The study of cellular metabolism in the context of high-throughput -omics data has allowed us to decipher novel mechanisms of importance in biotechnology and health. To continue with this progress, it is essential to efficiently integrate experimental data into metabolic modeling.in-silico framework to infer relevant metabolic pathways for a particular phenotype under study based on its gene/protein expression data. This framework is based on the Carbon Flux Path (CFP) approach, a mixed-integer linear program that expands classical path finding techniques by considering additional biophysical constraints. In particular, the objective function of the CFP approach is amended to account for gene/protein expression data and influence obtained paths. This approach is termed integrative Carbon Flux Path (iCFP). We show that gene/protein expression data also influences the stoichiometric balancing of CFPs, which provides a more accurate picture of active metabolic pathways. This is illustrated in both a theoretical and real scenario. Finally, we apply this approach to find novel pathways relevant in the regulation of acetate overflow metabolism in Escherichia coli. As a result, several targets which could be relevant for better understanding of the phenomenon leading to impaired acetate overflow are proposed.We present here an Escherichia coli.A novel mathematical framework that determines functional pathways based on gene/protein expression data is presented and validated. We show that our approach is able to provide new insights into complex biological scenarios such as acetate overflow in  Systems biology models biological processes at different hierarchical levels, ranging from genetic mechanisms to metabolic events as well as associated interactions and synergies. This “systems” perspective has allowed us to go beyond classical biology and chemistry, obtaining valuable insights regarding different fields: medicine ,2, pharmIn particular, the work presented here is focused on the metabolic layer. Metabolism has been intensively studied for the last decade due to its close relationship with cellular phenotype and, in consequence, with several diseases and biotechnological processes. The study of metabolism from a systems biology viewpoint has been accelerated by the publication of several genome-scale networks relating metabolites, enzymes and genes -14. As dGene expression can be quantified at the transcriptome level using DNA microarray technology , which pet al.,[Pathway analysis tools can be classified into two general groups depending on how they are obtained, namely via manual curation ,28 or viet al., has exteet al., presenteet al.,. In thiset al.,. This apet al.,. The veret al.,.The vast majority of approaches project gene expression data into a pre-calculated template containing a large number of pathways, regardless of the actual physiological scenario ,33,40. TE. coli[E. coli growth on glucose at high specific growth rates [et al. 2010 [acetyl-CoA synthetase and subsequent disruption of the phosphotransacetylase-acetyl-CoA synthetase (PTA-ACS) node. Despite decades of study, the regulation mechanisms and all pathways involved have not been unequivocally determined making this metabolic phenomenon a very attractive one to study and test with the new mathematical modeling method developed in the current work.Finally, we apply our new approach to examine key pathways and find novel metabolic pathways involved in the regulation of acetate overflow metabolism in E. coli. Acetateth rates ,44 and sth rates ,46. Seveth rates . Recentlal. 2010  proposedet al.[With the CFP method  as a staet al., the calH of highly/over expressed reactions, the set L of lowly/down expressed reactions and the set M = R-H-L of medium expressed/invariant reactions (where R is the complete set of reactions). Further details as to how we classified reactions based on gene and protein expression data can be found in the Methods section.We first classify each metabolic reaction as Highly/Over expressed, Lowly/Down expressed or Medium expressed/Invariant based on gene and protein data sets, as done in . For thiL; then minimize the total flux associated with reactions in M; then minimize the length of the flux path from source to target metabolite. Note here how our model integrates gene expression data  via the objective functions in our three-stage minimization. This allows gene expression data to influence the flux scenario without directly constraining it. In particular, we guide the flux through the reactions in H as a result of minimizing flux in L and M. Note that the third-stage optimization is a path length minimization step, as is common in the literature [iCFP (Integrated gene and protein expression Carbon Flux Paths).Our model is constrained to have non-zero flux between the source and target metabolites forcing an additional set of reactions to be active so as to balance the path. Given this constraint, our optimization model is a three-stage minimization: firstly, minimize the total flux associated with reactions in terature . ConclusE. coli. Finally, we apply iCFP for finding relevant metabolic pathways involved in acetate overflow metabolism in E. coli.We first highlight the inherent advantages of the iCFP approach with respect to existing approaches by means of a theoretical example. Afterwards, this theoretical example is reinforced in a more realistic scenario, namely in degradation of L-Alanine during biofilm formation of a genetically modified strain of L = {9,10,11}, M = {1,6,7,12} and H = {2,3,4,5,8}.We explain our iCFP approach by the example presented in Figure H). Even though this path is highly expressed, it needs flux in reactions 10 and 11 to achieve stoichiometric balance. This fact diminishes the relevance of the path since reactions 10 and 11 are lowly expressed. This demonstrates precisely how off-path reactions may alter the pathway enrichment structure.Classical path finding approaches, where stoichiometry is neglected, only take into account the reactions on the path. For the example shown in Figure H (reactions 2 and 4) and M (reactions 6 and 7), but no reactions from L when balancing the path.On the contrary, when iCFP is applied , the solution is as shown in Figure By means of this toy example, we show that iCFP, in contrast to classical approaches in the literature, obtains relevant paths with a consistent stoichiometric balance according to expression data. This theoretical example is now extended to a real metabolic scenario.L-Alanine (L-Ala) degradation pathways to Pyruvate (Pyr) during biofilm formation of a genetically modified strain of E. coli.In this subsection, using a more realistic scenario, we reinforce the importance of considering stoichoimetric balance when selecting relevant paths based on expression data. In particular, we analyze D-Amino acid dehydrogenase (DAAD), consumes and produces Flavin adenine dinucleotide oxidized (FAD) and Flavin adenine dinucleotide reduced (FADH2), respectively. We show below that the balancing of these two compounds plays a crucial role, confirming that the example presented in the previous section goes beyond a hypothetical scenario.Details as to the reactions constituting the backbone of L-Ala degradation pathway can be found in EcoCyc , which aE. coli (GSE14203). Several gene expression comparisons between cultures are presented in terms of log2 fold changes. In particular, we focus on the comparison between wild type and mqsR mutant E. coli (GSM355066/GSM355065). As in [2 (fold change) > 0.5850) are considered highly/over expressed genes; on the contrary, genes with a fold change below -1.5 (log2 (fold change) < -0.5850) are included within the set of lowly/down expressed genes. If a gene is neither highly nor lowly expressed, it is classified as a medium expressed gene.We took data from . Kim andE. coli in [H, M and L. We used the same growth medium (Lysogeny Broth (LB)) in our simulations as in [We incorporate gene expression data into the genome-scale metabolic network of  coli in . Followi coli in , we obtans as in .L and M, especially between the first three  and Pyridoxamine 5′-phosphate (PyAm5P) for Figure L, while for Pydx5P and PyAm5P only reactions in M and H are required. For this reason the pathway in Figure With the conditions described above, when iCFP is applied between L-Ala and Pyr, three paths are obtained Figure B, C, D be Figure B, C, D ae Figure A pathwayD-amino acid dehydrogenase (DAAD) in Figure M and L, respectively. If this differentiation had been considered, we can only expect a worse  position in the ranking for the canonical pathway in Figure As a technical note, consider that FAD/FADH2 is tightly bound to a protein, in our case to In summary, with this example of L-Ala degradation pathways into Pyr, we show that iCFP successfully accounts for stoichiometric balancing when selecting pathways based on gene expression data. This constitutes a progress over the state of the art, as it is typically neglected in classical path finding methods.E. coli from a systems biology viewpoint using advanced steady-state continuous cultivation  down-regulation results in decreased assimilation of acetate via phosphotransacetylase (PTA). We apply iCFP using the gene and protein expression data from [E. coli presented in [L/M/H sets is included in the Methods section.Recently, Valgepea et al., 2010  studied  (A-stat ) by inteata from  and the ented in  to gain Acetate (Ac) [Glucose (D-Glc), the unique external input of carbon in the experimental growth medium in [i.e. ignoring the first two stages in iCFP). Only 10 of 100 paths were the same in both scenarios, namely with and without gene expression data. In other words, our results here indicate that incorporating expression data in the manner described above does significantly alter the set of paths found. A detailed analysis that discusses the effect of incorporating expression data and the resulting paths can be found in Additional file Since the consequence of acetate overflow metabolism is the excretion of ate (Ac) , we set edium in , as the Acetyl Coenzyme A (AcCoA) as an intermediate metabolite. This is in line with a previous hypothesis for acetate overflow metabolism [Of the 100 paths from D-Glc to Ac determined using iCFP, we observed that 98 had tabolism ,58, whicAcetyl phosphate (AcP) as an intermediate metabolite. In particular, AcP is transformed to Ac through the highly expressed reaction acetate kinase (ACKr). Further details regarding this mechanism can be found in Valgepea et al., [The first path appearing in the ranking is one half of the PTA-ACS cycle recently proposed by Valgepea et al.,  as being et al., .Acetaldehyde (AcAld). In addition, note that the enzyme producing Ac from AcAld ) is over-expressed  and L-Glutamate 5-semialdehyde (Glu5sA), respectively. Since extracellular accumulation of Glu5sA was not detected in these experiments, we will analyze two mechanisms to balance this by-product:iCFP also provides the pathway in Figure y Figure C consumeProline (L-Pro) in a two-reaction pathway  (G5SADs) and pyrroline-5-carboxylate reductase (P5CR)). This pathway represents the canonical biosynthetic mechanisms of L-Pro [1. On one hand, Glu5sA can be consumed to produce of L-Pro . This is1-pyrroline-5-carboxylate dehydrogenase (P5CD), Glu5sA can be consumed to produce the required molecule of L-Glu consumed in ACGS reaction so that the pathway in Figure 2. On the other hand, by means of another two-enzyme pathway constituted by G5SADs and i.e. P5CR for L-Pro production and P5CD in the case of L-Glu, respectively. Interestingly, P5CR is classified as belonging to the set L, while P5CD is classified as belonging to the set M . Then, OAA is degraded by means of L-Threonine (L-Thr) metabolism [citrate synthase (CS) and citrate lyase (CITL). It is important to note that there is no effective carbon exchange between AcCoA and OAA through these two reactions. In contrast, it can be verified that Ac is fully composed by carbon atoms from OAA. As OAA can be produced by several mechanisms, such as glycolysis, effective carbon exchange is guaranteed between D-Glc and OAA, and in consequence between D-Glc and Ac. In this light, OAA can be produced from the glycolytic product phosphoenolpyruvate (PEP) by phosphoenolpyruvate carboxylase (PPC), which is classified as belonging to the set H.Finally, we discuss the pathway in Figure tabolism  so that Threonine aldolase (THRAi), L-Thr is degraded into glycine (Gly) and AcAld, which is then converted into Ac via aldehyde dehydrogenase (ALDD2x). Given that this pathway was obtained as differentially (highly) expressed with increasing specific growth rate, Gly and L-Thr should be synthesized at a high rate and, therefore, this pathway is likely to occur also at a high rate. The use of less efficient pathways also emphasizes increasing carbon wasting with faster growth as experimentally observed (mainly associated with acetate overflow). It is interesting to note in the context of acetate overflow that higher biomass yields are observed by supplementing minimal medium with Gly and L-Thr in addition to glucose [We are aware that this pathway is not particularly efficient for Ac production, as it requires 10 enzymatic steps and the consumption of 2 molecules of ATP and NADPH, respectively. However, this long path involves a well-known route for the biosynthesis of L-Thr, an essential amino acid for biomass synthesis. In addition, in the reaction catalyzed by  glucose , which iThere is a large volume of gene expression data available through different public databases. In addition, due to the rapid advancement in proteomics technologies, protein abundance data is increasing day by day, as well as absolute quantitative –omics data. In order to exploit this valuable information, we require models and efficient algorithms to extract biological conclusions.E. coli. In order to take into account the balancing in the obtained paths, the use of MILP is required. Efficient MILP solvers exist and therefore the application of iCFP to genome-scale metabolic networks is feasible, as shown in the Results section.Genome scale networks have shown that cellular metabolism underlies a wide number of phenotypes. As discussed elsewhere , represeAs posed above, genome scale metabolic networks may present assorted phenotypes. In order to correctly represent those phenotypes, it may be necessary to pre-calculate an extremely large number of metabolic pathways. The framework presented here overcomes this issue by directly calculating the best metabolic pathways under a given gene/protein expression phenotype. The effect of gene and protein expression data in our approach is clearly observed, particularly leading to two significantly different sets of pathways depending upon whether this information is included or not.H, M or L sets. We defined these sets based on a standard procedure (see Methods section). However, this problem deserves further research, as complex effects may arise. For instance, reducing the concentration of a particular protein may not lead to a proportional decrease in its catalytic activity [In iCFP, reactions are classified into activity ,63. TherAcetaldehyde is of particular interest, which to our knowledge has not been previously related to acetate overflow. In addition, we also discussed two other over-expressed metabolic pathways possibly relevant for acetate overflow. The evidence of acetate production directly linked to Gly and L-Thr seems plausible, as their importance for increasing biomass yield per substrate has been previously reported and explored in the context of acetate overflow. We are aware that overflow metabolism is a more general problem and additional studies, which complement the one presented here, need to be carried out to design an optimized strain, as if acetate secretion is blocked, the strain may still export other compounds such as lactate at a high rate.Based on the above, we proposed novel metabolic pathways involved in acetate overflow metabolism, which might be targets to possibly mitigate acetate overflow and help to understand the regulation of this phenomenon. The role of E. coli genome-scale metabolic network [As future lines for research into acetate overflow, note that our insights were obtained from the metabolic model presented in , as carb network  would ne network  and prot network , it seemE. coli acetate overflow is one of these complex scenarios. With our iCFP approach, we were able to develop non-trivial insights that contribute to the better understanding of this phenomenon relevant for industry biotechnology.In conclusion, the increasing efficiency of high-throughput technologies in association with novel computational frameworks lays the foundations for future discoveries. Overall, methods exploiting -omics data, as is the case of iCFP presented in this article, are essential to unravel the complex biological scenarios that exist in different areas of health science and biotechnology. The data presented in  contains2), which is particularly common in the literature [E. coli metabolic networks are removed  change in expression between the initial and final condition increases or decreases by 50%. Note that these probabilities are calculated for gene and protein expression data and, therefore, two sets of p-values are calculated , the set corresponding to the Lowly/Down expressed (L) and the set of Medium expressed/Invariant reactions (M) based on the Boolean rules included in [H, L and M sets, respectively.Finally, following the strategy adopted in previous works , we clasluded in . OverallC contains all the metabolites present in the metabolic network whilst R contains all the reactions. Reversible reactions contribute two different reactions to the metabolic network, so all fluxes (flux vr for reaction r) take positive values. Scr is the stoichiometric coefficient associated with metabolite c in reaction r. The source and target metabolites are α and β, respectively. A flux path is a simple path from α to β able to operate in steady-state. Arcs in the path are given by the zero-one (binary) variable uij = 1 if the arc i→j linking metabolite i to metabolite j is active in the path, 0 otherwise. The zero-one (binary) variable zr is equal to 1 if reaction r has non-zero flux, 0 otherwise. The coefficient dijr is one if there exists effective carbon exchange between input metabolite i (Sir < 0) and output metabolite j (Sjr > 0) in reaction r (and is zero otherwise). The constraints in the model are:Drawing directly on , the modα and one arc enters β. Equation (2) that no arc enters α and no arc leaves β. Equation (3) ensures that the number of arcs entering a metabolite k is equal to the number leaving. Equation (4) ensures that a metabolite cannot be revisited in the path. Equation (5) is the steady-state condition for the set (I) of internal metabolites. Equation (6) ensures that metabolites not involved in a specific growth medium (Em) can be produced, but cannot be consumed. Equation (7) relates each flux with a binary variable that indicate whether the reaction is active or not, where N is the maximum flux and the minimum (non-zero) flux is 1. Equation (8) prevents a reaction and its reverse both being active where B = {| reaction λ and reaction μ are the reverse of each other}. Equation (9) ensures that if a carbon exchange arc is used in the path, then some reaction involving such arc is active.Equation (1) ensures that one arc leaves R of reactions has been subdivided into three mutually exclusive sets L/M/H of lowly/medium/highly expressed reactions. Our model is a three-stage optimization model. In the first-stage optimization, we minimize the total flux associated with lowly expressed reactions, namely those included in the set L:With expression data, the set Subject to Equations (1)-(9)This is a mixed-integer linear problem and algorithmically such problems are solved by linear programming based tree search. Modern software packages to perform this task, such as IBM ILOG CPLEX which we used, are well developed and highly sophisticated. IBM ILOG CPLEX was run in a Matlab environment Version 7.5 (R2007b).V1, then, in the second-stage optimization, we minimize the total flux associated to reactions in M, i.e. medium expressed reactions:If the optimal objective function value is Subject to Equations (1)-(9) and V1) for This ensures that we retain the optimal value -(9) and This ensures that we retain the optimal values achieved at the first two stages.Note here that the three stage optimization model can be converted into a single step optimization by introducing suitable weighting factors for each term in equations (10)-(12), namely very large for the first objective, less large for the second, etc. However, this introduces numerical issues as the different terms in the objective vary widely in value. So, for the sake of numerical stability, we opted for the three step optimization model.Uijk is the binary solution value for the uij variable in the k-shortest path (and we have K such paths) then the elimination constraint is:Since the last stage here is to minimize the number of reactions, we can enumerate path solutions in ascending (staged) objective value order, so first find the CFP that best minimizes our three-stage objective; then the next best solution; then the next best solution; etc. To do this, after finding a path solution, we simply add a constraint eliminating it from the solution space. If 1Pyr5c: 1-Pyrroline-5-carboxylate; 3MOB: 3-Methyl-2-oxobutanoate; ACALD: Acetaldehyde dehydrogenase (acetylating); ACGK: Acetylglutamate kinase; ACGS: N-acetylglutamate synthase; ACKr: Acetate kinase; AGPR: N-acetyl-g-glutamyl-phosphate reductase; ALAR: Alanine racemase; ALATA_D2: D-alanine transaminase; ALATA_L2: Alanine transaminase; ALDD2x: Aldehyde dehydrogenase ; ASAD: Aspartate-semialdehyde dehydrogenase; ASPK: Aspartate kinase; ASPTA: Aspartate transaminase; Ac: Acetate; AcAld: Acetaldehyde; AcCoA: Acetyl-CoA; AcG5P: N-Acetyl-L-glutamyl 5-phosphate; AcG5sA: N-Acetyl-L-glutamate 5-semialdehyde; AcGlu: N-Acetyl-L-glutamate; AcP: Acetyl phosphate; CFP: Carbon Flux Path; CITL: Citrate lyase; D-Ala: D-Alanine; D-Glc: D-Glucose; DAAD: D-Amino acid dehydrogenase; EFM: Elementary Flux Mode; FAD: Flavin adenine dinucleotide oxidized; FADH2: Flavin adenine dinucleotide reduced; FDR: False discovery rate; G5SADs: L-glutamate 5-semialdehyde dehydratase (spontaneous); Glu5sA: L-Glutamate 5-semialdehyde; Gly: Glycine; HSDy: Homoserine dehydrogenase (NADPH); HSK: Homoserine kinase; L-Ala: L-Alanine; L-Asp: L-Aspartate; L-AspsA: L-Aspartate 4-semialdehyde; L-Glu: L-Glutamate; L-Hom: L-Homoserine; L-PHom: O-Phospho-L-homoserine; L-Pro: L-Proline; L-Thr: L-Threonine; L-Val: L-Valine; LB: Lysogeny Broth; MILP: Mixed-Integer Linear Programming; NACODA: N-acetylornithine deacetylase; NH4: Ammonium; OAA: Oxaloacetate; P5CD: 1-pyrroline-5-carboxylate dehydrogenase; P5CR: Pyrroline-5-carboxylate reductase; PDH: Pyruvate dehydrogenase; PEP: Phosphoenolpyruvate; PPC: Phosphoenolpyruvate carboxylase; PTA: Phosphotransacetylase; PyAm5P: Pyridoxamine 5′-phosphate; Pydx5P: Pyridoxal 5′-phosphate; Pyr: Pyruvate; THRAi: Threonine aldolase; THRS: Threonine synthase; VPAMT: Valine-pyruvate aminotransferase; iCFP: Integrative carbon flux path.The authors declare that they have no competing interests.JP conceived, developed and implemented the method, wrote the manuscript; KV developed the biological validation and wrote the manuscript; AR, designed the statistical analysis and wrote the manuscript; JEB developed the method and wrote the manuscript; FJP conceived the study, developed the method and wrote the manuscript. All authors discussed the results, and read, commented and approved the final manuscript.Details for pathways between Glucose-Acetate and Acetyl Coenzyme A-Acetate.Click here for fileEffect of gene expression data and sensitivity analysis on T value.Click here for file


Optimization procedures to identify gene knockouts for targeted biochemical overproduction have been widely in use in modern metabolic engineering. Flux balance analysis (FBA) framework has provided conceptual simplifications for genome-scale dynamic analysis at steady states. Based on FBA, many current optimization methods for targeted bio-productions have been developed under the maximum cell growth assumption. The optimization problem to derive gene knockout strategies recently has been formulated as a bi-level programming problem in OptKnock for maximum targeted bio-productions with maximum growth rates. However, it has been shown that knockout mutants in fact reach the steady states with the minimization of metabolic adjustment (MOMA) from the corresponding wild-type strains instead of having maximal growth rates after genetic or metabolic intervention. In this work, we propose a new bi-level computational framework--MOMAKnock--which can derive robust knockout strategies under the MOMA flux distribution approximation.In this new bi-level optimization framework, we aim to maximize the production of targeted chemicals by identifying candidate knockout genes or reactions under phenotypic constraints approximated by the MOMA assumption. Hence, the targeted chemical production is the primary objective of MOMAKnock while the MOMA assumption is formulated as the inner problem of constraining the knockout metabolic flux to be as close as possible to the steady-state phenotypes of wide-type strains. As this new inner problem becomes a quadratic programming problem, a novel adaptive piecewise linearization algorithm is developed in this paper to obtain the exact optimal solution to this new bi-level integer quadratic programming problem for MOMAKnock.E. coli core metabolic network and a large-scale iAF1260 E. coli metabolic network. The derived knockout strategies are compared with those from OptKnock. Our preliminary experimental results show that MOMAKnock can provide improved targeted productions with more robust knockout strategies.Our new MOMAKnock model and the adaptive piecewise linearization solution algorithm are tested with a small  Metabolic engineering has become an important environment friendly process in modern biotechnology, providing new potential solutions to many global problems, including energy and environmental crisis -5. Metabin silico whole-cell simulations possible to identify globally optimal targets for metabolic engineering with accurately predicted phenotypes In silicoramework  has beenResearchers have proposed different metabolic engineering methods based on these metabolic approximation models and typically the improved strains are sequentially modified based on FBA with multiple mutants. However, sequential metabolic engineering strategies do not have the guarantee of the optimality. In , the autIn this paper, we propose a bi-level programming framework for the identification of optimal genetic manipulations under the MOMA assumption. With the new MOMA assumption to approximate the condition to maintain the cell liveness as the essential phenotypic constraints, the inner optimization problem becomes a quadratic programming (QP) problem rather than the linear programming (LP) problem in OptKnock. To address the raised computational complexity, we develop a novel adaptive solution algorithm to solve this new bi-level optimization problem. The new algorithm under the minimizing flux adjustment assumption is tested on metabolic networks and our preliminary experimental results show that our framework can generate more practical and robust knockout strategies compared to OptKnock.S is 0: i ≤ N, in which we assume that the network model has M reactions and N metabolites in total; ij Sis the stoichiometric coefficient of metabolite i in reaction j; and j vdenotes the flux value of reaction j. For wild-type strains, as mentioned above, a common assumption is that their steady-state flux values follow an optimal distribution that maximizes the biomass production rate. The steady-state flux distribution is approximately solved as a LP problem to maximize the biomass production flux: biom vsubject to the FBA stoichiometry constraints, in which biom vis defined by summing up the metabolite precursors that contribute to the biomass production in FBA [j vto zero with the resulting knockout flux distribution maintaining biomass maximization assumption.Before introducing our new bi-level programming problem to identify optimal metabolic genes or reactions to delete for the maximization of targeted bio-productions, we first review the mathematical foundations of FBA  and MOMAn in FBA . In OptKL2 distance between the knockout flux values to wild-type steady-state flux values:As stated in , engineej vrepresents the flux value of reaction j in mutant strains and j wis the corresponding flux value in wild-type strains. The flux value for biomass production biom vis similarly defined as mentioned earlier. In addition, the glucose flux value glc vdenotes the glucose consumption rate, which is often set to a fixed value glc_uptakev. Finally, jv, which are determined by the availability of nutrients or the maximal fluxes that can be supported by enzymatic pathways [where pathways .Following the modeling strategy in OptKnock , we aim j y∈ {0, 1}, 1 ≤ j ≤ M, denoting gene or reaction knockout strategies in which reaction j either is knocked out (j y= 0) or remains active (j y= 1). The identification of optimal knockout strategies j yunder MOMA requires to solve the following bi-level programming problem:Mathematically, we introduce binary variables K is the allowed maximum number of knockouts and chemical vcorresponds to the reaction that produces the desired biochemical production target. Note that we do not count in the flux change for the target reaction in the inner problem as it would contradicts to our primal optimization for maximal biochemical overproduction.in which j vin knockout strains. As this nested inner problem is convex, we can still get its dual problem and the strong duality condition still holds for the inner primal and dual problems. Following the similar direction of [We emphasize that the nested inner optimization problem is a QP problem with respect to flux allocation ction of , we can new bi-level model into a single-level problem based on the LP strong duality. For the linearized problem, we can obtain the optimal solution similarly as in [To derive efficient solution algorithms for our new bi-level programming gene knockout problem, we adopt a novel adaptive linearization solution strategy to tackle the computational complexity introduced by the inner QP problem. Specifically, we propose to adaptively represent the quadratic terms in the objective function of the inner problem using a set of linear functions as illustrated in Figure ly as in  by solviv1, which can be represented by a convex combination of endpoints of piecewise segments for a given piecewise linearization. The corresponding quadratic objective function value at v1 is denoted by M1, which can be approximated linearly by The basic idea of adaptive piecewise linearization is illustrated in Figure With this basic understanding of our new bi-level model and adaptive piecewise linearization solution strategy, we describe the detailed algorithm in the following sections.jw) in MOMA, is the key obstacle to derive the efficient solution strategy. We propose to use piecewise linear functions to approximate this quadratic objective function. The basic idea of piecewise linearization is to assume that each reaction flux value j vcan be discretized into a finite number of segments, each of which is precisely defined by its corresponding consecutive endpoints j vcan then be represented by a convex combination of these endpoints:The quadratic objective function of the inner problem, denoting the metabolic adjustment to wild-type steady-state flux allocations  that produce nadph (Nicotinamide adenine dinucleotide phosphate - reduced) is removed. When K = 4, one Glycolysis reaction (dhap → gap) that produces most portion of gap is removed. Both nadph and gap are important precursors in the biomass reaction. Removing these reactions causes the reduction of biomass flux values.Based on the results from OptKnock in Table K = 4, in which OptKnock derives an impractical strategy. The optimal succinate flux value from MOMAKnock suggested deletions can improve at least 37.5 percent compared to OptKnock in the MOMA flux distribution. In addition, both the succinate and biomass reaction flux values change smoothly for MOMAKnock strategies. Finally, as expected due to the L2 distance based phenotypic constraints in the inner level of MOMAKnock, we can see that the optimal knock flux distributions from MOMAKnock is always closer to the wild-type flux distribution compared to OptKnock suggested knockouts.Table K is 2, one of the suggested knockout reactions is to eliminate the reaction that decompose the succinate (suc), and another one is to remove the reactions that involve competing byproduct metabolism for succinate such as 6-Phospho-D-gluconate (6pg) and Ribulose 5-phosphate (ru5p). With K = 3, MOMAKnock adds one additional knockout reaction to the previously identified ones based on the K = 2 case, which leads to the increase of succinate production to 32.15% of its theocratical maximum value. When K = 4, besides the reactions that consume succinate and the competing reactions, the reaction that decompose Phosphoenolpyruvate (pep) is also detected. This increases the succinate to 37.9% of the thearetical maximum. Finally, when K increases to 5, one more reaction is knocked out, which lead to 53.31 mmol/gDW · hr succinate produce rate. While as mentioned above, this reaction can convert biomass product to biomass precursor, so the deletion causes the reduction of the biomass flux rate.Biologically, it is interesting to note that our MOMAKnock indeed identifies relevant reactions as suggested knockout reactions. For example, when the knockout number K, both the targeted succinate flux values and biomass values change smoothly contrasting to the abrupt changes in OptKnock, which may also serve as an evidence that MOMAKnock can help derive more robust knockout strategies with predictable performance.Based on these preliminary results on this core network model, even though the OptKnock takes the maximizing biomass production as the inner cellular objective, the derived knockout strategies do not always achieve high biomass production when we simulate these knockout strategies under the MOMA objective. Sometimes, these knockout strategies cannot even guarantee the minimum biomass requirement. The reason for this is that the inner optimization in the bi-level framework of OptKnock serves as the additional constraint for the outer optimization problem. The derived optimization procedure first considers the outer problem as the primary objective and then the inner problem is optimized. The simulated low targeted chemical production rates for OptKnock suggested knockouts in the MOMA flux distribution and the abrupt biomass level changes in OptKnock illustrate that the biomass maximization assumption to approximate cellular objectives may not provide robust and reliable metabolic reaction deletion strategies. On the other hand, MOMAKnock approximates the inner cellular objective by the MOMA assumption which assumes that knockout strains stay closer to the corresponding wild-type strains. If this is guaranteed, knockout strains also can achieve appropriate biomass flux values. In fact, as shown in Tables E. coli metabolic network, it is clear that our MOMAKnock may suggest more practical and robust knockout strategies for optimal bio-productions under phenotypic constraints.By comparison with OptKnock on this core E. coli metabolic network model--iAF1260 [mmol/gDW · hr, and the minimum biomass is also set to 5 mmol/gDW · hr. All of our experiments are still based on the aerobic environment and all of the data are also from the OptKnock software package [K = 3, 4, 5. Figure K = 5.We further test MOMAKnock on a large -iAF1260 , which h package . Table 3K = 5, the succinate production can achieve as high as 79.73% of the theocratical maximum rate (83.62mmol/gDW · hr), which demonstrates that our MOMAKnock can serve as a computational tool for deriving potentially effective and robust knockout strategies.From Figure mmol/gDW · hr, which is the minimum biomass value set in the all of the tests. However, in this large network, the theoretical maximum biomass is 9.657mmol/gDW · hr. Experiments shows that if the the succinate dehydrogenase reaction (SUCDi) is recovered from knockouts, we can get higher biomass but the succinate can drop to as low as 10 mmol/gDW · hr. As shown in Figure We notice that in Table E. coli metabolic network model [E. coli metabolic network model [in silico metabolic engineering to help derive effective genetic or metabolic intervention strategies through genome-scale network dynamic analysis based on the FBA framework. To better approximate the phenotypic constraints for knockout strains, we have take the MOMA assumption instead of the maximal growth assumption as in OptKnock to model the underlying cellular objective. Based on the obtained results on two network models, it is clear that MOMOKnock derives improved knockout strategies under the MOMA objective, which are more robust and practical.In this paper, we have proposed a new bi-level programming optimization framework to identify optimal knockout strategies for maximum targeted bio-productions under the phenotypic constraints approximated by the MOMA assumption. A novel adaptive piecewise linearization solution strategy has been developed to efficiently solve this new mixed integer quadratic bi-level programming problem. The preliminary experiments on both the core rk model  and the rk model  have demin silico metabolic engineering. In addition to that, according to different cellular objective assumptions, we can formulate different inner problems as phenotypic constraints in this bi-level programming framework to derive optimal intervention strategies under different conditions. Our future research will focus on developing and testing such new models for large-scale metabolic networks. For example, as ROOM [L2 distance objective function in MOMA by either L0 or L1 norm, which will lead to different bi-level optimization problems. We will develop corresponding solution strategies to solve this category of bi-level problems for large-scale networks and compare their performances with respect to the efficacy and robustness of the correspondingly derived intervention strategies.Our new bi-level MOMAKnock model can serve as an alternative method with slightly higher computational complexity to OptKnock for  as ROOM  suggestsThe authors declare that they have no competing interests.Conceived and designed the experiments: XQ. Designed and Implemented the algorithm: SR, BZ, XQ. Performed the experiments: SR. Analyzed the results: SR, BZ, XQ. Wrote the paper: SR, BZ, XQ.The publication costs for this article were funded by the corresponding author's institution.BMC Bioinformatics Volume 14 Supplement 2, 2013: Selected articles from the Eleventh Asia Pacific Bioinformatics Conference (APBC 2013): Bioinformatics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/14/S2.This article has been published as part of


Non-alcoholic fatty liver disease comprises a broad spectrum of disease states ranging from simple steatosis to non-alcoholic steatohepatitis. As a result of increases in the prevalences of obesity, insulin resistance, and hyperlipidemia, the number of people with hepatic steatosis continues to increase. Differences in susceptibility to steatohepatitis and its progression to cirrhosis have been attributed to a complex interplay of genetic and external factors all addressing the intracellular network. Increase in sugar or refined carbohydrate consumption results in an increase of insulin and insulin resistance that can lead to the accumulation of fat in the liver. Here we demonstrate how a multidisciplinary approach encompassing cellular reprogramming, transcriptomics, proteomics, metabolomics, modeling, network reconstruction, and data management can be employed to unveil the mechanisms underlying the progression of steatosis. Proteomics revealed reduced AKT/mTOR signaling in fibroblasts derived from steatosis patients and further establishes that the insulin-resistant phenotype is present not only in insulin-metabolizing central organs, e.g., the liver, but is also manifested in skin fibroblasts. Transcriptome data enabled the generation of a regulatory network based on the transcription factor SREBF1, linked to a metabolic network of glycerolipid, and fatty acid biosynthesis including the downstream transcriptional targets of SREBF1 which include LIPIN1 (LPIN) and low density lipoprotein receptor. Glutathione metabolism was among the pathways enriched in steatosis patients in comparison to healthy controls. By using a model of the glutathione pathway we predict a significant increase in the flux through glutathione synthesis as both gamma-glutamylcysteine synthetase and glutathione synthetase have an increased flux. We anticipate that a larger cohort of patients and matched controls will confirm our preliminary findings presented here. Non-alcoholic fatty liver disease (NAFLD) is on the rise in the western population affecting one-third of adults and an increasing proportion of children  in the liver. In histological sections of liver parenchyma TG droplets are visible in the cytoplasm of hepatocytes. This accumulation of TG in small or large vesicles impairs hepatic function and makes the liver more susceptible to other injuries. It has been shown that patients with NAFLD have a worse outcome when compared with an age and sex-matched general population, stemming from cardiovascular and liver related mortality  or are synthesized in vivo. Congenital generalized lipodystrophy, glycogen storage disease type 1a, and citrin deficiency all cause severe hepatic steatosis even in the absence of obesity or insulin resistance. Several mutations and genetic defects are known to influence TG accumulation in the liver, like mutations of ATGL which prevent mobilization of TG from the hepatocytes or mutations in apolipoprotein B which impair TG secretion to the bloodstream  and hepatocellular carcinoma (HCC), finding genetic determinants of susceptibility to NAFLD might help in the prevention of NASH and HCC.Studying NAFLD in human patients is limited by the need for liver biopsies to determine the stage of NAFLD and by the strong influence of the individual environment of the patient with respect to food intake and exercise. Nevertheless some genome wide association studies have recently been performed and identified genetic variants that have distinct effect on metabolic traits  cells can thus present a model system in which the response of naïve hepatocytes with the patient’s genome to experimental stimuli can be tested and analyzed. These unique features of iPS cell-derived hepatocytes could help in elucidating the genetic networks that are responsible for the different reaction of the patients hepatocytes to the environmental stimulus presented to them by the patient’s lifestyle.in silico analysis which is made up of data integration, pathway reconstruction, and modeling. A description of the consortium can be found at http://www.erasysbio.net/index.php?index=264.In this manuscript we illustrate how a multi-disciplined systems biology approach can be used to study the etiology of steatosis. The scheme presented in Figure Starting in 2009, patients with morbid obesity were offered to participate in the Multidisciplinary Obesity Research (MORE) project at the Medical University of Graz, Austria. This study aims to take an interdisciplinary approach in research on obesity and related diseases like insulin resistance, liver steatosis and metabolic syndrome and includes the Institutes of Gastroenterology, Pathology, Surgery and Experimental Hepatology at the University Hospital of Graz. Patients with morbid obesity admitted into hospital for treatment by bariatric surgery  were invited to participate in the study and to sign the informed consent. In addition to obese patients we also included three healthy individuals in the study to obtain control dermal fibroblasts. If patients were admitted for bariatric surgery, samples of blood, skin and a liver biopsy were taken during the operation. The skin samples were immediately transported to the Institute of Pathology and fibroblast cultures were initiated. The liver biopsy was divided into one part for fixation and histology and a second part for liquid nitrogen storage while the blood sample was separated into mononuclear cells and plasma. Subsequently the disease phenotype characterization comprised detailed analysis of glucose  and lipid metabolism, hypertension, atherosclerosis, and histopathological evaluation of the liver biopsy. In addition, HOMA IR (homeostatic model assessment insulin resistance) was calculated for patients. The average normal-weight individual has an insulin resistance of 1, while for obese patients recruited for the study the HOMA IR index is within the range from 2.82 to 4.86, indicating the insulin resistance phenotype. Moreover, other diseases had to be excluded, and a detailed evaluation of the lifestyle  was performed from the structured questionnaires and, where applicable, biomarkers and physical measurements.Primary fibroblasts were derived by standard tissue culture practice. In brief, the skin surface was sterilized with ethanol and subcutaneous fat was removed. The sample was divided into small pieces (2 mm × 2 mm) and each of these pieces was minced in a culture dish, supplemented with 1 mL of DMEM, 10% FCS with antibiotics, and covered with a sterile coverslip. After approximately 2 weeks skin fibroblasts started to grow out from the skin pieces and after approximately 4 weeks the fibroblasts were harvested and expanded.Primary fibroblasts from two patients with steatosis  and a non-obese control (H0002) were established.After washing with PBS two times cells were directly lysed in M-PER buffer (Pierce) with EDTA free complete inhibitors (Roche) and phosphostop (Roche). Lysates were immediately frozen at −80°C. Further samples were allowed to thaw on ice and incubated on a rotating shaker at 4°C for 20 min. To clear protein samples, lysates were centrifuged at maximum speed for 30 min at 4°C. The protein supernatant was collected and protein concentration was estimated using a micro BCA method  and the p-value from Welch’s test (t2p < 0.05). The Bioconductor package q-value  of at least one of the control or treatment group and the false discovery rate were usually 0.05.Gene expression analysis was performed using the Illumina HumanRef-8 v3 BeadChip. The bead-level data generated by the image scanner software were summarized and saved without background correction and normalization in the Illumina BeadStudio. Bead-summary data was analyzed via R/Bioconductor  databaseGene-set enrichment and over-representation analysis (ORA) was achieved utilizing the unique pathway database integration system ConsensusPathDB . The modified model consists of the methionine cycle, glutathione synthesis and detoxification, ophthalmic acid synthesis, and the gamma-glutamyl cycle. We are more confident in the ability of this model to make quantitatively accurate predictions as this model was fitting on more experimental data (than the previous model) and is partially validated . However we have not validated the model for predicting the effect in changes in Vmax and therefore the predictions made here will offer limited certainty.For this work an updated version of the glutathione model previously published was used , patient H0007, and H0008 are presented in Table This model represents the control data set discussed in this study. The difference in expression between control (H0002) and steatosis  was first calculated and then used to calculate the Vmaxs was also incorporated in the control model based on the range of expression data of the control patient. Three models were then analyzed: the control (H0002) model, H0007, and H0008. In order to calculate the fluxes/concentrations of the model, the model selected a Vmax value for each enzyme within the new calculated range and the steady state was calculated. This was repeated 5,00,000 times using Condor-COPASIVmax values randomly chosen by COPASI. The average and standard deviation (SD) of flux/concentration was calculated from this data.Similarly, a range of Pathomorphological assessment of the histology of the liver biopsies Figure  allowed Retroviruses encoding the reprogramming factors, OCT4, SOX2, KLF4, and c-MYC were transduced into fibroblasts cells from patient with steatosis (H0008) and control (H0002) using well established protocols  and 33 year, female, BMI 51.3 (H0007)] and control fibroblasts  were analyzed. The clustering of the respective transcriptomes is shown in Figure p-value < 0.01) in steatotic  and control (H0002) fibroblasts. The Venn diagram in Figure First we considered all genes expressed  and cell cycle . Furthermore, ORA suggests that members of the NEST set related to glutathione metabolism  show jointly significant increase in transcription levels in steatosis patients. The NEST analysis is based on the genes glutathione S-transferase A4 (GSTA4), glutathione S-transferase theta-1 (GSTT1), and glutathione S-transferase mu 1 and 2 , which were identified as parts of the glutathione metabolism pathway, and the analysis identified a NEST gene-set that is centered around cytochrome CYP4F22  is up-regulated in the steatosis patients and catalyzes the formation of the oxidized form of glutathione  from glutathione (GSH). GPX4 is the only antioxidant enzyme known to directly reduce phospholipid hydroperoxides within membranes and lipoproteins, acting in conjunction with alpha-tocopherol to inhibit lipid peroxidation  compared to control (H0002) are enriched with respect to specific pathways, ORA was carried out with ConsensusPathDB. Using the hypergeometric test statistic of ConsensusPathDB on genes that are differentially expressed in the patients we were able to assign each pathway a LPIN and LDLR which encode lipin and low density lipoprotein receptor (LDLR), respectively, are downstream targets of SREBF1 , which is a decamer flanking, e.g., the LDL receptor gene (LDLR) and other genes involved in sterol biosynthesis. In addition to the transcriptional activation of LPIN and LDLR, Fajas et al. reported that SREBF1 regulates the expression of PPARG during adipocytes differentiation. Ishimoto et al.  regulating the transcription of genes encoding lipid droplet (LD) proteins such as PLIN1, PLIN2, PLIN3, PLIN5, FSP27, and HIG2, known to be expressed in steatotic liver. LD proteins play a role in the pathophysiology of fatty liver disease where hepatocytes enriched with LD proteins contain vast amounts of lipids.In addition, SREBF1 regulates the expression of the gene Reverse phase protein arrays (RPPA) were used to analyze protein extracts derived from skin fibroblast cell lines obtained from two steatosis patients  and from a healthy lean control (H0002). In detail, the expression level of 109 proteins and phosphoproteins was quantified and several target proteins as well as phosphoproteins were identified as differentially regulated. For example, phosphorylation levels of major signaling proteins such as AKT, PRAS40, p70S6K (T389), ERK1/2 were strongly and significantly reduced in both patient-derived fibroblast cell lines whereas expression and phosphorylation of insulin receptor beta (IRβ) and its key adaptor protein IRS1 were comparable in control cells.In animal models of diabetes as well as in human type 2 diabetes, a down-regulation of AKT/mTOR signaling indicates impaired insulin signaling  ORA and KEGG pathway analysis, which selected for the implementation of a kinetic model for glutathione synthesis and metabolism . We acknowledge clinical projects are often challenged with unbiased sample numbers, so we compared each patient data with control independently.The flux through glutathione synthetase  was observed to be higher in H0008 than control. The flux through glutathione synthetase was not significantly different between H0007 and control. The flux through glutamylcysteine synthetase (second to last step in glutathione synthesis) was higher in H0008 than in the control. The average concentration of glutathione was higher in H0008 compared to control but lower in H0007 than in control . To see whether the model predicts different effects on the pathway flux/behavior compared to using the mRNA data alone, we investigated whether there is a correlation between change in the Vmax and change in the flux through the pathway. To test this we investigated two enzymes/reactions which had the largest change in expression; 5-oxoprolinase and DNA (cytosine-5-)-methyltransferase 1 and then one with smaller changes of expression; glutathione synthetase.One of the assumptions of analyzing expression data was that a change in mRNA expression is correlated to changes in the pathway flux/behavior. The advantage of combining the mRNA expression data to the kinetic model would be to investigate how a change in expression might affect flux  have an advantage over analyzing mRNA expression data alone to synthesize information.The expression of 5-oxoprolinase was 1.86 times higher in H0008 than control (H0002). The increase in flux though 5-oxoprolinase  and control (H0002).Microarray-based transcriptome data of human hepatocellular xenograft models  and control (H0002) has unveiled the following novel results.The ribosomal protein PRAS40, abundant in steatosis patients has been identified as a crucial regulator of insulin signaling and also as mediator of AKT signals to mTOR  is based on a minimal network comprising a regulatory network based on the transcription factor SREBF1, linked to a metabolic network of glycerolipid and fatty acid biosynthesis including the downstream transcriptional targets of SREBF1, LIPIN1 (LPIN), and LDLR Figure . The ince et al.  reportedFor subsequent model analysis we will employ a Monte Carlo-based simulation strategy, in which the kinetic parameters are repeatedly sampled from specific probability distributions and used for multiple parallel simulations (Wierling et al., We aimed as well to generate a metabolic model of steatotic phenotype. Interestingly among the pathways enriched in steatosis vs. control was the glutathione pathway. Subsequently our investigation involved the mapping of mRNA expression data to a kinetic model of the glutathione synthesis pathway. Adopting this pathway-oriented approach we focused on a subset of complete pathways, rather than all genes of the genome. We have extended this approach to fatty acid biosynthesis, fatty acid metabolism, bile acid pathway, gluconeogenesis, urea cycle, glycolysis, tricarboxylic acid cycle, and glyoxalate shunt; all of which have vital roles in regulation, and are either directly or indirectly involved in the successful functioning of the liver.Upon data acquisition from all patient-specific samples, various cellular networks will be perturbed by changing environmental conditions of iPS-derived, diseased and healthy cells. The impact of the perturbation will be assessed by constructing key networks from the most significantly dysregulated genes to evaluate any changes as measured and the consequences of these changes as they present themselves throughout the cellular networks.in vitro model will also lay the foundation for identifying drug targets that may be used for treatment of steatosis. Subsequently, hepatocyte-like cells derived from healthy and diseased individuals would allow us to demonstrate the differences in response to clinically relevant drugs and enable to predict hepatotoxicity. We anticipate that the knowledge gained from this minimal number of patient samples will prove advantageous and beneficial to our ongoing analysis with a larger sample of steatosis patients and healthy controls. Undoubtedly, the application of high-throughput transcriptomics, metabolomics proteomic, and modeling to study healthy and stressed liver, will broaden our meager knowledge of steatosis thus, providing deeper insights into the mechanisms underlying the progression to steatohepatitis.Ultimately hepatocyte-like cells derived from patient-specific iPS cells will be challenged by free fatty acid overload and high glucose which resemble environmental conditions during the progression of steatosis. Moreover this The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


We postulate that the cell assumes one or a combination of these objectives and search for enzyme levels necessary to achieve this. We call the subspace of feasible enzyme levels the feasible enzyme space. Once this space is constructed, we can study how different objectives may (if possible) be combined, or evaluate the conditions at which the cells are faced with a trade-off among those. We apply FA to the experimental scenario of long-term carbon limited chemostat cultivation of yeast cells, studying how metabolism evolves optimally. Cells employ a mixed strategy composed of increasing enzyme levels for glucose uptake and hexokinase and decreasing levels of the remaining enzymes. This trade-off renders the cells specialized in this low-carbon flux state to compete for the available glucose and get rid of over-overcapacity. Overall, we show that FA is a powerful tool for systems biologists to study regulation of metabolism, interpret experimental data and evaluate hypotheses.Understanding cellular regulation of metabolism is a major challenge in systems biology. Thus far, the main assumption was that enzyme levels are key regulators in metabolic networks. However, regulation analysis recently showed that metabolism is rarely controlled via enzyme levels only, but through non-obvious combinations of hierarchical (gene and enzyme levels) and metabolic regulation . Quantitative analyses relating changes in metabolic fluxes to changes in transcript or protein levels have revealed a remarkable lack of understanding of the regulation of these networks. We study metabolic regulation via feasibility analysis (FA). Inspired by the constraint-based approach of Flux Balance Analysis, FA incorporates a model describing kinetic interactions between molecules. We enlarge the portfolio of objectives for the cell by defining three main physiologically relevant objectives for the cell:  In their natural habitat, most microbes are exposed to constantly changing physical and chemical environments. To perform optimally in these conditions, they must finely regulate their metabolism. Understanding how microbes regulate metabolism to achieve a desired objective, or how they adapt to changing conditions, is a major challenge top-down and bottom-up. The top-down approach employs genome-wide constraint-based modeling techniques, such as Flux Balance Analysis (FBA), to find viable intracellular flux distributions based on measured external fluxes and thermodynamical considerations. Constraint-based models have been shown useful in exploring cellular capabilities of biological systems and have enabled in silico characterization of several phenotypic features, such as growth yield under gene knockouts , combining elements of bottom-up and top-down approaches. FA starts from an explicit kinetic model describing the interactions between enzymes and metabolites. Inspired by the well-established constraint based approach of FBA, it then defines a number of physicochemical constraints on the cell, as well as three physiologically relevant objectives: function, robustness and temporal responsiveness, for which quantitative measures are introduced. Assuming that the cell follows one or a combination of these objectives, FA then searches for (a) set(s) of enzyme levels necessary to achieve these. Given the problem of inversion of general non-linear kinetic models, FA uses a straightforward sampling-based method, commonly used for various computational biology purposes, e.g. for ensemble modeling A similar approach of using physiological constraints to find feasible sets of enzyme levels was successfully applied to identify the required changes in gene expression in yeast upon heat shock ab initio selected set of kinetic interactions; subsequently, parameter values are set or estimated by fitting to a  number of measurements. The feasible enzyme spaces found by FA can also be used to enhance currently available kinetic models. These models are usually derived starting from an In this paper, we first describe FA in detail, listing a number of constraints and introducing quantitative measures for the proposed objectives. We then exemplify the approach using two cases: (1) an illustrative small model with tractable kinetics and (2) a larger dynamic model of yeast glycolysis feasibility analysis (FA), considering physiological constraints and a list of potential objectives. We first describe these constraints and objectives and then apply FA to analyze two illustrative cases, a toy model and a model describing the glycolysis in yeast.Biological systems constantly adapt to their environment and regulate their metabolism for optimal performance. In this paper, we study this regulation at a system level and use feasible enzyme space, this set should further be constrained. However, direct measures that can be applied as constraints are generally available for the physiological space only. In theory, since the enzyme space is mapped to the physiological space with the kinetic model, constraints in one space can be translated into the other by simply inverting the kinetic model. Yet, this inversion is generally not possible in practice due to the non-linear nature of the system. To solve this, similar to We start by considering a large range of enzyme levels as the initial enzyme space. To construct the We first apply the “hard constraints”  on the physiological space and, via the kinetic model, on the enzyme space. These constraints yield the viable enzyme and physiological space. Next, we evaluate each feasibility criterion for each of the viable physiological states. The labels “feasible” or “infeasible” are thus assigned to each state and the feasible space is constructed (panel fig:Feasibility-Feasible). Finally, hierarchical regulation analysis can be applied to inspect where metabolism is regulated mainly hierarchically or metabolically, allowing to study the relation between physiological objectives and type of regulation (see Methods for more details).thermodynamic, stability, kinetic, capacity and total protein constraints into account . We start by demanding that every biochemical reaction should obey thermodynamic laws. In general, this is formulated as discrete irreversibility constraints for fluxes through reactions operating far from equilibrium. When measurements on metabolites are available, thermodynamic properties such as Gibbs free energy can be calculated a priori be computed by calculating the eigenvalues of the jacobian of the system at a selected steady state and requiring that all should have negative real parts. Then, owing to the available kinetic model, we take kinetic constraints into account. The relation between an enzyme, the metabolites and the rate for any reaction is constrained by its kinetic law. When extracellular fluxes are known and the entire network is considered, if any two of either enzyme, independent metabolite or intracellular flux levels are fixed, the third can be deduced using this set of laws for each enzyme in the network. Next, the capacity constraints provide upper limits for fluxes. Lastly, we assume that the cell economizes the change in total enzyme levels, so that when adapting to a new environment, the total enzyme level is kept within limited range. We also note that, though the total enzyme level is constrained, individual enzyme levels can vary independently within the allowed range.The first step in FA is the application of the hard constraints. We take function, robustness, homeostasis and temporal responsiveness are proposed .FA continues by further constraining the viable space to obtain the feasible space, by considering a number of quantitiative, physiologically relevant objectives. Three feasibility objectives related to Biological systems have evolved to function optimally in a given environment. Within FBA, this optimal function is considered to be a flux towards a pathway, usually the growth rate; yet alternative optimality criteria such as minimization of uptake rate or redox potential provide adequate prediction of flux distribution robustness as a property of systems that maintain their function under perturbations and uncertainty, and homeostasis as maintaining the state via coordinated physiological processes. Despite detailed qualitative descriptions ad hoc defined metrics, . Where MCA's control coefficients quantify relative change in one variable (state or function) upon change in another variable (perturbation), co-response coefficients measure the ratio of relative change in two different variables (state and function) in a network, resulting from a change in a third variable (perturbation). This coefficient is especially interesting to measure robustness and homeostasis of the network, since all three entities can be in different parts of the network.Robustness and homeostasis are two fundamental characteristics of biological systems cs, e.g. , a generWe consider a set of enzyme levels to be feasible with respect to robustness if the function is maintained  upon a change of level of any of the enzymes in this set. In that case, metabolite levels are expected to change, resulting in a small co-response coefficient for robustness  the system, such as the response time. From an evolutionary perspective, it is likely that certain pathways or cell types are selected based on their fast (or slow) response to changes in their environment. The key importance of dynamic properties for the cell to adapt to external stimuli has been exemplified for metabolic 4 enzyme level triplets , we modified the initial network and inspected the changes in the objectives and feasible enzyme space. We added a feedback inhibition of The effect of this additional feedback inhibition on homeostasis feasibility is presented in ange of  . For theWe also considered a possible feedforward activation of Next, we applied FA to study yeast glycolysis under evolutionary pressure, especially focusing on trade-offs between alternative objectives. We used a model describing glycolysis in yeast We first consider the scenario, where yeast cells were grown in a carbon limited chemostat for In MC sampling the enzyme levels, we appended fermentor balances to the model in We first evaluated the hypothesis that the cells, under constant carbon influx, would economize the enzyme levels while coping with the constant carbon flux, as proposed in To explain this increase, we consider the homeostasis objective, and check the co-response coefficient of extracellular glucose and uptake flux for both enzymes . To takeTo illustrate the advantage of considering the trade-off between enzyme economy and competitive ability, we designed a synthetic competition experiment. Four organisms differing by their enzyme levels are grown in a carbon limited chemostat, and the time course for each organism during this competition is simulated. The organisms are (1) wild-type, (2) only considering enzyme economy, (3) only considering competitive ability and (4) considering the trade-off between these two. The enzyme levels, relative to wild type and the time course of each organism are given in Interestingly, we also predict that the evolved state is prompter for ATP response, i.e. in the evolved strain, ATP responds quicker to perturbations . This isTo demonstrate how FA can serve to study metabolic (re)design, we consider the so-called turbo design in glycolysis. The turbo design is a general strategy followed by many catabolic pathways, consisting of first activating a substrate in a reaction that requires ATP, after which further metabolism yields a surplus of ATP To illustrate the case, we consider the scenario where the cells are in glucose-rich conditions and inspect the homeostasis criterion of hexoses and ethanol flux , or introducing T6P inhibition of HK. The first option is costly for the cell since the capacities of all enzymes in the storage pathway have to be increased. The second option is economical and homeostatically feasible, as already illustrated with the FA on the toy model. We finally speculate that evolution pushes cells to acquire this inhibition, in order to adapt to conditions where glucose levels significantly change. Note that such a “brake” system is not present for less-favorable carbon sources , or by constructing a kinetic model with detailed molecular interactions (bottom-up) and applying theorems of MCA or BST. Our method aims to combine elements of the two approaches and allows to study objectives other than flux. Furthermore, as we propose quantitative objectives for feasibility, rather than studying “viable” or “lethal” changes, we can study sub-optimality and trade-offs. By exploring the feasible enzyme space, FA allows evaluating alternative hypotheses and interpreting experimental data.FA assumes the availability of a kinetic model. Despite a long list of challenges, e.g. a high degree of non-linearity, lack of sufficient experimental data, coexistence of multiple time scales etc. ad hoc definitions. This allows us to consider objectives other than thresholds on fluxes or concentrations, such as robustness/homeostasis and temporal responsiveness.Previous computational efforts to understand regulation of metabolism include searching for design principles using optimization principles In this paper, we addressed the following question: being under constraints and evolutionary pressure, why and how is metabolism regulated? To answer this question, we took a top-down approach and speculated that cells, under physiological constraints, are regulated to optimize (one or a combination of) a number of objectives and can hence only assume enzyme levels falling in a so-called feasible enzyme space. We further analyzed how metabolism should be designed from a feasibility perspective, i.e. we addressed the question what are the necessary kinetic interactions in order for cells to attain an objective. Unique to our approach, we proposed quantitative metrics to measure proposed cellular objectives.One of the fundamental characteristics of biological systems, homeostasis, requires globally coordinated regulation of enzyme levels. An interesting observation for homeostatically feasible states is that these fall in two distinct sub-regimes: a low-flux regime, where all enzymes are downregulated, and are less costly for the cell; and a high-flux regime, where all enzymes are upregulated, therefore costly for the cell. The actual regime chosen by the cell is defined with respect to the environment. In the prolonged chemostat scenario, the cell optimizes the enzyme levels for function, since the carbon influx is externally kept constant. From an enzyme budget point of view, the ubiquitously present feedback inhibition is an economical way to ensure homeostasis. This is especially important for keeping metabolite levels within limits upon a wide range of fluctuations in the environment.In contrast to homeostasis, maintaining robustness requires a local metabolic effect, meaning that the function can still be maintained by locally adjusting the metabolite levels around specific perturbed enzymes. In line with our findings, Sauer and co-workers recently showed in yeast that alterations in enzyme capacity are buffered by converse changes in substrate metabolite concentration, thereby minimizing the difference in metabolic flux caused by the alteration Temporal responsiveness reflects a dynamic property of the system. We speculate that this objective is especially applicable to networks whose dynamic properties are of evolutionary importance, e.g. ultrasensitivity, response time etc. As an example, for signaling pathways the effect of network structure on dynamic properties has already been discussed Taken together, we see that FA quantitatively evaluates alternative hypotheses, shows trade-offs between the available objectives and provides an intuitive platform to integrate the proteome information (enzyme space) with information on metabolome and fluxome . Such an integrative approach is indispensable to analyse and interpret the increasingly available multi-omics data on regulation of metabolic networks especially when considering optimal performance or adaptation in response to external stimuli. We illustrated quantitatively via FA that there is a very limited set of enzyme set that are feasible for all the considered objectives. Similar to theoretically possible physiological states, and call this the viable enzyme space. We then select a feasible subspace based on the pre-defined criteria and analyze the properties of this subspace. Overall, we construct the feasible enzyme space 0”. This way, all defined entities can be measured with respect to this reference state, much like the elasticity parameters or control coefficients in MCA literature. Conventionally, the reference state can be chosen as the steady state that cells achieve when they are grown under constant, substrate limited conditions and is usually characterized by intracellular fluxes, metabolites and enzyme levels.To construct the feasible enzyme space, we first quantitatively formulate the constraints and the objectives for physiological states. Then we calculate the range of http://bioinformatics.tudelft.nl/. The interface takes as input a dataset, calculates the feasibility criteria for selected objectives of the cell and visualizes by (selected) slices.The feasible enzyme space constructed is, in fact, a sampling of a multidimensional space containing enzymes, metabolites and fluxes. We visualize this space by slices, i.e. 2D cross-sections. A GUI written in Matlab, and supporting functions as well as the datasets mentioned in this paper can be downloaded at: Thermodynamic constraints are formulated as irreversibility constraints for fluxes through reactions operating away from equilibrium:0”). This criterion demands that the total enzyme level stays nearly constant  should be small, i.e. homeostasis is considered. A state is called feasible if upon enzyme perturbation, metabolite levels do not change significantly Focusing on Temporal responsiveness of metabolite levels in a metabolic network in response to perturbations is defined using the turn-over time of metabolitesAn important question is to what extent metabolic fluxes are regulated by gene expression or by metabolic regulation. In line with the convention in regulation analysis We consider cases where To illustrate feasibility analysis, we use a small example model with tractable kinetics, where a substrate abolites . The modSaccharomyces cerevisiae0 is the corresponding entity at the reference state.To study feasibility analysis applied on a real problem, we used a previously published model of glycolysis in For the feasibility analysis of alternative metabolic redesign in yeast glycolysis, the new kinetic expression for the HK reaction is:Lastly, in order to illustrate the competition in the fermentor, we added a growth equation for this and expressed the growth rate with simple monod-growth kinetics as:In using experimental data, there were multiple measurements for a specific time point. Since the data available was insufficient to assume and fit a parametric model, we used non-parametric Gaussian kernel regression


To validate our approach, we conduct in vitro measurements of ETC complex I, III, and IV activities using rat heart homogenates, and construct an estimation procedure for the parameter values directly from these measurements. In addition, we show the theoretical connections of our approach to the existing models, and compare the predictive accuracy of the rate law with our experimentally fitted parameters to those of existing models. Finally, we present a complete perturbation study of these parameters to reveal how they can significantly and differentially influence global flux and operational thresholds, suggesting that this modeling approach could help enable the comparative analysis of mitochondria from different systems and pathological states. The procedures and results are available in Mathematica notebooks at http://www.igb.uci.edu/tools/sb/mitochondria-modeling.html.Mitochondrial bioenergetic processes are central to the production of cellular energy, and a decrease in the expression or activity of enzyme complexes responsible for these processes can result in energetic deficit that correlates with many metabolic diseases and aging. Unfortunately, existing computational models of mitochondrial bioenergetics either lack relevant kinetic descriptions of the enzyme complexes, or incorporate mechanisms too specific to a particular mitochondrial system and are thus incapable of capturing the heterogeneity associated with these complexes across different systems and system states. Here we introduce a new composable rate equation, the chemiosmotic rate law, that expresses the flux of a prototypical energy transduction complex as a function of: the saturation kinetics of the electron donor and acceptor substrates; the redox transfer potential between the complex and the substrates; and the steady-state thermodynamic force-to-flux relationship of the overall electro-chemical reaction. Modeling of bioenergetics with this rate law has several advantages: (1) it minimizes the use of arbitrary free parameters while featuring biochemically relevant parameters that can be obtained through progress curves of common enzyme kinetics protocols; (2) it is modular and can adapt to various enzyme complex arrangements for both  This use of the pmf as an intermediate driving force in the overall conversion of energy is the essence of the chemiosmotic theory Throughout the mitochondria inner membrane are many energy-transducing protein complexes that help transform the chemical energy from the cell's metabolic intake into various useful forms of energy for the cell. Some of these complexes use the free-energy extracted froms reduction-oxidation (redox) reactions to transport proton across the membrane and establish a proton gradient, while others use this proton gradient in combination with the membrane potential, the proton-motive-force  may not be transformable into another set of condtions (e.g. in vivo system). Thus, if a model of mitochondrial bioenergetics is to be both extensive and flexible, it must be able to adapt to the mechanism of a particular system, and be applicable to various system conditions. Furthermore, the dynamics of a system depend on the system's component descriptions, yet a system is often more than just the sum of its components. Thus, to establish a functional relationship between the dynamics of a system and its components, it is also necessary to allow for potential unobserved properties to emerge from the synergistic network interactions that vary from system to system. Only then can a model effectively capture the system's response to a perturbation across multiple scales.The internal mechanisms of the complexes have been modeled with explicit elementary reaction steps to track intermediate metabolic species or reaction byproducts such as the generation of ROS in vitro assay protocols for the various ETC complexes in the OXPHOS pathway. The resulting in vitro reference dynamics can be transformed into dynamics of an in vivo system through changes in the system's constituent thermodynamic forces, the reaction equilibrium constant, and the rate constants in a systematic way. One particularly useful type of transformation is a “slippage”, or a loss in the thermodynamic efficiency of the complex's energy transduction process, which can be used to generally express a perturbation on the complex's reaction rate.In this paper, we present a new modeling approach for mitochondrial bioenergetics that addresses the problems of cross-mechanism adaptability, cross-conditional applicability, and cross-scale analysis through a novel composable kinetic rate equation that we term the chemiosmotic rate law. The rate law incorporates three configurable modulating factors that, via steady-state and rapid-equilibrium approximations, separately encapsulate the binding kinetics, the redox transfer potential, and the non-linear thermodynamic force-to-flux relationship of a prototypical energy transduction complex in the OXPHOS pathway. The separation of the factors allows one to selectively configure the mechanistic scheme of the complex, while the six biochemically relevant kinetic parameters of the rate law allow one to completely specify its kinetic properties at a particular reference condition. The kinetic parameters are estimated from data obtained experimentally using simple extensions to the standardized To validate the rate law, we conduct six parallel sets of assays on rat muscle homogenates, two for each complex. For each complex, we obtain kinetic parameter estimates from one of the two sets of assays, and then compare the progress curve predicted by our model to the experimental progress curve from the other set of assays. In addition, we formally establish the theoretical connections between our approach and other models, and compare their predictive accuracies. Finally, by using the optimized parameters for complex I, III, and IV, we perform a complete perturbation study of these parameters to reveal how they can significantly and differentially influence global flux and operational thresholds, suggesting that this modeling approach could help enable the comparative analysis of mitochondria from different systems and pathological states.In this section, we first describe the derivation of the new chemiosmotic rate law based on the common properties of the ETC complexes. We then describe the experimental protocols that can be used to obtain the various biochemical parameters appearing in the rate law.NADH-ubiquinone oxidoreductase) and complex II (succinate-ubiquinone oxidoreductase) respectively; (2) both complexes then catalyze the reduction-oxidation transfer of electrons to ubiquinol-cytochrome c oxidoreductase), which transfers the electrons to cytochrome c; and (4) cytochrome c binds to complex IV (cytochrome c- oxidoreductase), which transfers the electrons to oxygen, the final electron acceptor The ETC consists of four integral membrane protein complexes that facilitate a sequence of electron transfer steps: (1) Electrons from citric-acid cycle products Each type of ETC complex is structurally unique, has diverse catalytic and binding rates, and responds to different inhibitors. Much of the detailed mechanisms of each complex remain to be determined. Nevertheless, all ETC complexes and their supercomplex assemblies share the same general characteristic of binding to both a donor and an acceptor electron carrier (EC) species, and facilitating the flow of electrons through the complex by a series of intermediate internal redox steps To provide quantitative descriptions for the ETC complexes, an abstract prototypical ETC complex model, the electron-proton pump  of the products and substrates of a chemical reaction reaches a constant value defined as the equilibrium constant for that reaction. For any other ratio, there exists a disequilibrium in the chemical potentials, which provides the thermodynamic force thermodynamic force function), then coupling the net steady-state flux to the average transition rate between the internal redox centers (redox state function), and finally superimposing the kinetics of the binding reactions that produces the active ternary saturation function). In addition, we extend the derivation to allow adaptation of the flux expression to different system settings by means of transformations of rate constants and/or equilibrium constants from a reference system. We then utilize the transformation framework to introduce the chemiosmotic free energy transduction mechanism of the In the following subsections, we use a combination of statistical steady-state thermodynamics, macroscopic equilibrium thermodynamics, and kinetics theories to derive a quantitative expression of the reaction flux Let us first consider the redox mechanism of the At equilibrium By applying detailed balance on the individual elementry transition rates between the states in a cyclic system, a flux-to-force relationship is found that equates the steady-state ratio of the forward and reverse cyclic flux to the exponential of the thermodynamic force Combining Equation 7 with the definition of After establishing the flux-to-force relationship of the general redox reaction of the Microscopically, es  and   and 5. Ation see . SimilarTaking into account that each The probability for the forward state can then be taken asSubstituting for the probabilities in the net transition flux (Equation 12) and redefine The equilibrium constant for the reaction is found by noting that This allows one to approximate the total thermodynamic force spanning from D to A,Using the algebraic constraints in Equation 13 and Equation 14, the relationships between the probabilities of each internal state and the concentrations of the external reactants are expressed as:Here shorthand notations for the ratios of the redox states are introduced:Substituting the relationships in Equation 26 and 27 for all the internal probability values in Equation 19 and Equation 20 allows the reaction flux to be expressed in terms of the external reactant species:Noting the exponential form of Here the redox state function is introduced as:Equation 32 describes the flux as a forward rate coefficient At steady-state, the electron exchange reactions between the ECs and the Using the nomenclature of Cleland These constraints reduce the schemes in Equation 35 and 36 to pseudo-isomerization reactionsIn the situation where only one of the two substrate EC concentrations is varying while the other is held constant, one can essentially consider the effects of the individual variation separately as two separate and parallel reactions, each with a single substrate-enzyme binding stepAt steady-state, such independent binding reactions can always be expressed using the Michaelis-Menten-type kinetics One important point in using Equation 46 is that although Equations 8 and 9 are equivalent over the entire domain of in vitro experiments. Thus, the reference system represents a fundamental basis from which the rate law for many other in vitro or in vivo systems can be derived through the transformation framework of Equation 50.The standard form of the chemiosmotic rate law in Equation 46 considers only the redox thermodynamic force pmf), pmf,The incorporation of proton gradient can be viewed as such a transformation. The free energy transduction reaction in a chemiosmotic complex is driven by an overall thermodynamic force Since the equilibrium constant can be expressed as the ratio of the rate constants, the equilibrium constant of the transformed reaction with respect to the reference reaction is thenIn general, a transformation such as the addition of the Note that because the transformed thermodynamic force is a fraction of http://lbbma.univ-angers.fr/lbbma.php?id=58). The principle technique for these assays involves the use of an ultraviolet/visible (UV/VIS) absorption spectrophotometer in which the time-course conversion of a redox substrate species to its product species by an ETC complex in a closed system (inside a curvette) is recorded to obtain the initial velocity (rate) of the reaction. Homogenated tissues and isolated mitochondria (where membranes are fragmented in both cases) are specifically used as the reference system since its pmf can be neglected due to the absence of an intact mitochondrial inner membrane. Thermodynamic constants from the literature and saturation reactant concentrations used in our experiments are listed in The standard form of the chemiosmotic rate law, assuming a parallel binding mechanism (Equation 43), contains a total of six kinetic parameters: apparent maximum forward and reverse velocities nstants  . ThroughNonlinearModelFit function, which produces least-squares fits that are defined to minimize the quantity http://www.igb.uci.edu/tools/sb/mitochondria-modeling.html.In the following sections, procedures to determine the five kinetic parameters from the time-series data are described. With the exception of the kinetics for the final electron acceptor in complex IV, where both oxygen and water are reactants that are open to the bulk concentration, these procedures apply to all kinetic parameters in the ETC complexes. All of the fitting and subsequent simulations of the rate equation are performed using Mathematica 8′s As indicated in the derivation of Equation 45, out of the three modulating function, the For each assay set, an initial velocity is approximated for each of the initial concentrations by calculating the change in concentration over a starting time period (dependent on the amount of enzymes used). The initial velocity-concentration pairs are then used as data points for calculating the residuals in the least-square fitting of the corresponding Michaelis-Menten-like factor in Equation 43 to obtain the associated saturation parameter . A finalBeyond the transient period of the instantaneous initial velocity, both Given all other parameter values are fixed or determined, the values for Note that the ratios In Korzeniewski's rate equation follows Onsager's linear force-to-flux relationship, based on the local equilibrium approximation where thermodynamic forces vary slowly ation 46 .Least-square fitting of the three rate equations over the entire range of the experimental time-series data from an isolated complex I kinetics assay allow a comparison of how well the equations can reproduce the original time-series through their respective parameters. equation , one canThis can be explained by noting that when both An important result emerging from these comparisons is that the new chemiosmotic rate law represents a more general formulation of the kinetic and thermodynamic behaviors for a chemiosmotic ETC complex; one which encompasses both Korzeniewski's and Beard's formulations, and can smoothly interpolate between them to cover a spectrum of biochemical behaviors. Therefore, for a given overall reaction and a thermodynamic force definition, the chemiosmotic rate law can replace the existing ETC rate equations in the OXPHOS models of Korzeniewski and Beard to incorporate biochemically relevant kinetic parameters which allows a more accurate specification of an ETC complex.This section covers the analysis of the kinetic parameter values and their sensitivity in the chemiosmotic rate law applied to each of the ETC complexes in isolation. A summary of the experimental values obtained for all the kinetic parameters of complex I, III, and IV are given in In summary, the sensitivity of ubstrate . On the Whereas a Lastly, from Equation 55, it is clear that the effect of e system .in vitro homogenate environment, but with all the ETC complexes functioning in tandem. The simple pathway model of the ETC used for this analysis consists of interactions between a driving dehydrogenase reaction, the three main electron transfer complexes , and the three electron carrier redox pairs  This expanded definition expresses quantitatively the effect that small variations in the parameter Although FCCs can show the relative control of the individual components in a pathway network, they have little predictive value as each FCC is computed for a single steady-state. As the state of the system changes, the values of the FCCs, may change as well. To overcome the single steady-state limitation of the FCC, we next apply threshold curve analysis An example is shown in The same threshold profile can be observed irrespective of which kinetic parameter is used to establish the relationship between that of pmf, has an excess in the redox potential, and that only ten percent of the redox potential is necessary to drive the redox reaction of complex I. After dropping below 0.1, the threshold curve quickly approaches that of a straight line, showing direct effect on the flux. The corresponding threshold profile in The threshold profile of the system is consistent for all kinetic parameters because the threshold curve relates the relative changes instead of the absolute changes in In short, we have developed a framework for modeling mitochondria bioenergetics using a new chemosmotic rate law to represent each ETC complex in the OXPHOS pathway. This modular framework subsumes and generalizes several previous approaches and relies on a small set of biochemically relevant parameters. We have conducted enzymatic assays to derive those kinetic parameters for complex I, III, and IV, and validated the predictions of the model against experimental concentration time-series. These results, together with detailed sensitivity analyses, show that the parameters, originally derived from a simple reference system, provide a good breadth of feasible physiological responses of the complexes. In particular, threshold curves relating the flux of one complex to the global flux of the system show how each parameter in each complex can have a differential effect on the threshold curve of the corresponding complex, as well as on the overall threshold profile of the system  and 16.