Lactobacillus plantarum failed, as this lactic acid bacterium produces lactate, even under glucose-limited chemostat conditions, where FBA predicted mixed acid fermentation as an alternative pathway leading to a higher yield. In this study we tested, however, whether long-term adaptation on an unusual and poor carbon source (for this bacterium) would select for mutants with optimal biomass yields. We have therefore adapted Lactobacillus plantarum to grow well on glycerol as its main growth substrate. After prolonged serial dilutions, the growth yield and corresponding fluxes were compared to in silico predictions. Surprisingly, the organism still produced mainly lactate, which was corroborated by FBA to indeed be optimal. To understand these results, constraint-based elementary flux mode analysis was developed that predicted 3 out of 2669 possible flux modes to be optimal under the experimental conditions. These optimal pathways corresponded very closely to the experimentally observed fluxes and explained lactate formation as the result of competition for oxygen by the other flux modes. Hence, these results provide thorough understanding of adaptive evolution, allowing in silico predictions of the resulting flux states, provided that the selective growth conditions favor yield optimization as the winning strategy.In the study of metabolic networks, optimization techniques are often used to predict flux distributions, and hence, metabolic phenotype. Flux balance analysis in particular has been successful in predicting metabolic phenotypes. However, an inherent limitation of a stoichiometric approach such as flux balance analysis is that it can predict only flux distributions that result in maximal yields. Hence, previous attempts to use FBA to predict metabolic fluxes in  Saccharomyces cerevisiae and Escherichia coli, we have previously observed that the approach fails in Lactobacillus plantarum, a lactic acid bacterium that clearly does not optimize its yield on glucose but “wastes” glucose by producing lactic acid. In the current study we provide evidence that L. plantarum does optimize its yield when grown under a poor carbon condition, i.e., when grown on glycerol as its main carbon source. The study provides new insight in when the application of in silico optimization techniques can be expected to be predictive.Being able to predict the metabolic fluxes and growth rate of a microorganism is an important topic in microbial systems biology. One approach, constraint-based modeling, uses a reconstructed metabolic network and optimization techniques to make such predictions. Although widely used, the success of this approach depends on a number of important assumptions. First, it assumes that evolutionary forces have shaped the metabolism towards optimality of, in most cases, growth rate. Second, through the nature of the modeling approach, it assumes that microorganisms maximize the growth rate through optimizing the yield on the growth substrate. Despite successes of the approach in model organisms such as  The role of mathematical modeling in the study of microbial physiology has increased considerably by the development of genome-scale metabolic models In many of these studies flux balance analysis (FBA) was used. FBA uses optimization of an objective function to find a subset of optimal states in the large solution space of possible states that is shaped by mass balance and capacity constraints Lactobacillus plantarum, a lactic acid bacterium that is found in nutrient-rich niches such as decomposed plant material, does not conform to this consensus et al found good predictions with FBA for E. coliSaccharomyces cerevisiaeEscherichia coliHowever, we have recently demonstrated that To fully appreciate these results, it is crucial to understand very precisely what FBA assumes and what it predicts. Under many conditions, especially in the laboratory and under adaptive evolution protocols, growth rate is a good proxy for fitness. Thus, an optimization with respect to growth rate seem an appropriate modeling strategy, and that is what FBA does: it most often optimizes the growth rate Methods.μ is the specific growth rate (units h−1), in,substrateV is the uptake rate of the growth substrate (units mmol h−1 gDW−1), and Y is the yield of biomass with respect to the substrate (units gDW mmol−1). If in Eq 1 we want to predict the growth rate, we have to specify the input rate. FBA simply finds the highest yield Y such that the growth rate is maximal at the specified input rate. So, even though a rate is maximized in FBA (μ in Eq. 1), it is through the yieldY that this is achieved. Indeed, maximizing yield or rate using stoichiometric models is essentially the same, as illustrated in Where There are potentially other strategies that lead to enhanced fitness. These include a very fast  consumption of substrates, closely related to high growth rates L. plantarum cannot be predicted by FBA under the usual, sugar-rich conditions, we may be able to force L. plantarum to choose a strategy of high yield by adapting it under poor nutrient conditions. If so, FBA should become predictive. By serial dilution of L. plantarum on a medium containing glycerol as main carbon source, we selected strains that were able to grow an order of magnitude faster than the wild type on glycerol. FBA analysis proved to be able to predict the metabolic behavior of a selected adapted strain as a function of the oxygen uptake rate. To gain a more thorough insight into the characteristics of the observed metabolic behavior, elementary flux mode analysis was applied and it was shown that the organism was able to select the 3 optimal flux modes out of almost 2700 possible modes to maximize the ATP yield on the given substrates.In line with the above reasoning, we realized that even though the behavior of These results provide a thorough understanding of the limitations to methods that predict flux states after adaptive evolution based on FBA, but also provides handles (in the forms of specific cultivation conditions) to overcome them.L. plantarum with glycerol as main carbon source in chemically defined medium has not been described, but our previously developed genome-scale metabolic model predicted that growth on such medium should be possible (using the Minimization of Metabolic Adjustment (MoMA) algorithm. In this algorithm, the state in the new situation (growth on glycerol) is sought that minimizes the distance with the original state (growth on glucose), thereby mimicking the immediate effect of a medium change bd-cytochrome oxidase Glycerol is a substrate that is not unknown to lactobacilli, but often is co-fermented with a fermentable carbon source L. plantarum on glycerol medium, we observed very slow growth under anaerobic conditions . Under aerobic or respiratory conditions, no growth was observed at all. We attribute the failure to grow initially under aerobic conditions to oxygen-related stress, the cost of which was apparently higher than the potential benefit in terms of ATP yield (we will come back to this later). By applying the widely-used serial dilution protocol When we tried to cultivate 600 values increased substantially . At this point we tried to grow the adapted culture again under respiratory conditions, and growth was now possible and higher OD's were reached . Hence, from this point on the adaptive evolution protocol was continued under respiratory conditions. −1, which was more than a magnitude higher than the initial growth rate. After 150 days and approximately 500 generations, no further increase in final OD and growth rate was observed. At this point single colonies were picked from the adapted culture, and growth rate was assessed in the individual strains. The variation in growth rate among different strains was not different from that of a single one (data not shown). Moreover, sequencing of the promoter region of the glpKDF-operon (encoding the glycerol catabolic genes necessary for glycerol usage) of six clones identified the same point mutation (results not shown), indicating that the culture was homogeneous. We have therefore picked one of these six colonies (designated NZ1405) for further detailed physiological characterization. In After two months and approximately 20 generations, the OD−1 and 0.23±0.04 h−1 for shake flask and fermentor, respectively). Based on the fact that (i) the amounts of substrates consumed and products formed per unit of biomass were very similar in both settings, and (ii) the dissolved oxygen during the measurements in the fermentor was above 20%, which is generally more than enough for oxygen concentration not to be limiting for the oxygen consumption rate, the oxygen uptake rate in the fermentor and the shake flask should be very similar  in the reverse direction than usual, the usual direction being cleavage of the xylulose 5-P as phosphoketolase is involved in pentose catabolism in many lactic acid bacteria in silico that aerobically, glycerol can be completely converted to CO2 and water, yielding as much as 3 moles of ATP per mole of glycerol by substrate level phosphorylation alone. Clearly, the experimental results do not comply with an active phosphoketolase cycle. The observed behavior was therefore dismissed as an interesting future target for perhaps synthetic biological improvements, and the model was adjusted to prevent the phosphoketolase cycle from running  efficiency of some metabolic processes may have improved that have not been incorporated in the model. These relate to possible changes in efficiency of transport systems, to efficiencies in proton leakage and/or ATPase stoichiometry (number of protons pumped per ATP molecule), or to the assembly of biomass precursors into new cells (the growth-related ATP coefficient). Improvements in these processes, however, would lead to lower in silico growth rate predictions. The second possibility is that the ATP requirement for maintenance has changed. Based on the observation that respiratory growth on glycerol was not possible for the wild type (see above), we reasoned that the ATP maintenance under respiratory conditions could be higher than the maintenance measured previously −1 gDW−1 was required to obtain the observed growth rate of 0.26 h−1 under experimentally determined flux constraints. This maintenance coefficient is in the same order as that measured for respiratory organisms, such as 5.87 mmol h−1 gDW−1 for E. coliL. plantarum cannot grow on glycerol .The complex medium that was used, created another problem that needed to be solved: How to set reasonable constraints on the many medium components present in the medium? When all measured fluxes, including the amino acid fluxes, were set as constraints and biomass yield was optimized, a growth rate of 0.324 hReassuring as this result may be, the main question is, whether we can predict fluxes. Since we have so many input fluxes, the issue is what the minimum set of input fluxes is that needs to be fixed by experimental observations in order to prevent the system from becoming unbounded. This issue has not been specifically addressed before, as most studies have been performed on organisms that grow on minimal salts media with one carbon source To tackle this problem, we first dismissed a major impact of amino acid metabolism in the light of the smaller fluxes compared to primary metabolism . Hence, We found 2669 EFMs and 531 different overall EFM stoichiometries with net ATP production . To understand this result from the robustness analysis, we went back to the elementary flux modes. We applied a constraint-based optimization on all 2669 EFMs to ask which combination of EFMs would lead to the highest ATP production yield, given the measured fluxes of oxygen, glycerol and citrate as constraints see . The ana2 uses the citrate, EFM3 the oxygen and EFM174 consumes the glycerol. This analysis provides an explanation for the lactate production being optimal under these conditions . When 1 mole of oxygen were available, anaerobic conversion of glycerol into lactate (EFM174), and conversion of the ethanol into acetate using the oxygen (EFM3), yields more ATP than using the oxygen for the most efficient glycerol conversion into acetate (EFM190: 1.5 o2+5 adp+4 h+5 pi+glyc = 5 atp+7 h2o+co2+ac). Please note that the fact that lactate is predicted by FBA under respiratory growth is not in contradiction with the inability of FBA to predict lactate under anaerobic growth on glucose: the conditions are completely different.Thus, EFMs 2, 3 and 174 (numbered according to the table in the 2) was unexpected, since no succinate is being formed experimentally. Rather, acetoin was formed, a well known product from citrate metabolism implied in pH homeostasis 2258 and EFM2383) that convert citrate into acetoin and acetate with a slightly lower yield (2 atp per citrate compared to 3 atp for citrate plus ethanol in EFM 2). Since in the optimization EFM2 contributes only 6% of the total ATP yield, the small difference between EFM2 and EFM2258 or EFM2383 may have provided too small a selection pressure to allow selection of a strain which reprogrammed the network from acetoin to succinate production.The anaerobic conversion of citrate and ethanol into acetate and succinate  and observed fluxes is striking and does strongly suggest yield optimality under these conditions. In any case, yield optimality was useful to interpret and predict the observed fluxes.In summary, the constraint-based EFM analysis provided valuable additional mechanistic insight in the many options that the metabolic network had (as a function of the oxygen consumption rate), and strikingly, the adapted strain selected out of the 2669 possibilities the 3 EFMs that were  optimal for ATP production under oxygen uptake limitation. The use of elementary flux modes to decipher the optimal use of multiple nutrients is a new application of the EFM concept, underpinning the optimality of the solution used by the adapted strain. Under these relatively poor conditions, realizing an optimal yield appears the best strategy to win the battle of fitness. However, this is provided a restricted uptake of oxygen, and so one may argue that yield maximization is still not predictive in our case, the optimum being full oxidation to acetate. Indeed, the situation is analogous to the suboptimal acetate production by E. coliin silico predicted maximal growth rates and corresponding optimal consumption rates  do not always fit the data. Although the authors discuss the observations, they do not, however, provide an explanation when to expect a deviation from optimality. The case of L. plantarum described in ref This study shows how rational design of the selection conditions can be used to steer the strategy for fitness of an organism into a desired direction. It appears, at first glance, that this conclusion was reached before, especially by the studies of the Palsson group on A clear understanding of the difference will be crucial in appreciating the usefulness and limitations of optimization techniques in systems and synthetic biology. For example, the recently developed OptKnock strategy took the idea of being able to predict adaptive evolution one step further by identifying knockout targets that would result in the alignment of growth rate and byproduct formation as optimization objective, thereby forcing cells to increase fitness by increasing product formation Lactobacillus plantarum WCFS1. Cells were grown on Chemically Defined Medium (CDM) as described previously 600 reached in 24–48 hours).The bacterial strain used in this study was 600 and growth rates were compared. Subsequently, one isolate (NZ1405) was chosen for further characterization.At the end of the adaptation process, judged by no further growth improvements, 96 single colony isolates were isolated from a CDM agarose plate  and grown overnight in a KC Junior, micro-titer reader  shaken with intermediate intensity at 37°C. OD600 of 0.7), samples were collected for analysis of amino acids, organic compounds and dry weight, as described previously La) in this setting was determined directly before inoculation of the fermentor by a switch to N2 gas and back to air and recording the change in dissolved oxygen. The determined value (11.4±0.4 h−1) was used to calculate the oxygen diffusion rate  during growth:Strain NZ1405 was cultivated in Erlenmeyer shake flasks as described above (4 independent experiments). At mid-log phase (OD600 0.2–1) was used.The average oxygen consumption rate during mid-exponential growth , i.e. the corresponding compounds could only be produced. At each oxygen consumption rate, the growth rate was maximized according to Eq 3. At the resultant maximal growth rate, the exchange fluxes for lactate, glycerol, ethanol, acetate and citrate were maximized and minimized to test for alternative optimal solutions The robustness analysis presented in Elementary flux modes were calculated using Metatool 4.9 iα is the flux going through EFMi, atp,in the yield of ATP of EFMi . The optimization procedure thus provides the set of α's that maximize the total ATP yield given a set of capacity constraints on input fluxes (the summations). The optimizations were performed using glpk (http://www.gnu.org/software/glpk/).Here, Dataset S1Model capacity constraints. This file contains amino acid uptake rates and corresponding flux constraints used in the model, and more details on model results.(0.24 MB DOC)Click here for additional data file.Table S1L. plantarum model v3 used in this study.Model details. This file contains the abbreviations, reactions and the gene-protein-reaction associations of (0.28 MB XLS)Click here for additional data file.Text S1L. plantarum.The phosphoketolase cycle. This file contains more information on the phosphoketolase cycle that was discovered in the network of (1.60 MB DOC)Click here for additional data file.Text S2Elementary Flux Mode analysis. This file contains the Metatool input file used for elementary flux mode analysis, and the resulting EFMs.(0.66 MB DOC)Click here for additional data file.


Direct visualization of data sets in the context of biochemical network drawings is one of the most appealing approaches in the field of data evaluation within systems biology. One important type of information that is very helpful in interpreting and understanding metabolic networks has been overlooked so far. Here we focus on the representation of this type of information given by the strength of regulatory interactions between metabolite pools and reaction steps.E. coli network.The visualization of such interactions in a given metabolic network is based on a novel concept defining the regulatory strength (RS) of effectors regulating certain reaction steps. It is applicable to any mechanistic reaction kinetic formula. The RS values are measures for the strength of an up- or down-regulation of a reaction step compared with the completely non-inhibited or non-activated state, respectively. One numerical RS value is associated to any effector edge contained in the network. The RS is approximately interpretable on a percentage scale where 100% means the maximal possible inhibition or activation, respectively, and 0% means the absence of a regulatory interaction. If many effectors influence a certain reaction step, the respective percentages indicate the proportion in which the different effectors contribute to the total regulation of the reaction step. The benefits of the proposed method are demonstrated with a complex example system of a dynamic The presented visualization approach is suitable for an intuitive interpretation of simulation data of metabolic networks under dynamic as well as steady-state conditions. Huge amounts of simulation data can be analyzed in a quick and comprehensive way. An extended time-resolved graphical network presentation provides a series of information about regulatory interaction within the biological system under investigation. Research projects in systems biology produce large amounts of data that usually spread over various 'omics' domains, are time dependent or belong to different organisms and physiological conditions. Irrespectively of whether these data are produced in a wet lab or on a computer, the evaluation requires visualization techniques representing as much information as possible in an intuitive way. Clearly, the direct visualization of data sets in the context of a biochemical network drawing is one of the most appealing approaches in this field.This contribution is concerned with data visualization in the context of metabolic networks. It focuses on the representation of an important type of information given by the strength of regulatory interactions between metabolite pools and reaction steps. The following brief survey of visualization methods for metabolomic and fluxomic data shows that up to now metabolite pool size and flux data have been represented mainly in a network context whereas appropriate concepts to visualize regulatory information are missing.In general, a metabolic network is drawn as a directed graph where the nodes represent metabolite pools and the edges represent chemical reaction steps. As biochemical reaction steps can have multiple substrates and products, a hypergraph with multi-source multi-target edges is commonly used -4. AlterSome other conceptional differences found in the literature are whether metabolites or fluxes are allowed to be duplicated in order to avoid edge intersections or whether cometabolites are distinguished optically from reaction substrates and products. In any case, the data to be visualized can be linked directly to the nodes or edges of the network. This can basically be achieved in the following ways.• The most primitive way to represent data in the network context is to annotate the nodes or edges with textual tags  data have to be displayed. One option then is to show time course plots along with the network nodes or edges  of an interaction. The goal of this contribution is to develop such a definition which is suitable for the intuitive interpretation of data under dynamic and steady-state conditions. Clearly, such a definition can only be reasonably given for the case of simulated data because some information on the reaction kinetics of the involved steps is needed to establish a meaningful RS definition.E. coli.This contribution is organized as follows. A novel concept for the determination of the RS of effectors in enzyme-catalyzed reactions is presented in the first two sections. Next, a general definition for the RS is given followed by a description of the visualization approach. Finally, we provide an example to demonstrate the practical significance of the proposed method, where the whole concept is applied to a relevant dynamic model system of Before explaining in detail how the RS for metabolite pools influencing reaction steps is defined, a list of properties is given that should be reasonably fulfilled by the new concept. The driving force behind these properties is to ensure a maximum of intuitive interpretability and to avoid an overload of information for the user.(i) A RS is defined for all effectors (i.e. inhibitors or activators) of a reaction step which are not contained in the set of substrates or products. These effectors can be identified immediately from the corresponding reaction kinetic expression.(ii) One numerical RS value should be associated to any effector edge contained in the network. Thus, it is possible to visualize RSs directly in the network context. Any of the already-mentioned visualization techniques for pool sizes and fluxes might be used for this purpose.(iii) The RS of an effector with respect to a reaction step has to be calculated from the momentary values of pool sizes and fluxes in the network with the additional knowledge of the respective reaction kinetic formula and parameters. Consequently, RS is a time-dependent quantity which does not depend on the history of a current system state.(iv) The RS should express how strong an influence a reaction step has on a given reaction rate. Moreover, it should distinguish between activation relations (positive sign) and inhibition relations (negative sign). In the visualization the cases can be distinguished between easily by using different colors.(v) The RS should be approximately interpretable on a percentage scale where 100% means the maximal possible inhibition or activation and 0% means the absence of a regulatory interaction.(vi) If many effectors have an influence on a certain reaction step, the respective percentages should indicate the proportion in which the different effectors contribute to the total regulation of the reaction step.Here some comments concerning the reasoning and rationale behind these properties are appropriate. Referring to item (i), the definition of RSs for reaction substrates and products (reversible reaction only) is, in principle, possible. However, the obtained values would not indicate any metabolic regulation, but rather how strong the reaction is driven by the availability of substrates and products, respectively. This information can be directly represented by the visualization of metabolic pool sizes and fluxes. In most cases the effectors of an enzymatic reaction are not consumed by the reaction step itself. The only exceptions are substrate and product inhibition mechanisms which are explicitly denoted in the reaction kinetic formula and would also, therefore, be covered by the RS definition.It is reasonable to quantitate the effector influence by exactly one RS value, otherwise the multitude of visualized information is likely to become confusing (item (ii)). Moreover, for the RS calculation, the general assumption is made that a certain effector molecule modulating an enzymatic reaction step is instantaneously available and distributed equally over the whole cell (item (iii)).The properties in items (iv)–(vi) are important for a meaningful and intuitive interpretation of RSs. In particular, the distinction between activators and inhibitors is of fundamental importance with respect to the underlying effect of a metabolic enzyme regulation. With regards to the practical implementation of RS values, the definitions of lower and upper bounds are indispensable. In addition, applying a percentage scale facilitates the reception of information.When trying to construct a RS measure that fulfills these conditions it became clear that different approaches are possible and some decisions have to be made. Moreover, it turns out in the following that the above-mentioned requirements are not completely free of contradictions so that some compromises are necessary. However, it is important to note that the precise value of a displayed quantity plays no role in a graphical visualization, but, rather, it is the rough order of magnitude that is important. Thus, contradictions are not important if they can be resolved by sacrificing some numerical precision.k values, where k is the number of effectors. This immediately indicates the implicit assumption that activators and inhibitors act independently in a reaction. In contrast, it is well known from enzyme kinetics that this is not always the case. However, if correlations between the influences of different effectors have to be taken into account, further coefficients of higher order are needed to characterize this correlation. Clearly, this would prevent us from implementing an intuitive network-based visualization.One conceptional difficulty with the introduction of a RS is that the activation or inhibition state of a reaction step in relation to the state where all activators or inhibitors are absent must be quantitated by exactly A well-known family of methods that assigns exactly one coefficient to each effector are the sensitivity-based methods from which the elasticities defined in metabolic control theory are the best-known example. Although elasticities play an important role in metabolic control theory, they are certainly not the right quantities to be used for visualization in the way specified above. This can be explained easily with an inhibitory relationship expressed by a multiplicative hyperbolic term in a reaction kinetic expression (here S is substrate concentration and I is inhibitor concentration):r/∂I tends to zero with increasing inhibitor concentration which would erroneously indicate that the inhibitor has no effect on reaction flux. Obviously, the opposite is true and RS should tend to -100% in this case. Consequently, when used for network visualization, elasticities rather produce non-intuitive results.In this example the sensitivity ∂Likewise, the use of flux control coefficients is not appropriate because these scaled sensitivities reflect the global network regulation  and thus cannot be interpreted locally for one isolated reaction step in the network.In this contribution the aim is now to find a quantity expressing how strong a reaction step is up- or down-regulated compared with the completely non-inhibited or non-activated state, respectively. As a first example, in Equation 1 the RS can be reasonable defined bybecause this is the percentage by which the non-inhibited flux (I = 0) is down-regulated.n binding sites for an activator:In some cases enzymatic reactions are described by kinetic expressions that show no saturation behavior, i.e. the flux continuously increases with increasing activator concentration. As an example for a mechanistic enzyme description with these properties the following kinetic expression is given, describing an allosterically activated enzyme covering The determination of the RS for the activator A using the formular. Consequently, this results in a value of zero for νA. For this reason the definition of an upper bound Amax for the activator concentration is suitable. This boundary should be chosen according to the expected physiological concentration range of the respective effector metabolite.will not succeed, because the limit calculation for arbitrary high activator concentrations (A → ∞) leads to infinitely high reaction rates emax for all effectors are defined. However, in the case of kinetics with saturation behavior, the maximum effector concentrations need not be limited to a finite value for RS calculability.Moreover, the corresponding simulated values must also be restricted to this range. In order to derive a general definition for the RS, upper bounds It turns out that for arbitrary reaction kinetic formulae the definition of a RS is not as simple and straightforward as in the example from Equation 1. For this reason, the concept of RS is defined in the following in a step-by-step approach that starts with simple standard reaction kinetic formulae and successively generalizes the introduced concepts to the most general case. At the end it will be possible to apply the concept to any mechanistic reaction kinetics.As an instructive example, consider an enzymatic reaction where the conversion of one substrate is regulated by two effectors . The velocity equation is then given byα → ∞), non-competitive  and partial competitive  inhibition T and e = T comprise the substrate and effector concentrations included in the Pk reaction. The necessary steps for determining the RS values of all effectors are as follows.The vectors a = T and inhibitors i = [ATP].1. Separation of all effectors into activators amax = T and imax = [ATPmax].2. Definition of maximal effector concentrations 3. Determination of the resulting RS:with4. Determination of single effector influences:with5. Determination of scaled single effector influences:withνres) of all effectors. Clearly, without any influence of the inhibitor ATP, the Pk reaction is activated. With increasing ATP concentration the RS of this effector also increases leading to a decrease of the Pk flux. At the same time the proportions  in which the two activators contribute to the total regulation are reduced. Above a value of ATP = 5 mM the regulation is solely determined by the inhibitor (i.e. νres = νATP = 100%).Figure In order to allow for an automatic calculation of all RSs of a given metabolic network, the whole method is implemented in a Matlab GUI  providing a direct interface to the MMT2 software package that is used for the simulation of dynamic network models . Before Along with this contribution, the network-based visualization tool MetVis has been extended for the visualization of RS data. MetVis was introduced in  as a tooA new feature of MetVis is the visualization of RS by edges connecting metabolite pools and reactions which are represented by nodes in a bipartite manner. Once the precise meaning of RS has been defined, the respective data can be generated by a simulation tool and used for visualization.The results of simulations are usually delivered as a CSV structured file, containing information about the concentrations of metabolites, flows of reactions and the RS values of effectors. In the case of time-varying simulation data, the dynamic metabolic behavior contained in these data is expressed visually with an animation showing changing metabolite pool sizes and changing fluxes represented by differently filled boxes and varying arrow widths, respectively. Motivated by the fact that the metabolite concentrations and flux values can vary greatly, an adequate scaling of the input data is performed. This can be achieved in different ways depending on the scope of the study .To visualize effectors using MetVis, additional edges representing the inhibition or activation effect connecting metabolites with reactions (enzymes) need to be inserted into the designed network. These connecting edges are visualized with a red circle for inhibition and a green circle for activation and are placed next to the affected reaction. The dynamic behavior of effector influences (i.e. the RS data) is displayed by changing the size of the corresponding circles indicating the level of the respective activation and inhibition.E. coli, originally published in [In the following the benefits and practical significance of the proposed visualization approach are illustrated with the help of a complex dynamic network model of the central carbon metabolism of ished in . This neThe dynamic model is a system of ordinary differential equations consisting of 18 mass balance equations, 30 reaction rates and 7 analytical functions approximating the measured concentration values of cometabolites . All reactions steps are described by mechanistic enzyme kinetics resulting in a total number of 116 parameters. The special feature of this model is the high number of kinetic expressions describing regulatory interactions between metabolites and reactions  as the resulting effector influence (νres) shows a strong activation. The same holds true for the G1pat reaction .• The low activity of the Pk enzyme is solely determined by the generally small value of the maximal reaction rate . In this case the animation underlies a global scaling, where the metabolite pools and flux values are divided by the overall time maximum of all concentration and flux values, respectively [Figure ectively .Thus, an empty metabolite pool indicates a concentration value of zero. Conversely, a full pool box indicates a global concentration maximum regarding all metabolite concentrations, e.g. the high intracellular concentration of G6P. This representation is very useful in comparing the global flux distribution in the network and the absolute changes in metabolite pool sizes. In this manner the identification of flux controlling or flux limiting steps in the network becomes possible.Referring to the visualized network the high glucose uptake flux via the Pts, which dominates the system directly after the glucose pulse, can be recognized. Obviously, at this early time point after the pulse the G6P converting enzymes, Pgi and G6pdh, are the most limiting steps for the further conversion of the glucose after uptake.As already mentioned, the Pts reaction is strongly inhibited by its product G6P and thus limits the glucose uptake in the following time course .The first two enzymes, G6pdh and Pgdh, catalyzing the entrance reaction into the pentose phosphate pathway are inhibited by NADPH, leading to a 50% decrease in their reaction rates. Despite the low concentration of NADPH in comparison with the second inhibitor ATP of the Pgdh enzyme, the inhibitory effect is dominated by NADPH. The reason for this effect is the very low affinity of Pgdh towards ATP  also have to be defined.One limitation of mechanistic enzyme descriptions is the large number of model parameters that have to be estimated according to given experimental data. This has led to the development of approximative kinetic rate equations based on the linlog or power law approach. Despite their broad applicability, one major drawback of these kinetic formats is their indeterminacy for concentration values of zero caused by the logarithmization of pool sizes. Clearly, to utilize the RS concept to the linlog and power law approaches, lower boundaries for the effector pools  will help us to gain an insight into the complex cellular network in its entirety. In the case of mechanistic descriptions for the reactions taking place in the different 'omics' levels, the concept of RS can be applied to quantify and visualize all regulatory interactions.E. coli, Escherichia coli; FBP, fructose-1,6-bisphosphate; G1pat, glucose-1-phosphate adenyltransferase; G6P, glucose-6-phosphate; G6pdh, glucose-6-phosphate dehydrogenase; GUI, graphical user interface; MetVis, Metabolic Visualizer; MMT2, Metabolic Modeling Tool 2; NAD, nicotinamide-adenindinucleotide; NADPH, nicotinamide-adenindinucleotide-phosphate; OAA, oxaloacetate; PEP, phosphoenolpyruvate; PEPCx, phosphoenolpyruvate carboxylase; Pfk, phosphofructokinase; Pgdh, 6-phosphogluconate dehydrogenase; Pgi, phosphoglucose isomerase; Pk, pyruvate kinase; Pts, phosphotransferase system; Pyr, pyruvate; RS, regulatory strength.ADP, adenosindiphosphate; AMP, adenosinmonophosphate; ATP, adenosintriphosphate; CSV, character separated values; a, b, rate constant factors; emax, vector of maximal effector concentrations; Ki, affinity constant for a certain metabolite i; rj, reaction rate of enzyme j; rmax, maximum reaction rate; remin, , remax, , minimal and maximal reachable flux for variable effector pools; re0, , flux in the completely non-regulated state; S, A, I, substrate, activator and inhibitor pool; s, e, a, i, vector of substrate, effector, activator and inhibitor pools; α, β, affinity constant factors; νres, resulting RS; ω, scaling factor.Effector: metabolite that modulates an enzyme-catalyzed reaction leading to an acceleration (activator) or deceleration (inhibitor) of the reaction rate.Regulatory strength: measure for the strength of an up- or down-regulation of a reaction step compared with the completely non-inhibited or non-activated state.Competitive inhibition: metabolites which are not substrates of the enzyme can bind to the active site and compete with the substrate.Non-competitive inhibition: an inhibitor can bind to the enzyme substrate complex (ES) forming a complex (ESI) that reduces the amount of active enzyme.Allosteric inhibition or activation: the enzyme has additional binding sites for specific inhibitors or activators which can change the conformation resulting in a change of the enzyme activity.The author(s) declares that there are no competing interests.SN developed the methods for the RS determination in metabolic networks. EQ developed the MetVis tool and helped to extend its functionality for the visualization of RS data. AW and WW conceived of the RS concept, participated in its elaboration and helped to draft the manuscript. All authors read and approved the manuscript.


Cancer cells simultaneously exhibit glycolysis with lactate secretion and mitochondrial respiration even in the presence of oxygen, a phenomenon known as the Warburg effect. The maintenance of this mixed metabolic phenotype is seemingly counterintuitive given that aerobic glycolysis is far less efficient in terms of ATP yield per moles of glucose than mitochondrial respiration.Here, we resolve this apparent contradiction by expanding the notion of metabolic efficiency. We study a reduced flux balance model of ATP production that is constrained by the glucose uptake capacity and by the solvent capacity of the cell's cytoplasm, the latter quantifying the maximum amount of macromolecules that can occupy the intracellular space. At low glucose uptake rates we find that mitochondrial respiration is indeed the most efficient pathway for ATP generation. Above a threshold glucose uptake rate, however, a gradual activation of aerobic glycolysis and slight decrease of mitochondrial respiration results in the highest rate of ATP production.Our analyses indicate that the Warburg effect is a favorable catabolic state for all rapidly proliferating mammalian cells with high glucose uptake capacity. It arises because while aerobic glycolysis is less efficient than mitochondrial respiration in terms of ATP yield per glucose uptake, it is more efficient in terms of the required solvent capacity. These results may have direct relevance to chemotherapeutic strategies attempting to target cancer metabolism. Since its original discovery by Warburg  it has bThe actual molecular mechanisms that lead to the enhanced aerobic glycolysis are increasingly well understood. Under physiological condition, for example, PI3K/Akt signaling pathways play a critical role in promoting aerobic glycolysis in activated T cells in response to growth factors or cytokines stimulation . In the The evidence accumulated so far thus indicates that the presence of aerobic glycolysis is a common characteristics of rapidly proliferating cells, and that it may offer a growth advantage to rapidly proliferating normal cells, e.g., during development and tissue regeneration, and to cancer cells in tumor formation. Yet, a system-level interpretation of the origin of this growth advantage has not yet been formally provided. To start addressing this issue, here we introduce a reduced flux balance model of ATP generation that incorporates a glucose uptake capacity constraint and the limited solvent capacity of the cell cytoplasm. The model allows us to uncover the existence of two different energetically favorable metabolic regimes within proliferating cells, a finding that is congruent with experimental results.fG) is partitioned into the flux of aerobic glycolysis (fL), and to oxidative phosphorylation (fM). Aerobic glycolysis represents glycolysis, converting glucose into pyruvate, and then pyruvate reduction by lactate dehydrogenase (LDH) in the cytosol, resulting in the end product lactate that is then excreted to the extracellular millieau. ATP generation through oxidative phosphorylation is decomposed into the generation of pyruvate through glycolysis, followed by the oxidation of pyruvate in the TCA cycle and the respiratory chain, the latter two processes taking place in the cell's mitochondria. There is also a third component for glucose utilization accounting for the production of precursor metabolites needed in anabolic processes (fP)  is proportional with the rate of anabolic processes (fP), and use fATP as a surrogate for the overall cell metabolic rate , which e Figure .Aerobic glycolysis includes the glycolysis pathway producing pyruvate and the pyruvate reduction producing lactate. The overall reaction for aerobic glycolysis isAerobic glycolysis has a yield of 2 moles of ATP per mole of glucose, resulting from the conversion of glucose into lactate. On the other hand, oxidative phosphorylation includes the glycolysis pathway producing pyruvate, the oxidation of pyruvate in the TCA cycle, and the respiratory chain. The overall reaction of for oxidative phosphorylation isOxidative phosphorylation yields 38 moles of ATP per mole of glucose, two of which are produced in glycolysis and the other 36 during the oxidation of pyruvate in the TCA cycle coupled to respiratory chain activity. Summing up these differential contributions we obtain the rate of ATP productionfG = fL + fM.where the second equality was obtained using fL, fM) that provides maximum ATP production rate given the cell's metabolic constraints. Of these, the first metabolic constraint is associated with the maximum glucose uptake rate (or glucose uptake capacity)Our aim is to determine the optimal flux distribution , the lactate excretion rate (2fL) and the mitochondrial ATP production rate (36fM) are proportional to the mass of glycolytic enzymes, lactate dehydrogenase and mitochondria respectively, with fG = rGMG/V, 2fL = rLML/V and 36fM = rMMM/V, where rG is the glycolytsis rate per unit of glycolytic enzyme mass, rL is the rate of lactate production per mass of LDH, rM is the mitochondrial ATP production rate per unit of mitochondrial mass, and V is the cell volume. In these equations the product by the mass and the division by the cell volume takes into account that the rates r are commonly reported in the literature per unit of dry weight, while the pathway rates f are reported per unit of cell volume. Because of the interdependency of volume, mass and reaction rate, the volume constraint (5) can be translated to the metabolic constraintwhere ϕ ATP = VATP/V is the total volume fraction of the cell cytoplasm occupied by glycolytic enzymes, LDH and mitochondria, and the crowding coefficients aG = vG/rG, aL = 2 vL/rL and aM = 36 vM/rM quantify the occupied volume fractions per unit of glycolytic, lactate excretion and mitochondrial respiration rate, respectively. Using empirical data reported in the literature we have estimated the crowding coefficients (see Methods). We obtain aG ≈ 0.0027 (min/mM), aL≈ 0.00023 (min/mM) and aM ≈ 0.10 (min/mM), indicating that the mitochondria contributes about 5 and 50 times more to molecular crowding than glycolytic enzymes and lactate dehydrogenase, respectively. The unexpected consequences derived from this fact will be uncovered below.where VATP, rG, rL and rM are constant parameters that can be obtained from experimental estimates. Note though, that this is an approximation, as there may be regulatory mechanisms that under certain environmental or developmental conditions are capable of altering the amount of intracellular space allocated to ATP producing pathways and the activity of glycolytic enzymes, LDH and mitochondria.In our modeling we assume that Taken together the relevant metabolic optimization problem is as follows: maximize the ATP production rate (3) under the metabolic capacity constraints (4) and (6).fG = FG) to maximize the ATP production rate. However, there are some differences in the flux distribution along aerobic glycolysis and oxidative phosphorylation depending on the glucose uptake rate. Specifically,In the following we discuss the model-predicted utilization of aerobic glycolysis and oxidative phosphorylation pathways in the context of the cell's glucose uptake rate. It becomes obvious that the glycolysis rate matches the maximum glucose uptake capacity , when mitochondria occupy the entire cell volume fraction (VATP) available for ATP production pathways . Beyond this threshold value the concentration of mitochondria, and therefore the rate of ATP production through oxidative phosphorylation, cannot be increased further. However, additional glucose uptake may be diverted toward pathway(s) less efficient in terms of ATP yield per mole of glucose. Indeed, the optimal solution predicts a metabolic switch, in which glucose uptake rates larger than f1 leads to a linearly increasing lactate production , and therefore a component of the ATP production is now derived from aerobic glycolysis. Furthermore, the increase in lactate production is accompanied by a slight, gradual reduction of the oxidative phosphorylation flux  by fM ≈ f1 and fL ≈ fG-f1. Therefore, we predict that the plots of fM and fL versus fG are rather universal, with a distinctive change of behavior at fG = f1.The optimal solution is graphically illustrated in Figure f1 (9), where the metabolic switch takes place, and the shape of the lactate excretion, oxidative phosphorylation and ATP production plots as a function of the glucose uptake rate. To start addressing the validity of these predictions we first utilized previously reported experimental data for mouse LS cells [The model described above results in two experimentally testable predictions: the glucose uptake threshold LS cells  , which overlaps with the ranges reported above for the different cell lines. This agreement is surprisingly good given that they were estimated using independent data.The glucose uptake threshold, i.e., the uptake rate where the switch takes place, can be estimated from a linear fit to the lactate excretion plot in the regime where lactate is excreted  [To further test the model's predictive capability, we have next examined human BJ fibroblast cells serially transduced with the catalytic domain of human telomerase (hTERT), SV40 large T (LT) and small T (ST) antigens, and an oncogenic allele of H-ras H-RasV12) . Previou . Previo12) , and tha12) . Previou12) . In cont12) .To confirm and extend the above findings, we experimentally tested the extracellular lactate levels, mitochondrial mass and mitochondrial membrane potential (a proxy for respiratory chain activity) of CL1-4 cells at 48 h, 72 h, 96 h, and 120 h after seeding the same amount of cells, along with cell counts and total protein concentration measurements. We find that the tumorigenic CL4 cells have the highest extracellular lactate levels compared to the other cell types at all time points tested that correlates with a concomitant change in LDH enzyme activity but not protein expression level  production per its own  mass than the rate of ATP production by respiration per mitochondrial mass.By developing a reduced flux balance model of ATP production Figure  here we f1~0.7-3.8 mM/min predicted by our model. Also, experimental data reported here and previously [Comparisons to experimental data reported for LS mouse cells , hybridoeviously  on lactaeviously  are congeviously , and tumeviously -24.The model described here has its limitations and does not account for all the potential population-level advantages of the Warburg effect. First, decreased mitochondrial respiration also contributes to the mitochondrial depolarization that is frequently observed in tumor cells, and that is thought to dampen the cell's propensity for mitochondrially-induced apoptosis . SecondlE. coli and S. cerevisiae cells also exhibit aerobic glycolysis, resulting in the excretion of the metabolic byproducts, acetate and ethanol, respectively. More importantly, both for E. coli and for S. cerevisiae a shift toward aerobic glycolysis follows a switch from limited to rich nutrient conditions [E. coli the maximum growth rates in different carbon sources and the mode of utilization of mixed carbon sources are in agreement with those allowed by the limited solvent capacity of the cell's cytoplasm [From a more general perspective, it is evident that aerobic glycolysis is not unique to cancer- and rapidly proliferating normal mammalian cells but is present in unicellular organisms, as well. For instance, under aerobic growth conditions fast growing nditions ,29, and ytoplasm . Taken tThese results may have relevance to cancer therapy as well. It has been previously proposed that the Warburg effect may represent a unique property of cancer cell metabolism . PharmacaG: The glycolysis rate per total mass of glycolytic enzyme is about rG = 0.29 mmol glucose uptake/min/g at 30°C [vG = 0.79 mL/g. Putting these two values together we obtain aG = vG/rG = 0.0027 (min/mM). aL: The rate of lactate production per mass of lactate dehydrogenase is about rL = 7 mmol lactate/min/g at 30°C [vL = 0.79 mL/g. Putting these two values together we obtain aL = 2vL/rL = 0.00023 (min/mM). aM: The mitochondrial rate of ATP generation per mitochondrial mass is in the range of 0.1-1.0 mmol ATP/min/g [rM = 1.0 mmol ATP/min/g. The mitochondrial specific volume has been reported to be 3.15 mL/g in mammalian liver [vM = 2.9 mL/g. Putting these two values together we obtain aM = 36vM/rM = 0.10 (min/mM). ϕATP: The cell volume fraction occupied by cellular components involved in the generation of ATP was estimated from the cell volume fraction occupied by mitochondria. Literature reports for the latter range between 0.07 in some cell lines to 0.38 in muscle cells , aG = 0.0027 (min/mM) and ϕATP = 0.07-0.38 into (9) we obtain f1 = 0.7-3.8 mM/min. at 30°C , obtaine at 30°C ), divide at 30°C ), and di at 30°C , thus we at 30°C , obtaine at 30°C ), divide at 30°C ). The sp at 30°C , thus weTP/min/g -35. We uan liver  and 2.6 an liver , respectμmole per 106 cells per day. The experimental value of f1 was estimated from a linear fit to the data points in the region where there is a significant lactate production  (CL4) [2 incubator with 5% CO2 and 95% air. The experimental protocols for Western blots, Ras activity measurements cell growth and soft agar assays, LDH activity measurements, and for measuring mitochondrial mass and membrane potential are provided in the Additional file Genetically-manipulated derivatives of human BJ fibroblasts, transfected with the catalytic subunit of human telomerase (hTERT) (CL1), hTERT+ SV40 large T (LT) CL2), hTERT+LT+ SV40 small T (ST)(CL3), and hTERT+LT+ST+ Vs) (CL4)  were gen, hTERT+LThe authors declare that they have no competing interests.AV and ZNO conceived the project. AV developed the metabolic model and analyzed the literature data. JL and YZ performed the reported experiments. AV and ZNO wrote the manuscript. All authors read and approved the final manuscript.Supplementary tables. Supplementary Tables S2, S3 and S4Click here for fileSupplementary information. Experimental protocols and results (p2-4), Table S1 for parameter estimates (p6), Supplementary Figures (p6-8).Click here for file


Saccharomyces cerevisiae, little is known about their impact on the physiology of Pichia pastoris. We have recently reported a beneficial effect of hypoxia on recombinant Fab secretion in P. pastoris chemostat cultivations. As a consequence, a systems biology approach was used to comprehensively identify cellular adaptations to low oxygen availability and the additional burden of protein production. Gene expression profiling was combined with proteomic analyses and the 13C isotope labelling based experimental determination of metabolic fluxes in the central carbon metabolism.Yeasts are attractive expression platforms for many recombinant proteins, and there is evidence for an important interrelation between the protein secretion machinery and environmental stresses. While adaptive responses to such stresses are extensively studied in P. pastoris to hypoxia showed distinct traits in relation to the model yeast S. cerevisiae. There was a positive correlation between the transcriptomic, proteomic and metabolic fluxes adaptation of P. pastoris core metabolism to hypoxia, yielding clear evidence of a strong transcriptional regulation component of key pathways such as glycolysis, pentose phosphate pathway and TCA cycle. In addition, the adaptation to reduced oxygen revealed important changes in lipid metabolism, stress responses, as well as protein folding and trafficking.The physiological adaptation of P. pastoris strain. Remarkably, the integration of data from three different levels allowed for the identification of differences in the regulation of the core metabolism between P. pastoris and S. cerevisiae. Detailed comparative analysis of the transcriptomic data also led to new insights into the gene expression profiles of several cellular processes that are not only susceptible to low oxygen concentrations, but might also contribute to enhanced protein secretion.This systems level study helped to understand the physiological adaptations of cellular mechanisms to low oxygen availability in a recombinant  Over the last two decades significant progress has been made in heterologous protein production, particularly due to the initiation of the genomics era. The acquisition of profound knowledge and entire genome sequences of a number of expression platforms has lead to considerable success for the production of many pharmaceutical proteins or industrial enzymes -3. NeverSaccharomyces cerevisiae -COSY spectra were acquired for both aliphatic and aromatic resonances as described -COSY, for the calculation of relative abundances, f-values  and 34 (oxygen-limiting and hypoxic conditions) metabolic reactions. The measured uptake and excretion rates and the rates of metabolic precursor depletion to biosynthesis, as determined from the composition of ondition , were co13C constrained flux balancing approach similarly to a previous approach [The determined metabolic flux ratios were used as additional constraints for solving the metabolic network following a approach . Using tmit originating from Oaacyt, that is, Oaacyt transport into the mitochondria (only applicable to oxygen-limiting and hypoxic conditions):the fraction of Oaax24 (flux of Oaacyt into the mitochondria) was calculated as a net Oaa transport flux across the mitochondrial membrane , annotated as x24*. Also, the labelling patterns of Pep instead of Pyrcyt were considered (see Results section) and, therefore, the corresponding anaplerotic flux ratio was defined as:Under normoxic conditions, reaction cyt originating from Pyrcyt, that is, the anaplerotic flux ratio (only applicable to oxygen-limiting and hypoxic conditions):the fraction of Oaathe fraction of Pep from PPP assuming a maximal contribution of PPP:The following linear constraint equations (equations 5 to 8) were derived from the flux ratio equations:Equations 5 to 7 were added to the stoichiometric model as a submatrix F, obtaining the complete metabolic model to solve the metabolite mass balances:lsqlin using equation 8 as an additional constrain. Irreversibility was assumed for several intracellular fluxes and for the depletion of precursors to biosynthetic reactions, that is, only positive values were allowed for these fluxes , c is a column vector with either 0 for internal reactions or the corresponding value for each one of the input/output rates and × is the vector of fluxes. Solution of the resulting linear system was obtained using the MATLAB function Confidence intervals for the optimized fluxes were calculated up on the determination of their standard deviation using the Fisher Information Matrix approach (FIM)  as:Calculation of FIM was performed as:ij corresponds to:where C is the variance-covariance matrix of the measurements (assumed independent) and W is a parameter sensitivity matrix where each element of wi (e.g. a measurement) due to an infinitesimal change in parameter pj (a flux).which describes an infinitesimal change of the variable xpj can be derived [Confidence intervals for the estimated fluxes  derived  from:® 7.4 (v2007b) for Windows.where 13C-labeling experiments and macroscopic data processing, performed the metabolic flux calculations and participated in interpretation of results. MD assisted in the design and performance of proteome and microarray experiments. ABG participated in the design and bioinformatics analysis of the microarrays. JS carried out the protein identification by liquid chromatography-tandem mass spectrometry. PJ and HM performed the 2D-NMR and METAFoR analyses and participated in the subsequent interpretation of results. BG helped with the conceptual design of the study and with the interpretation of omics data. JA designed the 13C-constrained MFA approach, and participated in analysis and interpretation of MFA results. DM participated in the overall conceptual and experimental design of this study and interpretation of results. PF participated in the conceptual and experimental design, interpretation of results and in drafting the manuscript. All authors read and approved the final manuscript.KB performed bioreactor cultivations, proteomics and microarray experiments, quantitative real-time PCR, data analysis and interpretation of the results, and drafted the manuscript. MC carried out the Design 2D DIGE Gels. An example of the experimental design for the acquisition of statistical data on differences between samples taken from normoxic (21%), oxygen-limiting (11%) and hypoxic (8%) setpoints. Replica of 2 independent experiments  were labelled with either Cy5 or Cy3 . A pool of all samples served as reference and was labelled with Cy2 (= pooled standard).Click here for file2D DIGE Gels. Representative gel image from a 2D gel electrophoresis experiment with proteins obtained from the Fab-expressing strain. We identified 45 out of 81 proteins with a different expression pattern when comparing high and low oxygen experiments. Green spots show proteins downregulated and pink ones show those upregulated under hypoxia.Click here for fileIdentified protein spots. List of the 45 identified proteins with different abundances comparing hypoxic (8) and normoxic (21) conditions in the P. pastoris expressing and control strain. Proteins were identified by MALDI-TOF MS and grouped into 6 different biological processes. The protein name, short name and accession number, theoretical Mw and pI are reported together with the percentage of peptide coverage and number of identified peptides. Average ratios and 1-ANOVA (DeCyder) are given and only not indicated where no spot could be matched. p.i. = previously identified on other 2D gelsClick here for fileRelative abundances of intact carbon fragments in proteinogenic amino acids. Relative abundances of intact C2 and C3 fragments  in proteinogenic amino acids describing the conservation of carbon chain fragments in P. pastoris Fab-producing and control strains growing in glucose-limited chemostats at a D = 0.1 h-1 in different oxygenation conditions.Click here for fileMetabolic fluxesMetabolic fluxes in the central carbon metabolism of P. pastoris Fab-producing and control strain in glucose-limited chemostats at a D = 0.1 h-1 in different oxygenation conditions. The standard deviations of each net flux are given.Click here for fileqRT-PCR primer sequences used in this study. Table showing the primer sequences of the genes analyzed by qRT-PCR and characteristics of the corresponding amplicons. Calculated copy numbers and the dilution factors in order to obtain 109 copies μl-1 are indicated (see main text for explanation).Click here for fileP. pastorisMetabolic network model of the central carbon metabolism of . Bioreaction network model of the central carbon metabolism of P. pastoris used in the 13C-metabolic flux analysis for the determination of net fluxes under the different oxygenation conditions. Fluxes are represented as net fluxes and the directions of the arrows indicate the directions of the positive net fluxes. The metabolites consumed or produced by extracellular fluxes (shown as dashed arrows) are denoted with (E).Click here for fileP. pastorisStoichiometric model of the central carbon metabolism of . Reactions in the stoichiometric model of the central carbon metabolism of P. pastoris applied in the 13C-MFA determination of the metabolic fluxes under different oxygenation conditions; it also includes anabolic reactions from metabolic intermediates to biosynthesis, transport reactions across the mitochondrial membrane and uptake and excretion reactions. Note that O2, CO2, energy and redox cofactor mass balances were not included in the mass balance constraints for 13C-MFA.Click here for file


Protein translation is a multistep process which can be represented as a cascade of biochemical reactions , 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.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  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  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.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 , 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. 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 . The 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. . HistoriConcurrently, 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 -20. ThisMost 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 . IndepenThus, the data on the exact step of translational inhibition are clearly contradictory. Taking also into account the data about mRNA degradation and P-bodies 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  and [60S] are the concentrations of free 40S and 60S ribosomal subunits, [eIF4F] 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.where  ribosomal subunit in the steady state as x, in which we have neglected the terms of smaller order of magnitude, based on conditions (10):and the equation to determine δ > 1, γ << 1 and, if k1 >>k4/[eIF4F]0 then α <<β. From these remarks it follows that the constant term γ(1 -β) of the equation (13) should be much smaller than the other polynomial coefficients, and the equation (13) should have one solution close to zero and two others:From the inequalities on the parameters of the model, we have α << |1 - δ| or α << |1 - β|. In the expression for x1 we cannot neglect the term proportional to a, to avoid zero values in (13).provided that x2 is always negative, which means that one can have one positive solution x0 << 1 if x0 and x1 if x1 > 0 then x0 does not correspond to a positive value of [eIF4F]s. This means that for a given combination of parameters satisfying (10) we can have only one steady state (either x0 or x1).The solution x = x0 and x = x1 correspond to two different modes of translation. When, for example, the amount of the initiation factors [eIF4F]0 is not enough to provide efficient initiation (x = x1) then most of the 40S and 60S subunits remain in the free form, the initiation factor [eIF4F] being always the limiting factor. If the initiation is efficient enough x = x0 << 1 when almost all 60S ribosomal subunits are engaged in the protein elongation, and [eIF4F] being a limiting factor at the early stage, however, is liberated after and ribosomal subunits recycling becomes limiting in the initiation .The two values Let us notice that the steady state protein synthesis rate under these assumptions iseIF4F]0 microRNA action would be efficient only if it affects k2 or if it competes with eIF4F for binding to the mRNA cap structure  With higher concentrations of [eIF4F]0, other limiting factors become dominant: [60S]0  and k4 . Interestingly, in any situation the protein translation rate does not depend on the value of k1 directly , but only through competing with eIF4F (which makes the difference with the simplest linear protein translation model).This explains the numerical results obtained in : with loeIF4F] translation initiation factor enhances protein synthesis but its effect is abruptly saturated above a certain level.Equation (15) explains also some experimental results reported in : increasNpolysome is defined by mRNA] is the concentration of mRNA. However, [mRNA] is not an explicit dynamical variable in the model, it is implicitly included in other model constants, such as k1, 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 stateIt 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 Npolysome changes in the same way as the protein synthesis steady state value.and k1 = k2 = 2, k3 = 5, k4 = 1, [40S]0 = 100, [60S]0 = 25, [eIF4F]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 β = 0.48 < 1 and this corresponds to the simpler one-stage "inefficient" initiation scenario. To illustrate the alternative situation, we changed the value of k4 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).It was proposed to use the following model parameters in : k1 = k2ers from , β = 0.4Simulations of the protein translation model with these parameters and the initial conditionsare shown on the Fig. S] >> [eIF4F], [60S] >> [AUG]. During this stage, the two non-linear reactions 40S + eIF4F → mRNA : 40S and AUG + 60S → 80S can be considered as pseudo-monomolecular ones: eIF4F → 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 . 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.1) Stage 1: Relatively fast relaxation with conditions [402) Transition between Stage 1 and Stage 2.S] >> [eIF4F], [60S] << [AUG]. During this stage, the reactions 40S + eIF4F → mRNA : 40S and AUG + 60S → 80S can be considered as pseudo-monomolecular eIF4F → mRNA : 40S and 60S → 80S. This stage is characterized by two local quasi-steady states established in the two network reaction cycles (formed from R1-R2 and R3-R4 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 R1-R2 fluxes) relatively fast.3) Stage 2: Relaxation with the conditions [404) Transition between Stage 2 and Stage 3.S] << [eIF4F], [60S] << [AUG]. During this stage, the reactions 40S + eIF4F → mRNA : 40S and AUG + 60S → 80S can be considered as pseudo-monomolecular 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.5) Stage 3: Relaxation with the conditions [40Stages 1-3 can be associated with the corresponding dominant systems  which arThe dominant system of the Stage 1 Fig.  can be mk1 · [40S], k3 · [60S] and we consider that at this stage the changes of 40S and 60S are relatively slow. This system has simple approximate solution, taking into account constraints on the parameters k2 <<k4 <<k2, also assuming k2 << |where we have~ min(1/k4) 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 [80S] << [60S] condition is not satisfied anymore, i.e., following our convention to consider "much smaller" as difference in one order of magnitude, at S] is included, but from the time point S]0 >> [eIF4F]0 and [40S]0 > [60S]0 we have min = t'. This means that the parameters [40S], [60S] of the  steady states for [eIF4F] and [AUG] should slowly (at the same rate as [80S]) change from the time point t' . In other words, after t = t' the Stage 1 solution (20) should be prolonged asFrom this solution, one can conclude that the relaxation of this model goes at several time scales  will be violated, which happens at times ~ S]0 < [40S]0).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  are independently balanced:The Stage 2 is characterized by conditions [S]:Then (6) is simplified and, using the conservation laws, we have a single equation on [40A = [40S](t) + [AUG](t) is a constant quantity conserved accordingly to the quasi-steady state approximation (see "Quasi steady-state and quasiequilibrium asymptotics" section of Methods). Equation (23) can be already integrated but let us further simplify it for our analysis. Having in mind k4 <<k3 and assuming that at the beginning of the Stage 2 [AUG] >>where S] >> [AUG] let us approximate the right-hand side of the equation by a piecewise-linear functionand, further, assuming that at the beginning of the Stage 2 we have [40S]|t=t" is the amount of 40S at the beginning of the Stage 2. Then the descent of [40S](t) can be separated into linear and exponential phases:where s2 is the quasi-steady state value of [40S] at the end of the Stage 2:where S](t) asOther dynamic variables are expressed through [40eIF4F will not be the limiting factor in the initiation and the condition [40S] >> [eIF4F] will be violated. We can estimate this time as 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 R3 - R4, where the relaxation equations are simplyDuring the Stage 3 all fluxes in the network become balanced and the translation arrives at the steady state. From Fig. t"' is the time when the Stage 3 of the relaxation starts. This relaxation goes relatively fast, since k3[AUG]|t=t''' is relatively big. So, during the Stage 3, one can consider the cycle R1 - R4 equilibrated, with [80S] = [80]s, [60S] = [60]s values.where R1 - R2 ). Using the pseudo-linear approximation of this stage . [40]s and [mRNA : 40S]s are the steady-state values of the corresponding variables, see (12). The values [60S]|t = t''', [eIF4F]|t = t''', [AUG]|t = t''' and [mRNA : 40S]|t = t″' can be estimated from (28), using the [40S]|t = t''' value. The relaxation time at this stage equalswhere k2 <<k1[eIF4F]|t = t''' or k2 >>k1 [eIF4F]|t = t'''.The solution for the Stage 3 can be further simplified if 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.S]|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'''.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 0 or [eIF4F]0) or by changing the critical kinetic parameters (k2 or k4). For example, changing k4 from 1  about the parameters. An example of the approximate solution is given on Fig. k3, so if microRNA acts on the translation stage controlled by k3 then no microRNA effect could be observed looking at the system dynamics .One of the obvious predictions is that the dynamics of the system is not sensitive to variations of k4  then we should consider two cases of efficient (β > 1) and inefficient (β < 1) initiation. In the first case the steady state protein synthesis rate is controlled by k4  and any effect on k4 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 k4. Instead, the relaxation time is affected, being ~ k4 increases the β parameter, hence, this changes "inefficient initiation" scenario for the opposite, hence, making k4 critical for the steady state protein synthesis anyway when k4 becomes smaller than k4 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 k4 by several orders of magnitude does not change the steady state rate of protein synthesis.If microRNA acts on the translation stage controlled by see Fig. . This isk2 can convert the "efficient" initiation scenario into the opposite after the threshold value k2 in the following way. 1) in the case of "efficient" initiation k2 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 k2 leads to elongation of all dynamical stages duration , while the relaxation time is not affected  in tsee Fig. .k1 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 [eIF4F]0 value , which is a critical parameter of the model (see 15).MicroRNA action on nal work  cap-depek3 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.The effect of microRNA on various mechanism and in various experimental settings  is recapitulated in Table 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 .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 . These data should be taken into account when constructing more realistic and quantitative models of microRNA action on protein translation.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 -3.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).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 ,46,47. TIn 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 . Due to 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   54] or tdominant 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.Recently, we proposed a new theory of dynamic and static limitation in multiscale reaction networks ,48. ThisThe 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 non-critical. Parameters of dominant systems  indicate putative targets to change the behavior of the large network.A + B → ..., if concentrations of species A and B  are well separated, say cA >>cB then we can consider this reaction as B → ... with rate constant proportional to cA which is practically constant, because its relative changes are small in comparison to relative changes of cB. We can assume that this condition is satisfied for all but a small fraction of genuinely non-linear reactions . 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 [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 . Second, for binary reactions ented by . The ideented by .Below we give some details on the approaches used in this paper to analyze the models by Nissan and Parker .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.A1,...An. Each elementary reaction is represented by its stoichiometric equationThe list of components is just a list of symbols (labels) s enumerates the elementary reactions, and the non-negative integers αsi, βsi are the stoichiometric coefficients. A stoichiometric vector γs with coordinates. γsi = βsi- αsi is associated with each elementary reaction.where extensive variable Ni ≥ 0, amount of Ai, is associated with each component Ai. It measures "the number of particles of that species" . The concentration of Ai is an intensive variable: ci = Ni/V, where V is volume. In this paper we consider the volume (of cytoplasm) to be constant. Then the kinetic equations have the following formA non-negative real T is the temperature, ws 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 → Ai and Ai → Ø.where kinetic law of elementary reactions is the mass action law for perfect systems:The most popular ks is a "kinetic constant" of the reaction s.where Quasiequilibrium approximation uses the assumption that a group of reactions is much faster than other and goes fast to its equilibrium. We use below superscripts 'f' and 's' to distinguish fast and slow reactions. A small parameter appears in the following formTo separate variables, we have to study the spaces of linear conservation law of the initial system (35) and of the fast subsystemb1(c),...bn(c), where b1(c),...bm(c) is the basis of the linear conservation laws for the initial system, and b1(c),...bm+l(c) is the basis of the linear conservation laws for the fast subsystem. Then bm+l+1(c),...bn(c) are fast variables, bm+1(c),...bm+l(c) are slow variables, and b1(c),...bm(c) are constant. The quasiequilibrium manifold is given by the equations ε it serves as an approximation to a slow manifold.If they coincide, then the fast subsystem just dominates, and there is no fast-slow separation for variables . But if there exist additional linearly independent linear conservation laws for the fast system, then let us introduce new variables: linear functions 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 cs ("slow" or basic components) and cf ("fast intermediates"). For catalytic reactions there is additional balance for cf, 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 cf and cs to the compatible amounts. After that, the fast and slow time appear and we could write cs = Ws, ε is small parameter, and functions Ws, Wf 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 Wf = 0.The The QE approximation is also extremely popular and useful. It has simpler dynamical properties .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.S + E ↔ SE → P + E with reaction rate constants k2. The only possible quasiequilibrium appears when the first equilibrium is fast: κ±/ε. The corresponding slow variable is Cs = cS + cSE, bE = cE + cSE = const.The simplest demonstration of these two approximation gives the simple reaction: CSE(Cs), and the slow equation reads Ċs = -k2cSE(Cs), Cs + cP = bS = const.For the QE manifold we get a quadratic equation bE<<bS, we have fast intermediates E and SE. For the QSS manifold there is a linear equation cSE(cS): For the QSS approximation of this reaction kinetics, under assumption 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  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  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)  [Ai, edges correspond to reactions Ai → Aj with kinetic constants kji > 0. For each vertex, Ai, a positive real variable ci (concentration) is defined.digraph) : verticeAi → Ø with constants ki0. Production reactions can be represented as Ø → Ai with rates ki0. The kinetic equation for the system is"Pseudo-species" (labeled Ø) can be defined to collect all degraded products, and degradation reactions can be written as K0 + Kc. Solution of this system can be reduced to a linear algebra problem: let us find all left (li) and right (ri) eigenvectors of K, i.e.:or in vector form: ċ = li, ri) = δij, where δij is Kronecker's delta. Then the solution of (36) iswith the normalization (cs is the steady state of the system (36), i.e. when all c(0) is the initial condition.where kij 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 . 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 =  we have either kI >>kJ or kJ <<kI. In this case we say that the system is hierarchical with timescales  totally separated.If all reaction constants 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 li k are 0, 1 and the possible values of ri are -1, 0, 1 with high precision. In previous works, we provided an algorithm for finding non-zero components of li, ri, based on the network topology and the constants ordering, which gives a good approximation to the problem solution [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 solution ,48,50.A1 → A2 → ...An, with reaction rate constants ki (for Ai → Ai+1), gives the first example of limiting steps. Let the reaction rate constant kq be the smallest one. Then we expect the following behavior of the reaction chain in time scale ≳1/kq: all the components A1,...Aq-1 transform fast into Aq, and all the components Aq+1,...An-1 transform fast into An, only two components, Aq and An are present , and the whole dynamics in this time scale can be represented by a single reaction Aq → An with reaction rate constant kq. This picture becomes more exact when kq becomes smaller with respect to other constants.A linear chain of reactions, The kinetic equation for the linear chain isK of these equations is very simple. It has nonzero elements only on the main diagonal, and one position below. The eigenvalues of K are -ki  and 0. The left and right eigenvectors for 0 eigenvalue, l0 and r0, are:The coefficient matrix l0 are equal to 1, the only nonzero coordinate of r0 is r0 in row.all coordinates of 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:A1,... An). 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.Reduced form means that in reality some of these reaction are not monomolecular and include some other components , all the wi are equal: wi = w. This common rate w we call the cycle stationary rate, andwhere by definition b = ∑ici is the conserved quantity for reactions in constant volume. Let one of the constants, kmin, be much smaller than others (let it be kmin = kn):where In this case, in linear approximationThe simplest zero order approximation for the steady state givesThis 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.the stationary rate of a cycle is determined by the smallest constant, kmin, if it is much smaller than the constants of all other reactions (43):So, the cycle has a limiting step with constant kmin.In that case we say that cn = b, c1 = c2 =... = cn-1 = 0. The smallest rate constant kq gives the smallest positive eigenvalue, the relaxation time is τ = 1/kq. The corresponding approximation of eigenmode (right eigenvector) r1 has coordinates: kq can be represented by a single reaction Aq → An with reaction rate constant kq. The left eigenvector for eigenvalue kq has approximation l1 with coordinates lumping on time scale ≳1/kq. Let us introduce a new variable clump = ∑ilici, i.e. clump = c1 + c2 +... + cq. For the time scale ≳1/kqwe can write clump + cn ≈ b, dclump/dt ≈ -kqclump, dcn/dt ≈ kqclump.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 r0 for zero eigenvalue  has one simple zero eigenvalue that corresponds to the conservation law ∑ci = b and n - 1 nonzero eigenvaluesThe characteristic equation for a cycle with limiting step .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 The relaxation time of a stable linear system (41) is, by definition, τ = 1/min{Re(-λi)} (λ ≠ 0). For small kn, τ ≈ 1/kτ, kτ = min{ki}, . In other words, τ is the second slowest rate constantfor a cycle with limiting step, k: kmin <<kτ ≤ ....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.Analytical analysis of the case of very inefficient cap structure. In this text we derive an asymptotical solution for the case when k1 is very small corresponding to the case of very inefficient translation initiation Click here for file