ Ø with constants _{0i
}. Production reactions can be represented as Ø → _{
i
}with rates _{
i0}. The kinetic equation for the system is

or in vector form: ċ = _{0 }+ ^{
i
}) and right (^{
i
}) eigenvectors of

with the normalization (^{
i
}, ^{
i
}) = _{
ij
}, where _{
ij
}is Kronecker's delta. Then the solution of (36) is

where ^{
s
}is the steady state of the system (36), i.e. when all

If all reaction constants _{
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
}>> _{
J
}or _{
J
}<< _{
I
}. In this case we say that the system is **hierarchical **
_{
ij
},

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 _{
i
}k are 0, 1 and the possible values of _{
i
}are -1, 0, 1 with high precision. In previous works, we provided an algorithm for finding non-zero components of _{
i
}, _{
i
}, based on the network topology and the constants ordering, which gives a good approximation to the problem solution

Dominant system for a simple irreversible catalytic cycle with limiting step

A linear chain of reactions, _{1 }→ _{2 }→ ..._{
n
}, with reaction rate constants _{
i
}(for _{
i
}→ _{
i+1}), gives the first example of limiting steps. Let the reaction rate constant _{
q
}be the smallest one. Then we expect the following behavior of the reaction chain in time scale ≳1/_{
q
}: all the components _{1},..._{
q-1 }transform fast into _{
q
}, and all the components _{
q+1},..._{
n-1 }transform fast into _{
n
}, only two components, _{
q
}and _{
n
}are present (concentrations of other components are small), and the whole dynamics in this time scale can be represented by a single reaction _{
q
}→ _{
n
}with reaction rate constant _{
q
}. This picture becomes more exact when _{
q
}becomes smaller with respect to other constants.

The kinetic equation for the linear chain is

The coefficient matrix _{
i
}(^{0 }and ^{0}, are:

all coordinates of ^{0 }are equal to 1, the only nonzero coordinate of ^{0 }is ^{0 }in row.

The catalytic cycle is one of the most important substructures that we study in reaction networks. In the reduced form the catalytic cycle is a set of linear reactions:

Reduced form means that in reality some of these reaction are not monomolecular and include some other components (not from the list _{1},... _{
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.

For the constant of elementary reaction _{
i
}→ we use the simplified notation _{
i
}because the product of this elementary reaction is known, it is _{
i+1 }for _{1 }for _{
i
}= _{
i
}
_{
i
}, where _{
i
}is the concentration of _{
i
}. The kinetic equation is:

where by definition _{0 }= _{
n
}, _{0 }= _{
n
}, and _{0 }= _{
n
}. In the stationary state (_{
i
}= 0), all the _{
i
}are equal: _{
i
}=

where _{
i
}
_{
i
}is the conserved quantity for reactions in constant volume. Let one of the constants, _{min}, be much smaller than others (let it be _{min }= _{
n
}):

In this case, in linear approximation

The simplest zero order approximation for the steady state gives

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.

So, _{min}, if it is much smaller than the constants of all other reactions (43):

In that case we say that **the cycle has a limiting step with constant **
_{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 _{
n
}= _{1 }= _{2 }=... = _{
n-1 }= 0. The smallest rate constant _{
q
}gives the smallest positive eigenvalue, the relaxation time is _{
q
}. The corresponding approximation of eigenmode (right eigenvector) ^{1 }has coordinates: _{
q
}can be represented by a single reaction _{
q
}→ _{
n
}with reaction rate constant _{
q
}. The left eigenvector for eigenvalue _{
q
}has approximation ^{1 }with coordinates _{
q
}. Let us introduce a new variable _{lump }= ∑_{
i
}
_{
i
}
_{
i
}, i.e. _{lump }= _{1 }+ _{2 }+... + _{
q
}. For the time scale ≳1/_{
q
}we can write _{lump }+ _{
n
}≈ _{lump}/d_{
q
}
_{lump}, d_{
n
}/d_{
q
}
_{lump}.

In the example of a cycle, we approximate the steady state, that is, the right eigenvector ^{0 }for zero eigenvalue (the left eigenvector is known and corresponds to the main linear balance

If _{
n
}/_{
i
}is small for all _{1 }→ _{2 }→ ..._{
n
}, which we obtain after cutting the limiting step. The characteristic equation for an irreversible cycle, _{
n
}→ 0.

The characteristic equation for a cycle with limiting step (_{
n
}/_{
i
}<< 1) has one simple zero eigenvalue that corresponds to the conservation law ∑_{
i
}=

where _{
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 _{
i
}≡

**The relaxation time **of a stable linear system (41) is, by definition, _{
i
})} (_{
n
}, _{
τ
}, _{
τ
}= min{_{
i
}}, (_{
τ
}is the second slowest rate constant_{min }<< _{
τ
}≤ ....

Authors' contributions

AZ and AG have written the main body of the manuscript. NM, NN and AHB provided the critical review of miRNA mechanisms and contributed to writing the manuscript. AG and AZ developed the mathematical methodology for identifying dominant systems. AZ performed the analytical computations and numerical simulations. All authors have participated in discussing the results and model predictions. All authors read and approved the final manuscript.

Acknowledgements

We acknowledge support from Agence Nationale de la Recherche (project ANR-08-SYSC-003 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 LSHG-CT-2006-037900. We thank Vitaly Volpert and Laurence Calzone for inspiring and useful discussions.