Combinatorial complexity of pathway analysis in metabolic networks

Elementary flux mode analysis is a promising approach for a pathway-oriented perspective of metabolic networks. However, in larger networks it is hampered by the combinatorial explosion of possible routes. In this work we give some estimations on the combinatorial complexity including theoretical upper bounds for the number of elementary flux modes in a network of a given size. In a case study, we computed the elementary modes in the central metabolism of Escherichia coli while utilizing four different substrates. Interestingly, although the number of modes occurring in this complex network can exceed half a million, it is still far below the upper bound. Hence, to a certain extent, pathway analysis of central catabolism is feasible to assess network properties such as flexibility and functionality.     

基元模式分析在生物网络和途径分析中的应用

基元模式分析是应用最广泛的代谢途径分析方法。基元模式分析的研究对象从代谢网络发展到信号传导网络;研究尺度从细胞到生物反应器,甚至生态系统;数学描述从稳态分解到动态解析;研究领域从微生物代谢到人类疾病。以下综述了基元模式分析的算法和软件开发现状,以及其在代谢途径与鲁棒性、代谢通量分解、稳态代谢通量分析、动态模型与生物过程模拟、网络结构与调控、菌株设计和信号传导网络等方面的应用。开发新的算法解决组合爆炸问题,探索基元模式与代谢调控的关系以及提高菌株设计算法效率是今后基元模式的重要发展方向。     

ACoM: A classification method for elementary flux modes based on motif finding

Elementary flux mode analysis is a powerful tool for the theoretical study of metabolic networks. However, when the networks are complex, the determination of elementary flux modes leads to combinatorial explosion of their number which prevents from drawing simple conclusions from their analysis. To deal with this problem we have developed a method based on the Agglomeration of Common Motifs (ACoM) for classifying elementary flux modes. We applied this algorithm to describe the decomposition into elementary flux modes of the central carbon metabolism in Bacillus subtilis and of the yeast mitochondrial energy metabolism. ACoM helps to give biological meaning to the different elementary flux modes and to the relatedness between reactions. ACoM, which can be viewed as a bi-clustering method, can be of general use for sets of vectors with values 0, +1 or −1.     

Pathway classification of TCA cycle

The structural analysis of large metabolic networks exhibits a combinatorial explosion of elementary modes. A new method of classification has been developed [called aggregation around common motif (ACoM)], which groups elementary modes into classes with similar substructures. This method is applied to the tricarboxylic acid cycle and metabolite carriers. The analysis of this network evidences a great number of elementary flux modes (204) despite the low number of reactions (23). The ACoM is used to class these elementary modes in a low number of sets (8) with biological meanings.     

Exploring the pathway structure of metabolism: decomposition into subnetworks and application to Mycoplasma pneumoniae

MOTIVATION: Reconstructing and analyzing the metabolic map of microorganisms is an important challenge in bioinformatics. Pathway analysis of large metabolic networks meets with the problem of combinatorial explosion of pathways. Therefore, appropriate algorithms for an automated decomposition of these networks into smaller subsystems are needed. RESULTS: A decomposition algorithm for metabolic networks based on the local connectivity of metabolites is presented. Interrelations of this algorithm with alternative methods proposed in the literature and the theory of small world networks are discussed. The applicability of our method is illustrated by an analysis of the metabolism of Mycoplasma pneumoniae, which is an organism of considerable medical interest. The decomposition gives rise to 19 subnetworks. Three of these are here discussed in biochemical terms: arginine degradation, the tetrahydrofolate system, and nucleotide metabolism. The interrelations of pathway analysis of biochemical networks with Petri net theory are outlined.     

Detection of elementary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering

Rational metabolic engineering requires powerful theoretical methods such as pathway analysis, in which the topology of metabolic networks is considered. All metabolic capabilities in steady states are composed of elementary flux modes, which are minimal sets of enzymes that can each generate valid steady states. The modes of the fructose-2,6-bisphosphate cycle, the combined tricarboxylic-acid-glyoxylate-shunt system and tryptophan synthesis are used here for illustration. This approach can be used for many biotechnological applications such as increasing the yield of a product, channelling a product into desired pathways and in functional reconstruction from genomic data.     

FluxAnalyzer: exploring structure,pathways, and flux distributions in metabolic networks on interactive flux maps

MOTIVATION: The analysis of structure, pathways and flux distributions in metabolic networks has become an important approach for understanding the functionality of metabolic systems. The need of a user-friendly platform for stoichiometric modeling of metabolic networks in silico is evident. RESULTS: The FluxAnalyzer is a package for MATLAB and facilitates integrated pathway and flux analysis for metabolic networks within a graphical user interface. Arbitrary metabolic network models can be composed by instances of four types of network elements. The abstract network model is linked with network graphics leading to interactive flux maps which allow for user input and display of calculation results within a network visualization. Therein, a large and powerful collection of tools and algorithms can be applied interactively including metabolic flux analysis, flux optimization, detection of topological features and pathway analysis by elementary flux modes or extreme pathways. The FluxAnalyzer has been applied and tested for complex networks with more than 500,000 elementary modes. Some aspects of the combinatorial complexity of pathway analysis in metabolic networks are discussed. AVAILABILITY: Upon request from the corresponding author. Free for academic users (license agreement). Special contracts are available for industrial corporations. SUPPLEMENTARY INFORMATION: http://www.mpi-magdeburg.mpg.de/projects/fluxanalyzer.     

Thermodynamics-based Metabolite Sensitivity Analysis in metabolic networks

The increasing availability of large metabolomics datasets enhances the need for computational methodologies that can organize the data in a way that can lead to the inference of meaningful relationships. Knowledge of the metabolic state of a cell and how it responds to various stimuli and extracellular conditions can offer significant insight in the regulatory functions and how to manipulate them. Constraint based methods, such as Flux Balance Analysis (FBA) and Thermodynamics-based flux analysis (TFA), are commonly used to estimate the flow of metabolites through genome-wide metabolic networks, making it possible to identify the ranges of flux values that are consistent with the studied physiological and thermodynamic conditions. However, unless key intracellular fluxes and metabolite concentrations are known, constraint-based models lead to underdetermined problem formulations. This lack of information propagates as uncertainty in the estimation of fluxes and basic reaction properties such as the determination of reaction directionalities. Therefore, knowledge of which metabolites, if measured, would contribute the most to reducing this uncertainty can significantly improve our ability to define the internal state of the cell. In the present work we combine constraint based modeling, Design of Experiments (DoE) and Global Sensitivity Analysis (GSA) into the Thermodynamics-based Metabolite Sensitivity Analysis (TMSA) method. TMSA ranks metabolites comprising a metabolic network based on their ability to constrain the gamut of possible solutions to a limited, thermodynamically consistent set of internal states. TMSA is modular and can be applied to a single reaction, a metabolic pathway or an entire metabolic network. This is, to our knowledge, the first attempt to use metabolic modeling in order to provide a significance ranking of metabolites to guide experimental measurements.     

The geometry of the flux cone of a metabolic network

The analysis of metabolic networks has become a major topic in biotechnology in recent years. Applications range from the enhanced production of selected outputs to the prediction of genotype-phenotype relationships. The concepts used are based on the assumption of a pseudo steady-state of the network, so that for each metabolite inputs and outputs are balanced. The stoichiometric network analysis expands the steady state into a combination of nonredundant subnetworks with positive coefficients called extremal currents. Based on the unidirectional representation of the system these subnetworks form a convex cone in the flux-space. A modification of this approach allowing for reversible reactions led to the definition of elementary modes. Extreme pathways are obtained with the same method but splitting up internal reactions into forward and backward rates. In this study, we explore the relationship between these concepts. Due to the combinatorial explosion of the number of elementary modes in large networks, we promote a further set of metabolic routes, which we call the minimal generating set. It is the smallest subset of elementary modes required to describe all steady states of the system. For large-scale networks, the size of this set is of several magnitudes smaller than that of elementary modes and of extreme pathways.     

An improved algorithm for stoichiometric network analysis: theory and applications

MOTIVATION: Genome scale analysis of the metabolic network of a microorganism is a major challenge in bioinformatics. The combinatorial explosion, which occurs during the construction of elementary fluxes (non-redundant pathways) requires sophisticated and efficient algorithms to tackle the problem. RESULTS: Mathematically, the calculation of elementary fluxes amounts to characterizing the space of solutions to a mixed system of linear equalities, given by the stoichiometry matrix, and linear inequalities, arising from the irreversibility of some or all of the reactions in the network. Previous approaches to this problem have iteratively solved for the equalities while satisfying the inequalities throughout the process. In an extension of previous work, here we consider the complementary approach and derive an algorithm which satisfies the inequalities one by one while staying in the space of solution of the equality constraints. Benchmarks on different subnetworks of the central carbon metabolism of Escherichia coli show that this new approach yields a significant reduction in the execution time of the calculation. This reduction arises since the odds that an intermediate elementary flux already fulfills an additional inequality are larger than when having to satisfy an additional equality constraint.     

Analysis of optimal phenotypic space using elementary modes as applied to Corynebacterium glutamicum

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.     

An environmental perspective on metabolism

In principle the knowledge of an organism's metabolic network allows to infer its biosynthetic capabilities. Handorf et al. [2005. Expanding metabolic networks: scopes of compounds, robustness, and evolution. J. Mol. Evol. 61, 498-512] developed a method of network expansion generating the set of all possible metabolites that can be produced from a set of compounds, given the structure of a metabolic network. Here we investigate the inverse problem: which chemical compounds or sets of compounds must be provided as external resources in order to sustain the growth or maintenance of an organism, given the structure of its metabolic network? Although this problem is highly combinatorial, we show that it is possible to calculate locally minimal nutrient sets that can be interpreted in terms of resource types. Using these types we predict broad nutritional requirements for 447 organisms, providing clues for possible environments from the knowledge of their metabolic networks.     

Analyzing molecular reaction networks

Pathways are typically the central concept in the analysis of biochemical reaction networks. A pathway can be interpreted as a chain of enzymatical reactions performing a specific biological function. A common way to study metabolic networks are minimal pathways that can operate at steady state called elementary modes. The theory of chemical organizations has recently been used to decompose biochemical networks into algebraically closed and self-maintaining subnetworks termed organizations. The aim of this paper is to elucidate the relation between these two concepts. Whereas elementary modes represent the boundaries of the potential behavior of the network, organizations define metabolite compositions that are likely to be present in biological feasible situations. Hence, steady state organizations consist of combinations of elementary modes. On the other hand, it is possible to assign a unique (and possibly empty) set of organizations to each elementary mode, indicating the metabolites accompanying the active pathway in a feasible steady state.     

Expression-level optimization of a multi-enzyme pathway in the absence of a high-throughput assay

Engineered metabolic pathways often suffer from flux imbalances that can overburden the cell and accumulate intermediate metabolites, resulting in reduced product titers. One way to alleviate such imbalances is to adjust the expression levels of the constituent enzymes using a combinatorial expression library. Typically, this approach requires high-throughput assays, which are unfortunately unavailable for the vast majority of desirable target compounds. To address this, we applied regression modeling to enable expression optimization using only a small number of measurements. We characterized a set of constitutive promoters in Saccharomyces cerevisiae that spanned a wide range of expression and maintained their relative strengths irrespective of the coding sequence. We used a standardized assembly strategy to construct a combinatorial library and express for the first time in yeast the five-enzyme violacein biosynthetic pathway. We trained a regression model on a random sample comprising 3% of the total library, and then used that model to predict genotypes that would preferentially produce each of the products in this highly branched pathway. This generalizable method should prove useful in engineering new pathways for the sustainable production of small molecules.     

Integrating cybernetic modeling with pathway analysis provides a dynamic, systems-level description of metabolic control

Cybernetic modeling strives to uncover the inbuilt regulatory programs of biological systems and leverage them toward computational prediction of metabolic dynamics. Because of its focus on incorporating the global aims of metabolism, cybernetic modeling provides a systems-oriented approach for describing regulatory inputs and inferring the impact of regulation within biochemical networks. Combining cybernetic control laws with concepts from metabolic pathway analysis has culminated in a systematic strategy for constructing cybernetic models, which was previously lacking. The newly devised framework relies upon the simultaneous application of local controls that maximize the net flux through each elementary flux mode and global controls that modulate the activities of these modes to optimize the overall nutritional state of the cell. The modeling concepts are illustrated using a simple linear pathway and a larger network representing anaerobic E. coli central metabolism. The E. coli model successfully describes the metabolic shift that occurs upon deleting the pta-ackA operon that is responsible for fermentative acetate production. The model also furnishes predictions that are consistent with experimental results obtained from additional knockout strains as well as strains expressing heterologous genes. Because of the stabilizing influence of the included control variables, the resulting cybernetic models are more robust and reliable than their predecessors in simulating the network response to imposed genetic and environmental perturbations.     

Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems

The elucidation of organism-scale metabolic networks necessitates the development of integrative methods to analyze and interpret the systemic properties of cellular metabolism. A shift in emphasis from single metabolic reactions to systemically defined pathways is one consequence of such an integrative analysis of metabolic systems. The constraints of systemic stoichiometry, and limited thermodynamics have led to the definition of the flux space within the context of convex analysis. The flux space of the metabolic system, containing all allowable flux distributions, is constrained to a convex polyhedral cone in a high-dimensional space. From metabolic pathway analysis, the edges of the high-dimensional flux cone are vectors that correspond to systemically defined "extreme pathways" spanning the capabilities of the system. The addition of maximum flux capacities of individual metabolic reactions serves to further constrain the flux space and has led to the development of flux balance analysis using linear optimization to calculate optimal flux distributions. Here we provide the precise theoretical connections between pathway analysis and flux balance analysis allowing for their combined application to study integrated metabolic function. Shifts in metabolic behavior are calculated using linear optimization and are then interpreted using the extreme pathways to demonstrate the concept of pathway utilization. Changes to the reaction network, such as the removal of a reaction, can lead to the generation of suboptimal phenotypes that can be directly attributed to the loss of pathway function and capabilities. Optimal growth phenotypes are calculated as a function of environmental variables, such as the availability of substrate and oxygen, leading to the definition of phenotypic phase planes. It is illustrated how optimality properties of the computed flux distributions can be interpreted in terms of the extreme pathways. Together these developments are applied to an example network and to core metabolism of Escherichia coli demonstrating the connections between the extreme pathways, optimal flux distributions, and phenotypic phase planes. The consequences of changing environmental and internal conditions of the network are examined for growth on glucose and succinate in the face of a variety of gene deletions. The convergence of the calculation of optimal phenotypes through linear programming and the definition of extreme pathways establishes a different perspective for the understanding of how a defined metabolic network is best used under different environmental and internal conditions or, in other words, a pathway basis for the interpretation of the metabolic reaction norm.     

Metabolic pathway analysis of a recombinant yeast for rational strain development

Elementary mode analysis has been used to study a metabolic pathway model of a recombinant Saccharomyces cerevisiae system that was genetically engineered to produce the bacterial storage compound poly-beta-hydroxybutyrate (PHB). The model includes biochemical reactions from the intermediary metabolism and takes into account cellular compartmentalization as well as the reversibility/irreversibility of the reactions. The reaction network connects the production and/or consumption of eight external metabolites including glucose, acetate, glycerol, ethanol, PHB, CO(2), succinate, and adenosine triphosphate (ATP). Elementary mode analysis of the wild-type S. cerevisiae system reveals 241 unique reaction combinations that balance the eight external metabolites. When the recombinant PHB pathway is included, and when the reaction model is altered to simulate the experimental conditions when PHB accumulates, the analysis reveals 20 unique elementary modes. Of these 20 modes, 7 produce PHB with the optimal mode having a theoretical PHB carbon yield of 0.67. Elementary mode analysis was also used to analyze the possible effects of biochemical network modifications and altered culturing conditions. When the natively absent ATP citrate-lyase activity is added to the recombinant reaction network, the number of unique modes increases from 20 to 496, with 314 of these modes producing PHB. With this topological modification, the maximum theoretical PHB carbon yield increases from 0.67 to 0.83. Adding a transhydrogenase reaction to the model also improves the theoretical conversion of substrate into PHB. The recombinant system with the transhydrogenase reaction but without the ATP citrate-lyase reaction has an increase in PHB carbon yield from 0.67 to 0.71. When the model includes both the ATP citrate-lyase reaction and the transhydrogenase reaction, the maximum theoretical carbon yield increases to 0.84. The reaction model was also used to explore the possibility of producing PHB under anaerobic conditions. In the absence of oxygen, the recombinant reaction network possesses two elementary modes capable of producing PHB. Interestingly, both modes also produce ethanol. Elementary mode analysis provides a means of deconstructing complex metabolic networks into their basic functional units. This information can be used for analyzing existing pathways and for the rational design of further modifications that could improve the system's conversion of substrate into product.     

Application of hybrid cybernetic model in simulating myeloma cell culture co-consuming glucose and glutamine with mixed consumption patterns

A dynamic model called hybrid cybernetic model (HCM) based on structured metabolic network is established for simulating mammalian cell metabolism featured with partially substitutable and partially complementary consumption patterns of two substrates, glucose and glutamine. Benefiting from the application of elementary mode analysis (EMA), the complicated metabolic network is decomposed into elementary modes (EMs) facilitating the employment of the hybrid cybernetic framework to investigate the external and internal flux distribution and the regulation mechanism among them. According to different substrate combination, two groups of EMs are obtained, i.e., EMs associated with glucose uptake and simultaneous uptake of glucose and glutamine. Uptake fluxes through various EMs are coupled together via cybernetic variables to maximize substrate uptake. External fluxes and internal fluxes could be calculated and estimated respectively, by the combination of the stoichiometrics of metabolic networks and fluxes through regulated EMs. The model performance is well validated via three sets of experimental data. Through parameter identification of limited number of experimental data, other external metabolites are precisely predicted. The obtained kinetic parameters of three experimental cultures have similar values, which indicates the robustness of the model. Furthermore, the prediction performance of the model is successfully validated based on identified parameters.