Optimization methods allow designing changes in a system so that specific goals are attained. These techniques are fundamental for metabolic engineering. However, they are not directly applicable for investigating the evolution of metabolic adaptation to environmental changes. Although biological systems have evolved by natural selection and result in well-adapted systems, we can hardly expect that actual metabolic processes are at the theoretical optimum that could result from an optimization analysis. More likely, natural systems are to be found in a feasible region compatible with global physiological requirements.Saccharomyces cerevisiae. We next introduce a method to identify the feasibility parametric regions that allow a system to meet a set of physiological constraints that can be represented in mathematical terms through algebraic equations. This technique is based on applying the outer-approximation based algorithm iteratively over a reduced search space in order to identify regions that contain feasible solutions to the problem and discard others in which no feasible solution exists. As an example, we characterize the feasible enzyme activity changes that are compatible with an appropriate adaptive response of yeast Saccharomyces cerevisiae to heat shockWe first present a new method for globally optimizing nonlinear models of metabolic pathways that are based on the Generalized Mass Action (GMA) representation. The optimization task is posed as a nonconvex nonlinear programming (NLP) problem that is solved by an outer-approximation algorithm. This method relies on solving iteratively reduced NLP slave subproblems and mixed-integer linear programming (MILP) master problems that provide valid upper and lower bounds, respectively, on the global solution to the original NLP. The capabilities of this method are illustrated through its application to the anaerobic fermentation pathway in Our results show the utility of the suggested approach for investigating the evolution of adaptive responses to environmental changes. The proposed method can be used in other important applications such as the evaluation of parameter changes that are compatible with health and disease states. The emergence of design in biological systems was an unsolved problem until natural selection was established as the driving force for the evolution of such systems -4. At thIn unicellular organisms, adaptive capability is specially important as they lack an internal milieu than could buffer the environmental changes. In this context, the adaptive responses to different stress conditions, , have been extensively investigated using yeast as a biological model -26. In get al. [In this paper we develop a new approach that focuses on identifying feasible parametric regions that contain solutions for system parameters so that a set of physiological constraints are satisfied. With that method we will be able of identifying the possible evolutionary solutions that are expected to contain the actual adaptive response. First, we present a global optimization procedure that capitalizes on the properties of a particular class of nonlinear models, the GMA  models based on the power-law formalism. This method improves the method recently proposed by Polisetty et al.  and takeet al. -30. Secop processes that can contribute to the change in the concentration of the pool of any of the n internal metabolites, the basic mathematical representation isGenerally speaking, a metabolic network can be characterized by its processes, internal metabolites, and control  variables. Details on the regulatory effects (for instance feed-back inhibition) are important for a complete description. Furthermore, kinetic details are required for computing the response of the system to different inputs and environmental changes. For a system with μir is a stoichiometric factor that indicates how many molecules of Xi are produced or used by the process vr; it is a positive integer if the flux r produces Xi and it is a negative integer, if the flux r depletes the pool of Xi.Here, power-law formalism is one of the most convenient. Details for this choice can be found elsewhere [Each velocity can be represented by different functional forms that may include various parameters. The choice of a particular mathematical form is not trivial and can either facilitate or hinder the task of analyzing the optimal design and the adaptive responses to different environmental changes ,32. Fromlsewhere . In thisXj accounts for the concentration of metabolite j, γr is an apparent rate constant for flux r, and frj is the kinetic order of variable Xj in reaction r. Each kinetic order quantifies the effect of the metabolite Xj on flux r and corresponds to the local sensitivity of the rate vr to Xj evaluated at the operating point indicated by the subscript 0, that is:In this representation, Xj has no direct influence on the rate of reaction r, the kinetic order is zero. If it directly activates the flux of reaction r, the kinetic order is positive. If it inhibits the flux of reaction r, then the kinetic order is negative.If γr parameters. This is so because, in most cases, the kinetic-order for the enzyme is 1, as velocities are linearly dependent on the enzyme. Of course, if necessary, enzymes can also be included in the model as control variables. On the other hand, if the model includes dynamic changes in enzymes, for instance through gene regulation, they can also be considered as dependent variables.Without loss of generality, enzyme levels can be considered embedded in the Using this representation, a Generalized Mass Action (GMA) model is defined by expressing each velocity in (1) using its power-law form (2) :(4)m ≥ 0 control  variables. Alternatively, an S-system model is obtained by aggregating the different processes in a net process of synthesis Here, we assume that there are Using the power-law representation for each aggregated process, we obtain:g and h, and the rate constants α and β for convenience. The different parameters of models (4) and (6) can be obtained, for instance, via estimation from dynamic data [In this last formulation, the kinetic orders are called mic data ,35. If tmic data ,36. Furtmic data ,38. GMA mic data . Here weOptimization in biotechnological applications: Given a model that represents the reference state of the system, we are interested in finding the parameter changes (engineering design) that optimize a given objective function .1. Find feasibility regions in evolution studies: Given a model that represents the normal metabolic state of a cell, find the admissible changes at the level of enzyme activities that are compatible with a set of physiological and functional effectiveness criteria (evolutive emergence of design).2. Optimization of biological processes is a very important goal in biotechnological applications -44. In gOptimization based on nonlinear models defined with the power-law formalism was first proposed by Voit . By usinγr in GMA models), so that a given goal is attained (for instance maximize a flux) and a set of constraints are satisfied In practical biotechnological applications, we are interested in obtaining the best set of changes in enzyme levels  problem of the following form:ONLP(S) denotes the optimal objective value over S. The set S is called the set of feasible solutions, and contains all the values of vr, γr and Xj that satisfy a set of constraints on the GMA equations. In matrix notation:where v, γ, X denote the (column) vectors of continuous variables; S may also include additional equations others than those involved in the GMA representation to reflect specific biological conditions. Problem ONLP(S) can also be expressed in a more detailed way as follows:In this representation, S includes the steady-state equations as a basic constraint. Thus, optimization is run to find a steady-state solution that optimizes the objective function and that fulfills the considered constraints. As commented before, the optimization model may also incorporate other equations that impose certain conditions on the values of the variables vr, γr, Xj. For instance, we can consider a constraint X1 + X2 + X6 ≤ k that forces the summation of certain concentrations of metabolites to be lower than a desired upper limit k. This second type of constraints may represent specific physiological properties that the solution sought should satisfy.Here the feasible set U(·) and satisfies the equations  defined upon the biological system. There are currently many strategies available to solve NLP problems like ONLP. These methods are typically implemented in software packages that allow the solution of models with thousands of variables and constraints . UnfortTo overcome this drawback, it is necessary to resort to a specific type of mathematical methods known as global optimization techniques (for a detailed review see ), which ONLP to global optimality. The method introduced relies on hierarchically decomposing the original problem into two levels, an upper level master problem CMILP an a lower level slave problem RNLP, between which the algorithm iterates until a termination criterion is satisfied.In this paper, following this general idea, a novel deterministic global optimization method inspired on the works of Bergamini and co-workers -52 is prONLP , and therefore predicts a valid lower bound on its global optimum. Such a model is constructed by replacing the non-linearities in ONLP by valid over and under estimators. Particularly, in our method, supporting hyper-planes and piecewise linear functions are used to approximate the original search space of ONLP. In the lower level, the original problem is locally optimized in a reduced search space, thus yielding an upper bound on its global solution. The upper and lower level problems are solved iteratively until the bounds converge. For clarity, technical details of the main features of the proposed algorithm are commented in the Appendix section.The master level of our algorithm entails the solution of a mixed-integer linear programming (MILP) problem, which is a relaxation of ONLP. In the master problem, both, the objective function and search space of the original problem are convexified in order to avoid local optima. This is done by introducing auxiliary binary variables that allow to linearize the nonlinear terms. This leads to a reformulated master problem that can be solved to global optimality by standard optimization methods. As discussed on the Appendix, the convexification of the original problem takes advantage of the mathematical structure of the GMA representation. Note, however, that the solution found by the master problem does not necessarily have to satisfy all the constraints in ONLP, since some of them may have been relaxed to reformulate the nonconvexities.Figure Hence, the prediction made by the upper level must be checked in the lower level, so the solution is guaranteed to be feasible in the original search space. Specifically, in our algorithm, the solution of the master model is used as a starting point in the lower level problem, which is solved in a reduced search space that is defined according to the output of the master level. The lower level solution is then employed to tighten the search space of the master problem. As a result, the new modified master problem predicts better lower bounds that are closer to the global solution. Furthermore, tightening the search space of the master problem also improves the quality of both, the starting points and reduced search spaces passed to the lower level. This can be observed in Figure The detailed steps of the proposed decomposition strategy are as follows  satisfies the finalization criterion .• If CMILP and go to step 2.• Otherwise, update the grid in Saccharomyces cerevisiae presented in Polisetty et al.  as a beudy 1 in ). The moThe algorithm was implemented in GAMS interfacing with CPLEX 9.0 and SNOPT 6.2 as main MILP/NLP optimization packages, respectively, on an Intel 1.2 GHz machine. The upper level of the algorithm was constructed using 50 supporting hyper-planes and 9 piece-wise terms. These upper and lower estimators were updated during the execution of the algorithm by defining new linearizations and terms of the piece-wise approximation in the middle points of the active subintervals in the solution of the master problems. The tolerance error  was set to 0.2%.-1 is found when all enzymes can be changed. This is a theoretical situation that, in practice, can be difficult, if not impossible, to implement. In practice, enzyme manipulation may be restricted to only few enzymes. Thus, we explored a more realistic situation and obtain the optimal changes that would be necessary for maximizing ethanol production if only two different enzymes were to be manipulated at once. As in [-1 can be attained. Thus, a realistic strategy may reach about a 66% of the best theoretical solution in that case. In the extreme case that only one enzyme can be manipulated, the best solution is found by increasing glucose intake. This is not surprising as glucose intake is a bottleneck for obtaining a final increase in ethanol production. As illustration of the method performance, we also include in the lower part of Table In the upper part of Table e. As in , our met-1, whereas in [-1. This means that our method assures convergence to the global minimum within an optimality gap of 0.19%, whereas in the solution reported in [In Table ereas in  it is 12orted in  this gapAs a motivation for the feasibility approach that we shall develop in the next section, we shall obtain the optimal enzyme activity changes that would correspond to different goals in the adaptive response of yeast to heat shock. This problem was first analyzed by intensive computation by Vilaprinyo et al. . By usinWe shall obtain the optimal changes that optimize different goals: (i) Maximization of the rate of trehalose synthesis, (ii) Maximization of ATP synthesis, (iii) Maximization of NADPH synthesis, and (iv) Minimization of cost related to changing gene expression. We consider two different scenarios. First, we run the optimization procedure without physiological constraints. Second, we consider the constraints in Table -1 by increasing the activity of HTX, PYK, and TPS+GOL, while decreasing most of the other activities. In doing that, two of the involved metabolites (G6P and F16P) accumulates. Furthermore, synthesis of glycerol is reduced to zero and synthesis of NADPH is decreased. This optimal solution cannot be considered a good adaptive solution as it compromises the cellular inner milieu by accumulating metabolites and challenges other processes by decreasing some critical fluxes. In that sense, this solution optimizes a metabolic goal (vTRE) but does not match other important physiological constraints. A similar result is obtained when the goal is to maximize the synthesis of NADPH or ATP. Particularly striking is the unrealistic concentration of F16P that is obtained in the optimal strategy for maximizing ATP synthesis. In the case of minimization of Cost, as no further constraints are imposed, we find the trivial solution that corresponds to maintain the basal conditions.Trehalose synthesis can be increased up to 2.08 mM min-1. This is well below the maximum obtained without constraints, but now the solution is reasonable in the terms imposed by the physiological constraints. Similar results are obtained for the other objective functions. We also obtain the solution that minimizes the overall cost. In that case, cost can be lowered about a 50% with respect the other cases. Minimization of cost undertakes values of the fluxes that are lower than in the optimal solutions for the other objective functions. From the implementation point of view, it is interesting to notice that in all the cases the algorithm was able to provide near optimal solutions  in few CPU seconds. Similarly, as in the previous example, the master problem provided very tight relaxations, which led to few iterations.Considering the set of constraints identified by Vilaprinyo et al. Table , we run Optimization techniques seek finding the best strategy for changing control variables so that the system can reach a given goal. Thus, such methods are important in biotechnology and metabolic engineering where the scientist fixes the goal and searches for the best strategy in changing the underlying processes. The situation is different if we analyze the evolution of natural systems. As discussed above, natural systems are (in some sense) optimized by the evolutive process by natural selection that acts as a purifying process. Although the exact contribution of this mechanism is still discussed , it is gThis part of the paper introduces a method that aims at providing an approximate characterization of the feasible space of a biological problem rather than identifying a single optimal solution. The tool presented is intended to provide valuable insights on the set of changes in enzyme activities that would be required for adaptation to an environmental challenge.ONLP and exploits the specific structure of the GMA representation. The use of this particular representation allows to perform the feasibility study by slightly modifying the algorithm described in the previous section. Specifically, our strategy relies on solving the original problem ONLP iteratively over a reduced search space. At each iteration the method identifies a region that contains a feasible solution to the problem. This region is then removed from the search space, and the optimization problem is resolved in the reduced domain. The algorithm proceeds in this manner until there is no feasible solution in the remaining regions of the search space.The approach introduced is based on the NLP formulation defined in ONLP. At the lower level, the prediction made by the master problem is checked by solving the original problem in a reduced search space. If a feasible solution is found, then integer cuts are added to the master problem in order to exclude the region containing such a feasible point. Otherwise, the master model is updated by refining its grid, until either a feasible solution is obtained in the lower level or an infeasibility is detected in the higher level problem. The main features of the algorithm are outlined next, whereas more technical details can be found in the Appendix.The overall method, which relies on the global optimization approach introduced before, comprises two different levels. At the upper level, a master problem is solved to identify a region that may contain a feasible solution to q ≠ q') such that their union contains the feasible space S of ONLP  are unfeasible in any hyper-rectangle contained in the search space of the master problem . In our approach, this property is indeed exploited to terminate the algorithm.Model The detailed steps of the proposed algorithm are as follows :The mixed-integer formulation based on the convex hull reformulation  is as foThen, by combining A.5 and A.8, we get:l points. Equation A.6 is therefore rewritten as follows:On the other hand, the logarithmic term in constraint A.6 is approximated by a linear outer-approximation. This is accomplished by adding supporting hyper-planes, which are obtained by performing first order linearizations at a set of ONLP. The overall master problem can therefore be expressed as follows:Since the logarithmic function is concave, these hyper-planes strictly over estimate its value and thus do not chop off any feasible solution of the original model CMILP takes the form of a mixed-integer linear programming (MILP) problem. This type of model can be efficiently solved via standard branch and bound techniques [Model chniques .Note that the master problem can also be expressed as follows:RS is a relaxation of R  and it is defined as follows:where the set where RNLP), which is obtained from the original nonconvex formulation ONLP by imposing lower and upper bounds on the values of the velocity terms.The lower level of the algorithm is represented by a reduced NLP  represents the set of velocity pairs  associated with each metabolite i in TE. Note that in equation A.16, μir and μir' have opposite signs, so it is possible to perform a logarithmic transformation on both sides of the equation: where By combining equations A.4 and A.17, we finally get:which is a linear constraint.CMILP can be updated in different ways. A possible strategy to perform the updating consists of including in it the middle points of the active subintervals in the solution of the master problem CMILP. Therefore, if the solution of CMILP is such that h is active), then the grid corresponding to vr is modified by adding the new point RNLP.• The grid in problem RNLP.• In each iteration, additional hyper-planes can be added in the master problem in a similar way as it is done with the grid updating. Thus, the logarithmic terms can be linearized either at the middle points of the active subintervals or at the optimal values obtained in the lower level problem γr and/or the concentrations of metabolites Xj. These conditions can be expressed via the following constraints:• The approach presented can easily handle the case in which lower and upper bounds are imposed on the apparent rate constants which can easily be converted into the following linear inequalities:γr and Xj. This can be accomplished by adding the following inequalities:• The approach presented also allows to fix upper bounds on the summation of SG and SX denote the upper bounds on the summations of γr and Xj, respectively. These inequalities can be equivalently written as follows:Here, r and Xj leads to nonconvex terms. In this latter case, additional piecewise estimators are required to preserve the convexity of the model.Constraints A.25 and A.26 are convex, and hence can be linearized in a similar way as was done with equation A.6. Note, however, that the definition of lower bounds on the summation of γ• Let us finally note that different types of piecewise functions could be applied in the master problem .r falls into the subinterval T terms:The problem of identifying if the variable γThe mixed-integer formulation based on the convex hull reformulation , is as fzrt is a new binary variable that takes a value of one if γr lies in the subinterval t, and it is zero otherwise. Figure r = 2 and t = 4. Note that the binary variables zrt indicate the membership of a solution to a specific hyper-rectangle. In the example presented, each variable is divided into 4 intervals, which leads to 16 different hyper-rectangles, each of which is represented by a different binary solution z*.where v, γ and X according to a predefined criterion over the search space The task of identifying optimal values of FMINLP define a non convex feasible region. However, it is possible to convexify the model by applying the same algebraic transformations described before. Finally, the reformulated MILP problem can be expressed as follows:The equations in SP or not.In the context of our algorithm, this master problem is employed to predict whether a feasible solution exists in the search space RFNLP that is obtained from the original nonconvex formulation ONLP by imposing lower and upper bounds on the continuous variables vr and γr.The modified slave problem is a reduced NLP model RFNLP, the lower and upper bounds imposed to γr are given by the values of the binary variables zrt in the master problem, whereas the values of CFMILP.Note that in model At each iteration, the search space is reduced by removing all the sub-regions  containing feasible solutions that have been identified so far. This is accomplished by making use of integer cuts, which are mathematically expressed as follows:s component of the vector of binary variables in the feasible solution identified in the outer iteration b. The sets b, and are employed to derive integer cuts that are added cumulatively to the master problem CFMILP.where z* that correspond to the feasible hyper-rectangles. This procedure is repeated until an infeasibility is detected in the master problem, which implies that there is not any feasible solution to ONLP in the remaining hyper-rectangles of the search space.Figure tol, the algorithm can also determine the optimal solution in the region SP explored at each iteration within the predefined optimality gap.• Note that the algorithm only requires solving the problems to feasibility, i.e., a feasible solution of the problems is sufficient for the algorithm goal. However, by defining a small tolerance error CFMILP, which leads to an increase in its size.• The integer cuts in equation A.31 are added cumulatively at each iteration to the upper-level model Model details and physiological constraints. Detailed description of the models used in the main text and the physiological constraints considered in the optimization and feasibility examples.Click here for file


Natronomonas pharaonis is an archaeon adapted to two extreme conditions: high salt concentration and alkaline pH. It has become one of the model organisms for the study of extremophilic life. Here, we present a genome-scale, manually curated metabolic reconstruction for the microorganism. The reconstruction itself represents a knowledge base of the haloalkaliphile's metabolism and, as such, would greatly assist further investigations on archaeal pathways. In addition, we experimentally determined several parameters relevant to growth, including a characterization of the biomass composition and a quantification of carbon and oxygen consumption. Using the metabolic reconstruction and the experimental data, we formulated a constraints-based model which we used to analyze the behavior of the archaeon when grown on a single carbon source. Results of the analysis include the finding that Natronomonas pharaonis, when grown aerobically on acetate, uses a carbon to oxygen consumption ratio that is theoretically near-optimal with respect to growth and energy production. This supports the hypothesis that, under simple conditions, the microorganism optimizes its metabolism with respect to the two objectives. We also found that the archaeon has a very low carbon efficiency of only about 35%. This inefficiency is probably due to a very low P/O ratio as well as to the other difficulties posed by its extreme environment. Natronomonas pharaonis is one that has been able to adapt to both high salt concentration and an alkaline pH. In this study, we investigate the chemical reactions that occur within the microorganism, collectively referred to as its metabolic network, that allow it to convert the nutrients in its environment to biomass and energy. Specifically, we reconstructed the network by collecting evidence for the existence of reactions from the literature, and then supplemented them with computational approaches, for example by searching the genome of Natronomonas pharaonis for genes that could potentially encode analogs of known enzymes from other organisms. Finally, with the network in hand, we developed a computational model which we used to simulate growth. Among other results, we found indications that Natronomonas pharaonis regulates its metabolism such that energy production and growth are maximized. Despite this however, we also found that Natronomonas pharaonis is only able to incorporate a very small fraction of the total carbon that it consumes (approximately 35%), likely in no small part due to the difficulties posed by its environment.Extremophiles are organisms that thrive in physically or geochemically extreme conditions that are detrimental, even lethal, to the majority of life on Earth.  Natronomonas pharaonis is a polyextremophilic archaeon that can be isolated from soda lakes, where it has to cope with two extreme conditions: high salt concentration and an alkaline pH. The microorganism thrives at an optimal pH of 8.5, and remains viable up to a pH of about 11. Two strains of Natronomonas pharaonis have been described so far: strain Gabara from lake Gabara in Egypt (DSM 2160) Halobacterium salinarum. This greatly simplifies several experimental protocols, in particular those based on stable isotope labeling. The combination of this capability, a completely sequenced genome Natronomonas pharaonis in the study of extremophilic life.The metabolic network of an organism can be reconstructed at the genome scale through the combination of genomic, biochemical and physiological data, using bioinformatics methods and literature review E. coli metabolic mutant phenotypes up to an accuracy of 86% E. coli's secreted metabolites under various nutrient and oxygenation conditions M. barkeriL. plantarumHalobacterium salinarumNatronomonas pharaonis, and report on the development and use of a constraints-based model from it. This is the third curated reconstruction for a microorganism belonging to the archaeal domain of life; the first being for M. barkeriHalobacterium salinarumGenome-scale models of metabolism have been constructed and analyzed for various organisms Natronomonas pharaonis metabolic network is composed of 683 reactions and 597 distinct metabolites. It covers 654 genes, not including those with known transport function but with unclear substrate specificity. Reactions were added to the network based on either genetic  or literature  evidence. Natronomonas pharaonis, its literature base, particularly for subjects relevant to metabolism, is still quite small. This is the reason why most reactions in the network only have bioinformatic support. Nevertheless, at least 168 (24%) of the reactions are associated with literature  evidence from related haloarchaeal species . Note that reactions with neither genetic nor literature support were nevertheless added to the network in order to fill pathway “gaps”.The reconstructed Natronomonas pharaonis genome annotation found in Halolex Methanocaldococcus jannaschii does not use the classical precursors erythrose 4-phosphate and phosphoenolpyruvate, but rather proceeds via an alternative pathway that begins with the condensation of 6-deoxy-5-ketofructose 1-phosphate with aspartate 4-semialdehyde Natronomonas pharaonis. This alternative pathway was still not in KEGG when we did our reconstruction. Another example is the modified mevalonate biosynthesis pathway that was also demonstrated in Methanocaldococcus jannaschiiNatronomonas pharaonis. The reconstructed network is provided as supplementary information, both in tabular  from KEGG  tabular  and SBML tabular .Natronomonas pharaonis are encoded by NP4044A (beta subunit) and NP4046A . Accordingly, the reaction was assigned the gene-logic formula NP4044ANP4046A, because, presumably, both subunits are required in order for the enzyme to function. We should note, however, that given the current lack of appropriate genomic data, such as genome-scale single deletion lethality assays, the logical relationships were mostly inferred only from the genome annotation . Accordingly, this information should be taken with a grain of salt. For example, four genes are currently annotated in the genome as “probable aspartate aminotransferase (EC 2.6.1.1)”, and accordingly appear as disjuncted terms in the gene-logic of the reaction catalyzed by the enzyme EC 2.6.1.1:Although in silico gene deletion simulations were not part of our analysis, information on putative logical relationships between genes and reactions was nevertheless included in the reconstruction. For example, the oxidation of pyruvate to acetyl-CoA and COHowever, even if all four genes are indeed aspartate transaminases, there is a good probability that not all of them actually catalyze the reaction defined above. Some could work with an acceptor different from 2-Oxoglutarate, and as such are not true isoenzymes.All of the reactions in the reconstructed network are mass and charge balanced, except for 10. The latter could not be balanced because some reactants are still unknown. Seven of the reactions are involved in cofactor biosynthesis, and the rest belong to amino acid degradation pathways. The ionization state of the metabolites reflected in the network is that of the most abundant microspecies under a pH of 9.0.Natronomonas pharaonis is an approximation of the microorganism's average biomass composition, at least under the conditions used. For this we determined the total organic carbon content of samples taken from cultures at different population levels . The data is summarized in Halobacterium salinarumIn flux balance analysis, growth is typically simulated through the use of a growth reaction, which is a pseudo reaction where the reactants are the components of the biomass in the proper ratios and the product is a unit of population . From these measurements, we were able to observe good linear correlations between the amino acids and the optical density, which implies that the average cellular amino acid composition remains reasonably constant throughout growth. This observation reinforces the validity of using a static definition of growth in flux balance models, such as the one used in this study. Note that due to experimental limitations, measurements for aspartate and glutamate are already inclusive of asparagine and glutamine, respectively, and cysteine and tryptophan could not be reliably determined. For these amino acids, statistical analysis of the predicted Natronomonas pharaonis proteome was used to specify coefficients in the growth reaction (see Growth on Acetate).Determination of the total organic carbon content of the biomass is only the first step. The next is to decompose it into the different building blocks. Given that proteins typically account for a large fraction, if not a majority, of the organic mass of most microorganisms, we began our characterization of the Halobacterium salinarum (data not shown).The amount of amino acids in the biomass added up to about 335 mg/ODNatronomonas pharaonisNatronomonas pharaonis under other conditions Detection of 2-Sulfotrehalose) we were not able to detect any dissacharide.2-Sulfotrehalose has been demonstrated to be an osmolyte used by several haloalkaliphilic archaea, including Halobacterium salinarum model In addition to amino acids, the cellular biomass is also made up of other molecules such as lipids , sugars , cofactors , and other small molecules. Due to lack of experimental data, the stoichiometric coefficients that we used for these molecules in the growth reaction were simply taken from the approximations used in the Natronomonas pharaonis would require the presence of leucine because of a disruption in the 2-isopropylmalate synthase gene (NP2206A) Natronomonas pharaonis is capable of growing  on a single carbon source. Indeed, although the rest of this study deals with growth on acetate, we were able to find other possible substrates, including glutamate and pyruvate.It was previously believed that any growth medium for Computational Analysis for details on the calculations), it turns out that what is critical to the first two parameters with respect to the analysis is just the ratio between the two, i.e., the ratio between acetate and oxgyen usage. Scaling the actual values simply results in a similar scaling of the output of the model . Accordingly, for most of the analysis we could simplify the calculations by reducing the parameter space to just two instead of three; namely, the acetate to oxygen consumption ratio and the maintenance energy.In order to simulate aerobic growth on acetate using the constraints-based model, three parameters needed to be further defined: (1) the consumption rate of acetate, (2) the consumption rate of oxygen, and (3) the maintenance energy. Under certain assumptions however . Results of the stable isotope experiments showed that no net fixation seems to occur (data to appear in a separate publication). Accordingly, the amount of acetate that disappears from the medium is a direct measure of the total carbon consumption of the cells . By correlating this consumption, which is represented as “total uptake” (red borken curve) in Analysis of biomass composition), we find that only about 35% of the total carbon consumed was incorporated into the cells  incorporated into the biomass; (2) secreted as the respiratory byproduct COThe respiratory exchange ratio (RER) is the ratio between the production of COThe sum of fates 1 (incorporation) and 2  means that we now have a haloarchaeon for which genetic manipulation is possible and labeling experiments are relatively simple.We presented a genome-scale, manually curated metabolic reconstruction for the polyextremophile Natronomonas pharaonis, we experimentally determined several physiological parameters. These include a characterization of the biomass composition, and a quantification of carbon and oxygen consumption under typical conditions. The data was integrated with the metabolic reconstruction to create a computational model that we used to analyze the behavior of Natronomonas pharaonis when grown on a single carbon source. Among other results, we found that the archeaon, when grown aerobically on acetate, uses an acetate to oxygen consumption ratio that is theoretically near-optimal with respect to growth and energy production. This supports the hypothesis that, under simple conditions, Natronomonas pharaonis optimizes its metabolism with respect to these two objectives. We also found that the archaeon has a very low carbon efficiency of only about 35%, likely due to a very low P/O ratio and the other difficulties brought about by the haloalkaliphilic character of its environment.In order to complete a constraints-based model for Natronomonas pharaonisStable isotope labeling of small-molecules . The reader interested in more information on metabolic reconstructions is directed to the reviews by Reaction reversibilities were taken from literature sources whenever available. Particularly helpful in this regard is the BRENDA database A metabolic network can be conveniently represented by a stoichiometric matrix Analysis of biomass composition). Specifically, the acetate∶oxygen consumption ratio was allowed to vary from 1∶9 to 9∶1, and the maintenance energy was allowed to vary from 0 to 100 In the analysis of growth with respect to the acetate∶oxygen consumption ratio and the maintenance energy, Equation 1 was solved for different combinations of the two parameters using the growth reaction as objective function (see Natronomonas pharaonis strain Gabara (DSM 2160) was grown under aerobic conditions using chemically-defined media prepared according to For the analysis of growth, ormation .Due to experimental limitations, measurements of the aspartate and glutamate content of the biomass were already inclusive of asparagine and glutamine, respectively. For each of these pairs, specific coefficients in the growth reaction were assigned by dividing the combined total according to the relative abundance of the correspoding amino acids in all protein-coding genes . Similarly, the individual contributions of cysteine and tryptophan to the biomass were approximated using their relative abundance compared to the rest of the amino acids (for which measurement was possible).Natronomonas pharaonis. In particular, we tested different carbon sources, including all of the amino acids, acetate, glycerol, citrate cycle intermediates, and combinations thereof, in different concentrations using 96-well microtiterplates (Greiner bio-one).The formulation of the growth medium (Table S4) used in the analysis of aerobic growth is the result of optimizing previous synthetic media used for Natronomonas pharaonis biomass was determined by adapting a procedure typically used to determe the TOC of soils. The following solutions were used: Solution 1 was prepared by dissolving 12.5 grams of silver sulfate in 40 ml of HSolution 2 was prepared by dissolving 98.08 grams of KSolution 3 was prepared by dissolving 11.2 grams of FeSOSolution 1 was added in order to precipitate ClSolution 2 to the vial, the solution was heated to 135Solution 2), which oxidizes the organic carbon in the biomass, was then determined via titration with Solution 3. This value is inversely related to the amount of carbon in the biomass. Titration was carried out using the TitroLine easy automatic titrator from Schott Instruments .The total organic carbon (TOC) content of the Natronomonas pharaonis was grown in larger volumes (200 ml) to mid-log phase (OD 0.3–0.4), after which cells were harvested by centrifugation . Cell pellets were washed three times with 3.4 M NaCl, and then dried in a speed vac centrifuge. The pellets were suspended in 1.5 ml 70% (v/v) EtOH and vortexed for 5 min. Sonication for 5 min in a water bath followed, before centrifugation at 11 000 g for 10 min at 4A Table S1Reconstructed network in tabular format.(0.39 MB XLS)Click here for additional data file.Text S1Supplementary information.(0.09 MB PDF)Click here for additional data file.Text S2Reconstructed network in SBML format.(0.51 MB XML)Click here for additional data file.


Constraint-based flux analysis of metabolic network model quantifies the reaction flux distribution to characterize the state of cellular metabolism. However, metabolites are key players in the metabolic network and the current reaction-centric approach may not account for the effect of metabolite perturbation on the cellular physiology due to the inherent limitation in model formulation. Thus, it would be practical to incorporate the metabolite states into the model for the analysis of the network.iAF1260 in silico metabolic model of Escherichia coli, we demonstrated that this novel concept complements the conventional reaction-centric analysis.Presented herein is a metabolite-centric approach of analyzing the metabolic network by including the turnover rate of metabolite, known as flux-sum, as key descriptive variable within the model formulation. By doing so, the effect of varying metabolite flux-sum on physiological change can be simulated by resorting to mixed integer linear programming. From the results, we could classify various metabolite types based on the flux-sum profile. Using the Metabolite flux-sum analysis elucidates the roles of metabolites in the network. In addition, this metabolite perturbation analysis identifies the key metabolites, implicating practical application which is achievable through metabolite flux-sum manipulation in the areas of biotechnology and biomedical research. Cellular metabolism is more often than not represented and analysed based on a stoichiometric modelling framework under the stationary assumption of the metabolic network . Such stMetabolites and biochemical reactions in the metabolic network can be graphically represented by nodes and edges connecting the nodes respectively. Based on this graphical representation, it is obvious that there can be two approaches in the analysis of the network, focusing on the flow of materials through either the nodes (metabolite-centric approach) or edges (reaction-centric approach). Typical FBA can be conducted based on a reaction-centric approach where constraints were introduced to restrict the range of reaction flux values so as to define a feasible solution space . This anE. coli system using i AF1260 in silico metabolic model , H2O [p] and H2O [e] indicate water found in three different compartments. The cellular objective was assumed to be the maximization of biomass production. The reaction reversibility constraints were set as given by [Escherichia coli in glucose minimal medium. All these settings were based on previous observation for glucose and oxygen uptake rates and experimentally determined ATP requirement for maintenance [The genome-scale given by . The nonntenance ,21. The iAF1260 model by solving (P1), resulting in 4.20 and 23.9 for average (μFS) and standard deviation (sd) of the flux-sum values respectively. About 70% of the metabolites have zero flux-sum in the base case while metabolites with high , allowed us to identify different types of metabolites. Firstly, we found blocked metabolites with maximum flux-sum equal to zero. The consumption and production of these metabolites were blocked due to reaction pathway dead-ends  with constraint (C1) for these 394 basal active metabolites, thereby giving rise to the flux-sum attenuation profile  with constraint (C2), we generated the flux-sum intensification profile .Click here for fileEscherichia coli.Demonstration of flux-sum analysis for increasing succinate production in Click here for file


Improvement of a PHB fermentation requires good understanding and optimization under the realistic conditions of large bioreactors.Although the production of poly-Ralstonia eutropha and Lactobacillus delbrueckii generate better fermentation efficiencies than R. eutropha alone. In large bioreactors, incomplete dispersioin and perturbations in the dissolved oxygen (DO) concentration, both of which affect the fermentation, have to be considered. This study analyzes the effect of DO fluctuations on bioreactor performance for both ideal and optimally dispersed broths.Laboratory-scale studies have shown that co-cultures of Response coefficient analysis was employed to obtain quantitative information on the effect of DO perturbations on different variables. Three values of the Peclet number (Pe) cheracterized three levels of dispersion: Pe = 0.01 for nearly complete dispersion, Pe = 20 for optimum dispersion and Pe = 60 for insufficient dispersion. The response coefficients (RCs) of the pairs of bacterial concentrations and the main substrates, glucose and ammonium chloride, showed contrasting variations with time. Lactate, a critical intermediate, and PHB had similar RC profiles but those of lactate were one to two orders of magnitude larger than other RCs. Significantly, the optimum Pe also resulted in the largest RCs, suggesting a balance between productivity and reactor stability.R. eutropha requires oxygen for its growth whereas L. delbrueckii does not, fluctuations in the DO concentartion have a strong influence on the fermentation. Apart from this, the mechanism of PHB biosynthesis indicates that control of lactate is a critical determinant of fermentation efficiency. The RC profiles indicate that, under non-ideal conditions, a compromise may be required between PHB formation and reactor stability, especially in the latter half of the process.Since  There is growing commercial interest in PHB since many of its physical, chemical and rheological properties are comparable to those of widely used polymers such as polyethylene and polypropylene [Under conditions adverse to cell growth, many bacteria synthesize polyhydroxyalkanoates (PHAs) as energy storage devices. Poly-ropylene . While tropylene . These aropylene ,4.Ralstonia eutropha , Alcaligenes latus and Azotobacter vivelandii may be induced to synthesize PHB by imposing a chemical stress. This is normally done by depriving the organism of a nutrient such as nitrogen or phosphorus or sulfur, which are required for cell growth [R. eutropha [Bacteria such as l growth -4. Of thl growth -8, but rl growth  points tl growth  and proml growth . In addieutropha . TherefoR. eutropha alone have been varied either through on-line control based on glucose or PHB or the CO2 evolution rate [The complexity of the metabolic network ,3 and thion rate ,12 or thion rate -8, the rion rate , a key iWhile the effects of manipulating carbon and nitrogen supply have been analyzed adequately -9,11, thR. eutropha is the most widely used organism for PHB production because it is easy to cultivate, its metabolism is well understood and it can accumulate large amounts of PHB  insidell mass  and promell mass . Therefoell mass -9.Fructose and glucose are the common carbon sources, and either ammonium chloride or ammonium sulfate provides nitrogen. The fermentation is aerobic and the oxygen content of the broth influences PHB formation. A low DO concentration leads to an excess of reduced co-enzymes (NADH and NADPH), thus enabling a higher carbon flux directed toward PHB synthesis for reoxidation of these co-enzymes . HoweverR. eutropha in conjunction with another organism arises from the observation that R. eutropha is sluggish in metabolizing fructose and glucose but can utilize organic acids such as acetate, butyrate and lactate more easily [Lactococcus lactis [Lactobacillus delbrueckii [R. eutropha. Such a two-culture system can generate higher concentrations of PHB than R. eutropha alone. The present analysis is based on Tohyama and Shimizu [L. lactis-R. eutropha combination required two stages of cultivation [Thus, control of DO concentration is as critical as that of carbon and nitrogen. This is even more important when a pair of complementary organisms are used in place of one. The rationale for using e easily . So, soms lactis  or Lactobrueckii  to conve Shimizu  because tivation .L. delbrueckii is anaerobic whereas R. eutropha is aerobic. In a fed-batch system, L. delbrueckii is introduced at the start so as to utilize glucose and produce lactate. This requires a low concentration of DO. After sufficient amount of lactate has been generated, the reactor is inoculated with R. eutropha. At this stage there are conflicting requirements. R. eutropha requires a high DO concentration to metabolize lactate, where production of lactate by L. delbrueckii requires a low DO concentration. Thus, control of lactate concentration by manipulating DO becomes a critical factor.Now, The DO concentration is usually controlled by varying the stirring speed or the flow rate of the gas ,6,11,12.L. delbrueckii and R. eutropha, studied by Tohyama et al. [L. delbrueckii and then increasing this (3.0 ppm) after inoculation by R. eutropha. Thereafter, since both bacteria have to function in tandem, the DO was alternated between the two levels every hour. The fermentation was run for 30 h.The analysis presented here is based on the fed-batch fermentation with a et al. ,22 and mIn the absence of flow terms and for a completely homogeneous broth, Tohyama et al. ,22 propoL. delbrueckii isThe rate of growth of R. eutropha has a similar form:and that of L. delbrueckii at the rateGlucose is utilized by R. eutropha, so its net rate of formation isLactate is the product of glucose consumption and it is the carbon substrate for The specific rates in Eqs. (1)–(4) have the forms given below.β, and a growth-related component, αμ1. This arises because glucose is utilized by L. delbrueckii for growth as well as lactate synthesis. The constitutive rate has the formNote that the specific rate of lactate formation, Eq. (7), has a constitutive component, Similar to Eqs. (1)–(3), the rate of consumption of the nitrogen source isand that of PHB formation isThe specific rates in Eqs. (11) and (12) followandR. eutropha and not by L. delbrueckii. This may appear implausible. In fact, Eq. (11) is a practical approximation based on the experimental observations [L. delbrueckii (X1) compared to the changes generated by R. eutropha (X2). This observation is also reflected in the absence of a nitrogen term for μ in Eq. (5). Equations (8) and (13) similarly express the observations that cell growth increased with ammonium concentration while PHB was favored by reducing the ammonium concentration.While Tohyama et al.  have disrvations ,22 that L. delbrueckii and R. eutropha. As the model shows, the carbon substrate (glucose) and DO concentrations control the production of lactate whereas ammonium sulfate and DO control its consumption. This establishes the central role of DO and consequently the importance of any perturbations in this concentration.Lactate is the critical intermediate linking the growth of The degree of dispersion in a stirred reactor may be characterized by the Peclet numberwhere the 'characteristic length' L is usually the diameter of the vessel. For finite dispersion, the kinetics of Eqs. (1)–(4), (11) and (12) may be incorporated into the mass balances for a fed-batch bioreactor ,24 to obS + FN is the total inflow rate. The total inflow enters as a term in all equations since it dilutes the concentrations appropriately. However, only Eqs. (20) and (21) contain the inflow (or feed) concentrations because the flow stream contains only the carbon (S) and nitrogen (N) substrates.Here F = FEquations (16)–(22) are subject to the following initial and boundary conditions:Given that z = 0 corresponds to the central axis of the reaction vessel and z = 1 the periphery at the point of introduction of the feed stream, Eq. (24) expresses symmetry around the impeller shaft.R. eutropha alone [In their experiments, Tohyama et al.  applied ha alone . Here thDe → ∞ and hence Pe → 0. When there is no dispersion, called segregated or plug flow, De → 0 and Pe → ∞. Both these are idealized limits. Since production-scale reactors have finite non-zero values of Pe, the effect of Pe on the sensitivity is part of the current analysis. Given a value of Pe, the mass balances for a fed-batch bioreactor may be written by incorporating the kinetics presented above.For a fully dispersed broth, l, u denote the (time-dependent) outputs of a variable y at the lower, the average and the upper values of DO.The complete model was solved numerically for Pe = 0.01, 20 and 60, using the parameter data in Table The effect of a disturbance in the DO level on an output variable y may be quantified by the response coefficient, defined as Using the perturbed values mentioned above, Eq. (27) may be approximated as where h is the distance between two consecutive values of the DO concentration. Since Eqs. (27) and (28) generate dimensionless numbers, it becomes possible to compare the responses of different variables. Obviously, the higher the coefficient the more sensitive is that variable.R. eutropha, the productivity of PHB was maximum for dispersion corresponding to Pe ≈ 20 [L. delbrueckii. Ralsonia, the oxygen-dependent partner in the mixed culture, is less active during this phase.Previous studies, for both PHB  and othe Pe ≈ 20 . Therefo Pe ≈ 20 , while tL. delbrueckii grows in the absence of oxygen, R. eutropha is an aerobe and thus requires DO. For quantitative comparisons, the minimum and maximum coefficients for each concentration variable and each Peclet number have been compiled in Table L. delbrueckii and R. eutropha also differ by an order of magnitude, and this difference is continued between the two principal substrates, glucose and ammonium sulfate  DO and ammonium have similar but complex metabolic effects [E. coli fermentations [While the response coefficients of lactate Figure  and PHB a et al. ,22 obser effects ,13,28, tntations , where sIn addition to regulating the effect of DO concentration on the dynamics of lactate production and consumption, the degree of dispersion is also an important determinant of the balances between two other pairs of processes: (i) The synthesis and degradation of PHB and (ii) the formation and consumption of acetate. Under strong dispersion (Pe → 0), the nitrogen and carbon substrates are freely available throughout the broth. Easy access to nitrogen is detrimental to PHB synthesis but favorable to acetate formation ,3,31; acKlebsiella oxytoca cultivated on glucose and lactose [These considerations imply that maximization of PHB yield requires an optimum finite dispersion, which is attained at Pe = 20 . However lactose , suggestL. delbrueckii and R. eutropha. Since the former is anaerobic and the latter is aerobic, dissolved oxygen (DO) has a crucial role in the fermentation.In view of its strong potential for many applications where petroleum-based chemically synthesized polymers are currently used, the microbial product of PHB by fed-batch fermentation was analyzed. This was done through a response coefficient analysis of a co-culture of In large bioreactors the broth is less than perfectly dispersed and perturbations in oxygen supply are possible. Their effects were studied through the response coefficients of six main variables: glucose, ammonium sulfate, lactate, PHB and the bacteria. The coefficients differ widely among the variables. In particular, the differences between the two substrates and between the two organisms have similarities that may be related to the mechanistic basis of PHB formation.Lactate is a critical intermediate in a two-culture fermentation for PHB. Its response coefficients were one to two orders of magnitude larger than those of other concentrations. Like lactate, ammonium sulfate also had large response coefficients, suggesting that small concentrations are more sensitive to a DO perturbation than large concentrations.Results for three different degrees of dispersion showed that the dispersion at which PHB production is highest and the lactate concentration lowest (Pe = 20)  also makDe effective dispersion coefficient (cm2 h-1)S inflow rate of glucose (L h-1)FN inflow rate of ammonium sulfate (L h-1)Fi inhibition constant for μ2 (g L-1)KN reaction rate constant for PHB (g L-1)kN Monod constant for μ2 with respect to ammonium (g L-1)KP Monod constant for μ2 with respect to lactate (g L-1)KS Monod constant for μ1 (g L-1)KL characteristic dimension of bioreactor (cm)n empirical exponent (-)-1)N concentration of ammonium sulfate (g Lf feed concentration of ammonium sulfate (g L-1)NO concentration of dissolved oxygen (ppm)-1)P concentration of lactate (g Lm limiting concentration of lactate (g L-1)PPe Peclet number (-)-1)Q concentration of PHB (g Lm maximum specific PHB production rate (h-1)q1 rate of growth of L. delbrueckii (g L-1 h-1)r2 rate of growth of R. eutropha (g L-1 h-1)rN rate of consumption of ammonium sulfate (g L-1 h-1)rP net rate of formation of lactate (g L-1 h-1)rQ rate of formation of PHB (g L-1 h-1)rS rate of consumption of glucose (g L-1 h-1)r-1)S concentration of glucose (g Lf feed concentration of glucose (g L-1)St elapsed time (h)T duration of the fermentation (h)-1)u mean velocity of fluid in bioreactor (cm hV volume of the broth (L)1 concentration of L. delbrueckii (g L-1)X2 concentration of R. eutropha (g L-1)XQ/P yield coefficient for PHB with respect to lactate (g g-1)YP/S yield coefficient for lactate with respect to glucose (g g-1)YX1/S yield coefficient for L. delbrueckii with respect to glucose (g g-1)YX2/N yield coefficient for R. eutropha with respect to ammonium (g g-1)YX2/P yield coefficient for R. eutropha with respect to lactate (g g-1)Yα empirical constant (-)β constitutive component of σ1 (h-1)βm maximum value of β(h-1)μ1 specific rate of growth of L. delbrueckii (h-1)μ2 specific rate of growth of R. eutropha (h-1)μm1 maximum value of μ1 (h-1)μm2 maximum value of μ2 (h-1)ν1 specific rate of consumption of glucose (h-1)ν2 specific rate of consumption of lactate by R. eutropha (h-1)ν3 specific rate of consumption of ammonium sulfate (h-1)σ1 specific rate of production of lactate by L. delbrueckii (h-1)σ2 specific rate of formation of PHB (h-1)Based on their experimental results, Tohyama et al.  proposedThe values of the empirical constants are given in Table


In the present article, we propose a method for determining optimal metabolic pathways in terms of the level of concentration of the enzymes catalyzing various reactions in the entire metabolic network. The method, first of all, generates data on reaction fluxes in a pathway based on steady state condition. A set of constraints is formulated incorporating weighting coefficients corresponding to concentration of enzymes catalyzing reactions in the pathway. Finally, the rate of yield of the target metabolite, starting with a given substrate, is maximized in order to identify an optimal pathway through these weighting coefficients.The effectiveness of the present method is demonstrated on two synthetic systems existing in the literature, two pentose phosphate, two glycolytic pathways, core carbon metabolism and a large network of carotenoid biosynthesis pathway of various organisms belonging to different phylogeny. A comparative study with the existing extreme pathway analysis also forms a part of this investigation. Biological relevance and validation of the results are provided. Finally, the impact of the method on metabolic engineering is explained with a few examples.The method may be viewed as determining an optimal set of enzymes that is required to get an optimal metabolic pathway. Although it is a simple one, it has been able to identify a carotenoid biosynthesis pathway and the optimal pathway of core carbon metabolic network that is closer to some earlier investigations than that obtained by the extreme pathway analysis. Moreover, the present method has identified correctly optimal pathways for pentose phosphate and glycolytic pathways. It has been mentioned using some examples how the method can suitably be used in the context of metabolic engineering. Metabolism is a complex process that takes place for producing energy and forms the driving force for cellular activity. It involves a large number of chemical reactions/conversions carried out by living organisms as they feed, grow and reproduce. A cascade of such reactions/conversions form a highly branched network. A metabolic network consists of many reactions and transport processes associated with the production and depletion of cellular metabolites. Metabolic pathways are defined as coordinated series of biochemical reactions in which the product of one reaction is the reactant of the subsequent one in the chain. Examples of metabolic pathways include Glycolysis, the Krebs cycle and the Pentose phosphate pathways.S. cerevisiae. Moreover, modeling and analysis of metabolic networks may be useful to perform rational drug design  denotes the weighting factor corresponding to this reaction Rk. ck indicates the level of concentration of the enzyme catalyzing the reaction Rk. ck = 1 indicates that the required amount of the enzyme catalyzing the reaction Rk is present. On the other hand, ck → 0 indicates that sufficient amount of enzyme is not present to carry out the reaction. Higher the value of ck, higher is the concentration of the enzyme and vice-versa. The term vk is considered to be positive if the reaction Rk yields the metabolite B, otherwise it is negative. A reversible reaction is considered as two separate reactions corresponding to forward and backward reactions. It is to be mentioned here that the role of ci in Eq. (1) in extreme pathway analysis T that cannot be obtained easily as the full dynamics is not known or becomes intrackable in most of the scenarios. In order to overcome this situation, we now propose an algorithm for generating flux vectors that satisfy approximately the quasi-steady state condition n × n, then γij = δijci, where δij is the Kronecker delta. Note that ci is the weighting factor corresponding to the enzyme catalyzing ith reaction in the network, irrespective of whether the reaction involves the metabolite B or not.Here z is maximized with respect to c, subject to satisfying the constraint given in Eq. (6).Thus the problem of determining a metabolic pathway yielding maximum rate of production of a metabolite B starting from a substrate A, boils down to a maximization problem, where Combining Eq. (1) and Eq. (6), we can reformulate the objective function asci for all i. The term Λ = T is the regularizing parameter. For the sake of simplicity, we have considered here λ1 = ... = λm = λ (say). Minimization of y can be carried out in various ways. Here we have adopted the gradient descent technique  and the others are very close to zero. We consider a metabolic path to be an optimal one, if y is minimum and the ci-values of all the enzymes catalyzing the reactions in that path are greater than zero. Otherwise, a low c-value (close to zero) corresponding to an enzyme catalyzing an intermediate reaction may result in an insufficient product. This will reduce the rate of the next reactions and hence the amount of the target metabolite. In other words, these non-zero ci-values indicate an optimal pathway through which the rate of yield of metabolite B, being grown on the substrate A, becomes maximum. The major differences of the method from the existing extreme pathway analysis  is the minimum concentration of the enzyme E that is required to obtain the maximal rate (Vmax) of product formation. Thus an estimate of c may be taken aswhere S and P stand for substrate and product respectively. The terms Vmax, and the rate constants can be estimated in vitro [Emin can also be determined through [where [E] is the concentration level of the enzyme E which is required to get an optimal path. Note that the values of in vitro . Thus Em through in vitro, then the theoretical value of c (obtained by the proposed method) can be verified with the experimental value.and thereby [E] using c-value obtained by our method. On the other hand, if [E] can be determined RKD has conceived the study, formulated the methodology, made partial analysis of the results and has prepared the manuscript. MD has implemented the algorithms, made partial analysis of the results and has modified some parts of the manuscript. SM has made partial analysis of the results and given some fruitful suggestions while preparing the manuscript.Analysis of the results. Analysis of the results for the synthetic system, pentose phosphate pathway, glycolytic pathway and the carotenoid biosynthesis pathway for different organisms.Click here for file


Our results help explain existing experimental data on intracellular flux measurements and the usage of latent pathways, shedding new light on microbial evolution, robustness, and versatility for the execution of specific biochemical tasks. In particular, the identification of optimal reaction activity provides rigorous ground for an intriguing knockout-based method recently proposed for the synthetic recovery of metabolic function.Metabolic reactions of single-cell organisms are routinely observed to become dispensable or even incapable of carrying activity under certain circumstances. Yet, the mechanisms as well as the range of conditions and phenotypes associated with this behavior remain very poorly understood. Here we predict computationally and analytically that any organism evolving to maximize growth rate, ATP production, or any other linear function of metabolic fluxes tends to significantly reduce the number of active metabolic reactions compared to typical nonoptimal states. The reduced number appears to be constant across the microbial species studied and just slightly larger than the minimum number required for the organism to grow at all. We show that this  Helicobacter pylori, Staphylococcus aureus, Escherichia coli, and Saccharomyces cerevisiae to show that a cell seeking to optimize a metabolic objective, such as growth, has a tendency to spontaneously inactivate a significant number of its metabolic reactions, while all such reactions are recruited for use in typical suboptimal states. The mechanisms governing this behavior not only provide insights into why numerous genes can often be disabled without affecting optimal growth but also lay a foundation for the recently proposed synthetic rescue of metabolic function in which the performance of suboptimally operating cells can be enhanced by disabling specific metabolic reactions. Our findings also offer explanation for another experimentally observed behavior, in which some inactive reactions are temporarily activated following a genetic or environmental perturbation. The latter is of utmost importance given that nonoptimal and transient metabolic behaviors are arguably common in natural environments.Cellular growth and other integrated metabolic functions are manifestations of the coordinated interconversion of a large number of chemical compounds. But what is the relation between such whole-cell behaviors and the activity pattern of the individual biochemical reactions? In this study, we have used flux balance-based methods and reconstructed networks of  In complex metabolic networks of single-cell organisms, there is mounting evidence in the experimental and non-optimal conditions in terms of the number of active reactions . We implement our study within a flux balance-based framework Escherichia coli. For any typical non-optimal state T of these fluxes, including the fluxes of n internal and transport reactions, as well as nex exchange fluxes for modeling media conditions. Under the constraints imposed by stoichiometry, reaction irreversibility, substrate availability, and the assumption of steady-state conditions, the state of the system is restricted to a feasible solution spacev∈M.We model cellular metabolism as a network of metabolites connected through reaction and transport fluxes. The state of the system is represented by the vector n space . Within v∈M, all the metabolic reactions are active for almost all v∈M, i.e., for any typical state chosen randomly from M (n+(v) of active reactions in a typical non-optimal state is constant, i.e.,We can prove that, with the exception of the reactions that are inactive for all E. coli reconstructed model involves oxidized glutathione, but because there is no other metabolic reaction that can balance the flux of this metabolite, the reaction cannot be active in any steady state. We characterize such reactions uniquely by a linear relationship between vectors of stoichiometric coefficients . The phenomenon is caused by growth maximization via reaction irreversibility and cascading of inactivity.We now turn to mechanisms underlying the observed reaction silencing, which is spread over a wide range of metabolic subsystems  by making this pathway active in the opposite direction, which creates a metabolic cycle equivalent to a combination of the pyruvate kinase reaction and the transport of protons out of the cell. The pyruvate kinase itself does not contribute to the increase in biomass production (it is inactive under maximum growth condition), but the cycle would provide a more efficient transport of protons to balance the influx of protons accompanying the ATP synthesis, which in turn would increase biomass production.We identify three different scenarios in which reaction irreversibility causes reaction inactivity under maximum growth. The simplest case is when the reaction is part of a parallel pathway structure. While stoichiometrically equivalent pathways lead to alternate optima transverse pathway that, were it reversible, could be used to increase growth by redirecting metabolic flow from “non-limiting” pathways to those that limit the production of biomass precursors. This includes transverse reactions that establish one-way communication between pathways that lead to different building blocks of the biomass (when one of them is more limiting than the others). In the E. coli model, for example, isocitrate lyase in the glyoxylate bypass is predicted to be inactive under maximum growth, as shown in A different silencing scenario is identified when no clear parallel pathway structure is recognizable. In this scenario there is a E. coli under maximum growth, we computationally predict that this transport reaction is inactive. This indicates that E. coli growing maximally in the given glucose medium wastes no acetate by excretion, which is consistent with experimental observation in glucose-limited culture at low dilution rate more transport reactions, and hence excrete larger number of metabolites than a growth-optimized cell. This experimentally testable prediction can be further supported by our single-reaction knockout computations considered in a subsequent section  to model suboptimal response to perturbation.A third scenario for the irreversibility mechanism arises when a transport reaction is irreversible because the corresponding substrate is absent in the medium. For example, since acetate, a possible carbon and energy source, is absent in the given medium, the corresponding transport reaction is irreversible; acetate can only go out of the cell (Note 3). For Mopt must be part of the boundary of MMopt is characterized by a set of binding constraints, defined as inequality constraints  satisfying two conditions: the equality holds (iv = iβ) for all v∈Mopt and Mopt is sensitive to changes in the constraints (changes in iβ). In two dimensions, for example, Mopt would be an edge of M, characterized by a single binding constraint, or a corner of M, characterized by two binding constraints. In general, at least d – dopt linearly independent constraints must be binding, where d and dopt are the dimensions of M and Mopt, respectively. Since many metabolic reactions are subject to the irreversibility constraint (iv≥0), this is expected to lead to many inactive reactions (iv = 0). Indeed, by excluding the k constraints that are not associated with reaction irreversibility (those for the ATP maintenance reaction and exchange fluxes), we obtain an upper bound on the number of active reactions n+(v):We interpret these inactivation mechanisms involving reaction irreversibility as a consequence of the linear programming property that the set of optimal solutions v∈Mopt is due to cascading of inactivity. On one hand, if all the reactions that produce a metabolite are inactive, any reaction that consumes this metabolite must be inactive. On the other hand, if all the reactions that consume a metabolite are inactive, any reaction that produces this metabolite must be inactive to avoid accumulation, as this would violate the steady-state assumption. Therefore, the inactivity caused by the irreversibility mechanism triggers a cascade of inactivity both in the forward and backward directions along the metabolic network. In general, there are many different sets of reactions that, when inactivated, can create the same cascading effect, thus providing flexibility in potential applications of this effect to the design of optimal strains The remaining set of reactions that are inactive for all v∈Mopt, the inactivity of other reactions can depend on the specific growth-maximizing state, whose non-uniqueness in a given environment has been evidenced both theoretically Mopt :E. coli, these results are consistent with a previous study comparing reaction utilization under a range of different growth conditions E. coli metabolic models n+(v) within the upper and lower bounds is singular in that the upper bound is attained for almost all optimal states:Mopt. This behavior is expected, however, under the concurrent optimization of additional metabolic objectives, which generally tend to drive the flux distribution toward the boundary of Mopt.While the irreversibility and cascading mechanisms cause the inactivity of many reactions for all Although we have focused so far on maximizing the biomass production rate, the true nature of the fitness function driving evolution is far from clear Mopt consists of a single point, which must be a “corner” of M, termed an extreme point in the linear programming literature. In this case, dopt = 0, and Eq. (2) becomesWe first note the fact . For the 44 fluxes in the S. cerevisiae metabolism experimentally measured by Daran-Lapujade et al.p<10−7). This higher probability for reduced fluxes in irreversible reactions is consistent with our theory and simulation results . This correlation indicates that irreversible fluxes tend to be larger in non-optimal metabolic states, consistently with our predictions.Additional evidence for our results is derived from the inspection of 18 intracellular fluxes experimentally determined by Emmerling transient suboptimal states are the leading cause of latent pathway activation. Since we predict that a large number of reactions are inactive in optimal states but active in typical non-optimal states, many reactions are expected to show temporary activation if we assume that the state following the perturbation is suboptimal and both the pre-perturbation and post-adaptation states are near optimality. To demonstrate this computationally for the E. coli model, we consider the idealized scenario where the perturbation to the growth-maximizing wild type is caused by a reaction knockout and the initial response of the metabolic network—before the perturbed strain evolves to a new growth-maximizing state—is well approximated by the method of minimization of metabolic adjustment (MOMA) Altogether, our results offer an explanation for the temporary activation of latent pathways observed in adaptive evolution experiments after environmental much more than just a superposition of the reactions that are active in the pre- and post-perturbation optimal states, thus explaining the pronounced burst in gene expression changes observed to accompany media changes and gene knockouts E. coli in glucose minimal medium, temporary activation is predicted for the Entner-Doudoroff pathway after pgi knockout and for the glyoxylate bypass after tpi knockout, in agreement with recent flux measurements in adaptive evolution experiments Another potential application of our results is to explain previous experimental evidence that antagonistic pleiotropy is important in adaptive evolution Combining computational and analytical means, we have uncovered the microscopic mechanisms giving rise to the phenomenon of spontaneous reaction silencing in single-cell organisms, in which optimization of a single metabolic objective, whether growth or any other, significantly reduces the number of active reactions to a number that appears to be quite insensitive to the size of the entire network. Two mechanisms have been identified for the large-scale metabolic inactivation: reaction irreversibility and cascade of inactivity. In particular, the reaction irreversibility inactivates a pathway when the objective function could be enhanced by hypothetically reversing the metabolic flow through that pathway. We have demonstrated that such pathways can be found among non-equivalent parallel pathways, transverse pathways connecting structures that lead to the synthesis of different biomass components, and pathways leading to metabolite excretion. Although the irreversibility and cascading mechanisms do not require explicit maximization of efficiency, massive reaction silencing is also expected for organisms optimizing biomass yield or other linear functions (of metabolic fluxes) normalized by uptake rates Our study carries implications for both natural and engineered processes. In the rational design of microbial enhancement, for example, one seeks genetic modifications that can optimize the production of specific metabolic compounds, which is a special case of the optimization problem we consider here and akin to the problem of identifying optimal reaction activity E. coli and S. cerevisiae have shown that only a small fraction of their genes are essential for growth under standard laboratory conditions E. coli in glucose medium, we predict that 638 out of the 931 reactions can be removed simultaneously while retaining the maximum growth rate (Note 4), which is comparable to 686 computed in a minimal genome study in rich media In particular, our results open a new avenue for addressing the origin of mutational robustness S. cerevisiae has shown that the single deletion of almost any non-essential gene leads to a growth defect in at least one stress condition even if the pathways are not active in any single fixed environmental condition. In light of our results, this hypothesis can be tested experimentally in medium-perturbation assays by measuring the change in growth or other phenotype caused by deleting the predicted latent pathways in advance to a medium change.A recent study on network activity and biological function.We conclude by calling attention to a limitation and strength of our results, which have been obtained using steady-state analysis. Such analysis avoids complications introduced by unknown regulatory and kinetic parameters, but admittedly does not account for constraints that could be introduced by the latter. Nevertheless, we have been able to draw robust conclusions about dynamical behaviors, such as the impact of perturbation and adaptive evolution on reaction activity. Our methodology scales well for genome-wide studies and may prove instrumental for the identification of specific extreme pathways In addition, under steady-state conditions in the media considered in this study, more than 77% of the reversible reactions become constrained to be irreversible, rendering a total of more than 92% of all reactions “effectively” irreversible.This reaction is regarded in the biochemical literature as irreversible under physiological conditions in the cell, and is constrained as such in the modeling literature Similar effective irreversibility is at work for any transport or internal reaction that is a unique producer of one or more chemical compounds in the cell.For single-reaction knockouts, MOMA predicts that 662 out of the 931 deletion mutants grow at more than 99% of the wild-type growth rate. Among these 662 reactions, 95% are predicted to be inactive under maximum growth.in silico metabolic systems used in our study are available at http://gcrg.ucsd.edu/In_Silico_Organisms. For H. pylori 26695 S. aureus N315 E. coli K12 MG1655 S. cerevisiae S288C E. coli and 1 mmol/g DW-h for S. aureus and S. cerevisiae.All the stoichiometric data for the S is the m×N stoichiometric matrix and v = T include the fluxes of n internal and transport reactions as well as nex exchange fluxes, which model the transport of metabolites across the system boundary. Constraints of the form iv≤iβ imposed on the exchange fluxes are used to define the maximum uptake rates of substrates in the medium. Additional constraints of the form iv≥0 arise for the reactions that are irreversible. Assuming that the cell's operation is mainly limited by the availability of substrates in the medium, we impose no other constraints on the internal reaction fluxes, except for the ATP maintenance flux for S. aureus, E. coli, and S. cerevisiae (see Strains and media section above). The set of all flux vectors v satisfying the above constraints defines the feasible solution spaceUnder steady-state conditions, a cellular metabolic state is represented by a solution of a homogeneous linear equation describing the mass balance condition,cTv within the feasible solution space M, which is formulated as a linear programming problem:iα = −∞ if iv is unbounded below and iβ = ∞ if iv is unbounded above. For a given objective function, we numerically determine an optimal flux distribution for this problem using an implementation of the simplex method c is taken to be parallel to the biomass flux, which is modeled as an effective reaction that converts metabolites into biomass.The flux balance analysis (FBA) To find a set of reactions from which none can be removed without forcing zero growth, we start with the set of all reactions and recursively reduce it until no further reduction is possible. At each recursive step, we first compute how much the maximum growth rate would be reduced if each reaction were removed from the set individually. Then, we choose a reaction that causes the least change in the maximum growth rate, and remove it from the set. We repeat this step until the maximum growth rate becomes zero. The set of reactions we have just before we remove the last reaction is a desired minimal reaction set. Note that, since the algorithm is not exhaustive, the number of reactions in this set is an upper bound and approximation for the minimum number of reactions required to sustain positive growth.Text S1Mathematical Results(0.06 MB PDF)Click here for additional data file.


Obligate endoparasites often lack particular metabolic pathways as compared to free-living organisms. This phenomenon comprises anabolic as well as catabolic reactions. Presumably, the corresponding enzymes were lost in adaptation to parasitism. Here we compare the predicted core metabolic graphs of obligate endoparasites and non-parasites (free living organisms and facultative parasites) in order to analyze how the parasites' metabolic networks shrunk in the course of evolution.Core metabolic graphs comprising biochemical reactions present in the presumed ancestor of parasites and non-parasites were reconstructed from the Kyoto Encyclopedia of Genes and Genomes. While the parasites' networks had fewer nodes (metabolites) and edges (reactions), other parameters such as average connectivity, network diameter and number of isolated edges were similar in parasites and non-parasites. The parasites' networks contained a higher percentage of ATP-consuming reactions and a lower percentage of NAD-requiring reactions. Control networks, shrunk to the size of the parasites' by random deletion of edges, were scale-free but exhibited smaller diameters and more isolated edges.The parasites' networks were smaller than those of the non-parasites regarding number of nodes or edges, but not regarding network diameters. Network integrity but not scale-freeness has acted as a selective principle during the evolutionary reduction of parasite metabolism. ATP-requiring reactions in particular have been retained in the parasites' core metabolism while NADH- or NADPH-requiring reactions were lost preferentially. Entamoeba histolytica and Giardia duodenalis lack mitochondria, and hence oxidative phosphorylation and respiration [Trypanosoma brucei possess mitochondria, but these are only functional in the insect stages; the bloodstream-stages rely on substrate-level phosphorylation to generate ATP [Plasmodium falciparum [Treponema pallidum [Mycoplasma genitalium [Toxoplasma gondii [Unicellular endoparasites are the causative agents of a plethora of human diseases: malaria, sleeping sickness, Chagas' disease, toxoplasmosis, leishmaniasis, amoebic dysentery and many more, particularly in the tropics. Obligate endoparasitism among the protozoa is of polyphyletic origin. Nevertheless, the different parasites exhibit striking similarities regarding their metabolism, which is reduced - or streamlined - compared to free-living organisms. The parasites have lost metabolic functions in the course of evolution , presumapiration . Trypanorate ATP ; the samlciparum . The trepallidum  or Mycopnitalium ,9, whichnitalium ). Alternnitalium . Proof oa gondii .E. coli was reconstructed that integrates information on gene expression and metabolic flux [Legionella defies categorization. For eukaryotes, the distinction between obligate endoparasites, facultative parasites, and free-living organisms is more straightforward. With the completion of the genome projects for a number of obligate endoparasitic protozoa, their metabolic networks can be reconstructed in silico from the predicted proteomes [Traditionally, the metabolism of a cell has been represented as a network of interlinked pathways . More relic flux . This inlic flux . Howeverroteomes -30. HerePathway database and used to build a reference graph of 907 nodes. Species-specific enzyme lists for the core metabolism were downloaded from KEGG Organism via LinkDB. The obligate endoparasites considered here for analysis of their metabolic networks were P. falciparum, T. brucei, T. cruzi, L. major, T. parva, E. cuniculi, C. hominis and E. histolytica. The following species were included for comparison: the free-living S. pombe, S. cerevisiae, D. discoideum, D. melanogaster and C. elegans, the facultative parasites E. coli and C. albicans, and the mammalian host H. sapiens  more isolated edges than those of either parasites or non-parasites Table , and eveE. coli . In contTo further identify potential factors that shaped core metabolism in the parasites, we tested the hypothesis that the frequency distribution of the number of links per node may have exerted a selective pressure: possibly, scale-freeness had to be maintained in order to preserve the robustness of the metabolic networks.The parasites and non-parasites exhibited roughly parallel, nearly scale-free frequency distributions of links per node, provided the nodes of connectivity 1 were ignored Figure . The samE. cuniculi and C. hominis, had the highest percentages of ATP-consuming reactions . Contrary to expectation though, the parasites' core-metabolism contained a significantly higher percentage of ATP-requiring enzymes compared to the non-parasites . Those parasites with the smallest metabolic graphs in terms of number of nodes or edges, s Figure . In facts Figure . Note thCandida albicans (which is not a model organism) clustered with the free-living eukaryotes, while the obligate endoparasite Trypanosoma brucei  clustered with the parasites , 18% were essential for growth on rich glucose medium. Of the genes encoding enzymes that catalyze ATP-consuming reactions (n = 87), 22% were essential while for reactions involving NADH or NADPH (n = 87), the fraction of essential genes was 14%. However, the differences were not statistically significant . In summary, focusing on core pathways for the reconstruction of metabolic graphs has permitted comparative genomics between obligate endoparasites and free-living (or facultative parasitic) eukaryotes and has identified the preferred retention of ATP-consuming reactions, and the enhanced loss of NADH- or NADPH- utilizing reactions, as potential selective forces which may have acted during the reductive evolution of parasitic metabolism.To study in an evolutionary context the convergence in reduction of metabolic complexity among different parasites, we focused on metabolic pathways that must have been present in the free-living ancestor of all the organisms included Table . Core mes Figure . Thus wes Figure  while cos Figure . The tots Figure  and posis Figure . This in Project -41 and fFor all of the selected core metabolic pathways, educt-product pairs of reactions and the corresponding enzymes were retrieved manually from the Reference pathway maps of the KEGG database  and suppBN performed the reconstruction of metabolic networks, their comparative analysis, and drafted the manuscript. DN calculated the quantitative network parameters. PM conceived the study, performed the statistical analysis, and finalized the manuscript. All authors read and approved the final manuscript.Reconstructed core metabolic networks in tabular form. MS Excel file with one spreadsheet per organism, containing predicted enzymes (EC number), corresponding educts, products, reaction, and pathway maps (KEGG numbers).Click here for file