Analysis of optimal phenotypic space using elementary modes as applied to Corynebacterium glutamicum
- Kalyan Gayen^{1} and
- KV Venkatesh^{1, 2}Email author
DOI: 10.1186/1471-2105-7-445
© Gayen and Venkatesh; licensee BioMed Central Ltd. 2006
Received: 05 May 2006
Accepted: 12 October 2006
Published: 12 October 2006
Abstract
Background
Quantification of the metabolic network of an organism offers insights into possible ways of developing mutant strain for better productivity of an extracellular metabolite. The first step in this quantification is the enumeration of stoichiometries of all reactions occurring in a metabolic network. The structural details of the network in combination with experimentally observed accumulation rates of external metabolites can yield flux distribution at steady state. One such methodology for quantification is the use of elementary modes, which are minimal set of enzymes connecting external metabolites. Here, we have used a linear objective function subject to elementary modes as constraint to determine the fluxes in the metabolic network of Corynebacterium glutamicum. The feasible phenotypic space was evaluated at various combinations of oxygen and ammonia uptake rates.
Results
Quantification of the fluxes of the elementary modes in the metabolism of C. glutamicum was formulated as linear programming. The analysis demonstrated that the solution was dependent on the criteria of objective function when less than four accumulation rates of the external metabolites were considered. The analysis yielded feasible ranges of fluxes of elementary modes that satisfy the experimental accumulation rates. In C. glutamicum, the elementary modes relating to biomass synthesis through glycolysis and TCA cycle were predominantly operational in the initial growth phase. At a later time, the elementary modes contributing to lysine synthesis became active. The oxygen and ammonia uptake rates were shown to be bounded in the phenotypic space due to the stoichiometric constraint of the elementary modes.
Conclusion
We have demonstrated the use of elementary modes and the linear programming to quantify a metabolic network. We have used the methodology to quantify the network of C. glutamicum, which evaluates the set of operational elementary modes at different phases of fermentation. The methodology was also used to determine the feasible solution space for a given set of substrate uptake rates under specific optimization criteria. Such an approach can be used to determine the optimality of the accumulation rates of any metabolite in a given network.
Background
Application of metabolic engineering towards strain improvement involves detailed quantitative evaluation of cellular physiology [1–3]. Determination of intracellular metabolic fluxes helps in gaining valuable insights into the functioning of the active cellular metabolism, the knowledge of which aids in the development of rational strategies for strain improvement [4]. Theoretical methods have been developed for predicting key aspects of network functionality for a given metabolic network [5–8]. Experimental methods in tandem with theoretical analysis are key strategies for successful application of metabolic engineering to optimize the productivity of a native strain [9].
Most theoretical methods, such as metabolic flux analysis (MFA), are based on stoichiometric reactions involving various metabolites in a metabolic network [10–13]. The reaction network is used in conjunction with measured accumulation rates of certain metabolites as a constraint to determine the fluxes. Linear programming is used to maximize an objective function in the presence of stoichiometric constraints, which is also used in flux balance analysis (FBA) [7, 14]. These methods have gained popularity among many researchers as seen through its application to various microbial systems. Recently, the reaction details in a metabolic network have been used to determine elementary modes, which are minimal set of enzymes connecting the external metabolites [6, 15, 16]. Certain advantages have been associated with analysis involving elementary modes such as ease in evaluating maximum yields of metabolites and the flux distribution inherently ensuring the directionality of reactions. Elementary modes have also been used to determine the fluxes of a metabolic network using matrix algebra [17]. However, both the methodologies (i.e. FBA and elementary modes analysis) require experimentally determined rates. Typically, the measurements of the extracellular metabolites are used in the analysis assuming pseudo steady state levels of the intracellular metabolites. It is relevant to raise the question regarding the minimum number of accumulation rates of extracellular metabolites obtained through experiments, which are necessary for proper assessment of fluxes in a network. We address this issue by analyzing the flux distribution of Corynebacterium glutamicum for the production of amino acids (lysine) using elementary modes.
Flux distribution in the metabolic network of C. glutamicum, which is used for lysine production, is well demonstrated. Batch growth of C. glutamicum for lysine production can be represented through four phases. Phase-I represents balanced growth with little or no product formation and is dependent on threonine concentration. Phase-II represents high lysine synthesis and biomass production rate with constant respiration rate. In phase-III, lysine production continues at a high rate while biomass productions saturates with a decrease in respiration rate. Phase-IV sees a gradual reduction in lysine synthesis and redirection of lysine to other byproducts such as pyruvate, acetate, lactate etc. [18]. Therefore, phase-II and phase-III are the relevant phases for lysine synthesis. Previous studies have demonstrated that pentose phosphate pathway (PPP) and phosphoenolpyruvate carboxylase (PPC) shunt support substantial flux during lysine synthesis. Vallino and Stephanopoulos [10] have performed flux and nodal analysis to demonstrate that the branch point around the phosphoenolpyruvate (PEP) node is rigid, indicating a tight control of the lysine yield. This rigidity was due to the activation of PEP to OAA (oxaloacetate) reaction by AcCoA and inhibition of the same by aspartate. However, it should be noted that by eliminating rigidity around the PEP node, the rigidity might shift to other branch point such as glucose-6-phosphate (G6P) [10]. Thus, it is now clear that substantial alteration in flux partition at the three principle nodes namely PEP, pyruvate (PYR) and G6P in the metabolic pathway of the C. glutamicum is necessary to achieve high lysine yields. Although, flux analysis of C. glutamicum is well demonstrated for different fermentation stages, the effect of nitrogen and oxygen uptake rate on the productivity has not been quantified. We address this issue by analyzing the performance of the network at various oxygen and nitrogen uptake rates.
In the present work, we analyze the metabolic network of C. glutamicum using linear programming with the coefficients of the elementary modes as constraints. Further, we evaluate a feasible solution space for a given objective function of the network through a search in the space characterized by uptake rates of oxygen and nitrogen.
Results
The metabolic network of C. glutamicum included the glucose phosphotransferase system, reactions of glycolytic pathway, reactions of TCA cycle and PPP as the core metabolism [see Additional file 1]. Also, carboxylation reaction for connection of PEP to OAA was included as it plays an important role for lysine synthesis. The ammonia assimilation was through the amino acid synthesis. Further, the oxidative phosphorylation accounted for NADH recycling with ATP synthesis, while the biomass formation was included as a stoichiometric reaction involving the internal metabolites [18]. It is reported that pyruvate carboxylase, pyruvate decarboxylase, malic enzyme and PEP decarboxykinase are also present in C. glutamicum [19]. In vitro studies indicate that the activities of the anaplerotic enzymes are negligible except that of PEP carboxylase, which is active in presence of glucose as the sole carbon source in the media [18, 20–22]. Further, in vivo studies also indicate that these enzymes are inactive while growing on glucose alone [23]. These reactions are shown to be operational in the presence of lactate in the medium [19, 20, 24] due to pyruvate overflow in the system. Moreover, the glyoxylate shunt is active only in the presence of acetate in the media [25, 26]. It is becoming clear that these important anaplerotic bioreactions are regulated in C. glutamicum in presence of lactate/acetate in the media. Since we analyze the system in presence of glucose alone, we do not include these anaplerotic reactions. Elementary modes represent the overall stoichiometry of the metabolic network in terms of inter conversion of external metabolites. This implies that the internal metabolites are at pseudo steady state levels and the accumulation rates of external metabolites can be used to evaluate the fluxes of the elementary modes (see method).
Quantification of fluxes of elementary modes for Corynebacterium glutamicum
Normalized and absolute (within bracket) extracellular metabolites accumulation rates
Extracellular metabolites | Accumulation rates (mM/h) | ||
---|---|---|---|
11.5 h | 13.5 h | 15.8 h | |
Ac | 0.16 (0.04) | 0.028 (0.01) | 0.35 (0.1) |
Ala | 0.68 (0.16) | 0.056 (0.02) | 0.14 (0.04) |
Biomass | 90.43 (22.7) | 50.56 (17.9) | 23.5 (6.7) |
CO_{2} | 251.7 (63.2) | 218.6 (77.4) | 257.9 (73.5) |
Glc | -100.0 (-25.1) | -100.0 (-35.4) | -100.0 (-28.5) |
Lac | 0.24 (0.06) | 0.056 (0.02) | 0.53 (0.15) |
Lysi | 0.16 (0.04) | 30.23 (10.7) | 29.37 (8.37) |
NH_{3} | -67.7 (-17.0) | -96.9 (-34.3) | -78.25 (-22.3) |
O_{2} | -230.67 (-57.9) | -177.4 (62.8) | -221.4 (-63.1) |
Pyr | 0.52 (0.13) | 0.169 (0.06) | 0.46 (0.13) |
Trehal | 2.51 (0.63) | 2.54 (0.9) | 5.96 (1.7) |
Val | 0.24 (0.06) | 0.565 (0.2) | 2.1 (0.6) |
Comparison of experimental data and predicted values of accumulation rates of various metabolites at different time points
Accumulation rate (mM/h) | Biomass | Trehalose | CO_{2} | |||
---|---|---|---|---|---|---|
Time (h) | Expt. | Predicted | Expt | Predicted | Expt | Predicted |
11.5 | 90.43 | 91.55 | 2.51 | 2.7 | 251.79 | 249.0 |
13.5 | 50.53 | 49.51 | 2.54 | 2.45 | 218.6 | 217.46 |
15.8 | 23.5 | 26.5 | 5.96 | 6.32 | 257.89 | 256.0 |
Comparison of experimental data and predicted values of accumulation rates of various metabolites at different time points
Accumulation rate (mM/h) | NH_{3} | O_{2} | CO_{2} | |||
---|---|---|---|---|---|---|
Time (h) | Expt. | Predicted | Expt | Predicted | Expt | Predicted |
11.5 | -67.72 | -66.88 | -230.68 | -237.1 | 251.79 | 255.2 |
13.5 | -96.89 | -97.67 | -177.4 | -172.48 | 218.6 | 212.74 |
15.8 | -78.24 | -76.02 | -221.4 | -236.77 | 257.89 | 270.8 |
Simulation of feasible phenotypic space
As discussed above, the fluxes through individual modes were evaluated using the experimentally determined accumulation rates of glucose, ammonia, oxygen and lysine. However, the network can also be simulated by considering only the three uptake rates of substrates (i.e. glucose, ammonia and oxygen) as decision variables with a maximization criterion (maximization of biomass or that of lysine) to study the metabolic capability of the organism at various nutrient uptake rates. Thus, the network of C. glutamicum was simulated with the normalized glucose uptake rate equal to 100 moles/Lh with various combinations of uptake rates of ammonia and oxygen.
Discussion
We have quantified the metabolic network of C. glutamicum using elementary modes by linear optimization. The analysis was used to determine the feasible set of flux values for the elementary modes. The individual stoichiometric combination of various substrates towards specific product can be evaluated using this methodology. The linear optimization was used to determine the fluxes of elementary modes using accumulation rates obtained through experiments. In case of C. glutamicum, only four measurement values were essential for the closure of molar balance. In such a situation, the solution for predicting the accumulation rates of remaining external metabolites was independent of the objective function. This, however, did not ensure a unique flux distribution for the various elementary modes and provided a set of feasible ranges. By utilizing lesser number of accumulation rates in the linear optimization strategy, one could determine the capability of the network towards achieving a specific objective function. We used the uptake rates of three substrates (i.e. glucose, ammonia and oxygen) to evaluate the capability of the network to produce maximum biomass or lysine. The solution space bounds the accumulation rate of a metabolite for a specific substrate combination. The stoichiometry of the elementary modes also enforces a constraint on the uptake rates of the substrates. For example, in C. glutamicum for a given glucose (100) and ammonia (85) uptake rates, the oxygen uptake rate was bounded within a range of 153 – 302. This also reflects the constraint of the aerobic and anaerobic routes that may be used for balancing the oxidative potential (i.e. balancing of NADH/NAD^{+}). Similarly, the demand for balancing of NADPH/NADP^{+} may switch on the elementary modes containing the pentose phosphate pathway. In C. glutamicum, this was observed in the modes involving lysine synthesis. Also, for a given glucose and oxygen uptake rate, the ammonia uptake gets bounded, resulting in a constraint on the productivity of lysine. The feasible solution space was equivalent to the phenotypic phase plane as reported by Edwards et al. [30], where a phenotypic phase plane was evaluated for the growth of E. coli on glucose and acetate at various oxygenation levels. Here, we have similarly attempted to obtain optimal growth and lysine synthesis rates at various oxygenation levels and ammonia uptake rates.
The methodology presented here is useful in deciphering the capability of an organism for a given substrate combination. For example, the maximum yield coefficient (ratio of accumulation rate of a product and that of biomass) can be evaluated for a specific nutrient combination. Further, the analysis can yield the operational elementary modes at different points of the feasible solution space. In case of C. glutamicum, the biomass formation rate was highest (124) at ammonia and oxygen uptake rates of 91 and 146, respectively. In this case, elementary modes numbered as '5', '6', '7', were mainly functional and operated through the routes using glycolysis and TCA cycle with carbon dioxide as only the byproduct. Interestingly, we obtained the same result with the objective function of maximization of lysine. This implies that a minimum oxygen uptake of 146 is essential during the metabolism of C. glutamicum. A similar behavior in the flux distribution of each elementary mode was observed at the extreme points of the feasible set of oxygen and ammonia consumption rate. It was noted that for a maximum feasible ammonia uptake rate (127), the lysine synthesis rate was at a maximum with the elementary mode '13' being functional. This elementary mode included reactions involving PPP, PPC, TCA cycle and lysine synthesis.
Evaluation of the fluxes of elementary modes can therefore provide functional insights about any metabolic network. The operational limitation caused by ATP and NADH balancing was inherently captured using the elementary modes. The stoichiometric distribution towards a targeted product using a specific elementary mode provides insights into the route that needs engineering to enhance productivity. Further, the flux distribution through the elementary modes can yield overall capability of the organism in terms of growth, product formation and substrate uptake rates for a given media composition.
Conclusion
We have demonstrated the use of elementary modes with the help of linear optimization in quantifying metabolic networks. This methodology was used to quantify the network of C. glutamicum for lysine synthesis. Analysis in conjunction with experimental accumulation rates gives insight into the contribution of various elementary modes towards the accumulation rates of extracellular metabolites. In C. glutamicum, the elementary modes associated with biomass formation were operational at the initial experimental growth phase, while modes associated with lysine synthesis switch on at later phase of fermentation. The methodology was also used to determine the feasible solution space for a given set of substrate uptake rates. Such an approach is generic in nature and can be used to determine the optimality of the accumulation rates of a metabolite in any given system.
Methods
Elementary modes for the network were obtained using 'ScrumPy' software [31]. The input file of the reaction set to the software is provided as a .spy file [see Additional file 4]. The accumulation rates of extracellular metabolites can be represented through the fluxes in the elementary modes with equivalent stoichiometry of the elementary modes. The matrix representation of the same can be given as
S·E = M (1)
where, M is vector representing the accumulation rates of the external metabolites and S is the matrix, indicating the stoichiometry of the elementary modes. The elements of E{e_{1}, e_{2}, e_{3},.........e_{ n }} represents the vector of unknown fluxes of the elementary modes. The elements of vector E can be evaluated for a given set of measurements of accumulation rates (elements of M). Due to the paucity of measurements, a linear optimization technique can be employed to converge onto a feasible solution. Mathematically, the linear optimization formulation can be represented as:
Max(M_{ i })
such that,
S*E = M ^{•}
0 ≤ e_{ i }≤ ∞ for all the elements (2)
M_{ i }is the accumulation rate of a specific metabolites (such as biomass or carbon dioxide). Also, S* is stoichiometric matrix and M* is know vector of the accumulation rates of extracellular metabolites, wherein the rows of the i^{ th } external metabolite are eliminated from S and M, respectively. It should be noted that the above approach of using linear programming to evaluate fluxes is similar to the methodology used in FBA. FBA uses the stoichiometry of the original network for the analysis, while we use the stoichiometries of the elementary modes. The methodology has been explained using an illustrative example [see Additional file 5]
Declarations
Acknowledgements
KVV acknowledges financial support for the research from Swarnajayanti fellowship, Department of Science and Technology, India. Authors are thankful to Dr. M. G. Poolman and Prof. D. A. Fell (Oxford Brooks University) for providing the "ScrumPy" software. Authors are also thankful to Prof. Sharad Bhartiya and Mr. P. K. Vinod for their useful suggestions.
Authors’ Affiliations
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