Flux-balance analysis of mitochondrial energy metabolism: consequences of systemic stoichiometric constraints

Mitochondrial metabolism is a critical component in the functioning and maintenance of cellular organs. The stoichiometry of biochemical reaction networks imposes constraints on mitochondrial function. A modeling framework, flux-balance analysis (FBA), was used to characterize the optimal flux distributions for maximal ATP production in the mitochondrion. The model predicted the expected ATP yields for glucose, lactate, and palmitate. Genetic defects that affect mitochondrial functions have been implicated in several human diseases. FBA can characterize the metabolic behavior due to genetic deletions at the metabolic level, and the effect of mutations in the tricarboxylic acid (TCA) cycle on mitochondrial ATP production was simulated. The mitochondrial ATP production is severely affected by TCA-cycle mutations. In addition, the model predicts the secretion of TCA-cycle intermediates, which is observed in clinical studies of mitochondriopathies such as those associated with fumarase deficiency. The model provides a systemic perspective to characterize the effect of stoichiometric constraints and specific metabolic fluxes on mitochondrial function.     

METABOLIC FLUX IS A DETERMINANT OF THE EVOLUTIONARY RATES OF ENZYME‐ENCODING GENES

Relationships between evolutionary rates and gene properties on a genomic, functional, pathway, or system level are being explored to unravel the principles of the evolutionary process. In particular, functional network properties have been analyzed to recognize the constraints they may impose on the evolutionary fate of genes. Here we took as a case study the core metabolic network in human erythrocytes and we analyzed the relationship between the evolutionary rates of its genes and the metabolic flux distribution throughout it. We found that metabolic flux correlates with the ratio of nonsynonymous to synonymous substitution rates. Genes encoding enzymes that carry high fluxes have been more constrained in their evolution, while purifying selection is more relaxed in genes encoding enzymes carrying low metabolic fluxes. These results demonstrate the importance of considering the dynamical functioning of gene networks when assessing the action of selection on system‐level properties.     

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.     

Characterizing Escherichia coli DH5alpha growth and metabolism in a complex medium using genome-scale flux analysis

Genome-scale flux analysis of Escherichia coli DH5alpha growth in a complex medium was performed to investigate the relationship between the uptake of various nutrients and their metabolic outcomes. During the exponential growth phase, we observed a sequential consumption order of serine, aspartate and glutamate in the complex medium as well as the complete consumption of key carbohydrate nutrients, glucose and trehalose. Based on the consumption and production rates of the measured metabolites, constraints-based flux analysis of a genome-scale E. coli model was then conducted to elucidate their utilization in the metabolism. The in silico analysis revealed that the cell exploited biosynthetic precursors taken up directly from the complex medium, through growth-related anabolic pathways. This suggests that the cell could be functioning in an energetically more efficient manner by reducing the energy needed to produce amino acids. The in silico simulation also allowed us to explain the observed rapid consumption of serine: excessively consumed external serine from the complex medium was mainly converted into pyruvate and glycine, which in turn, led to the acetate accumulation. The present work demonstrates the application of an in silico modeling approach to characterizing microbial metabolism under complex medium condition. This work further illustrates the use of in silico genome-scale analysis for developing better strategies related to improving microbial growth and enhancing the productivity of desirable metabolites.     

High-throughput metabolic flux analysis based on gas chromatography-mass spectrometry derived 13C constraints

13C-constrained flux balancing analysis based on gas chromatography-mass spectrometry data is presented here as a simple and robust method for the estimation of intracellular carbon fluxes. In this approach, the underdetermined system of metabolite balances deduced from stoichiometric relations and measured extracellular rates is complemented with 13C constraints from metabolic flux ratio analysis. Fluxes in central carbon metabolism of exponentially growing Escherichia coli were estimated by 13C-constrained flux balancing from three different 13C-labeled glucose experiments. The best resolution of the network was achieved using 13C constraints derived from [U-13C]glucose and [1-13C]glucose experiments. The corresponding flux estimate was in excellent agreement with a solution that was independently obtained with a comprehensive isotopomer model. This new methodology was also demonstrated to faithfully capture the intracellular flux distribution in E. coli shake flasks and 1-ml deep-well microtiter plates. Due to its simplicity, speed, and robustness, 13C-constrained metabolic flux balancing is promising for routine and high-throughput analysis on a miniaturized scale.     

Modular metabolic control analysis of large responses

Deciphering the laws that govern metabolic responses of complex systems is essential to understand physiological functioning, pathological conditions and the outcome of experimental manipulations of intact cells. To this aim, a theoretical and experimental sensitivity analysis, called modular metabolic control analysis (MMCA), was proposed. This field was previously developed under the assumptions of infinitesimal changes and/or proportionality between parameters and rates, which are usually not fulfilled in vivo. Here we develop a general MMCA for two modules, not relying on those assumptions. Control coefficients and elasticity coefficients for large changes are defined. These are subject to constraints: summation and response theorems, and relationships that allow calculating control from elasticity coefficients. We show how to determine the coefficients from top-down experiments, measuring the rates of the isolated modules as a function of the linking intermediate (there is no need to change parameters inside the modules). The novel formalism is applied to data of two experimental studies from the literature. In one of these, 40% increase in the activity of the supply module results in less than 4% increase in flux, while infinitesimal MMCA predicts more than 30% increase in flux. In addition, it is not possible to increase the flux by manipulating the activity of demand. The impossibility of increasing the flux by changing the activity of a single module is due to an abrupt decrease of the control of the modules when their corresponding activities are increased. In these cases, the infinitesimal approach can give highly erroneous predictions.     

Metabolic control analysis of xylose catabolism in Aspergillus

A kinetic model for xylose catabolism in Aspergillus is proposed. From a thermodynamic analysis it was found that the intermediate xylitol will accumulate during xylose catabolism. Use of the kinetic model allowed metabolic control analysis (MCA) of the xylose catabolic pathway to be carried out, and flux control was shown to be dependent on the metabolite levels. Due to thermodynamic constraints, flux control may reside at the first step in the pathway, i.e., at the xylose reductase, even when the intracellular xylitol concentration is high. On the basis of the kinetic analysis, the general dogma specifying that flux control often resides at the step following an intermediate present at high concentrations was, therefore, shown not to hold. The intracellular xylitol concentration was measured in batch cultivations of two different strains of Aspergillus niger and two different strains of Aspergillus nidulans grown on media containing xylose, and a concentration up to 30 mM was found. Applying MCA showed that the first polyol dehydrogenase (XDH) in the catabolic pathway of xylose exerted the main flux control in the two strains of A. nidulans and A. niger NW324, but the flux control was exerted mainly at the first enzyme of the pathway (XR) of A. niger NW 296.     

Interplay between Constraints,Objectives, and Optimality for Genome-Scale Stoichiometric Models

High-throughput data generation and genome-scale stoichiometric models have greatly facilitated the comprehensive study of metabolic networks. The computation of all feasible metabolic routes with these models, given stoichiometric, thermodynamic, and steady-state constraints, provides important insights into the metabolic capacities of a cell. How the feasible metabolic routes emerge from the interplay between flux constraints, optimality objectives, and the entire metabolic network of a cell is, however, only partially understood. We show how optimal metabolic routes, resulting from flux balance analysis computations, arise out of elementary flux modes, constraints, and optimization objectives. We illustrate our findings with a genome-scale stoichiometric model of Escherichia coli metabolism. In the case of one flux constraint, all feasible optimal flux routes can be derived from elementary flux modes alone. We found up to 120 million of such optimal elementary flux modes. We introduce a new computational method to compute the corner points of the optimal solution space fast and efficiently. Optimal flux routes no longer depend exclusively on elementary flux modes when we impose additional constraints; new optimal metabolic routes arise out of combinations of elementary flux modes. The solution space of feasible metabolic routes shrinks enormously when additional objectives---e.g. those related to pathway expression costs or pathway length---are introduced. In many cases, only a single metabolic route remains that is both feasible and optimal. This paper contributes to reaching a complete topological understanding of the metabolic capacity of organisms in terms of metabolic flux routes, one that is most natural to biochemists and biotechnologists studying and engineering metabolism.     

Combining flux and energy balance analysis to model large-scale biochemical networks

Stoichiometric Network Theory is a constraints-based, optimization approach for quantitative analysis of the phenotypes of large-scale biochemical networks that avoids the use of detailed kinetics. This approach uses the reaction stoichiometric matrix in conjunction with constraints provided by flux balance and energy balance to guarantee mass conserved and thermodynamically allowable predictions. However, the flux and energy balance constraints have not been effectively applied simultaneously on the genome scale because optimization under the combined constraints is non-linear. In this paper, a sequential quadratic programming algorithm that solves the non-linear optimization problem is introduced. A simple example and the system of fermentation in Saccharomyces cerevisiae are used to illustrate the new method. The algorithm allows the use of non-linear objective functions. As a result, we suggest a novel optimization with respect to the heat dissipation rate of a system. We also emphasize the importance of incorporating interactions between a model network and its surroundings.     

Identification of distributed metabolic objectives in the hypermetabolic liver by flux and energy balance analysis

A methodology for inferring distributed metabolic objectives from time series flux data is developed by combining metabolic flux analysis, pathway identification, free energy balances, and nested optimization. This methodology is used to investigate the metabolic response of the rat liver to burn injury-induced whole body inflammation. Gibbs free energy changes were computed for stoichiometrically balanced sequences of reactions, or pathways, rather than individual reactions, to account for energetic coupling between reactions. Systematic enumeration of pathways proceeded by elementary flux mode (EFM) analysis. Together with stoichiometric balances and external metabolite flux measurements, the DeltaG(PATH)(o) criterion provided sufficient constraints to solve a series of nested optimization problems on the metabolic goal functions and associated flux distributions of fasted livers during the first-week time course of burn injury. The optimization results suggest that there is a consistent metabolic goal function for the liver that is insensitive to the changing metabolic burdens experienced by the liver during the first-week time course. As defined by the goal function coefficients, the global metabolic objective was to distribute the metabolic resources between amino acid metabolism and ketone body synthesis. These findings point to a role for the time-invariant structure of the metabolic reaction network, expressed as stoichiometric and thermodynamic constraints, in shaping the cellular metabolic objective.     

Quantification of intracellular metabolic fluxes from fractional enrichment and 13C-13C coupling constraints on the isotopomer distribution in labeled biomass components

A method for the quantification of intracellular metabolic flux distributions from steady-state mass balance constraints and from the constraints posed by the measured 13C labeling state of biomass components is presented. Two-dimensional NMR spectroscopy is used to analyze the labeling state of cell protein hydrolysate and cell wall components. No separation of the biomass hydrolysate is required to measure the degree of 13C-13C coupling and the fractional 13C enrichment in various carbon atom positions. A mixture of [1-13C]glucose and uniformly labeled [13C6]glucose is applied to make fractional 13C enrichment data and measurements of the degree of 13C-13C coupling informative with respect to the intracellular flux distribution. Simulation models that calculate the complete isotopomer distribution in biomass components on the basis of isotopomer mapping matrices are used for the estimation of intracellular fluxes by least-squares minimization. The statistical quality of the estimated intracellular flux distributions is assessed by Monte Carlo methods. Principal component analysis is performed on the outcome of the Monte Carlo procedure to identify groups of fluxes that contribute major parts to the total variance in the multiple flux estimations. The methods described are applied to a steady-state culture of a glucoamylase-producing recombinant Aspergillus niger strain.     

Identification of flux control in metabolic networks using non-equilibrium thermodynamics

A method is presented to identify flux controlling reactions in metabolic networks using experimentally determined flux distributions. The method is based on the application of Ziegler's principle for the maximization of entropy production. According to this principle a metabolic network tends to maximize the entropy production rate while satisfying mass balances and maximal rate constraints. Experimental flux data corresponding to four different metabolic states of Saccharomyces cerevisiae were used to identify the corresponding flux controlling reactions. The bottleneck nature of several of the identified reactions was confirmed by earlier studies on over-expression of the identified target genes. The method also explains the failure of all the previous trials of increasing the glycolysis rate by direct over-expression of several glycolytic enzymes. These findings point to a wider use of the method for identification of novel targets for metabolic engineering of microorganisms used for sustainable production of fuels and chemicals.     

Genome-scale modeling using flux ratio constraints to enable metabolic engineering of clostridial metabolism in silico

ABSTRACT: BACKGROUND: Genome-scale metabolic networks and flux models are an effective platform for linking an organism genotype to its phenotype. However, few modeling approaches offer predictive capabilities to evaluate potential metabolic engineering strategies in silico. METHODS: A new method called "flux balance analysis with flux ratios (FBrAtio)" was developed in this research and applied to a new genome-scale model of Clostridium acetobutylicum ATCC 824 (iCAC794) that contains 707 metabolites and 794 reactions. FBrAtio was used to model wild-type metabolism and metabolically engineered strains of C. acetobutylicum where only flux ratio constraints and thermodynamic reversibility of reactions were required. The FBrAtio approach allows solutions to be found through standard linear programming. RESULTS: Five flux ratio constraints were required to achieve a qualitative picture of wild-type metabolism for C. acetobutylicum for the production of: (i) acetate, (ii) lactate, (iii) butyrate, (iv) acetone, (v) butanol, (vi) ethanol, (vii) CO2 and (viii) H2. Results of this simulation study coincide with published experimental results and show the knockdown of the acetylacetyl-CoA transferase increases butanol to acetone selectivity, while the simultaneous over-expression of the aldehyde/alcohol dehydrogenase greatly increases ethanol production. CONCLUSIONS: FBrAtio is a promising new method for constraining genome-scale models using internal flux ratios. The method was effective for modeling wild-type and engineered strains of C. acetobutylicum.     

Species-Specific Effects on Ecosystem Functioning Can Be Altered by Interspecific Interactions

Biological assemblages are constantly undergoing change, with species being introduced, extirpated and experiencing shifts in their densities. Theory and experimentation suggest that the impacts of such change on ecosystem functioning should be predictable based on the biological traits of the species involved. However, interspecific interactions could alter how species affect functioning, with the strength and sign of interactions potentially depending on environmental context (e.g. homogenous vs. heterogeneous conditions) and the function considered. Here, we assessed how concurrent changes to the densities of two common marine benthic invertebrates, Corophium volutator and Hediste diversicolor, affected the ecological functions of organic matter consumption and benthic-pelagic nutrient flux. Complementary experiments were conducted within homogenous laboratory microcosms and naturally heterogeneous field plots. When the densities of the species were increased within microcosms, interspecific interactions enhanced effects on organic matter consumption and reduced effects on nutrient flux. Trait-based predictions of how each species would affect functioning were only consistently supported when the density of the other species was low. In field plots, increasing the density of either species had a positive effect on organic matter consumption (with no significant interspecific interactions) but no effect on nutrient flux. Our results indicate that species-specific effects on ecosystem functioning can be altered by interspecific interactions, which can be either facilitative (positive) or antagonistic (negative) depending on the function considered. The impacts of biodiversity change may therefore not be predictable based solely on the biological traits of the species involved. Possible explanations for why interactions were detected in microcosms but not in the field are discussed.     

Metabolic flux estimation--a self-adaptive evolutionary algorithm with singular value decomposition

Metabolic flux analysis is important for metabolic system regulation and intracellular pathway identification. A popular approach for intracellular flux estimation involves using ¹³C tracer experiments to label states that can be measured by nuclear magnetic resonance spectrometry or gas chromatography mass spectrometry. However, the bilinear balance equations derived from ¹³C tracer experiments and the noisy measurements require a nonlinear optimization approach to obtain the optimal solution. In this paper, the flux quantification problem is formulated as an error-minimization problem with equality and inequality constraints through the ¹³C balance and stoichiometric equations. The stoichiometric constraints are transformed to a null space by singular value decomposition. Self-adaptive evolutionary algorithms are then introduced for flux quantification. The performance of the evolutionary algorithm is compared with ordinary least squares estimation by the simulation of the central pentose phosphate pathway. The proposed algorithm is also applied to the central metabolism of Corynebacterium glutamicum under lysine-producing conditions. A comparison between the results from the proposed algorithm and data from the literature is given. The complexity of a metabolic system with bidirectional reactions is also investigated by analyzing the fluctuations in the flux estimates when available measurements are varied.     

The genome-scale metabolic extreme pathway structure in Haemophilus influenzae shows significant network redundancy

Genome-scale metabolic networks can be characterized by a set of systemically independent and unique extreme pathways. These extreme pathways span a convex, high-dimensional space that circumscribes all potential steady-state flux distributions achievable by the defined metabolic network. Genome-scale extreme pathways associated with the production of non-essential amino acids in Haemophilus influenzae were computed. They offer valuable insight into the functioning of its metabolic network. Three key results were obtained. First, there were multiple internal flux maps corresponding to externally indistinguishable states. It was shown that there was an average of 37 internal states per unique exchange flux vector in H. influenzae when the network was used to produce a single amino acid while allowing carbon dioxide and acetate as carbon sinks. With the inclusion of succinate as an additional output, this ratio increased to 52, a 40% increase. Second, an analysis of the carbon fates illustrated that the extreme pathways were non-uniformly distributed across the carbon fate spectrum. In the detailed case study, 45% of the distinct carbon fate values associated with lysine production represented 85% of the extreme pathways. Third, this distribution fell between distinct systemic constraints. For lysine production, the carbon fate values that represented 85% of the pathways described above corresponded to only 2 distinct ratios of 1:1 and 4:1 between carbon dioxide and acetate. The present study analysed single outputs from one organism, and provides a start to genome-scale extreme pathways studies. These emergent system-level characterizations show the significance of metabolic extreme pathway analysis at the genome-scale.     

Metabolic flux control analysis of branch points: an improved approach to obtain flux control coefficients from large perturbation data

An overview of published approaches for the metabolic flux control analysis of branch points revealed that often not all fundamental constraints on the flux control coefficients have been taken into account. This has led to contradictory statements in literature on the minimum number of large perturbation experiments required to estimate the complete set of flux control coefficients C(J) for a metabolic branch point. An improved calculation procedure, based on approximate Lin-log reaction kinetics, is proposed, providing explicit analytical solutions of steady state fluxes and metabolite concentrations as a function of large changes in enzyme levels. The obtained solutions allow direct calculation of elasticity ratios from experimental data and subsequently all C(J)-values from the unique relation between elasticity ratio's and flux control coefficients. This procedure ensures that the obtained C(J)-values satisfy all fundamental constraints. From these it follows that for a three enzyme branch point only one characterised or two uncharacterised large flux perturbations are sufficient to obtain all C(J)- values. The improved calculation procedure is illustrated with four experimental cases.     

Genome-scale model for Clostridium acetobutylicum: Part II. Development of specific proton flux states and numerically determined sub-systems

A regulated genome-scale model for Clostridium acetobutylicum ATCC 824 was developed based on its metabolic network reconstruction. To aid model convergence and limit the number of flux-vector possible solutions (the size of the phenotypic solution space), modeling strategies were developed to impose a new type of constraint at the endo-exo-metabolome interface. This constraint is termed the specific proton flux state, and its use enabled accurate prediction of the extracellular medium pH during vegetative growth of batch cultures. The specific proton flux refers to the influx or efflux of free protons (per unit biomass) across the cell membrane. A specific proton flux state encompasses a defined range of specific proton fluxes and includes all metabolic flux distributions resulting in a specific proton flux within this range. Effective simulation of time-course batch fermentation required the use of independent flux balance solutions from an optimum set of specific proton flux states. Using a real-coded genetic algorithm to optimize temporal bounds of specific proton flux states, we show that six separate specific proton flux states are required to model vegetative-growth metabolism and accurately predict the extracellular medium pH. Further, we define the apparent proton flux stoichiometry per weak acids efflux and show that this value decreases from approximately 3.5 mol of protons secreted per mole of weak acids at the start of the culture to approximately 0 at the end of vegetative growth. Calculations revealed that when specific weak acids production is maximized in vegetative growth, the net proton exchange between the cell and environment occurs primarily through weak acids efflux (apparent proton flux stoichiometry is 1). However, proton efflux through cation channels during the early stages of acidogenesis was found to be significant. We have also developed the concept of numerically determined sub-systems of genome-scale metabolic networks here as a sub-network with a one-dimensional null space basis set. A numerically determined sub-system was constructed in the genome-scale metabolic network to study the flux magnitudes and directions of acetylornithine transaminase, alanine racemase, and D-alanine transaminase. These results were then used to establish additional constraints for the genome-scale model.