Incorporating metabolic flux ratios into constraint-based flux analysis by using artificial metabolites and converging ratio determinants

One of the well-established approaches for the quantitative characterization of large-scale underdetermined metabolic network is constraint-based flux analysis, which quantifies intracellular metabolic fluxes to characterize the metabolic status. The system is typically underdetermined, and thus usually is solved by linear programming with the measured external fluxes as constraints. Thus, the intracellular flux distribution calculated may not represent the true values. (13)C-constrained flux analysis allows more accurate determination of internal fluxes, but is currently limited to relatively small metabolic networks due to the requirement of complicated mathematical formulation and limited parameters available. Here, we report a strategy of employing such partial information obtained from the (13)C-labeling experiments as additional constraints during the constraint-based flux analysis. A new methodology employing artificial metabolites and converging ratio determinants (CRDs) was developed for improving constraint-based flux analysis. The CRDs were determined based on the metabolic flux ratios obtained from (13)C-labeling experiments, and were incorporated into the mass balance equations for the artificial metabolites. These new mass balance equations were used as additional constraints during the constraint-based flux analysis with genome-scale E. coli metabolic model, which allowed more accurate determination of intracellular metabolic fluxes.     

Batch culture characterization and metabolic flux analysis of succinate-producing Escherichia coli strains

This study presents an in-depth analysis of the anaerobic metabolic fluxes of various mutant strains of Escherichia coli overexpressing the Lactococcus lactis pyruvate carboxylase (PYC) for the production of succinate. Previously, a metabolic network design that includes an active glyoxylate pathway implemented in vivo increased succinate yield from glucose in an E. coli mutant to 1.6 mol/mol under fully anaerobic conditions. The design consists of a dual succinate synthesis route, which diverts required quantities of NADH through the traditional fermentative pathway and maximizes the carbon converted to succinate by balancing the carbon flux through the fermentative pathway and the glyoxylate pathway (which has a lower NADH requirement). Mutant strains previously constructed during the development of high-yield succinate-producing strains were selected for further characterization to understand their metabolic response as a result of several genetic manipulations and to determine the significance of the fermentative and the glyoxylate pathways in the production of succinate. Measured fluxes obtained under batch cultivation conditions were used to estimate intracellular fluxes and identify critical branch point flux split ratios. The comparison of changes in branch point flux split ratios to the glyoxylate pathway and the fermentative pathway at the oxaloacetate (OAA) node as a result of different mutations revealed the sensitivity of succinate yield to these manipulations. The most favorable split ratio to obtain the highest succinate yield was the fractional partition of OAA to glyoxylate of 0.32 and 0.68 to the fermentative pathway obtained in strains SBS550MG (pHL413) and SBS990MG (pHL413). The succinate yields achieved in these two strains were 1.6 and 1.7 mol/mol, respectively. In addition, an active glyoxylate pathway in an ldhA, adhE, ack-pta mutant strain is shown to be responsible for the high succinate yields achieved anaerobically. Furthermore, in vitro activity measurements of seven crucial enzymes involved in the pathways studied and intracellular measurements of key intermediate metabolite pools provided additional insights on the physiological perturbations caused by these mutations. The characterization of these recombinant mutant strains in terms of flux distribution pattern, in vitro enzyme activity and intracellular metabolite pools provides useful information for the rational modification of metabolic fluxes to improve succinate production.     

Metabolic analysis of the synthesis of high levels of intracellular human SOD in Saccharomyces cerevisiae rhSOD 2060 411 SGA122

The synthesis of human superoxide dismutase (SOD) in batch cultures of a Saccharomyces cerevisiae strain using a glucose-limited minimal medium was studied through metabolic flux analysis. A stoichiometric model was built, which included 78 reactions, according to metabolic pathways operative in these strains during respirofermentative and oxidative metabolism. It allowed calculation of the distribution of metabolic fluxes during diauxic growth on glucose and ethanol. Fermentation profiles and metabolic fluxes were analyzed at different phases of diauxic growth for the recombinant strain (P+) and for its wild type (P-). The synthesis of SOD by the strain P+ resulted in a decrease in specific growth rate of 34 and 54% (growth on glucose and ethanol respectively) in comparison to the wild type. Both strains exhibited similar flux of glucose consumption and ethanol synthesis but important differences in carbon distribution with biomass/substrate yields and ATP production 50% higher in P-. A higher contribution of fermentative metabolism, with 64% of the energy produced at the phosphorylation level, was observed during SOD production. The flux of precursors to amino acids and nucleotides was higher in the recombinant strain, in agreement with the higher total RNA and protein levels. Lower specific growth rates in strain P+ appear to be related to the decrease in the rate of synthesis of nonrecombinant protein, as well as a decrease in the activities of the pentose phosphate (PP) pathway and TCA cycle. A very different way of entry into the stationary phase was observed for each strain: in the wild-type strain most metabolic fluxes decreased and fluxes related to energy reserve synthesis increased, while in the P+ strain the flux of 22 reactions (including PP pathway and amino acids biosynthesis) related to SOD production increased their fluxes. Changes in SOD production rates at different physiological states appear to be related to the differences in building blocks availability between respirofermentative and oxidative metabolism. Using the present expression system, ideal conditions for SOD synthesis are represented by either active growth during respirofermentative metabolism or transition from a growing to a nongrowing state. An increase in SOD flux could be achieved using an expression system nonassociated to growth and potentially eliminating part of the metabolic burden.     

Effect of amino acid supplementation on titer and glycosylation distribution in hybridoma cell cultures—Systems biology‐based interpretation using genome‐scale metabolic flux balance model and multivariate data analysis

Genome‐scale flux balance analysis (FBA) is a powerful systems biology tool to characterize intracellular reaction fluxes during cell cultures. FBA estimates intracellular reaction rates by optimizing an objective function, subject to the constraints of a metabolic model and media uptake/excretion rates. A dynamic extension to FBA, dynamic flux balance analysis (DFBA), can calculate intracellular reaction fluxes as they change during cell cultures. In a previous study by Read et al. (2013), a series of informed amino acid supplementation experiments were performed on twelve parallel murine hybridoma cell cultures, and this data was leveraged for further analysis (Read et al., Biotechnol Prog. 2013;29:745–753). In order to understand the effects of media changes on the model murine hybridoma cell line, a systems biology approach is applied in the current study. Dynamic flux balance analysis was performed using a genome‐scale mouse metabolic model, and multivariate data analysis was used for interpretation. The calculated reaction fluxes were examined using partial least squares and partial least squares discriminant analysis. The results indicate media supplementation increases product yield because it raises nutrient levels extending the growth phase, and the increased cell density allows for greater culture performance. At the same time, the directed supplementation does not change the overall metabolism of the cells. This supports the conclusion that product quality, as measured by glycoform assays, remains unchanged because the metabolism remains in a similar state. Additionally, the DFBA shows that metabolic state varies more at the beginning of the culture but less by the middle of the growth phase, possibly due to stress on the cells during inoculation. © 2016 American Institute of Chemical Engineers Biotechnol. Prog., 32:1163–1173, 2016     

Determination of metabolic fluxes in a non-steady-state system

Estimation of fluxes through metabolic networks from redistribution patterns of (13)C has become a well developed technique in recent years. However, the approach is currently limited to systems at metabolic steady-state; dynamic changes in metabolic fluxes cannot be assessed. This is a major impediment to understanding the behaviour of metabolic networks, because steady-state is not always experimentally achievable and a great deal of information about the control hierarchy of the network can be derived from the analysis of flux dynamics. To address this issue, we have developed a method for estimating non-steady-state fluxes based on the mass-balance of mass isotopomers. This approach allows multiple mass-balance equations to be written for the change in labelling of a given metabolite pool and thereby permits over-determination of fluxes. We demonstrate how linear regression methods can be used to estimate non-steady-state fluxes from these mass balance equations. The approach can be used to calculate fluxes from both mass isotopomer and positional isotopomer labelling information and thus has general applicability to data generated from common spectrometry- or NMR-based analytical platforms. The approach is applied to a GC-MS time-series dataset of (13)C-labelling of metabolites in a heterotrophic Arabidopsis cell suspension culture. Threonine biosynthesis is used to demonstrate that non-steady-state fluxes can be successfully estimated from such data while organic acid metabolism is used to highlight some common issues that can complicate flux estimation. These include multiple pools of the same metabolite that label at different rates and carbon skeleton rearrangements.     

Metabolic flux analysis for serine alkaline protease fermentation by Bacillus licheniformis in a defined medium: effects of the oxygen transfer rate.

The metabolic fluxes through the central carbon pathways in the bioprocess for serine alkaline protease (SAP) production by Bacillus licheniformis were calculated by the metabolic flux-based stoichiometric model based on the proposed metabolic network that contains 102 metabolites and 133 reaction fluxes using the time profiles of citrate, dry cell, organic acids, amino acids, and SAP as the constraints. The model was solved by minimizing the SAP accumulation rate in the cell. The effects of the oxygen-transfer rate (OTR) on the metabolic fluxes were investigated in a defined medium where citrate was used as the sole carbon source. The central pathways were active for the growth and the SAP synthesis in all the periods of the bioprocess at low (LOT), medium (MOT), and high (HOT) oxygen-transfer conditions. The flux partitioning in the TCA cycle at alpha-ketoglutarate towards glutamate group and at oxalacetate (OA) toward aspartic acid group amino acids were dependent on the OTR. The flux of the anaplerotic reaction that connects the TCA cycle either from malate or OA to the gluconeogenesis pathway via the main branch point pyruvate (Pyr) was also influenced by the OTR. With the decrease in the OTR, the intracellular flux values after glycerate 3-phosphate (PG3) in the gluconeogenesis pathway and the specific growth rate decreased. The total ATP-generation rate increased with the increase in OTR. The pathway towards the aspartic acid family amino acids which is important for sporulation that precedes the SAP synthesis were all active throughout the bioprocess. Metabolic flux analysis results at LOT, MOT, and HOT conditions encourage the design of an oxygen-transfer strategy in the bioreactor; moreover, asparagine synthetase or aspartate kinase could be the potential metabolic engineering sites due to the low value of the flux from the branch point aspartate toward asparagine.     

Metabolic flux analysis in a nonstationary system: fed-batch fermentation of a high yielding strain of E. coli producing 1,3-propanediol

Metabolic fluxes estimated from stable-isotope studies provide a key to understanding cell physiology and regulation of metabolism. A limitation of the classical method for metabolic flux analysis (MFA) is the requirement for isotopic steady state. To extend the scope of flux determination from stationary to nonstationary systems, we present a novel modeling strategy that combines key ideas from isotopomer spectral analysis (ISA) and stationary MFA. Isotopic transients of the precursor pool and the sampled products are described by two parameters, D and G parameters, respectively, which are incorporated into the flux model. The G value is the fraction of labeled product in the sample, and the D value is the fractional contribution of the feed for the production of labeled products. We illustrate the novel modeling strategy with a nonstationary system that closely resembles industrial production conditions, i.e. fed-batch fermentation of Escherichia coli that produces 1,3-propanediol (PDO). Metabolic fluxes and the D and G parameters were estimated by fitting labeling distributions of biomass amino acids measured by GC/MS to a model of E. coli metabolism. We obtained highly consistent fits from the data with 82 redundant measurements. Metabolic fluxes were estimated for 20 time points during course of the fermentation. As such we established, for the first time, detailed time profiles of in vivo fluxes. We found that intracellular fluxes changed significantly during the fed-batch. The intracellular flux associated with PDO pathway increased by 10%. Concurrently, we observed a decrease in the split ratio between glycolysis and pentose phosphate pathway from 70/30 to 50/50 as a function of time. The TCA cycle flux, on the other hand, remained constant throughout the fermentation. Furthermore, our flux results provided additional insight in support of the assumed genotype of the organism.     

Genealogy profiling through strain improvement by using metabolic network analysis: metabolic flux genealogy of several generations of lysine-producing corynebacteria

A comprehensive approach of metabolite balancing, (13)C tracer studies, gas chromatography-mass spectrometry, matrix-assisted laser desorption ionization-time of flight mass spectrometry, and isotopomer modeling was applied for comparative metabolic network analysis of a genealogy of five successive generations of lysine-producing Corynebacterium glutamicum. The five strains examined (C. glutamicum ATCC 13032, 13287, 21253, 21526, and 21543) were previously obtained by random mutagenesis and selection. Throughout the genealogy, the lysine yield in batch cultures increased markedly from 1.2 to 24.9% relative to the glucose uptake flux. Strain optimization was accompanied by significant changes in intracellular flux distributions. The relative pentose phosphate pathway (PPP) flux successively increased, clearly corresponding to the product yield. Moreover, the anaplerotic net flux increased almost twofold as a consequence of concerted regulation of C(3) carboxylation and C(4) decarboxylation fluxes to cover the increased demand for lysine formation; thus, the overall increase was a consequence of concerted regulation of C(3) carboxylation and C(4) decarboxylation fluxes. The relative flux through isocitrate dehydrogenase dropped from 82.7% in the wild type to 59.9% in the lysine-producing mutants. In contrast to the NADPH demand, which increased from 109 to 172% due to the increasing lysine yield, the overall NADPH supply remained constant between 185 and 196%, resulting in a decrease in the apparent NADPH excess through strain optimization. Extrapolated to industrial lysine producers, the NADPH supply might become a limiting factor. The relative contributions of PPP and the tricarboxylic acid cycle to NADPH generation changed markedly, indicating that C. glutamicum is able to maintain a constant supply of NADPH under completely different flux conditions. Statistical analysis by a Monte Carlo approach revealed high precision for the estimated fluxes, underlining the fact that the observed differences were clearly strain specific.     

Dynamics of glutamate synthesis and excretion fluxes in batch and continuous cultures of temperature-triggered Corynebacterium glutamicum

Corynebacterium glutamicum 2262 strain, when triggered for glutamate excretion, experiences a rapid decrease in growth rate and increase in glutamate efflux. In order to gain a better quantitative understanding of the factors controlling the metabolic transition, the fermentation dynamics was investigated for a temperature-sensitive strain cultivated in batch and glucose-limited continuous cultures. For non-excreting cells at 33°C, increasing the growth rate resulted in strong increases in the central metabolic fluxes, but the intracellular glutamate level, the oxoglutarate dehydrogenase complex (ODHC) activity and the flux distribution at the oxoglutarate node remained essentially constant. When subjected to a temperature rise to 39°C, at both high- and low-metabolic activities, the bacteria showed a rapid attenuation in ODHC activity and an increase from 28% to more than 90% of the isocitrate dehydrogenase flux split towards glutamate synthesis. Simultaneously to the reduction in growth rate, the cells activated a high capacity export system capable of expelling the surplus of synthesized glutamate.     

Comparison and analysis of objective functions in flux balance analysis

Flux balance analysis (FBA) is currently one of the most important and used techniques for estimation of metabolic reaction rates (fluxes). This mathematical approach utilizes an optimization criterion in order to select a distribution of fluxes from the feasible space delimited by the metabolic reactions and some restrictions imposed over them, assuming that cellular metabolism is in steady state. Therefore, the obtained flux distribution depends on the specific objective function used. Multiple studies have been aimed to compare distinct objective functions at given conditions, in order to determine which of those functions produces values of fluxes closer to real data when used as objective in the FBA; in other words, what is the best objective function for modeling cell metabolism at a determined environmental condition. However, these comparative studies have been designed in very dissimilar ways, and in general, several factors that can change the ideal objective function in a cellular condition have not been adequately considered. Additionally, most of them have used only one dataset for representing one condition of cell growth, and different measuring techniques have been used. For these reasons, a rigorous study on the effect of factors such as the quantity of used data, the number and type of fluxes utilized as input data, and the selected classification of growth conditions, are required in order to obtain useful conclusions for these comparative studies, allowing limiting clearly the application range on any of those results. © 2014 American Institute of Chemical Engineers Biotechnol. Prog., 30:985–991, 2014     

13C-Labeled metabolic flux analysis of a fed-batch culture of elutriated Saccharomyces cerevisiae

This study addresses the question of whether observable changes in fluxes in the primary carbon metabolism of Saccharomyces cerevisiae occur between the different phases of the cell division cycle. To detect such changes by metabolic flux analysis, a 13C-labeling experiment was performed with a fed-batch culture inoculated with a partially synchronized cell population obtained through centrifugal elutriation. Such a culture exhibits dynamic changes in the fractions of cells in different cell cycle phases over time. The mass isotopomer distributions of free intracellular metabolites in central carbon metabolism were measured by liquid chromatography-mass spectrometry. For four time points during the culture, these distributions were used to obtain the best estimates for the metabolic fluxes. The obtained flux fits suggested that the optimally fitted split ratio for the pentose phosphate pathway changed by almost a factor of 2 up and down around a value of 0.27 during the experiment. Statistical analysis revealed that some of the fitted flux distributions for different time points were significantly different from each other, indicating that cell cycle-dependent variations in cytosolic metabolic fluxes indeed occurred.     

Elucidation of metabolism in hybridoma cells grown in fed‐batch culture by genome‐scale modeling

Genome‐scale modeling of mouse hybridoma cells producing monoclonal antibodies (mAb) was performed to elucidate their physiological and metabolic states during fed‐batch cell culture. Initially, feed media nutrients were monitored to identify key components among carbon sources and amino acids with significant impact on the desired outcome, for example, cell growth and antibody production. The monitored profiles indicated rapid assimilation of glucose and glutamine during the exponential growth phase. Significant increase in mAb concentration was also observed when glutamine concentration was controlled at 0.5 mM as a feeding strategy. Based on the reconstructed genome‐scale metabolic network of mouse hybridoma cells and fed‐batch profiles, flux analysis was then implemented to investigate the cellular behavior and changes in internal fluxes during the cell culture. The simulated profile of the cell growth was consistent with experimentally measured specific growth rate. The in silico simulation results indicated (i) predominant utilization of glycolytic pathway for ATP production, (ii) importance of pyruvate node in metabolic shifting, and (iii) characteristic pattern in lactate to glucose ratio during the exponential phase. In future, experimental and in silico analyses can serve as a promising approach to identifying optimal feeding strategies and potential cell engineering targets as well as facilitate media optimization for the enhanced production of mAb or recombinant proteins in mammalian cells. Biotechnol. Bioeng. 2009;102: 1494–1504. © 2008 Wiley Periodicals, Inc.     

Response of fluxome and metabolome to temperature-induced recombinant protein synthesis in Escherichia coli

The response of the central carbon metabolism of Escherichia coli to temperature-induced recombinant production of human fibroblast growth factor was studied on the level of metabolic fluxes and intracellular metabolite levels. During production, E. coli TG1:plambdaFGFB, carrying a plasmid encoded gene for the recombinant product, revealed stress related characteristics such as decreased growth rate and biomass yield and enhanced by-product excretion (acetate, pyruvate, lactate). With the onset of production, the adenylate energy charge dropped from 0.85 to 0.60, indicating the occurrence of a severe energy limitation. This triggered an increase of the glycolytic flux which, however, was not sufficient to compensate for the increased ATP demand. The activation of the glycolytic flux was also indicated by the readjustment of glycolytic pool sizes leading to an increased driving force for the reaction catalyzed by phosphofructokinase. Moreover, fluxes through the TCA cycle, into the pentose phosphate pathway and into anabolic pathways decreased significantly. The strong increase of flux into overflow pathways, especially towards acetate was most likely caused by a flux redirection from pyruvate dehydrogenase to pyruvate oxidase. The glyoxylate shunt, not active during growth, was the dominating anaplerotic pathway during production. Together with pyruvate oxidase and acetyl CoA synthase this pathway could function as a metabolic by-pass to overcome the limitation in the junction between glycolysis and TCA cycle and partly recycle the acetate formed back into the metabolism.     

Simultaneous minute by minute determination of unidirectional and net water fluxes in frog urinary bladder. A reexamination of the two barriers in series hypothesis.

Unidirectional and net water fluxes were simultaneously estimated in frog urinary bladder. The minute by minute tritiated water (3HOH) transepithelial flux and the net volume of fluid traversing the tissue were employed. It was observed that: (1) the time course of the increase in the 3HOH flux induced by antidiuretic hormone had a very similar pattern to that reported for the increase in the net movement. (2) Unstirred layers strongly affected the magnitude of the antidiuretic hormone-induced increase in 3HOH fluxes while the time course of the response was almost non-affected. In non-stimulated bladders 3HOH fluxes were poorly modified by medium stirring. New steady-state conditions for 3HOH fluxes were established 1 min after stirring rate modifications. (3) The simultaneously determined net water flux was not affected by a modification in the unstirred layers, indicating that the variations in the measured net water fluxes are a good estimation of the changes in the mucosal border permeability. (4) The presence of an osmotic gradient during hormonal challenge (implying net water fluxes, cell swelling and dilation of the intracellular spaces) did not modify the time course of 3HOH movements. These results suggest that the time course of the increase in water permeability is an intrinsic characteristic of the experimental system that could result from the addition of permeability units that increase in number during the development of the hormonal action.     

On-line heat flux measurements improve the culture medium for the growth and productivity of genetically engineered CHO cells

With the increasingly competitive commercial production of target proteins by hybridoma and genetically engineered cells, there is an urgent requirement for biosensors to monitor and control on-line and in real time the growth of cultured cells. Since growth is accompanied by an enthalpy change, heat dissipation measured by calorimetry could act as an index for metabolic flow rate. Recombinant CHO cell suspensions producing interferon-γ were pumped to an on-line flow calorimeter. The results showed that an early reflection of metabolic change is size-specific heat flux obtained from dividing heat flow rate by the capacitance change of the cell suspension, using the on-line probe of a dielectric spectroscope. Comparison of heat flux with glucose and glutamine fluxes indicated that the former most accurately reflected decreased metabolic activity. Possibly this was due to accumulation of lactate and ammonia resulting from catabolic substrates being used as biosynthetic precursors. Thus, the heat flux probe is an ideal on-line biosensor for fed-batch culture. A stoichiometric growth reaction was formulated and data for material and heat fluxes incorporated into it. This showed that cell demand for glucose and glutamine was in the stoichiometric ratio of ∼3:1 rather than the ∼5:1 in the medium. It was demonstrated that the set of stoichiometric coefficients in the reaction were related through the extent of reaction (advancement) to overall metabolic activity (flux). The fact that this approach can be used for medium optimisation is the basis for an amino-acid-enriched medium which improved cell growth while decreasing catabolic fluxes. This revised version was published online in July 2006 with corrections to the Cover Date.     

Metabolic flux analysis of CHO cell metabolism in the late non‐growth phase

Chinese hamster ovary (CHO) cell cultures are commonly used for production of recombinant human therapeutic proteins. Often the goal of such a process is to separate the growth phase of the cells, from the non‐growth phase where ideally the cells are diverting resources to produce the protein of interest. Characterizing the way that the cells use nutrients in terms of metabolic fluxes as a function of culture conditions can provide a deeper understanding of the cell biology offering guidance for process improvements. To evaluate the fluxes, metabolic flux analysis of the CHO cell culture in the non‐growth phase was performed by a combination of steady‐state isotopomer balancing and stoichiometric modeling. Analysis of the glycolytic pathway and pentose phosphate pathway (PPP) indicated that almost all of the consumed glucose is diverted towards PPP with a high NADPH production; with even recycle from PPP to G6P in some cases. Almost all of the pyruvate produced from glycolysis entered the TCA cycle with little or no lactate production. Comparison of the non‐growth phase against previously reported fluxes from growth phase cultures indicated marked differences in the fluxes, in terms of the split between glycolysis and PPP, and also around the pyruvate node. Possible reasons for the high NADPH production are also discussed. Evaluation of the fluxes indicated that the medium strength, carbon dioxide level, and temperature with dissolved oxygen have statistically significant impacts on different nodes of the flux network. Biotechnol. Bioeng. 2011; 108:82–92. © 2010 Wiley Periodicals, Inc.     

Continuous modeling of metabolic networks with gene regulation in yeast and in vivo determination of rate parameters

A continuous model of a metabolic network including gene regulation to simulate metabolic fluxes during batch cultivation of yeast Saccharomyces cerevisiae was developed. The metabolic network includes reactions of glycolysis, gluconeogenesis, glycerol and ethanol synthesis and consumption, the tricarboxylic acid cycle, and protein synthesis. Carbon sources considered were glucose and then ethanol synthesized during growth on glucose. The metabolic network has 39 fluxes, which represent the action of 50 enzymes and 64 genes and it is coupled with a gene regulation network which defines enzyme synthesis (activities) and incorporates regulation by glucose (enzyme induction and repression), modeled using ordinary differential equations. The model includes enzyme kinetics, equations that follow both mass-action law and transport as well as inducible, repressible, and constitutive enzymes of metabolism. The model was able to simulate a fermentation of S. cerevisiae during the exponential growth phase on glucose and the exponential growth phase on ethanol using only one set of kinetic parameters. All fluxes in the continuous model followed the behavior shown by the metabolic flux analysis (MFA) obtained from experimental results. The differences obtained between the fluxes given by the model and the fluxes determined by the MFA do not exceed 25% in 75% of the cases during exponential growth on glucose, and 20% in 90% of the cases during exponential growth on ethanol. Furthermore, the adjustment of the fermentation profiles of biomass, glucose, and ethanol were 95%, 95%, and 79%, respectively. With these results the simulation was considered successful. A comparison between the simulation of the continuous model and the experimental data of the diauxic yeast fermentation for glucose, biomass, and ethanol, shows an extremely good match using the parameters found. The small discrepancies between the fluxes obtained through MFA and those predicted by the differential equations, as well as the good match between the profiles of glucose, biomass, and ethanol, and our simulation, show that this simple model, that does not rely on complex kinetic expressions, is able to capture the global behavior of the experimental data. Also, the determination of parameters using a straightforward minimization technique using data at only two points in time was sufficient to produce a relatively accurate model. Thus, even with a small amount of experimental data (rates and not concentrations) it was possible to estimate the parameters minimizing a simple objective function. The method proposed allows the obtention of reasonable parameters and concentrations in a system with a much larger number of unknowns than equations. Hence a contribution of this study is to present a convenient way to find in vivo rate parameters to model metabolic and genetic networks under different conditions.     

Integrating flux balance analysis into kinetic models to decipher the dynamic metabolism of Shewanella oneidensis MR-1

Shewanella oneidensis MR-1 sequentially utilizes lactate and its waste products (pyruvate and acetate) during batch culture. To decipher MR-1 metabolism, we integrated genome-scale flux balance analysis (FBA) into a multiple-substrate Monod model to perform the dynamic flux balance analysis (dFBA). The dFBA employed a static optimization approach (SOA) by dividing the batch time into small intervals (i.e., ～400 mini-FBAs), then the Monod model provided time-dependent inflow/outflow fluxes to constrain the mini-FBAs to profile the pseudo-steady-state fluxes in each time interval. The mini-FBAs used a dual-objective function (a weighted combination of "maximizing growth rate" and "minimizing overall flux") to capture trade-offs between optimal growth and minimal enzyme usage. By fitting the experimental data, a bi-level optimization of dFBA revealed that the optimal weight in the dual-objective function was time-dependent: the objective function was constant in the early growth stage, while the functional weight of minimal enzyme usage increased significantly when lactate became scarce. The dFBA profiled biologically meaningful dynamic MR-1 metabolisms: 1. the oxidative TCA cycle fluxes increased initially and then decreased in the late growth stage; 2. fluxes in the pentose phosphate pathway and gluconeogenesis were stable in the exponential growth period; and 3. the glyoxylate shunt was up-regulated when acetate became the main carbon source for MR-1 growth.