Flux balance analysis (FBA) together with its extension, dynamic FBA, have proven instrumental for analyzing the robustness and dynamics of metabolic networks by employing only the stoichiometry of the included reactions coupled with adequately chosen objective function. In addition, under the assumption of minimization of metabolic adjustment, dynamic FBA has recently been employed to analyze the transition between metabolic states.Here, we propose a suite of novel methods for analyzing the dynamics of  metabolic networks and for quantifying their robustness with limited knowledge of kinetic parameters. Following the biochemically meaningful premise that metabolite concentrations exhibit smooth temporal changes, the proposed methods rely on minimizing the significant fluctuations of metabolic profiles to predict the time-resolved metabolic state, characterized by both fluxes and concentrations. By conducting a comparative analysis with a kinetic model of the Calvin-Benson cycle and a model of plant carbohydrate metabolism, we demonstrate that the principle of regulatory on/off minimization coupled with dynamic FBA can accurately predict the changes in metabolic states.Our methods outperform the existing dynamic FBA-based modeling alternatives, and could help in revealing the mechanisms for maintaining robustness of dynamic processes in metabolic networks over time. Systems biology paradigm has provided insights in the maintenance of robustness for biological processes involving a multitude of interconnected elements  . In addiMetabolic flux analysis (MFA) has propelled the development of computational methods for analysis of metabolic networks ,12. FluxE. coli [Based on the hypothesis that fluxes in metabolic networks, altered by removal of a reaction, undergo a minimal redistribution compared to those of the wild type, minimization of metabolic adjustment (MOMA)  and reguE. coli .The analysis of metabolic network dynamics has traditionally been facilitated by models based on ordinary differential equations (ODEs) which require a large amount of information for simulating the temporal metabolic changes ,10. To t2 and water conditions [On the other hand, dynamic FBA (DFBA) offers the alternative to predict time-resolved metabolic profiles with limited knowledge of enzyme kinetics -8. Unliknditions . M-DFBA temporal changes in the metabolic state, characterized by the metabolite concentrations and flux rates, are principal to the notion of robustness, as used with DFBA-based approaches. For instance, the posited mechanism in M-DFBA is that metabolic networks operate to minimize the fluctuations in metabolic concentrations over time. This suggests alterations to the definition of robustness as a property that maintains system function in the case of external perturbations [internal perturbations due to changes in the dynamic state of the system's elements. External perturbations arise due to changing environmental conditions, such as stress conditions , but also by changes in the structure of a metabolic networks . In contrast, internal perturbations are due to temporal changes of the metabolic state characterized by the coupling of metabolite concentrations and flux distributions. Considering mass actions kinetics, which states that the flux rate of a reaction is proportional to the product of the concentrations of the participating reactants [We point out that the mechanisms describing the rbations ,25 to ineactants . As a reDriven by the idea of mutually influencing system elements, central to the systems biology paradigm, we argue that minimal fluctuations of metabolic profiles may only represent one possible explanation of the temporal changes in metabolic state. In this study, we design the ROOM-based DFBA approach (abbreviated to R-DFBA) by coupling the principle of ROOM with DFBA. R-DFBA extends the premise of the ROOM approach, which relies on significant flux changes, by considering the minimization of the total number of significant changes of metabolite concentrations. Furthermore, the coupling of ROOM and DFBA renders it possible to use the advantages of ROOM compared to MOMA in a dynamic setting.seven proposed methods based on different optimization functions and programming formulations. Including classical DFBA and the two known variants of M-DFBA [In addition, we modify and extend the proposed R-DFBA and the existing M-DFBA approaches to consider minimizing fluctuations not only in concentrations but also in flux levels, which result in f M-DFBA , a totalHere we describe the seven proposed methods and the comparison of their performance with two kinetic models--of the Calvin-Benson cycle and of the plant carbohydrate metabolism. Since the proposed methods build upon the DFBA approaches, we provide a brief overview of the mathematical apparatus used in their formulation. The suite of proposed methods and their relation to the existing DFBA approaches are depicted in Figure We point out that DFBA-based methods and kinetic modeling constitute independent approaches. The DFBA-based methods involve performing a constrained optimization over a time period to approximate the dynamics of a system with stoichiometric constraints only, which include relative quantities of reactants and products of the particular reaction. In contrast, a kinetic model requires all kinetic parameters to simulate the dynamics of the system. However, kinetic modeling is often hampered by incomplete knowledge of the underlying enzyme-kinetics and their associated parameter values .In addition, it is necessary to determine the extent to which the dynamics of both metabolite profiles as well as flux predictions can be successfully predicted from DFBA-based approaches. The outcome of testing this hypothesis will then suggest the most likely mechanism which describes the robustness of the system to internal perturbations over time. We argue that such approach goes beyond the analysis of steady states typical for metabolic control analysis (MCA)  and struDFBA overcomes the main drawback of the classical FBA which precludes the analysis of the dynamic behavior of a network-the steady-state assumption. Mahadevan and coworkers introduced two DFBA formulations--static and dynamic . The stax and v are vectors of metabolite concentrations and reaction fluxes over time, S denotes the stoichiometric matrix, with rows corresponding to metabolites and columns to reactions of the metabolic network described by S, and t is the time. The minimum and maximum allowable fluxes of each reaction and metabolite concentrations are defined by vmin and vmax and xmin and xmax, respectively. The vector X0 gives the initial concentration for the set of metabolites. The formulation of the DOA approach results in a nonlinear program (NLP) if nonlinear constraints and/or a nonlinear objective function are included.where t to obtain time-resolved flux rates and metabolite concentrations. The optimization is rendered computationally feasible by parameterizing the dynamic equations with the help of orthogonal collocation on finite elements [finite elements. Furthermore, the metabolite concentrations x and flux levels v are parameterized at the roots of an orthogonal polynomial  within each finite element [The DOA involves optimization over the entire time period elements . To this element . For reaWe point out that the number of variables in each optimization step of SOA is smaller compared to that of DOA, allowing scalability to larger networks. However, SOA does not allow dynamic formulation, and the remainder of the methods focus on DOA.N is the number of metabolites in the network and xi,j represents the concentration of metabolite i at the time point given by the orthogonal root j. Hence, the general M-DFBA optimization problem is defined as follows:Luo and coworkers developed an approach called M-DFBA which combines MOMA with DFBA . By emplM representing the number of orthogonal roots, and δ, the Dirac delta function  flux changes from the wild-type flux distribution. In our approach, we combine ROOM with DFBA, naming this modeling approach R-DFBA. Here, the ROOM hypothesis is extended from a minimization of significant flux changes to a minimization of fluctuation of the profile of metabolite concentrations, according to the assumption of the basic M-DFBA approach. Unlike M-DFBA, in R-DFBA this is realized by minimizing the significant concentration changes at adjacent orthogonal roots. In contrast to M-DFBA, which captures smooth changes over time (by the Euclidean distance), R-DFBA prevents a large number of small (significant) changes over time. Instead, it allows a large concentration change at a few time points  and a relatively constant concentration of the metabolites elsewhere, which is bounded by thresholds determining the significant change. Finally, there are two possibilities to capture concentration changes between two time points--the continuous, specified by the usage of a distance measure , and the binary, based on an appropriate definition of significant change. The proposed formulation of R-DFBA captures the binary case, as the remaining alternative to specify dynamic changes. From a biological perspective, the possibility for large concentration changes in a small time range, as specified in R-DFBA, may be necessary to capture the difference in concentrations between two steady states in a bi-stable system -32.Two different types of the R-DFBA approach are considered: The first includes the basic ROOM approach with integer constraints, whereas the second relies on relaxing the integer constraints. Due to the relaxation of the integer constraints, the programming problem in ROOM becomes linear in comparison to mixed-integer linear program (MILP) in the basic ROOM approach . The firwu, wl are the thresholds determining significant changes of metabolite concentrations  and γx, εx are the relative and absolute ranges of tolerance, respectively.where Due to the constraints defining the thresholds, the programming problem becomes a mixed-integer nonlinear programming problem (MINLP). These problems are difficult to solve precisely as they couple the combinatorial nature of mixed-integer programming (MIP) problem and the computational complexity of solving NLP problems .Consequently, we formulate a second R-DFBA approach based on relaxed integer constraints, described as follows:γx, and εx) are set to be zero, due to the relaxed integer constraints for yi,j.Compared to the first formulation of R-DFBA, the relative and absolute ranges of tolerance  and γv, εv are the relative and absolute ranges of tolerance, respectively. Due to the inclusion of the flux rates and metabolite concentrations in the description of the thresholds, R-DFBACF is solved based on mixed-integer nonlinear programming, by the first R-DFBA formulation, and based on nonlinear programming, according to the second. This extension minimizes fluctuations of both, metabolite concentrations and flux rates, over time, similarly to M-DFBACF. In comparison, in R-DFBACF large concentration and flux rate changes between two adjacent orthogonal roots for some metabolites and/or reactions are possible, which is precluded in M-DFBACF.where Additionally, we consider a third objective function for the R-DFBA approach, whereby only fluctuation of the profile of flux levels of the network over time is minimized:CF allow a comparison of the necessity of stabilizing the system's profile of concentration, flux or a combination of both to predict dynamics in a metabolic network.The proposed optimization functions of the MINLP and NLP variants of R-DFBA, flux-based R-DFBA and R-DFBAATP and redox elements (NADP/NADPH) and supply of CO2: (1) carboxylation, during which the enzyme RuBisCo adds CO2 to ribulose-1,5-bisphosphate (RuBP) to get two molecules of phosphoglycerate (PGA), (2) reduction, converting the obtained PGA into 1,3-diphosphoglycerate (DPGA) and glyceraldehyde-3-phosphate (GAP), and (3) regeneration, which recovers RuBP after several intermediate steps from ribulose-5-phosphate (Ru5P) [2 are assumed to be positive and unbounded, yielding the simplified equations shown in Table The Calvin-Benson cycle consists of three phases relying on energy supply in form of e (Ru5P) ,35. The e (Ru5P) . In thise (Ru5P)  to descr of CO2:  carboxyl of CO2:  carboxyl1-v7) are investigated over a time period of ten seconds, which is sufficient due to the fast settling of the system in a steady state [With the simplified model of the Calvin cycle, we examine and compare the performance of the proposed R-DFBA approach together with its extensions, the existing M-DFBA approach and our proposed extensions, as well as the classical DFBA. In total, ten different approaches, based on different constraint-based formulations, are used in the analysis of the dynamics of metabolite and flux profiles in the Calvin cycle Figure . With eady state .For the analysis, the time period is divided into five finite elements, and the variables (metabolite concentrations and reaction rates) are parameterized at the roots of the fifth order Legendre polynomial. Maximizing the sink production is assumed as systemic objective for the DFBA approach, capturing the utilization of PGA and GAP to build the metabolite precursors necessary for sucrose and starch synthesis. For the remaining methods, the objective is modified to minimizing the fluctuation of the profile of metabolite concentrations/fluxes, according to the hypotheses of M-DFBA and R-DFBA.-1; PGA = 2.4 mmol l-1; DPGA, GAP, Ru5P, sink = 1.0 mmol l-1). In addition, for the DFBA-based approaches the limits for the GAP concentration over time are set to the outcome of the kinetic model with an added tolerance of +/- 0.1 mmol l-1. The solutions to MINLP formulation of R-DFBA (and its extensions) are obtained with γx = 0.4, γv = 0.2, and εx = 0.01, εv = 0.05, representing the relative and absolute ranges of tolerance, respectively. For each approach, we determine the residual sum of squares (RSS), quantifying the correspondence between the constraint-based and kinetic modeling approaches. Furthermore, we determine the temporal evolution of Kendall τ for the metabolic states of the constraint-based and kinetic modeling approaches in order to qualitatively examine the coupling between metabolic profiles and distribution of fluxes for the two types of modeling approaches.The results obtained by the different methods are compared to the outcome of a kinetic model of the Calvin cycle, described in . For theFigure The effect of minimizing the flux changes is especially pronounced in the two variants of the flux-based R-DFBA. Here, the results for the DPGA concentration are similar to those of the basic M-DFBA and its extensions. The behavior of the DPGA concentration predicted by both flux-based R-DFBA variants at the beginning of the simulation is very close to the kinetic modeling results, which can be explained by the coupling between fluxes and concentrations giving rise to particular temporal profiles.CF, which combines the basic and flux-based variant of R-DFBA. In the variants of R-DFBACF, the changes in flux rates and metabolite concentrations are weighted equally, as given by the objective function: CF could avoid the loss of the observed behavior. This suggest that the objective function may have a more intricate form  R-DFBAThe results from the DFBA approach for the DPGA and RuBP concentration over time are very similar, with small changes in the first three seconds, leading to almost zero concentrations at the end of the simulation. The predicted RuBP concentrations of the basic M-DFBA and its extensions are nearly constant over time; on the other hand, with the proposed NLP-based R-DFBA approach, changes in concentrations are obtained in the first half of the simulated time interval, followed by a constant concentration. The concentration of this metabolite at the end of the simulation differs between the proposed MINLP-based R-DFBA variants. We point out that experimental measurements in isolated cells and chloroplasts have resulted in low levels of RuBP -41.CF. To quantify these observations, we determine the residual sum of squares, by calculating the distance between kinetic modeling results and predicted results of the different approaches, see Table Comparing these results with the outcome of kinetic modeling shows that for these two metabolites MINLP-based R-DFBA predicts the most accurate temporal profiles. The constant concentration in the second half of the simulated interval of the NLP-based R-DFBA is very close to that of kinetic modeling, for both, the basic R-DFBA and R-DFBA-1 except for the flux-based variant, which also predict a nearly consumed Ru5P after four seconds similar to the kinetic-based results. Surprisingly, the predictions based on the MINLP-based R-DFBACF are inconsistent with all other approaches. Here, the minimization of significant fluctuation changes of flux levels and metabolite concentrations results in a short decrease followed by an increase of the Ru5P concentration. We believe that this behavior of the R-DFBACF variants is due to the over-constraining of both fluxes and concentrations.The simulated results of the concentrations of the other four metabolites are depicted in Figures S1 - S4 in Additional file In addition, the PGA concentration closest to the kinetic modeling results is predicted by the MINLP-based R-DFBA approach which minimizes significant flux changes. Contradictory to the results from all other methods, the flux-based R-DFBA, formulated as NLP, predicts an increase of the PGA concentration.Finally, we analyze the concentration of sink over time. Intriguingly, the sink concentrations are almost constant in the predictions of all NLP-based variants of M-DFBA and R-DFBA. However, for all MINLP-based R-DFBA variants as well as for the DFBA approach, an increase of the sink concentration is observed, which is in line with the kinetic modeling results. Quantitatively the basic MINLP-based R-DFBA approach predicts the best simulation results for the concentration of sink.Taken together, the results of the RSS-based analysis demonstrate that MINLP-based R-DFBA outperforms M-DFBA in four out of six cases in predicting metabolite concentrations over time.CF and R-DFBACF. Furthermore, it can be observed, that after four seconds for all NLP-based R-DFBA variants the reaction rates are close to zero.The flux levels over time, also known as time-resolved flux predictions, obtained by the different methods are depicted in Figures S5 - S11 in Additional file 1, v3, v4, v6 and v7 the predicted reaction rates by DFBA are bounded by the corresponding vmax; the latter do not constrain the predictions in the rest of DFBA-based methods.Interestingly, the reaction rates results obtained by DFBA are very close to the kinetic modeling results, as indicated by the RSS values in Table CF are closer to the kinetic results than all M-DFBA approaches, as quantified by the RSS.We argue that to maintain robustness in case of an internal perturbation, the system may opt to optimize additional objectives which are captured by the DFBA-based extensions. Indeed, we observe that the predicted rates obtained by the MINLP-based R-DFBAi.e., that the metabolite concentrations and flux levels change independently of each other. However, it is possible that other optimal solution may reside in the vicinity of the results from the kinetic modeling.In the results of the MINLP-based R-DFBA variants, we observe slight local fluctuation which may be attributed to the presence of a large null space in the process of searching an optimal solution of the objective function. A higher-order Legendre polynomial and, therefore, more orthogonal roots in each finite element may reduce these fluctuations. Altogether, these results indicate the decoupling of metabolite concentrations and flux rates in the determined solutions, CF based on MINLP as well as M-DFBACF.To qualitatively examine the extent to which both metabolic concentrations and flux distributions are in agreement with the outcome of kinetic modeling over time, we determine the Kendall τ correlation for the corresponding metabolic states. The results in Figure CF, outperform the rest of the investigated approaches.In addition, these results demonstrate that while the NLP-based R-DFBA methods yield good predictions of metabolite concentrations, the predicted flux distributions over time are decoupled from the metabolite profiles. Interestingly, the same observations for the decoupling of fluxes from metabolite concentrations hold for all M-DFBA-based approaches, except the flux-based M-DFBA. These findings indicate that some results of DFBA-based approaches warrant caution in biological interpretation and applications to metabolic engineering via flux control. Due to the highly coupled regulation of flux and concentrations in metabolic networks , it is nC3-plants with the help of the ten implemented methods. In parallel to the wild type, analyses are performed for the inv4 mutant, which is defective in the vacuolar invertase gene, AtβFruct4 (AT1G12240). Like in the previous section, the performance of the methods to accurately simulate the dynamics of the metabolite concentrations and fluxes is determined based on the RSS values between the results of the constrained-based approaches and the kinetic models for the wild type plant and the inv4 mutant .The models for both, wild type and mutant, include six metabolites, namely: glucose, fructose, sucrose, sugar phosphates, and starch, as well as a combined sink export, which are interconverted through seven reactions , used to obtain the aforementioned predications, remains to be established with the help of experimental data of in vivo metabolite concentrations and intracellular reaction fluxes. However, designing and conducting experiments from which non-stationary metabolite concentrations and reaction fluxes are to be measured is an extremely challenging task. For instance, decades of experimental work for quantifying the absolute concentrations of photosynthetic intermediates  and end-Escherichia coli includes 2,251 metabolic reactions, and 1,136 unique metabolites [i.e., they belong to the class of NP-hard problems [The analysis of genome-scale networks is the focus of current modeling strategies with many different applications, including metabolic engineering and drug targeting . Furtherabolites . An accuproblems ,48). TheIn the present work, we proposed a new approach to analyze the dynamic adjustment of metabolic networks, called R-DFBA. R-DFBA combines the dynamic FBA with regulatory on/off minimization by minimizing the total number of significant metabolite concentration changes in comparison to flux changes in the classical approach. Additionally, we extended this method and the M-DFBA approach by considering not only the fluctuation of the metabolite concentration profiles but also those of the flux levels in the network. We introduced seven new approaches and, in total, ten constraint-based approaches were implemented and their accuracy was compared with the outcome of two kinetic models--for the Calvin-Benson cycle and for the plant carbohydrate metabolism.For the models of the Calvin-Benson cycle and the plant carbohydrate metabolism, we demonstrated that our proposed approaches yielded results in accordance with predictions from kinetic modeling, specifically for the case of metabolite concentrations. In addition, we demonstrated that the (MI)NLP-based R-DFBA approach captures the dynamic coupling of reaction fluxes and metabolite concentrations. Therefore, constraint-based approaches in combination with collocation on finite elements offer a promising framework for analysis of metabolic network dynamics without specifying the details of enzyme kinetics. The proposed approaches outperform the existing variants in several cases and are suitable for positing model-based hypotheses for the dynamics of metabolic pathways when little enzymatic details are available. Finally, our findings suggest that minimizing the combination of flux and metabolite concentration fluctuations is the mechanism most likely responsible for maintaining the metabolic network robustness due to internal perturbations.i.e., metabolite and reaction flux) is defined as follows:The results of applying the described methods are compared with the outcome from kinetic modeling with the help of residual sum of squares (RSS). The RSS for each system element (yj is the result of the kinetic modeling and f(xj) of the compared approaches at the orthogonal root j.where The Kendall rank correlation coefficient, denoted by τ, evaluates the degree of similarity between two sets of ranks given to a same set of objects . This coAll mathematical programming approaches are implemented in MATLAB 7.11.0, R2010b with the optimization platform TOMLAB v7.7 . We use SK and ZN designed the study and developed the mathematical formulation of the methods. SK carried out the implementation and computational analysis. SK and ZN performed the statistical and comparative analyses, and were both involved in writing the manuscript. All authors read and approved the final manuscript.Tutorial Orthogonal Collocation on Finite Elements. Tutorial for the orthogonal collocation on finite elements is included. An illustrative example demonstrating the usage of this method in approximating solutions to ODEs is also presented.Click here for fileAdditional Figures. Modeling results of concentrations of Ru5P, PGA, GAP and sink as well as results of reaction rates of v1-v7 in the simplified model of the Calvin cycle by the different approaches.Click here for fileAdditional results Photosynthesis Model. Results from the RSS analysis of the proposed methods for a lumped model of C3-plant carbohydrate metabolism are presented. The model includes 6 metabolites and 7 reactions, as depicted in Figure S12. In addition, the ODEs for the kinetic model used in establishing the RSS results are presented.Click here for file


The results of this study confirm that in organisms which are highly adapted to their environment robustness to mutations may decrease compared to other organisms.Robustness is the key feature of biological networks that enables living organisms to keep their homeostatic state and to survive against external and internal perturbations. Variations in environmental conditions or nutrients and intracellular changes such as genetic mutations have the potential to change stability and efficiency of an organism. Structural robustness helps biological systems to choose alternative routes of adaptation to varying conditions. In this study, in order to estimate the structural robustness in metabolic networks we presented a novel flux balance-based approach inspired by bond percolation theory. Fourteen  During the last years, many researchers have studied the robustness of metabolic networks against random mutations . The puRobustness is defined as the “insensitivity” of a system to parametric variations . VariatiIn the present work, we introduce a novel approach to the analysis of metabolic network robustness. We study the resistance of metabolic networks to deletion of reactions by removing reactions until no flux can pass through the network. We show that eukaryotes and free-living prokaryotes show much higher mutational robustness compared to organisms which are highly adapted to their habitats.The genome-scale metabolic network models of 14 species are used in this study, including 3 eukaryotes (group 1), 6 “free-living” prokaryotes (group 2), and 5 prokaryotes with highly specific growth conditions (group 3) , 28, 30.v, the metabolic concentrations do not change during time. In a metabolic network with m metabolites and n reactions, this assumption is equivalent to the following equation:S is an m × n matrix representing stoichiometric coefficients of metabolites in the reactions, v is the vector of the n steady-state fluxes, and 0 is an m-dimensional zero vector. Blocked reactions [i, we have vi = 0 subject to stoichiometric constraints (S · v = 0) and reversibility constraints . Finding blocked reactions is typically the first step of flux coupling analysis [We used constraint-based analysis of metabolic networks in our study . In thieactions  are thosanalysis , 33. In analysis  for thisOur algorithm is inspired by the concept of percolation. For more information, the interested reader may refer to , 36. HerIn each iteration of the experiment, a steel wire (a “bond”) is cut . The electric conductance of the lattice gradually decreases by cutting the wires. The idea is to cut steel wires randomly until no electrical current can pass through the mesh.P be the ratio of unblocked bonds to the total number of bonds. On average, when bonds are cut, at a critical value, say PC, conductivity of the lattice vanishes to zero [PC is a random variable which can be estimated by repeating the experiment several times.Let  to zero . TherefoThe method used in present study is based on solving a sequence of linear programming (LP) problems. In our algorithm, we used F2C2  to studyPC values were computed for each of the metabolic network models.The experiment is repeated 100 times for each network, and average in silico deletion of reactions. In each iteration, a reaction is deleted from the network and the sensitivity of the growth rate to the reaction deletion is modeled. We used the core reductive algorithm [We also compared our results with a classical measure of metabolic network robustness  based onlgorithm , 9, 40 fR package (http://www.r-project.org/) was used for statistical analyses. In order to compare the PC distributions in two organisms, one-sided two-sample t-test was used. To investigate the correlation between PC values and the number of reactions in the models, Pearson's product-moment correlation test was applied.The PC was defined as the critical ratio of the fluxes to be removed such that a metabolic network becomes entirely blocked  are significantly greater than PC values of free-living prokaryotes (group 2) and whether the PC values of free-living prokaryotes (group 2) are significantly greater than PC values of prokaryotes with highly specific growth conditions. The results are summarized in To investigate the significance of differences between PC is positive (R2 = 0.26), it is not statistically significant . Therefore, there is only a weak, if any, relationship between the number of reactions and network robustness. This finding emphasizes the importance of the network “structure” and “wiring” (as an intrinsic property in each metabolic network) in shaping the network mutational robustness.We also tested whether the number of unblocked reactions (and not the network structure) determines the network robustness. We found that although the correlation between the number of reactions and S. cerevisiae, A. nidulans, and A. thaliana). It is well known that S. cerevisiae and A. nidulans can adapt to a wide range of growth conditions. Moreover, A. thaliana is a multicellular organism with different tissues. For these reasons, in group 1 we expect a large number of alternative metabolic pathways, which in turn results in great robustness values. It should be noted that, in the metabolic model of A. thaliana, only 672 unblocked reactions are included. However, due to the large number of metabolic enzymes involved in the plants, this number is greatly underestimated. One expects that addition of the missing metabolic pathways to this model will greatly enhance the robustness of this network.In this study, group 1 includes eukaryotic species . The results confirm that our novel robustness measure is comparable with the classical measures of robustness, but with the advantage that no additional assumption is required for its computation.In order to find the relationship between our novel robustness measure and the FBA-based measure of robustness, we used our recent implementation of the core-reductive algorithm  to obtai


Especially, the combined analysis and visualization of data from different high-throughput technologies remains a key challenge in bioinformatics. We present here the MarVis-Graph software for integrative analysis of metabolic and transcriptomic data. All experimental data is investigated in terms of the full metabolic network obtained from a reference database. The reactions of the network are scored based on the associated data, and sub-networks, according to connected high-scoring reactions, are identified. Finally, MarVis-Graph scores the detected sub-networks, evaluates them by means of a random permutation test and presents them as a ranked list. Furthermore, MarVis-Graph features an interactive network visualization that provides researchers with a convenient view on the results. The key advantage of MarVis-Graph is the analysis of reactions detached from their pathways so that it is possible to identify new pathways or to connect known pathways by previously unrelated reactions. The MarVis-Graph software is freely available for academic use and can be downloaded at:  High-throughput technologies notoriously generate large datasets often including data from different omics platforms. Each dataset contains data for several thousand experimental markers, e.g., mass-to-charge ratios in mass spectrometry or spots in DNA microarray analysis. An experimental marker is associated with an intensity profile which may include several measurements according to different experimental conditions . UsuallyArabidopsis thaliana in the KEGG database  consists of vertices V and edges L = V × V. The vertices represent the following groups of biochemical entities:To represent full metabolic networks, especially in combination with experimental data, a flexible and expandable data structure is required. MarVis-Graph models the metabolic network as an undirected graph in which each entity  is represented by a single vertex. Mathematically, the graph •M: experimental markers for metabolites (metabolite markers),•C: metabolites,•R: reactions,•E: enzymes,•H: genes,•T: experimental markers for transcripts (transcript markers),•P: pathways.A minimal example of a graph containing only one vertex of each subset is shown in v1, v2 is added to the graph if the vertices are related within the metabolic network, e.g., a metabolite is a substrate or product of a reaction. Edges are represented as tuple l =  and the set of edges is restricted to An edge between two vertices ions see .MarVis-Graph has a built-in support for the creation of metabolic networks from the KEGG database and the BioCyc collection (source databases). The networks can either consist of all reactions from a database (reference network) or only reaction from a specific organism. Data from the KEGG database is downloaded directly via the KEGG API . Databas® Excel® spreadsheet format  files (for an example see marker annotation) may also be provided.Experimental markers can be imported from tabular data files, e.g., comma-separated-values (.csv) or Microsoftp-Value for each experimental marker based on its abundances in the different conditions of the experiment. For simplicity, weights for MarVis-Graph can be calculated based on these p-Values with A critical step is the calculation of weights for the experimental markers because of their influence to the extraction of sub-networks (see “Identification of sub-networks”). Usually, statistical analysis tools calculate a m/z-value) and specific isotope- or adduct-corrections . Note that matching metabolite markers to metabolites by mass-comparison is error-prone and may result in many false matches. Therefore, it is preferable to calculate the exact compound using sophisticated technologies  is specific for a gene and can therefore be used for annotation. For other technologies an annotation has to be provided by external tools.In MarVis-Graph, each reaction is scored initially based on the associated experimental data . This initial scoring is refined (see “Refining the scoring”) and afterwards reactions with a score below a user-defined threshold are removed. The network is decomposed into subsequent high-scoring reactions that constitute the sub-networks.The weight of each experimental marker  is equally distributed over all metabolites and genes associated with the metabolite marker or transcript marker, respectively. For all vertices, this is repeated as illustrated in Datasets from high-throughput technologies might not cover all metabolites or transcripts, e.g., if•they are not measurable: specific metabolites may not be detected by mass spectrometry analysis. Furthermore, several metabolites exist only for a short period of time within a protein complex that catalyze more than one reaction step,•they are filtered out by statistical analysis: transcripts may be equally expressed throughout all experimental conditions if the corresponding products are required all the time. This is especially true for enzymes that are not rate-limiting. The amount of a metabolite might not change across the different conditions when it is metabolized immediately.random walk with restart (RWR) algorithm (reaction graph) is constructed from the metabolic network. Two reactions in this graph are connected if they share a metabolite as substrate or product. Often, reactions are connected solely by “hub metabolites”, e.g., NADP and ATP, that take part in a high number of reactions . After convergence of the algorithm, reactions with a score lower than the user-defined threshold t  are removed from the reaction network. During the removal process, the network is decomposed into pairwise disconnected sub-networks containing only high-scoring reactions.The initial reaction scores are used as input scoring for the random walk algorithm. The algorithm is performed as described by G′ =  with V′ = M′∪C′∪R′∪E′∪G′∪T′∪P′.In the following, a resulting sub-network is denoted by a prime: The identified sub-networks are ranked with one of the following scoring methods and presented in a sorted list.The size of a sub-network is the total number of contained reactions: G′, the diameter is the maximum distance between all pairs of reactions ir, jr∈R′ whereas the distance d is the length of the shortest path: The graph diameter is an intuitive description of the dimension of a network. For a given sub-network The weights of the experimental markers are distributed to the reactions as described in the previous section. The sub-network score is the sum of the reaction scores in the sub-network: XSwap permutation  and false-discovery-rate (FDR) are calculated for each originally identified sub-network.A. thaliana wound response. Data from a metabolite fingerprinting  knock-out mutant dde-2-2  that cannot directly be estimated. In this case-study, the parameters for MarVis-Graph were set to•r = 0.8.restart-probability The restart-probability of the RWR algorithm controls the amount of score that is distributed equally to the neighboring reactions, i.e., •t = 0.2.score-threshold The score-threshold determines the reactions that are considered for sub-network construction after performing the RWR algorithm and directly depends on the weights of the experimental markers given on import. When using a restart-probability of 0.8, a score-threshold of 0.2 keeps only nodes that are high-scoring by themselves, have a very high-scoring neighbor, or are enclosed by several high-scoring neighbors see .•c = 10.hub metabolite-threshold The hub metabolite-threshold was chosen based on expert knowledge: a threshold of 10 was just low enough to eliminate known hub metabolites, e.g., ATP, in the AraCyc database.Ss, diameter Sd, and sum-of-weights Ssow scores  appeared as top candidate. Therefore, it was investigated further and discussed in detail in this study. This sub-network is constituted of reactions from different pathways related to fatty acids. In all rankings, the sub-network jasmonic acid (jasmonate). The presence of this pathway is very well established because of its central role in mediating the plants wound response -jasmonic acid methyl ester  is a volatile, airborne signal mediating wound response between plants -hydroperoxy-9(Z),11(E),15(Z)-octadecatrienoate, is shared with the traumatin biosynthesis. In the traumatin biosynthesis, linoleate (see paragraph “Poly-hydroxy fatty acids biosynthesis”) is degraded too. Both, linoleate and α-linolenate, are metabolized to traumatin and the reactions are likely to occur in parallel because 13S-lipoxygenase enzymes catalyze both reactions.The first reaction of the 12-fatty acid dehydrogenaseΔ2http://pmn.plantcyc.org/ARA/NEW-IMAGE?&object=1.14.99.33-RXNAvailable at  (accessed 14 May 2013). does not exists in A. thaliana, but the ω-3-fatty acid desaturase (3http://pmn.plantcyc.org/ARA/NEW-IMAGE?&object=1.14.99.33-RXNAvailable at  (accessed 14 May 2013). from which this metabolic network was built. Furthermore, the reaction’s product, crepenynic acid, does not exists. The ω-3-fatty acid desaturase should catalyze a reaction from linoleate to α-linolenate. Metabolite markers that match the mass of crepenynic acid do also match α-linolenate because both molecules have the same sum-formula and monoisotopic mass.The enzyme saturase  is annotsaturase  as it ca12-fatty acid dehydrogenaseΔ prior to the release of AraCyc version 11.0.As mentioned above, MarVis-Graph compiled the metabolic network for this study from the AraCyc database version 10.0. On June 4th, a curator changed the database to remove the A. thaliana wound response resulted in a convenient graphical representation of high-throughput data which allows the analysis of the complex dynamics in a metabolic network.The presented new software tool MarVis-Graph supports the investigation and visualization of omics data from different fields of study. The introduced algorithm for identification of sub-networks is able to identify reaction-chains across different pathways and includes reactions that are not associated with a single pathway. The application of MarVis-Graph in the case study on http://marvis.gobics.de/marvis-graph.The MarVis-Graph tool is implemented in Java 7, released under the terms of the GPL v3.0  and can 10.7717/peerj.239/supp-1Data O1p-values and to rank the markers of both datasets. The datasets were cut at 25 percent (metabolite markers) and 10 percent (transcript marker) respectively.The metabolic and transcriptomic datasets were preprocessed with MarVis-Filter to obtain Click here for additional data file.10.7717/peerj.239/supp-2Data S2Metabolic network model compiled from the AraCyc 10 database with the top 10 percent metabolite marker and top 25 percent transcript marker.Click here for additional data file.10.7717/peerj.239/supp-3Data S3The installer needs to be extracted and the executed. The installer was tested on Ubuntu Linux 13.04 and Windows 7.Click here for additional data file.10.7717/peerj.239/supp-4Table S4All detected sub-networks were scored by size, diameter, and sum-of-weight. Additionally, the number of metabolite marker and transcript marker within the sub-networks are listed.Click here for additional data file.10.7717/peerj.239/supp-5Figure F5Some common reaction-scoring patterns in the reaction graph (see section “Refining the scoring”) before (drawn in the nodes if non-zero) and after (below nodes) application of the random-walk-with-restart (RWR) algorithm. The parameters were set to r = 0.8, t = 0.2 (see section “Resulting sub-networks”). The resulting sub-networks are marked by a rectangle. a) A single reaction with a score of one, e.g., from the weight of one transcript marker, will not be enough to shift its neighbors above the threshold. b) Two reactions with a score of one are still not enough to close the gap. c) If one of the two reactions has additional evidence, e.g., from a metabolite marker with several annotations, the two reactions can shift the gapping reaction above threshold. d) If two reactions with a score of one enclose a reaction with a low score , these reaction will constitute a sub-network.Click here for additional data file.


Whilst this assumption may be valid in the case of microorganisms growing under certain conditions, it is likely invalid in general, and especially for multicellular organisms, where cellular objectives differ greatly both between and within cell types. Moreover, for the purposes of biotechnological applications, it is normally the flux to a specific metabolite or product that is of interest rather than the rate of production of biomass Saccharomyces cerevisiae cultures under two growth conditions, the method outperforms traditional approaches for predicting experimentally measured exometabolic flux that are reliant upon maximisation of the rate of biomass production.An alternative objective function is presented, that is based upon maximising the correlation between experimentally measured absolute gene expression data and predicted internal reaction fluxes. Using quantitative transcriptomics data acquired from Due to its improved prediction of experimentally measured metabolic fluxes, and of its lack of a requirement for knowledge of the biomass composition of the organism under the conditions of interest, the approach is likely to be of rather general utility. The method has been shown to predict fluxes reliably in single cellular systems. Subsequent work will investigate the method’s ability to generate condition- and tissue-specific flux predictions in multicellular organisms. The applications of genome-scale metabolic modelling have increased over recent years, as have the number of metabolic models available and the diversity of organisms that such reconstructions cover . Traditiet al.  and iMAT [14].Click here for fileGene expression mapper for enzymatic complexes (A and B).Helper Matlab function that is used in mapping gene expression data to individual reactions. Called by analysis function.Click here for fileGene expression mapper for isoenzymes (A or B).Helper Matlab function that is used in mapping gene expression data to individual reactions. Called by analysis function.Click here for file


Saccharomyces cerevisiae is a promising host for cost-effective biorefinary processes due to its tolerance to various stresses during fermentation, the metabolically engineered S. cerevisiae strains exhibited rather limited production of higher alcohols than that of Escherichia coli. Since the structure of the central metabolism of S. cerevisiae is distinct from that of E. coli, there might be a problem in the structure of the central metabolism of S. cerevisiae. In this study, the potential production of higher alcohols by S. cerevisiae is compared to that of E. coli by employing metabolic simulation techniques. Based on the simulation results, novel metabolic engineering strategies for improving higher alcohol production by S. cerevisiae were investigated by in silico modifications of the metabolic models of S. cerevisiae.While E. coli strains is derived from the flexible behavior of their central metabolism. Reducing this flexibility by gene deletion is an effective strategy to restrict the metabolic states for producing target alcohols. In contrast, the lower yield using S. cerevisiae originates from the structurally limited flexibility of its central metabolism in which gene deletions severely reduced cell growth.The metabolic simulations confirmed that the high production of butanols and propanols by the metabolically engineered S. cerevisiae was improved by the introduction of E. coli genes to compensate the structural difference. This suggested that gene supplementation is a promising strategy for the metabolic engineering of S. cerevisiae to produce higher alcohols which should be the next challenge for the synthetic bioengineering of S. cerevisiae for the efficient production of higher alcohols.The metabolic simulation demonstrated that the poor productivity of  Escherichia coli strains that produce butanols and propanols. Due to insufficient genetic tools for the natural 1-butanol producer clostridia [E. coli has been engineered for 1-butanol fermentation by introducing the CoA-dependent clostridial pathway [The bioproduction of higher alcohols with more than 3 carbon atoms is desirable because they have the preferred properties of renewable liquid fuels and wide applications in commodity chemicals. In designing fermentation processes for mass production, the development of recombinant microbial strains is a critical first step . The pioostridia , E. coli pathway . The pro pathway ,6,8. The pathway ,10. The  pathway -22. This pathway . In thesSaccharomyces cerevisiae). Yeast is a promising host for cost-effective biorefinary processes due to its tolerance to various stresses during fermentation. Tolerance to high concentrations of alcohol should be useful for the industrial production of higher alcohols [S. cerevisiae expressing enzymes involved in the CoA-depending clostridial pathway yielded only 2.5 mg/L of 1-butanol [S. cerevisiae. The structure of the central metabolism of S. cerevisiae is distinct from that of E. coli in terms of the anaplerotic pathways, cytosolic acetyl-CoA (AcCoA) biosynthesis, fermentation products, and occurrence of mitochondria . FBA is a mathematical approach for analyzing the flow of metabolites through a metabolic network [in silico analysis was applied for the backbone models of both E. coli and S. cerevisiae to highlight differences between their metabolic networks. The comparative analyses revealed that the central metabolism of S. cerevisiae has limited potential for the efficient production of higher alcohols in comparison with that of E. coli. It is also predicted that its poor productivity can be improved by extension of the central metabolic network.In this study, the potential production of higher alcohols by  network -32. The  network ,34. Alth network ,33. In tE. coli and S. cerevisiae were employed for this analysis. The backbone metabolic models, which describe the essential structures of the central metabolism, were constructed from information from both the literature [E. coli and S. cerevisiae are hereafter designated as iBKEco52  and iBKSce50, respectively. In iBKSce50, the metabolic network is compartmentalized in the cytosol and mitochondria. Although the backbone models contain only 50 or 52 reactions, their behavior was similar to that of genome-scale models of E. coli (iJR904) and S. cerevisiae (iMM904)[E. coli and S. cerevisiae.The backbone metabolic models of terature -41 and dterature -44 ,46 . In the analysis, reactions instead of genes were removed from the metabolic models. The simulation results were verified by comparisons with literature data. The metabolic engineering of E. coli demonstrated that the product yield of a recombinant E. coli strain expressing genes for 1-butanol or isobutanol biosynthesis was enhanced by deletions of genes such as ΔadhE, ΔldhA, ΔfrdBC, and Δpta as well as ΔadhE, ΔldhA, ΔfrdBC, and Δpfl, respectively [Cmol/Cmol glucose), respectively, by the deletions of the corresponding reactions.Via this methodology, the production of higher alcohols by ectively ,8. The rE. coli backbone model. Among the 8,668 viable and 1-butanol-producing models, an elite 347 models were considered "proper strains" in which all deletions contributed to an improvement in the 1-butanol yield [E. coli and S. cerevisiae backbone models were 34,880 and 690 respectively, suggesting that the behavior of the central metabolism of S. cerevisiae is more restricted than that of E. coli ,36,56. Bli Table .E. coli and S. cerevisiae backbone models [E. coli and S. cerevisiae backbone models were 0.117 and 0.092, respectively, indicating that the metabolites in the E. coli network are more densely connected by the metabolic reactions  and iBKSce50+7Δpgi  were able to produce 1-butanol with better yields than the most of E. coli models  and the merged compartment (iBKSce50Δmit) had excellent production capabilities for the target alcohols [S. cerevisiae (iMM904)[http://gcrg.ucsd.edu. The backbone models of E. coli (iBKEco52) and S. cerevisiae (iBKSce50) were reconstructed using the information from the literature [http://ecocyc.org/)[http://www.comp-sys-bio.org/yeastnet/)[The genome-scale models of  GSM/GPR) and S. c (iMM904) were dowterature -41 and tcyc.org/) and YEASeastnet/),44 (Suppeastnet/).-1 h-1 in all analyses. (2) To avoid indefinite results, the flux for target production is minimized under the maximum growth rate condition. (3) The growth rates and target productivities were determined at 16 different oxygen uptake rates . Those averages were considered the metabolic performance of the metabolic models. When the levels of growth speed were more than 1% of that of wild type, those deletion mutants were considered viable. The target-producing strains indicated the viable mutants producing target alcohols in yields exceeding YCmol/Cmol glucose = 0.01. Because most deletions in the target-producing strains are silent or have adverse effects, we defined "proper" strains as target-producing strains in which all deletions contributed to the improvement of product yield. The network density and the characteristic path length were determined with the aid of Network Analyzer version 2.7 [The FBA-based metabolic simulation was performed via previously described methods  using MAsion 2.7 .The authors declare that they have no competing interests.FM and CF performed the model construction and the metabolic simulations. FM, JI, and TK analyzed the data. SH and AK designed the study. JI, TK, and FM wrote the paper. All authors read and approved the final manuscript.The E. coli backbone model iBKEco52 and the S. cerevisiae backbone model iBKSce50.Click here for filein silico metabolic modelsFermentation profiles of the .Click here for filePathways for higher alcohol biosyntheses inserted into the backbone models iBKEco52 and iBKSce50.Click here for fileE. coli (iBKEco52) and S. cerevisiae The numbers of viable, target-producing, and proper deletion mutants obtained from all single, double, triple, and quadruple deletion mutants generated from the backbone models of .Click here for fileE. coli (iBKEco52) and S. cerevisiae Metabolic simulations of all single, double, triple, and quadruple deletion mutants generated from the backbone models of .Click here for fileA comparison of the predicted growth rates of all single-reaction-deleted mutants with the experimental data.Click here for file


In order to reduce time and efforts to develop microbial strains with better capability of producing desired bioproducts, genome-scale metabolic simulations have proven useful in identifying gene knockout and amplification targets. Constraints-based flux analysis has successfully been employed for such simulation, but is limited in its ability to properly describe the complex nature of biological systems. Gene knockout simulations are relatively straightforward to implement, simply by constraining the flux values of the target reaction to zero, but the identification of reliable gene amplification targets is rather difficult. Here, we report a new algorithm which incorporates physiological data into a model to improve the model’s prediction capabilities and to capitalize on the relationships between genes and metabolic fluxes.via “GR constraints”. This method scans changes in the variabilities of metabolic fluxes in response to an artificially enforced objective flux of product formation. The gene amplification targets predicted using this method were validated by comparing the predicted effects with the previous experimental results obtained for the production of shikimic acid and putrescine in Escherichia coli. Moreover, new gene amplification targets for further enhancing putrescine production were validated through experiments involving the overexpression of each identified targeted gene under condition-controlled batch cultivation.We developed an algorithm, flux variability scanning based on enforced objective flux (FVSEOF) with grouping reaction (GR) constraints, in an effort to identify gene amplification targets by considering reactions that co-carry flux values based on physiological omics data E. coli, in addition to the comparison with the previously reported experimental data. The FVSEOF strategy with GR constraints will be generally useful for developing industrially important microbial strains having enhanced capabilities of producing chemicals of interest.FVSEOF with GR constraints allows identification of gene amplification targets for metabolic engineering of microbial strains in order to enhance the production of desired bioproducts. The algorithm was validated through the experiments on the enhanced production of putrescine in  The advent of high-throughput technologies and the completion of genome sequencing for many organisms have led to an explosion of systems-wide biological data [One of the most ambitious goals in metabolic engineering is the design of biological systems based on cal data ,2. Genomcal data ,4. Some cal data ,6 and chcal data -9; othercal data -12. In mEscherichia coli[Increasing the expression levels of the relevant genes has also been successfully employed for the overproduction of target chemicals ,14. To achia coli. These achia coli,28. One Con/off) and the flux scale  of the metabolic reactions accurately predicted flux distributions in gene knockout mutant strains [In this study, in order to systematically handle the large flux solution spaces, as also revealed in the implementation of FSEOF , we cons strains .E. coli genome-scale metabolic model. The model then facilitated the scanning of changes in the variability among metabolic fluxes using FVA in response to the enforced enhancement of the fluxes toward a target chemical. This newly developed method, called flux variability scanning based on enforced objective flux (FVSEOF) with GR constraints, was employed in this study to identify gene amplification targets for the production of target chemicals. FVSEOF with GR constraints was first validated based on amplification targets reported for the production of shikimic acid and putrescine in E. coli, and then further validated by actually engineering E. coli for the enhanced production of putrescine based on new amplification targets.Based on these analyses, the grouping reaction (GR) constraints that constrain reactions to co-carry fluxes altogether regardless of the condition were incorporated into the E. coli metabolic network model, iJR904 [E. coli biomass composition [EcoMBEL979 was used throughout this study , which i, iJR904 . EcoMBELposition .E. coli genome-scale metabolic model were balanced under the pseudo-steady state assumption. The balanced reaction model was almost always underdetermined in calculations of the flux distribution due to insufficient measurements of the extracellular fluxes. Thus, the unknown fluxes within the metabolic reaction network were calculated by linear programming-based optimization using an objective function that maximized the growth rate, subject to constraints pertaining to mass conservation and reaction thermodynamics [The stoichiometric relationships among the metabolites and the reactions of the dynamics , This opSij represents the stoichiometric coefficient for metabolite i in reaction jνj is the flux of reaction jJ is the set of all reactions, and bi is the net transport flux of metabolite i. If this metabolite is an intermediate, bi is equal to zero. αj and βj are the lower and upper bounds of the flux of reaction j, respectively. Herein, the flux of any irreversible reaction is considered to be positive; a negative flux indicates the reverse direction of a reaction.where Con/off) can be described as follows:The algorithm introduced in this study, FVSEOF with GR constraints, starts with formulation of GR constraints, which are based on the genomic context and flux-converging pattern analyses Figure . Brieflyy( v1) and y( v2) indicate binary variables (on or off) of a certain reaction 1 and 2, respectively.where CxJy index, determined by flux-converging pattern analysis. Cx and Jy denote the total number of carbon atoms in metabolites that participate in each reaction and the number of passing flux-converging metabolites, respectively. Here, it should be noted that cofactors were not considered because the flux scales are controlled by the carbon number of primary metabolites, not cofactors, according to 13C-based flux analysis [Jy, the flux-converging metabolites indicate metabolites at which two pathways split by another metabolite converge. Jy has four types, including JAJBJC, and JD, depending on the characteristics of flux-converging metabolites. Subscript of Jy denotes the passing number of flux-converging metabolites, counting zero, one, two, or three times for the flux coming from a carbon source. In some cases, the subscript E is placed next to the subscripts of A, B, C, or D to indicate the fluxes derived from pyruvate, which causes more complex changes in flux distributions. The values of CxJy for each reaction were assigned based on possible flux routes reaching from glucose, and are partitioned by a slash. Based on this analysis, another constraint Cscale, indicating the flux scale of a reaction, can be given to the metabolic reactions. First, terms used to describe the flux scale of the reaction are as follows:Each reaction is then given a analysis . For Jy,jj, and NC,Rj the total number of carbon of primary metabolites without cofactors in reaction j.where Cscale is applied to these two reactions, which is defined as follows:If reaction 1 and 2 were predicted to be in the same functional unit according to the genomic context analysis, and their vn1 and vn2 are the normalized flux of reaction 1 and 2, obtained by dividing each reaction flux by the carbon source uptake rate, such as glucose. δ is the constant defining the flux level of reactions in this functional unit; the value of δ is recommended as 0.3.where Once grouping reaction constraints are defined, FVSEOF with GR constraints is subsequently performed as follows Figure . First, Min/Max li and ui are the lower and upper bound for the net transport flux of metabolite i, respectively, and where Z( vbiomass), was maximized while gradually increasing the target product formation rate from its initial  flux value to its near theoretical maximum: nth of the difference between the k steps.Second, the cell growth rate, 1 (n≥10) . The vtaZ( vintracellular reaction), with additional constraints: the enforced production rate of the target bioproduct, which varied from its initial to maximum values in 10 steps, and 95% optimal cell growth rate, vbiomass = 0.95 · Z( vbiomass)opt, for each step. The attainable flux ranges of intracellular reactions for each step were subsequently subjected to the targeting criteria introduced in the following section.Third, FVA was carried out with GR constraints by maximizing or minimizing the fluxes of all intracellular reactions, FVSEOF with GR constraints was calculated using mixed integer nonlinear programming with the DICOPT solver, subject to the constraints including GR constraints, mass conservation and reaction thermodynamics.Vavg), its slope (qslope) and flux capacity (lsol) were employed as targeting criteria for the initial set of gene amplification targets predicted from FVSEOF with GR constraints  into the site between the EcoRI and SacI restriction enzyme sites of pTac15K. The plasmid contained a kanamycin resistance selective marker. Cells were grown in Luria–Bertani (LB) broth or on plates containing appropriate antibiotics at 37°C for the construction of strains and plasmids. Antibiotics were added at following concentrations: 50 μg/mL ampicillin, 25 μg/mL kanamycin, and 35 μg/mL chloramphenicol.The e strain . E. coliPfu DNA polymerase was purchased from Solgent . Restriction enzymes and T4 DNA ligase were obtained from New England Biolabs  and Roche , respectively. The genomic DNA of E. coli W3110 was amplified to overexpress the target genes using the Pfu polymerase and PCR primers 2SO4. The R/2 medium (pH 6.8) contained (per liter): 2 g (NH4)2HPO4, 6.75 g KH2PO4, 0.85 g citric acid, and 0.7 g MgSO4·7H2O. In addition, 5 mL/L of a trace metal stock solution [4·7H2O, 2.25 g ZnSO4·7H2O, 1 g CuSO4·5H2O, 0.5 g MnSO4·5H2O, 0.23 g Na2B4O7·10H2O, 2 g CaCl2·2H2O, and 0.1 g (NH4)6Mo7O24. One milliliter of the overnight culture was transferred into a 300 mL Erlenmeyer flask containing 50 mL of the R/2 medium at 37°C and spun at 220 rpm in a shaking incubator (JEIOTech. Co. SI-900R). After obtaining an initial OD600 of 0.3, the seed cultures (200 mL) were introduced into the bioreactor for batch cultivation. The culture pH was maintained at 6.8 by the addition of 6 M KOH. The dissolved oxygen concentration was maintained at 20% air saturation by automatically adjusting the agitation speed. Under the comparable batch culture conditions, the single gene-overexpressing strains based on the E. coli XQ52 strain harboring p15SpeC, denoted as XQ52 (p15SpeC), with each target gene were tested by flask cultivation in duplicate using R/2 medium supplemented with 10 g/L glucose at 37 °C.Batch cultivation was conducted at 37°C in a 6.6 L jar fermentor  containing 2 L R/2 medium supplemented with 10 g/L glucose and 3 g/L (NHsolution  was adde600) using an Ultrospec 3000 spectrophotometer . Glucose concentrations were measured using a glucose analyzer . The concentrations of glucose and organic acids were determined by high-performance liquid chromatography  equipped with UV/visible light  and refractive index  detectors. A MetaCarb 87H column  was eluted isocratically with 0.01 NH2SO4 at 60°C at a flow rate of 0.4 mL/min.Cell growth was estimated by measuring the optical density at 600 nm , and the o-phthaldialdehyde derivative was detected by high-performance liquid chromatography  with UV detection, as described previously [The putrescine concentration was determined by derivatizing putrescine with eviously . The OPAeviously ,36,37. Feviously . The derCon/off) and flux scale  of the metabolic reactions, based on FBA with GR constraints, the flux distributions in gene knockout mutants were accurately predicted [Functionally related reactions can be grouped by genomic context and flux-converging pattern analyses . Severalredicted . In thisFVSEOF with GR constraints was implemented as follows . First, the theoretical minimal and maximal flux values for the target product formation were calculated using constraints-based flux analysis by minimizing and maximizing the target product formation rate with GR constraints. Second, again with GR constraints, the cell growth rate was maximized while gradually increasing the constraint value for the target product formation rate  from a minimal to a theoretical maximum, as calculated from the first step. Finally, FVA was conducted by maximizing and minimizing the fluxes of all intracellular reactions under additional constraints, including GR constraints, the enforced production rate of the target chemical varied from a minimal to a maximal value, and a 95% optimal growth rate constraint for each step. The attainable flux ranges for the intracellular reactions were calculated under the imposed constraints for each of the three steps.qslopeVavg, and lsol . The flux bias (Vavg) and flux capacity (lsol) indicate an average value for the maximal and minimal flux values and the length of the attainable flux ranges for a reaction, respectively [qslope) of the Vavg flux for each metabolic reaction using linear regression analysis . Changes in the patterns of the reaction fluxes in response to incrementally increasing fluxes toward a target product were categorized into nine types based on combinations of positive and negative changes in Vavg and lsol in order to facilitate the identification of amplification targets . Types 1, 2, and 3 displayed positive correlations with the amplification candidates for the production of a target chemical; types 4, 5, and 6 displayed negative correlations with the amplification candidates. Finally, types 7, 8, and 9 displayed no clear correlations with the production of a target chemical, based on Vavg . The reaction sets that were positively correlated  were initially selected as amplification candidates.Initial simulation results for FVSEOF with GR constraints were filtered based on rational criteria in an effort to select only the most effective amplification targets. The most important criterion was to identify gene amplification targets, the fluxes of which increased with the flux directed toward the target chemical. This procedure was implemented with quantitative values of ectively . FinallyVavg and a positive qslope in response to the enhanced production of a target chemical, whereas weakly positive reactions show the same pattern, except for the presence of a partially negative qslope . Shikimic acid is a key metabolic intermediate in the aromatic amino acid biosynthetic pathway. Shikimic acid and its derivatives are industrially important starting compounds for the production of several chemicals, such as phenols, herbicides, antibacterial agents, and the neuramidase inhibitor Tamiflu used for the treatment of influenza infections [glk and pps), pentose phosphate pathway , and the shikimic acid biosynthetic pathway  were potential amplification targets. The amplification of aroBaroDaroEaroFaroGaroHtalABtktAglk, and pps genes [pps gene as an amplification target. FSEOF results without FVA and GR constraints did not show notable fluxes among metabolic reactions controlled by the pps gene in response to the enforced shikimic acid production rate; however, the FVSEOF method with GR constraints correctly predicted the pps gene as one of the amplification targets beneficial for the accumulation of phosphoenolpyruvate, an important precursor for the production of shikimic acid from pyruvate. This consistency partly demonstrated the power of utilizing FVA and GR constraints for predicting reliable amplification targets by FSEOF. In practice, the overexpression of phosphoenolpyruvate synthase encoded by the pps gene also increased the yield of precursors for the production of shikimic acid [E. coli, in accordance with previous literature reports.FVSEOF with GR constraints was employed to identify gene amplification targets for the enhanced production of an important aromatic chemical shikimic acid in fections ,39. FVSEps genes -46, whicmic acid ,45. ThusE. coli. Putrescine  is an important industrial precursor for the synthesis of polymers, pharmaceuticals, surfactants, and certain additives [E. coli XQ52 (p15SpeC) strain [enopgm, gapA, fbaAB, tpiA, pgk, pykAF, and glk), TCA cycle , putrescine biosynthesis , and other pathways ( ackA and ppc). The predicted amplification targets  involved in the putrescine biosynthetic pathway were consistent with the mutations introduced in the E. coli XQ52 (p15SpeC) strain, as described in the previous report [The general applicability of FVSEOF with GR constraints was examined by applying the method to putrescine production in dditives . We conf) strain . FVSEOF s report .E. coli XQ52 (p15SpeC) strain (see Methods). Among these genes, the glk, acnA, acnB, ackA, and ppc gene, five out of the sixteen target genes, were found to attain increased putrescine yield when they were individually amplified in the E. coli XQ52 (p15SpeC) strain  strain additionally expressing the glk, acnA, acnB, ackA, and ppc genes resulted in the production of 2.23, 1.90, 1.89, 2.04, and 2.06 g/L of putrescine, respectively, which are 20.5 % more than that (1.68 g/L) produced by the control strain on average  for the production of shikimic acid and putrescine in E. coli. In conclusion, the strategy reported here should be generally useful for developing industrial strains that display enhanced production of a target chemical.FVSEOF with GR constraints, which is an upgraded version of the FSEOF method, allows for the FBA, Flux balance analysis; FSEOF, Flux scanning based on enforced objective flux; FVA, Flux variability analysis; FVSEOF, Flux variability scanning based on enforced objective flux; GR, Grouping reaction.The authors declare that they have no competing interests.JMP, TYK, and SYL generated the ideas. JMP, HMP, WJK, TYK, and SYL designed the research. JMP, HMP, and WJK performed the research. JMP and HMP performed the analytical experiments. JMP, HMP, WJK, and HUK analyzed the data. JMP, HMP, WJK, HUK, and SYL wrote the paper. All authors read and approved the final manuscript.Bacterial strains and plasmids used in this study. (PDF 142 kb)Click here for fileOligonucleotides used in this study. (PDF 101 kb)Click here for fileGenomic context and flux-converging pattern analyses for shikimic acid and putrescine production inEscherichia coli. (PDF 143 kb)Click here for fileAnalysis of flux patterns with partial variations. (PDF 84 kb)Click here for filePutrescine production yield (g putrescine/g glucose) for the single gene-overexpressing strains based onE. coliXQ52 (p15SpeC) strain by flask cultivation on R/2 medium supplemented with 10 g/L glucose at 37 °C. (PDF 147 kb)Click here for file


However, even these qualitative network models are not yet complete. As our understanding improves so do we recognise more clearly the need for a systems (poly)pharmacology.Metabolism represents the ‘sharp end’ of systems biology, because changes in metabolite concentrations are necessarily amplified relative to changes in the transcriptome, proteome and enzyme activities, which can be modulated by drugs. To understand such behaviour, we therefore need (and increasingly have) reliable consensus (community) models of the human metabolic network that include the important transporters. Small molecule ‘drug’ transporters are in fact metabolite transporters, because drugs bear structural similarities to metabolites known from the network reconstructions and from measurements of the metabolome. Recon2 represents the present state-of-the-art human metabolic network reconstruction; it can predict  It is clearly not news that the productivity of the pharmaceutical industry has declined significantly during recent years Arguably the two chief hallmarks of the systems biology approach are: (i) that we seek to make mathematical models of our systems iteratively or in parallel with well-designed ‘wet’ experiments, and (ii) that we do not necessarily start with a hypothesis •Assessing whether the model is accurate, in the sense that it reflects – or can be made to reflect – known experimental facts.•Establishing what changes in the model would improve the consistency of its behaviour with experimental observations and improved predictability, such as with respect to metabolite concentrations or fluxes.•Analyzing the model, typically by some form of sensitivity analysis •Hypothesis generation and testing, enabling one to analyse rapidly the effects of manipulating experimental conditions in the model without having to perform complex and costly experiments (or to restrict the number that are performed).As set out previously in silico before trying a smaller number of possibilities experimentally; indeed for combinatorial reasons it is often the only approach possible In particular, it is normally considerably cheaper to perform studies of metabolic networks Why we choose to model metabolic networks more than transcriptomic or proteomic networks comes from the recognition – made particularly clear by workers in the field of metabolic control analysis in situ, the commonest type of modelling, as in the spread of substances in ecosystems 13–1014 bacteria  that allow metabolite transformation and cross-feeding within the prokaryotic group and to our gut epithelia; it is also noteworthy that, although antibiotics have an obvious effect here, other human-targeted pharmaceuticals will also undergo microbial drug transformation Although one could seek to understand the time-dependent spatial distribution of signalling and metabolic substances within individual cellular compartments Kcat and Km). Given such information, it is then possible to provide a stochastic http://sbml.org/) In the first, qualitative stage we list all the reactions that are known to occur in the organism or system of interest. It is increasingly possible to automate this Given their topology, which admits a wide range of parameters for delivering the same output effects and thereby reflects biological robustness Saccharomyces cerevisiaeSalmonella typhimuriumhttp://humanmetabolism.org/). In this work Since 2007 The uptake and excretion of metabolites between cells and their macrocompartments requires specific transporters and in the order of one third of ‘metabolic’ enzymes N-acetyltaurine 2-isoprostanes l-homoarginine d-2-hydroxyglutarate N-methyl nicotinamide O-phosphoethanolamine d-ribonucleoside triphosphate inter alia) for assessing the ‘exposome’ To illustrate the ‘unfinished’ nature even of Recon2, which concentrates on the metabolites created via enzymes encoded in the human genome, and leaving aside the more exotic metabolites of drugs and foodstuffs and the ‘secondary’ in vivo (in Escherichia coli) concentrations of K+, Na+, Mg2+, phosphate, glutamate, sulphate and Cl− significantly modulated the activities of several enzymes tested relative to the ‘usual’ assay conditions Another combinatorial problem In addition, some metabolites are known to affect drug transportation into cells; a well known example of this occurs with grapefruit Armed with the metabolic network models, it is possible to predict metabolic fluxes directly. This can be done in a ‘forward’ direction , in an ‘inverse’ direction  detects metabolites in the range from μm to mm and liquid chromatography (LC)–MS significantly lower in the nm to μm levels Since its modern beginnings Novel methods of data analysis also remain very important de novo protein folding and interaction studies What is becoming increasingly clear, as we recognise that to understand living organisms in health and disease we must treat them as systems What this leads us to in drug discovery is the need to develop and exploit a ‘systems pharmacology’


We review and extend the work of Rosen and Casti who discuss category theory with regards to systems biology and manufacturing systems, respectively.f = f (f). This relationship is derived from some boundary conditions that, in molecular systems biology, can be stated as the cardinality of the following molecular sets must be about equal: metabolome, genome, proteome. We show that this conjecture is not likely correct so the problem of self-referential mappings for describing the boundary between living and nonliving systems remains an open question. We calculate a lower and upper bound for the number of edges in the molecular interaction network (the interactome) for two cellular organisms and for two manufacturomes for CMOS integrated circuit manufacturing.We describe anticipatory systems, or long-range feed-forward chemical reaction chains, and compare them to open-loop manufacturing processes. We then close the loop by discussing metabolism-repair systems and describe the rationality of the self-referential equation We show that the relevant mapping relations may not be Abelian, and that these problems cannot yet be resolved because the interactomes and manufacturomes are incomplete. Systems biology is a domain that generally encompasses both large-scale, organismal systems , and smaThere is significant other research focusing on defining the difference between living and nonliving matter. These including: category theory ,4, genetSeveral disparate mathematical and analytical techniques have been brought to bear on defining and analyzing molecular network systems ,11. For relational biology.The Petri net approach to systems biology is reasonable and draws on analogies from manufacturing systems -17. ArmbRelational biology, as defined by Rosen , is a maEscherichia coli is shown in Figure At a fundamental level cells, like factory production systems contain anticipatory systems, and much of the mathematics associated with factories can be exploited for systems biology. We start by analyzing the feed-forward system known as the coherent feed-forward loop described by Alon  and MangE, sends signals to the system, ∑. The model, M, reads the state of the system. The controller, C, sends signals to the system and the model. Causality is preserved by the fact that the past influences the prediction.Figure i Ais in the reaction sequence at i. The rate of the chemical reaction, or conversion of i Ato Ai+1, is given by ki+1, and i = 1,2, ...,n are the individual molecular substrates. The reaction from A0 → n Ais known as a forward activation step. Concentration of A0 activates the production of nA. In other words, concentration of A0 at t predicts concentration of n Aat t + τ. Essentially then, n k= n k(A0) and we leave all other i kconstant.As an example of an anticipatory system consider the chemical reaction network shown in Figure The reaction rates for the system can now be written as:An-1 in the face of environmental fluctuations to the initial substance, A0. This stabilization is achieved through the relation:The forward activation step stabilizes the level of substrate A0, and we can write the rate equation for this as kn-1 An-2 = n k(A0) An-1. This relationship can be achieved by the linear system:This shows that stabilization is independent of K1 and K2 are functions of the rate constants, ik, i = 1,...,n - 1. This clearly shows that A0 determines the values of An-1 and An-2 at future times. The control condition for n k(A0) must show that the rate for any step at any given time point be determined by the value of A0 at a prior time:In this system, An-1. This is known as open-loop control, in engineering, because the system controller feeds into the process to be controlled without any signals feeding back from the process to the controller. The hazard in this type of control is it can result in global system failure.Given the fact that there is some production time associated with any given protein (i.e. kinetics), this model provides insight into a possible system stabilization mechanism, in the face of either environmental fluctuations and/or gene expression variability. This could explain the reason that "higher" organisms have a longer signaling cascade than bacteria. In this model homeostasis is preserved by the anticipation or prediction of N subsystems. The following input/output relation can give the behavior of any one subsystem i S:To describe the weakness of open-loop control, or feed-forward control, assume our system, ∑ (e.g. factory or cell) is composed of i uand spans a real m-space. The output is represented as i yand spans a real p-space. The output from the subsystem is, of course a future time, represented as t+h, and the input occurs at time t. The subsystem can receive inputs either from other subsystems or from external sources.The input is represented as i Soperates according to the function i φ(ui (t), yi (t + h)), and is behaving well when the input and outputs are within the specified space  ∈ m × RpR.The subsystem ν ∈ n Rand output(s) ω ∈ q Rrelations that exist in some space Ω ⊂ n R× qR. In order to evaluate the health of the system  there are four logical possibilities:Analogously, the overall system ∑ has its own inputs, i Sis operating optimally, therefore the global system ∑ is operating optimally.1. Each subsystem 2. The global system is operating optimally, therefore each subsystem is operating optimally.3. Any subsystem failure gives rise to global system failure.4. The health of a subsystem is not related to the health of the global system.iS.The fourth possibility we will reject as being unreasonable for real-world systems. The third possibility is valid only if there are no redundancies in the global system; again not realistic for either cells or real world factories. The first possibility is the opposite of possibility number two, which we will describe in detail and is referenced in Figure E, is from the environment. The model output, the predicted input for the process, is sent to the controller. The output from the controller, r, is the control vector and is sent to the process, as are other inputs from other subsystems. It is important to realize that the process, i Swhich processes its input, i u(t + h) at a later time t + h.The input to the model, M. If the model is not updated from internal process signals then at some point the model will no longer be correct. Real world processes will have subsystems that degrade. This will result mean that the controller, and thus the model, are no longer commensurate with reality. In general there will be a time, T, at which this is no longer the case. M will effectively drift away from ideal behavior because there are no updates to the model. At this point the process i φis said to be incompetent. For a linear anticipatory system this will lead to ∑ system failure.Correct behavior of the global system ∑, indicates that the inputs and outputs lie within an acceptable region of Ω. For proper functioning of the global system, A represent a set of environmental inputs to the cell and B represent a set of output products. Then the set of physically realizable metabolisms is given by H. We can write the metabolic map as f : A → B. We assume for the sake of argument that this map is bijective, so elements of the two sets map to each other a ⟼ b.Biological cells are excellent examples of systems that contain internal models of themselves. Biology adapts to this lack of model fidelity in feed-forward networks by a repair function. Basically, a cell has two related process, metabolism and repair. Let R, is designed to restore metabolism f, when a particular environmental variable, a is a fluctuating time-series. This may involves synthesis of several enzymes and/or promoters to induce gene expression. Since we are assuming bijection and a ⟼ b, then the subsystem output y must also be a fluctuating time-series. When the overall system is operating correctly the metabolism function, f operates on the time-series of all inputs A to produce the relevant time-series output set B. If the input does not fluctuate from the evolved basal metabolism, the "design space," then the repair function essentially produces more of the same: R: B → H . This says that the repair function uses output Y from prior steps to produce a new metabolic map H. The boundary conditions for the metabolism and repair system are: R(f (a)) = R (b) = f. The repair operation is thus to stabilize any fluctuations in inputs or metabolism. The repair system, R is an error correcting mechanism. But when it fails the biological solution to the problem is to reproduce a new cell and destroy the broken one.Biology solves the model fidelity problem either by subsystem repair, or in some cases apoptosis - discard the system and start over. The repair operation i Swithin the global system ∑ fails, then the cell signals to begin replication. This affectively solves the open-loop control problem of model drift. The cell's genome receives information about the metabolic system, f and builds a copy of repair system, R. This reproduction mapping relation is given by: β : H → H). This is summarized as:If a critical subsystem Through metabolism, environmental signals are converted into cellular outputs and subsystem outputs. These signal the translation apparatus to begin building a new metabolism system. These "self-referential" systems are known as metabolism-repair systems (M-R) systems and can be described with category theory.E. coli [Among others, real biological examples of the anticipatory systems include the flagella motor expression in E. coli  and partE. coli .Life Itself, A Comprehensive Inquiry into the Nature, Origin and Fabrication of Life. There, he used extensively a branch of mathematics known as category theory, a theory involving mappings of sets and functions. To describe an M-R system we consider a simple model consisting of metabolism and repair "components." Each i Mand i Ris a considered as a closed black box. Figure Rosen  summarizThe concept of non-repairable molecular components in cells of course is not new. Hillenmeyer et al.  preformeβ: H → H) should not represent reproduction, per se, but rather re-synthesis, and the diagram in Figure For M-R systems the equation M}. While the entire set of inputs and outputs are represented as {A} and {B} respectively. We thus have the mapping relationship M : A → B representing all possible mappings from inputs to outputs.Adapting some terminology from Letelier et al. ,30, we wM, diamonds. Real enzymes degrade or need to be replaced. In Rosen's terminology, the broken or failing M units are repaired. Each i Mhas associated with it a repair unit, iR, so there is an entire set of repair units, {R}. In biological systems the repair would simply be replacement. This replacement is how biological systems circumvent the open-loop control found in so many subsystems (or subnetworks). We represent the i Runits as network icons to remind us that the actual repair or replacement comes about as a result of a network of subreactions. This entire M and R system comprises the  systems analyzed by Rosen [Figure by Rosen  and are B}, so we have Φ: B → M. The repair comes about at the expense of output from the metabolism and uses metabolism components. An example mapping would formally be written: b ⟼ Φ (b) = f, where f ∈ M . We now haveIn order to understand the function of the repair operation, it is important to realize that the domain of the repair is the set {orf (f) = f. This mapping, of course can also be written as f = f (f) so it is said to be Abelian. But as Cardenas et al. [our familiar equation derived from anticipatory systems analysis, and can be shown as the commutative mapping in Figure s et al.  point ouan obviously more acceptable equation. It says that molecules acting on molecules produces molecules.M : A → B represents all possible mappings from inputs to outputs. We impose restrictions, or boundary conditions. First, notice that the set of metabolites {M}, and repair-operations {Φ} need to be restricted.To avoid the infinite regress we need to recall that the mapping We impose the additional boundary conditions:Letelier et al.  has suggA| ≈ |B| ≈ |M | ≈ |S|. This says that the number of reactants | A |, is about equal to the number of products | B |, and is about equal to the number of enzymes | M |, and is about equal to the number of repair operators | S |. When we consider the enzymes as the processing machines for the metabolism, then we must also recall that enzymes are produced by the metabolism system. The genome, proteome, metabolome cannot be separated. It is a complex molecular network, and as we will show below the relation |A| ≈ |B| ≈ |M| ≈ |S| is not likely valid.|i Mneeds to be repaired, essentially that means there is insufficient quantity of that molecular species for it to participate as a catalyst. The insufficient quantity triggers a threshold to induce some gene to begin a reaction to produce more . FurtheA| ≈ |B| ≈ |M | ≈ |S| and the system dynamics is more complicated than shown in Figure Though the PPI network graph is not directed, we can still conclude that the mapping is obviously not Abelian because, as shown in the degree distribution, there are some very large hubs. This scale-free observation, which is common for many types of networks, suggests that protein machines are being recruited for more than one metabolic reaction. Biology is a little more complicated than implied by |A| ≈ |B| ≈ |M | ≈ |S| hold, and that the network graph be scale free. The biological MR-system shown in Figure A second test of the MR-systems theory would be to assemble an autocatalytic set of reactions in a simulation not unlike those by Palsson . Here ho2/2 ≈ 107.If we let percolation threshold in the network, A} and {B} are the inputs and outputs to the processing machines. Both biological and manufacturing systems are materially and thermodynamically open. Both are self-regulating, self-repairing dynamical systems. Of course the cell is also a self-replicating system, and as Drexler [To expand our parallel analysis of factories and biological cells consider that from a manufacturing perspective, the sets { Drexler  pointed  Drexler .For cellular systems biology we can view the system as a network of interacting molecular species, with one of the major time lags being diffusion and Brownian motion. Processes can take place reasonably rapidly and Le Chatelier's principle can drive the system dynamics. On the organism level, diffusion and other transport processes can be major time delays, and the dynamics of the organism can be minutes to days to weeks. Similarly, the time lag in manufacturing is far greater between sensing a manufacturing processing component failure (mean time to failure) and actual repair time (mean time to repair). This gives rise to a hysteresis . In the Above we described anticipatory systems and M-R systems from mostly a biological perspective, here we will draw further analogies with manufacturing and systems biology. Casti ,21 explof is the set {A} and the output is the set {B}. At the cellular biology scale, the seemingly infinite recursion of the compact version of this map, f (f) = f, can be explained as the fact that the genome, proteome, metabolome are all interrelated. Components and machine parts from the proteome are used in processing the metabolome. Components and machine parts from the proteome are used in resynthesis of the proteome components all the while making use of the metabolome and genome.In the mapping In biology the network of interacting proteins, commonly called the interactome, really consists of enzymes and protein inputs/outputs to the metabolism. If we could remove from the protein-protein interactome the inputs and outputs leaving a connected graph or a time sequence list of the enzymes that participate in cell cycle and/or cellular manufacturing, then we would essentially have the following type of linear network:W (it) represents the metabolites or materials to be processed by Pi,j during the time period between i and j. Biochemically Pi,j would be enzymes. In a manufacturing environment, it would be the processing machine.Where Now if we constructed what is called the edge graph for the linear network shown above, would have:a network of enzymes, or processing machines, as they are used in sequence.Pn-2,n-1 → Pn-1,n and could be laid out linearly; but since the same machine is used for similar processes, hubs are created. Obviously it is a directed graph, information not usually available for interactomes. The figure also shows the degree distribution for this rather small graph, N = 27.1k -1.2; R2 = 0.94. Notice there are large hubs; the most prominent being the inspection step. The next largest steps are expose stepper and develop/bake. These are lithography steps used to define the regions for transistor location and interconnection. Like the PPI network in Figure A|/| M | ≈ | B |/| M | ≈ 5.28. Keeping in mind that the full manufacturome represents other processes, using M = 28, the upper bound for the number of edges is 6272. The conjecture |A| ≈ |B| ≈ |M | ≈ |S| is valid only when the nodes are replicated the appropriate number of times to capture the dynamics. We never see this in biological interactomes, or manufacturomes because of protein/workstation reuse which embeds the larger cardinality in the edge-count, so the validity of this conjecture remains open, mainly because of incomplete information at this time.We have been drawing several parallels between manufacturing and systems biology. Since manufacturing networks are completely known we have an opportunity to explore algebraic graph theory and test algebraic and group theory hypothesis on manufacturing networks that are not possible yet with incomplete biological interactomes. Here we want to point out some network similarities. Figure N = 101.3k -1.3; R2 = 0.44, and given M = 259, we get the number of edges, E = 5.366 × 105.Figure in situ signals from the process being analyzed as X-bar  and R  charts [It is also important to realize that this manufacturome is not an autocatalytic set and the network diagram in Figure ) charts . These pMycoplasma genitalium, contains, 471 coding genes [M = 256), the upper bound for the number of edges in the full molecular network would have about E = 5.24 × 105.As a final comparison we look at the number of coding genes in the smallest known genome to our CMOS fab. The organism in question, ng genes . These bng genes  through In summary, the development of main equations for anticipatory systems and metabolism-repair systems are similar for manufacturing systems and cellular biology. The fact that these two disparate domains are so tightly coupled by similar mathematics suggests these concepts are indeed at the boundary of living and nonliving. Of course the coupling could also be a coincidence because of the "unreasonable effectiveness of mathematics in the natural sciences" .f = f (f) and found that the boundary condition |A| ≈ |B| ≈ |M | ≈ |S| can't be valid, but is likely only a lower bound. We suggest that the upper bound is the percolation threshold for the adjacency matrix of the molecular network, and compute these lower and upper bounds for S. cerevisiae, M. genitalium and two integrated circuit manufacturing lines. Further the relation f = f (f) is not likely Abelian so theoretical understanding of metabolic closure of living cells remains and open question.We reviewed the basis for the self-referential relation The authors declare that they have no competing interests.EAR did the research, did most of the calculations and wrote the first draft. JZC provided data for the manufacturome and assisted in analysis of the IBM data. JAT assisted in presenting and integrating the material into the manuscript and coordinating the project. All authors read and approved the final manuscript.ADP: Adenosine diphosphate; ATP: Adenosine triphosphate; CMOS: Complementary metal-oxide-semiconductor; DNA: Deoxyribonucleic acid; DRAM: Dynamic random access memory; Fru6P: Fructose 6-phosphate; G6P: Glucose-6-phosphate; HXK: Hexokinase; M-R: Metabolism-repair; MAP: Mitogen-activated protein; PGI: Phosphoglucose isomerase; PPI: Protein-protein interaction.