Processes in living cells are the result of interactions between biochemical compounds in highly complex biochemical networks. It is a major challenge in biology to understand causes and consequences of the specific design of these networks. A characteristic design feature of metabolic networks is the presence of hub metabolites such as ATP or NADH that are involved in a high number of reactions. To study the emergence of hub metabolites, we implemented computer simulations of a widely accepted scenario for the evolution of metabolic networks. Our simulations indicate that metabolic networks with a large number of highly specialized enzymes may evolve from a few multifunctional enzymes. During this process, enzymes duplicate and specialize, leading to a loss of biochemical reactions and intermediary metabolites. Complex features of metabolic networks such as the presence of hubs may result from selection of growth rate if essential biochemical mechanisms are considered. Specifically, our simulations indicate that group transfer reactions are essential for the emergence of hubs. Computer simulations show how the complex organization of metabolic networks can arise from selection for a simple trait such as growth rate. P(k), of metabolites participating in k reactions is given by P(k)−γ∼k, where γ is a constant coefficient [Metabolic networks belong to the best-studied biochemical networks. They consist of metabolites and of biochemical reactions transforming these metabolites into each other. In contrast to other biochemical networks such as signal transduction networks and genetic networks, complete network topologies can be easily obtained from annotated genomes. It has recently been reported that the connectivity of metabolic networks follows approximately a power law, i.e., the frequency, fficient –4. This Using computer simulations we here study whether hubs emerge in a widely accepted scenario for the evolution of metabolic networks. This scenario has originally been proposed by Kacser and Beeby  and consA number of alternative scenarios for the evolution of novel enzymes and metabolic pathways have been proposed . One sceWe simulate the evolution of metabolic networks based on the following assumptions:Because of the importance of group transfer in metabolic networks we assume that metabolic networks are group transfer networks . MetabolWe assume that initially all enzymes and all transporters are unspecific and allow small mutations in their kinetic properties. These mutations increase the rate constant of a single reaction while decreasing the rate constants of all other reactions of an enzyme or transporter. Alternatively they decrease the rate constant of a single reaction while increasing the rate constants of the other reactions. In addition to mutations affecting kinetic properties, we assume that there are gene duplications (or deletions) that increase (decrease) the dosage of enzymes and transporters. Changing the dosage of an enzyme has two opposing effects on fitness: An increased dosage increases the fluxes in the network, but it is also associated with increased metabolic costs of biomass formation. Duplications also enable enzymes to diverge and specialize on different reactions. Note that if a multifunctional enzyme has only a small impact on biomass formation, duplication can be detrimental provided that the metabolic costs of the additional copy are larger than the benefits in terms of increased growth rate. Consequently, enzymes and transporters do not necessarily specialize completely over the course of evolution. On the other hand, duplications can be beneficial if enzymes or transporters are already completely specialized. Therefore multiple identical copies of completely specialized enzymes and transporters can evolve provided they have a large impact on biomass formation.In our simulations, we assume that there are seven different biochemical groups that characterize each metabolite. Because a metabolite can either carry one or no copies of each group, we have an initial network of 128 metabolites. We initiate the simulations with seven unspecific enzymes that catalyze the transfer of one of the biochemical groups from any donor to any acceptor, and with a single unspecific transporter that catalyzes the transport of all metabolites. Sixteen randomly chosen metabolites are involved in biomass formation. For the production of biomass, it is necessary to have at least one donor and acceptor of each group as external metabolites. We therefore assume that a minimal set of two metabolites, namely X0 and X127, is present in the environment. For reasons of computational tractability, we let the network evolve by fixation of the most beneficial mutations rather than random mutations. Duplications and deletions, because we assume that they are rare compared to mutations that alter the kinetic properties, are only considered if no further beneficial kinetic mutations are possible. A detailed description of the computer simulations is given in A sample simulation is shown in Escherichia coli, each containing between 24 and 40 metabolites. The overall connectivity of these networks is shown in p ≍ 0.085; see To obtain more precise data for the connectivity distribution, we analysed 40 replicate simulations. The results of these simulations are similar to the sample simulation shown in To analyze whether group transfer plays a major role in the emergence of hubs, we performed additional simulations with monomolecular reaction networks. In contrast to the bimolecular group transfer networks described above, we assumed that groups are not transferred between metabolites, but are simply added or removed by enzymes. The topology of the resulting initial network is similar to the group transfer network see C and 1D.Our simulation results indicate that group transfer plays a key role in the emergence of hubs. Within the evolution of group transfer networks, hubs emerge because some metabolites become a preferential donor or acceptor of a specific group, i.e., they have a particularly high group transfer potential. Therefore it is of advantage that an enzyme that transfers this specific group increases its specificity towards this donor (or acceptor). Once a preferential donor (or acceptor) emerges and the corresponding enzyme starts to specialize towards this metabolite, the remaining enzymes in the network are selected for further increasing the group transfer potential of this specific metabolite. Thereby a single metabolite may “monopolize” this group transfer reaction. It serves as the only donor (or acceptor) of a specific group towards a number of different acceptors (donors) in the network. As a consequence it participates in a number of different reactions. In the absence of group transfer reactions, i.e., in monomolecular reaction networks, this mechanism does not work. Therefore, hubs do not emerge in monomolecular reaction networks.In summary, our simulations demonstrate that metabolic networks may have emerged according to the scenario of Kacser and Beeby . StartinOn the basis of our model, we expect that multifunctional enzymes can be maintained if they have only little impact in biomass formation and are therefore present as single copies. Alternatively, multifunctional enzymes can be maintained if there is no trade-off between the different functions. Recent results on the evolution of multifunctional enzymes indicate that this may at least be sometimes the case .Although the overall connectivity of the networks emerging in our simulations differs clearly from a power-law distribution, it is consistent with the connectivity distribution derived from natural subnetworks of similar size. Our simulations indicate that the scenario of Kacser and Beeby leads to the emergence of hub metabolites. Additionally, our results predict that group transfer is essential for the emergence of hubs in these networks. This prediction is in line with the observation that most hubs in natural networks such as ATP, NADH, glutamate, and coenzyme A are key compounds in the transfer of biochemical groups. The high connectivity of these metabolites results mainly from transferase reactions rather than other reaction types (such as isomerase reactions).In our simulations, hubs emerge as a consequence of selection for growth rate. In addition, along with a number of further publications given below, our simulations therefore support the view that hubs are not necessarily a result of selection for robustness as has been suggested previously ,14. Firs7 different metabolites that are characterized by a binary string of length 7, where “1” at position x denotes the presence of group x, whereas “0” denotes the absence of the group. Metabolites are associated with a random free energy that is taken from a uniform distribution between zero and one, and is required to specify thermodynamic properties of the biochemical reactions. For the production of biomass, it is necessary to have at least one donor and acceptor of each group as external metabolites. Thus a minimal set of two metabolites, namely X0 (characterized by the string 0–0–0–0–0–0–0) and X127 (characterized by the string 1–1–1–1–1–1–1), is assumed to be present in the environment. Sixteen randomly chosen metabolites are involved in biomass formation.Metabolites are assumed to consist of seven different biochemical groups. Each biochemical group is either present once or is absent. Thus there are 2iE(1) and iE(0), with iE(1)i+ E(0)i ,= E where iE is the total dosage of enzyme i,iE(0) is the concentration of the enzyme without its group being bound to it, and iE(1) is the concentration of enzyme with its group being bound to it. The free energy difference between both states of an enzyme is assumed to be a random value taken from a uniform distribution ranging from zero to one. We further assume that in principle all metabolites that contain a specific biochemical group  can serve as a donor for group transfer, whereas all metabolites that do not contain the group can in principle serve as an acceptor. We assume linear kinetics for the transfer of a group to an enzyme, given by ij v = k(iE(0)Xij)((1)i−E(1)Xij)((0)ij/q), where ijk is the rate constant of the reaction j of an enzyme i,Xij)((1) is the concentration of the donor of the reaction, Xij)((0) is the concentration of the corresponding acceptor, and ijq is the equilibrium constant of the reaction resulting from the free energies of the reactants. For the transfer of a group from a donor to an acceptor, two half-reactions need to be coupled. This results in Michaelis-Menten–like kinetics and implies that functional enzymes need to maintain nonzero rate constants for at least two reactions. We assume that initially the enzymes are unspecific and transform groups from each donor to each acceptor with the same rate constant. The initial dosage of enzymes is i =E 1.Enzymes catalyze the transfer of a specific biochemical group. We assume that groups are transferred by a “ping-pong mechanism” see : A donoritijv = T(j−XjextX), where iT is the dosage of the transporter i,ijt is the rate constant of the transport of metabolite j,jX is the metabolite concentration in the cell, and jextX is the metabolite concentration in the environment. We assume that in the initial network there is a single transporter that transports all metabolites with the same rate constant. The initial dosage of the transporter is T0= 1.We further assume that transporters transport metabolites passively across the cell membrane. The rate of transport is given by BM = kBMΠiXiv over all metabolites iX that are involved in biomass formation. The rate constant is set to BM =k 1 in all simulations. We assume that the formation of biomass leads to growth. The growth rate is given by W = 1BM //VdV/dt = v(C0EE+CTT+C), where C0 is the amount of biomass that is required for structural compounds , EC is the amount of biomass per enzyme, TC is the amount of biomass per transporter, E is the total dosage of enzymes, and T is the total dosage of transporters [C0, EC, and TC are set to 50, 1, and 1, respectively. Note that due to cell growth, metabolites are constantly diluted at a rate equal to the growth rate. The fitness of a network is assumed to be proportional to the steady-state growth rate.Biomass is formed by the condensation of specific metabolites. The rate of biomass formation follows linear kinetics given by the product sporters . The parj kij/α1 and Σj tij/α1 over all rate constants ijk or ijt of an enzyme or transporter, i, respectively, is constant. For values of α > 1 increasing the rate constant for a single reaction has an over-proportional effect on all other rate constants. In our simulations we use Σj kij/α1= 1, Σj tij/α1= 1, and α= 2. This implies, for example, that a transporter catalyzing the transport of a single metabolite has a four times higher rate constant for this reaction than a transporter that is specialized on the transport of two metabolites.In line with the scenario of Kascer and Beeby, we assume that reactions can either catalyze a large number of reactions but with low activity for each enzyme, or a lower number or reactions with improved catalytic activities. Specifically, we assume that the sum Σijk/α1 or ijt/α1, respectively, for a single reaction is either increased or decreased by a small value of m = 0.01, while the rate constants of the other reactions are decreased, or increased appropriately. Gene deletions decrease the dosage of an enzyme. Duplications increase the dosage of an enzyme. In our simulations we calculate the effect of all possible mutations of the current network on the steady-state growth rate to obtain the mutant with maximal increase in fitness. This mutation is then assumed to become fixed and the resulting network is used to search for the next mutations. The steady states are calculated using the NAG library function c05nbc (see http://www.nag.co.uk/numeric/CL/CLdocumentation.asp for documentation) for root finding of systems of nonlinear equations. The solver may fail (or may obtain steady states with negative concentrations) if the initial approximation of the steady state is far away from the actual steady state of the system. In this case we integrated the system towards its steady state (using the NAG library function d02ejc) until the root finder succeeds. To calculate the effect of a mutation on the steady state, we solve the mutated system using the metabolite concentrations of the non-mutated system as initial approximation, because for small mutations, the steady state of the mutated systems can be expected to be close to the non-mutated system. Note, however, that in a system of nonlinear equations there may be multiple steady states, unstable steady states, and saddle points. We therefore test whether the steady state of the most beneficial mutation obtained by the above procedure  is reached by integration from initial metabolite concentrations close to zero. This test failed in ten out of 50 simulations. These simulations are excluded from further analysis. Gene duplications and deletions are assumed to be rare compared to mutations affecting the catalytic properties of enzymes and transporters. We therefore consider these types of mutation only if none of the mutations affecting kinetic properties are beneficial. A simulation ends if there are no beneficial kinetic mutations nor gene duplications or deletions.We assume that there are two types of mutations: (1) mutations that change the kinetic properties of an enzyme and (2) mutations that change the number of copies of an enzyme, i.e., gene deletions and duplications. For the first type of mutation we assume that the value of E. coli downloaded from the KEGG database (ftp.genome.ad.jp/pub/kegg/ pathways/eco/) on the basis of the annotation in the reaction database (ftp.genome.ad.jp/ pub/kegg/ligand/reaction). Metabolites participating in fewer than two reactions are excluded. For comparison with the networks emerging in our simulation, we included 22 subnetworks of similar size, i.e., with between 24 and 40 metabolites. To compare the connectivity distributions, we use Pearson's chi square test. Data from the tail of the distributions (k > 11) are put into a single class.We analyzed the connectivity in metabolic subnetworks of i kijv = E(Xij)((1)−Xij)((0)ij/q), where ijk is the rate constant of the reaction j of an enzyme i, iE is the dosage of the enzyme, Xij)((1) is the concentration of the donor, Xij)((0) is the concentration of the corresponding acceptor, and ijq is the equilibrium constant of the reaction resulting from the free energies of the reactants. All other assumptions are equal to those used in the simulations of the bimolecular group transfer networks.For simulating the evolution of monomolecular networks, we assume that enzymes add or remove biochemical groups, thereby transforming a donor of a group into its corresponding acceptor and vice versa. In contrast to bimolecular reactions, it is not required to couple two half-reactions. Completely specialized enzymes catalyze a single reaction rather that two half-reactions. The rate of a reaction is given by


Many computer studies have employed either dynamic simulation or metabolic flux analysis (MFA) to predict the behaviour of biochemical pathways. Dynamic simulation determines the time evolution of pathway properties in response to environmental changes, whereas MFA provides only a snapshot of pathway properties within a particular set of environmental conditions. However, owing to the large amount of kinetic data required for dynamic simulation, MFA, which requires less information, has been used to manipulate large-scale pathways to determine metabolic outcomes.Here we describe a simulation method based on cooperation between kinetics-based dynamic models and MFA-based static models. This hybrid method enables quasi-dynamic simulations of large-scale metabolic pathways, while drastically reducing the number of kinetics assays needed for dynamic simulations. The dynamic behaviour of metabolic pathways predicted by our method is almost identical to that determined by dynamic kinetic simulation.The discrepancies between the dynamic and the hybrid models were sufficiently small to prove that an MFA-based static module is capable of performing dynamic simulations as accurately as kinetic models. Our hybrid method reduces the number of biochemical experiments required for dynamic models of large-scale metabolic pathways by replacing suitable enzyme reactions with a static module. Recent progress in high-throughput biotechnology -3 has maAnother approach, such as metabolic flux analysis (MFA) using stoichiometric matrices, has been employed for large-scale analyses of metabolism ,15,16. AHere we propose a method for dynamic kinetic simulation of cell-wide metabolic pathways by applying the kinetics-based dynamic method to parts of a metabolic pathway and the MFA-based static method to the rest. Because the static module does not require any kinetic properties except the stoichiometric coefficients, this method can drastically reduce the number of enzyme kinetics assays needed to obtain the dynamic properties of the pathway. We have evaluated the accuracy of the hybrid method in comparison to a classical dynamic kinetic simulation using small virtual pathways and an erythrocyte metabolism model.-16 of the maximum for the reaction rates. All the metabolite concentrations in the hybrid model were identical to those in the dynamic model  the static module involved enzymes of which the reactions were bottlenecks of dynamic behaviour, i.e. were not sufficiently susceptible to the boundary reaction rates, and (ii) a boundary reaction rate underwent a large change in response to changes in substrate concentrations. For example, the hybrid model of the hypothetical pathway with an over-determined static module exhibited approximately 10-fold higher one-step errors in the reaction rates of the static module when the rate constants of a boundary reaction E_BC were altered from kRelationships between kinetic properties and one-step errors were examined in depth using a simple linear pathway at a steady state Figure  and 2a gThe same analysis was performed using an erythrocyte metabolism model  to evalu-4, was observed in the reaction rate of 6-phosphogluconate dehydrogenase (6PGODH) , which has a larger elasticity, the time courses calculated by the hybrid model were different from those produced by the dynamic model.In the simulation experiments using hypothetical pathways and an erythrocyte model, the discrepancies between the dynamic and the hybrid models were sufficiently small to prove that an MFA-based static module is capable of performing dynamic simulations as accurately as a kinetic model. The key idea behind our method is to distinguish between dependent and independent variables (reactions). Although independent reactions can be affected by other dependent/independent reactions through effectors such as ADP in the phosphofructokinase reaction, the time evolution of adjacent reaction rates are mainly determined by independent reactions which constitute bottlenecks of dynamic behaviour in the metabolic network. Therefore, static modules should consist of only such dependent reactions, whereas dynamic modules can include both independent and dependent reactions. Our hybrid method reduces the number of biochemical experiments required for dynamic models of large-scale metabolic pathways by replacing suitable enzyme reactions with a static module. The optimal conditions for this method are (a) a system with few bottleneck reactions in order to enlarge the static modules, (b) small fluctuations in the reaction rates in static modules, and (c) accurately identifiable bottleneck reactions. How can such enzymes be identified? One obvious criterion for the enzymes to be suitably modelled by a static module is not to incorporate a bottleneck reaction in a transient state. Thus, the enzymes should not reach the maximum velocity quickly or be restrained at lower activities by allosteric regulation. Although the model comprising dynamic and static modules as a whole can represent transient states, it is assumed that the reactions in the static modules achieve or nearly achieve steady states within one numerical integration step. The existence of one or more bottleneck reactions in the static module may cause inconsistencies, because the hybrid method solves algebraic equations for static modules under a steady state assumption, although metabolites will be accumulated or depleted in real cells. Therefore, bottleneck reactions must be excluded from static modules. Another situation that should be avoided involves reaction rates in static modules that are affected by changes in enzyme concentration, such as those caused by changing levels of transcriptional/ post-transcriptional control. Such reactions should be included in dynamic modules.A similar cause of inconsistency is the reversibility of reactions. Since the hybrid method assumes that reactions in the static module are reversible, inclusion of an irreversible step may cause inconsistencies, particularly in the presence of a perturbation downstream of the irreversible step (data not shown).The accuracy of the calculation can also be affected by a time lag. In the static module of the hybrid model, time lags between the upstream and downstream reactions are not represented because the boundary reactions affect all subsequent reactions in the static module within one integration step regardless of the number of enzyme reactions. Depending on the simulation time scale, the static module should be limited to minimize the influence of time lags. This influence can be estimated by the ratio of elasticities, which can be an important criterion for including a reaction in the static module.v, that can be calculated from v = S#b in as little as one numerical integration step. This allows the hybrid model to produce results that are in agreement with the dynamic model when the boundary reactions weaken perturbation.The correlation between elasticity and one-step error Figure  indicateThe results we obtained when we relocated the boundary of the static module in the erythrocyte model support the importance of elasticity ratios. When G6PDH was included inside the static module, PGI became the new boundary reaction instead of G6PDH. The elasticity of PGI is large (elasticity = -452.496) compared to its neighbour G6PDH (elasticity = 0.0955). The relocated boundary is therefore composed of a pair of reactions that might produce unacceptable calculation errors, and in fact led to inconsistencies between the hybrid and dynamic models. Thus, the analytical conclusion presented in Eqs. (2) and (3) also holds for complex pathways, and elasticity provides a criterion for identifying groups of enzymes that can be approximated with sufficient accuracy by static modules. However, a large amount of experimental data is still required to determine the elasticities of all enzymatic reactions. In addition, the demarcation of the static module using elasticities determined by conventional biochemical experiments is unrealistic with respect to their throughput. Hence, the comprehensive determination of bottleneck reactions is the key task in the construction of large-scale metabolic pathway models using the hybrid method. Recent advances in flux measurement, quantitative metabolomics and proteomics allow large-scale measurement of flux distributions , intraceE. coli and human red blood cells. Pulse-chase analyses using 13C labeled molecules and the CE-MS/LC-MS high-throughput system have also been used successfully by the same metabolomics group to determine fluxes in the E. coli central carbon pathway.Recently, a method for high-throughput metabolomic analyses using capillary electrophoresis assisted by advanced mass spectrometry (CE-MS) and LC-MS/MS has been developed by the metabolomics group at our institute -27. ThisSeveral approaches have been proposed to quantify elasticity and other coefficients of metabolic control analysis from experimental data such as flux rates, metabolite concentrations or enzyme concentrations -31. ThusUsing this hybrid method, the cost of developing large-scale computer models can be greatly reduced since precise modelling with dynamic rate equations and kinetic parameters is limited to bottleneck reactions. This drastically reduces the number of experiments needed to obtain the kinetic properties required for the dynamic simulation of metabolic pathways.1, V2, V9, and V10 in Figure 3, V4, V5, V6, V7, and V8) is derived from the balance equation Sv = b, where S denotes the stoichiometric matrix, v the flux distribution, and b the rates of the dynamic exchange reactions at the system boundary  that are calculated in step (i); and (iii) the concentrations of the metabolites (X1-X13) are determined by numerical integration of the reaction rates calculated in steps (i) and (ii). All the reactions in the static module are assumed to be reversible.The hybrid method works within one numerical integration step as follows: (i) all the reaction rates in the dynamic module are calculated from dynamic rate equations , the Moore-Penrose pseudo-inverse is employed to calculate the reaction rate distribution of the static module at each numerical integration step. This should result in a smaller computational cost than linear programming, which is commonly used to determine the flux distribution of the underdetermined system. When the linear equation ribution  which miSv = b to minimize the error of the ideal reaction rate distribution specified by the user. The optimal solution vbest is represented in Eq. (1) below , [B], εv1A and εv1B denote the reaction rate v2 in the dynamic model, v2 in the hybrid model, concentration of metabolite A, concentration of metabolite B, elasticity of v1 with respect to metabolite A, and elasticity of v2, respectively. The variables with Δ are increments after a small time step Δt. The parameter p represents a ratio of the reaction rate in the static module to the influx, as in Δv2h = pΔv1. The ratio p is determined by the stoichiometric matrix of the pathway.where v2 to v1. When εv2B satisfies the relationship below, v2 in the hybrid model exhibits identical time evolution to the dynamic model:In Eq. (2), the left bracket term on the right-hand side indicates the magnitude of the perturbation transmitted to the static module. This term indicates that the error between the hybrid and dynamic models is proportional to the increment of metabolites and the elasticity of the boundary reactions. The right bracket describes the susceptibility of the reaction rate vSince a small Δt (<<1.0s) is usually employed for accurate simulations of metabolic pathways, Eq. (3) implies that a reaction with large elasticity can be appropriately replaced by a static module.For more complex pathways, such a theoretical analysis is not practical because large numbers of variables and parameters might impede clear discussions. Instead, simulation experiments were performed to compare the accuracy of hybrid models with dynamic models by numerical methods.The accuracy of the hybrid model was evaluated numerically in comparison with a conventional kinetic model of the same metabolic pathway under two conditions: a steady-state condition and a time evolution after a two-fold increase of metabolites that are catalyzed by boundary reactions. The errors under steady-state conditions were employed as controls to evaluate discrepancies in dynamic behaviour. These computer simulations were performed using the E-Cell Simulation Environment version 1.1 or 3.1.102 for RedHat Linux 9.0/i386. The errors of reaction rates and metabolite concentrations were measured as below:-d and vh denote either the reaction rates or the concentrations in the dynamic and hybrid models, respectively. The values of vd and vh were taken at the first numerical integration step, in which the concentration increase influences the initial steady-state values of the reaction rates and metabolite concentrations. In this article, this is termed "one-step error". We used one-step errors to represent the discrepancies between the two simulation methods in transient dynamics.where vf[S]-kr[P] where v, kf, kr, [S], and [P] are a reaction rate, a forward rate constant, a reverse rate constant, a substrate concentration and a product concentration, respectively. In the pathway with the over-determined static module, the rate constants were kf = 0.05s-1 and kr = 0.091s-1 for E_CD and E_CE, kf = 0.1s-1 and kr = 0.091s-1 for E_DF and E_EG, and kf = 0.01s-1 and kr = 0.001-1 for the other reactions in the pathway of Figure f = 0.01s-1 and kr = 0.001s-1 for E_AB, E_BC, and E_FG; kf = 0.1s-1 and kr = 0.098s-1 for E_CD and E_DF; kf = 0.1s-1 and kr = 0.097s-1 for E_CE and E_EF; and kf = 0.1s-1 and kr = 0.96s-1 for E_CF. The steady-state flux distribution was employed for the ideal reaction rate distribution in the static module; the ideal reaction rates were 2 μM/s for E_CD and E_DF, 3 μM/s for E_CE and E_EF, and 4 μM/s for E_CF. All the initial metabolite concentrations were 1.0 mM. The concentration of metabolite A was increased two-fold to evaluate the error.The one-step errors were evaluated using two simple pathways; the static module of one is determined, while the other is underdetermined Figure  and 2b. Elasticity is a coefficient used to quantify the sensitivity of the enzyme to its substrates and is defined as below in the context of metabolic control analysis :f = 0.01s-1 and kr = 0.009s-1 for E_AB, E_BC, and E_FG and kf = 0.1s-1 and kr = 0.099s-1 for E_CD, E_DE, and E_EF. All the initial metabolite concentrations were 1.0 mM. The two rate constants of reactions E_BC, E_CD, E_DE, and E_DF were altered within the range 0.01<kf<1.0. The value of kr was determined to satisfy kf-kr = 0.01 to sustain the initial steady-state concentrations. The concentration of metabolite A was increased two-fold to evaluate the errors. For error measurements in the glycolysis model, each enzymatic reaction was replaced, one by one, with a static module. The substrate concentrations of the boundary reactions were increased three-fold.where [S] and v denote the substrate concentration and the reaction rate of the enzyme, respectively. Correlation between the one-step errors and elasticities of each enzyme at a steady state was examined using a linear pathway and a glycolysis model ,20  was increased three-fold to measure the errors in dynamic behaviours.where The author(s) declare that they have no competing interests.Yugi contributed to the development and implementation of the hybrid method into the E-Cell system, and developed methods for analyzing errors at a steady state. Nakayama provided the concept of hybrid method and directed the project. Kinoshita contributed to the development of simulation models and the analyses, and Tomita is a project leader.Derivations of equations (Eqs. (1) and (2)), supplementary tables (Table 4 and Table 5) and figure .Click here for file


Elementary mode analysis of metabolic pathways has proven to be a valuable tool for assessing the properties and functions of biochemical systems. However, little comprehension of how individual elementary modes are used in real cellular states has been achieved so far. A quantitative measure of fluxes carried by individual elementary modes is of great help to identify dominant metabolic processes, and to understand how these processes are redistributed in biological cells in response to changes in environmental conditions, enzyme kinetics, or chemical concentrations.Selecting a valid decomposition of a flux distribution onto a set of elementary modes is not straightforward, since there is usually an infinite number of possible such decompositions. We first show that two recently introduced decompositions are very closely related and assign the same fluxes to reversible elementary modes. Then, we show how such decompositions can be used in combination with kinetic modelling to assess the effects of changes in enzyme kinetics on the usage of individual metabolic routes, and to analyse the range of attainable states in a metabolic system. This approach is illustrated by the example of yeast glycolysis. Our results indicate that only a small subset of the space of stoichiometrically feasible steady states is actually reached by the glycolysis system, even when large variation intervals are allowed for all kinetic parameters of the model. Among eight possible elementary modes, the standard glycolytic route remains dominant in all cases, and only one other elementary mode is able to gain significant flux values in steady state.These results indicate that a combination of structural and kinetic modelling significantly constrains the range of possible behaviours of a metabolic system. All elementary modes are not equal contributors to physiological cellular states, and this approach may open a direction toward a broader identification of physiologically relevant elementary modes among the very large number of stoichiometrically possible modes. Biological research in the twentieth century has been dominated by the reductionist approach, providing valuable information about the properties and functions of individual cellular components. But the behaviour of complex systems of interacting components cannot be comprehended by the sole characterisation of their individual components or pair-wise relations, because new properties emerge from the interaction of large numbers of components. Technological developments now allow to gain more and more knowledge about the various interaction networks that govern cellular properties, including protein-protein interactions, metabolic pathways, regulatory and signalling networks. Therefore systems-based approaches are now needed with the aim of modelling and understanding the complex cellular processes producing biological functions, and in the long term providing an integrated and predictive description of a complete cell .Metabolic pathways are an essential key to the systemic behaviour of a biological cell. Two types of approaches are generally possible for the study of cellular metabolism. The first type involves detailed modelling of the dynamical features of biochemical networks. Several tools have been created for simulating metabolic and signalling networks in biological cells -5. Howevelementary modes and extreme pathways have been introduced in recent years 14] showen genes.  presentehrocytes . Other ehrocytes , the phyhrocytes , the invhrocytes , and therevisiae .The two concepts of elementary modes and extreme pathways are very similar but not identical, the crucial difference being that extreme pathways are defined as systemically independent, which means that no extreme pathway can be represented as a non-negative linear combination of other extreme pathways. Comparisons between both concepts and discussions about their advantages and drawbacks have been published -23. If oAlthough the space generated by elementary modes or extreme pathways contains all possible steady-state flux distributions, not necessarily all these states may actually be reached by a real biological organism. Several approaches have already been proposed in order to further characterise the space of allowable states. Regulatory constraints repressing gene expression have been modelled, and it has been shown that they strongly reduce the number of allowed extreme pathways in specific environmental conditions . Singulaα-spectrum [α-spectrum describes the range of possible weightings a particular mode can take in the decomposition. This range is determined using linear optimisation to maximise and minimise the weighting of a particular extreme pathway in the reconstruction. A drawback of that description is that the range of allowable weights for a given extreme pathway is not necessarily independent of the weight value taken by any other extreme pathway.A first attempt to understand how physiological steady states can be reconstructed from a network's extreme pathways lead to the concept of spectrum . Any fluIn a previous communication, we presented a different decomposition approach that assigns each elementary mode a unique weight ,28. Beca-1·l-1, and concentrations are expressed in mmol·l-1. Abbreviations used for enzyme and compound names are listed in Table Our approach is illustrated using a model of yeast glycolysis presented by . This moThe system and its eight elementary modes are shown in Figure As shown earlier , decompoIt should be noted that the cases of EM1 and EM2 are particular. These elementary modes depend entirely on the trehalose and glycogen branches respectively. These two branches retain constant fluxes in the model, therefore the fluxes carried by EM1 and EM2 always remain constant as well. As a result, these modes cannot be further analysed and will not be further discussed. Nevertheless, we chose to keep them in our analysis to maintain consistency with the model from , and becL0 parameter in the PFK kinetic relation can be found in the literature: a value of 3342 was used in [In this section, we studied variations in elementary mode fluxes induced by variations of individual kinetic parameters, all other parameters of the model being kept at their reference values. This case may apply to the situation where a given parameter has not been experimentally measured or a large uncertainty exists for its value. For example, very different values of the  used in , while 0 used in .The Gepasi software  was usedThe obtained standard deviations of the distributions of elementary mode fluxes are given in Table In most cases, the effect of variation of a single kinetic parameter on the distribution of elementary mode fluxes is weak. Several parameters have almost no effect: the most striking examples are the parameters of ALD Figure  and PDC.Km·Ace and Km·Et of ADH, Km·G6P and Km·Glc of GLK .We additionally looked at the effects of variations in concentrations of external compounds on elementary mode fluxes. Only two external concentrations are defined in the glycolysis model: glucose and ethanol. The effect of a variation of glucose concentration is not very different from what was observed for some kinetic parameters, such as LK Table . For lowThe distributions of elementary mode fluxes obtained for 400 random glucose and ethanol concentration values are shown in Figure We subsequently attempted to span the entire space of kinetically attainable steady states by allowing all kinetic parameters of all enzymes to vary simultaneously in the ranges defined in Table Figure The distributions of EM3, EM4, EM5 and EM7 all show very little variation and are concentrated at low values. Only EM8 shows a broader distribution spanning most of the possible flux values, with an important peak around 80. A majority of steady states thus represent distributions where most of the flux in the system is consumed by EM8, with very little use of the other elementary modes. Interestingly, this dominant state is not identical to the experimental steady state, where EM8 is still the most important mode but EM7 consumes a non-negligible proportion of flux.Correlations between the different elementary mode fluxes were computed as in the previous section Figure . EM5 andThe approach followed in the previous sections presents resemblances to methods developed in the framework of Metabolic Control Analysis (MCA) . The majIn order to clarify the relations between both approaches, we nevertheless applied MCA to the most significant state for this system: the experimental steady state determined by . Two typp is the perturbation parameter and v the rate for the isolated enzyme. Second, control coefficients describe the effects of a given enzyme on system fluxes, by comparing a rate variation induced by a perturbation for the isolated enzyme to the variation of system flux induced by the same perturbation. Control coefficients can be defined in a similar way for elementary mode fluxes:where v is the rate of the isolated enzyme and w is an elementary mode flux in the system. It should be noted that control coefficients of a given reaction are independent on the choice of the perturbation parameter, therefore only one control coefficient exists characterising the effect of a given enzyme onto a given systemic flux [where mic flux . ElasticVmax and Keq of G3PDH have very different elasticities, while their effects on elementary mode fluxes are in the same range. As was shown in Figure Elasticity coefficients of parameters of the glycolysis model are shown in the last column of Table δw and w are equal to zero for these modes. For the remaining elementary modes, standard deviations over large intervals cannot be inferred from local control coefficients in general. For example, GLK and PGI produce similar large-range effects but have very different local control coefficients on EM8. On the opposite, G3PDH and GAPDH have both strong large-range effects and high local control coefficients on several modes. The overall correlation between EM5 and EM7 can be observed locally, as values of all control coefficients are very similar for these two modes. In order to check that the summation relationship of flux control coefficients also applies to elementary mode fluxes, we added to Table Control coefficients cannot be calculated at the experimental steady state for EM3, EM4, and EM6, because both The motivation for providing a decomposition of flux distributions onto elementary modes was to provide a quantitative measure of the utilisation of each elementary mode in a metabolic system under given conditions. Each elementary mode represents a particular route of transformation of some substrates into some products, and can therefore be viewed as an elementary possible biological function of a metabolic pathway. Having a measure of elementary mode utilisation is important for two reasons. First, it makes it possible to observe which elementary modes are significantly active in a biological system and which ones are not. This capability greatly enhances the biological interpretability of elementary mode based pathway analyses, and allows to concentrate further investigations on the physiologically active processes among the usually very large number of stoichiometrically possible processes. Second, a quantitative measure makes it possible to quantify the effects of changes in a system  on the redistribution of metabolic processes in that system. Such analyses may in turn be used in a biotechnology perspective for identifying components which have the strongest effect on some desirable physiological process. In our example, several kinetic parameters were shown to have very little effect on the steady state of the glycolytic system, while a smaller set of parameters can account for significant variations. The same approach may be relevant when attempting to model particular biochemical reactions based on experimental measurement of kinetic parameters, as the most important parameters could be identified in order to concentrate experimental efforts on them.i.e. minimising the norm of the elementary mode flux vector, was introduced independently by two different teams based on two different approaches. We previously justified this choice from a geometrical and biological point of view [best decomposition should assign maximum weights to the modes that are closest to the actual state of the system, i.e. the modes that best describe it biologically. The authors of [It is particularly interesting that the same choice for a relevant decomposition,  of view ,28. Our thors of  in turn Escherichia coli based on linear programming revealed that the overall activity of metabolism is dominated by a high-flux backbone, while most other reactions have low fluxes [In the glycolytic system, our analysis found the space of kinetically feasible steady states to be significantly smaller than the space of stoichiometrically feasible steady states. When only stoichiometry is taken into account, all linear combinations of reversible elementary modes and non-negative linear combinations of irreversible elementary modes are valid steady states. But when a kinetic model is applied, it appears that a significantly smaller part of the steady state space is actually attainable, even when large ranges of values are allowed for all kinetic parameters. Most elementary mode fluxes in the glycolytic system were constrained to narrow intervals, and only two elementary mode fluxes could span large intervals of values. These results are consistent with other findings on cellular metabolism. A system-wide analysis of metabolic fluxes in w fluxes . This rew fluxes . A numbew fluxes , indicatA genome-wide identification of such modes by the approach presented here remains a long process though, and further efforts should be directed toward developing automatic procedures combining such kinetic and elementary mode analyses. The decomposition algorithm itself is fast and converges in a fraction of a second for the glycolysis example. Quadratic programming problems defined by positive-definite Hessian matrices (as is the case here) can be solved in polynomial time , but perWe showed how a combination of kinetic modelling and elementary mode based flux decompositions makes it possible to analyse the distribution of metabolic processes in physiological cellular states, and to assess the effects of kinetic changes onto the balance of utilisation of different elementary modes. Similar approaches as the one presented here could be applied to understand the metabolic response to external perturbations or genetic changes, to identify possible genetic alterations for the optimisation of metabolic processes of particular interest, or to quantitatively measure and compare the effect of different drugs on cellular functions. Elementary modes indeed represent elementary possible metabolic functions, and the set of elementary modes provides a basis for the functional interpretation of biochemical systems. We believe that further efforts should be directed toward fully exploiting this potential, including the systematic computation and functional annotation of elementary modes in complete metabolic systems.m elementary modes and n reactions. The elementary modes are represented by vectors E1, E2...Em, each of them being of length n. A flux distribution in this system is represented by a vector v of length n. A decomposition of this flux distribution onto the set of elementary modes is defined by a vector w of length m such that:We consider a metabolic network with Ej are dimensionless, those of w are expressed in units of flux. We previously used to refer to wj values as elementary mode weightings [elementary mode fluxes. This terminology furthermore highlights the difference between wj values and α-weightings as introduced by [α-weightings are normalised to unity by dividing fluxes through the limiting Vmax of each extreme pathway, and are therefore dimensionless values representing the fractional usage of each extreme pathway compared to its maximum capacity.Given that the elements of ightings , but in ightings , we refeduced by . α-weighwj of w depends on the nature of elementary mode Ej. If all reactions composing Ej are reversible, then elementary mode Ej is reversible as well and its flux wj may be positive or negative. On the opposite, if Ej contains at least one irreversible reaction, then Ej is an irreversible elementary mode and wj can only be positive:The sign requirement for an element wj ≥ 0 if Ej is irreversible     (4)j if extreme pathways are used instead of elementary modes. Extreme pathways as defined by [It should be noted that condition (4) holds for all fined by  are alwaIn general, there are more elementary modes than reactions in a system, and conditions (3) and (4) do not define a unique solution but a continuous convex space of possible solutions. We thus introduced a third condition constraining the system to a unique solution:quadratic programming problem.Relations (3), (4) and (5) together define a non-linear optimisation problem, also known as Elementary modes are only unique up to a scaling factor, and the non-linearity of equation (5) makes the solution dependent on the rule decided for scaling elementary modes. It is therefore important to note that numerical values of elementary mode fluxes depend on the scaling strategy. No general rule for scaling elementary modes could be derived from the literature, as this point becomes crucial only with quantitative analysis. Nevertheless, most authors implicitly scale elementary modes to the smallest possible scale allowing all the stoichiometric coefficients they contain to remain integers, and the same rule was therefore applied here.active set algorithm, which uses an iterative process. In each iteration, some inequality constraints are set to equality while the remaining constraints are temporarily disregarded, until the correct set of active constraints is found. The main steps of our algorithm are described below, while a detailed mathematical demonstration of the active set algorithm can be found in [Our implementation for solving the quadratic programming problem is based on the classical found in .feasible solution, that is a vector w(0) which fulfils conditions (3) and (4) but is not in general the optimal solution. This step is equivalent to a linear programming problem on its own, and is solved by the simplex method. The obtained solution w(0) is used as the first feasible point in step 2, together with an empty active set.1a. The algorithm is initialised with a v do not strictly verify flux conservation. This may happen because of rounding approximations in numerical computations of flux values, and turned out to occur frequently when steady-state flux distributions were computed by the Gepasi software [1b. Difficulties in finding a feasible solution may appear if the values of software . Differesoftware . In our - the null space of the system is computed by a diagonalisation of the matrix of elementary modes,- the null space is multiplied through the flux vector to verify flux conservation,- if fluxes are not strictly conserved, a subset of systemically independent fluxes is selected and the remaining fluxes are adjusted to re-establish exact flux conservation.The previous operations furthermore enable us to reduce the dimension of the quadratic programming problem. In the following steps, only the last subset of systemically independent constraints is kept. This process first guarantees that no degeneracy exists in the constraint set, and second leads to a reduction in computation time that can be worthy for large systems.wk) (and a set of active constraints A. This is achieved by shifting the origin to wk) (and applying the method of Lagrange multipliers. A correction δk)     (= wk) (+ δk)  , then the quadratic problem has a unique global solution. Therefore any solution found in step 3 is the unique global minimum. Successful termination of the active set algorithm has been proved in general , excludiet al. in [The approach followed by Poolman t al. in  for const al. in  and we iHowever, the system shown in the previous example contains only reversible fluxes, and thus includes no sign requirement as defined by (4). As a matter of fact, the specificity of the active set algorithm precisely lays in its rigorous treatment of inequality constraints. We therefore cannot exclude that differences between both algorithms would arise when irreversible reactions are present. In that case, when the optimal solution found by Moose-Penrose inversion assigns negative fluxes to some irreversible elementary modes, these modes are assigned zero fluxes in , and theJMS initiated the project, carried out the analysis, and wrote the manuscript. MK supervised the project. All authors read and approved the final manuscript.


Through genetic engineering it is possible to introduce targeted genetic changes and hereby engineer the metabolism of microbial cells with the objective to obtain desirable phenotypes. However, owing to the complexity of metabolic networks, both in terms of structure and regulation, it is often difficult to predict the effects of genetic modifications on the resulting phenotype. Recently genome-scale metabolic models have been compiled for several different microorganisms where structural and stoichiometric complexity is inherently accounted for. New algorithms are being developed by using genome-scale metabolic models that enable identification of gene knockout strategies for obtaining improved phenotypes. However, the problem of finding optimal gene deletion strategy is combinatorial and consequently the computational time increases exponentially with the size of the problem, and it is therefore interesting to develop new faster algorithms.Saccharomyces cerevisiae. Potential metabolic engineering targets for improved production of succinic acid, glycerol and vanillin are identified and underlying flux changes for the predicted mutants are discussed.In this study we report an evolutionary programming based method to rapidly identify gene deletion strategies for optimization of a desired phenotypic objective function. We illustrate the proposed method for two important design parameters in industrial fermentations, one linear and other non-linear, by using a genome-scale model of the yeast We show that evolutionary programming enables solving large gene knockout problems in relatively short computational time. The proposed algorithm also allows the optimization of non-linear objective functions or incorporation of non-linear constraints and additionally provides a family of close to optimal solutions. The identified metabolic engineering strategies suggest that non-intuitive genetic modifications span several different pathways and may be necessary for solving challenging metabolic engineering problems. Microorganisms are widely used for producing antibiotics, therapeutic proteins, food and feed ingredients, fuels, vitamins and other chemicals. Currently there is an increasing trend to replace chemical synthesis processes with biotechnological routes based on microbial fermentations. In order to economically produce desired compounds from microbial cell factories it is, however, generally necessary to retrofit the metabolism, since microorganisms are typically evolved for maximizing growth in their natural habitat. Retrofitting of microbial metabolism has traditionally been done through classical strain improvement that involved random mutagenesis and screening, whereas in later years rational design strategies based on genetic engineering have been applied with an increasing success – often referred to as metabolic engineering. In metabolic engineering many experimental and mathematical tools have been developed for introducing directed genetic modifications that will lead to desirable metabolic phenotypes resulting in improved production of desirable compounds or in reduced production of by-products ,2. UntilMicrobial metabolism is often subjected to tight regulation and is constrained by mass and energy conservation laws on a large number of intracellular metabolites, and this makes it difficult to predict the effects of introducing genetic modifications in a given cell. Moreover, as metabolic pathways and related regulatory processes form complex molecular and functional interaction networks ,4, it isin silico metabolic engineering. However, despite of their potential, genome-scale stoichiometric models have been scarcely used for metabolic engineering purposes.Genome-scale stoichiometric models represent the integrated metabolic potential of a microorganism by defining flux-balance constraints that characterizes all feasible metabolic phenotypes under steady state conditions. Because of the large number of reactions occurring in cellular metabolism, the dimensions of the solution space (or the number of feasible metabolic phenotypes) defined by genome-scale models ,11 is veet al. . Although, this quantity will be equal to the substrate-specific productivity under the assumption of a fixed substrate uptake rate, we will refer to this term as Biomass-Product Coupled Yield (BPCY) rather than the productivity as this may cause confusion . BPCY represents an interesting example of a non-linear objective function that can be optimized by using OptGene.We illustrate the principles and utility of OptGene algorithm by using three interesting metabolic engineering problems with the yeast Saccharomyces cerevisiae which is a food grade organism. Since vanillin is not produced naturally by S. cerevisiae, the corresponding reactions were inserted into the model as suggested by Pharkya et al[in silico up to 90 % of the theoretical limit by deleting only 2 reactions (pyruvate decarboxylase and glutamate dehydrogenase), while keeping the growth rate at 60% of the parental strain. A similar strategy was predicted for a mutant with the maximum BPCY. The suggested strategy diverts the pyruvate flux going to ethanol towards vanillin where NADH is oxidised back to NAD+. Furthermore, deletion of glutamate dehydrogenase results in an increased availability of NADPH needed for vanillin biosynthesis. Increasing the allowable number of deletions did not result in substantial improvement in the yield or BPCY.Vanillin is a natural flavor compound extracted from plants and is widely used as a food ingredient. There is some commercial interest in producing vanillin by using recombinant microorganisms and in particular kya et al. Then weSaccharomyces cerevisiae naturally produces glycerol in small quantities during anaerobic fermentation or under osmotic stress. Moreover, glycerol plays an important role in maintaining the cytosolic redox balance under anaerobic conditions and it is therefore interesting to study the effects of gene-deletions on yield and productivity of glycerol. We applied the OptGene algorithm to identify gene deletions leading to improved yield and BPCY of glycerol under aerobic conditions, where the maximum theoretical yield of glycerol is much higher as opposed to anaerobic fermentation.Currently glycerol is mainly recovered as a by-product from soap manufacturing or produced from propylene and is widely used to synthesize several products ranging from cosmetics to lubricants . AlternaTDH1, TDH2 and TDH3). This strategy makes intuitive sense as reactions that branch the flux away from dihydroxyacetone phosphate (the precursor for glycerol) are deleted  and genes encoding Glyceraldehyde-3-phosphate dehydrogenase  and THR1 (homoserine kinase) is predicted to result in a succinic acid yield of 0.018 g/g glucose, with a 10% reduction in the growth rate. Flux re-distribution leading to this improvement in the double-deletion mutant is quite interesting and non-intuitive. Deletion or inactivation of the SDH-complex prevents the conversion of mitochondrial succinate to fumarate, while simultaneous deletion of THR1 forces threonine synthesis via glycine, which may be formed from glyoxylate. Consequently there is increased flux through ICL1 , thus creating surplus succinate that is secreted by the cell. Moreover, this flux re-distribution is also associated with an increased flux through the pentose phosphate (PP) pathway for increased NADPH availability. We note that in the mutant with only the SDH-complex deleted, threonine is synthesized via aspartate, which is optimal route for maximizing biomass production. The same double deletion mutant was also predicted to show maximum BPCY (4.5 mg/g glucose-hr).Firstly, we note that the maximum theoretical yield of succinic acid is 0.506 g/g glucose (Note 2) when no biomass is being produced, and that no succinic acid can be produced at optimal biomass growth rate. Moreover, no single gene deletion strategy resulted in succinic acid production. For a double gene deletion strategy, deletion of the SDH-complex, ZWF1 (Glucose-6-phosphate dehydrogenase) and PFK2 (Phosphofructokinase). Although this resulted in increased prediction of succinate yield (0.21 g/g glucose), the corresponding growth rate is very low (96% reduction in growth rate), making this solution unattractive. However, a triple deletion mutant with maximum BPCY (16 mg/g glucose-hr) was found to have 76 % of the wild-type growth rate and a succinate yield of 0.07 g/g glucose. The corresponding solution suggested deletion of SER3 in addition to the double deletion strategy discussed above. Deletion of SER3 blocks the synthesis of L-Serine via 3-Phospho-D-glycerate, which increases the demand on glycine production via glyoxylate. Overall, it leads to a further increase in the flux through ICL1 ensuring a higher flux towards succinate while maintaining a high growth rate. This increase is also associated with a further increase in the flux through the PP pathway.The search for a triple deletion mutant with maximum succinate yield suggested deletion of the SDH-complex, ZWF1, PDC6 (pyruvate decarboxylase) and AGP3 (glutamate permease). Deletion of ZWF1 increases the flux through glycolysis and deletion of PDC6 increases conversion of pyruvate to oxaloacetic acid via PYC1. This flux could be directed towards glutamate production and into the TCA cycle. But since the SDH-complex is deleted the flux through TCA cycle is limited, while deletion of secretion reaction for surplus glutamate forces malate formation from oxaloacetic acid. The flux through malate is then directed to succinate via fumarate. We also searched for a quadruple deletion strategy for maximum BPCY and the algorithm suggested the same deletion strategy as for the maximum yield, with a corresponding BPCY of 29 mg/g glucose-hr. This BPCY shows a substantial increase over the BPCY obtained with the triple deletion strategy.In spite of a slow growing triple deletion mutant for improved yield, the algorithm found a quadruple deletion mutant with not only improved yield (0.36 g/ g glucose), but also with much higher growth rate, as compared to the triple deletion mutant table , and thein vivo due to a variety of reasons, e.g. regulatory constraints, orphan reactions etc. However OptGene provides not only the optimal solution found, but also generates a family of "good" solutions and thus provides many strategies that can be further analyzed manually before experimental verification. Some of such alternative solutions are also reported in table Results of a further search allowing more gene deletions, for improvement in yield and BPCY, are summarized in table Echerichia coli have been shown to be better approximated by assuming that the fluxes tend to have a minimum distance from wild-type flux distribution, which may not correspond to the flux distribution for maximum growth [The examples discussed above use FBA as scoring function to evaluate fitness of an individual in the GA. However, as noted before, the flux distribution of mutants of m growth . Neverthm growth ,27.FUM1 (fumarase) and PDA1 (pyruvate dehydrogenase). This strategy is different from that suggested by using FBA, and it also predicts a better yield (0.11 g/g glucose) for a double deletion mutant. In case of BPCY the MOMA approach yielded the same productivity, although with different genes (RPE1 and an orphan reaction in mitochondria). However, an effective comparison of FBA and MOMA for multiple deletion mutants can only be done after experimental evaluation.To check whether the two approaches for evaluating flux distributions (namely FBA and MOMA) result in different predictions for multiple deletion mutants, we used OptGene to search for double and triple deletion mutants with improved succinic acid yield and BPCY. The double deletion strategy for obtaining maximum yield with MOMA includes deletion of Parameterization of stochastic optimization methods like evolutionary algorithms is recognized as a difficult task and for this particular problem only an empirical study of the effect of different parameters was conducted. The main purpose of this parameterization was to be able to obtain a global optimum within a reasonable computation time.Different sizes of the population were tested, and it was found that an increase beyond 125 individuals did not improve the results significantly. Furthermore, a mutation rate of 1/(genome size) was found to be optimal for both representations , thaThe differences on the representation of individuals in both approaches used in this work originate different requisites in terms of the formation of building blocks: as in the binary representation the order of the genes in the individuals follows closely that of the stoichiometric model (where genes are grouped according to the main pathways they integrate), only related genes can be a part of the building blocks. On the other hand, for the integer representation any subset of unrelated genes can form a building block. A natural conclusion of this observation is that the more the genes in the metabolic model follow a biological meaningful order, the more similar the binOptGene optimization approach is to a biological evolution of microorganisms under a given selective pressure.Additionally, we observed that if the limitation on the number of genes to be deleted in binOptGene is imposed by using penalty functions after evaluation of individuals, the number of invalid individuals in the population at a given generation is very large and consequently negatively affects the convergence.Nevertheless, in spite of the described differences, and although it is known that usually Genetic Algorithms do not perform very good for problems of the size found for the binary implementation, similar results were obtained for both approaches, after parameterization. In fact, for the majority of the runs, and with both representations, there was a clear convergence to an optimum Figure , and theIn case of succinate yield optimization, the optimality of the solution found by OptGene was verified using exhaustive search with up to 4 deletions. In case of BPCY, although the optimal solution for 3 deletions represented a global optimum, for a 4 deletion case OptGene found a sub-optimal solution. However, this solution was quite close to the global optimal solution . With 5 deletions the optimal solution found reached quite close to the maximum possible BPCY value. We hereby note that in cases where global optimality can not be directly verified, a good estimate for closeness to the global optimal solution can be found by using a curve similar to that presented in Supplementary Figure The computational cost of OptGene  was found to be 0.03 % of that found by using exhaustive search for 4 gene deletion case (succinate yield) and 0.33 % for succinate BPCY case. However, we did not observe any direct correlation between the number of deletions and the computational cost. Supplementary table 4 see  summarizSince the flux distribution obtained using FBA is not necessarily unique, the objective function value obtained in the fitness evaluation routine may not be unique as well. This is usually due to the possibility of other by-products being formed instead of the desired product  was found using OptGene by searching only 0.03% of the total solution space. For a higher number of deletions, the OptGene search space represents considerably lower fraction of the total solution space that increases exponentially. As a consequence of an exponential increase in the search space, a detailed study of the correlation between the OptGene search space and the total solution space was not feasible. Nevertheless, as discussed in the results section, it is possible to estimate the closeness to the global optimal solution by comparing the results with the plots as reported in Supplementary Figure S. cerevisiae include more than 1000 reactions. In case of simple minimal media that we used in our simulations, this set of 1000 reactions can be reduced to 240 reactions as described in the algorithm. This number can still be large for solving quadruple deletion problem using exhaustive search algorithms.We report a GA based framework termed OptGene for designing microbial strains a priori selection of candidate targets might lead to sub-optimal solution, and it is desirable to consider the whole model. Moreover, with the recent advances on the experimental front, it is feasible to construct mutants with many knockouts in real time. It should also be noted that we might often need to recalculate the results in case of changes/errors in the model, e.g. after including regulatory information or addition of a new reactions. Speed of calculations can be a key factor in such cases. OptGene can serve to provide a quick hint to whether a particular function of interest can be improved at all or up to what extent. The ability of OptGene to optimize for non-linear objective functions opens new opportunities for designing microbial strains with tailor-made metabolic phenotype, e.g. a strain with high BPCY of x and low yield of y.The metabolic engineering strategies reported in this work suggest that non-intuitive genetic modifications spanning several different pathways may be necessary for solving challenging metabolic engineering problems. Consequently in silico designing of microbial strains by using genome scale models.The GA formulation can provide us with multiple solutions, and thus an opportunity to choose from many good solutions. This is of interest as many of the predicted solutions might be difficult to realize due to complex biological regulation, which is difficult to account for in scoring function models. Moreover, the GA framework is very flexible and thus can easily be changed to use different scoring functions depending on the problem and system under investigation. In conclusion, OptGene represents a computationally efficient, flexible and natural tool for S. cerevisiae reported by Förster et al. [Genome scale reconstruction of r et al.  was usedr et al. ., while MOMA calculations were performed by using an Object oriented quadratic programming package [FBA simulations were performed using the GNU linear programming kit  package ..The genetic algorithm was implemented as a C++ program using the GAlib package As discussed in the main text, FBA (and other steady state models) can not simulate "rate" without specification of the specific substrate uptake rates (substrate uptake per unit biomass per unit time). Consequently the reported growth rates for the mutants should be more correctly interpreted as biomass yields.+ was balanced. In case where H+ is regarded as unbalanced  metabolite, maximum yield is 0.98 g/g glucose. This difference is very high and hence can result in big differences in the predictions reported. However, the choice is not trivial since the exact mechanism by which succinic acid is transported out of cell is unknown. Moreover, in case where H+ is not balanced, certain contradictions with the experimental observations were found under anaerobic conditions. For this reason we chose to use a conservative estimate for the maximum theoretical yield. We also note that the theoretical yields were calculated with the constraints for maintenance cost, and no CO2 uptake. Thus the reported yields are slightly lower than the stoichiometric yields (1.124 g/g glucose in case of succinate).The maximum theoretical yield of succinic acid reported in this study is calculated using FBA, whereas external HThe flux distributions, model reactions and other data related to this article can be obtained for non-profit research purposes by contacting the corresponding author (JN).KRP, IR and JN designed the research. KRP performed the simulations. KRP, IR, JF and JN analyzed the results. All authors read and approved the final manuscript.All Supplementary figures and tables.Click here for file


The integrated model is able to reproduce experimental observations on ATP, ADP, CrP, and inorganic phosphate levels in canine hearts over a range of workload and during coronary hypoperfusion and predicts that cytoplasmic inorganic phosphate level is a key regulator of the rate of mitochondrial respiration at workloads for which the rate of cardiac oxygen consumption is less than or equal to approximately 12 μmol per minute per gram of tissue. At work rates corresponding to oxygen consumption higher than 12 μmol min−1 g−1, model predictions deviate from the experimental data, indicating that at high work rates, additional regulatory mechanisms that are not currently incorporated into the model may be important. Nevertheless, the integrated model explains metabolite levels observed at low to moderate workloads and the changes in metabolite levels and tissue oxygenation observed during graded hypoperfusion. These findings suggest that the observed stability of energy metabolites emerges as a property of a properly constructed model of cardiac substrate transport and mitochondrial metabolism. In addition, the validated model provides quantitative predictions of changes in phosphate metabolites during cardiac ischemia.Observations on the relationship between cardiac work rate and the levels of energy metabolites adenosine triphosphate (ATP), adenosine diphosphate (ADP), and phosphocreatine (CrP) have not been satisfactorily explained by theoretical models of cardiac energy metabolism. Specifically, the in vivo stability of ATP, ADP, and CrP levels in response to changes in work and respiratory rate has eluded explanation. Here a previously developed model of mitochondrial oxidative phosphorylation, which was developed based on data obtained from isolated cardiac mitochondria, is integrated with a spatially distributed model of oxygen transport in the myocardium to analyze data obtained from several laboratories over the past two decades. The model includes the components of the respiratory chain, the F To function properly over a range of work rates, the heart must maintain its metabolic energy level within a range that is narrow relative to changes in the rate of energy utilization. Decades of observations have revealed that in cardiac muscle cells, the supply of adenosine triphosphate (ATP)—the primary currency of intracellular energy transfer—is controlled to maintain intracellular concentrations of ATP and related compounds within narrow ranges. Yet the development of a mechanistic understanding of this tight control has lagged behind experimental observation. This paper introduces a computational model that links ATP synthesis in a subcellular body called the mitochondrion with ATP utilization in the cytoplasm, and reveals that the primary control mechanism operating in the system is feedback of substrate concentrations for ATP synthesis. In other words, changes in the concentrations of the products generated by the utilization of ATP in the cell (adenosine diphosphate and inorganic phosphate) effect changes in the rate at which mitochondria utilize those products to resynthesize ATP. Over 30 years ago, Neely et al.  showed t0F1-ATPase, adenine nucleotide translocase, and the mitochondrial phosphate transporter. The mitochondrial model is integrated with a model of ATP, ADP, AMP, and CrP metabolism in the cytoplasm, including the reactions of adenylate kinase, creatine kinase, and ATP consumption. Oxygen transport is governed by advection of blood in capillaries, passive permeation between blood and interstitial fluid and between interstitial fluid and cardiomyocytes, and metabolic consumption at Complex IV of the respiratory chain. An empiric relationship between cardiac perfusion and ATP consumption rate is determined by comparing model simulations to data published by Katz et al. c increases from 0.02 mM at 0 workload to 0.36 mM at MVO2 = 6 μmol min−1 (g of tissue)−1 to 3.1 mM at MVO2 = 15 μmol min−1 (g of tissue)−1. Over the range for which the model best fits the observed data on ADP and CrP/ATP [MVO2 = 1.68 to 12 μmol min−1 (g of tissue)−1], the model predicts that Pi concentration increases from 0.02 mM to 1.6 mM. While it is difficult to absolutely quantify Pi concentration in vivo in the heart from 31P-NMR spectroscopy, it is possible to estimate changes in the amount of Pi relative to other phosphate metabolites. Zhang et al. [H+]. The application of this equation to experimental data on [ATP] and [CrP]/[Cr] is based on two assumptions: 1) that the measured levels of both ATP and CrP represent concentrations in the cytoplasm and 2) that the creatine kinase (CK) reaction is maintained in equilibrium. This standard assumption is invoked, for example, in the studies of Balaban and coworkers  and to increase to the millimolar range at high work rates [1.0 mM at MVO2 of 10 μmol min−1 (g of tissue)−1].The findings of the current study are consistent with the observations of Jeneson et al. ,25 that,2 greater than 7.8 μmol min−1 (g of tissue)−1 or approximately 200 μl O2 min−1 (g of tissue)−1. For lower values of oxygen consumption, the P/O ratio is less than 2, indicating that oxidative ATP synthesis is significantly less efficient at moderate work rates than at maximal work rates in the heart. This behavior is due to the fact that the predicted oxygen flux necessary to compensate for proton leak is nearly constant. Thus the relative contribution of proton leak to oxygen consumption increases with decreasing MVO2.The oxidative phosphorylation model predicts that the cardiac mitochondrial P/O ratio exceeds the value of 2 for MVO2 is approximately expressed as P/O = 2.5  is the total oxygen concentration in the blood (free plus hemoglobin-bound oxygen), G is blood flow to the tissue in volume per unit time per mass of tissue, ρ is the tissue density in mass per unit volume, L is the length of the capillary, V1 is the volume of the capillary region, α1 is the oxygen solubility coefficient in blood, PS12 is the permeability of the surface area of capillary wall, and P1 and P2 are the oxygen partial pressures in blood and interstitial fluid, respectively. The capillary oxygen concentration CO,1 is a function of the distance along the capillary, x, and time, t.Oxygen transport in the capillary is governed by advection of the blood, oxygen–hemoglogin binding and unbinding, and passive permeation between the blood and the interstitial space. The governing equation for oxygen in the capillary isHct is the hematocrit, CHb is the concentration of oxygen binding sites in red blood cells, and SHb is the hemoglobin saturation in red blood cells. The hemoglobin saturation curve is assumed to be governed by a Hill equation,P50,Hb is half the partial pressure of O2 saturation in hemoglobin and Hn is the Hill exponent.The total oxygen concentration of the capillary region is related to the partial pressure byCO,2 is the oxygen concentration in interstitial region, V2 is the volume of the interstitial region, PS23 is the permeability surface-area product for passive permeation between interstitium region and the parenchymal cell region, and P3 is the partial pressure of the oxygen in the cell. Oxygen concentration and partial pressure in the interstitial fluid are related by CO,2 = α2P2, where α2 is the oxygen solubility coefficient in the interstitial region.Oxygen transport in the interstitial region is governed by passive permeation of oxygen between the blood and the interstitial fluid and between the interstitial fluid and the myocyteCO,3 is the oxygen concentration in the cellular region, V3 is the volume of the cellular region, RMitoCell is the ratio of mitochondrial volume to total cell volume, and JC4 is the flux through Complex IV in the respiratory chain in units of mass per time per unit mitochondrial volume. The factor of ½ in the second term on the RHS of JC4 in the mitochondrial model (described below) represents the flux of electron pairs through the respiratory chain: one O2 molecule is consumed for each two pairs of electrons transferred through the system. In the cellular region, oxygen transport is governed by passive permeation and consumption of oxygen in mitochondriaCMb is the myoglobin concentration in the cell and α3 is the oxygen solubility in the myocyte. The equilibrium myoglobin saturation SMb is expressedP50,Mb is the half-saturation partial pressure for oxygen–myoglobin binding.Total oxygen in the cell is the sum of free oxygen plus myoglobin-bound oxygenThe differential equations for species other than oxygen areIn the above set of equations, the subscripts x, i, and c denote mitochondrial matrix, intermembrane (IM) space, and cytoplasm, respectively. All of the variables in this set of equations are defined in tot, Qtot, cytCtot, and Atot are the total concentrations of NAD(H), ubiquinol, cytochrome c, and adenine nucleotide in the matrix, respectively, and CRtot is the total creatine plus creatine phosphate concentration in the cytoplasm.In addition to the state variables treated in Parameters that appear in the above equations are described in detail below. The fluxes that appear on the right-hand side of the governing equations are tabulated in Flux expressions for the mitochondrial model (based on a previously developed model ) are lisDehydrogenase flux:GH = FΔΨ + RT1n([H+]c/H+]x).Complex I flux:Complex III flux:2] = α3P3 is the free oxygen concentration in the cell.Complex IV flux:0F1-ATPase flux:Fx, [fADP]x, [fATP]i, and [fADP]i denote magnesium unbound ATP in the matrix, ADP in the matrix, ATP in the IM space, and ADP in the IM space, respectively.Magnesium binding fluxes:Substrate transport fluxes:Adenine nucleotide translocase (ANT) flux:2i = [H+]i[Pi]i/([H+]i + kdH) and [H2x = [H+]x[Pi]x/([H+]x + kdH).The phosphate–hydrogen cotransporter flux:Mitochondrial adenylate kinase flux:Proton leak flux:2] and [Q] reach 0.Potassium–hydrogen ion exchange:ented in , with thFour biochemical processes are modeled in the cytoplasm—the adenylate kinase reaction, the creatine kinase reaction, ATP hydrolysis, and binding of magnesium ions to ADP and ATP.c and [fADP]c denote magnesium unbound ATP and ADP in the cytoplasm. Similarly, the cytoplasmic adenylate kinase is analogous to the mitochondrial reaction.The binding of magnesium to ATP and ADP in the cytoplasm takes the same form as the binding fluxes in the mitochondria:KAK is the equilibrium constant for the reaction 2 ADP ATP + AMP, and XAK is the enzyme activity, which is set to a large enough value so that the reaction is effectively maintained in equilibrium.In XCK is set to a large enough value so that the equilibrium KCK = ([ATP]c[Cr]c/[ADP]c[CrP]c[H+]c)eq is maintained during simulations.The creatine kinase flux is modeled using the expression:JAtC is defined as the flux through the reaction ATP → ADP + Pi. Mathematical models for the ATP consumption flux are considered in the Results section.The flux TPP. The total exchangeable phosphate pool is computed asVcyto = 0.472 ml (g of tissue)−1, Vmito = 0.200 ml (g of tissue)−1, and Vcell = 0.694 ml (g of tissue)−1 are the volume densities of cytoplasm, mitochondria, and total cell space in cardiac tissue [TPP = 15 mM is chosen as the value most consistent with the experimental observations.With the exception of one adjustable parameter, all parameters in the model are fixed at values justified by previous studies. The adjustable parameter is the total pool of exchangeable phosphate in the cell, which is a constant denoted by c tissue . By compThe parameter values listed in PS product. Similarly, the structure/volume parameters are available from experimental estimations published in the literature. The effective capillary length is set to the mean capillary path length for left ventricular tissue reported by Kassab and Fung [The oxygen transport parameters, including solubilities, permeabilities, and oxyhemoglobin and oxymyoglobin binding parameters, are obtained from the literature, as indicated in the table. The effective permeability–surface area product for the capillary wall is obtained by converting the mass transfer coefficient reported in  and usedand Fung . The majand Fung . Values and Fung .0F1-ATPase reaction is maintained near chemical equilibrium in the model parameterization that was published in [0F1-ATPase activity XF1 is set to the arbitrarily high value of 1,000 mol s−1 M−1 (l mito)−1. In addition, the updates to the functional forms for the Complex I and III fluxes require estimation of values for the activities of XC1 and XC3. To account for these changes, the full set of mitochondrial model parameters has been refined by repeating the fits to data from isolated mitochondrial function of Beard [Oxidative phosphorylation model parameters have been adjusted to account for modifications from , as indiished in , here thof Beard . The agrof Beard  and the of Beard .TPP), which is estimated in the Results section. Binding constants are obtained from the literature; enzyme activities for reactions maintained near equilibrium are set to arbitrarily high values.All values for concentrations of pooled metabolites are set according to values reported in previous studies, with the exception of the total pool of exchangeable phosphate . Obtaining a single steady-state solution requires approximately 30 min of computing time on a single processor. A total of 150 steady-state solutions are computed to construct