E. coli and triacetic acid lactone (TAL) in S. cerevisiae revealed that the identified interventions tend to cause less dramatic rearrangements of the flux distribution so as not to violate concentration bounds. In some cases the incorporation of kinetic information leads to the need for additional interventions as kinetic expressions render stoichiometry-only derived interventions infeasible by violating concentration bounds, whereas in other cases the kinetic expressions impart flux changes that favor the overproduction of the target product thereby requiring fewer direct interventions. A sensitivity analysis on metabolite concentrations shows that the required number of interventions can be significantly affected by changing the imposed bounds on metabolite concentrations. Furthermore, k-OptForce was capable of finding non-intuitive interventions aiming at alleviating the substrate-level inhibition of key enzymes in order to enhance the flux towards the product of interest, which cannot be captured by stoichiometry-alone analysis. This study paves the way for the integrated analysis of kinetic and stoichiometric models and enables elucidating system-wide metabolic interventions while capturing regulatory and kinetic effects.Computational strain design protocols aim at the system-wide identification of intervention strategies for the enhanced production of biochemicals in microorganisms. Existing approaches relying solely on stoichiometry and rudimentary constraint-based regulation overlook the effects of metabolite concentrations and substrate-level enzyme regulation while identifying metabolic interventions. In this paper, we introduce k-OptForce, which integrates the available kinetic descriptions of metabolic steps with stoichiometric models to sharpen the prediction of intervention strategies for improving the bio-production of a chemical of interest. It enables identification of a minimal set of interventions comprised of both enzymatic parameter changes (for reactions with available kinetics) and reaction flux changes (for reactions with only stoichiometric information). Application of k-OptForce to the overproduction of L-serine in  Computational strain design procedures aim at assisting metabolic engineering efforts by identifying metabolic interventions leading to the targeted overproduction of a desired chemical using network models of cellular metabolism. The effect of metabolite concentrations and substrate-level enzyme regulation cannot be captured with stoichiometry-only metabolic models and analysis methods. Here, we introduce k-OptForce, an optimization-based strain design framework incorporating the mechanistic details afforded by kinetic models, whenever available, into a genome-scale stoichiometric-based modeling formalism. The resulting optimization problems pose significant computational challenges due to the bilevel nature of the formulation and the nonconvex terms in the constraints. A tractable reformulation and solution procedure is introduced for solving the optimization problems. k-OptForce uses kinetic information to (re)apportion reaction fluxes in the network by identifying interventions comprised of both direct enzymatic parameter changes (for reactions with available kinetics) and reaction flux changes (for reactions with only stoichiometric information). Our results show that the introduction of kinetic expressions can significantly alter the identified interventions compared to those identified with stoichiometry-alone analysis. In particular, additional modifications are required in some cases to avoid the violation of metabolite concentration bounds, while in other cases, the kinetic constraints yield metabolic flux distributions that favor the overproduction of the desired product thereby requiring fewer direct interventions. Bio-production is emerging as a competitive strategy for the production of a wide range of chemicals ranging from biofuels, precursor chemicals and bioactive molecules  is typically solved to obtain the time-dependent variation in metabolite concentrations and reaction fluxes. Different forms of mechanistic expressions have been used extensively such as Michaelis-Menten or Hill Kinetic expressions The introduction of genome-scale models of metabolism Stoichiometry-based strain design algorithms are often structured as bilevel mixed integer linear optimization problems (MILP) et alet alet alE. coli model (comprised of 76 equations) to constrain an FBA simulation. Similarly, Cotton et alThe need to integrate the mechanistic detail (whenever available) of kinetic expressions with the genome-scale scope of stoichiometric models has been recognized early by the community. Dynamic Flux Balance Analysis (dFBA) E. coliS. cerevisiaeE. coli, and triacetic acid lactone (TAL) in S. cerevisiae. For the former, k-OptForce identified key regulatory bottlenecks in upper and lower glycolysis that must be overcome to redirect more flux towards L-serine, which regular OptForce fails to pinpoint. In addition, k-OptForce removed interventions identified by regular OptForce that resulted in kinetically infeasible flux re-distributions. Application of the k-OptForce for the microbial overproduction of TAL in S. cerevisiae revealed the impact of additional kinetic constraints in alleviating a severe worst-case simulation of regular OptForce, resulting in a higher prediction of TAL yield from fewer interventions as compared to regular OptForce predictions. The introduction of kinetic expressions in strain design can significantly affect the identified interventions in sometimes non-intuitive ways. In some cases additional modifications are needed to substitute interventions that cause enzyme saturation or concentration bound violations. The mechanism of action of these modifications is often subtle by alleviating substrate inhibition or draining away cofactors from competing pathways. In other cases, kinetic expressions shape flux distributions so as to favor the overproduction of the desired product requiring fewer direct interventions.Here, we introduce k-OptForce, which extends the previously developed OptForce procedure kinJ = {j|j = 1,2,…,kinN} and reactions with only stoichiometric information stoicJ = {j|j = 1,2,…,stoicN}  nonlinear ordinary differential equations (ODEs) denoting the non-steady-state balance for each metabolite. For a square system of ODEs integration yields steady-state metabolite concentrations and fluxes. The allowable metabolic phenotype  of the remaining portion of the network stoicJ is inferred so as to be consistent with the predicted fluxes of the reactions in kinJ.k-OptForce aims at utilizing the available kinetic information within a larger genome-scale stoichiometric model to more accurately characterize all possible metabolic phenotypes of the reference and engineered strains. The procedure builds upon the OptForce procedure oic} see . MetabolThe k-OptForce procedure is composed of the following steps:kinJ. The phenotypic space of the reference strain is then identified by iteratively maximizing and minimizing the flux of each reaction in stoicJ while keeping the fluxes of reactions in kinJ fixed at their steady-state values, and restricting the flux of any other reaction for which any experimental data is available at their experimentally determined values or ranges .The flux ranges of the strain consistent with a desired overproduction target are similarly constructed by successively maximizing and minimizing the flux of network reactions subject to network stoichiometry, overproduction target, while also incorporating the kinetic expressions for reactions in UMUST), down-regulated (LMUST), or be knocked out X) manipulations selected from within the MUST sets whose direct manipulation  ensures production of the desired chemical beyond a desired target even under the worst-case scenario where fluxes are re-apportioned to drain flux away from the target product.L,kiny and U,kiny associated with the LMUST and UMUST sets of reactions in kinJ respectively, are used to control the effect of engineering modifications to the corresponding L,kiny = 1 then U,kiny = 1 then stoicJ. A separate set of binary variables L,stoicy and U,stoicy identify interventions in stoicJ required to guarantee a non-zero yield of the target chemical consistent with the flux distribution in kinJ. It is important to note that the metabolic fluxes in kinJ remain unaffected by the worst-case simulation of the reactions in stoicJ in the inner problem.The worst-case scenario is mathematically described by extending the bilevel optimization problem used for original OptForce jv. The fitness function of the inner problem is imposed setting the objectives of the primal and the dual equal to each other. It is to be noted here that the dualization of the flux variable for each reaction in kinJ introduces a bilinear term to the single-level formulation. This can be avoided by imposing the Karush-Kuhn-Tucker (KKT) conditions of complementary slackness between the primal constraints and their dual variables. This leads to the introduction of a binary variable for each constraint in the primal problem which is generally more tractable than the original bilinear constraints (see Supplementary This bilevel formulation is converted into a single level mixed integer nonlinear problem (MINLP) using the conditions of strong duality. We construct the dual of the inner problem , and add all the dual constraints, along with those of the primal, to the outer optimization problem. Since all the nonlinear kinetic expressions are present in the outer problem, the inner  problem is linear in the reaction fluxes kinJ and subsequently within stoicJ  as in regular OptForce kinJ and stoicJ. It is to be noted that while the illustrated formulation only targets changes in kinJ the same analysis could be applied for the modulation of other kinetic parameters  in the model.Due to the nonconvex nature of the kinetic expressions and the large number of binary variables the resulting MINLP equivalent representation of the bilevel optimization problem may become computationally intractable. For these cases, we exploit the natural hierarchy of the model by first selecting interventions within n target . Keepingmaxv of one of the reactions has hit its upper bound, and, (5) analyze all alternate intervention strategies to identify common metabolites and/or enzymes that are being targeted.Once the FORCE set of interventions are identified , it is important to manually curate the results to understand the underlying metabolic reason behind each intervention. This is necessary since k-OptForce makes use of not just stoichiometry which imposes straightforward connections between reactants and products but also kinetics that introduce complex nonlinear couplings often between distal reactions through metabolite pools. While it is not possible to put forth an invariant set of rules for all applications, the following checks can be useful in general: (1) check whether any metabolite participating in affected reactions is hitting lower or upper concentration bounds, (2) identify if a common metabolite is fixing the branching ratio of fluxes in two pathways, (3) resolve whether a metabolite is limiting the flux of a reaction through substrate-level inhibition, (4) confirm if the flux in a pathway has been restricted because the E. coli and S. cerevisiae, respectively. The goal was to assess how the addition of kinetic information into stoichiometric models changes the list of identified interventions and more importantly what are the underlying reasons for the new interventions.We contrasted k-OptForce with the original OptForce Escherichia coli and Corynbacterium consists of a three-step pathway branching out of the glycolytic intermediate 3-phosphoglycerate (3pg). 3pg is converted to 3-phosphohydroxypyruvate (3php) by phosphoglycerate dehydrogenase , and phosphoserine transaminase  catalyzes the conversion of 3-phosphohydroxypyruvate (3php) to L-phosphoserine (pser-L) using L-glutamate as the amino acid donor. In the last step, phosphoserine phosphatase  catalyzes the final conversion to L-serine  flux ranges were identified by finding the maximum flux variability in the entire network while keeping the fluxes in kinJ fixed at the steady-state values obtained by solving the system of ODEs for the kinetic model . The L-serine overproducing network flux ranges were calculated for a target of 90% maximum theoretical yield . The minimum biomass production was kept at 10% of its maximum achievable. The maximum enzyme activity maxv of reactions in kinJ was allowed to vary from zero to two-fold up-regulation of its reference activity . Also, the concentration of metabolites in kinI was allowed to vary within a two-fold range from their steady-state concentrations in the reference strain.The reference strain  in oxidative pentose phosphate (PP) pathway consistent with the imposed L-serine overproduction ranged from 0 to 136 mmol gDW−1hr−1 when constrained by only stoichiometry and from 0 to 62 mmol gDW−1hr−1 when imposing also kinetic information. This range reduction is due to the limitations of the maximum enzyme activity as well as metabolic concentrations of glucose-6-phosphate (g6p) regulating G6PDH flux. The restrictions implied by kinetics also propagate throughout the stoichiometric part of the network leading to an on average ten percent range contraction for reactions in stoicJ. For example, the fluxes ranges in 2-oxogluterate dehydrogenase (AKGDH) and succinate dehydrogenase (SUCD) in the TCA cycle decreased 7.4% and 6%, respectively as a direct consequence of the flux range reduction for pyruvate dehydrogenase (PDH) in kinJ which supplies acetyl-CoA (accoa) to TCA cycle. As a result of the tighter flux ranges in the overproducing network (and characterization of base strain) many more reactions are identified that must depart from their original ranges  compared to regular OptForce both in UMUST (38 vs. 3) and LMUST (293 vs. 108) sets. For example, up-regulation of glucose-6-phosphate isomerase (PGI) in upper glycolysis supplies more flux towards 3pg and L-serine production. The flux range for PGI in the overproducing phenotype (−36–100 mmol gDW−1hr−1) was wide enough to overlap with its reference flux value (35 mmol gDW−1hr−1) suggesting PGI up-regulation is not necessary for L-serine overproduction. In contrast, using k-OptForce we find that the flux range of PGI in the overproducing phenotype is restricted to 38–98 mmol gDW−1hr−1 which does not include the reference value of 35 mmol gDW−1hr−1. This implies that it is impossible to produce L-serine at 90% theoretical yield without directly (or indirectly) increasing the flux through PGI which becomes a member of UMUST. We also observe a significant increase in the number of reactions in LMUST. This is because the kinetic expressions in kinJ fix the branching ratios for fluxes emanating from metabolites in kinI. As a result, many reactions in stoicJ involving metabolites participating in reactions from kinJ appear as down-regulations. For example, k-OptForce identifies that the acetyl-CoA carboxylase (ACCOAC) flux, which branches away from pyr and accoa towards membrane lipid metabolism must decrease  as it goes from (9–527) mmol gDW−1hr−1 in the reference strain to (2.1–2.3) mmol gDW−1hr−1 in the overproducing strain.et al−1 hr−1) and pyruvate dehydrogenase (PDH) (93 mmol gDW−1 hr−1) reactions. The high concentrations of 6pgc and pep both prevent the up-regulation of upper glycolysis and the down-regulation of lower glycolysis. To alleviate this, k-OptForce suggests removal of PDH coupled with an at least three-fold down-regulation in the activity of either transaldolase (TALA) or transketolase (TKT1) reactions in PP pathway. Removal of PDH also allows the concentration of pep to be reduced from 2.8 to 1.43 mM thus alleviating its inhibitory effect on PFK. Likewise, down-regulation of the PP pathway activity reduces the flux in GND lowering the concentration of 6pgc (from 0.8 to 0.4 mM). Removal of substrate inhibition on PFK and PGI by pep and 6pgc leads to increased flux towards L-serine. The original OptForce procedure cannot identify such interventions, as substrate inhibition is not captured through stoichiometric modeling.in vivo−1 hr−1 and the PP flux is down-regulated further (from 22 to 20 mmol gDW−1 hr−1) to provide the extra flux for L-serine (results not shown here). As a result, k-OptForce suggests down-regulation of GHMT by at-least 3 folds from its reference flux instead of its reversal. All other interventions remain un-altered.The inhibitory effect of 6pgc and pep cannot be completely removed due to their prescribed lower limits in concentration (0.4 mM and 1.43 mM respectively). Moreover, the upper limits on concentration of metabolites involved in upper glycolysis put an upper bound on the amount of flux that can be carried by upper glycolysis. Therefore, additional interventions are needed to meet the L-serine target yield by modulating pathways away from glycolysis. k-OptForce suggests the reversal of glycine hydroxymethyltransferase (GHMT) thus converting glycine to L-serine see . In cont−1 hr−1) of either citrate synthase (CS) or succinate dehydrogenase (SUCD) reactions to reduce the TCA cycle activity which arrests ATP production in the network. This prevents the conversion of L-serine to acetaldehyde whose activity requires five units of ATP per unit of flux. The original OptForce achieves the same goal by simply down-regulating the transport of oxygen and up-regulating the transport of ammonium into the cell. These interventions were not chosen by k-OptForce as they lead to flux values that are inconsistent with the kinetic expressions in kinJ.The remaining interventions suggested by k-OptForce aim at preventing the drain of metabolic flux from L-serine. Removal of L-serine deaminase (SERD_L) prevents the conversion of L-serine to pyruvate. This is followed by an at least six-fold down-regulation .Consistent with k-OptForce predictions, metabolic engineering studies on In addition to suggesting intervention strategies consistent with the kinetic constraints in the network, k-OptForce also pinpoints which ones and how original OptForce interventions violate network kinetics. For example, the original OptForce framework suggested the reversal of lower glycolytic reactions that converge to 3pg. This is accomplished by removing PGI and either GND, TKT1 or TALA in PP pathway to reroute the metabolic flux toward 3pg and pyruvate through E-D pathway by using 2-dehydro-3-deoxy-phosphogluconate aldolase (EDA). Reversal of enolase (ENO) and phosphoglycerate mutase (PGM) in lower glycolysis converts pyruvate to 3pg. k-OptForce finds that this redirection is not feasible. Reversible reactions PGM and ENO rely on the relative concentrations of its reactants and products to inform their directionality. Their reversal, to the extent suggested by the original OptForce procedure requires the respective metabolic levels of 2-phosphoglycerate (2pg) and pep to increase to 1.162 and 6.05 mM respectively, beyond the imposed upper limits of (0.856 and 4.726 mM respectively). Therefore, k-OptForce provides both a check on stoichiometry-only derived interventions and more importantly quantifies the impact of flux redirections on metabolite concentrations and required enzyme levels.E. coli kinetic model by Chassagnole et al−1hr−1 to 70 mmol gDW−1hr−1), as the glycolytic pathway was inhibited by increased 6pgc concentration. As a result, down-regulation of GND was suggested as an additional intervention to reduce the increased PP activity and route more flux from glycolysis towards L-serine. In all other cases metabolite concentrations and fluxes in the kinetic model were minimally affected by perturbations in enzyme activity.Previous reports E. coli and S. cerevisiae has been explored E. coli nor S. cerevisiae can natively synthesize TAL, production routes for TAL rely on the heterologous expression of non-native enzymes such as 2-pyrone synthase (2-PS) (found in Gerbera hybrida) fasBS. cerevisiae of only up to 6% of the theoretical maximum (with a titer of 1.8 g/l) in glucose medium. S. cerevisiae. We used the iAZ900 model S. cerevisiae as the stoichiometric network of metabolism. The kinetic expressions for reactions in central metabolism were imported from the kinetic model of central metabolism of S. cerevisiae described by van Eunen et alet alS. cerevisiae central metabolism from Gombert et al4-Hydroxy-6-methyl-2-pyrone, commonly known as TAL, is a precursor for the production of phloroglucinol maxv from their wild-type values. Contrary to the previous example, here the kinetic expressions do not restrict further the flux ranges as the ABC metric (see Methods) for all kinJ and stoicJ fluxes is zero. As a result, we find no difference in the UMUST (19 reactions) and LMUST (61 reactions) sets for kinetic and original OptForce. This is due to the relatively few fluxes with kinetic expressions and the already fairly tight flux ranges achieved by stoichiometry-alone. For example, the PGI flux varies within the narrow range between 91 and 97 mmol gDW−1hr−1 for a 90% yield for TAL even when no kinetic expressions are used. This is because the imposed high production target for TAL largely fixes the flow in glycolysis. As a consequence of negligible contraction in flux ranges due to the kinetic constraints, no difference in predicted MUST sets by regular and k-OptForce is observed.As in the first example, we allowed for up to two-fold changes in the metabolite concentrations and kinJ. −1 hr−1). This eliminates all flux through G6PDH and PP (from 19 to 0 mmol gDW−1 hr−1) to maintain the steady-state metabolite balance of g6p. Without the benefit of any kinetics, the original OptForce suggests the removal of G6PDH reaction as a requirement for down-regulating the PP pathway. k-OptForce also requires that the concentrations of glyceraldehyde 3-phosphate (g3p) and 3pg be elevated by 6% and 31%, respectively from their reference states to up-regulate the lower glycolysis flux from 168 to 192 mmol gDW−1 hr−1. This re-direction in glycolysis prevents the loss of metabolic flux towards glycerol synthesis. Instead, the original OptForce procedure suggests the removal of glycerol-3-phosphatase (G3PT) in the glycerol synthesis pathway to serve the same purpose and channel the flux towards lower glycolysis and TAL.−1 hr−1) in the flux of aldehyde dehydrogenase (ALDD) which converts acetaldehyde towards acetate. The major routes of acetaldehyde production in S. cerevisiae are either from direct decarboxylation of pyruvate through pyruvate decarboxylase (PYRDC) or through the alternate threonine synthesis pathway followed by the cleavage of threonine by threonine aldolase (THRD) to acetaldehyde and glycine  resulting in a decrease in nadph levels. The original OptForce does not arrive at this intervention as the kinetic control on the fluxes of PYRDC and ALCD is not captured. It instead suggests the removal of cytosolic isocitrate dehydrogenase (ICDHy) to reduce the nadph production, and thus arrests fatty acid synthesis.The fatty acid synthase is in direct competition to TAL production. It uses the same precursors as TAL  to form medium chain fatty acids. Activity of fatty acid synthase requires the cofactor nadph for the reductive steps in the pathway. Not surprisingly, both kinetic and original OptForce identify strategies to lower the availability of nadph. k-OptForce achieves this by suggesting a 20 fold up-regulation . However, when a kinetic model is integrated with a stoichiometric genome-scale model, reactions in stoicJ that involve metabolites present in kinI add in effect additional degrees of freedom to the square-system of equations thus decoupling metabolite concentrations from enzyme activities. As a result metabolite concentrations can change in such a way that fluxes are altered without requiring any enzymatic interventions as observed in the TAL overproduction example. In the absence of kinetic expressions for all reactions associated with the metabolites in kinI a number of degrees of freedom remain for the metabolite concentrations. To avoid drastic concentration changes in response to the overproduction goal we explored penalizing deviations of metabolite concentrations from their reference steady-state values using a weight penalty factor ε. This posture in essence imposes a homeostasis term in the optimization objective function. The outer objective function of the bilevel formulation for identifying FORCE sets is thus modified as follows:kinI , the objective function for the first step is modified as follows:kinJ, scaled by the total number of reactions in kinJ identified in MUST sets (kinMUST). No other changes are made in the formulation of the algorithms. We tested this modified formulation on TAL production in S. cerevisiae.The steady-state balances of metabolites in a kinetic model kinJ for the overproduction of TAL in S. cerevisiae. We varied ε from 0.1  to 0.9  on the identified interventions. Up to a ε value of 0.6, the penalty is not high enough to require direct interventions instead of concentration changes. For a ε value of 0.7 k-OptForce identified up-regulation of enolase (ENO) while maintaining the average deviation in concentration to 0.1053. Note that without the use of the penalty term the concentrations for 3pg and 2pg have to be elevated by 31% and 27% respectively, from their reference levels to redirect more flux through the lower glycolytic reactions of glyceraldehyde-3-phosphate dehydrogenase (PGM) and ENO. For ε = 0.8 the metabolite concentrations remain even closer to their reference levels (average deviation is 0.0754) thereby requiring the up-regulation of glyceraldehyde-3-phosphate dehydrogenase (GAPD) in addition to ENO. For ε equal to or greater than 0.9 at least 6 additional enzymatic interventions in kinJ are necessary to increase glycolytic flux while concentrations remain very close to their reference values. By increasing the value of the penalty ε enables drawing trade-offs between allowable concentration changes and minimality of needed interventions for an overproduction goal while also providing a prioritization strategy for implementing engineering interventions.E. coli in S. cerevisiae that directly converts pyruvate to acetyl-CoA. Note that S. cerevisiae has a pyruvate dehydrogenase activity in mitochondria but not in cytosol. The PDH complex in E. coli uses nad as the cofactor, however, an nadp-dependent PDH enzyme  in yeast follows a long and tightly regulated path involving the intermediate production of acetaldehyde and acetate E. coli proposed in Chassagnole et al.The kinetic expression for PDH was extracted from the kinetic model of k-OptForce integrates kinetic relations (whenever available) with stoichiometry based models to identify genetic perturbations that are consistent with enzyme expressions and metabolite concentrations. The resulting optimization problems pose significant computational challenges due to the bilevel nature of the formulation and the nonconvex terms in the objective function and constraints. We introduced tractable solution workflows for recasting the problems as equivalent single-level mixed-integer nonlinear optimization problems (MINLP) solved using the global optimization solver BARON to optimality. A hierarchical decomposition approach is also introduced for first identifying interventions within the kinetic part of the model followed by the interventions in the stoichiometry-only part of the model. As with other computational algorithms that make use of kinetics the results can be dependent upon the kinetic model structure and parameterization.Computational results show that the introduction of kinetic expressions in strain design can significantly affect the identified interventions in sometimes non-intuitive ways. In some cases additional modifications are needed to substitute interventions that cause enzyme saturation or concentration bound violations. The mechanism of action of these modifications is often subtle by alleviating substrate inhibition or draining away cofactors from competing pathways. In other cases, kinetic expressions shape flux distributions so as to favor the overproduction of the desired product requiring fewer direct interventions. Uncertainties in both the accuracy of the kinetic models and allowable concentration ranges imply that predicted interventions need to be carefully scrutinized to pinpoint the reasons for their inclusion. An important finding in this study was that concentration ranges have a very significant effect on the identified interventions. By penalizing departures of concentrations from the reference strain values substantial re-arrangements in the predicted interventions are observed. Each one of these changes can be analyzed and the underlying reason for its inclusion can be identified.E. coli provides an example of how k-OptForce can be used to both identify interventions and trace the reason(s) for the exclusion of others. k-OptForce revealed that inhibition of upper glycolysis by pep and 6pgc must be alleviated to route more flux towards L-serine. This is achieved through removal of PDH and down-regulation of TALA or TKT1 respectively. Flux analysis on single gene mutant strains of E. coli show that deletion of either tala or tktA increases the flux through PGI lpdA mutant encoding the PDH enzyme in E. colikinJ. Alternatively, since lpdA also encodes for the activity of ICDHy and the glycine cleavage system (GLYCL), its removal could be have a combined effect on down-regulating the flux in upper glycolysis, which is not captured by the kinetic model. k-OptForce, however, correctly predicts that PDH removal down-regulates lower glycolysis which is observed in the lpdA mutant strain cra), determines the directionality of both upper  and lower glycolysis  A key contribution of kinetic descriptions is that they can attribute performance bottlenecks to specific concentration bounds and/or enzymatic parameter ranges bottlenecks revealing avenues for model improvement and strain optimization. The case study for L-serine overproduction in S. cerevisiaeald6 (which encodes for the ALDD enzyme) increases flux towards acetyl-CoA. However, a fraction of the flux from pyruvate to acetaldehyde was routed through threonine degradation without the requirement of any additional interventions. This direct intervention-free flux redistribution may be an artifact of the kinetic model and may require direct manipulations to engineer. Metabolome studies on single-gene mutant analysis in E. coliet alkinI that do not have a kinetic expression (i.e. not part of kinJ). This would restore the square-system of equations in kinJ and recouple all metabolite concentrations with enzyme activities.k-OptForce may require fewer direct interventions if the kinetic expressions shape fluxes so as to favor the desired overproduction product as observed for the production of TAL where up-regulation of ALDD was suggested to redirect flux from pyruvate to acetyl-CoA. This is consistent with an experimental study for isoprenoid overproduction in E. coli metabolism by Lee et alThe k-OptForce procedure is versatile enough to incorporate additional omics information, whenever available, to further improve prediction fidelity. For example, MFA data for reactions can be included as additional constraints to further tighten flux ranges. k-OptForce can also capture other types of metabolic regulation and select from a wider palette of direct interventions  such as the dynamic hybrid model of Text S1E. coli and S. cerevisiae, supplementary figures for steady-state flux distribution and metabolite concentrations for the kinetic models, and flux ranges for the reference and overproducing phenotypes for L-serine and TAL production.Optimization formulation and solution procedure of k-OptForce, kinetic models of central metabolism for (PDF)Click here for additional data file.


Constrained minimal cut sets (cMCSs) have recently been introduced as a framework to enumerate minimal genetic intervention strategies for targeted optimization of metabolic networks. Two different algorithmic schemes  have been proposed to compute cMCSs from elementary modes. However, in their original formulation both algorithms are not fully comparable.Here we show that by a small extension to the integer program both methods become equivalent. Furthermore, based on well-known preprocessing procedures for integer programming we present efficient preprocessing steps which can be used for both algorithms. We then benchmark the numerical performance of the algorithms in several realistic medium-scale metabolic models. The benchmark calculations reveal (i) that these preprocessing steps can lead to an enormous speed-up under both algorithms, and (ii) that the adapted Berge algorithm outperforms the binary integer approach.Generally, both of our new implementations are by at least one order of magnitude faster than other currently available implementations. Mathematically, this problem is also known as dualization of a (hyper-)graph, a fundamental problem in discrete mathematics , then this reaction is considered to be essential and cannot be knocked out. A reaction i is essential if |D|−si<n, with D|=5 and n=3. R1 and R7 are essential reactions, as for them |D|−si=5−3=2<3=n, which indicates that knocking out R1 or R7 will kill more desirable modes than allowed. We note that if |D|=n, all active reactions are essential.Consider the example in Table n is large , as this results in the maximum number of essential reactions. Removing all essential reactions from the system is a critical step that opens the possibility to apply several other preprocessing procedures.In general, the more essential reactions we find, the more the system can be reduced. Consequently, it is beneficial if T and to remove them from the system.The removal of all essential reactions results in an important change of the system. By definition EMs are non-decomposable, i.e. an EM is not a subset of any other EM. However, if the essential reactions are removed then the residual EMs may become subsets or duplicates of other modes. Hence, the next step is to find all duplicate modes in Step 3. Next, we screen T to find and remove residual EMs that are supersets of other residual EMs in T. Consider the target set , T, shown in Table t1 can be removed by knocking out reaction R2. However, knocking out reaction R2 also kills t2, as t2 is a superset of t1.t3 has a norm of 2 and is killed either by knocking out reaction R4 or reaction R7. As both reactions are part of t4 and t5, they are certainly removed if mode t3 is killed.The same procedure can be applied to the other modes as well. Mode Step 4. In a final preprocessing step, we remove duplicate reactions across all EMs in both sets, D and T. Using the illustration in Table t2, t4, and t5 are removed. In this step it is not possible to analyze D and T separately. Reactions need to be identical in both sets, D and T, in order to be removed.http://www.gurobi.com/ for solving the BIP problem. The adapted Berge algorithm was implemented in C. The software is available from the authors on request. The simulations were all carried out on an Intel Xeon CPU X5650 @ 2.67GHz under a Linux operating system.We implemented the BIP algorithm in C using Gurobi Optimizer 5.0, E. coli core model, E0,  but drops dramatically whenever the norm of the solution decreases. Note that a decreasing norm means an increase in the number of required knockouts. In this model the computation time significantly drops after solution number 11, 53, 116, and 156. At these instances the number of required deletions changes from 11 to 12, to 13, to 14, and to 15, respectively. In all these cases the constraint in equation (7e) decreases too and introduces a tighter bound on the system. This allows the solver’s internal preprocessing to more efficiently compress the system, which in turn brings down the computation time. If, however, the norm of the solution does not change then the computation time scales approximately exponentially with the number of MCSs. This behavior is expected as each solution adds a new constrain to the system, which makes it harder to solve.Recently cMCSs have been introduced to predict optimal intervention strategies in order to achieve an arbitrary metabolic objective . Two algIt may appear as a surprise that the Berge algorithm performs so well even for the large cases presented in this study, especially since the Berge method is known for its unfavorable performance in huge networks . HoweverThe importance of preprocessing in the calculation of MCSs has been stressed earlier . The preD=∅) and T contains all EMs of a network. This situation corresponds to unconstrained MCSs and thus to a full dualization of the hypergraph spanned by the target modes. Except for step 4, none of our preprocessing routines then provides an advantage and other solvers may be more appropriate [n/|D| should be close to one. This means that many essential reactions will be removed from the system, and as a result of that many supersets will be detected, too. However, in practice it may suffice if only a few EMs out of the set of desired modes survive, i.e. n/|D|≪1. Still, preprocessing provides a significant performance gain as indicated in Table The efficiency of preprocessing is dependent on the imposed design criterion. In the worst case the set of desired modes is empty . We investigated the numerical performance of these approaches. Both methods significantly profited from the enhanced preprocessing procedures developed here. Under the tested conditions, our implementation of Berge’s algorithm performed best even outperforming other, currently available software.BIP: Binary integer program; cMCS: Constrained minimal cut set; CNA: CellNetAnalyzer; EM: Elementary mode; MCS: Minimal cut set; PP: Preprocessing.The authors declare that they have no competing interests.CJ implemented the algorithms and participated in the design of the study. GN coded the models and helped in running the analysis. SK participated in the design of the study. JZ conceived of the study, and participated in its design and coordination. CJ, SK and JZ wrote the manuscript. All authors read and approved the final manuscript.tar gzipped archive of the SBML files for the models E0, E1, and E2.Click here for fileTotal runtime with and without preprocessing for the Berge algorithm.Click here for file


Solanum tuberosum) as running example. The intention of the paper is to give a compact presentation of known steady-state concepts from a graph-theoretical viewpoint in the context of network decomposition and reduction and to introduce the application of Q-modularity to metabolic Petri net models. Functional modules of metabolic networks are essential for understanding the metabolism of an organism as a whole. With the vast amount of experimental data and the construction of complex and large-scale, often genome-wide, models, the computer-aided identification of functional modules becomes more and more important. Since steady states play a key role in biology, many methods have been developed in that context, for example, elementary flux modes, extreme pathways, transition invariants and place invariants. Metabolic networks can be studied also from the point of view of graph theory, and algorithms for graph decomposition have been applied for the identification of functional modules. A prominent and currently intensively discussed field of methods in graph theory addresses the Q-modularity. In this paper, we recall known concepts of module detection based on the steady-state assumption, focusing on transition-invariants (elementary modes) and their computation as minimal solutions of systems of Diophantine equations. We present the Fourier-Motzkin algorithm in detail. Afterwards, we introduce the Q-modularity as an example for a useful non-steady-state method and its application to metabolic networks. To illustrate and discuss the concepts of invariants and Q-modularity, we apply a part of the central carbon metabolism in potato tubers ( The knowledge of biochemical networks, in particular of metabolic networks, increases daily with the capabilities of new upcoming high-throughput technologies to measure all the participating molecules and the relations between them. This enables us to construct large and complex models for many pathways of different species. In particular, modeling of metabolism helps us to understand biological function. A prerequisite for a quantitative model is the complete knowledge of metabolite concentrations and reaction constants and/or rates or, at least, a critical amount of them. However, in most cases, quantitative data in sufficient amounts and of high quality are rare and only available for rather small metabolic systems. This situation motivated the development of qualitative methods, which enable us to analyze statements on functional behavior and dynamic properties of the system without any knowledge of the kinetic parameters. modules , in the incidence matrix is given by dfwpt and determines the change of the token number in a place, p, after the firing of a transition, t; see C : P ⊗ T → ℤ. C, of the PN in Generally, an t8 (sucrose input) produces a new token of sucrose on p1; see t, has solely one nonzero component, #t8 =1, and we yield:C, to compute the changes of token (metabolite) numbers, resulting from the firing sequence of transitions (reactions) .t4 and t5, form a transition-invariant. One of these two reactions, let us say, t4, may fire spontaneously and drive the systems away from the steady state, but the other reaction, t5, of the transition invariant can compensate for the effect of firing of t4. Such stochastic fluctuation is a natural and inherent property of a metabolic reaction system, in which all reactions are constantly active and the time-dependent state of the system fluctuates around an ideal steady state. The firing of transition t, and name the solution vector, transition-invariant or t-invariant. The solution, t, has to be an integer, because we consider discrete objects—the tokens, and nonnegative—because any sequence of firing transitions (reactions), s =, gives rise to an integer and nonnegative Parikh vector. Parikh vectors with negative components are senseless in the biological context. Let us now consider the invariant properties of the system. The invariants hold in every system state reachable from an arbitrary, initial marking. We define invariant properties for the active and the passive part of the system. Considering the active part and the equation system:CT be the transposed incidence matrix. Considering the passive part, we define the nontrivial, nonnegative integer solution, p, of the equation system:place-invariant or p-invariant. Let i.e., infinite. However, we are interested in a finite solution set, from which we can compute all possible solutions by positive integer linear combinations of the solution vectors. Such a set is given by all minimal solutions of the invariant equations, where minimal means: for an invariant, x, there exists no invariant, z, whose support is part of the support of x:x is one. The support of x, written as supp(x), contains the set of the nonzero entries of a vector, x. In the following, we consider minimal, nontrivial, nonnegative t-and p-invariants, shortened as t-invariants or TI and p-invariants or PI, respectively. The solution space of such linear equation systems is, in general, unbounded, Invariant properties have important applications in systems biology. P-invariants represent a set of places whose weighted sum remains always constant, thus representing a conservation of substances. T-invariants describe a cyclic firing behavior, because the firing of all transitions of a t-invariant leads back to the initial marking, forming a cycle in the RG. The TI represents basic pathways in biochemical networks at steady state and describes, thus, the basic network behavior. Before explaining the application of invariants in more detail, we first want to discuss their computation. i.e., the computation of t-invariants. The working principle of the FM can easily be demonstrated for the network in T|×|T| identity matrix: t1,t2,...,t9) and one column for the change of metabolites in each place . We can read the table as follows: Each line describes a sequence of reactions and its effect on the metabolites. The first line corresponds to the sequence (t1), because the entry for reaction, t1, is one, and the entries for all other reactions are zero. This sequence of reactions (t1) removes one metabolite of p1 and adds one metabolite of p2 and one metabolite of p3, because in the first line, the entry is −1 for p1, and the entries for p2 and p3 are both +1. The interpretation of the second to the eighth line is analog. The basic idea of the algorithm is to combine lines in such a way that the entries for all metabolites become zero. In the first step, we have to select a metabolite, let us say p1, and to construct the combinatorial diversity of all sequences of reactions that yield Δm1 =0. Checking the column for p1, we find an entry, −1, in the first line (t1) and an entry, +1, in the eighth line (t8). Each of the reactions, t1 and t8, influence the metabolite, p1, and the only possible combination of these reactions that yields Δm1 =0 is reaction t1 plus reaction t8. Consequently, we add the first line to the eighth line to get one new line with an entry zero for p1. We append this new line  to the table and delete the lines utilized to construct that new line. We get the new table: p1 are zero now. The new line  in the table has an entry, one, in the column for t1 and an entry, one, in the column for t8. Hence, the new line corresponds to the reaction sequence . According to the FM, we still have to check whether the new sequence of reactions is minimal, because the support of a new sequence may be a superset of the support of another sequence. In this case, the new sequence would not correspond to a minimal solution and would have to be eliminated from the table. In the particular case of reaction , the candidate solution is minimal. The FM algorithm proceeds with another metabolite that has nonzero entries in its column. For large networks, the metabolites should be chosen according to an advantageous heuristic [p2, in the table above. In the column for p2, we find the entry, −1, in the line, (t2), and the entry, one, in the line, . Again, we can construct only one new combination that gives p2 =0. We construct the new line, , by adding the first line, (t2), to the eighth line, . Then, we delete the utilized lines and append the new line: p3, two lines are utilized to append a new  line, : t7), with the fifth line, (t9), zeroes all entries for the metabolite on p6, and we yield: p4 shows that the number of lines does not have to decrease for each step. Proceeding with metabolite p4, we find two positive and two negative entries in the column for p4. The algorithm has to construct the combinatorial diversity of all solutions, which zeroes the entries in the column for p4. Such combinations are the first line, (t4), with the second line, (t5), the first line, (t4), with the third line,(t6), the forth line, , with the second line, (t5), and the forth line, , with the third line, (t6). In this way, we construct four new  lines and get the new table: p5 results in the table: p7 gives the final table: p1 to p8, have all to be zero in the final table. The entries for the transitions describe two t-invariants of the network. The t-invariant, {t4,t5}, represents a simple reversible reaction, and is called a trivial t-invariant. The more complex t-invariant, {15 t1, 15 t2, 13 t3, 2 t5, 28 t6, 15 t7,t8, 28 t9}, describes the basic functional pathway of the network in The Fourier-Motzkin elimination method (FM) ,37 is a cannot be compensated by other reactions have to be discussed carefully for their biological relevance. Such reactions are strong indicators for missing reactions or errors in the model. The identification of a reaction that contradicts a steady-state behavior is a computational challenge for large metabolic models. Standard approaches are based on the computation of a minimal generator set of all TI. In general, the computation of all TI requires exhausting resources in terms of computer time and memory [et al. [Reactions that d memory . Severald memory , the nuld memory , the cond memory , and a p [et al. . Even wi [et al. . This lecovered by t-invariants (CTI). Accordingly, we call a PN to be covered by p-invariants (CPI), if each place is a member of at least one p-invariant. The CPI property can be used to decide boundedness, i.e., the finite number of tokens for all places. Only for bounded PN, a finite reachability graph can be generated. Though the CPI property is important for many questions, we will not consider it in more detail in this paper. We want to define another property, which is helpful, in particular, to verify biochemical systems. This property represents a completeness condition which may be applied in network verification. If each transition belongs to at least one t-invariant, we say that the PN is one integer solution, t, of the equation:ti ≥ 1,i=1, 2,..., |T|) or to exclude the existence of such a solution. Thus, to decide the CTI question without computing all TI, a less expensive strategy would be beneficial. Despite the fact that the knowledge of an even huge number of t-invariants is valuable and represents a prerequisite for more advanced analytical techniques, we want to decide whether a network is CTI without computing all TI. Since the set of all t-invariants describes a minimal set of all functional modes of the system at steady state, each transition should belong to at least one t-invariant. To show the CTI property for a PN, we have to find s, e, {v1,v2,...,vn}), of dimension k, the reachability problem reads: do vectors w1,...,wm ∈ℕk exist, such that:k, be the number of places, (k = |P|), theincremental change vectors,  ∈ ℤ k, the column vectors of the incidence matrix, C, the end vector, e ∈ ℕk, sufficiently large and the starting vector, s ∈ ℕk, the sum of the end vector, e ∈ ℕk. Then, the change of metabolites, resulting from firing each transition once, is:yi =1,i=1, 2,... |T|). Any solution of this vector addition problem represents a solution of Equation (9) and shows the CTI property of the network. Lipton  gives thet al. [elementary flux modes or elementary modes (EM), which correspond to the TI [, of all solutions of Equation (6) is a pointed convex cone. A network is CTI if and only if the effect of firing of all reactions can be compensated. Firing of all reactions in our example network in s, in the convex cone, S, that compensates for Δm, i.e., s = −Δm. Geometrically, the network is CTI if the vector:S, and the Parikh vector, y, has all components equal to one . We have to check whether b is inside (b ∈ S) or outside (b ∉ S) of the cone. Here, the Lemma of Farkas [b, is either inside of the convex cone, S, or it is possible to find a hyperplane, S⊥, that separates b from the convex cone. Such a separating hyperplane must be a tangent hyperplane to the convex cone, S. Without loss of generality, we can choose a surface normal, s⊥, that points into the same direction as the cone, i.e., the angle between s⊥ and all vectors in the cone is not greater than 90°:I is the identity matrix and ν is an arbitrary vector with nonnegative integer components. The surface normal, s⊥, determines a tangent hyperplane, S⊥ = {s ∈ ℤk|sTs⊥ =0}. The CTI question and the concept of TI are closely related to the theory of convex cones. In this context, Schuster et al.  defined o the TI . It is of Farkas  providesb, is located on the “wrong” side of the hyperplane, i.e., opposite of the convex cone, S. It turns out that the vector, b, is located opposite of the convex cone if a solution, , of Equation (17) with nonzero positive components of ν ≥ 0 exists [ν identify the reactions not covered by TI. Applying this strategy to the network in s⊥,ν) of the dual system (17):s⊥,ν), of this system of the Diophantine Equation (17). In general, solutions of Equation (17) with zero components, ν =0, are called place-invariants in PN formalism [Now, we have to prove whether the vector, 0 exists . The nonormalism  and descormalism ; see Secormalism . Functional modules are important for representing, understanding, reducing and verifying general networks. This is true, in particular, for biochemical networks, which are big and complex and for which an experimental validation can be difficult or is even not possible. Several definitions of functional modules have been proposed in various scientific fields. Definitions inspired by biology are mainly manually derived induced by biological knowledge. They often rely on the experience of the individual researcher. With the growing amount of data, the automatic detection of modules becomes of great interest. All known definitions are at least implicitly based on graph-theoretical properties. For biochemical systems, we distinguish between module definitions that are based on the steady-state assumption and definitions that ignore it. Both types of definitions are advantageous to solve specific biological questions. The reactions (transitions) of each EM (TI) and the metabolites (places) in between, including the corresponding edges between them, build connected subnetworks that stand for a certain biological function. Thus, a subnetwork defined by a TI can be understood as a functional module. The careful evaluation of the biological interpretation of functional modules, often manually done, is part of proving the model for its correctness. There are many studies that provide exactly this kind of analysis. Some of them report the detection of new pathways that have been later experimentally validated. An example is the prediction of the glyoxylate pathway ,49 and iminimal cut sets. Methods based on the support vector do not explicitly take into account the integer numbers of the Parikh vector and, thus, implicitly ignore the stoichiometric relations. Instead, we consider the binary information of whether a reaction or enzyme (transition) is a member of a TI or not. An example of such a method to define modules are Minimal Cut Sets (MCS) [ts (MCS) : MCS hasts (MCS) . The next two module definitions are suitable for large networks. Since our running example is too small to illustrate the usefulness of these definitions, we refer to examples in ,54,55. Maximal Common Transition sets (MCT-sets) [m}, in which each pair of reactions, ti and tj, with i,j∈ 1, ...,m, occurs in exclusively the same TI, such that:X denotes the set of all TI x, with i, j ∈ 1, ...,m, and χ{0} denotes the characteristic binary function, indicating if its argument equals zero. This grouping leads to maximal sets of transitions, where each set of transitions, ϑ, fulfills:i.e., they do not necessarily form consecutive firing sequences. CT-sets) ,57: InspT-clusters [overlapping subnetworks with a broader, specific biological function. We define t-clusters based on hierarchical clustering methods, such as UPGMA or NEIGHBOR JOINING. As a distance measure, we use the Tanimoto coefficient [ti and tj, is then:supp(ti) and supp(tj) denote the support vectors of the t-invariants, ti and tj. The pair-wise similarity, sij, expressed by this coefficient is transformed into a distance measure for dissimilarity, dij [clusters ,58: Wherfficient . The simity, dij :dij =1 example, .best number of clusters, which is a fundamental problem in unsupervised classification, is implemented as a user-defined parameter. Additionally, cluster validity measures can be applied to identify the number of clusters which “best” represents the intrinsic grouping of the data [silhouette width [S, for an individual data sample, i, is defined as:ai denotes the average distance between i and all the data samples in the same cluster and bi denotes the average distance between i and all data samples in the nearest other cluster. In contrast to MCT-sets, subnetworks based on t-clusters can overlap. The definition of the the data . The silte width , which ii.e., the expression behavior of the participating genes should be similar. MCT-sets and t-clusters have been applied to metabolic systems, but also to signal transduction pathways  and geneACoM (Aggregation around Common Motif) [common motif defined as the set of transitions that belong to all TI as a seed, it will be extended according to specific rules. This seed motif is of determined length and is successively extended, until a certain threshold is reached. Similar to t-clusters, overlapping aggregations of common motifs were defined. n Motif) : StartinElementary Flux Patterns [Patterns : The conEnzyme subsets (ES) [monofunctional units or super-enzymes [i.e., the ratios of (nonzero) frequencies of the reactions, have to be identical. This requirement represents a restrictive criterion for the definition of functional modules. ets (ES) : Enzyme -enzymes ,69. In mQ-modularity, for a partition into communities is defined by:m unweighted undirected edges and communities, C [Aij, over all pairs of vertices, i,j, in the same community. The Kronecker delta, δ, becomes one, if both vertices, i andj, are in the same community and zero, otherwise. The summation over Aij gives the number of edges inside of all communities and a number that cannot exceed the total number of edges times two. The pre-factor, 1/(2m), guarantees a value which is equal to or less than one. Each entry, Aij, is reduced by a probability, Pij, to find the edge, , by chance in an appropriately chosen statistical null model. A random network with identical distribution of the vertex degree leads to a simple sum over nc modules:lc is the number of edges in the module, c, and dc is the total sum of the vertex degrees in the module, c. This formula for the Q-modularity is not directly applicable to metabolic networks because metabolic networks are bipartite graphs with directed edges and edge weights. An easy way to apply the formula above would be to transform a metabolic network into a unipartite network with undirected and unweighted edges. This can be done in different ways, but a loss of crucial information  cannot be avoided. Communities play a prominent role in a broad range of scientific fields, including, e.g., social science, economics, computer science, engineering, politics, and biology. Examples of communities are friends in a school class, readers of books sharing similar interests, electronic components to be placed together on a layout of a solid-state circuit board, co-authors of scientific articles, interacting proteins or words with similar associations. For an excellent review, we refer to the work of Fortunato . Communiities, C . The forCi, with i =1,2,...,nc. The vertices of a module can be transitions and/or places. An appropriate formula for Q-modularity of metabolic networks has to consider the direction of edges within modules and between modules in a bipartite metabolic network [Q reaches its maximum is an NP-hard problem [Q-modularity increases from generation to generation and reaches a maximum after a sufficient number of steps. A partition of a PN is given by disjoint modules,  network . Note th problem . We applSaccharomyces cerevisiae with 63 metabolites and 117 reactions considered in Jevremovic et al. [explode with the increasing number of network components. This explosion problem is a well-known drawback in practical computations. However, it is instructive to see how the explosion problem can be circumvented for networks using special networks properties: metabolic networks are usually expected to be scale-free; reaction chains appear often; there are super-hubs of metabolites playing an essential role for most reactions ; many reactions are reversible and most likely have a small number of one or two input metabolites. Such properties make metabolic networks special and well-distinguishable from random networks or technical networks. We have already discussed the complexity class of various methods for analyzing qualitative properties of metabolic networks. The search for the best possible partition in modules is an NP-hard task, and the CTI question is EXPSPACE-hard. For example, the rather medium-sized metabolic network of c et al.  has abouc et al. . The comnetwork reduction enables insights into coarse-grained structural properties of the network [conservative reduction techniques are favorable to decide the CTI question for large networks. For most biological networks, a significant reduction of the computational complexity is possible. A typical kind of a reduction step is inspired by MCT-sets [enzyme subsets [common transition pairs (CTP). A CTP is a local structure of a place that has exactly one pre-transition and one post-transition. Intuitively, the pre-transition produces tokens on a place that can be removed by the post-transition only. Another local structure useful in this context is the invariant transition pair (ITP). An ITP is a reversible reaction, consisting of a forward and backward reaction. It may, for example, be possible to reduce the computational effort to answer the CTI question by transforming a network into a smaller one. Thus,  network ,21,22,72MCT-sets  or enzym subsets  , which can be further decomposed by several approaches into disjunctive or overlapping subnetworks. We introduce Petri nets as a widely used and suitable formalism to model systems with concurrent processes. In the context of PN, we define the system’s invariants, which give us insight into the dynamic behavior of the system without any kinetic knowledge. To illustrate the idea, we provide a detailed example for the computation of t-invariants (EM) using the Fourier-Motzkin method. From the geometric point of view, t-invariants are equivalent to the extreme rays of a convex cone. We consider the CTI question, which is important to verify a biochemical model. Using the proof by Lipton, we show that this question corresponds to the reachability problem for vector addition systems and is EXPSPACE-hard. To consider connectivity-based methods, we define communities. We introduce the Q-modularity measure to verify the partitioning by these algorithms. In addition, we illustrate the methods described, using a small metabolic network, and discuss the development of new methods for the structural analysis of metabolic systems. Network reduction plays an important role, in particular, in handling genome-scale networks. We explain how common transition pairs (CTP) and invariant transition pairs (ITP) enable us to compute t-invariants of large networks, even if we will not get a minimal set of t-invariants. Finally, we shortly discuss network verification with respect to kinetic analysis techniques.


Cyanothece sp. ATCC 51142. The resulting metabolic reconstruction, iCce806, consists of 806 genes associated with 667 metabolic reactions and includes a detailed representation of the ETC and a biomass equation based on experimental measurements. Both computational and experimental approaches were used to investigate light-driven metabolism in Cyanothece sp. ATCC 51142, with a particular focus on reductant production and partitioning within the ETC. The simulation results suggest that growth and metabolic flux distributions are substantially impacted by the relative amounts of light going into the individual photosystems. When growth is limited by the flux through photosystem I, terminal respiratory oxidases are predicted to be an important mechanism for removing excess reductant. Similarly, under photosystem II flux limitation, excess electron carriers must be removed via cyclic electron transport. Furthermore, in silico calculations were in good quantitative agreement with the measured growth rates whereas predictions of reaction usage were qualitatively consistent with protein and mRNA expression data, which we used to further improve the resolution of intracellular flux values.Genome-scale metabolic models have proven useful for answering fundamental questions about metabolic capabilities of a variety of microorganisms, as well as informing their metabolic engineering. However, only a few models are available for oxygenic photosynthetic microorganisms, particularly in cyanobacteria in which photosynthetic and respiratory electron transport chains (ETC) share components. We addressed the complexity of cyanobacterial ETC by developing a genome-scale model for the diazotrophic cyanobacterium,  e.g., expression data), numerical simulations can provide experimentally testable predictions of carbon fluxes and reductant partitioning to different biosynthetic pathways and macromolecular synthesis. This work is the first to computationally explore the interactions between components of photosynthetic and respiratory systems in detail. In silico predictions obtained from model analysis provided insights into the effects of light quantity and quality upon fluxes through electron transport pathways, alternative pathways for reductant consumption and carbon metabolism. The model will not only serve as a platform to develop genome-scale metabolic models for other cyanobacteria, but also as an engineering tool for manipulation of photosynthetic microorganisms to improve biofuel production.Cyanobacteria have been promoted as platforms for biofuel production due to their useful physiological properties such as photosynthesis, relatively rapid growth rates, ability to accumulate high amounts of intracellular compounds and tolerance to extreme environments. However, development of a computational model is an important step to synthesize biochemical, physiological and regulatory understanding of photoautotrophic metabolism  at a systems level, to make metabolic engineering of these organisms tractable. When integrated with other genome-scale data ( Cyanothece spp. are unicellular, diazotrophic cyanobacteria that temporally separate light-dependent oxygenic photosynthesis and glycogen accumulation from N2 fixation at night Cyanothece sp. strain ATCC 51142 (thereafter Cyanothece 51142) cells can accumulate significant amounts of storage polymers including glycogen, polyphosphates, and cyanophycin Cyanothece 51142 is also of interest for bioenergy applications due to its ability to evolve large quantities of H2. Remarkably, H2 production in this organism can occur under light conditions in the presence of O2 and is mediated by nitrogenase Cyanothece 51142 genome Synechocystis sp. PCC 6803 Rhodobacter sphaeroidesChlamydomonas reinhardtiiSequencing of the Cyanothece 51142 and used a combination of computation and experimental approaches to investigate how photosynthetic and respiratory fluxes affect metabolism. Discrete representation of PS II and PS I and their integration with multiple respiratory pathways enabled modeling of photon fluxes and electron flux distributions under conditions of variable light quality and intensity. The predicted changes in growth rates of Cyanothece 51142 in response to changes in light input were experimentally tested using a photobioreactor with controlled sources of monochromatic 630 and 680 nm light. We also carried out computational and experimental analyses of light- and nitrogen-limited chemostat growth of Cyanothece 51142 and used mRNA and protein expression data to constrain model-predicted flux distributions. Both in silico and experimental data suggest that respiratory electron transfer plays a significant role in balancing the reductant (NADPH) and ATP pools in the cells during photoautotrophic growth. This study is a first step towards a systems-level analysis of cyanobacterial metabolism, as it integrates information into a genome-scale reconstruction to understand metabolism qualitatively and quantitatively through a constraint-based analysis in silico simulations and data.In this work, we developed the first genome-scale metabolic model of Cyanothece 51142, a genome-scale metabolic network was reconstructed using the genome annotation and data from NCBI iCce806 network contains 806 genes and 667 metabolic and transport reactions , glycolate conversion to serine, and glycerol catabolism are also included. Photosynthetic electron transfer associated with the thylakoid membrane is represented as a set of four separate reactions, including light capture by photosystem II (PS II) and photosystem I (PS I), electron transfer between the two photosystems, and cyclic electron transfer around PS I. Similarly, respiratory electron transfer is represented by reactions catalyzed by terminal cytochrome c oxidase (COX), quinol oxidases , NADH dehydrogenases , and succinate dehydrogenase. In addition, two reactions (NADP+- and ferredoxin- requiring) for flavin-dependent reduction of O2  were included. A simplified scheme of the photosynthetic and respiratory electron transfer reactions in iCce806 is shown in To build a constraint-based metabolic model of iCce806 to predict growth under photoautotrophic (using light and fixing CO2), heterotrophic (using glycerol in the dark), and photoheterotrophic (using glycerol and light) conditions with different nitrogen sources. In silico calculated biomass yields, which simulated carbon or light- limited growth (Cyanothece 51142 Cyanothece's ciracadian cycle 2) that could be fixed and stored in the dark, by maximizing cyanophycin production from glycogen. Although H2 is an obligate co-product of the nitrogenase reaction, no H2 was produced in the initial simulations under dark N2-fixing conditions, contradicting experimental observations. Model examination revealed that all of the nitrogenase-generated H2 was utilized by hydrogenases to reduce NAD(P) and ferredoxin, which ultimately increased cyanophycin production. When the three hydrogenase reactions  were eliminated from the model, the predicted ratio of fixed N2 to consumed glycogen depended on the non-growth associated ATP requirement (NGAR), and was estimated to be 0.3 (NGAR = 2.8) or 0.67 (NGAR = 0) mole N2/mole glycogen, which was in accordance with an experimentally measured value of 0.51 2 production would have same yields as fixed N2 (0.3 to 0.67 mole H2/mole glycogen) due to the stoichiometry of the nitrogenase reaction.For initial testing, we examined the ability of the constraint-based model of d growth , qualita2-fixing dark conditions. With glycogen being the sole source of reductant for both ATP-generating oxidative phosphorylation and N2 reduction, a balance between fluxes through respiratory pathways and nitrogenase reaction is needed. In the absence of the hydrogenase reactions, the model predicted that O2 reduction via COX, QOX, or Mehler reactions are required to consume NADH resulting from glycogen catabolism (2 fixation rate since it generates more ATP than the QOX or Mehler pathways (∼9 O2 are needed per N2 fixed). This is consistent with the results from recent proteomic studies showing the CoxB1 (cce_1977) subunit of COX is more predominant during the dark 2-fixing conditions not only to generate an intracellular anaerobic environment for nitrogenase, but also to provide ATP for nitrogenase activity.We also evaluated how fluxes through electron transfer reactions are affected by the nitrogenase flux under Ntabolism . The modiCce806 model. To perform model robustness analysis, we computationally explored the impact of key photosynthetic and respiratory pathways on growth rate and intracellular flux distributions under varying photon uptake flux for PS I, while the photon uptake flux for PS II was fixed at 20 mmol·g−1 AFDW·h−1  through each photosystem. The resulting phenotypic phase plane (PhPP) contained three distinct regions . By adding artificial ATP or NADPH generating reactions (ADP+HPO4+H→ATP+H2O and NADP+H→NADPH) to the model and analyzing changes in predicted maximal growth rates, we were able to identify that in regions 1 and 3 growth was NADPH/reductant-limited, while in region 2 it was limited by energy supply  was predicted to be active in region 2, but optional in regions 1 and 3. Closer examination of in silico calculated electron flux distributions revealed that, in addition to FNR, the model utilized a cycle involving glutamine synthetase, glutamate synthase and transhydrogenase, resulting in ATP-driven NADPH production. In regions 1 and 3, the model predicts there is excess ATP, and so this cycle can be used instead of FNR to move electrons from ferredoxin to NADPH. However, this cycle is unlikely to be of any physiological relevance since there has been no experimental data supporting this route for making NADPH, and FNR is essential for photoautotrophic growth in unicellular cyanobacteria such as Synechococcus 7002 6c, or Pq) generated from PS II. Similarly, in region 3 under PS II flux limitation, excess electron carriers  must be reduced via NDH-1 or –2 or ferredoxin-dependent cyclic electron transfer (FdPq). Conversely, due to ATP limitation in region 2, the model favored reactions with higher proton pumping capacities and so both the QOX and FdPq reactions were inactive. The usage of COX was optional in region 2 and depended on photon uptake rates .To analyze the effect of photon uptake rates on electron flux distributions, we calculated the flux ranges using flux variance analysis (FVA) for all photosynthetic and respiration reactions within each PhPP region . In thisCyanothece 51142 cells exposed to different intensities and ratios of 630 and 680 nm light , i.e., either 630 or 680 nm, Cyanothece 51142 cells displayed a similar trend with higher growth rates observed at higher photon flux intensities. The predicted growth rates were within 7% of the experimentally measured values, except for the two cases where single 630 nm wavelength irradiances were used (in vivo.The model predictions  were comnm light . When Cyere used . The rea−1 AFDW·h−1), and the y-intercept is NGAR (2.8 mmol·g−1 AFDW·h−1). The estimated GAR value is significantly higher than those reported from other bacteria Cyanothece 51142 this value was reported to be between ∼70–100% for photoautotrophic growth Cyanothece ATP requirements were most sensitive to reductions in quantum efficiency and the amount of ATP generated by photosynthesis and respiration , iron acquisition , respiratory electron transport  as well as peptide transport, synthesis, and protein repair . Among the most highly expressed genes in ammonium-limited Cyanothece 51142 cells was the one encoding 6-phosphogluconate dehydrogenase (cce_3746), a key PPP enzyme. Under light limitation, the major changes in the transcriptome of Cyanothece 51142 included upregulation of genes encoding: components of the photosynthetic apparatus and electron transport chain ; pigment biosynthesis ; CO2 uptake and fixation machinery ; ATP synthase , and protein synthesis machinery (cce_ 4016–4030)  .Cyanothece 51142 corroborated the shifts in gene expression (Cyanothece 51142 proteins involved in photosynthesis and carbon fixation under light-limited conditions (b6 (cce_1383), displayed elevated peptide abundances in light-limited cells.Global proteome profiles of pression . The abunditions . Notably2 production we used FVA to identify required (flux must be non-zero), optional (flux may or may not be zero), or inactive (flux must be zero) reactions under light- and ammonium-limited growth conditions. As our initial simulations (2 of mRNA expression level is less than 8). The model constrained by TPD predicted that the majority of reactions in central metabolism would be active under both chemostat conditions . Flux spans for reactions in central metabolism . Other single deletions with significant effects included FNR and NDH-1, which are involved in photosynthesis and respiration. The reaction deletions shown in −1 AFDW·h−1. The in silico analysis of double reaction deletions did not yield any new double deletions that would reduce the average central metabolic flux span significantly. However, some double deletions strategies did reduce flux spans of individual reactions.While the number of optional reactions was reduced by incorporating TPD into the model, the flux spans  of individual fluxes was still large (>30 mmol·gtabolism  were thetabolism . Single Synechocystis PCC 6803) have been published, which represented photosynthesis as two lumped reactions 6fb, PSI, and FNR) and cyclic (PS I and Cyt 6fb) pathways. In this study, we modeled photosynthesis as a larger set of separate reactions Synechocystis sp. PCC 6803, serving as a sink for excess electrons Cyanothece 51142, since the associated proteins were detected in the proteomic data , or small organic compounds  were predicted to be necessary in order to balance the electrons and ATP needed to support growth. In the presence of linear and cyclic photosynthesis reactions, these products must also be produced unless significant amounts of cyclic photosynthesis occurs (>3 times the amount of linear photosynthesis). Since H2 and small organic compounds are not generally produced under photoautotrophic conditions with excess ammonium, any additional energy is most likely supplied by cytochrome oxidase activities that reduce photosynthetically produced O2. Interestingly, in the absence of cytochrome oxidase activities in the model, the PS I fluxes must always be greater than or equal to the PS II fluxes. It was shown that the marine cyanobacteriium Synechococcus has a PS I/PS II protein ratio >1, which has been explained as a mechanism to protect PS II from photo-damage 2 concentrations to limit photorespiration.In the absence of cyclic photosynthesis, other products including water , Hvia NADH dehydrogenase- or ferredoxin-dependent routes have also been experimentally demonstrated to play important roles in balancing the amount of NADPH and ATP produced via photosynthesis. Synechocystis 6803 mutants lacking ndhD genes (encoding subunits of NDH-1) had significantly lower cyclic photosynthesis activity c6) and protect against photo-damage. Cyclic electron flow has been experimentally shown to help protect the photosynthetic apparatus from photo-damage Similarly, cyclic electron flow 2 production rates suggests that reductant was being directed towards O2via the Mehler, QOX, and/or COX reactions. In both chemostat conditions, the model predicted that steady-state growth rate could have been achieved using lower photon uptake rates by decreasing the amount of reductant generated by PS II that was predicted to reduce O2.In addition to studying the interactions between components of the photosynthetic and respiratory components computationally, we also experimentally evaluated cells grown under continuous light conditions in light- and ammonia-limited chemostats. The measured 630 nm and 680 nm photon uptake and O2 (a product of the proline oxidase reaction) could be reoxidized to FAD. After experimental confirmation that proline can be used as a nitrogen source (implying activity of proline oxidase) by Cyanothece 51142 (data not shown), a FADH2 recycling reaction was included in the final iCce806 model.A limitation to flux balance analysis is that a wide range of flux values may be consistent with the constraints in the computational model. An iterative application of computational and experimental methods is an important strategy to improve the comprehensive understanding of cyanobacterial metabolism. We have begun to apply this iterative approach, by including mRNA and protein expression datasets as additional constraints beyond biomass composition and physiological rate measurements. Experimentally-measured TPD were successfully used to further constrain the model, and thereby reduce uncertainty and increase the number of required  reactions . Howeverin silico can be used to identify knockout mutants that can potentially improve the resolution of intracellular flux distributions. A flux that is well resolved would have a small span meaning we can more definitively state its value. If the mutants show no growth defects then the corresponding reactions may not be used under the conditions tested, or alternative pathways not included in the model may occur. Either way, this information could be used to better resolve the intracellular flux distribution or improve the metabolic model. For Cyanothece 51142, this would require development of a genetic system  in 1-L Roux bottles sparged with CO2-enriched air (0.3% vol/vol). Culture purity was confirmed by plating on DIFCO Bacto Tryptic Soy Broth and DIFCO Luria-Bertani solid media  as well as by phase contrast or acridine orange fluorescent microscopy.Cyanothece 51142 were grown in a 7.5-L borosilicate glass vessel operated at 5.5-L working volume under the control of New Brunswick Bioflo 3000 bench-top bioreactor . The vessel was housed in a custom-made black anodized aluminum enclosure equipped with light-emitting diodes (LED) generating 680 nm and 630 nm light for the preferential excitation of chlorophyll a and phycobilin pigments, respectively. Built-in sensors allowed for automatic adjustment of incident and transmitted light intensities using custom-designed control module. Both hardware and software components of the LED enclosure and the control module were developed at Pacific Northwest National Laboratory . All experiments were carried out under continuous illumination in modified ASP2 medium sparged with CO2-enriched argon (Ar) (0.2% vol/vol). Agitation, temperature, pH, and gas flow rates were maintained at 250 rpm, 30°C, 7.4, and 2.8 L/min, respectively. Incoming and off-gas composition was constantly monitored by an in-line mass spectrometry based gas analyzer MGA iSCAN . Cell density was monitored spectrophotometrically at 625, 678, and 730 nm.Controlled batch and chemostat cultures of Cyanothece 51142 cells and maintained as a batch culture under 630 nm and 680 nm illumination at 40 and 70 µmol photon·m−2·s−1, respectively, until the culture reached late logarithmic stage. Chemostat mode was initiated by continuous inflow of medium at a dilution rate of 0.05 hr−1 that resulted in a steady-state optical density (OD730) of 0.20. Similarly, a nitrogen-limited continuous culture of Cyanothece 51142 was established using low-nitrogen ASP medium containing 0.75 mM NH4+. The ammonium-limited chemostat was maintained under identical operating conditions in regard to the culture dilution rate and optical density under incident light at 38.5 and 73.5 µmol photon·m−2·s−1 for 630 nm and 680 nm wave lengths, respectively).To establish a light-limited chemostat culture, the photobioreactor was inoculated with 10 mL of mid-log phase −1 AFDW·h−1) were determined by multiplying the light consumption rates (µmol photon m−2s−1) by the surface area of cell culture exposed to light (m2) and dividing by the amount of biomass in the reactor (g AFDW). The light consumption rates were determined by subtracting the transmitted light intensity from the values of incident light intensity after corrections were made for the abiotic consumption of light to account for the gas bubbles and probes in the reactor. Cells in the 5.5 L working volume were assumed to be equally exposed to the light at all times. Based on the inner diameter and height of the liquid culture at working volume, the surface area was 0.1403 m2. The amount of biomass in the reactor was determined from the working volume and biomass concentrations.The light uptake fluxes (mmol·gBiomass ash-free dry weight (AFDW) was measured using centrifuged  cell pellets as described previously Cyanothece 51142 Previously developed whole-genome oligonucleotide microarrays of Cell lysis and tryptic digestion followed a previously described “global protein preparation” scheme Cyanothece 51142 proteome Cyanothece 51142 samples using the AMT tag approach To build an accurate mass and time (AMT) tag database, SEQUEST analysis software was used to match the MS/MS fragmentation spectra to sequences from the annotation of the P-value cutoff of 0.05 to further remove outlying peptides. For increased confidence in protein identifications, each protein was required to be identified by at least 2 unique peptides, resulting in a total of 865 proteins. The minimum observed relative protein abundance value  was imputed as a crude surrogate for missing data for statistical calculations. Statistical differences between the two samples  were determined using ANOVA with a P-value cutoff of 0.05 (q<0.03) in DAnTE Data from the AMT output were imported into the software MDART , for filtering using a mass error tolerance of <5 parts per million, delta match score >0 (a measure of peptide uniqueness), match score >−1, and absolute normalized elution time error <10,000. The resulting 7450 peptides were imported into the software tool DAnTE Cyanothece 51142 was reconstructed in SimPheny  using a previously described automated model-building process Cyanothece 51142 and genes in other modeled organisms. Additional reactions were added as necessary to produce known biomass constituents or utilize known nutrients; detailed literature, database, and BLAST searches were then carried out to find genes encoding these reactions in Cyanothece 51142 genome. This resulted in several new GPR associations that were incorporated into the reconstruction.A draft metabolic network of Cyanothece 51142 was developed as described in S is a stoichiometric matrix for the reaction network, v is a flux vector, and α and β are parameters that limit the capacity and directionality of individual reactions. Flux balance analysis (FBA) uses these constraints to identify a flux distribution which maximizes or minimizes an objective function, such as growth rate Based on the metabolic reconstruction, a constraint-based metabolic model for et al.E  and were constrained to their measured values. Reactions associated with detected proteins were included in the high reaction set (RH). Reactions associated with undetected proteins and genes with low mRNA expression levels (whose mRNA expression was less than the lowest mRNA expression of detected proteins) were included in the low reaction set (RL). The method finds a flux distribution that maximizes the number of active reactions (v≠0) and inactive reactions (v = 0) in reaction sets RH and RL, respectively. For reactions in set RH, binary variables x and y indicate whether a reaction is active, meaning its flux is greater than a positive threshold ε (x = 0 and y = 1), or smaller than a negative threshold -ε (x = 1 and y = 0) for reversible reactions. If both x and y are zero then the reaction is inactive and its flux value is zero. Likewise, a binary variable z is used for reactions in set RL such that if z = 1 then the reaction is inactive (v = 0). The original method To further constrain the models based on mRNA or protein expression data, a modified version of the method developed by Shlomi x, y, and z) were further constrained by their optimal values  found in the original problem . In this study, all model simulations were performed in GAMS software .Dataset S1SBML file of the model iCce806 which stores reactions and GPR associations.(XML)Click here for additional data file.Figure S1In silico predictions for biomass yields under photoautotrophic, heterotrophic and photoheterotrophic conditions. Comparison of maximal biomass yields per g of C substrate when different nitrogen sources are used. Under photoautotrophic conditions CO2 uptake flux was fixed at 1 mmol.g−1 AFDW.h−1 and photon uptake fluxes at PSI and PSII were fixed at 10 mmol•g−1 AFDW•h−1. In the heterotrophic simulations glycerol was the limiting nutrient. Glycerol uptake was fixed at 1 mmol•g−1 AFDW•h−1 and maximal biomass yields were calculated under dark conditions. In photoheterotrophic simulations both glycerol and light were limiting (so an increase in either would improve growth rates). In this case, glycerol uptake rate was fixed at 1 mmol•g−1 AFDW•h−1, while photon uptake fluxes for PSI and PSII were both fixed at 10 mmol•g−1 AFDW•h−1. Since light was limiting in the photoheterotrophic condition CO2 was predicted to be secreted and not used as an additional carbon source.(EPS)Click here for additional data file.Figure S2Effects of distribution of fluxes through electron transport chains (ETC) on nitrogenase flux. Nitrogen fixation (nitrogenase) flux was varied while fluxes through ETC reactions were maximized and minimized under dark N2-fixing condition with all hydrogenase reactions eliminated from the model. Under this condition the amount of H2 produced is equal to the nitrogenase flux. A glycogen demand reaction was added to the model  and its flux was limited to 0.171 mmol•g−1 AFDW•h−1. A) Effects of distribution of fluxes through cytochrome c oxidases (COX) and cytochrome-quinol oxidases (QOX) on nitrogenase flux. B) Effects of distribution of total flux through COX and QOX, and flux through Mehler reactions on nitrogenase flux. C) Effects of distribution of fluxes through NADH dehydrogenase reactions (NDH) and Fd-dependent cyclic reaction (FdPq) on nitrogenase flux. Shaded regions indicate ETC reactions can have multiple values for a particular nitrogenase flux.(EPS)Click here for additional data file.Figure S3Estimating ATP requirements from batch data. Average growth rates and photon uptake fluxes from batch experiments were used to constrain the model. The maximum ATP hydrolysis flux (flux through the ATPM reaction) was calculated using these measurement constraints. The data points represent the calculated maximal ATP hydrolysis values for different batch experiments. The growth-associated ATP requirement  and non-growth associated ATP requirement  were estimated by linear regression of these data.(EPS)Click here for additional data file.Table S1Reactions in the genome-scale metabolic network iCce806.(XLSX)Click here for additional data file.Table S2Metabolites in the genome-scale metabolic network iCce806.(XLSX)Click here for additional data file.Table S3Genome-scale metabolic network statistics for iCce806. Number of genes, reactions, metabolites, and network gaps in iCce806.(XLSX)Click here for additional data file.Table S4Effects of changing simulation conditions on ATP requirement parameters, GAR and NGAR.(XLSX)Click here for additional data file.Table S5Biomass compositions under ammonia-limited (AL), and light-limited (LL) chemostat conditions.(XLSX)Click here for additional data file.Table S6Experimental measurements for two chemostat conditions.(XLSX)Click here for additional data file.Table S7Transcriptomic data. Fold change in mRNA expression levels between light-limited and ammonia-limited chemostats.(XLSX)Click here for additional data file.Table S8Proteomic data. Protein expression levels measured in light-limited and ammonia-limited chemostats.(XLSX)Click here for additional data file.Text S1Supplemental methods for A) biomass calculations and B) batch growth rate calculations.(PDF)Click here for additional data file.


Analyzing the detected pathways in detail we found that their energetic requirements potentially limit their capacity. This study has many other biochemical implications: effect of starvation, sports physiology, practically carbohydrate-free diets of inuit, as well as survival of hibernating animals and embryos of egg-laying animals. Moreover, the energetic loss associated to the usage of gluconeogenesis from fatty acids can help explain the efficiency of carbohydrate reduced and ketogenic diets such as the Atkins diet.The question whether fatty acids can be converted into glucose in humans has a long standing tradition in biochemistry, and the expected answer is “No”. Using recent advances in Systems Biology in the form of large-scale metabolic reconstructions, we reassessed this question by performing a global investigation of a genome-scale human metabolic network, which had been reconstructed on the basis of experimental results. By elementary flux pattern analysis, we found numerous pathways on which gluconeogenesis from fatty acids is feasible in humans. On these pathways, four moles of acetyl-CoA are converted into one mole of glucose and two moles of CO th century. With the discovery of the glyoxylate shunt that allows this conversion in some bacteria, plants, fungi and nematodes it has been considered infeasible in humans since the corresponding enzymes could not be detected. However, by this finding only a single route for gluconeogenesis from fatty acids has been ruled out. To address the question whether there might exist alternative routes in humans we searched for gluconeogenic routes from fatty acids in a metabolic network comprising all reactions known to take place in humans. Thus, we were able to identify several pathways showing that this conversion is indeed feasible. Analyzing evidence concerning the detected pathways lends support to their importance during times of starvation, fasting, carbohydrate reduced and ketogenic diets and other situations in which the nutrition is low on carbohydrates. Moreover, the energetic investment required for this pathway can help to explain the particular efficiency of carbohydrate reduced and ketogenic diets such as the Atkins diet.That sugar can be converted into fatty acids in humans is a well-known fact. The question whether the reverse direction, i.e., gluconeogenesis from fatty acids, is also feasible has been a topic of intense debate since the end of the 19 A related question biochemistry students are often asked in their exams is whether the reverse route is also feasible, that is, whether the human body can convert fatty acids back into glucose. As for even-chain fatty acids, the expected answer is “No” . This summarizes the result of a debate that dates back to the late 19th century. However, it was not until the 1950s that such a conversion could be monitored using In summary, there is a general consent that carbohydrates cannot be produced from even-chain fatty acids in humans although the opposite conversion is feasible. In consequence, this statement can be found throughout prominent biochemistry textbooks et alRecently, we re-investigated the problem in question in a small model of human central metabolism et al.The question whether fatty acids can be converted into carbohydrates has many biochemical implications: effect of starvation  from acetyl-CoA in the network of human metabolism described in In order to analyze this route in more detail and to determine whether alternative pathways exist, we determined the global pathway connecting acetyl-CoA and G6P using a procedure described previously Next, we checked which reactions are essential on the above pathway using an algorithm described in After having identified essential reactions of the gluconeogenic pathway we aimed to elucidate the parts of the pathway on which variable routes could be used. Since these alternative routes have to connect the previously identified sequences of essential reactions we could deduce that they perform one of the following biochemical functions:Conversion of mitochondrial acetyl-CoA into mitochondrial acetoacetateConversion of cytosolic acetol into mitochondrial pyruvateConversion of mitochondrial oxaloacetate into cytosolic phosphoenolpyruvateThe latter mainly concerns well-known routes for the transport of oxaloacetate and other intermediates of the mitochondrial TCA cycle into the cytosol. Therefore, we will concentrate on conversions (1) and (2) in the following.Using the procedure described in Overall we detected nine possible pathways for the conversion of acetyl-CoA into mitochondrial acetoacetate see . In acetIn addition to the work of + or oxidation of NADPH. Overall, 1–3 moles of NADPH are oxidized and 0–4 moles of NAD+ are reduced during the conversion of one mole of acetone into pyruvate . The most efficient pathway in terms of ATP consumption proceeds via the oxidation of methylglyoxal to pyruvate, oxidizes two moles of NADPH and reduces two moles of NADH . Moreover, we provided a detailed analysis of the energetic balance of these pathways, which explains their limited capacity and their contribution to the particular efficiency of carbohydrate reduced and ketogenic diets.4, Fe2+, Fe3+, H+, K+, Ca2+, Na+, Cl−, phosphate, O2, and water. To detect pathways converting even-chain fatty acids into glucose, we furthermore added an influx of mitochondrial acetyl-CoA and an outflow of cytosolic glucose 6-phosphate and mitochondrial coenzyme A. Furthermore, we added several reactions of amino acid degradation that were not present in the model and replaced reactions of the respiratory chain by a net reaction since the model could produce ATP without consumption of any other metabolite otherwise. Moreover we constrained the flux through several reactions allowing the unconditional production of ATP to zero .Furthermore, we call a flux pattern elementary if this set of reactions cannot be written as set union, of other flux patterns. For more details see Kaleta Elementary flux patterns can be used to elucidate all possible pathways consuming a certain compound and producing another. This process builds upon a successive expansion of the subsystem under study to reactions that belong to alternative pathways. It will be outlined by way of a small example network comprising glycolysis and the pentose phosphate pathway . Within We start with a subsystem encompassing the inflow reaction of Glc and the outflow of R5P . Two eleNext, we need to determine an elementary mode through the entire system using exactly the reactions of the elementary flux pattern in the subsystem. Such an elementary mode can be obtained using linear programming In order to determine reactions that belong to alternative pathways, we include the reactions of the first detected pathway into the subsystem of the next step. As noted above, we need not to add essential reactions and consequently, the subsystem of the second step comprises three reactions : the inf2O, CO2, Pi, Ppi, protons, O2, ATP, ADP, AMP, GTP, GDP, NADH, NAD+, NADPH, NADP+, coenzyme A and HCO3+ to external status to obtain elementary flux modes differing in the underlying carbon conversion pathway rather than in co-factor balancing reactions.We computed the 100 shortest elementary flux modes producing glucose 6-phosphate from acetyl-CoA using a previously described method We calculated the Gibbs free-energy changes of the described pathways. We performed these calculations for two scenarios: gluconeogenesis from acetyl-CoA and from palmitate. For the first scenario, we assumed that additionally to the energetic balance as depicted in Table S1List of metabolite and enzyme abbreviations.(PDF)Click here for additional data file.Text S1Supplementary material and methods.(PDF)Click here for additional data file.