Metabolic flux analysis (MFA) is an indispensable tool in metabolic engineering. The simplest variant of MFA relies on an overdetermined stoichiometric model of the cell’s metabolism under the pseudo-steady state assumption to evaluate the intracellular flux distribution. Despite its long history, the issue of model error in overdetermined MFA, particularly misspecifications of the stoichiometric matrix, has not received much attention. We evaluated the performance of statistical tests from linear least square regressions, namely Ramsey’s Regression Equation Specification Error Test (RESET), the F-test, and the Lagrange multiplier test, in detecting model misspecifications in the overdetermined MFA, particularly missing reactions. We further proposed an iterative procedure using the F-test to correct such an issue. Using Chinese hamster ovary and random metabolic networks, we demonstrated that: (1) a statistically significant regression does not guarantee high accuracy of the flux estimates; (2) the removal of a reaction with a low flux magnitude can cause disproportionately large biases in the flux estimates; (3) the F-test could efficiently detect missing reactions; and (4) the proposed iterative procedure could robustly resolve the omission of reactions. Our work demonstrated that statistical analysis and tests could be used to systematically assess, detect, and resolve model misspecifications in the overdetermined MFA. The ability of biological systems to produce highly complex molecules at high enantiomeric excess has pushed metabolic engineering that relies on directed alterations of the cell’s biochemical reactions through recombinant DNA technology to the center stage of biotechnology ,2. The u13C isotopic labeling experiments has emerged and matured over the past two decades. The experimental, analytical, and computational procedures for 13C-based MFA have now been standardized [A simple strategy of MFA, from here on referred to as the overdetermined MFA, uses a reduced stoichiometric model of the cell’s metabolism such that the estimation of the metabolic fluxes is mathematically well-posed, i.e., the flux estimation involves an (over)determined system. For larger and more realistic metabolic models, the flux estimation in the MFA often becomes underdetermined as the number of unknown fluxes exceeds the number of balance equations. There has been a flurry of activity in the development of MFA methods for large metabolic models based on linear programming  , which gdardized .13C isotopic labeling experiments, the overdetermined MFA continues to be used in practice, thanks to its simple formulation and numerical implementation [t-test) [Due to the experimental and computational complexity in the application of MFA using genome-scale metabolic models and entation ,7. A comentation . In consentation ,10. The entation  or by prentation . Finallyt-tests [The test for goodness of fit or statistical significance of regression in the overdetermined MFA could fail because of several reasons, including (1) incorrect assumptions abuot the characteristics of data noise  and (2) model specification errors ,15. In st-tests . In this work, we adapted statistical analysis and tools for model misspecifications commonly used in the field of linear least square regression to address the issue of missing reactions in the overdetermined MFA. We posited that the simplification of the stoichiometric model, either manually or using a numerical algorithm , to genec is the vector of m metabolite concentrations, v denotes for the vector of n metabolic fluxes, and S denotes the S and the flux vector v into: The MFA is based on the mole balance equation for the intracellular metabolites under a pseudo steady state assumption, as follows:y denotes the vector of measured response variables, X denotes the (non-random) design matrix containing the values of explanatory variables,The estimation of the internal metabolic fluxes By invoking the first order necessary condition for optimality  and the least square regression problem in Equation 4), we could set , we coult-test, and simulated flux values to show that model errors could lead to gross deviations in flux estimates that are not statistically significant [Due to the rank condition on nificant . In the u denotes the vector of measurement errors for the true model. Without a loss of generality, in order to illustrate the impact of omitting reactions, we could consider the OLS estimate of the internal fluxes u has zero mean, the flux estimate In the following, we considered the problem of missing or omitted reactions in the stoichiometric matrix. We assume that the metabolic network model in the MFA given in Equation (1) is incomplete and that the true mole balance is governed by:S contains reactions that are non-existent in the actual system, would not introduce any bias to the OLS estimate of the fluxes. However, having additional reactions would artificially increase the variance and mean square error of the flux estimate [The derivation above shows how the omission of or failure to include reactions in the stoichiometric model could cause a bias in the flux estimate in the overdetermined MFA. In addition to the flux bias, missing reactions would also result in a lower variance, i.e., the variance of the OLS estimate of ion (11) . Under cion (11) . On the estimate .Xβ is a good description of the response variable y. While tests for checking the validity of constant variance or the normality of the error variables in the least square regression also exist, we do not deal with these so-called misspecifications of higher moments. Interested readers are referred to the article by Long and Trivedi [p order RESET test is based on the following linear least square problem:p powers of 0: y, K is the dimension of the unknown parameterp and There exist several tests to detect model misspecification in a linear least square regression. In this study, we focus on tests of the misspecification of regression mean; whether  Trivedi . The fir Trivedi , which dZ denotes the design matrix of the candidate missing variables; namely, the stoichiometric matrix of the possible missing reactions in the overdetermined MFA. The next two misspecification tests apply to the scenario in which the stoichiometry of the candidate missing reactions are known. Given the current extensive knowledge on metabolic reactions, including the complete maps of metabolic networks for many major organisms and the availability of extensive and curated biochemical reaction databases  , MetaCyc0: o is the number of possible missing variables SF−tZ in Equation (18), denoted by denoted by X. In addition, the LM test also incorporates a heteroscedasticity-consistent (HC) covariance matrix to accommodate the situation when the variance of the measurement error is not constant. Following the procedure formulated in Long and Trivedi [1 is the (column) vector of 1s with the same row dimension as the matrix X. The test statistic is given by:0: o degrees of freedom. The rejection of the null hypothesis due to a  significant correlation between the residual The third and last misspecification test in this work is derived from the Lagrange multiplier (LM) test, as proposed by Davidson and Mackinnon . The pre Trivedi , we firsEv, the stoichiometric matrices ES and IS, and the possible missing reaction stoichiometric matrix AS, we formulate the linear least square regression problem with Given the exchange fluxes Z constructed from every k-tuple combination of the columns (reactions) of AS. Compute k-tuple combination(s) satisfying AS to IS.Identify the AS to IS, that is, until the remaining set of k-tuple reaction combinations satisfying Repeat steps 2 to 3 until no more reactions can be moved from The rejection of the null hypothesis in the tests above does not immediately point to the identity of the missing or omitted reactions. Given the set of possible missing reactions, we could nevertheless apply the F-test or LM test to determine whether the addition of a reaction or a set of reactions among the possible missing reactions would significantly improve the linear regression. Reactions that are positively identified by the tests could therefore be added into the stoichiometric matrix. The procedure above can be repeated until no more reactions can be added. More precisely, we propose the following iterative procedure for correcting misspecifications in the stoichiometric matrix for the overdetermined MFA using the F-test:k-tuple combination of reactions. In the case study, we performed the procedure, first using k = 1 (single reactions) and using k = 1 followed by k = 2 (pairs of reactions). If desired, the LM test could also be used in place of the F-test.The procedure above is written generally for any IS has a dimension of 49 metabolites and 47 internal fluxes with a full column rank. IS in the MFA. The complete list of reactions, the measured uptake flux values, and the standard deviations reported by Sokolenko et al. [In the demonstration of the specification bias caused by missing reactions and the iterative procedure for resolving the stoichiometric matrix misspecification, we employed a metabolic network model previously created for the flux analysis of Chinese hamster ovary (CHO) batch cultivation data ,27. The o et al.  are giveIS of various sizes (m = 50 to 200 metabolites and n = 55 to 220 reactions) and with different numbers of exchange fluxes (between 25 and 100). More precisely, for each data vector y, we employed the RMBNToolbox (MATLAB) [IS with the desired dimensions and a full column rank, wherein each species participated in at least one reaction. For the specified number of exchange fluxes Em, we assigned the first Em metabolites (rows) of IS as the species whose exchange fluxes were measured, while the remaining species only existed intracellularly (without exchange fluxes). More specifically, we used the matrix IS into:Em rows of the measured exchange fluxes and Iv using a linear combination of the kernel of Iv in Step (1) according to Equation (22) and contaminated the exchange flux vector with a vector of independent realizations of Gaussian distributed random numbers with zero mean and for different coefficients of variation (CoV of 1 to 20%). Note that since the noise standard deviation scales with the noise-free y  from the matrix For a large-scale evaluation of the model misspecification tests, we generated in silico data (MATLAB)  to generThe performance of each misspecification test was judged based on the rate of true positive (TP), true negative (TN), false positive (FP), and false negative (FN) results. For the Ramsey RESET test, the true positives and false negatives were determined by applying the test to the generated data using the stoichiometric matrix y as described above and applied the tests using the stoichiometric matrix Z came from the set of actual reactions that were removed from the matrix Z containing a distinct set of linearly independent reactions of the same size as the set of the omitted reactions. While the arrangement for determining the FP and TN rates does not reflect the null hypotheses in the F- and LM tests, it reflects better the typical scenario in practice (where the true model is unknown). The rates of TP, FN, FP, and TN were computed from 1000 repeated runs of the above tests. Finally, in all of these tests, we ensured that the rank condition for the OLS estimation was always satisfied.In order to determine the TP and FN rates of the F-test and LM test, we generated in silico data In this case study, we considered the CHO metabolic model in ,GLS see . Figure gression . p values from the ANOVA are given in p value of 0.05 as the threshold to reject the GLS regression; any p values higher than the threshold indicate a poor regression outcome. viations . For eacp < 0.05. On average, the median, mean, and maximum specification biases in the flux estimates were higher for the removal of reactions that caused a poor regression (p > 0.05). The two highest p values expectedly came from the removal of reactions with the two largest fluxes, and each expectedly had large specification biases. There were nevertheless exceptions where a poor regression resulted from removing a reaction with a moderately low flux value . On the other hand, many of the cases with a significant regression (p < 0.05) were associated with high maximum specification biases. In fact, several of the cases among the lowest p values  had a mean bias of >30% and a maximum bias of above 800%. Equally important, the removal of several reactions with a low flux magnitude led to large mean and maximum flux biases (mean bias >150%), as highlighted in The individual removal of roughly 3/4 of the reactions (26 out of 36 reactions) still produced a significant regression with IS. As outlined in Materials and Methods, we determined the rates of TP, TN, FP, and FN using randomly generated pairs of data IS with missing reactions. For the F- and LM tests, we used the information on the actual missing reactions, as well as a distinct set of reactions, as the design matrix of the missing variables Z. The results using the baseline MFA problems with 100 metabolites (m), 60 unknown internal reactions  in IS was closer to the true system and could more accurately capture the flux balance. Therefore, for the F-test to correctly detect a misspecification of IS, the missing reactions would need to cause a significant deterioration in the data fitting, a scenario that was less likely to occur as the number of missing reactions became lower. Meanwhile, the FP rates were not a strong function of the noise level. The FP rates improved with a lower number of missing reactions, albeit only slightly. In general, the results in m = 200), the rate of correctly detecting a misspecification with two missing reactions was slightly lower than 60%. Fortunately, the TP rates for detecting five or more missing reactions were still high (>88%), and the FP rates depended weakly on the size of the networks and the number of missing reactions and remained relatively low, between 10% to 15%, in most of the cases in our study  . Since tThe LM test performed better than the RESET test but produced lower TP rates than the F-test, particularly at the highest number of missing reactions  k = 1 and (2) k = 1 followed by k = 2. In the last case study, we evaluated the performance of the proposed iterative procedure to resolve stoichiometric matrix misspecifications in the overdetermined MFA . Here, we returned to the flux analysis of the CHO metabolic network in mate see  and contAS that were not positively identified by the iterative procedure to be included in the stoichiometric matrix. As an indication of good performance, the number of omitted reactions (extra reactions) that remained should be low (high). The results in k = 1 was able to correctly detect and incorporate almost all of the omitted reactions, while keeping the incorrect inclusion of extra reactions low. As expected, performing an additional run with k = 2 after finishing the procedure with k = 1 led to a higher incorporation rate of the omitted reactions, but such a strategy came at the cost of a higher rate of incorrect addition of the extra reactions. Due to the small size of the CHO model and the number of missing reactions considered, a higher k  led to the incorporation of all omitted and extra reactions . In this work, we adapted statistical analysis and tests for model misspecifications in linear least square regressions. To the best of our knowledge, such analysis and tests have not yet been applied to the flux estimation problem of the overdetermined MFA. Here, we first derived a simple formula to evaluate the specification bias in the flux estimate due to missing reactions in the stoichiometric model. Using a stoichiometric model of the CHO metabolism, we showed that the significance of the regression is not a sufficient indicator of a low flux bias. In particular, the omission of reaction(s) that results in a high bias (error) in the flux estimate could still produce a statistically acceptable regression. Furthermore, we demonstrated that the removal of reactions with a low flux value could also cause disproportionately large specification biases. In practice, the significance of the regression and the prior information on the magnitude of reaction fluxes are often used for the curation of a reduced order stoichiometric model for the overdetermined MFA. Our findings from the first case study clearly motivated computing the potential flux specification bias during the removal of reaction(s). k reactions from the set of candidate missing reactions that would  significantly improve the linear regression. The combinations that passed a certain p value threshold are then incorporated in the stoichiometric model. The application of this iterative procedure to the CHO stoichiometric model indicated that the simple implementation using k = 1 gave the most robust performance. The computational cost of performing the iterative procedure scales linearly with the number of candidate reactions when using k = 1 or combinatorically when using k > 1. Fortunately, since the F-test involves only computationally efficient matrix inversions and matrix-vector multiplications, each iteration of this procedure takes little time to finish. For example, each iteration in Case Study III, with eight missing reactions, using k = 1 finished in roughly Among the statistical tests that we evaluated for detecting stoichiometric matrix misspecifications, the results from random metabolic networks clearly demonstrated the superiority of the F-test. The F-test could provide high TP rates and low FP rates for detecting missing reactions for most problem sizes (except when there were only very few missing reactions). Based on these findings, we proposed an iterative procedure using the F-test to detect and correct missing reactions. In each iteration, we used the F-test to identify a combination of Finally, there exist several obvious limitations in the statistical tests evaluated in this work. First, these tests rely on the assumption of normality for the noise. The validity of this assumption could be checked by standard normality tests such as the Lilliefors or Shapiro-Wilk tests  or usingIn this work, we addressed the misspecification of the stoichiometric matrix in the overdetermined MFA, particularly the omission of reactions. For this purpose, we adapted statistical analysis and tools from linear least square regression to quantify, detect, and resolve the issue of missing reactions. In particular, we derived a simple formula to evaluate the flux bias caused by missing reactions in the overdetermined MFA. We further assessed the performance of several model misspecification tests, namely Ramsey’s RESET test, the F-test, and the Lagrange multiplier test, in detecting missing reactions for overdetermined stoichiometric models. Finally, we proposed an iterative procedure for resolving the issue of missing reactions based on applying the F-test to the candidate missing reactions one at a time and incorporating reactions that pass the significance test. The application of these techniques to the CHO metabolic model and random metabolic networks provided several important conclusions. First of all, the significance of the regression, a common metric for assessing the data-model consistency in the overdetermined MFA, does not guarantee a low bias in the flux estimates. In addition, a high flux bias could result from the removal of a reaction with a low flux magnitude that would be typically deemed unimportant in the construction of the stoichiometric matrix. Therefore, the potential flux bias due to any removal of reaction(s) should be computed during the construction of the stoichiometric model. When the stoichiometry of the candidate missing reactions is known, the F-test provides a robust means, with a high TP rate (nearly 100% for many cases) and a relatively low FP rate (<15%), to detect model misspecifications in the overdetermined MFA. Upon a positive detection of model misspecification in the overdetermined MFA, the proposed iterative procedure in this study gives an effective and robust systematic approach to resolve this issue.


Biologically meaningful metabolic pathways are important references in the design of industrial bacterium. At present, constraint-based method is the only way to model and simulate a genome-scale metabolic network under steady-state criteria. Due to the inadequate assumption of the relationship in gene-enzyme-reaction as one-to-one unique association, computational difficulty or ignoring the yield from substrate to product, previous pathway finding approaches can’t be effectively applied to find out the high yield pathways that are mass balanced in stoichiometry. In addition, the shortest pathways may not be the pathways with high yield. At the same time, a pathway, which exists in stoichiometry, may not be feasible in thermodynamics. By using mixed integer programming strategy, we put forward an algorithm to identify all the smallest balanced pathways which convert the source compound to the target compound in large-scale metabolic networks. The resulting pathways by our method can finely satisfy the stoichiometric constraints and non-decomposability condition. Especially, the functions of high yield and thermodynamics feasibility have been considered in our approach. This tool is tailored to direct the metabolic engineering practice to enlarge the metabolic potentials of industrial strains by integrating the extensive metabolic network information built from systems biology dataset. E. coli1, S. aureus2, H. pylori3, M. barkeri4, S. cerevisiae5, B. subtilis6, and so on. The pseudo-steady state condition (PSSC) refers to the main assumption that the concentration of internal compounds keeps invariable over time. Thus, internal compounds satisfy dxc/dt = 0 where xc is the concentration of compound C7.Metabolic network, the pseudo-steady state condition (PSSC): Genome-scale metabolic network (directed graph) is used to model the metabolism of biological systems, such as microorganisms. A few of models have been published including Source and target, external and internal compounds, exchange reactions: For a genome-scale metabolic network, exchange reactions are transport reactions through which cells exchange materials with the environment. External compounds are the compounds in the extracellular environment, but they enter the cell through exchange reactions and then play a rule. Source and target are respectively the start and the end of the pathways we hope to find.8. Within a graph representation of a metabolic network, there may be multiple pathways. From the source compound to the target compound, there is a directed path with no cycles and in a particular determined metabolic pathway, and we refer to this directed path as metabolic path9. Of course and especially, when the pathway is branched, it may not be unique for this path. The metabolic pathway contains all the compounds and reactions involved in the pathway, all the internal compounds must be mass balanced in PSSC. Non-decomposability condition means that a pathway can’t be separated into two or more independent pathways.Pathway and path: A metabolic pathway (a subset of the whole metabolic network) is a set of reactions by which a living organism transforms a source compound into a target compound10.Smallest pathway in large-scale metabolic networks: For a metabolic network, many pathways may have no biological meaning and if we can find experimentally determined pathways, this may provide in-depth knowledge for biomedical or biotechnological applications. So methodologies on metabolic pathway will devote to discover biologically meaningful metabolic pathways in metabolic networks. There may be many pathways between a source and a target in a large-scale network, and it would be computationally impracticable to completely enumerate all these pathways. Thus, pathway finding methods should focus on finding a set of pathways which were defined by the stoichiometric constraints and could be able to span the complete solution space of pathways. The smallest pathway is defined as the pathway with least reactions which convert the source compound to the target compound. Although pathway research should not come down only to the smallest pathway, the smallest pathway is an important aspect of biological meaning12 provided a method to discover a set of genetically independent pathways and their work represented the first stoichiometric methodology for the computation of metabolic pathways. But their algorithm required big computational effort, so could deal with only metabolic networks of relatively small size. In addition, their approach was based on the assumption that the relationship in gene-enzyme-reaction was a one-to-one unique association. 2) Improved genetically independent pathways (IGIP): Mavrovouniotis15 developed the algorithm of Seressiotis and Bailey, and used it to deal with pathways which comprised multiple targets and sources. His approach can be applied to a moderate size of the metabolic network. 3) Elementary flux modes: Elementary flux modes (EFMs), i.e. non-decomposable pathways at PSSC, were named by Schuster and co-workers16. With the increase in the size of the metabolic network, the number of EFMs entails combinatorial fashion17. In order to overcome this combinatorial explosion, different strategies have been adopted20. 4) Extreme pathways: Extreme pathways (EPs), a refined set of EFMs, were proposed by Schilling et al.21. Apart from the non-decomposability condition and the PSSC defined above, the systemic independence condition must be satisfied by the set of EPs, i.e. no EP can be written as a non-trivial nonnegative linear combination of other EPs8. As for EFMs, when applied to large-scale networks, computing all the EPs will suffer a combinatorial explosion. But enumerating special EFMs or EPs, such as from a substrate to a product in a given large-scale metabolic network, is computationally feasible. 5) k-shortest EFMs and flux paths: Figueiredo and Planes have presented a method to find the shortest elementary flux modes in genome-scale metabolic networks with integer programming10. By examining carefully the paper and doing computational practice, we found that this method did not consider ATP maintenance and the yield from substrate to product, and at the same time it did not provide the actual flux distribution in the identified pathways.Pathway finding and approaches: As we mentioned above, pathway finding approaches aim to find a set of pathways that should satisfy the stoichiometric constraints, so they can be called stoichiometric approaches. 1) Genetically independent pathways (GIP): Seressiotis and Bailey10. We will show the difference and give a comparison in the result section of this paper. At the same time, although a pathway exists in stoichiometry, it may not always be feasible in thermodynamics. If we regard a pathway as an overall reaction and if we hope it is able to proceed spontaneously, it should satisfy the requirement of free energy change. Moreover, if there are several pathways which satisfy the condition, which one is more probable to occur in the cell?High yield and thermodynamics feasibility for a pathway: In the area of industrial biotechnology, improving bacterium is an important task and a high yield from substrate to product is the first target. For the construction of microorganism, we should utilize the pathway with high yield. The shortest EFMs may not be pathways with high yield and they are not equivalent to each other. But the two sets of pathways usually were regarded as equivalent, as stated in the literatureMotivation and our contribution: At present, the modeling and simulation method for the genome-scale metabolic network is constraint-based method which satisfies steady-state criteria. For the reason of inadequacy assumption , computational difficulty , or ignoring the yield (k-shortest EFMs and flux paths), previous pathway finding approaches as we stated above can’t effectively design optimal pathways to direct the metabolic engineering practice. In this work, by using MIP (Mixed Integer Programming) strategy we put forward an algorithm to identify the smallest balanced pathways (SBPs) which convert the source compound to the target compound in large-scale metabolic networks. Under PSSC, the resulting SBPs of our method can well satisfy the stoichiometric constraints and non-decomposability condition; Multiple pathways which meet the above-mentioned criteria can be found and provided as candidate design; In addition, high yield is a new function; Especially, thermodynamics feasibility has been considered in our approach. The smallest pathways founded by our method can provide good references in the pathway design for the industrial microorganism. Our model can be easily solved by existing optimization software.S, to describe genome-scale metabolic network and the elements in S are the coefficients of reactions22. Under steady-state criteria, the time derivatives of metabolite concentrations are zero7, i.e. those internal metabolites should satisfy mass balance, so the equations of mass balance for all the metabolites can be represented as followsS is the stoichiometric matrix, and αi and βi define the bounds through each reaction vi, R is the set of reactions.Usually, we can use a stoichiometric matrix, E. coli_iJO13661, there are more than 300 exchange reactions and we can choose any two as source and target.As for metabolic networks in genome-scale, the fluxes within a cell usually can be computed with flux balance analysis (FBA) that can give optimal growth phenotypes, though not unique. In mathematics, FBA is an equivalent to a large-scale linear programming (LP). In our algorithm, we confine source and target compounds to be external compounds, i.e. there are exchange reactions related to them. For example, for the genome-scale metabolic network of y of the same number of continuous variable v to indicate the absence or presence of a reaction vi.In order to find the smallest balanced pathways in large-scale metabolic networks, MIP strategy is used as the mathematic model. We introduce binary variable yi = 0 then vi = 0 and If yi = 1 then αi ≤ vi ≤ βi, we can express this idea as a constraint:If The source and target nodes should be external nodes, and there are exchange reactions connected to them. In order to give a connected pathway, two bounds are added.vt is to let vi of other reactions in the pathway not be beyond their constraints, and here constant1 and constant2 are positive values. Equation . The SBPs is smaller than the null space of the stoichiometric matrix.i = 0; In other case, certain reactions must be reversible, we can preset vmin = −1000, vmax = 1000. These could be achieved by setting the boundaries of the reactions. Then the solution of smallest balanced pathways is within the scope of the given conditions.For this model, we can easily preset the metabolic network to meet the requirement of the specific situations. For example, certain reactions must not be appearing, or some genes are to be disrupted, we just preset v24 here. With a statistic of the fluxes which are not zero in absolute value  or which yi is 1 (the two ways are consistent), we can determine those reactions which should appear, and further, we can obtain the smallest balanced pathways.For MIP, some existing software can be used to find its solution and we use Gurobi25. The approach of Bender’s cut is that iteration is used from an existing solution, at the same time in each iteration to exclude an existing solution by adding the following binary cutSometimes, there exist different states of integer variables but the objective value is the same, i.e. a MIP may have multi integer solutions. Up to date, as we know, there does not exist optimization tool which can give directly multi integer solutions for a MIP. Here we utilize a method proposed by Balas and Jeroslow, named Combinatorial Bender’s cutAll the multi integer solutions will be got by this way.chem of this chemical as the objective and we will get the theoretical ratio Vmax. Then we can use 95% of the value of highest yield Vmax as a constraint in our MIP model. Finally, we will get all SBPs with a high yield which is larger than 95% of the value of highest yield.SBPs have the least number of reactions but may not have the high yield of a chemical which the microorganism produces. High yield means a high amount of desired product and little or no by-product which might make the downstream complicated, costly, and polluted. In another word, high yield means cost-saving. Sometimes, high yield is our interesting aspect, so it is best to find SBPs with high yield. In order to estimate the reachable high yield of the chemical, we can use FBA with the exchange reaction rate vE. coli27. The thermodynamic data of E. coli model was calculated by Group Contribution Method28. There is a range of free energy change for every reaction and it is calculated by min/max delta G. The range of delta G could be used to decide the reversibility of a reaction.Although a pathway exists in stoichiometry, it may not always be feasible in thermodynamics. If we regard a pathway as an overall reaction and if we hope it is able to proceed spontaneously, it should satisfy the requirement that each reaction in the pathway must be thermodynamically feasible individually, i.e. the flux and the free energy change of this reaction must have opposite signs or the reaction is reversible. The data of free energy change for a microbe is not rich in literature and the first one is E. coli. By using our algorithm, we computed out the smallest balanced pathways from glucose  to succinic acid  in the genome-scale metabolic network of E. coli (its SBML model is iJO13661). In the process of computation, we restrict the input rate of glucose (vs = −100 mmol/g(Dw)h) and the output rate of succinic acid (vt ≥ 0.01 mmol/g(Dw)h). The given conditions are that we restrict the input and output of the cell to be only five compounds, i.e. glucose, succinic acid, CO2, H2O, and H. The reason for restricting only these five compounds is that succinic acid can be synthesized by them. There are 12 alternative solutions for this model, seeing Supplementary Material In this example, we hope to know how succinic acid is synthesized by glucose with One of these pathways, the first solution, was illustrated in Fig. From this pathway, we can know clearly the pathway to synthesize succinic acid and the balanced proportions among fluxes through every reaction in this pathway. We know also how these enzymes (or reactions) cooperate with each other to synthesize succinic acid. This means these reactions are equally important to fulfill the overall function of succinic acid production. The pathways including the reactions, genes, and fluxes should be regarded as ideal references to guide strain engineering activity. This will greatly reduce the scope of targets to enhance genes in order to accelerate the speed of producing succinic acid.E. coli27, seeing Supplementary Material With the data of free energy change of each reaction for E. coli is selected for the study. Threonine is an important chemical in industry, which can be produced by E. coli with glucose. We have computed all the threonine SBPs with highest yield in E. coli with glucose as substrate, and get 16 SBPs. All the SBPs have 50 step-reactions and the yield is 1.248:1 in a molar ratio which is near the theoretical yield. All the SBPs are in Supplementary Material In the above succinic acid case, the yield 171.43:100 has almost been the theoretical value. In order to show the difference of those with high yield and SBPs without high yield, threonine production with E. coli_iJO1366, illustrated in Supplementary Material In order to make a quantitative comparison with conventional methods, the method of k-shortest EFMs was selected as it is the nearest approach to ours. We use this algorithm to compute out all the shortest EFMs from glucose to succinate in the genome-scale metabolic network of E. coli can produce many other chemicals such as lactic acid, formic acid, fumaric acid and so on. In the model iJO1366, there are 324 exchange reactions. Only 25 reactions have low bounds which less than 0, while all the up bounds equal to 1000. We use glucose (start or source) as the input, at the same time use all these 324 reactions except for glucose as the output (target) respectively, and calculate the SBPs with maximum productivity for every chemical. For a given chemical, if E. coli can’t produce it, i.e. the maximum productivity for it is zero, the algorithm will not return its SBPs.In addition to succinic acid and threonine that we mentioned above, E. coli can produce and found that in many cases, the number of SBPs is less than 10 and that those cases which are larger than 100 only take a very small proportion, as shown in Fig. We have made a statistics for all the SBPs to a variety of chemicals which Up to date, modeling a genome-scale metabolic network in dynamics is still beyond the access of most laboratories, so the best way to make use of flooding metabolic network information to direct the metabolic engineering practice is the constraint-based approach which satisfies the pseudo-steady state condition (PSSC). A pathway that converts a given source compound to a given target compound should satisfy the stoichiometric constraints and non-decomposability condition. EFMs and EPs are two pathway finding approaches, but calculating the set of EPs or enumerating all the EFMs will suffer a combinatorial explosion when applied to large networks. Existed approaches of k-shortest EFMs and flux paths are not the methods of considering the yield from substrate to product which is highly biotechnologically relevant.In this work, by utilizing Mixed Integer Programming (MIP), we present an approach for pathway finding. Our algorithm has a number of good features: 1) It is a method of thorough stoichiometry. It can identify the balanced pathways in the genome-scale metabolic network. The balance here means that the mass of internal compounds is balanced, i.e. stoichiometric balance. The smallest means that the pathway identified has least reactions. 2) The pathways found by this approach are usually short enough, which simplify the metabolic engineering practice and also save the cellular energy consumption for synthesizing proteins for the reactions. It is well-known that protein synthesis is the most energy-intensive process. 3) Our algorithm can return all the alternative solutions, and this can provides more choices in industrial stain design. 4) High yield can be added as required condition, which is important for biotechnology purpose. 5) Thermodynamics data are integrated to allow the thermodynamics feasibility analysis.We recognized that although the short pathway has the advantages as mentioned above, the shortest pathway may not necessarily be biologically feasible and some biological pathways are not the shortest one in nature. Our algorithm is to break the evolutionary barrier and eventually help to create artificial cell factory. Furthermore, by just simply modifying our code, we can easily found all balanced pathway with the length of shortest plus 1, 2, …, and so on.The model of our approach comes down to a MIP and MIP is an essentially combinational problem. Computational complexity will be proportional to the scale of the problem. But for a large scale problem, existing solving software can solve it in not long time. Such as our case study with several thousand of variables, the computation time will take just several minutes by an HPC (high-performance computer) with 48 cores.In industrial stain design, high yield from the source substrate to the target product is the first important aspect. To implement this, biologically feasible and high yield pathways should be utilized. The smallest balanced pathway with high yield can provide an ideal reference to guide metabolic engineering practice. In particular, the balanced pathways tells many co-dancing reactions which are beyond the sight of normal biological knowledge. The metabolic bottleneck may not necessarily locate on the traditional biochemical pathway. The accessory reaction which is responsible to recycle the cofactors, intermediate or to supply the precursors is shown to be as important as the reactions in the known biochemical pathway. Meanwhile, the relative strength of the fluxes of different reactions also tells the metabolic engineer how to fine-tune the relative activities of different reactions. Integrating with the experimental determination of intracellular transcriptome, proteome and even metabolome, metabolic engineering should be able to identify the potential rate-limiting reactions which they need to put effort on.E. coli, the input rate of glucose is 100 mmol/g(Dw)h and the output rate of succinic acid is 171 mmol/g(Dw)h, so this pathway almost reaches the theoretical ratio of glucose/succinic acid in E. coli. If we fulfill the pathway in E. coli, it can make a good utilization of glucose in producing succinic acid. SBPs with high yield from glucose to a variety of chemicals, which E. coli can produce, have been calculated by our algorithm. Especially, we can make decisions on the thermodynamics feasibility by integrating the data of free energy change.In our first case study, producing succinic acid with Supplementary material aSupplementary material b/c/d


Recent developments in metabolic engineering have led to the successful biosynthesis of valuable products, such as the precursor of the antimalarial compound, artemisinin, and opioid precursor, thebaine. Synthesizing these traditionally plant-derived compounds in genetically modified yeast cells introduces the possibility of significantly reducing the total time and resources required for their production, and in turn, allows these valuable compounds to become cheaper and more readily available. Most biosynthesis pathways used in metabolic engineering applications have been discovered manually, requiring a tedious search of existing literature and metabolic databases. However, the recent rapid development of available metabolic information has enabled the development of automated approaches for identifying novel pathways. Computer-assisted pathfinding has the potential to save biochemists time in the initial discovery steps of metabolic engineering. In this paper, we review the parameters and heuristics used to guide the search in recent pathfinding algorithms. These parameters and heuristics capture information on the metabolic network structure, compound structures, reaction features, and organism-specificity of pathways. No one metabolic pathfinding algorithm or search parameter stands out as the best to use broadly for solving the pathfinding problem, as each method and parameter has its own strengths and shortcomings. As assisted pathfinding approaches continue to become more sophisticated, the development of better methods for visualizing pathway results and integrating these results into existing metabolic engineering practices is also important for encouraging wider use of these pathfinding methods. E. coli and S. cerevisiae, to separate the supply of these therapeutics from the plants they were sourced from. At the root of these successes is the identification of the requisite pathways and the systematic transfer of these pathways to a microbial host.Metabolic engineering is the scientific process of manipulating the metabolism of a microorganism to produce valuable compounds. Engineering microbial production involves the disruption of endogenous genes or adding genes from heterologous organisms to form pathways that tap into the natural metabolic network. There have been numerous successes of metabolic engineering, including the well publicized biosynthesis of artemisinic acid, a precursor to the antimalarial drug artemisinin , and theMetabolic pathfinding has clear applications to the first step in the design-build-test-learn cycle for developing biosynthetic pathways . We defiTraditionally, researchers have manually searched existing literature and databases to design pathways. However, the rapidly growing body of metabolic information makes it difficult to effectively survey and utilize all available resources. Computational approaches have been developed to enable researchers to take advantage of these growing resources. For example, pathways for production of 1,4-butanediol, a non-natural compound, were discovered with the assistance of a pathway-identification algorithm . ThousanAssisted metabolic pathfinding aims to solve two main challenges—the challenge of efficiently speeding up the pathway search process and the challenge of selectively finding biologically feasible, novel pathways. This paper focuses primarily on the approaches of pathfinding algorithms that address these two challenges. However, improvements in the search algorithms alone are not sufficient to solve these challenges, as the quality of the pathway results is also heavily dependent on the metabolic resources utilized by the search algorithm. Advancements in metabolic pathfinding rely on advancements in techniques for expanding the metabolic search space. For example, retrosynthesis-based approaches , 10 can E. coli. However, this approach is not yet able to computationally scale to very large metabolic networks [The metabolic pathfinding problem itself can be further divided into two different approaches: graph-based pathfinding and constraint-based pathfinding. This review will focus on graph-based pathfinding, which highlights the connections between compounds and reactions in the metabolic network. Graph-based approaches represent a metabolic pathway as a path that consists of an ordered series of intermediate compounds and reactions that transform some defined starting compound(s) to some defined target compound(s). Graph-based pathfinding utilizes a very well-studied data structure to represent the metabolic network, abstracting away more complicated interactions between compounds and enzymes in the cell. This abstraction enables graph-based methods to readily scale with larger metabolic networks spanning multiple organisms. However, since much of the underlying metabolic network is abstracted by the graph representation, there is a greater chance for graph-based approaches to return pathways without biological significance unless relevant parameters and heuristics are introduced to guide the search. Constraint-based methods  highlignetworks . Though networks , 21, we A metabolic network can be described as connections between compounds and the enzymes catalyzing reactions between compounds, which lends itself well to graph representation. There are many different ways a metabolic network can be represented as a graph Fig. b. One ofThis review covers the techniques supporting graph-based metabolic pathfinding algorithms and the heuristics that guide pathway discovery from networks, enzymatic reactions, and chemical structures to a specific host organism context Fig. b. We wilProperties of the metabolic network representation can be used to guide and constrain the search problem and rank the resulting pathways. The properties that have been used in the literature are the connectivity of the network and the length of pathways found. The individual compounds and reactions of a pathway can also be assigned weights based on biochemical and network-based properties.The graph-based representation of the network makes it intuitive to gravitate towards graph-based features and constraints, particularly graph connectivity Fig. a. Many aIn Faust et al. , differeIn MetaRoute , the weiThe connectivity of a graph is very simple to compute, and it is no surprise that it has been used by several metabolic pathfinding approaches. Despite its simplicity, connectivity can be used to effectively infer some biochemical information about the metabolic network. However, excluding features of the metabolic network based on connectivity alone may not reflect known biochemical properties. For example, excluding highly connected compounds to avoid currency compounds may also exclude compounds that play a significant role in pathways . Unlike other algorithms, M-path by Araki et al.  uses hubk-shortest paths [k-shortest paths) and used to organize pathway results. In order to distinguish pathfinding methods that actively include path length as a constraint or heuristic from methods that only use path length to rank pathway results, the latter cases were not marked as using path length in Table Pathfinding algorithms often optimize for pathways with the smallest number of enzymatic steps, as these pathways tend to require less manipulation in a metabolic engineering context Fig. b. Many pst paths , 42). RaMost network representations include both structures of compounds and reactions, along with parameters that give additional information on both these parts. The chemical structure of compounds in the metabolic network can be useful in inferring the existence of a biochemical reaction between compounds, as biochemical reactions tend to have products that structurally resemble one or more reactants. Structural information can be represented at different levels of detail, which introduces a trade-off between the accuracy of the similarity measure and the computational complexity of the overall metabolic pathfinding problem.k-shortest paths. The MetaRoute algorithm [At the finest level of detail, algorithms can track changes on atomic level Fig. c. Retainlgorithm  also uselgorithm  uses a hlgorithm , the patlgorithm , pathwaylgorithm , any realgorithm  to insurlgorithm  maximizelgorithm . Insteadlgorithm . Even soIf two compounds have similar chemical structures, there is a decent chance that these compounds can be connected by a common reaction. Several approaches have used different representations of chemical structure as a way of guiding, constraining, and ranking the search.Several approaches use chemical fingerprints and Tanimoto coefficients  to measuOther approaches rely on the graph representation of chemical compounds. Metabolic Tinker  uses a hUnlike chemical fingerprints, where a pre-determined set of chemical characteristics are used to compare two compounds, the graph comparison approach directly compares the chemical structure of two compounds against each other. The graph comparison approach tends to be more accurate in calculating structural similarity but is more computationally expensive . In GEM-Calculating compound similarity is not as computationally expensive as atom mapping and serves as a check that the reactions included in pathways are biochemically feasible. However, compound similarity falls short in the cases where two compounds share many common structural components but are not biochemically related.In addition to information about the compounds involved in the network, graph-based searches also include information on reactions. This information can be used to both constrain and expand the search to find novel pathways.Building off the idea of structural similarity, some algorithms introduced reaction rules, or more general transitions between compounds based on changes in chemical structure. Two enzymatic reactions may involve different reactants and products; however, if the same structural change occurs between reactants and products in these reactions , these reactions may both fall under the same reaction rule. Reaction rules allow new, potentially feasible pathways to be found by introducing reactions that may not yet have been added to metabolic databases. These rules can both be used to (1) create a metabolic network without directly requiring information on enzymatic reactions from metabolic databases and (2) help expand an existing metabolic network created based on a metabolic database.Reaction rules are based heavily on structural representations of compounds. In BNICE , 14, coml et al. , the molAnother common feature taken into account by pathfinding algorithms is thermodynamic feasibility of the reactions in pathways. Almost all algorithms that include thermodynamics use the component contribution method  for calcicTinker , missingIn BNICE , the \doo et al. , enzymesmentclasspt{minimathFinder  uses \doE. coli, compared to a graph-based search without stoichiometric constraints. Introducing stoichiometric constraints allows carbon flux paths to distinguish between oxic and anoxic conditions in E. coli, which was not possible in previous graph-based algorithms. However, this pathfinding method was only tested within the metabolic network of a single well-studied organism (E. coli) and, like constraint-based methods, is not easily scalable to large multi-organism networks.Graph-based pathfinding methods can incorporate reaction stoichiometry to limit the number of biologically irrelevant pathway results. The carbon flux paths algorithm proposed by Pey et al. , 43 intrEnzymes can have different reaction rates, depending on how efficient an enzyme is in converting the substrate to product. On the other hand, promiscuous enzymes can catalyze reactions which may not be found in existing databases and may be used to expand the metabolic pathfinding search. In Cho et al. , bindingMany algorithms give the user the ability to select an organism of interest. Arita et al.  mention Pathfinding is a critical and preliminary step in the development of novel biosynthetic pathways. Pathfinding is often done manually, though there are many existing tools that can enumerate putative pathways with minimal input from the user. After a pathway has been identified, much time and effort goes into building, testing, troubleshooting, and optimizing the biological system, and not the initial pathway discovery . This isThere has been a recent push to expand searches to non-native compounds using reaction rules, building on BNICE , 14, becAlthough the cumulative information that is available across all metabolic databases is extensive, manually searching, gathering, and compiling information from different databases is a challenging task. Each database often has its own representation and set of ID numbers for identifying components like compounds and reactions, in addition to its own organization schema, suited specifically for the intended purposes of the database. These differences make it challenging to determine the exact links and relationships between information in different databases. There have been a few recent efforts to integrate different metabolic databases and create a less redundant, more comprehensive, and more accessible resource for metabolic information (e.g., BKM-react , MetRXN As the pathfinding capabilities improve, so do the number of solutions that can potentially be generated, and with it the challenge of providing the user with tools to explore the solutions that can number in the thousands and identify pathways of interest. Because of this, there is an increasing amount of user interaction built into pathfinding webservers (see MRSD , BioSyntIn addition to visualizations, a well-developed interface could integrate suggestions for genes based on enzyme activity and evidence of heterologous gene expression so that the user can seamlessly transition from pathway discovery to the initial build phase. Databases, such as BRENDA , have exAlmost all pathfinding algorithms are limited to producing linear pathways with a few exceptions , 69. BraUltimately, the best pathfinding algorithm is the one that suits the user’s needs and is paired with an interface that facilitates pathway discovery. Pathfinding webservers can assist with the design of novel, feasible, and hopefully improved pathways, but as discussed, pathfinding needs to become more highly integrated with the entire process of metabolic engineering. This survey of the available features and future directions aims to increase adoption of existing pathfinding tools while advocating for advancements that will increase their utility.


Pseudomonas putida KT2440 and identified the discrepancies leading to inconsistent pathway calculation results. The mistakes in the models were corrected based on information from literature so that all the calculated synthesis and uptake pathways were the same. Subsequently we built a pathway-consensus model and then further updated it with the latest genome annotation information to obtain modelPpuQY1140 for P. putida KT2440, which includes 1140 genes, 1171 reactions and 1104 metabolites. We found that even small errors in a GSMN could have great impacts on the calculated optimal pathways and thus may lead to incorrect pathway design strategies. Careful investigation of the calculated pathways during the metabolic network reconstruction process is essential for building proper GSMNs for pathway design.Over 100 genome-scale metabolic networks (GSMNs) have been published in recent years and widely used for phenotype prediction and pathway design. However, GSMNs for a specific organism reconstructed by different research groups usually produce inconsistent simulation results, which makes it difficult to use the GSMNs for precise optimal pathway design. Therefore, it is necessary to compare and identify the discrepancies among networks and build a consensus metabolic network for an organism. Here we proposed a process for systematic comparison of metabolic networks at pathway level. We compared four published GSMNs of  GSMNsput flux . It is dSalmonella Typhimurium LT2 [To address this problem, efforts have been made to create a consensus metabolic network by merging GSMNs for the same organism. The first consensus metabolic network, Yeast 1, was created from available yeast networks iMM904 and iLl672 by using a jamboree approach [rium LT2  and Humarium LT2  which harium LT2 . A drawbrium LT2 .Pseudomonas putida KT2440 based on four previously published metabolic network models, iJN746 [One of the important applications of GSMNs is to design optimal pathways for the production of biochemicals. Using methods like flux-balance analysis (FBA) ,15, rese, iJN746 , iJP815 , iJN746 , PpuMBEL, iJN746  and iJP9, iJN746 . ActuallP. putida models were quite different. iJP815 and iJP962 were reconstructed by the same research group and had smaller differences in the growth yield (92.4 gDCW·molGlc-1vs 101.1 gDCW·molGlc-1) [P. putida KT2440 and further improved with updated genome annotation information to obtain model PpuQY1140 for P. putida KT2440. Compared with previous models, the simulation results from the new model are in better agreement with published experimental data.The original analysis results from the four olGlc-1) . By compolGlc-1) ,21. As tPseudomonas putida KT2440 models through a process summarized in P. putida KT2440.The pathway-consensus metabolic network reconstruction was bulit by taking the systematic comparison of published P. putida KT2440, iJN746, iJP815, PpuMBEL1071 and iJP962, have been reconstructed by three research groups. The optimal growth rate can be calculated by performing FBA with the biomass production reaction as objective function. The optimal ATP production rate can be calculated by using the ATP maintenance reaction as objective function [Four metabolic networks of function . The sim-1 compared to 0.46 h-1). More surprisingly, the obtained ATP production rate from PpuMBEL1071 was 999999 mmol·gDCW-1·h-1 which was actually the upper boundary of the reactions in that model. The ATP production rate remained constant at the value of the upper boundary even when we set glucose uptake rate to zero. This indicated that ATP was generated infinitely in PpuMBEL1071 without substrate consumption, which was obviously unreasonable for a biological network model. ATP plays an important role in cell metabolism as energy carrier. The generation and consumption of most metabolites are linked with energy metabolism, and thus mistakes in ATP production may also lead to incorrect calculation results for pathways to other metabolites. Therefore, we first analyzed the optimal ATP production pathways from the four models to identify the mistakes and correct them.It can be seen that there was nearly a two-fold difference between the highest and lowest growth rates H generation loops as shown in In view of the incorrect reaction loops caused by incorrect reaction equations, we further investigated if this kind of loops can be detected by using the method proposed by Gevorgyan et al  for checP. putida KT2440 [2 could be oxidized through the respiratory chain  for the production of 1 gram dry cell weight. j.After revising the models to obtain consistent ATP production rates, we further calculated the optimal growth rates of the four models  and then to PAPS (adenosine-5-phosphosulfate) [In above-mentioned example, the energy consumption for sulfate transport and metabolism was different in the four models . Two sulsulfate) . PAPS is+. The numbers of reactions corrected for all the three types of errors in the four models were shown in -1. The calculated sum of squared errors indicated that the results from iJP815 and iJP962 were close to those from the consensus network while iJN746 and PpuMBEL1071 had more errors in the model.As described in the ATP production pathway analysis section, incorrect reaction equation led to unlimited ATP production. Here by investigating the precursor synthesis pathways, we found more mistakes in reaction equations in all four models. For example in PpuMBEL1071 the reaction equation of R0749 was Folate + NADPH → Tetrahydrofolate + NADPP. putida KT2440 is well known for its ability to grow on many amino acids and organics acids as carbon source [P. putida [In all the above analysis glucose was used as the sole carbon source. Whereas n source ,32. Corr. putida ,31,33–36. putida . By comp2-dehydro-3-deoxy-phosphogluconate aldolase.However, there are still quantitative differences on the optimal growth rates calculated from different models. Actually we found that for 36 substrates the growth rate calculated with PpuMBEL1071 was higher than those obtained with other models. By investigating the pathways we revealed that the main reason was the incorrect reversibilty information of the reaction ED_edd  catalyzeThe two reactions were mistakenly set to be reversible and thus enabled the formation of D-Ribulose 5-phosphate by the opposite direction of Entner-Doudoroff (ED) pathway (G3P + Pyr → 6-Phospho-D-gluconate → D-ribulose 5-phosphate) which consumed less energy for some substrates rather than pentose phosphate (PP) pathway. We corrected this error and recalculated the growth rates in PpuMBEL1071 for further comparison. The process was repeated untill the growth rates from all 48 substrates were the same in all four models. The number of added and modified reactions during this process were also shown in Four updated models of iJN746, iJP815, PpuMBEL1071 and iJP962 with more reactions and genes were obtained –S6 FilesP. putida KT2440 in 2002 [P. putida KT2440 from NCBI and submitted it to the RAST annotation server [3-1→H+ + Pi + ADP + Oxaloacetate) was linked with an nonexist gene PP_5437 in the updated iJP962 model. From the annotation information we found the right gene should be PP_5347 annotated as pyruvate carboxyl transferase subunit A (EC 6.4.1.1). Altogether we added 155 genes and 74 reactions. The final pathway-consensus metabolic model PpuQY1140 contains 1140 genes, 1171 reactions and 1104 metabolites . The results shown in gnuK deletion strain which grew even faster than the wild type strain. This phenomenon might be caused by complex regulatory mechanisms and thus can not be explained by metabolic network analysis alone. Generally the simulation results from the consensus model are also better than those from other models, especially iJN746 and PpuMBEL1071. This is in agreement with the biomass precusor synthesis pathway simulation results as shown in P. putida KT2440 and its deficient mutants growing on glucose minimal medium, the growth rates of wild type P. putida KT2440 and two deficient mutants (gcd and glk) growing on toluene M9 minimal medium were also tested by del Castillo T et al [To test the accuracy of the updated model PpuQY1140, we compared the simulated growth rates from different models with the experimentally measured values reported by del Castillo T et al . They meP. putida KT2440, all synthesis pathways for biomass building blocks and substrate uptake pathways were made consensus. Comparative analysis of the models revealed that small errors in reaction equations and reaction reversibility could greatly affect the pathway calculation results, sometimes even led to infeasible pathways such as the unlimited NADH generation loops. Therefore the pathway examination step should be included in the metabolic network reconstruction process to make sure that the quality of network is high enough for reliable pathway design.We proposed a novel pathway-consensus process for identifying and solving the discrepancies between metabolic network reconstructions for the same organism by pathway level comparison. By applying this approach to four GSMNs of This pathway-consensus approach should be regarded as complementary to the network reconciliation approach proposed by Oberhardt et al . The netP. putida GSMNs in SBML format were obtained from supplementary materials of the publications [-1·h-1. The uptake rates of nitrogen source, oxygen, phosphate and sulfur were not constrained. Exchange reactions were added to PpuMBEL1071 because there were no reactions for the utilization of different substrates and nutritions in this model. The growth-associated and non-growth-associated energy maintenance (GAM) and (NGAM) [Four ications –19. The d (NGAM)  parameteFlux-balance analysis (FBA)  is a widThe P/O ration represents the number of ATP molecules (P) formed from per oxygen atom (O) . Here thThe four reactions were combined to get the overall reaction  by elimiS1 Table(XLSX)Click here for additional data file.S2 Table(XLSX)Click here for additional data file.S3 Table(XLSX)Click here for additional data file.S4 Table(XLSX)Click here for additional data file.S5 Table(XLSX)Click here for additional data file.S6 Table(XLSX)Click here for additional data file.S7 Table(XLSX)Click here for additional data file.S1 File(DOCX)Click here for additional data file.S2 File(DOCX)Click here for additional data file.S3 File(XML)Click here for additional data file.S4 File(XML)Click here for additional data file.S5 File(XML)Click here for additional data file.S6 File(XML)Click here for additional data file.S7 File(XLSX)Click here for additional data file.S8 File(XML)Click here for additional data file.


Oryza sativa japonica, no GSM with gene-protein-reaction association exist for Oryza sativa indica. Here, we report a GSM, OSI1136 of O.s. indica, which includes 3602 genes and 1136 metabolic reactions and transporters distributed across the cytosol, mitochondrion, peroxisome, and chloroplast compartments. Flux balance analysis of the model showed that for varying RuBisCO activity (Vc/Vo) (i) the activity of the chloroplastic malate valve increases to transport reducing equivalents out of the chloroplast under increased photorespiratory conditions and (ii) glyceraldehyde-3-phosphate dehydrogenase and phosphoglycerate kinase can act as source of cytosolic ATP under decreased photorespiration. Under increasing light conditions we observed metabolic flexibility, involving photorespiration, chloroplastic triose phosphate and the dicarboxylate transporters of the chloroplast and mitochondrion for redox and ATP exchanges across the intracellular compartments. Simulations under different enzymatic cost conditions revealed (i) participation of peroxisomal glutathione-ascorbate cycle in photorespiratory H2O2 metabolism (ii) different modes of the chloroplastic triose phosphate transporters and malate valve, and (iii) two possible modes of chloroplastic Glu–Gln transporter which were related with the activity of chloroplastic and cytosolic isoforms of glutamine synthetase. Altogether, our results provide new insights into plant metabolism.To combat decrease in rice productivity under different stresses, an understanding of rice metabolism is needed. Though there are different genome scale metabolic models (GSMs) of  Oryza sativa) is one of the major food crops cultivated worldwide and it provides nearly 20% of the caloric intake for humans. Several environmental conditions like drought, flood, nutrient availability etc. govern the rice yield. In order to understand the effect of these factors on rice production it is important to study rice cellular physiology which includes metabolism under various conditions. One of the approaches toward understanding rice metabolism is reconstruction and analysis of its genome scale metabolic model (GSM). GSMs represent a list of all reactions encoded in the genome of the target species and associated metabolites. GSMs of several plants have been reconstructed including Arabidopsis thaliana (japonica) was reported in 2013 which was used to study rice metabolism under varying light intensity   to stress conditions (Vc/Vo < 3). Also, plant leaves are constantly exposed to different intensities of sunlight. Such variations in light intensity may cause readjustments in plant metabolism. Previous studies have simulated the dependence of photosynthetic metabolism of prokaryotes, algae and plants on light energy  . We perft energy . These st energy . Furthermilation . The difmilation . Additiomilation .O.s. indica (OSI1136) which represents a photosynthetic leaf tissue, with 1136 reactions, 1330 metabolites and gene protein reaction (GPR) association for 3602 unique genes. The reconstructed model was simulated and analyzed under three conditions – (i) varying flux ratio of carboxylation to oxygenation of RuBisCO (Vc/Vo) from 1:1 to 5:1 (ii) varying light intensity (light scan) and (iii) different enzymatic costs by assigning random weights to the reactions in the model. Vc/Vo < 3 corresponds to increased rate of photorespiration which is representative of drought stress in plants  based ScrumPy metabolic modeling package . The biomass transporters were set to carry fixed flux  were left unconstrained, unless specified. A cell maintenance cost was assumed to 0.1 flux unit through a generic ATPase reaction , FVA was performed. FVA calculates the maximum and minimum allowable flux values of each of the reactions of the solution space, while satisfying the optimality of a given objective function. The optimal objective value of the primary objective function  was computed for synthesis of all the biomass components in experimentally defined proportions. For each reaction in the solution space, the reaction flux was maximized and minimized to get the allowable range of flux values.To find the allowable range of flux of each of the reactions under drought like and normal air conditions (v = vv) was varied, starting at the point at which solution was obtained till changes in reaction-fluxes remained linear. Thus, the photon flux was varied from 0.33 to 10.00 flux unit. The lower limit indicated the photon flux below which there existed no feasible solution for biomass synthesis under defined constraints. The upper limit indicated the photon flux beyond which the flux plot of the reactions remained uniform. An additional constraint was that the flux of cyclic photophosphorylation could not exceed that of non-cyclic.In addition to the above-mentioned constraints, the photon flux (Vc/Vo) of RuBisCO from 1:1 to 5:1. The RuBisCO catalyzed carboxylation and oxygenation reactions were combined into a single reaction according to the Vc/Vo investigated  was reported in 2013 (indica we started with its protein and we used the updated rice japonica model (CK1721) as a reference for gap filling. Recently, a rice japonica model (iOS2164) was reported . These unique reactions are mainly involved in secondary metabolism such as jasmonic acid synthesis pathway , phytoalexin synthesis pathway  fatty acid beta oxidation  etc. . Compared to iOS2164 the indica model has lesser number of reactions firstly, because the former has considered reactions associated with the thylakoid, vacuole and ER compartments. Secondly, instead of including all the steps of cyclic and non-cyclic photophosphorylation (as considered in iOS2164), in OSI1136, we represented them as single reactions which were obtained from elementary modes analysis . At the same time fluxes of cyclic and non-cyclic photophosphorylation showed that the total ATP and NADPH generated was maximum at Vc/Vo = 1 and minimum at Vc/Vo = 5 . Despite the changes in total ATP and NADPH, the ATP:NADPH ratio was found to be same from Vc/Vo = 2 to Vc/Vo = 5 . While the non-cyclic photophosphorylation was always active the cyclic photophosphorylation carried flux only at Vc/Vo = 1. The higher amount of ATP generated in the chloroplast at Vc/Vo = 1 was found to be partly used in photorespiratory ammonia fixation in the chloroplast  and the rest of it got utilized in the recycling of glycolate  which is formed from phosphoglycolate (Pgly) produced in the RuBisCO oxygenase catalyzed reaction. Accordingly, the flux of these reactions was maximum at Vc/Vo = 1 . Most of the other ATP consuming reactions in the chloroplast did not show variation in the flux except chl_Ru5Pk (EC 2.7.1.19) and chl_PGK (EC 2.7.2.3). The former is the source of ribulose bisphosphate (RuBP) in the chloroplast which participates in RuBisCO carboxylase/oxygenase reaction to generate 3-phosphoglycerate (3-PGA or PGA). PGA in turn gets phosphorylated to 1,3-bisphosphogycerate (BPGA) by phosphoglycerate kinase (PGK). Both RuBP generation (by chl_Ru5Pk) and phosphorylation of PGA to BPGA (by chl_PGK) were found to be maximum to sustain the C3 cycle and to synthesize the same amount of biomass under limited carbon conditions at Vc/Vo = 1 . The other major source of ATP (besides cyclic and non-cyclic photophosphorylation) is the mitochondrial ATP synthase (Complex_V) which also carried maximum flux at Vc/Vo = 1 . Thus, the ATP generated in the mitochondrion by Complex V and ATP released by the mitochondrial ATP transporter to cytosol was highest at Vc/Vo = 1. This was because, while chloroplastic ATP met increased PR demands, ATP demand for rest of cellular metabolism was met by mitochondrial ATP. In addition, we also calculated the quantum demand  from Vc/Vo = 1 to Vc/Vo = 5. As expected, the fall in the QD from Vc/Vo = 1 to Vc/Vo = 5 was similar to that observed for photon flux . The higher photon flux ultimately reflects the requirement to synthesize the same amount of biomass under limited carbon conditions where the efficiency of photosynthetic triose phosphate production reduces and where carbon recycling by photorespiration becomes primary.We calculated the photon flux needed to produce ATP and NADPH which ultimately are required for biomass synthesis in fixed proportion and cellular maintenance. There was a decrease in photon flux from Vc/Vo < 3) when the chloroplastic NADPH generation is high (as described in the previous section) the activities of Mal-OAA transporter and NADP–MDH were also high; and with decrease in photorespiration  the flux through the Mal–OAA and NADP–MDH decreased . The observation suggested that the malate valve is activated to transport excess reducing equivalents out of chloroplasts under drought stress and at higher rates of photorespiration. This observation was supported by a previous experiment on tobacco plants which reported reduced expression of NADP–MDH at decreased rate of photorespiration (Plant membranes are broadly impermeable to NAD(P) and NAD(P)H, so they possess specific transporters for the exchange of Mal and OAA which enable the indirect transport of reducing equivalents between different cellular compartments. NADPH dependent malate dehydrogenase (NADP–MDH) in the chloroplast converts OAA to Mal, facilitating the regeneration of the electron acceptor NADP in the chloroplasts. The enzyme (MDH) is known to play an important role in maintaining redox balance in plants under changing environmental conditions . The ‘mapiration .Vc/Vo.The glycolytic pathway in plant’s cytosol consists of alternative ways to metabolize GAP to PGA. One way occurs via the GAPDH (EC 1.2.1.12) and phosphoglycerate kinase  generating NADH and ATP; the second is a single step catalyzed by non-phosphorylating glyceraldehyde-3-phosphate dehydrogenase  and generating NADPH . The stuVc/Vo . However, GAPDH and PGK were inactive at Vc/Vo = 1 , active at Vc/Vo = 3  and the flux through these reactions increased from Vc/Vo = 2 to Vc/Vo = 5 . The export of GAP from the chloroplast increased at Vc/Vo = 3 which was used in GAPN and GAPDH. At the same time the flux through the mitochondrial Complex V decreased. Consequently, the ATP generation in the mitochondrion and its release decreased . Under this condition the need of ATP and NADH in the cytosol were fulfilled by the combined activity of GAPDH and PGK. On the other hand, the ATP generated in the mitochondrion by Complex V was maximum at Vc/Vo = 1 . Thus, the mitochondrion serves to play an important role in supplying the cytosolic ATP under drought stress as discussed earlier. Cytosolic GAPDH and malate valve are important for maintaining cellular redox homeostasis in plants  and normal air condition (Vc/Vo = 3). The minimum and maximum flux obtained from FVA  for chloroplastic Mal-OAA shuttle and malate dehydrogenase showed decrease in their activities at Vc/Vo = 3 compared to Vc/Vo = 1 which supported our results of FBA . Also, the flux range for GAPDH and PGK catalyzed reactions showed that at Vc/Vo = 1 both the reactions carried zero flux and were operative at Vc/Vo = 3. This also supported our FBA predictions as described in Figure 3. While some reactions like chl_61118RXN, chl_DXSRXN, HISTALDEHYD-RXN showed FBAval = FVAmin = FVAmax , other reactions like mit_AconDHatase, mit_AconHydr etc. always showed FBAval = FVAmin < FVAmax . Non-cyclic modes of TCA cycle can be operative if the demand for ATP is low or if alternative sources of ATP are available . Glycine oxidation and MalDH are sources of mitochondrion NADH. The flux through part of TCA cycle which operated to support biomass synthesis remained constant. As such any fluctuation in mitochondrion ATP production was associated with changes in MalDH and GDC activity. Although MalDH activity was minimum at Vc/Vo = 1, the flux of mitochondrion complex V was maximum . Thus, under such conditions photorespiratory glycine oxidation was important for ATP supply.The TCA cycle and the Calvin cycle form a part of the central metabolic pathways in plants. The flux analysis showed increase in the flux through the Calvin cycle under stress condition. For example, variations in the fluxes through some of the reactions associated with Calvin cycle are shown in oportion . Only a vailable . Flux anncreases . Here, tVc/Vo = 1 corresponds to increased photorespiration, accordingly our results showed increase in the ammonia generated in the mitochondrion via glycine decarboxylase (GDC) . The analysis of flux values through ammonia transporters showed that an equivalent amount of mitochondrial ammonia released into the cytosol got imported into the chloroplast where it was completely utilized in the GS/GOGAT pathway . It is well established that the photorespiratory ammonia generated in the mitochondrion is transported to plastids, which is then re-assimilated into glutamate by GS and GOGAT .nd GOGAT . AccordiVc/Vo = 1 both chloroplastic ATP and NADPH were maximum , the cellular ATP demand was fulfilled mainly through mitochondrial glycine oxidation. Concomitantly, ammonia generated by glycine oxidation was refixed by GS2. Refixation of ammonia, the glycerate kinase reaction (photorespiratory metabolism) and necessary recycling of RuBP utilized the excess chloroplastic ATP . The excess reductant was transported out to the cytosol by the malate valve; a part of this excess reductant was used in HPR1 to sustain PR and the rest was utilized to support other cellular processes. Thus, at Vc/Vo = 1 we observed coordinated metabolism wherein PR served as a sink for excess chloroplastic ATP and redox. At Vc/Vo = 3, ATP generation in the mitochondrion and its release to the cytosol decreased . Simultaneously, the export of GAP from the chloroplast increased which was used in GAPN and GAPDH. Under this condition the need of ATP and NADH in the cytosol were fulfilled by the combined activity of GAPDH and PGK. Such plasticity in plant metabolic network across different cell organelles is useful in maintaining cellular ATP and NAD(P)H homeostasis, further discussed below.The chloroplast is the site of cyclic and non-cyclic photophosphorylation for ATP and NADPH generation. However, depending on environmental conditions metabolic adjustments particularly in mitochondrial metabolism becomes important in maintaining cellular ATP and redox homeostasis . AlthougPhotosynthetic metabolism is light dependent. Studies have reported that varying light conditions affect gene expression and therefore enzyme activity . In-siliFigure 6A. This observation supports the view that photorespiratory metabolism may serve to dissipate excess light energy and mitigate photoinhibition . With further increase in light, the chloroplastic triose phosphate exchange reactions became active to fulfill the increasing reductant demand of HPR1 . GAP, formed through PGK and GAPDH catalyzed reactions of the Calvin cycle was exported to the cytosol. In the cytosol, GAP to PGA conversion took place and the PGA thus produced was recycled back to the chloroplast. Chloroplastic reductants and ATP were made available to the cytosol through this recycling between PGA and GAP. This reductant was exported to the peroxisome and made available for HPR1 through Mal-OAA shuttle and malate dehydrogenase catalyzed reactions of the cytosol and peroxisome compartments. Thus, photorespiratory metabolism facilitates export of ATP and reducing equivalents from the chloroplast. Increased light stress leads to generation of reactive oxygen species in photosynthetic systems and photorespiration alleviates this stress especially in the chloroplast . This is because the AOX component of the ETC decouples ATP generation from redox dissipation . Simultaneously, the fluxes of chloroplastic MalDH  and malate valve increased to supply reductant to the cytosol which was finally used in the HPR1 catalyzed reaction . The chloroplastic triose phosphate and Mal-OAA transporters are important shuttle systems for transporting reducing equivalents out of the chloroplast. In a study on NADP-MDH Arabidopsis mutants, the NADP-GAPDH was the only alternative that operated to supply reductants to the cytosol . Although at low and medium light we found that glycine oxidation was coupled to oxidative phosphorylation . On the other hand, reaction fluxes correlated with light intensity over the range 0.0–1.0 . Figure 7C shows the comparison of reactions that were found to be strongly correlated either with RuBisCO carboxylase/oxygenase or light intensity or both. Forty-seven reactions were found to be strongly correlated with both RuBisCO carboxylase/oxygenase activity and light intensity. This included the reactions of photorespiration in the chloroplast, mitochondrion and peroxisome and the reactions associated with Calvin cycle. A study showed that genes associated with Calvin cycle and photorespiration are co-expressed  were also found to be highly correlated with RuBisCO carboxylase/oxygenase activity and light intensity. Some of the reactions that showed high correlation only with RuBisCO carboxylase/oxygenase were found to be moderately correlated with light intensity . This was because the fluxes of these reactions namely mitochondrial complex V (mit_Complex_V), chloroplastic transporters for Mal-OAA , GAP (chl_GAP_tx), and PGA (chl_PGA_tx) showed significant variations with increase in light intensity Figure 6. Cytosolic glutaminase reaction (GLUTAMIN-RXN) and chloroplastic Glu-Gln transport (chl_Gln_Glt_tx) were found to be highly correlated with light intensity but not with Vc/Vo . With increase in light intensity, flux through cyclic photophosphorylation (chl_LightCyc), and thus ATP generated in the chloroplast increased. Cytosolic glutaminase reaction (GLUTAMIN-RXN) and glutamate-glutamine transporter (chl_Gln_Glt_tx) operated along with chloroplastic glutamine synthetase (GS2) and the ammonia transporter across the chloroplast to utilize the ATP.Reaction correlation with xpressed . Also, aVc/Vo) was fixed at 3:1 and Case 2 – RuBisCO carboxylase/oxygenase activity was not fixed in any ratio. Plants have acquired several adaptive mechanisms including flexibility in metabolic pathways to combat different environmental stress conditions  . Variation in the activity of the enzymes of malate valve is known to occur in plants under different conditions like high light, high CO2, cold stress etc . This activity of the malate valve which acts as an indirect export system for reducing equivalents serves to balance the ATP/NADPH ratio in the chloroplast (When or NADP) . Most ofor NADP) . The resress etc . Our resoroplast . Fifty-soroplast . The resVc/Vo fixed at 3:1) we found that different modes of chloroplastic Glu–Gln transporter operated. It operated maximum number of times  to import Glu into the chloroplast and export Gln out of the chloroplast . At other times  it operated in reverse direction, i.e., Glu was exported out of the chloroplast and Gln entered into the chloroplast  except once where it was found to be inactive. In, Mode A, most of the times low photon flux was required (≈ 0.33–0.33 flux units) and in Mode B the photon flux was in the range of 0.47–0.48 flux units . Concomitantly, the ATP:NADPH ratio was less for Mode A than for Mode B . These observations were related to changes in the cytosolic (GS1) and chloroplastic (GS2) glutamine synthetase activities. The relative amounts of these isoenzymes GS1 and GS2 . GS2 is the predominant isoform of GS in C3 species The increased activity of the malate valve under drought stress indicates its possible role in transporting reducing equivalents out of the chloroplast.(2)Vc/Vo increases .GAPDH and PGK can also act as source of cytosolic ATP when (3)For the light conditions investigated, we have presented a coherent mechanism of photorespiratory metabolism in conjunction with other reactions namely mitochondrion ETC, glycolysis, and intracellular transporters which function to maintain cellular ATP and NAD(P)H homeostasis.(4)Under different enzymatic cost conditions there exists variation in participation of peroxisomal glutathione-ascorbate cycle and different modes of the chloroplastic TPTs and the malate valve may become operative. Also, chloroplastic Glu–Gln transporter may show different modes of activity related with the functioning of GS1 and GS2, which may serve as alternative routes for stress adaptation in plants.AC and BH reconstructed the model. AC, BH, and RS carried out simulation and analysis. AC and BH wrote the manuscript. SK supervised the work.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.