New analysis methods are being developed to integrate data from transcriptome, proteome, interactome, metabolome, and other investigative approaches. At the same time, existing methods are being modified to serve the objectives of systems biology and permit the interpretation of the huge datasets currently being generated by high-throughput methods.Transcriptomic and metabolic data from chemostat fermentors were collected with the aim of investigating the relationship between these two data sets. The variation in transcriptome data in response to three physiological or genetic perturbations  was investigated using linear modelling, and open reading-frames (ORFs) whose expression changed significantly in response to these perturbations were identified. Assuming that the metabolic profile is a function of the transcriptome profile, expression levels of the different ORFs were used to model the metabolic variables via Partial Least Squares (Projection to Latent Structures – PLS) using PLS toolbox in Matlab.The experimental design allowed the analyses to discriminate between the effects which the growth medium, dilution rate, and the deletion of specific genes had on the transcriptome and metabolite profiles. Metabolite data were modelled as a function of the transcriptome to determine their congruence. The genes that are involved in central carbon metabolism of yeast cells were found to be the ORFs with the most significant contribution to the model. Saccharomyces cerevisiae were reviewed recently [After the completion of the genomic sequencing of organisms, integrative post-genomic studies and the systems biology approach have emerged with the aim of developing a more complete understanding of cell physiology. Attempts at data integration for the model organism, recently . ExperimIt is possible to design systems biology experiments in a hypothesis-driven manner, such that the designed perturbations provide the information of interest. Alternatively, question-driven discoveries may be made by observing the effects of an intuitively chosen modification and making use of the extracted information to generate new ideas and hypotheses .S. cerevisiae growing in chemostats on a glucose medium under carbon, nitrogen, phosphorus or sulphur limitation allowed detection of the genes that were affected by the different nutrient limitations [Transcriptome data from itations . The genitations . Responsitations .Perturbations can also be introduced by genetic, rather than physiological, means – e.g. by gene deletions. Yeast cells carrying gene deletions have been investigated for various purposes: (a) functional analysis based on discrimination of mutants via metabolic fingerprints  or footpHAP4 over-expression [hap4Δ deletion mutants [hap4Δ deletion mutants were reported to be respiratory deficient [HAP4 causes down-regulation of respiration-related genes. In contrast, such genes were expressed at higher levels in HAP4-overexpressing strains growing under aerobic conditions. Moreover, an increase in yeast's respiratory capacity was observed due to over-expression of HAP4 [Hap4p was reported to have a function in the regulation of respiration-related genes on the basis of transcriptome data collected during batch growth of yeast cells on glucose, followed by diauxic shift from the fermentation of glucose to the respiratory metabolism of ethanol . The actpression  and the  mutants  have alseficient  and dele of HAP4 .hap4 and ho) were the perturbations studied. Transcriptome profiles, biomass, glucose and ethanol concentrations of samples from chemostats operating under steady-state conditions were analysed to show the applicability of the Partial Least Squares (Projection to Latent Structures – PLS) method in the integration of transcriptome and metabolite data.In the present study, three types of perturbations that were expected to have an impact on yeast central metabolism, were investigated in chemostat cultures. Changes in growth medium (C- and N-limitations), growth rate (dilution rates) and gene deletions  expression levels of 6361 ORFs were modelled and p-values were calculated to decide on the significance of the effects of the factors on the expression of the ORFs. For most of the ORFs, the constructed models did not predict a variation more significant than the expected level of random error, thus these ORFs were not included in further analyses. A p-value of 0.05 was used as the threshold, in order to include all ORFs that were affected significantly by the three factors considered in this study.324 out of 6361 models estimated that at least one of the factors was affecting the expression of the modelled ORF. The growth medium is the factor with most effect on the expression of most of the ORFs (62.1%), followed by dilution rate (26.2%); while gene deletion is the most effective factor for only 11.7% of the ORFs.hap4Δ/hap4Δ deletion mutant, the production rates of both biomass and ethanol are higher under ammonium limitation than under glucose limitation. In contrast, for the standard strain (hoΔ/hoΔ), only the rate of ethanol production is significantly higher under ammonium limitation as compared to glucose limitation. For both strains, the rate of glucose consumption is elevated under ammonium limitation.The biomass production rates obtained in chemostats operating at steady state are presented in Figure PLS is unable to discriminate between the effects of the different perturbations on the transcriptome; this is because it fails to filter out those transcripts that show significant changes from the vast majority of transcripts that do not change at all (results not shown). Hence, in order to model the metabolic variables as a function of the transcriptome data, and to identify the ORFs that mediate the effects of the perturbations, the partial least squares (PLS) method was applied to the transcriptome of 324 ORFs (X) and metabolic data (Y). Dimensions of the matrices X and Y were 8 × 324 and 8 × 3, respectively, and both matrices were mean-centred and scaled to variance 1 prior to PLS regression. In Table i.e., in each step, one of the samples was not included in the model and metabolic profile of the left-out sample was predicted using the model constructed, then the prediction error of that sample was calculated, and the procedure was repeated until all samples had been left out once. The model improves for all three response variables  as more latent variables are included  are plotted to evaluate the prediction power and limitations of the constructed model Fig. . A 'leavded Fig. . The onlIn order to see whether the distribution of the samples is linear according to Eq. 5, the scores for the transcriptome (t) and the metabolic data (u) on each LV were plotted against each other Fig. . The dishoΔ/hoΔ samples are positive on LV2, while scores of transcriptome data from hap4Δ/hap4Δ samples are negative on LV2. Similar discrimination applies for scores of metabolome on LV1  and metabolic data (u) on the first four LVs are plotted in Figure  LV1 Fig ; howeverThe variation generated by the change in dilution rate was represented by LV3 in transcriptome data . For instance, the projection of samples onto LV1 represents the change that was generated in the sample by ammonium limitation when compared to glucose limitation. The direction of each new variable (LV) in the space is a linear combination of the original variables, i.e. ORFs and metabolites. The direction of an LV is dominated by the variables that respond more than the others and the direction of their response. Thus, an LV can be interpreted as a new composite variable that is the only affected feature in the cell when a certain perturbation is applied. As an exception, for the dilution rate change, two latent variables are needed in order to discriminate the metabolic samples from two different dilution rates.hap4Δ/hap4Δ deletion, which is indeed the case  on LVs are listed in Table The ORFs with significant positive or negative loadings  are involved in hexose transport, these ORFs are up-regulated under ammonium limitation in comparison to carbon limitation. Up-regulation of the hexose transport pathway may be the first step of the mechanism to increase biomass production, ethanol production and glucose consumption rates under nitrogen limitation. The ORFs that are down-regulated under ammonium limitation as compared to glucose limitation are the genes that are active in oxidative phosphorylation, generation of precursor metabolites and energy (LV1-). The high glucose concentration in the ammonium-limited culture should repress the expression of genes acting on the respiratory pathways (oxidative phosphorylation). Repression of respiration, in turn, would cause the fermentation pathway to be activated and ethanol production to be enhanced.hap4Δ/hap4Δ deletion when compared to standart strain hoΔ/hoΔ mainly have roles in respiration and phosphate metabolism (LV2+). Consequently, the hap4Δ/hap4Δ deletion causes respiratory deficiency under the conditions studied, and fermentation is the only route for glucose metabolism. The higher glucose consumption and ethanol production rates achieved provide further confirmation that Hap4p plays a major role in the switch mechanism from respiration to fermentation. The genes that were up-regulated in response to the hap4Δ/hap4Δ deletion (LV2-) were involved in regulation of carbohydrate biosynthesis, indicating the propensity of the cells to convert excess carbon into storage molecules if the carbon source cannot be respired. While this theory explains the results from glucose-limited case perfectly, the effect of hap4Δ/hap4Δ deletion is not apparent in ammonia-limited conditions as the high glucose levels in the N-limited medium repress respiration, quite independently from the respiratory deficiency caused by the hap4Δ/hap4Δ deletion. Thus the metabolic variables behave similarly in hoΔ/hoΔ and hap4Δ/hap4Δ mutants growing under ammonium limitation, and the insignificant variation in these variables cannot be estimated by the model, as discussed previously.The genes that were down-regulated due to the -2); however, up-regulation of these mechanisms at the lower growth rate is an interesting phenomenon that remains to be explained.The ORFs up-regulated at the higher dilution rate (LV3+) are the genes that act in ribonucleotide metabolism. Up-regulation of these ORFs mediates the increase in biomass production rate, and the consequent increases in ethanol production and glucose consumption rates. The GO terms common among the ORFs down-regulated at the lower dilution rate (LV3-) are related to reproduction mechanisms. The significance of these terms is quite low (p ~10-12). The unknown ORFs that appear in the same group as these genes may also be members of the functional categories denoted by the over-represented GO terms.The number of genes given in groups LV1- and LV2+ Table  that areThe use of formal experimental design allowed the analyses that we performed to discriminate between the effects that the growth medium, dilution rate, and the deletion of specific genes had on the transcriptome and metabolite profiles. The PLS method was applied to metabolic and transcriptomic data to gain insight into the changes in metabolism due to three factors . The method enabled extraction of the following information from these data sets:1. Discrimination of the effects of the above factors on transcriptome and metabolic data.2. Modelling of metabolic data as a function of transcriptome data and elucidation of the extent of congruence between these two data sets.3. Identification of ORFs that mediate the changes in metabolic data in response to perturbations.In cases where the number of variables in the metabolic data is much higher, the PLS method will help in the identification of metabolites that are affected by the conditions applied and the genes that mediate the effects of the conditions. The unknown genes can be annotated using this methodology and studies towards product maximization can be conducted by identifying the genes and pathways that are responsible for the changes in formation of metabolic products.S. cerevisiae with genomic background BY4743 (MATa/MATα his3Δ1/his3Δ1 leu2Δ0/leu2Δ0 lys2Δ0/LYS2 MET15/met15Δ0 ura3Δ0/ura3Δ0) from the Yeast Genome Deletion Project library [hoΔ/hoΔ deletant is commonly used as a standard strain in control experiments since the deletion has no measurable impact on either flux  or the th rate, . The abs2PO4 (2 g/l), MgSO4·7H20 (0.55 g/l), NaCl (0.1 g/l), CaCl2·2H2O (0.09 g/l), Uracil (0.02 g/l), L-Histidine (0.02 g/l), L-Leucine (0.1 g/l), ZnSO4·7H2O (0.7 × 10-4 g/l), CuSO4·5H2O (0.1 × 10-4 g/l), H3BO3 (0.1 × 10-4 g/l), KI (0.1 × 10-4 g/l), FeCl3·6H2O (0.5 × 10-4 g/l), inositol (0.12 g/l), thiamine/HCl (0.014 g/l), pyridoxine (0.004 g/l), Ca-pantothenate (0.004 g/l), biotin (0.0003 g/l).Mineral media, supplemented with trace elements and vitamins were used . The com4)2SO4 and 2.5 g/l glucose were added to the medium described above. For ammonium-limited medium 0.46 g/l (NH4)2SO4 and 20 g/l glucose were added to medium described above.For glucose-limited medium, 3.13 g/l  were grown overnight in G418-containing YPD medium and used to inoculate the fermentors. The medium was fed at a constant flow rate. Temperature and pH of chemostats with 1L working volume were kept constant at 30°C and 4.5 respectively, and the oxygen content was maintained at saturation.hoΔ/hoΔ and hap4Δ/hap4Δ deletion strains were grown both in glucose-limited and ammonium-limited media in separate experiments. The experiments were started at a dilution rate of 0.1 h-1 and, after samples had been collected, the dilution rate was shifted to 0.2 h-1. The samples were collected at steady state after three residence times, and total RNA extraction was carried out. Yeast Genome S98 arrays were used for hybridizations as described by the manufacturer . SupernSamples (5 ml) were centrifuged in pre-weighed tubes, dried at 80°C overnight and re-weighed to determine the dry weight of biomass.k experiments are needed to cover all possible combinations. The 23 factorial design used in this work is given in Table Factorial experimental design is used to reveal the effects of various factors on the output of a system. For an experiment set with "a" levels of "k" factors, ahoΔ/hoΔ and hap4Δ/hap4Δ mutants deletion strains are "hoΔ" and "hap4Δ", respectively. The "G" and "N" are the abbreviations for the glucose and ammonium-limited cases, while "1" and "2" are used for dilution rates 0.1 h-1 and 0.2 h-1, respectively. In the Figures, "Δ" is omitted and the abbreviations ho and hap4 represent the deletion mutants.The abbreviations used for the homozygous The linear model for a factorial experiment with three factors is as follows :yijk = μ + τi + βj + γk + εijk     (1)2 basis.where i, j, k: indices of the levels of the factors; i = 1, 2, ..., a; j = 1, 2, ..., b; k = 1, 2, ..., c; μ: mean of the outputs; y: simulated value of the output variable; τ, β, γ: effects of the factors; ε: random error. In the present experimental design, each factor has two levels; thus, a = b = c = 2. The output variable y represents the expression level of an ORF on a log The linear model in Eq.(1) was used to estimate the coefficients to describe the expression level of each ORF. The coefficients obtained in each case are the "effects" of the factors on the expression of the ORF modelled. A positive effect made by a factor indicates that the ORF is up-regulated at Level (+) when compared to Level (-) of that factor. Similarly, a negative effect indicates down-regulation of the ORF at Level (+) when compared to Level (-).P-values of the factors were calculated using the ratio of variation sum of squares to error sum of squares in order to indicate the significance of the correlation between the gene expression and the factor.In industrial processes, large sets of process data are collected by computerized control and monitoring systems. Multivariate data analysis methods have emerged to compensate for the need for data reduction towards understanding the nature of the process and fault diagnosis. Partial least squares (projection to latent structures – PLS) is a statistical method that was proposed for process analysis, monitoring and diagnosis -28. LateIn PLS methodology, the independent "cause" matrix X and the dependent "response" matrix Y are regressed and modelled simultaneously. The columns of these matrices represent the variables  and rows represent the samples. The linear model is:Y = XB + E     (2)where B is the regression vector and E is the residual matrix.Projection of the original data set X to a new space with reduced dimensions is made by the loading matrix (p) and the observations are represented by the score matrix (t) in the new space. Decomposition of the data matrix X into the score matrix (t), the loading matrix (p) and the residual matrix (e) is as follows:t + e     (3)X = tpt" denotes the transpose of the matrix p. Columns of p and t matrices correspond to the latent variables (LVs), which lie in the direction of the maximum variation that remains in the data after removal of the variation explained by the previous LV. The residual matrix (e) represents the variation that remains unrepresented in the t and p matrices. Similarly, the response matrix Y is decomposed as:where the superscript "t + f     (4)Y = uqThe score vectors (vectors of u) and the loading vectors (vectors of q) correspond to LVs. The residuals are given by the f matrix. A linear inner relation also exists between the matrices t and u, where ū denotes the matrix of estimated values of u:ū = bt     (5)The optimal number of LVs to be included in the model depends on the amount of variation explained by the LVs which are in descending order of the variation they explain. One criterion for the selection of an optimal number of LVs is to set a threshold value for the variation. Then, a sufficient number of LVs is included in the model to represent the threshold variation, and the rest of the variation remains in the residual matrix. Cross-validation is another criterion where the analysis is performed with a subset of the data and the rest of the data set is used to determine the prediction power of the model. Then, the number of LVs that results in minimum prediction error sum of squares (PRESS) is selected.The biological problem was conceived by BK and SGO; the experiments were executed by PP and AH. The PLS approach was suggested by KOU who assisted PP in the analyses. The manuscript was written by PP, BK, KOU and ZIO, and was revised by SGO. All authors read and approved the final version.


We use a biogeochemical model of the terrestrial biosphere to differentiate the effects of temperature and precipitation on net primary production (NPP) and heterotrophic respiration (Rh) during the two largest anomalies in atmospheric CO2 increase during the last 25 years. One of these, the smallest atmospheric year-to-year increase (largest land carbon uptake) in that period, was caused by global cooling in 1992/93 after the Pinatubo volcanic eruption. The other, the largest atmospheric increase on record (largest land carbon release), was caused by the strong El Niño event of 1997/98.The amount of carbon dioxide in the atmosphere steadily increases as a consequence of anthropogenic emissions but with large interannual variability caused by the terrestrial biosphere. These variations in the CO2 flux.We find that the LPJ model correctly simulates the magnitude of terrestrial modulation of atmospheric carbon anomalies for these two extreme disturbances. The response of soil respiration to changes in temperature and precipitation explains most of the modelled anomalous COObserved and modelled NEE anomalies are in good agreement, therefore we suggest that the temporal variability of heterotrophic respiration produced by our model is reasonably realistic. We therefore conclude that during the last 25 years the two largest disturbances of the global carbon cycle were strongly controlled by soil processes rather then the response of vegetation to these large-scale climatic events. Atmosph to year . Additio regions . In comp regions , defores regions .Uncertainty remains, however, regarding the relative influence of the driving climatic anomalies  on the most prominent terrestrial carbon processes, namely vegetation growth (NPP) and soil decomposition (Rh). Numerical models of the land carbon cycle allow investigations of these relationships.2 growth rate during the last 25 years are related to two large climatic disturbances – the increased planetary albedo after the eruption of Mount Pinatubo in 1991 and the strong El Niño event of 1997/98. The Pinatubo eruption was an extraordinary event because of the large amount of aerosols that were injected into the lower stratosphere where they were distributed around the globe, leading to a world-wide cooling of about 0.5°C [The two largest anomalies of atmospheric CO2 growth rate variability is controlled by varying soil organic matter decomposition rather than changing plant productivity. This sheds additional light on previous studies that highlighted connections between the interannual variability of atmospheric CO2 growth and net primary productivity [We use the LPJ model of terrestrial carbon and water cycles ,10 to exuctivity -13.2 variability controlled by terrestrial ecosystem physiology we used state-of-the-art estimates of carbon flux anomalies from oceans, land-use change, anthropogenic emissions, and fires and reduced the measured atmospheric CO2 growth rate anomalies accordingly ..2 increaThe physiological response of the land biosphere during the strong 1997/98 El Niño event was diametrically opposite to that of the 1992/93 period. A large anomalous flux of 2.82 GtC into the atmosphere was measured in 1998 of which 1.62 GtC can be assigned to vegetation growth and soil decomposition processes of the terrestrial biosphere  were removed in the simulations, i.e. monthly values of temperature and precipitation were replaced by their corresponding 1979–2003 averages. These simulations help to clarify how relevant the changes in temperature and precipitation were for the carbon budgets of of vegetation and soils. We find that if the global cooling after the Pinatubo eruption is removed (NEE T const.) the terrestrial carbon sink declines together with Rh, and higher values of NPP are achieved  using an interpolated climatology of monthly values of temperature, precipitation and radiation ,29. A laThe author(s) declare that they have no competing interests.Tim Erbrecht carried out the LPJ simulations and participated in the analysis and interpretation of modeling results, and wrote the manuscript. Wolfgang Lucht participated in data analysis and interpretation.


Hence, only single parameter sets can be considered at a time. Consequently, it is impossible to conclude that the "best" solution was identified or that no good solution exists, because parameter spaces typically cannot be explored in a reasonable amount of time.Current approaches to parameter estimation are often inappropriate or inconvenient for the modelling of complex biological systems. For systems described by nonlinear equations, the conventional approach is to first numerically integrate the model, and then, in a second We introduce a novel approach based on semidefinite programming to directly identify consistent steady state concentrations for systems consisting of mass action kinetics, i.e., polynomial equations and inequality constraints. The duality properties of semidefinite programming allow to rigorously certify infeasibility for whole regions of parameter space, thus enabling the simultaneous multi-dimensional analysis of entire parameter sets.Our algorithm reduces the computational effort of parameter estimation by several orders of magnitude, as illustrated through conceptual sample problems. Of particular relevance for systems biology, the approach can discriminate between structurally different candidate models by proving inconsistency with the available data. Systems biology is a framework integrating diverse disciplines such as molecular biology and genetics with mathematical modelling and simulation, where computational models assume an increasingly important role in recent years. Simulations are an essential element for quantitative understanding because the highly nonlinear behavior of complex biological systems can sometimes be counterintuitive ,2. Systelinlog kinetics, respectively [a priori neglects the dynamics of enzyme complexes. Thus, in many cases the use of mass-action kinetics in the form of elementary reactions becomes necessary, for instance when some of the enzyme kinetics do not follow Michaelis-Menten [Most approaches to mathematical modelling of biological systems either avoid parametrization by focusing on static steady-state models of reaction stoichiometry or topology ,5, approectively ,9. Whileectively , the unds-Menten , a steads-Menten , transpos-Menten  or the ns-Menten ,15.wrapping algorithm, is of particular importance [The systems of equations that arise in time-dependent models based on chemo-physical kinetics of any kind are usually hard to parameterize, since their components are tightly coupled and there is limited information about the time evolution of the concentrations of all the species. Several approaches for efficient system identification have been developed, ranging from reduction of system dimensionality  to decouportance  and was portance . All of portance  and furtsemidefinite programming (SDP) [a posteriori as in most current methods.We present here a conceptually novel approach based on techniques from numerical convex optimization, in particular ng (SDP) , that casets of parameters. Our algorithm therefore provides rigorous proofs for the classification of the parameter space into feasible and infeasible regions, thus reducing the number of points needed for consideration by several orders of magnitude.Previous applications used SDP either for rational experimental design based on covariance analysis  or for tThe natural starting point in the analysis of dynamical systems in biology is the determination of a steady state equilibrium of the model, since it represents the reference point for any kind of perturbation introduced to the system. It is therefore of utmost importance that the species' concentrations at an equilibrium point, which is henceforth synonymously referred to as steady state, are realistic and validated against as much data as possible.y and the kinetic constants k. At an equilibrium point, the corresponding system of n ODEs reduces to a set of polynomial equationsIn the present study we consider models in the form of mass action kinetics, hence all reaction rates are direct functions of the concentrations fi  = 0,     i = 1, ..., n.     (1)inequality constraints also arise naturally. Consider for instance the general mass balance equations, here described by a matrix M, which when multiplied with the array of state variables y satisfyBesides the equations above, the state concentrations also have to satisfy various kinds of constraints based on experimental data, e.g., formation rates or overall mass balances. Since these will in practice inevitably include certain errors, M·y = b,     M ∈ ℝm × n,     (2)b ∈ ℝm is the total amount of each species. Hence, Eqn. (2) represents overall mass balances while Eqn. (1) comprises the mass action kinetics, i.e., the formation and consumption rates of each single species. Since components can exist both unbound or bound in a complex, respectively, there are in general fewer mass balance equations than differential equations (n ≥ m). An experimental error ε can be taken into account by relaxing the exact mass balance equations to inequalities wherewhere ε)·b ≤ M·y ≤ (1 + ε)·b.     (3) is an arbitrary objective function, e.g., the overall model error, the equality constraints fi  represent the steady state condition and g  is a set of inequality constraints, e.g., the mass balances.that depends on the steady state concentrations fi are quadratic and the gi are linear, although our methods extend to the fully polynomial case; see [In the following, we will introduce a reformulation of the generalized parameter estimation problem (4) based on semidefinite programming (SDP). SDP is a specific kind of convex optimization problem ,23,24, wase; see  for detax, X in terms of the original state variables y by:For the SDP relaxation we define new variables Based on these definitions, the original problem (4), including both steady state equations and data consistency inequalities, can be rewritten as:Qobj defines an objective function , that selects a specific solution out of the possibly many equilibria. The symmetric matrices Qi correspond to the set of ODEs at steady state (Eqn. (1)) and the matrix L to linear inequality constraints derived, for instance, from approximate mass-balance equations (Eqn. (3)). Note, that in general the matrices Qi are a function of the set of  rate parameters kj, while L, which represents the mass balances, is not. The basic convex relaxation described in the Appendix then yields the following SDP relaxation of (6):Here, the symmetric matrix X must be positive semidefinite, i.e., all its eigenvalues should be greater than or equal to zero. The vector e1 is an all-zero vector, except for its first entry, which equals one to enforce X11 = 1. The set of feasible solutions, i.e., the set of matrices X that satisfy the constraints, is always a convex set. Recall that the matrix X as defined in Eqn. (5) is by construction a rank one matrix. This rank condition, however, is not guaranteed for the X obtained from the optimization, and thus this property has to be checked independently after a solution to the SDP is computed. This is necessary because the relaxed problem formulation (7) is less strict then the original form (6), hence the set of feasible solutions becomes larger. Note, that introduction of additional nonlinear constraints helps to reduce the set of " false positive" solutions, an aspect discussed in more detail in an additional section below. Finally, in the particular case of Qobj = 0, the problem reduces to whether or not the inequality can be satisfied for some matrix X. In this case, the SDP is referred to as a feasibility problem, where we are interested in proving the mere existence of solutions rather than finding any particular one.where Tr is the trace operator, which adds up the diagonal elements, and the inequality in the last line indicates that the matrix duality theory . For every SDP of the form (7) , there is another associated SDP, called the dual problem, which can be derived via Lagrangian duality:The convexity of SDP has made it possible to develop sophisticated and reliable analytical and numerical methods to solve them . A very r ≥ 0, S ≥ 0, Sii = 0. The dual variables γ, λ, r, S are Lagrange multipliers associated to the different constraints in the primal problem. In the case of feasibility problems , the dual problem can be used to certify the nonexistence of solutions of the primal. This property will be crucial in our developments.where the constraint represents a matrix inequality with Identifying a satisfactory equilibrium point of a set of ODEs requires the solution of a system of algebraic equations (those that define the steady-state conditions) subject to additional equality or inequality constraints . This can be time-demanding, because the system must often be simulated from a given set of initial conditions until it settles to an equilibrium ,26. Thisconvex relaxation of the generalized parameter estimation problem at steady state (4). Our techniques apply whenever the model presentation is in polynomial form, since in this case the resulting system can be relaxed into a semidefinite programming problem (6), which in turn can be solved using efficient interior-point methods [As opposed to these "indirect" techniques, our method is a conceptually novel direct approach based on a  methods -30.We illustrate the application of our approach with the following nonlinear system given byA and B form a complex, A • B, that transforms into a modified form, A • B*, and finally decays into its building blocks. This system yields a set of four ODEs that at steady state reduces to a polynomial system:Here, two components A and B one third of them is unbound while the rest is associated in one of both complexes, we can write mass balances asIf it is furthermore known from experimental data that in equimolar concentrations of ε = 20%.which have to be kept within a certain experimental accuracy The example above is thus a system of nonlinear polynomial relations based on mass action kinetics that comprises both equalities (Eqns. (10)) and inequalities (Eqns. (11)). As explained earlier, we define the state variablesyT = ,     (12)X according to Eqn. (5). To write the SDP problem, the equality and inequality constraints, Eqns. (10) and Eqns. (11), respectively, have to be rewritten into the form of Eqn. (6). For example, the steady state equation for species B becomeswhich are then used to produce a new matrix variable L corresponding to the mass balance inequalities for component A arewhile the rows of k. In contrast, the SDP-based approach bypasses this time consuming step: if the problem (7) is infeasible the parameters k are inconsistent with the experimental data, and if a feasible solution of rank one can be found, the given parameters are valid.The feasible set of the relaxed SDP problem (7) is always convex and includes all the equilibria of the original problem. Thus, SDP optimization enables the possibility of directly solving the underlying nonlinear system of algebraic equations, or of proving its inconsistency. Furthermore, this property makes possible a direct consistency check based on feasibility or infeasibility of the SDP optimization. Typically, a parameterization approach would require identification of a stable steady state, either by root-finding or numerical integration, at which point the simulation can be verified or falsified for a given set of parameters k1 = 3, k2 = 1 and k3 = 1, respectively, for which the SDP is feasible. Let the solution beFor our running example, consider a set of parameters, X to scale the corresponding eigenvector yieldswhich is rank one. Using the largest eigenvalue of A simple check shows that X in Eqn. (5) is by construction a rank one matrix. This property of X, however, is not necessarily guaranteed when solving the SDP relaxation (7) (including this constraint would destroy convexity). Hence, it has to be verified independently once the optimization is completed. From our numerical experience, a very limited number of solutions fails the rank one condition such that the corresponding results have to be discarded. A simple but effective combination of constraints, however, allows to reduce the number of these "false positive" solutions: since most of the inequality constraints are linear , they can be used to tighten the description of the feasible set by using redundant constraints. By exhaustive multiplication of m pairs of linear inequality constraints, i.e. each upper bound is multiplied with every possible lower bound, L·X·LT appearing in (7). Note, that even the product of a lower and an upper bound of the same constraint can be helpful, because the thus generated matrix has nonzero diagonal  elements. These additional nonlinear algebraic constraints improve the performance of the algorithm significantly, because the set of feasible solutions that do not meet the rank one condition becomes much smaller.The decision variable single sets of parameters, that have to be provided by an external parameter estimation algorithm. Since in general the parameter space can be quite large, it would be very helpful to be able to discard a priori large regions of the space, where we know for sure that no consistent rates can be found. As we will see, we can achieve this goal in a very efficient way by considering the dual optimization problem (8) presented earlier.The approach presented in the previous paragraphs can directly handle nonlinear algebraic equations in polynomial form. However, it considers only As in general convex optimization, infeasibility of the primal problem can be proven by the existence of a dual feasible solution, and conversely, dual infeasibility follows from a primal solution. Thus, the existence of dual variables satisfying the dual constraints in (8) directly proves primal infeasibility. This criterion can be used to yield an efficient procedure for the analysis of the parameter space. Therein, large regions can be detected where the given model is inconsistent with the experimental data. The feasible region will usually represent a small fraction in the overall parameter space. Therefore an efficient search of the parameter space will focus rather on negative (inconsistent) than positive (consistent) regions.Qi, i.e., the mass action kinetics, are actually dependent on the rate parameters k. We thus used a bisection approach in which the current parameter set is an n-dimensional box. Since this is a polytope, it follows that if the conditions are valid on the corners then they are also valid on the interior, and thus can be neglected for further analysis. In particular, we determine a factor η by which each parameter can be perturbed such that dual feasibility holds. We remark that the choice of cuboid-like regions is only a matter of convenience, the results can be easily extended in a direct fashion to any other polyhedral partition scheme .The use of SDP-based parametrizations allows to obtain exactly this information. Once dual feasibility has been proven for a given set of parameters, this point in the solution space is associated to specific values of the dual variables. This information can then be used to extend the feasibility check to larger regions in the parameter space. Recall that only the matrices Appendix. The pseudo-code of the complete algorithm for an efficient classification of the parameter space in feasible and infeasible regions is shown in Table A formal proof of the guaranteed inconsistency of whole regions is provided in the Appendix). In the figure, the contour plot representing the feasible and infeasible regions in the 3-dimensional parameter space was obtained by exhaustive evaluation of sets of parameters. Particular solutions obtained with the SDP-based algorithm that prove infeasibility of whole regions are indicated in the graph. The respective regions of feasibility and infeasibility are shown, as well as a few cuboids that illustrate how we can discard the existence of solutions on whole regions of parameter space. Each box is obtained from the solution of a single small SDP optimization problem. It is clear that the SDP-based search in the infeasible region of the parameter space provides a powerful analysis tool, as it allows the efficient exploration of whole areas instead of the time consuming consistency check at single points. The figure also shows how the size of the boxes varies depending on the relative position  is shown. Finally, since the systems of equations and inequalities are all linear in k, the feasible region is invariant under non-negative scalings , as can be seen in the three-dimensional plot in Fig. The results in Fig. ion Fig. . In otheEi). However, while the reaction scheme for the pathway from A to C via B is well established, it may not be known whether the conversion from B to D is catalyzed by enzyme E3 or whether component A exerts some cooperative effect  of nonlinear optimization problems in polynomial form. It is thus possible to establish the possible infeasibility of steady state concentrations for a given set of parameters under direct consideration of experimental constraints. Our approach avoids the time-consuming numerical identification of a stable steady state, where feasibility can only be validated in retrospective ,26. BesiThe possibility of discarding whole regions of the parameter space from further exploration is extremely valuable for model discrimination, where determination of model consistency or inconsistency with experimental data is the desired goal . This apη, and this could be used to determine the direction of the next step .Consider a set of quadratic equations and linear inequalities for the vector xT ·Qi·x = 0,     L·x ≥ 0.     (16)X = x·xT, we find that these equations can be rewritten as affine expressions in the matrix X. A semidefinite relaxation of Eqns. (16) is then obtained by replacing the nonconvex constraint X = x·xT with the weaker (but convex) alternative:Defining X11 = 1,     X ≥ 0,yielding the systemQi·X) = 0, L·X·e1 ≥ 0,     (17)Tr) and the new (Eqns. (17)) formulations are equivalent only if the matrix A typical SDP problem in primal form isC·X)min     Tr(Ai·X) = bi, i = 1, ..., m     (18)s.t     Tr, and the vector z =  contains the dual decision variables.where X and z be any two feasible solutions of the primal and dual problems respectively. Then we have the following inequality:The key relationship between the primal and the dual problem is the fact that feasible solutions of one can be used to bound the values of the other problem. Indeed, let C·X) ≥ bT z.     (20)Tr.This property is known as Qi explicitly depend on the unknown rates k in an affine way. Thus, we want to be able to disprove the consistency of not just one particular fixed value of the rate constants, but to simultaneously discard whole sets of rates at the same time. In other words, we want to find solutions  of the dual form (8) that work for all rate constants k on a given set.If a feasible dual solution can be found, Eqn. (8) will be satisfied. As noticed earlier, the matrices Qi on the rate constants k is affine. Assume the nominal value and deviation of rate constant j is kj0 ± Δkj. Then, we can writeNotice that the dependence of the matrices δj := (kj - kj0)/Δkj, and |δj| ≤ 1. To guarantee that the dual form (8) holds for all allowable values of the rate constants, we can use the following result:where Lemma 1 Consider the linear matrix inequality given by:i, such thatIf there exist η, Wfor i = 1,..., n, then Eqn. (21) holds for all δ such that |δi| ≤ η.The lemma follows easily from the identity:η, Wi, we can hence directly use this lemma to obtain guaranteed regions of inconsistency.Since the expressions in (22) are affine in the unknowns Additional example 1:A is converted into component B and vice versa:Consider a simple two-element linear reaction, where component A and B that are kept within an accuracy of 25%. The two-dimensional contour plot including feasible and infeasible regions is shown in Fig. We assume unimolar concentrations of species Additional example 2:mRNA binds to a ribosome to form a protein P, which decays at a certain rate:Consider another simple nonlinear system, where mRNA, ribosome and P are all equal to 1 and have to be kept within an accuracy of ε = 25%. The overall parameter space is 4-dimensional. Fig. k3 - k4 plane.For simplicity, the overall concentrations of


In recent years, constrained optimization – usually referred to as flux balance analysis (FBA) – has become a widely applied method for the computation of stationary fluxes in large-scale metabolic networks. The striking advantage of FBA as compared to kinetic modeling is that it basically requires only knowledge of the stoichiometry of the network. On the other hand, results of FBA are to a large degree hypothetical because the method relies on plausible but hardly provable optimality principles that are thought to govern metabolic flux distributions.Escherichia coli (931 reactions). Our calculations show that increasing network complexity entails increasing sensitivity of predicted flux distributions to variations of standard Gibb's free energy changes and metabolite concentration ranges. We demonstrate the usefulness of our method for assessing critical concentrations of external metabolites preventing attainment of a metabolic steady state.To augment the reliability of FBA-based flux calculations we propose an additional side constraint which assures thermodynamic realizability, i.e. that the flux directions are consistent with the corresponding changes of Gibb's free energies. The latter depend on metabolite levels for which plausible ranges can be inferred from experimental data. Computationally, our method results in the solution of a mixed integer linear optimization problem with quadratic scoring function. An optimal flux distribution together with a metabolite profile is determined which assures thermodynamic realizability with minimal deviations of metabolite levels from their expected values. We applied our novel approach to two exemplary metabolic networks of different complexity, the metabolic core network of erythrocytes (30 reactions) and the metabolic network iJR904 of Our method incorporates the thermodynamic link between flux directions and metabolite concentrations into a practical computational algorithm. The weakness of conventional FBA to rely on intuitive assumptions about the reversibility of biochemical reactions is overcome. This enables the computation of reliable flux distributions even under extreme conditions of the network  where metabolite concentrations may be drastically altered. E. coli under conditions of optimal bacterial growth. Here we go one step further to include information on metabolite concentrations directly into the calculation of the flux distribution. Our algorithm considers the optimization of two different objectives: On one hand a functionally optimal and thermodynamically feasible flux distribution is demanded an on the other hand the calculated metabolite concentrations are required to deviate as little as possible from set-point values prescribed on the basis of biochemical knowledge. In the following we outline the method and provide applications to two different metabolic networks: (i) the energy- and redox metabolism of red blood cells for which a detailed kinetic model has been established  is the active concentration (activity) of the metabolite M, S and P denote the sets of substrates and products of the reaction, respectively. R is the universal gas constant and T is the absolute temperature. The change of the standard Gibb's energy is related to the equilibrium constant Keq of the reaction bywhere [r values for all reactions of the network, S is the stoichiometric matrix of the system, where rows refer to reactions and columns refer to metabolites. A positive or negative matrix element of S represents the stoichiometric coefficient with which the metabolite indicated by the column number appears as a product or substrate of the reaction indicated by the row number. Changes of the standard Gibb's energies of reactions can be additively composed of changes of the standard Gibb's energies of the formation of their reactants [It has to be noted that standard Gibb's energy changes depend on temperature, pH value, and ion strength and thus may significantly differ from those determined under specific in vitro conditions. For a metabolic network, (eq. 2) reads in vector notation eactants :Owing to the first law of thermodynamics the values of the standard Gibb's free energy changes are not independent from each other but have to obey the principle of micro-reversibility dictating the sum of standard free energy values in a closed system to be zero. In several flux balance studies ,17,29-32where ||E|| is the 2-norm of the vector E, and In case that the metabolite concentrations might assume arbitrary non-negative values it would be always possible to let a chemical reaction proceed in either forward or backward direction. Hence, including information on metabolite concentrations as additional constraints in FBA makes only sense if the concentration of the metabolites can be restricted to a feasible range. If the concentration of a metabolite is known, we use this value as set-point which should be approximated as best as possible by the calculated metabolite concentration. Thus, we add the termm for which a set-point (logarithmic) concentration value sm is available and cm is the component of C related to metabolite m. If the concentration of a metabolite is not exactly known but can be restricted to a narrower concentration range based on metabolite profiling  binding of metabolites to macromolecular structures lowering their effective free concentrations inside the cell. Therefore, concentrations lying outside the range of the soft bounds are allowed in our algorithm but are penalized in the optimization criterion:to the objective function where W denotes the set of metabolites l et al. ) we use c is the (logarithmic) active concentration of the metabolite. The penalty function for the whole concentration vector isHere As in Henry et al.  we intro2O where R-OOH represents a large group of not further specified hydroperoxides that can be detoxified by the glutathione system. For these lumped reactions it is impossible to give a realistic E. coli [Metabolic network models may contain reactions which are simplified in a way that reactants are dropped from the reaction formula. For example, the oxidation of glutathione (GSH) to glutathione disulfide (GSSG) is usually written as an overall reaction 2GSH → GSSG. Actually, this reaction should read 2GSH + R-OOH → GSSG + R-OHH E. coli ). For suIn FBA, formulation of the optimization problem requires to define the following three elements: (i) a physiologically meaningful scoring function to evaluate flux distributions, (ii) the steady-state conditions for all internal metabolites valid for the time-scale of interest  and (iii) further constraints taking into account biochemical knowledge as, for example, maximal enzyme capacities limiting the flux rates  or thermS' derives from the full stoichiometric matrix S of the network upon deletion of all columns referring to those metabolites which are exchanged with the external environment and thus need not to be balanced. According to the principle of minimal fluxes [Lj, j ∈ J to all output fluxes, which are directly linked to cellular functions, the so-called target fluxes, the set of which is denoted by J. Adding the weighted terms (eq. 6) and (eq. 8) to the scoring function and including the constraint (eq. 9) the complete optimization problem is written aswhere l fluxes ,9 we setvj is the j-th component of V. λ1, λ2 ∈ ℝ+ are empirical factors weighting the relative contribution of the various penalty scores relative to the scoring function of fluxes.  n is the number of reactions, and for any 1 ≤ j ≤ n, dj is a binary variable. α is set to a positive number which is larger than any possible flux value and larger than any possible Gibb's energy value, and it can easily be shown that the constraints 0 ≤ vj + αdj ≤ α and 0 ≤ -αdj ≤ α are equivalent to vj ≠ 0 → sgn(vj) = -sgn  thermodynamically realizable and refer to it in the following as 2G, reaction #9) required to modulate oxygen affinity of hemoglobin, (ii) ATP-utilization , which is mostly spent on the Na+/K+-ATPase to build up the Na+/K+-gradient across the plasma membrane, (iii) oxidation of GSH  to prevent oxidative damage of cellular proteins and lipids, (iv) synthesis of PRPP  required for the salvage of adenine nucleotides. The magnitude of these 4 target reactions depends on the specific external conditions of the cell as, for example, osmolarity of the blood or preservation medium, oxidative stress caused by reactive oxygen species, or lowering of the oxygen tension during hypoxia. In our calculations the flux values for these 4 target reactions were chosen as reported for the normal in vivo state of erythrocytes: DPGM = 0.49 mmol/h, ATPase = 2.38 mmol/h, GSHox = 93 μmol/h, PRPPS = 26 μmol/h. With these values for the target fluxes, the comprehensive kinetic model [The method described above was applied to a metabolic network of human red blood cells ,33 for wic model ,33 yieldUsing the same values of standard Gibb's free reaction energy changes as used in the kinetic model and putting the set-point values of the metabolite concentrations to the 'observed' ones, the TR-fluxmin solution of the optimization problem turns out to be identical with the flux-minimized solution determined by our previous approach . A detaiμM...100 mM (glucose), 0.1 μM...25 mM (CO2 and phosphate), and 0.01 μM...10 mM (28 remaining compounds). Calculation of the flux distribution with randomly altered set-point concentration values was repeated in 1000 trials. Surprisingly, for smaller perturbations the reference solution  was retained in all trials. But perturbations with a standard deviation of 2  resulted in a second, slightly different, flux distribution in some trials. For perturbations with standard deviation of 6  this second alternative flux distribution dominated and on top a third alternative flux distributions was obtained in a significant number of trials  this pathway is not used at all whereas in the third flux distribution (glucose uptake 1.4972 mmol/h) it is used in backward direction producing 25.8 μmol/h ribulose-5-phosphate. Interestingly, the latter flux distribution is also obtained for the unperturbed network if the maximization of biomass production used as flux criterion. Notably, all three different flux distributions obtained as solution of the minimization problem (eq. 10) for randomly altered thermodynamic parameters and set-point concentrations are feasible from the kinetic view point, i.e. the kinetic model of the erythrocyte metabolism yields a stable stationary solution.The three alternative flux distributions obtained in the perturbation studies differ in the uptake flux of glucose  and the fluxes in the non-oxidative pentose phosphate pathway converting ribulose-5-phosphate (Ru5P) into fructose-5-phosphate (Fru6P) and glyceraldehyde 3-phosphate (GraP). The reference solution predicts this pathway to proceed in forward direction thereby forming 20.7 + and NADH tended towards the hard bounds to ensure the flux through the lactate dehydrogenase (EC:1.1.1.27) to be directed towards formation of lactate. Our find of a metabolic threshold effect with respect to blood lactate levels corresponds well with clinical observations. At lactate levels higher than 4 mM a reduced deformability of erythrocytes is observed, which may account for the exercise-induced arterial hypoxemia occuring in athletes [Our algorithm allows assessing how the predicted flux distributions are affected by changes in the concentration of external metabolites. In vivo, such a situation may occur if some essential fuels for the cellular metabolism are depleted, for example, due to a reduced blood flow through vessels with severe atherosclerotic stenoses, or some end products of the cellular metabolism accumulate because of a reduced excretion capacity of the body. For example, in case of strong physical exercise the concentration of lactate in human blood may rise to values as high as 19.5 mM (in blood plasma) respectively 7.0 mM (in erythrocyte cytoplasm)  indicatiathletes . DecreasE. coli reconstructed by Palsson and co-workers [E. coli has been determined by Emmerling et al. [E. coli [To check the applicability of our algorithm to genome-scale metabolic networks comprising hundreds of reactions and metabolites, we performed the same type of analysis as described above with respect to the metabolic network iJR904 of the bacterium -workers . In thisg et al.  which cog et al.  onto the[E. coli .) The th[E. coli ,20,23,32[E. coli  we use cμM-0.5 mM (soft bounds), 5 μM-2 mM (hard bounds). Further details of the model are given in the supplement [see Additional file These values were obtained by a slightly modified version of the group contribution method ,41. PhysWe calculated the flux distribution in this network according to the proposed optimization principle (eq. 10) using as flux objective the maximization of the biomass production. No a priori assumptions were made with respect to the directionality of reactions with two exceptions: The direction of the exchange fluxes was fixed according to the experimental conditions  and the In a third calculation we again omitted the condition of thermodynamic realizability from the optimization algorithm but instead used the heuristic classification of reactions into reversible and irreversible ones as outlined in  . One important restriction of fluxes in the network arises from thermodynamics. Reactions associated with a decrease of free energy larger than 30 kJ/mol are generally thought to be irreversible. This condition can be used as an additional constraint on feasible flux distributions. Kümmel and co-workers  have rec2 [E. coli the realization of a specific flux distribution assuring optimal bacterial growth. In contrast to this approach, the algorithm proposed in this work aims at employing reliable information on metabolite concentrations to restrain the solution space of FBA.For example, under hypoxic conditions, the cellular concentration of oxygen may become so low that the respiratory chain – usually thought to carry electrons from hydrogen to oxygen in a strictly irreversible manner – may indeed operate in the reverse direction, i.e. reducing NAD to NADH2 . Thus, i2  who inclE. coli network is still obtained if the directionality of fluxes is a priori defined based on biochemical conventions. This is obviously due to the fact that – owing to the lack of reliable experimental data – the soft bounds for the metabolite concentration used in our method have been too generously chosen. Indeed, enlarging systematically the physiologically feasible concentration ranges one eventually obtains a network without any constraint of flux directionalities. Hence, the usefulness of the proposed method essentially depends upon the availability of reliable information on values of free energy changes and metabolite concentrations. As long as this information is not available, the benefit of our method consists mostly in the generation of alternative flux distributions by varying the values of standard Gibb's free energy changes and/or in the physiologically relevant concentration ranges of metabolites. Applying such perturbation analysis to two networks of different complexity has provided evidence that the larger the network is, the more alternative flux distributions occur, even at relatively modest variation of energy values of about 5 kJ/mol. Inspection of such alternative flux distributions reveals critical reactions for which fluxes are largely undetermined by the FBA approach. In this respect, our method represents a useful complement to the thermodynamic evaluation method recently proposed by Kümmel et al. to identify putative regulatory sites by network-embedded thermodynamic analysis of metabolome data [Hence, depending on reported ranges of metabolite concentrations, our algorithm may yield different flux distributions. In other words, in our approach we do not ask for metabolite concentrations that are compatible with a given flux distribution but in contrast ask for the flux distribution that is compatible with a given metabolite profile. As demonstrated for two exemplary metabolic networks, even if no measurements of metabolite concentrations are available restriction of concentrations to physiologically feasible ranges alone allows the prediction of reliable flux distributions if no a priori assumptions are made on the reversibility of the reactions. As demonstrated for the erythrocyte network, our approach may provide valuable information about alterations in the external conditions of a cell that may result in a metabolic dysfunction. Of course, FBA cannot assess whether a stable steady state may exist at very high concentrations of external lactate because this is determined by kinetic regulation. Possibly the metabolite concentrations may vary even in a larger interval than imposed in our calculations. This problem can be addressed better by a comprehensive kinetic network model. Nevertheless, our method may provide valuable information on external conditions causing metabolic problems merely for thermodynamic reasons. Concerning the predictive capacity of our method it must be critically noted that – based on a comparison with a relatively small number of measured fluxes – the most reliable flux distribution for the ome data .The basic idea of the presented theoretical concept was developed by HGH. AH and SH prepared of the network models. AH developed the algorithms and carried out the computations. All authors jointly contributed to the manuscript.This document shows the proof that thermodynamic realizability implies that there is no net flux in a closed loop – a consequence of the generalization of Kirchhoff's loop law.Click here for fileThis archive contains three files relating to the erythrocyte network. The file "Ery-model.txt" gives the definition of the network as tab-separated text file which is organized in the sections: 'Metabolites', 'Reactions', 'Reactions excluded from the TR property', 'Equilibrium constants', 'Targetfluxes', 'Fixed concentrations', The file "Ery.sbml" gives the network description in SBML format. The file "Ery-solutions.txt" contains three solutions: the thermodynamically realizable fluxmin solution (TR-fluxmin) together with the associated metabolite concentrations and the two alternative solutions obtained in the perturbation analysis.Click here for fileE. coli iJR904 computations: "FullyReversible" and "HeuristicIrreversibilities". Both contain the file "Ecoli-model.txt", a tab-separated text file, as the definition of the network which is organized in the sections: 'Metabolites', 'Reactions', 'Reactions excluded from the TR property', 'Equilibrium constants', 'Targetfluxes', 'Concentration bounds', and 'Setpoint concentrations'. Both directories also contain the file "Ecoli.sbml", the network description in SBML format. The file "Biomax.txt" also residing in both directories contains the solution of the biomass maximization without the constraint of thermodynamic realizability as a tab-separated text file assigning a reaction identifier with a flux value if it is not zero.. Additionally the directory "FullyReversible" contains the file "TR-Biomax.txt" as the solution to the thermodynamically realizable biomass maximization together with the hypothetic metabolite concentrations compatible with the flux directions.This archive includes two directories relating to the Click here for file


Corynebacterium glutamicum. The feasible phenotypic space was evaluated at various combinations of oxygen and ammonia uptake rates.Quantification of the metabolic network of an organism offers insights into possible ways of developing mutant strain for better productivity of an extracellular metabolite. The first step in this quantification is the enumeration of stoichiometries of all reactions occurring in a metabolic network. The structural details of the network in combination with experimentally observed accumulation rates of external metabolites can yield flux distribution at steady state. One such methodology for quantification is the use of elementary modes, which are minimal set of enzymes connecting external metabolites. Here, we have used a linear objective function subject to elementary modes as constraint to determine the fluxes in the metabolic network of C. glutamicum was formulated as linear programming. The analysis demonstrated that the solution was dependent on the criteria of objective function when less than four accumulation rates of the external metabolites were considered. The analysis yielded feasible ranges of fluxes of elementary modes that satisfy the experimental accumulation rates. In C. glutamicum, the elementary modes relating to biomass synthesis through glycolysis and TCA cycle were predominantly operational in the initial growth phase. At a later time, the elementary modes contributing to lysine synthesis became active. The oxygen and ammonia uptake rates were shown to be bounded in the phenotypic space due to the stoichiometric constraint of the elementary modes.Quantification of the fluxes of the elementary modes in the metabolism of C. glutamicum, which evaluates the set of operational elementary modes at different phases of fermentation. The methodology was also used to determine the feasible solution space for a given set of substrate uptake rates under specific optimization criteria. Such an approach can be used to determine the optimality of the accumulation rates of any metabolite in a given network.We have demonstrated the use of elementary modes and the linear programming to quantify a metabolic network. We have used the methodology to quantify the network of  Application of metabolic engineering towards strain improvement involves detailed quantitative evaluation of cellular physiology -3. DeterCorynebacterium glutamicum for the production of amino acids (lysine) using elementary modes.Most theoretical methods, such as metabolic flux analysis (MFA), are based on stoichiometric reactions involving various metabolites in a metabolic network -13. The C. glutamicum, which is used for lysine production, is well demonstrated. Batch growth of C. glutamicum for lysine production can be represented through four phases. Phase-I represents balanced growth with little or no product formation and is dependent on threonine concentration. Phase-II represents high lysine synthesis and biomass production rate with constant respiration rate. In phase-III, lysine production continues at a high rate while biomass productions saturates with a decrease in respiration rate. Phase-IV sees a gradual reduction in lysine synthesis and redirection of lysine to other byproducts such as pyruvate, acetate, lactate etc. [C. glutamicum is necessary to achieve high lysine yields. Although, flux analysis of C. glutamicum is well demonstrated for different fermentation stages, the effect of nitrogen and oxygen uptake rate on the productivity has not been quantified. We address this issue by analyzing the performance of the network at various oxygen and nitrogen uptake rates.Flux distribution in the metabolic network of ate etc. . Therefoate etc.  have perate etc. . Thus, iC. glutamicum using linear programming with the coefficients of the elementary modes as constraints. Further, we evaluate a feasible solution space for a given objective function of the network through a search in the space characterized by uptake rates of oxygen and nitrogen.In the present work, we analyze the metabolic network of C. glutamicum included the glucose phosphotransferase system, reactions of glycolytic pathway, reactions of TCA cycle and PPP as the core metabolism [see C. glutamicum [In vitro studies indicate that the activities of the anaplerotic enzymes are negligible except that of PEP carboxylase, which is active in presence of glucose as the sole carbon source in the media [in vivo studies also indicate that these enzymes are inactive while growing on glucose alone [C. glutamicum in presence of lactate/acetate in the media. Since we analyze the system in presence of glucose alone, we do not include these anaplerotic reactions. Elementary modes represent the overall stoichiometry of the metabolic network in terms of inter conversion of external metabolites. This implies that the internal metabolites are at pseudo steady state levels and the accumulation rates of external metabolites can be used to evaluate the fluxes of the elementary modes (see method).The metabolic network of lism see . Also, clism see . It is rutamicum . In vitrhe media ,20-22. Fse alone . These rse alone ,20,24 duse alone ,26. It i'ScrumPy', which was developed by M. G. Poolman & D. A. Fell . A total of 542 elementary modes were observed by incorporating all the possible extracellular metabolites of C. glutamicum. Experimental evidence suggested that the accumulation of lactate, acetate, pyruvate, alanine and valine were negligible during phase-I, II & III of lysine fermentation. These metabolites together accounted for less than 1.8 %, 0.87 % and 3.5 % of the glucose uptake rate at 11.5 h, 13.5 h and 15.8 h, respectively. Therefore, these metabolites were not considered as extracellular metabolites for the analysis. This modification resulted in only fourteen elementary modes [see Figure Elementary modes were generated using the public domain python based software named Table To study the effect of the maximization criteria, the network was analyzed using other objective functions. The accumulation rates of glucose, lysine, trehalose and biomass were used with maximization of carbon dioxide as the objective function. Table The ranges of the allowable flux values for an individual elementary modes was used to determine its contribution towards uptake rates of substrates  and formation of products . Figure Figure C. glutamicum was simulated with the normalized glucose uptake rate equal to 100 moles/Lh with various combinations of uptake rates of ammonia and oxygen.As discussed above, the fluxes through individual modes were evaluated using the experimentally determined accumulation rates of glucose, ammonia, oxygen and lysine. However, the network can also be simulated by considering only the three uptake rates of substrates  as decision variables with a maximization criterion (maximization of biomass or that of lysine) to study the metabolic capability of the organism at various nutrient uptake rates. Thus, the network of Figure C. glutamicum do not tend to maximize lysine synthesis. However, it was observed that on the edges of the surface of the three dimensional solution space, the form of the objective function did not matter indicating that solution obtained was independent of the criteria of maximization. In such a condition, the reaction from PEP to PYR was not operational and carbon for glycolysis and PPP was channeled through PEP carboxylase to OAA. This implies that TCA cycle and lysine synthesis reactions were operational through PEP carboxylase at points lying on the edges of the solution space. The carboxylase enzyme, which was present in five elementary modes , would not be operational on the edges of the solution space. However, for a non-edge point in the solution space, the maximization criteria yielded different set of fluxes. For example, at glucose, ammonia and oxygen uptake rates of 100, 67 and 332, respectively with the criteria of maximization of biomass yielded a biomass formation rate of 28.6 and lysine formation rate of 23. In this case, the flux from PEP to PYR was highly active (around 60) with no flux through pentose phosphate pathway, while with the criteria of maximization of lysine, biomass formation rate was zero and lysine synthesis rate was higher at 33.5. In this non-edge solution, flux from PEP to PYR and PPP were operational to maximize lysine synthesis, which was required for NADPH balancing.Figure C. glutamicum using elementary modes by linear optimization. The analysis was used to determine the feasible set of flux values for the elementary modes. The individual stoichiometric combination of various substrates towards specific product can be evaluated using this methodology. The linear optimization was used to determine the fluxes of elementary modes using accumulation rates obtained through experiments. In case of C. glutamicum, only four measurement values were essential for the closure of molar balance. In such a situation, the solution for predicting the accumulation rates of remaining external metabolites was independent of the objective function. This, however, did not ensure a unique flux distribution for the various elementary modes and provided a set of feasible ranges. By utilizing lesser number of accumulation rates in the linear optimization strategy, one could determine the capability of the network towards achieving a specific objective function. We used the uptake rates of three substrates  to evaluate the capability of the network to produce maximum biomass or lysine. The solution space bounds the accumulation rate of a metabolite for a specific substrate combination. The stoichiometry of the elementary modes also enforces a constraint on the uptake rates of the substrates. For example, in C. glutamicum for a given glucose (100) and ammonia (85) uptake rates, the oxygen uptake rate was bounded within a range of 153 – 302. This also reflects the constraint of the aerobic and anaerobic routes that may be used for balancing the oxidative potential . Similarly, the demand for balancing of NADPH/NADP+ may switch on the elementary modes containing the pentose phosphate pathway. In C. glutamicum, this was observed in the modes involving lysine synthesis. Also, for a given glucose and oxygen uptake rate, the ammonia uptake gets bounded, resulting in a constraint on the productivity of lysine. The feasible solution space was equivalent to the phenotypic phase plane as reported by Edwards et al. [E. coli on glucose and acetate at various oxygenation levels. Here, we have similarly attempted to obtain optimal growth and lysine synthesis rates at various oxygenation levels and ammonia uptake rates.We have quantified the metabolic network of C. glutamicum, the biomass formation rate was highest (124) at ammonia and oxygen uptake rates of 91 and 146, respectively. In this case, elementary modes numbered as '5', '6', '7', were mainly functional and operated through the routes using glycolysis and TCA cycle with carbon dioxide as only the byproduct. Interestingly, we obtained the same result with the objective function of maximization of lysine. This implies that a minimum oxygen uptake of 146 is essential during the metabolism of C. glutamicum. A similar behavior in the flux distribution of each elementary mode was observed at the extreme points of the feasible set of oxygen and ammonia consumption rate. It was noted that for a maximum feasible ammonia uptake rate (127), the lysine synthesis rate was at a maximum with the elementary mode '13' being functional. This elementary mode included reactions involving PPP, PPC, TCA cycle and lysine synthesis.The methodology presented here is useful in deciphering the capability of an organism for a given substrate combination. For example, the maximum yield coefficient (ratio of accumulation rate of a product and that of biomass) can be evaluated for a specific nutrient combination. Further, the analysis can yield the operational elementary modes at different points of the feasible solution space. In case of Evaluation of the fluxes of elementary modes can therefore provide functional insights about any metabolic network. The operational limitation caused by ATP and NADH balancing was inherently captured using the elementary modes. The stoichiometric distribution towards a targeted product using a specific elementary mode provides insights into the route that needs engineering to enhance productivity. Further, the flux distribution through the elementary modes can yield overall capability of the organism in terms of growth, product formation and substrate uptake rates for a given media composition.C. glutamicum for lysine synthesis. Analysis in conjunction with experimental accumulation rates gives insight into the contribution of various elementary modes towards the accumulation rates of extracellular metabolites. In C. glutamicum, the elementary modes associated with biomass formation were operational at the initial experimental growth phase, while modes associated with lysine synthesis switch on at later phase of fermentation. The methodology was also used to determine the feasible solution space for a given set of substrate uptake rates. Such an approach is generic in nature and can be used to determine the optimality of the accumulation rates of a metabolite in any given system.We have demonstrated the use of elementary modes with the help of linear optimization in quantifying metabolic networks. This methodology was used to quantify the network of 'ScrumPy' software [Elementary modes for the network were obtained using software . The inpsoftware . The accS·E = M     (1)M is vector representing the accumulation rates of the external metabolites and S is the matrix, indicating the stoichiometry of the elementary modes. The elements of E{e1, e2, e3,.........en} represents the vector of unknown fluxes of the elementary modes. The elements of vector E can be evaluated for a given set of measurements of accumulation rates (elements of M). Due to the paucity of measurements, a linear optimization technique can be employed to converge onto a feasible solution. Mathematically, the linear optimization formulation can be represented as:where, Mi)Max0 ≤ Mi is the accumulation rate of a specific metabolites (such as biomass or carbon dioxide). Also, S* is stoichiometric matrix and M* is know vector of the accumulation rates of extracellular metabolites, wherein the rows of the ith external metabolite are eliminated from S and M, respectively. It should be noted that the above approach of using linear programming to evaluate fluxes is similar to the methodology used in FBA. FBA uses the stoichiometry of the original network for the analysis, while we use the stoichiometries of the elementary modes. The methodology has been explained using an illustrative example [see mple see in silico experiments. KG & KVV conceptualized the manuscript. All authors have read and approved the final manuscript.KG and KVV conceived the methodology and designed the experiments. KG performed the various C. glutamicum. Lists all the reactions in the metabolic network used for the analysis.Reactions involved in the metabolic network of Click here for fileC. glutamicum. Lists the elementary modes obtained using "ScrumPy" software.Elementary modes obtained for the network of Click here for fileAbbreviations used for various metabolites. Lists the abbreviations used for the metabolites present in the network.Click here for fileInput file used to generate the elementary modes. The input .spy file used in "ScrumPy" software to obtain elementary modesClick here for fileDemonstration of the methodology using a simple illustrative example. The optimization methodology is applied to an illustrative example to demonstrate the steps used for obtaining the solution.Click here for file


Escherichia coli and Mycobacterium tuberculosis, and studied the effects of inhibition of various enzymes in the M. tuberculosis model.Targeting persistent tubercule bacilli has become an important challenge in the development of anti-tuberculous drugs. As the glyoxylate bypass is essential for persistent bacilli, interference with it holds the potential for designing new antibacterial drugs. We have developed kinetic models of the tricarboxylic acid cycle and glyoxylate bypass in E. coli to validate the pathway-modeling protocol and showed that changes in metabolic flux can be estimated from gene expression data. The M. tuberculosis model reproduced the observation that deletion of one of the two isocitrate lyase genes has little effect on bacterial growth in macrophages, but deletion of both genes leads to the elimination of the bacilli from the lungs. It also substantiated the inhibition of isocitrate lyases by 3-nitropropionate. On the basis of our simulation studies, we propose that: (i) fractional inactivation of both isocitrate dehydrogenase 1 and isocitrate dehydrogenase 2 is required for a flux through the glyoxylate bypass in persistent mycobacteria; and (ii) increasing the amount of active isocitrate dehydrogenases can stop the flux through the glyoxylate bypass, so the kinase that inactivates isocitrate dehydrogenase 1 and/or the proposed inactivator of isocitrate dehydrogenase 2 is a potential target for drugs against persistent mycobacteria. In addition, competitive inhibition of isocitrate lyases along with a reduction in the inactivation of isocitrate dehydrogenases appears to be a feasible strategy for targeting persistent mycobacteria.We used We used kinetic modeling of biochemical pathways to assess various potential anti-tuberculous drug targets that interfere with the glyoxylate bypass flux, and indicated the type of inhibition needed to eliminate the pathogen. The advantage of such an approach to the assessment of drug targets is that it facilitates the study of systemic effect(s) of the modulation of the target enzyme(s) in the cellular environment. Tuberculosis is an ancient disease that has plagued humans for centuries, and presently there is an urgent need for new drugs to combat drug-resistant tuberculosis and shorten the time of tuberculosis therapy. Tuberculosis treatment is lengthy because of a population of persistent bacilli that is not effectively eliminated by current drugs.isocitrate lyase (ICL) and malate synthase (MS), essential for the bacterium; in its absence there will be no net formation of the intermediates required for synthesizing cellular materials. Inhibition of both ICL1 (prokaryotic-like isoform) and ICL2 (eukaryotic-like isoform) has been shown to block the growth of M. tuberculosis in macrophages and in mice  and [M. tuberculosis model-1 and model-2]) were either obtained from publicly available databases, namely CyberCell Database (CCDB) [-12 ml [-10 mg [M was not available, usually for a reverse reaction, it was assumed to be equal to 10 × KM of the substrate from which that product was formed . The metabolites acetyl-CoA, oxaloacetate and CoA were considered as boundary metabolites, so their concentrations were fixed in the simulations. The initial concentration of each variable metabolite was taken as 2 × KM for the reaction for which that metabolite is a substrate   and BRENe (CCDB) , or extr [-12 ml  and the  [-10 mg . We wereE. coli model, the carbon flux through the pathway was predicted under two growth conditions, viz. growth on glucose and acetate as carbon sources. Most enzyme kinetic parameters are available for E. coli grown on glucose, but it is also necessary to estimate the enzyme kinetic parameters for the acetate condition. The changes in E. coli gene expression when growth shifts from glucose to acetate were described by Oh et al. [In the h et al. . Assumina and g1g respectively. Therefore, the fold change when growth shifts from glucose to acetate is n = g1a/g1g. Taking account of the assumption that a change in mRNA level leads to a proportional change in protein level,Let, the expression levels of a gene g1 under the acetate and glucose conditions be g1a/p1g = g1a/g1g = n     equation (5)p1where p1 is the amount of the protein encoded by g1 and the subscripts 'a' and 'g' denote its level in acetate and glucose respectivelySince Vmax = kcat × E  and kcat is a constant, Vmax α Ea/Vmaxg = nTherefore, from equation (5), Vmaxa, Vmaxg = Vmax of the enzyme in acetate and glucose respectively) and 0.0341 times (for acetate condition) the rate of the ICD reaction in the mentally . Owing tondition , the conSimulations were performed by writing scripts for Jarnac 2.14 . First, E. coli model with experimental findings [x = (50)/(4.187 mM/min). Using this conversion factor (x), all the fluxes computed from the model were converted to the units in which experimental fluxes were expressed.The fluxes computed from the models were expressed in mM/min. To compare the steady state fluxes of the findings , they wex min/mM) = 40.5.Example: flux through α-ketoglutarate dehydrogenase (KDH) reaction step in the model = 3.394 mM/min = (3.394 mM/min) ×  declare that they have no competing interests.VKS has contributed in developing the models, analysis and interpretation of data, and writing the manuscript. IG was involved in the overall design of this study, critical analysis and interpretation of the data, and revision of the draft of the manuscript.E. coli modelKinetic constants of the enzymes in . E. coli model.Click here for fileM. tuberculosis model-1 and model-2Kinetic constants of the enzymes in . M. tuberculosis models.Click here for file


The required experience, the complicated network models, large and heterogeneous data sets, and the time-consuming set-up of highly controlled experimental conditions largely restricted metabolic flux analysis to few expert groups. A conceptual simplification of flux analysis is the analytical determination of metabolic flux ratios exclusively from MS data, which can then be used in a second step to estimate absolute in vivo fluxes.Quantitative knowledge of intracellular fluxes is important for a comprehensive characterization of metabolic networks and their functional operation. In contrast to direct assessment of metabolite concentrations, in vivo metabolite fluxes must be inferred indirectly from measurable quantities in 13C-pattern in protein-bound amino acids. Predefined fragmentation patterns are automatically identified and appropriate statistical data treatment is based on the comparison of redundant information in the MS spectra. In the second module, absolute intracellular fluxes may be calculated by a 13C-constrained flux balancing procedure that combines experimentally determined fluxes in and out of the cell and the above flux ratios. The software is preconfigured to derive flux ratios and absolute in vivo fluxes from [1-13C] and [U-13C]glucose experiments and GC-MS analysis of amino acids for a variety of microorganisms.Here we describe the user-friendly software package FiatFlux that supports flux analysis for non-expert users. In the first module, ratios of converging fluxes are automatically calculated from GC-MS-detected 13C-substrates, labeling mixtures, and organisms.FiatFlux is an intuitive tool for quantitative investigations of intracellular metabolism by users that are not familiar with numerical methods or isotopic tracer experiments. The aim of this open source software is to enable non-specialists to adapt the software to their specific scientific interests, including other  The preconfigured analytes are the TBDMSTFA-derivatized proteinogenic amino acids that are typically detected by robust GC-MS analysis [The set of calculable flux ratios is a function of the biochemical reaction network, the carbon substrates and their corresponding s yeasts ,32, Eschhia coli , Bacillusubtilis , and othsubtilis  for growanalysis . Notably13C-constrained flux balancing [13C-derived equality or inequality constraints; i.e. flux ratios that are exactly determined or for which only reaction bounds are available, respectively [fmincon function. This approach permits to simultaneously integrate equality and reaction bound constraints in the calculation, and weight the constraints with the experimental uncertainty [The second module (NETTO) addresses the estimation of absolute in vivo (net) fluxes through a reaction network. This is achieved by global material balances derived from a stoichiometric model and accounting for the withdrawal of precursors during growth Figure . Becausealancing ,20, addiectively . Dependiectively . Exactlyertainty . Confideertainty . If the In NETTO, metabolic models can be constructed from scratch and error-prone operations such as introduction or modification of reactions are executed by the software. In a text file, the user provides a list of reactions, ratios, and biomass composition with a user-friendly syntax Figure . The infBoth modules offer functions to save all variables and recover work at a later point. Results are visualized directly on the graphical user interface and can be stored to text files or to Microsoft Excel spreadsheets.13C-constrained flux balancing as in FiatFlux or with global isotopomer balancing was demonstrated previously [FiatFlux is the first publicly available software for flux ratio analysis from MS data and, consequently, no comparison can be done with other programs. The scientific value and accuracy of FiatFlux-calculated flux ratios has already been discussed extensively ,26,36,37eviously . NotablyM from the MDVA. Since the residuals are graphically represented on the graphic user interface of RATIO, bad fragments are rapidly identified and excluded with a single click. Also when the quality of the fragments has to be diagnosed in detail, and MDVM fitting and flux ratios estimation have therefore to be iterated several times, a correct estimate is obtained within some minutes. Using FiatFlux, a typical user with moderate experience will be able to determine intracellular net fluxes for hundreds of samples per day from previously generated MS data.In FiatFlux, user supervision is necessary only when MS-signals are low, saturated, or overlapping. This affects the ion statistics of the corresponding fragment and results in relatively high residuals after inferring MDVThe open source nature of FiatFlux, and in particular of the RATIO module, permits to modify and extend its capabilities beyond the predefined features. Although the necessary skills strongly depend on the functionalities to be modified, fundamental biochemical knowledge of the reactions investigated is paramount for every user to understand initial assumptions and critically interpret outcomes. Provided that metabolism of a new organisms to be investigated is similar to that of any of the 4 preconfigured models,, very few adaptations are necessary and the task is manageable by any biochemically-trained biologist. In fact, in previous works we already demonstrated the analysis of about 20 different species with the 4 core models ,32. The Finally, introduction of new GC methods or derivatization procedures is very simple, and can be attained by users with basic familiarity with the Matlab environment. In principle, the same applies to implementing other separation techniques, such liquid-phase systems. Currently, RATIO is not compatible with MS/MS product ion scans.FiatFlux condenses the know-how developed over years in our lab and has become our workhorse for quantitative analyses of microbial central carbon metabolism. The software is preconfigured for the most widely used substrate (glucose), the most frequently used (and informative) tracer mixtures, and several model microbes. While this covers about 80% of all current flux applications, it is, of course, not complete. The aim of this open source software is to enable non-specialists to adapt the software to their specific scientific interests, including other substrates and or labeling mixtures. In particular, we aim at biologists that are not familiar with numerical methods or isotopic tracer experiments. In fact, with the availability of this software, the only burden for such studies remains the access to a GC-MS instrument. We hope that this transparent and flexible framework will support further developments.Project name: FiatFluxOperating system: preferably Microsoft Windows. Some minor problems were encountered using Matlab's graphic user interface with Linux.Programming language: Matlab R14 (The Mathworks).Other requirements: Matlab Optimization ToolboxLicense: source code is freely available from the authors for academic purposes.Any restriction to use by non-academics: license required.α Mass distribution vector of analyteMDVA Carbon-specific mass distribution vector of analyteMDVM Mass distribution vector of metaboliteMDVN-(tert-butyldimethelsylil)-N-methyl-trifluoroacetamideTBDMSTFA NZ and EF developed the software. US supported the work. NZ and US wrote the manuscript. All authors read and approved the final version.