Development of cells with minimal metabolic functionality is gaining importance due to their efficiency in producing chemicals and fuels. Existing computational methods to identify minimal reaction sets in metabolic networks are computationally expensive. Further, they identify only one of the several possible minimal reaction sets.Escherichia coli and Saccharomyces cerevisiae. In case study 1, the proposed method identified three minimal reaction sets each containing 38 reactions in Escherichia coli central metabolic network with 77 reactions. Analysis of these three minimal reaction sets revealed that one of them is more suitable for developing minimal metabolism cell compared to other two due to practically achievable internal flux distribution. In case study 2, the proposed method identified 256 minimal reaction sets from the Saccharomyces cerevisiae genome scale metabolic network with 620 reactions. The proposed method required only 4.5 hours to identify all the 256 minimal reaction sets and has shown a significant reduction (approximately 80%) in the solution time when compared to the existing methods for finding minimal reaction set.In this paper, we propose an efficient graph theory based recursive optimization approach to identify all minimal reaction sets. Graph theoretical insights offer systematic methods to not only reduce the number of variables in math programming and increase its computational efficiency, but also provide efficient ways to find multiple optimal solutions. The efficacy of the proposed approach is demonstrated using case studies from Identification of all minimal reactions sets in metabolic networks is essential since different minimal reaction sets have different properties that effect the bioprocess development. The proposed method correctly identified all minimal reaction sets in a both the case studies. The proposed method is computationally efficient compared to other methods for finding minimal reaction sets and useful to employ with genome-scale metabolic networks. The depletion of fossil fuels and increasing concerns over environmental changes are key driving factors for the development of sustainable bioprocesses to produce chemicals and fuels from renewable resources . Today, Synthetic biology provides the tools and techniques to design and construct artificial cells with minimal functionality containing a minimal genome, but with all the essential genes for survival in a defined environment and possessing replication capabilities . Such miThe presence of redundant pathways in metabolic networks results in alternate optimal solutions and consequently create mismatch between model predictions and experimental observations ,20. ReduN metabolites and M reactions is mathematically represented as [The metabolic network of a given microorganism with ented as :S captures interactions among reactions where Sij is the stoichiometric coefficient of the ith metabolite in the jth reaction and vj is the flux (rate) of reaction j. The zero in the right hand side is due to the steady-state assumption generally considered in metabolic network analysis. The mathematical representation of metabolic networks enables analysis of the metabolism using optimization methods to identify internal flux distribution, metabolic capabilities, and strain improvement strategies through gene knock-out or insertion of non-native reactions [The Stoichiometric matrix eactions ,24-26. Ieactions :vjmin and vjmax represent the lower and upper bounds for the flux through reaction j. A binary variable yj is associated with each reaction with ‘1’ indicating the presence/activation of the reaction and ‘0’ its absence/deactivation. Cellular objectives are incorporated as constraints, for example, the objective in Eq. (2) is to produce at least Here, eactions .depth is associated with each node and arc in the network starting with the extracellular metabolites and primary uptake reactions which are deemed to be of depth 1. The depth of every other metabolite and reaction is assigned as an increment over its predecessor’s. There are two phases in the hybrid approach  are required for the cell to meet biological objectives and hence they are the part of every minimal set. Extraneous reactions (XRs) are not necessary for the cell. Indeterminate reactions (IRs) primarily consist of substitutable reactions i.e. reactions which can be substituted with other reactions to achieve the cellular objectives. These IRs are holistically analysed in the subsequent Phase 2, using a MILP with the same structure as that in (2) but smaller than the monolithic one. Through this, a subset of IRs called Additional reactions (ARs) necessary for the minimal metabolism cell are identified which together with SRs identified in Phase 1 form the minimal reaction set. A substantial reduction in the computational time (~66%) required to identify one solution could be achieved through the hybrid approach compared to solving Eq. (2) directly. In this paper, we extend the above hybrid approach to identify all minimal reaction sets. The theoretical basis of the proposed approach is discussed first.We previously reported an efficient approach that combines graph theory with math programming to solve this problem . In ourlinear and flux dependency. Linear dependency arises between reactions due to the structure of the network where the product(s) of a reaction feed into exactly one other reaction. When a set of reactions are all linearly dependent, they can be considered to form a linear pathway. Instances of several reactions forming a linear pathway in metabolic networks are common. For example, in the sample metabolic network shown in Figure A_ext and B_ext enter into the cell and biomass is produced from them through the reaction network. Two linear pathways can be identified in this sample network {r1,r3, r4,r5} and {r2,r6}. In linear pathways, under steady-state assumption mentioned above, the flux through all the reactions has to be equal. Hence, deletion of any reaction in the linear pathway would result in the deletion of the whole pathway.Reactions in the metabolic network are dependent on each other since the network is an interconnected system designed to achieve the biological objectives of the cell. Two different kinds of dependencies can be identified – {r9, r10, r11} are not linearly dependent, they have flux dependencies, since reactions {r10, r11} consume the products of r9 and are dependent on each other through the flux balance requirement.Another kind of dependency, flux dependency, exists between reactions which are not structurally linear, but are required to co-exist to balance the fluxes. If a reaction produces two products which are then consumed by two different reactions, then these three reactions are dependent on each other, since at steady state, the metabolites produced by the first reaction have to be consumed by the two down-stream reactions. Deletion of any one of these reactions would make all other reactions incapable of carrying flux at steady state. In the sample network, although reactions S. The null-space represents all the possible steady-state flux distributions that satisfy Eq. 1 and the dependent reactions are the rows in this matrix with same values after normalization with no contradictions in the directionality for irreversible reactions. Since this procedure depends on the directionality of reactions without considering the structural features of metabolic netwotk, it may not identify some dependent reaction groups due to imperfect assignment of reaction directionality in the metabolic networks. Also, identification of dependent reaction groups strictly based on flux distributions results in groups of structurally unrelated reactions which hinders interpretation. We have developed a graph based algorithm that exploits the structure of metabolic network to identify groups of dependent reactions as described next.The linear and flux dependency among reactions in a metabolic network can be exploited to assemble reactions into groups for analysis, rather than analyzing them individually. Since deletion of any reaction would force all the dependent reactions to be excluded from the network, for network optimization using a MILP, it is sufficient to associate a single binary variable with each group of reactions. Reduction of the number of binary variables reduces the search space and consequently reduces the computational cost of finding solutions. Identification of dependent reactions and simplification of metabolic networks using the reaction dependency has been reported in literature ,27,28). ,28. 27,2i.e. reaction with the lowest depth. Dependent reactions are then added to this group step-by-step by searching the metabolic network in a breath-first manner. In each step, all the reactions at the current depth +1 are collected and tested for linear or flux dependency. Reactions are added to the group if they dependent on other reactions that are already present in the group. Specifically, a single reaction that receives its reactants exclusively from another reaction in that group is deemed as linearly dependent. Similarly, multiple reactions are deemed to have flux dependency if all their reactants originate from one reaction already in the group. The search continues until al the reactions are evaluated i.e., the highest depth in the network is reached or no dependent reaction can be found at a given depth. Once a group of dependent reactions is identified, all these reactions are removed from the reaction list. In the subsequent iteration, the algorithm continues with the creation of a new group with the first reaction in the updated reaction list. The algorithm terminates when the reaction list becomes empty.Given a metabolic network where the depth has been assigned to reactions as described in MILP for finding minimal reaction set, the algorithm first creates a list of reactions, sorted in the ascending order of their depth. Reactions are grouped from this list using an iterative procedure. In each iteration, a new group is created starting with the first reaction in the list, r1 (depth 1), identifies reaction r3 as linearly dependent at depth 2 in the first step, and then reaction r4 in the second step, and r5 in the third step. In the fourth step, r9 is added to the group since one of its two reactants is exclusively from r5. Reactions {r10,r11} are then identified as having flux dependency since they exclusively receive their reactants from r9. The search stops at this step since there are no further reactions in the list at higher depths. Thus the first group of dependent reactions is {r1,r3, r4,r5,r9,r10,r11}. The graph based algorithm thus groups reactions based on both linear and flux dependencies. The number of reactions in a group is called its norm. Thus, the norm of this group is 7. All these 7 reactions are removed from the reaction list. The second iteration starts with reaction r2 (since it has the lowest depth of the reactions in the updated reaction list), and identifies reaction r6 as its dependent. The search for dependent reactions stops here since two reactions, {r7, r8}, consume the product (D) of r6. Hence, the second group has 2 reactions {r2,r6}. Continuing in this fashion, two single reaction groups {r7} and {r8} are also identified. Thus, in total, there are four different groups in the sample network as shown in Figure We illustrate the algorithm using the sample network shown in Figure Once the groups of dependent reactions have been identified in the metabolic network, analysis can be carried out on these groups rather than on the individual reactions. If a reaction from a group is essential for the cell, all the reactions in that group become essential since they are dependent on each other. Similarly, all the reactions in the group become extraneous or indeterminate if one of the reactions in the group is extraneous or indeterminate, respectively. Hence, the minimal reaction set identification problem is reduced to identification essential reaction groups (SRGs), extraneous reaction groups (XRGs), and indeterminate reaction groups (IRGs).The proposed recursive MILP approach for identifying all the minimal reaction sets in metabolic network is shown in Figure wl is the norm of the group and yl is the binary variable associated with that group. The optimization procedure thus will identify the ARGs such that the total number of reactions is minimal. The ARGs together with the SRG from Phase 1 forms the first minimal reaction set. Once an optimal solution is found, a constraint is added to the model to exclude that solution from the search space (Step 2). Based on Lee et al. (2000), the following constraint is added to Eq. 2:The algorithm for finding all ARGs from IRGs formulated as a recursive MILP as shown in Figure NZ is the groups in the optimal solution, and yr is the binary variable associated with the groups in NZ. Eq. (3) means that at least one of the non-zero binary variable in the optimal solution is set to zero. Hence in the next recursion, NZ is excluded from the search space and the optimizer is forced to find a new optimal solution. This recursive procedure terminates when the optimizer returns a sub-optimal solution, i.e. a solution with more reactions than that in the first solution.where In principle, all the minimal reaction sets can be identified by recursively solving the MILP with a new constraint added to the model in each recursion. However, the grouping of reactions offers insights which enable the computational cost to be reduced significantly by generating additional solutions without solving the MILP, using group substitutability analysis (Step 3).r7 and r8 consume metabolite D and produce metabolite G. The presence of reactions with similar functions enables the cell to survive under different conditions, stress, and malfunction of genes through substitution of reaction for another inactive reaction. These reactions are considered substitutable since they result in alternate optima. In the minimal reaction set identification problem, substitutable reactions lead to multiple minimal reaction sets. The above recursive MILP approach can be employed to identify all minimal reaction sets. Alternatively, many candidate solutions can be generated more efficiently by simply substituting a reaction in an optimal solution.In metabolic networks, some reactions share similar cellular functions such as producing, consuming a metabolite, or recycling co-factors. For example, in the sample network shown in Figure In this paper, we perform this substitution analysis on groups to efficiently identify alternate optimal solutions. Two types of group substitution are possible – single and multi-group substitution. In single group substitution, a group is substituted with another group of the same norm. So, the total number of reactions in the optimal solution remains unchanged. For example, in Figure Group substitutability analysis is conducted in the proposed approach following the identification of a solution by the MILP and candidate solutions generated. Not every candidate solution identified by the qualitative approach would meet the cellular objectives. Therefore, it is essential to verify the candidate solutions to ensure that the predefined biological objectives are satisfied. This verification is conducted by solving a linear program (LP) with the objective of maximizing the cellular objective (Step 4), which is computationally efficient. Only candidate solutions that satisfy the objective are deemed as optimal solutions to the original MILP and appended to the set of optimal solutions (Step 5). Other candidate solutions are discarded. A new constraint is also added to the model for each optimal solution thus identified to eliminate their re-identification in future recursions (Step 6). The algorithm then continues with solving the MILP to find other optimal solutions.Escherichia coli and Saccharomyces cerevisiae.We illustrate the proposed method by identifying all minimal reaction sets that support predefined growth for two systems – E. coli metabolic network so as to meet cellular objective i.e. single reaction groups. Hence, the number of binary variables required for MILP is reduced from 77 to 62. The proposed recursive MILP method is then employed to identify all the minimal reaction sets from this network. Phase 1 of the proposed approach classified 14 groups (18 reactions) as essential reaction groups of which 4 groups with norm 2 the remaining 10 groups of norm 1. The method also denoted 8 groups (12 reactions) as extraneous. There were 40 groups (47 reactions) identified as indeterminate containing 1 group with norm 3, 5 groups with norm 2 and the remaining 34 with norm 1. Hence, the number of binary variables defined in Phase 2 is reduced from 47 to 40. Then the recursive MILP is employed to identify IRGs. The first optimal solution contains 18 groups (20 reactions) — 17 groups of norm 1 and 1 group with norm 3. The recursive MILP found two more optimal solutions  that meet the predefined cellular objective. In the fourth iteration, the optimizer found a sub-optimal solution with 21 ARS and hence is terminated. These three sets of additional reactions together with the 18 SRs from Phase 1 form the three different minimal reaction sets. To cross-validate the results, we also implemented the classical monolithic MILP approach with 77 binary variables. The monolithic MILP also identified the same three minimal reaction sets thus confirming the accuracy of the proposed approach.Here, we identify all the minimal reaction sets from the eactions . These 7E. coli metabolic network are shown in Figure E.coli and contains no coupled reactions (cycles); hence, it is a suitable reaction set for developing the minimal metabolism cell.The three minimal reaction sets identified in the E.coli in order to develop minimal metabolism cell based on each minimal reaction set using the graph theory based approach [Escherichia coli; 4 of these 6 reactions are the same in all cases. Hence, the minimal metabolism cell can be constructed by suitable blocking out the reactions corresponding to minimal reaction set 2. In summary, finding multiple minimal sets enables us to develop the best minimal metabolism cell by selectively deleting the remaining two reactions.We identify the number of reactions to be knocked-out from approach . In brieSaccharomyces cerevisiae containing 1061 metabolites and 1266 reactions [We now illustrate the computational efficiency of the proposed method by identifying all the minimal reaction sets for a genome-scale model of eactions . The celE. coli model, the Saccharomyces cerevisiae model has compartments. Out of the 370 indeterminate reactions, 52 reactions are involved in transporting metabolites among the compartments and inter-converting cofactor metabolites. These are deemed to be essential reactions. The remaining 249 groups containing 318 indeterminate reactions are further analyzed in Phase 2 using recursive MILP to find all additional reactions. The results are given in Table Phase 1 of the proposed approach identified 128 groups (213 reactions) as essential, 22 groups (37 reactions) as extraneous, and 301 groups (370 reactions) as indeterminate. The extraneous reactions are removed from further analysis. Unlike the Based on the first solution, 7 other minimal reaction sets that meet the predefined cellular objective are identified through group substitutability analysis leading to a total of 8 minimal reaction sets in the first iteration. These 8 optimal solutions were excluded from the search space through addition of new constraints. In the seconds iteration, 6 more minimal reaction sets were identified — 1 from MILP and 5 from substitution analysis. The algorithm then continues with next iteration. The results for each iteration are shown in Table To quantify the improvement achieved, we executed the MILP with 318 binary variables for 318 indeterminate reactions. The solver required 3,735,864 CPLEX iterations and 1038 seconds to find the optimal solution. The reduction of number of binary variables has resulted in a significant improvement with approximately 60% reduction in CPLEX iterations and 80% reduction in the time required for finding the optimal solution in Phase 2. We also compared the total time required for Phase 1 & 2 to find the first solution by the proposed method with monolithic MILP and graph theory based approach without grouping. The results are given in Table Escherichia coli and Saccharomyces cerevisiae. The proposed approach correctly identified all the minimal reaction sets in both the cases. We also proposed the concept of grouping dependent reactions to reduce the number of binary variables for MILP formulation. In the present study, several groups of dependent reactions are identified in Escherichia coli and Saccharomyces cerevisiae and exploited to reduce the number of binary variables and consequently the solution time. Since the use of binary variables is very common in metabolic network analysis for identifying strain improvement strategies [Development of cells with minimal metabolic functionality is increasingly gaining importance. The presence of redundant reactions in metabolic networks results in multiple minimal reactions sets that can meet the predefined cellular objectives. In this paper, we proposed a graph theory augmented recursive MILP approach to identify all the minimal reaction sets in a metabolic network. The proposed method has been demonstrated by finding all the minimal reaction sets for rategies ,25,27, tHere, we have developed a graph based algorithm that exploits the structure of the metabolic network to identify groups of dependent reactions. We now compare the groups of dependent reactions identified by the proposed method with the previously reported approach based on steady-state flux distribution. We used the METATOOL software  to find E.coli model in case study 1 and very high flux through co-factor recycling reaction in another solution. Thus, of the three, only one solution has practically achievable fluxes without any coupled reactions (cycles) and is suitable for developing the minimal metabolism cell. This clearly shows the importance of identifying all minimal reaction sets. In Case Study 2, the proposed method identified 256 solutions. While some of them use glycolysis pathway to produce DHAP others use pentose phosphate pathway. Other differences observed include use of different reactions for the consumption of DHAP and formate, recycling of NAD, NADH, NADP, and NADPH, and production of Ammonium. A preliminary analysis shows that some minimal reaction sets require complex set of reactions with impractical high fluxes compared to others. Further analysis may reveal the details of these pathways that qualify them to developing into minimal metabolism cells.Generally, different minimal reaction sets have different structural and functional properties in terms of robustness, predictable metabolic interactions, practically achievable metabolic fluxes, and thermodynamically non-favourable pathways and cycles. We identified one such cycle in one of the solutions identified in The authors declare that they have no competing interests.Both SJ and RS contributed to the concept and methodology development. SJ implemented the methodology and conducted the analysis of results and biological interpretation. RS supervised the study and assisted in implementation. SJ drafted the manuscript. Both authors read and approved the final manuscript.


Improving the synthesis rate of desired metabolites in metabolic systems is one of the main tasks in metabolic engineering. In the last decade, metabolic engineering approaches based on the mathematical optimization have been used extensively for the analysis and manipulation of metabolic networks. Experimental evidence shows that mutants reflect resilience phenomena against gene alterations. Although researchers have published many studies on the design of metabolic systems based on kinetic models and optimization strategies, almost no studies discuss the multi-objective optimization problem for enzyme manipulations in metabolic networks considering resilience phenomenon.S. cerevisiae and E. coli. Results show that the maximum synthesis rates of target products by genetic interventions are always over-estimated in metabolic networks that do not consider the resilience effects.This study proposes a generalized fuzzy multi-objective optimization approach to formulate the enzyme intervention problem for metabolic networks considering resilience phenomena and cell viability. This approach is a general framework that can be applied to any metabolic networks to investigate the influence of resilience phenomena on gene intervention strategies and maximum target synthesis rates. This study evaluates the performance of the proposed approach by applying it to two metabolic systems: Considering the resilience phenomena in metabolic networks can improve the predictions of gene intervention and maximum synthesis rates in metabolic engineering. The proposed generalized fuzzy multi-objective optimization approach has the potential to be a good and practical framework in the design of metabolic networks. Improving the synthesis rate of desired metabolites in metabolic systems is one of the main tasks in metabolic engineering. Two recent advancements in this area are promising to increase the performance of metabolic systems. The first factor is a significantly better understanding of the structure of metabolic networks and the kinetics and thermodynamics of biochemical reactions that take place in living cells. In many cases, this understanding is not merely qualitative but quantitative, and can be expressed in terms of kinetics equations. The second factor is the current advance in molecular biological techniques and the development of numerous useful vectors. This has enabled microbiologists to change the protein content in a given organism and alter its enzymatic profile, enhancing the synthesis of specific end-products or intermediates. The combination of these two factors permits the modification of the metabolic structure and the improvement of the synthesis rate of some desired metabolites in an organism.In the last decade, many researchers have used model-based optimization strategies to analyze and manipulate metabolic networks -12. The Selecting a proper genetic manipulation strategy for metabolic network optimization problem is a tedious task. The regulatory structure of metabolic networks can be determined by model-based optimization strategies ,16. ReseSaccharomyces cerevisiae and amino acid synthesis rates in Escherichia coli were employed to evaluate the applicability of the GFMOOP. Suitable membership functions are used to quantify the resilience effects and cell viability constraints. Results show that the maximum synthesis rates of target products by genetic interventions are always over-estimated in metabolic networks that do not consider the resilience effects.This study introduces a generalized fuzzy multi-objective optimization problem (GFMOOP) to determine the optimal enzymatic manipulations for metabolic network systems considering resilience effects. This study first formulates a multi-objective optimization problem that simultaneously considers the resilience effects and minimum set of manipulated enzymes by combining the concepts of MOMA and ROOM into an optimization framework. Since nonlinear kinetic models offer a more detailed description of metabolic networks than stoichiometric models and the gene manipulations, including gene repressions and over-expressions, in metabolic networks can directly correspond to the changes of maximum flux parameters and reaction rates in the kinetic models, this study uses a nonlinear kinetic model in the optimization formulation. Integer variables are also introduced to model gene over-expression and repression. Thus, the optimization formulation must be solved using mixed-integer nonlinear programming (MINLP) methods. The metabolic networks for ethanol production by Each following example solves two optimization problems and compares their results. The first problem is a primal optimization problem for determining the optimal enzyme manipulations, corresponding gene over-expression or repression, in metabolic networks without considering cell viability and metabolic adjustment. The second problem is the fuzzy optimization problem, which is similar to the primal optimization problem, but considers cell viability and metabolic adjustment.S. cerevisiae is still the most important microorganism for ethanol production to date. Researchers have developed many strategies to enhance ethanol productivity using yeast, and its metabolic network is well studied. Figure S. cerevisiae for anaerobic ethanol production. Curto et al. developed a GMA model for analyzing the anaerobic ethanol fermentation of S. cerevisiae at steady state  and [pep], and inhibited by [atp] so that the improved ratio of PYK vto its basal value can be expressed asTable PYKv, increases when the concentrations of metabolites [f6p] or [pep] increase and the concentration of metabolite [atp] decreases. Figure PYKv. This result makes sense from a biological viewpoint, since a lower [atp] level slows down cell growth and allows yeast to carry out the anaerobic fermentation required to produce ethanol. Although the exponent of [f6p] is positive, the small value causes an insignificant effect on the improved ethanol flux ratio. As a result, the maximum value of PYK vobtained by modulating enzymes HXT and TDH exceeds that by modulating enzymes HXT and PFK. Following the similar procedures, the best selected enzymes to be modulated for the resilience problem when at most three enzymes can be modulated are HXT, TDH, and ATPase. These results differ from those obtained from the primal optimization problem, as Tables The exponents in equation 1) indicate that the ethanol synthesis flux,  indicateEach rate equation contains some metabolite concentration information so that the kinetic model is able to account for important regulatory features. Such regulations depend on the model validity because model parameter values are estimated from a normal condition. As a result, the kinetic model can cope with dynamics of metabolic networks under a small perturbation on enzyme level. However, an optimal enzyme manipulation problem may yield an over-estimation productivity if a large-scale change for each enzyme level is allowed. For instance, this study sets the perturbation region for each enzyme to expand/shrink 5-fold based on its basal value. Such a large-scale perturbation may reach beyond the regulation capability of kinetic models. The resilience phenomena can be applied to compensate such a gap. We also solve the primal and resilience optimization problems using two another smaller perturbation regions for each enzyme, i.e., the lower and upper factors E. coli. The kinetic model for the network is complex and highly nonlinear. Chassagnole et al. developed a nonlinear dynamic model for part of central carbon metabolism of E. coli /[nadh] and [nad]/[nadh] in the redox metabolic network are equal to the ratios of their basal levels and can be considered as constants [In this metabolic system, we assume that concentrations of the seven co-metabolites are constant in the mathematical model. These adenine nucleotides stoichiometrically join in all of the metabolic networks of a living cell. In practical, they actively participate in the dynamics of the network and cannot be ignored. The value of the adenylate energy charges in onstants . Their ri care constants. This study also evaluated the effects of the assumption of constant co-metabolite concentrations by applying equations (2)-(4) to the optimization problems. Similar procedures were also used to obtain the optimal modulations for the primal and resilience optimization problems. The computational results can be found in Tables S1 and S2 of Additional File where The optimization of biological systems, which is a branch of metabolic engineering, has generated a lot of industrial and academic interest for a long time. The ultimate goal of this optimization is to find the optimal mutation strategy for improving productivity. Model-based optimization strategies have been applied to analyze and design metabolic networks during the last decade. The accuracy of optimization results depends heavily on the development of essential kinetic models of metabolic networks. Kinetic models can quantitatively capture the experimentally observed regulation data of metabolic systems and are often used to find the optimal manipulation of external inputs. To address the issues of optimizing the regulatory structure of metabolic networks, it is necessary to consider qualitative effects, e.g., the resilience phenomena and cell viability constraints. Combining the qualitative and quantitative descriptions for metabolic networks makes it possible to design a viable strain and accurately predict the maximum possible flux rates of desired products.S. cerevisiae and E. coli. The resulting optimal enzymatic manipulations for metabolic networks and the maximum flux ratios for desired metabolites are more justifiable based on biological knowledge. We could qualitatively speculate the priority of modulated enzymes obtained by iteratively solving the optimization problems using various allowable numbers of manipulated enzymes. These results can help microbiologists make a proper decision when genetically modifying a microorganism.This study introduces a generalized fuzzy multi-objective optimization approach to determine the optimal enzymatic manipulations for metabolic network systems to obtain the maximal flux ratios of desired metabolites of interest. The goal of this optimization problem is to find the maximal synthesis rates of desired metabolites and the minimum set of manipulated enzymes simultaneously based on kinetic models. The kinetic model was directly used for the optimization problem and MINLP solvers were applied to determine which enzymes to manipulate and how their corresponding activities changed. This study applies fuzzy equal and fuzzy inequality operations to the optimization problem to deal with the resilience phenomena and cell viability constraints. This study tests the practical utility of the proposed approach by applying it to two metabolic networks of The dynamics of a metabolic network can be represented generically using a set of nonlinear ordinary differential equations with the following structure:x ∈ ℝn is a vector of concentrations of metabolites or pools, e ∈ ℝm is a vector of enzyme levels corresponding to the enzyme activities, θ ∈ ℝp is a vector of parameters, v ∈ ℝm is a vector of reaction rates, and S ∈ ℝn×m is the stoichiometric matrix describing the interconnecting fluxes. The stoichiometry of biochemical reactions is constant. Kinetic aspects are used to capture the dynamics of a system and may change rather quickly as they are driven by the state of the system. The stoichiometry of a biochemical pathway determines the wiring diagram of the network, describes which fluxes enter or leave which pool, and ensures that mass is conserved in the process. The reaction rate can be expressed by the power-law functions or Michaelis-Menten-based rate laws in the field of biological systems.where The model outlined above can be combined with mathematical programming to support microbiologists in the biotechnological improvement of microorganisms in industrial applications . The optth iflux i v∈ v, ΣO ∈ ℕr is the set of indices of production rates to be maximized, r is the number of target fluxes to be maximized, and the binary variable j y∈ y indicates whether the th jenzyme should be modulated and is defined aswhere Equation (6) is a general formula for simultaneously maximizing a set of metabolite synthesis rates. Several researchers have introduced genetic manipulations to redistribute various metabolic fluxes in a metabolic network to enhance the desired synthesis rates ,27. EquaThis study includes the following additional constraints for each enzyme in the metabolic networks.i is not modulated, then its activity can have a small variance around its basal value th iun-modulated enzyme due to other enzyme alterations. A similar non-significant variance in enzyme activity discussed in the assumption of ROOM [where  of ROOM . Constraith metabolite i x∈ x.where An abnormally high protein or intermediate concentration in a metabolic system renders a cell non-viable. This is because the burden on cellular metabolism is too high for the cell to survive or the cellular osmolarity constraint is violated. Several researchers have introduced a constraint on the total enzyme concentration to overcome this issue and assure that it never reaches a value that is considered unacceptable for the cell viability ,6,10,13.x and ζe are the restriction factors for the constraints on total metabolite concentration and total enzyme concentration, respectively.where ζThe primal multi-objective optimization problem formulated by equations (5) to (12) is a multi-objective mixed-integer nonlinear programming problem. Many methods are capable of solving multi-objective optimization problems (MOOPs) to obtain the Pareto front -32 and gε-constraint, one of the generating methods, retains only one of the objective functions as the criterion and converts the others into inequality constraints. This approach is suitable for the MOOP with objective functions, which can easily assign the ε-values. The weighted infinite norm method is a reference-goal method that can conveniently determine a trade-off solution if the lower and upper bounds of each objective value are known in advance. This study combines the ε-constraint method and weighted infinite norm method to solve the primal multi-objective optimization problem. The objective function in equation (7) can be straightforwardly converted into an ε-constraint because the number of enzymes is an integer value. The primal multi-objective optimization problem can be transformed into a weighted infinite norm problem defined as follows:The original subject toε of the manipulated enzymes in advance, the lower bound i vonly, and the feasible set Ω consists of all feasible solutions that satisfy the material balance equations in the steady state and the constraints in equations (8)-(12).where the user provides the allowable number A strain may reflect resilience phenomena after genetic interventions. This study introduces fuzzy equal and fuzzy max operators to deal with the resilience phenomena and the maximization of target fluxes, respectively. The minimization of the number of enzymes to be modulated is still defined on a crisp domain. The primal multi-objective optimization problem is therefore extended to be a generalized fuzzy multi-objective optimization problem (GFMOOP) that can be expressed as follows:X ∈ ℕn is the set of metabolite indices and ΣE ∈ ℕm is the set of enzyme indices. Here, the symbols, ≿ and ≈ denote a relaxed or fuzzy version of the ordinary inequality "≥" and equality "=", respectively. The fuzzy maximization, "th iflux ratio exceeds its upper bound th iflux ratio is less than the lower bound j xand enzyme activity k eshould be restored to a state that is as close to the wild-type as possible. Equation (18) is the crisp objective function as same as Equation (7).where ΣThe cell viability constraints in equations (11) and (12) are crisp limits, indicating that all cells die when any of the inequality constraints is violated. These constraints are not so strict in practical situations according to the growth patterns and kinetics of cells in culture . In genewhere the symbol "≾" denotes a fuzzy version of the ordinary inequality "≤". Here, ε-constraints, abbreviated as ε-FMOOP, by transforming the crisp objective function to an ε-constraint. To solve the ε-FMOOP, each fuzzy objective functions for maximizing synthesis rate can be quantified by eliciting the following membership function:The objective functions of GFMOOP are defined on the fuzzy and crisp domains. Almost no studies discuss how to obtain an optimal Pareto solution of the GFMOOP. This study converts the GFMOOP into a fuzzy multi-objective optimization problem with i dis a strictly monotonically increasing function for evaluating the degree of satisfaction. The maximum synthesis rate becomes somewhat acceptable if its objective value lies between the lower and upper bounds. The membership function value is zero if the synthesis rate is less than the desired lower bound. Conversely, the grade of membership function is one when the synthesis rate exceeds its upper bound.where The membership function for each fuzzy equal objective function in equation (16) is defined as follows:th jmetabolite, and the user-provided functions where The membership function value is zero if the metabolite concentration exceeds the bounded interval. When the metabolite concentration lies within the bounds, the metabolite concentration should be as close as possible to its basal value. The membership function for the fuzzy equal objective function for each enzyme in equation (17) can be defined as equation (22). The fuzzy inequality constraint in equation (19) can be quantified by eliciting the following membership function:where Sakawa proposed five types of membership functions to evaluate membership grades: linear, exponential, hyperbolic, inverse, and piecewise linear functions . For conε-FMOOP can be expressed as the goal attainment problem:Having elicited the membership functions for fuzzy objective functions, fuzzy equal objectives, and fuzzy inequality constraints, the O ∪ ΣX ∪ ΣE, and D ηdenotes an aggregation function defined on the crisp domain Ω, which consists of the feasible solutions satisfied equation (5), the crisp bounds in equations (8)-(9), and the ε-constraint in equation (14). Sakawa introduced several aggregation functions in which the value of the aggregation function can be interpreted as an overall degree of satisfaction with user's fuzzy goals [δ is a small positive value within 10-3 - 10-5. The fuzzy goal attainment approach can directly find a satisfactory solution in the Pareto set without yielding the Pareto frontier of the problem.where zy goals . This stThe authors declare that they have no competing interests.WHW implemented the optimization methods, performed the computational experiments, and contributed to the analysis of the experimental data. FSW conceived of the study, participated in its design and coordination, and analyzed the results. WHW and FSW wrote the manuscript. MSC assisted in developing the optimization methods and finalizing the manuscript. All authors read and approved the final manuscript.S. cerevisiaeMathematical model of anaerobic fermentation in . This file includes the set of differential equations and rate equations for anaerobic fermentation in S. cerevisiae, all of the relevant definitions of state and independent variables, and the nominal values of parameters appearing in the differential equations.Click here for fileEscherichia coliMathematical model of central carbon metabolism in . This file includes the set of differential equations and rate equations for central carbon metabolism in E. coli and all of the nominal values of parameters appearing in the differential equations.Click here for fileEscherichia coli considering energy and redox conservation for co-metabolitesResults of multi-synthesis maximization by . This file includes the results of multi-synthesis maximization by Escherichia coli under the conservation of co-metabolites. The suggested modulated enzymes and the optimal synthesis rates considering resilience phenomena or not are shown for comparison.Click here for file


S. cerevisiae BY4709 (wild type) cultured on 3 different C-sources  and sampled along 10 time points with 5 biological replicates. In order to analyze the dynamic mass spectrometry data, we developed the novel analysis methods that allow us to perform correlation analysis to identify metabolites that significantly correlate over time during growth on the different carbon sources. Both positive and negative electrospray ionization (ESI) modes were performed to obtain the complete information about the metabolite content. Using sparse principal component analysis (Sparse PCA), we further identified those pairs of metabolites that significantly contribute to the separation. From the list of significant metabolite pairs, we reconstructed an interaction map that provides information of how different metabolic pathways have correlated patterns during growth on the different carbon sources.Metabolic footprinting offers a relatively easy approach to exploit the potentials of metabolomics for phenotypic characterization of microbial cells. To capture the highly dynamic nature of metabolites, we propose the use of dynamic metabolic footprinting instead of the traditional method which relies on analysis at a single time point. Using direct infusion-mass spectrometry (DI-MS), we could observe the dynamic metabolic footprinting in yeast  During the past years the complexity and dynamics nature of the metabolites have become increasingly apparent +.The DI-MS analysis showed that amino acids were best detected in positive mode, probably due to the amino group. The typical base peak ion was [M + H]After conversion and alignment of the data according to Hansen and Smedsgaard , each ofThe first cherry-picking criteria identified a list of single metabolites that showed the proper change over time. After this step, the number of columns in the matrix was reduced to 145 metabolites. Two examples of identified metabolites can be seen in the top two figures in R2, and the P value of the regression line for each metabolite were stored in a matrix of dimension as the number of metabolites.As the next criteria, the degree of covariance over time was calculated. In order to evaluate the information present in the matrix of metabolite pairs, we initially analyzed the variation in yields by PCA to identify the key metabolites which are significantly different between the sample groups and to how well they can separate growth on the different carbon source from each other.The upper triangular part of the matrix of the yields was extracted and unwrapped. Based on the matrix columns as shown with the cherry-picking method , we calcSparse PCA was used to obtain sets of Sparse vectors for weights (loadings) in the linear combinations while explaining most of the variance present in the data. As can be seen from the Sparse PCA, a few pairs of metabolites can separate the data into distinct carbon sources. Next, we wish to investigate those metabolites in further detail. From the reconstruction of the correlation networks it was found that pairs from PC1 are most important as they generate the largest network. PC1 segregates the signal from glucose from the other two carbon sources (ethanol and glycerol). The size of key metabolites network represents the level of differences in dynamic pattern among the sample groups , 21.m/z there are three potential candidates for the corresponding metabolite, namely, 3-(4-hydroxyphenyl) pyruvate, glucose, and myo-inositol. In this particular case glucose is the most likely metabolite and it was chosen, but cases where there could not be made a clear assignment both metabolite names were used.Pairs of MS ions that correlated over time were identified as those the Sparse PCA were assigned weights. Only the metabolite pairs from the loadings corresponding to the Sparse PC1 and Sparse PC2 were extracted. There were a total of 48 pairs, which corresponds to correlation between 44 ions as key metabolites: 24 pairs between 17 metabolites for PC1 and 3 pairs between 3 metabolites for PC2 in positive ESI mode see , and 17 PC1 ions primarily separate the glucose samples from the ethanol and glycerol samples and PC2 mainly separates the ethanol samples from the glycerol samples. The metabolite pairs from PC1 represent a network as illustrated in The PCA results, both ordinary PCA, and Sparse PCA showed that the main variance in the data is caused by the 3 different C-sources: glucose, ethanol, and glycerol. The Sparse PCA finds sets of Sparse vectors for use as weights in the linear combinations while still explaining most of the variance present in the data. It is built on the fact that PCA can be written as a regression-type optimization problem, with a quadratic penalty; the lasso penalty (via the elastic net) can then be directly integrated into the regression criterion, leading to a modified PCA with Sparse loadings . AccordiThe key metabolites network on PC1 which contain more metabolite nodes compared to that on PC2 show the higher difference between glucose and ethanol or glucose and glycerol cultures while there are less differences between glycerol and ethanol cultures. In other words, the dynamic pattern of the glucose culture is more unique than the rest. So, we can obviously see that there are largest dynamic changes in metabolite concentrations during growth on glucose, which can be explained by the fact that glucose is a fermentative carbon source whereas there is purely respiratory growth on ethanol and glycerol.The 4 main hubs of the network are glucose, cyclic AMP, cyclic dAMP, and nicotinate-D-ribonucleotide (NAMN) as illustrated in the key metabolites network on PC1 .Cyclic AMP and cyclic dAMP have been shown to regulate many different nutrient responses. The levels of these metabolites are linked to the biosynthesis of many amino acids and also affected by the concentrations of glucose and also amino acids . Since fSince low cAMP concentration stimulates the uptake of and L-leucine , we can In conclusion we here show that through dynamic metabolic footprinting it is possible to identify correlations between metabolites that can be used to provide new insight into possible regulatory structures. Furthermore, our correlation analysis may be used for identification of key biomarkers for specific phenotypes during dynamic growth conditions.


Escherichia coli, and they revealed that many chaperonin substrates are metabolic enzymes. This result implies the importance of chaperonins in metabolism. However, the relationship between chaperonins and metabolism is still unclear.Chaperonins are important in living systems because they play a role in the folding of proteins. Earlier comprehensive analyses identified substrate proteins for which folding requires the chaperonin GroEL/GroES (GroE) in We investigated the distribution of chaperonin substrate enzymes in the metabolic network using network analysis techniques as a first step towards revealing this relationship, and found that as chaperonin requirement increases, substrate enzymes are more laterally distributed in the metabolic. In addition, comparative genome analysis showed that the chaperonin-dependent substrates were less conserved, suggesting that these substrates were acquired later on in evolutionary history.This result implies the expansion of metabolic networks due to this chaperonin, and it supports the existing hypothesis of acceleration of evolution by chaperonins. The distribution of chaperonin substrate enzymes in the metabolic network is inexplicable because it does not seem to be associated with individual protein features such as protein abundance, which has been observed characteristically in chaperonin substrates in previous works. However, it becomes clear by considering this expansion process due to chaperonin. This finding provides new insights into metabolic evolution and the roles of chaperonins in living systems. Understanding metabolic activities in the body is important because metabolism is responsible for physiological functions and thus maintaining life. Metabolism can be defined as a series of chemical reactions involving enzymes as catalysts, and these reactions are often represented as a network  -3. In reProtein folding is an important aspect of metabolism because normal metabolism requires proper functioning of cellular enzymes . According to Anfinsen's dogma , the uniEscherichia coli, the chaperonin GroEL, together with its cofactor GroES, acts a chaperone system, which assists in protein folding in this organism, and is essential under several growth conditions (temperatures)  /2 indicates the maximum number of possible edges that can be drawn among i kneighbors.The clustering coefficient of node ki - 1)] ,47, wherIn Figure http://www.genome.jp/kegg/ko.html); thus, it is similar to the Clusters of Orthologous Group (COG) database [The degree of conservation is calculated based on the KEGG orthology (KO) database . The KO database . Howeveri/SStotal, where i Scorresponds to the number of species possessing at least 1 orthologous gene for the gene i coding the chaperonin substrate. Stotal denotes the total number of species that are available in the KO database, which is 1,368 (as of 19 January 2011).The degree of conservation is simply defined as The authors declare that they have no competing interests.KT designed this research, analyzed the data, and drafted the manuscript. TN and HT provided suggestions for data analysis, and they discussed the results with KT. All authors read and approved of the final manuscript.Description for each enzyme node and metabolic network data. This Excel file contains 2 sheets. One sheet includes the gene identifiers (b-numbers), the chaperonin substrate class, and the network measures, for each enzyme node. The other sheet includes the edge list  for the metabolic network of Escherichia coli.Click here for file


Signal transduction and gene regulation determine a major reorganization of metabolic activities in order to support cell proliferation. Protein Kinase B (PKB), also known as Akt, participates in the PI3K/Akt/mTOR pathway, a master regulator of aerobic glycolysis and cellular biosynthesis, two activities shown by both normal and cancer proliferating cells. Not surprisingly considering its relevance for cellular metabolism, Akt/PKB is often found hyperactive in cancer cells. In the last decade, many efforts have been made to improve the understanding of the control of glucose metabolism and the identification of a therapeutic window between proliferating cancer cells and proliferating normal cells. In this context, we have modeled the link between the PI3K/Akt/mTOR pathway, glycolysis, lactic acid production, and nucleotide biosynthesis. We used a computational model to compare two metabolic states generated by two different levels of signaling through the PI3K/Akt/mTOR pathway: one of the two states represents the metabolism of a growing cancer cell characterized by aerobic glycolysis and cellular biosynthesis, while the other state represents the same metabolic network with a reduced glycolytic rate and a higher mitochondrial pyruvate metabolism. Biochemical reactions that link glycolysis and pentose phosphate pathway revealed their importance for controlling the dynamics of cancer glucose metabolism. Growth factors and nutrients are required for cell growth and proliferation in multicellular organisms. As a consequence of growth factors withdrawal normal cells undergo apoptosis, while most transformed cells escape the regulatory mechanisms and acquire the ability to proliferate even in the absence of growth signals Edinger, . In bothThe correlation between signal transduction pathways and cellular metabolism is mediated by some key components of the growth factor-induced cascades; typically these elements are protein kinases at the core of physiology and disease. Several growth factor-induced signal transduction pathways have been characterized so far and, in particular, the phosphoinositide 3-kinase (PI3K) is a key component downstream of the receptor tyrosine kinases , are available for metabolic pathways. Therefore, these data can be carefully integrated to fill the lack of information, so that models can be optimized by means of various simulation and analysis tools  of the glucose metabolism: the glucose transport, the glycolytic reactions, the lactic acid production, the glycogen synthesis/degradation, and the pentose phosphate pathway  that consists in 25 ODEs and 3 algebraic equations to calculate ADP, NADH, and NADPH concentrations see . As prev2 and H2O through the tricarboxylic acid cycle  of the corresponding process, because this quantity is directly proportional to the total protein concentration  regulates the protein concentration, we increased (decreased) the value of the maximum rate of the forward reaction  on the model was performed to determine how robust the steady states under analysis were in relation to local perturbations in the parameter values. These sensitivity indexes represent a measure of the degree to which the steady states H and L are sensitive to the value of individual parameters. A high sensitivity value indicates that changing a given parameter determines significant changes to the considered dynamics. The averages (median) of the sensitivity indexes, of all the 29 adjustable parameters, are respectively 3.198e-03 and 7.320e-04 for conditions H and L , while the other (condition L) represented the same metabolic network with a reduced glycolytic rate, a reduced lactic acid production, but a higher MPM, as reported in literature in relation to a lower activity of PI3K/Akt/mTOR . Nevertheless, considering that model definition is related to the questions the modeler wants to answer, current models can be a useful tool for directing experimental efforts on a subset of all possible hypotheses. In relation to the crucial steps that emerged in this work for the dynamics of the metabolic network, it is worth to mention that, recently, several authors have recognized G6PDH and TKL as potential anticancer targets  workstations equipped with two 2.50 GHz INTEL(R) Quad-core Xeon(R) E5420 processors and 10 GB RAM. The results obtained were displayed using MATLAB.The DAE system representing the metabolic network was numerically integrated using MATLAB (2008b) and the stiff ode solver f1(x), f2(x)] were defined asf1) or enzyme Vmf (for f2), xi is the corresponding value used in the model, N is the number of elements (metabolites or enzymes), J* is the vector of experimental values of enzyme fluxes and J(x) are the respective model predictions obtained using x. The multi-objective optimization problem was solved using the Non-Dominated Sorting Genetic Algorithm II  Monte Carlo (MC) sampling.To overcome this issue, especially considering that the biological quantities represented with parameters  are subject to significant variations, in our study we carried out the SA calculating the DGSMs , n = 29, k = , pk ∈ , where each pk represents a possible perturbations of all the 29 Vf reference values (p) reported in Table quasi-random low discrepancy sequences . Hence, for each of the two conditions H and L, we obtained a total of 500 29-by-29 matrices of elementary effects that were used to calculate the 29-by-29 matrices of scaled sensitivity indexesij and sij are, respectively, the mean and sample standard deviation of the elementary effects ijke.To compute the DGSMs we generated, for each of the conditions H and L, a set of parameterizations s Sobol, . The parijr =  is the rank of the scaled sensitivity index ijg when ordering the elements  from the highest (strong influence of parameter j on flux i) to the lowest, and the superscript L or H indicates the condition. Hence, ijR will be positive (negative) for Vf parameters (and the corresponding reaction) exerting a higher (lower) control in condition H in relation to condition L; ijR will be higher for reactions having a higher control in both conditions.The normalized differential ranking of the reactions was calculated asijg in relation to the median i’m of all the scaled sensitivity indexes related to the same Vf parameter .The relative log sensitivity was calculated asijs for steady state flux i to perturbation of Vf of reaction j was obtained by calculating the elementary effects ijke using the respective values Vf listed in Table pk = p.For each of the two conditions H and L, the 29-by-29 matrix of the first-order local sensitivity indexes −6 and δ = 10−6; steady state fluxes iJ were obtained by numerical integration of the DAE system within the interval , followed by the identification of the steady state (see above the N(n + 1) numerical solutions of the model were required, where N = 500 represents the number of parameterizations, n = 29 is the number of the parameters, and the factor 2 is due to the two conditions H and L.We used ε = 5 · 10The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Mathematically, the sensitivity of FBA to these flux imbalances is captured by a native feature of linear optimization, the dual problem, and its corresponding variables, known as shadow prices. First, using recently published data on chemostat growth of Saccharomyces cerevisae under different nutrient limitations, we show that shadow prices anticorrelate with experimentally measured degrees of growth limitation of intracellular metabolites. We next hypothesize that metabolites which are limiting for growth (and thus have very negative shadow price) cannot vary dramatically in an uncontrolled way, and must respond rapidly to perturbations. Using a collection of published datasets monitoring the time-dependent metabolomic response of Escherichia coli to carbon and nitrogen perturbations, we test this hypothesis and find that metabolites with negative shadow price indeed show lower temporal variation following a perturbation than metabolites with zero shadow price. Finally, we illustrate the broader applicability of flux imbalance analysis to other constraint-based methods. In particular, we explore the biological significance of shadow prices in a constraint-based method for integrating gene expression data with a stoichiometric model. In this case, shadow prices point to metabolites that should rise or drop in concentration in order to increase consistency between flux predictions and gene expression data. In general, these results suggest that the sensitivity of metabolic optima to violations of the steady state constraints carries biologically significant information on the processes that control intracellular metabolites in the cell.Stoichiometric models of metabolism, such as flux balance analysis (FBA), are classically applied to predicting steady state rates - or fluxes - of metabolic reactions in genome-scale metabolic networks. Here we revisit the central assumption of FBA,  Cellular metabolism is composed of a complex network of biochemical reactions that convert environmental nutrients into biosynthetic building blocks and energetic currency. Genome-scale mathematical models of metabolic networks focus largely on trying to predict the rates – or fluxes - of these reactions. By assuming that the concentrations of intracellular metabolites are at steady-state , and invoking optimality, these constraint-based methods for modeling metabolism have offered abundant insight into how metabolic flux is routed through the cell. Here we ask how cellular growth would respond to deviations from steady state  of every possible intracellular metabolite. This question can be addressed through a sensitivity analysis inherent to linear optimization theory, known as duality. We show how some features of metabolite concentrations, such as their growth-limitation and their transient response, are captured by this sensitivity analysis. Our results suggest that, in addition to predicting fluxes, stoichiometric models offer a valuable route towards probing the metabolites themselves and their relevance to growth dynamics. Cells endure relentless variations in intra- and extra-cellular conditions. These perturbations propagate through the cell's metabolic and regulatory networks, leading to a diverse range of interdependent, transient responses in the abundance of metabolites, transcripts, and proteins One approach to this question is to use genome-scale, constraint-based models of metabolism  problem exchange reactions, i.e. source/sink reactions mediating the interaction between a cell and its surrounding environment. A canonical FBA calculation can be formally expressed as the following primal LP problem:S is the m (metabolites) by n (reactions) stoichiometric matrix, v is the vector of metabolic fluxes, LBv is a vector of lower bounds for all fluxes, UBv is a vector of upper bounds for all fluxes, b is the vector of the rates of accumulation/depletion of each metabolite, and c is the vector defining the contribution of different fluxes to the objective function.Flux balance analysis is a method for computing expected reaction rates in complex metabolic networks, and has been described in detail elsewhere ib in ib coefficients. Such a deviation implies a flux imbalance at metabolite i, and hence its accumulation or depletion. Importantly, this interpretation of Z. How can one quantify the sensitivity of the objective function to such flux imbalances? What is the biological significance of these sensitivities?For intracellular reactions, the right-hand-side coefficients shadow prices, iλ) specifically capture the change in the value of the objective function upon a unit change in the right-hand-side of a single constraint (ib). The general formulation of the dual problem can be found in any linear optimization textbook . In this case, a shadow price corresponds to the change in the biomass flux when one of the intracellular metabolites deviates from steady state. Importantly, if a metabolite has a negative shadow price, this means that allowing additional outflow from this metabolite (so that ib<0) will increase the maximal value of the biomass flux, implying that this metabolite is limiting for the biomass objective M  and phosphate limitation , where there were substantially more data points .We compared the growth-limitation measurements for each metabolite identified as significantly growth-limiting or non-growth-limiting/overflow in i.e. predicted to be growth-limiting) even though its concentration was experimentally observed to fall with increasing growth rate. The authors of strength of growth-limitation (as quantified by the magnitude of the shadow price) plays a role in the extent to which these metabolites regulate the rates of enzymatic reactions.In agreement with ρ = −0.87, p-value 2×10−8, Pearson r = −0.69, p-value = 8×10−5) and (ii) all three nutrient limitation conditions, together with auxotrophies . The results also remained valid when we only considered cytosolic metabolites, rather than metabolites from all compartments (see i.e. either zero or negative) could be used as a predictive binary classifier for whether a metabolite is growth-limiting or non-growth-limiting. To do so, we calculated the Matthews Correlation Coefficient (MCC) −8) and only cytosolic metabolites .We repeated the statistical analyses above for two “lumped” datasets containing data from (i) all three natural nutrient limitation conditions  re-calculations of the sensitivity of the objective function to deviations from each steady-state constraint. We then compared the incremental shadow price, decremental shadow price, and the shadow price obtained directly from the LP solver, and found no instances of degeneracy in our shadow price calculations.An important question in the above analysis, and in the calculation of shadow prices in general, is whether the possible alternative optima in the FBA optimization problem could give rise to degenerate shadow prices, and hence ambiguity in the comparison with experimental data. As described in detail in the nt bi in  is incrent bi in . The shaOur results so far indicate, in line with our intuition and with prior work on duality in FBA Escherichia coli to sudden perturbations which demonstrated that the growth rate of cells responds remarkably quickly to changes in environmental conditions. In two experiments E. coli lead to a 3.7-fold increase in growth rate less than a minute. Similar results were observed in Given the metabolite-specific associations between shadow prices and growth-limitation, we decided to investigate whether shadow prices could also aid in understanding other features of intracellular metabolites. In particular, we reasoned that if a metabolite is truly growth-limiting, then its concentration in the cell should be tightly controlled. If, in contrast, a growth-limiting metabolite's concentration is allowed to fluctuate or vary uncontrollably, this temporal variability would eventually propagate to growth rate and have potentially deleterious consequences. Our reasoning was further bolstered by recent studies of the metabolic response of E. coli for different glucose and nitrogen perturbations CV, the standard deviation of the time series, divided by its mean; see Based on our reasoning and on the two studies in a priori reason to expect linear correlations. The correlations were statistically significant for nitrogen upshift , as well as the 4 different carbon perturbations . These correlations were further substantiated using non-parametric permutation tests, described in the The results of our analysis are shown in i.e., metabolites with relatively large, negative shadow prices and high temporal variation) appeared in our results. Among the outliers under glucose limitation  for each of the two knockouts. As illustrated in i.e. becoming more growth-limiting). The most drastic changes were found for lipids and precursors, like undecaprenyl phosphate, and UDP-D-glucoronate, both of which showed a drop in shadow price of 1.29. In contrast, the ΔGDH knockout featured no metabolites with a substantial (greater than 0.1) drop in shadow price. This absence of new growth-limiting metabolites in ΔGDH is consistent with the observation in increase in shadow price in ΔGDH (corresponding to a relaxation in growth-limitation). Thus, while the “growth-limitation landscape” of the ΔGDH mutant, characterized by the most growth-limiting metabolites in the model, seemed relatively similar to that of the wild-type, the ΔGOGAT strain displayed substantially different growth limitations.To further corroborate our findings, we tested whether the differential dynamic behavior of mutant knockout strains could be captured through our analysis. We used additional metabolite time series available for the wild type and two knockout strains (ΔGOGAT and ΔGDH) of Second, we tried to recapitulate the primary qualitative finding of the knockout study from E. coli , its general validity and mechanistic basis across different organisms and types of perturbations will require further scrutiny and will be an important aspect of future work. In particular, the availability of specific mechanistic models for the metabolic response to perturbations, coupled with higher temporal resolution data, would allow one to obtain more precise estimates of temporal variability, and hence better quantitative comparisons with shadow prices.Thus, the shadow prices associated with individual intracellular metabolites provide information not only about the extent to which each metabolite is limiting for growth, but also about its overall temporal variation following a perturbation. Importantly, in the current framework, shadow prices do not provide quantitative predictions about the speed at which metabolites respond, or the new steady-state concentrations they reach. Hence, shadow price analysis should not be treated as a substitute to explicit predictions from kinetics. While our results seem to hold across different experiments in i.e. the specific objective function used in the FBA optimization. Does the analysis of shadow prices have a meaning and an application for stoichiometric problems with radically different objective functions, or is it biologically interpretable only for the growth maximization objective? To answer this question, we decided to explore the significance of shadow prices in a recently proposed optimization problem aimed at identifying genome-scale fluxes that minimize the inconsistency relative to a given set of gene expression data. This approach, pioneered with the GIMME algorithm Inconsistency Score, IS) is minimized across the entire metabolic network c is a vector of reaction penalties, and the reaction flux RMFv is a required metabolic functionality (RMF), some minimal, user-defined metabolic behavior which the model must reproduce . One of the reasons the RMF constraint is imposed is to avoid the trivial solution v = 0.So far, we have corroborated the notion that shadow prices are indicative of growth limitation, and demonstrated that shadow prices are even more broadly related to metabolite dynamic variability. As described above and in the the change in the inconsistency score IS i deviates from zerowhen the steady-state constraint on one metabolite b. It is reasonable to assume that some portion of the inconsistency between experimentally measured gene expression and TEAM's flux predictions is the result of imposing the steady state assumption in our model, while a metabolite may be truly accumulating or depleting during certain time intervals in the experiment. Allowing such a metabolite to violate the flux balance condition (either accumulate or deplete) should lower the inconsistency score. Then, if a metabolite's abundance is decreasing, we should expect the shadow price to be positive (Shadow Price = negative change in IS/negative change in abundance). Conversely, if the metabolite's abundance is increasing, then we should expect the shadow price to be negative (Shadow Price = negative change in IS/positive change in abundance). As illustrated in As depicted in ρ = −0.41, p-value = 9×10−6; Pearson r −0.31, p-value = 8×10−4; e.g. increase or decrease) in 55 of 63 metabolites in the dataset. We used this data in combination with the Matthews Correlation Coefficient . Interestingly, among the incorrect predictions, many were for amino acids . The failure of TEAM's shadow prices to predict changes in abundance for these compounds suggests that inconsistency with gene-expression data in pathways utilizing these metabolites may not be due to flux imbalances, and may instead indicate that other regulatory mechanisms are at play.To test whether TEAM's shadow prices indeed could predict changes in intracellular metabolite abundances, we re-analyzed a transcriptomics θ , within the range θ = 0.50–0.73 and θ = 0.78–0.88. This range of thresholds is substantially larger than the range of θ's found to accurately recapitulate experimental data in our studies of Shewanella oneidensis using TEAM (θ = 0.65 to θ = 0.72) θ. Thus, our analysis of flux imbalances in TEAM, a constraint-based approach based on an objective function radically different from the classical growth maximization of FBA, reveals that shadow prices have useful applications beyond conventional flux balance methods.Using the same sensitivity analysis developed in Constraint-based stoichiometric models of metabolism have become a widely used approach for characterizing and predicting cellular metabolic states The results we have presented may seem at first glance surprising. How can a steady state solution convey information about the dynamical changes of metabolite pools? The answer is that flux balance models are not simply steady state solutions to a dynamical system. Rather, they use constraints and optimality to predict how a cell should allocate its resources for maximal efficiency, given the underlying network architecture. It would be tempting to make the leap of inferring that the architecture itself truly constrains the dynamics, independent of parameters and regulation. Rather, we suggest that the stoichiometric architecture may dictate how regulation should evolve to guarantee robustness to temporary variations in the intracellular milieu. If the cell cannot allow itself to accumulate or deplete certain metabolites, without incurring a substantial penalty to growth, then the response to variations in these metabolite pools should be swift. This suggests that quick allosteric and post-translational metabolite-induced regulatory feedback should control the stability of these pools i.e. by moving to a different corner of the feasible space). In future research, it would be interesting to directly assess the potential existence of such discontinuities in the dynamical behavior of a perturbed metabolic network. In addition, the magnitude of the range of validity of a shadow price may be thought of as an additional tolerance metric for each individual metabolite, conveying the scale beyond which its response to a perturbation becomes unpredictable. Future models may test whether the extent to which a metabolite is regulated depends both on its shadow price, as well as this tolerance to large perturbations.A subtle but potentially important aspect of shadow prices and their biological interpretation in metabolic network models is the fact that they are defined only over a certain range, as dictated by the structure of the feasible space. These ranges capture how large a perturbation can be before the genome-scale optimal flux distribution changes sharply . In such circumstances, for every metabolite for which appropriate data is available, the right-hand-side of the corresponding steady-state constraint (ie.g. b) could be adjusted accordingly.Additionally, upon availability of comprehensive data on intracellular metabolite concentrations at multiple time steps, one could envisage implementing stoichiometric models that use explicit flux imbalances (rates of accumulation/depletion) as inputs to the constraint-based model. For example, our shadow price analysis with TEAM is readily extendible to cases where the rate of accumulation/depletion is known for one subset of metabolites, but unknown for another set . We associate with each set of constraints in the primal problem a single set of dual variables. For the steady state constraints, we assign variables m, the shadow prices which we use throughout this work), for the constraints on the lower bounds of each flux, we assign variables n), and for the constraints on the upper bounds of each flux, we assign variables n). Then, following any standard text on linear optimization . To do so, we used brute-force techniques to validate that each shadow price reported by the solver was indeed the sensitivity of the objective function to each steady-state metabolite constraint. This process thus simultaneously helped ensure that our results were robust to alternative dual optima.We implemented a number of measures to ensure that each shadow price used in our calculations was accurate and meaningful. In particular, we validated that the shadow prices obtained directly from the LP solver could not take on different values depending on whether a metabolite was accumulating or depleting (+G) and lower (−G) bounds for which these shadow prices are valid. These bounds indicate the maximum that the right hand side of each constraint may be perturbed while still maintaining the validity of each shadow price. First, we ensured that any calculated shadow prices had non-zero ranges of validity (range = +−G−G). Any shadow prices which did not exhibit a minimal range εrange = 10−6 were discarded. Other tested values of εrange in the range 10−3 to 10−6 led to qualitatively identical results.In addition to the primal solution , the Gurobi LP solver provides the corresponding dual solution to the FBA problem. The dual solution contains (i) the shadow price value relative to each metabolite steady-state constraint and (ii) the upper (+ (the change in the objective function when the right-hand-side of a constraint is increased) and the decremental shadow price λ− (the change in the objective function when the right-hand-side of a constraint is decreased) testi) in the model using a perturbation procedure. This calculation was implemented by solving the two following optimization problems, in which the steady state constraint is positively or negatively violated at each individual metabolite:p modulates how large we allow a steady-state constraint to be violated, while remaining in the range  = p can take value between 0 and 13 p = 0.5 # For every metabolite, check for degeneracy in the shadow price of the metabolite by changing one of the steady-state constraints from zero to a non-zero value within the range of validity4 # for i = 1…number of metabolites5 if SPUpRange(i) >0: # if we can perturb up6  Use a temporary variable7   RHSConstraintsPlus = RHSConstraints # Change one constraint8   RHSConstraintsPlus(i) = SPUpRange(i)*p # # NEXT: Solve FBA with new constraint 9   Run_FBA10 [FluxPlus SPPlus SPUpRangePlus SPDownRangePlus OptValPlus] = end #end if11  if SPDownRange(i) <0: # if we can perturb down12  13   RHSConstraintsMinus = RHSConstraints;14   RHSConstraintsMinus(i) = SPDownRange(i)*p;# NEXT: Solve FBA with new constraint 15   Run_FBA16   [FluxMinus SPMinus SPUpRangeMinus SPDownRangeMinus OptValMinus] = end # end if17  Compare manually calculated shadow prices (if they exist) to solver's18  # 19  SPPlus(i) = /SPPlusRange(i)*p20  SPMinus(i) = /SPMinusRange(i)*pif |SPPlus(i) – SP(i)|>tolerance OR |SPMinus(i) – SP(i)|>tolerance21  There is a degenerate shadow price22   return ERROR # end #end if23  end # end for24 Escherichia coli metabolic model In this work, all optimization problems were solved using the Gurobi optimization software E. coli, we used the metabolic network reconstruction iJO1366 For all simulations relating to In order to calculate temporal variation of metabolite, we use the coefficient of variation (CV):In the main text, we show that metabolites with large negative shadow prices exhibit little temporal variation, and metabolites with large temporal variation should exhibit small (or zero) shadow price. To further corroborate the significance of the anticorrelation between shadow prices and temporal variation illustrated in λ) and vector of temporal variation (CV) of each metabolite were calculated. Then, the mean temporal variation (Tm) and mean shadow price (Sm) for the experiment were determined using λ and CV, respectively. We then computed the number of metabolites, originalp, which exhibited a shadow price more negative than Sm and a temporal variation larger than Tm. These metabolites served as a proxy for the number of “incorrect” assignments made by our model.For each experiment, the vector of shadow prices . Then, we identified the proportion of permutations for which i<poriginalp , reported in We generated 10c quantifies the modeler's expectation that a reaction is metabolically active  to an extent that depends on the expression of its constituent genes. The c vector is calculated by assigning a penalty to each gene in the metabolic model, and then propagating these penalties to the reactions using the Boolean gene-to-reaction mapping provided in the model iMM904 ic for reaction i, the higher our confidence that the reaction is inactive. In contrast, reactions with c = 0 are expected to be active and carry flux. Importantly, each element of c is calculated using experimental measurements of gene expression.The penalty vector g in the metabolic model. Gene penalties are determined by comparing the expression value of a gene with a predefined threshold. For each gene g, we created a cumulative distribution function (CDF) of all expression measurements for that gene , we use the CDF to calculate (for each gene) the expression value corresponding to that percentile. This was the gene's penalty threshold gx. For the purposes of this article, the primary difference between TEAM unique penalty thresholds to each gene in the metabolic model, while GIMME assigns a common penalty threshold to each gene. In First, we describe how we assign a penalty to each gene g, gp, is calculated:gEXP is the expression of gene g. Thus, if a gene's expression is above the penalty threshold gx, that gene is assigned no penalty. In contrast, if its expression is found to be below the threshold, then its penalty is equal to the difference between the two. As described in p to a vector of reaction penalties c.Once the gene expression penalty thresholds have been calculated, the penalty for each gene in silico the environmental conditions of the experiments. We decided to use acetate secretion as our RMF flux. To do so, we first used FBA to identify the maximal amount of acetate that could be secreted by solving the optimization problemRMF,minv was set to some proportion p of this maximal secretion rate. We used p = 0.3, although other values of p yielded qualitatively similar results.An essential part of TEAM's formulation is a user-defined required metabolic functionality (RMF). The RMF is a metabolic behavior (such as growth or the secretion of a metabolite) that TEAM must reproduce. It was observed in Because the metabolomics and transcriptomics measurements were obtained from two distinct experiments in which the periods of the cycles were significantly different , we used dissolved oxygen measurements (DO) Click here for additional data file.Figure S1TEAM's shadow prices are predictive of metabolomics measurements for a large range of penalty thresholds. We evaluated the sensitivity of the predictive power of TEAM's shadow prices to the particular choice of penalty threshold θ. We calculated the Spearman correlation between shadow prices and observed changes in metabolite abundance for θ = 1% to θ = 99% . Expression data (hour 35 of (TIF)Click here for additional data file.Table S1Correlations between shadow prices and measures of growth-limitation from . For lumped datasets (e.g. “All Conditions”) we use non-normalized shadow prices (as in the main part of e.g. ATP) exist in several compartments, and the shadow prices in each compartment were used in the statistical test. While in most cases the shadow prices across compartments were identical, there were several instances where this was not the case (see Dataset S1 for data).(DOCX)Click here for additional data file.Table S2Results of permutation testing of shadow prices and temporal variation. For all experimental conditions, fewer than 5% of permuted shadow prices exhibited fewer incorrect predictions than the true shadow prices.(DOCX)Click here for additional data file.Text S1Supporting Text. Contains additional details for the calculations presented in the text.(DOCX)Click here for additional data file.