Structural Control of Metabolic Flux

Organisms have to continuously adapt to changing environmental conditions or undergo developmental transitions. To meet the accompanying change in metabolic demands, the molecular mechanisms of adaptation involve concerted interactions which ultimately induce a modification of the metabolic state, which is characterized by reaction fluxes and metabolite concentrations. These state transitions are the effect of simultaneously manipulating fluxes through several reactions. While metabolic control analysis has provided a powerful framework for elucidating the principles governing this orchestrated action to understand metabolic control, its applications are restricted by the limited availability of kinetic information. Here, we introduce structural metabolic control as a framework to examine individual reactions'' potential to control metabolic functions, such as biomass production, based on structural modeling. The capability to carry out a metabolic function is determined using flux balance analysis (FBA). We examine structural metabolic control on the example of the central carbon metabolism of Escherichia coli by the recently introduced framework of functional centrality (FC). This framework is based on the Shapley value from cooperative game theory and FBA, and we demonstrate its superior ability to assign “share of control” to individual reactions with respect to metabolic functions and environmental conditions. A comparative analysis of various scenarios illustrates the usefulness of FC and its relations to other structural approaches pertaining to metabolic control. We propose a Monte Carlo algorithm to estimate FCs for large networks, based on the enumeration of elementary flux modes. We further give detailed biological interpretation of FCs for production of lactate and ATP under various respiratory conditions.     

Towards kinetic modeling of genome-scale metabolic networks without sacrificing stoichiometric,thermodynamic and physiological constraints

Mathematical modeling is an essential tool for the comprehensive understanding of cell metabolism and its interactions with the environmental and process conditions. Recent developments in the construction and analysis of stoichiometric models made it possible to define limits on steady-state metabolic behavior using flux balance analysis. However, detailed information on enzyme kinetics and enzyme regulation is needed to formulate kinetic models that can accurately capture the dynamic metabolic responses. The use of mechanistic enzyme kinetics is a difficult task due to uncertainty in the kinetic properties of enzymes. Therefore, the majority of recent works considered only mass action kinetics for reactions in metabolic networks. Herein, we applied the optimization and risk analysis of complex living entities (ORACLE) framework and constructed a large-scale mechanistic kinetic model of optimally grown Escherichia coli. We investigated the complex interplay between stoichiometry, thermodynamics, and kinetics in determining the flexibility and capabilities of metabolism. Our results indicate that enzyme saturation is a necessary consideration in modeling metabolic networks and it extends the feasible ranges of metabolic fluxes and metabolite concentrations. Our results further suggest that enzymes in metabolic networks have evolved to function at different saturation states to ensure greater flexibility and robustness of cellular metabolism.     

Including metabolite concentrations into flux balance analysis: thermodynamic realizability as a constraint on flux distributions in metabolic networks

Background  

In recent years, constrained optimization – usually referred to as flux balance analysis (FBA) – has become a widely applied method for the computation of stationary fluxes in large-scale metabolic networks. The striking advantage of FBA as compared to kinetic modeling is that it basically requires only knowledge of the stoichiometry of the network. On the other hand, results of FBA are to a large degree hypothetical because the method relies on plausible but hardly provable optimality principles that are thought to govern metabolic flux distributions.     

Flux Imbalance Analysis and the Sensitivity of Cellular Growth to Changes in Metabolite Pools

Stoichiometric models of metabolism, such as flux balance analysis (FBA), are classically applied to predicting steady state rates - or fluxes - of metabolic reactions in genome-scale metabolic networks. Here we revisit the central assumption of FBA, i.e. that intracellular metabolites are at steady state, and show that deviations from flux balance (i.e. flux imbalances) are informative of some features of in vivo metabolite concentrations. Mathematically, the sensitivity of FBA to these flux imbalances is captured by a native feature of linear optimization, the dual problem, and its corresponding variables, known as shadow prices. First, using recently published data on chemostat growth of Saccharomyces cerevisae under different nutrient limitations, we show that shadow prices anticorrelate with experimentally measured degrees of growth limitation of intracellular metabolites. We next hypothesize that metabolites which are limiting for growth (and thus have very negative shadow price) cannot vary dramatically in an uncontrolled way, and must respond rapidly to perturbations. Using a collection of published datasets monitoring the time-dependent metabolomic response of Escherichia coli to carbon and nitrogen perturbations, we test this hypothesis and find that metabolites with negative shadow price indeed show lower temporal variation following a perturbation than metabolites with zero shadow price. Finally, we illustrate the broader applicability of flux imbalance analysis to other constraint-based methods. In particular, we explore the biological significance of shadow prices in a constraint-based method for integrating gene expression data with a stoichiometric model. In this case, shadow prices point to metabolites that should rise or drop in concentration in order to increase consistency between flux predictions and gene expression data. In general, these results suggest that the sensitivity of metabolic optima to violations of the steady state constraints carries biologically significant information on the processes that control intracellular metabolites in the cell.     

From annotated genomes to metabolic flux models and kinetic parameter fitting

Significant advances in system-level modeling of cellular behavior can be achieved based on constraints derived from genomic information and on optimality hypotheses. For steady-state models of metabolic networks, mass conservation and reaction stoichiometry impose linear constraints on metabolic fluxes. Different objectives, such as maximization of growth rate or minimization of flux distance from a reference state, can be tested in different organisms and conditions. In particular, we have suggested that the metabolic properties of mutant bacterial strains are best described by an algorithm that performs a minimization of metabolic adjustment (MOMA) upon gene deletion. The increasing availability of many annotated genomes paves the way for a systematic application of these flux balance methods to a large variety of organisms. However, such a high throughput goal crucially depends on our capacity to build metabolic flux models in a fully automated fashion. Here we describe a pipeline for generating models from annotated genomes and discuss the current obstacles to full automation. In addition, we propose a framework for the integration of flux modeling results and high throughput proteomic data, which can potentially help in the inference of whole-cell kinetic parameters.     

Using Bioconductor Package BiGGR for Metabolic Flux Estimation Based on Gene Expression Changes in Brain

Predicting the distribution of metabolic fluxes in biochemical networks is of major interest in systems biology. Several databases provide metabolic reconstructions for different organisms. Software to analyze flux distributions exists, among others for the proprietary MATLAB environment. Given the large user community for the R computing environment, a simple implementation of flux analysis in R appears desirable and will facilitate easy interaction with computational tools to handle gene expression data. We extended the R software package BiGGR, an implementation of metabolic flux analysis in R. BiGGR makes use of public metabolic reconstruction databases, and contains the BiGG database and the reconstruction of human metabolism Recon2 as Systems Biology Markup Language (SBML) objects. Models can be assembled by querying the databases for pathways, genes or reactions of interest. Fluxes can then be estimated by maximization or minimization of an objective function using linear inverse modeling algorithms. Furthermore, BiGGR provides functionality to quantify the uncertainty in flux estimates by sampling the constrained multidimensional flux space. As a result, ensembles of possible flux configurations are constructed that agree with measured data within precision limits. BiGGR also features automatic visualization of selected parts of metabolic networks using hypergraphs, with hyperedge widths proportional to estimated flux values. BiGGR supports import and export of models encoded in SBML and is therefore interoperable with different modeling and analysis tools. As an application example, we calculated the flux distribution in healthy human brain using a model of central carbon metabolism. We introduce a new algorithm termed Least-squares with equalities and inequalities Flux Balance Analysis (Lsei-FBA) to predict flux changes from gene expression changes, for instance during disease. Our estimates of brain metabolic flux pattern with Lsei-FBA for Alzheimer’s disease agree with independent measurements of cerebral metabolism in patients. This second version of BiGGR is available from Bioconductor.     

Basic concepts and principles of stoichiometric modeling of metabolic networks

Metabolic networks supply the energy and building blocks for cell growth and maintenance. Cells continuously rewire their metabolic networks in response to changes in environmental conditions to sustain fitness. Studies of the systemic properties of metabolic networks give insight into metabolic plasticity and robustness, and the ability of organisms to cope with different environments. Constraint-based stoichiometric modeling of metabolic networks has become an indispensable tool for such studies. Herein, we review the basic theoretical underpinnings of constraint-based stoichiometric modeling of metabolic networks. Basic concepts, such as stoichiometry, chemical moiety conservation, flux modes, flux balance analysis, and flux solution spaces, are explained with simple, illustrative examples. We emphasize the mathematical definitions and their network topological interpretations.     

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     

Investigation of metabolic objectives in cultured hepatocytes

Using optimization based methods to predict fluxes in metabolic flux balance models has been a successful approach for some microorganisms, enabling construction of in silico models and even inference of some regulatory motifs. However, this success has not been translated to mammalian cells. The lack of knowledge about metabolic objectives in mammalian cells is a major obstacle that prevents utilization of various metabolic engineering tools and methods for tissue engineering and biomedical purposes. In this work, we investigate and identify possible metabolic objectives for hepatocytes cultured in vitro. To achieve this goal, we present a special data-mining procedure for identifying metabolic objective functions in mammalian cells. This multi-level optimization based algorithm enables identifying the major fluxes in the metabolic objective from MFA data in the absence of information about critical active constraints of the system. Further, once the objective is determined, active flux constraints can also be identified and analyzed. This information can be potentially used in a predictive manner to improve cell culture results or clinical metabolic outcomes. As a result of the application of this method, it was found that in vitro cultured hepatocytes maximize oxygen uptake, coupling of urea and TCA cycles, and synthesis of serine and urea. Selection of these fluxes as the metabolic objective enables accurate prediction of the flux distribution in the system given a limited amount of flux data; thus presenting a workable in silico model for cultured hepatocytes. It is observed that an overall homeostasis picture is also emergent in the findings.     

A guide to metabolic flux analysis in metabolic engineering: Methods,tools and applications

The field of metabolic engineering is primarily concerned with improving the biological production of value-added chemicals, fuels and pharmaceuticals through the design, construction and optimization of metabolic pathways, redirection of intracellular fluxes, and refinement of cellular properties relevant for industrial bioprocess implementation. Metabolic network models and metabolic fluxes are central concepts in metabolic engineering, as was emphasized in the first paper published in this journal, “Metabolic fluxes and metabolic engineering” (Metabolic Engineering, 1: 1–11, 1999). In the past two decades, a wide range of computational, analytical and experimental approaches have been developed to interrogate the capabilities of biological systems through analysis of metabolic network models using techniques such as flux balance analysis (FBA), and quantify metabolic fluxes using constrained-based modeling approaches such as metabolic flux analysis (MFA) and more advanced experimental techniques based on the use of stable-isotope tracers, i.e. ¹³C-metabolic flux analysis (¹³C-MFA). In this review, we describe the basic principles of metabolic flux analysis, discuss current best practices in flux quantification, highlight potential pitfalls and alternative approaches in the application of these tools, and give a broad overview of pragmatic applications of flux analysis in metabolic engineering practice.     

Elementary Flux Modes Analysis of Functional Domain Networks Allows a Better Metabolic Pathway Interpretation

Metabolic network analysis is an important step for the functional understanding of biological systems. In these networks, enzymes are made of one or more functional domains often involved in different catalytic activities. Elementary flux mode (EFM) analysis is a method of choice for the topological studies of these enzymatic networks. In this article, we propose to use an EFM approach on networks that encompass available knowledge on structure-function. We introduce a new method that allows to represent the metabolic networks as functional domain networks and provides an application of the algorithm for computing elementary flux modes to analyse them. Any EFM that can be represented using the classical representation can be represented using our functional domain network representation but the fine-grained feature of functional domain networks allows to highlight new connections in EFMs. This methodology is applied to the tricarboxylic acid cycle (TCA cycle) of Bacillus subtilis, and compared to the classical analyses. This new method of analysis of the functional domain network reveals that a specific inhibition on the second domain of the lipoamide dehydrogenase (pdhD) component of pyruvate dehydrogenase complex leads to the loss of all fluxes. Such conclusion was not predictable in the classical approach.     

Estimating Metabolic Fluxes Using a Maximum Network Flexibility Paradigm

MotivationGenome-scale metabolic networks can be modeled in a constraint-based fashion. Reaction stoichiometry combined with flux capacity constraints determine the space of allowable reaction rates. This space is often large and a central challenge in metabolic modeling is finding the biologically most relevant flux distributions. A widely used method is flux balance analysis (FBA), which optimizes a biologically relevant objective such as growth or ATP production. Although FBA has proven to be highly useful for predicting growth and byproduct secretion, it cannot predict the intracellular fluxes under all environmental conditions. Therefore, alternative strategies have been developed to select flux distributions that are in agreement with experimental “omics” data, or by incorporating experimental flux measurements. The latter, unfortunately can only be applied to a limited set of reactions and is currently not feasible at the genome-scale. On the other hand, it has been observed that micro-organisms favor a suboptimal growth rate, possibly in exchange for a more “flexible” metabolic network. Instead of dedicating the internal network state to an optimal growth rate in one condition, a suboptimal growth rate is used, that allows for an easier switch to other nutrient sources. A small decrease in growth rate is exchanged for a relatively large gain in metabolic capability to adapt to changing environmental conditions.ResultsHere, we propose Maximum Metabolic Flexibility (MMF) a computational method that utilizes this observation to find the most probable intracellular flux distributions. By mapping measured flux data from central metabolism to the genome-scale models of Escherichia coli and Saccharomyces cerevisiae we show that i) indeed, most of the measured fluxes agree with a high adaptability of the network, ii) this result can be used to further reduce the space of feasible solutions iii) this reduced space improves the quantitative predictions made by FBA and contains a significantly larger fraction of the measured fluxes compared to the flux space that was reduced by a uniform sampling approach and iv) MMF can be used to select reactions in the network that contribute most to the steady-state flux space. Constraining the selected reactions improves the quantitative predictions of FBA considerably more than adding an equal amount of flux constraints, selected using a more naïve approach. Our method can be applied to any cell type without requiring prior information.AvailabilityMMF is freely available as a MATLAB plugin at: http://cs.ru.nl/~wmegchel/mmf.     

Genome-scale metabolic flux analysis of Streptomyces lividans growing on a complex medium

Constraint-based metabolic modeling comprises various excellent tools to assess experimentally observed phenotypic behavior of micro-organisms in terms of intracellular metabolic fluxes. In combination with genome-scale metabolic networks, micro-organisms can be investigated in much more detail and under more complex environmental conditions. Although complex media are ubiquitously applied in industrial fermentations and are often a prerequisite for high protein secretion yields, such multi-component conditions are seldom investigated using genome-scale flux analysis. In this paper, a systematic and integrative approach is presented to determine metabolic fluxes in Streptomyces lividans TK24 grown on a nutritious and complex medium. Genome-scale flux balance analysis and randomized sampling of the solution space are combined to extract maximum information from exometabolome profiles. It is shown that biomass maximization cannot predict the observed metabolite production pattern as such. Although this cellular objective commonly applies to batch fermentation data, both input and output constraints are required to reproduce the measured biomass production rate. Rich media hence not necessarily lead to maximum biomass growth. To eventually identify a unique intracellular flux vector, a hierarchical optimization of cellular objectives is adopted. Out of various tested secondary objectives, maximization of the ATP yield per flux unit returns the closest agreement with the maximum frequency in flux histograms. This unique flux estimation is hence considered as a reasonable approximation for the biological fluxes. Flux maps for different growth phases show no active oxidative part of the pentose phosphate pathway, but NADPH generation in the TCA cycle and NADPH transdehydrogenase activity are most important in fulfilling the NADPH balance. Amino acids contribute to biomass growth by augmenting the pool of available amino acids and by boosting the TCA cycle, particularly when using glutamate and aspartate. Depletion of glutamate and aspartate causes a distinct shift in fluxes of the central carbon and nitrogen metabolism. In the current work, hurdles encountered in flux analysis at a genome-scale level are addressed using hierarchical flux balance analysis and uniform sampling of the constrained solution space. This general framework can now be adopted in further studies of S. lividans, e.g., as a host for heterologous protein production.     

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.     

Automated Bayesian model development for frequency detection in biological time series

Background

The ability to perform quantitative studies using isotope tracers and metabolic flux analysis (MFA) is critical for detecting pathway bottlenecks and elucidating network regulation in biological systems, especially those that have been engineered to alter their native metabolic capacities. Mathematically, MFA models are traditionally formulated using separate state variables for reaction fluxes and isotopomer abundances. Analysis of isotope labeling experiments using this set of variables results in a non-convex optimization problem that suffers from both implementation complexity and convergence problems.

Results

This article addresses the mathematical and computational formulation of ¹³C MFA models using a new set of variables referred to as fluxomers. These composite variables combine both fluxes and isotopomer abundances, which results in a simply-posed formulation and an improved error model that is insensitive to isotopomer measurement normalization. A powerful fluxomer iterative algorithm (FIA) is developed and applied to solve the MFA optimization problem. For moderate-sized networks, the algorithm is shown to outperform the commonly used 13CFLUX cumomer-based algorithm and the more recently introduced OpenFLUX software that relies upon an elementary metabolite unit (EMU) network decomposition, both in terms of convergence time and output variability.

Conclusions

Substantial improvements in convergence time and statistical quality of results can be achieved by applying fluxomer variables and the FIA algorithm to compute best-fit solutions to MFA models. We expect that the fluxomer formulation will provide a more suitable basis for future algorithms that analyze very large scale networks and design optimal isotope labeling experiments.