Utilizing elementary mode analysis, pathway thermodynamics, and a genetic algorithm for metabolic flux determination and optimal metabolic network design
 Brett A Boghigian^{1},
 Hai Shi^{1},
 Kyongbum Lee^{1} and
 Blaine A Pfeifer^{1}Email author
DOI: 10.1186/17520509449
© Boghigian et al; licensee BioMed Central Ltd. 2010
Received: 24 July 2009
Accepted: 23 April 2010
Published: 23 April 2010
Abstract
Background
Microbial hosts offer a number of unique advantages when used as production systems for both native and heterologous smallmolecules. These advantages include high selectivity and benign environmental impact; however, a principal drawback is low yield and/or productivity, which limits economic viability. Therefore a major challenge in developing a microbial production system is to maximize formation of a specific product while sustaining cell growth. Tools to rationally reconfigure microbial metabolism for these potentially conflicting objectives remain limited. Exhaustively exploring combinations of genetic modifications is both experimentally and computationally inefficient, and can become intractable when multiple gene deletions or insertions need to be considered. Alternatively, the search for desirable gene modifications may be solved heuristically as an evolutionary optimization problem. In this study, we combine a genetic algorithm and elementary mode analysis to develop an optimization framework for evolving metabolic networks with energetically favorable pathways for production of both biomass and a compound of interest.
Results
Utilization of thermodynamicallyweighted elementary modes for flux reconstruction of E. coli central metabolism revealed two clusters of EMs with respect to their ΔG_{ p }°. For proof of principle testing, the algorithm was applied to ethanol and lycopene production in E. coli. The algorithm was used to optimize product formation, biomass formation, and product and biomass formation simultaneously. Predicted knockouts often matched those that have previously been implemented experimentally for improved product formation. The performance of a multiobjective genetic algorithm showed that it is better to couple the two objectives in a single objective genetic algorithm.
Conclusion
A computationally tractable framework is presented for the redesign of metabolic networks for maximal product formation combining elementary mode analysis (a form of convex analysis), pathway thermodynamics, and a genetic algorithm to optimize the production of two industriallyrelevant products, ethanol and lycopene, from E. coli. The designed algorithm can be applied to any smallscale model of cellular metabolism theoretically utilizing any substrate and applied towards the production of any product.
Background
Microorganisms are increasingly utilized to synthesize a variety of products [1–3], including fuels (bioalcohols [4–13] and biodiesels [14, 15]), specialty chemicals (amino acids [16–20]), therapeutic smallmolecules [21–25] (antibacterials, anticancer agents, and cholesterollowering agents), and biopharmaceuticals [26] (proteins, vaccines, and virus particles). A common challenge in developing highyield cellular production systems is that organisms have evolved to optimize growth rather than the formation of a particular endproduct. In principle, this challenge could be met by reprogramming the cellular objective using genetic modifications (such gene insertions, overexpressions, or deletions). In practice, the selection of appropriate gene modification targets can be a daunting task. Biomass formation as well as product synthesis requires building block precursors and cofactors provided through the concerted actions of a large number of interconnected metabolic pathways encoded by hundreds to thousands of genes. While purely empirical attempts at genetic modifications have in some cases led to impressive success [27], these cases have provided the exceptions rather than the rule. There is now considerable evidence that substantial improvements in productivity require manipulating the activities of multiple enzymes in different parts of cellular metabolism [28]. In this respect, optimizing biosynthetic productivity will almost certainly benefit from computational modeling tools that systematically and efficiently explore the consequences of gene or enzymelevel modifications across the breadth of cellular metabolism.
Currently, there exists a variety of methods for studying metabolic networks in both quantitative and qualitative manners: flux balance analysis (FBA) [29–31], ^{13}Clabeling based metabolic flux analysis (^{13}CMFA) [32], metabolic control analysis [33], elementary mode analysis (EMA) [34], extreme pathway analysis [35], cybernetic modeling [36, 37], and biochemical systems theory [38–40]. Many of these methods do not necessarily identify experimentally tractable metabolic engineering targets such as gene deletions. Whereas, some algorithms based on the aforementioned methods can be used to identify such targets including minimization of metabolic adjustment (MoMA) [41], regulatory on/off minimization (ROOM) [42], OptKnock [43], OptStrain [44], OptReg [45], and OptGene [46]. All six of these methods require solving an optimization problem to determine flux distributions as a means of evaluating the strain's (or mutant strain's) metabolic capabilities. Although these optimization approaches can accurately predict optimal growth and production fluxes in some cases [47], other experimental settings produce inaccurate predictions [48]. In addition, situations that require the removal of numerous genes to achieve high productivity will lead to mutant strains significantly different from wildtype systems, further weakening the assumptions behind FBA. OptKnock and OptStrain utilize a bilevel optimization for determining superior mutant strains. The mixed integer linear programming (MILP) framework used in these two algorithms optimizes for one objective within another competing one (a cellular objective (biomass production) within an engineering objective (chemical production)). However, the user must provide the number of knockouts that OptKnock and OptStrain can allow. In general, exhaustively searching genomic space for knockout candidates is computationally intractable even on smallscale metabolic models (less than 100 reactions), much less on current genomescale metabolic models (greater than 1000 reactions) due to prohibitive computation time. This situation coupled to the fact that two or three knockouts are likely not sufficient for generating a mutant capable of maximal productivity motivated the use of a genetic algorithm as demonstrated in OptGene.
Genetic algorithms (GAs) have been classically utilized as a search method for optimization of objective functions that are discontinuous, nondifferentiable, stochastic, or highly nonlinear. As the name implies, the underlying theory behind GAs is based on Darwinian evolution. GAs seek to evolve a population of potential solutions by crossover and mutation, and through multiple generations, the entire population will eventually "evolve" towards a global optimum (a "fitness score"). They have been used to some extent in modeling biological systems [46] and are used as a search technique in this study. Though OptGene utilizes a GA to efficiently explore genotypic space, the framework still requires the use of a metabolic assumption required to determine metabolic fluxes (such as FBA, MoMA, or ROOM). The goal of this study was to leverage the power of a GA without the need for a metabolic assumption. As such, a primary objective of this work was to identify a genotype with a high productivity phenotype strictly from the wildtype organism's metabolic network topology, utilizing thermodynamics.
An elementary mode (EM) is a nondecomposable set of reactions (encoded by a set of genes) that leads to a functional metabolic pathway. EMA is a method of enumeration of all of the EMs of a metabolic network. As such, an EMA presents a convex analysis problem from computational geometry, in which the extreme rays of the polyhedral cone (as defined by stoichiometry and reversibility) are the EMs of the metabolic network. As a result, an EM represents a single functional pathway within overall cellular metabolism, a linear combination of a cell's EMs can be used to describe any metabolic state achievable by the cell's stoichiometry. The algorithmic complexity of this problem has not been studied in detail and has therefore been classified as at least an NPhard (nondeterministic polynomialtime hard) problem [35]. Empirical observations have shown that the computation time of EMA algorithms grows approximately quadratically with respect to the number of EMs, and unfortunately, the number of EMs grows exponentially with respect to network size. As a result, the computation time increases greatly with respect to network size, limiting analysis to non genomescale metabolic networks. Nonetheless, EMA has been utilized to design strains of E. coli that are efficient at producing biomass from glucose [49] and ethanol from five and sixcarbon sugars [50]. In two cuttingedge applications, EMA was combined with linear programming to determine flux distributions from external measurements in lysineproducing Corynebacterium glutamicum[51], and to determine the metabolic fluxes of Lactobacillus rhamnosus growing on medium containing mixed substrates [52]. EMA has also been utilized to determine flux distributions in polyhydroxybutyrateproducing E. coli, mediated by a thermodynamic analysis of the EMs [53].
In this study, an algorithm based on EMA, pathway thermodynamics, and a GA has been constructed with the goal of redesigning a metabolic network towards maximally producing compounds of interest while simultaneously sustaining high biomass formation. This algorithm was applied to producing ethanol and heterologous lycopene through E. coli. We addressed issues of computation speed by coupling the EMA model with a GA to efficiently explore genotypic space. This algorithm presents a combination of a variety of traits that have been explored previously by themselves: 1) it is based solely on reaction stoichiometry and network topology, independent of any experimental flux measurements (although, if flux measurements are available, they can be used to constrain the problem); 2) by the utilization of a GA, it contains an efficient search method arriving at a solution within minutes on a singleprocessor notebook system; 3) by the utilization of a GA, it arrives at an optimal solution in a computationally tractable amount of time; and 4) it contains the option to use a multiobjective genetic algorithm (MOGA), only utilized very recently in analysis of metabolic networks, but still within the context of FBA [54].
Results & Discussion
Algorithm
Elementary Mode Analysis
Information on the ethanol and lycopene models, their corresponding EMs, and their reaction and pathway change in Gibbs free energy.
Ethanol Model  Lycopene Model  

Number of reactions  60  64 
Number of metabolites  47  50 
Mean EMA Computation Time (s) (± SD)  8.04 s ± 0.27 s  12.30 s ± 0.24 s 
Number of EMs  33,220  42,659 
Number of productproducing EMs (%)  8,389 (25.3%)  9,439 (22.1%) 
Number of biomassproducing EMs (%)  28,336 (85.3%)  32,763 (76.8%) 
Number of product and biomassproducing EMs (%)  7,156 (21.5%)  4,427 (10.4%) 
Δ G _{ r } ° Mean (kcal mol ^{ 1 } )  1.95  1.77 
Δ G _{ r } ° Skewness (kcal mol ^{ 1 } )  4.54  0.42 
Δ G _{ p } ° Mean (kcal mol ^{ 1 } )  111.87  88.72 
Δ G _{ p } ° Median (kcal mol ^{ 1 } )  71.71  50.40 
Number of thermodynamicallyinfeasible EMs (%)  242 (0.73%)  1487 (3.49%) 
Pathway Gibbs Free Energies
Previous analyses have shown that most metabolic reactions are near equilibrium (ΔG_{ r }≈ 0) [57]; whereas, metabolic pathways are energetically favorable (ΔG_{ p }< 0) [56], consistent with the trends shown in the panels of Figure 3. In this regard, the calculated ΔG_{ p }° values should offer qualitatively correct and quantitatively reasonable estimates of the thermodynamic favorability of metabolic pathways. The change in Gibbs free energy across an entire pathway (an EM) is much more likely to be negative than the change in Gibbs free energy across individual reactions within the pathway as shown in Figure 3c and Figure 3d. Therefore, there is a strong correlation between stoichiometric feasibility and energetic favorability at the level of the pathway. Moreover, thermodynamic favorability has already been used in the past to narrow the solution space or eliminate infeasible solutions when estimating or optimizing metabolic flux distributions [57, 58]. In the algorithm, we used the ΔG_{ p }° estimates to identify thermodynamically favorable reaction routes and enrich the mutant organism with these favored routes towards product and biomass production.
As stated previously, cellular metabolism (a flux vector) can be represented as a linear combination of the cell's EMs. It has been demonstrated previously that there exists a strong correlation between the standard change in entropy across an EM (ΔS_{ p }°) and the weighting factor of its contribution to the overall flux state in both a wildtype E. coli strain and a strain engineered for polyhydroxybutyrate production [53]. The resulting method of determining fluxes based on weighting by ΔS_{ p }° values was then compared to flux values reported in literature and showed a strong correlation (R^{2} = 0.85). In an analogous manner, we utilized Gibbs free energies for flux determination similar to previous efforts [57].
Comparison of Calculated Fluxes with Alternative Methods
Ethanol Case Study
Two case studies were performed to evaluate our algorithmic framework. In these case studies, the algorithm was tasked to identify gene knockouts that would result in the optimal product yield, biomass yield, or overall productivity (biomass yield × product yield). The two cases were: aerobic production of ethanol (a native compound) and aerobic production of lycopene (a heterologous compound), both in E. coli. The search space for the algorithm excluded reactions that computationally led to either no biomass formation or no product formation. For the ethanol case study, these reactions were BIO, FEM5, FEM6, GG1, PPP5r, TCA1, TCA2r, TCA3r, TCA4, TRA1, and TRA3 (see Additional File1 for more information). These reactions were identified by conducting a single reaction removal analysis on the wildtype model. The reactions were removed individually, the EMs were enumerated, and the EMs for each mutant were rank ordered based on their stoichiometric ethanol yield or biomass yield. If the maximal yield for either ethanol or biomass was zero, then these reactions were considered to be necessary. As a result, any strain that contained any one of these knockouts would produce either no ethanol or no biomass.
Whenever ethanol was being optimized (either by itself or with biomass), reactions for either NADH dehydrogenase I/ATP synthase (OPM1) or NADH dehydrogenase II (OPM4r; also known as NADH:ubiquinone oxidoreductase II) were removed. Both NADH dehydrogenase I and II are involved in driving electron flow, while NADH dehydrogenase I is driven by oxygen. NADH dehydrogenase II uses NADH exclusively and is repressed when E. coli is grown anaerobically [59]. The results predicted using either EMweighting method are consistent with what is known about ethanol production through E. coli, namely, that it is mainly produced anaerobically. Also for the majority of the cases, reactions for the pyruvate oxidase (coded by poxB) and phosphate acetyltransferase (coded by pta) were identified for removal, consistent with what has been previously reported for improving ethanol production from glucose through E. coli[49]. Depending on the weighting scheme, reactions identified for removal were fumarate reductase (coded by frdABCD), malate dehydrogenase (coded by sfcA and maeB), or lactate dehydrogenase (ldhA), all also identified by the pervious study as a near optimal producing genotype.
All of the solutions generated strains with between 16 and 24 EMs (over a three order of magnitude reduction in EMs); the different fitness functions produced mutant genomes that shared a number of similarities. Across many of the solutions, reactions in oxidative phosphorylation (as previously described) were removed. Many of the fermentative acid pathways were also removed (most notably for acetate and lactate production), which would limit the formation of undesired byproducts. Relative few reactions in glycolysis, the TCA cycle, and the PPP were removed in the different weighting schemes. It is also interesting to note that the main fermentative pathway preferred by E. coli (the acetic acid pathway) and the lactic acid pathway are chosen for removal over other secreted weak acid metabolites (formate, succinate, and pyruvate). It is therefore likely that the removal of other fermentative pathways would improve flux to the ethanologenic pathway. Taken together, the general results across many of the fitness functions suggest removal of certain fermentative and anapleurotic pathways improves cellular phenotype.
As stated, many of the knockouts identified by the algorithm presented here have been reported in E. coli. The frdA and ndh knockouts were also identified by an EMAbased algorithm and implemented in the laboratory to improve ethanol production [50]. The nuo and atp operons were neglected in the previous model because the model was restricted to anaerobic action, therefore as stated previously, the algorithm here correctly identifies these as knockout targets (akin to operating anaerobically). However, the nuo knockout increases glucose uptake and ethanol production, while decreasing acetate, succinate, lactate, and formate formation in an anaerobic chemostat with complex medium supplemented with glucose [60]. While the previous algorithm identified a variety of other knockouts predicted to improve production, the algorithm design was slightly different and exemplifies one of the challenges in both engineering and modeling biological systems. As has been shown here and in other places [61, 62], multiple genotypic states can lead to the same phenotypic state.
Lycopene Case Study
Next, the E. coli metabolic network was optimized for lycopene production. Lycopene is a C_{40} carotenoid natural product with antioxidant properties. Much work has been devoted to engineering lycopene biosynthesis in E. coli[63–72], due to the fact that it shares metabolic precursors (DMAPP and IPP) to other isoprenoid natural products with immense therapeutic value, such as the antimalarial sesquiterpene artemisinin and the anticancer diterpenoid paclitaxel [70].
Information on resulting strains after running the algorithm for the ethanol model.
EqualWeighted  ThermodynamicallyWeighted  

Fitness function^{a}  Ethanol  Biomass  Ethanol × Biomass  Ethanol  Biomass  Ethanol × Biomass 
Number of reactions removed  13  10  18  11  10  11 
Number of associated genes (or gene clusters) ^{ b }  16  8  18  12  10  9 
Gene list  aceA ackAB atpABCDEFGHI cyoABCD fbp frdABCD glpX ldhA mdh nuoAHJKLMNEFGBC pflB poxB pta talAB tdcE tktAB  aceA aceEF ldhA lpdA mdh pgi pntAB tktAB  adk atpABCDEFGHI cyoABCD eda fbp frdABCD glpX ldhA maeB mdh ndh pckA pflB poxB pps pta sfcA tdcE  atpABCDEFGHI cyoABCD fbaAB gnd maeB mdh ndh poxB pta rpe sfcA sucCD  atpABCDEFGHI cyoABCD frdABCD maeB pgmA pgml pntAB sfcA tktAB ytjC  aceA adk atpABCDEFGHI cyoABCD eda ndh pntAB pta sucCD 
Total number of EMs  23  23  23  16  17  24 
Biomass yield (fraction of theoretical)  0.000  0.839  0.456  0.000  0.861  0.483 
Ethanol yield (fraction of theoretical)  1.000  0.007  0.584  1.000  0.003  0.499 
(Information on resulting strains after running the algorithm for the lycopene model.
EqualWeighted  ThermodynamicallyWeighted  

Fitness function^{a}  Lycopene  Biomass  Lycopene × Biomass  Lycopene  Biomass  Lycopene × Biomass 
Number of reactions removed  17  14  13  10  12  10 
Number of associated genes (or gene clusters) ^{ b }  21  15  12  9  9  10 
Gene list  aceA adhE atpABCDEFGHI cyoABCD eda fbp fdhF frdABCD fumABC glpX hycBCDEFG ldhA maeB ndh aceEF pflB ppc pps pykAF sfcA tdcE  ackAB adhE adhP frdABCD lpdA maeB ndh aceAB pckA pgi poxB sfcA sucAB sucCD talAB  ackAB adhE adhP adk fbp fdhF glpX hycBCDEFG ldhA pckA ppc pta  adk eda frdABCD gnd ndh aceB rpe sucCD tktAB  aceA adhE adhP fbaAB ldhA maeB ndh rpe sfcA  aceAB adhE atpABCDEFGHI cyoABCD ldhA pgmA pgml poxB pta ytjC 
Total number of EMs  18  21  80  174  35  86 
Biomass yield (fraction of theoretical)  0.000  0.900  0.272  0.000  0.842  0.410 
Lycopene yield (fraction of theoretical)  0.968  0.015  0.698  0.646  0.001  0.418 
A pioneering study on applying computational methods (specifically, MoMA) for driving metabolic engineering studies focused on lycopene production from glucose minimal medium through E. coli[64]. The single knockout target search identified seven knockout targets (gdhA, cyoA, gpmAB/yjtC, ppc, glyA, eno, and aceE), two of which are not included in the model used here as they pertain to amino acid metabolic pathways (gdhA and glyA). Of the other five, all were identified in some capacity by our model when optimizing for lycopene or lycopene and biomass, some of them in multiple cases. Of these computationally identified knockouts, numerous single, double, and tripleknockout strains were constructed in the laboratory and showed improved lycopeneproducing phenotypes.
The fdhF knockout (identified here as a knockout candidate in the equal weighting cases, both with and without biomass production) improved specific lycopene production by 4%. Combining these two knockouts with a gdhA knockout (as identified by genomescale MoMA simulations, but not considered in the model presented here) resulted in the best triple knockout strain, improving specific lycopene production by 37% [64]. The nuo knockout improved specific lycopene production by 45% (from 1,100 ppm to 2,040 ppm) in complex medium supplemented with glucose. The other knockouts identified in this case have either not been reported with respect to improving lycopene production, or are lethal to the cell (as is the case with pgk). The aceE knockout was identified by MoMA simulations and implemented in the laboratory, improving specific lycopene production by 9% in minimal medium supplemented with glucose [64]. The pykAF doubleknockout improved specific lycopene production threefold (from approximately 5 to 15 mg gDCW^{1}) in complex medium [69]. While many of these knockouts have not been conducted in the same strain, there remain many opportunities to improve lycopene titers. Currently, lycopene production yields reported are well below the theoretical yield on glucose (316 mg/g glucose). For example, bioreactor cultivation of the overproducing ΔgdhA ΔaceE ΔfdhF triple knockout strain resulted in a lycopene yield of 2.15 mg g glucose^{1}[66], less than 1% of the theoretical yield. It is reasonable to assume that numerous additional knockouts would further aid efforts to reach this theoretical yield. Overall, the reported literature on metabolic engineering effort to improve lycopene production in E. coli strongly supports the validity of the algorithm developed here.
Multiobjective Genetic Algorithm
Given the dual objective nature of the system in question (product yield and biomass yield), it would be logical to also assess the performance of a multiobjective genetic algorithm (MOGA). MOGA maximizes/minimizes a vector of objective functions (in this case, a vector of length two) rather than a scalar objective, as was the case for the GA. As a result, there is no single, unique solution to this problem. Instead of identifying a single solution, a MOGA aims to identify a set of solutions in which an improvement in one objective requires a decrease in the other. Each solution is considered to be a noninferior solution and the entire set of noninferior solutions is referred to as the Pareto optima. The MOGA invoked here uses a controlled elitist genetic algorithm, a variant of the Nondominated Sorting Genetic AlgorithmII (NSGAII).
Though MOGAs have not been used for optimizing the structure of metabolic networks, there has been a recently reported example of using one for optimizing an industrial bioprocess (penicillin V production from Penicillium chrysogenum) [73]. In particular, a MOGA was used for 1) maximizing penicillin titer and maximizing penicillin yield from substrate, 2) maximizing penicillin titer and minimizing fermentation time, among other decision variables. While all of these were optimizing for two objectives, the authors invoked a triobjective GA yield for simultaneously optimizing penicillin titer, penicillin yield, and profit.
Conclusions
This article presents the development and application of a computationally tractable framework which combines elementary mode analysis, pathway thermodynamics, and a genetic algorithm. The framework was then used to efficiently redesign the E. coli metabolic network for maximal production of two industriallyrelevant products, ethanol and lycopene. Our results show that E. coli metabolism can be retailored quite efficiently for optimal or nearoptimal production of a product of interest (ethanol or lycopene were examples here), biomass, or coupled product and biomass. As discussed, many of the gene knockouts identified by the algorithm to improve production formation have been tested experimentally (however, most often individually and not in combination) and have been shown to improve product formation rates.
It has been shown that the contribution of an individual EM to overall cellular metabolism can be estimated from its pathway thermodynamics [53]. It has been proposed that this is a result of billions of years of evolution underlying the metabolic regulation and expression patterns of the genes within these pathways. As a result of this proposal, it can be assumed that a cell will attempt to reduce its overall free energy by favoring pathways (EMs) that have a more negative Gibbs free energy. Equivalently, pathways with a positive free energy are thermodynamically infeasible and are not assigned a weight in the analysis presented here (for the case of thermodynamic weighting). This allows flux determination based solely on reaction stoichiometry and thermodynamics from the EMs generated by EMA, rather than applying a metabolic assumption (maximizing growth rate) in optimization based studies (such as FBA). It is important to note that these weighting factors are not strictly predetermined but are determined within the context of the overall cellular network.
Generally speaking, equalweighting of the EMs was shown as a proofofprinciple demonstration of the algorithm. At the same time, this was used as a reference to determine whether certain gene knockouts were predicted under both weighting schemes. Ideally, a significant fraction of the gene knockouts identified would be consistent between equal and thermodynamic weighting of the modes. As a result, the incorporation of thermodynamic calculations was an integral part of this algorithm providing for more accurate flux distributions (as compared to FBAcalculated fluxes).
The utilization of a GA to search the solution space enables the identification of an optimal genotype in a computationally tractable amount of time. The number of reaction removals required to meet these predicted optimal values are well above what is computationally feasible through exhaustive searching. For example, even ten reaction removals (the smallest number for the ethanol case study) would require evaluating 2.74 × 10^{17}in silico organisms. With the genetic algorithm, the simulations here converged when evaluating only 2,500 in silico organisms (50 generations of 50 individuals).
Metabolic and genetic networks are highly connected with significant regulation across scales, even for microbial systems. A clear disadvantage of the model and algorithm presented, as well as most of stoichiometric modeling, is the lack of integrated regulatory information. Because these models are used to study steadystate behavior, the dynamic regulation of these systems is neglected. There have been efforts to reconstruct genomescale transcriptional and translational (TRTR) networks and transcriptional regulatory networks (TRNs) [74, 75]; however, the integration of these models with metabolic models has been somewhat limited [76–79]. Utilizing EMA for identifying knockout targets for improving ethanol production in E. coli allowed for simultaneous utilization of pentoses and hexoses in batch culture [50]. This shows that a strictly stoichiometric analysis using EMA can synthetically deregulate catabolite repression (perhaps the most wellstudied means of metabolic regulation).
A potential limitation of this method is the utilization of EMA, which is computationally intensive and currently cannot be applied to genomescale metabolic networks. As cited previously, the computation time of EMA algorithms grows approximately quadratically with respect to the number of EMs and the number of EMs grows exponentially with respect to network size. For example, an E. coli model of 110 reactions (28 of which were reversible) using any combination of glucose, succinate, glycerol, and acetate contained 507,632 EMs [80]. However, when making many smallmolecule products through E. coli, minimal medium with a single carbonsource is often used such that many of the reactions in E. coli metabolism would not acquire flux. Therefore, the engineering of highflux pathways (glycolysis, the TCA cycle, etc.), as represented in this smallscale model, would have more impact on product formation. Very recently, the concept of elementary flux patterns was introduced, where an elementary flux pattern is defined as a set of reactions within a subsystem of a larger network that represents the basic routes of each steadystate flux of the larger network through the subnetwork [81]. They are computed using MILP and as a result, this technique can be applied to genomescale networks, a quality mediated by the fact that computation time climbs only polynomially with respect to network size. Also very recently, an algorithm was developed to identify the Kshortest EMs within a genomescale metabolic network utilizing integer linear programming [82]. The algorithm here could be similarly applied to these two recently developed algorithms.
Methods
Model Construction
The two smallscale E. coli stoichiometric models utilized in this study were based on one previously developed [50]. Briefly, because the previous model was developed for the utilization of multiple five and sixcarbon sugars, all of the carbonsource utilization reactions besides the glucose utilization reaction were removed; glucose was assumed to be actively uptaken by the phosphoenolpyruvate sugar transferase system. In the original model, an additional reaction was included due to a heterologous pyruvate decarboxylase from Zymomonas mobilis; this reaction is not native to E. coli and was therefore also removed.
For the lycopene case study, the ethanol model previously described served as a basis with four additional reactions added. Lycopene biosynthesis was introduced into the model and coupled to the nonmevalonate pathway (native to E. coli) previously used to support heterologous carotenoid production [83, 84]. Whenever possible, linear pathways were combined into a single reaction to reduce the size of the model. The first reaction (encoded by dxs and ispCDEFGH) held the stoichiometry: glyceraldehyde3phosphate + pyruvate + 2 NADPH + ATP → dimethylallyl diphosphate (DMAPP) + CO2 + 2 NADP^{+} + ADP. The second reaction was for the reversible isomerization of DMAPP and isopentyl diphosphate (IPP), encoded by idi. The third reaction held the stoichiometry 4 IPP → geranylgeranyl diphosphate (GGPP) and is encoded by ispA and crtE. The last reaction was for lycopene biosynthesis and held the stoichiometry 2 GGPP + 8 NADPH → lycopene + 8 NADP^{+}, and is encoded by crtBI. To avoid the inclusion of a specific transport reaction, lycopene was not balanced in this reaction; however, it was taken into account for thermodynamic calculations. The final model included three more metabolites and four more reactions than the ethanol model.
Elementary Mode Enumeration
Elementary mode analysis (EMA) was undertaken utilizing the bit pattern tree method [85]. Developed recently, this algorithm is capable of enumerating 2,450,787 EMs over tentimes faster (on a fourthread system) than the latest release of METATOOL [86], and is therefore currently the fastest method for EM enumeration [85]. The mathematical rigor associated with the bit pattern tree method and other EMA algorithms has been described previously [86–88]. The code was acquired from Professor Jörg Stelling's website http://www.csb.ethz.ch/tools/efmtool and interfaced with The MathWorks™ MATLAB software (version 7.6.0.324).
Pathway Gibbs Free Energy Calculations
The group contribution method of Mavrovouniotis was used in this study to estimate the standard Gibbs free energy of reaction for all of the model reactions [55]. Briefly, the group contribution method estimates the ΔG_{ f }° of metabolites by decomposing a single molecular structure into a subset of smaller functional groups, each individually contributing to overall ΔG_{ f }° values. The ΔG_{ r }° is then known as a result of the known stoichiometry of the reaction in question. Although currency metabolites were not included in the stoichiometric model, they were accounted for in the Gibbs free energy calculations to ensure consistency with reported data. All of the metabolites used in the stoichiometric models utilized here had corresponding ΔG_{ f }° values reported recently [56].
Genetic Algorithm
Chromosomal representation of the metabolic genotype for passing to the genetic algorithm is binary in nature where a "1" indicates the reaction is included in the individual and "0" indicates that the reaction is not present. For simplicity's sake, a onetoone association between reactions in the network and genes in the GA's population was assumed. This onetoone association decreases computation time by utilizing fewer variables for optimization. This onetoone association does not present a significant problem experimentally, for the geneassociations with the enzymes catalyzing the reactions are wellknown for E. coli due to the organism's biochemical knowledge and sequenced genome [89–91]. A binary vector of length n therefore represents a single individual in the GA population.
Initialization of a population is a critical step for determining the success of the algorithm to find the global optimum. An initial population of fifty individuals containing between two and six knockouts was seeded to the algorithm (using MATLAB's "randerr" function). This was arrived at empirically as randomly seeding individuals with approximately 50% 0's resulted in mostly nonviable strains and did not allow for the GA to reach the optimal solution. Next, each individual in the population is evaluated and given a fitness score. A previous study on using GAs to optimize genotypic space for succinate, glycerol, and vanillin production used product flux determined by optimization (FBA and MoMA) as a scoring function [46]. As stated before, this approach relies on assumptions that may or may not be valid. Here, EMA was used as the method for scoring the individuals with fitness functions as described below.
Genetic algorithms use crossover of the chromosomes (mixing of two individuals in a population to create a new individual) and mutation (change a "0" to "1" and viceversa with a specified frequency) to evolve the solution population. The implementation here was interfaced with The MathWorks™ MATLAB software and its Genetic Algorithm & Direct Search Toolbox. For crossover, mutation, and selection of individuals, twopoint, uniform, and tournamentbased methods were used, respectively. These parameters were not optimized in this study. As stated, the population size was chosen as fifty individuals, with five of the top performing individuals automatically passed to the next generation of the GA. The selection function used in the GA was either roulette or tournamentbased. The GA always terminated as a result of being below the tolerance (of the MATLAB default, 10^{6}) which was always between 50 and 100 generations.
As a method to reduce the computation time of the GA optimization, the GA was forced to always include (through fixed inclusion of a "1" in the individual genotype) reactions that were determined to either 1) reduce maximal product yield to zero, or 2) reduce maximal biomass yield to zero (indicating a lethal knockout). This reduced the genotypic space from 60 to 49 variables in the ethanol case study and 64 to 52 variables in the lycopene case study.
Flux Determination & Fitness Function Selection
Here, the T represents for temperature, which was taken to be 310.15K (37°C, the optimal temperature for E. coli growth). To satisfy the constraint that the sum of the weighting vector must be equal to unity, the weighting vector is then divided by the sum of the weighting vectors.
The computation time for these GA simulations were between 520 min on a notebook equipped with an Intel^{®} Core™ 2 Duo T9300 CPU running at 2.50 GHz, 4.0 GB memory, and a 32bit version of Microsoft Windows Vista™ Ultimate.
Flux Balance Analysis
Here, S is the stoichiometric matrix as described as previously and v is the flux vector.
In this optimization framework, c is a row vector containing weighting factors for individual fluxes on the objective function, z. For FBA calculations, this objective is solely the biomass reaction flux. a_{ i }and b_{ i }are the lower and upper bounds, respectively, of each flux as determined by either thermodynamics or experimental measurements.
Multiobjective Genetic Algorithm
The computation time for these MOGA simulations was much greater than the singleobjective GA simulations, as expected. These simulations generally terminated after approximately 48 hours running on the same computer system described above.
Abbreviations
 ΔG_{ f }°:

standard change in Gibbs free energy of formation
 ΔG_{ p }°:

standard change in Gibbs free energy across a pathway/EM
 ΔG_{ r }°:

standard change in Gibbs free energy across a reaction
 ΔS_{ p }°:

standard change in entropy across a pathway/EM
 EM:

elementary mode
 EMA:

elementary mode analysis
 FBA:

flux balance analysis
 GA:

genetic algorithm
 MFA:

metabolic flux analysis
 MOGA:

multiobjective genetic algorithm
 MoMA:

minimization of metabolic adjustment
 NSGAII:

nondominated sorting genetic algorithmII
 ROOM:

regulatory on/off minimization
 TRN:

transcriptional regulatory network
 TRTR:

transcriptional and translational.
Declarations
Acknowledgements
BAB and BAP would like to thank the Tufts University Faculty Research Awards Committee for support. BAB would also like to thank Ryan Nolan and Mark Walker for their helpful discussions.
Authors’ Affiliations
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