In the process of constructing a dynamic model of a metabolic pathway, a large number of parameters such as kinetic constants and initial metabolite concentrations are required. However, in many cases, experimental determination of these parameters is time-consuming. Therefore, for large-scale modelling, it is essential to develop a method that requires few experimental parameters. The hybrid dynamic/static (HDS) method is a combination of the conventional kinetic representation and metabolic flux analysis (MFA). Since no kinetic information is required in the static module, which consists of MFA, the HDS method may dramatically reduce the number of required parameters. However, no adequate method for developing a hybrid model from experimental data has been proposed.In this study, we develop a method for constructing hybrid models based on metabolome data. The method discriminates enzymes into static modules and dynamic modules using metabolite concentration time series data. Enzyme reaction rate time series were estimated from the metabolite concentration time series data and used to distinguish enzymes optimally for the dynamic and static modules. The method was applied to build hybrid models of two microbial central-carbon metabolism systems using simulation results from their dynamic models.A protocol to build a hybrid model using metabolome data and a minimal number of kinetic parameters has been developed. The proposed method was successfully applied to the strictly regulated central-carbon metabolism system, demonstrating the practical use of the HDS method, which is designed for computer modelling of metabolic systems. Since a biochemical network is essentially a nonlinear, nonequilibrium, non-steady-state system, dynamic simulation is especially effective for analyzing or predicting its behaviour in a detailed and realistic manner. However, a large amount of experimental information, including reaction mechanisms of enzymes, kinetic constants, and initial concentrations of enzymes and metabolites, is required to construct a dynamic model of a metabolic pathway. Although a number of high-throughput technologies for obtaining comprehensive biochemical data have been developed -6, most et al. proposed a novel method for dynamic modelling of metabolism, the hybrid dynamic/static method (HDS method) [i.e. reaction rates of enzymes in the static module) is applied in performing the MFA, the stoichiometric coefficient matrix for the unknown variables does not have to be square and regular.Yugi  method) . The HDS method) ,12. Sinc method) , the HDS method) . Since t method) ,16 of thAlthough the HDS method has the aforementioned advantages, no method has been proposed for splitting a dynamic system into a dynamic module and a static module before completion of the initial model construction. Advanced measurement technologies have been developed that now enable researchers to obtain the metabolome, that is, comprehensive metabolite concentration data -19. It iEscherichia coli [Saccharomyces cerevisiae [The proposed method consists of two parts. First, the enzyme reaction rate time series are estimated from metabolite concentration time series data. The dynamic and static enzymes are distinguished using the estimated enzyme reaction rate time series. The purpose of this study was to confirm that the proposed method can be used to construct accurate hybrid models, with accuracy comparable to that of a fully dynamic model. Therefore, we used pseudo-experimental data obtained from preliminarily constructed fully dynamic models. Two models of microorganisms, hia coli  and Saccrevisiae , were uset al. [The hybrid dynamic/static method (HDS method) is described in Yugi et al. . Enzyme vstatic is the static module enzyme reaction rate vector, vmodule boundary is the module boundary enzyme reaction rate vector, Sstatic# is the Moore-Penrose pseudoinverse of the stoichiometric coefficient matrix for enzymes in the static module, and Smodule boundary is the stoichiometric coefficient matrix for module boundary enzymes. The HDS method aims to describe a system in which a quasi-steady state is attained in the static module at each instant, while the overall system (both the dynamic and the static modules) acts dynamically [where amically . A transet al. [To calculate reaction rates of enzymes from metabolite concentrations, we define a "system boundary enzyme" as an enzyme located on the border of the metabolic system and extending outside the system. The system boundary enzyme is not the same as the "module boundary enzyme" defined by Yugi et al. . A non sThe reaction rates of internal enzymes were calculated from the slopes of metabolite concentrations and the reaction rates of the system boundary enzymes. With the definitions of the system boundary enzyme and the internal enzyme, a mass balance equation of a metabolic system under a dynamic transition state can be expressed as follows :S is the stoichiometric coefficient matrix, diag(-1) is a diagonal matrix (column number = row number = metabolite number), vsystem boundary(t) is the system boundary enzyme reaction rate vector, vinternal(t) is the internal enzyme reaction rate vector, and C'(t) is the metabolite concentration slope vector.where C(t) and vsystem boundary(t) are known, the reaction rates of the internal enzymes can be estimated from Eq. (3), which is transformed from Eq. (2).If Sinternal# is the Moore-Penrose pseudoinverse of the stoichiometric coefficient matrix for internal enzymes, and Ssystem boundary is the stoichiometric coefficient matrix for system boundary enzymes [see Supplementary Text  is applied to an over-determined system, the least-squares estimation of vinternal is obtained [vinternal. This is a limitation of the current procedure.where obtained . In bothobtained . When EqThe accuracy of the estimated internal enzyme reaction rates was evaluated by means of the reproduced metabolite concentration time series, which were calculated from the estimated enzyme reaction rates. Since it is difficult to compare the true and estimated reaction rates, we compared the metabolic concentrations. If an enzyme catalyzes a reversible reaction, the sign of the sum of the forward and reverse reaction rates may change. Near such a sign change, the calculated relative error between the true reaction rate and the estimated reaction rate may at times be a very large value (see Eq. (4) below). When the value of a data point is close to zero, a large error will be obtained. However, in general, most metabolite concentrations have a sufficiently large positive value for the problem caused by a value close to zero to be avoided.The metabolite concentration time series slope was calculated from the reaction stoichiometric matrix and each estimated enzyme reaction rate time series. The metabolite concentration time series was calculated by numerical integration of the metabolite concentration slope time series obtained. The mean relative error (MRE)  between Ci,j data,is the true concentration of the j-th metabolite at the i-th sampling point, Ci,j estimated,is the estimated (reproduced) concentration of the j-th metabolite at the i-th sampling point, nmetabolite is the number of metabolites, and nsampling point is the number of sampling points.where In this study, the MRE between the true metabolite concentration data and the reproduced metabolite concentrations is called the "basal error".The genetic algorithm (GA)  is emploCi,j data,is true concentration of the j-th metabolite at the i-th sampling point, Ci,j estimated,is estimated concentration of the j-th metabolite at the i-th sampling point, nmetabolite is number of metabolites, nsampling point is number of sampling points, nenzyme is number of internal enzymes, nstatic enzyme is number of enzymes included in static module, and w is weighting coefficient.where The first term in the fitness function represents the average error of the metabolite concentrations. For the fitness function, for the same reason as in the evaluation of estimated enzyme reaction rates, the metabolite concentrations rather than the enzyme reaction rates themselves were used. The second term in the fitness function evaluates the ratio of static enzymes included in the metabolic system; this term was added to adjust the number of enzymes in the static modules. The second term is multiplied by an adjusting parameter, a weighting coefficient, to control the balance between the model error and the static enzyme ratio.w = 1.000, 0.750, 0.500, 0.250, 0.100, 0.075, 0.050, 0.025, and 0.010) were employed. The results of distinguishing dynamic and static enzymes were used to construct the hybrid models.9 different values of the weighting coefficient . Finally, in the process for distinguishing dynamic/static enzymes, the "basal error", which originated from the incompleteness of the estimation of the enzyme reaction rates and from the error of the numerical integration of the enzyme reaction rates, rather than from the HDS calculation, was subtracted from the MRE.E. coli model constructed by Chassagnole et al. [S. cerevisiae model constructed by Hynne et al. [E. coli model, starting from a steady state for which the extracellular glucose concentration was 5.56 × 10-2 mM, a glucose pulse was added. The concentration of the injected glucose pulse was 1.67 mM. In Chassagnole's original model, time series of nucleotides , NADP(H)) were expressed by time-dependent functions [S. cerevisiae model, starting from a steady state for which the glucose concentration in the feed solution was 2.50 mM, the glucose concentration was shifted to 5.00 mM. The metabolite concentrations in both models at the steady state – that is, the initial concentrations for the dynamic simulations – are shown in Table S1 ) and the reaction rates of the system boundary enzymes (vsystem boundary(t)) in a performance test of the method. The slopes of the metabolite concentrations were obtained by first-order differentiation of the interpolated metabolite concentration time series.Two microbial central-carbon metabolism models were chosen for testing: the e et al.  and the e et al. . For theunctions . HoweverTo evaluate the practical use of the proposed method, artificial noise was added to each pseudo-experimental metabolite concentration data point. The coefficient of variance (CV) was assumed to be 15%, and the standard deviation (SD) of each pseudo-experimental data point was calculated by multiplying the CV by the noise-free value. A normally distributed random number around the noise-free value was generated for each data point using the SD obtained. Five noise-added data points were generated for each noise-free data point as pseudo-replicated measurements. The average of the five noise-added data points was used in the following smoothing procedure.Each noise-added metabolite concentration time series pseudo data set was smoothed by fitting it to a polynomial or a rational function of time using the least-squares method.MATLAB Release 2006a (MathWorks) was used for all calculations. Ordinary differential equations were solved by the ODE15s algorithm . For intIn the HDS method, reaction rates of enzymes in a dynamic module are used to estimate reaction rates of enzymes in a static module. If the true reaction rates of all enzymes in a metabolic system are known, they can be used directly for discriminating dynamic and static enzymes. However, the true reaction rates of enzymes in a cell cannot be determined in most cases. Therefore, we tried to estimate the reaction rates of enzymes from metabolite concentrations, which can be experimentally measured by high-throughput metabolome technologies.E. coli and S. cerevisiae models to evaluate our method of estimating reaction rates. In this section, the noise-free pseudo-experimental data were used to obtain a clear assessment of the estimation method itself. In the true reaction rate time series of Tkb in E. coli, TA in E. coli, and AK in S. cerevisiae, some sign-changing points were observed . Even if the same data set is used, the result for distinguishing dynamic/static enzymes may vary for different w.Using enzyme reaction rate time series data, we can apply the HDS method to calculate the reaction rates of static enzymes from the reaction rates of dynamic enzymes. These calculated static enzyme reaction rates can then be compared with the original reaction rate data. The errors between the estimated static enzyme reaction rates and the static enzyme reaction rate data can be used to find an optimal pattern for distinguishing dynamic from static enzymes. In this study, a fitness function (Eq. (5)) consisting of two terms was used for the optimization. In Eq. (5), the second term is multiplied by an adjusting parameter, a weighting coefficient  were used to test this method for distinguishing enzymes, and the optimized patterns of dynamic and static enzymes shown in Table w. The dynamic/static enzymes displayed on the metabolic map are shown in Supplementary Figure S2 . This result indicates that the accuracy of the hybrid model constructed using the estimated distinguishing of dynamic/static enzymes exactly reflects the magnitude of the error estimated by processes for distinguishing dynamic/static enzymes. Therefore, the proposed method for distinguishing dynamic/static modules can be used to build a hybrid model.Figure S. cerevisiae with w = 0.250, a considerable degree of error enlargement was shown in the hybrid model. This result can be considered to have been caused by error propagation at each integration step, as expected.The relationship between w and the MRE of the metabolite concentration time series and that between w and the static enzyme ratio was examined .The error in the hybrid models was higher than that obtained by processes for distinguishing dynamic/static enzymes. In particular, in the distinguishing of dynamic/static enzymes of d Figure . The twoS. cerevisiae obtained using smoothed noisy data, when w < 0.250, the number of static enzymes tended to be larger than in the results obtained using noise-free data. In the results for E. coli, the same tendency was observed when w = 0.010. Because the smoothing process of the metabolite concentration time series might result in loss of the high-frequency component of the time series data, the smoothed data might apparently change more slowly than is actually the case. Thus, when smoothed noisy data are used, the number of required dynamic enzymes in a HDS model tends to be smaller than the number needed when noise-free data are used. Because more precise metabolite concentrations need to be calculated when w is small, this tendency might be enhanced.In the previous sections, we used noise-free values to obtain a clear evaluation of the proposed method itself. However, real experimental data of metabolite concentrations are generally noisy. For practical use of the proposed method, the effect of noise on the process for distinguishing dynamic/static enzymes should be evaluated. Thus, we added noise to the noise-free data and then smoothed the noisy data for use in distinguishing the dynamic/static enzymes. In this study, simple smoothing by fitting to a polynomial or rational function of time was employed. The smoothing functions that were used and their parameters are shown in Supplementary Tables S2 and S3 , even a slight error in the enzyme reaction rate had a large influence. In fact, the average errors between the true enzyme reaction rate time series and the estimated enzyme reaction rate time series for both GAPDH (PGP-producing enzyme) and PGK (PGP-consuming enzyme) in E. coli were adequately small, 2.44% and 1.46%, respectively. In the process for distinguishing dynamic/static enzymes, the average of the squared errors of all metabolite concentrations is used to calculate the fitness function (Eq. (5)); thus, an error in only one metabolite concentration has a limited effect. Actually, the results of distinguishing dynamic/static enzymes without the PGP time series (data not shown) were entirely the same as those shown in Table E. coli, errors for all metabolites for S. cerevisiae were adequately small, because the dynamics of the metabolic system in S. cerevisiae is relatively slow compared with the sampling frequency.As shown in Table Another difficulty in applying the proposed method is that we assume that the concentrations of all metabolites are measurable. It is expected that high-throughput measurement techniques for detecting a huge number of metabolites, such as capillary electrophoresis combined with mass spectrometry (CE-MS) -19, can After a process for distinguishing dynamic/static enzymes is completed, the MRE in the corresponding hybrid model can be estimated using the linear relationship between the MRE in the process for distinguishing dynamic/static enzymes and the MRE in the hybrid model Figure . This inw, observed in Figure w is greatly changed, the few potentially static enzymes would eventually start to be converted to static enzymes.The staircase pattern of the relationships between the error and static enzyme ratio with decreasing w) is not given a priori, we need to consider how to define the value.The weighting coefficient in the fitness function (Eq. (5)) is a tuning parameter. Since a suitable value for the weighting coefficient  ,38, may et al.  have proTo ensure the validity of the predicting performance of an HDS model, careful perturbation experiments should be carried out to obtain the metabolome time series data to be used for distinguishing dynamic/static enzymes. The metabolite concentration variations used should be those considered to be of the maximum possible magnitude under the modelled conditions. To reproduce a rapidly changing metabolite concentration time series by an HDS model, a larger number of dynamic enzymes is required. Thus, if the number of dynamic enzymes included in the model is defined by using data showing the maximum possible variation in magnitude, that is, the model is constructed with the maximum possible number of dynamic enzymes, then the model can calculate all probable states of the system. For instance, consider building a metabolic model of cultured cells in a reactor, where the model has no mechanism for calculating gene expression levels or the consequent changes in protein concentrations (most proposed metabolic models are of this type). A substrate-pulse injection experiment giving the maximal substrate concentration that does not cause changes in gene expression levels in the cells  is useful for distinguishing dynamic/static enzymes. To determine the maximal permitted substrate concentration, many preliminary experiments may be required, and this seems to decrease the value of the HDS method, which aims to reduce experimental efforts. However, fundamentally speaking, such evaluation of the limits of a model's parameters is absolutely necessary for maintaining the accuracy of calculations in any kind of modelling, not only in HDS modelling. Therefore, this requirement for experiments to determine the maximal possible variation is not a specific disadvantage of the HDS method.i.e., experimentally measurable variables, enables us to discriminate dynamic/static enzymes to construct a hybrid model. In this method, the enzyme reaction rate time series are estimated from metabolite concentration time series data. Since this estimation relies on only the mass balance in the system, no kinetic information about internal enzymes is required. Therefore, the aim of employing the HDS method – to reduce the experimental effort required to obtain enzyme kinetics information – can be achieved. Two microbial central-carbon metabolism models were used to evaluate our method. Central-carbon metabolism has many feedback loops and is rigidly controlled to maintain homeostasis of a living cell. Since our method was successfully applied for such a strictly regulated system, we believe it will have wide-ranging applicability to many types of metabolic systems. Furthermore, the analysis using noisy metabolite concentration data demonstrated that, for the most part, the proposed method tolerates noise well.The proposed method of using metabolite concentration time series,Metabolites2PG 2-phosphoglycerate3PG 3-phosphoglycerate6PG 6-phosphogluconateACA acetaldehyde, intracellularx acetaldehyde, extracellularACAo cyanide, mixed flowCNx cyanide, extracellularCNDHAP dihydroxyacetone phosphateE4P erythrose 4-phosphateEtOH ethanol, intracellularx ethanol, extracellularEtOHF6P fructose 6-phosphateFDP fructose 1,6-bisphosphateG1P glucose 1-phosphateG6P glucose 6-phosphateGAP glyceraldehyde 3-phosphateo glucose, mixed flowGlcx glucose, extracellularGlcGlyc glycerol, intracellularx glycerol, extracellularGlycPEP phosphoenolpyruvatePGP 1,3-bisphosphoglyceratePyr pyruvateRib5P ribose 5-phosphateRibu5P ribulose 5-phosphateSed7P sedoheptulose 7-phosphateXyl5P xylulose 5-phosphateEnzymes/reactionsADH acetaldehyde dehydrogenaseAK adenylate kinaseALDO aldolaseconsum ATP consumptiondifACA diffusion of acetaldehydedifEtOH diffusion of EtOHdifGlyc diffusion of glycerolENO enolaseG6PDH glucose-6-phosphate dehydrogenaseGAPDH glyceraldehyde-3-phosphate dehydrogenaseGlcTrans glucose transporterHK hexokinaselpGlyc lumped glycerol formation reactionlpPEP lumped PEP formation reactionPDC pyruvate decarboxylasePFK phosphofructokinasePGDH 6-phosphogluconate dehydrogenasePGI glucose-6-phosphate isomerasePGK phosphoglycerate kinasePGluMu phosphoglycerate mutasePGM phosphoglucomutasePK pyruvate kinaseR5PI ribose-phosphate isomeraseRu5P ribulose-phosphate epimeraseTA transaldolaseTIS triosephosphate isomeraseTKa transketolase, reaction aTKb transketolase, reaction bThe author(s) declare that they have no competing interests.NI contributed to the development of the proposed method and wrote this manuscript. YN provided the basic ideas and directed the project, and MT was the project leader. All authors read and approved the final manuscript.Supplementary information for "Distinguishing enzymes using metabolome data for the hybrid dynamic/static method". An example of the estimation of internal enzyme reaction rates (supplementary text), supplementary tables for conditions of simulation and smoothing  and supplementary figures of results .Click here for file


Cellular processes such as metabolism, decision making in development and differentiation, signalling, etc., can be modeled as large networks of biochemical reactions. In order to understand the functioning of these systems, there is a strong need for general model reduction techniques allowing to simplify models without loosing their main properties. In systems biology we also need to compare models or to couple them as parts of larger models. In these situations reduction to a common level of complexity is needed.κB pathway.We propose a systematic treatment of model reduction of multiscale biochemical networks. First, we consider linear kinetic models, which appear as "pseudo-monomolecular" subsystems of multiscale nonlinear reaction networks. For such linear models, we propose a reduction algorithm which is based on a generalized theory of the limiting step that we have developed in . Second,Our approach allows critical parameter identification and produces hierarchies of models. Hierarchical modeling is important in "middle-out" approaches when there is need to zoom in and out several levels of complexity. Critical parameter identification is an important issue in systems biology with potential applications to biological control and therapeutics. Our approach also deals naturally with the presence of multiple time scales, which is a general property of systems biology models. Model reduction techniques are used to reduce the dimensionality of complex dynamics. Applications of model reduction techniques in chemical engineering , in eTrajectory based techniques use the integration of the dynamical equations and look for a small number of reduced variables  and zero elsewhere.for κn ≪ κi, i ≠ n) then this expression in the first order is simplified toIf we have a system with well separated constants .which means that most of substance is concentrated just before the "bottleneck" κn ≪ κi, i ≠ n, it is sufficient to remove the slowest step of the cycle An → An-τ+1. After removing, we will have acyclic non-branching system of reactions with eigenvalues and eigenvectors that can be computed from the formulas in the previous section. These formulas give n - 1 eigenvector sets corresponding to n - 1 non-zero eigenvalues λi = -κi, i = 1..n - 1. For example, removing A8 → A6 step at Fig. To approximate the dynamics of the reaction network for Ai, let us define κi as the maximal kinetic constant for reactions Ai → Aj: κi = maxj{kji}. For correspondent j we use notation ϕ(i): ϕ(i) = arg maxj{kji}. The function ϕ(i) is defined under condition that for Ai outgoing reactions Ai → Aj exist. If there exist no such outgoing reactions then let us define ϕ(i) = i.Now let us consider an arbitrary linear reaction network with well-separated constants. For each Ai → Aϕ(i) with kinetic constants κi. The correspondent kinetic equation isAn auxiliary reaction network is the set of reactions i → Φ (i) that is used to compute the eigenvectors of the kinetic matrix. The auxiliary network can have several connected components. In each connected component the minimal sink is an attractor of the auxiliary dynamical system, hence it is either a node, or a cycle.The auxiliary network also defines a auxiliary discrete dynamical system 1) Let us consider a reaction network κi are known) or  to obtain the 0–1 asymptotics (if only the ordering of κi is known).2) If the auxiliary network does not contain cycles then the auxiliary network kinetics (16) approximates relaxation of the initial network C1, C2, ... with periods τ1, τ2, ... > 1.3) In general case, let the system Ci, we introduce a new vertex Ai. The new set of vertices is Ci and add vertices Ai).By "gluing" cycles into points, we transform the reaction network A → B  do not change.A → Ai (to the whole glued cycle) with the same constant.2. Reactions from the second group ("entering to cycles") transform into Ai → B with the rate constant renormalization: let the cycle Ci be the following sequence of reactions A1 → A2 → ... A1, and the reaction rate constant for Ai → Ai+1 is ki (A1). For the limiting reaction of the cycle Ci we use notation kilim . If A = Aj and k is the rate reaction for A → B, then the new reaction Ai → B has the rate constant kkilim /kj. This corresponds to a quasistationary distribution on the cycle (15). It is obvious that the new rate constant is smaller than the initial one: kkilim /kj <k, because ki lim <kj due to definition of limiting constant. If after gluing, several reactions Ai → B appear, then only the one with the maximal constant should be kept.3. Reactions of the third type ("exiting from cycles") change into Ai → Aj.4. The same constant renormalization is necessary for reactions of the fourth type ("between cycles"). These reactions transform into 5. Reactions of the fifth type ("inside cycles") are discarded.4) After the new network Ci can be neglected when they are dominated by a stronger flux of Ci are minimal sinks in A → B in A ∈ Ci, B ∉ Ci. Nevertheless, there may be such reactions in the initial network The algorithm produces an hierarchy of cycles. Notice that the algorithm is based on an asymmetry between entering reactions and outgoing reactions from cycles in the hierarchy. Indeed, some fluxes of ified in .Ai ∈ i = 1..n which is called dominant kinetic system. Dynamics of this acyclic system can be computed from . To construct the dominant kinetic system, the following algorithm is applied:Now we show how to find an approximation of the dynamics of the reaction network Let 1. For 2. For each glued cycle node τi,a) Recall its nodes b) Let us assume that the limiting step in c) Remove τi vertices d) Add e) Add to B in B with the same constant, i.e. outgoing reactions f) If there exists an outgoing reaction B → g) If there exists an incoming reaction in the form A → B, and substitute the reaction by B with the same constant, as for 3. If in the initial m ← m - 1, and repeat steps 1–4 until no glued cycles left.4. Let One has to notice that in the process of network preprocessing some reaction rates are substituted by monomials of the initial reaction constants, i.e. expressions in the form K and does not depend on the production reactions K0. In particular, relaxation times do not depend on the system being closed or not. However, the stationary distribution cs and the sequence of relaxation events depends on production reactions (see Eq.(2)).The dominant kinetic system fully describes the relaxation modes of the network. The construction of this system depends only on the matrix Kc = 0, therefore they are determined up to multiplication by positive constants. They form a k-dimensional cone where k is the multiplicity of the zero eigenvalue of the matrix K, also the number of minimal sinks of the network.For closed systems, steady states are solutions of the linear homogeneous equation k sink vertices of the auxiliary network Ai, i = 1..n be vertices in the initial network Let ith sink vertex 1. Let us take the x = b = 1, and a null vector bi ∈ Rn.2. Define x is not a glued cycle then it corresponds to a vertex Aj ∈ bi has components δij; stop.3. If x is a glued cycle then recall all its vertices x1, ..., xτ.4. If κlim = mins=1..τ {smin = arg mins=1..τ {5. Determine the limiting  rate constant xj of the cycle repeat:6. For each vertex j ≠ smin and a) Let xj corresponds to a simple vertex Ak then cj,b) if xj corresponds to a glued cycle then do recursively steps 4–6 with x = xj and b = cj.c) if Any possible stationary distribution has form In brief, the distribution of the concentrations on any cycle is approximated by the first order expression (15), and this procedure is applied recursively for the vertices that represent glued cycles. The state thus obtained is equally a good approximation of the steady state of the dominant kinetic system.c0 = 1. The corresponding reaction are ∅ → Ai and the constants are the production rates ki0. The normalization c0 = 1 is possible for all bounded steady states, because these are determined up to multiplication by a constant. Furthermore, all steady states are bounded, provided that the following topological condition holds: if there exists an oriented path from ∅ to Ai, then there exists an oriented path from Ai to ∅. We suppose this condition to be always fulfilled. Applying the algorithm to the closed system we obtain Open systems can be reduced to closed ones by considering that all production reactions originate from the node ∅ that has concentration Example. Let us consider an example of the network 2) An auxiliary reaction network A3 → A4 → A5 → A3 in the auxiliary reaction network is glued in one vertex A4 → A2, renormalize its rate to 3) The cycle A3 → A2 → A3 in the auxiliary reaction network 4) The cycle A3 → A2 → A3 and try to determine the limiting step in it, we have two possibilies: k32 and k32. Let us consider them separately:Now if we restore the cycle Case k32k32, then we remove the limiting step A2 and obtain 3.1.2).3.1.1) Since A2 → A2 → A3 in 3.1.3) We restore the glued cycle corresponding to A3 → A4 → A5 → A3 (it is A5 → A3) and as a result we obtain acyclic dominant kinetic system shown at Fig. 3.1.4) We remove the limiting step reaction in the cycle Case k32k32, then we remove the limiting step A2 → 3.2.1) Since A2 to the head of the limiting step in the cycle (it is A5 vertex); the rate of A5 → A2 is 3.2.3) We restore the glued cycle corresponding to A3 → A4 → A5 → A3 (it is A5 → A3) and as a result we obtain acyclic dominant kinetic system shown at Fig. 3.2.4) We remove the limiting step reaction in the cycle Dominant approximations of hierarchical linear reaction network allow us to introduce some new concepts important for the dynamics of multiscale systems.Piecewise affine dynamics has been widely used to approximate dynamics of gene regulatory networks -37 as a j0, ci (0) ~ λk an exponential vanishes in Eq. (2) and the state has a "jump" -rk  -cs). Let us consider that eigenvalues are ordered λ1 >> λ2 >> ... >> λn-1. Then, the sequence of right eigenvectors rk such that  -cs) ≠ 0 codes the dynamics starting in c(0). In other words, there is a sequence of well separated times τ1 = 1/λ1 <<λ2 = 1/λ2 << ... <<τn-1 = 1/λn-1 such that something happens (a state transition) between each one of these times. Left eigenvectors provides the lumping  and right eigenvectors provide the sequence of state transitions. On timescales τk <t <τk+1 one can observe a jump -rk in state space provided that  - cs) ≠ 0. On this timescale the dynamics is equivalent to the degradation of pseudospecies , λk .Indeed, zero-one approximation of eigenvectors in hierarchical linear systems justifies a discrete coding of dynamics. Suppose that initial state is concentrated in Our approach to dominant subsystems emphasizes some simple but important principles. First of all, dynamics of a hierarchical linear network can be specified if a) the topology of the network is given and if b) to each reaction we associate a positive integer representing order ; c) for cyclic topologies, some monomials grouping constants of several reactions have also to be ordered in the same manner .second slowest constant  is the reaction number. Unless reactants and products belong to compartments of different volumes αji, βjk are nonnegative integers (stoichiometric coefficients). Reactions involving components from different compartments have non-integer stoichiometry. For instance, a reaction translocating a molecule from nucleus to the cytosol has stoichiometry  where kv is the volume ratio of cytosol to nucleus.where Dynamics of nonlinear networks is described by a system of differential equations:νj = βj - αj is the global stoichiometric vector. S is the stoichiometric matrix whose columns are the vectors νj. The reaction rates c) are non-linear functions of the concentrations. For instance the mass action law reads There are no simple rules to relate timescales to reaction constants of nonlinear models. The units of the inverse constants of bimolecular reactions are concentration multiplied by time and one needs at least one concentration value in order to construct a timescale. Generally, timescales are functions of many reaction constants and concentration variables. These functions are not necessarily smooth. Near bifurcations , at least one timescale of the system diverges for finite changes of the reaction constants. However, nonlinear biochemical networks have wide distributions of time-scales, as can be shown by simple (Jacobian based) analysis of models.Various reduction methods of nonlinear models are based on projection of the dynamics on a lower dimensional invariant manifold -8. The rA major improvement in calculating dominant subsystems can be obtained by combining quasi-stationarity and averaging. Averaging techniques are widely used in physics and chemistry to simplify models by eliminating fast, oscillating (microscopic) variables -24. Our I be the set of indices of intermediate components, that will be eliminated. I. Rates of T be the set of indices of terminal species. Terminal species represent the frontier between the rest of the system and the subsystems made of intermediate species. Although instead of terminal the name boundary species could be more appropriate, the latter term has already been employed in systems biology with a different meaning, which is species whose concentrations are fixed in a simulation.Let f Temkin  are callS the columns corresponding to the reactions SI and the terminal stoichiometric matrix ST, respectively.Extracting from the matrix In multiscale biochemical systems, some components react much more rapidly to changes in the environment than others. The reasons for the existence of such fast species can be multiple. Thus, rapidly transformed or rapidly consumed molecules , or promoter sites submitted to rapid binding/unbinding processes are examples of fast species. Fast species are good candidates for intermediate species. Indeed, it is easy to prove that they can be eliminated by quasistationarity. When production rates are not weak, fast species are those whose concentrations are small and well separated from the concentrations of other species. Though straightforward, the precise condition connecting quasi-stationarity and smallness of concentrations can not be easily found in literature, hence we briefly discuss it below.ϵ be a small parameter, representing concentrations. Suppose for simplicity that reactions SI RI  = KI (cT) cI + cT), where RI is the restriction of the vector R to the intermediate species. An important assumption is cT) = Let Suppose that among the reactions cI = ϵ), it means that KI (cT) = ϵ). This leads to the following asymptotic:Because cI/ϵ, ϵKI = Ic is the complement of I designating species other that I. Intermediate species are fast and the system (19) can be reduced using Tikhonov's , but most of the properties of this probability distribution will not be used here. The only property that we will use is the following: if ki > kj, then ki/kj ≫ 1 (with probability close to one). It means that we can assume that ki/kj ≫ a for any preassigned positive value of a that does not depend on k values. One can interpret this property as an asymptotic one for α → -∞, β → ∞. This property allows us to simplify algebraic formulas. For example, ki + kjcan be substituted by max{ki, kj} , orIn linear hierarchical models, ensembles with well separated constants appear . We coua, b, c, d.for nonzero k1 + k2) - k1, if k1 ≫ k2? If we first simplify the expression in brackets, it is zero, but if we open brackets without simplification, it is k2. This is a standard difficulty in use of relative errors for round-off. If we estimate the error in the final answer, and then simplify, we shall avoid this difficulty. Use of o and k1 ≫ k2, then we can write (k1 + k2) = k1(1 + o(1)), and k1(1 + o(1) - k1 = k1o(1). The last expression is neither zero, nor absolutely small – it is just relatively small with respect to k1.Of course, some ambiguity can be introduced, for example, what is  1/kB <<kA. Both cases mean that degradation of A, B is weak and/or the propensity of complex formation is high. Case a) means also that B is in excess, the opposite being true in case b).Supposing A, B quasi-stationary we have to find the positive solutions of the equations kA - kB = kB and from the second nonlinear equation we obtain Rc = kcAB. Actually, the two solutions can be summarized by:Let us consider the case a). We consider that the order of A from the linear equation we remain with a quadratic equation for B) we can show that the min-funnel approximation (29) is valid under less restrictive conditions. The only separation condition that we need is min >> kdeg, A kdeg,B/kc. We can easily identify the critical parameters kA, kB and the non-critical ones kdeg,A, kdeg,B, kc. The validity of the expression (29) depends on order relations involving monomials of critical and non-critical parameters.Using the exact solution of the system .Averaging is an useful model reduction technique for high-dimensional clocks or for other types of oscillating molecular systems  has an attractive hyperbolic limit cycle x = ψ , of period T(y): ψ (τ + T(y), y) = ψ (τ = t/ϵ). Then, after a short transient, the slow variable satisfies the averaged equation:It is supposed that for any x = ψ describes damped oscillations, with damping time much larger than ϵ, i.e. when the fast dynamics (30) has a stable focus and the eigenvalues of the Jacobian ∂f/∂x calculated at the focus are of the form -λ ± iμ, 0 <λ <<μ = ϵ).The result can be extended to the case when y:The following averaged steady state equation allows to eliminate the slow species y are constant in time and can be considered to be conserved, which has two significant consequences.If (32) has a stable steady state, we always reach this situation. In this case, the slow non-oscillating variables First, Eq.(33) restores conservation. Slow variables are often the result of broken conservation laws. In fact, in biological open systems, nothing is conserved. Conservation laws result from balancing production and degradation either passively (slow processes) or actively (feed-back). Thus, we can ignore production and degradation of molecules whose level is rigorously controlled. Eq.(33) describes such a case.y reach stationarity, the only variables that change in time are the oscillating variables x. Eq.(30) describes the dynamics of x, considering that y satisfies (33). For oscillators, averaging provides a way to eliminate slow non-oscillating variables, which is formally equivalent to quasi-stationarity and represents a new case of applicability of Clarke's method. The difference between the two cases is that we eliminate fast variables by solving quasi-stationarity equations and we eliminate slow variables by solving averaged stationarity equations. Thus, intermediate non-oscillating variables are expressed in terms of only non-oscillating terminal variables. If there are no non-oscillating terminal variables, then non-oscillating intermediates become conserved quantities.Second, (33) are averaged steady state equations for the slow variables. If slow variables In this section, we demonstrate hierarchical model reduction, model comparison and critical parameter identification. Critical parameters are identified during the reduction procedure.n species, r reactions and p parameters is designated by M. Our conception about systems biology models is summarized by the following idea. Instead of providing a single model, it is better to provide a hierarchy of models, and the relations between them. Depending on the application, we can choose the most appropriate model in the hierarchy or couple several simple models into a larger model.Model reduction starts with a complex model, from which we obtain a hierarchy of reduced models by eliminating various intermediate species. The intermediate species are either quasi-stationary species , or non-oscillating species (for oscillators). The complexity of a model is quantified by three integers. A model with nk = 2nr + ni kinetic constants, where nr, ni are the numbers of reversible and irreversible reactions. Reactions with kinetic constants zero are not considered in the counting. Each one of the nc conservation laws adds an extra parameter, the value of the conserved quantity. These values follow from initial data and are important parameters for the dynamics. For multi-compartment models, the ratios of the compartments volumes  are extra parameters. Thus p = nk + nc + 1.The number of parameters in a model are obtained as follows. If the elementary reactions follow mass action law kinetics, there are Model comparison has a similar flowchart. By model comparison we understand a) mapping one model to another one by model reduction or mapping each model to a third one, closest in some sense to both; b) compare predictions of the models  for sets of parameters related one to the other by the mapping at a). In this case, the choice of intermediate species is dictated by the differences between the models to be compared.κB is involved in a wide diversity of domains such as the immune and inflammatory responses, cell survival and apoptosis, cellular stress and neuro-degenerative diseases, cancer and development. NF-κB is sequestered in the cytoplasm by inactivating proteins named IκB. Upon signalling, IκB molecules are phosphorylated by a kinase complex, then ubiquitinylated, and finally degraded by the proteasomal complex. NF-κB bound to IκB molecules is then transported to the nucleus to activate its target genes. There are known five members of the NF-κB family in mammals, Rel (c-rel), RelA (p65), RelB, NF-κB1 (p50 and its precursor p105) and NF-κB2 (p52 and its precursor p100). This generates a large combinatorial complexity of dimers, affinities and transcriptional capabilities. IκB family comprises seven members in mammals  , x8 = [A20], x13 = [IkBa : p50 : p65@csl], kp1 21= k3k9/k4, k21p2 = k15p2 = k15/k13, k15p2 = (k15k3k9)/(k4k13), k21p3 = k15p3 = k5/k4.where After reduction of the first module we obtain the model ℳ.The second module is situated in the nucleus and contains IkBa@ncl and IKBnp50:p65@ncl. Three intermediate reactions (translocations of inhibitor and of the complex and complex formation) are replaced by one simple submechanism describing the nuclear complex formation and translocation (IkBa@csl + p50:p65@ncl → IkBa:p50:p65@csl) whose dominant rate is:x7 = [p50 : p65@ncl], kp1 14= k23, where This reduction step leads to the model ℳ. The dynamics . Both period and phase changes result from the reduced delay on the negative feed-back loop containing A20. Reducing the species mRNAIkB has destabilizing effect on the oscillations. It is no longer possible to obtain self-sustained oscillations and damping times are generally smaller than for the non-reduced model. It is well known that delayed negative feed-back favors stable oscillations and that reducing the delay destabilizes oscillations. Our findings suggest that the delay along the IkBa negative feed-back loop is more important for the stability of the oscillations than the delay along the A20 loop.We have tested reduction of two more species that have small concentration but are not quasi-stationary. Reducing the species mRNAA20 leads to the model ℳ Intermediate reactions (representing the transcription/translation module) are replaced by a single one (production of protein), of parameter Model reduction allows to identify critical and non-critical parameters. Parameters of reduced models are monomials of parameters of the non-reduced models (see Eqs.(34),(35),(36)). Some parameters of the non-reduced model may not occur in these monomials; these are non-critical parameters. Among monomials, only some are critical. Critical monomials are detected by sensitivity studies  performeτ be the studied quantity and k the parameter . We say that k is critical if A > 0 is some fixed constant and k0 some central value of the parameter. The sensitivity study is presented in Fig. kp114, k18, kp20, kp121, k22, k26, C0. By changing these parameters, the oscillations can be modified from damped to self-sustained. The above parameters are the critical monomials from which we get the critical parameters (with respect to damping time) of the unreduced model: k23, k18, k16, k20, k17, k3, k9, k4, k22, k26, C0. The degrees of the critical monomials represent logarithmic sensitivities, therefore they provide both sign an strength of the influence of the critical parameters on the studied property. For instance, from kp1 21= k3k9(k4)-1 we can say that damping time can be increased (produce sustained oscillations) by reducing k3, or by reducing k9, or by increasing k4), see also Fig. As an example, we detect critical monomials in the simplest reduced model ℳ5, 8, 15), first with respect to damping time and then with respect to the period of the oscillations. Deciding rigorously what large sensitivity means is not easy. In  we propo, 8, 15, kp 20stands for increasing the NF-κB dependent A20 production . Increasing k26, k22 stands for increasing the NF-κB dependent IκB production. The latter effect has been exploited in , x11 = [IkBa@ncl], Where p50 : p65@csl.The fourth step is a min funnel simplification of the production of the complex R40, R47, R52 are replaced by ∅ → p50:p65@csl, of rate:This leads to the model ℳ.R53–54, R71–72, that produce and consume p50 : p65, the total amount of p50 : p65 (free or in complexes with other species) would be conserved. Considering that the degradation reactions R53–54, R71–72, R72 have the same constant k, the total amount of NF-κB  represented by the variableThe fifth step, justified by averaging, introduces a new conservation law  has no conservation law). Without the reactions C = [p50 : p65@csl] + [p50 : p65@ncl]/kv + [p50 : p65 : IkBa@csl] + ... satisfies the equation :k and a rapid timescale, the period of the oscillations  has two time scales, a slow timescale 1/where the average is over a period of the oscillations.R52, R53, R54, R71, R72 are eliminated. Initial conditions of the system are chosen such that initial total NF-κB is C0 .In the fifth reduction step, reactions κB transcription rate κB concentration x11 . This phenomenon is not taken into account in [R26 = k26x7.We obtain the model ℳ that has the same species and reactions as Lipniacki's model ℳ, but slightly more parameters. The difference in the number of parameters comes from the more complex expressions of the mRNA Icount in  where thC0 and k26 which are obtained by averaging (C0 is given by Eq.(42) and k26 is calculated in order to have equal average production rates of mRNAIκB in the two models, see Fig. κBa. Without signal, the quantity of inhibitor in ℳ is small (it is largely in excess in the other two models). With signal, the amplitude of the oscillations is higher in ℳ. These differences follow from the different kinetic laws for the transcription of IκBa. Basically, in M and ℳ, IκBa has some negative influence on its own production (see Eq.(39)).One important objective of model comparison is to obtain the parameter mapping. This allows to calculate the parameters of one model if the parameters of the other model are known. Then, dynamical properties of the models can be compared. In our example, all the parameters of ℳ should be equal to the corresponding parameters of ℳ except for κB total quantity. Although this is not important in normal situations (when C is conserved), it could become important if one wants to cope with strong perturbations of NF-κB activity.Our most complex model can account for phenomena that can not be studied by any of the conservative models ℳ, ℳ, namely it can take into account variations of the NF-Thus, using a more complex model depends on the experimental situation . The role of mathematics and modeling in quantitative biology is to predict the behavior of a system. Depending on which behavior, the simplest theory can change, and we want a hierarchy of models and model mapping methods. The process can go in both directions: reducing, or increasing the number of details.Model mapping also allows to identify non-critical and critical parameters. Let us give only two examples. The constants or reactions 13,14 (formation of the complex) are not critical and one does not need to know them with precision. Actually, variations by a factor 100 in these constants do not change the dynamics.C0  is a critical parameter. Reference [μm3. This means C0 = 0.06 μM. Nevertheless, cell volume estimate is not really precise and errors can easily shift the model from a damped oscillatory to a sustained oscillatory dynamics. In this case, it is the comparison between model prediction and theoretical observation that can fix the value of the critical parameter.The values that we use come from , who citeference  proposes with the following ids: MODEL7743656488, MODEL7743631122, MODEL7743608569, MODEL7743576806, MODEL7743528808, MODEL7743444866.The sequence of reduction steps described in this section is illustrated on Fig. 2S1–2S7 in Additional File We have presented a methodology for reducing and comparing systems biology models. We show how to produce a hierarchy of coarse grained models that can be used for understanding functioning of the biological systems. We show how models in the hierarchy can be mapped one onto another, thus allowing to decrease or to increase the number of details that are needed for the description of the system. Our method identifies the set of critical parameters of the system. This can be particularly useful for robustness studies (when robustness is understood as stability against parameter variability ) or for We did not approach aspects of multi-scale modeling that occur in multi-organ physiology, or spatial aspects. Relation with stochastic modelling has been only briefly discussed and will be presented in detail elsewhere . The reduction methods presented here can be applied to systems of biochemical reactions modeling cell physiology  and can A central idea in our treatment is the hypothesis that biological systems are hierarchial, involving many separated time scales. The reduction methods were adapted to exploit this situation . The hierarchical nature of the systems is not sufficiently exploited by more traditional approaches. For instance, singular perturbation copes with two time scales and eliminates the fastest. In biology, we are often interested in a "middle" time-scale, corresponding to a particular process that we study. We have shown how to eliminate both faster and slower variables. Another specificity of systems biology is the quest for critical parameters. Our approach offers naturally a solution: critical parameters are detected in the reduction process. It also extends the theory of limiting step to complex networks . ShowingAs future work we will improve our algorithms in order to propose fully automated reduction tools. At present, the automated sections of our methods are the calculation of dominant subsystems of pseudo-monomolecular subsystems and the calculation of simple sub-mechanisms stoichiometries and rates. The detection of quasi-stationary and non-oscillating species is semi-automated. The solutions of quasi-stationarity and averaged stationarity equations are not yet fully automated (except for the pseudo-monomolecular case).We also plan to consider other applications such as high dimensional switches .Concerning our model comparison methods, we would like to study hierarchies of kinetic models coming from various organisms, for which the conserved and the specific parts are the result of evolution.κB model. All authors drafted, read and approved the final manuscript.OR proposed the methodology to reduce nonlinear models. AG developed the general theory of multiscale linear system, together with OR and AZ. AZ and OR designed and implemented the algorithms. AL designed the NF-κB models. This file shows the hierarchy of models using Systems Biology Graphical Notation (SBGN) and the results of a study of the truly non-linear reactions.Hierarchy of NFClick here for file


In silico analyses provide valuable insight into the biology of obligately intracellular pathogens and symbionts with small genomes. There is a particular opportunity to apply systems-level tools developed for the model bacterium Escherichia coli to study the evolution and function of symbiotic bacteria which are metabolically specialised to overproduce specific nutrients for their host and, remarkably, have a gene complement that is a subset of the E. coli genome.Buchnera aphidicola (symbiont of the pea aphid) as a model for using systems-level approaches to discover key traits of symbionts with small genomes. The metabolic network is extremely fragile with > 90% of the reactions essential for viability in silico; and it is structured so that the bacterium cannot grow without producing the essential amino acid, histidine, which is released to the insect host. Further, the amount of essential amino acid produced by the bacterium in silico can be controlled by host supply of carbon and nitrogen substrates.We have reconstructed and analysed the metabolic network of the γ-proteobacterium B. aphidicola to the host and, together with the impact of host-derived substrates on the profile of nutrients released from the bacteria, point to a dominant role of the host in controlling the symbiosis.This systems-level analysis predicts that the fragility of the bacterial metabolic network renders the symbiotic bacterium intolerant of drastic environmental fluctuations, whilst the coupling of histidine production to growth prevents the bacterium from exploiting host nutrients without reciprocating. These metabolic traits underpin the sustained nutritional contribution of  In particular, insight into the metabolic capabilities of these bacteria can be obtained from the construction and analysis of metabolic models generated from the inventory of genes with function in metabolism. Of the various methods available, constraints-based modelling using flux balance analysis (FBA) has particular application because it reconstructs flux through metabolism without requiring kinetic or other detailed information on the function of individual metabolic enzymes . In E. coli iJR904 model. The limited number of metabolic pathways  is small, comprising 196 gene products, 240 compounds and 263 reactions, only 39% of the compounds and 27% of the reactions in the E. coli. In the reaction graph, the modal path length between every pair of compounds, the maximal path length and %-unreachable nodes are all higher for APS than for E. coli  to 50% (threonine) of the amount synthesised  and E. coli K-12 (iJR904).We also analysed single gene deletions by the linear minimisation of metabolic adjustment (linearMOMA) method . This anB. aphidicola are close to those of the postulated minimal metabolic network [B. aphidicola is intolerant, first, of mutations that eliminate metabolic reactions unless that loss is compensated for by enhanced metabolic support from the host, and, second, of drastic changes in conditions, suggesting that the environmental conditions in the symbiosis may be relatively uniform. The condition of B. aphidicola in the symbiosis is consistent with this interpretation. The bacterial cells are restricted to a single insect cell type, the bacteriocyte, the sole function of which appears to be to house and maintain the bacteria; the cytoplasm of the bacteriocyte is packed with bacterial cells, each of which is enclosed by an insect membrane, known as the symbiosomal membrane [These data demonstrate that APS does not conform to the generality that metabolic networks are complex and robust , and sug network . The framembrane . The metB. aphidicola is predicted to vary because the amino acid composition of the aphid diet of plant phloem sap is influenced strongly by environmental conditions and plant age and species [B. aphidicola. One hypothesis is that EAA production is controlled by the supply of carbon and nitrogen substrates from the host. In iGT196, the principal carbon sources are glucose and mannitol, and the nitrogen sources are the amino acids aspartate, glutamate and glutamine. To test this hypothesis, carbon-limited and nitrogen-limited models were generated by increasing supply of either nitrogen or carbon sources, respectively. Consistent with prediction, these changes to the inputs to the APS network had substantial impacts on the output of EAAs  could replicate the proximal truncation of the purine biosynthetic pathway  and herein potentially limit the full interpretation of this experiment, it is clear that, unlike the evolutionary loss of various other metabolic genes, the truncation of the purine pathway was an improbable evolutionary step. To our knowledge, this type of host-symbiont linkage has not been described previously for any symbiosis. There is, however, evidence for coupling of the metabolism of nitrogen-fixing rhizobia and the surrounding plant cells in legume root nodules, such that rhizobial access to host amino acids is dependent on the release of the nitrogen-fixation product, ammonia [B. aphidicola and its host may reveal similar couplings between nutrient supply to B. aphidicola and bacterial overproduction of EAAs. Such couplings would preclude the evolution of bacteria with reduced export of EAAs, despite the evidence that EAA export is a costly trait for B. aphidicola . The metabolic coupling is in the selective interest of the B. aphidicola. Because B. aphidicola is an obligately vertically transmitted microbe with a small effective population size, any short-term growth advantage of reduced EAA export to the host would rapidly translate into depressed fitness of both the host and its bacterial complement.To assess whether we could model the evolution of the reduced  network  and applway Fig. . Whilst  [et al. 06 could in silico analysis has shed light on the evolution and function of a symbiotic bacterium with a small genome. From the properties of the reconstructed metabolic network of B. aphidicola, adaptations for the symbiotic lifestyle can be identified. The fragile metabolic network suggests that the symbiotic environment is benign and not subject to drastic fluctuations, and host controls over bacterial metabolism are indicated by the responsiveness of the essential amino acid profile released from the bacteria to the host supply of carbon and nitrogen substrates. The coupling of purine synthesis  to the overproduction of histidine transferred to the host is potentially one route by which this bacterium is bound to the cooperative lifestyle. Systems level analysis of other taxa will establish the generality of these key traits of metabolic fragility, flexibility and coupling among symbiotic bacteria with reduced genomes.Systems level E. coli K-12 iJR904 model [apaH, gloB, lig, gltX, suhB, mutT) or representing isolated enzymes in missing pathways . Other reactions supported by experimental evidence were added , which has been detected in isolated extracts of B. aphidicola APS ; the true 'orphan' ; and one spontaneous reaction (HCO3E). Pathways and reactions were checked using the EcoCyc Pathway/Genome database [BuchneraBASE [ with accession number MODEL7434234848. The reconstruction is also available in BuchneraCyc  a BioCyc genome/pathway database created and adopted for this project using the Pathologic software [The iGT196 model comprised a subset of 04 model  derived database . Mappingsoftware . The reaction HCOE. PathwaE. coli K-12[B. aphidicola to the aphid  compared to E. coli K-12 (50.8%).The biomass reaction for the model was derived from that used in iJR904 which was based on experimental data for coli K-12. The coe. The 5.21 model provides glucose and mannitol as carbon sources and glutamine, glutamate and aspartate as nitrogen sources and precursors for biosynthesis  was performed using the Fluxor toolkit, which uses linear optimization algorithms provided by the GLPK and OOQP toolkits. The Fluxor software is freeware available at software .Buchnera was represented as a directed bipartite graph G = , where C is a set of nodes representing compounds (metabolites), R is a set of nodes representing reactions and L is a set of directed links, i.e. an ordered pair of nodes from one C to one R. Compound ci is involved in reaction rj if and only if there is link between node ci and rj. This directed graph is referred to as the "reaction graph". The reaction graph can also be simplified by removing all reaction nodes and replacing them with links connecting each substrate with all its products [ and is available from the authors asorokin@inf.ed.ac.uk upon request. The network diameter is the maximal length of the shortest path between two reachable nodes in the network. Calculation of the shortest path between two compounds using the reaction graph considers the two links that join two compounds to a reaction as a single link. We used a maximum likelihood approach [The metabolic network of products ,24. Thesproducts  to fit tapproach  to fit tgrowth of the resulting mutant  and nitrogen-limited . Uptake of nitrogen sources in the nitrogen-limited model and of carbon sources in the carbon-limited model were maximal.et al., (2006) with the biomass reactions from iGT196 and iJR904, both with riboflavin included in the biomass reaction ). Initial FBA analysis using iJR904 modified to have the biomass reaction from iGT196 did not produce a feasible solution because the biosynthetic pathways to biotin and thiamine diphosphate were incomplete. Both of these cofactors were added to iGT196 as 'cofactor constraints' and were absent from the original iJR904 biomass reaction which lacked these additional criteria for growth. Therefore we added an efflux reaction for S-adenosyl-4-methylthio-2-oxobutanoate (amob) and for 4-hydroxy-benzyl alcohol (4 hba) which are produced but not consumed in the biotin and thiamine diphosphate biosynthesis pathways, respectively, and also an uptake reaction for pimeloyl-CoA (pmcoa) which is consumed but not produced during biotin biosynthesis. The analysis was performed as described previously [The evolution of minimal networks were simulated by the methods of Pal The construction and analysis of the model was performed by GHT, JZ, SJM, AS and IG. AED, GHT and JZ conceived the study and designed and coordinated it. All authors contributed towards the writing of the manuscript.Data deposition The model is available in BioModels database (MODEL7434234848), and the reconstruction is available at . The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to GHT.Additional Methods. This file contains additional methods for the construction and validation of the model and the flux correlation method with references.Click here for fileB. aphidicola APS.Quantification of essential amino acid release from  These data demonstrate the contribution of B. aphidicola APS-derived essential amino acids to protein growth of 2-to-7-day-old larval pea aphids (clone LL01) on chemically-defined diets.Click here for fileDescription of the iGH196 model and fluxes from the 5.21 model. The Excel sheet describes the gene/reaction relationships in the model and the fluxes in the 5.21 model.Click here for fileAnalysis of the FBA simulations for non-essential genes. The file summarises the results from our analysis of the FBA simulations for non-essential genes from iGT196 by our own method and also by using the linearMOMA method.Click here for fileFlux correlation of the histidine and purine biosynthesis pathways. Pairwise scatterplots showing correlation of fluxes in reactions of the histidine and purine biosynthetic pathways of Buchnera aphidicola.Click here for fileList of APS genes and the frequency of their retention in evolved symbionts. This file lists the APS genes and the frequency of their retention in evolved symbionts from Pal et al,. (2006) and this study.Click here for file


One of the most important goals of the postgenomic era is understanding the metabolic dynamic processes and the functional structures generated by them. Extensive studies during the last three decades have shown that the dissipative self-organization of the functional enzymatic associations, the catalytic reactions produced during the metabolite channeling, the microcompartmentalization of these metabolic processes and the emergence of dissipative networks are the fundamental elements of the dynamical organization of cell metabolism. Here we present an overview of how mathematical models can be used to address the properties of dissipative metabolic structures at different organizational levels, both for individual enzymatic associations and for enzymatic networks. Recent analyses performed with dissipative metabolic networks have shown that unicellular organisms display a singular global enzymatic structure common to all living cellular organisms, which seems to be an intrinsic property of the functional metabolism as a whole. Mathematical models firmly based on experiments and their corresponding computational approaches are needed to fully grasp the molecular mechanisms of metabolic dynamical processes. They are necessary to enable the quantitative and qualitative analysis of the cellular catalytic reactions and also to help comprehend the conditions under which the structural dynamical phenomena and biological rhythms arise. Understanding the molecular mechanisms responsible for the metabolic dissipative structures is crucial for unraveling the dynamics of cellular life. Living cells are essentially dynamic metabolic systems, which are highly self-organized and formed by complex membranes surrounding a dense fluid mixture where millions of different biochemical elements interact to form self-assembled aggregates, a rich variety of supra-macromolecular functional structures and a great diversity of temporal metabolic behaviors.The enzymes are the most outstanding molecules of these surprisingly reactive systems. They are responsible for almost all the biomolecular transformations, which globally considered are called cellular metabolism. Likewise, the dynamic functional organization of the cellular metabolism acts as an intricate network of densely integrated biochemical reactions forming one of the most complex dynamical systems in nature ,2.i.e., lower entropy, which allows to diminish the number of chemical entities and to increase their dimension by means of biochemical interactions and molecular bonds, emerging highly ordered macro structures and complex functional dynamic behaviors ) = (Rate of input of [S]) − (Rate of degradation of [S]).The explicit equation for 1a is 1]) = (Rate of synthesis of [P1]) − (Rate of degradation of [P1]) and the corresponding equation is2a. (Rate of change of [P2]) = (Rate of synthesis of [P2]) − ; the equation is3a.  and by the dissociation constant of P1 for E1 (K P1) and of P2 for E2 (KP2); L1 and L2 are the allosteric constants of E1 and E2.where, α, β and γ denote the normalized concentrations of SOnce the different elements of the equations are normalized, the time-evolution of the metabolic subsystem is described in any instant by the following three differential equations:1 and σ2 correspond to the normalized maximum activity of the enzymes E1, E2, ; q1 = Km1/Kp1, q2 = Kp1/Kp2 and d = Kp1/Km2.σ2) was fixed as the control parameter of the multienzymatic instability-generating reactive system.Once the values for the parameters are specified and given initial values for the dependent variables  oscillate with the same frequency but different amplitudes.After the numerical integration, a wide range of different types of dynamic patterns can be evidenced as a function of the control parameter value. At very small ks values, the system (1) admits a single steady-state solution and when ks is increased, the steady state become unstable, leading to the emergence of a periodic pattern . In thisS ≤ 2.034 0. the metabolic subsystem exhibits the most interesting dynamical behaviors.In the interval 0.792 < kS ≤ 1.584 hard excitation emerges in the functional enzymatic association and two kinds of integral solutions coexist under the same control parameter value: a stable steady state and a stable periodic oscillation (bistability). The metabolic subsystem starts from a stable steady state but evolves to a stable periodic regime when the initial concentrations of S1, P1 and P2, exceed a determinate threshold value  the metabolic subsystem undergoes a reorganization of its dynamics and spontaneously presents a temporal structure characterized by the coexistence of two stable periodic behaviors under the same control parameter value (bistability). The integral solutions now settle on two regular oscillatory regimens depending on the initial conditions .At further increases in ks  and when the control parameter increases (1.99 < kS ≤ 2.034) the numerical solutions of the biochemical oscillator display a classical period-doubling cascade preceding chaos, i.e., when ks reaches a threshold, the oscillation of period-1 becomes unstable, which leads to the establishment of a new regular oscillation of period-2 (two maximums and two minimums per oscillation); when kS increases, a new instability provokes the emergence of regular oscillations of period-4; next a new bifurcation of period-8 appears, etc.; this cascade of bifurcations of period doubling continues successively ending in a chaotic response .Between 1.82 < k1, P1 and P2) present infinite transitions, modifying uninterruptedly their activity so that they never repeat themselves for arbitrarily long time periods  allows showing how metabolite concentrations of the biochemical oscillator, formed by only two irreversible enzymes, vary second by second following a notable diversity of dynamic patterns as a function of the control parameter values . In the numerical analyses, the feedback processes are the main sources of nonlinearity that favor the occurrence of instabilities which provoke the emergence of different dynamical patterns.In the course of time, open enzymatic systems that exchange matter and energy with their environment exhibit a stable steady state. This stationary non-equilibrium state is more ordered that the equilibrium state of the same energy. Once the enzymatic subsystem operates sufficiently far-from-equilibrium due to the nonlinear nature of its kinetics, the steady state may become unstable leading to the establishment of other dynamical behavior. New instabilities may originate the emergence of different biochemical temporal behaviors.All these dynamic patterns (including chaos) correspond to ordered motions in the system representing examples of non-equilibrium self-organizations and can therefore be considered as temporal dissipative structures.The emergence of quantitative behaviors belonging to different metabolic subsystems has been investigated in extensive studies, mainly carried out by means of systems of differential equations, e.g., in the Krebs cycle , amino aCell-cycle in eukaryotic cells is governed by a complex network of metabolic reactions controlling the activities of M-phase-promoting factors. These metabolic reactions belonging to a set of enzymes functionally associates can be self-organized in far-from-equilibrium conditions, exhibiting periodic oscillations which govern the cell-cycle.The network forms a metabolic subsystem that mainly involves enzymes of covalent regulation and protein kinases (Cdk) whose activities depend on binding to cyclins. More concretely, mitosis-promoting factor (MPF) has been identified as a dimmer of two distinct protein molecules: a cyclin subunit and a cyclin-dependent protein kinase (Cdc2), which is periodically activated and inactivated during the cell cycle. MPF activity is regulated by synthesis and degradation of cyclin subunit and by phosphorylation and desphosphorylation of the protein kinase Cdc2 at an activatory threonine (Thr) residue and an inhibitory tyrosine (Tyr) residue. When the active form of MPF is phosphorylated on Thr161, then M-phase begins by phosphorylating a suit of target proteins involved in the main events of mitosis .Xenopus oocytes  with δ = max {λ1,…, λr}, which involves the consideration, in the solution of the system, of the functions f0: → Rn called initial functions. It can be observed therefore that infinite degrees of freedom exist in the determination of the particular solutions.As 0 equal to any y(t), which, in particular, can be a periodic solution of the system for λ1 = … = λr = 0 and t ≤ t0.In the system described by (3), it is possible to take the initial function fδ(t): → Rn, with yδ(t) = y(t), ∀t ∈ .The initial function will be y1 > 0, λ2 = ... = λr = 0, the first component of the initial function will be y1δ (t) :→ R; to each λ1 corresponds a y1δ (t0 − λ1), which is the value of the first component of y in t0 − λ1; the parameter λ1 determines the initial function domain and, given that the solution is periodic, for each different domain of the initial function exists an ordinate value in the origin y1δ ( t0 − λ1) and a corresponding phase-shift. It is observed that y1δ ( t0 − λ1) is the value of the function y1δ ( t0 − λ1) evaluated in t0, where h is the initial function with a phase shift of λ1.In the particular case when λ1 affect the independent variable which represent time delays and can be related to the phase shifts and the domains of the initial functions. Let us see an example next.With this type of systems, it is possible to take into account dynamic behaviors related to parametric variations linked to the independent variable. The parametric variations λThe glycolysis continues to be the best known example of temporal self-organization in metabolic processes, and more than four decades ago the existence of variations in the phase shift values of different metabolites during the glycolytic oscillations was experimentally observed ,278.1 = … = λr = 0. And we can take this solution as the general initial function f0.In order to study the repercussion on the dynamic system of phase-shifts, it is suitable to utilize the systems described by differential equations with delay. For example, we can consider a particular ODE-solution to be equal to a periodic solution y(t) of the system (3) for λδ (t):→Rn.As we have seen, each delay time reflects a domain and a phase shift of the initial function. Different domains and phase shifts of the initial functions can be considered in system (3) for each value of the parameter λ linked to the independent variable; and so, particular phase shifted ODE-solutions can be made to correspond to phase-shifted initial functions yIn the integration of the delayed functional differential equations system, certain values can be considered for the dependent variables evaluated in t − λ, which correspond to a phase-shifted oscillation of the past. Therefore, it is possible to study if phase-shifted initial functions can be followed  by a mere final phase shift or by a variation in the dynamic behavior of the system.In this sense, several studies on phase shifts have been carried out in the yeast glycolytic subsystem by means of a delayed differential equations system –283 and In 1 (hexokinase) into the product P1 (glucose-6-phosphate). The enzymes E2 (phosphofructokinase) and E3 (pyruvatekinase) are allosteric, and transform the substrates P′1 (fructose 6-phosphate) and P2′ (phosphoenolpyruvate) in the products P2 (fructose 1-6-bisphosphate) and P3 (pyruvate), respectively. The step P2 → P2′ represents a particular catalytic activity, reflected in the dynamic system by means of a functional variable β′.In the multienzymatic instability-generating reactive system, it is shown how the metabolite S (glucose), brought into the system at constant speed, is transformed by the first enzyme E1 does not continue in the multienzymatic instability-generating reactive system, q1 being the first-order rate constant for the removal of P1; likewise q2 is the rate constant for the sink of the product P3 (which is related with the activity of pyruvate dehydrogenase complex).It is supposed that a part of P2 (phosphofructokinase)  where τ is an integer delay.This converts the dimensional vector x(t), into the m-dimensional vector X(t). The dimension m is known as the embedding dimension, and if m is great enough, the trajectory of X(t) converges to an attractor in the m-dimensional Euclidian space, which is, up to a continuous change of variable, the attractor of the subunit dynamical system.The election of embedding dimension (m) and delay (τ) is mostly a question of trial and error because although there are criteria, they are not clean-cut .For m election, the method of false nearest neighbors is appropriate. False neighbors are far points in the original phase space with near projections in lower dimensions  − T(x0′)| ≥ Tmin. Then trace system dynamics using initial points x0 and x0′. Then a distance ɛ 0′ between two trajectories will exceed some value ɛmax. Stop and fix the time of tracing T0 and the ratio ɛ0′/ɛ0. After that one should find another starting point x1″ which is close to x1 and shifted in the direction of the vector x1′ – x1. Let ||x1 – x1″|| = ɛ1. Trace the dynamics of the system using x1 and x1″ as initial points. Then a distance ɛ1′ between two trajectories will exceed ɛmax. Stop and fix the time of tracing T1 and the ratio ɛ1′/ɛ1, etc.The maximal Lyapunov exponent is very useful in testing the existence of chaos and the Wolf algorithm can be used for it . The ideThe Maximal Lyapunov exponent is estimated aswhere N is the iteration number.To calculate the maximal Lyapunov exponent, the software developed with MATLAB can be used.In order to study the presence of long-term correlations in metabolic chaotic data, first it is necessary to determinate whether the series is a fractional Gaussian noise (fGn) or a fractional Brownian motion (fBm).FGn is a stationary stochastic process with the property that the n-th autocorrelation coefficient is given bywhere H is the Hurst coefficient. On the contrary, fBm is a non-stationary, self-similar process, whose first differences form a fGn, that is, taking differences between points sampled at equal intervals a fGn is obtained .H(t) with 0 ≤ t ≤ ∞; when the independent variable t is sampled at equally spaced times obtaining a discrete fractional Brownian motion. Therefore, fBm is a generalization of Brownian motion in which the increments are normally distributed but they are no longer independent and consequently the process is correlated in time.Taking into account these concepts and 0)/fβ for adequate constants S(f0) and β. If −1 < β < 1, then the signal corresponds to an fGn. If 1 < β < 3, then the signal corresponds to a fBm ) as the maximum absolute difference between their respective scalar components and for each X(i) we count the number of j such that d < r, denoted as Nrm(i) is φm(r), which portrays the average frequency of the occurrence that all the m-point patterns in the sequence remain close to each other, and finallyThe average value of CThe idea is that ApEn measures the logarithmic likelihood that runs of patterns that are close (within r) for m contiguous observations remain close on subsequent incremental comparisons.Some Approximate Entropy works can be found in ,386–390.ApEn can be calculated with software developed with MATLAB.One of the most important goals of contemporary biology is to understand the elemental principles and quantitative laws governing the functional metabolic architecture of the cell.In this review, I mainly focused on some functional enzymatic structures which allow the temporal self-organization of their metabolic processes.My aim was to provide an overview of temporal metabolic behaviors, including new examples, some kinds of quantitative mathematical models and non-linear tools for the analysis of dissipative functional enzymatic associations (metabolic subsystems) where oscillatory behaviors may emerge.From the first studies in 1957 of oscillatory phenomena in fluorescent studies of yeast , the numOscillatory phenomena, apart from constituting a singular property that manifests itself at all levels of biological organization, present a great functional significance in enzymatic processes. As recalled in the previous sections, biochemical oscillations constitute in themselves a manifestation of the nonlineal characteristics involved in the metabolic regulation activity.In the light of the research in course, the biological rhythms that emerge in self-organized biomolecular structures constitute one of the most genuine properties of cellular dynamics; and the rigorous knowledge of their nature and significance may be an essential element in the comprehension of the biological fact at its most basic and elementary levels.The transition from simple periodic behavior to complex oscillatory phenomena including chaos is often observed in metabolic behaviors. In this sense, the relationship between chaotic patterns and long-term correlations (information correlated in time ) is a stAs mentioned above, different studies have evidenced that global cellular enzymatic activities are able to self-organize spontaneously, forming a metabolic core of reactive processes that remain active under different growth conditions while the rest of the metabolic subsystems exhibit structural plasticity. This global and stable cellular metabolic structure  appears to be an intrinsic characteristic common to all cellular organisms –165.The existence of chaotic patterns and long-term correlation properties in the activity of the metabolic subsystems integrated in a stable global functional structure may constitute a biological advantage.Chaotic patterns exhibit sensitive dependence on initial conditions. Sensitivity means that a small change in the initial state will lead to large changes in posterior system states and the fluctuations of the chaotic patterns are conditioned by the degree of perturbation of the initial conditions. These changes in the system states present exponential divergence, provoking fast separations in the chaotic orbits.For “slow dynamical systems” the typical time scale of the chaotic fluctuations is on the order of 1 μs ,392 and Furthermore, different studies have shown that chaos permits fast transmission of information and high efficiency .The existence of chaos (which exhibits long-term correlations) in some functional structures may constitute a biological advantage by allowing fast and specific responses during the adaptation of the metabolic system to environmental perturbations.For example, calcium plays an important role in the regulation of cell metabolism, modulating many physiological processes . In respIn this sense, numerous works have shown chaotic behaviors at cellular conditions e.g., in intracellular free amino acid pools , respiraSince a notable part of the biological temporary processes seem to be chaotic in cell conditions, it can be important to take into account these persistent phenomena in Systems Biology.A vast amount of new information on structural enzymatic organization and genome dynamics is currently being accumulated, and a great part of the network molecular interactions are perfectly established. Therefore, the real functional structure of the enzymatic associations and genome regulation is an open question to be elucidated through mathematical modeling and numerical analysis.Within the new area of Systems Biology, quantitative mathematical models, non-linear tools and computational approaches are particularly valuable for exploring dynamic phenomena associated with dissipative metabolic structures, and due to that, these methods will be crucial in making sense of the functional metabolic architecture of the cell.The comparison of experimental results with numerical analysis calls for more quantitative data on the self-organization of metabolic processes, and these methods will be able to provide an integrative knowledge of the organization of cooperating enzymes into macromolecular complexes and microcompartments with the emergence of temporal metabolic patterns.This new field fusions several concept and tools from many areas, including: computational intelligence, dynamical systems theory, stochastic processes, nonlinear dynamics and networks theory, among others.Day by day, Systems Biology is developed as a new methodology about metabolic dynamic processes, which allows explaining how higher-level properties of complex enzymatic processes arise from the interactions among their elemental molecular parts, forming complex spatial structures where singular temporal reactive behaviors emerge.System Biology will be crucial to the understanding of the functional architecture of the cell.